Journal of Applied Mathematics and Computation

Journal of Applied Mathematics and Computation Volume 2, Number 9, September 2018 (Serial Number 10)

Hill Publishing Group www.hillpublisher.com

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Assoc. Prof. Fateh Mebarek-Oudina, Department of Physics, Faculty of Sciences, University of 20 août 1955-Skikda,Route El-Hadaeik, B.P. 26, Skikda 21000, Algeria Assoc. Prof. Miranda Narmania, University of Georgia. 77, M. Kostava str., Tbilisi 0186, Georgia Assoc. Prof. Victor Nesterenko, Institute of mathematics, mechanics and computer Sciences, South Federal University, Rostov-on-Don, Russia Assoc. Prof. Vishnu Narayan Mishra, Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Madhya Pradesh 484 887, India Assoc.Prof. Luis A. Moncayo Martí nez, Department of Industrial Engineering and Operations Instituto Tecnológico Autónomo de México (ITAM), Mexico Asst. Prof. Jasvinder Pal Singh Virdi, Department of Physics VSS University of Technology, Sambalpur, India Asst. Prof. Dr. Ferit Gürbüz, Hakkari University, Hakkari, Turkey Asst. Prof. Dr. Erdoğan Şen, Tekirdag Namik Kemal University Faculty of Science and Letters, Department of Mathematics 59030, Merkez, Tekirdag, Turkey Asst. Prof. Mohammed Mazhar Ul Haque, Theem college of Engineering, Palghar, Mumbai, India Assist. Prof. Burcu Gürbüz, Üsküdar University,Faculty of Engineering and Natural Sciences, Department of Computer Engineering, Haluk Turksoy sok. No: 14 34662 Üsküdar/ Istanbul/ Turkey Assist. Prof. Ihsan Yanıkoğlu, Industrial Engineering Department Ozyegin University, Turkey Assist. Prof. Sanjeev Singh, Discipline of Mathematics Indian Institute of Technology Indore Khandwa Road, Simrol Indore 453552, M.P., India Dr. Dai Hai Nguyen, Bioinformatics Center, Institute for Chemical Research Kyoto University, Japan Dr. Sinan Deniz, Manisa Celal Bayar University, Faculty of Arts & Science, Department of Mathematics, Muradiye Campus, 45140,Yunusemre-Manisa, Turkey Dr. Robab Kalantari, Sharif University of Technoloy, Iran Dr. Naeem Saleem, Department of Mathematics, University of Management and Technology, C-II, Johar Town, Lahore, Pakistan Dr. Marzieh Shamsizadeh, Department of Applied Mathematics in Graduate University of Advanced Technology, Kerman, Iran Dr. Amirouche Berkani, Faculty of Mathematics & Informatics, University Mohamed El Bachir El Ibrahimi, Bordj Bou Arréridj, El-Anasser 34030, Algeria Dr. Zeki Kasap, Faculty of Education, Elementary Mathematics Education Department in Pamukkale University, Kinikli Campus, Denizli, Turkey Dr. Hasan Bulut, Department of Statistics, Ondokuz Mayıs University, Samsun, Turkey Dr. Yasemin Soylu, Department of Mathematics, Faculty of Arts and Sciences, Giresun University, Giresun, Turkey Dr. Tuğçem PARTAL, Yildiz Technical University, Department of Mathematical Engineering Davutpasa Campus 34210 Esenler- Istanbul, Turkey Dr. Esra Kirmizi Çetinalp, Karamanoglu Mehmetbey University, Kamil Özdag Science Faculty, Department of Mathematics, Yunus Emre Campus, 70100 Karaman, Turkey Dr. Melis Zeybek, Ege University, Faculty of Sciences, Department of Statistics, 35100, Bornova, İzmir, Turkey Dr. Snezana S. Djordevic, University of Nis, Faculty of Technology, Leskovac, Serbia Dr. Parbati Sahoo, Dept. of Mathematics, BITS-Pilani Hyderabad Campus, Hyderabad, India Dr. Tutku Tuncalı Yaman, Hamidiye Mah. Başak Konutları A207 D:11 Kağıthane / İstanbul, Turkey Dr. Yuriy Danyk, Chief (Rector) of Institute of Information Technologies, Ukraine Dr. V. Lysenko, Head of the Department of Automation and Robotics Systems of the National University of Life and Environmental Sciences of Ukraine Dr. Charis Ntakolia, Department of Applied Mathematics, National Technical University of Athens, Greece Dr. Igor Nikolayevich Skopin, Institute of Computational Mathematics and Mathematical Geophysics SB RAS (ICM&MG) Novosibirsk State University (NSU), Russia Dr. Raúl Santiesteban Cos, 700 Forest Hill, Apartment 205, Fredericton, New Brunswick, Canada Dr. Jean Paulo dos Santos Carvalho, Federal University of the Recôncavo of Bahia (UFRB), Brazil Dr. Ionescu Adela Janeta, Department of Applied Mathematics, University of Craiova, Romania Dr. Ringo Hok-Ling Lee, HSBC, Group Chief Control Office, Hongkong, China Dr. Mustafa Özkan, Trakya University, Faculty of Science, Department of Mathematics, Edirne, Turkey

Dr. Andrii Chepok, O. Popov Odessa national academy of Telecommunications (ONAT, Odessa, Ukraine), IT Dept. at the Faculty of Information Networks and Software Engineering, Ukraine Dr. Bolun Chen, Volen National Center for Complex Systems, Brandeis University Waltham, MA 02453, USA Dr. Francesco Rizzi, NexGen Analytics USA, Senior Computational Scientist Dr. Dayane S. Alvares, Department of Physics – Biomembrane Laboratory, UNESP/IBILCE, Brazil Dr. Maleafisha Joseph Pekwa Stephen Tladi, Department of Mathematics and Applied Mathematics University of Limpopo Polokwane, 0727, South Africa Dr. Oğuzhan Bahadır, Department of Mathematics, Faculty of Science and Letters, Kahramanmaras Sutcu Imam University, Kahramanmaras, TURKEY Dr. Nadiia Derevianko, Department of Theory of Functions Institute of Mathematics National Academy of Sciences of Ukraine Dr. Gayatri Pany, Post Doctoral Research Fellow Singapore University of Technology and Design,Somapah Road, Singapore Dr. Güzide ŞENEL, Assistant Professor of Mathematics University of Amasya, Amasya-TURKEY

Journal of Applied Mathematics and Computation (JAMC), 2018, 2(9), 357-455 http://www.hillpublisher.org/journal/jamc ISSN Online:2576-0645 ISSN Print:2576-0653

Table of Contents Volume 2 Number 9 September 2018

Study Method for Testing Image Quality of Optical Remote Sensing Satellite of Vietnam Nghiem Van Tuan1, Nguyen Minh Ngoc2, Tran Van Anh3, Do Thi Phuong Thao4

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New Hybrid Conjugate Gradient Method as A Convex Combination of HS and FR Methods Snezana S. Djordjevic

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Well-Posedness and Long-Time Dynamics of the Rotating Boussinesq and Quasigeostrophic Equations Maleafisha Joseph Pekwa Stephen Tladi

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Study Method for Testing Image Quality of Optical Remote Sensing Satellite of Vietnam Nghiem Van Tuan1, Nguyen Minh Ngoc 2, Tran Van Anh3, Do Thi Phuong Thao 4 1

National Remote Sensing Department, Ministry of Natural Resources and Environment, VietNam Space Technology Institute, VietNam Academy of Science and Technology, VietNam 3 Faculty of Geomatics and Land administration, Hanoi University of Mining and Geology, VietNam 4 Faculty of Geomatics and Land administration, Hanoi University of Mining and Geology, VietNam 2

How to cite this paper: Nghiem, V.T., Nguyen, M.N., Tran, V.A., Do, T.P.T.. (2018) Study Method for Testing Image Quality of Optical Remote Sensing Satellite of Vietnam. Journal of Applied Mathematics and Computation, 2(9), 357-365. Doi:10.26855/jamc.2018.09.001 *Correspondi ng author: Nguyen Minh Ngoc, Space Technology Institute, VietNam Academy of Science and Technology, VietNam Email: [email protected] m

Abstract VNREDSat-1 satellite had been launch on 7th May 2013 with 5 years life time designed and it opens up new opportunities for natural resources and environment monitoring in Vietnam. Precise information on the Earth surface which is extracted from remote sensing data including VNREDSat-1 depends on parameters such as spatial resolution, radiometric resolution, temporal resolution, and spectral resolution. Image quality is evaluated by spatial and radiometric resolution. MTF (Modulation Transfer Function) shows quality of spatial resolution and sharpness. Radiometric resolution is delineated through noise value characterized by signal to noise ratio (SNR). VNREDSat-1 is the first multi-spectral Vietnamese satellite that suitable image quality assessment methods should be built and possible applied for similar future satellites. MTF could be estimated by algorithms as general mathematical framework based on Wiener filter (1949), target based absolute MTF; or direct and indirect parametric models of MTF. SNR estimation could be performed by several methods which are single view for homogeneous and quasihomogeneous area or synthetic landscape such as the desert, snowy expans of Greenland or Antarctic, etc. Outputs of this work are methodology and algorithm for VNREDSat-1 image quality assessment and similar future satellites that is suitable with characteristics of Viet Nam test site.

Keywords Image quality, Optical satellite, VNREDSat-1.

1. Introduction. VNREDSat-1 satellite has been designed and used AstroSat100-bus of EADS Astrium, with Myriade platform and instrument as NAOMI-125, which is a high resolution pushbroom imager. It provides imagery of 2.5m in Panchromatic and 4 multispectral bands of 10m GSD (VAST, 2013a). The optical system in NAOMI-125 (Fig.1) is one of the most important sub-systems, built on Korsch telescope with a Three-mirror Anastigmat design to ensure lightness and provide good optical quality with three aspherical mirrors (Luquet et al, 2008). Once VNREDSat-1 has been operated, validation and calibration activities have been performed cyclically, but mainly for radiometric (DSNU and PRNU). Therefore, it is necessary to propose image quality assessment method for not only VNREDSat-1 but also similar satellites in the future. In this article, the authors focus on two main parts related to image quality: radiometric and geometric.

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Journal of Applied Mathematics and Computation(JAMC)

V.T. Nghiem, M .N. Nguyen, V.A. Tran, T.P.T. Do

M1, M2, M3: curved mirror FM1, FM2: flat folding mirror Source: EADS Astrium

Figure 1. The optical concept of the Korsch telescope

2. Material and Methods In orbit test (IOT), VNREDSat-1 has been validated radiometric and geometric parameters related image quality. During operational phase, calibration and validation activities should be continued. In that, related geometric is MTF and related radiometric are SNR and PRNU. 2.1. Geometric testing Modulation Transfer Function is defined as the modulus of the Fourier Transform of the Point Spread Function (Hearn, 2005):

There are some method to estimate MTF such as: general mathematical framework based on Wiener filters (Wiener, 1949), based on targets, bi-resolution, etc. a. MTF estimation method using general mathematical framework General mathematical framework based on Wiener (1949) is an interesting method. It allows not only MTF estimation but also the error of the MTF absolute estimation, without considering MTF/PSF/LSF or ESF, which reduce the impact of measurement noise and aliasing effects on the MTF absolute estimation (Hearn, 2005). However, this method only considers relationship between LSF, PSF and MTF in one dimension (without loss of generality). LSF is defined for a given orientation and corresponds to the integration of the PSF over the orthogonal to that direction. LSF’s two particular directions are important because it corresponds to the separable axis of the detector and the motions blur MTF. b. MTF estimation method using target Most of absolute MTF estimation is based on image analysis from acquisition of specific well known targets, they can be artificial objects such as bridges, buildings, etc. or dedicated targets such as painted surfaces, single or multiple spotlights, etc. or even natural objects such as fields and so on. In general, there are two main targets used for MTF estimation: the pulse target and edge target. According to the design, test site in Vietnam will be similar with edge target. Therefore this study did not consider pulse target.

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An edge target corresponds to high contrast Heaviside edge (Fig-2). The main parameters are: the differential radiance ΔL between the dark and bright part, the width of the target in the direction of MTF profile LW, the orientation angle α with respect to the direction of the MTF profile, and the height LH of the target in the orthogonal direction of the MTF profile (Hearn, 2005).

LT: The transition distance

Figure 2. The design of edge target for MTF estimation

Based on those principles, Vietnamese test site is designed and constructed which consists of two parts: - About MTF validation and measurement: in square shape, the longitudinal edge is tilted to a certain angle to the North, and is shaped bay a 2x2m chessboard, each box is painted black and white alternately. - About SNR validation and measurement which can also simultaneously calculate the dynamic range and radiance responds on the instrument: in the form of four consecutive squares, painted from white-light gray-strong gray-black, with the same side north. c. Bi-resolution MTF estimation method This method use spectrum ratios in the Fourier domain of two images of the same scene in order to estimate the ratio of their respective MTF. It requires specific pre-processing such as geometric registration and radiometric alignment. Change detection algorithm could be also used to determine temporally stable areas between the two acquisitions. There are two cases for this method. In the first case, using two unknown images and their respective GSD are comparable, so only relative MTF can be estimated. In the second case, using two images that are different GSD, one image has higher at least a factor 5 between GSDs (Li et al, 2015). In this case, even if MTF of higher resolution image is unknown, it will be assumed to correspond to a certain value at the scale of low resolution image. In other words, this case will be allowed estimation MTF of lower resolution image. d. MTF estimation method with specific on-board devices There are some specific devices today allow of MTF estimation without having the knowledge of the scene or object, named “blind” method. An example for this method is the specific on-board devices, which is the phase diversity method. Image processing technique for this method will infer the global MTF from a set of more than two associate images of the same extend object or rich natural scenes (Paxman et al, 1992). In fact, image processing process corresponds to the general estimation and repeat of deconvolution processing, taking into the account of the estimated MTF previously.

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And MTF taking into account the current estimation of the unknown object. These two processes are performed until convergence. 2.2. Radiometric calibration a. Photo Response Non-Uniformity (PRNU) Imagery for PRNU validation can be deserts for example in Lybia, Algeria, Arabia, Australia, etc. These are areas of uniform landscape, uniform radiance level, and less affected by the seasonal weather cycle to the measured radiance level. So they can be used to validate all year round. b. Signal-Noise-Ratio (SNR) assessment Signal-Noise-Ratio (SNR) is one of the factors that affect image quality, and it is characterized by radiometric noise. The image noise quantifies the difference in radiance levels for a homogeneous area is estimated by the formula 2 (Wang, et al., 2009): SNR = m/σ (2) Where, m is the mean of a series of radiance for uniform landscape and σ is the standard deviation of this series. Depend on the sensor of the instrument, it is an array or matrix that is subject to different interference effects (Kubik et al, 1998). If the sensor is an array, the noise sources of the image coming from two separated sources are colum-wise noise, also called instrument noise and line-wise noise, also called normalization noise. If the sensor is a matrix, then each detector has its own noise; because there are several normalization steps to equalize the detectors’ signal, the noise is assumed to be the same as the normalization noise. A SNR assessment method is usually created by three factors: site selection, method to compute the mean and noise, and the timing of application of assessment with respect to routine operations and other calibration operations. Site selection is important, affecting assessment results. There are usually two ways to select SNR test site: single view and synthetic landscape. Single view means taking a scene over a given site; synthetic landscape means taking several images and merging in order to create a synthetic landscape with requires properties. - For single view, there are two cases can be occurred: homogeneous area and quasihomogeneous area + Homogeneous area: The principle of this case is to find a homogeneous area representing for uniform landscape, to calculate mean value and standard deviation on this area, and finally to get the ratio of the mean and standard deviation. However, homogeneous area selection requires areas having average radiance value in the area that needs to be high enough. + Quasi-Homogeneous area: One of the most known methods using this case is the Fourier transform of a part of an image (Jenkins et al, 1969). Noise will appears at high frequencies, but chaotic operation of spectral density for high wave numbers as well as the presence of large scale trends can affect accuracy when estimating noise and mean. The requirement for reliable SNR assessment is using data capture of smooth landscape with low-energy intrinsic high frequencies. - For synthetic landscape: The limitation of a homogeneous area can be reduced by considering a synthetic landscape. It is generated by the fusion of actual single image (Wald, 1999), properties of synthetic landscape are more suitable than single view when assessing SNR. The Possible solution is using desert areas where having very stale reflectance in time, once corrected for bi-directional effects, such as Northern Africa area (Cosnefoy et al, 1996). A combining cloud image solution is also used to create a completely clouded composite scene.

3. Result and Discussion Validation activities have been performed every 3 months after satellite is launch. The parameter will be calculated and compared with the previous test. If the error between the two tests exceeds the threshold, the correction will be performed. Due to the differences between working conditions such as: temperature, thermal difference between compoDOI: 10.26855/jamc.2018.09.001

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nents, cosmic radiation, power supply noise, etc. calibration will be made as soon as possible after false detection. We will discuss some results as well as practical experiences in the operation VNREDSat-1 satellite. 3.1. Geometric validation a. MTF estimation method using general mathematical framework With the aim of providing an estimation method suitable not only for existing satellite systems but also for future satellites, the authors have not chosen this method, as the design of the test site is not in accordance with the algorithm used; it is lack of general when estimating MTF and considering the relationship between MTF, PSF, LSF. b. MTF estimation method using target In some studies, it has been shown that by using this method, the relative root mean square error of MTF estimation obtained by the Wiener filter will not be affected by aliasing. Nevertheless, this filter does not take into account neither the zero-crossing nor the radiometric noise (Helder, 2003, Helder et al, 2004, Helder and Choi, 2005). Therefore, with the design characteristics of the Vietnam test site, this method can be applied to estimate MTF for VNREDSat-1 satellite and other ones of future remote sensing satellites of Vietnam. We would like to propose MTF estimation method based on actual condition in VietNam for existing satellite and future one, as well as test site under constructing, which is MTF estimation using edge target. On that basis, we validated the MTF with VNREDSat-1 image, which taken on 18 July 2016 and 25 April 2016 at the Salon de Provence test site, France (Fig.3). The illustration in (Fig.4 and 5) shows that, MTF@ Nyquist frequency in image taken on 25 April is 0.22, and on 18 July is 0.25. These results is quite similar to the survey conducted by ADS after VNREDSat-1 was launched, is 0.21. The slight difference may be due to the different meteorological conditions at the Salon de Provence test site. The results in the image taken on 18 July, MTF curve has some point up. This phenomenon may be due to noise during sampling.

Figure 3. Salon de Provence test site on 25th April2016 (left) and on 18th July2016 (right)

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Figure 4. MTF@ Nyquist frequency in data taken on 18 th July2016

Figure 5. MTF@ Nyquist frequency in data taken on 25th April2016

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c. Bi-resolution MTF estimation method In fact, this direct application of the bi-resolution method suffered from an over-estimation of MTF due to aliasing effects; this has been noticed and confirmed by some scientists in previous studies (Viallefont-Robinet, 2003). In addition, applying

this method always requires having a higher resolution image than ones need to estimate MTF. This would be difficult in the case of new satellite system with very high resolution, which would be difficult to find at higher resolution even using aerial imagery. For example, the image needs to estimate MTF has resolution 0.5m, there must be images with higher resolution of 0.2m or 0.1m. If new satellite designed with resolution 0.1m, what would be happened? Thus, this method is highly applicable but must be selected and balanced before using. d. MTF estimation method with specific on-board devices This method is suitable for satellite systems with specific on-board devices to estimate MTF and is designed from beginning. For VNREDSat-1 currently operating, there are no such devices, so it is impossible to use this method for MTF estimation. 3.2. Radiometric validation a. PRNU validation PRNU has been calculated from imagery input data. This value is in LSB and averaged the dark signal noise on all the images (after excluding the abnormal image). PRNU has been calculated on all rows of image. Each image will be calculated PRNU, then use high pass filter to eliminate the low frequency variable components. The post-filter PRNU is multiplied by the PRNU value at low frequency that is measured on the ground to recreate PRNU value of the instrument. To determine whether calibrate the instrument or not, the dark signal and PRNU are taken as averages from all images, then compared with the standard value. If it is not exceed the threshold, it will not need to conduct calibration activities. Figure 6 shows result of SNR assessment. On the left is the original image and on the right is enhanced image. The figure shows that after enhancing, the noise streaks are more clearly expressed on the original data.

Figure 6: Original image and enhanced image

b. SNR validation As mentioned above, the SNR assessment can use the data in two ways: single view and synthetic landscape. Under conditions in Vietnam, the proposed SNR assessment methodology would be single view-homogeneous area, and the calculation model would follow the definition given in section 2. These results in Figure 6 show that SNR calculated on the Pan bands of VNRDSat-1 on 25/4/2016 and on 18/7/2016 correspondent 86.15 and 116.22, which are higher than reference value (average SNR =143 on L2 (VAST, 2013b)). They only reached the asymptotic level, due to the basic calculate model, and the small test site in compare with the whole DOI: 10.26855/jamc.2018.09.001

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scene. Also the radiance at test site is not accurately ver ified at L2. We will continue to refine the model more precisely, experimenting in large scale site such as desert areas, even pano ramic imaging is the oceans. SNR reference

SNR =86.15 Data on 25/4/2016

SNR =116.22 Data on 18/7/2016

Figure 6. SNR validated on 25th April2016 and 18th July2016

4. Conclusion Validation image quality of the instrument NAOMI-125 of the VNREDSat-1 satellite, in term of geometry and radiometry has been performed by using data on 25th April2016 and 18th July2016. Geometric validation is based on MTF method and using the test site at Salon de Provence, France. The results show that, the values of MTF@Nyquist frequency validated are higher than the one provided by ADS at IOT (0.22 and 0.25 compared with 0.21). For the radiometric validation, there are two parameters have been calculated: PRNU and SNR. PRNU and SNR validations have been performed by using VNREDSat-1 images taken in the Atlantic Ocean on 25/4 and 18th July2016. The SNR values are calculated correspondent 86.15 and 116.22, lower than result at IOT (SNR average is 143@L2). The results confirmed that the geometric validation using MTF method and radiometric validation using natural targets such as deserts, oceans are suitable to the conditions of VietNam. The results also show that the geometric and radiometric quality of the instrument NAOMI-125 still having the required stability, but also show slight changes in compare with at IOT. That signals show an aging of optical sensor, and need to be monitored and verifying more often. This is feasible when VietNam’s test site is available in the near future and the reliability of testing methods suitable for VietNam conditions has been verified through this study.

Acknowledgment The work has been supported by the Project: “Research on the development of method for validation and calibration image

quality of optical satellite of Vietnam” (ID: TNMT2016.08.02), under the Remote Sensing Program, period 2016-2020 of Ministry of Natural Resources and Environment of Vietnam.

References [1] Viallefont-Robinet, F,. 2003. Removal of aliasing effect on MTF measurement using bi-resolution images. In SPIE Conference Sensor, Systems, and Next-Generation Satellites. Vol. VII, pp.468-479. [2] Helder.D,. 2003. In-flight characterization of spatial quality of remote sensing imaging systems using point spread function estimation. International Workshop on Radiometric and Geometric Calibration, Gulfport, Mississippi, USA. [3] Helder,.D,. Choi, T., Rangaswamy, M,. 2004. In-flight characterization of spatial quality using point spread function. Post-lauch Calibration of Satellite Sensors. ISBN 9058096930, pp.151-170. [4] Helder.D,. Choi, T,. 2005. Generic sensor modeling using pulse method. NASA Technical Report. Retrieved Septembe 05, 2017 from https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050214150.pdf. [5] VAST,. 2013a. VNREDSat-1 Design Report. Report of VietNam Academy on Science and Technology. DOI: 10.26855/jamc.2018.09.001

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[6] VAST, 2013b. IOT Report. Report of VietNam Academy on Science and Technology. [7] Luquet, P,. Chikouche, A,. Benbouzid, A. B,. Arnoux, J. J,. Chinal, E,. Massol, C,. Rouchit, P,. de Zotti, S,. 2008. "NAOMI instrument: a product line of compact & versatile cameras designed for high resolution missions in Earth observation". In Proceedings of the 7th ICSO of International Conference on Space Optics, Toulouse, France, Oct. 14-17, 2008. [8] ESA,. 2014. Sentinel-2 Calibration and Validation plan for the Operational Phase. Document of ESA, Issu 1, Revision 6, Retrieved Septembe 05, 2017 from https://sentinel.esa.int/documents/247904/2047089/Sentinel-2_Cal-Val_Phase-E2. [9] Paxman, R. G., Schulz T., Fienup J. R., 1992. Joint estimation of object and aberration by using phase divesity. Journal of Optical Society of America, Vol 9, No.7, pp.1072-1085. [10] Kubik Ph., Breton E., Meygret A., Cabrieres B., Hazane Ph.,Leger D., 1998. SPOT4 HRVIR first in-flight image quality results. In Proceedings of the EUROPTO Conference on Sensors, System, and Next-Generation Satellite, Barcelona, Spain, SPIE vol. 3498, pp. 376-389. [11] Jenkins, G.M and D.G Watts,. 1969. Spectral Analysis and its Application. Holden-Day, San Francisco, pp 525. [12] Wald L., 1999. Some terms of preferences in data fusion. IEEE Trasactions on Geosciences and Remote Sensing, Vol.37, Issu.3, pp1190-1193. [13] Cosnefoy, H., Leroy, M,. and Briottet, X,. 1996. Selection and characterization of Saharan and Arabian desert sites for the calibration of optical satellite sensors. Journal of Remote Sensing of Environment, Vol.58, pp.101-114. [14] Hearn, R.H., 2005. Spatial calibration and imaging performance assessment of the advanced land imager, Lincoln Laboratory Journal, Vol. 15(2), pp.225–250. [15] Wiener N,. 1949. Extrapolation, Interpolation and Smoothing of Stationary Time Series. MIT Press Cambridge, MA. [16] Li, C. R,. Tang, L.L,. Ma, L. L ,. Zhou, Y. S,. Gao, C. X,. Wang, N,.. Li, X. H,. Wang, X. H,. Zhu, X. H,. 2015. A comprehensive calibration and validation site for information remote sensing. The International Archives of the Phot ogrammetry, Remote Sensing and Spatial Information Sciences, Volume XL-7/W3, 2015. 36th International Symposium on Remote Sensing of Environment, 11–15 May 2015, Berlin, Germany. [17] Wang, X.H, Tang, L.L., Li, C.R., Y. B., Zhu, B., 2009. A practical SNR estimation scheme for remotely sensed optical imagery. In: Proc. of SPIE, Vol.7384, pp.738434-1~738434-6.

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New Hybrid Conjugate Gradient Method as A Convex Combination of HS and FR Methods Snezana S. Djordjevic 1

Faculty of Technology, University of Nis, 16000 Leskovac, Serbia

How to cite this paper: Snezana, S.D. (2018) New Hybrid Conjugate Gradient Method as A Convex Co mb ination of HS and FR Methods. Journal of Applied Mathematics and Computation, 2(9), 366-378. DOI: 10.26855/jamc.2018.09.002 *Correspondi ng author : Snezana S. Djordjevic, Faculty of Technology, University of Nis, 16000 Leskovac, Serb ia Email: snezanadjordjevic1971@g mail.com

Abstract In this paper we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Hestenes-Stiefel conjugate gradient method and Fletcher-Reeves conjugate gradient method. The parameter  k is chosen in such a way that the search direction satisfies the condition of the Newton direction. The strong Wolfe line search conditions are used. The global convergence of new method is proved. Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one.

Keywords Conjugate gradient method; Convex combination; Newton direction. 2010 Mathematics Subject Classification: 90C30.

1. Introduction. We consider the nonlinear unconstrained optimization problem

min f ( x) : x  R n ,

(1.1)

n where f : R  R is a smooth function, and its gradient is available.

There exist many different methods for solving the problem (1.1). Here we are interested in conjugate gradient methods, which have low memory requirements and strong local and global convergence properties [12]. To solve the problem (1.1), starting from an initial point x0  R , the conjugate gradient method generates a sequence n

xk  R n such that xk 1  xk  tk dk ,

(1.2)

where tk  0 is a step size, received from the line search, and the directions d k are given by

d0   g0 , dk 1   gk 1  k dk .

(1.3)

In the last relation,  k is the conjugate gradient parameter, g k  f ( xk ) . Now, we denote

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yk  gk 1  gk .

(1.4)

An excellent survey of conjugate gradient methods is given by Hager and Zhang [20]. Different conjugate gradient methods correspond to different values of the scalar parameter  k , for example, Fletcher-Reeves method (FR) [16], Polak-Ribiere -Polyak method (PRP) [27], Hestenes-Stiefel method (HS) [21], Liu-Storey method (LS) [25], Dai-Yuan method (DY) [11], Conjugate-Descent method (CD) [15], etc. These methods are identical for a strongly convex quadratic function and the exact line search, but they behave differently for general nonlinear functions and the inexact line search. FR, DY and CD conjugate gradient methods are characterized by the strong convergence and bad practical behavior. By the other hand, LS, HS and PRP conjugate gradient methods are well-known per its good practical behavior and in the same time, they may not converge in general. A hybrid conjugate gradient method is a certain combination of different conjugate gradient methods, made in the aim to improve the behavior of these methods and to avoid the jamming phenomenon. Here we consider a convex combination of HS and FR conjugate gradient methods. The corresponding conjugate gradient parameters are T

 kHS 

g k 1 yk T d k yk

[21]

(1.5)

 kFR 

|| g k 1 || 2 || g k || 2

[16].

(1.6)

and

The first global convergence result of the FR method for an inexact line search was given by Al-Baali [1] in 1985. Under

1 , he proved that the FR method generates sufficient descent directions. 2 1 As a consequence, global convergence was established using the Zoutendijk condition. For   , d k is a descent 2 the strong Wolfe conditions with  

direction, but this analysis did not establish sufficient descent. Liu et al. [24] extended the global convergence proof of Al-Baali to the case  

1 . 2

Further, Dai and Yuan [9] showed that in the consecutive FR iterations, at least one iteration satisfies the sufficient descent property, i.e.

 g k d k  g k 1 d k 1 1 max{ , } . 2 2 || g k || || g k 1 || 2 T

T

On the other hand, HS method, as well as LS and PRP methods, possess a built-in restart feature that addresses the jamming problem: when the step xk 1  xk is small, the factor yk  g k 1  g k in the numerator of  k

HS

Hence, 

HS k

tends to zero.

becomes small and the new search direction d k 1 is essentially the steepest descent direction  g k 1 .

The HS method has the property that the conjugacy condition d k 1 yk  0 always holds, independently on the line T

search. Also,  k

HS

  kPRP holds for an exact line search.

So, the convergence properties of the HS method should be similar to the convergence properties of the PRP method.

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Especially, by Powell's example [29] the HS method with an exact line search may not converge for a general nonlinear function. The fact is that if the search directions satisfy the sufficient descent condition and if a standard Wolfe line search is employed, then the HS method satisfies Property (*). Similarly to the PRP+ method, we can let  kHS   max {  kHS ,0}, then it follows [20] that the HS+ method is globally convergent. Generally, the performance of HS, LS and PRP methods is better than the performance of methods with

g k 1

2

in the

numerator of  k .

2. Convex Combination The parameter  k in the presented method, denoted as  khyb , is computed as a convex combination of  kHS and

 kFR , i.e.

khyb  (1  k )kHS  k kFR .

(2.1)

d0   g0 , dk 1   gk 1  khybdk .

(2.2)

So, we can write The parameter  k is a scalar parameter which we have to determine. We use here the strong Wolfe line search, i.e., we are going to find a step length t k , such that:

f ( xk  tk dk )  f ( xk )  tk gkT dk

(2.3)

g ( xk  tk dk )T dk  gkT d k

(2.4)

and

Obviously, if  k  0 , then  k

hyb

  kHS , and if k = 1, then  khyb   kFR .

On the other side, if 0   k  1 , then  k

hyb

is a proper convex combination of the parameters  k

FR

and  k . HS

Having in view the relations (1.5) and (1.6), the relation (2.1) becomes

 k hyb  (1   k ) 

T

g k 1 yk T d k yk

(2.5)

|| g k 1 || 2  k  , || g k || 2

(2.6)

d0   g0 ,

(2.7)

so the relation (2.2) becomes 2

g gT y d k 1   g k 1  (1   k )  kT1 k  d k   k  k 1 2  d k d k yk gk

(2.8)

We shall choose the value of the parameter  k in such a way that the search direction d k is the Newton direction. We prove that under the strong Wolfe line search, our algorithm is globally convergent. According to the numerical experiments, our algorithm is efficient.

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The paper is organized as follows. In the next section, we find the value of the parameter k and we give the algorithm of our method. We also consider the sufficient descent property. In Section 3, we establish the global convergence of our method. The numerical results are given in Section 4.

3. A New Hybrid Conjugate Gradient Algorithm We remind to the fact that the Newton method has the quadratical convergence property for solving unconstrained optimization problems depending in part on its search direction. 2 1 So, assuming that  f ( x) exists at each iterative point for the objective function f , we are going to choose the

parameter  k such that the search direction d k 1 , defined by (2.8), satisfies the condition of the Newton direction, i.e.,

2 f ( xk 1 )1 gk 1   gk 1  khyb dk

(3.1)

This is the idea introduced in [6] and [8]. Using  khyb from (2.5)-(2.6) and multiplying (3.1) by skT  2 f ( xk 1 ) from the left, we get

 skT gk 1  skT  2 f ( xk 1 ) gk 1  (1  k )  kHS skT 2 f ( xk 1 )  dk  k kFR skT 2 f ( xk 1 )  dk . Further, we use the secant condition  f ( xk 1 ) sk  yk and we get 2

 sk gk 1   yk gk 1  (1  k )kHS yk dk T

T

T

. From (3.2)-(3.3) we get the value for  k , which we denote by 

 kNT 

NT k

(3.3)

:

 sk g k 1  yk g k 1   k yk d k FR HS T (  k   k ) yk d k T

(3.2)

T

HS

T

,

(3.4)

i.e.

 kNT 

( sk g k 1 ) || g k || 2 . T T ( g k 1 yk ) || g k || 2 ( yk d k ) || g k 1 || 2 T

(3.5)

HHSFR method Step 1. Select x0  Dom( f ),   0. Let k  0 . Calculate f 0  f ( x0 ), g0  f ( x0 ). Set d0   g0 , and the initial guess t0  1. Step 2. Test a criterion for stopping iterations, for example, if

|| g k ||  , …………

(3.6)

then stop. Step 3. Compute t k by the strong Wolfe line search, i.e., t k satisfies

f ( xk  tk dk )  f ( xk )  tk gk dk , …………

(3.7)

| g ( xk  tk dk )T dk | gk dk , …………

(3.8)

T

T

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where 0      0.5. Step 4. Let xk 1  xk  tk d k , g k 1  g ( xk 1 ). Compute sk and yk . Step 5. If ( gk 1T yk ) || gk ||2  || yk T dk gk 1 ||2  0, then  k  0, else compute  k   k

NT

.

Step 6. If  k  0, then compute  k   kHS . If  k  1,

then compute  k   kFR .

If 0   k  1, then compute  k by (2.5)-(2.6). Step 7. Generate d   gk 1   k dk . If the restart criterion of Powell

| gk 1 gk | 0.2 || gk 1 ||2 ………… T

(3.9)

is satisfied, then set d k 1   g k 1 , otherwise define d k 1  d . Set the initial guess tk  1. Step 8. k  k  1, go to Step 2. The next theorem claims that HHSFR method satisfies the sufficient descent condition. Theorem 3.1. Let the sequences {g k } and {d k } be generated by HHSFR method. Then the search direction satisfies the sufficient descent condition:

gk d k  c || gk ||2 , for all k , ………… T

where  

(3.10)

5 . 11

Proof. From HHSFR method, we know that if the restart criterion (3.9) holds, then d k   g k and (3.10) holds. So, we assume that (3.9) doesn’t hold. Then we have T

| gk 1 gk | 0.2 || gk 1 ||2 . …………

(3.11)

The following proof is by induction. For k  0, g0T d0   || g0 || 2 , so (3.10) holds. Next it holds

dk 1   gk 1  khybdk , i.e.

dk 1   gk 1  ((1  k )kHS  k kFR )dk . We can write

dk 1  (k gk 1  (1  k ) gk 1 )  ((1  k )kHS  k kFR )dk . It follows that

dk 1  k ( gk 1  kFR dk )  (1  k )( gk 1  kHS dk ), wherefrom

dk 1  k dkFR1  (1  k )dkHS1.

(3.12)

T

Multiplying (3.12) by g k 1 from the left, we get DOI: 10.26855/jamc.2018.09.002

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S.D. Snezana

gkT1dk 1  k gkT1dkFR1  (1  k ) gkT1dkHS1.

(3.13)

Firstly, let  k  0. Then d k 1  d KHS1. Remind that

d KHS1   gk 1  kHS dk .

g Let denote T 

T HS k 1 k 1

d

( g kT1 yk )( g kT1d k )   || g k 1 ||  . d kT yk 2

(3.14)

( g kT1 yk )( g kT1d k ) . d kT yk

From the second strong Wolfe line search condition, it holds

g kT1d k  gkT d k i.e.

gkT dk  gkT1dk  gkT dk . Now

ykT dk  gkT dk  gkT dk  (1   ) gkT dk . So,

g kT1d k    T yk d k 1 Now

T

 1

If (3.9) holds, then d k 1   g k 1 , so

g kT1 yk  T k 1 k 1

g d



|| g k 1 || 2 



g kT1 g k 

1 1 2   || gk 1 || , and so it is proved that d k 1 satisfies the sufficient

descent condition. If (3.9) doesn’t hold, then

T



g k 1  0.2 2

1



g k 1  1,2 2

1

 1

g k 1  2

Now it holds

g kT1d kHS1   g k 1 (1  1.2 2

So, it should hold

 1  2, 2 2 ) g k 1 . 1 1

1  2.2  0,

wherefrom



5 . 11

So, we have

gkT1d kHS1  c1 gk 1 , c1  2

1  2.2 ,  1

5 . 11

(3.15)

Now let  k  1. Then d k 1  d k 1. FR

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Let’s remind to the fact that the sufficient descent condition holds for FR method in the presence of the strong Wolfe conditions, and this fact is mentioned in [20]. So, there exists a constant c2  0, such that

g kT1d kFR1  c2 g k 1  2

(3.16)

Now suppose that

0   k  1, i.e., 0  a1  k  a2  1  From the relation (3.13), now we conclude

gkT1dk 1  a1gkT1dkFR1  (1  a2 ) gkT1dkHS1  Denote c  a1c2  (1  a2 )c1; then we finally get

gkT1d k 1  c gk 1 . 2

(3.18)

4. Convergence Analysis For further considerations, we need the following assumptions. (i)

n The level set S  {x R : f ( x)  f ( x0 ) } is bounded, i.e. there exists a constant B   , such that

x  B, for all x  S . (ii)

In a neighborhood N of S the function f is continuously differentiable and its gradient f (x) is Lipschitz continuous, i.e. there exists a constant

0  L   such that

f ( x)  f ( y)  L x  y for all x, y  N . Under these assumptions, there exists the constant Г  0 ,

(4.1)

such that

g ( x)  Г ,

(4.2)

for all x  S . [4]. Lemma 4.1. [11] Let assumptions (i) and (ii) hold. Consider the method (1.2), (1.3), where d k is a descent direction, and t k is received from the strong Wolfe line search. If

1

d k 1

 ,

(4.3)

g k =0.

(4.4)

2 k

then Theorem 4.1. Consider the iterative method, defined by HHSFR method. Let the conditions of Theorem 3.1. hold. Assume that the assumptions (i) and (ii) hold. Then either g k  0 , for some k , or

g k =0.

(4.5)

Proof. Suppose that g k  0 , for all k . Then we have to prove (4.5). Suppose, on the contrary, that (4.5) doesn’t hold. Then there exists a constant r  0 , such that

g k  r , for all k

(4.6)

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From (2.5)-(2.6) we get

 khyb   kHS   kFR g gT y  kT1 k  k 1 2 d k yk gk

(4.7)

2

(4.8)

Further, using the second Wolfe line search condition, we can get

ykT dk  gkT1dk  gkT dk

(4.9)

   1gkT dk  (1   ) gkT dk  0 .

(4.10)

So,

1 1 .  y d k  1   g kT d k

(4.11)

T k

But, all conditions of Theorem 3.1. are satisfied, so it holds:

g kT d k  c g k  g kT d k  c g k

1 1  T  gk dk c gk

2

(4.12)

2

(4.13) (4.14)

2

Now, using (4.6), we get

1 1  2 T  g k d k cr

(4.15)

So,

1 1 1    T y d k  (1   ) g k d k (1   )cr 2 T k

(4.16)

On the other hand, using Lipschitz condition (4.1), we have

g kT1 yk  ГL sk

(4.17)

 ГLD ,

(4.18)

where D is a diameter of the level set S . Now, we get

 khyb 

ГLD Г2   A. (1   )cr 2 r 2

(4.19)

Next, we are going to prove that there exists t*  0 , such that

tk  t*  0 , for all k .

(4.20)

Suppose, on the contrary, that there doesn’t exist any t* , such that t k  t *  0 . Then there exists an infinite subsequence tk  

jk

, k  K1 such that lim k  K tk  0  1

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(4.21)


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Then lim k  K 

j k 1

1

 0,

i.e. lim k  K jk 1   . 1

But, now we get

f ( xk   jk dk )  f ( xk )   jk gkT dk ,

(4.22)

f ( xk   jk 1dk )  f ( xk )  

(4.23)

j k 1

gkT dk .

Remind that   1. From (4.23), we have

f ( xk   j k 1d k )  f ( xk )



j k 1

 g kT d k .

(4.24)

From (4.24), we conclude that

gkT dk  gkT d k .

(4.25)

But, HHSFR method satisfies the sufficient descent, so g k d k  0 . T

Also,   1. So, the relation (4.25) is correct only if g k d k  0 . Then, from the second strong Wolfe condition, we get T

that g k 1d k  0 , and then it is the exact line search. So we have a contradiction. T

Now we can write

d k 1  gk 1   khyb d k . It holds sk  tk d k , so we can write d k 

(4.26)

sk . So, from (4.26), we have tk d k 1  Г  А

sk , tk

(4.27)

wherefrom

d k 1  Г  A

D  P. t*

(4.28)

,

(4.29)

Now we get

d k 1

1 2 k 1

so, applying Lemma 4.1, we conclude that

g k =0. This is a contradiction with (4.6), so we have proved (4.5).

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5. Numerical Experiments In this section we present the computational performance of a Mathematica 10 implementation of HHSFR algorithm on a set of unconstrained optimization test problems from [7]. Each problem is tested for a number of variables: n=1000, n=3000 and n=5000. The criterion used here is CPU time. We present comparisons with PRP [27], the algorithm from [22], which we call HuS here, the algorithm from [32], which we denote by LSCDMAX, the algorithm from [18], [19], here denoted by HzbetaN, using the performance profiles of Dolan and Moré[14]. The stopping criterion of all algorithms is given by (3.6), where   106 . Considering Figure 1, we can conclude that HHSFR is better than PRP method in the first part of Figure 1, and in the later part of Figure 1, it behaves similar to PRP method. In the last part of Figure 1, it coincides with PRP method. From Figure 2 we see that although there are intervals inside which HHSFR method isn’t the best method, there exists the part of the graphic with the best behavior of HHSFR method. Finally, looking into Figure 3, we can see that there exist significant parts of this graphic, inside which our method is better than the others or similar to them. Generally, from Figure 1, Figure 2 and Figure 3, which are presented below, we can conclude that HHSFR algorithm is comparable to the other algorithms.

Figure 1. (n = 1000) : Comparison between HHSFR and PRP CG methods

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Figure 2. (n = 3000) : Comparison among HHSFR, HZbetaN and HuS CG methods

Figure 3. (n = 5000) : Comparison among HHSFR, LSCDMAXand PRP CG methods

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References [1] M. Al-Baali, Descent property and global convergence of the Fletcher-Reeves method with inexact line search, IMA J. Numer. Anal., 5 (1985), 121-124. [2] N. Andrei, 40 Conjugate Gradient Algorithms for unconstrained optimization, A survey on their definition, ICI Technical Report, 13/08, 2008. [3] N. Andrei, New hybrid conjugate gradient algorithms for unconstrained optimization, Encyclopedia of Optimization, 2560-2571, 2009. [4] N. Andrei, A Hybrid Conjugate Gradient Algorithm with Modified Secant Condition for Unconstrained Optimization as a Convex Combination of Hestenes-Stiefel and Dai-Yuan Algorithms, STUDIES IN INFORMATICS AND CONTROL, 17, 4 (2008), 373-392. [5] N. Andrei, A hybrid conjugate gradient algorithm for unconstrained optimization as a convex combination of Hestenes-Stiefel and Dai-Yuan, Studies in Informatics and Control, 17, 1 (2008), 55-70. [6] N. Andrei, Another hybrid conjugate gradient algorithm for unconstrained optimization, Numerical Algorithms, 47, 2 (2008) , 143-156. [7] N. Andrei, An unconstrained optimization test functions, Advanced Modeling and Optimization. An Electronic International Journal, 10 (2008), 147-161. [8] N. Andrei, Accelerated hybrid conjugate gradient algorithm with modified secant condition for unconstrained optimization, Numer. Algorithms 54 (2010) 23-46. [9] Y.H. Dai, Y.Yuan, Convergence properties of the Fletcher-Reeves method, IMA J. Numer. Anal., 16 (1996), 155-164. [10] Y. H. Dai,, L. Z. Liao, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim. 43 (2001), 87-101. [11] Y. H. Dai, Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182. [12] Y.H.Dai, Han J.Y., Liu G.H., Sun D.F., Yin X., Yuan Y., Convergence properties of nonlinear conjugate gradient methods, SIAM Journal on Optimization, 10 (1999), 348-358. [13] S. S. Đorđević, New hybrid conjugate gradient method as a convex combination of FR and PRP methods, Filomat, 30:11 (2016), 3083-3100. [14] E. D. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. Programming, 91 (2002), 201-213. [15] R. Fletcher, Practical methods of optimization vol. 1: Unconstrained Optimization, John Wiley and Sons, Ne w York, 1987. [16] R. Fletcher and C. Reeves, Function minimization by conjugate gradients, Comput. J., 7 (1964), 149 -154. [17] J. C. Gilbert, J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM Journal of Optimization, 2 (1992), 21-42. [18] W. W. Hager, H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16, 1 (2003), 170-192. [19] W. W. Hager, H. Zhang, CG-DESCENT, a conjugate gradient method with guaranteed descent, ACM Transactions on Mathematical Software, 32, 1 (2006), 113-137. [20] W.W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pacific journal of Optimization, 2 (2006), 35-58. [21] M. R. Hestenes, E. L. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409-436. [22] Y. F. Hu and C. Storey, Global convergence result for conjugate gradient methods, J. Optim. Theory Appl., 71 (1991), 399-405. [23] J.K. Liu, S.J. Li, New hybrid conjugate gradient method for unconstrained optimization, Applied Mathematics and Co mputation, 245 (2014), 36-43. [24] G. H. Liu, J. Y. Han and H. X. Yin, Global convergence of the Fletcher-Reeves algorithm with an inexact line search, Appl. Math. J. Chinese Univ. Ser. B, 10 (1995), 75-82. [25] Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, Part 1: Theory, JOTA, 69 (1991), 129-137. [26] J. Nocedal, S. J. Wright, Numerical Optimization, Springer, 1999. [27] E. Polak, G. Ribiére, Note sur la convergence de méthodes de directions conjugués, Revue Française d'Informatique et de Recherche Opérationnelle, 16 (1969), 35-43. [28] B. T. Polyak, The conjugate gradient method in extreme problems, USSR Comp. Math. Math. Phys., 9 (1969), 94-112.

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[29] M. J. D. Powell, Nonconvex minimization calculations and the conjugate gradient method, in D.F. Griffiths, ed., Numerical Analysis Lecture Notes in Mathematics 1066 (Springer-Verlag, Berlin, 1984), 122-141. [30] M. J. D. Powell, Restart procedures of the conjugate gradient method, Mathematical Programming, 2 (1977), 241-254. [31] D. Touati-Ahmed, C. Storey, Efficient hybrid conjugate gradient techniques, J. Optim. Theory Appl., 64 (1990), 379-397. [32] X. Yang, Z. Luo, X. Dai, A Global Convergence of LS-CD Hybrid Conjugate Gradient Method, Advances in Numerical Analysis, 2013 (2013), Article ID 517452, 5 pages. [33] Y. Yuan, J. Stoer, A subspace study on conjugate gradient algorithm, Z. Angew. Math. Mech. 75 (1995), 69 -77. [34] P. Wolfe, Convergence conditions for ascent methods, SIAM Review, 11 (1969), 226-235. [35] P. Wolfe, Convergence conditions for ascent methods. II: Some corrections, SIAM Review, 11 (1969), 226 -235. [36] G. Zoutendijk, Nonlinear programming, computational methods, in Integer and Nonlinear Programming, J. Abadie, ed., North-Holland, Amsterdam, (1970), 37-86.

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Well-Posedness and Long-Time Dynamics of the Rotating Boussinesq and Quasigeostrophic Equations Maleafisha Joseph Pekwa Stephen Tladi 1

Department of Mathematics and Applied Mathematics, University of Limpopo, Private Bag 1106 Polokwane, 0727 South Africa

How to cite this paper: Tladi, M.J.P.S.. (2018) Well-Posedness and Long-Time Dynamics of the Rotating Boussinesq and Quasigeostrophic Equations. Journal of Applied Mathematics and Computation, 2(9), 379-455. DOI: 10.26855/jamc.2018.09.003 *Correspondi ng author : Maleafisha Joseph Pekwa Stephen Tladi, Depart ment of Mathemat ics and Applied Mathematics, University of Limpopo, Private Bag 1106 Polo kwane, 0727 South Africa Email: [email protected]

Abstract Most differential equations occurring in multiscale modelling of physical and biological systems cannot be solved analytically. Numerical integrations do not lead to a desired result without qualitative analysis of the behavior of the equation’s solutions. The authors study the quasigeostrophic and rotating Boussinesq equations describing the motion of a viscous incompressible rotating stratified fluid flow, which refers to PDE that are singular problems for which the equation has a parabolic structure (rotating Boussinesq equations) and the singular limit is hyperbolic (quasigeostrophic equations) in the asymptotic limit of small Rossby number. In particular, this approach gives as a corollary a constructive proof of the well-posedness of the problem of quasigeostrophic equations governing modons or Rossby solitons. The rotating Boussinesq equations consist of the Navier-Stokes equations with buoyancy-term and Coriolis-term in beta-plane approximation, the divergence-constraint, and a diffusion-type equation for the density variation. Thus the foundation for the study of the quasigeostrophic and rotating Boussinesq equations is the Navier-Stokes equations modified to accommodate the effects of rotation and stratification. They are considered in a plane layer with periodic boundary conditions in the horizontal directions and stress-free conditions at the bottom and the top of the layer. Additionally, the authors consider this model with Reynolds stress, which adds hyper-diffusivity terms of order 6 to the equations. This course focuses primarily on deriving the quasigeostrophic and rotating Boussinesq equations for geophysical fluid dynamics, showing existence and uniqueness of solutions, and outlining how Lyapunov functions can be used to assess energy stability. The main emphasis of the course is on Faedo-Galerkin approximations, the LaSalle invariance principle, the Wazewski principle and the contraction mapping principle of Banach-Cacciopoli. New understanding of quasigeostrophic turbulence called mesoscale eddies and vortex rings of the Gulf Stream and the Agulhas Current Retroflection could be helpful in creating better ocean and climate models.

Keywords Wave-Current Interactions, Atmospheric and Oceanic Dynamics, Dynamical Systems and Turbulence, Lagrangian and Eulerian Analysis, Rotating Stratified Fluid Flows.

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Content 1 Introduction 1.1 Observations of mesoscale eddies and vortex rings 1.2 Modelling of mesoscale eddies and vortex rings 1.3 Prototype models for rotating Boussinesq and quasigeostrophic flows 1.4 Limitations of standard techniques 1.5 The purpose of this research and innovation 2 Models for rotating Boussinesq equations 2.1 Models for rotating Boussinesq equations 2.1.1 Description of rotating Boussinesq equations 2.1.2 The initial-boundary value problems 3 Nonlinear analysis preliminaries 3.1 Elements of Hilbert spaces and Sobolev spaces 3.2 Properties of some functionals and operators 4 Well-posedness of solutions 4.1 Existence and uniqueness of solutions 4.1.1 Rotating Boussinesq equations 4.1.2 Rotating Boussinesq equations with Reynolds stress 5 Nonlinear stability of solutions 5.1 Aspects of stability and attraction 5.2 Rotating Boussinesq equations 5.3 Rotating Boussinesq equations with Reynolds stress 6 Quasigeostrophic flows 6.1 Description of quasigeostrophy 7 Conclusions 7.1 Discussion of results 7.2 New research and innovation directions

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CHAPTER 1 Introduction

1.1 Observations of mesoscale eddies and vortex rings Mesoscale eddies in marine science are of particular interest to meteorologists and oceanographers because satellite-tracked surface drifters and analysis of observed motion of floats suggest the structures are an important mechanism for transport of salinity, kinetic energy, available potential energy and enstrophy, the latter being the integrated squared vorticity. The meticulously detailed observations of Lutjeharms et al [74, 73] reveal that heat and salinity exchange around the Agulhas Current Retroflection takes place through mesoscale ring detachment with an associated volume transport of approximately 0.5  1.5Sv (1Sv  106 m3 s 1 ). The consequent transport within these currents is undisputedly complex as validated by observed motion of convoluted spaghetti floats plots called SOFAR and RAFOS. These include such mesoscale vortices as the Gulf Stream rings. In particular, it is evident from the investigations [68, 40, 18, 19, 55] that heat and salinity exchange around the Middle Atlantic Bight occurs through the mesoscale ring detachment of the Gulf Stream. The investigation of the flow patterns of such upwelling fronts are therefore of concern for biological studies of the highly productive ecosystems where nutrient budgets play a significant role. Furthermore, understanding the dynamics of exchange processes by mesoscale eddies is essential so that their effects in shelf-water transport can be accounted for, as observed by Joyce et al [45, 46, 47] in their utilization of hydrographic data and acoustic doppler current profiles to estimate total volume transport for a streamer of the Middle Atlantic Bight shelf water. Similarly, satellite-tracked dipole structure images of Hooker et al [40, 41, 42] provide additional details in the observation of mesoscale eddies. Specifically, advanced mesoscale observation techniques of Hooker et al [40, 41, 42] have shown long-period fluctuations in the ring eccentricity as well as vorticity and rotation rate of WCR82B, the Gulf Stream warm core ring that detached in February 1982. This important discovery serves as a paradigm that the ocean is full of mesoscale eddies that should be considered as part of a dynamically linked ring system. Nevertheless, the meager information available on the observations of mesoscale eddies using moored instruments and remote sensors made it possible to reveal various discrepancies between mathematical modelling of mesoscale eddies utilizing the system of nonlinear differential equations of geophysical fluid dynamics. 1.2 Modelling of mesoscale eddies and vortex rings Since its inception in the 19th century through the efforts of Poincare and Lyapunov, the theory of dynamical systems addresses the qualitative behaviour of dynamical systems as understood from models. From this perspective, the modelling of dynamical processes in applications requires a detailed understanding of the processes to be analyzed. This deep understanding leads to a model, which is an approximation of the observed reality and is often expressed by a system of ordinary/partial, underdetermined (control), deterministic/stochastic differential or difference equations. While models are very precise for many processes, for some of the most challenging applications of dynamical systems (such as climate dynamics), the development of such models is notably difficult. In the derivation of fluid models one tries to make models as well-posed as possible, building into them behavior designed to achieve manifestation of order-chaotic processes and Lorenz-like butterfly effects. We proceed to note that the modelling of the dynamics of the ring systems or mesoscale and synoptic scale eddies is imperative due to limitations of current oceanographic observations. Explorations using satellite-tracking of vorticity structures is usually masked by heating of the surface layer and may not always be detected by satellite measurements. In the description of strategies utilized in observing geophysical phenomena, unsatisfactory results from oceanographic and meteorological interest, is the fact that deployment of floats such as SOFAR DOI: 10.26855/jamc.2018.09.003

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and RAFOS is at the cutting edge of remote sensing technology and very expensive especially in the forecasting of mesoscale and synotic scale eddies. In this regard, the design and utilization of geophysical fluid dynamics models is imperative in supplementing costly observations and experiments described above in an effort of understanding the dynamics of mesoscale eddying coherent structures. Over the past decades, there have emerged several elegant complementary approaches driving current research in vortex dynamics work, using models from geophysical fluid dynamics. The first approach involved construction of closed form solutions which delivered a dynamically consistent theory of geophysical vortical coherent structures. For example, there are two distinct and robust developments of exact close-form vortex solutions called modons. The building blocks of modons are sinusoidal and Bessel functions. Researchers developing to these solutions proliferated in the past decades and include the translating modons group directed by Flierl et al [20, 17, 18, 79] and the rotating modons group driven by Mied et al [62, 54, 53, 68]. These exact close-form vortex solutions have been used to simulate oceanic ring systems. In Lipphardt et al [68, 18, 55, 19, 61], rotating modon solutions are utilized to investigate WCR82B. Although closed form results compare well with oceanographic observations, this approach has fallen short of elucidating the problem due to challenges associated with nonlinearities of the system of partial differential equations. Moreover, most feature models for mesoscale eddies are quasigeostrophic which we derive later and from observational evidence cannot acount for ageostrophic motions which we derive later are significant in the calculation of ring systems quantities not directly measured, such as ring energies, enstrophy, and Lagrangian transport. The baroclinic dissipative quasigeostrophic equations govern the evolution of the streamfunction, denoted by  , whenever the Rossby number is asymptotically small and are given by

q Ek  J ( , q )  q t Ro  1  J ( ,  )   t Ed  2 q   2 y z (1.1)    z  u y  v x where the nondimensional fields u , q ,  , and  are fluid velocity, potential vorticity, density and streamfunction, respectively. The geophysically relevant parameter Ro is the Rossby number, Ek is the Ekman number,  is the reference reciprocal Coriolis parameter. Here J (.,.) is the Jacobian operator and the operator

  J (.,.) represents t

advection along fluid particle trajectories. Indeed, the translating modons of Flierl et al [20, 17, 18, 79] and the rotating modons of Mied et al [62, 54, 53, 68] represent closed form vortex solutions of the above quasigeostrophic potential vorticity equation when for example dissipation terms are neglected. Other useful simplifying assumptions include the replacement of vertical coordinate by density so that the quasigeostrophic equations may lend themselves to discretization in the vertical resulting in layered models. We remark here that the introduction of two-layer quasigeostrophic

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models leads to the indepedence of horizontal velocity with respect to height and the phenomena is called barotropic which complement baroclinic, the change of horizontal velocity with height as in the above equations (1.1). In order to elucidate chaotic mixing of barotropic flows such as eddies and jets, Lagrangian tranport of the quasigeostrophic system is investigated utilizing the system

dx     ( x, y, t ), dt y dy    ( x, y, t ), dt x where  ( x, y, t ) is the streamfunction. Lagrangian transport, which is the lodestar in the analysis of transient and aperiodic geophysical phenomena, is defined with respect to the different regimes that arise in a flow. Fluid exchange between these regimes is examined by analyzing the unstable and stable invariant manifolds of certain distinguished parcel trajectories in the flow. By virtue of this technique, Pierrehumbert [80] analyzed solutions of the barotropic quasigeostophic equations with flow fields giving a large-amplitude meandering configuration and vortex motions. Numerically generated velocity vector fields of the barotropic quasigeostrophic equations using a pseudospectral scheme were developed by Flierl et al [21, 81]. Similarly, the flow fields of their numerical results yield a meandering jet with vortical coherent structures. In Miller et al [63, 64], a perturbed current evolves nonlinearly into a large-amplitude meandering configuration that propagates zonally. Although there are two dominant time scales, the flow is aperiodic and dissipative, characteristics that challenge standard techniques in the theoretical investigation of geophysical phenomena. Nevertheless, the meager information available on the system of nonlinear differential equations of geophysical fluid dynamics made it possible to reveal various discrepancies between mathematical modelling of jets and mesoscale vortices and observations of jets and mesoscale vortices using moored instruments and remote sensors. 1.3 Prototype models for rotating Boussinesq and quasigeostrophic flows In what follows, we briefly review the equations governing the dynamics of the ocean with emphasis on the rotating Boussinesq equations. The equations for the conservation of mass, energy, salt and momentum budget in a rotating framework of reference are simplified by the Boussinesq assumption which states that the effect of compressiblity is negligible in the balance equations, with the exception of the buoyancy term and equation of state. The resulting system of partial differential equations governs the flow of a viscous incompressible stratified fluid with the Coriolis force. Some preliminary remarks are offered on other geophysical fluid dynamics models that are derived from the equations of a viscous incompressible stratified fluid with the Coriolis force where use is made of a perturbation analysis of the flow fields with respect to a geophysically relevant parameter called the Rossby number, which compares advection to the Coriolis force, when we give a nondimensional formulation of the novel modelling system. At the zero order we obtain the geostrophic equations and in this limit the equations reduce to a balance between the Coriolis force and the pressure gradient. First-order terms in the Rossby number yield the quasigeostrophic equations that govern the evolution of geostrophic pressure. These remarks serve as a motivational proposal to name the flow of a viscous incompressible fluid with ambient rotation and density heterogeneity or stratification, the rotating Boussinesq equations. For reasons that will become clear in the sequel, mesoscale eddies and gyres describe geophysical eddying currents on the scale of at least the Rossby deformation radius. In the present context, Rossby deformation radius is defined as the distance covered by a wave travelling at a given speed during one inertial period. Prototype problems include dynamics of oceanic vortices, atmospheric vortex blocking, rings of the Agulhas Current, and the Gulf Stream [75, 74, 73, 68, 40, 18, 61] ring systems. As a first effort, it is instructive for mesoscale eddies and gyres to consider the beta-plane approximation. The validity of the  -plane approximation is based on scale analysis and the plausible notion of mapping the curved earth's surface

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onto a plane. Another consequence of the approximation is restriction to geophysical phenomena with length scales substantially smaller than the radius of the earth [8, 13, 77]. Thus, each approximation or assumption is suitable for only a certain range of geophysical situations and the treatment that follows will not exhaustively cover all the possible situations. The beta-plane approximation to the Coriolis f and the reciprocal Coriolis parameter f* is specified by the two-term Taylor series

f  f0  0 y f*  r0  0 

f0 y r0

where f 0  2 sin 0 is the Coriolis parameter at reference altitude southward,

0 ; y  (  0 )r0 is the coordinate oriented

 is the angular rate for a rotating framework of reference, r0 is the earth's radius, and

0  2 r cos 0 is the beta parameter. The inclusion of the distinguishing geophysical rotary term in the balance of 1 0

momentum asserts that the inertial acceleration of geophysical fluids is the decomposition of the relative acceleration and and the Coriolis acceleration due to the rotating framework of reference. The significance of the Coriolis rotary term is in the generation of planetary or Rossby waves that support geophysical jets and mesoscale vortices. Thus we study the rotating Boussinesq equations describing the motion of a viscous incompressible stratified fluid in a rotating system which is relevant, e.g., for Lagrangian coherent structures. These equations consist of the Navier-Stokes equations with buoyancy-term and Coriolis-term in beta-plane approximation, the divergence-constraint, and a diffusion-type equation for the density variation. They are considered in a plane layer with periodic boundary conditions in the horizontal directions and stress-free conditions at the bottom and the top of the layer. Additionally, we consider this model with Reynolds stress, which adds hyper-diffusivity terms of order 6 to the equations. 1.4 Limitations of standard techniques In the description of strategies utilized in modelling and observing geophysical phenomena, results are unsatisfactory due to challenges associated with nonlinearities of the systems from the mathematical analysis viewpoint, whereas from the oceanographic and meteorological interest, deployment of floats such as SOFAR and RAFOS is at the cutting edge of remote sensing technology and very expensive especially in the forecasting of mesoscale and synotic scale eddies. For example, since the quasigeostrophic equation is quadratic in the streamfunction, there is no general solution for this system of nonlinear partial differential equations. Additionally, the translating modons of Flierl et al [20, 17, 18, 79] and the rotating modons of Mied et al [62, 54, 53, 68] represent closed form vortex solutions of the quasigeostrophic model without dissipation, and as such the models have not accounted for the spin-down of mesoscale eddies. These limitations lead suggest the need for alternative perspectives for studying solutions of the models. One such perspective is that of Lagrangian transport in oceanic flows, which has benefited from progress in dynamical systems theory. By courtesy of the dynamical systems approach, the trajectories of particles of fluid corresponding to quasigeostrophic flows is determined by the kinematic problem

dX (t )  U ( X (t ), t ), X (0)  X 0 , dt where U is the velocity field satisfying the rotating Boussinesq equations and quasigeostrophic system of partial differential equations. Other techniques anchored on the dynamical systems approach entail the well-posedness and existence of Lipschitz invariant manifolds for the resulting system of partial differential equations from geophysical fluid dynamics. With this approach many properties of solutions to the feature models can be deduced without resorting to the cumbersome project DOI: 10.26855/jamc.2018.09.003

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of solving the rotating Boussinesq equations equations numerically. Moreover, well-posedness results anticipate that numerical approximations such as finite-difference schemes to the derivatives in the equations are convergent in the sense of the Lax equivalence theorem, i.e., a given finite-difference scheme to a well-posed initial-boundary value problem converges to the solution of of a partial differential equation with the rate of convergence specified by the order of accuracy of the finite-difference scheme. Existence and uniqueness of solutions for the inviscid rotating Boussinesq equations is tackled by Bourgeois and Beale [9], who also demonstrate that when the Rossby number is asymptotically small, the inviscid quasigeostrophic solutions are accurate approximations of solutions of the inviscid rotating Boussinesq equations for small initial data. The essential part of their analysis and results is in establishing necessary and sufficient criteria for the suppression of fast-scale motions in inviscid rotating Boussinesq equations Most feature models currently rely on quasigeostrophic dynamics which fortunately is amenable to slowly-modulated Hamiltonian systems theory. Thus, the splendid work of Bourgeois and Beale [9] on the inviscid ageostrophic dynamics serves as an improvement in those feature models that emphasize quasigeostrophic dynamics. For example, as by observed Joyce et al [45, 46, 47] in their utilization of moored instruments as well as Hooker et al [40, 41, 42] in the remote sensing of warm-core rings, the ring system has bimodal distribution of spin-down with short-lived mean of 54 days and a long-lived mean of 229 days. It is not clear, however, if such results qualitatively and quantitatively match those generated by dynamically consistent translating modons of Flierl et al [20, 17, 18, 79] and dynamically consistent rotating modons of Mied et al [62, 54, 53, 68]. In the splendid report of Lipphardt et al [68, 18, 55, 19, 61], it is argued that there is a need for dissipative eddy-permitting models for the examination of mesoscale eddies that spin-down under the action of viscosity. 1.5 The purpose of this research and innovation Concrete analysis of the equations for ocean dynamics can assist in the long range to build a synthesis of techniques in the framework of the theory of dynamical systems. In order to elucidate ageostrophic dynamics, it is desirable for the theory of dynamical systems to recognize and be be able to supplement, or in some cases even supplant, the expensive observational and experimental framework of oceanography developed. In this section we give the basic setup for our problems and state the main results. It should be noted at the outset, that the questions of well-posedness, stability, attractors and invariant manifolds for non-homogeneous Navier-Stoke type equations already commands a large body of work that will be utilized in the nonlinear analysis of the rotating Boussinesq equations. We address in a concrete way dynamical challenges for the rotating Boussinesq equations. The problem of rotating Boussinesq equations continue to be of interest for various purposes and the thesis aims are therefore as follows:  A sysnthesis of wellposedness, stability, attractors and invariant manifolds anticipated in ageostrophy;  The problem of ageostrophic flows is among noteworthy and accessible mathematical problems and continue to have oceanographic and meteorogical interest especially for the modelling and forecasting of mesoscale and synoptic scale eddies;  The improvement beyond quasigeostrophy is imperative since the Rossby number need not be asymptotically small especially in situations such as the Gulf Stream ring systems and the Agulhas Current Retroflection;  The rationale of the Boussinesq and the hydrostatic approximations as well as the assumption of  -plane approximation serve as paradigmatic models that validate and guide theoretical, computational and experimental research on dynamical systems. Further new accomplishments of this investigation entail the proof that the general three-dimensional space variables initial-boundary value problem for the rotating Boussinesq equations with Reynolds stress has a unique solution in the large that enjoys useful properties such as differentiability with respect to initial conditions. Furthermore, we illustrate that uniqueness and continuity with respect to initial conditions of the solution U (t ) of the initial-value problem genDOI: 10.26855/jamc.2018.09.003

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erate a dynamical system provided by continuous solution operators S (t ) , S (t ) , t  solution operators S (t ) , t  time t 





. We further prove that the

are injective and as a result we obtain the solution operators S (t ) defined for all

. The crucial properties of the solution operators which we establish are as follows:

S (t )U 0  U (t )  S (U 0 , t ) satisfying the group property

S (t )U ( s)  S ( s)U (t ) S (0)U  U  t , s  . The basic ingredients for the proofs are the Faedo-Galerkin method, the Lebesgue dominated convergence theorem and the mini-max principle. The estimates are presented in terms of geophysically relevant parameters. As regards to the nonlinear stability of the solutions of the initial-boundary value problem for the rotating Boussinesq equations, according to more refined a priori estimates that we develop, one of the criterion necessary and sufficient for nonlinear stability of the rest state is achieved with the flow energy and entropy production provided with

1 Ek 1 E (t )  {‖U‖2  ‖▽u‖2  ‖▽‖2  2 R  wdx}  2 Ro Ed which is specification of energy and the entropy production in the Sobolev H -norm whenever Ga  R  ( 1

1 Ro  2) Ro Fr

is valid. For instance, the stability criterion number Ga is given by Ga  80 as in the rigorous investigations of Galdi et al [28, 27]. It is crucial to note that the techniques employed in developing the nonlinear stability criteria have been completed without carrying out closed form wave solutions for mesoscale and synoptic eddies. The presence of stratification and Reynolds stress introduces some new considerations that need special attention since in this more general setting, exact close-form solutions are not easily accessible. Thus, some cases of oceanographic or meteorological interest such as vortex spin-down, the decay of energy in the the nonlinear stability criteria need not account for the dissipation of energy and enstrophy for mesoscale and synoptic wave motions. However, we consider the energetic stability criteria as a first effort towards elucidating such geophysical phenomena. Furthermore, we observe that nonlinear stability criteria may give the possibility of a more rigorous investigation of the numerical approximation of the solution in the sense of the Lax equivalence theorem. One of the advantages of the results presented in this thesis is that they may be considered as more refined generalization for the dissipation of the flow energy and the entropy production provided by

1 1 E (t )  ‖U (t ‖ ) 2   (u 2   2 )dx, 2 2  2 which is specification of energy and the entropy production in the L -norm. In the proof of results for the attractor of the general three-dimensional space variables initial-boundary value problem for the rotating Boussinesq equations with Reynolds stress we infer from more refined a priori estimates that the attractor is given by the  -limit set of Q  B2 2 ,

A   (Q)  s0 Cl (t s S (t )Q). where B2 2 denotes an open ball in phase space of radius 2  2 , which depends on geophysically relevant parameters. Utilitiy of the group property and continuity of of the solution operators S (t ) defined for all time $t\in \R$, gives the following invariance property of the above established attractor:

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Consequently, examining the expression for the invariance of the attractor we observe that the attractor for the general three-dimensional space variables initial-boundary value problem for the rotating Boussinesq equations with Reynolds stress is both positively and negatively invariant and consists of orbits or trajectories that are defined for all t  .

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CHAPTER 2 Models for rotating Boussinesq equations

2.1 Models for rotating Boussinesq equations 2.1.1 Description of rotating Boussinesq equations Rotating Boussinesq equations The objective of this investigation is to develop prototype geophysical fluid dynamics models as the first effort towards understanding the impact of rotating Boussinesq equations in the ocean. First, we review the fundamental assumptions and techniques involved in the derivation of a system of partial differential equations governing geophysical fluid flows [8, 13, 77]. The evolution equations for the quantities of interest, which may be scalar or vector or tensor valued are derived with the assistance of Reynolds' transport theorem, conservation laws and constitutive assumptions [5, 8, 32]. The basic fields in the description of the motion and states of geophysical phenomena are the velocity, the pressure, the density, the temperature and the salinity. The equations governing these fields consists of the mass balance or continuity equation, the momentum equation, the energy equation and the equation for salt budget. Consider a geophysical fluid occupying a region B  as depicted in figure 2.1 for the benchmark Lagrangian coherent structures utilizing the DsTool package program of Worfolk, Guckenheimer et al , [109] which greatly helped 3

codify the discipline.

Figure 2.1 Lorenz chaotic attractor For clarity, conventional vector, tensor and indicial notation are utilized. Also, for convenience in all that follows, we will adopt a Cartesian coordinate system on a spherical rotating earth. The x-axis is directed westward, the y-axis is southward and the z-axis is oriented upward. Heuristically, Reynolds' transport theorem asserts that the rate at which the integral of density for a geophysical fluid parcel over V (t ) is changing is equal to the sum of the rate performed with V (t ) fixed in its current position and the

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rate at which density is transported out of the fluid parcel V (t ) across its boundary. Consequently, the use of the divergence theorem to transform the surface integral into a volume integral gives the equation

  u.▽  ▽.u  0, t

(2.1)

for mass balance where u  (u, v, w) is Eulerian velocity of the fluid,  is density, t is time, and x  ( x, y, z) is the coordinate system. Equation 2.1 is the continuity equation for compressible fluids. Similarly, in order to obtain the equation of motion we utilize Reynolds' transport theorem, and the conservation hypothesis that the rate of change of momentum transported in a geophysical fluid parcel of volume V (t ) is balanced by the resultant of both body and surface forces. Concerning the contact or surface forces, as in the analysis of the relative motion of a fluid near a point and Cauchy's constitutive hypothesis, we represent surfaces forces by the stress tensor  . Further, it is instructive to decompose the velocity gradient ▽u into the superposition of symmetric rate-of-deformation tensor D and the skew-symmetric vorticity tensor  defined by

1 (▽u  ▽T u ) 2 1   (▽u  ▽T u ). 2 D

(2.2)

Next, we consider the Cauchy's stress tensor  using the constitutive assumption

1 3

   p  2 ( D  ▽.u )

(2.3)

for a Newtonian viscous compressible fluid. Here  is dynamic viscosity and p is pressure. In addition, consideration of the important ambient rotation of geophysical fluids and application of the simplifying effect of the continuity equation yield the following equation for the conservation of linear momentum customarily called the equation of motion

u  (2.4)  u.▽u  f  u  gk )   ▽p   u  (▽.u ), t 3 where g is gravitation, f  (0, f , f* ) is the earth's rotation, f is Coriolis parameter and f* is reciprocal

(

Coriolis parameter. Scale analysis ensures consideration of the meriodinal change in the Coriolis parameter and the reciprocal Coriolis parameter. According to the investigations [13, 77], nonlinear Rossby waves and vortex coherent structures such as alternating cyclones and anticyclones of the Gulf stream and Agulhas current span numerous degrees of latitude; and for them, it is imperative to consider the meridional change in the Coriolis parameter f and the reciprocal Coriolis parameter f* . The  -plane approximation to the Coriolis f and the reciprocal Coriolis f* is obtained by considering

y as sufficiently small and invoking the 2-term Taylor series r0 f  f 0   0 y, f*  r0  0 

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where f 0  2 sin 0 is the Coriolis parameter at reference altitude southward,

0 ; y  (  0 )r0 is the coordinate oriented

 is the angular rate for rotating framework of reference, r0 is the earth's radius and

0  2 r01 cos 0 is the beta parameter. We note that the  -plane approximation is valid whenever the term  0 y is small compared to the leading term f 0 . The Cartesian coordinate system where the beta term

 0 y is not retained is

called the f-plane approximation, that is,

f  2 sin 0 ,

(2.6)

f*  2 cos 0 .

The inclusion of the distinguishing geophysical rotary term in the balance of momentum asserts that the inertial acceleration of geophysical fluids is the decomposition of the relative acceleration and and the Coriolis acceleration due to the rotating framework of reference. The significance of the Coriolis rotary term is in the generation of planetary or Rossby waves that support geophysical jets and mesoscale vortices. And the validity of the approximation (2.5) is a consequence of restriction to geophysical phenomena with length scales substantially smaller than the radius of the earth [8, 13, 77]. Consequently, no appeal will be made in this investigation to the spherical geometry of the earth which in curvilinear coordinate system contain challenging extraneous curvature terms. Formal passage from the  -plane approximation to retain the extraneous curvature terms corresponding to the full geometry of the spherical rotating earth is treated in [8, 77]. The mathematical analysis of the resulting geophysical fluid equations with extraneous curvature terms is examined in Lions et al [105, 104]. Equations (2.1)-(2.4) are supplemented by equation of state, energy and salt budgets. In order to derive these results, utility of the first law of thermodynamics and Reynolds's transport theorem assert that the rate of change of the internal energy supplied to a geophysical fluid parcel is balanced by the heat out of the fluid parcel and the power or work done by the system against external forces. Thus, by consideration of the power produced by the surfaces forces and Fourier's consitutive hypothesis for the rate of heat and subsequent application of the continuity equation give the following temperature form of the energy equation

Cv (

T 2  u.▽T )  p▽.u  T T  ▽.(u.▽u )   (▽.u )2 t 3

(2.7)

An alternative derivation of the energy equation may be achieved by exploiting the enthalpy relation [5]. The equations must be completed with the addition of the state equation

  0 [1  T (T  T0 )   S (S  S0 )]

(2.8)

S  u.▽S   S S t

(2.9)

and the following salt budget

which asserts that ocean water conserve their salt content except in the presence of diffusion. In the evolution equations (2.7)-(2.9), $T$ is absolute temperature, Cv is heat capacity,  T is thermal conductivity, S is salinity, T0 , S0 , 0 are reference values of temperature, salinity and density,

T is coefficient of thermal expansion,  S is coefficient of

saline contraction and  S is coefficient of saline diffusion. The coupled equations (2.1)-(2.9) yield the following system of partial differential equations governing general heterogenous compressible geophysical fluid flows:

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  u.▽  ▽.u  0, t u   (  u.▽u  f  u  gk )  ▽p   u  (▽.u ), t 3 T 2  Cv (  u.▽T )  p▽.u  T T  ▽.(u.▽u )   (▽.u ) 2 , t 3   0 [1  T (T  T0 )   S ( S  S0 )],

(2.10)

S  u.▽S   S S . t Next, the equations for the conservation of mass, momentum, energy and salt budget are simplified by the Boussinesq assumption which states that the effect of compressiblity is negligible in the balance equation except in the buoyancy term and the equation of state. Employing the additive decomposition ´

´

  0   ( x, y, z, t ),  in (2.1) and retaining terms multiplied by

0

(2.11)

 0 , we obtain ▽.u  0

(2.12)

which is the continuity equation for an incompressible fluid. Physically, this statement means that conservation of mass has become conservation of volume. Another implication is the elimination of sound waves. Substitution in (2.10) of the additive decompositions (2.11) and ´

p  p0 ( z )  p( x, y, z, t ) with p0 ( z )  P0  0 gz being the hydrostatic pressure, taking into account the continuity equation (2.12) and retaining dominating terms, we obtain the set of equations for a general heterogeneous incompressible geophysical fluid flows:

▽.u  0, u 0 (  u.▽u  f  u )  g  k  ▽p   u , t T 0Cv (  u.▽T )  T T , t   0 [1  T (T  T0 )   S ( S  S0 )],

(2.13)

S  u.▽S   S S . t We set T 

T  , and   is the kinematic viscosity. 0Cv 0

In order to circumvent the challenges of directly accounting for molecular diffusion and oceanic salt finger pattern formations that result from the competitive effects of the diffusivities of heat and salt budgets, we examine the case where the salt and heat diffusivities are assumed to be equal to the eddy diffusivity, that is,

  T   S . The choice of the

eddy diffusivity [13, 77], which forms the basis of the model analyzed here, incorporates the ubiquitous geophysical phenomena that temperature, density and salinity structures of the ocean are influenced primarily by chaotic transport and mixing within jet streams and geophysical eddying currents on the scale of the Rossby deformation radius. The Rossby deformation radius is defined as the distance covered by a wave travelling at a given speed during one inertial DOI: 10.26855/jamc.2018.09.003

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period. The existence and persistence of jet streams and other geophysical coherent structures in the ocean make plausible that molecular diffusion is weak to be directly significant in the evolution equations governing the observed organized motions of a viscous incompressible stratified fluid with the Coriolis force. Application of (2.11) into the equation of state (2.8) and appying the operators

and

  u.▽ and taking (2.13) in t

consideration yield ´

´ ´   u.▽     . t

´

(2.14)

´

Thus, dropping the primes from  and p , the above Boussinesq approximation gives the following system of primitive partial differential equations

u  u.▽u  f  u )  g  k  ▽p   u , t ▽.u  0,

0 (

(2.15)

  u.▽    . t These equations govern the flow of a viscous incompressible stratified fluid with the Coriolis force. Some preliminary remarks are offered on other geophysical fluid dynamics models that are derived from the equations (2.15) where use is made of a perturbation analysis of the flow fields at the Rossby number on the order of unity or less. In particular, at the zero-order we obtain the geostrophic equations and in this limit the rotating Boussinesq equations reduces to a balance between the Coriolis force and the pressure gradient. Subsequently, first-order terms in the Rossby number yield the quasigeostrophic equations that govern the evolution of geostrophic pressure. Next, concerning the the boundary conditions we assume the fields to be periodic along the horizontal coordinates x and y with periods L1 and L2 , respectively. The flow domain is assigned on an open rectangular periodic region

B  {( x, y, z)  (0, L1 )  (0, L1 )  (0, h)} with boundary   {z  0, h} . The domain $B$ represents the flow region and the surfaces z  0 and z  h represent the lower and upper boundaries of the ocean, respectively. To system (2.15) we prescribe boundary conditions

u v   w  0 at , z z  ( z  0)  l

(2.16)

 ( z  h)  u where

l and u are constants. In order to eliminate rigid motions we consider the case

 udx   B

B

vdx  0.

We introduce the following conducting solutions

z 2

    l  ( u  l ) p  p  z l g 

g 2 z ( u  l ) 4

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u g 1  u.▽u  f  u   k   ▽p   u , t 0 0 ▽.u  0,

(2.17)

  u.▽  ( u  l )u.k    . t Rotating Boussinesq equations with Reynolds stress Since its inception in the 19th century through the efforts of Poincare and Lyapunov, the theory of dynamical systems addresses the qualitative behaviour of dynamical systems as understood from models. From this perspective, the modelling of dynamical processes in applications requires a detailed understanding of the processes to be analyzed. This deep understanding leads to a model, which is an approximation of the observed reality and is often expressed by a system of ordinary/partial, underdetermined (control), deterministic/stochastic differential or difference equations. While models are very precise for many processes, for some of the most challenging applications of dynamical systems (such as climate dynamics), the development of such models is notably difficult. We embark on the the following additive decomposition of flow fields for the rotating Boussinesq equations (2.15): ´

u  u  u , ´

p  p   p,

(2.18)

´

      . Here the time-averaging operator  .  is defined by

 u 

1



 u ( x)  ( x)dx

with  being a characteristic evolution time-scale and  ( x ) a probability density function. The terms in the splitting (2.18) represent coherent and incoherent flow fields, respectively. Furthermore, we assume that the incoherent velocity field satisfies ´

 u  0. The formal additive decomposition of the flow fields of the rotating Boussinesq equations into coherent and incoherent terms is justified partly by the realization that solutions to the primitive equations of geophysical fluid dynamics contain both fast and time scales with frequencies proportional to the Rossby number. Employing the decomposition (2.18) in the rotating Boussinesq equations (2.15) and invoking the time-averaging operator, the system

u  u.▽u  f  u )  g  k  ▽p   u  ▽. A t ▽.u  0

0 (

(2.19)

  u.▽     ▽.W . t for coherent flow fields in which A and W are Reynolds stress [77] fields. The presence of the effective dissipative terms ▽.A and ▽.W implies that although the incoherent flow fields have zero average, the momentum and density fluxes of the fluctions, which are quadratic in the incoherent fluctions, need not vanish when the time-averaging operator

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is employed. As far as the contributions of the incoherent flow fields are concerned, we observe that a prototypical fluid ´

´

parcel with fluctuation velocity u in the x-direction will transport z-momentum  0 w with a nonzero flux across the ´ ´

surface given by the quadratic relation  0  u w  Axz . It is similarly convenient to introduce the following quadratic protocols for other Reynolds stress fields ´´

Axx    0  u u , ´´

Ayy    0  v v , ´ ´

Azz    0  w w , ´´

Ayx    0  u v  Axy , ´ ´

Azx    0  u w  Axz ,

(2.20)

´ ´

Azy    0  v w  Ayz , ´ ´

Wx    0  u  , ´ ´

W y    0  v  , ´ ´

Wz    0  w   . It is useful from the oceanographic and meteorological perspectives to idealize the Reynolds stress fields as a representation of wind stress quantities. To be more specific, we consider the wind stress comprising primarily the trades that is communicated to ocean water through absorption of a shear stress and is responsible for driving oceanic circulations in the form of jets and eddying currents that impact the earth's weather [13, 77]. It is important to realize that the additive decomposition of the ageostrophic flow quantities into coherent and incoherent terms and consideration of the averaging operator to obtain the Reynolds stress fields or wind stress fields have resulted in a set of evolution equations that are not closed. We adopt the following Pedlosky closure protocols [77] for the Reynolds stress fields (2.20):

u , x v Ayy  2 0 5 , y w Azz  20 5 , z u v Ayx   0 5 (  )  Axy , x y w u Azx  0 5 (  )  Axz , z x w v Azy  0 5 (  )  Ayz , z y Axx  20

5

(2.21)

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 2u  x 2  2v Wy   ( 2  y Wx   (

2w Wz   ( 2  z

 ), x 5  ), y

5

5

(2.22)

 ), z

where  and  are parameters. The geophysical motivation of the choice of the closure protocols (2.21-2.22) will become clear in the sequel when we give a mathematical treatment of the resulting evolution equations. As a validation from a dynamical systems approach, the closure protocols lead to well-posedness in the sense of Hadamard and the existence of attractors for the solution of the system of partial differential equations. The closure protocols (2.21-2.22) are the simplest which allows existence of stability and attraction. Another validation desirable from the craft [35, 31] of stability and attraction follows from the fact that a, siginificant quantity in analyzing whether the solution of an evolution equation is bounded in some suitable norm is that of Lyapunov functional of the system with interpretations such as fictitious energy, enstrophy and entropy production which are decreasing along solutions. The equations governing the flow of ageostrophic equations with Reynolds stress (2.21-2.22) induce damping mechanisms in the nature of viscosity, diffusion, stratification and rotation effects that manifest themselves in the evolution equation with the existence of Lyapunov functionals which are equivalent to some norm induced by the inner product of solution. We now proceed with the derivation of our geophysical fluid dynamics model. Employing the Pedlosky closure hypothesis into the equations (2.19) for coherent flow fields gives the following primitive partial differential equations governing rotating Boussinesq equations with Reynolds stress:

u  u.▽u  f  u   t ▽.u  0,

0 (

  u.▽   t

6

6

u )  g  k  ▽p   u , (2.23)

   .

u and  is equivalent to stress due to the impact of the incoherent The presence of effective dissipative terms terms on the coherent flow. Alternatively, the dissipative terms account for the wind stress along the ocean surface. Next, concerning the boundary conditions, we assume the fields to be periodic along the coordinates x, y and z with peri6

6

ods L1 , L2 and L2 , respectively. The flow domain is assigned on an open rectangular periodic region

B  {( x, y, z)  (0, L1 )  (0, L2 )  (0, L3 )}. To system (2.23) we prescribe the following space-periodic boundary conditions for the flow fields

u ( x  Lei , t )  u ( x, t )

 ( x  Lei , t )   ( x, t )

(2.24)

p( x  Lei , t )  p( x, t ) where {e1 , e2 , e3} is the standard basis of

3

and with the simplifying assumption that the period L  L1  L2  L3 .

We further assume the derivatives of u ,  and p are also space-periodic. In order to eliminate rigid motions we consider the case where the average velocity and pressure vanish



B

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

B

p( x, t )dx  0.

Substitution in (2.23) reduce to the following:

u g  u.▽u  f  u  k   t 0

u

6

1

0

▽p   u ,

▽.u  0,

(2.25)

  u.▽  ( u  l )u.k   t

6

   .

2.1.2 The initial-boundary value problems Rotating Boussinesq equations In this section we introduce the nondimensional form of the equations governing the flow of a viscous incompressible stratified fluid under the Coriolis force without Reynolds stress. In order to achieve this objective, the system of partial differential equations and boundary conditions are made nondimensional with the following length and velocity scales of motions in which rotation and stratification effects are essential:

x  Lx , y  Ly , z  Hz , t  u  Uu , v  Uv , w 

L t U

UH w, L1  LL1 L

1 f 0 0UL , p  f 0 0ULp gH L cos 0 1 u  l  N 2  0 ,   g r0 Ro sin 0

L2  LL2 ,  

L represent the time scale, U , represent the horizontal velocity scale, W represent the vertical velocity U scale, L represent the length scale, H represent the vertical length scale, etc. Distinguishing attributes of geophysiwhere T 

cal fluid dynamics are due to the effects of rotation, density heterogeneity or stratification and scales of motion. To ensure the measure of the effect of variations of density defined by stratification frequency

N2  

g d 0 dz

is real(which implies static stability of the stratified fluid), we assume

d  0 for 0  z  h . It is known that an undz

stable density stratification leads to rapid convective motions. Employing the beta-plane approximation and substituting the above scaling into the system (2.17) and omitting bars from nondimensional quantities, we obtain the following nondimensional form of the equations governing the flow of a viscous incompressible stratified fluid with the Coriolis force:

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u  u.▽u )  (1  yRo 0 )u  k   k  ▽p  Ek u , t ▽.u  0, Ro(

Fr 2  Fr 2 (  u.▽ )  u.k  , Ro t Ro( Ed ) H to unity. The evolution equations have been scaled so that the relative order of each L U term is measured by the dimensionless parameter multiplying it. Here the parameter Ro  is the Rossby number Lf 0 where we set the aspect ratio

which compares the inertial term to the Coriolis force; Fr  portance of stratification; Ek 

 2

H f0

U is the Froude number which measures the imNH

measures the relative importance of frictional dissipation; Ed 

 f0 N 2

is the

Fr 2 nondimensional eddy diffusion coefficient. The ratio measures the significant influence of both rotation and Ro stratification to the dynamics of the flow. The corresponding initial-boundary conditions are specified as follows: We make the vertical density boundary conditions homogeneous using

1 z 0 N 2 g 1 2 p  p  z l  z 0 N 2 2g

    l 

Omitting bars the conditions (2.16) become

u v   w    0 at , z z

 udx   B

B

vdx  0.

And the flow domain reduces to B  {( x, y, z )  (0, L)  (0, L)  (0,1)} with boundary   {z  0,1} . Furthermore,velocity and density are given at initial time t  0

u ( x, 0)  u0 ( x )

 ( x, 0)  0 ( x ). We conclude this section with recapitulation of the set of equations that will be analyzed in this investigation. The evolutionary equation for the  -plane rotating Boussinesq equations governing geophysical fluid flows is the following initial-boundary value problem:

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u  u.▽u )  (1  yRo 0 )u  k   k  ▽p  Ek u , t ▽.u  0, Ro(

(2.26)

Fr 2  Fr 2 (  u.▽ )  u.k  , Ro t Ro( Ed ) u v   w    0 at , z z

 udx   B

B

(2.27)

vdx  0.

u ( x, 0)  u0 ( x )

 ( x, 0)  0 ( x ). Rotating Boussinesq equations with Reynolds stress In this section we introduce the nondimensional form of the equations governing the flow of a viscous incompressible stratified fluid under the Coriolis force with Reynolds stress. In order to achieve this objective, the system of partial differential equations and boundary conditions are made nondimensional with the following length and velocity scales of motions in which rotation and stratification effects are essential:

x  Lx , y  Ly , z  Hz , t  u  Uu , v  Uv , w 

L t U

UH w, L1  LL1 L

1 f 0 0UL , p  f 0 0ULp gH L cos 0 1 u  l  N 2  0 ,   . g r0 Ro sin 0

L2  LL2 ,  

Employing the beta-plane approximation and substituting the above scaling into the system (2.17) and omitting bars from nondimensional quantities, we obtain the following nondimensional form of the equations governing the flow of a viscous incompressible stratified fluid under the Coriolis force with Reynolds stress:

u  u.▽u )  (1  yRo 0 )u  k   k  Ek t ▽.u  0, Ro(

Fr 2  Fr 2 (  u.▽ )  u.k  Ro t Ed ( Ro) where we set the aspect ratio

6



6

u  ▽p  Ek u ,

Fr 2 , Ro( Ed )

H to unity. The evolution equations have been scaled so that the relative order of each L

term is measured by the dimensionless parameter multiplying it. The corresponding initial-boundary conditions are specified as follows: The space-periodicity of the fields and the vanishing of the average velocity and pressure are represented by

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u ( x  Lei , t )  u ( x, t ),

 ( x  Lei , t )   ( x, t ), p( x  Lei , t )  p ( x , t ),

 

B

B

u ( x, t )dx  0, p( x, t )dx  0,

where the flow domain B  {( x, y, z )  (0, L)  (0, L)  (0, L)} is the periodic region. We conclude this chapter with recapitulation of the set of equations that will be analyzed in this investigation. The evolutionary equation for the  -plane rotating Boussinesq equations with Reynolds stress is the following initial-boundary value problem:

u  u.▽u )  (1  yRo 0 )u  k   k  Ek t ▽.u  0,

6

Ro(

Fr  Fr (  u.▽ )  u.k  Ro t Ed ( Ro) 2

2

6



u  ▽p  Ek u , (2.29)

2

Fr , Ro( Ed )

u ( x  Lei , t )  u ( x, t ),

 ( x  Lei , t )   ( x, t ),

(2.30)

p( x  Lei , t )  p ( x , t ),

 

B

B

u ( x, t )dx  0, p( x, t )dx  0,

u ( x, 0)  u0 ( x )

 ( x, 0)  0 ( x ).

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(2.31)


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CHAPTER 3 Nonlinear analysis preliminaries

3.1 Elements of Hilbert spaces and Sobolev spaces This is an introductory chapter on the analysis of nonlinear partial differential equations using techniques from dynamical systems theory. The main goal is to present techniques and results for nonlinear PDE on unbounded domains in a self-contained format. This is achieved by developing the theory with an introduction to finite-dimensional dynamics defined by ordinary differential equations; next section is focused on dynamics in countably many dimensions, typically defined by PDE on bounded domains. Topics in ODE typically covered in a graduate introduction to the subject, including existence and uniqueness, linear systems, $\omega$-limit sets and attractors, chaos, stable/unstable/center manifold theory, and bifurcations. The author has used this as a nice way to introduce a certain perspective on nonlinear dynamics and begin to build a toolkit that will be utilized and further developed in later chapters on PDE. This is quite useful because often a proof technique appears for the first time in this finite-dimensional setting, and this base case can then be referred to when studying similar ideas in more complicated settings in the later chapters. This setting is used to introduce a variety of key ideas in PDE theory, including semigroup theory, Sobolev spaces, and nonlinear analysis, as well as to discuss important issues like smoothing and compactness. It is very helpful to see these ideas presented first in this countable context, because one can really see where the finite-dimensional theory starts to breaks down. This chapter is focused on the Navier-Stokes equation on a bounded domain, and includes basic results on local existence and uniqueness and a discussion of the important open millennium problem and of why standard methods fail to address it. Although this chapter seemed somewhat distinct from the overall theme of the article, we still appreciated its inclusion, since the Navier-Stokes equation is such a well known system and a natural place to apply many of the techniques developed in the article up to this point. The purpose of this introductory section is to develop the definitions of relevant function spaces and briefly examine propositions appropriate for analysis of the evolution equations established in the previous chapter. In the investigation of the initial-value problems, it should be proved that the solution of the equations in a variety of function spaces such as Hilbert spaces is well-posed using the auxiliary results available from this chapter. The proof of existence and uniqueness of solution to initial-value problems utilizing Hilbert spaces and Sobolev spaces techniques will be of central importance in this thesis. After the functional formulation of the problems is given, we proceed to investigate existence, uniqueness and nonlinear stability of solutions. We represents the flow region by a bounded domain denoted by the symbol B  {( x, y, z ) 

3

} of the three-dimensional Euclidean space. We assume B has a

locally Lipschitz boundary   B , that is,  is locally the graph of a Lipschitz function which holds when B is of class at least C 1 . The scalar-, vector-, and tensor-valued functions are assumed to be real and locally summable in the sense of Lebesgue, whereas their derivatives will be interpreted in the generalized sense of the theory of distributions of Schwartz. Thus, throughout this work we will use the standard Lebesgue spaces L ( B) , 1  p   , which consist of p

p -integrable functions on B with corresponding norms [84, 85]

‖u‖Lpp   |u ( x) | p dx, B

‖u‖L  supxB ess | u ( x) |

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3

where dx is a notation for the Lebesque measure on

and the assumption of identification through the a.e. equiva-

p

lence relation. The standard Lebesgue spaces L ( B) are complete normed linear spaces in the sense that every Cauchy 2

sequence in the space converges. We denote the inner product of the Hilbert space L ( B) by

(u , v )   u ( x).v ( x)dx B

and the associated norm is defined with

(u , u ) ‖u‖2L2 . The idea of an inner product can be more abstractly formalized, with (.,.) as a specific example. For a general set of elements u, v, w,

 L2 ( B) and scalars c1 , c2 ,

 C , an inner product by definition is a bilinear form that satisfies

the following conditions:

(u, v)  (v, u ) (u, v  w)  (u, v)  (u, w) (u, c1v)  c1 (u, v) (u, u )  0 with (u, u )  0  u  0. The norm of u  L ( B) , 2

‖u‖ (u, u)1/2 , in effect yields the distance of the element u from the origin of the Hilbert space. The main point to note is that the p

Cauchy-Schwarz inequality(H o lder inequality for L ( B) )

| (u, v) |‖u‖‖‖ v p

and the triangle inequality(Minkowski inequality for L ( B) )

‖u  v‖‖u‖‖‖  v are a consequence of Young's inequality

a p bq ab   , p q 1 1   1 , and the convexity of the function  (t )  t p on the interval [0, ) for 1  p   where a and b p q are nonnegative scalars. The function spaces of real continous functions on B are represented by the notation C ( B) . The spaces of infinitely continuous differentiable functions on B with a compact support in B is denoted by C0 ( B) . We define a function to be of compact support in the domain B if it is nonzero only on a bounded subdomain

Bs of the domain B with the subdomain lying at a positive distance from  . We also utilize the Sobolev spaces H k ( B) , k  1, 0,1, , with generalized derivatives up to order k belonging to L2 ( B) and norms‖ ‖ . k which are Hilbert spaces endowed with the inner product

((u, v ))

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k



( D u , D v )      

[ ] m

[ ] m

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where

  (1  n ) is a multi-index with each  i a nonnegative integer; |  | 1 

  n and D u denotes



the partial derivative | |u / x11  xn n . The Sobolev $k$-norm associated with the above inner product is defined by

‖u‖2k  ((u , u ))k whereas the H k -seminorm is given by

‖u‖2k 

‖D u‖ .   2 L2

| |k

For example, in the case of scalar-valued functions then the tedious and involved expression for the Sobolev k -norm of a function u simplifies to the following: k





‖u‖2k   |u ( x) |2 dx    | 

 1



 u ( x) |2 dx.  x

It follows that the concept of Sobolev spaces underlies the construction of a priori energy type estimates for suitable functions and their derivatives up to order k . Thus, in order to prove well-posedness and other properties of solution to 2

initial-value problems we resort to a priori energy type estimates which are derived from the bound of the L -norm of the solution in terms of the Sobolev k -norm of the initial condition. In the sequel we will require the Sobolev space

H k , per ( Bp ) of space-periodic functions in the periodic domain Bp  {( x, y, z )  (0, L1 )  (0, L2 )  (0, L3 )}. Let u ( x) represent a space-periodic function or, equivalently, the space-periodic extension of a function. Define the Fourier series expansion

u ( x) 

x

 u exp(2i k. L ) k

kZ 3

with uk  u k ,

 (1 | k | ) | u 2

k

kZ

|2  .

3

Furthermore, in the space-periodic case we assume that the flow average vanishes

1 u ( x)dx  0. | Bp | Bp k , per

k , per

The rigorous characterization of the Sobolev spaces H even ( Bp ) and H odd ( Bp ) of functions belonging to

H k , per ( Bp ) that are even and odd, respectively, in z may be found in the work of Bourgeois and Beale [9]. We will k , per

k , per

have need of the following useful result from [9] that characterizes the spaces H even ( Bp ) and H odd ( Bp ) : Lemma 3.1.1

Suppose here the flow domain is given by B  {( x, y, z )  (0, L1 )  (0, L2 )  (0,1)} with the bounda-

k , per ry   {z  0,1} . Consider a function u  H ( B) for a given integer k . Then u can be extended to H even ( Bp )

k

 all odd z derivatives of u with index less than k are zero on the boundary  of B . Also, u may be exk , per tended to H odd ( Bp )  all even z derivatives of u with index less than k including u itself are zero on the boundary  of B .

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Numerous a priori estimates and important properties have been developed for Sobolev spaces and are customarily referred to as the Sobolev's embedding theorem and the Rellich's compact injection theorem. These results are established ´

in the expos e [66, 92, 91, 90]. The Sobolev space

H 01 ( B) of functions in H 1 ( B) which vanish on  in the

1

1

sense of traces and its dual H ( B) will be employed. Duality pairing of two elements l  H ( B) and

u  H 01 ( B) is represented by the notation  l , u  . Often the interval I  (a, b) will be the interval (0, T ) ,

T  0 fixed. The following proposition from Temam [92, 91, 90] considers the sense in which integration by parts is valid in functional formulation of evolution equations: Lemma 3.1.2 Suppose Y is a given complete normed linear space with dual denoted by Y  and assume

u, f  L1 ( I ; Y ) . Then the following results are equivalent: (1) u is a.e. equal to the Lebesgue integral of f , that is, there is   Y such that for t  I a.e. t

u (t )     f ( s)ds. 0

(2) Given a test function   C (a, b) ,  0



b

a

b d u (t )dt    f (t ) (t )dt. a dt

(3) Given   Y  , then the following is valid in the sense of distributions on (a, b)

d (3.1)  u,  f ,  . dt du Whenever conditions (1)  (3) are satisfied, f  is considered the Y -valued distribution derivative of u and dt in this case u is a.e. equal to an element of C ( I ; Y ) . Next, we consider the following useful result from Temam [92, 91, 90] that will be utilized in the sequel: Lemma 3.1.3 Consider V  H  V  , where the injections are continuous and each Hilbert space is dense in the following one. If a function u  L ((a, b);V ) and its derivative 2

du 2 is an element of L ((a, b);V ) , then $u$ is a.e. dt

equal to an element of C ( I ; H ) as in the above lemma and the following

d du | u |2  2  ,u  dt dt

(3.2)

is valid in the scalar distribution sense on (a, b) since the functions t | u (t ) | and t  2

du (t ), u (t )  are both dt

1

elements of L (a, b) . ´

Now we proceed with other significant results from Sobolev spaces. By the Poincar e -Friedrichs inequality, the H

1

-seminorm | u |1 is equivalent to the Dirichlet norm

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which is a norm on H 01 ( B) . For brevity of notation, the function spaces of vector-, or tensor-valued functions which have components in one of the spaces defined above, will be denoted by the same symbol. In the investigation of the differentiability properties of generalized solutions to the initial-boundary value equations reformulated in variational form, the following Banach spaces are employed: given that I  (a, b)  is an open interval and Y is a complete normed space, then L ( I ; Y ( B))$,$1  p   , are spaces of functions from I into Y which are Banach spaces with corresponding norms p

b

‖f‖Lpp ( I ;Y ( B ))   ‖f (t ‖ ) Yp dt a

‖f‖L ( I ;Y ( B ))  suptI ess‖f (t ‖ ) Y. Thus, L ( I ; H ( B)) , 1  p   , are function spaces which consist of p -integrable functions with values in p

k

H k ( B) . Similar function spaces C ( I ; H k ( B)) are the spaces of vector functions u ( x, t ) such that u (., t ) is an element of H ( B) for all t  I and the function t  u (., t ) with values in H ( B) is continuous on I . Simik

k

k

k larly, Cb ( I ; H ( B)) denotes the space of vector functions u ( x, t ) such that u (., t ) is an element of H ( B) for

all t  I and the function t  u (., t ) with values in H ( B) is a bounded continuous function on I . k

3.2 Properties of some functionals and operators In the functional formulation of the initial-boundary value problems, the following function spaces of divergence-free or solenoidal vector functions in the sense of the theory of distributions will be utilized:

Q  {u  L2 ( B) : ▽.u  0, u satisfies Space  periodicity}, X  { :  ij  L2 ( B), ij   ji , ij  0, a.e.  B}  H  Closure of Q in L2 ( B) Space-periodicity H  1 Condition(2.27)  H1  H 0 V  Closure of Q in H 1 ( B) Space-periodicity V  1 Condition(2.27) V1  V0

‖ . V ‖ ‖ . 1 and ‖ which are Hilbert spaces equipped, respectively, with norms ‖ ‖ . H ‖ ‖ . L2 ,‖ ‖ . X ‖ ‖ . L2 corresponding to the product Hilbert structure. The rigorous construction and useful features of Hilbert spaces of divergence-free or solenoidal vector functions can be found in the noteworthy and accesible work of Ladyzhenskaya [66]. Here

H1  L2 ( B) and V1 is the space of functions in H 1 ( B) satisfying condition (2.27) which are Hilbert spaces en‖ . H1 ‖ ‖ . L2 and ‖ ‖ . V1 ‖ ‖ . 1 . And the function spaces H 0 and V0 are given by dowed, respectively, with norms ‖

H0  {u  L2 ( B) : ▽.u  0 in B, u.n  0 at  }, with n the unit outward normal on the boundary  and V0  {u V1 : ▽.u  0} ‖ . 1 . Additionally, we need the following function space of tenwhich are Hilbert spaces with norm denoted by ‖ sor-valued functions DOI: 10.26855/jamc.2018.09.003

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H k  H k ( B)  X . Let  

Ek 1 or . It is useful to set Ro Ek

As u   u , where u  D( As ) , the domain of the Stokes operator As , and  is the orthogonal projection of L ( B) onto H 0 . 2

Due to the presence of the boundary condition (2.27) and the flow region B  {( x, y, z )  (0, L)  (0, L)  (0,1)} with boundary   {z  0,1} , we have

 u  u  ▽ with  satisfying the Neumann problem corresponding to the Laplace operator:

  0 in B,   w at . z The domain of the Stokes operator As is defined by

D( As )  V0  H 2 ( B),

‖ . 2 of the Sobolev space and is endowed with the norm ‖u‖D ( As ) ‖As u‖L2 which is equivalent to the natural norm ‖ H 2 ( B) . Moreover, the Laplace's operator defined by Al     has domain given by

D( Al )  V1  H 2 ( B). In order to ensure the solution of an evolution equation is bounded in some suitable norm, we need to consider the following Hilbert space

{U   : (| u |2  |  |2 )dx  ,  , u satisfy (2.27)} B  D( L)   6 2 6 2 {U   : B (| u |  |  | )dx  , u satisfy (2.27)}. We recall that the Stokes operator As and the Laplace's operator Al may be regarded as ubounded self-adjoint positive linear operators, respectively, from D( As ) into H 0 and from D( Al ) into H1 defined by

( As u , v )  ((u , v ))  ((u , v ))1 , ( Al u , v )  ((u , v ))  ((u , u ))1 1 1 for all u , v  D( Al ) and D( As ) . Furthermore, Al and As are compact self-adjoint linear operators in H1 and

1

H 0 . The basic function space D( Ln ) is Hilbert space with norm ‖U‖D ( L ) ‖LnU‖. Further, the dual of D( L2 ) n

1

is denoted by D( L 2 ) . Under appropriate assumptions [92, 91, 90, 22], it is possible to establish existence of the following injections

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1 2

1 2

D( L )  V  D( L)    H  D( L ).

(3.3) The density and compactness of the injections will find applications in the sequel when we prove well-posedness of solutions to the initial-boundary value problems (2.29-2.31) and (2.26-2.28). In addition to the Stokes operator and Laplace operator, with the assistance of the orthogonal projection  we will need the bilinear mapping b(.,.) defined by

 (u.▽) v  b s (u , v ) Condition(2.27)  b(u , v )   (u.▽) v  b (u , v ) Space-periodicitycondition l  where u , v  D( As ) for the bilinear operator b s (u , v ) and u  D( As ) , v  D( Al ) for the bilinear operator

b l (u , v ) . The quantity of central importance in analyzing whether the solution of an evolution equation is bounded in a suitable norm is that of Lyapunov functional of the system. We will illustrate in the sequel a technique for constructing Lyapunov functionals with interpretations such as energy, enstrophy and entropy production which are decreasing along solutions. The equations governing the flow of an incompressible stratified fluid under the Coriolis force induce damping mechanisms in the nature of viscosity, diffusion, stratification and rotation effects that manifest themselves in the evolution equation with the existence of Lyapunov functionals which are equivalent to some norm induced by the inner product of solution. Specifically, the techniques used are spectral properties of the linear operator and a priori estimates. Now we recall ideas which are required in the formulation of the nonlinear stability criteria [27, 28] to be customized for use in rotating Boussinesq equations. A self-adjoint operator L is considered essentially dissipative if the following hold:

( LU ,U )  0  U  D( L) ( LU ,U )  0  U  0. The complement of essentially dissipative is called essentially non-dissipative. By the spectral theorem [27, 28, 50], essential dissipativity is equivalent to the spectrum of L being nonnegative and zero not an eigenvalue. For every ssentially dissipative operator L , the bilinear form

(U ,W ) L  ( LU ,W )  U ,W  D( L)

‖ . L given by defines a scalar product in D( L) . We denote by H L the completion of D( L) in the norm ‖ ‖U‖2L  ( LU ,U )  U  D( L). In fact the case of essential dissipativity of linear operators in Hilbert spaces developed here may be considered as an n

extension of advanced matrix theory in , the finite-dimensional space. The most crucial utility of essential dissipativity is to obtain exponential decay estimates for solutions of the initial-boundary value problems (2.29-2.31) and (2.26-2.28) that yield asymptotic stability of solutions. The ideas can be less abstractly formalized in the finite dimensional case. First, we observe that each basis of

n

renders a scalar product denoted by (.,.) . It is known that if L is

an n  n matrix for which

1  Re( )   2 for all

where

(3.4)

 in the spectrum  ( L) , of L then there is a basis of n in which 1 (u, u)  ( Lu, u)   2 (u, u)

(3.5)

1 and  2 are nonzero constants. Thus if we set u (t ) to be a solution of a system of linear ordinary differen-

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du  Lu, u (0)  u0  H , dt

(3.6)

with the matrix L satisfying (3.4-3.5)then we obtain

| u0 | exp(1t ) | u(t ) || u0 | exp( 2t ).

(3.7)

where | . | is the Euclidean norm. Consequently, if Re( )  0   ( L) we obtain asymptotic stability and attraction of the rest state and whenever Re( )  0   ( M ) then the rest state is unstable in the sense of Lyapunov. We emphasize that the results on linear stability (3.7) belong to a finite-dimensional space. Formal passage from the fin

nite dimensional space to the infinite-dimensional case of Hilbert spaces will be established in the sequel by the virtue of energetic stability criteria customized for solutions of initial-value problem. We consider the orbit or trajectory [91, 90, 14, 15] of motion for an initial-value problem starting at the initial condition $u_0\in H$ to be defined by the set

  {(u, t ) : u  H , t 



}  t  S (t )u0 .

Uniqueness and continuity with respect to initial conditions of the solution u (t ) generate a dynamical system prescribed by continuous solution operators S (t ) , t  ever the solution operators S (t ) , t 





which is a mapping of the phase space H into itself. When-

are injective then we denote by S (t ) the inverse mapping of S (t ) H

onto H . As a result we obtain the solution operators S (t ) defined for all time t 

. The crucial properties of the

solution operators which will be employed are as follows:

S (t )u0  u(t )  S (u0 , t ) satisfying the group property

S t  u  s   S  s  u t  S (0)u  u  t , s  .

That the solution operators S (t ) are continuous is a consequence of the continuity of u (t ) in time and in initial conditions. The group property is a consequence of the injectivity of the solution operators which is equivalent to the backward uniqueness of solution for the initial-value problem. In this work we shall adopt the definition that a set Q  H is an invariant set [14, 15, 91, 90, 23, 22] for the dynamical system if the following useful technique for the absorbing or trapping of trajectories in phase space is valid

S (t )Q  Q, t  . We remark that an invariant set is both positively and negatively invariant and consists of orbits or trajectories that are defined for all t  . It is known that of central significance in the analysis of dynamical systems is the asymptotic behavior of trajectories such as homoclinic and heteroclinic orbits[31, 35]. Geophysical fluid dynamical processes are identified with trajectories of a dynamical system in a suitable phase space and the investigation of asymptotic behavior is reduced to the structure of  -limit sets of these orbits. Invoking these ideas, we define the  -limit set of Q  H with

(Q)  s0 Cl (t s S (t )Q), where the closure is taken in the Hilbert space H. From the fact that the arbitrary intersection of closed sets is closed, we deduce that  -limit set of Q given by  (Q) is closed. An invariant set   H is defined as an attractor if there is a neighborhood  of Q such that

 ()  Q.

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Beside stability and attraction, it is important to note that other properties for the solution of the initial-boundary value problems are also possible. Differentiability of solution that lies in the fact that trajectories nearby a given trajectory yield a variation on the base trajectory which is approximated, to first order, by a linear nonautonoumous differential equation. Thus, the equation of variations is obtained by linearizing the nonlinear initial-value problem about a solution. The technique of proving injectivity and differentiability of the solution operators is of great importantance as it uses the linearized flow or the variational equation and this is a central geometric construction in dynamical systems. Consider a ´

given dynamical system u0  S (t )u0 generated by the initial-value problem. It is Fr e chet differentiable in a Hilbert space  with differential L(t , u0 ) :   (t ) given by the solution of the corresponding variational equation if the following is satisfied

‖S (t )v0  S (t )u0  L(t , u0 ).(v0  u0 ‖ )2  o‖ ( v0  u0‖) ‖v0  u0‖2 ´

as v0  u0 . The Fr e chet derivative has properties similar to the derivative in finite dimensional space: the chain rule holds and the mean value theorem is valid [35]. The assertion of the following Gronwall's inequality [36, 92, 91, 90] will obtain use in establishing a priori energy type estimates, for instance, on the difference between two solutions of the ´

nonlinear initial-value problems which as always is required in proving backward uniqueness and Fr e chet differentiability of solution. Lemma 3.2.1 Suppose f , g and u are nonnegative locally integrable functions on an interval I  (t0 , ) with the derivative of u ,

du , locally integrable on I  (t0 , ) and satisfying the differential inequality dt du  gu  f for t  t0 . dt

(3.8)

If in addition the estimates,



t 

t



t 

t



t 

t

hold for

g ( s)ds  1 , f ( s)ds   2 , u ( s)ds   3 ,

t  t0 where  , 1 ,  2 and  3 are nonnegative constants then the following Gronwall's inequality is val-

id:

u (t   )  (

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3   2 )exp(1 )  t  t0 . 

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CHAPTER 4 Well-posedness of solutions

4.1 Existence and uniqueness of solutions In the functional formulation of the initial-boundary value problems, the following function spaces of divergence-free or solenoidal vector functions in the sense of the theory of distributions will be utilized:

Q  {u  L2 ( B) : ▽.u  0, u satisfies Space  periodicity}, X  { :  ij  L2 ( B), ij   ji , ij  0, a.e.  B}  H  Closure of Q in L2 ( B) Space-periodicity H  1 Condition(2.27)  H1  H 0 V1  Closure of Q in H 1 ( B) Space-periodicity V  Condition(2.27) V1  V0 ‖ . V ‖ ‖ . 1 and ‖ which are Hilbert spaces equipped, respectively, with norms ‖ ‖ . H ‖ ‖ . L2 , ‖ ‖ . X ‖ ‖ . L2 corresponding to the product Hilbert structure. The rigorous construction and useful features of Hilbert spaces of divergence-free or solenoidal vector functions can be found in the noteworthy and accesible work of Ladyzhenskaya [66]. Here

H1  L2 ( B) andV1 is the space of functions in H 1 ( B) satisfying condition (2.27) which are Hilbert spaces endowed, respectively, with norms ‖ ‖ . H1 ‖ ‖ . L2 and ‖ ‖ . V1 ‖ ‖ . 1 . And the function spaces H 0 and V0 are given by

H0  {u  L2 ( B) : ▽.u  0 in B, u.n  0 at  }, with n the unit outward normal on the boundary  and V0  {u V1 : ▽.u  0} ‖ . 1 . Additionally, we need the following function space of tenwhich are Hilbert spaces with norm denoted by ‖ sor-valued functions

H k  H k ( B)  X . Let  

Ek 1 or . It is useful to set Ro Ek

As u   u , where u  D( As ) , the domain of the Stokes operator As , and  is the orthogonal projection of L ( B) onto H 0 . 2

Due to the presence of the boundary condition (2.27) and the flow region B  {( x, y, z )  (0, L)  (0, L)  (0,1)} with boundary   {z  0,1} , we have

 u  u  ▽ with  satisfying the Neumann problem corresponding to the Laplace operator:

  0 in B,

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  w at . z The domain of the Stokes operator As is defined by

D( As )  V0  H 2 ( B),

‖ . 2 of the Sobolev space and is endowed with the norm ‖u‖D ( As ) ‖As u‖L2 which is equivalent to the natural norm ‖ H 2 ( B) . Moreover, the Laplace's operator defined by Al     has domain given by

D( Al )  V1  H 2 ( B). In order to ensure the solution of an evolution equation is bounded in some suitable norm, we need to consider the following Hilbert space

{U   : (| u |2  |  |2 )dx  ,  , u satisfy (2.27)} B  D( L)   6 2 6 2 {U   : B (| u |  |  | )dx  , u satisfy (2.30)}. We recall that the Stokes operator As and the Laplace's operator Al may be regarded as ubounded self-adjoint positive linear operators, respectively, from D( As ) into H 0 and from D( Al ) into H1 defined by

( As u , v )  ((u , v ))  ((u , v ))1 , ( Al u , v )  ((u , v ))  ((u , u ))1 1 1 for all u , v  D( Al ) and D( As ) . Furthermore, Al and As are compact self-adjoint linear operators in H1 and

1 2

H 0 . The basic function space D( L ) is Hilbert space with norm ‖U‖D ( L ) ‖L U‖. Further, the dual of D( L ) n

n

n

1 2

is denoted by D( L ) . Under appropriate assumptions [92, 91, 90, 22], it is possible to establish existence of the following injections 1

1

D( L2 )  V  D( L)    H  D( L 2 ).

(4.1)

The density and compactness of the injections will find applications in the sequel when we prove well-posedness of solutions to the initial-boundary value problems (2.29-2.31) and (2.26-2.28). In addition to the Stokes operator and Laplace operator, with the assistance of the orthogonal projection  we will need the bilinear mapping b(.,.) defined by

 (u.▽) v  b s (u , v ) Condition(2.27)  b(u , v )   (u.▽) v  b (u , v ) Space-periodicitycondition l  where u , v  D( As ) for the bilinear operator b s (u , v ) and u  D( As ) , v  D( Al ) for the bilinear operator

b l (u , v ) .

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4.1.1 Rotating Boussinesq equations The purpose of this section is to develop results of existence, uniqueness and differentiability of solution for the initial-boundary value problem of  -plane rotating Boussinesq equations (2.26-2.28). We prove the significant aspects of well-posedness of solution utilizing the machinery developed above and additional accessible techniques constructed in this section. The functional formulation for the evolution equation of  -plane rotating Boussinesq equations (2.26-2.28) now takes the form of the following initial value problem in the Hilbert space  :

dU  LU  N (U )  0 dt U (0)  U 0

(4.2)

where U  (u, v, w,  ) . The linear operator LU and the nonlinear operator N (U ) are given by

N U   B U ,U   FU



Ro w) Ro Fr 2 LU  TU  SU  Ek   Ro  u   As u  TU       1    Al      Ed  FU  (0, 0, 

,

  u  ▽u   b s ( u , u )  B(U ,U )  B(U )      u ▽   bl (u ,  ) 

 0 1  1  y  Ro  1 0 S   ( ) 0 0 Ro  0 0

0 0  0 0  U  D( L)  D(T ). 0 0  0 0

This problem as the Navier-Stokes equations in the presence of stratification but without Coriolis terms, has been examined by Temam et al [92, 91, 23, 22] and cited works therein. In order to develop basic results concerning whether there exists a unique solution in the large for the general three-dimensional space variables for initial-value problem (4.2), we derive a priori energy type estimates useful in proving that (4.2) generate a dynamical system, denoted, U (t )  S (t )U 0 , where U (t ) is the unique solution uniformly bounded in finite-time and S (t ) is a group of continuous nonlinear solution operators. The principal result concerning the existence of such a unique solution will be proven by the utility of the Faedo-Galerkin technique, a priori energy type and Leray-Schauder principle. As for the linear operator LU of the initial-value problem (4.2), the following is valid

( LU ,U )  (TU ,U ) 

Ek 1 ‖▽u‖2  ‖▽‖2 Ro Ed

(4.3)

It follows from [27, 28, 66, 50] that the operator $T$ is self-adjoint and its spectrum  (T ) satisfies  (T )  [0, ) . The following result on the spectrum of L from [27, 28, 66, 50] will be utilized:

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Lemma 4.1.1  ( L)  [0, )  positive eigenvalues n  n (

, where

is either empty or an at most denumerable set consisting of isolated,

1 Ro  2 ) with finite multiplicity such that Ro Fr   1  2   n  ,

clustering at zero. Furthermore, whenever

1 Ro  2  Ga Ro Fr then

  whereas if 1 Ro  2  Ga Ro Fr

we have

  . Also, zero is not an eigenvalue whenever the following is satisified: 1 Ro  2  Ga. Ro Fr

Before giving the proof of the lemma, we make remarks and investigate the steady linear versions of the nonlinear initial-value problem. The examination demonstrate that the steady linear system has a unique solution and provide useful properties which are employed in the investigation of the nonlinear initial-value problem. We start by recalling that when  ( L)  (0, ) which hold if

1 Ro  2  Ga Ro Fr then we obtain

( LU ,U )  (TU ,U )  ‖U‖2  0 1 which implies the bilinear form induced by L is coercive and L is compact and self-adjoint. This establishes exist´

ence and uniqueness of solution to the steady linear case by the Riesz-Fr e chet representation theorem or the Lax-Milgram lemma. Through the utility a result from [92, 27, 28, 66, 50],  is the smallest eigenvalue of Laplace's operator and Stokes operator. Also, the symmetry of the bilinear form a(U ,W ) , together with the coerciveness of

a(U ,U ) imply the existence of a sequence of positive eigenvalues and a corresponding sequence of eigensolutions which yield an orthonormal basis of  such that LWi  iWi

0  1  2    i  , and

(4.5)

i   as i   . Moreover, 1

1

‖L2U‖ 12‖U‖ 1 2 1

1 2

(4.6)

‖LU‖  ‖L U‖ Next, we focus our attention to establishing the above lemma. The flavor of the proof is similar to that given in Kato [50] for the existence of eigensolutions and eigenvalues of perturbed elliptic operators Proof: According to a perturbation theorem [27, 28, 50, 66], if TU is closed and SU is relatively compact with respect to TU then  ( L  T  S ) and  (T ) differ by at most a denumerable number of isolated, positive eigenvalDOI: 10.26855/jamc.2018.09.003

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ues of finite multiplicity clustering at zero. Thus it suffices to show that S is relatively compact with respect to T since from [27, 28, 50, 66] we have  (T )  [0, ) . By definition, S is relatively compact with respect to T if for any sequence {U n }   ,

‖U n‖‖  TU n‖  , uniformly in n,

(4.7)

then there is a subsequence U ´  {U n } such that SU ´ is strongly convergent. With strong convergence, we can n

n

make strong statements. Indeed, the following inequality is valid

1  y  Ro 2 1 Ro 2 2 ‖SU ‖  [5( ) (  2 )2 ‖ ] U‖ ´ ´ n n Ro Ro Fr (4.8) 1  y  Ro 2 1 Ro 2 2  [5( ) (  ) ] Ro Ro Fr 2 and by the courtesy of Ascoli-Arzela theorem {SU ´} contains a uniformly convergent subsequence which is a Cauchy n

sequence in a complete normed space  . Thus SU ´ is strongly convergent which illustrates that S is relatively n

compact with respect to T and the claim of the lemma is achieved. We proceed put together a series of results that will be employed in establishing properties of solution to the initial-value problem (4.2). The following mini-max principle from [92, 23] will be utilized:

lim sup‖ (t ‖ ) 2 ‖‖

(4.9)

t 

where ‖‖ is the volume of  which represents the flow region   {( x, y, z ) 

3

} . Let   max{

Ek 1 , }. Ro Ek

By the orthogonal property of the nonlinear operator, we have

( B(U ),U )   [(u u  u )  (u  )  ]dx  0 

(4.10)

Taking the inner product of (4.2) with U , utilizing Cauchy-Schwarz inequality and Young's inequality gives the following:

d 2 ‖U‖2  ‖ 1 U‖ dt 1 d 2  ‖U‖  ‖L2U‖2 dt  Ro 1 2  1( 2  ) ‖‖2 .  Fr Ro

(4.11)

The consideration of Gronwall's inequality into (4.11) and invoking the mini-max principle (4.9) provides a priori estimate

‖U (t ‖ ) 2 ‖U 0‖2exp( t )  (

Ro 1 4 12  ) ‖‖(1  exp( t )) Fr 2 Ro  2

(4.12)

which is asymptotic stability with flow energy and entropy production provided by

1 1 E (t )  ‖U (t ‖ ) 2   (u 2   2 )dx 2 2 

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and exponential dissipation rate   max{

Ek 1 , } . Furthemore, in the limit as time goes to infinity we obtain Ro Ek

Ro 1 4 12 (4.13) lim sup‖U (t ‖ )  ( 2  ) 2‖‖ 12 t  Fr Ro  which shows U (t ) is uniformly bounded for all time in  . Next, by the mini-max principle (4.9) and uniform bound 2

 there exists T1  T1 (U 0 ,  ) such that for t  T1 then

(4.13), it follows that for any U 0   and

1 2

‖ (t ‖ ) ‖‖  

(4.14)

Integration of the second inequality in (4.11) and the courtesy (4.14) provide a priori energy type estimate



T1

1

‖L2U (t ‖ ) 2 dt 

1

{‖U 0‖2  (

1 Ro 1 2 1 2  ) ‖ (  ‖   )2 }  112 . 2 Fr Ro 

 Taking the inner product of (4.2) with LU and making use of Cauchy-Schwarz inequality, Young's inequality and 0

Sobolev inequality yield the estimates 1 1 d ‖L2U‖2  1‖L2U‖2 dt 1 d  ‖L2U‖2  ‖LU‖2 dt 1 c02 Ro 1 2 2  2 ( 2  ) ‖‖  ‖L2U‖6 . Fr Ro 4

By setting c  max{

1 c02 Ro 1 , 2 ( 2  ) 2 ‖ ( ‖2   ) 2 } then (4.15) and (4.14) imply that 4 Fr Ro 1 1 d 2 2 (1‖L U‖ )  c(1‖L2U‖2 )3 . dt

(4.15)

(4.16)

Invoking Gronwall's inequality in the differential inequality (4.16) gives 1

1

‖L2U‖2  1‖L2U 0‖2 1

3

( L2U 0‖)  for t  T2 ‖

1 2

(4.17)

. Consequently putting the pieces together we obtain

8c(1‖L U 0‖ ) 2

1 2

1 2

2 Supt[0,T2‖ ‖L U 0‖2  20 ] L U‖  1 2

(4.18)

which shows that U (t ) is bounded for finite-time T2 in D( L) . Integrating the differential inequality (4.16) provide a priori energy type estimate 1 c02 6 Ro 1 2 2 2 ) dt  {‖L U 0‖  2 ( 2  ) ‖ ( ‖   )  20 }  212 . 0 ‖LU (t‖  Fr Ro 4 T2

2

1

1 2

2

We now state existence, uniqueness, continuity and differentiability with respect to initial conditions of solution to the initial-value problem (4.2) and utilize the above a priori estimates to prove the results.

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Proposition 4.1.2 Under the above hypothesis for U 0   given, (4.2) generate a dynamical system U (t )  S (t )U 0 satisfying 1

U  C ([0, T ];  )  L2 (0, T ; D( L2 )) 1

for all T  0 and if U 0  D( L2 ) then 1

U  C ([0, T ]; D( L2 ))  L2 (0, T ; D( L)) for all T  0 where T  min{T1 , T2 } is finite-time. Proof: For fixed m , the Faedo-Galerkin approximation m

U m   gim (t )Wi i 1

of the solution U of (4.2) is defined by the finite-dimensional system of nonlinear ordinary differential equations for

gim (t ) given by dU m  LU m  Pm N (U m )  0 dt U m (0)  PmU 0

(4.19)

1

1

where Pm is the orthogonal projection in , D( L2 ), D( L 2 ) and D( L) onto the space spanned by the m eigenfunctions of L given in (4.5). Let X be the finite-dimensional Hilbert space spanned by the m eigenfunctions 2 of L given in (4.5) with inner product ((.,.)) induced by D( L) . Consider a closed ball of radius  21 contained in

D( L) . Integration of (4.19) and the above a priori energy type estimates provide a continuous mapping defined by the integral of (4.19) from the closed ball of radius  21 into itself. Thus, by the Leray-Shauder principle \cite{HALE,LADY} 2

the finite-dimensional system of nonlinear ordinary differential equations (4.19) has at least one solution U m inside the ball of radius  21 . The passage to the limit 2

m   and Tm  T follows from the following a priori estimates: Utilizing the differential inequality (4.11) yields

d 2 ‖U m‖2  ‖ 1 U m‖ dt  Ro 1  1 ( 2  ) 2‖ m‖2  Fr Ro

(4.20)

which together with a priori estimate ()4.12 and the subsequent one imply U m remains bounded in 

1 2

L ((0, T );  )  L (0, T ; D( L )). 2

This useful result combined with the weak compactness implies there is a subsequence also denoted by U m and 

1 2

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such that 1  U  L2 (0, T ; D( L2 )) weakly Um     weak-star. U  L (0, T ;  )

Invoking the same inequalities that let to the differential inequality (4.15), give the boundeness of N (U m ) and 2

1 2

Pm N (U m ) in L (0, T ; D( L )) . Furthermore, from (4.19), it follows that 2

dU m is also bounded in dt

1 2

L (0, T ; D( L )) . Utility of this result and weak compactness yield 1 dU m dU 2   L (0, T ; D( L 2 )) dt dt

weakly. The assertion of Lebesgue dominated convergence theorem as well as the above convergence results yield

U m  U  L2 (0, T ;  ) strongly. With strong convergence we can declare strong assertions. We pass the limit in (4.19) and obtain the required result (4.2). We obtain 1

U  C ([0, T ];  )  L2 (0, T ; D( L2 )). The result 1 2

U  C ([0, T ]; D( L ))  L2 (0, T ; D( L)) 1 2

We conclude that if U 0  D( L ) and by the Faedo-Galerkin technique 1

U  C ([0, T ]; D( L2 ))  L2 (0, T ; D( L)) for all T  0 . According to Gronwall's inequality, uniqueness and continuity with respect to initial conditions of solution U (t ) follows from considering the difference between two solutions of the initial-value problem (4.2) and employing a priori estimates (4.12-4.18). Uniqueness and continuity with respect to initial conditions of solution U (t ) generate a dynamical system which is described by continuous solution operators S (t ) , t 

defined by


S (t )U ( s)  S ( s)U (t ) S (0)U  U  t , s 

(4.21)

such that t  s  T . That the solution operators S (t ) are continuous is a consequence of the continuity of U (t ) in time and in initial conditions. The group property is a consequence of the injectivity of the solution operators which is equivalent to the backward uniqueness of solution for the initial-value problem (4.2). Next, we turn our attention to es´

tablish uniqueness, continuity and Fr e chet differentiability with respect to initial conditions of solution U (t ) for the initial-value problem (4.2) with the work of Temam [92, 91, 90] as an exploration upon which we build. The technique of the proof uses the linearization of the evolution equation (4.2) about the difference of two given solutions. Let u and

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v be solutions of the initial value problem (4.2) corresponding to the initial conditions u(0)  u0 and v(0)  v0 , respectively. Then the difference w  v  u satisfies dw  Lw  N (v)  N (u )  0 (4.22) dt w(0)  v0  u0 where

N  v   N  u   B  v, v   B  u , u   B  v, w   B  w, u 

(4.23)

 B  u, w   B  w, u   B  w, w  .

The linearization of the nonlinear equation (4.22) about a given solution reduces to the variational equation

d  L  l0 (t )  0 dt  (0)  v0  u0  

(4.24)

where l0 (t )w  B(u(t ), w)  B(w, u (t )) and l0 (t ) is a linear bounded operator from

1 2

1 2

D( L ) to D( L ) . Con-

sequently, the equation of variation is a linear nonautonomous ordinary differential equation. The difference   w   satisfies

d  L  l0 (t )  l1 (t ; w(t ))  0 dt  (0)  0

(4.25)

where l1 (t; w(t ))  B(w, w) . From the last equality of (4.23), we have N (v)  N (u)  l0 (t )w  l1 (t; w) . Proposition 4.1.3 If u0   and u is the unique solution of the nonlinear initial-value problem (4.2) then the corresponding variational equation (4.24) has a unique solution  satisfying 1

  C ([0, T ];  )  L2 (0, T ; D( L2 )) ´

for all T  0 where T  min{T1 , T2 } is finite-time. Moreover, the dynamical system u0  S (t )u0 is Fr e chet differentiable in  with differential L(t , u0 ) :   (t ) given by the solution of the variational equation. Proof: Existence and uniqueness of solution to the variational equation (4.24) is proved by the courtesy of the splendid Faedo-Galerkin technique and the above a priori energy type estimates. Taking the inner product of the equation (4.22) with w and application of Cauchy-Schwarz inequality, Young inequality and Sobolev inequalities yield 1 d 2 ‖w‖ ‖L2 w‖2  k 2‖w‖2 dt

where k  c1 21 and

 21 is given in the estimate (4.18). By the virtue of the Gronwall's inequality we obtain ‖v(t )  u(t ‖ ) 2  exp(k 2T ‖ ) v0  u0‖2

DOI: 10.26855/jamc.2018.09.003

417

(4.27)


M.J.P.S. Tlad i

t  (0, T ) . The sharp a priori estimate (4.27) illustrates forward uniqueness and continuity with respect to initial conditions of solution. Furthermore, invoking the estimates (4.26) yield 1

t

) 2 ds  exp(k 2T ‖ ) v0  u0‖2 . ‖L2 w(s‖ 0

Additionally, taking the inner product of equation (4.25) with  we obtain

c2 1 d 1 1 ‖‖2  ‖L2‖2  k 2‖‖2  1 ‖L2 w(t ‖ )3 dt 2 1

(4.28)

By the utility of Gronwall's inequality we obtain the estimate

‖ (t ‖ )2

c12

T

1

) 3ds  ‖L2 w(s‖

1

0

for all t [0, T ] . Consequently it follows that

‖ (t ‖ )2

c12

1

exp(3k 2T / 2)‖v0  u0‖

(4.29)

which gives the required result


as v0  u0 . Thus, the dynamical system S (t ) : u0  u (t ) is Fr e chet differentiable at u0 in the Hilbert space  . Next, we investigate the problem of proving backward uniqueness of solution for the initial-value equation (4.2) that establish the injectivity properties and group properties of the solution operators S (t ) . The backward uniqueness of ´

solution is proved in approximately the same way as Fr e chet differentiability result using the standard log-convexity method that has been employed in [3, 91]. In order to motivate this objective of the injectivity of the solution operators, consider two solutions u and v of the initial-value problem elaborated above such that at time t  t* both u and

v satisfy the equation (4.2) for t  (t* ò, t* ) and u(t* )  v(t* ) . Given that this hypothesis is valid, backward uniqueness of the solution operators is accomplished if u(t )  v(t ) for all time t  t* whenever the solutions are well-defined. Alternatively, suppose the solution operators satisfy

S (t )  u(t  s) t  0 s  then for   (0, ò) , the problem of backward uniqueness is whether the following hold:

S ( )u(t*   )  S ( )v(t*  )  u(t*   )  v(t*   )

(4.30)

which is the injectivity of the solution operators S ( ) . Proposition 4.1.4 Suppose the following hold 

1 2

u, v  L (0, T ;V  D( L )  L2 (0, T ; H  D( L)) then the difference w  v  u satisfies the differential equation (4.22) for t  (0, T ) and the solution operators S (t ) are injective. DOI: 10.26855/jamc.2018.09.003

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We give a sketch of the proof due to Temam [91]. Proof: We will use the notation $\varphi=\varphi(t)$ for the quotient

‖ (t ‖ ) V2 ‖ (t ‖ ) 2H

 

( L (t ),  (t )) . ( (t ),  (t ))

Invoking Cauchy-Schwarz inequality, Young's inequality, Sobolev inequalities, continuity properties of the nonlinear operator B( w, w) established above and the fact that

( Lw   w, w)  0 we obtain

dw , w)) ‖w‖V2 dw 1 d   dt 2  ( , w) 2 dt ‖w‖ ‖w‖4 dt 1 dw  ( , Lw   w) ‖w‖2 dt 1  ( N (v)  N (u )  Lw, Lw   w) ‖w‖V2 ((

(4.31)

‖Lw   w‖ 1   ( Lw, N (v)  N (u )) 2 ‖w‖ ‖w‖2 ‖Lw   w‖2 ‖N (v)  N (u ‖ )2   2‖w‖2 2‖w‖2 ‖Lw   w‖2   k 2 2 2‖w‖ where N (v)  N (u) is given in (4.23) and the function k (t )  k1 (t )  B(w, w) is defined by the mapping 2

1

1

1

1

k1 (t ) : t  c‖ ( u(t ‖ ) V2‖Lu(t ‖ ) 2 ‖v(t ‖ ) V2‖Lv(t ‖ ) 2 )  L2 (0, T ). Utility of Gronwall's inequality into (4.31) gives t

 (t )   (0)exp(2 k 2 (s)ds) t  (0, T ). 0

Thus, if the difference w  v  u satisfies the above conditions and

w( )  0  w(t )  0, t [0, T ]. We proceed by contradiction. Hence, suppose that ‖w(t0 )  0‖ for t0  [0, T ) . As a result by continuity we have

‖w(t )  0‖ on some open interval (t0 , t0  ò) and we employ the notation t*  T for the largest time for which

w(t )  0 t [t0 , t* ). ) is well-defined and the fact It follows that w(t* )  0 . In the interval [t0 , t* ) the mapping given by t  log‖w(t ‖ that w(t ) satisfies the differential equation (4.22) we obtain

d 1 log  2  k 2 . dt ‖w‖ DOI: 10.26855/jamc.2018.09.003

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Consequently for the time t  [t0 , t* ) the following is valid: T 1 1 log  log   2 (s)  k 2 (s)ds t0 ‖w‖ ‖w(t0 ‖ )

which illustrates the boundedness of

1 from above as the time t  t* from below and this is a contradiction. w(t )

4.1.2 Rotating Boussinesq equations with Reynolds stress The goal of this section is to establish results of existence, uniqueness and differentiability of the solutions for the initial-boundary value problem of  -plane rotating Boussinesq equations with Reynolds stress (2.29-2.31). We prove the significant aspects of well-posedness of solution utilizing the machinery developed above and additional accessible techniques constructed in this section. The functional formulation for the evolution equation of  -plane rotating Boussinesq equations with Reynolds stress (2.29-2.31) takes the following form of an initial value problem in the Hilbert space  :

dU  LU  N (U )  0 dt U (0)  U 0

(4.32)

where U  (u, v, w,  ) . The boundary conditions have been taken to be space-periodic in order to simplify the analysis. Concerning the linear operator LU and the nonlinear operator N (U ) we have

N U   B U ,U   CU  FU FU  (0, 0,



Ro w) Ro Fr 2 ,

 Ek   Ro u   Al u  CU       1    Al      Ed   u  ▽u   b l ( u , u )  B(U ,U )  B(U )      u ▽   bl (u ,  )  LU  TU  SU  Ek 6   Ro u   Al6 u  TU     6   1 6    Al      Ed 

 0 1  1  y  Ro  1 0 S  ( ) 0 0 Ro  0 0 DOI: 10.26855/jamc.2018.09.003

0 0  0 0  U  D( L)  D(T ). 0 0  0 0 420


M.J.P.S. Tlad i

In order to develop basic results concerning whether there exists a unique solution in the large for the general three-dimensional space variables for initial-value problem (4.32), we derive a priori energy type estimates useful in proving that (4.32) generate a dynamical system, denoted, U (t )  S (t )U 0 , where U (t ) is the unique solution uniformly bounded in time and S (t ) is a group of continuous nonlinear solution operators. The principal result concerning the existence of such a unique solution will be proven by the utility of the Faedo-Galerkin technique, a priori energy type estimates and Leray-Schauder principle. The linear operator LU of the initial-value problem (4.32) satisfies

( LU ,U )  (TU ,U )  a(U ,U )

(4.33)

where a(.,.) is symmetric bilinear form. It follows from [92, 27, 66, 50] that this implies L is self-adjoint. Additionally,

( LU ,U )  (TU ,U )  ‖U‖2  0

(4.34)

1

which implies the bilinear form induced by L is coercive and L is compact and self-adjoint. By the courtesy of an excellent result from [92, 27, 66, 50] ,  is the smallest eigenvalue of Laplace's operator and Stokes operator. Also, the symmetry of the bilinear form a(U ,W ) , together with the coerciveness of a(U ,U ) imply the existence of a sequence of positive eigenvalues and a corresponding sequence of eigensolutions which yield an orthonormal basis of  such that

LWi  iWi

(4.35)

0  1  2    i  , and

i   as i   . Moreover, 1 2

1 2 1

‖L U‖  ‖U‖ 1 2 1

(4.36)

1 2

‖LU‖  ‖L U‖ The following useful inequalities are utilized in the derivation of a priori energy type estimates:

( B(U ),U )   [(u u  u )  (u  )  ]dx  0

(4.37)



1 2

1 2 1

1 2

‖CU‖  ‖ 1 L U‖   ‖LU‖‖L U‖ 2

2

(4.38)

Since LC  CL , the inequality 1 2

2 ‖L CU‖2  ‖ 1 LU‖

(4.39)

holds. By the courtesy of the Sobolev inequality and continuity properties of the nonlinearity $B(U)$ the following bounds are satisfied 1

‖B(U ‖ ) 2  c02 (CU ,U )[‖CU‖2 ‖U‖2 ]  c02 (1  1/ 1 ‖ ) U‖‖L2U‖2‖LU‖

(4.40)

By the virtue of Cauchy-Schwarz inequality, Young's inequality and (4.40) the following is satisfied 1 2

1 2

1 | ( B(U )  CU , LU ) | 54(c ‖U‖‖L U‖  1 ‖ ) L U‖2  ‖LU‖2 4 2 where c1  c0 (1  1/ 1 ) . Also, from [22] the following a priori estimates is valid 4 1

2

1 2

2

‖L B(U ‖ ) 2  c‖LU‖2 . DOI: 10.26855/jamc.2018.09.003

421

(4.41)

(4.42) Journal of Applied Mathematics and Computation

M.J.P.S. Tlad i

Concerning the solution to the steady linear case we have

( LU  CU ,U )  (TU ,U )  (CU ,U ) (4.43)

1

 (  16 ‖ ) U‖2  0,

which implies L  C is positive and hence the bilinear form it induces is coercive. This establishes existence and ´

uniqueness of solution to the steady linear case by the Riesz-Fr e chet representation theorem or the Lax-Milgram lemma. The following mini-max principle from [92, 23] will be utilized:

lim sup‖F‖2  ( t 

Ro 1 2  ) ‖‖2 2 Fr Ro

(4.44)

where ‖‖ is the measure of  . Taking the inner product of (4.32) with $U$, and invoking the inequalites (4.43), (4.36) and Young's inequality yield the following differential inequality:

d 2 ‖U‖2  ‖ 1 U‖ dt 1 d  ‖U‖2  ‖L2U‖2 dt 1 Ro 1 2  ( 2 ) ‖‖2  Fr Ro

(4.45)

1

where     16 . The consideration of Gronwall's inequality in (4.45) provides a priori estimate

‖U (t ‖ ) 2 ‖U 0‖2exp( t ) 

1



2

(

Ro 1 2  ) ‖‖2 (1  exp( t )) 2 Fr Ro

(4.46)

which is asymptotic stability with flow energy and entropy production provided by

1 1 E (t )  ‖U (t ‖ ) 2   (u 2   2 )dx 2 2  1

and exponential dissipation rate     16 . Thus, we are able to assert that the following asymptotic estimate is valid

lim sup‖U (t ‖ )2 t 

1



2

(

Ro 1  ) 2‖‖2 2 Fr Ro

(4.47)

which shows U (t ) is uniformly bounded in time in  . Taking the inner product of (4.2) with LU and using inequalities (4.36) and (4.41), we find that 1 1 d 2 ‖L2U‖2  ‖ L U‖2 1 dt 1 d  ‖L2U‖2 ‖LU‖2 dt 1 1 Ro 1 2  2( 2  ) 2‖‖2  108(c14‖U‖2‖L2U‖4  ‖ L U‖2 ) 1 Fr Ro

(4.48)

We estimates (4.45) using the inequality t 1

1 2

lim sup  ‖L U (s‖ ) 2 ds  (1/  2  1/  3 )( t 

DOI: 10.26855/jamc.2018.09.003

t

422

Ro 1  )2‖‖2 2 Fr Ro Journal of Applied Mathematics and Computation

M.J.P.S. Tlad i

and by the virtue of Gronwall's inequality in (4.48) we obtain a priori estimate 1

‖L2U (t ‖ ) 2  12 where 12   1 2 , and the parameters

(4.49)

 1 and  2 are given by the following involved and tedious expressions, re-

spectively,

 1  (2  1071 (1/  2  1/  3 )  1/  2  1/  3 )(

Ro 1  ) 2‖‖2 2 Fr Ro

and

 2  exp(108c02 (1/  2  1/  3 )(

Ro 1 2  ) ‖‖2 ). 2 Fr Ro

Consequently from the last inequality the following is valid 1

lim sup‖L2U (t ‖ ) 2  12

(4.50)

t 

1

which shows U (t ) is uniformly bounded in time in D( L2 ) . Similarly, taking the inner product of (4.32) with L2U and making use of the inequality (4.36) and (4.42) yield

d 2 ‖LU‖2  ‖ 1 LU‖ dt 3 d 2  ‖LU‖ ‖L2U‖2 dt

(4.51)

2  3c 2‖LU‖4  3‖ 1 LU‖  3(

1 Ro 2 2  ) 1 Ro 2 Fr 4

Using the differential inequality (4.48) and invoking H o lder inequality we obtain a priori energy type estimate



t 1

t

‖LU (s‖ ) 2 ds   32

where  3  a1  a2 denotes the sum of the involved expressions 2

a1  1 (1/  2  1/  3 )(

Ro 1  )2‖‖2  12 2 Fr Ro

and

a2  108c02 (1/  2  1/  3 )2 (

Ro 1 4  ) ‖‖4 . 2 Fr Ro

Substitution of this a priori energy type estimate into (4.51) yields

‖LU (t ‖ ) 2  22

(4.52)

with  2 denoting 2

1 Ro2 2   3c   2  3( 2  4 ) 1 . Ro Fr 2 2

2

4 3

2 1 3

Consequently, putting the estimates together we obtain

lim sup‖LU (t ‖ ) 2  22 t 

DOI: 10.26855/jamc.2018.09.003

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(4.53)


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which shows U (t ) is uniformly bounded in time in D( L) . We now state existence, uniqueness, continuity and differentiability with respect to initial conditions of solution to the initial-value problem (4.32) and invoke the above a priori estimates to prove the result. 1 2

Proposition 4.1.5 Under the above hypothesis for F  D( L ) and U 0   given, (4.32) generate a dynamical system U (t )  S (t )U 0 satisfying 1

U  C ([0, T ];  )  L2 (0, T ; D( L2 )) 1

for all T  0 . Moreover, if U 0  D( L2 ) then 1 2

U  C ([0, T ]; D( L ))  L2 (0, T ; D( L)) for all T  0 . Proof: For fixed m , the Faedo-Galerkin approximation m

U m   gim (t )Wi i 1

of the solution U of (4.32) is defined by the finite-dimensional system of nonlinear ordinary differential equations for

gim (t ) given by dU m  LU m  Pm B(U m )  CU m  Pm F dt U m (0) m U 0

(4.54)

1

1

where Pm is the orthogonal projection in , D( L2 ), D( L 2 ) and D( L) onto the space spanned by the m eigenfunctions of L given in (4.35). Let X be the finite-dimensional Hilbert space spanned by the m eigenfunctions 2 of L given in (4.35) with inner product ((.,.)) induced by D( L) . Consider a closed ball of radius  2 given by

(4.53) contained in D( L) . Integration of (4.54) and the above a priori energy type estimates provide a continuous mapping defined by the integral of (4.54) from the closed ball of radius  2 given by (4.53) into itself. Thus, by the 2

Leray-Shauder principle [36, 66] the finite-dimensional system of nonlinear ordinary differential equations (4.54) has at 2 least one solution U m inside the ball of radius  2 . The passage to the limit

m   and Tm  T   follows from the following a priori estimates: Utilizing the differential inequality (4.54) provides

d 1 Ro 1 2 2 ‖U m‖2  ‖ ( 2 ) 1 U m‖  dt  Fr Ro

(4.55)

which implies U m remains bounded in 

1 2

L ((0, T );  )  L (0, T ; D( L )). 2

This useful result combined with the weak compactness implies there is a subsequence also denoted by U m and

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

1 2

U  L (0, T ;  )  L (0, T ; D( L )) 2

such that 1  U  L2 (0, T ; D( L2 )) weakly Um     weak-star. U  L (0, T ;  )

2

1 2

Using (4.37) and (4.40), we obtain the boundeness of B(U m ) and Pm B(U m ) in L (0, T ; D( L )) . Furthermore, 1

from (4.54),

dU m 2 is also bounded in L (0, T ; D( L 2 )) . Utility of this result and weak compactness yield dt 1 dU m dU   L2 (0, T ; D( L 2 )) dt dt

weakly. The assertion of Lebesgue dominated convergence theorem as well as the above convergence results yield

U m  U  L2 (0, T ;  ) strongly. We pass the limit in (4.54) and obtain the required result (4.32). We obtain 1 2

U  C ([0, T ];  )  L (0, T ; D( L )). 2

1 2

Similarly, from (4.48) and (4.51) we conclude that if U 0  D( L ) then by the Faedo-Galerkin technique 1

U  C ([0, T ]; D( L2 ))  L2 (0, T ; D( L)) for all T  0 . According to Gronwall's inequality, uniqueness and continuity with respect to initial conditions of solution U (t ) follows from considering the difference between two solutions of the initial-value problem (4.32) and employing a priori estimates (4.47-4.53). Uniqueness and continuity with respect to initial conditions of solution U (t ) generate a dynamical system which is prescribed by continuous solution operators S (t ) , t 

defined by


S (t )U ( s)  S ( s)U (t ) S (0)U  U  t , s  .

(4.56)

That the solution operators S (t ) are continuous is a consequence of the continuity of $U(t)$ in time and in initial conditions. The group property is a consequence of the injectivity of the solution operators which is equivalent to the backward uniqueness of solution for the initial-value problem (4.32). Next, we turn our attention to establish uniqueness, continuity and Fr$\acute{e}$chet differentiability with respect to initial conditions of solution U (t ) for the initial-value problem (4.32). As in the previous section, the technique of the proof uses the linearization of the evolution equation (4.32) about the difference of two given solutions. For the sake of completeness, we repeat the analysis here. Consider u and v satisfying the evolution equation (4.32) corresponding to the initial conditions u (0)  u0 and v(0)  v0 , respectively. The precise analysis of the solution difference w  v  w yields

dw  Lw  N (v)  N (u )  0 dt w(0)  v0  u0 DOI: 10.26855/jamc.2018.09.003

425

(4.57)


M.J.P.S. Tlad i

where

N  v   N  u   B  v, v   B  u, u   Cw  B  v, w   B  w, u   Cw

(4.58)

 B  u, w   B  w, u   B  w, w   Cw.

The linearization of (4.57) along the dynamical system trajectory satisfies the variational equation

d  L  l0 (t )  0 dt  (0)  v0  u0  

(4.59)

where l0 (t )w  B(u(t ), w)  B(w, u (t ))  Cw and l0 (t ) is linear bounded operator from

1

1

D( L2 ) to D( L 2 ) .

Consequently, the equation of variation is a linear nonautonomous ordinary differential system. We shall be concerned with the difference   w   and the following is valid

d  L  l0 (t )  l1 (t ; w(t ))  0 dt  (0)  0

(4.60)

where l1 (t; w(t ))  B(w, w) . The remarkable feature of the last equality is that from (4.58), we have

N (v)  N (u)  l0 (t )w  l1 (t; w) . We are now in a position to state the keystone result: Proposition 4.1.6 If u0   and u is the unique solution of (4.32) then the variational equation (4.59) has a unique solution  satisfying 1

  C ([0, T ];  )  L2 (0, T ; D( L2 )) ´

for all T  0 . Moreover, the dynamical system u0  S (t )u0 is Fr e chet differentiable in  with differential

L(t , u0 ) :   (t ) given by the solution of the variational equation. Proof: Existence and uniqueness of solution to (4.59) is a consequence the Faedo-Galerkin technique and the use of a priori estimates together with the Gronwall's inequality. Next, we exhibit a series of a priori estimates which will be required in establishing the above proposition. Taking the inner product of (4.57) with w and application of Cauchy-Schwarz inequality, Young's inequality and Sobolev inequalities yield 1 d ‖w‖2 ‖L2 w‖2  k 2‖w‖2 , dt

where k  c1 2 and

(4.61)

 2 is given in the estimate (4.52). According to the Gronwall's inequality the following is valid ‖v(t )  u(t ‖ ) 2  exp(k 2T ‖ ) v0  u0‖2

(4.62)

t  (0, T ) . The sharp a priori estimate (4.62) yield forward uniqueness and continuity with respect to initial conditions of solution. Furthermore, from (4.61) we obtain a priori energy type estimate t

1 2

) ds  exp(k T ‖ ) v ‖L w(s‖ 2

2

0

0

 u0‖2 .

By the utility of the inner product of (4.60) with  we obtain the differential inequality

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c12 12 d 1 12 2 2 2 2 ‖‖  ‖L ‖  k ‖‖  ‖L w(t ‖ )3 dt 2 1

(4.63)

We repeat the construction using Gronwall's inequality and find a priori estimate T

1

‖ (t ‖ ) 2   ‖L2 w(s‖ ) 3ds 0

t  (0, ) . Consequently we obtain the bound

‖ (t ‖ )  2

c12

1

exp(3k 2T / 2)‖v0  u0‖

(4.64)

which implies the needed result


as v0  u0 . Hence, the dynamical system S (t ) : u0  u (t ) is Fr e chet differentiable at u0   . Next, we examine the problem of proving backward uniqueness of solution for the initial-value equation (4.32) that establish the injectivity properties and group properties of the solution operators S (t ) . The backward uniqueness of solu´

tion is proved in approximately the same way as Fr e chet differentiability result using the standard log-convexity method as in the previous section. For the sake of completeness of presentation we cover the proof of the injectivity of solution operators here. In order to motivate this objective, consider two solutions u and v of the initial-value problem such that at time t  t* both u and v satisfy (4.32) for t  (t* ò, t* ) and u (t* )  v(t* ) . Given that this hypothesis is valid, backward uniqueness of the solution operators is accomplished if u(t )  v(t ) for all time t  t* whenever the solutions are well-defined. Alternatively, suppose the solution operators satisfy

S (t )  u(t  s) t  0 s  then for   (0, ò) , the problem of backward uniqueness is whether the following hold:

S ( )u(t*   )  S ( )v(t*  )  u(t*   )  v(t*   )

(4.65)

which is the injectivity of the solution operators S ( ) . The following significant result on the injectivity of the solution operators is valid: Proposition 4.1.7 Suppose the following hold 1

u, v  L (0, T ;V  D( L2 )  L2 (0, T ; H  D( L)) then the difference $w=v-u$ satisfies the differential equation (4.57) for t  (0, T ) and the solution operators S (t ) are injective. We give a sketch of an elegant proof due to Temam [91.] Proof: We will use the notation    (t ) for the quotient

‖ (t ‖ ) V2  ‖ (t ‖ ) 2H 

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( L (t ),  (t )) . ( (t ),  (t ))

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Invoking Cauchy-Schwarz inequality, Young's inequality, Sobolev inequalities, continuity properties of the nonlinear operator B(w, w)  Cw developed above and the fact that ( Lw   w, w)  0 we obtain

dw (( , w)) ‖w‖V2 dw 1 d   dt 2  ( , w) 2 dt ‖w‖ ‖w‖4 dt 1 dw  ( , Lw   w) ‖w‖2 dt 1  ( N (v)  N (u )  Lw, Lw   w) ‖w‖V2

(4.66)

‖Lw   w‖ 1   ( Lw, N (v)  N (u )) 2 ‖w‖ ‖w‖2 ‖Lw   w‖2 ‖N (v)  N (u ‖ )2   2‖w‖2 2‖w‖2 ‖Lw   w‖2   k 2 2‖w‖2 where N (v)  N (u) is given in (4.58) and the function k (t )  k1 (t )  B(w, w)  Cw is defined by the mapping 2

1

1

1

1

k1 (t ) : t  c‖ ( u(t ‖ ) V2‖Lu(t ‖ ) 2 ‖v(t ‖ ) V2‖Lv(t ‖ ) 2 )  L2 (0, T ). Utility of Gronwall's inequality into (4.66) gives t

 (t )   (0)exp(2 k 2 (s)ds) t  (0, T ). 0

Thus, if the difference w  v  u satisfies the above conditions and

w( )  0  w(t )  0, t [0, T ]. We proceed by contradiction. Hence, suppose that ‖w(t0 )  0‖ for t0  [0, T ) . As a result by continuity we have

‖w(t )  0‖ on some open interval (t0 , t0  ò) and we employ the notation t*  T for the largest time for which

w(t )  0 t [t0 , t* ). ) is well-defined and the fact It follows that w(t* )  0 . In the interval [t0 , t* ) the mapping given by t  log‖w(t ‖ that w(t ) satisfies the differential equation (4.22) we obtain

d 1 log  2  k 2 . dt ‖w‖ Consequently for the time t  [t0 , t* ) the following is valid: T 1 1 log  log   2 (s)  k 2 (s)ds t0 ‖w‖ ‖w(t0 ‖ )

which illustrates the boundedness of

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1 from above as the time t  t* from below and this is a contradiction. w(t )

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CHAPTER 5 Nonlinear stability of solutions 5.1 Aspects of stability and attraction In this section we consider nonlinear stability and attractors of solutions of rotating Boussinesq equations by adapting a priori estimates employed in establishing well-posedness and differentiability of solutions. One of the advantages of the results presented in this chapter is that they may be considered as more refined generalization of results such as (4.12) and (4.46) for the dissipation of flow energy and the entropy production provided by

1 1 E (t )  ‖U (t ‖ ) 2   (u 2   2 )dx, 2 2  2 which is specification of energy and the entropy production in the L -norm. Hence, the results of this section on nonlinear stability differ in form rather than in essence from their counterparts utilized in establishing well-posedness and differentiability of solutions. The problem of showing that the rest state of system (4.2) or (4.32) is nonlinearly stable is investigated using ideas developed in Galdi et al [27, 28]. An important quantity in analyzing whether the solution of an evolution equation is bounded in some suitable norm is that of Lyapunov functional or a priori energy type estimates of the system as we have already noted in establishing well-posedness. We will illustrate in the sequel a technique for constructing Lyapunov functionals with interpretations such as energy, enstrophy and entropy which are decreasing along solutions. The equations governing the flow of an incompressible stratified fluid under the Coriolis force induce damping mechanisms in the nature of viscosity, diffusion, stratification and rotation effects that manifest themselves in the evolution equation with the existence of Lyapunov functionals which are equivalent to some norm induced by the inner product of solution. Here we state a technique for constructing generalized Lyapunov functionals that furnish necessary and sufficient conditions for nonlinear stability. Specifically, the tools used are spectral properties of the linear operator and a priori estimates for the nonlinear operator. Now we recall definitions which are required in the formulation of the nonlinear stability criteria to be customized for use in the solutions of the rotating Boussinesq equations. Consider the following prototype initial-value problem

dU  LU  N (U ), U (0)  U 0 . dt

(5.1) ´

where LU is linear operator and N (U ) is nonlinear operator in the sense that its Fr e chet derivative at zero vanishes. A self-adjoint linear operator L for the initial-value problem (5.1) is called essentially dissipative if

( LU ,U )  0  U  D( L) ( LU ,U )  0  U  0

(5.2)

The complement of essentially dissipative is called essentially non-dissipative. By the spectral theorem [27, 50], essential dissipativity is equivalent to the spectrum of L being nonnegative and zero not an eigenvalue. For every essentially dissipative operator L , the bilinear form

(U ,W ) L  ( LU ,W )  U ,W  D( L)

(5.3)

‖ . L given by defines a scalar product in D( L) . We denote by H L the completion of D( L) in the norm ‖ ‖U‖2L  ( LU ,U ).

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According to the condition (5.2) on essential dissipativity of linear operators, we are in a position to state necessary and sufficient criteria for nonlinear stability using the Lyapunov functional

J U     LU ,U 

(5.4)

and generalized energy flow energy and entropy production

1 E (t )  {‖U‖2 ‖U‖2L } 2

(5.5)

for the paradigmatic example of initial-value problem (5.1). Proposition5.1.1

Suppose L is essentially dissipative and the nonlinear operator satisfies 2 2 ‖NU‖2  c‖U‖‖ ‖U‖2L ) L ( LU‖ 

(5.6)

where c  0 . Then the rest state of system (5.1) is asymptotically stable, that is,

1 1 4(1   ) J (U )  {2 E (0)(1  )exp[ ]}/ (t  1),



0

0

(5.7)

whenever

  (1  cE(0))  0, where   0 and 0 

(5.8)

1 . Next assume L is essentially non-dissipative and the nonlinear operator satisfies 2

‖NU‖2  cJ (U )(‖LU‖2 ‖U‖2 )

(5.9)

c  0 . Then the rest state is unstable in the sense of Lyapunov. This proposition is proved in approximately the same way as the analogous fact for the finite-dimensional linear problem with the assistance of a priori energy type estimates that dominates the nonlinearity by linear elliptic operators. Proof: Taking the inner product of the equation (5.1) with U yields

1 d (U ,U )  ( LU ,U )  ( NU ,U ). 2 dt Similarly, taking the inner product of the equation (5.1) with LU gives 1 d ( LU ,U )  ( LU , LU )  ( LU , NU ). 2 dt Putting the estimates together provide the following equation for the evolution the generalized energy functional (5.5)

dE  ( LU ,U )  ( NU ,U )  ( LU , LU )  ( LU , NU ). dt

(5.10)

Applying Cauchy-Schwarz inequality and Young's inequality in (5.10) yields

dE 3 1  ( LU ,U )  ‖LU‖2 ‖U‖‖NU‖ ‖NU‖2 dt 2 2 1   0 ‖ ( LU‖2  | U |2L )‖U‖‖NU‖ ‖NU‖2 . 2

(5.11)

Substitution of a priori estimate (5.6) in (5.11) and the courtesy of the generalized energy functional (5.5) provides the differential inequality

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M.J.P.S. Tlad i 1 1 dE c   0 ‖ ( LU‖2 ‖U‖2L )[1  c 2‖U‖L‖U‖‖ ( LU‖2 ‖U‖2L ) 2  ‖U‖2L ] dt 2 c   0 ‖ ( LU‖2 ‖U‖2L )[1  ‖U‖2L ] 2 2 2   0 ‖ ( LU‖ ‖U‖L )[1  cE ].

(5.12)

According to the inequality (5.12) and assumption (5.8) we obtain a priori energy estimate

0 ‖ ( LU‖2 ‖U‖2L )(1  cE (0))  

dE dt

which implies





0

0 ‖ ( LU‖2 ‖U‖2L )(1 

c

1

E (0))dt  2 E (0).

In this respect it should be noticed that the following a priori energy type estimate holds





0

‖ ( LU‖2 ‖U‖2L )dt 

2 E (0)

0

.

We proceed by recalling the useful property



1 d ( LU ,U )  ( LU , LU )  ( LU , NU ) 2 dt

and invoking the Cauchy-Schwarz inequality, Young's inequality and a priori estimate (5.6) which yield the following differential inequality

d ‖U‖2L  2c‖U‖2L ‖ ( LU‖2 ‖U‖2L ). dt

(5.13)

In order to establish the hypotheses of Gronwall's inequality we note that (5.13) may be restated equivalently as follows:

d 2c  h1 (t )  h2 (t ) dt 1

(5.14)

where

(t ) ‖U‖2L (t  1),

h1 (t ) ‖U‖2L , and

h2 (t ) ‖LU‖2 ‖U‖2L . Thus, the following holds t

t

s

0

0

0

(t )  exp[2c  h2 ( s)ds]{(0)   exp[2c  h2 ( )d ]h1 ( s)ds}. which is the assertion of Gronwall's inequality. Moreover, through the use of estimates from above we deduce the following a priori estimate that guarantee the hypotheses of the Gronwall's lemma are valid





0



h1 (t )dt   h2 (t )dt  0

2 E (0)

0

.

Another application of relation (5.8) yields a priori estimate

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

2c  h2 (t )dt  0

4cE (0)

0



4(1   )

0

.

Consequently, we have

exp[

2c

1





0

h2 (t )dt ]  exp[

4(1   )

0

].

Similarly, putting the above estimates together we obtain the following finite energy solution

(t )  2 E (0)(1  Substitution of

1

0

)exp[

4(1   )

0

].

(5.15)

(t ) ‖U‖12 (t  1) into (5.15) gives the desired nonlinear stability result (5.7) since

1 J (U )  ‖U‖12 whenever L is essentially dissipative. Hence, as a bonus we obtain the proof of Gronwall's lemma.



The instability result holds by following the proof given in (27, 28). For the rest states, asymptotic stability and attraction coincide. However, for more complicated invariant sets than rest states, an attractor may be considered as the stronger notion of stability than the one given above. The following proposition [93, 92, 91, 90, 23, 22] on the characterization of attractors and determining of the compactness of an  -limit set is required: Proposition 5.1.2 Suppose that for Q  H , Q   , and that for some $s>0$ the set t  s S (t )Q is relatively compact in H . Then

 -limit set of Q denoted by  (Q) is nonempty, compact and invariant.

Proof: The proposition is proven on page 20 in Temam [91] using properties of

 -limit sets.

The proposition is of central importance in the construction of attractors. In order to prove the major hypothesis that t  s S (t )Q is relatively compact in H , it is adequate to illustrate that this set is bounded in a spaceV compactly imbedded in H by employing the existence of the following injections 1 2

1 2

D( L )  V  D( L)    H  D( L ). The density and compactness of the injections will find applications in the sequel when we prove existence of attractors for the initial-value problems. 5.2 Rotating Boussinesq equations The goal of this section is to develop nonlinear stability results for  -plane rotating Boussinesq equations (2.26-2.28). In order to give nonlinear stability bounds that agree approximately with established paradigmatic example of initial-value problem (5.1), the functional formulation for the evolution equation of  -plane rotating Boussinesq equations is restated equivalently as follows:

dU  LU  N (U )  0 dt U (0)  U 0

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(5.16)


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where U  (u, v, w,  ) . Concerning the linear operator LU and the nonlinear operator N (U ) we have the following appropriate setup:

N U   B U ,U 

LU  TU  SU  FU  Ro FU  (0, 0,  , w) Ro Fr 2  Ek   Ro  u   As u  TU       1    Al      Ed    u  ▽u   b s ( u , u )  B(U ,U )  B(U )      u ▽   bl (u ,  ) 

 0 1  1  y  Ro  1 0 S   ( ) 0 0 Ro  0 0

0 0  0 0  U  D( L)  D(T ). 0 0  0 0

According to the above setup of the linear operator LU , we obtain the following useful result

( LU ,U ) 

Ek 1 1 Ro ‖▽u‖2  ‖▽‖2  (  2 )  wdx Ro Ed Ro Fr 

(5.17)

which is a slight modification of the Lyapunov functional (4.3) that was employed in establishing well-posedness of solution for the problem of  -plane rotating Boussinesq equations (2.26-2.28). With the attendant relation ()5.17 we are able to demonstrate a necessary and sufficient nonlinear stability criterion and hence the elegant power of energy methods is fully realized. The first effort is then to let R and I (u ,  ) (5.18) be defined as follows

1/ 2R  sup{ wdx}/ J (U )  supI (u ,  ) 

where the supremum is taken for U  D( L) . We shall compile various inequalities about the equation (5.16). Substituting

|  wdx | ‖ ( u‖2 ‖‖2 ) / 2 

´

into (5.18) and utilizing the Poincar e -Friedrichs inequality yields R  Ga  80 . As it is known, the nonlinear stability bounds often give conditions that agree with numerical approximations. Moreover, the adequacy cited above may be interpreted by noting that the size of the critical value Ga depends on the conditions of the numerical approximations and can be considerably improved by implementing more refined and accurate numerical schemes. In the asymptotic stability analysis we take the number Ga  80 and we note that we have reformulated the initial-boundary value problem for  -plane rotating Boussinesq equations in order to reflect the influence of Galdi et al [27, 28] investigations on energy stability theory. The above stability criterion number provides a great deal of light on the potentialities of DOI: 10.26855/jamc.2018.09.003

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the energy stability theory and validates this exposition from a benchmark point of view and offers the opportunity of invoking established results as a guide for nonlinear stability criteria. It is worth mentioning that the stability criterion number serve as an addition to available paradigmatic examples of stability criteria such as the Richardson number

Ri 

1 for stratified shear instability and the Rayleigh number Ra  1015 for penetrative convection [13, 77]. 4

The attendant relation (5.17) may be restated as follows

( LU ,U )  (1  (

1 Ro  2 ) I (u ,  )) J (U ), Ro Fr

(5.19)

with the Lyapunov functional J (U ) defined in (5.4) given by

Ek 1 ‖▽u‖2  ‖▽‖2 . Ro Ed 1 Ro It is appropriate at this point to note that if (  )  2 R then the following is valid Ro Fr 2 1 Ro ( LU ,U )  (1  (  )) J (U )  0, 2 RRo 2 RFr 2 and in this case  L is essentially dissipative. Additionally, the following inequalities hold 1 Ro 1 Ro (1  (  )) J (U ) ‖U‖2L  (1  ( 2  2 2 )) J (U ) 2 2 RRo 2 RFr  Ro  Fr J (U ) 

which shows that ‖U‖L is topologically equivalent to J (U ) . The next step is to note that if ( 2

(5.20)

(5.21)

(5.22)

1 Ro  2 )  2R Ro Fr

then there is a sequence {U n }   such that

lim I (un , n )  1/ 2 R. n 

1 1 Ro Hence given ò  ( (  2 )) , there exists $U$ such that 2 R Ro Fr 1 Ro 1 (5.23) ( LU ,U )  [1  (  2 )(  ò]J (U )  0, Ro Fr 2 R and in this case  L is essentially non-dissipative. The following result on the spectrum of  L from [27, 50, 66] will be utilized: Lemma5.1.3  ( L)  (,0]  positive eigenvalues n  n (

, where

is either empty or an at most denumerable set consisting of isolated,


clustering only at zero.

1 Ro  2 is due to the relation (5.23). Ro Fr According to a perturbation theorem [27, 50, 66], if $TU$ is closed and MU  SU  FU is relatively compact with respect to T then  ( L  (T  S  F )) and  (T ) differ by at most a denumerable number of isolated, positive eigenvalues of finite multiplicity clustering only at zero. Thus it suffices to show that M  S  F is relatively comProof: The validation of the dependence of the positive eigenvalues

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pact with respect to T since from [92, 27, 50, 66] we have  (T )  (,0] . By definition, M  S  F is relatively compact with respect to T if for any sequence {U n }   ,

‖U n‖‖  TU n‖  , uniformly in n,

(5.24)

then there is a subsequence U ´  {U n } such that MU ´ is strongly convergent. Indeed, the following inequality is n

n

valid

1  y  Ro 2 1 Ro 2 2 ‖MU ‖  [5( ) (  2 )2 ‖ ] U‖ ´ ´ n n Ro Ro Fr (5.25) 1  y  Ro 2 1 Ro 2 2  [5( ) (  ) ] Ro Ro Fr 2 and by the courtesy of Ascoli-Arzela theorem {MU ´} contains a uniformly convergent subsequence which is a Caun

chy sequence in a complete normed space  . Thus MU ´ is strongly convergent which illustrates that M  S  F n

is relatively compact with respect to T . It remains to estimate the nonlinearity N (U ) . Next, we exhibit a series of a priori estimates on the nonlinearity which will be needed in establishing necessary and sufficient nonlinear stability of the rotating Boussinesq equations (5.16). Consideration of the Sobolev inequality and continuity properties of the nonlinearity B(U ) yield the following uniform bounds

‖NU‖2  (TU ,U )c02 ‖ ( TU‖2 ‖U‖2 ) 1

1

2 2  c02‖T 2U‖(‖ ‖U‖2 ) 1 L U‖  1

1

2 2  c02  2‖TU‖(‖ ‖U‖2 ) 1 L U‖  1 2

1 2 1

1 2

1 2 1

 c   ‖L U‖( ‖LU‖ ‖U‖ ) 2 0

1

1

1

1

2

(5.26)

2

1

 c02 c 2 12‖L2U‖‖ ( LU‖2 ‖U‖2 )  c02 c 2 12 J (U )(‖LU‖2 ‖U‖2 ) 1

for the nonlinearity N (U )  B(U ) where c  max{12 ,1} . Consequently, a priori estimates (5.26) on the nonlinearity coincide with (5.9) which means that the nonlinear operator is suitably dominated by the linear operator LU . Thus, the purpose of proving Lyapunov instability aspect has been achieved. Moreover, when the following

1 Ro  2 )  2 R holds, we observe the topological equivalence of the Lyapunov functional J (U ) with the norm Ro Fr ‖U‖2L given in inequality (5.22). In this case, putting the estimates (4.36) and (5.26) together and invoking the equiva(

lence relation (5.22) gives the needed a priori estimate for the nonlinearity

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1

‖NU‖2  c02c 2 12 J (U )(‖LU‖2 ‖U‖2 ) 1 2

1 2 1

(5.27)

 c c c  ‖U‖ ‖ ( LU‖   ‖U‖ ) 2 1 0

2 L

2

that is necessary for stability with the parameter c1 denoting c1  (1  (

1 1

2 L

1 Ro  ))1 . According to the sharp a 2 RRo 2 RFr 2

priori estimate (5.27), aspect of nonlinear stability provided by the inequality (5.6) is satisfied. Hence, energetic stability criteria for necessary and sufficient nonlinear stability of the rest state f  -plane rotating Boussinesq equations (2.26-2.28) or equivalently (5.16) hold with the energy and entropy production provided by the following:

1 Ek 1 E (t )  {‖U‖2  ‖▽u‖2  ‖▽‖2  2 R  wdx}  2 Ro Ed 1 which is specification of energy and the entropy production in the Sobolev H -norm if the criterion 1 Ro Ga  R  (  2 ) is valid. In the asymptotic stability analysis we take the number Ga  80 and we note that Ro Fr we have reformulated the initial-boundary value problem for  -plane rotating Boussinesq equations in order to reflect the influence of Galdi et al [27, 28] investigations on energy stability theory. The above stability criterion number provides a great deal of light on the potentialities of the energy stability theory and validates this exposition from a benchmark point of view and offers the opportunity of invoking established results as a guide for nonlinear stability criteria. The result serve as a paradigm for constructing generalized Lyapunov functionals with interpretations such as energy and entropy production which are decreasing along solution of the initial-value problem. Next, we focus our attention to the development of results for the attractor of the $\beta$-plane rotating Boussinesq equations with Reynolds stress (2.29-2.31) for both the three-dimensional and two-dimensional space variables. We show existence of the attractor for the dynamical system and proceed by customizing the splendid techniques employed in [14, 15, 92, 91, 23, 22] which have been summarized in the above proposition. Next, we turn our attention to develop results on the attractor for  -plane rotating Boussinesq equations (2.26-2.28) in the case of two-dimensional space variables. We show existence of the attractor for the dynamical system and proceed essentially using the techniques employed in [14, 15, 92, 91, 23, 22] which have been summarized in the above proposition, but the calculations that we present in this examination are tedious and involved. Putting the estimate (4.12) and (4.11) together gives the bound specified by 1 t 1  Ro 1 ‖‖ 12 Ro 1 lim sup  ‖L2U (s‖ ) 2 ds  [ 2 ( 2  )4  1 ( 2  )2 ] t t    Fr Ro  Fr Ro Taking the inner product of the in the initial-value problem (4.2) with LU yields the following differential inequalities 1 1 d 2 ‖L2U‖2  ‖ L U‖2 1 dt 1 d 2  ‖L2U‖2 ‖LU‖2   22 dt

such that 22   0  1 2 follows successive application of Gronwall's inequality. And the parameters 2

(5.28)

 0 , 1 and  2

are given by

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3‖‖ Ro 1 ( 2  )2 ,  Fr Ro 27c 2  1  3  3 (a23  a22 )exp(a21 ), 4  ‖‖ Ro 1  2  2 ( 2  ) 4 (a13  a12 ) 2 exp(2a11 ),  Fr Ro 27c 1 2 Ro 1 2 1 Ro 1 2‖‖2 Ro 1 a11  3 (( ) ( 2  )  3 )( 2  ) ( 2  )4 , 2 4  Fr Ro  Fr Ro  Fr Ro Ro 1 a12  3‖‖2 ( 2  ) 2 , Fr Ro 1 Ro 1 1 Ro 1 a13  (( ) 2 ( 2  ) 2  3 )( 2  ) 4‖‖,  Fr Ro  Fr Ro 3 9c‖‖ 1 Ro 1 1 Ro 1 1 Ro 1 a21  (  ( 2  )7  2 2 )( 2 ( 2  ) 2  3 ) 2 ( 2  ) 4 , 5 2  Fr Ro  Fr Ro  Fr Ro 3 ‖‖ Ro 1 1 Ro 1 1 Ro 1 a22  4 ( 2  ) 4 ( 2 ( 2  ) 2  3 ) 2 ( 2  ) 4 , 2  Fr Ro  Fr Ro  Fr Ro 1 1 Ro 1 a23  (  2 2 )( 2  )6‖‖.    Fr Ro

0 

Consideration of the Gronwall's inequality in the differential inequality (4.11) provides a priori estimate 1

lim sup‖L2U (t ‖ )2 t 

222  22 

(5.29)

1 2

which shows U (t ) is uniformly bounded in time in D( L ) . Putting the estimate (5.28) and (5.29) together we obtain t 1

1

t



lim sup  ‖LU (s‖ ) 2 ds  (1  t 

) 22 .

(5.30)

Concerning the results for the attractor of the two-dimensional space variables initial-value problem for viscous  -plane ageostrophic equations without Reynolds stress we infer from the more refined a priori estimates (5.29) that the attractor for 4.2 is provided by the  -limit set of Q  B2 2 ,

A   (Q)  s0 Cl (t s S (t )Q). where B2 2 denotes an open ball in phase space of radius 2  2 , which depends on the geophysically relevant parameters. The closure is taken in the Hilbert space  . Utility of the group property and continuity of of the solution operators S (t ) defined for all time t 

, gives the following invariance property of the above established attractor:

S (t ) A  A, t  . Consequently, examining the expression for the invariance of the attractor we observe that the attractor for the two-dimensional space variables initial-value problem for viscous  -plane rotating Boussinesq equations without Reynolds stress is both positively and negatively invariant and consists of orbits or trajectories that are defined for all t .

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According to a priori estimates (4.12) and ()5.29, we have the following result which holds for all time. Proposition 5.1.4 If u0   and u is the unique solution of (4.2) for the case of two-dimensional space variables then the corresponding variational equation (4.24) has a unique solution  satisfying 1

  C ([0, T ];  )  L2 (0, T ; D( L2 )) T  0. 1

And if u0  D( L2 ) then 1

  C ([0, T ]; D( L2 ))  L2 (0, T ; D( L)) T  0. The proof of the proposition follows from the stronger a priori energy type estimate (5.29) and by repeating the arguments of the previous chapter. 5.3 Rotating Boussinesq equations with Reynolds stress The aim of this section is to establish nonlinear stability results for the rest state of rotating Boussinesq equations with Reynolds stress (2.29-2.31). In order to give nonlinear stability bounds that agree approximately with established paradigmatic example of initial-value problem (5.16), the functional formulation for the evolution equation of  -plane rotating Boussinesq equations with Reynolds stress is restated equivalently as follows:

dU  LU  N (U )  0 dt U (0)  U 0

(5.31)

where U  (u, v, w,  ) . The boundary conditions have been taken to be space-periodic as in the investigation of well-posedness of solution. Concerning the linear operator LU and the nonlinear operator N (U ) we have the following appropriate setup:

N U   B U ,U 

LU  TU  CU  SU  FU  Ro FU  (0, 0, , w) Ro Fr 2  Ek   Ro u   Al u  CU       1    Al      Ed   u  ▽u   b l ( u , u )  B(U ,U )  B(U )      u ▽   bl (u ,  ) 

 Ek  Ro TU     1   Ed

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 u  6  Al u    6  6    Al    6

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 0 1  1  y  Ro  1 0 S  ( ) 0 0 Ro  0 0

0 0  0 0  U  D( L)  D(T ). 0 0  0 0

From the above definition of the linear operator LU , we have

( LU ,U )  (TU ,U )  (CU ,U )  (

1 Ro  2 )  wdx. Ro Fr 

(5.32)

which is a modification of the bilinear form (4.33) that was employed in establishing well-posedness of solution for the problem of rotating Boussinesq equations with Reynolds stress. As in the previous section, with the attendant relation (5.32) we are able to demonstrate a necessary and sufficient nonlinear stability criterion and hence the elegant power of energy methods is fully realized. For the sake of completeness, we repeat the analysis in this section for rotating Boussinesq equations with Reynolds stress. The first effort is then to let R and (5.33) I (u ,  ) be defined as follows

1/ 2R  sup{ wdx}/ J (U )  supI (u ,  ) 

where the supremum is taken for U  D( L) . We shall compile various inequalities about the equation (5.31). Substituting

|  wdx | ‖ ( u‖2 ‖‖2 ) / 2 

´

into (5.33) and utilizing the Poincar e -Friedrichs inequality yields The attendant relation (5.32) may be restated as follows

( LU ,U )  (1  (

R  Ga .

1 Ro  2 ) I (u ,  )) J (U ), Ro Fr

(5.34)

with the generalized energy functional J (U ) defined in (5.4) given by the Lyapunov functional

J (U )  (TU ,U ) 

Ek 1 ‖▽u‖2  ‖▽‖2 . Ro Ed

1 Ro  2 )  2 R then the following is valid Ro Fr 1 Ro ( LU ,U )  (1  (  )) J (U )  0, 2 RRo 2 RFr 2

(5.35)

It is appropriate at this point to note that if (

(5.36)

and in this case  L is essentially dissipative. Additionally, the following inequalities hold

1 Ro 1 Ro  )) J (U ) ‖U‖2L  (1  ( 2  2 2 )) J (U ) 2 2 RRo 2 RFr  Ro  Fr 2 which shows that ‖U‖L is topologically equivalent to J (U ) . (1  (

The next step is to note that if (

(5.37)

1 Ro  2 )  2 R then there is a sequence {U n }   such that Ro Fr lim I (un , n )  1/ 2 R. n 

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1 1 Ro (  2 )) , there exists U such that 2 R Ro Fr 1 Ro 1 (5.38) ( LU ,U )  [1  (  2 )(  ò]J (U )  0, Ro Fr 2 R and in this case  L is essentially non-dissipative. We require the following lemma on the spectrum of  L whose Hence given ò  (

validity may be proved by using the technique of proof similar to those given above: Lemma 5.1.5  ( L)  (,0]  , where is either empty or an at most denumerable set consisting of isolated, positive eigenvalues n  n (


clustering only at zero. Next, we proceed to illustrate that suitable a priori estimates for the nonlinear operator N (U )) are available. Putting estimates (4.36) and (4.38) together into the inequality (4.40) yield the desired bounds

‖NU‖2  (CU , U )c02 ‖ ( CU‖2 ‖U‖2 ) 1 2

1 2

2  c ‖C U‖(‖ ‖U‖2 ) 1 L U‖  2 0

1 2

1 2

2  c  ‖CU‖(‖ ‖U‖2 ) 1 L U‖  2 0

1 2

1 2 1

1 2

1 2 1

 c   ‖L U‖( ‖LU‖ ‖U‖ ) 2 0

1 2

1 2 1

1 2

1 2 1

2

(5.39)

2

1 2

 c c  ‖L U‖‖ ( LU‖2 ‖U‖2 ) 2 0

 c c  J (U )(‖LU‖2 ‖U‖2 ) 2 0

1 2 1

for the nonlinearity N (U )  B(U ) where c  max{ ,1} . Consequently, a priori estimates (5.26) on the nonlinearity coincide with (5.9) which means that the nonlinear operator is suitably dominated by the linear operator LU . Thus, the purpose of proving Lyapunov instability aspect has been achieved. Moreover, when the following

1 Ro  2 )  2 R holds, we observe the topological equivalence of the Lyapunov functional J (U ) with the norm Ro Fr ‖U‖2L given by the inequality (5.37). In this case, putting the estimates (5.36) and (4.40) together and invoking the (

equivalence relation (5.37) gives the needed a priori estimate for the nonlinearity 1

1

‖NU‖2  c02c 2 12 J (U )(‖LU‖2 ‖U‖2 ) 1 2

1 2 1

(5.40)

 c c c  ‖U‖ ‖ ( LU‖   ‖U‖ ) 2 1 0

where c1  (1  (

2 L

2

1 1

2 L

1 Ro  ))1 . According to the sharp a priori estimate (5.40), aspect of nonlinear stability 2 2 RRo 2 RFr

provided by the inequality (5.6) is satisfied. Hence, energetic stability criteria for necessary and sufficient for nonlinear stability of the rest state of  -plane rotating Boussinesq equations with Reynolds stress (5.31) or equivalently (2.29-2.31) hold with DOI: 10.26855/jamc.2018.09.003

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the energy and entropy production provided by the following:

1 E (t )  {‖U‖2  J (U )  2 R  wdx}  2 which is specification of energy and the entropy production in the Sobolev H 1 -norm if the criterion

Ga  R  (

1 Ro  2 ) is valid. The result serve as the archetype for constructing generalized Lyapunov functionals Ro Fr

with interpretations such as energy and entropy which are decreasing along solution of the system. Furthermore, we observe that in the craft of nonlinear stability, the equations governing the flow of an incompressible stratified fluid under the Coriolis force induce damping mechanisms in the nature of viscosity, diffusion, combined stratification and rotation effects that manifest themselves in the evolution equation with the existence of Lyapunov functionals which are equivalent to some fictitious energy or enstrophy or entropy. Next, we focus our attention to the development of results for the attractor of the  -plane rotating Boussinesq equations with Reynolds stress (2.29-2.31) for both the three-dimensional and two-dimensional space variables. We show existence of the attractor for the dynamical system and proceed by customizing the splendid techniques employed in [14, 15, 91, 90, 23, 22] which have been summarized in the above proposition. Concerning the results for the attractor of the general three-dimensional space variables initial-value problem for viscous  -plane rotating Boussinesq equations with Reynolds stress we infer from the more refined a priori estimates (4.53) that the attractor for (4.32) is provided by the  -limit set of Q  B2 2 ,

A   (Q)  s0 Cl (t s S (t )Q). where B2 2 denotes an open ball in phase space of radius 2  2 , which depends on the geophysically relevant parameters. The closure is taken in the Hilbert space  . Utility of the group property and continuity of the solution operators

S (t ) defined for all time t 

, gives the following invariance property of the above established attractor:

S (t ) A  A, t  . Consequently, examining the expression for the invariance of the attractor we observe that the attractor for the three-dimensional space variables initial-value problem for viscous  -plane ageostrophic equations with Reynolds stress is both positively and negatively invariant and consists of orbits or trajectories that are defined for all t  . In the derivation of  -plane rotating Boussinesq equations with Reynolds stress, it was essential to note that the additive decomposition of the rotating Boussinesq equations quantities into coherent and incoherent terms and consideration of the averaging operator to obtain the Reynolds stress fields or wind stress fields resulted in a set of evolution equations that were not closed. In order to close the set of equations, we adopted the Pedlosky closure protocols.

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CHAPTER 6 Quasigeostrophic flows 6.1 Description of quasigeostrophy The baroclinic dissipative quasigeostrophic equations which we derive govern the evolution of streamfunction denoted  whenever the the Rossby number is considered asymptotically small. The quasigeostrophic system has been of oceanographic and meteorological interest especially for the modelling and forecasting of mesoscale and synotic scale eddies. For the sake of completeness, we give the description of the quasigeostrophic system which expresses the evolution of the zero-order potential vorticity of the flow. The standard derivation of the quasigeostrophic potential vorticity equations from the ageostrophic system, which is a set of primitive equations governing the flow of a viscous incompressible stratified fluid under the Coriolis force with or without the Reynolds stress, is accomplished by the systematic use of scaling arguments and asymptotic series in the Rossby number. The quasigeostrophic limit is useful from the fact that the translating and the rotating modons represent closed form vortex solutions of the quasigeostrophic potential vorticity equation when for example dissipation terms are neglected. Moreover, other simplifying assumptions include the replacement of vertical coordinate by density so that the quasigeostrophic equations may lend themselves to discretization in the vertical resulting in layered models. We allude here that the introduction of two-layer quasigeostrophic models lead to the indepence of horizontal velocity with height and the phenomena is called barotropic which complement baroclinic, the change of horizontal velocity with height. It is now possible to systematically utilize the presence of parameters in the evolution equations to embark on a perturbation analysis. The next aim is to derive geostrophic and quasigeostrophic models from the rotating Boussinesq equations for the atmosphere and ocean via a perturbation analysis of the flow fields at the Rossby number on the order of unity or less. In order to illustrate the the asymptotic expansion, consider

u  u0 ( x , t , Ro,  , Fr , Ek , Ed ,  )  Rou1 ( x , t , Ro,  , Fr , Ek , Ed ,  ) p  p0 ( x , t , Ro,  , Fr , Ek , Ed ,  )  Rop1 ( x , t , Ro,  , Fr , Ek , Ed ,  )

(6.1)

  0 ( x, t , Ro,  , Fr , Ek , Ed ,  )  Ro1 ( x , t , Ro,  , Fr , Ek , Ed ,  ). Fr 2  Ro . When the expansion (6.1) is inserted in the nondimensional system (2.26) and equating with assumption Ro terms of the same order in the Rossby number, Ro , at zero-order we obtain geostrophic equations

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p0 x p u0   0 y p 0   0 z ▽.u0  0 v0 

(6.2)

w0  0. In (6.2), horizontal velocity is divergence free and pressure plays the role of streamfunction. From consideration of the continuity equation and the realization that zero-order velocity is isobaric, that is, pressure of the system remain constant along streamlines, we obtain

w0  0 in addition of w0  0 . In order to derive equation for evolution of the geoz

strophic pressure, p0   , we equate first-order terms in the Rossby number which yield

u0 p Ek  u0 .▽u0  v1  y  0 v0   1  u0 t x Ro v0 p Ek  u0 .▽v0  u1   0u0  1   0  v0 t y Ro p   0u0  yu0  1   1 z ▽.u0  0

(6.3)

0 1  u0 .▽0  w1  0 . t Ed The next goal is to rewrite the equations (6.3) in an equivalent setup with emphasis on geostrophic pressure which we denote by

  p0 , the streamfunction. We take the two-dimensional curl of horizontal momentum in(6.3) to eliminate

the pressure gradient and utilize the fact that from (6.2) horizontal velocity is divergence free which together with suitable simplifying assumptions gives the following quasigeostrophic potential vorticity system

q Ek  J ( , q )  q t Ro  1  J ( ,  )   t Ed  2 q   2 y z    z  u y  v x DOI: 10.26855/jamc.2018.09.003

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(6.4)


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where the nondimensional fields u , q ,  , and  are fluid velocity, potential vorticity, density and streamfunction, respectively. The geophysically relevant parameter Ro is the Rossby number, Ek is the Ekman number,  is the reference reciprocal Coriolis parameter. Here J (.,.) is the Jacobian operator and the operator

  J (.,.) represents t

advection along fluid particle trajectories. Advances in our understanding of the dynamics of mesoscale eddies and vortex rings of the Agulhas Current Retroflection, and the Gulf Stream in the Middle Atlantic Bight have been based on modons which are nonlinear Rossby solitary wave solutions of quasigeostrophic equations. This research and innovation focuses primarily on geometric singular perturbation (GSP) theory and Melnikov techniques to address the problem of adiabatic chaos and transport for translating and rotating modons to the quasigeostrophic potential vorticity system which is relevant, e.g., a Lagrangian and Eulerian analysis of a geophysical fluid flow. A very general and central question is what hypotheses on the equations and singular solutions guarantee that the solutions approximate some solutions for the perturbed quasigeostrophic potential vorticity system. We present a geometric approach to the problem which gives more refined a priori energy type estimates on the position of the invariant manifold and its tangent planes as the manifold passes close to a normally hyperbolic piece of a slow manifold. We apply Melnikov technique to show that the Poincare map associated with modon equations has transverse heteroclinic orbits. We appeal to the Smale-Birkhoff Homoclinic Theorem and assert the existence of an invariant hyperbolic set which contains a countable infinity of unstable periodic orbits, a dense orbit and infinitely many heteroclinic orbits. The main object of this result gives geometric detection of adiabatic chaos and transport in singularly perturbed dynamical systems as depicted in figure \ref{tladi2} for the benchmark Lagrangian coherent structures utilizing the DsTool package program of Worfolk, Guckenheimer et al , [109] which greatly helped codify the discipline. In the Agulhas Current Retroflection, the information concerning adiabatic and chaotic advection in the mesoscale eddies and vortex rings is very recent and we thank Professor Geoff Brundrit who provided the material for introducing us to this rich physical oceanography programme. Turbulence, the splintering of smooth streams of fluid into chaotic vortices, does not just make for bumpy plane rides. It also throws a wrench into the very mathematics used to describe atmospheres, oceans and plumbing. Turbulence is the reason why the Navier-Stokes equations, the laws that govern fluid flow, are so famously hard that whoever proves whether or not they always work will win a million dollars from the Clay Mathematics Institute. But turbulence unreliability is, in its own way, reliable. Turbulence almost always steals energy from larger flows and channels it into smaller eddies. These eddies then transfer their energy into even smaller structures, and so on down. If you switch off the ceiling fan in a closed room, the air will soon fall still, as large gusts dissolve into smaller and smaller eddies that then vanish entirely into the thickness of the air. But when you flatten reality down to two dimensions, eddies join forces instead of dissipating. In a curious effect called an inverse cascade, which the applied mathematician Audrey Rogerson first fished out of the Navier-Stokes equations, turbulence in a flattened fluid passes energy up to bigger scales, not down to smaller ones. Eventually, these two-dimensional systems organize themselves into large, stable flows like vortexes or river-like jets. These flows, rather like vampires, support themselves by sucking away energy from turbulence, instead of the other way around. While the inverse cascade effect has been known for decades, a mathematical, qualitative prediction of what that final, stable flow looks like has eluded theorists. But a glimmer of hope came in 2017, when Stephen Tladi, now an applied mathematician at University of Limpopo, and his colleagues published a full description of the flow shape and speed under strict, specific conditions. Since then, new simulations, lab experiments and theoretical calculations published as recently as last month have both justified the team calculations and explored different cases where their prediction starts to break down. All this might seem like only a thought experiment. The universe is not flat. But geophysicists and planetary scientists have long suspected that real oceans and atmospheres often behave like flat systems, making the intricacies of two-dimensional turbulence surprisingly relevant to real problems. After all, on Earth, and especially on the gas giant planets like Jupiter and Saturn, weather is confined to thin, flattish slabs of atmosphere. Large patterns like DOI: 10.26855/jamc.2018.09.003

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hurricanes or the Gulf Stream, and Jupiter huge horizontal cloud bands and Great Red Spot, might all be feeding on energy from smaller scales. In the last few years, researchers analyzing winds both on Earth and on other planets have detected signatures of energy flowing to larger scales, the telltale sign of two-dimensional turbulence. They have begun mapping the conditions under which that behavior seems to stop or start. The hope, for a small but dedicated community of researchers, is to use the quirky but simpler world of two-dimensional fluids as a fresh entry point into processes that have otherwise proved impenetrably messy. They can actually make progress in two dimensions, said Audrey Rogerson, now an applied mathematician at Brown University, which is more than what we can say for most of our turbulence work. On September 30, 2003, the National Oceanic and Atmospheric Administration (NOAA) sent an aircraft into Isabel (El Nino), a Category 5 hurricane bearing down on the Atlantic Coast with winds gusting to 203 knots, the strongest readings ever observed in the Atlantic. NOAA wanted to get readings of turbulence at the bottom of a hurricane, crucial data for improving hurricane forecasts. This was the first, and last, time a crewed aircraft ever tried. At its lowest, the flight skimmed just 60 meters above the churning ocean. Eventually salt spray clogged up one of the plane four engines, and the pilots lost an engine in the middle of the storm. The mission succeeded, but it was so harrowing that afterward, NOAA banned low-level flights like this entirely. About a decade later, George Haller got interested in these data. Haller, an applied mathematician at the Swiss Federal Institute of Technology Zurich, had previously studied turbulent energy transfer in lab experiments. He wanted to see if he could catch the process in nature. He contacted Andrew Poje, an NOAA scientist who had been booked on the very next flight into Isabel (El Nino). By analyzing the distribution of wind speeds, the two calculated the direction in which energy was traveling between large and small fluctuations. Starting at about 150 meters above the ocean and leading up into the large flow of the hurricane itself, turbulence began to behave the way it does in two dimensions, the pair discovered. This could have been because wind shear forced eddies to stay in their respective thin horizontal layers instead of stretching vertically. Whatever the reason, though, the analysis showed that turbulent energy began flowing from smaller scales to larger scales, perhaps feeding Isabel (El Nino) from below. Their work suggests that turbulence may offer hurricanes an extra source of fuel, perhaps explaining why some storms maintain strength even when conditions suggest they should weaken. Haller now plans to use uncrewed flights and better sensors to help bolster that case. If we can prove that, it would be really amazing, he said. On Jupiter, a much larger world with an even flatter atmosphere, researchers have also pinpointed where turbulence switches between two-dimensional and three-dimensional behavior. Wind speed measurements taken by the Voyager probes, which flew past Jupiter in the 1970s, had already suggested that Jupiter large flows gain energy from smaller eddies. But in 2017, George Haller, and Andrew Poje, now his postdoc at the time, made a wind speed map using data from the space probe Cassini, which swung past Jupiter in 2000 on its way to Saturn. They saw energy flowing into larger and larger eddies, the hallmark of two-dimensional turbulence. But nothing about Jupiter is simple. On smaller scales, across patches of surface about the distance between New York and Los Angeles or less, energy dissipated instead, indicating that other processes must also be afoot. Then in March, the Juno spacecraft orbiting Jupiter found that the planet surface features extend deep into its atmosphere. The data suggest that not just fluid dynamics but magnetic fields sculpt the cloud bands. For Sanjeeva Balasuriya, who studies turbulence at the Ecole Normale Superieure (ENS) in Lyon, France, this is not too discouraging, since the two-dimensional models can still help. I do not think anybody believes the analogy should be perfect, he said. At the end of 2017, Balasuriya at ENS, sketched out his own theoretical account of how two-dimensional fluid flow can describe a rotating system such as the atmosphere of a planet. His work shows how flows built from smaller turbulence can match the enormous pattern of alternating bands visible on Jupiter through a backyard telescope. That makes it really relevant for discussing real phenomena, Balasuriya said. Balasuriya work relies on considering the statistics of the large-scale flows, which exchange energy and other quantities in a balance with their environment. But there is another path to predicting the form these flows will take, and it starts with those same obstreperous Navier-Stokes equations that lie at the root of fluid dynamics. For two totally fruitless years at the beginning of this decade, Denny Kirwan, a pen-and-paper theorist at Old Dominion University Center for Coastal Physical Oceanography (CCPO) in Norfolk, Virginia, stared at those equations. He tried to write out how the flow of energy would balance beDOI: 10.26855/jamc.2018.09.003

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tween small turbulent eddies and a bigger flow feeding on them in a simple case: a flat, square box. A single term, related to pressure, stood in the way of a solution. So Kirwan just dropped it. By discarding that troublesome term and assuming that the eddies in this system are too short-lived to interact with each other, Kirwan and his colleagues tamed the equations enough to solve the Navier-Stokes equations for this case of a modon. Then he tasked Bruce Lipphardt, his postdoc at the time, with running numerical simulations that proved it. It is always nice when you have an exact result in turbulence, Kirwan said. Those are rare. In the team 1996 paper, they found a formula for how the velocity in the resulting large flow, a big vortex ring, in this situation, would change with distance from its own center. And since then, various teams have filled in the theoretical rationale to excuse Kirwan lucky shortcut. Hoping for payoff in the pure mathematics of fluids and for insight into geophysical processes, physicists have also pushed the formula outside a simple square box, trying to figure out where it stops working. Just switching from a square to a rectangle makes a dramatic difference, for example. In this case, turbulence feeds river-like flows called jets in which the formula starts to fail. As of now, even the mathematics of the simplest case, the square box, is not totally settled. Kirwan formula describes the large stable vortex itself, but not the turbulent eddies that still flicker and fluctuate around it. If they vary enough, as they might in other situations, these fluctuations will overwhelm the stable flow. Just in May, though, two former members of Kirwan lab, published a paper describing the size of these fluctuations. It teaches a little bit what the limitations of the approach are, Lipphardt said. But their hope, ultimately, is to describe a far richer reality with the research focused on the dynamics of multi-pole eddies in the Mid-Atlantic Bight, comparing a quasigeostrophic model with satellite SST observations of cyclone-anticyclone pairs. For Haller, the pictures returned from Juno’s mission over Jupiter, showing a fantasyland of jets and tornadoes swirling like cream poured into the solar system largest coffee, are a driving inf luence. If it is something that I could help understand, that would be cool, he said. Since its inception in the 19th century through the efforts of Poincare and Lyapunov, the theory of dynamical systems addresses the qualitative behaviour of dynamical systems as understood from models. From this perspective, the modelling of dynamical processes in applications requires a detailed understanding of the processes to be analyzed. This deep understanding leads to a model, which is an approximation of the observed reality and is often expressed by a system of ordinary/partial, underdetermined (control), deterministic/stochastic differential or difference equations. While models are very precise for many processes, for some of the most challenging applications of dynamical systems (such as climate dynamics), the development of such models is notably difficult.

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Figure 6.1 Lagrangian coherent structures

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CHAPTER 7 Conclusions

7.1 Discussion of results This investigation represents the first effort in the theoretical and applied aspects of the quasigeostrophic and rotating Boussinesq equations. The results are striking and encouraging, and we point out that based on the manifestation of well-posedness and stability, it is possible to design numerical algorithms such as finite-difference schemes. Furthermore, we observe that the nonlinear stability criteria may give the possibility of a sharper examination of the numerical approximation of the solution in the sense of the Lax equivalence theorem: a finite-difference scheme to a well-posed system of partial differential equation converges to the solution with the rate of convergence specified by the order of accuracy of the scheme. The theory of the Navier-Stokes equations (NSE) constitutes a central problem in continuum physics. These equations are physically well accepted model for the description of very common phenomena, and much effort has been devoted to them by fluid mechanics engineers, meteorologists, and oceanographers; nevertheless many problems are still at the frontier of science. Turbulence, the splintering of smooth streams of fluid into chaotic vortices, does not just make for bumpy plane rides. It also throws a wrench into the very mathematics used to describe atmospheres, oceans and plumbing. Turbulence is the reason why the Navier-Stokes equations, the laws that govern fluid flow, are so famously hard that whoever proves whether or not they always work will win a million dollars from the Clay Mathematics Institute. On the mathematical side NSE are a model for the study of nonlinear phenomena and nonlinear equations which is itself in its infancy: the quasigeostrophic and rotating Boussinesq equations encompass four central problems in nonlinear equations, namely, well-posedness, oscillations, discontinuities, and nonlinear dynamics. The mathematical theory of the NSE started with the pioneering work of J. Leray, who on this occasion introduced for the first time the concept of weak formulation of partial differential equations before the development of the distribution theory by L. Schwartz, shortly before S.L. Sobolev systematically introduced the spaces which bear his name. Motivated by the NSE, the study addressed two aspects of the mathematical theory of the quasigeostrophic and rotating Boussinesq equations: well-posedness, i.e., existence, uniqueness and regularity of the solutions in various function spaces and a new approach to energy theory in the stability of fluid motion. Progress in NSE had a big influence on the development of the qualitative theory of parabolic PDE as we know it today. The dynamical system generated by NSE was shown in the late 1960s to have special properties, which were special cases of asymptotically smooth dynamical systems. With dissipation, these equations also have compact global attractors. Once these concepts were introduced, it led to many natural questions about the finite dimensionality of the attractor, the discussion of special types of equations for which one could describe the flow on the attractor, etc. The subjects essentially touched in this research include the transition to turbulence in relation with bifurcation, the relations of the NSE with models of turbulence, non-Newtonian flows, and numerical analysis (numerical algorithms and scientific computing with MATLAB and C/C++). The problem of the numerical solution of the NSE equations was initialized by von Neumann and his collaborators in the 1940s computing with FORTRAN to elucidate transonic fluid flow, which refers to partial differential equations that possess both elliptic and hyperbolic regions. The flow is supersonic in the elliptic region, while a shock wave is created at the boundary between the elliptic and hyperbolic regions. S.J. Friedlander chose Cathleen Morawetz article based on her 1980s Gibbs Lecture entitled The mathematical approach to the sonic barrier. This article combined state-of-the-art applied mathematics with practical issues concerning flight at close to the speed of sound. It is still a rich source of open questions. Cathleen Morawetz wrote this article in connection with The Josiah Willard Gibbs Lecture she presented at the American Mathematical Society meeting in San Francisco, California, January 7, 1981. This is a beautiful piece on a subject at the core of applied mathematical analysis and numerical methods motivated by the pressing engineering technology of the mid-twentieth DOI: 10.26855/jamc.2018.09.003

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century and the human urge to travel fast at efficient cost. From the mathematical viewpoint this problem comprises the understanding of models of nonlinear partial differential equations arising in compressible fluid mechanics, as much as understanding how to obtain numerical approximations to a model discretization that result both in finding numerically computed surfaces close to the model’s solutions (if such exists) but also in matching these computed model outputs to experiments from engineering or experimental observation viewpoints. With the considerable development of the power of computers during the past decades, numerical simulation has developed as subjects of its own, Computational Fluid Dynamics (CFD), at the interface with engineering and physics. 7.2 New research and innovation directions The author elucidate in a concrete way dynamical challenges concerning approximate inertial manifolds (AIMS), i.e., globally invariant, exponentially attracting, finite-dimensional smooth manifolds, for dynamical systems on Hilbert spaces. The goal of this theory is to prove the basic theorem of approximation dynamics, wherein it is shown that there is a fundamental connection between the order of the approximating manifold and the well-posedness and long-time dynamics of the rotating Boussinesq and quasigeostrophic equations. Abreast of these results, the author presents a new technique for the construction of AIMS that preserve the structure of attractors for the flow in the thermal convection of Oldroyd-B fluids. Although this article is motivated by challenges in partial differential equations, we consider a two-mode Faedo-Galerkin approximation (Ed Lorenz ansatz sense) given by a system of singularly perturbed ordinary differential equations. We note that the foundation for the study of the low-dimensional model of turbulence, which capture the dominant focus energy bearing scales, from the flow for the thermal convection of viscoelastic fluids with a differential constitutive law, is the Lorenz equations extended through singular perturbation:

x    y  x   w , y   x  x  y  xz, z   z  xy,

 w   x  w. It is shown that the equilibrium point at the origin is asymptotically stable and attractive by the LaSalle invariance principle. Utilizing center manifold calculations, we show existence of supercritical pitchfork bifurcation and Poincare-Andronov-Hopf bifurcation. In order to utilize geometric singular perturbation (GSP) theory and Melnikov techniques, we perturb the problem and carry the nonlinear analysis further to the question of the persistence of inclination-flip homoclinic orbits. We appeal to the Shilnikov-Rychlik Homoclinic Theorem and assert existence of the Lorenz chaotic attractor which contains a nested sequence of hyperbolic, transitive, compact invariant sets each of which is equivalent to a suspension over a finite Markov chain with positive topological entropy.

Acknowledgments This research project is dedicated to Prof.Dr. Natalia Copelovici Sternberg, who taught me mathematical modelling with differential equations (ordinary differential equations, partial differential equations, stochastic differential equations, and functional differential equations) at Clark University, Worcester, Massachusetts, USA. I am deeply grateful for her expressions of encouragement and support and her generosity of spirit and fundamental good nature have inspired me. This masterly exposition and encyclopedic presentation of the nonlinear dynamics of rotating stratified fluid flows was done in collaboration with my students at University of Limpopo and I thank them for their assistance on matters de modus vivendi. I am forever indebted to my Mom Seageng for instilling in me the idea of ambition and to my Dad Pekwa for the idea of intellectual achievement. Thank you Mom and Dad for your confidence, devotion, guidance and understanding in helping me to attain this most important milestone of my life...this expression of heartfelt thanks is affectionately inscribed. Wife, Raesibe, you are the greatest gift God could've ever given me. Thank you for finding me. Daily you

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teach me, stretch me, and cover me. I can never thank you enough. Thank you for hearing God and using your anointing to create in earth as it is in heaven. Sisters Mmafedi, Rammushi, Raesetja and brothers Makebe, Petja, Lepharathaba, Mampaka you've been my protector since day one. Thank you for carrying out your assignment with perfection. My children, Thapelo and Seageng, I'm the most blessed Dad in the world. You all are gifts from God to me and I cherish each of you. It is the joy of my life to watch you grow into all what God has called you to be... and even more of an honor to play a part in helping to cultivate your gifts. Mama Dimakatso, thank you for loving me and embracing me like your own. You are the best Mother in love. Your support and prayers push us towards greatness. I love you Grace, Kabelo, and Mojapelo. To my Pastors Rev.Dr. Chris and Ose Oyakhilome, thank you for your love, teaching and mentorship. You both are amazing gifts to my life. To all my aunts, uncles, cousins, and youthful friends (Nelson Sefara, Thama Duba, Motodi Maserumule, Daniel Mashao, Gelonia Dent, Monica Stephens and Mulalo Doyoyo), thank you all for being supportive and keeping me covered in prayer. A beautiful NASA image of von Karman vortex streets over the Cape Verde islands off the western coast of Africa. The Reynolds number of this flow is on the order of 10 billion (1e10).

References [1] D.J. ACHESON, Elementary fluid dynamics, Clarendon Press, Oxford, 1990. [2] R.A. ADAMS, Sobolev spaces, Academic Press, New York, 1975. [3] S. AGMON AND L. NIRENBERG, Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space, Comm. Pure Appl. Math, 20 (1967), 207-229. [4] V.I. ARNOLD, Mathematical methods of classical mechanics, Springer-Verlag, Inc., New York, 1989. [5] H. BAER AND K. STEPHAN. Heat and mass transfer, transl. by Janepark N., Springer-Verlag, Inc., New York, 1998. [6] S.H. BALASURIYA, Barriers and transport in unsteady flows: a Melnikov approach, SIAM, Philadelphia, 2017. [7] S.H. BALASURIYA, C.K.R.T. JONES, AND B. SANDSTEDE, Viscous perturbations of vorticity-conserving flows and separatrix splitting, Nonlinearity 11 (1998), 47-77. [8] G.K. BATCHELOR. Introduction to fluid dynamics. Cambridge University Press, Cambridge, 1967. [9] A.J. BOURGEOIS AND J.T. BEALE, Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean, SIAM J. Math. Anal. 25 (4) 1994:1023-1068. [10] M.G. BROWN AND R.M. SAMELSON, Particle motion in vorticity-conserving 2-dimensional incompressible flows, Physics of Fluids 6 (1994) 2875-2876. [11] C. CANUTO, M.Y. HUSSAINI, A. QUARTERONI, AND T.A. ZANG, Spectral methods in fluid dynamics, Springer-Verlag, Inc., New York, 1988. [12] A. CONSTANTIN, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS regional conference, SIAM, Philadelphia, 2011. [13] B. CUSHMAN-ROISIN, Introduction to geophysical fluid dynamics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1994. [14] C.M. DAFERMOS, Contraction semigroups and trend to equilibrium in continuum mechanics, Springer-Verlag, Inc., New York, 1976. [15] C.M. DAFERMOS, Hyperbolic conservation laws in continuum physics, 3rd ed., Springer-Verlag, Inc., New York, 2010. [16] F. DUMORTIER, H. KOKUBU, AND H. OKA, A degenerate singularity generating geometric Lorenz attractors, Ergod. Th. Dynam. Sys. 15 (1995) 833-856. [17] G. FLIERL, Isolated eddy models in geophysics, Ann. Rev. Fluid Mech. 19(1987) 493-530. [18] G. FLIERL, M. STERN, AND J. WHITEHEAD, The physical significance of modons: laboratory experiments and general physical constraints, Dyn. Atmos. Oceans 7(1983) 233-263. [19] G. FLIERL, M.E. STERN, AND A. WHITEHEAD, The physical significance of modons: laboratory experiments and general integral constraints. Dyn. Atmos. Oceans 7 (1983) 263-293.

DOI: 10.26855/jamc.2018.09.003

451


M.J.P.S. Tlad i

[20] G. FLIERL, V.D. LARICHEV, J.C. MCWILLIAMS, AND G.M. REZNIK, The dynamics of baroclinic and barotropic solitary eddies, Dyn. Atmos. Oceans 5 (1980) 1-41. [21] G. FLIERL, P. MALANOTTE-RIZOLLI, AND N. ZABUSKY N., Nonlinear waves and coherent vortex structures in barotropic -plane jets, J. Phys. Oceanogr. 17 (1987) 1408-1438. [22] C. FOIAS, O. MANLEY AND R. TEMAM, Attractors for the Bernard problem: Existence and physical bounds on their fractal dimension, Nonlinear Anal. 11 (1987), 939-967. [23] C. FOIAS, G. SELL AND R. TEMAM, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), 309-353. [24] S.J. FRIEDLANDER, Lectures on stability and instability of ideal fluid, Institute of Advanced Studies, Princeto n University, Princeton, 1999. [25] S.J. FRIEDLANDER, Introduction to the mathematical theory of geophysical fluid dynamics, North Holland, New York, 1980. [26] G.P. GALDI, An introduction to the mathematical theory of the Navier-Stokes equations, Springer-Verlag, Inc., New York, 1994. [27] G.P. GALDI AND M. PADULA, A new approach to energy theory in the stability of fluid motion, Arch. Rational Mech. Anal. 110 (1990), 187-286. [28] G.P. GALDI AND S. RIONERO, Weighted energy methods in fluid dynamics and elasticity, Springer-Verlag, Inc., New York, 1985. [29] J. GRUENDLER, Homoclinic solutions and chaos in ordinary differential equations with singular perturbations, Trans. Amer. Math. 350 (9) (1998) 3797-3814. [30] J. GRUENDLER, The existence of transverse homoclinic solutions for higher order equations, J. Differential Equations 130 (1996), 307-320. [31] J. GUCKENHEIMER AND P. HOLMES, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, Springer-Verlag, Inc., New York, 1983. [32] M.E. GURTIN, An introduction to continuum mechanics, Academic Press, Inc., San Diego, 1981. [33] J.K. HALE AND H. KOCAK, Dynamics and bifurcations, Springer-Verlag, Inc., New York, 1991. [34] J.K. HALE AND S.M. VERDUYN LUNEL, Introduction to functional differential equations, Springer-Verlag, Inc., New York, 1993. [35] J.K. HALE AND S.-N. CHOW, Methods of bifurcation theory, Springer-Verlag, Inc., New York, 1982. [36] J.K. HALE, Ordinary differential equations, Wiley-Interscience, New York, 1969. [37] G. HALLER, Chaos near resonance, Springer-Verlag, Inc., New York, 1999. [38] G. HALLER AND A.C. POJE, Finite time transport in aperiodic flows, Physica D 83 (1998) 353-380. [39] P. HARTMAN, Ordinary differential equations, SIAM, Philadelphia, 2002. [40] S.B. HOOKER, J.J. HOLDZKOM, AND A.D. KIRWAN, A comparison of a hydrodynamic lens model to observations of a warm core ring, J. Geophys. Res. 100 C8}(1995) 15889-15897. [41] S.B. HOOKER AND J.W. BROWN, Warm core ring dynamics derived from satellite imagery, J. Geophys. Res. 99 (1994) 25181-25194. [42] S.B. HOOKER AND D.B. OLSON, Center of mass estimation in closed vortices: A verification in principle and practice, J. Atmos. Oceanic Technol. {\bf 1} (1984) 247-255. [43] F. JOHN, Partial differential equations, 4th ed., Springer-Verlag, Inc., New York, 1971. [44] C.K.R.T. JONES, Session on dynamical systems: geometric singular perturbation theory, C.I.M.E. Lectures, 1994. [45] T.M. JOYCE, J.K.B. BISHOP AND O.B. BROWN, Observations of offshore shelf-water transport induced by a warm-core ring, Deep Sea Res. 39 (1992) 97-113. [46] T.M. JOYCE, Velocity and hydrographic structure of a Gulf Stream warm-core ring, J. Phys. Oceanogr. 14 (1984) 936-947. [47] T.M. JOYCE, Gulf Stream warm-core ring collection: an introduction, J. Geophys. Res. {\bf 90} (1985) 8801-8802. [48] T. KAPER AND G. KOVACIC, A geometric criterion for adiabatic chaos, J. Math. Phys. 35/3 (1994) 1202-1218. [49] T. KAPER AND S. WIGGINS, Lobe area in adiabatic Hamiltonian systems, Physica D 51 (1991) 205 -212. DOI: 10.26855/jamc.2018.09.003

452


M.J.P.S. Tlad i

[50] T. KATO, Perturbation theory for linear operators, Springer-Verlag, Inc. New York, 1966. [51] T. KATO AND G. PONCE, Well-posedness of the Euler and Navier-Stokes equations in Lebesgue spaces, Revista Mathematica Iberoamerican 2 (1986) 73-88. [52] R.E. KHAYAT, Chaos and overstability in the thermal convection of viscoelastic fluids, J. Non-Newtonian Fluid Mech. 53 (1994) 227-255. [53] A.D. KIRWAN, P.R. MIED, AND B.L. LIPPHARDT, Rotating modons over isolated topography in two -layer ocean, Z. Angwe. Math. Phys. 48 (1997) 535-570. [54] A.D, KIRWAN, P.R. MIED, AND G.J. LINDEMANN, Rotating modons over isolated topographic features, J. Phys. Oceanogr. 22 (1992) 1569-1582. [55] R.C. KLOOSTERZIEL, G.F. CARNEVALE AND D. PHILIPPE, Propagation of the barotropic dipoles over topography in a rotating tank, Dyn. Atmos. Oceans 19 (1993) 65-100. [56] A. MAJDA, Introduction to PDE and waves for the atmosphere and ocean, Courant Institute of Mathematical Sciences, New York University, New York, 2003. [57] A. MAJDA, Vorticity and the mathematical theory of incompressible flow, Comm. Pure Appl. Math. 39 (1986) S187 -S220. [58] A. MAJDA AND S. KLAINERMAN, Singular limits of quasilnear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 43 (1981) 481-524. [59] D. MARCHESIN, A.V. AZEVEDO, B.J. PLOHR, AND K. ZUMBRUN, Nonuniqueness of solutions of Riemann pro blems, Z. Angew. Math. Phys. 47 (1996) 977-998. [60] A. MATSUMURU AND T. NISHIDA, The initial-boundary value problem for the equations of motion of general fluids, North-Holland Publishing Co., New York 10 (1982). [61] J.C MCWILLIAMS, An application of the equivalent modons to atmospheric blocking, Dyn. Atmos. Oceans 5}(1980) 43-66. [62] P.R. MIED AND G.J. LINDEMANN, The birth and evolution of eastward propagating modons, J. Phys. Oceanogr. 12 (1982) 213-230. [63] P. MILLER, A. ROGERSON, C.K.R.T. JONES, AND L. PRATT, Quantifying transport in numerically generated velocity fields, Physica D 110 (1997) 105-122. [64] P. MILLER, A. ROGERSON, C.K.R.T. JONES, AND L. PRATT, Lagrangian motion and fluid exchange in a barotropic meandering jet, J. Phys. Oceanogr. 29 (1999) 2635-2655. [65] J. MOSER, Stable and random motions in dynamical systems, Princeton University Press, Princeton, 1973. [66] O.A. LADYZHENSKAYA, The mathematical theory of viscous incompressible flow, transl. by R.A. Silverman and J. Chu, Gordon and Breach Science Publishers, New York, 1969. [67] A.J. LICHTENBERG AND M.A. LIEBERMAN, Regular and chaotic dynamics, Springer-Verlag, Inc., New York, 1992. [68] B. L. LIPPHARDT, Dynamics of dipoles in the Middle Atlantic Bight, CCPO tech. 95 -07, Old Dominion University, Norfolk, 1995. [69] E.N. LORENZ, Deterministic non-periodic flow, J. Atm. Sci. 20 (1963) 130-141. [70] E.N. LORENZ,, Attractor sets and quasi-geostrophic equilibrium, J. Atm. Sci. 37 (1980) 1685-1699. [71] J.L. LUMLEY, G. BERKOOZ, AND P. HOLMES, Turbulence, coherent structures, dynamical systems and symmetry, Cambridge University Press, Cambridge, 1996. [72] J.L. LUMLEY, ed., Turbulence at the crossroads, Springer-Verlag, Inc., New York, 1990. [73] J. LUTJEHARMS, W. DE RUIJTER, A. BIASTOCH, S. DRIJFHOUT, R. MATANO, T. PICHEVIN, P. VAN LEEUWEN, AND W. WEIJER, Indian-Atlantic interocean exchange: dynamics, estimation and impact, J. Geophys. Res. 104 C9 (1999) 20885-20910. [74] J. LUTJEHARMS AND R. VAN BALLEGOOLLEN, The retroflection of the Agulhas current, J. Phys. Oceanogr. 18 (1988) 1570-1583. [75] D.B. OLSON AND R.H. EVANS, Rings of the Agulhas Current, Deep-Sea Reseach 33(1)(1996) 27-42. [76] J.M OTTINO, The kinematics of mixing, Cambridge University Press, Cambridge, 1989. DOI: 10.26855/jamc.2018.09.003

453


M.J.P.S. Tlad i

[77] J. PEDLOSKY. Geophysical fluid dynamics, 2nd ed., Springer-Verlag, Inc., New York, 1987. [78] H.-O. PEITGEN, H. JURGENS, AND D. SAUPE, Chaos and fractals: new frontiers of science, Springer-Verlag, Inc., New York, 1992. [79] R. PIERREHUMBERT, A family of steady, translating vortex pairs with distributed vorticity, J. Fluid. Mech. 99 (1980) 129-144. [80] R. PIERREHUMBERT, Chaotic mixing of tracer and vortcity by modulated travelling Rossby waves, Geophy. Astrophys. Fluid Dynamics 58 (1991) 285-319. [81] R. PIERREHUMBERT AND P. MALGUZZI, Forced coherent structures and local multiple equilibria in a barotropic atmosphere, J. Atmos. Sci. 41 (1984) 246-257. [82] [B.D. REDDY AND G.P. GALDI, Well-posedness of the problem of fiber suspension flows, J. Non-Newtonian Fluid Mech., 83 (1999) 205-230. [83] B.D. REDDY, Introductory functional analysis, Springer-Verlag, Inc., New York, 1998. [84] M. REED AND B. SIMON, Functional analysis, Academic Press, San Diego, 1980. [85] H.L. ROYDEN, Real Analysis, Macmillan Publishing Co., New York, 1988. [86] S. SMALE, Dynamics retrospective, Physica D 51 (1991) 267-273. [87] S. SMALE, Differentiable dynamical systems, Bull. Amer. Math. 73 (1967) 747-817. [88] E.A. SPIEGEL AND G. VERONIS, On the Boussinesq approximation for a compressible fluid, Astrophy. J. 131 (1960) 442-447. [89] S.H. STROGATZ, Nonlinear dynamics and chaos, Addison-Wesley, New York, 1994. [90] R. TEMAM, Navier-Stokes equations and nonlinear functional analysis, CBMS regional conference, SIAM, Philadelphia, 1983. [91] R. TEMAM, Infinite-dimensional dynamical systems in mechanics and physics, Springer-Verlag, Inc., New York, 1988. [92] R. TEMAM, Navier-Stokes equations: Theory and Numerical Analysis, AMS Chelsea Ed., Providence, 2001. [93] R. TEMAM, B. NICOLAENKO, C. FOIAS AND P. CONSTANTIN, Integral manifolds and inertial manifolds for diss ipative partial differential equations, Springer-Verlag, Inc., New York, 1988. [94] E.S. TITI AND C. CAO, Global well-posedness of the three-dimensional stratified primitive equations with partial vertical mixing turbulence diffusion, Communications in Mathematical Physics, 310 (2012) 537-568. [95] E.S. TITI AND C. CAO, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Annals of Mathematics, 166 (2007) 245-267. [96] E.S. TITI, On approximate inertial manifolds to the Navier-Stokes equations, J. Math. Anal. And Appl., 149 (1990) 540-557. [97] M.J.P.S. TLADI, Well-posedness and long-time dynamics of geophysical fluid flows, Journal of Applied Mathematics and Computation, 2 (8) 2018: 291-331. [98] M.J.P.S. TLADI, On the qualitative theory of the rotating Boussinesq and quasigeostrophic equations, Quaetiones Mathematicae 40 (6) 2017: 705-737. [99] M.J.P.S. TLADI, A geometric approach to differential equations, Lecture Notes, Department of Math. And Applied Math., University of Limpopo, 2009. [100] M.J.P.S. TLADI, Adiabatic chaos and transport in mesoscale eddies and vortex rings, CERECAM tech. 2005-01, University of Cape Town, Rondebosch, 2005. [101] M.J.P.S. TLADI, Well-posedness and long-time dynamics of -plane ageostrophic flows, Ph.D. Thesis, Department of Math. And Applied Math., University of Cape Town, 2004. [102] A. TSINOBER AND H.K. MOFFATT, eds., Topological fluid mechanics, Cambridge University Press, Cambridge, 1990. [103] S. WANG, Attractors for the 3D baroclinic quasigeostrophic equations of large scale atmosphere, J. Math. Anal. And Appl. 165 (1992) 266-283. [104] S. WANG, J.L. LIONS AND R. TEMAM, New formulations of the primitive equations of the atmosphere and applications, Nonlinearity 5 (1992), 237-288. [105] S. WANG, J.L. LIONS AND R. TEMAM, On the equations of the large-scale ocean, Nonlinearity 5 (1992), 1007-1053. DOI: 10.26855/jamc.2018.09.003

454


M.J.P.S. Tlad i

[106] S. WIGGINS, Chaotic transport in dynamical systems, Springer-Verlag, Inc., New York, 1992. [107] S. WIGGINS, Introduction to applied nonlinear dynamical systems and chaos, Springer-Verlag, Inc., New York, 1990. [108] P.A. WORFOLK AND W. CRAIG, An integrable normal form for water waves in infinite depth, Physica D 84 (1995) 513-531. [109] P.A. WORFOLK, J. GUCKENHEIMER, M. MYERS, F. WICKLIN AND A. BAK, DsTool: Computer assisted exploration of dynamical systems, Notices Amer. Math. Soc. 39 (1992), 303-309.

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