Laboratory experiments on the generation of internal waves on two ...

GEOPHYSICAL RESEARCH LETTERS, VOL. 39, L04602, doi:10.1029/2011GL049993, 2012

Laboratory experiments on the generation of internal waves on two kinds of continental margin Tao Wang,1 Xu Chen,1 and Wensheng Jiang1 Received 14 October 2011; revised 18 January 2012; accepted 18 January 2012; published 17 February 2012.

[1] We quantitatively study the relationship and differences between the internal waves generated on two kinds of continental margin in laboratory. Experimental results show that the steepness of the shelf break exerts a strong influence on the spatial structure of the internal waves generated on the continental margin. When the oscillatory barotropic tide flows over the continental margin with an abrupt shelf break and is not resonant with its slope, internal waves with three energy rays are generated, while on the continental margin with a gentle shelf break, only two rays can be found in the internal waves. Moreover, we compare the flux, width and kinetic energy flux of these rays and find that the phases of the internal waves generated by the interaction between the oscillatory barotropic tide and the shelf break lag behind the oscillatory barotropic tide. Citation: Wang, T., X. Chen, and W. Jiang (2012), Laboratory experiments on the generation of internal waves on two kinds of continental margin, Geophys. Res. Lett., 39, L04602, doi:10.1029/2011GL049993.

1. Introduction [2] The oceanic internal wave, also known as the internal gravity wave or the internal inertial-gravity wave, occurs in the interior of the sea where the seawater is stably stratified. The generation and breaking of internal waves can cause strong mixing, which plays a vital role in the energy transfer from the motions of seawater on a large scale to those on a small scale [Wunsch and Ferrari, 2004]. The studies in the past two decades have clearly shown that internal tides were substantially generated by the tidal currents over seamounts and continental margins [Munk and Wunsch, 1998]. Furthermore, internal waves on the continental margins have significant influences on biology, engineering and military activities. Therefore, it is very important to study the generation of internal waves on continental margins. [3] However, due to the limitation of in-situ instruments, the relevant oceanic observations are limited to the measurement at single points, which result in insufficient data and make it impossible to fully capture the spatial structure of internal waves. As a result, researchers could only give a brief description of internal waves [New and Pingree, 1992; Azevedo et al., 2006]. Although a number of numerical and theoretical studies were also done on the generation of internal waves on continental margins [Baines and Fang, 1985; Craig, 1987; Giese and Hollander, 1987; Holloway, 2001; Gerkema et al., 2004], their results depend on param-

eterization, and remain to be verified. Fortunately, the experimental methods can make up for the above disadvantages. So far, only a few experimental studies on the generation of internal waves on the continental margins [Chapman, 1984; Sutherland et al., 1999; Zhang et al., 2008] have been reported, but they are not able to measure the spatial structure of both the velocity field and the density field of internal waves simultaneously. Nowadays, the Particle Image Velocimetry (PIV) [Dalziel et al., 2007] and the Synthetic Schlieren [Dalziel et al., 2000; Sveen and Dalziel, 2005; Gostiaux et al., 2006] techniques have made this possible. [4] In stably stratified seawater, internal waves will be generated when the oscillatory tide flows over the continental margins, but the shape of the continental margin may influence their structure. Baines [1982] and St. Laurent et al. [2003], by using the theoretical method and the numerical simulation method respectively, demonstrated that internal waves with three rays would be generated when the barotropic tide flowed over the continental margins. However, only two rays were reported according to both the numerical results given by Gerkema et al. [2004] and the experimental results given by Gostiaux and Dauxois [2007]. Gostiaux and Dauxois [2007] explained that this may be due to the fact that there is a singular point in both Baines’ study [1982] and the numerical simulation of St. Laurent et al. [2003]. According to Gostiaux and Dauxois [2007], it is the very singular point that generates the third ray, but this explanation has not been proven. Therefore, the generation of the third ray remains to be clarified. [5] We think that the root cause of the difference between the results of Baines [1982] and Gostiaux and Dauxois [2007] is the difference in the shapes of their models. An abrupt shelf break exists in Baines’ [1982] model, while in Gostiaux and Dauxois’ [2007] model, there is a gentle shelf break. We think the steepness of the junction between the shelf and the slope, which is called the shelf break, has a strong influence on the spatial structure of internal waves. [6] In order to clarify this issue, we designed two continental margin models (Figure 1), one with an abrupt shelf break and the other with a gentle shelf break. Then we got the velocity fields of the internal waves by the PIV technique and their density fields by the Synthetic Schlieren technique. Using the velocity and density fields, we discussed the relationship and differences between the internal waves generated on the two different continental margins, and made a quantitative analysis of the flux, width and kinetic energy flux of the internal waves’ rays.

1 Physical Oceanography Laboratory, Ocean University of China, Qingdao, China.

2. Experimental Setup

Copyright 2012 by the American Geophysical Union. 0094-8276/12/2011GL049993

[7] The laboratory experiments were conducted in a rectangular tank, which is 5000 mm long, 400 mm high and

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Figure 1. (a) The experiment setup diagram, including the PIV and the Synthetic Schlieren equipment. (b) The continental margin with an abrupt shelf break. (c) The continental margin with a gentle shelf break. Both models are 150 mm wide, 150 mm high and 450 mm long. The horizontal length of the slope in either model is 150 mm. The angle between the slope and the horizon is 45° in Figure 1b, which is equal to the angle between the tangent line through the middle point of the slope and the horizon in Figure 1c.

150 mm wide (Figure 1). We studied the internal waves in the directions parallel to both the length and the height of the tank, so the results were two-dimensional. [8] In order to simulate the situation in which the oscillatory tide flows over the continental margin, we linked our model to the towing device by an L-shaped metal arm and made it oscillate back and forth by an eccentric wheel driven by an electromotor. The frequency of the electromotor could be adjusted. The oscillating amplitude was set to be far less than the length of the model, which fits the actual situation of the sea [Bell, 1975]. The velocity of the model is u¼


 6p 2p sin tþ’ T T

ð1Þ

where T is the oscillating period of the model, and j is related to the starting position of the model. The experimental water, which was 272 mm deep, was linearly stratified, with its surface and bottom density being 1003 g/mm3 and 1025 g/mm3 respectively (Figure 1). According to the Brunt-Väisälä frequency of the linearly stratified water, we adjusted the model’s oscillatory circular frequency to 0.42/s so that the theoretical angle between the ray of the internal wave and the horizon was about 28°. For the continental margin with an abrupt shelf break, the angle between its slope and the horizon was 45°. As a result, there was no resonance between the internal waves and the slope. The internal waves would be generated at the shelf break. On the other hand, for the continental margin with the gentle shelf break, the internal waves would be generated in the near-critical region. [9] We then obtained the velocity fields with the PIV technique and the density fields with the Synthetic Schlieren technique. Images obtained with a resolution of 2452  2054 pixels were analyzed initially by the DigiFlow software. The experiments showed that the internal waves would become stable after the oscillatory tide flowed over the margins for about 10 periods, so we began shooting the PIV and Synthetic Schlieren images after the models oscillated for 10

periods. We also measured the velocity field when these models oscillated in isopycnal water (barotropic tide).

3. Results and Analysis 3.1. Distribution of the Kinetic Energy and Density Perturbation [10] The PIV and Synthetic Schlieren techniques enable researchers to get the velocity fields and the density perturbation fields of the internal waves at any time, and the density perturbation fields, in turn, make it possible to get the density fields. We used Fast Fourier Transform (FFT) to find the main frequency of the velocity variance of the internal waves (Figure 2). [11] It can be seen in Figure 2 that the main frequency of the internal waves is identical to that of the oscillatory barotropic tide, which shows that the energy of the internal waves is mostly stored in the first harmonic wave, with only a small portion of its energy stored in the higher harmonic waves. So we filtered the velocity fields and the density perturbation fields by using the main frequency to get the velocity

Figure 2. Fast Fourier Transform. w represents the circular frequency of the harmonic waves and N represents the Brunt-Väisälä frequency of the linearly stratified water, which is constant. We can see that, the main circular frequencies of the internal waves generated by these two shelf breaks are same, which equal the oscillatory barotropic tide. The signals of the other frequencies are very weak.

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the steepness of the junction between the shelf and the slope. According to Baines [1973, equation (8.2)] r~ g^z ∂ui 1 QN 2 zh′ðxÞ þ f  ui þ rp þ ¼ sin wt^z
0
0 r r ∂t w h2

Figure 3. The amplitude of the kinetic energy field of the first harmonic wave.

amplitude fields and the density perturbation amplitude fields. Taking the horizontal velocity as an example, we defined the k-th harmonic wave as
 N 1 2 X 2p nk þ f huik ¼ uðx; z; nÞ cos N n¼0 N

ð2Þ

where N is the number of data points. The amplitude of the k-th harmonic 〈u〉k is the function of x and z. f is the starting phase related to the starting position of the model. In this experiment, we defined f = p/2. [12] We used the velocity fields and the density perturbation fields to get the kinetic energy fields of the harmonic waves, and found that the energy of the first harmonic wave was much stronger than that of the other harmonic waves, which coincided with the result of FFT. So we used the kinetic energy field of the first harmonic wave to describe our results (Figure 3). The equation of the kinetic energy is   1 E1 ¼ ðr0 þ rb Þ u21 þ v21 2

ð3Þ

where r0 represents the density field of the still water, and rb represents the density perturbation field acquired by the Synthetic Schlieren technique when the internal waves are generated, in g/mm3 , with u1 and v1 representing the amplitude of the horizontal and vertical velocity component of the first harmonic wave respectively, in mm/s. The amplitude of the kinetic energy is in g/(mm  s2). [13] In linearly stratified water, when the oscillatory barotropic tide flows over the continental margin, its energy is mostly stored in several narrow rays of the internal waves. From Figure 3, we can see that when the frequencies of the oscillatory barotropic tides are equal, the directions of the rays of the internal waves generated by the two models are equal and the measured angle between the rays and the horizon is almost equal to the theoretical angle of 28°. [14] By comparing the kinetic energy fields of the two models, we can see that 3 energy rays were generated on the continental margin with an abrupt shelf break, while on the continental margin with a gentle shelf break, only 2 energy rays were found. This result can also be proved by the density perturbation fields (see Figure 4). This confirms our opinion that the spatial structure of the internal waves is influenced by

ð4Þ

where the right-hand side represents the body force and h represents the water height from the surface to the bottom, which is a function of x only. We can see that when the junction is abrupt or nearly abrupt, h(x) is non-differentiable right at the point of the junction. That is to say, the body force at the junction, which is a small region, is large and decreases rapidly as it moves away from the junction. As a result, the generated internal waves are sensitive to the variations in stratification in this small region. When the oscillatory barotropic tide flows over this kind of junction, internal waves with three rays will be generated. This coincides with the results got by Baines [1982] and St. Laurent et al. [2003]. [15] However, when the junction is gentle enough, the body force decreases slowly, and the internal waves generated due to the resonant effect has only two rays. This has been verified by the results got by Gerkema et al. [2004] and Gostiaux and Dauxois [2007]. [16] In addition, from Figure 3, we can see that the third energy ray in the abrupt shelf break is the weakest among the three rays. It formed a special flowing pattern with the other two rays, but its velocity and density perturbation is lower than those of the other two. From Figure 3, we can also see that the rays tend to be the narrowest in the generating region and become wider as the internal waves propagate outwards. 3.2. The Flux, Width and Energy Flux of Each Ray [17] As illustrated in Figure 3, there are five lines, called a, b, c, d and e respectively, all of which are long enough, perpendicular to the direction of each corresponding ray, and of equal distance to their generating regions. With the raw data obtained by the PIV technique, we were able to get the time series of the velocity of each line. We then chose the effective points (v > 0.2  vmax) to calculate the flux of the rays. vmax refers to the maximum velocity amplitude in each line. We defined the velocity as positive when its horizontal component pointed to the right in Figure 3, otherwise, we defined it as negative. These approaches enabled us to get the chronological flux change of each ray over four

Figure 4. The amplitude of the density perturbation field of the first harmonic wave.

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equal to that of Ray b. But when the downward-propagating ray is near the slope, its energy decreases as a result of the mixing caused by the interaction between the ray and the slope, which explains why the kinetic energy flux of Ray d is smaller than that of Ray e.

4. Conclusions

Figure 5. (a–e) The flux of each ray. (f) The time series of the flux of the barotropic tide.

periods. Meanwhile, in order to make comparisons, we also got the time series of the flux from the velocity fields of the barotropic tide (Figure 5). [18] By comparing the three rays generated on the continental margin with an abrupt shelf break, we find that they are different in phase, with each of them lagging behind the phase of the oscillatory barotropic tide by p/2, p and 3p/2 respectively. That is to say, the phases of the internal tide are not synchronous with the oscillatory barotropic tide. Ray a, b and c are behind the barotropic tide by at least 1/4 of one period, 2/4 of one period, and 3/4 of one period respectively. Ray d and e also lag behind by at least 1/4 of one period and 2/4 of one period respectively. Therefore, in stably stratified seawater, when the oscillatory barotropic tide flows over continental margins and generates internal waves, the phases of the internal waves are not synchronous with the oscillatory barotropic tide and the phase differences vary among these rays. [19] In addition, the phase of Ray a equals that of Ray d, and the phase of Ray b is identical to that of Ray e. This means that the rays, when located in the same positions of the internal waves, are equal in the phases, whether they are generated in an abrupt or gentle shelf break. [20] In a way similar to the calculation of the flux, we got the width of the 5 rays at each time point over the 4 periods. We then averaged them to get the average widths for the 5 rays. The results show that the average widths of Ray a, b, c, d and e are 6.04 mm, 6.34 mm, 6.44 mm, 6.34 mm and 6.76 mm respectively, indicating they are almost equal in width. [21] In the same way, we calculated the average kinetic energy fluxes for all the rays. They are 9.3  103 g/(mm  s2), 9.8  103 g/(mm  s2), 5.3  103 g/(mm  s2), 1.15  102 g/ (mm  s2), and 1.57  102 g/(mm  s2) respectively. These results reveal that although the width of Ray c is close to those of the other rays, its kinetic energy is smaller, which indicates that its strength is not proportional to its width. The kinetic energy of Ray a is close to that of Ray b. It hints that when Ray a is far away from the slope, its energy is almost

[22] This experiment proves that the steepness of the shelf break has a strong influence on the spatial structure of the internal waves generated on the continental margin. This clearly explains the difference between Baines [1982] and Gostiaux and Dauxois [2007]. In linearly stratified seawater, when the oscillatory tide flows over the continental margin with a gentle shelf break, internal waves with two energy rays will be generated, while when the oscillatory tide flows over the continental margin with an abrupt shelf break and is not resonant with the slope, internal waves with three energy rays will be generated. Both the energy and the flux of the third ray are the weakest, but its width is close to that of the other two rays. Moreover, the phases of the internal waves lag behind the phase of the oscillatory barotropic tide, and the lagging phases for each ray are different, too. [23] Acknowledgments. This work is supported by the Natural Science Foundation of China (project 40906001 and J0730530). [24] The Editor would like to acknowledge two anonymous reviewers for their assistance in evaluating this manuscript.

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