A Comparative Study on the Modeling of Dynamics of ...

International Journal of Advances in Science and Technology, Vol. 5, No.6, 2012

A Comparative Study on the Modeling of Dynamics of the Jovian Atmosphere A. B. Bhattacharya1, S. Mondal2 and D. Halder 3 1

2

Department of Physics, University of Kalyani, Kalyani 741235, West Bengal, India [email protected]

Department of Physics, Darjeeling Govt. College, Darjeeling 734101, West Bengal, India [email protected] 3

Department of Physics, University of Kalyani, Kalyani 741235, West Bengal, India [email protected]

Abstract The meteorology of Jupiter plays a crucial role in its dynamic atmosphere. In this paper, we have examined Jupiter’s atmosphere with special emphasis on some distinct features like zones, belts, the Great Red Spot, the Little Red Spot, lightning etc. Not only their nature but their origin, transit and other remarkable changes have been considered including the lightning power distribution. Besides morphological observation, a comparative study has been made concerning the dynamic nature of its atmosphere. When examined critically, it is seen that none of them can reproduce exact nature of Jupiter’s atmosphere due to scarcity of known perturbation parameters. Those models when scrutinized in the light of some important mathematical theories it appears that Abelian-Higgs equation with proper boundary conditions has the potential to simulate for further studies. Keywords: Jovian atmosphere, Lightning, Modeling, Abelian-Higgs equation

1. Introduction Jupiter, being the largest planet (mass 1.8992 x 1027 kg and equatorial radius 71,398 km) of the solar system having average density 1.332 x 103 kg m-3, is mostly composed of few gaseous elements [1-5]. In this paper, we have studied the morphological structure of the vertically spanned atmosphere. Other than geomorphologic characteristics, topographically quite nature of Jupiter due to only 3o inclination to equator is cultivated here under the influence of several straight and narrow eastward jet streams in each hemisphere [6-9]. The nature of the Jovian storms and other discrete features can be predicted with the help of some mathematical theories about vortex and some basic plasma physics distribution laws [10, 11]. These models related to the dynamic nature of Jupiter’s atmosphere have been elaborately examined to provide additional input to the existing models and to find a new one.

2. Meteorology of Jupiter The average composition and vertical structure of the Jovian atmosphere is based partly on observation and partly on theory. Laser induced plasma spectroscopic discharge studies from the earth concluded that the Jupiter atmosphere is mostly molecular hydrogen (86%) and helium (10%), with lesser amounts of methane (CH4), ammonia (NH3) and a growing list of other gases. Detection and relative abundance of inert gases such as helium was determined from the relative strengths of the absorption spectra of hydrogen, methane and ammonia. At the very top there is an exosphere containing only H2 and He with a temperature of about 130 K [12-14]. Below that in the stratosphere, there are two layers of haze; the upper one is being responsible for methane bands observed in the spectra of this planet. In troposphere, under the topmost NH3 layer NH4 HS (NH3 and H2S) condensed into cloud particles. Below this layer

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International Journal of Advances in Science and Technology, Vol. 5, No.6, 2012 there is a layer of H2O cloud which covers water and liquid NH3 layer [15-17], that again covering up a solid surface of ice. Above all, there is the magnetosphere with a bow shock at about 3 x 106 km. On the top of that, heavy ions are formed by solar wind. Photographic observations of Jupiter by the Voyager 1, Voyager 2, Galileo and Cassini spacecrafts show the presence of extensive lightning activity [18-20]. The lightning strikes were concentrated in clusters, suggesting that several discrete storms produce multiple strikes during each of the exposures. Twenty-nine unique storms were documented in the two Galileo data sets [21] and for Cassini four lightning clusters were detected [22]. These storm clusters are associated with high levels of humidity [23] and deep level clouds [7]. The powers captured by Galileo and Cassini cameras were well distributed from 108 W to 109 W. Though they cannot be scaled in one single frame but comparative study of their power distribution has presented here in Figure 1 and Figure 2.

3

14

Number of flashes

Number of flashes

12 10 8 6 4 2 0

2

1

0 0.5

2.0 3.5 5.0 6.5 8.0 Lightning power in 108 W

1

2

3

4

5

Lightning power in 108 W

Figure 1. Histogram of lightning power observed by Galileo

Figure 2. Histogram of lightning power observed by Cassini for different storm clusters

When same polar-latitudinal regions of both hemispheres are taken into considerations using the data observed by New Horizons almost identical number of lighting were noted as evident from Figure 3. This proves that internal heat is the main driver of convection.

70° S-79° S

80° N-90° N

70° N-79° N 60° S-69° S

60° N-69° N

50° N-59° N Figure 3. A comparative display of lightning rate in polar region observed by New Horizons

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International Journal of Advances in Science and Technology, Vol. 5, No.6, 2012

Figure 4. Half Width Half Maxima of flashes in polar region showing that energy of lightning varies more as it appears towards polar region [24] According to observation, more and stronger flashes with higher energy were seen in the northern hemisphere than in the southern at non-polar latitudes as shown in Figure 4. This hemispherical difference may be due to slight differences in solar insolation or atmospheric stability [24].

3. Model Development In order to create model for wind structure of Jupiter atmosphere, three critical ingredients are: the correct geometry, turbulent convection and rapid rotation. The three-dimensional convection in Jupiter's deep atmosphere is a likely agent to drive the zonal flow. The bands of Jupiter represent a global system of powerful jets. Among them eastward equatorial jets are the main which are associated with other smaller-scale, higher-latitude jets flowing in alternating directions [6, 20]. Jupiter’s zonal flow depth is limited by increasing density and electrical conductivity in the molecular hydrogen–helium atmosphere towards the centre of the planet [25]. There are two types of planetary flow models which represent the Jovian jets: shallow-layer models reproduce multiple high-latitude jets, but not the equatorial flow system [26], whereas deep convection model only reproduces an eastward equatorial jet with two oppositely directed adjacent jets [27]. In 1979, Pioneer probes discovered that Jupiter radiate 1.67 times heat energy than that it received from the sun. It proves that Jupiter certainly has an internal core source of heat which is still settling down as 0.1 mm yr-1, through release of a large amount of gravitational potential energy [27]. This heat energy is being transported outwards by convection currents including flows in the metallic hydrogen, responsible for the origin of Jovian magnetic field (H = 4.3 G) making a 10o angle with its rotational axis. Spectroscopic observations reveal that under stratosphere the temperature decreases with altitude at the rate of about 2 K km-1. That rate is close to the adiabatic lapse rate, which draws an analogy to baroclinic model instead of barotropic model because probability of exchange of potential vorticity in horizontal plane becomes minimal due to vertical structure of thermal characteristics. So, geostrophic wind is not independent of height. The vorticity equation for an Euler fluid is expressed as, = −

.

−

−

−

.(

x

)

… (1)

where, is the absolute vorticity component and is density, is the pressure; u, v, and are the components of wind velocity, k is eddy diffusivity and h suffix denotes horizontal-component of that element. The terms on the RHS denote the positive or negative generation of absolute vorticity by divergence of wind, twisting of the axis of rotation, and baroclinity in pressure system. In the Red spots noted in Jupiter atmosphere, an adiabatic atmosphere parcel of gas moves vertically without exchanging heat or potential vorticity with neighbouring parallel parcels. This causes pressure change and as a result those parcels get cooler or warmer. Due to variable density i.e. ≠ 0, a source term appears in the vorticity equation whenever isopycnic surfaces and isobaric surfaces are not aligned or not in equilibrium [28]. The material derivative of the local vorticity is then given by

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+ ( ∙ )

=

= (

∙ ) −

( ∙ ) +

.(

x

)

… (2)

The last term in Eq. (2) contributes baroclinicity into atmosphere. So unless there is horizontal motion of winds across latitude, barotropic model cannot predict stability of zonal flows. Though the presence of the fast free surface gravity waves and cloud top elevation make an analogy to barotropic model but at deepest level it is baroclinic model, which carries greater similarities. Baroclinic model with weak shear can describe zonal jets at deepest levels. Baroclinic instability introduces a strong meridional variability in the velocity field. This instability acts as an energy source for the eddies which can produce substantial zonal flows that are stronger than the general eddy field flow at lower latitudes.

3.1. Charney-Hasegawa-Mima (CHM) Model In the absence of dissipation, the model of ion drift waves in a two-dimensional geometry transversal to a strong magnetic field reduces to a set of two differential equations for the stream function . From this, a further simplification of adiabatic density response leads to the Charney Hasegawa-Mima (CHM) equation and a nondissipative and purely fluid-dynamical model: ∆

where,

+ and

∆

−

+

∆

=0

… (3)

are constants depending upon length scale and fluid velocity respectively.

The CHM equation is accounted for the evolution in time of the deviation of the atmospheric depth from the mean. Here the motion of atmospheric elements is assumed to be confined in X-Y plane with a strong magnetic field in the Z-direction. Assuming uniform density and temperature, the unperturbed state is taken as a continuum of ions present in the atmosphere performing gyromotion. In this case, vorticity is connected to stream function by the Laplacian operator. ∆ +

ℎ (

ℎ − )=0

… (4)

where, p is a positive constant. The CHM equation preferably describes the effect of excited ion fluid motion over this back-ground. The alternative description of this system consists of a discrete set of point-like vortices interacting via the short range potential [29]. On the basis of the existence of a condensate of vorticity and the short range of the potential, a field theory model as a continuum version of the model of discrete vortices can be developed, which at self-duality reduces to a differential equation for the stream function responsible for storm origin [30]. When there are local perturbations which vanish at large distances, the vortex boundaries must smoothly match the constant vorticity of the asymptotic stationary states. Any storms must be seen as an excitation developed on this condensate of vorticity. Presence of the condensate of vorticity and the intrinsic finite length remove the space-scale invariance as well as induce a finite range of interaction between the elements present inside those storms by changing drift and translational velocities. Quasi-geostrophic (QG) and plasma drift turbulence within the CHM model was studied and confirmed that zonostrophy, which is an extra invariant in the CHM model, is well conserved and cascading the energy towards large zonal scales [29]. An important feature of QG method is inverse cascade, i.e. energy flows towards origin and it is constant over time. Other than energy, enstrophy are also exact invariants of the CHM model. Under the conditions of weak nonlinearity and random phases, zonostrophy is also conserved [28]. The energy cascade directed to the zonal scales, explain the formation of the zonal jets in Jovian atmosphere. The anisotropic inverse cascade mechanism is considered as most preferred zonal jets generation mechanism in Jupiter atmosphere [31]. According to this, energy moves from initial small-scale turbulence to the large-scale zonal flows in step-by-step transfer mechanism [30]. From different observations on the GRS and other strong winds, it was suggested that a second order extension to QG theory can be useful to explain Jupiter's atmosphere. Again strong height of non-linearity suggests that QG theory should include larger amplitudes or coriolis effect. From the spherical shallow water model [29], a potential vorticity equation can be derived, which can be reduced to QG and FP

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International Journal of Advances in Science and Technology, Vol. 5, No.6, 2012 equations. This potential vorticity equation is fixed to a parameter known as Rossby number which can be expressed as, … (5) = where, and are wind velocity and length scale and f is the coriolis parameter. Separating the atmospheric height into a static depth and surface perturbation with respect to small Rossby number leads to the quasi-geostrophic potential vorticity equation (QG-PVE) for which the surface perturbation is written as, ∑

( ∆

(−1)

−

) = 0

… (6)

where, i=0 stands for time part and 1, 2 for spatial part.

3.2. Hasegawa-Mima-Charney (HMC) Model Another popular model to describe vortex nature of Jupiter storms is Hasegawa-Mima-Charney model. The Hasegawa–Mima-Charney model for drift phenomena in a plasma fluid uses the same equations for the ion fluid, but there is an additional assumption that the isothermal electron fluid has enough time to maintain the Boltzmann density distribution. With constant background density the equation is, (∆ − ) +[ , ∆ − ] = 0

… (7)

This has exactly the same form as the quasi-geostrophic equation. The length scale is the plasma equivalent of the Rossby radius. Hasegawa–Mima equation is rather too simple for direct comparison with experiment [32]. Along field lines the Hasegawa–Mima equation is obtained for small scales and the stationary states exhibit dipolar structures of the order of the Larmor radius. On large scales the Korteweg-de Vries type scalar nonlinearity is prevailing and the only known structures are mono-polar. The mono-polar vortex has been considered in relation with the GRS.

3.3. Drift-Alfv´en Model Jupiter is considered basically a hot plasma bound planet for which the plasma pressure is small compared to the internal magnetic pressure. The compression of the fluid is small, so that compression perpendicular to the magnetic field is negligible. The zonal winds are considered as drift-Alfven´ waves and rather than magneto-acoustic waves and other compressional effects. The atmosphere fluid temperature is high, so collisional effects are negligible, assuming the electron temperature to be uniform throughout the atmospheric layer. The ions are so massive than the electron that they will be much slower than electrons, and their parallel motion as well as their thermal motion can be neglected. Then the thermal ion gyroradius is small compared to all other length scales in the system and we can put the ion temperature to zero. Charge separation effects are then negligible so that the plasma is quasi-neutral, i.e. the density of electrons equals the density of ions i.e. ne ≈ ni = n, where e and i subscripts are used for electron and ion respectively. The closed set of drift-Alfv´en fluid equations consists of the parallel electron momentum balance which can be written as, ( −

)+



[ ,

−

]=

−

ln

… (8)

Together with the electron continuity equation +



[ ,

]=



Ω

… (9)

and the ion equation − where,

Ω

∆↑

+



, ln −

= magnetic flux stream function,

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Ω

∆↑

=0

=electrostatic potential,

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… (10) = electron fluid temperature, J = parallel

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International Journal of Advances in Science and Technology, Vol. 5, No.6, 2012 =

current density,

, the electron inertial skin depth, e and me are the electron charge and mass respectively

and is the permeability of vacuum, B = magnetic field in Z direction, Ω is electron gyrofrequency,↑ denotes normal component. operates on any function f as, = + [ , ]. To simplify this set of equations to a two-dimensional model, one may assume the vortex centres with a constant upward velocity uz. i.e. = − , where = is inverse Alfv´en Mach number and vA = , the Alfv´en velocity. Those simplified equations can be cast in a Lagrangian form with three conserved vorticities by its own velocity field respectively. +[

,

]=0

α = +, −,0.

transported

… (11)

The relation between combined potentials and the generalized vorticities are given by, ∆( + λ ) = ∑

… (12a)

and )( + λ ) = − ∑

(∆ −

where,

=

… (12b)

In Eq. (12a) the potential ( +λ ) is the same as the equation for the stream function in the Euler equation. Eq. (12b) for the potential ( +λ ) is similar to the relation between vorticity and stream function of the Hasegawa–Mima equation and gives a short-range interaction in terms of a modified Bessel function. It is this combined nature of the relation between conserved fields and their streaming potentials which provides the current-vortex phenomena described by the drift-Alfv´en equations interpolating the dynamical behaviour [33]. In a two-dimensional model of the planetary atmosphere, the convective flow of vorticity represents a strong nonlinearity to drive the fluid toward a quasi-coherent vortical pattern. In the absence of dissipation the 2D models of the planetary atmosphere can be reduced to differential equations having the same structure as the Charney equation for the nonlinear Rossby waves, and the Hasegawa-Mima equation for drift wave turbulence. The problem of the atmosphere is, however, fundamentally different since owing to an intrinsic length, parameter, which can be seen from the fact that there is no space scale invariance, Rossby radius.

3.4. Flierl-Petviashvili Model The Flierl-Petviashvili equation can be explained in a simplified form as, ∆ = where,

=

−

… (13)

1−

… (14)

=

… (15)

( )

In Eq. (17), e and mi are charge and mass of ion, u is zonal velocity, is drift velocity, Larmor radius and L(t) the Lax operator. The Flierl-Petviashvili equation is extended to include topographic forcing. The initial value problem in which a monopole moves from a region of flat topography toward a region of variable topography and then back to flat topography is solved numerically. The numerical results show that topography can alter both the speed and the shape of the monopoles [34]. It is found that the Petviashvili monopole vortex is numerically stable when isolated in space. The solution is anticyclonic and axially symmetry and propagates slightly faster than the long wave Rossby wave phase speed. The Jovian banned structure can be achieved through imposing definite conditions on the FlierlPetviashvili equation to provide an isolated, finitely extended vortical solution, with smooth decay at boundaries as represented in Figure 5 [35].

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Figure 5. Periodic solution of the Flierl-Petviashvili equation [35]

4. Comparative Studies of Selected Models In this section we have made a comparison among some other important models developed mainly on the dynamics of Jovian atmosphere. The comparison is presented in Table 1. Table 1. A comparison among some selected models Models

Shallow-layer models [26]

Basic theory

Jets on Jupiter are driven A hypothetical Exchange of potential Moist buoyancy effects by small scale turbulent dynamical fluid column is vorticity in horizontal on Jupiter, with the (inverse cascade) which is joining the solid surface plane. atmosphere limited by in turn maintained by moist of Jupiter to the red cloud environ mental convection in stratosphere. at the top of the Jovian stratification and initial Here small vortices merge atmosphere. conditions yielded high to form larger ones. updraft velocities.

High-latitude jets, more likely strong retrograde (sub-rotating) jets, but fail to explain the prograde (super-rotating) equatorial jets. Importance When the largest vortices reach a certain size, the energy begins to flow into Rossby waves instead of larger vortices. Shallow models cannot explain atmospheric stability criteria. Applicable to

Taylor-Proudman Barotropic and Moist convection model column hypothesis [1] Baroclinic model [34] [36]

The Great Red Spot and equatorial jets.

Since Jupiter has no solid surface at any depth, the Taylor column hypothesis is not that much obliging. It assumed that the molecular hydrogen mantle is occupying only the outer 10% of Jupiter's radius. Most shallow layer models Galileo probe proved that Other information predict westward jets while the winds on Jupiter in reality they are eastward extend well below the and almost twice widen water clouds at 5–7 bars, than predicted values. which stands for deep circulation.

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Zonal jets.

Vertical temperature structure.

The actual lapse rate It produces layered is very much profiles in the vertical and less than the adiabatic a sub-saturated, stratified lapse rate but it is state but cannot explain not the case for detailed flow fields for Jupiter atmosphere charge separation and except the lightning [36]. stratosphere. Introduction of terms Lightning storms related to randomly and moist convection varying force and seem to occur in the dissipation of energy cyclonic belts due to attributable to the horizontal divergence on viscosity of the effect of eddy flux. atmosphere.

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International Journal of Advances in Science and Technology, Vol. 5, No.6, 2012 There are two other important models viz. Lewis' model and “one and half layer” model. The Lewis’ model basically states that for different layers of Jupiter atmosphere all the substances are at local chemical equilibrium. According to it, color of the cloud should be white but actually it is reddish. This model can be applicable to the vertical cloud structure. On the other hand, by the one and half layer model, effective height of cloud can be calculated by observing the variation of vorticity of the flow assuming potential vorticity as constant. In actual scenario, barotropic or baroclinic model cannot explain Jupiter atmosphere exclusively and that is why a combination of these two was developed in the one and half layer model. It basically represents the latitudinal variation in height up to observable levels [6].

5. Discussion Being a gaseous giant, Jupiter’s atmosphere cannot be totally described by these shallow water or quasi-geostrophic models as the observed winds violate the barotropic stability condition at some lower latitudes [1, 37]. This predicts that the modelled jets are weaker or wider than the actual ones. For a number of years the only theory that seemed capable of explaining the Great Red Spot was the Taylor-column hypothesis [1]. A Taylor column is the cylinder of stagnant fluid that was believed to join the solid object to the red cloud we see at the top of the Jovian atmosphere. Since Jupiter has no solid surface at any depth, the Taylor column hypothesis is not as compelling. The emergence and robustness of intense jets in Jovian atmospheres can be described by an approach of general statistical mechanics to potential vorticity patches. Potential vorticity mixes up and leads to the formation of a steady organized coarse-grained flow, corresponding to the statistical equilibrium state. Considering the quasi-geostrophic one and half layer model, the narrow jets can be obtained, which are scale of small Rossby radius of deformation. These jets can be either zonal, or closed into a ring bounding a vortex. Turbulent atmospheric layer into an ovalshaped vortex within a background shear as in the Great Red Spot and other storms can be explained as combined effect of planetary β effect due to coriolis force and a sub-layer deep shear flow of atmospheric fluid of Jupiter atmosphere. For planetary atmospheric vortices particularly for the GRS and the LRS of Jupiter, Korteweg-de Vries type nonlinearity scalar of lower differential degree plays an important role. The effect of the temperature gradient and the second order space derivatives of the density in the equation can be reduced at stable state to the Flierl-Petviashvili equation [38]. For temperature gradient, numerical stationary solution are possible but for density gradient, the existence of solitary vortices could not be proved numerically though the monopolar vortices with non-solitonic nature are possible when there is a higher degree density and temperature gradient. Assuming a drift wave ordering, the effective spatial scale changes can be expressed as,



=



1−

… (16)

are of same range. Since the drift velocity is From observation it appeared that zonal velocity u and drift velocity varying across the plasma section and is constant, so there must be a space varying scale of interaction in the pointlike vortex model [39]. Abelian-Higgs equation (AH), governing the vortices of the superconducting media of Higgs field ( ) is written as, ∆ ( )=

( )

−

… (17)

where, C is a constant. This equation has resembled to exact FP equation. By numerical simulation one can show that solution of Abelian-Higgs equation with a perturbation develops different vortices at different positions. This characteristic has resembles to exact nature of Jovian atmospheric wind structure. In Jupiter, we have seen few similarities among group of few storms [36, 40-42]. This type local common nature can be obtained from AH equation by introducing local parameter to the boundary condition.

6. Acknowledgments We are thankful to the Council of Scientific and Industrial Research for financial support in this work [Project No. 03(1153)/10/EMR-II, dated 26.04.2010]. S. Mondal is thankful to the Principal of Darjeeling Govt. College for constant encouragement and support. D. Halder is thankful to the CSIR for awarding him Net fellowship.

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International Journal of Advances in Science and Technology, Vol. 5, No.6, 2012

Authors Profile Prof. A. B Bhattacharya has published more than 200 research papers and 13 text books of high repute. He has conducted many research projects in the Department of Physics, Kalyani University. He is a reviewer of many scientific journals. His field of interest is Solar radio astronomy, radio wave propagation and atmospherics. He did his post doctoral work at the MIT, USA.

S. Mondal is now an Assistant Professor in Darjeeling Government College, West Bengal. He did his M.Sc. from Kanpur I.I.T. and has published a few papers in the field of Jovian Radio Signal.

D. Halder passed his M. Sc degree from University of Kalyani in 2010 and joined as a research fellow in July, 2010. He is now working as a Junior CSIR NET Fellow under the guidance of Prof. Bhattacharya in the Department of Physics, University of Kalyani.

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