Nonlinear Atmospheric Flow Patterns Confined to Zonal Cloud Bands

A. Constantin Faculty of Mathematics, University of Vienna, Vienna, Austria

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R. S. Johnson School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, United Kingdom

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Abstract

We derive, at leading order in the thin-shell parameter, a consistent set of nonlinear governing equations for the dynamics of flows confined to a zonal cloud band such as those on Jupiter, in a thin layer near the top of the planetary troposphere. Some exact solutions are provided in the material (Lagrangian) framework. The explicit specification of the individual particle paths enables a detailed study of these flows that model oscillations superimposed on a mean current. This approach is applied to Jupiter’s Great Red Spot and to the filamentary zonal flow at its southern boundary.

Significance Statement

We propose a new approach to the study of some flow patterns visible in zonal cloud bands on Jupiter. Motivated by observations showing that the dominant motions in the cloud bands on Jupiter are zonal and rotational, we provide some exact solutions to the governing equations for the leading-order dynamics. These solutions model rotating particle paths interacting with a straight-line flow. The approach offers detailed insight into basic features of the flow, highlighting the interplay between density variations and wind forcing. The exact solutions presented here are a useful starting point for a perturbation analysis. Current advances in computing methods enhance the feasibility of numerical simulations of perturbed flows that are designed to capture a wider range of effects, whose relevance can be ascertained by comparison with the exact solutions presented in this paper.

© 2024 American Meteorological Society. This published article is licensed under the terms of a Creative Commons Attribution 4.0 International (CC BY 4.0) License .

Corresponding author: Adrian Constantin, adrian.constantin@univie.ac.at

Abstract

We derive, at leading order in the thin-shell parameter, a consistent set of nonlinear governing equations for the dynamics of flows confined to a zonal cloud band such as those on Jupiter, in a thin layer near the top of the planetary troposphere. Some exact solutions are provided in the material (Lagrangian) framework. The explicit specification of the individual particle paths enables a detailed study of these flows that model oscillations superimposed on a mean current. This approach is applied to Jupiter’s Great Red Spot and to the filamentary zonal flow at its southern boundary.

Significance Statement

We propose a new approach to the study of some flow patterns visible in zonal cloud bands on Jupiter. Motivated by observations showing that the dominant motions in the cloud bands on Jupiter are zonal and rotational, we provide some exact solutions to the governing equations for the leading-order dynamics. These solutions model rotating particle paths interacting with a straight-line flow. The approach offers detailed insight into basic features of the flow, highlighting the interplay between density variations and wind forcing. The exact solutions presented here are a useful starting point for a perturbation analysis. Current advances in computing methods enhance the feasibility of numerical simulations of perturbed flows that are designed to capture a wider range of effects, whose relevance can be ascertained by comparison with the exact solutions presented in this paper.

© 2024 American Meteorological Society. This published article is licensed under the terms of a Creative Commons Attribution 4.0 International (CC BY 4.0) License .

Corresponding author: Adrian Constantin, adrian.constantin@univie.ac.at

1. Introduction

The gas giants of our planetary system—Jupiter and Saturn—are composed, in their outer layers, of bands of clouds of mainly hydrogen and helium. These cloud bands each move at slightly different speeds and, typically, contain waves and vortex structures. Our focus here is on the description of some of the flow patterns that are confined to bands in the surface clouds of the gas giants. The overall aim is to develop a systematic mathematical approach which can provide a theoretical basis for this type of phenomenon. The wave patterns and flow structures of interest here are those that are clearly visible, observed through a telescope or in a fly-by, sitting in the outermost layers of Jupiter (and Saturn). The most obvious of these is Jupiter’s Great Red Spot (GRS), but many other, somewhat smaller, filamentary wave structures are also visible on Jupiter (see Fig. 1).

Fig. 1.
Fig. 1.

Color map of Jupiter in cylindrical coordinates, constructed from images taken by the camera onboard NASA’s Cassini spacecraft on 11–12.XII.2000 (credit: NASA). The map captures in some detail the GRS and the filamentary zonal flow at its southern boundary.

Citation: Journal of the Atmospheric Sciences 82, 1; 10.1175/JAS-D-24-0038.1

In the case of the GRS, whose existence has been known since about 1665, we have a flow pattern that moves around the planet at a speed little different from the speed of the surrounding band of clouds in which it is embedded (designated as the system II coordinates). Thus, we may reasonably regard it as stationary in this rotating frame. It plainly has a strong vortex-like structure (which rotates counterclockwise) and is of immense size (although shrinking appreciably over the last century, changing its shape from an oval to nearly circular) but is thought to extend to a depth of no more than 300–500 km below the upper surface of the clouds. Its horizontal dimensions (currently) are about 16 500 km across, so it may be treated as a reasonably sized, but shallow, structure within Jupiter’s atmosphere. Furthermore, the GRS arises from the unique conditions that exist on Jupiter: a large Coriolis force generated by the planet’s rapid rotation; a thin, inviscid layer in the atmosphere where the weather predominantly resides; and wide, azimuthal, axisymmetric regions of cyclonic and anticyclonic shear on the boundaries of the strip containing the GRS. The modeling of the GRS as a coherent structure, with closed particle paths, and no driving mechanism other than that provided by its dynamical setting, was pioneered a half-century ago (see the discussion in Ingersoll 1973; Sagan 1971). Since then, several shallow-water models have been put forward, aiming to capture the main features of the GRS (see, e.g., the developments presented in Cho et al. 2001; Dowling and Ingersoll 1989; Marcus 1988; Suetin et al. 2023; Williams 1985). Data obtained from measurements of the outermost reaches of the Jovian atmosphere show that the cloud tops act like a frictionless, layered fluid linearly superposed on a fluid flow described by a time-independent streamfunction (see Sánchez-Lavega 2011). Thus, density stratification and steady flow are fundamental ingredients of the fluid motion in and around the GRS; these have been used to guide the modeling of the GRS. But we assert that, as compared with other available models, our systematic approach is better suited to capture the intrinsic and important nonlinear effects. So, for example, the shallow-water potential vorticity equation becomes easier to solve numerically when made advectively linear by invoking the quasigeostrophic approximation, but this procedure necessarily restricts nonlinear interactions. With all this in mind, we consider a model which comprises a flow of fairly limited meridional extent, sitting in a thin layer near the top of the visible atmosphere (which we might interpret as the corresponding upper troposphere). The aim here is to develop a set of suitable governing equations which can accommodate and describe such flows.

The analysis that we present uses the Lagrangian framework to provide a description of flows that do not extend far in the meridional direction and otherwise are restricted to a thin layer near the top of the planetary troposphere; the flows may extend to large distances in the azimuthal direction. This approach is based on the f-plane approximation and the thin-shell approximation and the construction of exact solutions of the resulting system of nonlinear equations. A careful development, which shows how the asymptotic method unfolds, enables higher-order approximations to be accessed, if required. The resulting, dominant equations are the familiar Euler equation, written in a rotating frame, in the f-plane and local Cartesian tangent-plane version, within the thin-shell approximation, and an equation of mass conservation; the vertical motion, in this asymptotic system, is absent at leading order. It is clear that the confinement of the GRS, and its associated flow structures, to a relatively thin zonal band justifies the use of the f-plane approximation as the basis for modeling (see the discussion in Suetin et al. 2023). While a β-plane approximation would clearly reveal more details, the discussion in section 5a shows that all the essential features of the GRS flow are captured within the f-plane framework. Our philosophy (Occam’s razor) is that simplicity ought to be the one key criterion for evaluating a model, providing that it is self-consistent and accords reasonably well with the available data. In light of this, the important observation that we exploit is that we can permit any (reasonable) density variation through the depth of the atmosphere and, coupled with the requirement to satisfy a consistency condition, we are able to find exact solutions, expressed in parametric form. But this raises another issue.

It might be argued that the Boussinesq approximation should play a role here. This approximation retains density variations only in the buoyancy term in the vertical momentum equation, replacing elsewhere the density by a constant mean value (see section 5.1.1 in Holton 2004); this leads to an incompressible framework that makes mathematical headway possible (see Sánchez-Lavega 2011). But the flow in and around the GRS occurs predominantly in layers, with the top of Jupiter’s troposphere being stably stratified and rapidly rotating; cf. Cho et al. (2001). In addition, the density variation in Jupiter’s upper atmosphere is significant (see Kaspi et al. 2009); thus, we conclude that the Boussinesq approximation is not an adequate model, nor relevant, for the description of these phenomena.

The upshot of all the above is that our model, and technical development, produces solutions that are readily constructed and interpreted, being relevant to the types of flows and wave patterns that are observed. Furthermore, since we invoke the ideal gas equation and an appropriate first law of thermodynamics, we are able to examine how the temperature gradients control the flows. In addition, with the velocity and temperature fields determined, we can use the first law to identify the heat source required to maintain the motion. With all this in mind, the plan for the paper is as follows.

In section 2, starting from the Navier–Stokes and mass conservation equations written in a rotating, Cartesian frame, we introduce a suitable nondimensionalization. We then derive, at leading order in the thin-shell parameter, a consistent set of equations. In particular, the inviscid setting being adequate, this generates an Euler equation, where the vertical component is the hydrostatic pressure law. As we show in section 3, the other equations admit an arbitrary vertical density profile, with the hydrostatic equation providing a consistency condition. This procedure accommodates equatorial traveling waves but requires that midlatitude flows are stationary—time independent—in the chosen rotating frame. This restriction nevertheless accords closely with some observed flows in the cloud bands on Jupiter, for which any motion relative to the rotating frame that contains the cloud formation is very small. In section 4, we derive the relevant version of this system of equations, now expressed in terms of a parameter (which is time, but for steady motion), and then, in section 5, we present two examples in detail. Section 6 is devoted to a brief summary of the results and a discussion, covering the relevance of the analysis and possible future work.

2. Governing equations

We introduce a set of (right-handed) Cartesian coordinates (x′, y′, z′), with associated unit vectors (e1, e2, e3) fixed in a suitable frame linked to a particular cloud band, where the tangent vector e1 points from west to east, the tangent vector e2 points from south to north, and e3 points in the direction of the outward normal (see Fig. 2); we avoid the poles since e1 and e2 are not well defined there. The corresponding velocity components in the fluid are (u′, υ′, w′) (≡u′). (We use primes, throughout the formulation of the problem, to denote physical/dimensional variables; these will be removed when we nondimensionalize.) We invoke the f-plane approximation, so that we are considering flows that are restricted to a relatively narrow zonal band, treating the Coriolis parameters f′ = 2Ω′ sinθ and f^=2Ωcosθ as constants, with θ denoting the fixed angle of latitude and Ω′ denoting the constant rate of rotation of the planet (or of a gaseous band). The flows that we describe here are those that exist in a thin layer near the top of the planet’s troposphere, visible to the external observer. The local tangential coordinates are then used to describe the position of points within the strip, with x′ measuring the distance in the azimuthal direction around the planet and y′ being the measure from the centerline of the strip.

Fig. 2.
Fig. 2.

The Cartesian coordinate system on the spherical planet, fixed in a suitable rotating frame, where r is the position vector of a point of latitude θ and longitude φ, relative to the center.

Citation: Journal of the Atmospheric Sciences 82, 1; 10.1175/JAS-D-24-0038.1

The Navier–Stokes equation, in this rotating frame, is written as
DuDt+(f^wfυ,fu,f^u)=1ρP+(0,0,g)+νh22u+νυuz2,
where νh and νυ are the (constant) horizontal and vertical kinematic eddy viscosities, respectively; ρ′ is the density and P′ is the pressure; and g′ is the constant acceleration of gravity throughout the depth of the strip. We have used the familiar shorthand notation for the material derivative:
DDtt+ux+υy+wz,
where t′ is the time variable, with
(x,y,z),222x2+2y2.
The equation of mass conservation is
DρDt+ρ(ux+υy+wz)=0.
In addition, we require an equation of state for the gas, which we treat as ideal, so
P=ρRT,
where T′ is the temperature and ℜ′ is the appropriate gas constant, together with the first law of thermodynamics:
cpDTDt+κ2T1ρDPDt=Q,
where cp is the specific heat, κ/cp is the thermal diffusivity, and Q′ is the heat source. The set (1)(4) constitutes the governing equations that we shall use to describe the various flows of interest, within the f-plane and thin-shell approximations. To this we add the vorticity, which will prove useful when we interpret the solutions that we obtain; this is
γ=u=(wyυz,uzwx,υxuy).
The next stage in the construction of the relevant system of equations is to nondimensionalize, using suitable scales. A fundamental length scale would be the average radius of the planet R′ which is the basis for measuring the length of the strip in the azimuthal direction. This same scale can be used as the scale length in the meridional direction, even though we expect the strip to be quite narrow relative to its length but not so narrow that this requires a separate parameter. The introduction of an additional parameter to describe the ratio of width to length of the strip overcomplicates the development. Indeed, the accepted asymptotic procedure is to use parameters and associated limiting processes, only if this is necessary to make analytical headway. Since the shape of the GRS is close to circular in the horizontal plane, with a maximum dimension of about 16 500 km across, a natural choice is to use the length scale L′ = 104 km in both azimuthal and meridional directions to investigate it. Its vertical dimension is very much less, the GRS being primarily a tropospheric storm that extends into the lower stratosphere, with an overall height of about 300–500 km. Thus, we write
(x,y)=L(x,y)and|r|=R+Hz,
so that
z=Hz,
leading to the introduction of the thin-shell parameter
ε=HR ;
here, H′ ≈ 100 km is the average height of the troposphere. As we will show, the asymptotic structure is driven by the thin-shell approximation and so we may, independently, choose a suitable model for the horizontal-flow configuration. Because the GRS lies in a narrow zonal band (about 6° of latitude wide), the f-plane approximation is a reasonable first choice (see section 2.3.1 in Vallis 2017). We opt for this setting here so that we are able to present the results in a particularly transparent form; we hope, in a future study, to remove this restriction. We also need a typical speed U′ of motions in the azimuthal direction, and, in keeping with the argument used for the scale length in the meridional direction, we use this same speed scale for speeds in this direction, so we have
(u,υ)=U(u,υ).
Since the speeds in the vertical direction are very much less, we introduce W′ = δU′ and write
w=Ww=δUw,
where δ is another small parameter (to be chosen later). We may now use L′ and U′ to define a suitable time scale for propagation in the azimuthal direction; we set
t=LUt.
The inclusion of an average density ρ¯ leads to the further nondimensionalization:
ρ=ρ¯ρandP=ρ¯U2P.
We now introduce the parameters
g=gHU2,ω=2ΩLU,f=ωsinθ,Reh=WHνh,Reυ=WHνυ,
the latter two because the natural choice for the Reynolds numbers (Reh, Reυ) is based on the scales associated with the vertical motion, which then leads to a consistent asymptotic structure. In Table 1, we present some relevant data for the two gas giants (Saturn is included for comparison), together with estimates for the parameters that we have introduced.
Table 1.

Field data for the upper troposphere of Jupiter and Saturn (Catling 2015; Ingersoll et al. 2004; Ingersoll 2020). Since the center of Jupiter’s Red Spot is located at 22°S, by (6), the corresponding value of the Coriolis parameter is f = −8.7.

Table 1.
With the normalization
T=U2RT
of the temperature, (1)(4) become (with the Navier–Stokes equation written in component form)
DuDt+f(δwcotθυ)=1ρPx+εδReh(2ux2+2uy2)+δ/εReυ2uz2,
DυDt+fu=1ρPy+εδReh(2υx2+2υy2)+δ/εReυ2υz2,
DwDtfδucotθ=1εδ(1ρPz+g)+εδReh(2wx2+2wy2)+δ/εReυ2wz2,
DρDt+ρ(ux+υy+δεwz)=0,
P=ρT,
cpDTDt+κ2T1ρDPDt=Q,
where
DDtt+ux+υy+δεwz,22z2+ε2(2x2+2y2),cp=cpR,κ=κRRUH2,Q=QRU3.
The system (7)(12) is to be interpreted under the limiting process ε → 0 and δ → 0, while keeping all the other parameters fixed. However, this produces a meaningful and well-defined result (as an asymptotic procedure) only if we specify how δ/ε behaves, and this is possible because the choice of δ can be used to characterize the type of flow being considered. In this case, since the motion in the vertical direction is very weak, we set δ = εk with k > 1 and we can readily construct a complete asymptotic solution in ε since the viscous contribution is unimportant in the upper troposphere. Extracting the dominant terms from (7)(10), the leading-order dynamics is governed by
ut+uux+υuyfυ=1ρPx,
υt+uυx+υυy+fu=1ρPy,
0=1ρPzg,
ρt+uρx+υρy+ρ(ux+υy)=0,
P=ρT,
cp(Tt+uTx+υTy)+κ2Tz21ρ(Pt+uPx+υPy)=Q.
If ρ = ρ(z), then (16) becomes
ux+υy=0.
Here we assume that the density ρ(z) is given, describing the upper troposphere of the planet, and then, (13), (14), and (19) involve the z variable only via a parametric dependence. If we can find the horizontal velocity field—the vertical velocity component being negligible in this asymptotic solution—and the associated pressure distribution [which, for consistency, must satisfy (15)], then (17) provides the temperature distribution. All this information can be used in (18) to give the heat source Q required to drive and maintain the motion. This interpretation of the system of equations has been used in a number of other atmospheric studies (Constantin 2023; Constantin and Germain 2022; Constantin and Johnson 2021, 2022; Henry 2024).
An important property of the system (13)(16) is its nonlinearity which, with the pressure gradients and, in the horizontal directions, contributions from the Coriolis forces and advection terms, together governs the leading-order dynamics of the flow. So, rather than simply expressing a geostrophic balance, this system arises in the asymptotic regime that corresponds to shallow-water models generated from these governing equations. The considerations in section 5a show that, without further approximations, this system admits an exact solution that captures all the salient features of the GRS. We conclude this section by presenting another view of the asymptotic procedure used to derive the system (13)(16) from the governing (1) and (2) in the f plane. For simplicity, we start from the outset with the inviscid setting (νh0 and νυ0). Introducing
(x,y)=L(x,y),z=Hz,(u,υ)=U(u,υ),w=δUw,g=gHU2,t=LUt,ρ=ρ¯ρ,f=2ΩLUsinθ,P=ρ¯U2P,
we obtain the nondimensional form of the inviscid counterpart of system (1) and (2):
ut+uux+υuy+δεwuz+δfwcotθfυ=1ρPx,υt+uυx+υυy+δεwυz+fu=1ρPy,δε{wt+uwx+υwy+δεwwz}+δfucotθ=1ρPzg,ρt+uρx+υρy+δεwρz+ρ{ux+υy+δεwz}=0,
where ε=H/L is the thin-shell parameter and δ = O(ε2). Given that f ≈ −8.7, we obtain the system (13)(16) in the limit ε → 0. In this formulation, δ = δ(ε) because the vertical movement is directly linked to the thin-shell approximation, and then the choice of this relation describes the class of problem under discussion (see Johnson 2005). However, f is independent of ε and so may take any fixed value as ε → 0. The parameter ε, indicative of a flow residing in a thin layer, is the sole basis for the construction of an asymptotic solution: The behavior in ε controls the structure of the solution. Indeed, we see that, to leading order, we retain all the expected terms, as well as the intrinsic nonlinearity of the flow: on the basis of this asymptotic formulation, we are not at liberty to ignore any of these terms.

3. Reformulation

To construct suitable exact solutions of (13)(18), we first recast them in a more tractable form by introducing a coordinate frame, moving in the azimuthal direction relative to the chosen rotating frame, by writing the equations in terms of ξ = xct, y, and z, where c is a constant. Thus, we obtain
(uc)uξ+υuyfυ=1ρPξ,
(uc)υξ+υυy+fu=1ρPy,
0=1ρPzg,
uξ+υy=0,
P=ρT,
cp[(uc)Tξ+υTy]+κ2Tz21ρ[(uc)Pξ+υPy]=Q.
We now invoke the scale-separation property underlying the derivation of the system (20)(23), seeking solutions of the form
[u(ξ,y,z),υ(ξ,y,z)]=(c,0)+ρm(z)U[(X,Y),V(X,Y)],(X,Y)=ρn(z)(ξ,y),
with m and n constants to be chosen. Since
ξρnX,yρnY,
zz+nρn1dρdz(ξX+yY)z+nρdρdz(XX+YY),
(20)(23) are transformed into
UUX+VUY=ρ12mPX+fρnmVUVX+VVY=ρ12mPYfρnmUcfρn2m,Pz+nρdρdz(XPX+YPY)=gρ,UX+VY=0.
To proceed, we write
P(ξ,y,z)=gz0zρ(s)dscfρ1nY+ραP(X,Y)+P0,
where α (to be chosen) and P0 are constants, and we have fixed the bottom of the layer at z = z0. Thus, we obtain
UUX+VUY=ρα12mPX+fρnmV,
UVX+VVY=ρα12mPYfρnmU,
[αP+n(XPX+YPY)]ρα1(1n)cfYρn=0,
UX+VY=0.
These equations, to be consistent, must not depend on z; so, from (30) and (31), we see that
m=n,α=12n,
while (32) shows that we must have either
n{0,1}
or
cf=0,
with (36) accommodating only equatorial waves (for f = 0) or, at midlatitudes, stationary flows in the rotating frame (with c = 0). We note that the choices n = 1 and cf = 0 produce the same reduced equation from (32); we will comment on this later.
Equation (33) shows that we may introduce a streamfunction Ψ(X, Y) with
ΨY=U,ΨX=V,
so that the system (30)(32) simplifies to
UUX+VUY=X(P+fΨ),
UVX+VVY=Y(P+fΨ),
with either
(12n)P+n(XPX+YPY)=0forn0
or
P=(1n)cfY.
At this stage, it is helpful to make two important observations. First, we write (40) in polar coordinates by setting
R=X2+Y2,ν={arccos(XR)ifY0,arccos(XR)ifY<0,for(X,Y)(0,0),
which gives
(12n)P+nRPR=0;
the general solution is P=0 for n = 0 or
P(X,Y)=P0(ν)R2(1/n)ifn0,
for some arbitrary function P0. On the other hand, the compatibility condition generated by the elimination of P+fΨ from (38) and (39) produces
ΨYX(2ΨX2+2ΨY2)ΨXY(2ΨX2+2ΨY2)=0.
Thus, throughout the regions where there are no stagnation points (Ψ/X,Ψ/Y)(0,0), the rank theorem (see Dieudonné 1969) yields
2ΨX2+2ΨY2=h(Ψ),
for some function h. Writing H(Ψ) for the primitive of h, we may, using (45), write (38) and (39) as
12[(ΨX)2+(ΨY)2]+H(Ψ)+P(X,Y)+fΨ=constant,
a relation analogous to Bernoulli’s hydrodynamical law (see Constantin 2011). With P given by (44), a solution to (46)—a stationary Hamilton–Jacobi equation (see Crandall and Lions 1986; Ishii 1986)—yields a solution to the system (20)(23) that describes the leading-order dynamics of flows in a zonal band. Since particle paths and streamlines are identical for steady flows, it is convenient to seek solutions to (46) within the Lagrangian framework. Note that the Lagrangian approach proved to be very useful in locating the edge of the terrestrial polar vortex (Serra et al. 2017).

4. The Lagrangian framework

Writing the particle path as [X(t), Y(t)], then Ψ is constant along the corresponding streamline and (46) simplifies to
12({U[X(t),Y(t)]}2+{V[X(t),Y(t)]}2)+P[X(t),Y(t)]=B0(Ψ).
We have derived (46) and (47) under the assumption that there are no stagnation points in the flow. We now show that the Lagrangian approach yields the validity of (47) even if there are stagnation points present. Along the particle path [X(t), Y(t)], we have
U(t)=dXdt,V(t)=dYdt,
and
ddtdXdtX+dYdtY.
Thus, (38) and (39) become, respectively,
dUdt=(P+fΨ)X,
dVdt=(P+fΨ)Y,
and then, we form
UdUdt+VdVdt=[U(P+fΨ)X+V(P+fΨ)Y]=[dXdt(P+fΨ)X+dYdt(P+fΨ)Y]=d(P+fΨ)dt,
so (47) follows since Ψ[X(t), Y(t)] = constant, a development that does not exclude the possibility that [U(t), V(t)] = (0, 0) at some time t.
An important reformulation, which enables us to make significant headway when seeking solutions, is to observe, from (49) and (50), that we have
PX=dUdtfΨX=dUdt+fV=d2Xdt2+fdYdt,
PY=dVdtfΨY=dVdtfU=d2Ydt2fdXdt,
and so (40) can be written as
2(12n)B0+(4n1)[(dXdt)2+(dYdt)2]nd2dt2(X2+Y2)+2nf(XdYdtYdXdt)=0,
an equation that must be satisfied by all the solutions that we develop. While (53) follows from (46), the equations are not equivalent since the particle-specific constant B0 = B0(Ψ) in (47) has to satisfy the constraint
B0(Ψ)+H(Ψ)+fΨ=constant.

5. Two exact solutions

The previous considerations have laid the foundations for the construction of some specific exact solutions that are relevant and useful in the description of phenomena observed in planetary atmospheres. The underlying system of nonlinear equations, (33) and (38)(40), based on the f-plane and thin-shell approximations (and with viscosity playing no role), has exact solutions obtained by making choices based on the trigonometric functions. The special solutions that we describe enable us to provide a firm foundation for the study of Jupiter’s Red Spot and for the filamentary zonal flow at its southern boundary. Then, knowing [X(t), Y(t)] and [U(t), V(t)], we may find the pressure distribution and hence, from (17), the temperature distribution. Indeed, we may investigate the behavior of the temperature and so find the conditions—which, in the planetary context, we regard as given—that may control the properties (or even the existence) of the wave motion. Knowing all this, we may then use the first law of thermodynamics, (18), to identify the heat sources required to maintain the motion.

a. Example 1: Jupiter’s Great Red Spot

The GRS is the most conspicuous feature of Jupiter’s visible cloud surface (see Fig. 1) and the largest storm in our solar system (Dowling and Ingersoll 1989; Ingersoll 2013; Wong et al. 2021). Given that the spot is nearly circular and that no systematic variation in the angular rotation velocity is evident within the accuracy of the available observations, it is natural to seek circular particle trajectories. Since relation (36) holds with c = 0 and f ≈ −8.7, we set
X(t)=Asin(μt)+Bcos(μt),Y(t)=Csin(μt)+Dcos(μt),
with μ = μ(Ψ), A = A(Ψ), B = B(Ψ), C = C(Ψ), and D = D(Ψ). Given that in a steady flow the particle paths coincide with the streamlines, Ψ[X(t), Y(t)] = constant, we obtain
U(t)=dXdt=μ[Acos(μt)Bsin(μt)],V(t)=dYdt=μ[Ccos(μt)Dsin(μt)].
Equation (53) becomes
2(12n)B0+μ2(4n1)(A2+C2)+2μ2n(B2+D2A2C2)+2μnf(BCAD)μ2(B2+D2A2C2)sin2(μt)+μ2(AB+CD)sin(2μt)=0,
which is satisfied for all t if and only if
B2+D2A2C2=0,
AB+CD=0,
2(12n)B0+μ2(4n1)(A2+C2)+2μ2n(B2+D2A2C2)+2μnf(BCAD)=0.
Equation (58) merely determines B0, given n, A, B, C, D, and μ. In terms of complex variables, (56) and (57) express the fact that the complex numbers ζ = A + iC and Υ = D + iB have the same modulus and their product is a real number, so that the complex conjugate of ζ must equal ±Υ: We have
A=±D,C=B.
The particle paths are therefore
X(t)=Asin(μt)+Bcos(μt),Y(t)=[Bsin(μt)Acos(μt)],
with μ dependent on the initial-data labels (A, B), and the velocity components are given by
U(t)=μ[Acos(μt)Bsin(μt)],V(t)=μ[Bcos(μt)+Asin(μt)],
so that we have closed circular paths with
[X(t)]2+[Y(t)]2=1μ2{[U(t)]2+[V(t)]2}=A2+B2.
The solution described by (60) and (61) is recovered from
U=dXdt=±μY,V=dYdt=μX.
Taking into account (37), from (63) we get
YΨX+XΨY=0,
which, using polar coordinates, shows that the streamfunction depends solely on the distance to the origin, that is, Ψ = Φ(R) with R=X2+Y2, for some function Φ. Now, (63) and (37) yield μR=Φ(R), so that
μ=Φ(A2+B2)A2+B2,
due to (62). Since the entire GRS storm system rotates counterclockwise with nondimensional period of about 5.832, corresponding by means of t=(L/U)t to about 4.5 Earth days (see Wong et al. 2021), and its outer rim R = R0 corresponds to R0 ≈ 0.825, using the lower sign in (60), we get
μ(R0)2π5.8321413,
an approximation that we use hereafter, for simplicity. From (53), (62), and (63), we obtain
2(12n)B0+μ(A2+B2)[(4n1)μ+2nf]=0.
We see now that n=1/2 is impossible since it would force μ = f ≈ 8.7, in contradiction to (64). Proceeding with n1/2, (65) yields
B0=μ2(2n1)[μ(4n1)+2nf](A2+B2).
From (47) and (62), we get
P[X(t),Y(t)]=nμ(μ+f)2n1(A2+B2)withn12.
Due to (62), the compatibility with (44) provides the dependence of μ on R:
nμ(μ+f)2n1=R(1/n),
for some constant ℵ. To ensure the limit μ(R) → 0 for R → 0, we require n < 0 [which confirms the interpretation c = 0 in (32)]. Now, (68) yields
μ(R)=f2±f24+(2n1)nR(1/n),
and (64) selects (with f < 0) the single root
μ(R)=f2f24+(2n1)nR(1/n),
with
=14(14+13f)169(2n1)R01/n<0,
so that
μ(R)=f2f24+14(14+13f)169(RR0)(1/n).
Since 14 + 13f < 0 (because f ≈ −8.7, see Table 1) and n < 0, we see that μ′(R) > 0 for R ∈ (0, R0), with μ(0) = 0 and μ(R0)=14/13. This is of relevance since from (63) we compute the vertical component of the vorticity,
VXUY=[2μ(R)+Rμ(R)],0<R<R0,
and, following the right-hand rule, the lower sign is appropriate for a counterclockwise rotation.
The relations (26), (29), (34), (67), and (68) yield the pressure
P(x,y,z)=gz0zρ(s)ds+14n(14+13f)169(2n1)×(x2+y2)1(1/2n)R01/n+P0,
showing that we have used a baroclinic model for the fluid (the surfaces of constant pressure and surfaces of constant density being transversal) and (24) gives the temperature
T(x,y,z)=1ρ(z){gz0zρ(s)ds+14n(14+13f)169(2n1)×(x2+y2)1(1/2n)R01/n+P0},
with vertical temperature gradient
Tz=g+1ρ2dρdz{gz0zρ(s)ds14n(14+13f)169(2n1)×(x2+y2)1(1/2n)R01/nP0}<g+1ρ2dρdz[gz0zρ(s)dsP0].
The nondimensional parameter P0 > 0 in (70) corresponds to the pressure at the base z = z0 of Jupiter’s Red Spot, which is about 0.7 bar (see Ingersoll et al. 2004) giving a value P0 ≈ 10−3. Since a realistic density distribution for Jupiter’s troposphere is ρ(z)=eb(zz0) where b ∈ (2, 5) is a constant (see in Catling 2015), given the sizes of the various parameters, relation (72) guarantees a decrease of the temperature with height throughout Jupiter’s Red Spot, while n gives us the appropriate power law for the behavior of the temperature across the GRS. Further, the above solution also captures the following observed features of Jupiter’s Red Spot (see Ingersoll et al. 2004):
  • the relations (60) and (64) ensure a nondimensional period of about 5.832 for the counterclockwise rotation of the vortex, corresponding to about 4.5 Earth days;

  • relation (62) shows that the central part of the vortex is quiescent and there is no eyewall structure as typically associated with terrestrial hurricanes, while the winds around the outer parts of the GRS blow at about 133 m s−1, corresponding to the nondimensional speed R0μ(R0);

  • relation (70) ensures that the vortex has a high pressure center, since at every fixed height z, the pressure P(x, y, z) is a decreasing function of (x2, y2);

  • relation (71) shows that the vortex has a warm core (within an otherwise cold storm system that averages −160°C);

  • relations (63) and (71) ensure
    uPx+υPy=uTx+υTy=0,
    so that by (25), and using the realistic density profile quoted above, we find that the heat source associated with the vortex is
    Q=κ2Tz2=κb(gbP0)eb(zz0)<0,
    because bP0 > 0 is small (and so gbP0 > 0). This indicates that heat flows out of the Red Spot into its environment. Note that recent observations show that the atmosphere above the Red Spot is much hotter than its surroundings.

One further property of our solution can be investigated, which is worthy of note: the aspect ratio of the vortex structure of the GRS. Our asymptotic solution is driven by the thin-shell approximation [requiring only the choice of a suitable δ(ε)], and so we have the opportunity to compare with available results without recourse to further simplifications. The aspect ratio a, the vertical half thickness to the horizontal length scale, of a large-scale astrophysical vortex reflects the competition between the opposing effects of rotation and stratification, as the homogenization of the flow along the axis of rotation tries to balance the restrained vertical motion. While a universal law relating the aspect ratio to flow characteristics in rotating, stratified fluids remains elusive, two models have gained some prominence. On the one hand, we have the generic law a=|f|/N, where N=(g/ρ)(ρ/z) is the Brunt–Väisälä frequency, attributed to Charney (1971), while on the other hand Gill (1981) proposed the relationship a=O[Ro(|f|/N)], where Ro=U/(fL) is the Rossby number (see Aubert et al. 2012). We may use our formulation and solution to test these options. The GRS field data yield N′ ≈ 0.0158 rad s−1 (see Shetty and Marcus 2010) and, since sin(22°) ≈ 0.374, we obtain (see Table 1) a0.006,Ro0.11 and |f|/N0.0083, so that neither scaling law is a close fit. These observations, derived from our specific solution that accommodates nonlinear dynamics on the reliable basis of a systematic asymptotic analysis, suggest that any simplistic model is likely to ignore some important elements needed to describe the flow. Significantly, we do not require any such modeling assumption in order to produce a suitable solution that captures the salient features of the GRS.

b. Filamentary zonal flow at the southern boundary of Jupiter’s Great Red Spot

Jupiter’s visible atmosphere is dominated by banded structures parallel to the equator. The light colored are called zones and the dark bands are called belts (in infrared, these are dark/bright, indicating cool/warmer clouds, respectively). Typically, zonal flow emerges in time averages of geophysical flows over time intervals from years to decades long (see Cheng and Mahalov 2013; Galperin et al. 2004; Galperin and Read 2019). Jupiter’s zonal jets are strongest on the boundaries between the alternating belts and zones. The flow in the zones is anticyclonic—rotating counterclockwise in the Southern Hemisphere—because of an eastward jet on the poleward side and a westward jet on the equatorial side, while belts are cyclonic. Jupiter’s Great Red Spot is confined to a zone bounded by a westward jet at 19.5°S with nondimensional speed λ = −0.46 (about 70 m s−1) and an eastward jet at 26.5°S with nondimensional speed λ = 0.33 (about 50 m s−1). These zonal jets interact with the circular vortex motion of the GRS. At the northern boundary, the outcome is somewhat irregular since the GRS is associated with a persistent indentation, known as the red spot hollow, in the southern boundary of the south equatorial belt, and this generates strong perturbations away from the GRS. However, at the southern boundary, the interaction results in filamentary flow (see Fig. 1). To describe the observed pattern, we seek trajectories of the form:
X(t)=α+λt+Asin(μt)+Bcos(μt),Y(t)=β+Acos(μt)Bsin(μt),
describing curves traced by a particle rolling in counterclockwise circular motion along a horizontal line (Brieskorn and Knörrer 2012), with λ (which is the nondimensional jet speed) fixed but all other parameters particle-specific. Since there should be 2 degrees of freedom, we regard these parameters as functions of the labeling variables α = α(Ψ) and β = β(Ψ). First, we show that if μ = f, then (73) yields a solution of (38), (39), and (41), where we identify λ = −c and so we have n = 0. Then, since the Hamiltonian constraint (37) holds precisely if, for every fixed time, the flow map [X(0), Y(0)] → [X(t), Y(t)] is symplectic, and symplectic planar flows are precisely those that are orientation and volume preserving (see Meyer et al. 2009), we require that the determinant
D(α,β)=|XαXβYαYβ|

is time independent (and nonzero). We then identify the corresponding time-independent Hamiltonian (the streamfunction).

From (73), we obtain
U(t)=dXdt=λ+μ[Acos(μt)Bsin(μt)],V(t)=dYdt=μ[Asin(μt)+Bcos(μt)],
and
d2Xdt2=μ2[Asin(μt)+Bcos(μt)]=μV(t),d2Ydt2=μ2[Acos(μt)Bsin(μt)]=μ[U(t)λ].
Since the left sides of (38) and (39) are accelerations, we see that (38) and (39) hold if μ = f and
PX=0,PY=fλ.
For c = −λ, the above relation ensures the validity of (41) with n = 0, as claimed. Setting μ = f in (73), we now verify (33) by computing
D(α,β)=|XαXβYαYβ|=|1+Aαsin(ft)+Bαcos(ft)Aβsin(ft)+Bβcos(ft)Aαcos(ft)Bαsin(ft)1+Aβcos(ft)Bβsin(ft)|=1+AβBαAαBβ+(AαBβ)sin(ft)+(Bα+Aβ)cos(ft).
Consequently, (73) describes the particle paths of a Hamiltonian system (i.e., D is time independent) if the Cauchy–Riemann equations,
Aα=Bβ,Aβ=Bα,
hold, with
1+AβBαAαBβ=1(Aα)2(Aβ)2>0.
Writing Ψ[X(t), Y(t)] = Φ(α, β), using (73) and (74) with μ = f, we have
Φα=ΨXXα+ΨYYα=V(t)Xα+U(t)Yα=f[Asin(ft)+Bcos(ft)][1+Aαsin(ft)+Bαcos(ft)]+[λ+fAcos(ft)fBsin(ft)][Aαcos(ft)Bαsin(ft)]=f(AAα+BBα)+(fAλBα)sin(ft)+(fB+λAα)cos(ft)
and
Φβ=ΨXXβ+ΨYYβ=V(t)Xβ+U(t)Yβ=f[Asin(ft)+Bcos(ft)][Aβsin(ft)+Bβcos(ft)]+[λ+fAcos(ft)fBsin(ft)][1+Aβcos(ft)Bβsin(ft)]=λ+f(AAβ+BBβ)(fB+λBβ)sin(ft)+(fA+λAβ)cos(ft).
Thus, a time-independent Hamiltonian is obtained if
fAλBα=fB+λAα=fA+λAβ=fB+λBβ=0.
Recalling (75), we get
A=ae(f/λ)βcos[fλ(α+α0)],B=ae(f/λ)βsin[fλ(α+α0)],
for some constants a and α0. With μ = f in (73), we obtain the solution
X(t)=α+λt+ae(f/λ)βsin[fλ(α+λt)+α0],Y(t)=β+ae(f/λ)βcos[fλ(α+λt)+α0],
where a ≠ 0, λ ≠ 0, and α0 are fixed constants. Since f < 0, a counterclockwise circular motion requires a > 0, while the constraint (76) imposes
|a|e(f/λ)β<|λ|f,
so that |λ|/f is the maximal amplitude of the horizontally propagating oscillations (77). Note that the parametric equations of the curtate trochoidal curve Y = ξ(X; h, q), representing the trajectory of a point located at the distance h ∈ (0, q) from the center of a disk of radius q rolling without slipping along a horizontal line, are s[qshsin(s),qhcos(s)] (see Brieskorn and Knörrer 2012). Consequently, (77) with a > 0 and f < 0 yields
Y(t)=βλfξ[X(t)+λfα0;ae(f/λ)β,|λ|f].
The function ξ in (79) is not elementary. Indeed, Kepler’s transcendental equation X = sh sin(s), linking the eccentric anomaly s, the mean anomaly X, and the eccentricity h of the elliptic planetary motion about a center located at the focus of the ellipse, plays an important role in celestial mechanics. Bessel obtained the solution to Kepler’s equation in the form of the Fourier series
s=b(X;h)=X+i1Ji(ih)isin(iX),
where Ji are the Bessel functions of the first kind of order i ≥ 1 (see Colwell 1993). We have
ξ(X;h,q)=b(Xq;hq).
The solution that we have derived, (77), captures some observed features of the filamentary flow at the southern boundary of the zonal band to which Jupiter’s Red Spot is confined, namely:
  • the amplitude of the westward-propagating oscillations to the northern boundary is larger than that of the eastward-propagating oscillations to the south, in accordance with (78);

  • far away from the vortex, the dynamical structure is significantly affected by other factors; however, in the proximity of the vortex, (77) is an oscillatory motion of a filamentary flow (see Fig. 3) which has a period 2π/|f|0.72, considerably less than the rotation period of the GRS.

Fig. 3.
Fig. 3.

Depiction of the trochoidal waves (77). The distortion from a sinusoidal wave profile is indicative of the nonlinear character of the flow pattern.

Citation: Journal of the Atmospheric Sciences 82, 1; 10.1175/JAS-D-24-0038.1

6. Discussion

The underlying thinking behind the work presented here is to show that, with careful analysis based on asymptotic methods, a model that accommodates many and varied flows that are of some importance can be derived. So we have employed the thin-shell approximation, coupled to the f-plane model, to describe flows that are observed in the cloud bands that make up our gas giants. The appropriate reduced, asymptotic equations are derived—so error terms can be identified—showing that we can reasonably use, at leading order, the inviscid (Euler) system in a suitable rotating frame. These equations are necessarily nonlinear, but with a constant Coriolis coefficient and no motion in the vertical direction. We have shown that the two-component equations in the horizontal (tangential) plane, together with the equation of mass conservation, admit solutions for arbitrary density variation ρ(z) in the vertical direction. The equation of hydrostatic equilibrium then provides a consistency condition involving this density function. The existence of solutions, for arbitrary ρ(z), is confirmed by invoking a scale-separation property of the equations. Then, to access this solution, we use a Lagrangian representation from which all the relevant details can be extracted. Further, because we treat the gas as ideal and we have a corresponding first law of thermodynamics, we are able to determine the temperature field and the heat source associated with the flows that sit in a narrow band of the outermost layer of the planet’s troposphere.

Although the general formulation leads to a familiar system of equations, the use of arbitrary ρ(z) and the method of solution (and interpretation) are combined in a novel way. First, we have shown that this approach is quite general, allowing many different solutions for the dynamics to be analyzed in considerable detail; we have presented two examples. Second, the connection to the thermodynamics allows us to determine how the temperature varies through the flow field (and temperature data are readily available, with a fair degree of certainty). This, in turn, provides the opportunity to identify the heat sources required to drive and maintain the motion. However, all this comes—perhaps not surprisingly—at a cost, but a fairly small one: the scale separation allowing arbitrary ρ(z) works only for steady motion in the chosen rotating frame or, exceptionally, in a close neighborhood of the equator where the Coriolis term can be ignored, or for a special version of the scale-separation property (n = 1). Thus, we are able to construct detailed descriptions of flow structures that move with a particular cloud band, which is the chosen rotating frame, or that barely move relative to this frame. On the other hand, this constraint does not apply—the flow structure may move relative to the rotating frame—close to the equator.

The method of solution requires that we seek a Lagrangian representation (so with time t as the parameter) based on the trigonometric functions. As is evident from our examples, there are many choices available; no doubt others could be constructed and, perhaps, not restricted to the elementary functions. The novelty of the approach is that, rather than seeking general solutions, we embed observed dynamical patterns into the framework of time-independent Hamiltonian systems, relying on symplectic flow maps. All this, we submit, lays the foundations for the investigation of many other flows; we briefly describe those that we have examined here.

The application of most interest (probably) is Jupiter’s GRS. We have shown that this flow can be derived as an exact solution of our system of equations, so the development is predicated on a robust mathematical approach. In the process, we have elucidated a number of the observed properties of the GRS, including the period of rotation of its vortex, its pressure and thermal structure, and the heat source required to maintain it. In addition to this example, we have shown that the filamentary flow on the southern edge of the GRS can be obtained as a solution that recovers some of its observed features.

These examples provide evidence that we have put in place a systematic approach which can accommodate various types of flows that are observed on the outermost layers of our gas giants. This provides, we argue, an excellent basis for further investigation. This might be, on the one hand, to seek other solutions and their properties and, on the other hand, and more significantly, to develop detailed representations of other phenomena that are observed on the gas giants and similar planets.

Acknowledgments.

This research was supported by the Austrian Science Fund (FWF) (Grant Z 387-N). The authors are grateful for helpful comments from the referees.

Data availability statement.

All data for this paper are properly cited and referred to: the relevant data found in Catling (2015), Ingersoll et al. (2004), Ingersoll (2020), and Sanchez-Lavega (2011).

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Save
  • Aubert, O., M. L. Bars, P. L. Gal, and P. S. Marcus, 2012: The universal aspect ratio of vortices in rotating stratified flows: Experiments and observations. J. Fluid Mech., 706, 3445, https://doi.org/10.1017/jfm.2012.176.

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    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Cheng, B., and A. Mahalov, 2013: Time-averages of fast oscillatory systems. Discrete Contin. Dyn. Syst. S, 6, 11511162, https://doi.org/10.3934/dcdss.2013.6.1151.

    • Search Google Scholar
    • Export Citation
  • Cho, J. Y.-K., M. de laTorre Juárez, A. P. Ingersoll, and D. G. Dritschel, 2001: A high-resolution, three-dimensional model of Jupiter’s Great Red Spot. J. Geophys. Res., 106, 50995105, https://doi.org/10.1029/2000JE001287.

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  • Constantin, A., and R. S. Johnson, 2021: On the propagation of waves in the atmosphere. Proc. Roy. Soc., 477A, 20200424, https://doi.org/10.1098/rspa.2020.0424.

    • Search Google Scholar
    • Export Citation
  • Constantin, A., and P. Germain, 2022: Stratospheric planetary flows from the perspective of the Euler equation on a rotating sphere. Arch. Ration. Mech. Anal., 245, 587644, https://doi.org/10.1007/s00205-022-01791-3.

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  • Fig. 1.

    Color map of Jupiter in cylindrical coordinates, constructed from images taken by the camera onboard NASA’s Cassini spacecraft on 11–12.XII.2000 (credit: NASA). The map captures in some detail the GRS and the filamentary zonal flow at its southern boundary.

  • Fig. 2.

    The Cartesian coordinate system on the spherical planet, fixed in a suitable rotating frame, where r is the position vector of a point of latitude θ and longitude φ, relative to the center.

  • Fig. 3.

    Depiction of the trochoidal waves (77). The distortion from a sinusoidal wave profile is indicative of the nonlinear character of the flow pattern.

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