Modeling Jupiter's cloud bands and decks

Icarus 200 (2009) 548–562

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Modeling Jupiter’s cloud bands and decks 1. Jet scale meridional circulations L.C. Zuchowski ∗ , Y.H. Yamazaki, P.L. Read University of Oxford, Clarendon Laboratory, Parks Road, OX1 3PU Oxford, UK

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 13 May 2008 Revised 18 November 2008 Accepted 19 November 2008 Available online 13 December 2008 Keywords: Jovian planets Jupiter, atmosphere Atmospheres, dynamics

We have investigated the formation of jet scale meridional circulation cells on Jupiter in response to radiative and zonal momentum forcing. In the framework of semi-geostrophic theory, the meridional streamfunction is described by an elliptic equation with a source term dependent on the sum of the latitudinal derivative of the radiative forcing and the vertical derivative of the zonal momentum forcing. Using this equation with analytic terms similar to the assumed forcing on Jupiter, we obtained two set of atmospheric circulations cells, a stratospheric and a tropospheric one. A possible shift in the overturning circulation of the high and deep atmosphere can be induced by breaking the latitudinal alignment of radiative heating with the enforced belt and zones. A series of numerical simulations was conducted with the Jovian GCM OPUS, which was initiated with observational data obtained from the Cassini CIRS temperature cross-section and a corresponding geostrophic zonal wind field. Newtonian forcing of potential temperature as well as zonal momentum was applied respectively towards latitudinally and vertically uniform equilibrium fields. In accordance with the analytic illustrations two rows of jet scale circulation cells were created. The stratospheric circulation showed the distribution of upwelling over zones and downwelling over belts, consistent with cloud observations. The tropospheric cells featured a partial reversal of the downward vertical velocity over the belts and a considerable reduction of the upward movement over the zones in the domain, consistent with recent detections of high water clouds and lightning in belts. We also used the modeled new forcing fields as source terms for the semigeostrophic Poisson equation to attribute the origin of the modeled secondary circulation. In this analysis, the stratospheric circulation cells observed in the model are primarily generated in response to radiative forcing, while momentum forcing induces the shifted configurations in the deep atmosphere. © 2008 Elsevier Inc. All rights reserved.

1. Introduction The atmospheres of the gas giants, Jupiter and Saturn, feature numerous zonal jet streams in association with the characteristic banded appearance of the planets. Eastward jets are thereby located at equatorial edges of the dark belts and at poleward edges of the bright zones (Limaye, 1985). Spectroscopic observations of Jupiter’s highest cloud deck at about 0.5 bar, which is thought to consist mostly of NH3 -ice (Baines et al., 2002), have shown that zones possess dense, elevated clouds while lower clouds are found in belts (West et al., 1986; Simon-Miller et al., 2001). From these observations, the existence of jet-scale meridional circulation cells, with upwelling branches over zones and downwelling branches over belts, has been proposed. Recent observations of elevated water clouds and lightning in belts (Banfield et al., 1998; Gierasch et al., 2000; Little et al., 1999; West et al., 2004) have

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Corresponding author. E-mail address: [email protected] (L.C. Zuchowski).

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also prompted the suggestion of a second set of counter-rotating cells in the lower troposphere. The upper troposphere/stratosphere belt/zone pattern of vertical velocity has been linked to differential heating in Jupiter’s stratosphere (Gierasch et al., 1986), whereby large scale net diabatic heating over zones and cooling over belts is assumed in association with upwelling and downwelling respectively. Various numerical models employing Newtonian radiative forcing towards an equilibrium temperature profile derived from parahydrogen and aerosol measurements have successfully reproduced the expected up- and downwelling distribution (Gierasch et al., 1986; Conrath et al., 1990; West et al., 1992; Moreno and Sedano, 1997; Lian and Showman, 2008). The possibility of secondary deep circulations has so far not been extensively tested by numerical modeling. No comparable observational evidence exists to illuminate the mechanism and influence of zonal jet forcing in Jupiter’s and Saturn’s atmosphere and the nature of this forcing is still discussed controversially. Two main hypotheses have evolved: shallow forcing models, which assume zonal velocity forcing to be restricted

Jet scale meridional circulation cells on Jupiter

to a shallow layer around the height of the NH3 -ice cloud deck, and deep forcing models, in which the jets are driven by deep convection (Vasavada and Showman, 2005, for a review). A correlation has been observed between the eddy momentum flux (EMF) on Jupiter and the latitudinal derivative of the zonal flow, which provides evidence for the hypothesis that the small scale eddies might transfer energy into the jets (Beebe et al., 1980; Ingersoll et al., 1981; Salyk et al., 2006). The vertical extent of the EMF transfer layer is unknown, but it has been estimated to persist at least as deep as 2.5 bar. The vertical wind profile measured by the Galileo probe points to a relatively deep (at least down to 20 bar) extent of the zonal jets themselves (Atkinson et al., 1997) but Showman et al. (2006) have shown that such deep jets can also result from shallow zonal momentum forcing. In numerical investigations of Jupiter’s jet scale circulation cells, zonal momentum forcing has so far mainly been included as a prescribed boundary profile at bottom level of the modeled domain (Conrath et al., 1990). In the current work we have investigated the formation of Jupiter’s jet scale meridional circulation cells both theoretically, in the framework of the semi-geostrophic approximation, and numerically with the Oxford Planetary Unified model System (OPUS), a sophisticated GCM solving an extended form of the hydrodynamic primitive equations. We show that the meridional streamfunction derived from the forced semi-geostrophic equations can be directly determined from the radiative and zonal momentum forcing terms, and that two sets of jet scale circulation cells are generated in response to Jupiter-like analytic forcing terms in the upper and lower atmosphere. We also consider the conditions under which these two circulations may differ in direction. We have also directly modeled the formation of jet scale circulation cells with OPUS, using initial and forcing fields computed from the temperature cross-section measured by the Cassini Composite InfraRed Spectrometer (CIRS) (Flasar et al., 2004) during the Cassini fly-by of Jupiter in December 2000. Under the influence of Newtonian radiative and zonal momentum relaxation forcing, the model develops two sets of jet scale circulation cells in the stratosphere and troposphere in association with two jet pairs. We are able to show that the stratospheric cells are primarily caused by radiative forcing and thus adhere to the belt/zone pattern of down/upwelling inferred from NH3 -ice cloud observations. In the deep atmosphere, direct momentum forcing by eddies is more likely to provide the predominant driving mechanism for zonal jets and our simulations suggest a partial reversal of vertical velocities over the belts as well as a reduction of upwelling over zones. We further analyzed the influence of each forcing component by solving the semi-geostrophic elliptic equation under inclusion of the simulated radiative and momentum forcing, to diagnostically infer circulations from the modeled forcing fields. These diagnostics capture the formation of four radiative cells in response to the radiative forcing in the stratosphere, and further show that momentum forcing can generate shifted cells in the deep atmosphere. A follow-up study to this work, presented in a companion paper, will employ a simple cloud scheme for OPUS to show that patterns of elevated and depressed NH3 -ice cloud bands, similar to observations, may be generated in response to these circulations, and will investigate the consequences of disconnected tropospheric cells for deeper cloud layers. In Section 2 the defining equation for the semi-geostrophic meridional streamfunction in the presence of diabatic and momentum forcing, is derived from the semi-geostrophic equations of motion (Holton, 2004) and illustrations are presented for a series of cases comparable to Jupiter. Section 3 introduces the numerical model, initial input fields and forcing parameterizations while Section 4 presents results from the OPUS simulations and the diagnostic attributions from the semi-geostrophic elliptic equation. We

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discuss the implications of our findings for the atmospheric motion of clouds and chemical tracers, and draw final conclusions in Section 5. 2. Forced semi-geostrophic theory 2.1. A Sawyer–Eliassen equation with forcing Since the Rossby number Ro in Jupiter’s atmosphere has been estimated to be much less than unity, it is usually assumed that the dynamics can be adequately described by a set of geostrophic or quasi-geostrophic equations (Hess, 1969). Here we assume the semi-geostrophic equations to be valid, which enforce geostrophic balance in the zonal direction but allow an ageostrophic meridional velocity component. The derivation of the semi-geostrophic streamfunction in this section closely follows that of Holton (2004) but additionally includes radiative (potential temperature) F rad and momentum forcing F mom terms in the hydrodynamic primitive equations. We have also replaced the Boussinesq-style uniform density approximation with a more general anelastic approximation, in order to account for the large density stratifications found on Jupiter. From the hydrodynamic primitive equations one can obtain the semi-geostrophic equations by defining geostrophic wind components as 1 ∂Φ

ug =

f ∂y

vg = −

(1)

,

1 ∂Φ f ∂x

(2)

,

where Φ = p /ρ is the pressure p scaled by the density ρ , f is the Coriolis parameter, and y and x are the horizontal coordinates. The horizontal velocities are then given by u = ug,

(3)

v = v g + va,

(4)

where v a denotes the ageostrophic velocity component. Inserting Eq. (3) and Eq. (4) into the hydrodynamic primitive equations, and linearizing the thermal energy equation, the resulting set of equations becomes Du g

− f v a = F mom ,

Dt g Dθ

g

+ w N2 = F rad , θ00 Dt θ00 ∂w ∂ va + = 0, ∂y ∂z

(5) (6) (7)

where g is the gravitational acceleration, w is the vertical velocity component, θ is the potential temperature deviation from an atmospheric reference profile θ0 ( z), t is the time coordinate and the covariant derivative D / Dt is given by D Dt

=

∂ ∂ ∂ ∂ ∂ + ug + vg + va +w . ∂t ∂x ∂y ∂y ∂z

(8)

The continuity equation (7) does not influence the following computations and is given in the approximated form used by Holton (2004). We use a vertical coordinate z defined by z = − H ln( p / p 0 ), where H = R T 0 / g is a constant atmospheric scale height, p 0 is an atmospheric reference pressure, R is the gas constant and T 0 is a constant atmospheric reference temperature. Note that Eq. (8) differs from the more often used quasi-geostrophic covariant derivative by the inclusion of the two ageostrophic advection terms.

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Following the simplifications by Holton (2004) we have scaled Eq. (6) by an atmospheric potential temperature reference value θ00 . The Brunt–Vaisala frequency is defined as N2 =

g ∂θ0 . θ00 ∂ z

(9)

Using Eqs. (1) and (2) as well as the hydrostatic approximation ∂p ∂ z = −ρ g one can derive an additional thermal wind relationship:

∂ug R =− ∂z Hf ∂vg R =− ∂z Hf

∂T g ∂θ g ∂θ =− ≈− , ∂y fθ ∂y f θ00 ∂ y ∂T g ∂θ g ∂θ = ≈− . ∂x f θ ∂x f θ00 ∂ x

(10) (11)

The substitution of the reference value θ00 for θ is equivalent to Holton’s use of the Boussinesq approximation by assuming a uniform density value in the definition of Φ . Cross-differentiating Eqs. (5) and (6) with respect to y and z, and defining a streamfunction Ψ M for the ageostrophic flow by

∂ΨM , ∂z ∂Ψ M , w= ∂y

va = −

(12) (13)

leads to the elliptic equation

∂ 2 ΨM ∂ 2 ΨM ∂ 2 ΨM + S2 + M2 ∂ z∂ y ∂ z2 ∂ y2 ∂ F rad f θ00 ∂ F mom = Q2 + + . g ∂z ∂y

F2

(14)

Equation (14) effectively constitutes a Sawyer–Eliassen equation with the coefficients M2 =

θ00 N 2 g

∂θ , ∂y  f θ00

+

∂θ , ∂z

(15)

S2 = − F2 =

g

(16)



f −

∂ug , ∂y

(17)

and the geostrophic forcing term Q2 =

∂ v g ∂θ ∂θ ∂ u g − . ∂ y ∂x ∂y ∂y

(18)

2.2. Simple analytic solutions with Jupiter-like forcing For a reduced case with ∂ u g /∂ y  f and ∂θ/∂ z  ∂θ0 /∂ z, Eq. (14) simplifies to f 2 θ00 ∂ 2 Ψ M g

∂ z2

= Q2 +

∂ 2 ΨM N 2 θ00 ∂ 2 Ψ M + ∂ z∂ y g ∂ y2 ∂ F mom ∂ F rad + . ∂z ∂y

+ S2

f θ00 g

(19)

Assuming ∂ w /∂ z  1 as well as zonally averaged states will remove the mixed differential and the geostrophic forcing term Q 2 in Eq. (19), thus leading to a simplified elliptic equation, N2

2 ∂ 2 ΨM ∂ F mom g ∂ F rad 2 ∂ ΨM + f =f + . ∂z θ00 ∂ y ∂ y2 ∂ z2

(20)

The streamfunction Ψ M is thus uniquely determined by the two forcing terms and the choice of appropriate boundary conditions. Assuming constant coefficients N 2 and f 2 , a redefinition of coordinates according to

1 y, N 1 z∗ → z f y∗ →

can be employed and one can thus reduce Eq. (20) to a simple Poisson equation

∂ 2 ΨM g ∂ F rad ∂ 2 ΨM ∂ F mom + = + . ∗ 2 ∂ z∗ N θ00 ∂ y ∗ ∂y ∂ z ∗2

(21)

In order to apply the forced semi-geostrophic theory derived above to Jupiter’s atmospheric circulations, it is necessary to specify the two forcing terms F rad and F mom . As we are aiming to obtain analytic solutions to Eq. (21) the terms will be greatly simplified but should nevertheless capture some essential characteristics of Jupiter’s forcing in order to illustrate some generic features. Considering the temperature structure of Jupiter’s atmosphere as derived from observational data over the last decades (SimonMiller et al., 2006), it is possible to identify two characteristic features which need to be represented in the radiative forcing. Firstly a stable increase of the potential temperature in the upper troposphere/stratosphere and an almost adiabatic temperature profile in the deep atmosphere seem to be characteristic. Secondly, above 0.5 bar latitudinal temperature variations aligned with the jovian belts and zones have been observed, leading to a decay of winds with height (Gierasch et al., 1986). In response to these hot and cold bands, heating over zones and cooling over belts can be expected. Previous parameterizations of Jupiter’s radiative forcing have usually been based on, or have implicitly included, these assumptions (Gierasch et al., 1986; West et al., 1992; Moreno and Sedano, 1997; Lian and Showman, 2008). As a suitable general expression for the radiative forcing in the upper atmosphere (z > z0 ) we have thus chosen the form F rad = F 0,rad e α z cos(ky ), where F 0,rad is the forcing amplitude in the deep atmosphere, α (>0) the increase of potential temperature forcing with height and the constant k embodies the latitudinal variations in heating. In accordance with the observed temperature structure we will assume F rad = F 0,rad in the deep atmosphere, which will not lead to a contribution in (21). The temperature profile as described above entails the existence of large differences between the buoyancy frequency of the upper atmosphere and the lower atmosphere, where a value very close to zero is usually assumed for Jupiter (e.g., Bosak and Ingersoll, 2002). Equation (21) cannot capture these variations, an inadequacy that we tolerate here for the sake of a simple discussion. The momentum forcing term F mom cannot be deduced directly from observations. At the height of the main visible cloud deck the possible EMF transfer into the jets has been estimated (e.g., Salyk et al., 2006). However, the vertical extent of the layer in which this transfer takes place is unknown and might be as deep as 2.5 bar. In previous numerical studies momentum forcing has usually taken the form of a prescribed zonal velocity profile at the lower boundary of the simulated domain (Conrath et al., 1990; Yamazaki et al., 2004). Showman et al. (2006) employed momentum forcing towards a banded jet profile in the upper part of their model. The vertical dependency of the momentum forcing term in Eq. (21) is intricately related to the underlying assumptions about the vertical extent of Jupiter’s zonal jets. In this paper we concur with the deep jet hypothesis as supported by the Galileo probe measurements (Atkinson et al., 1997) and assume momentum forcing towards a banded zonal wind profile throughout the domain. Physically such deep forcing could be provided by convection or EMF transfer or a combination of these two components. We will thereby test two cases, the first one assuming a vertical dependency of the momentum forcing inverse to A = F 0,mom e −β z sin(my ) and the radiative forcing profile with F mom

Jet scale meridional circulation cells on Jupiter

551

Fig. 1. (a) Simplified representation of Jupiter’s radiative forcing (in K/s). (b) Momentum forcing A: Constant forcing in the deep atmosphere, decaying forcing in the stratosphere. (b) Momentum forcing B: Momentum forcing decaying throughout the atmosphere. Both (a) and (b) are given in m/s2 . Dashed lines represent negative values and solid lines denote positive contours.

Fig. 2. Sample circulations for Case A. The plots show the streamline function Ψ M as defined by Eqs. (12) and (13) (in m2 /s). Stratospheric forcing is employed above p = 0.5 bar, below this height an unforced solution is assumed. (a) Circulations induced by momentum forcing only. (b) Circulations induced by radiative forcing only. F

(c) Fully forced case ( 0θ,rad 00 spaced logarithmically.

g

= F 0,mom = 1, α = β = 0.35). Dashed lines represent clockwise rotation and solid lines denote anti-clockwise circulations. The contours are

A F mom = F 0,mom sin(my ) in the deep atmosphere, where all quantities are defined analogue to the radiative ones. As the constant forcing in the lower atmosphere will not influence the right-hand side of Eq. (21). Case A corresponds to a meridionally unforced lower atmosphere, whose circulations are solely driven by the forcing configurations in the top layer. This is a scenario similar to the one studied by Showman et al. (2006). In Case B we assume B F mom = F 0,mom e −β z sin(my ) throughout the vertical domain. Perfect alignment of the radiative heating and cooling and mechanical forcing within the zones and belts entails m = k. Fig. 1 illustrates these three forcing profiles. Analytic solutions can be derived for both Case A and Case B. In Case A the equations

∇ 2 Ψ MU = −

gkF 0,rad α z e sin ky − F 0,mom β e −β z sin(my ), N θ00

∇ 2 Ψ ML = 0 have to be solved for the circulations in the upper (subscripted U ) and lower atmosphere (subscripted L), whereas the top boundary condition for the deep atmosphere case is given by the upper atmosphere circulations at z = z0 . The corresponding equations for case B are

∇ 2 Ψ MU = −

gkF 0,rad α z e sin ky − F 0,mom β e −β z sin(my ), N θ00

∇ 2 Ψ ML = − F 0,mom β e −β z sin(my ). Sample solutions to these equations are shown in Figs. 2 and 3, and will be discussed below. The exact solutions are given in Appendix A.

The exact form of the solutions in both cases is sensitive to the boundary conditions specified at the top and bottom level of the domain. Figs. 2c and 3c show the respective scenarios in Cases A and B under the prescription of Neumann boundary conditions ((∂Ψm /∂ z)z=0 = 0) at the bottom boundary. The bottom boundary is located at 100 bar, which lies just outside the plotted domain. For Dirichlet boundary conditions (e.g., Ψ M = 0) any atmosphere circulation cells present will be closed at the respective boundary. Fig. 4a displays the solution in Case B for Dirichlet boundary conditions at the top and bottom boundary. It can be seen that the circulations in Figs. 3c and 4a are essentially the same, despite the fact that the vanishing of the streamfunction at the boundaries enforces the formation of more compact, closed atmospheric cells. However, such a strict prescription of streamfunction values will lead to the generation of a boundary layer with large horizontal velocities (Fig. 4). On Jupiter, the cells will likely close eventually due to the configuration of forcing in the upper stratosphere. Conrath et al. (1990) found that the planetary scale variations of Jupiter’s Voyager measured temperature profile might produce an equator to pole circulation above a pressure level of 0.001 bar. While this region lies outside the domain modeled with OPUS, and we so do not expect to observe this effect in the numerical experiments, we have considered an analytical scenario in which the planetary variations of solar insolation are parameterized by an additional radiative forcing term F plan = F 0,plan e γ z sin(2 y /π ), where y has been scaled by the size of the planet. The constants are again defined similarly to the ones used in the diabatic radiative forcing. This term is only applied in the upper atmosphere above 0.5 bar. The equations to be solved are then given by

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L.C. Zuchowski et al. / Icarus 200 (2009) 548–562

Fig. 3. Sample circulations for Case B. The plots show the streamline function Ψ M as defined by Eqs. (12) and (13) (in m2 /s). Stratospheric forcing is employed above p = 0.5 bar, below this height a solution forced by momentum forcing only is assumed. (a) Circulations induced by momentum forcing only. (b) Circulations induced by radiative forcing only. (c) Fully forced case ( The contours are spaced logarithmically.

F 0,rad g

θ00

= F 0,mom = 1, α = β = 0.35). Dashed lines represent clockwise rotation and solid lines denote anti-clockwise circulations.

Fig. 4. Circulations for different boundary conditions in Case B (fully forced). The plots show the streamline function Ψ M as defined by Eqs. (12) and (13) (in m2 /s). (a) Dirichlet conditions (Ψ M = 0) enforced at both vertical boundaries. (b) Dirichlet boundary conditions with overlaying equator to pole circulation. The contours are spaced logarithmically.

gkF 0,rad α z e sin ky − F 0,mom β e −β z sin(my ) N θ00 g F 0,plan α z e sin(2 y /π ), − π N θ00

∇ 2 Ψ MU = −

∇ 2 Ψ ML = − F 0,mom β e −γ z sin(my ). Dirichlet conditions have again been assumed for both vertical boundaries. Fig. 4b shows the resulting circulations and they have been included in Appendix A. While blending into the equator to pole circulation in the upper stratosphere, the resulting configuration in the lower stratosphere and troposphere is virtually identical to the scenario displayed in Fig. 4b. Accordingly, it can be assumed that the jet scale circulations discussed here are fully compatible with a possible equator to pole circulation in the upper stratosphere. If m = k the circulation cells in the upper and lower atmosphere are of the same number and direction of circulation, regardless of differences in the z-dependent part of F rad and F mom . A quantitative difference in the modeled meridional circulations Ψ M can only derive from a mismatch between k and m. Fig. 5 shows solutions for Case A and Case B in which the wave number m has been chosen to be different from k. It can be seen that the deep circulations in Case B now behave with wave number m, while due to the increase in radiative forcing with height the upper ones still conform to k (Figs. 5c and 5d). The contrast between the two atmospheric regions implies that the lower atmosphere here is clearly dominated by the momentum forcing term while the upper

layer is mainly forced by radiation. In Case A the lower atmosphere is generally dominated by the smaller of the two wave numbers while the higher atmosphere develops circulations in accordance with the larger wave number (Figs. 5a and 5b). This is due to the fact, that in the unforced region the streamfunction’s amplitude e-folding with depth is proportional to the horizontal wave number, so that modes with a large wave number will be attenuated strongly below the stratospheric/tropospheric boundary. Such configurations with non-identical m and k can result in a change in the latitudinal distribution of up- and downwelling with height. On Jupiter the existence of counter-rotating meridional circulation cells in the stratosphere and deep troposphere has been proposed to explain observations of high water clouds in belts (Banfield et al., 1998; Gierasch et al., 2000). The solutions from the forced semi-geostrophic equations as discussed above imply that such a configuration would have to be associated with latitudinal changes in the wind and temperature fields between the stratosphere and troposphere. Since little is known about the jet forcing on Jupiter and Saturn, any attribution of such changes to a physical process are purely speculative at present. Conceivably, a change in forcing could be the result of different eddy configurations below the visible cloud deck, which could break the correspondence between the jets as seen at cloud level and the deep atmosphere EMF. The zonal wind cross-section constructed from Cassini CIRS retrievals displays quantitative changes of the flow structure such as jet narrowing and widening in the upper atmosphere (Flasar et al., 2004; Read et al., 2006), which could be associated with two

Jet scale meridional circulation cells on Jupiter

553

Fig. 5. Circulations for a difference in wave numbers k and m. The plots show the streamline function Ψ M as defined by Eqs. (12) and (13) (in m2 /s). (a) m = 0.75k for Case A. (b) m = 1.25k for Case A. (c) m = 0.75k for Case B. (d) m = 1.25k for case B. Dashed lines represent clockwise rotation and solid lines denote anti-clockwise circulations. The contours are spaced logarithmically.

sets of differential circulation cells as displayed in Fig. 5. We will investigate this hypothesis in the following section by using the observational data to initialize jovian GCM simulations. 3. Model set-up and configurations OPUS solves an extended form of the hydrodynamic primitive equations on a rotating sphere, including finite depth terms and all components of the Coriolis force (White and Bromley, 1995), on a scaled pressure (σ ) coordinate grid comprising 40 levels. A Richardson number based vertical diffusion procedure has been implemented in order to parameterize subgrid scale mixing. The GCM has been described in detail by Yamazaki et al. (2004), who employed OPUS to examine the hydrodynamic stability of various mid-latitude jets on Jupiter and to test possible generation mechanisms for the equatorial jet (Yamazaki et al., 2005; Yamazaki and Read, 2006). For the present study we conducted Southern hemisphere limited area simulations with a spatial resolution of 0.5◦ using 53 latitudinal rows and 179 longitudinal columns. The Northern edge of the domain was placed at 15◦ South while the zonal channel was periodic in longitude. No-slip conditions were employed at the North and South boundaries. This domain includes the Tropical and Temperate Belt/Zone pair as well as the location of the Great Red Spot and the White Ovals. We used a deep atmosphere model consisting of 40 logarithmically spaced levels from 0.01 bar to 100 bar. A temperature of 600 K was prescribed everywhere on the lowest level of the model, consistent with a steady heat flow from the planetary interior.

The model was initiated with a potential temperature crosssection which, down to 0.5 bar, was derived from CIRS retrievals (Flasar et al., 2004). The profile was the extended downwards adiabatically. By adjusting towards an adiabatic profile we assume that the deep atmosphere is dominated by convection. A corresponding thermally balanced zonal wind input field was computed. It can be seen that in the region under consideration the stratospheric jets have a slightly different latitudinal wave number than the flow in the deep atmosphere. This is due to the widening of the two retrograde jets in the stratosphere and the introduction of a third prograde stratospheric jet at the Northern boundary. The meridional velocity fields were initially set to zero. Radiative and mechanical forcing in the model are parameterized by Newtonian relaxation F rad =

θeq − θ

F mom =

τrad

(22)

,

u eq − u

τmom

,

(23)

where τrad and τmom are altitude dependent timescales, θ is the potential temperature, u is the zonal velocity, θeq is the thermal equilibrium profile and u eq is the zonal velocity equilibrium profile. The two equilibrium profiles were computed from the input fields; θeq = θeq (σ ) consists of the latitudinally averaged CIRS cross-section while u eq = u eq (λ) is given by the thermal wind input field, vertically averaged below 0.5 bar and uniformly extended upwards. As illustrated in Fig. 6c and listed in Table 1, we have tested three different forcing scenarios. In Run 1, both timescales τrad and

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Fig. 6. Input fields derived from Cassini CIRS retrievals. (a) Temperature input field (in K). (b) Zonal wind input field (in m/s). Dashed contours indicate eastward velocities and solid contours represent westward velocities. (c) Radiative (solid and dashed) and momentum (dotted) forcing timescales (in s). The identifiers of the runs using the profile are printed next to the curve.

4.1. OPUS results

Table 1 Summary of runs performed in this study. Run

τrad,0 [s]

τmom,0 [s]

dτrad /d(ln p ) [s]

dτmom /d(ln p ) [s]

1 2 3

5.1 × 105 5.1 × 105 5.1 × 105

5.1 × 106 5.1 × 106 5.1 × 106

0 4.8 × 105 4.8 × 105

0 0

−4.8 × 105

τmom are assumed to be uniform with height. Run 2 features a radiative timescale decreasing with height above 0.5 bar and in Run 3 we have additionally included a momentum timescale increase with height throughout the stratosphere, corresponding to a possible weakening of eddy forcing above the tropopause. Although simplified, the altitude dependent form of τrad is in general agreement with the heating profiles derived by more complete radiative calculations (e.g., Conrath et al., 1990; West et al., 1992). With magnitudes here of 103 s to 108 s these radiative time scales are considerably smaller than the estimates based on purely radiative thermal adjustment in Jupiter’s atmosphere, which are of the order of 108 s in the troposphere (e.g., Conrath et al., 1990; West et al., 1992). However, the use of larger thermal forcing constants in the model accelerated the establishment of a dynamical equilibrium state and thus greatly improved the computational economy of the simulations, making more complex computational studies possible. According to Eq. (6) the steady state vertical velocities scale directly with the radiative forcing and therefore inversely with the radiative timescale. The absolute numerical values of the modeled vertical velocities are therefore not quantitatively predictive. The momentum time constants τmom employed in the model have absolute values comparable in magnitude to estimates based on the EMF computations by Salyk et al. (2006). In our model the radiative time constant τrad remains smaller than the momentum time constant τmom throughout the deep atmosphere in all simulated scenarios. This would not be the case if more realistic, larger radiative time constants had been chosen for the deep atmosphere. However, due to the latitudinal uniformity of the CIRS temperature cross-section in this region we can assume that radiative forcing in the deep atmosphere will be small and will exert little influence on the generated meridional circulations. 4. Results In this section we first present results from the jovian flow simulations discussed in Section 3. In Section 4.2 we go on to use the semi-geostrophic theory developed above to diagnostically attribute the origins of the simulated meridional circulations respectively to the net radiative and mechanical forcings in the model.

4.1.1. Time development and flow instabilities All runs were integrated for 250 days. The evolution of the flow with integration time is illustrated in Fig. 7, which shows the development of Ertel’s potential vorticity on the 800 K isentropic surface in the stratosphere for Run 3. The potential temperature was computed with respect to a reference pressure of 100 bar and corresponds to a depth of about 0.2 bar. This flow is typical of the other two runs as well. The evolution of vortices and instabilities in the given domain from a zonally symmetric initial state similar to the one used here was already discussed by Yamazaki et al. (2004). As the present work focusses on the investigation of large scale circulations, the time development and instabilities will only be briefly discussed as supplementary information to the exploration of the large scale, time averaged structures. Instabilities in the flow started to appear after about 40 days in the form of jet meanders around 34◦ S. By 100 days a wave number 5 disturbance is clearly discernible at the location of the two major shear zones in the domain. Large scale vortices like the one shown in Fig. 7c appeared first after 120 days. These eddies seem to form at the latitudes formerly associated with the White Ovals on Jupiter and are advected with the flow at this location for several days before disintegrating. Occasionally migration of vortices to higher latitudes took place, but these events were short lived. Beyond 250 days the jet structure began to dissolve so that the flow became increasingly unrealistic. The vertical structure of a typical large scale vortex that developed spontaneously within the domain at latitude 34◦ S is illustrated in Fig. 8. In accordance with inferences from cloud observations for Jupiter’s large scale vortices (Simon-Miller et al., 2002), the eddy is characterized by increased upwelling in the center (Fig. 8b). The vortex features an anomalously cold core region and a hot collar (Fig. 8a), similar to what spectroscopic observations (Flasar et al., 1981) indicate for Jupiter’s Great Red Spot. This temperature anomaly extends to a depth of about 0.7 bar, after which it vanishes into the adiabatic interior. While meridional velocities are most intense in this stratospheric region as well (Fig. 8c), the barotropic response in the vortex center exists throughout the vertical domain. 4.1.2. Time averaged large scale circulations All fields shown in this section are equilibrated states and have been time averaged for 200 days (over the time span between 10 and 210 simulated days). As the model becomes highly turbulent after 250 days, they represent the flow in a temporarily steady configuration. Fig. 9 displays the temperature and zonal velocity adjustments resulting from Eqs. (22) and (23). Newtonian forcing

Jet scale meridional circulation cells on Jupiter

555

Fig. 7. Ertel’s potential vorticity on the θ = 800 K isentropic surface. The images show snap-shots from the evolution of Run 3, with a similar development being observed for the other two runs.

Fig. 8. Structure of the large scale vortex at 33◦ S in Run 3 (Fig. 7c). (a) Temperature anomaly (in K). (b) Vertical velocity anomaly (in m/s). (c) Meridional velocity anomaly (in m/s). The fields were computed by subtracting the zonally and time averaged fields from the instantaneous values at 150 days.

towards a horizontally uniform temperature field leads to stratospheric heating over zones and cooling over belts. The differential heating is most pronounced between 0.4 bar and 0.05 bar, which corresponds to the region of greatest latitudinal temperature contrasts in the input field (Fig. 6a). This pattern is clearly visible in all three cases but appears most well formed for runs with short radiative timescales in the stratosphere (Figs. 9b and 9c). The zonal momentum forcing as plotted in Fig. 9 also features a quantitative difference between stratospheric and deep tropospheric forcing. At latitudes lower than 20◦ S and around 30◦ S retrograde adjustments are made above 5 bar while prograde forcing exists in the deep atmosphere. These discrepancies are especially pronounced in the Run 3 (Fig. 9c), where momentum forcing decays throughout the stratosphere. The height differences in momentum forcing might be linked to the upper atmosphere flow variations in the initial wind profile and could represent the balancing of the introduction of the additional prograde jet at low latitudes as well as the widening of the retrograde jet at 30◦ S by the equilibrium profile uniform with height. Comparing the fields in Fig. 9 to the analytic forcing terms in Fig. 1 one can see that the modeled forcing is similar to Case B (with m = 5) discussed in Section 2. Accordingly we would expect to obtain comparable circulations in response to the adjustments. The resulting time-mean zonal velocities obtained with model simulations are shown in Fig. 10. The modeled zonal velocity profiles conform well with the input fields for all cases, although the

two retrograde flows have become significantly weaker than in the initial cross-section. Stratospheric widening of the retrograde jets can still be observed, while the intrusion of the prograde jet at the Northern edge of the domain has been almost entirely suppressed by momentum forcing. Similar to Lian and Showman (2008), we do not observe any decay of the jets in the upper atmosphere. In Run 3 the choice of an increasing time constant for momentum forcing in the upper atmosphere was expected to allow such jet dampening. However, even for further control simulations with even less stratospheric momentum forcing (not shown here) a general stratospheric velocity reduction with altitude did not occur. It thus appears that the horizontal temperature differences in the model are not sufficient to cause the jet decay described by Gierasch et al. (1986). This might also be due to the fact that the CIRS data, used for initialization of the model, does not contain such pronounced jet decay as in the Voyager data. In Fig. 10 the meridional circulations are represented by the velocity streamfunction as defined by Eqs. (12) and (13). In accordance with the configurations computed directly from the forced semi-geostrophic equations (Fig. 5), two sets of meridional circulation cells have been generated. Four stratospheric cells form above 0.2 bar and are well aligned with the position of the jet centers. Due to the fact that no damping is observed or explicitly applied at the upper boundary of the model, the cells are open to the overlying atmosphere. Between 0.2 bar and 0.5 bar a quiescent region exists. Below this depth a secondary set of tropospheric cir-

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Fig. 9. Forcing for three runs with different time constant profiles. Images (a)–(c) show the zonally averaged radiative forcing (in K/s) and images (d)–(e) the zonally averaged zonal velocity forcing (in m/s2 ) for Run 1 (a, d), Run 2 (b, e) and Run 3 (c, f). Solid contours denote positive and dashed contours indicate negative forcing.

culations cells is generated. There exist three clockwise rotating cells, such that the latitudinal wave number in this layer is slightly higher than in the upper atmosphere. Due to the increased number of cells, a general shift of the circulation pattern to the South is observed, leading to a configuration very similar to the analytic solutions displayed in Fig. 5d. It is also noticeable that the addition of the Northernmost clockwise cell seems to restrict the tropospheric extension of the two anticlockwise cells, which both end above the 30 bar level and partly bisect the intruding cell. Fig. 11 presents a comparison of the vertical velocities in the upper and lower atmosphere for Runs 2 and 3. It becomes imme-

diately obvious that the different numbers of circulation cells in the stratosphere and troposphere lead to a qualitative difference between the vertical velocity distributions. In the stratosphere, upwelling takes place over the South Tropical and South Temperate Zones, while the South Tropical and South Temperate Belts are characterized by downwelling (Figs. 11a and 11b). Northwards of 18◦ S and around 30◦ S, the stratospheric downwelling regions overlie tropospheric upwelling. Accordingly the zone/belt pattern of up and downwelling is no longer valid in the deep atmosphere. In addition, upwelling in the South Tropical Zone is greatly reduced in this region. Fig. 11 also shows the vertically-averaged latitudinal

Jet scale meridional circulation cells on Jupiter

557

Fig. 10. Images (a)–(c) show the zonally averaged zonal velocity (in m/s) and images (d)–(f) the zonally averaged meridional streamfunction (in m2 /s) for Run 1 (a, d), Run 2 (b, e) and Run 3 (d, f). Westward velocities are shown with solid lines and eastward velocities are shown with dashed lines. Dashed cells rotated clockwise and solid cells rotated anticlockwise.

profiles of the simulated radiative and momentum forcing. As expected from Eq. (6), the vertical velocity profile closely follows the variations of the radiative forcing in the upper atmosphere which, due to the equilibrium temperature variations, traces the expected stratospheric belt/zone pattern. In the deep atmosphere, radiative forcing is practically non-existent while momentum forcing continues to operate without reduction in strength (Fig. 11). Accordingly the vertical velocity will mainly be influenced by the structure of this term. This term forces towards a different wave number in

the deep atmosphere and thus induces the partially reversed vertical velocity pattern. Similar circulations are generated in all three forcing cases, but the stratospheric/tropospheric differences are most pronounced in Run 3, where we have intensified the relative importance of radiative and momentum forcing in the stratosphere and troposphere by prescribing an inverse development of the forcing timescales. In general the influence of the chosen timescale profiles seems to be minor compared to the importance of variations in tempera-

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Fig. 11. Vertical velocity (solid line) in the stratosphere and troposphere. The curve has been obtained by averaging over the vertical domain in question. Also plotted are the latitudinal profiles of the temperature (dashed) and momentum forcing (dotted) terms. (a) Run 2. (b) Run 3.

Fig. 12. Explicit eddy momentum flux in the model. (a) Zonally and vertically averaged EMF (solid) and EMF forcing profiles (dashed). (b) Zonally averaged EMF forcing (in m/s2 ). Solid lines denote positive forcing, dashed lines denote negative forcing.

ture and thermal wind input profiles. This was also confirmed in a series of further runs (not shown here) investigating the influence of different absolute strengths and slopes of τrad and τmom . The generation of the particular pattern of circulation cells is thus a solid feature in the model, appearing over a range of forcing variations. While the direct momentum forcing in the model can be seen as a parametrization of forcing through EMF, we have also explicitly diagnosed the EMF associated with the large and small scale eddies evolving in the simulation. Following the definition used in previous studies of EMF on Jupiter and Saturn (e.g., Salyk et al., 2006), we have computed the horizontal velocity deviations v  and

u  from the zonally averaged flow profiles and defined EMF = v  u  . The corresponding momentum forcing term is given by F eddy = −

∂(u  v  ) . ∂y

(24)

The time and vertically averaged profile of both the EMF and the momentum forcing, as given by Eq. (24), are shown in Fig. 12a. As can be seen in the diagram the explicit momentum forcing is very variable with latitude but features a distinct maximum around 22◦ S, the location of the Great Red Spot on Jupiter and that of several large anticyclones in our simulation. A secondary maximum exists around 32◦ S. The high spatial variability on a

Jet scale meridional circulation cells on Jupiter

very small scale is also apparent in the vertical cross-section of the forcing (Fig. 12b), although the general trends are clearly discernible at all heights. However, in comparison to the two other forcing fields illustrated in Fig. 9, the explicit forcing by eddies in the simulation is of negligible magnitude. It can thus be assumed that the circulations shown in Fig. 10 can be virtually entirely attributed to direct forcing. 4.2. Diagnostics from the forced semi-geostrophic theory In order to test the validity of Eq. (20), we have used this equation to re-calculate the circulations consistent with the zonally and time-averaged radiative and momentum forcing fields modeled by OPUS (displayed in Fig. 9). Specifying these two fields for the F mom and F rad terms respectively, Jacobi’s method was employed to discretize and numerically solve Eq. (20). The boundary conditions prescribed were the same as the ones used by OPUS; namely noslip conditions at the sides and open boundaries at the top and bottom. Comparison of the predicted fields to the actually modeled ones provides an estimate for how well the simplified theory in Section 2 approximates the full system of hydrodynamic primitive equations. In order to solve Eq. (20) values for the Brunt-Väisällä frequency N 2 need to be estimated. For simplicity we used a step-function profile with a constant value of N 2 = 5.2 × 10−5 s−2 above 0.5 bar and N 2 = 0 below. The upper atmosphere frequency value corresponds to the average buoyancy frequency in the model in this layer. Specification of a very small but finite value in the deep atmosphere did not alter the obtained results. While the chosen profile is a simplification of the actual buoyancy frequency that developed in the model, it retained the defining characteristic of the expected profile (Bosak and Ingersoll, 2002) and allowed convergence of the numerical scheme. The Coriolis parameter f was determined from f = 2Ω sin φ , where φ is the planetographic latitude. Fig. 13 shows the computed circulations for Runs 2 and 3. In a way similar to the discussion of the analytic solutions in Section 2 we are now able to consider the effects of the radiative and momentum forcing separately. The solutions include the four stratospheric circulation cells modeled in the upper stratosphere (Fig. 10). It now becomes immediately apparent that these are induced by the stratospheric radiative forcing. A slight southward tilt, as seen in the model, is already induced by the radiative forcing. In the purely radiatively forced case the stratospheric jet-scale cells will extend throughout the atmosphere, which is in correspondence with both the analytic cases discussed in Section 2 (Figs. 2a and 3a), as well as with the simulations presented by Lian and Showman (2008). Zonal momentum forcing in itself generates a large anti-clockwise cell in the stratosphere and the three clockwise cells at shifted positions in the deep atmosphere. The deep atmosphere cells thus show the wave number 5 variations expected from the model. If both momentum forcing and radiative forcing operate the momentum induced circulation distorts the radiatively forced jet scale cells in the troposphere. The additional clockwise cell at the Northern boundary pushes the first radiatively induced positive cell southwards, while the circulations from the large clockwise cell are continued in the second deep cell. Accordingly the predicted circulations contain all the defining features of the modeled ones, although the extent of the second anti-clockwise circulation cell is overestimated, while the one of the first is underestimated. This, as well as the failure of the two southward cells to join up, might be due to the fact that the predictions do not take into account the effects of sub-gridscale mixing and diffusion. Inclusion of the explicit eddy momentum forcing field illustrated in Fig. 12 does not alter the results, due to the extremely small magnitude of

559

this term. On Jupiter, where vortices are significantly more persistent than in our model, EMF forcing has been found to provide a sizeable contribution to the energy flux and might well constitute one of the major driving mechanisms for the zonal jets (e.g., Salyk et al., 2006). Furthermore, the fields presented in Section 4.1 are imperfect steady state solutions, while the diagnostic attributions assume a truly equilibrated flow. Corresponding to what we observe for the OPUS results, the obtained configurations are similar for all three runs. It is noteworthy that the damping of the momentum forcing field in Run 3 (Fig. 13b) does not significantly alter the final circulations generated by this component in comparison to the ones formed under vertically uniform forcing (Fig. 13a). 5. Discussion and conclusions We have investigated the formation of jet scale meridional circulation cells on Jupiter in response to radiative and zonal momentum forcing. In the framework of forced semi-geostrophic theory, we have shown that the meridional streamfunction is approximately described by an elliptic equation with a source dependent on the sum of the latitudinal derivative of the radiative forcing and the vertical derivative of the zonal momentum forcing. On solving this equation for idealized forcing terms that follow a simple sinusoidal relationship in latitude in a configuration that favors a predominance of radiative forcing in the upper stratosphere and of momentum forcing in the lower stratosphere, we obtained the formation of two sets of jet scale circulation cells, one in the stratosphere and another one in the deep troposphere. If perfect alignment of heating and cooling with the mechanically forced jovian belts and zones occurs, the cells will retain their sense of circulation throughout the atmosphere. A possible shift in the overturning circulation of the deep atmosphere may occur if the latitudinal alignment of radiative heating with the mechanically enforced belt and zones is broken. Our theoretical results thus support the assumption of jet scale meridional cells in Jupiter’s stratosphere and include the possibility of circulations with a different up- and downwelling distribution in the deep atmosphere, depending on the form of momentum forcing in the deep troposphere. We have conducted a series of numerical model simulations with the Jovian GCM OPUS, which were initiated with observational data from the Cassini CIRS temperature cross-section and Newtonian forcing of potential temperature and zonal momentum. The circulations modeled within these simulations confirmed the theoretically obtained configurations and also featured two separate stratospheric and tropospheric sets of jet scale circulation cells. Since previous numerical studies of Jupiter’s meridional circulations were either restricted to the stratosphere (Gierasch et al., 1986; Conrath et al., 1990; West et al., 1992; Moreno and Sedano, 1997) or did not include both radiative and mechanical adjustment processes (Showman et al., 2006; Lian and Showman, 2008), the second set of deep circulations has not been generated in previous numerical simulations. Changes in the direction of vertical transport have been enforced by confinement of active forcing to a finite layer and the application of drag on the zonal jets below and above this region (Williams, 2003; Showman et al., 2006). In accordance with previous numerical results, the stratospheric circulation cells showed the expected distribution of upwelling over zones and downwelling over belts. We did not obtain the planetary scale poleward transport in the upper stratosphere found by Conrath et al. (1990) as our zonal velocity forcing was applied throughout the domain. Instead the upper layer circulation cells were open to the overlying atmosphere. The latitudinal vertical velocity distribution in the stratosphere was predominately influenced by the potential temperature forcing in

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Fig. 13. Circulations diagnostically attributed by Eq. (20) for the modeled forcing fields. The plots show the streamline function Ψ M as defined by Eqs. (12) and (13) (in m2 /s). The circulations associated with the radiative forcing term, the momentum forcing term and the two of them combined are shown for (a) Run 2 and (b) Run 3. Dashed lines represent clockwise rotation and solid lines denote anti-clockwise circulations.

the model, which consisted of heating over zones and cooling over belts. Our numerical results thus provide strong support for the hypothesis that jovian stratospheric jet scale circulation cells can be induced by differential radiative heating acting in response to the tropospheric jets. The deep atmosphere set of circulation cells followed a wave number 5 pattern instead of k = 4 in the stratospheric circulations over the domain studied. Consequently the cells are generally shifted to the South in comparison to their upper counterparts. This leads to a partial reversal of the downward vertical velocity over the two belts and to a considerable reduction of the upward movement over the South Tropical Zone. As there is virtually no radiative forcing in the deep atmosphere, this difference in circulation has to be caused by a misalignment of the deep atmosphere momentum forcing with the stratospheric belt/zone heating and cooling. Our results thus suggest the possibility of atmospheric upwelling at the water cloud base level within belts as deduced from observational results (Banfield et al., 1998; Gierasch et al., 2000). In order to link the theoretical predictions and OPUS results we have numerically solved a simplified form of the characteristic elliptic equation (20) for the modeled radiative and momentum forcing fields. This procedure can be viewed as creation of a diagnostic attribution for the expected circulations in response to the given forcing. It also offers the possibility to examine the consequences of each forcing component separately. We could then show that the four stratospheric jet scale cells are induced by heating and cooling in the stratosphere, while the application of deep momentum forcing generates the three clockwise cells in the deep atmosphere. The diagnostics thus adequately represented the wave number 4/5 relationship observed in the model, although in the combined forcing case the influence of momentum forcing was weaker than in the GCM. This is likely due to the non-representation of sub-gridscale mixing and diffusion in the equation. Consequently the severity of the shift between the up-

and downwelling branches is not fully represented in the diagnostics. The current work will be followed by a study of vertical and latitudinal condensate distributions generated in response to the simulated meridional circulations. We will thereby employ a simple cloud scheme for OPUS modeling the four major jovian cloud species and thus investigate the extent to which the meridional circulations obtained here can account for the observed cloud formations. Further work will also be focused on including the effects of moist convection into both the theoretical and numerical analysis as well as on the investigation of the meridional circulations on Saturn in comparison to Jupiter. Appendix A. Solving the forced semi-geostrophic Sawyer–Eliassen equation The governing equation for the upper atmosphere in both Case A and Case B is given by

∇ 2 ΨM = −

gkF 0,rad α z e sin ky − F 0,mom β e −β z sin my . N θ00

(A.1)

Using the superposition principle Eq. (A.1) can be split into to equations to be solved independently

∇ 2 Ψ Mk = −

gkF 0,rad α z e sin ky , N θ00

∇ 2 Ψ Mm = − F 0,mom β e −β z sin my , where Ψ M = + are thus given by

Ψ Mk

Ψ Mk = − Ψ Mm = −

(A.2) (A.3)

Ψ Mm .

The resulting upper atmosphere solutions

gkF 0,rad

1

N θ00

α 2 − k2

e α (z−z0 ) sin ky ,

F 0,mom β −β(z−z0 ) e sin my , β 2 − m2

(A.4) (A.5)

Jet scale meridional circulation cells on Jupiter

and the full solution is

ΨM = −

Ψ MU =

gkF 0,rad

1

N θ00

α 2 − k2

F 0,mom β −β(z−z0 ) e α (z−z0 ) sin ky − e sin my .

β 2 − m2

N  

A n e −nz + B n enz C n sin(ny ) + D n cos(ny )

Ψ M0

=−

gkF 0,rad

1

N θ00

α 2 − k2

sin ky −

F 0,mom β

β 2 − m2

sin my ,

Ψ MA = −

qkF 0,rad ek(z0 −z)

θ00 N α

2

− k2

sin ky −

F 0,mom β em(z0 −z)

β 2 − m2

sin my .

(A.8)

In Case B the harmonic m does not have to conform to the Laplace equation but continues to follow Eq. (A.5), thus that we find

Ψ MB

=−

qkF 0,rad ek(z0 −z)

F 0,mom β −β z sin ky − 2 e sin my . α 2 − k2 β − m2

θ00 N

(A.9)

If additional boundary conditions are employed, terms from the superimposed homogeneous solution need to be invoked to match those. The solutions for Neumann (∂Ψ M /∂ z = 0) or Dirichlet (Ψ M = 0) at the top (z = ztop ) and bottom (z = 0) in Case A are given by

Ψ MU =

k

N θ00

α 2 − k2

+ Ψ ML

=



g F 0,rad

N θ00

−

α

α 2 − k2 k

F 0,mom β

β2



k



− m2

β m

e



(k+α )ztop

−1 

e (m−β)ztop + 1

(A.10)

ekz + e −kz ekz0 + e −kz0

emz + e −mz emz0 + e −mz0

sin(ky )

sin(my )

(A.11)

and

Ψ MU =

g F 0,rad N θ00

k

α

2





e (k+α )ztop −kz − e α z sin(ky )

− k2

 F 0,mom β  (m−β)ztop −mz e − e −β z sin(my ), + 2 2 β −m Ψ ML =

g F 0,rad N θ00

k

α

2

− k2



 ekz − e −kz sin(ky ) ekz0 − e −kz0  emz − e −mz − 1 mz sin(my ). e 0 − e −mz0

(A.12)

e (k+α )ztop − 1

F 0,mom β  (m−β)ztop + 2 e β − m2

In Case B the same scenarios lead to

m



e (k+α )ztop −kz − e α z sin(ky )

 emztop − e −β ztop  −mz e + emz emztop − e −mztop





k

α 2 − k2 k

N θ00

F 0,mom β β

β2

α

− m2



m

(A.14)



e (k+α )ztop − 1

ekz + e −kz ekz0 + e −kz0

sin(ky )

 emztop − e −β ztop  −mz e + emz emztop − e −mztop



(A.15)

β

and

Ψ MU = Ψ M = +

Ψ ML =

g F 0,rad N θ00

F 0,mom β

β2



− m2

k

α

2

− k2





e (k+α )ztop −kz − e α z sin(ky )

 e −β ztop − emztop  −mz emz + −mz e − emz top − emztop e

 − e −β z sin(my ),

(A.16)

 ekz − e −kz sin(ky ) N θ00 α 2 − k2 ekz0 − e −kz0   F 0,mom β mz e −β ztop − emztop  −mz e + −mz e − emz + 2 mztop 2 top β −m e −e  − e −β z sin(my ).

(A.17)

g F 0,rad

k



e (k+α )ztop − 1

In the case of an overlaying equator to pole circulation the solutions for Case B will be

Ψ MU = (A.12) +

g F 0,rad

π

N θ00 2(α 2 − ( π2 )2 )

  × e (π /2+α )ztop −π z/2 − e α z sin( y π /2), Ψ ML = (A.13) +

g F 0,rad

(A.18)

π

N θ00 2(α 2 − ( π2 )2 )

  e π z/2 − e −π z/2z sin(π y /2). × e (π /2+α )ztop − 1 π z /2 e 0 − e −π z0 /2

(A.19)

In Case A the same factors are added to Eq. (A.12) and Eq. (A.13) respectively.



e (k+α )ztop −kz − e α z sin(ky )

 F 0,mom β  (m−β)ztop −mz e − e −β z sin(my ), 2 2 β −m

g F 0,rad



m − emz − e −β z

(A.7)

which is equivalent to the upper boundary condition for the lower atmosphere solutions. In Case A the lower atmosphere circulations have to obey Laplace equation and will thus be of the general form given above, where the constants A n , B n , C n and D n have to be determined by boundary conditions. At the upper boundary (z = z0 ) Ψ M needs to equal Eq. (A.7). The resulting solution is then given by

− m2

g F 0,rad

+

n=0

can always be added and Eq. (21) is still fulfilled. This property can be used to meet imposed boundary conditions. However, as shown below such a superposition of a further solution can significantly alter the circulation. At z = z0 , which we take as the lower boundary of the upper atmosphere we have

F 0,mom β β

β2

α

α 2 − k2 k

N θ00

β

Ψ ML =







k

m − emz − e −β z ,

It should be noticed that this is only one possible solution, a general solution to the Laplace equation of the form

Ψ=

g F 0,rad

+

(A.6)

561

(A.13)

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