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Numerical simulations of Jupiter’s zonal jets are presented, which are generated with realistic and hyper energetic source. The models are three dimensional and nonlinear, applied to a gas that is convective, stratified and compressible. Two solutions are presented, one for a shallow 0.6% envelope, the other one 5% deep. For the shallow model (SM), Jupiter’s small energy flux was applied with low kinematic viscosity. For the deep model (DM), the energy source and viscosity had to be much larger to obtain a solution with manageable computer time. Alternating zonal winds are generated of order 100 m/s, and the models reproduce the observed width of the prograde equatorial jet and adjacent retrograde jets at 20
° latitude. But the height variations of the zonal winds differ markedly. In SM the velocities vary radially with altitude, but in DM Taylor columns are formed. The dynamical properties of these divergent model results are discussed in light of the computed meridional wind velocities. With large planetary rotation rate Ω, the zonal winds are close to geostrophic, and a quantitative measure of that property is the meridional Rossby number, Ro
_{m}. In the meridional momentum balance, the ratio between inertial and Coriolis forces produces Ro
_{m} = V
^{2}/
ΩLU, U zonal, V meridional winds, L horizontal length scale. Our analysis shows that the meridional winds vary with the viscosity like
*ν*
^{1/2}. With much larger viscosity and meridional winds, the Rossby number for DM is much larger, Ro
_{m}(DM) >> Ro
_{m}(SM). Compared to the shallow model with zonal winds varying radially, the deeper and more viscous model with Taylor columns is much less geostrophic. The zonal winds of numerical models in the literature tend to be independent of the energy source, in agreement with the present results. With 10
^{4} times larger energy flux, the zonal winds for DM only increase by a factor of 3, and the answer is provided by the zonal momentum budget with meridional winds, VU/L =
ΩV, yielding U =
ΩL, independent of the source. The same relationship produces the zonal Rossby number, Roz = U/ΩL, of Order 1, which is commonly used as a dimensionless measure of the zonal wind velocities.

The alternating wind bands observed on Jupiter (e.g., Smith et al. [

In this paper, we present simulations of Jupiter’s alternating wind bands generated by convection. The numerical models are nonlinear and three dimensional, applied to a gas that is stratified and compressible. Two solutions are discussed, one for a shallow envelope 0.6% of planetary radius, the other one 5% deep more commensurate with reality. For the shallow model (SM), the small planetary energy flux from the interior was applied together with the corresponding low kinematic viscosity. For the deep model (DM), the applied energy and viscosity had to be much larger to achieve sufficient fast thermal relaxation with manageable computer time, conceptually similar to the deep convective models that have appeared in print.

The mean zonal winds generated with SM and DM differ markedly, varying radially with altitude and aligned along Taylor columns, respectively, and they are discussed in light of the computed meridional winds.

The numerical models discussed are based on a series of earlier studies (Chan and Sofia [

The numerical code employs the transformed spectral procedure with associated vector spherical harmonics and solves the time-dependent 3D nonlinear Navier Stokes equations (Chan et al. [

The shallow model (SM) extends into the convection region with ∆r = 0.6% depth of the planetary radius. Jupiter’s energy flux, F = 5.4 W/m^{2}, is applied at the bottom boundary, but the smaller Solar input is ignored. The kinematic viscosity, ν = 5.6 m^{2}/s, is employed, which produces the dimensionless Ekman number, E = ν/Ω∆r^{2} = 1.7 × 10^{−7}, with planetary rotation rate, Ω = 1.778 × 10^{−4} rad/s (9.8 hours). SM applies triangular truncation of spherical harmonics up to degree 20 (T20) with limited latitudinal resolution, and 68 radial grid levels for the 0.6% (430 km) shell. With low kinematic viscosity and small Ekman number, the model ran 1 year to reach thermal relaxation, and we present in the following the time average zonal mean variations of the computed zonal wind velocities.

The Taylor-Proudman theorem predicts that for a fluid that is geostrophic and incompressible, the zonal winds become aligned along Taylor columns (TC), and this is the prevailing picture of convective models of the Jupiter atmosphere (e.g., Christensen [

The zonal winds of SM are not aligned along the rotation axis but vary radially with altitude, and the question is whether this property will survive in deeper models with realistic planetary parameters and sufficient low viscosity. Short of the results from such a computationally demanding study, it is instructive to examine a simulation from a numerical model, stratified and compressible, which is much deeper but employs a much larger energy source and viscosity.

Considering Ohmic dissipation associated with Jupiter's magnetic field and measured conductivity, Liu et al. [

Following CAOH, a deep model (DM) was constructed with relative depth Δr = 5%, which was presented by Chan and Mayr [^{4} W/m^{2}, a factor of 1.4 × 10^{4} larger than that of Jupiter. For the kinematic viscosity the value ν = 2.3 × 10^{5} m^{2}/s was chosen, which produces the Ekman number E = ν/ΩΔr^{2} = 1.0 × 10^{−4}. With this large viscosity, the model ran 2 months to produce the numerical results.

Analogous to ^{o} wide in agreement with Voyager observations. But unlike the latitudinal variations of the computed zonal winds, on Jupiter the equatorial jet dominates.

In contrast to SM, the zonal winds from DM clearly show the pattern of Taylor columns. Illustrated in

For model simulations of a convective Jovian atmosphere that is both stratified and compressible, it is remarkable that the resulting alternating wind bands are formed with such different altitude patterns. In the shallow model the zonal velocities vary radially, but in the deeper model the variations are aligned along the rotation axis to form Taylor columns. Apart from the different vertical domains, the applied energy source and related viscosity must come into play.

The Taylor-Proudman theorem applies if the zonal winds are in geostrophic balance. For the shallow model (SH), geostrophy was explicitly demonstrated by comparing the meridional pressure gradient with the Coriolis force (Chan and Mayr [_{m}, which is defined as the ratio between inertial and Coriolis forces. Among the nonlinear inertial accelerations in the meridional momentum equation that describes the mean zonal wind, U, the term V∂V/∂θ is the largest, θ latitude and V mean horizontal meridional wind. Compared with the Coriolis force term ΩU, the meridional Rossby number then can be estimated, Ro_{m} = V^{2}/LΩU, with the characteristic horizontal length scale, L = λ/2π (λ, horizontal wavelength) that is related to the planetary radius, r.

at the top of the domain (black versus green/red), where the ambient densities are much smaller (identical in both models). Apart from the contrasting wind patterns, the meridional winds feature large differences in magnitude. For SM at the top of the domain, the maximum wind velocities are very small less than 0.10 m/s, in contrast to DM with velocities close to 30 m/s.

Given the wind velocities, the corresponding model parameters are listed in _{m} << 1, demonstrating that both model results are approximately in geostrophic balance. But Ro_{m} is much larger for DM with Taylor columns.

Our model results reveal an intriguing relationship between the viscosity and meridional winds. As shown in ^{2}/νΩ differ only by a factor of two, which is remarkable considering that the input parameters for the energy source and viscosity differ by orders of magnitude. The chosen viscosity apparently determines the magnitude of the meridional wind, V varying with (νΩ)^{1/2}.

Another intriguing property of the numerical results is the invariance of the zonal velocities in relation to the energy source. As shown in ^{4} larger. This trend is observed in planetary atmospheres. And the Jupiter models in the literature, with energies far exceeding the planetary value, all feature zonal wind velocities comparable to those observed.

Mayr et al. [^{2}, with K the eddy viscosity and L the horizontal scale of the circulation. Applying mixing length theory, K = VL, one obtains U = ΩL, which produces for the Jovian circulation zonal winds of order 100 m/s.

Essentially, the solution is provided by the nonlinear zonal momentum budget. For the zonal-mean circulation, the inertial force, V∂U/∂λ, dominates, and the balance with the Coriolis force yields, VU/L = ΩV, to produce U = ΩL, independent of the energy source. The same relationship produces the zonal Rossby number, Ro_{z} = U/ΩL, of order 1, which is commonly used as dimensionless measure of the zonal wind velocities.

Δr (m) | F (W/m^{2}) | ν (m^{2}/s) | U (m/s) | V (m/s) | E = ν/ΩΔr^{2} | Ro_{m} = V^{2}/rΩU | V^{2}/νΩ | |
---|---|---|---|---|---|---|---|---|

Shallow SM | 4.3 × 10^{5} | 5.4 | 5.6 | 70 | 0.1 | 1.7 × 10^{−7} | 1.1 × 10^{−8} | 10.0 |

Deep DM | 3.6 × 10^{6} | 7.5 × 10^{4} | 2.3 × 10^{5} | 200 | 30 | 1.0 × 10^{−4} | 3.4 × 10^{−4} | 22.0 |

In geostrophic balance, the zonal winds are produced by latitudinal temperature/pressure variations. But the source that produces the temperature variations is also generating the meridional circulation that redistributes or dissipates the kinetic energy. The zonal velocities thus tend to be independent of the energy source.

Numerical simulations of Jupiter’s zonal jets are discussed, which are generated with models that are three dimensional and fully nonlinear, applied to a gas that is convective, stratified and compressible. Solutions are presented for shallow and deep atmospheric envelopes, generated with realistic and hyper-energetic source. In the shallow model (SM) with realistic energy source, the zonal winds vary radially with altitude, in contrast to the energetic deep model (DM) where the winds are aligned along the rotation axis to form Taylor columns (TC). In agreement with observations, both models produce prograde equatorial jets of order 100 m/s. Both models also reproduce the observed width of the equatorial jet with adjacent retrograde jets at 20˚ latitude—a natural outcome for SM, but determined by the chosen 5% depth of DM with TC.

The dynamical properties of these divergent model results are discussed in light of the meridional winds, which are small in magnitude compared with the zonal winds. But unlike the rotational zonal winds, the meridional winds have divergence, and thus are involved with energy and momentum transport, which is of central importance for understanding the zonal mean circulation.

The Rossby number, Ro_{m}, for the meridional momentum balance is the quantitative measure of geostrophy, and it is a quadratic function of the meridional winds. For DM with large viscosity and TC, the meridional winds and Ro_{m} are orders of magnitude larger compared to SM. DM is much less geostrophic. Ranking geostrophy cannot explain the difference between SM and DM. For Taylor columns to form, Taylor-Proudman requires that the gas is also incompressible, in addition to geostrophic. But both models treat the atmosphere as compressible. In models like DM with large viscosity, the enhanced energy transport by the meridional winds has the capacity to reduce the vertical variations in the latitudinal temperature distribution to produce a barotropic environment that favors the formation of TC. The question is whether deeper models, with realistic energy flux and low viscosity, will produce zonal winds that vary radially with altitude like SM.

The numerical results presented highlight an important property of planetary atmospheres, the invariance of the zonal winds in relation to the energy source. With 10^{4} times larger source, the velocities of DM increase only by a factor of 3. And the Jupiter models in the literature with energies far exceeding the planetary value all feature zonal winds comparable to those observed. Following up on an earlier paper (Mayr et al. [

Funded by the Science and Technology Development Fund, Macau SAR (File No. 0045/2018AFJ). The reviewer’s comments contributed significantly to improve the presentation of the paper. This work was supported by the State Key Laboratory for Lunar and Planetary Sciences, Macau University of Science and Technology.

The authors declare no conflicts of interest regarding the publication of this paper.

Mayr, H.G. and Chan, K.L. (2019) Convective Models of Jupiter’s Zonal Jets with Realistic and Hyper-Energetic Excitation Source. International Journal of Astronomy and Astrophysics, 9, 292-301. https://doi.org/10.4236/ijaa.2019.93021