A Simplified Radiative-Dynamical Model for the Static Stability of Rotating Atmospheres

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  • 1 Center for Earth and Planetary Physics, Harvard University, Cambridge, Mass. 02138
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Abstract

In order to obtain estimates of the static stability in rotating atmospheres without performing numerical integrations of the equations of motion, a simple model is developed in which the radiative flux of heat is assumed to be balanced by the fluxes of sensible beat and potential energy due to large-scale eddies. The radiative flux divergence is modeled by a linearization about the radiative equilibrium state and the dynamical fluxes are modeled by calculating correlations from stability theory and by assuming that the amplitudes are limited by nonlinear effects. From the energy equation a single algebraic equation is derived for the mean equilibrium value of the Richardson number, Ri, in the troposphere. The radiative equilibrium state is assumed to be known. Once the solution for RI is found, the mean vertical and meridional gradients of potential temperature, 〈∂θ/∂z〉 and 〈∂θ/∂y〉, and the main properties of the mean zonal wind and eddies can be easily calculated. Even though many important fluxes are left out of the model, it is capable of giving good qualitative results because of the strong feedback in the dynamical fluxes.

When applied to the earth, the model yields the values 〈∂θ/∂z〉≈+2K km−1, 〈∂θ/∂y〉≈−0.4K (100 km)−1, Ri≈30. These values are much more realistic than can he obtained from the traditional assumption of radiative-convective equilibrium, although the static stability is still about one-half the observed value because of neglected fluxes. When applied to Mars the model yields the values 〈∂θ/∂z〉≈+2K km−1, 〈∂θ/∂y〉≈−1.2K (100 km)−1, Ri≈10. which are in good agreement with the Mariner observations and with the numerical results of Leovy and Mintz. The destabilization of the Martian atmosphere compared to the earth's (i.e., the smaller value of Ri) is due to the much shorter radiative relaxation time on Mars, which makes the radiative fluxes more efficient at destabilizing the atmosphere. When applied to Jupiter the model yields the values 〈∂θ/∂z〉≈10−4K km−1, 〈∂θ/∂y〉≈−2K (10,000 km)−1. The destabilization of Jupiter's atmosphere compared to Mars’ and the earth's is caused by the small horizontal temperature gradients, which in turn are due to both the large scale of Jupiter and the presence of an internal heat source. These small gradients lead to relatively weak large-scale motions, so that the dynamical fluxes are less efficient at stabilizing the atmosphere. The value of Ri found for Jupiter is subject to large error because of its sensitivity to the values of the external parameters. It may lie anywhere in the range O<Ri<20. Consequently, Jupiter's dynamical regime cannot be specified with certainty, but the results do show that Jupiter's mean temperature structure will in any case be very near radiative-convective equilibrium.

The model is also used to study how an atmosphere will adjust to deviations from equilibrium. In the case of the earth and Mars the deviations from equilibrium trace out damped oscillations, while in the case of Jupiter they are simply damped. The c-folding time for the damping is 34 days for the earth, 3 days for Mars, and 16 years for Jupiter. The adjustment time in all three cases is primarily determined by the radiative relaxation time.

Abstract

In order to obtain estimates of the static stability in rotating atmospheres without performing numerical integrations of the equations of motion, a simple model is developed in which the radiative flux of heat is assumed to be balanced by the fluxes of sensible beat and potential energy due to large-scale eddies. The radiative flux divergence is modeled by a linearization about the radiative equilibrium state and the dynamical fluxes are modeled by calculating correlations from stability theory and by assuming that the amplitudes are limited by nonlinear effects. From the energy equation a single algebraic equation is derived for the mean equilibrium value of the Richardson number, Ri, in the troposphere. The radiative equilibrium state is assumed to be known. Once the solution for RI is found, the mean vertical and meridional gradients of potential temperature, 〈∂θ/∂z〉 and 〈∂θ/∂y〉, and the main properties of the mean zonal wind and eddies can be easily calculated. Even though many important fluxes are left out of the model, it is capable of giving good qualitative results because of the strong feedback in the dynamical fluxes.

When applied to the earth, the model yields the values 〈∂θ/∂z〉≈+2K km−1, 〈∂θ/∂y〉≈−0.4K (100 km)−1, Ri≈30. These values are much more realistic than can he obtained from the traditional assumption of radiative-convective equilibrium, although the static stability is still about one-half the observed value because of neglected fluxes. When applied to Mars the model yields the values 〈∂θ/∂z〉≈+2K km−1, 〈∂θ/∂y〉≈−1.2K (100 km)−1, Ri≈10. which are in good agreement with the Mariner observations and with the numerical results of Leovy and Mintz. The destabilization of the Martian atmosphere compared to the earth's (i.e., the smaller value of Ri) is due to the much shorter radiative relaxation time on Mars, which makes the radiative fluxes more efficient at destabilizing the atmosphere. When applied to Jupiter the model yields the values 〈∂θ/∂z〉≈10−4K km−1, 〈∂θ/∂y〉≈−2K (10,000 km)−1. The destabilization of Jupiter's atmosphere compared to Mars’ and the earth's is caused by the small horizontal temperature gradients, which in turn are due to both the large scale of Jupiter and the presence of an internal heat source. These small gradients lead to relatively weak large-scale motions, so that the dynamical fluxes are less efficient at stabilizing the atmosphere. The value of Ri found for Jupiter is subject to large error because of its sensitivity to the values of the external parameters. It may lie anywhere in the range O<Ri<20. Consequently, Jupiter's dynamical regime cannot be specified with certainty, but the results do show that Jupiter's mean temperature structure will in any case be very near radiative-convective equilibrium.

The model is also used to study how an atmosphere will adjust to deviations from equilibrium. In the case of the earth and Mars the deviations from equilibrium trace out damped oscillations, while in the case of Jupiter they are simply damped. The c-folding time for the damping is 34 days for the earth, 3 days for Mars, and 16 years for Jupiter. The adjustment time in all three cases is primarily determined by the radiative relaxation time.

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