Tidal Torques are the Answer

Angular momentum conservation places severe restrictions on the possible steady-state differential rotation of any body. The only terms that survive the time-averaging of the zonal momentum equation (which expresses this conservation rule) are due to tidal torques, Lorentz (magnetic) forces, and friction. These terms are all much smaller than the canceling terms due to Coriolis forces and Reynolds stresses. So numerical models which do not solve the equations of motion exactly will inevitably give erroneous results. On the other hand, the time-average equations are straightforward to solve analytically. The results are that the balance between tidal torques and friction determines the differential rotation; and the tidal dissipation and friction together determine the diabatic heating which maintains the required baroclinicity (horizontal temperature gradients). In conducting regions, the dynamo equation and the time-average zonal momentum equation form a nonlinear, but dissipative, system which is mathematically equivalent to the famous Lorenz system. The behavior of the resulting αω dynamo can be quite various, but depends on only a few parameters (the strength of the α-effect [α], the strength of the differential rotation [ω], and the magnetic Prandtl number [ν/λ] being the most important). It is totally independent of the Coriolis force, which does not appear in the time-average zonal balance. Equilibrium is reached when there is a balance between the Lorentz force and friction with B²R² ~ ρλν(D - D_min), where the dynamo number D ~ αωR²/λ². The time and spatial dependence of the dynamo is a function of D as indicated in the Table. In particular, there is a critical value of D above which the steady-state solutions of the dynamo are unstable, resulting in possible chaotic behavior. The jets and thermal structure of the neutral atmosphere principally depend on the tidal forcing frequency, the (vertical) thermal structure of the troposphere, the viscosity, and the Prandtl number. These parameters can be determined by straightforward assimilation of the observed winds and temperatures into a nearly linear model based on the strengths of tidal modes. This simple approach explains all of the observed differential rotations and hydromagnetic dynamos in the solar system.t;