Hydromagnetic constraints on deep zonal flows in the giant planets

The observed zonal flows of the giant planets will, if they penetrate below the visible atmosphere, interact significantly with the planetary magnetic field outside the metallized core. The appropriate measure of this interaction is the Chandrasekhar number Q = H^2 /4πρνα^2 λ (H = radial component of the magnetic field, ν = eddy viscosity, λ = magnetic diffusivity, α^-1 = length scale on which λ varies); at depths where Q ≳ 1, the velocity will be forced to oscillate on a small length scale or decay to zero. We estimate the conductivity due to semiconduction in H_2 (Jupiter, Saturn) and ionization in H_(2)0 (Uranus, Neptune) as a function of depth; the value λ ≈ 10^10 cm^2 s^-1 needed for Q = 1 is readily obtained well outside the metallic core (where A ≈ 10^2 cm^2 s^-1). These assertions are quantified by a simple model of the equatorial zonal jet in which the flow is assumed uniform on cylinders concentric with the spin axis, and viscous and magnetic torques on each cylinder are balanced. We solve this "Taylor constraint" simultaneously with the dynamo equation to obtain the velocity and magnetic field in the equatorial plane. With this model we reproduce the widely differing jet widths of Jupiter and Saturn (though not the flow at very high or low latitudes) using v = 2500 cm^2 s^-1, consistent with the requirement that viscous dissipation not exceed the specific luminosity. A model Uranian jet consistent with the limited Voyager data can also be constructed, with appropriately smaller v, but only if one assumes a two-layer interior. We tentatively predict a wide Neptunian jet. For Saturn (but not Jupiter or Uranus) the model has a large magnetic Reynolds number where Q = 1 and hence exhibits substantial axisymmetrization of the field in the equatorial plane. This effect may or may not persist at higher latitudes. The one-dimensional model presented is only a first step. Variation of the velocity and magnetic field parallel to the spin axis must be modeled in order to answer several important questions, including: (1) What is the behavior of flows at high latitudes, whose Taylor cylinders are interrupted by the region with Q > 1? (2) To what extent is differential rotation in the envelope responsible for the spinaxisymmetry of Saturn's magnetic field?