  The atmospheres of Jupiter and Saturn exhibit strong and stable zonal winds.
How deep the winds penetrate unabated into each planet is unknown. Our
investigation favors shallow winds. It consists of two parts.
  The first part makes use of an Ohmic constraint; Ohmic dissipation associated
with the planet's magnetic field cannot exceed the planet's net luminosity.
Application to Jupiter (J) and Saturn (S) shows that the observed zonal winds
cannot penetrate below a depth at which the electrical conductivity is about
six orders of magnitude smaller than its value at the molecular-metallic
transition. Measured values of the electrical conductivity of molecular
hydrogen yield radii of maximum penetration of 0.96R_J and 0.86R_S, with
uncertainties of a few percent of R. At these radii, the magnetic Reynolds
number based on the zonal wind velocity and the scale height of the magnetic
diffusivity is of order unity. These limits are insensitive to difficulties in
modeling turbulent convection. They permit complete penetration along cylinders
of the equatorial jets observed in the atmospheres of Jupiter and Saturn.
  The second part investigates how deep the observed zonal winds actually do
penetrate. Truncation of the winds in the planet's convective envelope would
involve breaking the Taylor-Proudman constraint on cylindrical flow. This would
require a suitable nonpotential acceleration which none of the obvious
candidates appears able to provide. Accelerations arising from entropy
gradients, magnetic stresses, and Reynolds stresses appear to be much too weak.
These considerations suggest that strong zonal winds are confined to shallow,
stably stratified layers, with equatorial jets being the possible exception.
