The outer areas of Jupiter and Saturn have multiple zonal winds,
reaching the high latitudes, that penetrate deep into the planets'
interiors, as suggested by gravity measurements. These characteristics
are finally replicable in numerical simulations by including both a
shallow stably stratified layer, below a convecting envelope, and
increasing electrical conductivity. A dipolar magnetic field, assumed to
be generated by a dynamo below our model, is imposed. We find that the
winds' depth into the stratified layer depends on the local product of
the squared magnetic field strength and electrical conductivity. The key
for the drop-off of the zonal winds is a meridional circulation which
perturbs the density structure in the stable layer. In the stable region
its dynamics is governed by a balance between Coriolis and
electromagnetic forces. Our models suggest that a stable layer extending
into weakly conducting regions could account for the observed deep zonal
wind structures.