  During the past few years, both the Cassini mission at Saturn and the Juno
mission at Jupiter, provided measurements with unprecedented accuracy of the
gravity and magnetic fields of the two gas giants. Using the gravity
measurements, it was found that the strong zonal flows observed at the
cloud-level of the gas giants are likely to extend thousands of kilometers deep
into the planetary interior. However, the gravity measurements alone, which are
by definition an integrative measure of mass, cannot constrain with high
certainty the exact vertical structure of the flow. Taking into account the
recent Cassini magnetic field measurements of Saturn, and past secular
variations of Jupiter's magnetic field, we obtain an additional physical
constraint on the vertical decay profile of the observed zonal flows on these
planets. Our combined gravity-magnetic analysis reveals that the cloud-level
winds on Saturn (Jupiter) extend with very little decay, i.e., barotropically,
down to a depth of around 7,000~km (2,000~km) and then decay rapidly in the
semiconducting region, so that within the next 1,000 km (600~km) their value
reduces to about 1\% of that at the cloud-level. These results indicate that
there is no significant mechanism acting to decay the flow in the outer neutral
region, and that the interaction with the magnetic field in the semiconducting
region might play a central role in the decay of the flows.
