Dynamics of stratified regions in Saturn

Saturn has a rather peculiar magnetic field in that it is highly spin axisymmetric.
Evidence for the decay time of a magnetic field on the scale of Saturn would
suggest that a dynamo operates deep within its interior. As a consequence, the
observed field would be in violation of Cowling’s theorem. It is believed that a
stably stratified layer under the influence of a thermal shear is the reason for the
observed axisymmetric field. This stable layer is believed to be formed from helium
sedimentation deep within Saturn, with the thermal shear driven by pole-equator
temperature differences. The combined effects of shearing and the stable layer attenuate
the non-axisymmetric field components leaving only the axisymmetric field
at the surface.
Motivated by the influence of this stable stratification, we follow on from initial work
by Stevenson (1982b) by first considering the linear problem with variable conductivity
and looking at the consequences of increasing the parameter that controls the
strength of the thermal wind as mentioned in his paper.
In subsequent chapters the analysis concentrates on the nonlinear contributions by
including the momentum equation into our calculations. We present asymptotic
analysis of such a system and show that the geostrophic flow, found by satisfying
Taylor’s constraint, is singular for an inviscid interior solution in the limit of small
Rm, where Rm, the magnetic Reynolds number, controls the strength of the shearing
effect within the layer.
Numerical treatment of the system of equations for a viscous system are also considered.
The results of exploring the parameter space for Rm and Ha, the Hartmann
number, lead to further asymptotic analysis in which viscosity is considered. A
boundary layer solution is found, which is validated by the numerical solution.
The latter part of the thesis looks at the numerical solution with the inclusion of
a horizontal field, the motivation for which will become apparent in the analysis of
the inviscid regime.