  Barotropic rotation and radiative equilibrium are mutually incompatible in
stars. The issue is often addressed by allowing for a meridional circulation,
but this is not devoid of theoretical complications. Models of rotation in the
Sun which maintain strict radiative equilibrium, making use of the observation
that the Sun is not in a state of barotropic rotation, have recently been
suggested. To investigate the dynamical behaviour of these solutions, we study
the local stability of stratified, weakly magnetized, differentially rotating
fluids to non-axisymmetric perturbations. Finite heat conductivity, kinematic
viscosity, and resistivity are present. The evolution of local embedded
perturbations is governed by a set of coupled, ordinary differential equations
with time-dependent coefficients. Two baroclinic models of rotation for the
upper radiative zone and tachocline are studied: (i) an interpolation based on
helioseismology data, (ii) a theoretical solution directly compatible with
radiative equilibrium. The growth of the local Goldreich-Schubert-Fricke
instability appears to be suppressed, largely because of the viscosity. An
extensive exploration of wavenumber space is carried out, with and without a
magnetic field. Although we easily find classical local instabilities when they
ought formally to be present, for the Sun the analysis reveals neither unstable
solutions, nor even solutions featuring a large transient growth. We have not
ruled out larger scale or nonlinear instabilities, nor have we rigorously
proven local stability. But rotational configurations in close agreement with
observations, generally thought to be vulnerable to the classic local
Goldreich-Schubert-Fricke instability, do appear to be locally stable under
rather general circumstances.
