  We present a fully nonlinear hydrodynamic 'shallow water' model of the solar
tachocline. The model consists of a global spherical shell of differentially
rotating fluid, which has a deformable top, thus allowing motions in radial
directions along with latitude and longitude directions. When the system is
perturbed, in the course of its nonlinear evolution it can generate unstable
low-frequency shallow-water shear modes from the differential rotation,
high-frequency gravity waves, and their interactions. Radiative and overshoot
tachoclines are characterized in this model by high and low effective gravity
values respectively. Building a semi-implicit spectral scheme containing very
low numerical diffusion, we perform nonlinear evolution of shallow-water modes.
Our first results show that, (i) high latitude jets or polar spin-up occurs due
to nonlinear evolution of unstable hydrodynamic shallow-water disturbances and
differential rotation, (ii) Reynolds stresses in the disturbances together with
changing shell-thickness and meridional flow are responsible for the evolution
of differential rotation, (iii) disturbance energy primarily remains
concentrated in the lowest longitudinal wavenumbers, (iv) an oscillation in
energy between perturbed and unperturbed states occurs due to evolution of
these modes in a nearly dissipation-free system, and (v) disturbances are
geostrophic, but occasional nonadjustment in geostrophic balance can occur,
particularly in the case of high effective gravity, leading to generation of
gravity waves. We also find that a linearly stable differential rotation
profile remains nonlinearly stable.
