  Global magnetohydrodynamic (MHD) instabilities are investigated in a
computationally tractable two-dimensional model of the solar tachocline. The
model's differential rotation yields stability in the absence of a magnetic
field, but if a magnetic field is present, a joint instability is observed. We
analyze the nonlinear development of the instability via fully nonlinear direct
numerical simulation, the generalized quasilinear approximation (GQL), and
direct statistical simulation (DSS) based upon low-order expansion in
equal-time cumulants. As the magnetic diffusivity is decreased, the nonlinear
development of the instability becomes more complicated until eventually a set
of parameters are identified that produce a previously unidentified long-term
cycle in which energy is transformed from kinetic energy to magnetic energy and
back. We find that the periodic transitions, which mimic some aspects of solar
variability -- for example, the quasiperiodic seasonal exchange of energy
between toroidal field and waves or eddies -- are unable to be reproduced when
eddy-scattering processes are excluded from the model.
