  We consider quasi-geostrophic (Q-G) models in two- and three-layers that are
useful in theoretical studies of planetary atmospheres and oceans. In these
models, the streamfunctions are given by (1+2) partial differen- tial systems
of evolution equations. A two-layer Q-G model, in a simpli- fied version, is
dependent exclusively on the Rossby radius of deformation. However, the f-plane
Q-G point vortex model contains factors such as the density, thickness of each
layer, the Coriolis parameter, and the constant of gravitational acceleration,
and this two-layered model admits a lesser number of Lie point symmetries, as
compared to the simplified model. Finally, we study a three-layer oceanography
Q-G model of special inter- est, which includes asymmetric wind curl forcing or
Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we
obtain solutions pertaining to the wind-driven double-gyre ocean flow for a
range of physi- cally relevant features, such as lateral friction and the
analogue parameters of the f-plane Q-G model. Zero-order invariants are used to
reduce the partial differential systems to ordinary differential systems. We
determine conservation laws for these Q-G systems via multiplier methods.
