Solutions of quasi-geostrophic turbulence in multi-layered configurations

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.