Solutions of Quasi-Geostrophic turbulence in multi-layered configurations

Abstract

We consider quasi-geostrophic (QG) 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 differential systems of evolution equations. A two-layer QG model, in a simplied version, is dependent exclusively on the Rossby radius of deformation. However, the ƒ-plane QG 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 simplied model. Finally, we study a three-layer oceanography QG model of special interest, 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 physically relevant features, such as lateral friction and the analogue parameters of the ƒ-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.

Mathematics Subject Classication (2010): 35Q86, 35L65, 76M60.

Key words: Symmetries, quasi-geostrophic equations, conservation laws.