Symmetry analysis and exact solutions of multi-layer quasi-geostrophic problem

Abstract

We carry out an extended symmetry analysis of the multi-layer quasi-geostrophic problem. This model is given by a system of an arbitrary number of coupled barotropic vorticity equations. Conservation laws and a Hamiltonian structure for the general case of the model are correctly described for the first time. Using original methods, we compute the maximal Lie invariance algebra and the complete point-symmetry pseudogroup of the model. After classifying one- and two-dimensional subalgebras of the Lie invariance algebra, we exhaustively study codimension-one, -two and -three Lie reductions. Notably, among invariant submodels of the original nonlinear model, we obtain uncoupled systems of well-known linear equations, including the Helmholtz, modified Helmholtz, Laplace, Klein-Gordon, Whittaker, Bessel and linearized Benjamin-Bona-Mahony equations. Integration of these systems significantly depends on spectral properties of the model's vertical coupling matrix, which we also revisit in detail. As a result, we construct wide families of exact solutions, including rediscovered representations of stationary and travelling baroclinic Rossby waves, coherent baroclinic eddies, hetons and localized dipolar vortices. We illustrate the physical relevance of obtained solutions using real-world geophysical data for a three-layer ocean model.