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

Serhii D. Koval, Alex Bihlo, Roman O. Popovych

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.

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.

Paper Structure

This paper contains 43 sections, 19 theorems, 224 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Multi-layer quasi-geostrophic problem
  3. Mathematical model
  4. Vector representation of the model
  5. Properties of the vertical coupling matrix
  6. Barotropic and baroclinic modes
  7. Numerical example
  8. Conservation laws and Hamiltonian structure
  9. Lie invariance algebra and Hamiltonian symmetries
  10. Point symmetry pseudogroup
  11. Equivalence groupoid, pseudogroup and algebra
  12. Classification of subalgebras
  13. Codimension-one Lie reductions
  14. Subalgebra family 1.1
  15. Reduced system
  16. ...and 28 more sections

Key Result

Lemma 1

If the physical constraints of positivity on the essential components $f_{i+1,i}$ and $f_{i,i+1}$, $i=1,\dots,m-1$, of the matrix $\mathsf F$ is imposed, the coefficients of its characteristic polynomial $P(\lambda):=\det(\lambda\mathsf E-\mathsf F)$ are positive, except the zero-degree one, which i

Figures (9)

  • Figure 1: Three-layer model
  • Figure 2: Baroclinic simple plane waves \ref{['eq:Red11PlaneWaves']}.
  • Figure 3: Coherent baroclinic eddy \ref{['eq:Red11Eddy']}.
  • Figure 4: Coherent baroclinic heton \ref{['eq:Red11Heton']}.
  • Figure 5: Superposition of two baroclinic Rossby waves \ref{['eq:Red11SuperpositionRossby']}.
  • ...and 4 more figures

Theorems & Definitions (46)

  • Lemma 1
  • proof
  • Corollary 2
  • proof
  • Remark 3
  • Remark 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 36 more