Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem

The quasi-geostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Where possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.Comment: Extended version, 24 pages, 1 figur

Symmetry Analysis of Barotropic Potential Vorticity Equation

Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, $F$ and $\beta$, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth' angular rotation, respectively. In the present paper it is shown that in the case $F\ne 0$ there exists a well-defined point transformation to set $\beta = 0$. The classification of one- and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination $F\ne 0$ and $\beta = 0$. Based upon this classification, distinct classes of group-invariant solutions is obtained and extended to the case $\beta \ne 0$.Comment: 6 pages, release version, added reference for section

Variational integrator for the rotating shallow-water equations on the sphere

We develop a variational integrator for the shallow‐water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler–Poincaré reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler–Poincaré equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and excellent conservational properties of the discrete variational integrator

Selective decay for the rotating shallow-water equations with a structure-preserving discretization

Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical fidelity of such simulations also depends on properly preserving conserved quantities, such as energy, of the system. To address this apparent paradox, we develop a variational integrator for the shallow water equations that conserves energy, but dissipates potential enstrophy. Our approach follows the continuous selective decay framework [F. Gay-Balmaz and D. Holm. Selective decay by Casimir dissipation in inviscid fluids. Nonlinearity, 26(2):495, 2013], which enables dissipating an otherwise conserved quantity while conserving the total energy. We use this in combination with the variational discretization method [D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. Marsden and M. Desbrun. Structure-preserving discretization of incompressible fluids. Physica D: Nonlinear Phenomena, 240(6):443-458, 2011] to obtain a discrete selective decay framework. This is applied to the shallow water equations, both in the plane and on the sphere, to dissipate the potential enstrophy. The resulting scheme significantly improves the quality of the approximate solutions, enabling long-term integrations to be carried out