Archives of ALGA, Volume 9, 2012

Applications of Lie group analysis in Geophysical fluid dynamics

by Nail Ibragimov, Ranis Ibragimov

Series on Complexity, Nonlinearity and Chaos - Vol. 2, World Scientific Publishing Co Pte Ltd and Higher Education Press, 2011, ISBN:978-981-4340-46-5

This book introduces an effective method for seeking local and nonlocal conservation laws and exact solutions for nonlinear two-dimensional equations which provide a basic model in describing internal waves in the ocean. The model consists of non-hydrostatic equations of motion which uses the Boussinesq approximation and linear stratification. The Lie group analysis is used for constructing non-trivial conservation laws and group invariant solutions. It is shown that nonlinear equations in question have remarkable property to be self-adjoint. This property is crucial for constructing physically relevant conservation laws for nonlinear internal waves in the ocean. The comparison with the previous analytic studies and experimental observations confirrms that the anisotropic nature of the wave motion allows to associate some of the obtained invariant solutions with uni-directional internal wave beams propagating through the medium. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams are considered. The behavior of the invariant solutions near the critical latitude is investigated.