Non-linear internal gravity waves in a slightly stratified atmosphere

Abstract

Internal gravity waves in a stratified atmosphere of unbounded inviscid incompressible fluid are considered. A class of non-linear waves is found in the Boussinesq approximation, whereby the inertial variation of density is neglected but the buoyancy is not, by reduction of the equations of motion to ordinary differential equations. To find similar non-linear waves when the atmosphere is slightly stratified, i.e. when the inertial variation of density is small but not entirely negligible, the equations of motion are first expressed in Lagrangian variables and derived from a variational principle. The Lagrangian variables are transformed to systematize the Boussinesq approximation. Finally, the properties of the non-linear waves in a slightly stratified atmosphere are found by Whitham's method of averaging. In terms of the transformed Lagrangian variables, strong non-linearity affects the linearized solution only by adding to the pressure a term proportional to the square of the wave amplitude. It follows that the amplitude of the waves is inversely proportional to the density of the atmosphere, even where the amplitude is not small and the linear approximation becomes invalid.