Interaction equations for Rossby waves derived from Hamilton’s principle in the Eulerian description

Abstract

Variational principles provide a useful tool for deriving the evolution equations of a resonant triad of hydrodynamic waves, but they are often considered in the Lagrangian description, as Eulerian variational principles require the introduction of extra variables, related to the particle displacements. To show that a variational approach remains possible with these extra variables, the derivation of interaction equations is presented in detail for Rossby waves on a β -plane, using a variational principle based on the Clebsch representation of the velocity field.