On the Discrete Normal Modes of Quasigeostrophic Theory

TL;DR

This paper develops a complete set of normal modes for quasigeostrophic theory by incorporating boundary buoyancy gradients, revealing new wave solutions and enabling advanced analysis of geostrophic turbulence.

Contribution

It introduces a complete basis of quasigeostrophic modes considering boundary buoyancy gradients, addressing previous incompleteness and uncovering novel wave properties.

Findings

Identified stationary step-wave solutions of Rossby waves.

Constructed a complete basis capable of representing any quasigeostrophic state.

Derived a basis for vertical velocity modes and potential vorticity expansion.

Abstract

The discrete baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. Using this complete basis, we are able to obtain a differentiable series expansion to the potential vorticity of Bretherton (with Dirac delta contributions). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes topography into account.