Agradient velocity and balanced vortical motion by Georgi

A new approach to consider balance dynamics for modelling slow vortical motion is suggested within the rotating shallow water primitive equations. Instead of a traditional approach using divergence and ageostrophic vorticity as fast variables, the velocity is exactly expressed as a sum of the gradient part, described by a Bernoulli function, and the rest, an agradient part, proportional to the velocity tendency, which is used as the fast variables slaved to potential vorticity. The source of the flow evolution is expressed in a form of a Jacobian operator on the Bernoulli function and potential vorticity that allows for direct estimation of the slow evolution rate. Momentum equations for the agradient velocity are derived for accurate representation of both slow vortical motion and fast inertia-gravity waves. This approach allows for the construction of balance conditions for vortical dynamics and a potential vorticity inversion scheme even for moderate Rossby and Froude numbers, assuming that the angle between the isolines of Bernoulli function and potential vorticity is small. This approach is found to be limited by the ellipticity condition for balanced tendencies which incorporates both known criteria for formal stability: the gradient velocity modified by the local Rossby wave speed should be subcritical. A numerical example of a symmetric dipole evolution on the f -plane is considered.