Vorticity coordinates and balanced models

Balanced atmospheric models approximate the equations of motion by selectively introducing a dynamical balance between the wind and mass fields. Both the semigeostrophic and balanced vortex models also use a coordinate transformation which renders the unbalanced part of the flow implicit. In both cases the coordinates are vorticity coordinates, in the sense that the Jacobian of coordinate transformation is the corresponding absolute vorticity. While useful, these models cannot accurately represent asymmetric flows with large curvature vorticity.

To formulate a balanced model which does not suffer this limitation, we first consider the problem of defining vorticity coordinates. Of the many possible approaches available, requiring the coordinate displacement to be irrotational gives a simple formulation in which both the forward and inverse coordinate transformations can be computed by solving a Monge-Ampere equation. Efficient multigrid methods for this problem are described.

The transformation to vorticity coordinates replaces advection by the actual velocity with advection by a balanced velocity which is nondivergent in the transformed coordinate. Combining the conservation of potential vorticity with an appropriate balance approximation then yields a balanced model. Here we consider a general balance assumption which neglects only the local time derivatives in the momentum equations. The resulting model consists of a single predictive equation and an invertibility relation; the semigeostrophic and balanced vortex models are recovered in the appropriate limits.