A new method for solving the nonlinear balance equation

The nonlinear balance equation (NBE) can be a very useful dynamic constraint for mesoscale data assimilation because it links the wind or streamfunction field with the mass field more accurately than the geostrophic balance on the mesoscale. However, retrieving the streamfunction from the mass field constrained by the NBE remains to be very challenging and largely unsolved since the early attempts traced back to 1950's. It is well known mathematically that the NBE is a special case of the Monge-Amptre's differential equation. If the geostrophic vorticity is larger than -f/2 for a constant Coriolis parameter f, then the NBE is of the elliptic type with two and only two solutions for given streamfunction boundary values. If the geostrophic vorticity is smaller than -f/2 in a local area, then the NBE becomes locally hyperbolic and this complicates the solution. Because of this complication, there has not been a method of solution for the NBE with the geostrophic vorticity decreased below or even close to -f/2 in a local area. In this study, a new method is developed to attack this problem by rearranging the NBE into a multi-step iterative form based on the leading order balance in the semi-balance model [see (2.13) of Xu 1994, J. Atmos. Sci., 51, 953–970]. The method can solve the NBE efficiently even when the vorticity of the balanced flow becomes smaller than -f/2 (so the geostrophic vorticity is even more negative), but the absolute vorticity should not become negative as required by the inertial stability of balanced flow. The effectiveness and performance of the method will be demonstrated by idealized and real-data examples.