Solutions of the Eliassen balance equation for inertially and/or symmetrically stable and unstable vortices

Abstract

AbstractTwo methods for solving the Eliassen equation for the corresponding balanced secondary circulation of a numerically simulated, high‐resolution tropical cyclone vortex are compared. In idealized calculations for a symmetrically stable vortex, both methods (successive overrelaxation [SOR] and multigrid) converge and the solutions are broadly similar. In more typical cases, where the vortex has regions of inertial or symmetric instability, it is necessary to coarsen the data from the numerical simulation to determine the balanced secondary circulation. A convergent solution can be obtained with the multigrid method for a finer grid spacing than with the SOR method. However, the multigrid method fails to converge when the vertical grid spacing is similar to that of the numerical simulation. Results using both methods confirm the inability of the balance formulation to capture the strong inflow and resulting tangential wind spin‐up in the frictional boundary layer during a period of rapid intensification. Typical tropical cyclone simulations show an inflow layer just beneath the upper‐level outflow layer, and the corresponding balanced secondary circulation may show such an inflow layer also. However, caution is called for in attributing this inflow layer to a balanced flow response driven by the distribution of diabatic heating and tangential momentum forcing. Our study suggests that this inflow layer is likely an artifact of the ad hoc regularization procedure that is necessary to keep the Eliassen equation globally elliptic in regions of inertial and/or symmetric instability.