Consequences of regularizing the Sawyer–Eliassen equation in balance models for tropical cyclone behaviour

The consequences of regularizing the Sawyer–Eliassen equation to calculate the stream function for the axisymmetric secondary circulation of a tropical cyclone are explored. Regularization is an ad hoc procedure in which the coefficients of the equation are suitably modified to replace negative values of the discriminant by small positive values, thereby ensuring that the equation is globally elliptic. The consequences of the procedure may be understood in terms of the analogue behaviour of a stretched membrane subject to a particular force distribution. Several regularization procedures are assessed by comparing the azimuthally averaged radial flow from a three-dimensional numerical simulation of a tropical cyclone with that from an axisymmetric balance calculation of the Sawyer–Eliassen equation, forced by diabatic and frictional terms diagnosed from the simulation. The comparison shows that the largest challenge for regularization occurs in regions of inertial instability, especially when the diagnosed forcing overlaps with such regions. In the example shown, the diagnosed balanced flow is sensitive to the particular regularization procedure and none of the procedures examined give a flow that is structurally and quantitatively close to that obtained from the numerical solution in and near the region of regularization. The flow in regions of large vertical shear that are common in the lower part of the boundary layer is less sensitive to the regularization procedure, even though such regions are ones in which there is (frictional) forcing. Nevertheless, there are comparatively large differences between the low-level inflow in the azimuthally averaged numerical solution and the axisymmetric balance solution. These differences can be attributed to the intrinsic lack of balance in the boundary layer. This finding, together with the issues associated with regularization, is further confirmation that balance dynamics is unable to adequately capture the flow in the boundary layer, contrary to recent claims