The vorticity method: Extension to mesoscale vertical velocity and validation for tropical storms

The vorticity method infers vertical velocity from vorticity variations in space and in time based on the vorticity equation. The viability of the large-scale vorticity method has been demonstrated in previous studies based on the quasi-geostrophic vorticity equation. However, the extension of the vorticity method to the mesoscale vorticity equation, including the vertical stretching and tilting terms, is hindered by inseparable variables when formulated in Cartesian coordinates. In this study, the separation of variables for the mesoscale vorticity method is achieved by formulating the mesoscale vorticity equation in Lagrangian coordinates in terms of the characteristic line composed of three nonlinear ordinary differential equations. A numerical method based on backward integration is also presented to solve the vertical velocity along the characteristic of the mesoscale vorticity equation. Observing-system simulation experiments were undertaken to quantify both the numerical accuracy of the vorticity method, and the sensitivities of the kinematic and vorticity methods to errors present in horizontal wind fields. With perfect horizontal winds, the vertical velocities derived from the kinematic and vorticity methods are identical to the degree of accuracy governed by the truncational errors. When error perturbations are included. the vertical motions derived from the vorticity method are less susceptible to errors in horizontal winds than the kinematic method. This conclusion is consistent with the analytical analysis and numerical results with the large-scale vorticity method shown in previous studies. The mesoscale vorticity method with the tilting term is shown to be more accurate than the large-scale vorticity method in deriving vertical velocities in tropical storms. Numerical experiments demonstrate that the mesoscale vorticity method can benefit from high-temporal- resolution data observed by Doppler radars to infer the vertical velocity in tropical storms, where the inner circulation is dominated by the rotational winds.