Dual-Doppler wind analysis of convective storms using the vertical vorticity equation

The dual-Doppler analysis of the vertical velocity field in convective storms has long been fraught with difficulty. One of the main problems is that the lowest scan of the radars may be a kilometer or more above the surface of the earth, where the impermeability condition could safely be applied. Our work focusses on a new method of dual-Doppler wind analysis, with an emphasis on improving the retrieval of the vertical velocity field in the presence of substantial low-level data voids. The analysis proceeds in a three-dimensional variational (3DVAR) framework with the anelastic form of the vertical vorticity equation imposed along with traditional constraints of mass conservation and smoothness. The local derivative term in the vertical vorticity equation is estimated with the Taylor frozen turbulence hypothesis (space to time conversion). The method is general enough to include data from multiple radars but tests have thus far been restricted to two radars. Results are first presented with analytical data of a checkerboard grid of counter-rotating updrafts and downdrafts (a Beltrami flow solution of the Navier-Stokes equations). Several sources of error and limitations are considered with this analytical test case: random observational error, error in the assumed pattern-translation components used in the space to time conversion, evolution effects, and notably, rejection of all data beneath an altitude of 1.5 km. Comparisons are made between (i) a traditional analysis in which the impermeability condition is imposed but the vorticity constraint is not considered, (ii) a vorticity-equation based procedure in which the impermeability condition is not imposed, and (iii) a combined vorticity equation and impermeability condition approach. It is found that regardless of sector-volume scan rate (considered from 1 min to 5 min), the best results are obtained with the combined vorticity constraint and impermeability approach, and among these results the best results are obtained with faster scan rates. The latest results from real-data tests with the 8 May 2003 tornadic supercell in central Oklahoma will also be presented.