Use of a Vertical Vorticity Equation in Variational Dual-Doppler Wind Analysis

Abstract

The utility of the anelastic vertical vorticity equation in a weak-constraint (least squares error) variational dual-Doppler wind analysis procedure is explored. The analysis winds are obtained by minimizing a cost function accounting for the discrepancies between observed and analyzed radial winds, errors in the mass conservation equation, errors in the anelastic vertical vorticity equation, and spatial smoothness constraints. By using Taylor’s frozen-turbulence hypothesis to shift analysis winds to observation points, discrepancies between radially projected analysis winds and radial wind observations can be calculated at the actual times and locations the data are acquired. The frozen-turbulence hypothesis is also used to evaluate the local derivative term in the vorticity equation. Tests of the analysis procedure are performed with analytical pseudo-observations of an array of translating and temporally decaying counterrotating updrafts and downdrafts generated from a Beltrami flow solution of the Navier–Stokes equations. The experiments explore the value added to the analysis by the vorticity equation constraint in the common scenario of substantial missing low-level data (radial wind observations at heights beneath 1.5 km are withheld from the analysis). Experiments focus on the sensitivity of the most sensitive analysis variable—the vertical velocity component—to values of the weighting coefficients, volume scan period, number of volume scans, and errors in the estimated frozen-turbulence pattern-translation components. Although the vorticity equation constraint is found to add value to many of these analyses, the analysis can become significantly degraded if estimates of the pattern-translation components are largely in error or if the frozen-turbulence hypothesis itself breaks down. However, tests also suggest that these negative impacts can be mitigated if data are available in a rapid-scan mode.

Vertical air motion retrievals in deep convective clouds using the ARM scanning radar network in Oklahoma during MC3E

“…We required that (a) both KVNX and CSAPR-I7 be available (except 11 May 2011) and initiate a volume scan 2 min or less apart, and (b) any complementary XSAPR input initiate from a volume scan 2 min or less from either KVNX or CSAPR-I7. These criteria are designed to mitigate the errors associated with unaccounted advection and evolution of the cloud system (e.g., Gal-Chen, 1982;Shapiro et al, 2009). …”

“…L is a length scale inserted to unify the dimensions and magnitude of J c with J o (e.g., Legler and Navon, 1991;Bousquet and Chong, 1998;Shapiro et al, 2009). For this study we set L = 250 m, which is the grid spacing.…”

Impact of Observation Operators on Low-Level Wind Speed Retrieved by Variational Multiple-Doppler Analysis

“…This can be achieved via the retrieval of three-dimensional wind velocity using dual-or multiple-Doppler wind analysis. This method was first proposed by Armijo (1969) and various multiple-Doppler wind analysis procedures have been developed and evaluated subsequently (e.g., Ray et al 1975Ray et al , 1978Ray et al , 1980Doviak et al 1976;Bousquet and Chong 1998;Shapiro and Mewes 1999;Gao et al 1999Gao et al , 2004Liou and Chang 2009;Shapiro et al 2009;Liou et al 2012;Potvin et al 2012a, b, c). Potvin et al (2012c) reported that the variational multiple-Doppler wind analysis proposed by Gao et al (1999) is more accurate than the multiple-Doppler wind analysis that explicitly integrates the continuity equation vertically, such as that by Ray et al (1980).…”

“…The variational multiple-Doppler wind analysis using radial velocity based on the PPI format (hereafter, PPI-VAR) has been used by Gao et al (1999Gao et al ( , 2004, Shapiro et al (2009), andPotvin et al (2012a, b, c). Gao et al (1999) mentioned that one advantage of PPI-VAR is that it is a single-step method that combines interpolation and analysis.…”

Characteristics of Wind Fields Derived from the Multiple-Doppler Synthesis and Continuity Adjustment Technique (MUSCAT)

“…To avoid the iterative procedure and its associated disadvantages, most recent formulations of multipleDoppler wind analysis have been based on variational principles (Gao et al 1999;Shapiro and Mewes 1999;Liou and Chang 2009;Shapiro et al 2009;Liou et al 2012). These methods determine the wind components in a volume, and include the anelastic mass continuity equation as a constraint.…”