THREE-DIMENSIONAL VARIATIONAL ANALYSIS OF DOPPLER WIND OBSERVATIONS
John F. Gamache
Frank D. Marks, Jr.
Wen-Chau Lee (NCAR)
To employ a variational analysis scheme to improve Doppler wind
analyses and to incorporate all data without numerical instability.
Most former analyses of three-dimensional wind fields from Doppler observation
have begun by assuming the vertical wind in the region of analysis was
zero. Then by vertically integrating horizontal divergence, a new vertical
wind was estimated. That vertical wind was then used to correct the horizontal
winds. This process was iterated to convergence. Unfortunately the analyses do
not converge when nearly vertical Doppler observations are included. Much of the
upper-level Doppler data in small hurricane eyewalls are nearly vertical. The
three-dimensional analysis developed recently allows us to include all Doppler
data and thus to improve our analyses of upper-level hurricane
Doppler observations from two or more independent directions are interpolated
to a three dimensional grid. A cost function is developed that places a high
weight upon the difference between the projection of the analysis upon the
Doppler radials and the Doppler radial velocities themselves, and also a high
weight upon the solution of the continuity equation. These become
virtually strong constraints. Much lower weights are given to minimizing
the difference between the Doppler radials of the analysis and the Doppler
radial velocities actually observed, and between the precipitation
terminal fallspeeds of the analysis and those estimated from radar
reflectivity. The difference between the vertical wind analyzed at
the surface and the value zero is also added to the cost function at a
fairly low weight. Finally, the Laplacian of the three wind components are
also added to the cost function with a weight much smaller than all the
above cost functions.
The variational scheme has been applied to airborne Doppler data obtained
both in and out of hurricanes. The application to two cases in TOGA-COARE
has resulted in many recent improvements to the code.
Analyses in Hurricane
Olivia of 25 September 1994 have allowed the detailed observation of
changes in the upper-level wind structure as the hurricane was being sheared
and was filling (increased minimum surface pressure). Before this new
scheme was developed the detailed analysis of the upper levels would be
impossible in regions near the flight track.
The analysis has also been applied to a case in TOGA-COARE where Doppler
observations were included that were both nearly vertical and nearly horizontal.
The continuity equation was integrated from the bottom upward, and yet
the vertical and horizontal winds near echo top appear quite reasonable.
A cross-section through this analysis is shown in the figure. There is very
good agreement between the reflectivity field and the Doppler wind field.
While satisfying the continuity equation to .000001/sec rms error, the analysis
also matches the Doppler observations to an rms difference under 2 m/s.
Further studies will be performed to test how the analysis works on prescribed
wind fields following known analytical functions.
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