It is also possible to fit the radial velocity measurements to simple (e.g., linear) models of the wind field. In the radar community, the most common variations on these techniques are the velocity azimuth display (VAD) and the volume velocity-processing (VVP) methods (i.e., Browning and Wexler 1968; Waldteufel and Corbin 1979; Koscielny et al. 1982). Since only radial velocity measurements are used, information concerning the vertical vorticity can not be obtained by these methods. A different approach to obtaining the Cartesian components of the wind are to estimate the velocity from the movement of the reflectivity fields. Such techniques are well suited to clear air returns and can produce estimates of the winds even when the Doppler velocities may not be reliable. Studies of this type include Zadawazki (1973), Rinehart (1979), Symthe and Zrnic/' (1983), Tuttle and Foote (1990) focused on statistical approach through correlating radar echoes at different times and assuming that the movement of the echoes was the approximate movement of the wind field. The movement of the reflectivity field can be obtained by finding the maximum in the cross-correlation of features in sequential radar scans or tracking echo centroids. The general impression that one obtains from this work and the other related studies is that there is favorable agreement when comparisons are made to independent observations but that the techniques tend to not to capture the fine-scale features of the flow. Recently Shapiro et al. (1995) applied reflectivity conservation, allowing for changes due to raindrop fallspeeds, with the condition of incompressibility and a temporal constraint of either velocity stationarity or Taylor's hypothesis to produce an over-determined system of equations that simplifies to solving a Poisson equation derived from a least squares formulation. The technique is diagnostic and compares well with observations in a clear boundary layer and with a microburst downdraft.
An alternate approach is to use some form of data assimilation where the model data is ingested into a numerical model such as the modern forms of data assimilation discussed in the previous chapter. Since it is generally accepted that nudging is not a valid approach at these small scales, efforts by Kapitza (1991), Sun et al. (1991), Sun and Crook (1994) and Sun (1994) have concentrated on using the adjoint method. Essentially a solution is found by use methods of optimal control theory to adjust the initial state variables to minimize the difference between the observations and the retrieved values in a series of radar scans. Both winds and thermodynamic fields are retrieved. In the work of Sun and Crook (1994) a Boussinesq system of equations for dry and shallow flow were applied to single Doppler radar measurements of a gust front which included the equations of motion, a thermodynamic equation, a continuity equation and an equation for the radar reflectivity. Experiments were conducted with and without the conservation equation for radar reflectivity and tests were undertaken into the sensitivity of the method to boundary conditions, number of radar volumes, time interval between volumes, amount of smoothing of the reflectivity field and other factors. The meteorological situation was a gust front moving across the domain. An example from one of their tests with both wind and reflectivity information input is shown in (Fig. 13.15) where the single Doppler velocity shows a value quite close to that of a dual-Doppler derived wind field:
The thermodynamic fields showed an agreement with mesonet data that was not as favorable as the wind fields. In this example, the radar volumes were 1.83 minutes. When the scan spacing was doubled, the thermodynamic fields also showed a clear gust front (Fig. 13.16) that compared well to observations:
It is worth noting that while favorable results were found with only two volumes, additional volumes improved the retrieval results. While these results are exciting, thus far the technique has not been extended to precipitating events. The technique is very computationally demanding but the convergence of the technique is hastened by using the dynamic retrieval to provide an initial guess if the horizontal velocities are available from either a multiple Doppler analysis or a single Doppler retrieval (Sun and Crook 1996). The relative advantages of the adjoint and dynamic retrieval techniques are discussed in Sun and Crook (1996) with the adjoint in general being less sensitive to observational errors but this result depends to some degree on the nature of the error.
It is also possible to create a simpler adjoint retrieval system (Qin
and Xu 1992; Laroche and Zawadsky 1993; Gal-Chen and Zhang 1993; Xu et
al 1994) through focusing on using an equation for reflectivity conservation
and/or combining it with some simple assumptions about behavior of the
radial velocity field. The solution techniques are thus based on simple
prognostic equations rather than the full set of governing equations used
for example in some of the previously discussed retrieval techniques. The
solution techniques are some variations of the adjoint methods but the
simple equations means that the wind can be retrieved at a lower computational
cost.
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Figure 2: Schematic of the temporal error that can result solely due to the advection of a steady state wind field across a radar volume.
Figure 3: Non-dimensional profiles of temperature perturbations within the boundary layer from the aircraft data of Caughey and Palmer (1979) as indicated with crosses, the simulations of Deardroff (1974) indicated with filled circles and the retrieval results of Gal-Chen and Kropfli (1984).
Figure 4. Non-dimensionalized pressure perturbations within the boundary layer retrieved from three different radar volumes (from Gal-Chen and Kropfli 1984).
Figure 5: Results of a thermodynamic retrieval in a microburst downdraft in eastern Colorado created from taking the thermodynamic perturbations near the microburst center at different analysis times. a) Contribution to negative buoyancy from water loading. b) Virtual temperature contributions. (from Parsons and Kropfli 1990).
Figure 6: Average cross-frontal characteristics obtained from thermodynamic retrieval for a surface cold front observed in central California by multiple Doppler radar. a) Buoyancy, b) virtual temperature and cloud water loading, and c) pressure deviations relative to a sounding taken just ahead of the front. (from Parsons et al. 1987).
Figure 7: Vertical cross-section of retrieved perturbation pressure through the mesocyclone containing the Del City-Edmond tornadic thunderstorm.
Figure 8: Vertical cross-sections of two-dimensionally averaged a) winds, b) induced buoyancy, and pressure perturbations derived by the techniques described by Roux (1993) applied to a cold front observed over western Europe.
Figure 9: A selected vertical cross-sections of a) winds, b) induced buoyancy, and pressure perturbations derived by the techniques described by Roux (1985) applied to a convective squall line observed over west Africa.
Figure 10: Schematic of an early microphysical retrieval simulation from Rutledge and Hobbs (1983). The seeder region where snow is generated and the feeder region where precipitation content is enhanced is shown along with the mean Doppler derived vertical velocity in the feeder region.
Figure 11: Comparison between retrieved and known fields of rain and cloud water in g kg-1 a) Retrieved values, b) known field, and c) difference field obtained from subtracting the known from the retrieved values. (from Ziegler 1985)
Figure 12: Comparison of cloud liquid water content (g kg-1) from the model retrieval (solid curve) with in situ values obtained by a sailplane. Retrieval results were interpolated to the aircraft locations. (from Ziegler et al. 1991).
Figure 13: Vertical cross-sections of the microphysical retrieval of Hauser and Amayenc (1986) in the west African squall line. Comparison can be made to Fig. 9 which contains the results of the dynamic retrieval by Roux (1985). a) Winds and contours of rain water (g kg-1), b) cloud water mixing ratio (g kg-1) and saturation deficit (negative values shaded and contoured in g kg-1), and c) deviations of potential temperature (o C) with respect to the environment.
Figure 14: Rainfall rate in the convective region of the west African squall line obtained from the raingauges, from applying a Z-R relationship to the radar reflectivity, and from the results of a thermodynamic-microphysical retrieval. From Hauser et al. (1988).
Figure 15: a) A horizontal cross-section at z = 0.7 km of the horizontal velocity predicted from a single Doppler retrieval using both radial velocity and reflectivity data. A gust front is clearly evident. b) Difference vectors between the retrieved results and a dual-Doppler synthesis. In this test the two Doppler volumes were 1.86 minutes apart. From Sun and Crook (1994).
Figure 16: A horizontal cross-section at a height of z
= 0.1 km showing retrieved a) perturbation potential temperature
(K) and b) pressure perturbation (divided by constant density to give units
of m2 s-2. The cool air associated
with the gust front is clearly evident. In this test the two volumes were
3.67 minutes apart. From Sun and Crook (1994).