Applications of Dynamic Retrieval  Dynamic Retrievals and Related  Doppler Radar Analysis

Dynamic Retrieval

Doppler studies provided a time history of the three-dimensional winds in a variety of atmospheric phenomena that resulted in a number of new findings, especially for early radar studies that took place while high resolution modeling of the atmosphere was in its infancy. For example, early studies showed for the first time that intense horizontal gradients can occur at the leading edge of cold fronts (e.g., Browning and Harrold 1970), led to new insight into the three-dimensional circulations within heated boundary layers (Frisch and Clifford 1974), and also provided knowledge of the processes that lead to the formation of tornadoes (e.g., Burgess et al. 1975). When these wind fields were combined with radar reflectivity data a better understanding of the precipitation processes in convective storms was also obtained (e.g., Browning and Harrold 1970; Kropfli and Miller 1976). However, despite the success of these and subsequent studies, the physical insight that could be obtained from Doppler radar analysis has always been limited by a lack of measurements of thermodynamic properties at the same high spatial and temporal resolutions. Gal-Chen (1978) and Hane and Scott (1978) proposed a solution to this problem through proposing the dynamic retrieval technique explored in this chapter.

In our description of the technique, we begin by assuming that a complete three-dimensional field of the velocity is available and is accurate enough to permit mathematical manipulations (like numerical integration, numerical differentiation, multiplication, etc.). Refining the definition of (u,v) and (w) to be the velocity vectors in the (x,y) and (z) directions respectively and defining $p^\prime$ as the pressure perturbation from a reference state or more conveniently an atmospheric sounding within the domain, we can follow Gal-Chen (1978) and rewrite Newton's second law for horizontal motions as

$$\rho_{0}(z)\left({\partial u\over \partial t}+u{\partial u\ov......r \partialz}+f_x\right)=-{\partial p^\prime \over \partial x}$$ (13.5)

  

$$\rho_{0}(z)\left({\partial v\over \partial t}+u{\partial v\ov......artial z}+f_y\right)=-{\partial p^\prime \over \partial y} \ .$$ (13.6)

 

In this formulation, f_x and f_y stand for the horizontal components of other type of forces that could be prescribed from knowledge of the wind, specifically the Coriolis force and turbulent friction. Thus the r.h.s. of (13.5) and (13.6) contain components of the pressure force which is the only unknown in the formulation (i.e., it cannot be calculated from the input Doppler velocities).

For the vertical equation of motion, there are two types of forces that are unknown. These forces are the vertical pressure gradient and the buoyancy. With this in mind the vertical equation of motion can be written as

$$\rho_{0}(z)\left ({\partial w\over \partial t}+u{\partial w\o......ht)=-{\partial p^\prime \over \partial z}-{\rho}^\prime g \ .$$ (13.7)

 

Here f_z stands for the vertical component of forces like turbulent friction and Coriolis force, which again could be prescribed from knowledge of the velocity field, ${\rho}^\prime$ is the density deviation from the reference density $\rho_{0} (z)$, therefore $-{\rho}^\prime g$ is the buoyancy force. For an air parcel without clouds or precipitation, the density fluctuation is approximately given by

$${\rho}^\prime =-\rho_{0}(z) {{\theta}^\prime_{v} \over \theta_{0}}$$ (13.8)

 

where $\theta_{0}$ is some reference potential temperature (usually at the ground) and ${\theta}^\prime_{v}$ is the fluctuation (from a given sounding) of the virtual potential temperature (i.e the effect of water vapor is included). In the presence of clouds and/or precipitation, the buoyancy is also a function of the amount of liquid water, ice, etc.; whose explicit form will be given later.

As already stated, the task at hand is to deduce the pressure and buoyancy from the given estimate of the velocities. To accomplish this task note that the horizontal momentum equation (13.5) and (13.6) can be rewritten as

$${\partial p^\prime \over \partial x}=F\ ; \ {\partial p^\prime \over \partial y}=G$$ (13.9)

 

where F and G are obtained from the left sides of (13.5) and (13.6) and are presumed measured and therefore known.

The usual temptation is to try to integrate (13.9) using some straightforward numerical solver. This is doomed to fail when F and G are computed from real data and will therefore contain measurement errors and errors in the subsequent processing (i.e., interpolation errors, errors due to evolution and advection, etc.) To see why this approach will fail, it is sufficient to note that a solution exists if and only if

$${\partial F\over \partial y} = {\partial G\over \partial x} \ .$$ (13.10)

 

There is no guarantee that real data will satisfy constraint (13.10) above and therefore in general the differential equations (13.9) have no solution consistent with the data.

In practice the system is solved in the least squares sense by minimizing the integral

$$\int\int \Big[({\partial p^\prime \over \partial x}-F)^2+({\partial p^\prime \over\partial y}-G)^2\Big]dxdy=Min.$$ (13.11)

  

The solution to this minimization problem can be shown to be a Poisson equation for the pressure (Courant and Hilbert, 1953, pp. 164-274), specifically

$${\partial^2 p\over\partial x^2}+{\partial^2 p\over\partial y^2}={\partialF\over \partial x}+{\partial G\over\partial y} \ .$$ (13.12)

  

The boundary conditions for this equation are also part of the minimization problem. If the pressure on the boundary is known then one can simply specify it as a boundary condition, but in general the pressure is not known and the mathematics of the minimization problem dictate a so-called natural boundary condition given by

$$\left ({\partial p^\prime \over\partial x}\right )n_x+\left ({\partial p^\prime \over\partialy}\right )n_y = Fn_x+Gn_y$$ (13.13)

 

where n_x and n_y are the direction cosines to the normal. Physically the left side of (13.13) is the derivative of the pressure in the direction normal to the boundary.

It is not difficult to see that the solution of the Poisson equation (13.12) with the Neumann type boundary condition (13.13) is not unique. Whatever solution one obtains, it is possible to add a constant to it and the new solution will still satisfy (13.11)-(13.13). There are now "canned" routines (e.g., Swarztrauber and Sweet 1975) that "solve" the above Poisson equation (13.12) with Neumann boundary conditions. Typically these "solvers" start by assigning an arbitrary number to the pressure at a selected point. Once this is done a solution for the pressure is found.

It is not unusual for different solvers to give different solutions which differ from each other by arbitrary constants. To avoid such ambiguity Gal-Chen (1978) suggested defining a new perturbation pressure $p^{\prime\prime}$ by

$$p^{\prime\prime} = p^\prime -(1/A)\int\int p^\prime dxdy$$ (13.14)

 

where A is the area of integration. Gal-Chen (1978) showed that even though the solution for $p^\prime$ may not be unique (different solvers will give different answers) $p^{\prime\prime}$ is unique. Of course, it is however important to realize that removing the mean for each level of data creates a situation where the perturbations are no longer defined relative to a specified sounding or to the ambient environment.

Having obtained the pressure deviation (from the horizontal average) one can then calculate the vertical pressure deviation gradient. To do so (13.7) is modified by subtracting the horizontal average (defined by the symbol $<\ >$). The net result is an explicit equation for the density fluctuations in terms of known quantities:

$$-({\rho}^\prime-<{\rho}^\prime>)g={\partial (p^\prime-<p^\prime>)\over \partial z}+ H-<H>$$ (13.15)

 

where

$$H = \rho_0\left ( {\partial w\over \partial t}+u{\partial w\......\partial y} +w{\partial w\over\partial z}\right ) +f_{z} \ .$$ (13.16)

 

From inspection of (13.16) it is evident that the density fluctuations should be noisier than the pressure fields since the derivation of the buoyancy fields involve an additional level of derivatives. In contrast the solver used to derive the pressure effectively acts to reduce the noise through a least squares fit.

Having obtained an estimate of the buoyancy and pressure fluctuations the issue of accuracy and verification of the results is important. Direct measurement of the pressure and buoyancy fluctuations on these spatial and time scales is extremely difficult with current technology. Even if measurement technology existed to obtain measurements over large areas with these space and time scales, accuracy issues would still arise. For example, typical pressure fluctuations in the convective boundary layer rarely exceeds 0.1 mb. One way to assess the reliability of the pressure fluctuations is to derive a consistency check to the least squares solution. Examining (13.11)-(13.13) and noting that (13.12) is the least square solution to (13.11), one can calculate a quantity E_r defined by,

$$E_r={\int\int \left({({\partial p\over \partial x}-F )}^{2} +......l y}-G )}^{2}\right)dxdy\over \int\int (F^{2} +G^{2})dxdy} \ .$$ (13.17)

 

E_r may be thought of as the goodness of the fit as discussed for example in Gal-Chen and Kropfli (1984). If E_r=0 the fit is perfect and it can be shown that a value of 0.5 can be obtained if F and G are generated by white noise, i.e. an E_r value of 0.5 is an indication that the retrievals are suspect. The number 0.5/E_r can be defined as the signal to noise ratio. A signal to noise ratio of 2 means that the signal variance is twice the noise variance.

In practice some thought is required if one wishes to use this ratio to decide whether it is reasonable to make physical inferences from a data set. For example, Kingsmill and Wakimoto (1991) showed that introducing small systematic errors into an error free flow field (E_r = 0) could leave the pressure field qualitatively intact while creating relatively large values of E_r. Kingsmill and Wakimoto (1991) also discussed the large values of E_r that can often occur with extensive regions of very weak gradients of pressure, which can often be of little interest to a researcher. In such instances other levels or locations of interest might still contain valuable information even when the areas of weak overall signal may create an unfavorable ratio. Of course, the reverse situation can also occur where the ratio appears favorable, while the areas of physical interest are poorly represented. For this reason it is important to consider applying other criteria to evaluate the accuracy. For example, a more subjective way to judge the reliability is to examine pressure fluctuations from several radar scans and to examine the scan-to-scan correlation in time as such a correlation would not be expected with random noise. Comparisons with in-situ data at common grid points or tests for physical plausibility should also be included whenever possible. 

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