Microphysical retrievals  Three-dimensional Extensions of the  Combined use of the

Incorporation of the thermodynamic energy equation

An alternative to the previously described retrieval approaches is to include an additional equation as a constraint to eliminate the vertical variation in the unknown integration constant in the dynamic retrieval method proposed by Gal-Chen (1978) or the difficulty in obtaining accurate boundary conditions in the method of Brandes (1984). One possibility is to employ the thermodynamic energy equation as an additional constraint. Such an approach has been explored in a series of papers by Roux et al. (1984), Roux (1985), Roux (1988), and Roux et al. (1993). The technique is a great deal more complicated than those techniques discussed thus far so we will only outline the general concepts. In this technique, the starting point is the anelastic equations of motion defined previously. From the three components of the equations of motion it is straightforward to solve for the spatial gradients of the pressure terms resulting in three equations for $\pi$ similar to the starting point for the basic dynamic retrieval methods.

A second set of equations is then derived to obtain an analogous expression for buoyancy fluctuations. Following the general derivations used by Roux and colleagues, we can take the equations of motion and derive two equations for unmeasured portion of the buoyancy equations through ignoring the Coriolis terms and taking the horizontal gradients of the vertical equation of motion resulting in

$${{\partial \beta_{um}} \over {\partial x}} = {{cp \theta_{0} ......cdot {\bf\nabla}) u / (c_{p} \theta_{v0})} \over {\partialz}}$$ (13.24)

 

and

$${{\partial \beta_{um}} \over {\partial y}} = {{cp \theta_{0} ......cdot {\bf\nabla}) v / (c_{p} \theta_{v0})} \over {\partialz}}$$ (13.25)

 

where $\beta_{um}$ is the unmeasured portion of the buoyancy field defined using our previous notation as $\beta^\prime - q_{r}$. A third equation for this unmeasured buoyancy equation can be obtained from considering the thermodynamics. Ignoring radiation, surface fluxes and other processes not associated with phase changes, Roux and colleagues obtain

$$({\bf V} \cdot {\bf\nabla}) \beta_{um} = - {{L_{v/s}} \over {......{dz}} - \theta_{0} ({\bf V \cdot \nabla}) (q_{c} - 0.61 q_{v})$$ (13.26)

 

where L_v/s is the latent heat of vaporization or sublimation appropriately applied, L_f the latent heat of fusion, $\delta_{s}$ is equal to 1 when air is saturated and equal to 0 when it is not, C is the condensation rate, E the rate of evaporation and M the melting rate. The solution procedure for this technique is to solve the two sets of equations, three equations each for $\pi$ and for $\beta_{um}$, through least square techniques with two functions to be minimized in a variational approach. The conjugate gradient approach (Polak 1971) was used to solve the resulting matrix of equations. Following the suggestions of Gal-Chen (1978), this method also has a consistency check for the accuracy of the derived fields. From these equations it is easy to see where the complexities of this technique arise. First it is evident that knowledge of where transitions occur between water and ice and where the air is saturated or unsaturated must be obtained or somehow parameterized. In this general approach we have also introduced the need for continuity equations for water substances which themselves have unknowns such as transformation of cloud water into precipitation through collection and autoconversion. These complexities can be parameterized from knowledge of the atmosphere gained through the Doppler derived air flows (saturation occurs with upward motion for example), radar reflectivities (to estimate precipitation rates and changes in these rates), and thermodynamic information derived from soundings (to estimate the characteristics of inflow air to determine such characteristics as condensation rates). Hence, while the assumptions appear to be reasonable for the situations investigated by Roux and colleagues, it is clear in this approach that the results will be more strongly influenced by the assumptions about the state of the atmosphere than in simple dynamic retrieval.

In this series of papers, Roux and co-workers have studied convective systems over west Africa and a cold frontal system over western Europe. Since parameterizations are more extensively employed in this method, it is important to critical examine the consistency checks for the derived fields. In general the results are encouraging with mean values over the domain typically well below the 0.5 critical value discussed earlier for the convective events studied. In contrast, the values for the frontal case (Roux et al. 1993) are quite close to the 0.5 critical value for the three dimensional analysis, which is less promising than their study of convective systems. However, it should be noted that a two dimensional application of the technique to the frontal event did produce more acceptable consistency checks of order 0.3.

Vertical cross-sections of wind, the unmeasured buoyancy fluctuations and pressure fluctuations are shown from the two dimensional study of the cold front (Roux 1993) (Fig. 13.8)

 

Figure 13.8: 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.

and an example from one of the west African convective systems (Roux 1985) are shown in (Fig. 13.9):

 

Figure 13.9: 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.

These examples are intended to provide the reader with some insight into the types of results that can be obtained with this technique. Also several simple points can be made from comparing the results from these two systems. For example, in both cases there is some evidence of an upward directed pressure force near the base of the updraft at the leading edge of a pool of cooler air. As noted in one of our earlier examples, this pressure force can itself result in very large updrafts in cold fronts. Perhaps surprisingly the cold air mass produced has a much larger temperature deficit in the convective system. This stronger cold air mass was produced by melting and evaporation of precipitation in the convective event. In these examples, as expected, the positive temperature fluctuations associated with the updraft are much larger in the west African convective systems than in the middle latitude cold front. In the convective system this warming leads to a broad pressure minimum below the updraft.

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