Incorporation of the thermodynamic  Three-dimensional Extensions of the  Three-dimensional Extensions of the

Combined use of the three equations of motion

One three-dimensional extension of the dynamic retrieval concept was discussed by Brandes (1984). In this approach, a three-dimensional elliptic equation for the perturbation pressure is derived by taking the three-dimensional divergence of (13.20) and substituting the continuity equation (13.21) to obtain   

$$\left(\nabla^{2} \pi^\prime + {\partial {ln (\rho_{0}(z) \theta_{......_{v0}}}\bf {\nabla} \cdot \rho_{0} (z) \bf {f_{i}}\right) \quad\quad\quad\quad$$

$$\quad\quad = {1 \over c_{p} \theta_{v0}}\left({-1 \over {\rho_{...... \over \partial z} [ \rho_{0} (z) ({\beta \over\theta_{0} - q_{c})} ]\right).$$ (13.22)

  

Solutions to (13.22) are again found by relaxation. This method has a number of advantages over the two dimensional approach. For example, while the solutions are not unique and are known only to within a constant, there is only one constant of integration for the entire volume rather than a series of constants that will vary with height. In addition, there are no time dependent terms in the equations avoiding the time resolution problems discussed earlier in terms of the study of microbursts. The method also has distinct disadvantages such as the pressure field being dependent upon the cloud and precipitation fields. As stated earlier, the precipitation field can be diagnosed to a reasonable accuracy from the radar reflectivity. Presumably the cloud water field must be parameterized. While the Neumann lateral boundary conditions can be used, there are significant difficulties in deriving suitable vertical boundary conditions for this technique and some aspects of the boundary condition used at the lower boundary in this study (the vertical gradient of the horizontal velocities and dw/dt all are equal to zero) are difficult to defend.

In this approach, buoyancy can be computed using the derived pressure fields and the vertical equation of motion in an iterative manner. Instead Brandes (1984) derived an expression for the buoyancy fields through first cross-differentiating the momentum equations to obtain a three-dimensional vorticity equation and subsequently dot-multiplying the curl of the vorticity equation by $\bf {k}$ to obtain

  

$$\left({g \over \theta_{0}} \delta_{H} ({\theta_{v} \over \theta_{......t}}- (\omega {\bf\cdot \nabla}) {\bf V} + ({\bf\nabla \cdot} {\bf V}) \omega ]$$

$$\quad\quad\quad\quad\quad - {\bf k} \cdot {\bf\nabla} {\bf X} {\b......_{i}} + {\bf k g q_{r}} +{\bf k g q_{c}} - c_{p} \theta_{v0} {\bf\nabla} \pi).$$ (13.23)

  

From (13.23) it is evident that the retrieval of the buoyancy field in this formulation is two-dimensional and the buoyancy field depends on the pressure field (although this dependence is weak and can be ignored). If the radar domain extends to include portions of the environment than a rather reasonable lateral boundary conditions of zero buoyancy at the edges of the domain can be invoked. Brandes (1984) was able to provide new insight into our understanding of the lifecycle of a tornadic storm finding that the intensification of the parent cyclone to a large tornado was associated with deepening pressure deficits at lower levels reducing and eventually reversing the upward directed pressure forces in the vicinity of the parent cyclone (Fig. 13.7). This study also showed that a strong downdraft that often forms near the parent cyclone was the result of the intensification process rather than the cause of it. Comparisons between the observations and a simulation of the same event by Klemp et al. (1981) and Klemp and Rotunno (1983) were also promising.

 

Figure 13.7: Vertical cross-section of retrieved perturbation pressure through the mesocyclone containing the Del City-Edmond tornadic thunderstorm.

NCAR Advanced Study Program
http://www.asp.ucar.edu