The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 51

fluctuations. In addition, the wind field must be low-pass filtered to reduce the
effect of noise, since spatial derivatives are computed.
Perturbation pressure p 0 in theory may be calculated at each level by inverting
the Laplacian operator, subject to appropriate boundary conditions; if the storm is
localized, then we can use p 0 ¼ 0 on the boundaries (Dirichlet boundary con-
ditions) because the lateral boundaries are far from the storm; if they are
relatively near the storm, then the horizontal pressure gradient at the lateral
boundaries can be solved from the wind field using the horizontal equations of
motion, such that at the lateral boundaries (Neumann boundary conditions)

p 0

@ =@ x ¼ Du = Dt ð 2 : 77 Þ

p 0

@ =@

y ¼ D v=

ð 2 :

78 Þ Dt

If there is no detectable radar echo outside the storm, however, it is not possible
to use (2.77) and (2.78) because there are no Doppler wind data there.
Because there are instrument and sampling errors in the determination of the
Doppler wind field, solution of the perturbation pressure field exactly using (2.76)
is not possible, especially when

Dt Þ . An alternative
approach is to solve for perturbation pressure using variational analysis by
minimizing the cost function

@=@ y ð Du = Dt Þ6¼@=@ x ð D v=

ðð

p 0

2

p 0

2

J ¼ f½@ =@ x ð Du = Dt Þ þ½@ =@ y ð D v= Dt Þ g dx dy ð 2 : 79 Þ

where the domain of the integral is over the analysis region at each level. This
procedure amounts to fitting a pressure field to the wind field by minimizing the
difference between the fitted pressure gradient and the equation of motion. The
resulting Euler equation as shown by Gal-Chen, however, is identical to (2.76).
Inverting the horizontal Laplacian operator in (2.76) effectively filters out noise as
it fits the pressure field to the wind field.
The vertical variation of p 0 along with the wind field are used to solve for
buoyancy B via the vertical equation of motion (2.70), without distinguishing
between p 0 d and p 0 b as follows:
B ¼ Dw p 0

=

= Dt þ 1 @ =@ z ð 2 : 80 Þ

It is assumed that the perturbation pressures at each level represent true deviations
from the average, an assumption that may not be entirely valid and must be
checked. A measure of how well retrieved perturbation pressures may be estimated
is by the ''consistency check'' quantity E, where

ðð

p 0

2

p 0

2

f½ 1 = @ =@ x Du = Dt þ½ 1 = @ =@ y D v= Dt g dx dy

ðð

E ¼ ð 2 : 81 Þ

2 2

f½ Du = Dt þ½ D v= Dt g dx dy

and the integrals are computed over the entire radar domain at each level. The
values of E are then averaged over all levels for which there are Doppler radar