The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 58

divergence equation like (2.136) must be solved in order to determine the initial
pressure field based on the initial wind and buoyancy fields.

2.6 ERTEL'S POTENTIAL VORTICITY

Another method for analyzing convective storm dynamics involves the use of
Ertel's potential vorticity Z, which is given as follows:
Z ¼ 1

=½ð JT

v Þ EJ s

ð 2 :

137 Þ

where s is the specific entropy, s ¼ C p ln . (If the air is saturated and/or has water

substance suspended in it, then must be modified.) If diabatic heating and molec-

ular and turbulent viscosity are negligible or zero, Ertel's potential vorticity is
conserved (e.g., see pp. 265-267 of Bluestein, 1992 for a derivation), so that
DZ = Dt ¼ 0 ¼ D = Dt ½ð JT v Þ EJ ð 2 : 138 Þ

Conservation of Ertel's potential vorticity for a fluid is like conservation of
angular momentum for rigid bodies: when the gradient of potential temperature
decreases ( surfaces spread farther apart), the fluid contracts and spins up about

the axis defined by the potential temperature gradient vector, and vice versa.
Equation (2.138) is based on the equations of motion, the equation of continuity,
and the thermodynamic equation. It can be used to estimate the future three-
dimensional distribution of Z, from which, under certain conditions and using
appropriate boundary conditions, it is possible to retrieve the temperature and
winds fields. In severe convective storms, diabatic heating plays a prominent role,
so that Z is not conserved. However, if it is assumed that the latent heat of con-
densation from the formation of cloud material is absorbed by the air parcel (i.e.,
as in the moist-adiabatic process), then e , the equivalent

potential temperature, which is conserved for moist-adiabatic processes. In
addition, turbulent mixing may render Z not conserved. For some pedagogical
purposes, though, we sometimes treat moist Ertel's potential vorticity as if it were
conservative.

may be replaced by

2.7 THE EXNER FUNCTION AS A VERTICAL COORDINATE,
POTENTIAL TEMPERATURE AS A THERMODYNAMIC
VARIABLE, AND THE PSEUDO-INCOMPRESSIBLE
CONTINUITY EQUATION

Derivation of the equations of motion (2.13) and (2.7) in terms of density involves
approximations because density appears in the denominator in the RHS: in par-
ticular, 1

0 0 is much less than

and the p 0

=ð þ Þ is approximated because =

p
term in the expression for Archimedean buoyancy B must be dealt with. It
was shown earlier that the p 0

=

p term can be neglected in comparison with T 0

=

T
for largely subsonic flow and that nonlinear perturbation terms are neglected.
A commonly used alternative to pressure as a vertical coordinate is the Exner