The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 40

wavelength of the radar radiation, the case in the Rayleigh scattering region) is
added as a third prognostic variable, the bulk parameterization scheme is then
known as a ''three-moment'' scheme. It has been found that improvements in the
realism of simulations when a one-moment scheme is changed to a two-moment
scheme are greater than when a two-moment scheme is changed to a three-
moment scheme.
The student and researcher are referred to Jerry Straka's 2009 book Cloud and
Precipitation Microphysics for more details on bin and bulk microphysics schemes
used in models and Pruppacher and Klett's 1998 book Microphysics of Clouds and
Precipitation for more detailed derivations of governing equations, etc. The main
point the student should come away with from this discussion of microphysical
parameterization is that it is dicult to represent accurately the thermodynamic
consequences of cloud and precipitation formation and dissipation. Since the
amount of cloud and precipitation material affect buoyancy (cf. (2.23)), which
appears in the vertical equation of motion (2.7) and therefore affects the kine-
matics of the convective cloud, cloud microphysics can significantly affect the
dynamics and behavior of convective storms.

2.4 THE VORTICITY AND CIRCULATION EQUATIONS

The dynamics of severe convective storms are often elucidated by using ''derived''
forms of the equations of motion (2.13) and (2.7). By applying the curl operator
to the three-dimensional equation of motion represented by the combination (2.13)
and (2.7), the following time-dependent, prognostic vorticity equation is found:

p 0 Þþ JT ð Bk Þ D = Dt ð JT v Þ¼½ð JT v Þ EJ v JT ð 1 = J ð 2 : 49 Þ

where k is a unit vector pointing upward; and v is the three-dimensional

vorticity. The first term on the RHS of (2.49) is the stretching and tilting term, the
second term is the solenoidal term, and the third term is the baroclinic term. In a
Boussinesq atmosphere the solenoidal term vanishes because density is treated as a
constant (since it comes from the pressure gradient term in the equation of
motion). In (2.49), Earth's vorticity is not included, but could be added to the
vertical component of vorticity to account for the behavior of convective phenom-
ena that persist for a relatively long time (i.e., when the Coriolis force becomes
significant).
The vertical component of (2.49) is

JT

p 0 D = Dt ¼ @=@ t

| {z }
1

þ v h EJ þ w z

| {z }
3

@=@ ¼

|{z}
1

þ k w Þ

| {z }
2

E ð@

v =@ z þ k Þ

| {z }
3

E ½ J TJ ð 1

=

TJ

| {z }
2

ð 2 : 50 Þ

¼ k

v ¼ @v=@ x @

=@

where EJT u y is the vertical component of vorticity;

¼ @ =@

x þ@v=@

y is horizontal divergence; x and y are coordinate axes that
point to the east and north, respectively; and u and v are the components of the

u