The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 52

Figure 2.9. Spherical coordinates. At point P, the coordinates are given in terms of the distance
r (radius) from the origin (center of the sphere), latitude ( ) measured from the equator, which
lies in the x-y-plane, and longitude ( ) measured relative to the þ x-axis.

@

atmosphere is at rest; also, the T =@ z term does not appear in the latter because

at t ¼ 0, w ¼ 0.
In spherical coordinates ( Figure 2.9 ), r,

represent the radial distance
from the center of the sphere, the latitude measured from the equator (which lies
in the x-y-plane) and the longitude measured from some reference (e.g., the x-
axis). The z-axis (vertical) is oriented from

, and

¼= ¼ =

2to 2. We will neglect

any variations in longitude, so that @=@ ¼ 0. Thus, when looking down on the

sphere from above there is symmetry about the z-axis. Also, we will neglect any
motions in the longitudinal direction, so that D Dt ¼ 0.

The two non-trivial components of the equation of motion in spherical
coordinates subject to longitudinal symmetry are as follows:

=

@=@ t ð rD = Dt Þ¼ 1 = r @ P =@þ B ð k z E j Þ ð 2 : 86 Þ

@=@ t ð Dr = Dt Þ¼@ P =@ r þ B ð k z E k Þ ð 2 : 87 Þ

where the former is the j (latitudinal) component and the latter is the k (radial)
component, respectively, of the equation of motion.
The equation of continuity is

@=@ð r 2 cos

r ð r 2 cos

=

Dt Þþ@=@

=

Dt Þ¼ 0

ð 2 :

88 Þ D Dr

and the thermodynamic equation is still

@ B =@ t ¼ 0 ð 2 : 89 Þ

We define the radial and latitudinal components of the wind as

u ¼ Dr = Dt ð 2 : 90 Þ

and

v ¼ rD = Dt ð 2 : 91 Þ

t, P, and T) 3 The four governing equations at t ¼ 0 in four unknowns (

@ =@ @v=@

u t,

3 The fourth unknown is really B. But since T is computed from B, T can also be regarded as
the fourth unknown.