I oM . 10M.
V M = - - - I + - - J
acos(J oA
a 0(J
a
(7.114)
KTw
aps
2
2
• oT
oa
V 0
ot
' "
U
0
o
where
oä
-(In Ps) = -
ot
2
a(1 - /L ) oA
(ln Ps) - --(ln Ps) - V X - - , (7.115)
a o/L
oa
380
7. Physically Insignificant Fast Waves
where a is the mean radius of the Earth. Then using the fonnula for the horizontal
divergence in spherical coordinates,
= a(1 -
I
/L2)1/2
(OM. oA I + (I - /L )a;J ,
2 oM .)
and the relations (4.73)-(4.76), the prognostic equations for the a -coordinate system may be expressed in the form
oV
- - = 'H(B , -A) - 2Q (U - - - /LV 2) 1/!
2X
ot
a
- V 2 ( 4J + U
2 + V
2(1 - /L )
--=-'H(A,B)-2Q oV21/!
(V -+/LV 2 X ) ,
ot
) ,
(7.112)
(7.113)
oT'
-
= -'H(UT , VT) + T V X - a - + - ,
U = ucos(J = (1- /L2)'H(X , -1/!),
V = ucos e' = (I - /L2)'H(1/!, X),
2
.0V
RT'
2 0
A = UV 1/! + a - + - ( 1 - /L )-(lnps)'
oa
a
o/L
2
. 0U
B = VV 1/! - a - - --(lnps).
Ba
a B):
RT' 0
The preceding system of equations is formulated using In Ps instead of Ps as
the prognostic variable to make the tenn (RT / Ps)VCI Ps into a binary product
of the prognostic variables and thereby facilitate the alias-free evaluation of the
pressure-gradient force via the spectral transform method.
At each a level, the unknown functions 1/!, X, T' , and 4J are approximated using a truncated series of spherical harmonics. The unknown function In Ps is also
approximated by a spherical-harmonic expansion. Expressions for the time tendencies of the expansion coefficients for each spherical-harmonic are obtained
using the transform method in a manner analogous to that for the global shallowwater model described in Section 4.4.4. As an example, suppose that the stream
function and velocity potential at a given a level are expanded in spherical harmonics as in (4.81) and (4.82). Then, using the notation defined in Seetion 4.4.4,
the equation for o1/!m,n/ot is onee again given by (4.86) exeept that Am and E