7.6 Primitive Equation Models
383
exeept in the half-Iayer between the lowest o level and the surface, where
l/JN -l/Js _ -RT
InaN
-
N·
Defining CXN = -In o» and CXk = !In(ak+I!aÜ for 1 :::: k < N. the discrete
analogue of (7.109) becomes
(7.122)
Finally, as suggested by Corby et al. (1972). the vertical diseretization for the co
equation (7.107) is chosen to preserve the energy-eonservation properties of the
vertically integrated eontinuous equations . Sueh eonservation is aehieved if
(7.123)
7.6.4 Energy Conservation
Why does (7.123) give better energy-conservation properties than the simpler formula that would result ifboth cxk and CXk-1 were replaeed by !:!..ak/(2ak)? In order
to answer this question it is necessary to review the energy-eonservation properties of the continuous a-coordinate primitive equations. Dur foeus is on the vertieal discretization, so it is helpful to obtain a conservation law for the vertically
integrated total energy per unit horizontal area. Using the hydrostatic equation
(7.91) and the definition o Ps = p, the vertical integral of the sum of the kinetic''
and internat energy per unit volume may be expressed as
1
00
P (U U
--+cvT ) dz=-- I 1 0 (U U ) dp
--+cvT
zs
2
g Ps
2
= g
Ps
10 r (U' U )
-2- +cvT da .
Using the hydrostatic equation twice and integrating by parts, the vertical integral
of the potential energy becomes
1
00 pgzdz =
lpSl/J -dp = -
I l<1>sps d(l/Jp)- l<1>sP -dl/J = - +
l/JsPs l pS P
-dp.
Zs
0
g
g 0
00
g
g
0
gp
9Note that as a consequenceof the primitive-equation approximation,the verticalvelocitydoes not
appear as part of the kinetic energy.