358
7. Physically Insignificant Fast Waves
zoidal differences; in this case (7.57) and (7.58) become
b n+1 + IltN 2 w n+1 = ib,
v n + 1 +llt(vpn+l-kbn+l) =G,
(7.61)
where
ib = b n- I - Ilt [N 2 w n- 1 + 2v
n • Vb n] ,
G= v n- I - Ilt [vpn-I - kb n - I + 2v n. vvnJ .
The implicit coupling in the resulting semi-implicit system can be reduced to a
single Heimholtz equation for pn+1 as folIows. Let G= ts«. gv, gw); then using
(7.61) to substitute for b n + 1 in the vertical-momentum equation, one obtains
( 1 + (N Ilt)2 ) w
apn+1
n + 1 + Ilta;- = gw + ibllt.
(7.62)
Using the horizontal-momentum equations to climinate u and v from (7.59) yields
2
a
As the final step, w n +I is eliminated betwecn the two preceding equations to
obtain
2]
+ (N /lt)2) ( a
[( I
ax
2
2 + a y2 - (c
I)
sllt)2
+ a pn+J
az 2
= (I + (N Ilt)2) [_1 Ilt
(a
gU
ax
+ a
gv
8y
) _
h
(c sllt)2
] + 8z
(g w + ib) .
Ilt
After this elliptic equation is solved for pn+ I , then u and v are updated using
the horizontal momentum equations, w is updated using (7.62), and finally, b is
updated using (7.61) .
Notice that the vertical advection of density in (7.61) is split between a term
involving the mean vertical density gradient (N
2 ) , which is treated implicitly, and
a term involving the gradient of the perturbation density field (8bj8z), which is
treated explicitly. As discussed in Section 7.2.3, when terms are split between a
reference state that is treated implicitly and aperturbation that is treated explicitly,
stability considerations demand that the term treated implicitly dominate the term
treated explicitly. Thus, in most atmospheric applications the reference stability is
chosen to be isothermal, thereby ensuring that N 2 8bj8z. When semi-implicit
differencing is used to integrate the complete Euler equations, the terms involving
the pressure gradient and velocity divergence must also be partitioned into implicitly differenced terms involving a reference state and the remaining explicitly dif-
7. Physically Insignificant Fast Waves
zoidal differences; in this case (7.57) and (7.58) become
b n+1 + IltN 2 w n+1 = ib,
v n + 1 +llt(vpn+l-kbn+l) =G,
(7.61)
where
ib = b n- I - Ilt [N 2 w n- 1 + 2v
n • Vb n] ,
G= v n- I - Ilt [vpn-I - kb n - I + 2v n. vvnJ .
The implicit coupling in the resulting semi-implicit system can be reduced to a
single Heimholtz equation for pn+1 as folIows. Let G= ts«. gv, gw); then using
(7.61) to substitute for b n + 1 in the vertical-momentum equation, one obtains
( 1 + (N Ilt)2 ) w
apn+1
n + 1 + Ilta;- = gw + ibllt.
(7.62)
Using the horizontal-momentum equations to climinate u and v from (7.59) yields
2
a
As the final step, w n +I is eliminated betwecn the two preceding equations to
obtain
2]
+ (N /lt)2) ( a
[( I
ax
2
2 + a y2 - (c
I)
sllt)2
+ a pn+J
az 2
= (I + (N Ilt)2) [_1 Ilt
(a
gU
ax
+ a
gv
8y
) _
h
(c sllt)2
] + 8z
(g w + ib) .
Ilt
After this elliptic equation is solved for pn+ I , then u and v are updated using
the horizontal momentum equations, w is updated using (7.62), and finally, b is
updated using (7.61) .
Notice that the vertical advection of density in (7.61) is split between a term
involving the mean vertical density gradient (N
2 ) , which is treated implicitly, and
a term involving the gradient of the perturbation density field (8bj8z), which is
treated explicitly. As discussed in Section 7.2.3, when terms are split between a
reference state that is treated implicitly and aperturbation that is treated explicitly,
stability considerations demand that the term treated implicitly dominate the term
treated explicitly. Thus, in most atmospheric applications the reference stability is
chosen to be isothermal, thereby ensuring that N 2 8bj8z. When semi-implicit
differencing is used to integrate the complete Euler equations, the terms involving
the pressure gradient and velocity divergence must also be partitioned into implicitly differenced terms involving a reference state and the remaining explicitly dif-