and
7.2 The Serni-irnplicit Method
at + ax
p + c
2
S
[au +
ax
az - r) wJ = 0,
351
where
r = ap +.!.
2p az
(7.37)
and
In an isothermal atmosphere at temperature To,
If To = oDe, r is 2.7 x 10- 5 ms- I , implying that the vertical derivative in the
operators
will exceed the tenn involving r by an order of magnitude in all waves with
vertical wavelengths shorter than 100 km. The terms involving r can therefore be
neglected in most applications," in which case the transformed Euler equations
become
a
a)
( at
-+Uax
er
u+- =0,
ax
(7.38)
a
a)
( at
-+Uax
s»
w+-=b,
az
( - + U -
a
a) b + N
2
w = 0,
at
ax
a
a)
( at
-+Uax
P+c 2 (au - + -
aw) =0.
s
ax az
(7.39)
(7.40)
(7.41)
The preceding simplified compressible system (7.38)-(7.41) can also be regarded as the linearization of a "compressible Boussinesq" system in which
P - p(z)
p = p - Ji(z) ,
2
g ap
b =-g
Po
,
Po
N = - - -
PO az
(7.42)
As in the standard Boussinesq approx imation, the compressible Boussinesq system neglects the influence of density variations on inertia while retaining the in4The isothermal atmosphere does support a free wave (known as the Lamb wave, see Section 7.5)
that disappears in the limit r --+ 0, but it is not necessary to account for the Lamb wave in this
discussion of semi-implicit differencing.