18
I. Introduction
The basic-state vertical velocity is zero to ensure that the basic state is a steady
solution to the nonlinear equations. Substituting these expressions for u, w, n , and
() into the two-dimensional versions of (1.33) , (1.37), and (1.38), and neglecting
second-order terms in the perturbation variables under the assumption that the
perturbations are small-amplitude, one obtains the linear system
( 0
-
ot
+ U - 0), -orr'
ox
U + c p () -
ox
= 0,
( 0 0),
-+Uot
ox
-orr' ()'
+cp()-=g=,
oz
o
W
( 0 0),
-+U- ()
»,«,
ot
ox
+-N w =0,
g
-
0 +U- rr +w -+ -
0 ) ,
,07f
R7f (OU' -+ - OW') =0,
( ot
OX
oz C v ox oz
(1.41)
(1.42)
(1.43)
(1.44)
where
2
g de
N = = -
() dz
is the square of the Brunt-Väisälä frequency.
Suppose that the reference state is isothermal. Then N 2 and the speed of sound
Cs = (cpRT /c v ) I /2 are constant, and the preceding system can be simplified by
removing the influence of the decrease in the mean density with height via the
transformation
_ (/3)1/2, ii = (/3 )1/ 2 cpe n',
u = -
u,
Po
_)1 /2
Po
(1.45)
w ,
9 = (
w=
-
-)1 /2
P
-
,
Cs
(1.46)
:e()'.
(
Note that 9 represents a scaled buoyancy and ir a scaled pressure. Let
Po
v = ( u ÜJ 9 ir )T;
then the transformed equations have the form
ov
ov
ov
+AI- +A2- +Bv = 0,
-
ot
OX
oz
(1.47)
in which
0
0
0
(},
0
=
U 0
AI
0 U o '
0 U
0
0
0
-N
)
o '
B =
N
0
S
0
0
=
S =
A2
0
0 0
0 0 o '
C s 0 0
Cs [
0/3 + l oe).
2/3 oz () OZ
0
0