1
1.2 Wave Equations in Geophysical FluidDynamics
17
The Euler equations are a quasi-linear system of first-order partial differential equations. The fundamental character of the smooth solutions to this system
can be determined by Iinearizing these equations about a horizontally uniform
isothermally stratified basic state. Simpler basic states can be obtained by neglecting gravitational forces and the density stratification (Gustafsson et aI. 1995,
p. 136; see also Problem 3), but the isothermal basic state is of more geophysical
relevance. As a preliminary step, P and p can be eliminated from (1.31)-(1.33)
by introducing the nondimensional Exner function pressure defined as
(1.36)
It follows that
-'Vp = cp()'VTl,
P
so the momentum equation may written
dv
dt + cp()'VTl = -gk.
(1.37)
It also follows from (1.35) and (1.36) that
R
Tl = -p()
( Po
)R/CV ;
thus
-ln(Tl) = -
d
R[d -ln(p) + -ln«()) d ]
dt
or, using (1.32) and (1.33),
Cu
dt
dt
dn
-
RTl
+ -'V·v=O.
dt
Cu
,
(1.38)
Equations (1.33), (1.37), and (1.38) constitute a closed system of five equations
in the five unknown variables, (), Tl, and the three components of v.
The essential properties of this system can be more simply examined in a twodimensional context. Let x and z be the horizontal and vertical coordinates, and
decompose the thermodynamic fields into a vertically varying basic state and a
perturbation such that
Tl(x, Z, t) = ]fez) + Tl'(x, Z, r),
()(x, Z, t) = O(z) + ()'(x, Z, t),
-d]f
(1.39)
c p () -
= -g o
dz
u(x, Z, t) = U + u'(x, Z, t),
w(x, Z, t) = w'(x, Z, t).
(1.40)