16
1. Introduction
that Rossby waves do not propagate along the characteristics, because the tenns
involving the undifferentiated functions of u and v play no role in the determination of the characteristics of (1.28) , yet those same tenns are essential for the
maintenance of the Rossby waves.
The Euler equations goveming inviscid isentropic motion in a density stratified
fluid provide another example of a hyperbolic system that supports a type of wave
whose propagation is completely unrelated to the characteristics. The Euler equations for the inviscid isentropic motion of a perfect gas can be expressed in the
form
(1.31)
(1.32)
(1.33)
dv
I
-+-Vp=-gk,
dt
P
ap
-
at
+ V . (pv) = 0,
dO
-=0,
dt
where Coriolis forces have been neglected,
d( ) = a( ) + v . V( ),
dt
at
v is the three-dirnensional velocity vector, p is density, p is pressure, g is the
gravitational acceleration, k is a unit vector directed opposite to the gravitational
restoring force, and 0 is the potential temperature, which is related to the entropy,
S, such that
S = Cp In 0 + constant.
Conservation of momentum is required by (1.31), conservation of mass by (1.32),
and conservation of entropy by (1.33) .
As written above, the Euler equations constitute a system of five equations
involving six unknowns. In atmospheric applications, the system may be closed
using the equation of state for a perfect gas
p=pRT
(1.34)
and the definition of the potential temperature
to arrive at the diagnostic equation
P = PO ( -pO R
i"
PO
(1.35)
In the preceding, T is the temperature, Po is a constant reference pressure, R is
the gas constant for dry air, c p is the specific heat at constant pressure, and C v is
the specific heat at constant volume.