Geography Reference
In-Depth Information
several types of waves in the atmosphere. In Chapter 8 the perturbation theory is
used to study the development of synoptic-wave disturbances.
7.1
THE PERTURBATION METHOD
In the perturbation method, all field variables are divided into two parts, a
basic
state
portion, which is usually assumed to be independent of time and longitude,
and a perturbation portion, w
hi
ch is the local deviation of the field from the basic
state. Thus, for example, if u designates a time and longitude-averaged zonal
velocity and u
is t
h
e deviation from that average, then the complete zonal velocity
field is u(x, t)
u
(x, t). In that case, for example, the inertial acceleration
u∂u/∂x can be written
=
u
+
∂x
=
u
u
∂x
u
u
=
u
∂u
∂u
∂x
u
∂u
∂
u
+
+
∂x
+
The basic assumptions of perturbation theory are that the basic state variables
must themselves satisfy the governing equations when the perturbations are set
to zero, and the perturbation fields must be small enough so that all terms in the
governing equations that involve products of the perturbations c
an
be neglected.
The latter requirement would be met in the above example if
u
/u
|
|
1 so that
u∂u
∂x
u
∂u
∂x
If terms that are products of the perturbation variables are neglected, the nonlin-
ear governing equations are reduced to linear differential equations in the pertur-
bation variables in which the basic state variables are specified coefficients. These
equations can then be solved by standard methods to determine the character and
structure of the perturbations in terms of the known basic state. For equations
with constant coefficients the solutions are sinusoidal or exponential in charac-
ter. Solution of perturbation equations then determines such characteristics as the
propagation speed, vertical structure, and conditions for growth or decay of the
waves. The perturbation technique is especially useful in studying the stability
of a given basic state flow with respect to small superposed perturbations. This
application is the subject of Chapter 8.
7.2
PROPERTIES OF WAVES
Wave motions are oscillations in field variables (such as velocity and pressure)
that propagate in space and time. In this chapter we are concerned with linear
sinusoidal wave motions. Many of the mechanical properties of such waves are
also features of a familiar system, the linear harmonic oscillator. An important
