Geography Reference
In-Depth Information
8.1
HYDRODYNAMIC INSTABILITY
A zonal mean flow field is said to be hydrodynamically unstable if a small dis-
turbance introduced into the flow grows spontaneously, drawing energy from the
mean flow. It is useful to divide fluid instabilities into two types: parcel instability
and wave instability. The simplest example of a parcel instability is the convective
overturning that occurs when a fluid parcel is displaced vertically in a statically
unstable fluid (see Section 2.7.3). Another example is inertial instability, which
occurs when a parcel is displaced radially in an axisymmetric vortex with negative
absolute vorticity in the Northern Hemisphere or positive absolute vorticity in the
Southern Hemisphere. This instability was discussed in Section 7.5.1. A more gen-
eral type of parcel instability, called
symmetric instability
, may also be significant
in weather disturbances; this is discussed in Section 9.3.
Most of the instabilities of importance in meteorology, however, are associated
with wave propagation; they cannot be related easily to the behavior of individual
fluid parcels. The wave instabilities important for synoptic-scale meteorology gen-
erally occur in the form of zonally asymmetric perturbations to a zonally symmetric
basic flow field. In general the basic flow is a jetstream that has both horizontal
and vertical mean-flow shears.
Barotropic instability
is a wave instability associ-
ated with the horizontal shear in a jet-like current. Barotropic instabilities grow by
extracting kinetic energy from the mean-flow field.
Baroclinic instability
, however,
is associated with vertical shear of the mean flow. Baroclinic instabilities grow by
converting potential energy associated with the mean horizontal temperature gra-
dient that must exist to provide thermal wind balance for the vertical shear in the
basic state flow. In neither of these instability types does the parcel method provide
a satisfactory stability criterion. A more rigorous approach is required in which a
linearized version of the governing equations is analyzed to determine the structure
and amplification rate for the various wave modes supported by the system.
As indicated in Problem 2 of Chapter 7, the traditional approach to instability
analysis is to assume that a small perturbation consisting of a single Fourier wave
mode of the form exp[
ik
(
x-ct
)] is introduced into the flow and to determine the
conditions for which the phase velocity c has an imaginary part. This technique,
which is called the
normal modes
method, is applied in the next section to analyze
the stability of a baroclinic current.
An alternative method of instability analysis is the
initial value
approach. This
method is motivated by the recognition that in general the perturbations from
which storms develop cannot be described as single normal mode disturbances,
but may have a complex structure. The initial growth of such disturbances may
strongly depend on the potential vorticity distribution in the initial disturbance.
On the time scale of a day or two, such growth can be quite different from that of
a normal mode of similar scale, although in the absence of nonlinear interactions
the fastest-growing normal mode disturbance must eventually dominate.
