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included in the equations. One way to filter gravity waves is thus to divide the
horizontal velocity into irrotational and nondivergent parts as discussed in Sec-
tion 11.2 and to neglect the irrotational part of the motion in the velocity tendency
term. The resulting prognostic equation is the barotropic vorticity equation (11.14).
The stream function for the nondivergent flow can then be diagnostically related
to the geopotential field by the nonlinear balance equation (11.15). For extratropi-
cal synoptic scale motions, (11.15) can be replaced approximately by geostrophic
balance on the midlatitude β-plane, for which ψ
/f
0
.
The barotropic vorticity equation can be used to compute the evolution of the
flow at a single level in terms of the nondivergent flow at that level alone. No vertical
coupling is involved. Thus, no explicit predictions for levels above or below the
assumed nondivergent level are possible. The only “predictions” of surface weather
possible with this model are those based on climatological relationships between
the surface flow and flow at the nondivergent level. For extratropical flows it is
possible to explicitly model the vertical structure and still filter gravity waves by
utilizing the conservation of quasi-geostrophic potential vorticity (6.24), which
is related to the geopotential field through the elliptic boundary value problem
(6.25). Sections 13.4 and 13.5 utilize the barotropic vorticity equation to illustrate
the basic methodology of numerical weather prediction.
=
13.3
NUMERICAL APPROXIMATION OF THE EQUATIONS
OF MOTION
The equations of motion are an example of a general class of systems known
as
initial value problems
. A system of differential equations is referred to as an
initial value problem when the solution depends not only on boundary conditions,
but also on the values of the unknown fields or their derivatives at some initial
time. Clearly, weather forecasting is a primary example of a
nonlinear
initial value
problem. Due to its nonlinearity even the simplest forecast equation, the barotropic
vorticity equation, is rather complicated to analyze. Fortunately, general aspects of
the numerical solution of initial value problems can be illustrated using linearized
prototype equations that are much simpler than the barotropic vorticity equation.
13.3.1 Finite Differences
The equations of motions involve terms that are quadratic in the dependent variables
(the advection terms). Such equations generally cannot be solved analytically.
Rather, they must be approximated by some suitable discretization and solved
numerically. The simplest form of discretization is the
finite difference
method.
To introduce the concept of finite differencing, we consider a field variable ψ(x),
which is a solution to some differential equation in the interval 0
≤
x
≤
L. If this
