Geography Reference
In-Depth Information
It is verified readily that if ψ satisfies periodic boundary conditions there is a
cancellation of terms when (13.30) is summed over the domain. Thus,
M
N
F m,n =
0
(13.31)
m =
n =
1
1
Therefore, mean vorticity is conserved (except for errors introduced by time dif-
ferencing) when (13.30) is used as the finite difference form of the advection term.
This form also conserves mean kinetic energy (see Problem 2). Enstrophy is not
conserved in this difference formulation, and it is conventional to add a small
diffusion term in order to control any numerically generated increase in enstrophy.
The procedure for preparing a numerical forecast with the barotropic vorticity
equation can now be summarized as follows:
(1) The observed geopotential field at the initial time is used to compute the
initial stream function ψ m,n (t
=
0) at all grid points.
(2)
F m,n is evaluated at all grid points.
δt) is determined using centered differencing except at the first
time step when a forward difference must be used.
(4)
(3)
ζ m,n (t
+
δt).
(5) The predicted array of ψ m,n is used as data, and steps 2-4 are repeated until
the desired forecast period is reached. For example, a 24-h forecast with 30-min
time increments would require 48 cycles.
The simultaneous set (13.29) is solved for ψ m,n (t
+
13.5
THE SPECTRAL METHOD
In the finite difference method, the dependent variables are specified on a set
of grid points in space and time, and derivatives are approximated using finite
differences. An alternative approach, referred to as the spectral method, involves
representing the spatial variations of the dependent variables in terms of finite
series of orthogonal functions called basis functions . For the Cartesian geometry
of a midlatitude β-plane channel, the appropriate set of basis functions is a double
Fourier series in x and y. For the spherical earth, however, the appropriate basis
functions are the spherical harmonics.
A finite difference approximation is local in the sense that the finite difference
variable m,n represents the value of ψ(x, y) at a particular point in space, and the
finite difference equations determine the evolution of the m,n for all grid points.
The spectral approach, however, is based on global functions, that is, the individual
components of the appropriate series of basis functions. In the case of Cartesian
geometry, for example, these components determine the amplitudes and phases of
the sinusoidal waves that when summed determine the spatial distribution of the
Search WWH ::




Custom Search