Geography Reference
In-Depth Information
Fig. 13.2
A portion of a two-dimensional (x, y) grid mesh for solution of the barotropic vorticity
equation.
given array ζ m,n . This set can be solved by standard methods of matrix
inversion.
Before expressing the advection term F(x, y, t) in finite difference form, it is
worth noting that if ψ is taken to be constant on the northern and southern bound-
aries of the β-plane channel, it is easily shown by integration over the area of the
channel that the average value of F is zero. This implies that the mean vorticity is
conserved for the channel. It is also possible, with a little more algebra, to show
that the mean kinetic energy and the mean square vorticity (called the enstrophy )
are conserved.
For accuracy of long-term integrations, it is desirable that any finite differ-
ence approximation to F satisfy the same conservation constraints as the original
differential form; otherwise the finite difference solution would not be conser-
vative. The mean vorticity, for example, might then drift systematically in time
purely due to the nature of the finite difference solution. Finite difference schemes
that simultaneously conserve vorticity, kinetic energy, and enstrophy have been
designed. They are, however, rather complex. For our purposes it is sufficient
to note that by writing the advection in the flux form (13.27) and using cen-
tered space differences we can conserve both mean vorticity and mean kinetic
energy:
2d u m + 1,n ζ m + 1,n
u m 1,n ζ m 1,n
1
F m,n =
+ v m,n + 1 ζ m,n + 1
v m,n 1 ζ m,n 1 +
βv m,n
(13.30)
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