Geography Reference
In-Depth Information
13.4
THE BAROTROPIC VORTICITY EQUATION IN FINITE
DIFFERENCES
The simplest example of a dynamical forecast model is the barotropic vorticity
equation (11.14), which for a Cartesian β plane can be written in the form
∂ζ
∂t
=−
F (x, y, t)
(13.26)
where
∂x
u
ψ
ζ
+
∂y
v
ψ
ζ
+
∂
∂
F (x, y, t)
=
V
ψ
·∇
(ζ
+
f )
=
βv
ψ
(13.27)
2
ψ . We have here used the fact
that the horizontal velocity is nondivergent (∂u
ψ
/∂x
and u
ψ
=−
∂ψ/∂y, v
ψ
=
∂ψ/∂x, and ζ
=∇
0) to write the
advection term in flux form. The advection of absolute vorticity F(x, y, t) may
be calculated provided that we know the field of ψ(x, y, t). Equation (13.26)
can then be integrated forward in time to yield a prediction for ζ . It is then
necessary to solve the Poisson equation ζ
+
∂v
ψ
/∂y
=
2
ψ to predict the stream-
=
∇
function.
A straightforward solution method is the leapfrog scheme discussed in Sec-
tion 13.3.2. This requires writing (13.27) in finite difference form. Suppose that
the horizontal x,y space is divided into a grid of (M
1) points
separated by distance increments δx and δy. Then we can write the coordinate
position of a given grid point as x
m
=
+
1)
×
(N
+
mδx, y
n
=
=
nδy, where m
0, 1, 2,...,M
and n
0, 1, 2,...,N. Thus any point on the grid is uniquely identified by the
indices (m, n). A portion of such a grid space is shown in Fig. 13.2.
Centered difference formulas of the type (13.4) can then be used to approximate
derivatives in the expression F(x, y, t). For example, if we assume that δx
=
=
δy
≡
d
u
m,n
=−
ψ
m,n
+
1
−
ψ
m,n
−
1
/2d
u
ψ
≈
v
m,n
=+
ψ
m
+
1,n
−
ψ
m
−
1,n
/2d
(13.28)
v
ψ
≈
Similarly, with the aid of (13.5) we find that the horizontal Laplacian can be
approximated as
≈
ψ
m
+
1,n
+
4ψ
m,n
/d
2
2
ψ
∇
ψ
m
−
1,n
+
ψ
m,n
+
1
+
ψ
m,n
−
1
−
=
ζ
m,n
(13.29)
The finite difference form of the Laplacian is proportional to the difference
between the value of the function at the central point and the average value at
the four surrounding grid points. If there are (M
−
1)
×
(N
−
1) interior grid
points, then (13.29) yields a set of (M
1) simultaneous equa-
tions, which together with suitable boundary conditions determine ψ
m,n
for a
−
1)
×
(N
−
