Geography Reference
In-Depth Information
13.11.
The
Dufort-Frankel
method
for
approximating
the
one-dimensional
diffusion equation
K
∂
2
q
∂x
2
can be expressed in the notation of Section 13.3.2 as
∂q
∂t
=
r
ˆ
q
m
+
1,s
−
ˆ
q
m,s
−
1
+ˆ
q
m
−
1,s
q
m,s
+
1
=ˆ
ˆ
q
m,s
−
1
+
q
m,s
+
1
+ˆ
2Kδt/(δx)
2
. Show that this scheme is an explicit differencing
scheme and that it is computationally stable for all values of δt.
where r
≡
13.12.
Starting with the assumed solution (13.61), obtain the normal mode solu-
tions to the linearized versions of (13.58)-(13.60) and hence verify (13.62)
and (13.63).
13.13.
Show that the projection of the observations onto the Rossby normal mode
(13.66) is equivalent to requiring that the quasi-geostrophic potential vor-
ticity of the Rossby mode be proportional to the sum of the observed
relative vorticity minus the observed geopotential multiplied by the factor
(f/c
2
).
Hint
: Linearize (4.26) and assume that f is constant.
13.14.
Derive the best estimate of T given in (13.70) based on minimization of
the cost function of (13.69).
MATLAB EXERCISES
M13.1.
The MATLAB script
finite diff1.m
can be used to compare the cen-
tered difference first derivative formula (13.4) for the function ψ (x)
=
sin (πx/4) to the analytic expression dψ/dx
=
(π/4) cos (πx/4) for
various numbers of grid intervals in the domain
4. Graph the
maximum error as a function of number of grid intervals,
ngrid
,inthe
range of 4 to 64. Carry out a similar analysis for the second derivative.
(Note that first-order differencing is used next to the boundaries.)
−
4
≤
x
≤
M13.2.
Modify the script of M13.1 to evaluate the error in the first derivative of
the function tanh (x) using the same domain in x.
M13.3.
The MATLAB script
advect 1.m
demonstrates the leapfrog differencing
scheme for the one-dimensional advection equation as given in (13.8).
By running the script for various grid intervals, find the dependence of
phase error (degrees per wave period) on the number of grid intervals per
wavelength for the range 4 to 64 and compare your results with Table 13.1.
