Geoscience Reference
In-Depth Information
(a)
(b)
(c)
(d)
(e)
(f)
Figure 12.4.
Solution of the nonlinear shelf wave equation (12.28) using parameters that correspond to the reference experiment
in Figure 12.3. The thin solid lines show the positions of the inner and outer walls of the annulus, and the dashed line indicates the
width of the protrusion. The thick solid line indicates the position of the PV front at (a)
t
= 0 s, (b)
t
=17s,(c)
t
=29s,(d)
t
=45s,
(e)
t
= 66 s, and (f)
t
= 104 s.
12.5. NUMERICAL SOLUTION OF INVISCID
QUASI-GEOSTROPHIC EQUATIONS
In this section we outline our algorithm for solving
the QG model equations numerically. This approach is
motivated in part by the apparent failure of the long-
wave approximation employed in Section 12.4 to describe
the behavior of our laboratory experiments. Additionally,
these numerical solutions allow us to identify and explain
deviations of the experimental results from QG theory
and thus extrapolate to the behavior of coastal currents
in the real ocean.
To ensure stability, we modify the QG model equations
(12.6) to include a numerical viscosity term that smooths
vorticity gradients at the scale of the numerical grid,
∂q
∂t
=
Figure 12.5.
Solution at
t
= 8.7 s of the nondispersive wave
equation corresponding to
γ
= 0 in (12.28) using parameters
appropriate to the reference experiment.
this is formally a less accurate approximation. Figure 12.5
shows the computed solution to this equation on a grid
of 7200 points in azimuth at
t
= 8.7s, around which time
the interface
R(θ
,
t)
forms a shock. This solution resem-
bles the experimental wave shown in Figure 12.3c, but the
wave breaking takes place much earlier in the evolution
of the PV front. This may be due to the approximation of
the slope as a step, which leads to a larger mean relative
vorticity within the theoretical wave envelope.
2
ζ
,
−
J(ψ
,
q)
−
κζ
+
A
n
∇
(12.34a)
fh
H
.
2
ψ
=
q
∇
−
(12.34b)
Here
J
is the two-dimensional Jacobian operator,
q
and
ζ
are related via (12.3), and
A
n
measures the numerical
