Geoscience Reference
In-Depth Information
In dimensionless variables, for the dissipation-free case
κ = 0, this implies
it has crossed the shelf line [ Stewart , 2010]. Instead, we
employ a crude representation of bottom friction to pro-
vide a source of dissipation in (12.28). We modify the PV
equation (12.6) to an exact material conservation law,
r = R w
= 2 π
0
∂ψ (n)
∂r
d
dt (n)
=0, (n)
w
R w ,
(12.29)
w
ζ e κt + fH s
H
= 0.
D
Dt
(12.33)
for all n
. Substituting (12.24) and (12.26)
into (12.29) yields evolution equations for ψ ( 0 )
0
∈{
0,1, ...
}
and ψ ( 1 )
0
,
respectively. For example, for n = 0 we obtain
This representation possesses identical conservation laws
to (12.6) and (12.2) for the total vorticity and total energy
in the annulus. In fact, its dynamics are identical to (12.6)
away from the shelf line, where vorticity simply decays
exponentially with rate κ . Fluid columns crossing the shelf
line acquire a relative vorticity of constant magnitude
2 π
ln (R b /R)
ln (R b /R w ) RR t
2 π
( 0 )
0
dt
0
=
Q
.
(12.30)
ln (R b /R w )
|
under (12.6), whereas under (12.33) they acquire a relative
vorticity of magnitude
|
Q
0
We omit the corresponding expression for ψ ( 1 ) for the sake
of brevity. Equation (12.30) describes the tendency of net
radial vorticity fluxes to modify the along-channel trans-
port. It is a consequence of our channel's finite length that
the wave equation (12.28) acquires contributions that are
dependent on the global behavior of the solution, whereas
that of Clarke and Johnson [1999] in an infinite channel
did not.
To complete the evolution of ψ 0 (t) , we must also deter-
mine its initial condition following Section 12.3.2. Using
R(θ ,0 ) = R h , equation (12.24) approximates the initial
azimuthal velocity as
e κt .
All of the results discussed in this section may be
rederived using (12.33). The nonlinear shelf wave equa-
tion acquires an additional factor of e κt multiplying the
right-hand side of (12.28), while the azimuthal transport
equation (12.30) is unchanged. The latter may seem con-
tradictory, as the azimuthal transport must decay due to
bottom friction. This is because ψ is a stream function for
the modified vorticity ζ e κt ,andso ψ 0 differs from the true
along-channel transport by a factor of e κt .
|
Q
|
t =0
12.4.5. Comparison with Experimental Flows
2 f r + ψ ( 0 )
( 0 ) + ( 1 / 4 ) f (R b
R w )
∂ψ ( 0 )
∂r
1
0
=
.
In Figure 12.4 we plot snapshots of a solution to (12.28)
using parameters that correspond to the reference experi-
ment shown in Figure 12.3. We solve (12.28) numerically
by discretizing R on an azimuthal grid of 400 equally
spaced points covering θ
r ln (R b /R w )
(12.31)
To a consistent order of approximation, we must omit the
radial velocity ( u 2 ) term in (12.12b), so the initial energy
of the flow is
E ( + ) = 2
A
∈[
0,2 π) . We evaluate azimuthal
derivatives spectrally via the fast Fourier transform, and
we integrate (12.28) forward in time using third-order
Adams-Bashforth time stepping. We employ an exponen-
tial Fourier filter [ Hou and Li , 2007] to damp the mild
instability that arises at high wave numbers due to aliasing
error [ Boyd , 2001].
Having assumed that the PV front remains a single-
valued function of θ , it is perhaps unsurprising that
the solution shown in Figure 12.4 is unable to capture the
wave breaking shown in Figure 12.3. A large-amplitude
wave develops in the lee of the bump in the outer wall,
but instead of breaking it develops a dispersive wave train
that spreads clockwise around the tank due to the strong
retrograde mean flow. Although the mean flow decelerates
rapidly due to bottom friction, the volume of water trans-
ported across the shelf line continues to increase even at
t = 104s, and the wave envelope does not collapse as in
Figure 12.3.
The wave breaking shown in Figure 12.3c is actually
captured more accurately by the nondispersive wave equa-
tion, corresponding to γ = 0 in (12.28), even though
t =0
2 (f + f )r + ∂ψ ( 0 )
2
dA . (12.32)
∂r
Thus we may determine ψ ( 0 0 ( 0 ) by substituting (12.31)
into (12.32) and solving E ( + ) = E ( ) . For example, in a
regular annulus with R b (θ)
R c , equation (12.31) gives
the exact initial stream function ( ψ ( 0 )
| t =0
ψ
| t =0 ), and
1
4 f (R c
R w ) . The corre-
this procedure yields ψ 0 ( 0 ) =
1
sponding azimuthal velocity v
2 f r has uniform
angular velocity, which is the expected response of the
flow to a change in the tank's rotation rate.
| t =0 =
12.4.4. Parameterizing Bottom Friction
Our neglect of bottom friction in (12.28) causes its
solutions to diverge substantially from our experimen-
tal results. It is not possible to introduce κ exactly in
our nonlinear wave theory because the vorticity of each
fluid column depends sensitively on the times at which
 
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