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conservation of total energy across
t
= 0, i.e.,
E
(
−
)
=
E
(
+
)
,
where
E
(
−
)
=
1
2
12.4. NONLINEAR SHELF WAVE THEORY
In interpreting the results of our laboratory experi-
ments, it is instructive to compare with the predictions
of the established nonlinear shelf wave theory [
Haynes
et al.
, 1993;
Clarke and Johnson
, 1999;
Johnson and Clarke
,
1999, 2001]. This theory approximates the experimen-
tal/continental slope as a step, equivalent to
W
s
1
2
fr
2
dA
,
(12.12a)
A
u
2
+
1
2
f
+
f
r
+
v
2
dA
. (12.12b)
E
(
+
)
=
1
2
0in
Figure 12.1, and describes the evolution of a fluid inter-
face that lies initially over the shelf line, corresponding
to the dye line in Figure 12.3. In this section we adapt
this approach to an annular channel, improving upon the
derivations of
Stewart
[2010] and
Stewart et al.
[2011] to
utilize the initial and boundary conditions described in
Sections 12.3.2 and 12.3.3, respectively.
→
A
Thus,
ψ
0
(
0
+
)
must be chosen such that (12.12a) and
(12.12b) are equal. For example, in a regular annulus
(
W
b
R
w
, which
corresponds to the intuitive result that the fluid acquires a
uniform angular velocity opposite to the direction of the
tank's acceleration. In general,
E
(
−
)
=
E
(
+
)
must be solved
numerically, but the long-wavelength asymptotic analysis
described in Section 12.4 provides a very accurate approx-
imate solution for the slowly varying channel shown in
Figure 12.2.
4
f
R
c
−
0), this yields
ψ
0
(
0
+
)
=
1
≡
12.4.1. Quasi-Geostrophic Dynamics Over a Step
0), the
height of the bottom topography in our annular channel
becomes
In the limit of a vanishingly narrow slope (
W
s
→
12.3.3. Boundary Conditions
h
=
H
s
H
−
We apply no-flux boundary conditions at the inner
and outer walls of the channel, requiring that both be
streamlines of
ψ
. Without loss of generality we choose
(r
R
h
)
,
(12.16)
where
denotes the Heaviside step function. Neglect-
ing the influence of bottom friction, the PV
q
is con-
served exactly on fluid columns, and the dynamics may be
described completely by the position of the interface that
lies initially above the shelf line. With a view to describing
waves whose length is much greater than their ampli-
tude, we make the a priori assumption that this interface
remains a single-valued function of azimuth and so may
be denoted
r
=
R(θ
,
t)
.At
t
=0,wehave
R(θ
,0
)
=
R
h
by
definition, and from (12.3) and (12.10) the initial PV is
H
ψ
(
R
w
,
θ
,
t)
=0,
ψ (R
b
,
θ
,
t)
=
ψ
0
(t)
,
(12.13)
so
ψ
0
corresponds to the counterclockwise along-channel
transport. In a regular annulus (
W
b
0) it is possible
to derive an analytical evolution equation for
ψ
0
(t)
that
excludes contributions from the unknown lid pressure
.
This is not possible when
W
b
≡
= 0, because the bump in
the outer wall may support an azimuthal pressure gradi-
ent that modifies the along-channel transport
ψ
0
. Instead,
we determine additional boundary conditions by consid-
ering the circulations
w
and
c
around the inner and
outer walls of the tank, respectively [
McWilliams
, 1977].
It follows from (12.2) that
d
dt
c
=
−
H
−
q(r
,
θ
,0
)
=
f
+
Q
(r
R
h
)
,
(12.17)
where
Q
=
fH
s
/H
. Thereafter, material conservation of
q
ensures that there is always a jump in PV at
r
=
R
,
c
=
q(r
,
θ
,
t)
=
−
f
+
Q
H
(r
−
R)
.
(12.18)
−
κ
c
,
u
·
d
r
,
(12.14a)
Using (12.3) and the definition of the relative vorticity
(12.6), this may be rearranged as a Poisson equation for
the stream function,
r
=
R
b
w
=
d
dt
w
=
−
κ
w
,
u
·
d
r
.
(12.14b)
r
=
R
w
2
ψ
=
Q
[
∇
−
α
+
H
(r
−
R)
−
H
(r
−
R
h
)
] ,
(12.19)
In fact, we only need to ensure that either (12.14a) or
(12.14b) is satisfied, because by Stokes's theorem
where we define
α
=
f
/Q
. Given the position of the
interface
r
=
R
, inverting (12.19) yields a complete descrip-
tion of the flow in the annulus at any time
t
subject to the
boundary equations described in Section 12.3.3. The evo-
lution of the interface position
R(θ
,
t)
is determined by
the requirement that particles on the interface remain on
the interface, i.e.,
(D/Dt)(r
w
=
A
d
dt
(
c
−
c
−
ζ dA
⇒
w
)
=
−
κ (
c
−
w
)
.
(12.15)
The second equation in (12.15) follows from integrating
(12.3) over the annulus. Condition (12.14b) determines the
evolution of the outer wall stream function
ψ
0
(t)
. We out-
line separate asymptotic and numerical strategies to solve
this problem in Sections 12.4 and 12.5, respectively.
−
R)
= 0. This condition may
be rewritten as
∂R
∂t
=
1
R
∂
∂θ
ψ(R(θ
,
t)
,
θ
,
t)
.
−
(12.20)
