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+
2
|
(f /H)
=
∂
u
∂t
+
q
annulus. However, repeated measurements are not avail-
able to provide an accurate parameterization of
κ
, nor to
explain the apparent discrepancy in
κ
between deeper and
shallower waters.
2
ˆ
|
e
z
×
u
+
∇
u
−
−
κ
u
, (12.2)
−∇ ×
ψ
ˆ
ˆ
ˆ
where
u
=
e
θ
is the depth-
independent horizontal velocity and
is the density-
normalized, ageostrophic pressure at the rigid lid,
z
=
H
.
The PV is defined as
e
z
=
u
e
r
+
v
12.3.2. Initial Conditions
q
=
ζ
+
fh(r)
As described in Section 12.2, we initiate our experiments
by changing the tank's Coriolis parameter from
f
H
,
(12.3)
f
to
f
. This acceleration is relatively rapid, requiring only
1-2s, so for the purposes of our model we treat it as
an instantaneous modification of the fluid velocity in the
rotating frame between
t
=0
−
and
t
=0
+
. This avoids
reformulating the QG equations for a frame rotating with
variable velocity.
Relinquishing the QG approximation momentarily, we
note that at
t
=0
−
the fluid is in solid-body rotation with
the tank and so adheres exactly to columnar motion. The
absolute vertical vorticity
ζ
a
of any fluid column is then
equal to that of the tank,
−
where
h(r)
is the prescribed height of the bottom topog-
raphy, given by
⎧
⎨
1
0,
r
≤
R
h
−
2
W
s
,
r
+
2
W
s
−
R
h
h(r)
=
H
s
(12.4)
1
,
|
r
−
R
h
|≤
2
W
s
,
⎩
W
s
R
h
+
2
W
s
≤
1,
r
,
as shown in Figure 12.1. Finally,
is the first-order
transport stream function
H
u
ag
−
h(r)
u
=
−∇ ×
ˆ
e
z
,
(12.5)
where
u
ag
is the (unknown) ageostrophic correction to the
velocity. Taking the curl of (12.2) yields a material con-
servation law for PV, as modified by friction proportional
to
κ
,
ζ
a
|
t
=0
−
=
f
−
f
.
(12.8)
The acceleration of the tank modifies the absolute vor-
ticity of its walls and base but leaves the absolute vortic-
ity of the fluid instantaneously unchanged. Formally, we
require that
Dq
Dt
=
2
ψ
.
−
κζ
,
ζ
=
∇
(12.6)
ζ
a
|
t
=0
+
=
f
+
ζ
|
t
=0
+
=
ζ
a
|
t
=0
−
,
(12.9)
Here D
/
D
t
≡
∂/∂t
+
u
·∇
is the advective derivative and
ζ
is the relative vorticity.
Motivated by our experimental observations, we have
neglected any lateral viscous dissipation in (12.2) because
bottom friction removes energy from the flow much more
rapidly. The action of bottom friction is represented by
a linear drag with the constant rate
κ
set by QG theory
[
Pedlosky
, 1987],
so our initial condition for the QG relative vorticity is
ζ(r
,
θ
,0
+
)
=
−
f
.
(12.10)
That is, the fluid acquires a relative vorticity that is every-
where equal to the change in the Coriolis parameter. One
could follow a similar line of reasoning under the QG
approximation, but the small-Rossby-number approxima-
tion of the PV (12.3) would introduce an
κ
=
A
v
f
H
O
(H
s
/H)
error
.
(12.7)
in (12.10).
Although (12.10) provides an initial condition for
ζ
,
and therefore for
q
, the initial stream function
ψ(r
,
θ
,0
+
)
is not uniquely determined by (12.10) alone. To invert
(12.6) for
ψ
at
t
=0
+
, we require the stream function
ψ
0
(
0
+
)
on
r
=
R
b
, which corresponds to the along-channel
transport. We obtain
ψ
0
under the QG approximation
by considering the total kinetic energy of the fluid in an
inertial frame,
E(t)
=
1
2
The vertical viscosity
A
v
is here simply the molecular vis-
cosity
ν
=1
10
−
6
m
2
/
s.
Ibbetson and Phillips
[1967]
found that the damping of Rossby waves in a similarly
sized annulus was accurately described by (12.7).
We have separately conducted experiments without the
bump in the outer wall, in which we released dye along a
line of constant azimuth and then reduced
f
from 1.5r to
1.3rad
/
s. This bestows upon the fluid a uniform angular
velocity that may be expected to decay with an
e
-folding
time of
κ
−
1
. We estimate that, off the shelf,
κ
is around
1.8 times smaller than the theoretical value (12.7), while
on the shelf it is around 1.5 times larger than (12.7). It
is not clear why the variation in
κ
should be so large;
simply replacing
H
by the actual depth
H
×
u
2
+
(V
+
v)
2
dA
,
(12.11)
A
where
V
is the azimuthal velocity due to the rotation of
the tank and
A
denotes the area of the annulus. In our
model, only bottom friction can extract energy from the
fluid, and we assume that this may be neglected during
the rapid acceleration of the tank. We therefore require
h
in (12.7)
cannot account for this. Nonetheless, (12.7) is a reason-
able approximation to the bottom friction over the whole
−
