Geoscience Reference
In-Depth Information
viscosity. We set
A
n
=
Qd
2
so that numerical diffusion
becomes comparable to advection at the scale of one grid
box
d
over a dynamical time scale of
Q
−
1
. Thus
A
n
is a
purely numerical contribution to (12.34a), because
A
n
→
0as
d
and replacing derivatives with second-order central differ-
ences. We approximate the Laplacian operator in (12.34a)
and (12.34b) using straightforward second-order central
differencing in
ρ
,
φ
coordinates.
Our numerical scheme is simplified by the fact that
the initial vorticity (12.10) is uniform. As the numeri-
cal viscosity acts only in the interior, (12.34a) states that
the vorticity remains constant along the boundary but
decays exponentially with time scale
κ
−
1
. This serves as
a boundary condition for the relative vorticity, which may
be written in
ρ
,
φ
coordinates as
0. The numerical viscosity is not intended to
represent the fluid's molecular viscosity, and (12.34a) and
(12.34b) describe inviscid flow with numerical diffusion
that acts only on the interior vorticity field. We therefore
retain free-slip, rather than no-slip, boundary conditions
at the channel walls.
→
12.5.1. Wall-Following Coordinates
dζ
0
dt
ζ (R
w
,
φ
,
t)
=
ζ (R
c
,
φ
,
t)
=
ζ
0
(t)
,
=
−
κζ
0
. (12.38)
The protrusion (12.1) in the outer wall of our annu-
lus means that we cannot discretize the channel using a
regularly spaced grid in
r/θ
coordinates without adapta-
tion. We therefore transform the QG equations (12.34a)
and (12.34b) into coordinates that follow the walls of the
annulus,
ρ
=
R
w
+
(r
with
ζ
0
(
0
)
=
f
. To simplify the presentation of our
numerical scheme, we will assume for the moment that the
stream function
ψ
0
(t)
on the outer wall is also a known
function of time. We will explain how we evolve
ψ
0
in
Section 12.5.3.
We evolve the PV
q
m
,
n
and stream function
ψ
m
,
n
for-
ward in time as follows. Given
q
m
,
n
at all grid points at
any time
t
, we first invert (12.34b) iteratively via succes-
sive overrelaxation to determine
ψ
m
,
n
at all interior points
subject to
−
−
R
w
)(R
c
−
R
w
)
,
φ
=
θ
.
(12.35)
R
b
−
R
w
Here
ρ
is simply a rescaled radial coordinate that satisfies
ρ
=
R
w
on
r
=
R
w
and
ρ
=
R
c
on
r
=
R
b
. Derivatives with
respect to
r
and
θ
may be transformed to
ρ
,
φ
space by
writing
q
=
q(ρ(r
,
θ)
,
φ(θ))
and applying the chain rule,
θ
φ
ψ
1,
n
(t)
=0,
ψ
N
ρ
,
n
(t)
=
ψ
0
(t)
,
n
=1,
...
,
N
φ
. (12.39)
∂q
∂r
=
R
c
−
R
w
∂q
∂ρ
,
(12.36a)
R
b
−
R
w
We then compute the right-hand side of (12.34a) at
all interior grid points utilizing the vorticity boundary
conditions
r
ρ
φ
∂q
∂θ
=
∂q
∂φ
ρ
−
R
w
dW
dφ
∂q
∂ρ
+
.
(12.36b)
R
b
−
R
w
For example, the Jacobian operator in (12.34a) becomes
ζ
1,
n
(t)
=
ζ
N
ρ
,
n
(t)
=
ζ
0
(t)
,
n
=1,
...
,
N
φ
,
(12.40)
∂ψ
∂ρ
,
(12.37)
1
r(ρ
,
φ)
R
c
−
R
w
∂q
∂φ
−
∂ψ
∂φ
∂q
∂ρ
J(ψ
,
q)
=
to evaluate derivatives in the grid rows
m
=2and
m
=
N
ρ
R
b
−
R
w
−
1. This yields the time derivative of
q
at all inte-
rior grid points, which we use to step
q
m
,
n
(t)
forward in
time using the third-order Adams-Bashforth scheme. We
ensure that the fixed time step
t
satisfies the advective
CFL condition throughout the integration.
For the purpose of comparison with our laboratory
experiments, we track the position of the PV front that
lies initially above the center of the slope. We accomplish
this by advecting
M
passive tracer particles at
(ρ
i
,
φ
i
)
for
i
=1,
...
,
M
using the computed stream function
ψ
m
,
n
(t)
.
These particles are initially spread evenly around the shelf
line, so
where
r(ρ
,
φ)
may be obtained by inverting (12.35). The
Laplacian operator
2
is considerably more complicated
and includes a second-order cross-derivative in
ρ
and
φ
.
We omit this expression for brevity.
∇
12.5.2. Numerical Integration
We discretize our dependent variables
ψ
,
ζ
,and
q
on
agridof
N
ρ
N
φ
points with regular spacings
ρ
and
φ
, respectively. We denote their positions as
ρ
m
for
m
=1,
...
,
N
ρ
and
φ
n
for
n
=1,
...
,
N
φ
,where
ρ
1
=
R
w
and
ρ
Nρ
=
R
c
. We denote variables stored at
(ρ
m
,
φ
n
)
as, for example,
ψ
m
,
n
(t)
, and for now we retain
a continuous dependence on time for notational con-
venience. We approximate the Jacobian (12.37) using a
second-order energy-conserving discretization [
Arakawa
,
1966], equivalent to rewriting (12.37) as a flux form
×
ρ
i
(
0
)
=
R
w
+
(R
h
−
R
w
)(R
c
−
R
w
)
,
φ
i
(
0
)
=
2
π i
M
.
(12.41)
R
b
[
φ
i
(
0
)
]−
R
w
Thereafter, at any time
t
the particle evolution is deter-
mined by
