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12.4.2. Asymptotic Solution
where
ψ ( 0 )
0
Q
4 α R b
R w
1
ln (R b /R w )
+ 1
In general, (12.19) is not analytically tractable, so
instead we obtain an asymptotic solution under the
assumption of slow variations in azimuth and time. We
first nondimensionalize (12.19) using the channel width
L = R c
F =
2 R 2 ln R b
ln R b
R h
.
R 2
R h + 1
+ 1
4
1
2 R h
R
R w as a length scale and Q 1 as a time scale,
(12.25)
Substituting (12.24) and (12.25) into (12.20) yields a
nondispersive nonlinear wave equation for R(θ , t) , anal-
ogous to that studied by Haynes et al. [1993]. Solutions
of this equation rapidly form shocks, so following Clarke
and Johnson [1999], we continue the asymptotic solution
to introduce dispersive terms,
ψ ( 1 )
Q
r , t = Q 1
q , ψ = QL 2
ψ .
t , q = Q
r = L
ˆ
ˆ
(12.21)
Here hats denotes dimensionless variables. We then rescale
under the assumption that azimuthal variations are char-
acterized by 2 πR h , the length of the channel at the
shelf line. The parameter μ = (L/ 2 πR h ) 2 , assumed to
be asymptotically small, measures the ratio of radial to
azimuthal length scales. We further assume that the flow
evolves on a time scale that is
6 F θθ ln 2 r
2 R θ ln 2 r
1
1
=
H
(r
R)
R w
R
R) + G ln r
R w
,
6 (RR θ ) θ ln 3 r
1 / 2 ) longer than Q 1 ,
consistent with a velocity scale of QL and a length scale
of 2 πR h . This motivates rescaling θ and t as
+ 1
O
H
(r
R
(12.26)
where
ψ ( 1 )
0
Q
6 F θθ ln 2 R b
θ = μ 1 / 2 φ ,
t = μ 1 / 2 τ .
(12.22)
1
ln (R b /R w )
+ 1
G =
R w
2 R θ ln 2 R b
6 (RR θ ) θ ln 3 R b
. (12.27)
Under this scaling, (12.19) and (12.20) become
+ 1
1
R
R
r + ψ
+ μ
ˆ
Higher-order corrections may be obtained by continuing
the asymptotic solution, but the calculus becomes pro-
hibitively complicated. Substituting (12.24)-(12.27) into
(12.20) yields the nonlinear shelf wave equation,
∂R
∂t =
ˆ
r
ψ
r 2 ψ φφ =
H
(r
R)
H
(r
R h )
α ,
(12.23a)
r
ˆ
ˆ
r
ˆ
∂φ ψ R(φ , τ) , φ , τ , (12.23b)
∂ R
∂τ =
1
R
R w ) + F ln R
R w
Q
R
∂θ
1
4 α(R 2
2 R h ln R
1
4 (R 2
R h ) + 1
The asymptotic parameter μ does not enter (12.23b) and
only multiplies the azimuthal derivative in (12.23a).
We proceed by posing an asymptotic expansion of the
stream function, ψ = ψ ( 0 ) + μ ψ ( 1 ) +
R h )
H
(R
H
(R
R h )
R h
+ γ
6 F θθ ln 2 R
+ G ln (R/R w ) .
1
. We solve (12.23a)
at successive orders in μ subject to (12.13) in the form
ψ ( 0 ) =
···
R w
(12.28)
Here γ is simply a switch for the dispersive terms due
to the first-order stream function (12.26). Setting γ =0
recovers the leading-order nondispersive wave equation,
while γ = 1 yields the full dispersive wave equation.
0 (t) , ψ ( 1 ) =
ψ ( 1 0 (t) on r = R b . For notational simplicity we present
the solution in dimensional variables, writing ψ = ψ ( 0 ) +
ψ ( 1 ) +
ψ ( 1 ) =0on r = R w and
ψ ( 0 ) =
ψ ( 0 )
···
. The leading-order stream function is
R w ) + F ln r
1
4 α(r 2
12.4.3. Azimuthal Transport
ψ ( 0 ) /Q =
R w
The nonlinear shelf wave equation (12.28) is closed
except for the stream function ψ 0 = ψ ( 0 )
2 R 2 ln r
+ 1
1
4 (r 2
R 2 )
0 + ψ ( 1 ) 0 on
the outer boundary, or equivalently the along-channel
transport. We choose to constrain the transport using
(12.14b) because (12.14a) is complicated by the bump
in the outer wall. Under our asymptotic expansion, the
stream function must satisfy (12.14b) at every order in μ .
H
(r
R)
H
(r
R)
R
1
4 (r 2
R h )
H
(r
R h )
ln r
R h
+ 1
2 R h
H
(r
R h ) ,
(12.24)
 
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