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characteristic scale of the inversion and
L
x
,
L
z
are the characteristics of a ducted
gravity wave with initial amplitude
A
.If
N
−
0
is the fundamental time scale
and
h
is the fundamental length scale, the following independent dimensionless
parameters then characterize the problem:
L
x
=
L
z
=
δ
=
N
1
/
N
0
,
L
x
/
h
,
L
z
/
h
,
A
N
0
h
2
, which is the initial dimensionless amplitude of the stream
and,
=
A
/
function,
ϕ
. The nondimensional model is then
∂
∂
=−
∂θ
∂
2
2
4
t
∇
⊥
ϕ
+
ϕ,
∇
⊥
ϕ
x
+
μ
∇
⊥
ϕ
∂
∂
)
∂ϕ
∂
N
2
t
θ
+[
ϕ, θ
]=
(
z
x
θ
where
is the potential temperature,
]=
∂
A
∂
x
∂
B
∂
−
∂
B
∂
x
∂
A
∂
[
A
,
B
z
z
and
is a dimensionless kinematic viscosity.
For the numerical solutions, Seyler used a smooth approximation to a sharp
profile,
N
2
μ
2
z
4
(
z
)
=
δ
+
exp
(
−
)
, where
δ
=
N
1
/
N
0
. The initial conditions on
ϕ
and
θ
were chosen to be of the form
A
sin
2
sin
n
x
L
x
π
z
L
z
π
ψ(
x
,
z
,
0
)
=
,
n
=
1
,
2
,
and,
sin
2
sin
n
,
2
π
AN
1
L
z
x
L
x
π
π
z
θ(
x
,
z
,
0
)
=−
n
=
1
,
2
,
L
z
This initial condition is related to a ducted wave since it is a superposition
of oblique right-upward and right-downward propagating gravity waves with a
frequency below the background cutoff frequency
N
1
.
Table 7.1 provides a summary for 8 of the numerical simulations investigated.
All of the runs listed in the table form bores, although in run 8, a single peak
was formed and, strictly speaking, this should be considered a solitary wave.
The wavelength
λ
x
was determined approximately
3
4
of the way through the
runs and the phase speed was averaged over the last half of the runs. The initial
wavelength at the time of peak formation is considerably smaller.
Figure 7.25a is a set of contour plots of the stream function given at four uni-
formly spaced times for run 1. In the upper left panel at time 200, the distortion
of the initial “ducted” gravity wave is evident. At time 400, shown in the upper
right panel, three peaks have formed. Peak formation continues to occur in the
bottom left and right panels at times 600 and 800, respectively. A maximum of
8 peaks was generated for this case. The propagation is from left to right and
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