Surface current data file
An animation of the surface current evolution can be viewed by clicking the
image below
![surface current evolution](shallow02b/shallow02_current-poster.jpg)
The soliton propagates at nearly constant speed and amplitude and the
surface current follows suit. A very slight dispersive leading edge
develops and propagates out in front of the main surface current pulse.
The particle vertical displacement and density perturbation profiles at
the center of the soliton (where the displacements are maximized) are shown
in the following two plots.
![perturbation density profile](shallow02b/rho_case1.png)
Comparing with the Study 3 results, we see that the phase speed, the maximum
surface current, and the shape of the surface current pulse are all fairly
similar. The main difference is in the width of the surface current pulse.
The pulse in Study 3 has a half maximum half width of approximately 190 m,
whereas Study 4 has a half width of about 1680 m. This is larger by a factor
of 8.8. This substantial difference is related to differences in the
shapes of the stability ducts for the two cases. The plot below shows a
comparison of the buoyancy frequencies for the two cases.
![buoyancy frequency](compare_N.png)
Indeed there are marked differences between the two ducts. The Yang winter
density data is fit well with a function that produces a symmetric duct
sandwiched between regions of neutral stability. The Case 1 duct, on the
other hand, is quit asymmetric and contains no region of neutral stability
on either side. The gradual reduction in buoyancy frequency with depth
for Case 1 results in an effective duct with that is much greater than
for Yang's data.
According to KDV theory the soliton horizontal wavelength is given by
where
![](phi.png)
is the modal function for vertical displacement,
![](eta.png)
is the maximum displacement, and
![](D.png)
is
the depth of water containing the soliton. The modal function satisfies
the equation
and thus depends only on the buoyancy frequency,
![](bvf.png)
, and
the boundary conditions. The soliton phase speed,
![](speed.png)
,
appears as an eigenvalue. The above two equations demonstrate that the
soliton horizontal wavelength depends on the maximum displacement amplitude
as well as the indicated moments of the modal function, with the latter
depending on the buoyancy frequency profile. Since the Yang winter profile
soliton and the Case 1 soliton have the same target maximum displacement
amplitude (30 m), differences in the horizontal wavelength must arise from
differences in the moments of the modal functions. In order to investigate
this line of reasoning further, the modal functions and their derivatives
are plotted below.
![modal function derivative cubed](yang02a/dmodal3.png)
These plots demonstrate that the most significant difference between the two
soliton cases is in the
![](deriv3.png)
profile, which appears in the
denominator of the expression for the horizontal wavelength. The Yang winter
profile soliton has much larger
![](deriv3.png)
values, which result
in a significantly smaller horizontal wavelength as compared to the Case 1
soliton. In simpler terms, the relatively narrow Yang winter stability duct
results in a more compact modal function, thus producing a larger first
derivative, which leads to a smaller horizontal wavelength.
The above analysis demonstrates that the horizontal wavelength is a fairly
non-linear function of the stability duct width.
References