34
R. Grimshaw and C. Yuan
the leading wave would have an amplitude of 2U 0 = 12 in this simulation, whereas
we see that instead the largest wave in the undular bore has an amplitude of about
10. Nevertheless, when taking this complication onto account, we see the formation
ahead of the undular bore of two solitary wave trains, and each has the expected
linear profile (22). At the end of the simulation at í µí¼− = í µí¼ a the leading wave has
an amplitude about 12, consistent with the asymptotic estimate 10í µí»¼
1∕3
a
= 11.45. We
interpret the presence of two wave solitary wave trains as due to the truncation at
the initial front causing some ambiguity about the which is the leading wave. But
we note that the leading wave in the second solitary wave train has an amplitude of
about 11, consistent with an asymptotic estimate of 9.5í µí»¼a 1∕3 where we see that 9.5
is the amplitude of the second highest wave in the initial undular bore.
In the second case, when í µí»¼ decreases to zero and changes sign after a critical
point, we show a typical numerical simulation of (13) with the initial condition (25)
in Fig. 4. As for the previous case the truncation at the head of the undular bore
does have an effect. Nevertheless, up to the critical point (middle panel) we see similar behaviour to that shown in Fig. 3 except that now the amplitudes in the emerging solitary wave trains decrease. After the critical point, the leading wave in these
emerging solitary wave trains has deformed in a manner similar to that described
above in Fig. 2 for a single solitary wave passing through a critical point. We can
clearly see the formation of an elevation rarefaction wave, and some still quite small
solitary waves of depression riding on this pedestal. On the other hand the rear of the
undular bore essentially retains its shape, and shows little evidence of any change in
amplitude. Essentially it is behaving as a linear wave. A more detailed analysis of
this and the previous case in Fig. 3 can be found in [16]. Hence the essential dynamics as described above on section “Numerical Simulations of a Process Model” is
not changed when expressed in terms of the original physical variables, although in
particular the amplitude magnitudes may be altered significantly.
Discussion
Our purpose in this brief review article is to show how the variable-coefficient
Korteweg-de Vries equation (7) can be used to model the propagation of internal
undular bores over the continental slope. As we noted in the Introduction there have
been many studies on the application of the variable-coefficient KdV equation (7),
or the transformed equation (13). to model the propagation of internal solitary waves
propagating over variable topography, see the reviews by [14, 15]. However, since
observations of oceanic internal solitary waves often show that they occur as part of
a wave train, it is desirable to consider the dynamics of an internal undular bore as a
whole. This was undertaken recently by [16] who used the variable-coefficient KdV
equation (13) to simulate the behaviour of internal undular bores propagating over
variable topography, and in this article we have reviewed this application in a briefer
and less detailed manner.
R. Grimshaw and C. Yuan
the leading wave would have an amplitude of 2U 0 = 12 in this simulation, whereas
we see that instead the largest wave in the undular bore has an amplitude of about
10. Nevertheless, when taking this complication onto account, we see the formation
ahead of the undular bore of two solitary wave trains, and each has the expected
linear profile (22). At the end of the simulation at í µí¼− = í µí¼ a the leading wave has
an amplitude about 12, consistent with the asymptotic estimate 10í µí»¼
1∕3
a
= 11.45. We
interpret the presence of two wave solitary wave trains as due to the truncation at
the initial front causing some ambiguity about the which is the leading wave. But
we note that the leading wave in the second solitary wave train has an amplitude of
about 11, consistent with an asymptotic estimate of 9.5í µí»¼a 1∕3 where we see that 9.5
is the amplitude of the second highest wave in the initial undular bore.
In the second case, when í µí»¼ decreases to zero and changes sign after a critical
point, we show a typical numerical simulation of (13) with the initial condition (25)
in Fig. 4. As for the previous case the truncation at the head of the undular bore
does have an effect. Nevertheless, up to the critical point (middle panel) we see similar behaviour to that shown in Fig. 3 except that now the amplitudes in the emerging solitary wave trains decrease. After the critical point, the leading wave in these
emerging solitary wave trains has deformed in a manner similar to that described
above in Fig. 2 for a single solitary wave passing through a critical point. We can
clearly see the formation of an elevation rarefaction wave, and some still quite small
solitary waves of depression riding on this pedestal. On the other hand the rear of the
undular bore essentially retains its shape, and shows little evidence of any change in
amplitude. Essentially it is behaving as a linear wave. A more detailed analysis of
this and the previous case in Fig. 3 can be found in [16]. Hence the essential dynamics as described above on section “Numerical Simulations of a Process Model” is
not changed when expressed in terms of the original physical variables, although in
particular the amplitude magnitudes may be altered significantly.
Discussion
Our purpose in this brief review article is to show how the variable-coefficient
Korteweg-de Vries equation (7) can be used to model the propagation of internal
undular bores over the continental slope. As we noted in the Introduction there have
been many studies on the application of the variable-coefficient KdV equation (7),
or the transformed equation (13). to model the propagation of internal solitary waves
propagating over variable topography, see the reviews by [14, 15]. However, since
observations of oceanic internal solitary waves often show that they occur as part of
a wave train, it is desirable to consider the dynamics of an internal undular bore as a
whole. This was undertaken recently by [16] who used the variable-coefficient KdV
equation (13) to simulate the behaviour of internal undular bores propagating over
variable topography, and in this article we have reviewed this application in a briefer
and less detailed manner.