Internal Undular Bores in the Coastal Ocean
27
constants, and the latter yields the well-known adiabatic expression a 3 ∝ í µí»¼. However, this requires taking the limit k → 0 which is inconsistent with the assumption
that the width scale of the wave should be much less than that of the variable medium
defined here by the variation in í µí»¼. Instead, as is now well-known, see the reviews by
[10, 14, 15], a multi-scale asymptotic expansion for the solitary wave should be
used, which confirms the adiabatic expression a
3
∝ í µí»¼ due to conservation of wave
action flux, but also reveals that the deforming solitary wave is accompanied by a
trailing shelf needed to conserve the total mass. Assuming here that í µí»¼ > 0, so that
the solitary wave is one of elevation, the trailing shelf has an amplitude at the solitary wave location proportional to í µí»¼ −8∕3 í µí»¼ í µí¼ and hence is an elevation or depression
shelf according as í µí»¼ is increasing or decreasing, í µí»¼ í µí¼ > 0 or í µí»¼ í µí¼ < 0. An analogous
result holds when í µí»¼ < 0. The essential difference between the solitary wave and the
solitary wave train is that in the latter, the mass is represented by the independent
parameter d, whereas the solitary wave has only one parameter, say the amplitude a
whose variation is already determined, while the solitary wave mass is 2a∕í µí»¾ and is
then not a constant and varies as |í µí»¼| −1∕3 .
Of special interest is the case when there is a change of polarity, that is, there
is a critical point í µí¼ = í µí¼ c where í µí»¼ changes sign, say from í µí»¼ > 0 to í µí»¼ < 0, implying
that solitary waves are waves of elevation for í µí¼ < í µí¼ c and waves of depression for
í µí¼ > í µí¼ c . The behaviour of a single solitary wave as it passes through this critical point
is now well understood, see the reviews by [10, 14, 15] and the recent study by
[16]. As the solitary wave approaches the critical point, its amplitude decreases as
í µí»¼
1∕3 but at the same time the amplitude of the trailing shelf of depression grows as
í µí»¼ −8∕3 . Close to the critical point, when the solitary wave and the trailing shelf have
comparable amplitudes, the adiabatic behaviour breaks down. The whole structure
passes through the critical point and in í µí¼ > í µí¼ c generates a depression rarefaction
wave connected to the original zero level by an undular bore of elevation waves. The
modulation theory for a solitary wave train presented above can be used to describe
both processes before and after the critical point.
When í µí»¼ is a constant, taken here as positive without loss of generality, a full representation of an undular bore can be found by seeking a similarity solution of the
Whitham modulation equations (30, 31, 32), where the modulus m depends only on
X∕í µí»¼U 0 í µí¼, see [6, 18, 29] and the review by [4]. This describes an expanding wave
train connecting a zero level at the front where m → 1 to a mean level U 0 > 0 at the
rear where m → 0. At the front the leading wave is a solitary wave of amplitude 2U 0
and at the rear the waves are linear sinusoidal waves. However, in a variable medium,
when as here í µí»¼ = í µí»¼(í µí¼), although the Whitham modulation equations are again available, see the Appendix, it would seem that no such simple wave solution is available
to describe the evolution of an undular bore. A recent study by [5] of a water wave
undular bore propagating up a slope (that is í µí»¼ > 0 increases) demonstrated that the
deformation at the front of the undular bore is essentially non-adiabatic. Briefly, it
is argued that if the undular bore retains its structure as a single-phase wave train,
then the jump U 0 is preserved, and so then the leading solitary wave would have a
constant amplitude 2U 0 . But this is inconsistent with the result that the leading solitary wave amplitude should behave as í µí»¼
1∕3 . The resolution of this inconsistency is
27
constants, and the latter yields the well-known adiabatic expression a 3 ∝ í µí»¼. However, this requires taking the limit k → 0 which is inconsistent with the assumption
that the width scale of the wave should be much less than that of the variable medium
defined here by the variation in í µí»¼. Instead, as is now well-known, see the reviews by
[10, 14, 15], a multi-scale asymptotic expansion for the solitary wave should be
used, which confirms the adiabatic expression a
3
∝ í µí»¼ due to conservation of wave
action flux, but also reveals that the deforming solitary wave is accompanied by a
trailing shelf needed to conserve the total mass. Assuming here that í µí»¼ > 0, so that
the solitary wave is one of elevation, the trailing shelf has an amplitude at the solitary wave location proportional to í µí»¼ −8∕3 í µí»¼ í µí¼ and hence is an elevation or depression
shelf according as í µí»¼ is increasing or decreasing, í µí»¼ í µí¼ > 0 or í µí»¼ í µí¼ < 0. An analogous
result holds when í µí»¼ < 0. The essential difference between the solitary wave and the
solitary wave train is that in the latter, the mass is represented by the independent
parameter d, whereas the solitary wave has only one parameter, say the amplitude a
whose variation is already determined, while the solitary wave mass is 2a∕í µí»¾ and is
then not a constant and varies as |í µí»¼| −1∕3 .
Of special interest is the case when there is a change of polarity, that is, there
is a critical point í µí¼ = í µí¼ c where í µí»¼ changes sign, say from í µí»¼ > 0 to í µí»¼ < 0, implying
that solitary waves are waves of elevation for í µí¼ < í µí¼ c and waves of depression for
í µí¼ > í µí¼ c . The behaviour of a single solitary wave as it passes through this critical point
is now well understood, see the reviews by [10, 14, 15] and the recent study by
[16]. As the solitary wave approaches the critical point, its amplitude decreases as
í µí»¼
1∕3 but at the same time the amplitude of the trailing shelf of depression grows as
í µí»¼ −8∕3 . Close to the critical point, when the solitary wave and the trailing shelf have
comparable amplitudes, the adiabatic behaviour breaks down. The whole structure
passes through the critical point and in í µí¼ > í µí¼ c generates a depression rarefaction
wave connected to the original zero level by an undular bore of elevation waves. The
modulation theory for a solitary wave train presented above can be used to describe
both processes before and after the critical point.
When í µí»¼ is a constant, taken here as positive without loss of generality, a full representation of an undular bore can be found by seeking a similarity solution of the
Whitham modulation equations (30, 31, 32), where the modulus m depends only on
X∕í µí»¼U 0 í µí¼, see [6, 18, 29] and the review by [4]. This describes an expanding wave
train connecting a zero level at the front where m → 1 to a mean level U 0 > 0 at the
rear where m → 0. At the front the leading wave is a solitary wave of amplitude 2U 0
and at the rear the waves are linear sinusoidal waves. However, in a variable medium,
when as here í µí»¼ = í µí»¼(í µí¼), although the Whitham modulation equations are again available, see the Appendix, it would seem that no such simple wave solution is available
to describe the evolution of an undular bore. A recent study by [5] of a water wave
undular bore propagating up a slope (that is í µí»¼ > 0 increases) demonstrated that the
deformation at the front of the undular bore is essentially non-adiabatic. Briefly, it
is argued that if the undular bore retains its structure as a single-phase wave train,
then the jump U 0 is preserved, and so then the leading solitary wave would have a
constant amplitude 2U 0 . But this is inconsistent with the result that the leading solitary wave amplitude should behave as í µí»¼
1∕3 . The resolution of this inconsistency is