28
R. Grimshaw and C. Yuan
the formation of a solitary wave train ahead of the undular bore. This solitary wave
train can be described by the reduced set of modulation equations (18, 19, 21), and
there is a region where the rear of the solitary wave train interacts with the undular
bore, forming a two-phase wave interaction. The rear part of the undular bore retains
its shape, where pertinently we note that in the similarity solution for the constant
coefficient case, í µí»¼ occurs only in the variable x∕í µí»¼U 0 í µí¼, implying that the effect of a
variable í µí»¼ is in some sense equivalent to adjusting the time scale. This leading solitary wave train can be described by a similarity solution of the modulation equations
(18, 19, 21),
d = 0 , a =
3í µí»¼ 1∕3 X
í µí¼
, X m (í µí¼) < X < X M (í µí¼) , í µí¼ =
í µí¼
∫
0
í µí»¼
4∕3
(í µí¼
′
) dí µí¼
′
.
(22)
At the head X = X M (í µí¼) of the wave train, the amplitude is a M = 3í µí»¼ 1∕3 X M ∕í µí¼, while
at the rear X = X m (í µí¼) > 0 of the wave train, the amplitude is a m = 3í µí»¼
1∕3 X m ∕í µí¼. The
speed of each solitary wave is V = í µí»¼a∕3 (17) which yields the asymptotic position
of each wave as X = Cí µí¼ where C is a “constant” which depends on the initial position of each solitary wave. This must be matched to the following undular bore to
determine X M,m (í µí¼), and the details of that are described by [5]. But note that if it
is assumed that the leading wave in the following undular bore has amplitude 2U 0
then we may expect that a m ≈ 2U 0 and a M ≈ 2U 0 í µí»¼
1∕3 , assuming that initially í µí»¼ = 1.
The dynamics of an undular bore as it passes through a critical point was examined
recently by [16]. Before the critical point, the behaviour of the leading solitary wave
train can again be described by the modulation theory as above, where now í µí»¼ > 0
decreases towards zero. The behaviour after the critical point is described below.
Numerical Simulations of a Process Model
Numerical simulation of an internal undular bore propagating up a slope were
reported by [16] using the transformed vKdv equation (13). Here we review and
supplement those results. For the numerical simulation we use a pseudo-spectral
method where the nonlinear term is evaluated in physical space.
We consider a process model to detect interesting dynamics, and choose the coefficient í µí»¼ in the transformed equation (13) to model internal waves propagating up a
slope, in two cases, one when the nonlinear coefficient í µí»¼ increases, and one when it
decreases with change of sign, causing a polarity change. Note there is no loss of generality in choosing the initial value of í µí»¼ > 0, as otherwise one one can make the transformation U → −U. Thus we set í µí»¼ = í µí»¼(í µí¼) varying monotonically from í µí»¼ = 1, í µí¼ = 0
to some constant value í µí»¼ = í µí»¼ a , í µí¼ ≥ í µí¼ a . Then there are two cases, either í µí»¼ a > 1, or
í µí»¼ a < 0 (a change of polarity). Specifically the coefficient í µí»¼ is given by
R. Grimshaw and C. Yuan
the formation of a solitary wave train ahead of the undular bore. This solitary wave
train can be described by the reduced set of modulation equations (18, 19, 21), and
there is a region where the rear of the solitary wave train interacts with the undular
bore, forming a two-phase wave interaction. The rear part of the undular bore retains
its shape, where pertinently we note that in the similarity solution for the constant
coefficient case, í µí»¼ occurs only in the variable x∕í µí»¼U 0 í µí¼, implying that the effect of a
variable í µí»¼ is in some sense equivalent to adjusting the time scale. This leading solitary wave train can be described by a similarity solution of the modulation equations
(18, 19, 21),
d = 0 , a =
3í µí»¼ 1∕3 X
í µí¼
, X m (í µí¼) < X < X M (í µí¼) , í µí¼ =
í µí¼
∫
0
í µí»¼
4∕3
(í µí¼
′
) dí µí¼
′
.
(22)
At the head X = X M (í µí¼) of the wave train, the amplitude is a M = 3í µí»¼ 1∕3 X M ∕í µí¼, while
at the rear X = X m (í µí¼) > 0 of the wave train, the amplitude is a m = 3í µí»¼
1∕3 X m ∕í µí¼. The
speed of each solitary wave is V = í µí»¼a∕3 (17) which yields the asymptotic position
of each wave as X = Cí µí¼ where C is a “constant” which depends on the initial position of each solitary wave. This must be matched to the following undular bore to
determine X M,m (í µí¼), and the details of that are described by [5]. But note that if it
is assumed that the leading wave in the following undular bore has amplitude 2U 0
then we may expect that a m ≈ 2U 0 and a M ≈ 2U 0 í µí»¼
1∕3 , assuming that initially í µí»¼ = 1.
The dynamics of an undular bore as it passes through a critical point was examined
recently by [16]. Before the critical point, the behaviour of the leading solitary wave
train can again be described by the modulation theory as above, where now í µí»¼ > 0
decreases towards zero. The behaviour after the critical point is described below.
Numerical Simulations of a Process Model
Numerical simulation of an internal undular bore propagating up a slope were
reported by [16] using the transformed vKdv equation (13). Here we review and
supplement those results. For the numerical simulation we use a pseudo-spectral
method where the nonlinear term is evaluated in physical space.
We consider a process model to detect interesting dynamics, and choose the coefficient í µí»¼ in the transformed equation (13) to model internal waves propagating up a
slope, in two cases, one when the nonlinear coefficient í µí»¼ increases, and one when it
decreases with change of sign, causing a polarity change. Note there is no loss of generality in choosing the initial value of í µí»¼ > 0, as otherwise one one can make the transformation U → −U. Thus we set í µí»¼ = í µí»¼(í µí¼) varying monotonically from í µí»¼ = 1, í µí¼ = 0
to some constant value í µí»¼ = í µí»¼ a , í µí¼ ≥ í µí¼ a . Then there are two cases, either í µí»¼ a > 1, or
í µí»¼ a < 0 (a change of polarity). Specifically the coefficient í µí»¼ is given by