Internal Undular Bores in the Coastal Ocean
31
and at 𝜏 = 𝜏 a the leading solitary wave has reached the zero level at the head of
the rarefaction wave. In this region where 𝛼 < 0, the solitary wave train equations
(18, 21) have a similarity solution
d =
X − X 0
𝜂
, 𝜂 =
𝜏
∫
𝜏 c
𝛼(𝜏
′
) d𝜏
′
, X < X 0 ,
(26)
A = −
1
𝜂
{
3(X − X 0 )
𝜂𝜉
}
3∕2
, 𝜉 =
𝜂
∫
−∞
|𝛼(𝜂 ′ )| 1∕3 d𝜂 ′
|𝜂 ′ | 5∕3
, A = {
a
3
𝛼
}
1∕2
.
(27)
Note that here 𝛼 < 0 and so 𝜂 < 0, ensuring that the rarefaction wave d > 0 in
X < X 0 . The determination of X 0 requires a detailed matching with the solution at
the critical point, beyond the scope of this present review. The rarefaction wave
(26) can only extend to a point X − X 0 = −L r (𝜂) where L r (𝜂) is likewise undetermined. But the mass of the rarefaction wave is then −L
2
r (𝜂)∕2𝜂 and this can be
approximately equated to the initial solitary wave mass 2U 0 ∕𝜅 = 2(12U 0 ) 1∕2 (24),
thus giving an approximate expression for L r (𝜂). The expression (27) for the solitary wave amplitude a holds on the domain −L r (𝜂) < X − X 0 < −L s (𝜂) where the
upper bound L s (𝜂) determines the amplitude of the leading solitary wave, that is
a s = −3|𝛼|
1∕3 L s ∕|𝜂|
5∕3
𝜉. The values of a s , L s are undetermined and requires matching with the solution at the critical point, beyond the scope of this review article.
However, an approximate estimate can be based on the assumption that since the
emerging solitary wave train is the leading edge of an undular bore resolving the
jump at the rear of the rarefaction wave, and then a s = 2L r ∕𝜂, where in turn L r is estimated from conservation of mass, as above. Note that when 𝛼 < 0 is a constant, then
as 𝜏 → ∞, 𝜂 ∼ 𝛼𝜏, 𝜉 ∼ 3∕2|𝛼|
1∕3
𝜏
2∕3 and then a ∼ −2(X − X 0 )∕𝛼𝜏, whose envelope
is a rarefaction wave. The outcome from the numerical simulation in Fig. 2 shows
qualitative agreement with all the above features. For the parameters in this simulation U 0 = 4, and so the initial mass is 13.86, which yields a value L r ≈ 5.26|𝜂
1∕2
|
and then a s ≈ 10.52|𝜂| −1∕2 ; at 𝜏 = 𝜏 a = 100, 𝜂 = −62 (26), and then a s ≈ 1.34, in
good agreement with the numerical simulation.
Next we examine how an undular bore behaves when propagating on a slope. As
for the case of a solitary wave, there are two main situations, one where the nonlinear
coefficient 𝛼 increases, and the other where 𝛼 decreases and changes sign at a critical
point. In the first case, which is similar to the study by [5] of a surface wave train propagating up a slope, we expect from the discussion in section “Solitary Wave Train”
that the leading waves in the undular bore will form a solitary wave train ahead of the
undular bore, with amplitudes deforming adiabatically as 𝛼 1∕3 . A typical numerical
simulation of (13) using the undular bore initial condition (25) is shown in Fig. 3.
First, note that the effect of the front end of the enclosing envelope ENV(X), which
of numerical necessity is a smoothed out version of a rapid change, is to truncate
the amplitudes of the leading waves in the initial undular bore. Without the envelope