Internal Undular Bores in the Coastal Ocean
29
𝛼 = 1 + (𝛼 a − 1) tanh (K𝜏) ,
(23)
where K, 𝜏 a , K𝜏 a ≥ 1 are chosen so that 𝛼 varies smoothly and slowly from 1 at 𝜏 = 0
to 𝛼 a at 𝜏 = 𝜏 a . There are two cases, either 𝛼 a > 1 for propagation up a slope, or
𝛼 a < 0 for propagation up a slope and through a critical point of polarity change.
The initial condition U(X, 0) = U ic (X) is either (1) a KdV solitary wave or (2) a
modulated cnoidal wave representation of an undular bore in the constant coefficient
KdV equation evolving from a step of height U 0 > 0 at time 𝜏 = −𝜏 1 , as developed
by [6, 18, 29] and subsequently used by many authors.
(1) ∶ U ic (X) = U 0 sech
2
(𝜅X) , U 0 = 12𝜅
2
,
(24)
(2) ∶ U ic (X) = U 0 ENV(X){2mcn
2
(𝜅(X − V𝜏 1 ); m) + 1 − m} ,
(25)
−U 0 𝜏 1 < X <
2U 0 𝜏 1
3
, V =
U 0
3
{1 + m} , U 0 = 6𝜅
2 k
2
,
X =
U 0 𝜏 1
3
{1 + m −
2m(1 − m)K(m)
E(m) − (1 − m)K(m)
} .
In case (1) the evolving solitary wave has a time scale of (𝜅V) −1 where the speed
V = 4𝜅 2 , and so to be slowly varying we choose K ≪ 𝜅V = 4𝜅 3 . In case (2) the
initial undular bore occupies a domain of length L ub = 5U 0 𝜏 1 ∕3 and we choose 𝜏 1 so
that L ub ≫ 1. The envelope ENV(X) is chosen to be very close to a box of height 1,
and of very long length L > L ub , occupying a domain to contain the initial undular
bore. The front end of the box is placed precisely at the front end of the undular bore
X = 2U 0 𝜏 1 ∕3, but the rear end is chosen far away from the rear end of the bore. As it
evolves in the constant coefficient KdV equation (that is 𝛼 = 1) the leading wave is
a solitary wave of amplitude 2U 0 and so we again choose K so that K ≪ 4𝜅
3 where
here U 0 = 6𝜅 2 k 2 . Note that the wavelength 2𝜋∕k is a free parameter.
A numerical simulation of (13) using the solitary wave initial condition for
the case when 𝛼 increases is shown in Fig. 1. As expected, this exhibits adiabatic
behaviour with the amplitude increasing from the initial value U 0 = 4 to U 0 𝛼
−1∕3
a
=
4.58 at 𝜏 = 𝜏 a . At same time a trailing shelf of very small amplitude can be seen.
Then in Fig. 2 we show a case with the same solitary wave initial condition, but with
a polarity change in which 𝛼 varies from 𝛼 = 1 to 𝛼 = −1. As the solitary wave of
elevation approaches the critical point, its amplitude decreases as U = U 0 𝛼 1∕3 but
at the same time the amplitude of the trailing shelf of depression grows as 𝛼 −8∕3 .
Close to the critical point, when the solitary wave and the trailing shelf have comparable amplitudes, the adiabatic behaviour breaks down, and the whole structure
passes through the critical point into the region 𝜏 > 𝜏 c , generating a depression rarefaction wave connected to the original zero level by an undular bore of elevation
waves. The waves in the undular bore section have a shorter length scale than the rarefaction wave, giving the appearance of a solitary wave train of depression solitary
waves riding on an elevation pedestal. These solitary waves ride down the pedestal,