24
R. Grimshaw and C. Yuan
I𝜆 =
0
∫
−h
𝜌 0
(
c − u 0
) 2 𝜙
2 dz ,
(5)
I = 2
0
∫
−h
𝜌 0
(
c − u 0
) 𝜙
2
z
dz .
(6)
The KdV equation (1) is integrable, and the outcome of a localised initial condition is
a finite set of rank-ordered solitary waves and some small-amplitude dispersing radiation, see [1, 29]. However, here we are concerned with the undular bore solution,
which can be found as the outcome of a step initial condition by using the Whitham
modulation equations, see [18], or more generally is a representation of the development of a solitary wave train from a broad initial condition. Such internal undular
bores are also generated by transcritical flow over topography, see [9, 11, 13].
When the depth h, and background current u 0 , density 𝜌 0 vary slowly in the horizontal direction with x, the KdV equation (1) is replaced by a variable-coefficient
KdV (vKdV) equation. first derived in the Boussinesq approximation in the absence
of a background current by [26], and then in the general case by [8], see also [14,
15, 30]. It has the same form as (1) with two extra terms,
𝜂 t + c𝜂 x +
cQ x
2Q
𝜂 + 𝜇𝜂𝜂 x + 𝜆𝜂 xxx + 𝜎𝜂 = 0 ,
(7)
Q = c
2 I , I𝜎 = −
𝜂 0
∫
−h
𝜙𝜙 z F 0z dz , F 0 = 𝜌 0 (u 0 u 0x + w 0 u 0z ) + p 0x .
(8)
Here the modal equation depends also on x parametrically, that is 𝜙 = 𝜙(z ∶ x), c =
c(x), and hence the coefficients 𝜇, 𝜆, Q also depend (slowly) on x. The coefficient Q
ensures conservation of wave action flux Q𝜂 2 at the linear long wave order, and 𝜎
arises due to the presence of a body force in the basic horizontal momentum equation,
needed in general whenever the basic density field and basic current vary in the
horizontal direction. In (8) p 0 is the basic pressure, such that p 0z = −g𝜌 0 , and w 0 is
the basic vertical velocity such that u 0x + w 0z = 0.
It is convenient to transform this to the “spatial” evolution form, asymptotically
equivalent to (7)
X =
x
∫
x 0
dx
c
− t , T =
x
∫
x 0
dx
c
,
(9)
𝜂 T +
Q T
2Q
𝜂 + 𝜈𝜂𝜂 X + 𝛿𝜂 XXX + 𝜎𝜂 = 0 ,
(10)