Internal Solitary Waves in a Layered Weakly Stratified Flow
57
where N j = const is the Brunt—Väisälä frequency in j-th layer, and constants 𝜌 1 and
𝜌 2 are such that 𝜌 2 < 𝜌 1 . The special case N j = 0 is related to the ordinary two-fluid
system with a piecewise constant density 𝜌 = 𝜌 j in j-th layer, but we specify N j ≠ 0
for the wave model to be constructed.
Further, we consider a steady non-uniform flow, hence we have 𝜂 t = 0 and u t =
v t = 𝜌 t = 0 in Eq. (1). We introduce the stream function 𝜓 by standard formulae
u = 𝜓 y , v = −𝜓 x , so the mass conservation implies the following dependence
𝜌 = 𝜌(𝜓), and the pressure p can be found from the Bernoulli equation
1
2
(𝜓
2
x + 𝜓
2
y ) +
1
𝜌(𝜓)
p + gy = b(𝜓).
(4)
We are seeking solitary-wave solutions, we require that fluid velocity (u, v) to attain
the upstream velocity (u j , 0) as x → −∞. In this case, kinematic boundary conditions
at the bottom, at the interface and at the lid take the form
𝜓 = −u 1 h 1 (y = −h 1 ),
𝜓 = 0 (y = 𝜂),
𝜓 = u 2 h 2 (y = h 2 ),
(5)
respectively.
It is known [26] that stationary system (1) can be reduced to the non-linear
Dubreil-Jacotin—Long (DJL) equation for stream function:
𝜌(𝜓) (𝜓 xx + 𝜓 yy ) + 𝜌
′
(𝜓)
(
gy +
1
2
𝜓
2
x +
1
2
𝜓
2
y
)
= H
′
(𝜓).
(6)
Here, function H(𝜓) = 𝜌(𝜓)b(𝜓) involves the Bernoulli function b(𝜓) and the density function 𝜌(𝜓), so that H is specified by the upstream condition. More exactly,
the density function is determined by relation 𝜌(𝜓) = 𝜌 0 (𝜓∕u j ) in j-th layer, and the
Bernoulli function is defined by the formula
b(𝜓) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1
2
u
2
1
+ g
𝜓
u 1
+
g 2
N
2
1
(
1 − e
N 2
1
𝜓
gu 1
)
, −h 1 < y < 𝜂(x),
1
2
u
2
2 + g
𝜓
u 2
+
g 2
N
2
2
(
1 − e
N 2
2
𝜓
gu 2
)
, 𝜂(x) < y < h 2 .
As a consequence, we can rewrite the DJL equation (6) as follows:
𝜓 xx + 𝜓 yy =
N
2
j
gu j
{
g
(
y −
𝜓
u j
)
+
1
2
(
𝜓
2
x + 𝜓
2
y − u
2
j
) }
,
(7)
where j = 1 should be taken in the lower layer, and j = 2 in the upper layer. Similarly
non-linear terms also appear in the boundary condition
[𝜌(𝜓)(𝜓
2
x + 𝜓
2
y + 2gy − 2b(𝜓)] = 0,
y = 𝜂(x),
(8)