Internal Solitary Waves in a Layered Weakly Stratified Flow
61
𝜓
(0)
= ry − r𝜂
sin 𝛼 2 (y)
sin 𝛼 2 (𝜂)
(𝜂 < y < 𝜋∕r),
where is denoted
𝛼 1 (y) = 𝜆 1 (𝜋 + y),
𝛼 2 (y) = 𝜆 2 (𝜋 − ry).
The dispersive term 𝜓
(1) is much more complicated, it has the form
𝜓
(1)
=
𝜂(𝜂−y)
2
sin 𝛼 1 (y)
sin 𝛼 1 (𝜂)
+
+
sin 𝛼 1 (y)
2𝜆 1
(
𝜂
sin 𝛼 1 (𝜂)
)
𝜉𝜉
{
(𝜋 + 𝜂) cot 𝛼 1 (𝜂) − (𝜋 + y) cot 𝛼 1 (y)
}
+
+
𝜂 2
6
{
sin 𝜆 1 (y−𝜂)−sin 𝛼 1 (y)
sin
3
𝛼 1 (𝜂)
+
1+ sin
2
𝛼 1 (y)
sin
2
𝛼 1 (𝜂)
−
sin 𝛼 1 (y)
sin 𝛼 1 (𝜂)
}
in lower layer, and
𝜓
(1)
=
r 2 𝜂(𝜂−y)
2
sin 𝛼 2 (y)
sin 𝛼 2 (𝜂)
+
+
sin 𝛼 2 (y)
2𝜆 2
(
𝜂
sin 𝛼 2 (𝜂)
)
𝜉𝜉
{
(y − 𝜋∕r) cot 𝛼 2 (y) − (𝜂 − 𝜋∕r) cot 𝛼 2 (𝜂)
}
+
+
r
2
𝜂
2
6
{
sin 𝜆 2 r(𝜂−y)−sin 𝛼 2 (y)
sin
3
𝛼 2 (𝜂)
+
1+ sin
2
𝛼 2 (y)
sin
2
𝛼 2 (𝜂)
−
sin 𝛼 2 (y)
sin 𝛼 2 (𝜂)
}
in the upper layer. Now we substitute power expansion (16) of function 𝜓 into integral relation (14) and truncate the terms with powers higher than the first degree of
𝜎. Hence, system (11)–(14) reduces to the first-order ordinary differential equation
for the wave elevation 𝜂(x) having the following form
(
d𝜂
dx
) 2
= 𝜂
2 D(𝜂; F 1 , F 2 )
Q(𝜂; F 1 , F 2 )
.
(17)
Here, function D is given by the formula
D(𝜂; F 1 , F 2 ) =
√
𝜋F 1 cot 𝛼 1 (𝜂) +
√ 𝜋F 2 cot 𝛼 2 (𝜂) +
1
3
(1 − r)𝜂 − 1
where 𝛼 1 and 𝛼 2 should be taken as