60
N. Makarenko et al.
and the dimensionless version of the integral relation (9) takes the form
𝜂
∫
−𝜋
e
−𝜎 1 𝜓
{ 𝜇F
2
1
2
(
𝜓
2
y − 𝜓
2
x + 1
)
+
1 + 𝜇
𝜋
(
𝜓 − y −
e
𝜎 1 𝜓
− 1
𝜎 1
)}
dy +
(14)
+
𝜋∕r
∫
𝜂
e
−𝜎 2 𝜓
{ 𝜇F
2
2
2r 3
(
𝜓
2
y
− 𝜓
2
x
+ r
2
)
+
1
𝜋r
(
𝜓 − ry −
e 𝜎 2 𝜓 − 1
𝜎 2
)}
dy = C
with constant
C = 𝜋𝜇
(
F
2
1
+
F
2
2
r 2
)
+ (1 + 𝜇)
e 𝜎 1 𝜋 − 1 − 𝜎 1 𝜋
𝜋(𝜆
2
1
+ 𝜎
2
1
)
+
1 − 𝜎 2 𝜋 − e −𝜎 2 𝜋
𝜋r 2 (𝜆
2
2
+ 𝜎
2
2
)
.
Constant C is chosen here so that the upstream horizontal flow is described by the
solution 𝜂 = 0 and 𝜓 = y (−𝜋 < y < 0), 𝜓 = ry (0 < y < 𝜋∕r).
The Non-linear Long-Wave Model
The Boussinesq parameters 𝜎 1 , 𝜎 2 and 𝜇 are small in the case of extremely weak
stratification in the abyssal water. We assume here that these parameter are of the
same order, so we can use a single small parameter 𝜎 by setting
𝜎 = 𝜎 1 = 𝜎 2 = 𝜇.
(15)
In accordance with this hypothesis, the derivation procedure of non-linear long-wave
model should involve slow horizontal variable 𝜉 =
√ 𝜎 x, as it was demonstrated by
[1] in the case of weak linear stratification. We expand the stream function into power
series with respect to 𝜎 as
𝜓 = 𝜓
(0)
(𝜉, y) + 𝜎 𝜓
(1)
(𝜉, y) + ⋯
(16)
where the leading-order term 𝜓
(0) defines the hydrostatic mode, and coefficient 𝜓
(1)
provides the correction due to non-linear dispersion. All these coefficients 𝜓 (k) can
be uniquely determined from equations (11) and (12) (with fixed Long’s numbers 𝜆 1
and 𝜆 2 ) under kinematic boundary condition (13). Thus, we obtain
𝜓
(0)
= y − 𝜂
sin 𝛼 1 (y)
sin 𝛼 1 (𝜂)
(−𝜋 < y < 𝜂),
and