Large Internal Solitary Waves in Shallow Waters
97
Fr
2
s
− h + h
2
> 0.
Therefore, the admissible intervals for h are
0 < h 0 ≤ h ≤ h
−
, h
+
≤ h ≤ h 0 < 1,
(11)
where
h
±
=
1 ±
√
1 − 4Fr 2
s
2
.
(12)
Naturally, the necessary condition for solitary wave existence reads
Fr s ≤
1
2
.
(13)
The profile of the symmetric solitary wave can be found from quadratures
x = x
±
(h) = ±
h
∫
h m
ds
√
G(s).
(14)
Here h m = h + for a wave of depression and h m = h − for a wave of elevation. The solution of (14) for Fr s = 0.48, h 0 = 0.15 is shown by solid curves in Fig. 4 (in dimensional variables).
Remark 4 It is important to underline that solutions (14) can be effectively applied to
large amplitude internal wave simulation for different scales of flow not only at interfaces (the second mode symmetric waves), but also to subsurface waves of depression
and to bottom waves of elevation.
Solitary Waves in BLM and ULM
In similar manner, solitary wave solutions for BLM and ULM may be represented in
quadratures. Consider (7) with 𝛽 − = 1, 𝛽 + = 0. The governing equations describing
the long subsurface waves of depression (BLM) take the form
1
2
u
2
+ bh + ̄
b𝜂 + p −
1
3
h
2 uu
′′
+
1
2
h
2
(u
′
)
2
=
1
2
u
2
0
+ bh 0 + ̄
b𝜂 0 + p 0 = J
−
,
1
2
w
2
+ p =
1
2
u
2
0
+ p 0 = J
+
,
1
2
v
2
+ ̄
b(h + 𝜂) + p =
1
2
u
2
0
+ ̄
b(h 0 + 𝜂 0 ) + p 0 = ̄
J,
hu = h 0 u 0 = Q
−
, 𝜁w = 𝜁 0 u 0 = Q
+
, 𝜂v = 𝜂 0 u 0 = ̄
Q, h + 𝜂 + 𝜁 = H. (15)