Large Internal Solitary Waves in Shallow Waters
93
quickly (Fig. 5). Nevertheless, for special choice of stratification the nonsymmetric
solitary wave can be generated, which is able to propagate along the channel keeping
its permanent form (Fig. 6). Solid lines shown in Figs. 3, 4, 5 and 6 represent the
solutions of the mathematical model developed in next sections.
Mathematical Models
Basic Model
As a basic model for simulation of nonlinear internal waves, we consider the three–
layer shallow water equations for large amplitude, weakly nonhydrostatic long waves
introduced in [19]. The model is a version of multi-layer strongly nonlinear equations
derived in [2] with the following simplifications:
1. Boussinesq approximation |𝜌 − 𝜌 0 | ≪ 𝜌 0 is used, where 𝜌, 𝜌 0 are the density and
reference density, correspondingly [14].
2. The pressure distribution in the intermediate layer is supposed to be hydrostatic.
The first condition is well justified due to small density variation in the ocean.
The second one will be used in two cases:
(i) to describe large amplitude internal waves of the first mode with relative small
thickness of the interlayer
(ii) to simulate interface solitary waves of the second mode, in which wave velocity
and particle velocity differ only slightly.
In dimensional variables the governing equations take the form
h t + (hu) x = 0, 𝜂 t + (𝜂v) x = 0, 𝜁 t + (𝜁 w) x = 0,
u t + (
1
2
u
2
+ b(h + z) + ̄
b𝜂 + p) x +
𝛽
−
3h
(h
2
d
2
1
h
dt 2 ) x = f
−
,
v t + (
1
2
v
2
+ ̄
b(h + 𝜂 + z) + p) x = ̄
f ,
w t + (
1
2
w
2
+ p) x +
𝛽 +
3𝜁
(𝜁
2
d
2
2
𝜁
dt 2 ) x = f
+
,
d 1
dt
=
𝜕
𝜕t
+ u
𝜕
𝜕x
,
d 2
dt
=
𝜕
𝜕t
+ w
𝜕
𝜕x
,
b = (𝜌
−
− 𝜌
+
)g∕𝜌
+
, ̄
b = ( ̄
𝜌 − 𝜌
+
)g∕𝜌
+
.
(1)
Here g is the gravity acceleration, p is the modified pressure, 𝜌
+
, ̄
𝜌, 𝜌
− are the densities, 𝜁, 𝜂, h are the thicknesses, w, v, u are the mean horizontal velocities in the upper,
intermediate and lower layers, correspondingly. The function z = z(x) describes the
bottom profile. We suppose that the bottom changes very smoothly, so we don’t