Internal Solitary Waves in a Layered Weakly Stratified Flow
59
Namely, the ratio h 1 ∕𝜋 is used as an appropriate length scale for x, y, 𝜂, and normalized volume discharges u j h j ∕𝜋 serve as the units for the stream function 𝜓 considered
separately in lower- (j = 1) or upper layer (j = 2). The value 𝜋 is introduced here
only due to the specific form of trigonometric modal functions, which are typical for
the exponential density distribution (3). Scaling procedure with this density profile
uses the Boussinesq parameters 𝜎 1 , 𝜎 2 and 𝜇 defined by the formulae
𝜎 j =
N
2
j
h j
𝜋g
(j = 1, 2),
𝜇=
𝜌 1 − 𝜌 2
𝜌 2
.
(10)
Here, constants 𝜎 j characterize the slope of density profile in continuously stratified
layers, and the parameter 𝜇 determines the density jump at the interface.
Following [22], we introduce densimetric (or internal) Froude number
F j =
u j
√
g j h j
(j = 1, 2)
which presents scaled fluid velocity u j in j-th layer, defined with reduced gravity
acceleration g j = (𝜌 1 − 𝜌 2 )g∕𝜌 j . In addition to the Froude numbers F j , it is also
convenient to use the pair of the Long’s numbers 𝜆 j given by the following formula:
𝜆 j =
N j h j
𝜋u j
(j = 1, 2).
The Long’s numbers 𝜆 j are coupled with the Boussinesq parameters 𝜎 1 , 𝜎 2 , 𝜇 and
the Froude numbers F j by the following relations
𝜆
2
1
=
𝜋𝜎 1 (1 + 𝜇)
𝜇F
2
1
, 𝜆
2
2
=
𝜋𝜎 2
𝜇F
2
2
.
Finally, we introduce the ratio of undisturbed thicknesses of the layers r = h 1 ∕h 2 . By
that notation, the bottom is located at y = −𝜋, and relation y = 𝜋∕r defines the rigid
lid. Thus, we obtain the equations for scaled stream function 𝜓 and wave elevation
𝜂 as follows:
𝜓 xx + 𝜓 yy + 𝜆
2
1
(𝜓 − y) =
1
2
𝜎 1
(
𝜓
2
x
+ 𝜓
2
y
− 1
)
(−𝜋 < y < 𝜂(x))
(11)
𝜓 xx + 𝜓 yy + 𝜆
2
2
r
2
(𝜓 − ry) =
1
2
𝜎 2
(
𝜓
2
x
+ 𝜓
2
y
− r
2
)
(𝜂(x) < y < 𝜋∕r). (12)
The kinematic boundary conditions (5) can be rewritten now as follows:
𝜓(x, −𝜋) = −𝜋,
𝜓(x, 𝜂(x)) = 0,
𝜓(x, 𝜋∕r) = 𝜋,
(13)