a reduced model for internal waves interacting with submarine ...

demands that
η t + u i η x = w i , p 1 = p 2 ,
namely, a kinematic condition for the material curve and no pressure jumps allowed.
At the top we impose a rigid lid condition,
w 1 (x, h 1 , t)=0,
commonly used in ocean and atmospheric models, while at the irregular impermeable
bottom
− h 2
l
h ′ ( x
l
)
u 2 + w 2 = 0.
Introducing the dimensionless dispersion parameterβ= ( )
h 1 2,
L it follows from
the shallowness of the upper layer that
O (√ β ) = O
( )
h1
≪ 1.
L
From the continuity equation for i=1we have,
w 1
u 1
= O
( )
h1
= O (√ β ) .
L
Let U 0 = √ gh 1 be the characteristic shallow layer speed. According to these
scalings, physical variables involved in the upper layer equations are non-dimen-
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