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

function f (x, z, t), let its associated mean-layer quantity f be
f (x, t)= 1 ∫ 1
f (x, z, t) dz.
1−η
η
By averaging we will reduce the 2D Euler equations to a 1D system.
Letη 1 = 1−η. From the horizontal momentum equation we have
η 1 u 1t +η 1 u 1 u 1x +η 1 w 1 u 1z =−η 1 p 1 x. (2.3)
We need to express each of these mean-layer quantities in terms of u 1 andη.
The difficulty at this stage is breaking up the mean of square, and other general
quadratic terms, into individually averaged terms. To begin with, note that
∫ 1
(η 1 u 1 ) t = u 1t dz−η t u 1 (x,η, t),
η
=η 1 u 1t −η t u 1 ,
where u 1 is evaluated at the interface (x, z, t)=(x,η(x, t), t). So,
η 1 u 1t = (η 1 u 1 ) t +η t u 1 . (2.4)
Similarly
η 1 u 1x = u 1 η x + (η 1 u 1 ) x , (2.5)
and
2η 1 u 1 u 1x =η x u 2 1 + (η 1 u 2 1
)
x
. (2.6)
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