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

which together with
η tt = ( (1−η)u 1
)
xt =( (1−η)u 1
)ξt /M(ξ)
leads to
√
√ β
βφtx | z=0 =
M(ξ) T[ ( ) ]
[
β
(1−η)u1
ξt +
M(ξ) T η
M(˜ξ) T[ ( )] ]
(1−η)u1
ξ
+ O(β 3 2 ),
ξt
and as we saw
layer is
β( ∣
φ
2 ∣∣z=0
)
2
=β x x 2
=
=
⎛{ 1
⎜⎝
β
2M(ξ)
β
2M(ξ)
M(ξ) T[ ] } 2 ⎞
M(˜ξ)η t + O(β 2 ⎟⎠x
3 ),
⎛{ 1
⎜⎝
M(ξ) T[ ] } 2 ⎞
M(˜ξ)η t + O(β ⎟⎠ξ
3 2 ),
⎛{ 1
⎜⎝
M(ξ) T[ ( ) (1−η)u1 ξ] } 2 ⎞ ⎟⎠ + O(β 3 2 ).
ξ
Summarizing, the higher order pressure term connecting the top and lower
√ β
M(ξ) T[ ( ) (1−η)u1
ξt]
+
+ β [
M(ξ) T η
M(ξ) T[ ( )] ]
(1−η)u1
ξ
+
ξt
⎛{ β 1
+ ⎜⎝
2M(ξ) M(ξ) T[ ( ) (1−η)u1 ξ] } 2 ⎞ ⎟⎠ +
ξ
(
ηη tt + 1 2 η2 t)⎞⎟⎠ + O(β 3 2 ).
P x (x, t)=− ρ 2
ρ 1
⎛⎜⎝ η x+
+ β
M(ξ)
This order of approximation is compatible with the nonhydrostatic corrections
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