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

For the weakly nonlinear model of orderβ 3 2 introduceη ∗ , u ∗ 1 such that
η=αη ∗ , u 1 =αu 1 ∗ ,
withα=O ( √ β
)
, a typical scaling used for solitary waves. After dropping the
asterisks we have
⎧
⎪⎨
⎪⎩
η t = 1 [ ] (1−αη)u1
ξ
M(ξ)
,
(
1− ρ )
2
η ξ = ρ 2
ρ 1 ρ 1
u 1t + α
M(ξ) u 1 u 1ξ + 1
M(ξ)
( )
β 1
+
3M(ξ) M(ξ) u 1ξt + O(β 2 3 ).
ξ
√ β
M(ξ) T[ (1−αη)u 1
]
ξt +
The higher-order weakly nonlinear model has exactly the same form as the
lower-order strongly nonlinear model when the last term, namely,
( )
β 1
3M(ξ) M(ξ) u 1ξt
of the weakly nonlinear model is neglected. This implies that the weakly nonlinear
higher-order model should serve as a good model for moderate amplitude internal
waves in a deep water configuration. Furthermore, when the additional term
from the upper layer is included, the linear dispersion relation for the higher-order
weakly nonlinear model becomes the closest to the exact linear dispersion relation
when compared to lower-order models, as will be shown in the next Section. In
other words, the weakly nonlinear higher-order model might have a large domain
of validity so that the model can be used for a moderate (although still large) ra-
ξ
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