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| Titel |
Extended modified Korteweg - de Vries equation for internal gravity waves in a symmetric three-layer fluid |
| VerfasserIn |
Oxana Polukhina, Andrey Kurkin, Ekaterina Vladykina |
| Konferenz |
EGU General Assembly 2010
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 12 (2010) |
| Datensatznummer |
250034063
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| Zusammenfassung |
| Three-layer stratification is proved to be a proper approximation of sea water density and
background current profiles in some basins in the World Ocean with specific hydrological
conditions. Such a medium is interesting from the point of view of internal gravity wave
dynamics, because, in the symmetric about mid-depth case (equal thicknesses of the
lower and the upper layers, equal small density jumps on the interfaces), it leads to
disappearing of quadratic nonlinearity when described in the framework of weakly
nonlinear evolutionary models, which are derived through the asymptotic expansion
in small parameters of nonlinearity and dispersion. In this situation the nonlinear
transformation of the internal wave disturbances, as is customary, is determined by the
influence of the next-order – cubic – nonlinear term in asymptotic series, and for
three-layer fluid model the cubic nonlinearity coefficient can have either sign depending
on the layer depths (in contrast to traditional two-layer approximation, for which
cubic nonlinearity is always negative). Appropriate nonlinear evolutionary equation
is modified Korteweg – de Vries equation (mKdV). It is well-known integrable
equation of KdV-type, providing solitary wave and breather solutions for positive cubic
nonlinearity. The property of sign change for cubic nonlinear coefficient in the
mKdV for internal gravity waves in symmetric three-layer fluid requires taking into
account next-order nonlinear term (or terms), therefore higher-order extensions of
mKdV equation are necessary to provide improved description of internal wave
processes.
In the present study we derive nonlinear evolution equations for both interfaces in
symmetric three-layer fluid (under Boussinesq approximation) up to the fourth order in small
parameters of nonlinearity (ε) and dispersion (μ). Applying mKdV-scaling for ratio of these
parameters (μ = ε2) we obtain high-order mKdV equations for interfaces (they have different
signs of even-power nonlinear and nonlinear dispersion terms). These equations include
additional terms of nonlinearity (fourh and fifth power), nonlinear dispersion and linear
dispersion (fifth derivative). Coefficients of these equations are found in the explicit form as
functions of medium parameters (layer depths, densities), their signs are analyzed.
But the equations derived are too complex for the analysis of the nonlinear wave
dynamics, therefore a simplifying asymptotic nonlinear transformation of wavefield is
suggested, which reduces these equations to a simpler equation having a form of
mKdV equation with additional fourth and fifth power nonlinear terms. But for the
particular case of three-layer symmetric fluid fourth power nonlinear correction
has zero coefficient, and final equation has only one additional term: fifth power
nonlinearity. Its coefficient after the transformation is negative for symmetric three-layer
stratification. It is worth to notice that equations for both interfaces are reduced to the
same equation, but the asymptotic transformations for the displacements of the
interfaces differ by the sign of one term proportional to the square of the displacement
amplitude. Thus, the asymptotic transformation introduces an asymmetry of the
interfacial displacements in a clear, explicit form in contrast to complex high-order
equations.
Next, we considered the equation obtained after transformation (we call it “extended
mKdV”), and found its one-soliton solutions for positive cubic nonlinearity. These solitary
waves can have either polarity, as well as solitons of mKdV equation, but they have amplitude
limit, and while their amplitude grows up to this limit, solitons become wider in much the
same manner as the solitons of Gardner equation (extended KdV equation with quadratic and
cubic nonlinear terms) in a case of negative cubic nonlinearity (corresponding to internal
waves in two-layer fluid).
The solitary wave solutions of the improved weakly nonlinear theory are compared to the
fully nonlinear solitary-like waves numerically simulated in the framework of Euler equations
for slightly smoothed symmetric three-layer fluid with small density jumps. Qualitatively
both solutions are in good agreement, but quantitative estimates show that improved weakly
nonlinear theory underestimates limiting amplitude of solitary waves for about
30%. |
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