Abstract
A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al. (Phys. Rev. Lett., vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.
Highlights
The interaction of waves with a mean flow is a fundamental and longstanding problem of fluid mechanics with numerous applications in geophysical fluids (see e.g. Mei, Stiassnie & Yue (2005), Bühler (2009) and references therein)
We show that in both cases, the wave–mean flow interaction exhibits two adiabatic invariants of motion that govern the variations of the wavenumber and the amplitude in the linear wavetrain, and prescribe its transmission or trapping inside the hydrodynamic state: either a rarefaction wave (RW) or a dispersive shock wave (DSW)
It is surprising that the interaction of a plane wave (PW) with a non-uniform, unsteady hydrodynamic state does not change the PW amplitude, which is in sharp contrast with the classical case of the interaction of a surface water wave with a counter-propagating steady current where the amplitude varies following the inhomogeneities of the current in (1.2)
Summary
The interaction of waves with a mean flow is a fundamental and longstanding problem of fluid mechanics with numerous applications in geophysical fluids (see e.g. Mei, Stiassnie & Yue (2005), Bühler (2009) and references therein). The dynamics of the small-amplitude, short-wavelength wave is dominated by dispersive effects while the large-scale mean flow variation is a nonlinear process In this scenario, the modulation system (1.1) for the linear wave couples to an extra nonlinear evolution equation for the mean flow. We show that in both cases, the wave–mean flow interaction exhibits two adiabatic invariants of motion that govern the variations of the wavenumber and the amplitude in the linear wavetrain, and prescribe its transmission or trapping inside the hydrodynamic state: either a rarefaction wave (RW) or a dispersive shock wave (DSW). This system consists of the two usual modulation equations (1.1) that describe conservation of wavenumber and wave action, which are coupled to the simple wave evolution equation describing mean flow variations in the RW/DSW.
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