Near-critical reflection of internal waves

Abstract

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity. In this paper, we prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the incident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.

Linear boundary layer analysis of the near-critical reflection of internal gravity waves with different sizes of viscosity and diffusivity

“…The physical phenomenon which is analyzed here is the reflection of those internal waves from a sloping flat boundary of an arbitrary but fixed angle γ. As widely discussed in [1], since the notion of propagation does not make sense for a single plane wave, we work with wave packets. Our typical wave packet is a simple linear superposition of plane waves, as defined in [1].…”

“…As widely discussed in [1], since the notion of propagation does not make sense for a single plane wave, we work with wave packets. Our typical wave packet is a simple linear superposition of plane waves, as defined in [1]. More precisely, the definition of the wave packet hitting the boundary y = 0 (incident wave packet) reads as follows…”

“…A(k, m)X k,m e −iω k,m t+ikx+imy dk dm, (1.4) where (1) the eigenvector X k,m ∈ R 3 is given by (1.3), where the pressure can be recovered for instance using the first equation of (1.1) (with ν = 0), which in Fourier reads…”

“…ζ ∼ 0. In the presence of (small) viscosity and diffusivity as in (1.7) with ν > 0, κ > 0, the reflected wave is replaced by boundary layers (of high amplitude), see [4,1]. The physical literature on the near-critical reflection of internal waves is quite vast: the first investigation goes back to 1999 and it is due to Dauxois & Young [4] (earlier physical experiments were realized by the group of Leo Mass, [11]), while this is still an active direction of research in physics, see for instance [9].…”

“…An additional assumption is crucially adopted in [4] (and consequently in [1]): both those studies rely on the scaling assumption ν ∼ κ ∼ ε 6 , i.e. viscosity and diffusivity have the same size, which is given in terms of the size of the criticality parameter ζ ∼ ε 2 .…”