Kinetic limits for waves in a random medium

Bal, Guillaume; Komorowski, Tomasz; Ryzhik, Lenya

doi:10.3934/krm.2010.3.529

Cited by 60 publications

(67 citation statements)

References 54 publications

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“…For weak random flows as assumed here, the ray equations can be analysed asymptotically using methods developed for noisy Hamiltonian systems (e.g. Bal et al 2010) to show that the IT wavevector diffuses along the constant frequency circle |k| = const, consistent with the kinetic-equation description; see Müller (1976Müller ( , 1977 for early treatments in this spirit.) Isotropisation is most effective when κ has an order of magnitude similar to |k|: for the chosen energy spectrum, isotropisation is fastest for κ ≈ 6 × 10 −5 m −1 corresponding to a flow correlation length of about 50 km.…”

Section: Predicted Behaviourmentioning

confidence: 99%

“…In this appendix we apply the formalism of Ryzhik et al (1996) to derive a kinetic equation for the equivalent shallow-water system (2.2) (see also Bal et al (2010) and Powell & Vanneste (2005)). This formalism exploits the weakness of the flow as measured by the small Rossby number, as well as the scale separation between wavelength and flow scale on the one hand, and the scale of variation of the wave amplitude on the other hand.…”

Section: Appendix B Kinetic Equation Derivationmentioning

confidence: 99%

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Scattering of internal tides by barotropic quasigeostrophic flows

Savva

Vanneste

2018

J. Fluid Mech.

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Received xx; revised xx; accepted xx)Oceanic internal tides and other inertia-gravity waves propagate in an energetic turbulent flow whose lengthscales are similar to the wavelengths. Advection and refraction by this flow cause the scattering of the waves, redistributing their energy in wavevector space. As a result, initially plane waves radiated from a source such as a topographic ridge become spatially incoherent away from the source. To examine this process, we derive a kinetic equation which describes the statistics of the scattering under the assumptions that the flow is quasigeostrophic, barotropic, and well represented by a stationary homogeneous random field. Energy transfers are quantified by computing a scattering cross section and shown to be restricted to waves with the same frequency and identical vertical structure, hence the same horizontal wavelength. For isotropic flows, scattering leads to an isotropic wavefield. We estimate the characteristic time and length scales of this isotropisation, and study their dependence on parameters including the energy spectrum of the flow. Simulations of internal tides generated by a planar wavemaker carried out for the linearised shallow-water model confirm the pertinence of these scales. A comparison with the numerical solution of the kinetic equation demonstrates the validity of the latter and illustrates how the interplay between wave scattering and transport shapes the wave statistics.

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Section: Predicted Behaviourmentioning

confidence: 99%

Section: Appendix B Kinetic Equation Derivationmentioning

confidence: 99%

Scattering of internal tides by barotropic quasigeostrophic flows

Savva

Vanneste

2018

J. Fluid Mech.

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“…Additionally, we assume the covariance function K q (x, y) to be smooth out of the diagonal, which means that the long distance interactions depends smoothly on their locations; we also assume the average roughness (or smoothness) of q to remain unchanged for every sub-domain of D. However, we allow the size of this roughness to change in different sub-domains of D. The local strength of the potential measures or controls these different sizes. These assumptions 1 can be rigorously introduced, assuming that the covariance operator C q is a classical pseudodifferential operator (see for example [31]) of order −m with m > n − 1,and such that, C q has (2) σ(x, ξ) = µ(x) |ξ| m as a principal symbol, with µ a smooth non-negative function supported on D-called the the local strength of the potential. As we will see in Definition 2.1, this is to say that q is a Gaussian microlocally isotropic random field.…”

Section: Introductionmentioning

confidence: 99%

“…We will later see that q ∈ L p −s (R n ) almost surely for any 1 < p < ∞ and −s < (m−n)/2. The local strength µ determines the roughest component of q in the sense that if C q is a properly classical pseudodifferential operator of order −m having the same principal symbol (2) as the operator C q , then q = (C q ) 1/2 W is also a microlocally isotropic Gaussian random field of order −m such that q − q ∈ L p −s+1 (R n ) almost surely for any 1 < p < ∞ and −s < (m − n)/2, that is, q − q is one degree smoother than q.…”

Section: Introductionmentioning

confidence: 99%

Inverse scattering for a random potential

2019

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In this paper we consider an inverse problem for the n-dimensional random Schrödinger equation (∆ − q + k 2 )u = 0. We study the scattering of plane waves in the presence of a potential q which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential q, we uniquely determine the principal symbol of the covariance operator of q. Especially, for n = 3 this result is obtained for the full non-linear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance. 35 4.4. Non-zero-mean potentials 37 Appendix A. Scaling regimes 40 A.1. Two-scale model of a non-smooth scatterer 40 A.2. Inverse scattering with the two-scale model 43 A.3. Effects from the higher order scattering 44 Appendix B. Random variables with Gaussian probability laws 47 References

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“…It is a consequence of the Kolmogorov extension theorem [28,Thm. 6.16] that there is an unique probability distribution 1] in R N , satisfying µ J (V j 1 , . .…”

Section: Now By Identitymentioning

confidence: 99%

Correlation imaging in inverse scattering is tomography on probability distributions

et al. 2018

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Scattering from a non-smooth random field on the time domain is studied for plane waves that propagate simultaneously through the potential in variable angles. We first derive sufficient conditions for stochastic moments of the field to be recovered from correlations between amplitude measurements of the leading singularities, detected in the exterior of a region where the potential is almost surely supported. The result is then applied to show that if two sufficiently regular random fields yield the same data, they have identical laws as function-valued random variables.

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Kinetic limits for waves in a random medium

Cited by 60 publications

References 54 publications

Scattering of internal tides by barotropic quasigeostrophic flows

Scattering of internal tides by barotropic quasigeostrophic flows

Inverse scattering for a random potential

Correlation imaging in inverse scattering is tomography on probability distributions

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