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Frequency diffusion of waves by unsteady flows

Published online by Cambridge University Press:  04 November 2020

Wenjing Dong Open the ORCID record for Wenjing Dong [Opens in a new window] ,
Oliver Bühler Open the ORCID record for Oliver Bühler [Opens in a new window]  and
K. Shafer Smith Open the ORCID record for K. Shafer Smith [Opens in a new window]
Wenjing Dong*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY10012, USA
Oliver Bühler
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY10012, USA
K. Shafer Smith
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY10012, USA
*
†Email address for correspondence: wd583@nyu.edu
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Abstract

The production of broadband frequency spectra from narrowband wave forcing in geophysical flows remains an open problem. Here we consider a related theoretical problem that points to the role of time-dependent vortical flow in producing this effect. Specifically, we apply multi-scale analysis to the transport equation of wave action density in a homogeneous stationary random background flow under the Wentzel–Kramers–Brillouin approximation. We find that, when some time dependence in the mean flow is retained, wave action density diffuses both along and across surfaces of constant frequency in wavenumber–frequency space; this stands in contrast to previous results showing that diffusion occurs only along constant-frequency surfaces when the mean flow is steady. A self-similar random background velocity field is used to show that the magnitude of this frequency diffusion depends non-monotonically on the time scale of variation of the velocity field. Numerical solutions of the ray-tracing equations for rotating shallow water illustrate and confirm our theoretical predictions. Notably, the mean intrinsic wave frequency increases in time, which by wave action conservation implies a concomitant increase of wave energy at the expense of the energy of the background flow.

JFM classification

Acoustics: Waves in random media Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , R3
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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