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Wave-induced mean flows in rotating shallow water with uniform potential vorticity

Published online by Cambridge University Press:  01 February 2018

Jim Thomas Open the ORCID record for Jim Thomas [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]
Jim Thomas
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Oliver Bühler*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
K. Shafer Smith
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
*
†Email address for correspondence: [email protected]
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Abstract

Theoretical and numerical computations of the wave-induced mean flow in rotating shallow water with uniform potential vorticity are presented, with an eye towards applications in small-scale oceanography where potential-vorticity anomalies are often weak compared to the waves. The asymptotic computations are based on small-amplitude expansions and time averaging over the fast wave scale to define the mean flow. Importantly, we do not assume that the mean flow is balanced, i.e. we compute the full mean-flow response at leading order. Particular attention is paid to the concept of modified diagnostic relations, which link the leading-order Lagrangian-mean velocity field to certain wave properties known from the linear solution. Both steady and unsteady wave fields are considered, with specific examples that include propagating wavepackets and monochromatic standing waves. Very good agreement between the theoretical predictions and direct numerical simulations of the nonlinear system is demonstrated. In particular, we extend previous studies by considering the impact of unsteady wave fields on the mean flow, and by considering the total kinetic energy of the mean flow as a function of the rotation rate. Notably, monochromatic standing waves provide an explicit counterexample to the often observed tendency of the mean flow to decrease monotonically with the background rotation rate.

JFM classification

Geophysical and Geological Flows: Quasi-geostrophic flows Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Shallow water flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , pp. 408 - 429
Copyright
© 2018 Cambridge University Press 

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