Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-03T06:44:34.712Z Has data issue: false hasContentIssue false
  • English
  • Français

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: obuhler@cims.nyu.edu
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alford, M. H., MacKinnon, J. A., Simmons, H. L. & Nash, J. D. 2016 Near-inertial internal gravity waves in the ocean. Ann. Rev. Marine Sci. 8, 95–123.CrossRefGoogle ScholarPubMed
Andrews, D. G., Holton, J. R. & Leovy, C. B. 1987 Middle Atmosphere Dynamics. Academic Press.Google Scholar
Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
Babin, A., Mahalov, A. & Nicolaenko, B. 1997 Global splitting and regularity of rotating shallow-water equations. Eur. J. Mech. B 16 (5), 725–754.Google Scholar
van den Bremer, T. S. & Sutherland, B. R. 2014 The mean flow and long waves induced by two-dimensional internal gravity wavepackets. Phys. Fluids 26 (10), 106601.CrossRefGoogle Scholar
Bretherton, F. P. 1969 On the mean motion induced by internal gravity waves. J. Fluid Mech. 36, 785–803.CrossRefGoogle Scholar
Bühler, O. 1998 A shallow-water model that prevents nonlinear steepening of gravity waves. J. Atmos. Sci. 55, 2884–2891.Google Scholar
Bühler, O. 2000 On the vorticity transport due to dissipating or breaking waves in shallow-water flow. J. Fluid Mech. 407, 235–263.CrossRefGoogle Scholar
Bühler, O. 2014 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Bühler, O. & Holmes-Cerfon, M. 2009 Particle dispersion by random waves in rotating shallow water. J. Fluid Mech. 638, 5–26.CrossRefGoogle Scholar
Bühler, O. & Jacobson, T. E. 2001 Wave-driven currents and vortex dynamics on barred beaches. J. Fluid Mech. 449, 313–339.Google Scholar
Bühler, O. & McIntyre, M. E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343.Google Scholar
Bühler, O. & McIntyre, M. E. 2003 Remote recoil: a new wave–mean interaction effect. J. Fluid Mech. 492, 207–230.CrossRefGoogle Scholar
Eliassen, A. & Palm, E. 1961 On the transfer of energy in stationary mountain waves. Geofys Publ. 22 (3), 1–23.Google Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Majda, A. J. & Embid, P. 1998 Averaging over fast gravity waves for geophysical flows with unbalanced initial data. Theoret. Comput. Fluid Dynam. 11 (3), 155–169.CrossRefGoogle Scholar
McIntyre, M. E. 1988 A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves. J. Fluid Mech. 189, 235–242.CrossRefGoogle Scholar
Plumb, R. A. 1977 The interaction of two internal waves with the mean flow: implications for the theory of the quasi-biennial oscillation. J. Atmos. Sci. 34, 1847–1858.Google Scholar
Thomas, J., Smith, K. S. & Bühler, O. 2017 Near-inertial wave dispersion by geostrophic flows. J. Fluid Mech. 817, 406–438.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press.Google Scholar
Wagner, G. L. & Young, W. R. 2015 Available potential vorticity and wave-averaged quasi-geostrophic flow. J. Fluid Mech. 785, 401–424.CrossRefGoogle Scholar
Wagner, G. L. & Young, W. R. 2016 A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic. J. Fluid Mech. 802, 806–837.Google Scholar
Xie, J.-H. & Vanneste, J. 2015 A generalised-lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech. 774, 143–169.Google Scholar