Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-19T22:14:14.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial and barotropic instabilities of a free current in three-dimensional rotating flow

Published online by Cambridge University Press:  14 May 2013

G. F. Carnevale ,
R. C. Kloosterziel  and
P. Orlandi
G. F. Carnevale*
Affiliation:
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093, USA
R. C. Kloosterziel
Affiliation:
School of Ocean and Earth Science and Technology, University of Hawaii, Honolulu, HI 96822, USA
P. Orlandi
Affiliation:
Dipartimento di Meccanica e Aeronautica, University of Rome, ‘La Sapienza’, via Eudossiana 18, 00184 Roma, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A current in a homogeneous rotating fluid is subject to simultaneous inertial and barotropic instabilities. Inertial instability causes rapid mixing of streamwise absolute linear momentum and alters the vertically averaged velocity profile of the current. The resulting profile can be predicted by a construction based on absolute-momentum conservation. The alteration of the mean velocity profile strongly affects how barotropic instability will subsequently change the flow. If a current with a symmetric distribution of cyclonic and anticyclonic vorticity undergoes only barotropic instability, the result will be cyclones and anticyclones of the same shape and amplitude. Inertial instability breaks this symmetry. The combined effect of inertial and barotropic instability produces anticyclones that are broader and weaker than the cyclones. A two-step scheme for predicting the result of the combined inertial and barotropic instabilities is proposed and tested. This scheme uses the construction for the redistribution of streamwise absolute linear momentum to predict the mean current that results from inertial instability and then uses this equilibrated current as the initial condition for a two-dimensional simulation that predicts the result of the subsequent barotropic instability. Predictions are made for the evolution of a Gaussian jet and are compared with three-dimensional simulations for a range of Rossby numbers. It is demonstrated that the actual redistribution of absolute momentum in the three-dimensional simulations is well predicted by the construction used here. Predictions are also made for the final number and size of vortices that result from the combined inertial and barotropic instabilities.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , pp. 117 - 151
Copyright
©2013 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

Billant, P. & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.Google Scholar
Bouchut, F., Ribstein, B. & Zeitlin, V. 2011 Inertial, barotropic, and baroclinic instabilities of the bickley jet in two-layer rotating shallow water model. Phys. Fluids 23, 126601.Google Scholar
Carnevale, G. F., Kloosterziel, R. C., Orlandi, P. & van Sommeren, D. D. J. A. 2011 Predicting the aftermath of vortex breakup in rotating flow. J. Fluid Mech. 669, 90–119.Google Scholar
Carnevale, G. F., McWilliams, J. C., Pomeau, Y., Weiss, J. B. & Young, W. R. 1991 Evolution of vortex statistics in two-dimensional turbulence. Phys. Rev. Lett. 66, 2735–2737.Google Scholar
Carnevale, G. F., McWilliams, J. C., Pomeau, Y., Weiss, J. B. & Young, W. R. 1992 Rates, pathways, and end states of nonlinear evolution in decaying two-dimensional turbulence: scaling theory versus selective decay. Phys. Fluids A 4, 1314–1316.CrossRefGoogle Scholar
Charney, J. G. 1973 Lecture notes on planetary fluid dynamics. In Dynamic Meteorology (ed. Morel, P.). pp. 97–351. Kluwer.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Eldevik, T. & Dysthe, K. B. 2002 Spiral eddies. J. Phys. Oceanogr. 32, 851–869.Google Scholar
Flierl, G. R., Malanotte-Rizzoli, P. & Zabusky, N. J. 1987 Nonlinear waves and coherent vortex structures in barotropic $\beta $ -plane jets. J. Phys. Oceanogr. 17, 1408–1438.2.0.CO;2>CrossRefGoogle Scholar
Gallaire, F. & Chomaz, J. M. 2003 Three-dimensional instability of isolated vortices. Phys. Fluids 15 (8), 2113–2126.Google Scholar
Griffiths, S. D. 2008 The limiting form of inertial instability in geophysical flows. J. Fluid Mech. 605, 115–143.Google Scholar
Holton, J. R. 1979 An Introduction to Dynamic Meteorology. Academic.Google Scholar
Kloosterziel, R. C. & Carnevale, G. F. 2008 Vertical scale selection in inertial instability. J. Fluid Mech. 594, 249–269.Google Scholar
Kloosterziel, R. C., Carnevale, G. F. & Orlandi, P. 2007a Inertial instability in rotating and stratified fluids: barotropic vortices. J. Fluid Mech. 583, 379–412.Google Scholar
Kloosterziel, R. C. & van Heijst, G. J. F. 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.Google Scholar
Kloosterziel, R. C., Orlandi, P. & Carnevale, G. F. 2007b Saturation of inertial instability in rotating planar shear flows. J. Fluid Mech. 583, 413–422.Google Scholar
Markowski, P. & Richardson, Y. 2010 Mesoscale Meteorology in Midlatitudes. Wiley-Blackwell.Google Scholar
Matthaeus, W. H., Stribling, W. T., Martinez, D., Oughton, S. & Montgomery, D. 1991 Decaying two-dimensional turbulence at very long times. Physica D 51, 531–538.Google Scholar
Melander, M. V., Zabusky, N. J. & McWilliams, J. C. 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303–340.Google Scholar
Montgomery, D., Matthaeus, W. H., Stribling, W. T., Martinez, D. & Oughton, S. 1992 Relaxation in two dimensions and the ‘sinh-poisson’ equation. Phys. Fluids A 4, 3–6.CrossRefGoogle Scholar
Munk, W., Armi, L., Fischer, K. & Zachariasen, F. 2000 Spirals on the sea. Proc. R. Soc. Lond. 456, 1217–1280.Google Scholar
Orlandi, P. & Carnevale, G. F. 1999 Evolution of isolated vortices in a rotating fluid of finite depth. J. Fluid Mech. 381, 239–269.Google Scholar
Plougonven, R. & Zeitlin, V. 2009 Nonlinear development of inertial instability in a barotropic shear. Phys. Fluids 21, 106601.Google Scholar
Poulin, F. J. & Flierl, G. R. 2003 The nonlinear evolution of barotropically unstable jets. J. Phys. Oceanogr. 33, 2173–2192.Google Scholar
Ribstein, B., Plougonven, R. & Zeitlin, V. 2013 Inertial vs baroclinic instability of the bickley jet in continuously stratified rotating fluid, J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Shen, C. Y. & Evans, T. E. 1998 Inertial instability of large Rossby number horizontal shear flows in a thin homogeneous layer. Dyn. Atmos. Oceans 26, 185–208.Google Scholar
Shen, C. Y. & Evans, T. E. 2002 Inertial instability and sea spirals. Geophys. Res. Lett. 29, 1–19.Google Scholar
Smyth, W. D. & McWilliams, J. C. 1998 Instability of an axisymmetric vortex in a stably stratified, rotating environment. Theor. Comput. Fluid Dyn. 11, 305–322.CrossRefGoogle Scholar
Winter, Th. & Schmitz, G. 1998 On divergent barotropic and inertial instability in zonal-mean flow profiles. J. Atmos. Sci. 98, 758–776.2.0.CO;2>CrossRefGoogle Scholar