Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-19T07:54:14.610Z Has data issue: false hasContentIssue false
  • English
  • Français

New intermediate models for rotating shallow water and an investigation of the preference for anticyclones

Published online by Cambridge University Press:  10 September 2009

MARK REMMEL  and
LESLIE SMITH
MARK REMMEL
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
LESLIE SMITH*
Affiliation:
Departments of Mathematics and Engineering Physics, University of Wisconsin, Madison, WI 53706, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

New intermediate models for the rotating shallow water (RSW) equations are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The such simplest model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbers Ro in the range 0.1 < Ro < 1 (away from the QG parameter regime Ro → 0). A hierarchy of models is explored to determine the effects of wave-vortical and wave–wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting them a priori to any particular parameter regime) and (ii) insight into the physical and mathematical consequences of vortical–wave interactions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 635 , 25 September 2009 , pp. 321 - 359
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Allen, J. S. 1991 Balance equations based on momentum equations with global invariants of potential enstrophy and energy. J. Phys. Oceanogr. 21, 265–276.2.0.CO;2>CrossRefGoogle Scholar
Allen, J. S. 1993 Iterated geostrophic intermediate models. J. Phys. Oceanogr. 23 (1), 2447–2461.2.0.CO;2>CrossRefGoogle Scholar
Allen, J. S., Barth, J. A. & Newberger, P. A. 1990 a On intermediate models for barotropic continental shelf and slope flow fields. Part i. Formulation and comparison of exact solutions. J. Phys. Oceanogr. 20, 1017–1042.2.0.CO;2>CrossRefGoogle Scholar
Allen, J. S., Barth, J. A. & Newberger, P. A. 1990 b On intermediate models for barotropic continental shelf and slope flow fields. Part iii. Comparison of numerical model solutions in periodic channels. J. Phys. Oceanogr. 20 (2), 1949–1973.2.0.CO;2>CrossRefGoogle Scholar
Allen, J. S. & Newberger, P. A. 1993 On intermediate models for stratified flow. J. Phys. Oceanogr. 23 (2), 2462–2485.2.0.CO;2>CrossRefGoogle Scholar
Babin, A., Mahalov, A. & Nicolaenko, B. 2000 Fast singular oscillating limits and global regularity for the three-dimensional primitive equations of geophysics. Math. Model. Num. Anal. 34, 201–222.CrossRefGoogle Scholar
Babin, A., Mahalov, A. & Nicolaenko, B. 2002 Singular oscillating limits of stably-stratified three-dimensional euler and Navier–Stokes equations and a geostrophic wave fronts. In Large-Scale Atmosphere-Ocean Dynamics I (ed. Norbury, J. & Rhoulstone, I.). Cambridge University Press.Google Scholar
Bartello, P. 1995 Geostrophic adjustment and inverse cascades in rotating stratified turbulence. J. Atmos. Sci. 52, 4410–4428.2.0.CO;2>CrossRefGoogle Scholar
Barth, J. A., Allen, J. S. & Newberger, P. A. 1990 On intermediate models for barotropic continental shelf and slope flow fields. Part ii. Comparison of numerical model solutions in doubly periodic domains. J. Phys. Oceanogr. 20, 1044–1076.2.0.CO;2>CrossRefGoogle Scholar
Charney, J. G. 1948 On the scale of atmospheric motions. Geofysiske Publikasjoner 17 (2), 1–17.Google Scholar
Embid, P. F. & Majda, A. J. 1996 Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Comm. Partial Diff. Eq. 21 (3–4), 619–658.CrossRefGoogle Scholar
Embid, P. F. & Majda, A. J. 1998 Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers. Geophys. Astrophys. Fluid Dyn. 87 (1–2), 1–50.CrossRefGoogle Scholar
Farge, M. & Sadourny, R. 1989 Wave-vortex dynamics in rotating shallow water. J. Fluid Mech. 206, 433–462.CrossRefGoogle Scholar
Ford, R., McIntyre, M. E. & Norton, W. A. 2000 Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57, 1236–1254.2.0.CO;2>CrossRefGoogle Scholar
Hakim, G. J., Snyder, C. & Muraki, D. J. 2002 A new surface model for cyclone–anticyclone asymmetry. J. Atmos. Sci. 59, 2405–2420.2.0.CO;2>CrossRefGoogle Scholar
Julien, K., Knobloch, E., Milliff, R. & Werne, J. 2006 Generalized quasi-geostrophy for spatially anisotropic rotationally constrained flows. J. Fluid Mech. 555, 233–274.CrossRefGoogle Scholar
Kraichnan, R. H. 1973 Helical turbulence and absolute equilibrium. J. Fluid Mech. 59, 745–752.CrossRefGoogle Scholar
Kundu, P. K. & Cohen, I. M. 2004 Fluid Mechanics. Academic Press.Google Scholar
Kuo, A. C. & Polvani, L. M. 1999 Wave–vortex interaction in rotating shallow water. I. One space dimension. J. Fluid Mech. 394, 1–27.CrossRefGoogle Scholar
Kuo, A. C. & Polvani, L. M. 2000 Nonlinear geostrophic adjustment, cyclone/anticyclone asymmetry, and potential vorticity rearrangement. Phys. Fluids 12 (5), 1087–1100.CrossRefGoogle Scholar
Lorenz, E. N. 1960 Energy and numerical weather prediction. Tellus 4, 364–373.CrossRefGoogle Scholar
Majda, A. & Embid, P. 1998 Averaging over fast gravity waves for geophysical flows with unbalanced initial data. Theor. Comput. Fluid Dyn. 11, 155–169.CrossRefGoogle Scholar
Majda, A. 2003 Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, vol. 9. New York University, Courant Institute of Mathematical Sciences.CrossRefGoogle Scholar
Maltrud, M. E. & Vallis, G. K. 1990 Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence. J. Fluid Mech. 228, 321–342.Google Scholar
Martin, Jonathan E. 2006 Mid-Latitude Atmospheric Dynamics: A First Course. Wiley.Google Scholar
McIntyre, M. E. & Norton, W. A. 2000 Potential vorticity inversion on a hemisphere. J. Atmos. Sci. 57 (9), 1214–1235.2.0.CO;2>CrossRefGoogle Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.CrossRefGoogle Scholar
McWilliams, J. C. & Gent, P. R. 1980 Intermediate models of planetary circulations in the atmosphere and ocean. J. Atmos. Sci. 37 (8), 1657–1678.2.0.CO;2>CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2001 Hierarchies of balance conditions for the f-plane shallow water equations. J. Atmos. Sci. 58, 2411–2426.2.0.CO;2>CrossRefGoogle Scholar
Muraki, D. J. & Hakim, G. J. 2001 Balanced asymmetries of waves on the tropopause. J. Atmos. Sci. 58, 237–252.2.0.CO;2>CrossRefGoogle Scholar
Muraki, D., Snyder, C. & Rotunno, R. 1999 The next-order corrections to quasigeostrophic theory. J. Atmos. Sci. 56, 1547–1560.2.0.CO;2>CrossRefGoogle Scholar
Obukhov, A. M. 1949 On the problem of geostrophic wind. Izv. Geogr. Geophys. 13, 281–306 (in Russian).Google Scholar
Pedlosky, J. 1986 Geophysical Fluid Dynamics. Springer-Verlag.Google Scholar
Polvani, L., Mcwilliams, J., Spall, M. & Ford, R. 1994 The coherent structures of shallow-water turbulence: deformation radius effects, cyclone/anticyclone asymmetry and gravity-wave generation. Chaos 4, 177–186.CrossRefGoogle ScholarPubMed
Reznik, G. M., Zeitlin, V. & Ben Jelloul, M. 2001 Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model. J. Fluid Mech. 445, 93–120.CrossRefGoogle Scholar
Rotunno, R., Muraki, D. & Snyder, C. 2000 Unstable baroclinic waves beyond quasigeostrophic theory. J. Atmos. Sci. 57, 3285–3295.2.0.CO;2>CrossRefGoogle Scholar
Sadourny, R. 1975 The dynamics of finite-difference models of the shallow-water equations. J. Atmos. Sci. 32, 680–689.2.0.CO;2>CrossRefGoogle Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.CrossRefGoogle Scholar
Smith, L. M. & Waleffe, F. 2002 Generation of slow, large scales in forced rotating, stratified turbulence. J. Fluid Mech. 451, 145–168.CrossRefGoogle Scholar
Spall, M. A. & McWilliams, J. C. 1992 Rotational and gravitational influences on the degree of balance in the shallow-water equations. Geophys. Astrophys. Fluid Dyn. 64, 1–29.CrossRefGoogle Scholar
Stegner, A. & Zeitlin, V. 1995 What can asymptotic expansions tell us about large-scale quasi-geostrophic anticyclonic vortices? Nonlinear Process. Geophys. 2 (3/4), 186–193.CrossRefGoogle Scholar
Sukhatme, J. & Smith, L. 2008 Vortical and wave modes in three-dimensional rotating stratified flows: random large-scale forcing. Geophys. Astrophys. Fluid Dyn. 102 (19), 437–455.CrossRefGoogle Scholar
Vallis, G. K. 1996 Potential vorticity inversion and balanced equations of motion for rotating and stratified flows. Q. J. Royal Meteorol. Soc. 122, 291–322.CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2006 The transition from geostrophic to stratified turbulence. J. Fluid Mech. 568, 89–108.CrossRefGoogle Scholar
Warn, T. 1986 Statistical mechanical equilibria of the shallow water equations. Tellus Ser. Dynamic Meteorol. Oceanogr. 38, 1–11.CrossRefGoogle Scholar
Warn, T., Bokhove, O., Shepherd, T. G. & Vallis, G. K. 1995 Rossby number expansions, slaving principles and balance dynamics. Q. J. Royal Meteorol. Soc. 121, 723–739.CrossRefGoogle Scholar
Yavneh, I. & McWilliams, J. C. 1994 Breakdown of the slow manifold in the shallow-water equations. Geophys. Astrophys. Fluid Dyn. 75, 131–161.CrossRefGoogle Scholar
Yuan, L. & Hamilton, K. 1994 Equilibrium dynamics in a forced-dissipative f-plane shallow-water system. J. Fluid Mech. 280, 369–394.CrossRefGoogle Scholar