Hostname: page-component-f554764f5-fr72s Total loading time: 0 Render date: 2025-04-10T10:45:36.915Z Has data issue: false hasContentIssue false
  • English
  • Français

A study of surface semi-geostrophic turbulence: freely decaying dynamics

Published online by Cambridge University Press:  04 March 2016

Francesco Ragone  and
Gualtiero Badin
Francesco Ragone*
Affiliation:
Institut für Meereskunde, Universität Hamburg, Bundesstr. 53, 20146 Hamburg, Germany Laboratoire de Physique, École Normale Supérieure de Lyon, 46 allée d’Italie, 69007 Lyon, France
Gualtiero Badin
Affiliation:
Institut für Meereskunde, Universität Hamburg, Bundesstr. 53, 20146 Hamburg, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this study we give a characterization of semi-geostrophic turbulence by performing freely decaying simulations for the case of constant uniform potential vorticity, a set of equations known as the surface semi-geostrophic approximation. The equations are formulated as conservation laws for potential temperature and potential vorticity, with a nonlinear Monge–Ampère type inversion equation for the streamfunction, expressed in a transformed coordinate system that follows the geostrophic flow. We perform model studies of turbulent surface semi-geostrophic flows in a domain doubly periodic in the horizontal and limited in the vertical by two rigid lids, allowing for variations of potential temperature at one of the boundaries, and we compare the results with those obtained in the corresponding surface quasi-geostrophic case. The results show that, while the surface quasi-geostrophic dynamics is dominated by a symmetric population of cyclones and anticyclones, the surface semi-geostrophic dynamics features a more prominent role of fronts and filaments. The resulting distribution of potential temperature is strongly skewed and peaked at non-zero values at and close to the active boundary, while symmetry is restored in the interior of the domain, where small-scale frontal structures do not penetrate. In surface semi-geostrophic turbulence, energy spectra are less steep than in the surface quasi-geostrophic case, with more energy concentrated at small scales for increasing Rossby number. The energy related to frontal structures, the lateral strain rate and the vertical velocities are largest close to the active boundary. These results show that the semi-geostrophic model could be of interest for studying the lateral mixing of properties in geophysical flows.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Geostrophic turbulence Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 740 - 774
Check for updates
Copyright
© 2016 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.)

Article purchase

Temporarily unavailable

References

Andrews, D. G. & Hoskins, B. J. 1978 Energy spectra predicted by semi-geostrophic theories of frontogenesis. J. Atmos. Sci. 35, 509–512.2.0.CO;2>CrossRefGoogle Scholar
Arakawa, A. 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys 1, 119–143.Google Scholar
Badin, G. 2013 Surface semi-geostrophic dynamics in the ocean. Geophys. Astrophys. Fluid Dyn. 107, 526–540.CrossRefGoogle Scholar
Badin, G. 2014 On the role of non-uniform stratification and short-wave instabilities in three-layer quasi-geostrophic turbulence. Phys. Fluids 26, 096603.CrossRefGoogle Scholar
Badin, G., Tandon, A. & Mahadevan, A. 2011 Lateral mixing in the pycnocline by baroclinic mixed layer eddies. J. Phys. Oceanogr. 41, 2080–2101.Google Scholar
Badin, G., Williams, R. G., Holt, J. T. & Fernand, L. 2009 Are mesoscale eddies in shelf seas formed by baroclinic instability of tidal fronts? J. Geophys. Res. 114, C10021.Google Scholar
Badin, G., Williams, R. G. & Sharples, J. 2010 Water-mass transformation in the shelf seas. J. Mar. Res. 68, 189–214.Google Scholar
Badin, G., Williams, R. G., Jing, Z. & Wu, L. 2013 Water-mass transformations in the Southern Ocean diagnosed from observations: contrasting effects of air–sea fluxes and diapycnal mixing. J. Phys. Oceanogr. 43, 1472–1484.Google Scholar
Benamou, J.-D., Froese, B. D. & Oberman, A. M. 2010 Two numerical methods for the elliptic Monge–Ampère equation. ESAIM: Math. Model. Numer. Anal. 44, 737–758.Google Scholar
Blender, R. & Badin, G. 2015 Hydrodynamic Nambu mechanics derived by geometric constraints. J. Phys. A: Math. Theor. 48, 105501.Google Scholar
Blumen, W. 1978 Uniform potential vorticity flow: Part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35, 774–783.Google Scholar
Blumen, W. 1979 Unstable nonlinear evolution of an Eady wave in time-dependent basic flows and frontogenesis. J. Atmos. Sci. 36, 3–11.Google Scholar
Blumen, W. 1981 The geostrophic coordinate transformation. J. Atmos. Sci. 38, 1100–1105.Google Scholar
Boyd, J. P. 1992 The energy spectrum of fronts: time evolution and shocks in Burgers’ equation. J. Atmos. Sci. 49, 128–139.Google Scholar
Callies, J. & Ferrari, R. 2013 Interpreting energy and tracer spectra of upper-ocean turbulence in the submesoscale range (1–200 km). J. Phys. Oceanogr. 43, 2456–2474.Google Scholar
Capet, X., Klein, P., Hua, B. L., Lapeyre, G. & McWilliams, J. C. 2008a Surface kinetic energy transfer in surface quasi-geostrophic flows. J. Fluid Mech. 604, 165–174.Google Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008b Mesoscale to submesoscale transition in the California Current System. Part I: Flow structure, eddy flux, and observational tests. J. Phys. Oceanogr. 38, 29–43.Google Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008c Mesoscale to submesoscale transition in the California Current System. Part II: Frontal processes. J. Phys. Oceanogr. 38, 44–64.Google Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008d Mesoscale to submesoscale transition in the California Current System. Part III: Energy balance and flux. J. Phys. Oceanogr. 38, 2256–2269.CrossRefGoogle Scholar
Constantin, P., Lai, M. C., Sharma, R., Tseng, Y. H. & Wu, J. 2012 New numerical results for the surface quasi-geostrophic equation. J. Sci. Comput. 50, 1–28.CrossRefGoogle Scholar
Constantin, P., Majda, A. J. & Tabak, E. 1994 Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533.CrossRefGoogle Scholar
Craig, G. C. 1993 A scaling for the three-dimensional semigeostrophic approximation. J. Atmos. Sci. 50, 3350–3355.Google Scholar
Cullen, M. J. P. 2006 A Mathematical Theory of Large-Scale Atmospheric and Oceanic Flows. Cambridge University Press.Google Scholar
Cullen, M. J. P., Norbury, J. & Purser, R. J. 1991 Generalized Lagrangian solutions for atmospheric and oceanic flows. SIAM J. Appl. Maths 51, 20–31.CrossRefGoogle Scholar
Cullen, M. J. P. & Purser, R. J. 1984 An extended Lagrangian theory of semi-geostrophic frontogenesis. J. Atmos. Sci. 41, 1477–1497.Google Scholar
D’Asaro, E., Lee, C., Rainville, L., Harcourt, R. & Thomas, L. 2011 Enhanced turbulence and energy dissipation at ocean fronts. Science 332, 318–322.Google Scholar
Davies, H. C. & Mueller, J. C. 1988 Detailed description of deformation-induced semi-geostrophic frontogenesis. Q. J. R. Meteorol. Soc. 114, 1201–1219.Google Scholar
Delhaies, S. & Roulstone, I. 2010 Hyper-Kähler geometry and semi-geostrophic theory. Proc. R. Soc. Lond. A 466, 195–211.Google Scholar
Eliassen, A. 1948 The quasi-static equations of motion. Geofys. Publ. 17, 5–44.Google Scholar
Fjortoft, R. 1962 On the integration of a system of geostrophically balanced prognostic equations. In Proceedings of the International Symposium on Numerical Weather Prediction, Tokyo, Japan, pp. 153–159. Meteorological Society of Japan.Google 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
Held, I. M., Pierrehumbert, R. T., Garner, S. T. & Swanson, K. L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.Google Scholar
Hoskins, B. J. 1975 The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32, 233–242.Google Scholar
Hoskins, B. J. 1976 Baroclinic waves and frontogenesis. Part I: Introduction and Eady waves. Q. J. R. Meteorol. Soc. 102, 103–122.Google Scholar
Hoskins, B. J. & Bretherton, F. 1972 Atmospheric frontogenesis models: mathematical formulation and solutions. J. Atmos. Sci. 29, 11–37.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B. J. & Draghici, I. 1977 The forcing of ageostrophic motion according to the semi-geostrophic equations and in an isentropic coordinate model. J. Atmos. Sci. 34, 1859–1867.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B. J., Draghici, I. & Davies, H. C. 1978 A new look at the ${\it\omega}$ -equation. Q. J. R. Meteorol. Soc. 104, 31–38.Google Scholar
Hoskins, B. J. & West, N. V. 1979 Baroclinic waves and frontogenesis. Part II: Uniform potential vorticity jet flows – cold and warm fronts. J. Atmos. Sci. 36, 1663–1680.Google Scholar
Hou, T. Y. & Li, R. 2007 Computing nearly singular solutions using pseudo-spectral methods. J. Comput. Phys. 226, 379–397.CrossRefGoogle Scholar
Juckes, M. N. 1994 Quasigeostrophic dynamics of the tropopause. J. Atmos. Sci. 51, 2756–2768.Google Scholar
Juckes, M. N. 1998 Baroclinic instability of semi-geostrophic fronts with uniform potential vorticity. I: An analytic solution. Q. J. R. Meteorol. Soc. 124, 2227–2257.Google Scholar
Kerr, R. M. 1990 Velocity, scalar and transfer spectra in numerical turbulence. J. Fluid Mech. 211, 309–332.Google Scholar
Klein, P., Hua, B. L., Lapeyre, G., Capet, X., Le Gentil, S. & Sasaki, H. 2008 Upper ocean turbulence from high-resolution 3D simulations. J. Phys. Oceanogr. 38, 1748–1763.Google Scholar
Kushner, P. J. 1995 A generalized Charney–Stern theorem for semi-geostrophic dynamics. Tellus A 47, 541–547.Google Scholar
Kushner, P. J. & Shepherd, T. G. 1995a Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1. Pseudomomentum-based theory. J. Fluid Mech. 290, 67–104.Google Scholar
Kushner, P. J. & Shepherd, T. G. 1995b Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 2. Pseudoenergy-based theory. J. Fluid Mech. 290, 105–129.CrossRefGoogle Scholar
LaCasce, J. H. 2012 Surface quasigeostrophic solutions and baroclinic modes with exponential stratification. J. Phys. Oceanogr. 42, 569–580.CrossRefGoogle Scholar
LaCasce, J. H. & Mahadevan, A. 2006 Estimating subsurface horizontal and vertical velocities from sea-surface temperature. J. Mar. Res. 64, 695–721.CrossRefGoogle Scholar
Lapeyre, G. 2009 What vertical mode does the altimeter reflect? On the decomposition in baroclinic modes and on a surface-trapped mode. J. Phys. Oceanogr. 39, 2857–2874.Google Scholar
Lapeyre, G. & Klein, P. 2006 Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory. J. Phys. Oceanogr. 36, 165–179.Google Scholar
Lapeyre, G., Klein, P. & Hua, B. L. 2006 Oceanic restratification forced by surface frontogenesis. J. Phys. Oceanogr. 36, 1577–1590.Google Scholar
Levy, M., Klein, P. & Treguier, A. M. 2001 Impacts of submesoscale physics on phytoplankton production and subduction. J. Mar. Res. 59, 535–565.CrossRefGoogle Scholar
Liu, L., Peng, S., Wang, J. & Huang, R. 2014 Retrieving density and velocity fields of the ocean’s interior from surface data. J. Geophys. Res. 119, 8512–8529.CrossRefGoogle Scholar
Mahadevan, A. & Tandon, A. 2006 An analysis of mechanisms for submesoscale vertical motion at ocean fronts. Ocean Model. 14, 241–256.Google Scholar
Malardel, S., Thorpe, A. J. & Joly, A. 1997 Consequences of the geostrophic momentum approximation on barotropic instability. J. Atmos. Sci. 54, 103–112.Google Scholar
McIntyre, M. E. & Roulstone, I. 2002 Are there higher-accuracy analogues of semigeostrophic theory? In Large-Scale Atmosphere–Ocean Dynamics: Vol. II, Geometric Methods and Models (ed. Norbury, J. & Roulstone, I.), Cambridge University Press.Google Scholar
Nagai, T., Tandon, A. & Rudnick, D. L. 2006 Two-dimensional ageostrophic secondary circulation at ocean fronts due to vertical mixing and large-scale deformation. J. Geophys. Res. 111, C09038.Google Scholar
Okubo, A. 1970 Horizontal dispersion of floatable trajectories in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17, 445–454.Google Scholar
Oliver, M. 2006 Variational asymptotics for rotating shallow water near geostrophy: a transformational approach. J. Fluid Mech. 551, 197–234.Google Scholar
Oliver, M. 2014 A variational derivation of the geostrophic momentum approximation. J. Fluid Mech. 751, R2.CrossRefGoogle Scholar
Pedder, M. A. & Thorpe, A. J. 1999 The semi-geostrophic diagnosis of vertical motion. I: Formulation and coordinate transformations. Q. J. R. Meteorol. Soc. 125, 1231–1256.Google Scholar
Pinot, J.-M., Tintoré, J. & Wang, D.-P. 1996 A study of the omega equation for diagnosing vertical motions at ocean fronts. J. Mar. Res. 54, 239–259.CrossRefGoogle Scholar
Plougonven, R. & Zeitlin, V. 2005 Lagrangian approach to the geostrophic adjustment of frontal anomalies in a stratified fluid. Geophys. Astrophys. Fluid Dyn. 99, 101–135.Google Scholar
Purser, R. 1993 Contact transformations and Hamiltonian dynamics in generalized semigeostrophic theories. J. Atmos. Sci. 50, 1449–1468.Google Scholar
Purser, R. 1999 Legendre-transformable semigeostrophic theories. J. Atmos. Sci. 56, 2522–2535.2.0.CO;2>CrossRefGoogle Scholar
Purser, R. & Cullen, M. J. P. 1987 A duality principle in semigeostrophic theories. J. Atmos. Sci. 44, 3449–3468.Google Scholar
Ren, S. 1998 Linear stability of the three-dimensional semigeostrophic model in geostrophic coordinates. J. Atmos. Sci. 55, 3392–3402.Google Scholar
Ren, S. 1999 Linear stability theorems for shallow water semi-geostrophic dynamics. Geophys. Astrophys. Fluid Dyn. 90, 189–227.CrossRefGoogle Scholar
Ren, S. 2000a Evolution of disturbances and singular vectors in the shallow-water semi-geostrophic model. Q. J. R. Meteorol. Soc. 126, 2487–2509.Google Scholar
Ren, S. 2000b Finite-amplitude wave-activity invariants and nonlinear stability theorems for shallow water semigeostrophic dynamics. J. Atmos. Sci. 57, 3388–3397.Google Scholar
Ren, S. 2005 Instability of zonal flows in a two-layer shallow water semi-geostrophic model. Q. J. R. Meteorol. Soc. 131, 1441–1459.Google Scholar
Roubtsov, V. & Roulstone, I. 1997 Examples of quarterionic and Kähler structures in Hamiltonian models of nearly geostrophic flow. J. Phys. A Math. Gen. 30, 63–68.Google Scholar
Roubtsov, V. & Roulstone, I. 2001 Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. Lond. A 457, 1519–1531.Google Scholar
Roullet, G. & Klein, P. 2010 Cyclone–anticyclone asymmetry in geophysical turbulence. Phys. Rev. Lett. 104, 1–4.Google Scholar
Salmon, R. 1983 Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431–444.Google Scholar
Salmon, R. 1985 New equations for nearly geostrophic flow. J. Fluid Mech. 153, 461–477.Google Scholar
Salmon, R. 1988 Semi-geostrophic theory as Dirac bracket projection. J. Fluid Mech. 196, 345–358.Google Scholar
Schär, C. & Davies, H. C. 1990 An instability of mature cold fronts. J. Atmos. Sci. 47, 929–950.Google Scholar
Shcherbina, A. Y., D’Asaro, E. A., Lee, C. M., Klymak, J. M., Molemaker, M. J. & McWilliams, J. C. 2013 Statistics of vertical vorticity, divergence, and strain in a developed submesoscale turbulence. Geophys. Res. Lett. 40, 1–6.Google Scholar
Shcherbina, A. Y., Sundermeyer, M. A., Kunze, E., D’Asaro, E., Badin, G., Birch, D., Brunner-Suzuki, A.-M. E. G., Callies, J., Kuebel Cervantes, B. T., Claret, M. et al. 2015 The LatMix summer campaign: submesoscale stirring in the upper ocean. Bull. Am. Meteorol. Soc. 96, 1257–1279.Google Scholar
Smith, K. S. & Bernard, E. 2013 Geostrophic turbulence near rapid changes in stratification. Phys. Fluids 25, 046601.Google Scholar
Smith, K. S. & Vanneste, J. 2013 A surface-aware projection basis for quasigeostrophic flow. J. Phys. Oceanogr. 43, 548–562.Google Scholar
Snyder, C., Skamarock, W. C. & Rotunno, R. 1991 A comparison of primitive-equation and semigeostrophic simulations of baroclinic waves. J. Atmos. Sci. 48, 2179–2194.Google Scholar
Thomas, L. 2005 Destruction of potential vorticity by winds. J. Phys. Oceanogr. 35, 2457–2466.Google Scholar
Thomas, L. N. & Joyce, T. M. 2010 Subduction in the northern and southern flanks of the Gulf Stream. J. Phys. Oceanogr. 40, 429–438.Google Scholar
Thorpe, A. J. & Pedder, M. A. 1999 The semi-geostrophic diagnosis of vertical motion. II: Results for an idealized baroclinic wave life cycle. Q. J. R. Meteorol. Soc. 125, 1257–1276.Google Scholar
Tricomi, G. F. 1923 Sulle equazioni lineari alle derivate parziali di 2 ordine di tipo misto. Rend. Fis. Accad. Lincei 14, 134–247.Google Scholar
Tulloch, R. & Smith, K. S. 2006 A new theory for the atmospheric energy spectrum: depth-limited temperature anomalies at the tropopause. Proc. Natl Acad. Sci. USA 103, 14690–14694.Google Scholar
Tulloch, R. & Smith, K. S. 2009a A note on the numerical representation of surface dynamics in quasigeostrophic turbulence: application of the nonlinear Eady model. J. Atmos. Sci. 66, 1063–1068.Google Scholar
Tulloch, R. & Smith, K. S. 2009b Quasigeostrophic turbulence with explicit surface dynamics: application to the atmospheric energy spectrum. J. Atmos. Sci. 66, 450–467.Google Scholar
Viudez, A. & Dritschel, D. 2004 Potential vorticity and the quasigeostrophic and semigeostrophic mesoscale vertical velocity. J. Phys. Oceanogr. 34, 865–887.Google Scholar
Wang, J., Flierl, G. R., LaCasce, J. H., McClean, J. L. & Mahadevan, A. 2013 Reconstructing the ocean’s interior from surface data. J. Phys. Oceanogr. 43, 1611–1626.Google Scholar
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294.CrossRefGoogle Scholar