# Surface Quasi-Geostrophy

## Abstract

**:**

## 1. Introduction

## 2. General Formulation of Surface Quasi-Geostrophy (SQG)

#### 2.1. General Quasi-Geostrophic (QG) Theory

#### 2.2. Inversion of Potential Vorticity ($PV$) Equation

#### 2.3. Surface Quasi-Geostrophy (SQG) Formulation

#### 2.4. Invariants

#### 2.5. Relation between the Active Tracer and Streamfunction

## 3. Coherent Structure Dynamics

#### 3.1. Exact Surface Quasi-Geostrophy (SQG) Solutions

#### 3.2. Contour Dynamics

#### 3.3. Instabilities

## 4. Turbulent Cascades

#### 4.1. Buoyancy Variance Spectra

#### 4.2. Physical Properties of the Cascade

#### 4.3. Role of Meridional Gradients and Linear Damping

#### 4.4. Energy Transfers

## 5. Surface Quasi-Geostrophy (SQG) from a Mathematical Point of View

#### 5.1. Analogy with 3D Euler Equations

#### 5.2. Regularity of Surface Quasi-Geostrophy (SQG) Solutions

## 6. Passive Tracers

## 7. Surface Quasi-Geostrophy (SQG) and the Ocean Dynamics

#### 7.1. Horizontal Motions from Surface Quasi-Geostrophy (SQG) Equilibrium

#### 7.2. Relation between Surface and Interior Dynamics

#### 7.3. Vertical Motions from Surface Quasi-Geostrophy (SQG) Equilibrium

#### 7.4. Frontogenesis

#### 7.5. Impact on Tracers

## 8. Surface Quasi-Geostrophy (SQG) and the Atmosphere Dynamics

## 9. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Charney, J.G. Geostrophic turbulence. J. Atmos. Sci.
**1971**, 28, 1087–1095. [Google Scholar] [CrossRef] - Pedlosky, J. Geophysical Fluid Dynamics; Springer: Berlin, Germany, 1987. [Google Scholar]
- Vallis, G.K. Atmospheric and Oceanic Fluid Dynamics; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Eady, E.T. Long waves and cyclone waves. Tellus
**1949**, 1, 33–52. [Google Scholar] [CrossRef] - Hakim, G.J. Developing wave packets in the north Pacific storm track. Mon. Weather Rev.
**2003**, 131, 2837. [Google Scholar] [CrossRef] - LaCasce, J.H.; Mahadevan, A. Estimating sub-surface horizontal and vertical velocities from sea surface temperature. J. Mar. Res.
**2006**, 64, 695–721. [Google Scholar] [CrossRef] - Lapeyre, G.; Klein, P. Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory. J. Phys. Oceanogr.
**2006**, 36, 165–176. [Google Scholar] [CrossRef] - Blumen, W. Uniform potential vorticity flow: Part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci.
**1978**, 35, 774–783. [Google Scholar] [CrossRef] - Held, I.M.; Pierrehumbert, R.T.; Garner, S.T.; Swanson, K.L. Surface quasi-geostrophic dynamics. J. Fluid Mech.
**1995**, 282, 1–20. [Google Scholar] [CrossRef] - Constantin, P.; Majda, A.J.; Tabak, E. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity
**1994**, 7, 1495–1533. [Google Scholar] [CrossRef] - Hoskins, B.J.; McIntyre, M.E.; Robertson, A.W. On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc.
**1985**, 111, 877–946. [Google Scholar] [CrossRef] - Bretherton, F.P. Critical layer instability in baroclinic flows. Q. J. R. Meteorol. Soc.
**1966**, 92, 325–334. [Google Scholar] [CrossRef] - Schneider, T.; Held, I.M.; Garner, S.T. Boundary effects in potential vorticity dynamics. J. Atmos. Sci.
**2003**, 60, 1024–1040. [Google Scholar] [CrossRef] - Charney, J.G.; Stern, M.E. On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci.
**1962**, 19, 159–172. [Google Scholar] [CrossRef] - Hua, B.L.; Haidvogel, D.B. Numerical simulations of the vertical structure of quasi-geostrophic turbulence. J. Atmos. Sci.
**1986**, 43, 2923–2936. [Google Scholar] [CrossRef] - Smith, K.S.; Vallis, G.K. The scales and equilibration of midocean eddies: Freely evolving flow. J. Phys. Oceanogr.
**2001**, 31, 554–571. [Google Scholar] [CrossRef] - Smith, K.S.; Boccaletti, G.; Henning, C.C.; Marinov, I.N.; Tam, C.Y.; Held, I.M.; Vallis, G.K. Turbulent diffusion in the geostrophic inverse cascade. J. Fluid Mech.
**2002**, 469, 14–47. [Google Scholar] [CrossRef] - Juckes, M. Instability of surface and upper-tropospheric shear lines. J. Atmos. Sci.
**1995**, 52, 3247–3262. [Google Scholar] [CrossRef] - Dritschel, D.G. An exact steadily-rotating surface quasi-geostrophic elliptical vortex. Geophys. Astrophys. Fluid Dyn.
**2011**, 105, 368–376. [Google Scholar] [CrossRef] - Castro, A.; Cordoba, D.; Gomez-Serrano, J.; Zamora, A.M. Remarks on geometric properties of SQG sharp fronts and alpha-patches. Discret. Contin. Dyn Syst.
**2014**, 34, 5045–5059. [Google Scholar] [CrossRef] - Castro, A.; Córdoba, D.; Gómez-Serrano, J. Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations. Duke Math. J.
**2016**, 165, 935–984. [Google Scholar] [CrossRef] - Lim, C.C.; Majda, A.J. Point vortex dynamics for coupled surface/interior QG and propagating heton clusters in models for ocean convection. Geophys. Astrophys. Fluid Dyn.
**2001**, 94, 177–220. [Google Scholar] [CrossRef] - Muraki, D.J.; Snyder, C. Vortex dipoles for surface quasigeostrophic models. J. Atmos. Sci.
**2004**, 61, 2961–2967. [Google Scholar] [CrossRef] - Carton, X.; Ciani, D.; Verron, J.; Reinaud, J.; Sokolovskyi, M. Vortex merger in surface quasi-geostrophy. Geophys. Astrophys. Fluid Dyn.
**2015**, 110, 1–22. [Google Scholar] [CrossRef] - Zabusky, N.J.; Hugues, M.H.; Roberts, K.V. Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys.
**1979**, 30, 96–106. [Google Scholar] [CrossRef] - Cordoba, D.; Fefferman, C.; Rodrigo, J.L. Almost sharp fronts for the surface quasi-geostrophic equation. Proc. Natl. Acad. Sci. USA
**2004**, 101, 2687–2691. [Google Scholar] [CrossRef] [PubMed] - Fefferman, C.L.; Rodrigo, J.L. Construction of almost-sharp fronts for the surface quasi-geostrophic equation. Arch. Rational Mech. Anal.
**2015**, 218, 123–162. [Google Scholar] [CrossRef] - Rodrigo, J.L. The vortex patch problem for the surface quasi-geostrophic equation. Proc. Natl. Acad. Sci. USA
**2004**, 101, 2684–2686. [Google Scholar] [CrossRef] [PubMed] - Gancedo, F. Existence for the α-patch model and the QG sharp front in Sobolev spaces. Adv. Math.
**2008**, 217, 2579–2598. [Google Scholar] [CrossRef] - Cordoba, D.; Fontelos, M.A.; Mancho, A.M.; Rodrigo, J.L. Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA
**2005**, 102, 5249–5252. [Google Scholar] [CrossRef] [PubMed] - Scott, R.K.; Dritschel, D.G. Numerical simulation of a self-similar cascade of filament instabilities in the Surface quasigeostrophic System. Phys. Rev. Lett.
**2014**, 112, 144505. [Google Scholar] [CrossRef] [PubMed] - Gancedo, F.; Strain, R.M. Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci. USA
**2014**, 111, 635–639. [Google Scholar] [CrossRef] - Dritschel, D.G.; Haynes, P.H.; Juckes, M.N.; Shepherd, T.G. The stability of a two-dimensional vorticity filament under uniform strain. J. Fluid Mech.
**1991**, 230, 647–665. [Google Scholar] [CrossRef] - Harvey, B.J.; Ambaum, M.H.P. Instability of surface temperature filaments in strain and shear. Q. J. R. Meteorol. Soc.
**2010**, 136, 1506–1513. [Google Scholar] [CrossRef] - Scott, R.K. A scenario for finite-time singularity in the quasigeostrophic model. J. Fluid Mech.
**2011**, 687, 492–502. [Google Scholar] [CrossRef] - Carton, X. Instability of surface quasi-geostrophic vortices. J. Atmos. Sci.
**2009**, 66, 1051–1062. [Google Scholar] [CrossRef] - Harvey, B.J.; Ambaum, M.H.P. Perturbed Rankine vortices in surface quasi-geostrophic dynamics. Geophys. Astrophys. Fluid Dyn.
**2010**, 105, 377–391. [Google Scholar] [CrossRef] - Harvey, B.J.; Ambaum, M.H.P.; Carton, X. Instability of shielded surface temperature vortices. J. Atmos. Sci.
**2011**, 68, 964–971. [Google Scholar] [CrossRef] - Friedlander, S.; Shvydkoy, R. The unstable spectrum of the surface quasi-geostrophic equation. J. Math. Fluid Mech.
**2005**, 7, S81–S93. [Google Scholar] [CrossRef] - Friedlander, S.; Pavlovic, N.; Vicol, V. Nonlinear instability for the critically dissipative quasi-geostrophic equation. Commun. Math. Phys.
**2009**, 292, 797–810. [Google Scholar] [CrossRef] - Rotunno, R.; Snyder, C. A generalization of Lorenz’s model for the predictability of flows with many scales of motion. J. Atmos. Sci.
**2008**, 65, 1063–1076. [Google Scholar] [CrossRef] - Tran, C.V.; Blackbourn, L.A.K.; Scott, R.K. Number of degrees of freedom and energy spectrum of surface quasi-geostrophic turbulence. J. Fluid Mech.
**2011**, 684, 427–440. [Google Scholar] [CrossRef] - Kraichnan, R.H. Inertial ranges in two-dimensional turbulence. Phys. Fluids
**1967**, 10, 1417–1423. [Google Scholar] [CrossRef] - Hoyer, J.M.; Sadourny, R. Closure modeling of fully developed baroclinic turbulence. J. Atmos. Sci.
**1982**, 39, 707–721. [Google Scholar] [CrossRef] - Tran, C.V.; Bowman, J.C. Large-scale energy spectra in surface quasi-geostrophic turbulence. J. Fluid Mech.
**2005**, 526, 349–359. [Google Scholar] [CrossRef] - Tran, C.V. Diminishing inverse transfer and non-cascading dynamics in surface quasi-geostrophic turbulence. Physica D
**2006**, 213, 76–84. [Google Scholar] [CrossRef] - Pierrehumbert, R.T.; Held, I.M.; Swanson, K.L. Spectra of local and nonlocal two-dimensional turbulence. Chaos Solitons Fractals
**1994**, 4, 1111–1116. [Google Scholar] [CrossRef] - Celani, A.; Cencini, M.; Mazzino, A.; Vergassola, M. Active and passive fields face to face. New J. Phys.
**2004**, 6, 72. [Google Scholar] [CrossRef] - Sukhatme, J.; Pierrehumbert, R.T. Surface Quasi-Geostrophic turbulence: The study of an active scalar. Chaos
**2002**, 12, 439–450. [Google Scholar] [CrossRef] [PubMed] - Capet, X.; Klein, P.; Hua, B.L.; Lapeyre, G.; McWilliams, J.C. Surface kinetic and potential energy transfer in SQG dynamics. J. Fluid Mech.
**2008**, 604, 165–174. [Google Scholar] [CrossRef] - Watanabe, T.; Iwayama, T. Unified scaling theory for local and non-local transfers in generalized two-dimensional turbulence. J. Phys. Soc. Jpn.
**2004**, 73, 3319–3330. [Google Scholar] [CrossRef] - Constantin, P. Energy spectrum of quasigeostrophic turbulence. Phys. Rev. Lett.
**2002**, 89, 184501. [Google Scholar] [CrossRef] [PubMed] - Burgess, B.H.; Scott, R.K.; Shepherd, T.G. Kraichnan-Leith-Batchelor similarity theory and two-dimensional inverse cascades. J. Fluid Mech.
**2015**, 767, 467–496. [Google Scholar] [CrossRef] - Tobias, S.M.; Cattaneo, F. Dynamo action in complex flows: The quick and the fast. J. Fluid Mech.
**2008**, 601, 101–122. [Google Scholar] [CrossRef] - Bernard, D.; Boffetta, G.; Celani, A.; Falkovich, G. Inverse turbulent cascades and conformally invariant curves. Phys. Rev. Lett.
**2007**, 98, 024501. [Google Scholar] [CrossRef] [PubMed] - Venaille, A.; Dauxois, T.; Ruffo, S. Violent relaxation in two-dimensional flows with varying interaction range. Phys. Rev. E
**2015**, 92, 011001. [Google Scholar] [CrossRef] [PubMed] - Teitelbaum, T.; Mininni, P.D. Thermalization and free decay in surface quasigeostrophic flows. Phys. Rev. E
**2012**, 86, 016323. [Google Scholar] [CrossRef] [PubMed] - Rhines, P.B. Waves and turbulence on a β-plane. J. Fluid Mech.
**1975**, 69, 417–443. [Google Scholar] [CrossRef] - Sukhatme, J.; Smith, L.M. Local and nonlocal dispersive turbulence. Phys. Fluids
**2009**, 21, 056603. [Google Scholar] [CrossRef] - Watanabe, T.; Iwayama, T. Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence. Phys. Rev. E
**2007**, 76, 046303. [Google Scholar] [CrossRef] [PubMed] - Scott, R.B.; Wang, F. Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry. J. Phys. Oceanogr.
**2005**, 35, 1650–1666. [Google Scholar] [CrossRef] - Constantin, P.; Majda, A.J.; Tabak, E. Singular front formation in a model for quasi-geostrophic flow. Phys. Fluids
**1994**, 9, 6–11. [Google Scholar] - Yudovich, V.I. On the loss of smoothness of the solutions of the Euler equations. Dyn. Contin. Media
**1974**, 16, 71–78. (In Russian) [Google Scholar] - Marchioro, C.; Pulvirenti, M. Mathematical theory of incompressible nonviscous fluids. In Applied Mathematical Sciences; Springer: New York, NY, USA, 1994; Volume 96. [Google Scholar]
- Yudovich, V.I. Loss of smoothness and inherent instability of 2D inviscid fluid flows. Commun. Partial Diff. Equal.
**2000**, 33, 943–968. [Google Scholar] - Dutton, J.A. The nonlinear quasi-geostrophic equation: Existence and uniqueness of solutions on a bounded domain. J. Atmos. Sci.
**1974**, 31, 422–433. [Google Scholar] [CrossRef] - Bennett, A.F.; Kloeden, P.E. The periodic quasigeostrophic equations: Existence and uniqueness of strong solutions. Proc. R. Soc. Edinb.
**1982**, 91A, 185–203. [Google Scholar] [CrossRef] - Chae, D. The global regularity for the 3D continuously stratified inviscid quasi-geostrophic equations. J. Nonlinear Sci.
**2015**, 25, 959–966. [Google Scholar] [CrossRef] - Beale, J.T.; Kato, T.; Majda, A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys.
**1984**, 94, 61–66. [Google Scholar] [CrossRef] - Ohkitani, K.; Yamada, M. Invicid and invicid-limit behavior of a surface quasigeostrophic flow. Phys. Fluids
**1997**, 9, 876–882. [Google Scholar] [CrossRef] - Denisov, S.A. Double exponential growth of the vorticity gradient for the two-dimensional Euler equation. Proc. Natl. Acad. Sci. USA
**2015**, 143, 1199–1210. [Google Scholar] [CrossRef] - Cordoba, D. On the geometry of solutions of the quasi-geostrophic and Euler equations. Proc. Natl. Acad. Sci. USA
**1997**, 94, 12769–12770. [Google Scholar] [CrossRef] [PubMed] - Cordoba, D. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. Math.
**1998**, 148, 1135–1152. [Google Scholar] [CrossRef] - Constantin, P.; Nie, Q.; Schörghofer, N. Front formation in an active scalar. Phys. Rev. E
**1999**, 60, 2858–2863. [Google Scholar] [CrossRef] - Wu, J. Quasi-geostrophic-type equations with initial data in Morrey spaces quasi-geostrophic. Nonlinearity
**1997**, 10, 1409–1420. [Google Scholar] [CrossRef] - Ju, N. Global solutions for the two dimensional quasi-geostrophic equation with critical or super-critical dissipation. Math. Ann.
**2006**, 334, 627–642. [Google Scholar] [CrossRef] - Resnick, S.G. Dynamical Problems in Non-Linear Advective Partial Differential Equations. Ph.D. Thesis, University of Chicago, Chicago, IL, USA, January 1995. [Google Scholar]
- Constantin, P.; Wu, J. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal.
**1999**, 30, 937–948. [Google Scholar] [CrossRef] - Ju, N. Dissipative 2D quasi-geostrophic equation: Local well-posedness, global regularity and similarity solutions. Indiana Univ. Math. J.
**2007**, 56, 187–206. [Google Scholar] [CrossRef] - Wu, J. Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. Nonlinear Anal.
**2007**, 67, 3013–3036. [Google Scholar] [CrossRef] - Chae, D.; Lee, J. Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys.
**2003**, 233, 297–311. [Google Scholar] [CrossRef] - Cordoba, A.; Cordoba, D. A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys.
**2004**, 249, 511–528. [Google Scholar] [CrossRef] - Constantin, P.; Cordoba, D.; Wu, J. On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J.
**2001**, 50, 97–107. [Google Scholar] [CrossRef] - Kiselev, A.; Nazarov, F.; Volberg, A. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math.
**2007**, 167, 445–463. [Google Scholar] [CrossRef] - Caffarelli, L.A.; Vasseur, A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math.
**2010**, 171, 1903–1930. [Google Scholar] [CrossRef] - Constantin, P.; Tarfulea, A.; Vicol, V. Long time dynamics of forced critical SQG. Commun. Math. Phys.
**2015**, 335, 93–141. [Google Scholar] [CrossRef] - Lesieur, M.; Sadourny, R. Satellite-sensed turbulent ocean structure. Nature
**1981**, 294, 673. [Google Scholar] [CrossRef] - Scott, R.K. Local and nonlocal advection of a passive scalar. Phys. Fluids
**2006**, 56, 122–125. [Google Scholar] [CrossRef] - Batchelor, G.K. The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. Ser. A
**1952**, 213, 349–366. [Google Scholar] [CrossRef] - Wirth, V.; Borth, H.; López, J.F.; Panhans, W.G.; Riemer, M.; Szabo, T. Dynamics in the extratropical tropopause region: A case of transition between dynamically active and passive tracer advection? Q. J. R. Meteorol. Soc.
**2005**, 131, 247–257. [Google Scholar] [CrossRef] - Klein, P.; Hua, B.L.; Lapeyre, G.; Capet, X.; Gentil, S.L.; Sasaki, H. Upper ocean turbulence from high 3-D resolution simulations. J. Phys. Oceanogr.
**2008**, 38, 1748–1763. [Google Scholar] [CrossRef] - Capet, X.; McWilliams, J.C.; Molemaker, M.; Shchepetkin, A. Mesoscale to submesoscale transition in the California current system. Part III: Energy balance and flux. J. Phys. Oceanogr.
**2008**, 38, 2256–2269. [Google Scholar] [CrossRef] - Shcherbina, B.A.Y.; Sundermeyer, M.A.; Kunze, E.; D’Asaro, E.; Badin, G. The LATMIX summer campaign. Bull. Am. Meteor. Soc.
**2015**, 96, 1257–1279. [Google Scholar] [CrossRef] - Johnson, E.R. Topographically bound vortices. Geophys. Astrophys. Fluid Dyn.
**1978**, 11, 61–71. [Google Scholar] [CrossRef] - Isern-Fontanet, J.; Lapeyre, G.; Klein, P.; Chapron, B.; Hecht, M.W. Three-dimensional reconstruction of oceanic mesoscale currents from surface information. J. Geophys. Res.
**2008**, 113. [Google Scholar] [CrossRef] - Qiu, B.; Chen, S.; Klein, P.; Ubelmann, C.; Fu, L.L.; Sasaki, H. Reconstructability of three-dimensional upper-ocean circulation from SWOT sea surface height measurements. J. Phys. Oceanogr.
**2016**, 46, 947–963. [Google Scholar] [CrossRef] - Klein, P.; Lapeyre, G.; Roullet, G.; Le Gentil, S.; Sasaki, H. Ocean turbulence at meso and submesoscales: Connection between surface and interior dynamics. Geophys. Astrophys. Fluid Dyn.
**2011**, 105, 421–437. [Google Scholar] [CrossRef] - Lapeyre, G. What vertical mode does the altimeter reflect? On the decomposition in baroclinic modes and on a surface-trapped mode. J. Phys. Oceanogr.
**2009**, 39, 2857–2874. [Google Scholar] [CrossRef] - Isern-Fontanet, J.; Chapron, B.; Lapeyre, G.; Klein, P. Potential use of microwave Sea surface temperatures for the estimation of ocean currents. Geophys. Res. Lett.
**2006**, 33. [Google Scholar] [CrossRef] - Gonzalez-Haro, C.; Isern-Fontanet, J. Global ocean current reconstruction from altimetric and microwave SST measurements. J. Geophys. Res.
**2014**, 119, 3378–3391. [Google Scholar] [CrossRef] - Wunsch, C. The vertical partition of oceanic horizontal kinetic energy. J. Phys. Oceanogr.
**1997**, 27, 1770–1794. [Google Scholar] [CrossRef] - Stammer, D. Global characteristics of ocean variability estimated from regional TOPEX/POSEIDON altimeter measurements. J. Phys. Oceanogr.
**1997**, 27, 1743–1769. [Google Scholar] [CrossRef] - Smith, K.S.; Vanneste, J. A surface-aware projection basis for quasigeostrophic flow. J. Phys. Oceanogr.
**2013**, 43, 548–562. [Google Scholar] [CrossRef] - Roullet, G.; McWilliams, J.C.; Capet, X.; Molemaker, M.J. Properties of steady geostrophic turbulence with isopycnal outcropping. J. Phys. Oceanogr.
**2012**, 42, 18–38. [Google Scholar] [CrossRef] - Ponte, A.; Klein, P. Reconstruction of the upper ocean 3D dynamics from high-resolution sea surface height. Ocean Dyn.
**2013**, 63, 777–791. [Google Scholar] [CrossRef] - Tulloch, R.; Smith, K.S. A note on the numerical representation of surface dynamics in quasigeostrophic turbulence: Application to the nonlinear Eady model. J. Atmos. Sci.
**2009**, 66, 1063–1068. [Google Scholar] [CrossRef] - Tulloch, R.; Smith, K.S. Quasigeostrophic turbulence with explicit surface dynamics: Application to the atmospheric energy spectrum. J. Atmos. Sci.
**2009**, 66, 450–467. [Google Scholar] [CrossRef] - Wang, J.; Flierl, G.R.; LaCasce, J.H.; McClean, J.L.; Mahadevan, A. Reconstructing the ocean’s interior from surface data. J. Phys. Oceanogr.
**2013**, 43, 1611–1626. [Google Scholar] [CrossRef] - Lacasce, J.H.; Wang, J. Estimating subsurface velocities from surface fields with idealized stratification. J. Phys. Oceanogr.
**2015**, 9, 2424–2435. [Google Scholar] [CrossRef] - Chavanne, C.P.; Klein, P. Quasigeostrophic diagnosis of mixed layer dynamics embedded in a mesoscale turbulent field. J. Phys. Oceanogr.
**2016**, 46, 275–287. [Google Scholar] [CrossRef] - Callies, J.; Flierl, G.; Ferrari, R.; Fox-Kemper, B. The role of mixed-layer instabilities in submesoscale turbulence. J. Fluid Mech.
**2016**, 788, 5–41. [Google Scholar] [CrossRef] - Boccaletti, G.; Ferrari, R.; Fox-Kemper, B. Mixed layer instabilities and restratification. J. Phys. Oceanogr.
**2007**, 37, 2228–2250. [Google Scholar] [CrossRef] - Hoskins, B.J.; Draghici, I.; Davies, H.C. A new look at the ω-equation. Q. J. R. Meteorol. Soc.
**1978**, 104, 31–38. [Google Scholar] [CrossRef] - Klein, P.; Isern-Fontanet, J.; Lapeyre, G.; Roullet, G.; Danioux, E.; Chapron, B.; Le Gentil, S.; Sasaki, H. Diagnosis of vertical velocities in the upper ocean from high resolution Sea Surface Height. Geophys. Res. Lett.
**2009**, 36, L12603. [Google Scholar] [CrossRef] - Ponte, A.; Klein, P.; Capet, X.; Le Traon, P.Y.; Chapron, B.; Lherminier, P. Diagnosing surface mixed layer dynamics from high-resolution satellite observations: Numerical insights. J. Phys. Oceanogr.
**2013**, 43, 1345–1355. [Google Scholar] [CrossRef] - Garrett, C.; Loder, J. Dynamical aspects of shallow sea fronts. Philos. Trans. R. Soc. Lond.
**1981**, A94, 563–581. [Google Scholar] [CrossRef] - Hakim, G.J.; Snyder, C.; Muraki, D.J. A new surface model for cyclone-anticyclone asymmetry. J. Atmos. Sci.
**2002**, 59, 2405–2420. [Google Scholar] [CrossRef] - Petterssen, S. Contribution to the theory of frontogenesis. Geofys. Pub.
**1936**, 11, 5–27. [Google Scholar] - Williams, R.T.; Plotkin, J. Quasi-geostrophic frontogenesis. J. Atmos. Sci.
**1968**, 25, 201–206. [Google Scholar] [CrossRef] - Sawyer, J.S. The vertical circulation at meteorological fronts and its relation to frontogenesis. Proc. R. Soc. Lond. Ser. A
**1956**, A234, 346–362. [Google Scholar] [CrossRef] - Eliassen, A. On the vertical circulation in frontal zones. Geofys. Publik.
**1962**, 24, 147–160. [Google Scholar] - Hoskins, B.J.; Bretherton, F.P. Atmospheric frontogenesis models: Mathematical formulation and solution. J. Atmos. Sci.
**1972**, 29, 11–37. [Google Scholar] [CrossRef] - Lapeyre, G.; Klein, P.; Hua, B.L. Does the tracer gradient vector align with the strain eigenvectors in 2-D turbulence? Phys. Fluids A.
**1999**, 11, 3729–3737. [Google Scholar] [CrossRef] - Gower, J.F.R.; Denman, K.L.; Holyer, R.J. Phytoplankton patchiness indicates the fluctuation spectrum of mesoscale oceanic structure. Nature
**1980**, 288, 157–159. [Google Scholar] [CrossRef] - Lapeyre, G.; Klein, P. Impact of the small-scale elongated filaments on the oceanic vertical pump. J. Mar. Res.
**2006**, 64, 935–951. [Google Scholar] [CrossRef] - Perruche, C.; Rivière, P.; Lapeyre, G.; Carton, X.; Pondaven, P. Effects of Surface Quasi-Geostrophic turbulence on phytoplankton competition and coexistence. J. Mar. Res.
**2011**, 69, 105–135. [Google Scholar] [CrossRef] - Juckes, M. Quasigeostrophic dynamics of the tropopause. J. Atmos. Sci.
**1994**, 51, 2756–2768. [Google Scholar] [CrossRef] - Smith, K.S.; Bernard, E. Geostrophic turbulence near rapid changes in stratification. Phys. Fluids
**2013**, 25, 046601. [Google Scholar] [CrossRef] - Plougonven, R.; Vanneste, J. Quasi-geostrophic dynamics of a finite-thickness tropopause. J. Atmos. Sci.
**2010**, 67, 3149–3163. [Google Scholar] [CrossRef] - Rivest, C.; Davis, C.A.; Farrell, B.F. Upper-tropospheric synoptic-scale waves. Part I: Maintenance as Eady normal modes. J. Atmos. Sci.
**1992**, 49, 2108–2119. [Google Scholar] [CrossRef] - Tomikawa, Y.; Sato, K.; Shepherd, T.G. A diagnostic study of waves on the tropopause. J. Atmos. Sci.
**2006**, 63, 3315–3332. [Google Scholar] [CrossRef] - Wirth, V.; Appenzeller, C.; Juckes, M. Signatures of induced vertical air motion accompanying quasi-horizontal roll-up of stratospheric intrusions. Mon. Weather Rev.
**1997**, 125, 2504–2519. [Google Scholar] [CrossRef] - Nastrom, G.D.; Gage, K.S. A climatology of atmospheric wavenumber spectra of wind and temperature observed by commercial aircraft. J. Atmos. Sci.
**1985**, 42, 950–960. [Google Scholar] [CrossRef] - Lindborg, E. The effect of rotation on the mesoscale energy cascade in the free atmosphere. Geophys. Res. Lett.
**2005**, 32. [Google Scholar] [CrossRef] - Hamilton, K.; Takahashi, Y.O.; Ohfuchi, W. Mesoscale spectrum of atmospheric motions investigated in a very fine resolution global general circulation model. J. Geophys. Res.
**2008**, 113. [Google Scholar] [CrossRef] - Tulloch, R.; Smith, K.S. A new theory for the atmospheric energy spectrum: Depth-limited temperature anomalies at the tropopause. Proc. Natl. Acad. Sci. USA
**2006**, 103, 14690–14694. [Google Scholar] [CrossRef] [PubMed] - Asselin, O.; Bartello, P.; Straub, D.N. On quasi-geostrophic dynamics near the tropopause. Phys. Fluids
**2016**, 28, 026601. [Google Scholar] [CrossRef] - Morss, R.; Snyder, C.; Rotunno, R. Spectra, spatial scales, and predictability in a quasigeostrophic model. J. Atmos. Sci.
**2009**, 66, 3115–3130. [Google Scholar] [CrossRef] - Greenslade, M.D.; Haynes, P.H. Vertical transition in transport and mixing in baroclinic flows. J. Atmos. Sci.
**2008**, 65, 1137–1157. [Google Scholar] [CrossRef] - Bracco, A. Boundary layer separation in the surface quasigeostrophic equations. Il Nuovo Cimento
**2000**, 23, 487–505. [Google Scholar] - Charney, J.G. The dynamics of long waves in a baroclinic westerly current. J. Meteor.
**1947**, 4, 135–162. [Google Scholar] [CrossRef] - Green, J.S.A. A problem in baroclinic instability. Q. J. R. Meteorol. Soc.
**1960**, 86, 237–251. [Google Scholar] [CrossRef] - Heifetz, E.; Methven, J.; Hoskins, B.J.; Bishop, C.H. The counter-propagating Rossby-wave perspective on baroclinic instability. II: Application to the Charney model. Q. J. R. Meteorol. Soc.
**2004**, 130, 233–258. [Google Scholar] [CrossRef] - Held, I.M. The vertical scale of an unstable baroclinic wave and its importance for eddy heat flux parameterizations. J. Atmos. Sci.
**1978**, 35, 572–576. [Google Scholar] [CrossRef] - Venaille, A.; Vallis, G.K.; Griffies, S.M. The catalytic role of the beta effect in barotropization processes. J. Fluid Mech.
**2012**, 709, 490–515. [Google Scholar] [CrossRef] - Perrot, X.; Reinaud, J.N.; Carton, X.; Dritschel, D.G. Homostrophic vortex interaction under external strain, in a coupled QG-SQG model. Reg. Chaot. Dyn.
**2010**, 15, 66–83. [Google Scholar] [CrossRef] - Reinaud, J.N.; Dritschel, D.G.; Carton, X. Interaction between a surface quasi-geostrophic buoyancy filament and an internal vortex. Geophys. Astrophys. Fluid Dyn.
**2016**, 110, 461–490. [Google Scholar] [CrossRef] - Williams, R.T. Atmospheric frontogenesis: A numerical experiment. J. Atmos. Sci.
**1967**, 24, 627–641. [Google Scholar] [CrossRef] - Hoskins, B.J. The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci.
**1975**, 32, 233–242. [Google Scholar] [CrossRef] - Badin, G. Surface semi-geostrophic dynamics in the ocean. Geophys. Astrophys. Fluid Dyn.
**2013**, 107, 526–540. [Google Scholar] [CrossRef] - Ragone, F.; Badin, G. A study of surface semi-geostrophic turbulence: Freely decaying dynamics. J. Fluid Mech.
**2016**, 792, 740–774. [Google Scholar] [CrossRef] - Lapeyre, G.; Klein, P.; Hua, B.L. Oceanic restratification forced by surface frontogenesis. J. Phys. Oceanogr.
**2006**, 36, 1577–1590. [Google Scholar] [CrossRef] - Juckes, M. The structure of idealized upper-tropospheric shear lines. J. Atmos. Sci.
**1999**, 56, 2830–2845. [Google Scholar] [CrossRef] - Bembenek, E.; Poulin, F.J.; Waite, M.L. Realizing surface-driven flows in the primitive equations. J. Phys. Oceanogr.
**2015**, 45, 1376–1392. [Google Scholar] [CrossRef] - Snyder, C.; Muraki, D.J.; Plougonven, R.; Zhang, F. Inertia-gravity waves generated within a dipole Vortex. J. Atmos. Sci.
**2007**, 64, 4417–4431. [Google Scholar] [CrossRef]

**Figure 1.**Cascade of filament instabilities near the time of singularity (infinite curvature). On the bottom left (box with white background), two elliptical vortices are elongating a thin filament. Each panel in clockwise order shows a close-up of the filament. Finer and finer scale structures can be observed with the rolling-up of the filament. Reprinted with permission from [31]. Copyright (2014) by the American Physical Society .

**Figure 2.**(

**a**) Surface buoyancy in a freely-decaying simulation at a resolution of $1024\times 1024$. Note that the vortices at all scales develop from filament instability. (

**b**) Spectral fluxes ${\mathrm{\Pi}}_{\theta}$ (continuous curve), ${\mathrm{\Pi}}_{u}$ (dashed curve) and ${\mathrm{\Pi}}_{a}$ (dash and dotted curve). See the text for the definition. Panel (b) is from [50], with permission of Cambridge University Press.

**Figure 3.**(

**a**) Surface buoyancy $\theta +\beta y$ in a forced Surface Quasi-Geostrophy (SQG) simulation in the presence of a β effect; (

**b**) corresponding relative vorticity. Only a subdomain is shown. The model is forced at wavenumber ${k}_{f}=40$ for a domain size of $[0,\text{}2\pi ]\times [0,\text{}2\pi ]$ and a resolution of $512\times 512$.

**Figure 4.**Validation of the SQG balance in the Gulf Stream region. (

**a**) Sea surface temperature (shadings) and sea surface height (white contours); (

**b**) corresponding velocity field, deduced from altimetry (blue arrows) and from SQG balance (red arrows). Note the good agreement between the two velocity fields. From [99], with permission of John Wiley and Sons.

**Figure 5.**Validation of the SQG balance at the tropopause. (

**a**) Geopotential height contours of the Potential Vorticity ($PV$) $=-2$ surface (500-m intervals, heavy contour 10 km). The shading is the region where potential temperature lies between 310 and 330 K. Simulations taken from a General Circulation Model. (

**b**) Contoured scatterplot of potential temperature anomalies (from zonal mean) versus geopotential height anomalies on the tropopause ($PV$$=-2$$PV$ units). In panel (a), there is a clear correspondence between geopotential height contours and potential temperature shadings. In panel (b), we clearly see the linearity between geopotential height and potential temperature at the tropopause. From [127]. ©American Meteorological Society. Used with permission.

© 2017 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lapeyre, G.
Surface Quasi-Geostrophy. *Fluids* **2017**, *2*, 7.
https://doi.org/10.3390/fluids2010007

**AMA Style**

Lapeyre G.
Surface Quasi-Geostrophy. *Fluids*. 2017; 2(1):7.
https://doi.org/10.3390/fluids2010007

**Chicago/Turabian Style**

Lapeyre, Guillaume.
2017. "Surface Quasi-Geostrophy" *Fluids* 2, no. 1: 7.
https://doi.org/10.3390/fluids2010007