Skip to main content
    • Aa
    • Aa

The quasi-geostrophic theory of the thermal shallow water equations

  • Emma S. Warneford (a1) and Paul J. Dellar (a1)

The thermal shallow water equations provide a depth-averaged description of motions in a fluid layer that permits horizontal variations in material properties. They typically arise through an equivalent barotropic approximation of a two-layer system, with a spatially varying density contrast due to an evolving temperature field in the active layer. We formalize a previous derivation of the quasi-geostrophic (QG) theory of these equations, by performing a direct asymptotic expansion for small Rossby number. We then present a second derivation as the small Rossby number limit of a balanced model that projects out high-frequency dynamics due to inertia-gravity waves. This latter derivation has wider validity, not being restricted to mid-latitude $\beta $ -planes. We also derive their local energy conservation equation from the QG limit of a thermal shallow water pseudo-energy conservation equation. This derivation involves the ageostrophic correction to the leading-order geostrophic velocity that is eliminated in the usual derivation of a closed evolution equation for the QG potential vorticity. Finally, we derive the non-canonical Hamiltonian structure of the thermal QG equations from a decomposition in Rossby number of a pseudo-energy and Poisson bracket for the thermal shallow water equations.

Corresponding author
Email address for correspondence:
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. Adcroft & R. Hallberg 2006 On methods for solving the oceanic equations of motion in generalized vertical coordinates. Ocean Model. 11, 224233.

D. G. Andrews 1981 A note on potential energy density in a stratified compressible fluid. J. Fluid Mech. 107, 227236.

J. A. Barth 1994 Short-wavelength instabilities on coastal jets and fronts. J. Geophys. Res. 99, 1609516115.

R. Bleck , C. Rooth , D. Hu & L. T. Smith 1992 Salinity-driven thermocline transients in a wind- and thermohaline-forced isopycnic coordinate model of the North Atlantic. J. Phys. Oceanogr. 22, 14861505.

F. Bouchut , J. Lambaerts , G. Lapeyre & V. Zeitlin 2009 Fronts and nonlinear waves in a simplified shallow-water model of the atmosphere with moisture and convection. Phys. Fluids 21, 116604.

R. Camassa , D. D. Holm & C. D. Levermore 1996 Long-time effects of bottom topography in shallow water. Physica D 98, 258286.

H. A. A. C. Carbonel & N. C. A. Galeao 2007 A stabilized finite element model for the hydrothermodynamical simulation of the Rio de Janeiro coastal ocean. Commun. Numer. Meth. Engng 23, 521534.

J. G. Charney & N. A. Phillips 1953 Numerical integration of the quasi-geostrophic equations for barotropic and simple baroclinic flows. J. Meteorol. 10, 7199.

P. J. Dellar 2002 Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics. Phys. Plasmas 9, 11301136.

P. J. Dellar 2003 Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields. Phys. Fluids 15, 292297.

P. J. Dellar 2011 Variations on a beta-plane: derivation of non-traditional beta-plane equations from Hamilton’s principle on a sphere. J. Fluid Mech. 674, 174195.

P. J. Dellar & R. Salmon 2005 Shallow water equations with a complete Coriolis force and topography. Phys. Fluids 17, 106601.

C. Eckart 1960 Variation principles of hydrodynamics. Phys. Fluids 3, 421427.

T. Eldevik 2002 On frontal dynamics in two model oceans. J. Phys. Oceanogr. 32, 29152925.

B. F. Farrell & P. J. Ioannou 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5, 26002609.

B. F. Farrell & P. J. Ioannou 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 21012118.

B. F. Farrell & P. J. Ioannou 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 36523665.

FRAM Group 1991 An eddy-resolving model of the Southern Ocean. EOS, Trans. Amer. Geophys. Union 72, 169170.

Y. Fukamachi , J. P. McCreary & J. A. Proehl 1995 Instability of density fronts in layer and continuously stratified models. J. Geophys. Res. 100, 25592577.

P. J. Gierasch , A. P. Ingersoll , D. Banfield , S. P. Ewald , P. Helfenstein , A. Simon-Miller , A. Vasavada , H. H. Breneman , D. A. Senske & I. Galileo Team 2000 Observation of moist convection in Jupiter’s atmosphere. Nature 403, 628630.

A. E. Gill 1982 Atmosphere Ocean Dynamics. Academic Press.

P. A. Gilman 1967 Stability of baroclinic flows in a zonal magnetic field: Part I. J. Atmos. Sci. 24, 101118.

P. A. Gilman 2000 Magnetohydrodynamic ‘shallow water’ equations for the solar tachocline. Astrophys. J. Lett. 544, 7982.

A. E. Green & P. M. Naghdi 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237246.

T. Guillot 2005 The interiors of giant planets: Models and outstanding questions. Annu. Rev. Earth Planet. Sci. 33, 493530.

D. Holliday & M. E. McIntyre 1981 On potential energy density in an incompressible, stratified fluid. J. Fluid Mech. 107, 221225.

D. D. Holm 1986 Hamiltonian formulation of the baroclinic quasigeostrophic fluid equations. Phys. Fluids 29, 78.

D. D. Holm , J. E. Marsden , T. Ratiu & A. Weinstein 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1116.

B. J. Hoskins , M. E. McIntyre & A. W. Robertson 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. 111, 877946.

V. O. Ivchenko , A. E. Krupitsky , V. M. Kamenkovich & N. C. Wells 1999 Modelling the antarctic circumpolar current: a comparison of FRAM and equivalent barotropic model results. J. Mar. Res. 57, 2945.

M. Juckes 1989 A shallow water model of the winter stratosphere. J. Atmos. Sci. 46, 29342956.

P. D. Killworth 1992 An equivalent-barotropic mode in the fine resolution antarctic model. J. Phys. Oceanogr. 22, 13791387.

P. D. Killworth & C. W. Hughes 2002 The Antarctic Circumpolar Current as a free equivalent-barotropic jet. J. Mar. Res. 60, 1945.

A. Krupitsky , V. M. Kamenkovich , N. Naik & M. A. Cane 1996 A linear equivalent barotropic model of the Antarctic Circumpolar Current with realistic coastlines and bottom topography. J. Phys. Oceanogr. 26, 18031824.

H. L. Kuo 1959 Finite-amplitude three-dimensional harmonic waves on the spherical Earth. J. Meteorol. 16, 524534.

J. H. LaCasce & P. E. Isachsen 2010 The linear models of the ACC. Prog. Oceanogr. 84, 139157.

J. Lambaerts , G. Lapeyre , V. Zeitlin & F. Bouchut 2011 Simplified two-layer models of precipitating atmosphere and their properties. Phys. Fluids 23, 046603.

R. L. Lavoie 1972 A mesoscale numerical model of lake-effect storms. J. Atmos. Sci. 29, 10251040.

C. E. Leith 1980 Nonlinear normal mode initialization and quasi-geostrophic theory. J. Atmos. Sci. 37, 958968.

E. N. Lorenz 1955 Available potential energy and the maintenance of the general circulation. Tellus 7, 157167.

A. Majda & X. Wang 2006 Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press.

J. B. Marston , E. Conover & T. Schneider 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65, 19551966.

J. P. McCreary , Y. Fukamachi & P. K. Kundu 1991 A numerical investigation of jets and eddies near an eastern ocean boundary. J. Geophys. Res. 96, 25152534.

J. P. McCreary & P. K. Kundu 1988 A numerical investigation of the Somali Current during the Southwest Monsoon. J. Marine Res. 46, 2558.

J. P. McCreary & Z. Yu 1992 Equatorial dynamics in a $2\frac{1}{2} $-layer model. Prog. Oceanogr. 29, 61132.

M. E. McIntyre & T. G. Shepherd 1987 An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and Arnol’d’s stability theorems. J. Fluid Mech. 181, 527565.

J. C. McWilliams 1977 A note on a consistent quasigeostrophic model in a multiply connected domain. Dyn. Atmos. Oceans 1, 427441.

J. Miles & R. Salmon 1985 Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519531.

A. R. Mohebalhojeh & D. G. Dritschel 2001 Hierarchies of balance conditions for the f-plane shallow-water equations. J. Atmos. Sci. 58, 24112426.

P. J. Morrison 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467521.

P. J. Morrison & R. D. Hazeltine 1984 Hamiltonian formulation of reduced magnetohydrodynamics. Phys. Fluids 27, 886897.

D. J. Muraki , C. Snyder & R. Rotunno 1999 The next-order corrections to quasigeostrophic theory. J. Atmos. Sci. 56, 15471560.

G. Neumann 1960 On the dynamical structure of the Gulf Stream as an equivalent-barotropic flow. J. Geophys. Res. 65, 239247.

J. Pedlosky 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.

L. M. Polvani , D. W. Waugh & R. A. Plumb 1995 On the subtropical edge of the stratospheric surf zone. J. Atmos. Sci. 52, 12881309.

J. N. Reinaud , D. G. Dritschel & C. R. Koudella 2003 The shape of vortices in quasi-geostrophic turbulence. J. Fluid Mech. 474, 175192.

M. Remmel & L. Smith 2009 New intermediate models for rotating shallow water and an investigation of the preference for anticyclones. J. Fluid Mech. 635, 321359.

P. Ripa 1993 Conservation laws for primitive equations models with inhomogeneous layers. Geophys. Astrophys. Fluid Dyn. 70, 85111.

P. Ripa 1995 On improving a one-layer ocean model with thermodynamics. J. Fluid Mech. 303, 169201.

P. Ripa 1996a Linear waves in a one-layer ocean model with thermodynamics. J. Geophys. Res. 101, 12331245.

P. Ripa 1999 On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution. Dyn. Atmos. Oceans 29, 140.

L. P. Røed 1997 Energy diagnostics in a $1\frac{1}{2} $-layer, nonisopycnic model. J. Phys. Oceanogr. 27, 14721476.

L. P. Røed & X. B. Shi 1999 A numerical study of the dynamics and energetics of cool filaments, jets, and eddies off the Iberian Peninsula. J. Geophys. Res. 104, 29,817–29,841.

M. L. Salby 1989 Deep circulations under simple classes of stratification. Tellus A 41, 4865.

R. Salmon 1982 The shape of the main thermocline. J. Phys. Oceanogr. 12, 14581479.

R. Salmon 1983 Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431444.

R. Salmon 1988a Hamiltonian fluid mechanics. Ann. Rev. Fluid Mech. 20, 225256.

R. Salmon 1988b Semigeostrophic theory as a Dirac-bracket projection. J. Fluid Mech. 196, 345358.

P. S. Schopf & M. A. Cane 1983 On equatorial dynamics, mixed layer physics and sea-surface temperature. J. Phys. Oceanogr. 13, 917935.

W. H. Schubert , R. K. Taft & L. G. Silvers 2009 Shallow water quasi-geostrophic theory on the sphere. J. Adv. Model. Earth Syst. 1, 2.

R. K. Scott & L. M. Polvani 2008 Equatorial superrotation in shallow atmospheres. Geophys. Res. Lett. 35, L24202.

R. L. Seliger & G. B. Whitham 1968 Variational principles in continuum mechanics. Proc. R. Soc. Lond. A 305, 125.

T. G. Shepherd 1990 Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Adv. Geophys. 32, 287338.

T. G. Shepherd 1993 A unified theory of available potential energy. Atmos.-Ocean 31, 126.

X. B. Shi & L. P. Roed 1999 Frontal instabilities in a two-layer, primitive equation ocean model. J. Phys. Oceanogr. 29, 948968.

K. Srinivasan & W. R. Young 2012 Zonostrophic instability. J. Atmos. Sci. 69, 16331656.

A. L. Stewart & P. J. Dellar 2010 Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech. 651, 387413.

P. H. Stone 1966 On non-geostrophic baroclinic stability. J. Atmos. Sci. 23, 390400.

C. H. Su & C. S. Gardner 1969 Korteweg–de Vries equation and generalizations III. Derivation of the Korteweg–de Vries equation and Burgers equation. J. Math. Phys. 10, 536539.

R. Tailleux 2013 Available potential energy and exergy in stratified fluids. Annu. Rev. Fluid Mech. 45, 3558.

E. Tassi , C. Chandre & P. J. Morrison 2009 Hamiltonian derivation of the Charney–Hasegawa–Mima equation. Phys. Plasmas 16, 082301.

J. Theiss & A. R. Mohebalhojeh 2009 The equatorial counterpart of the quasi-geostrophic model. J. Fluid Mech. 637, 327356.

J. Thuburn & V. Lagneau 1999 Eulerian mean, contour integral, and finite-amplitude wave activity diagnostics applied to a single-layer model of the winter stratosphere. J. Atmos. Sci. 56, 689710.

S. M. Tobias , K. Dagon & J. B. Marston 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727, 127.

G. K. Vallis 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.

N. G. Van Kampen 1985 Elimination of fast variables. Phys. Rep. 124, 69160.

W. T. M. Verkley 2009 A balanced approximation of the one-layer shallow-water equations on a sphere. J. Atmos. Sci. 66, 17351748.

Á Viúdez & D. G. Dritschel 2004 Optimal potential vorticity balance of geophysical flows. J. Fluid Mech. 521, 343352.

T. Warn , O. Bokhove , T. G. Shepherd & G. K. Vallis 1995 Rossby number expansions, slaving principles, and balance dynamics. Q. J. R. Meteorol. Soc. 121, 723739.

A. Weinstein 1983 Hamiltonian structure for drift waves and geostrophic flow. Phys. Fluids 26, 388390.

A. A. White 2002 A view of the equations of meteorological dynamics and various approximations. In Large-Scale Atmosphere-Ocean Dynamics 1: Analytical Methods and Numerical Models (ed. J. Norbury & I. Roulstone ), pp. 1100. Cambridge University Press.

K. B. Winters , P. N. Lombard , J. J. Riley & E. A. D’Asaro 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.

W. R. Young 1994 The subinertial mixed layer approximation. J. Phys. Oceanogr. 24, 18121826.

W. R. Young & L. Chen 1995 Baroclinic instability and thermohaline gradient alignment in the mixed layer. J. Phys. Oceanogr. 25, 31723185.

V. Zeitlin 2007 Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances. Elsevier.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 44 *
Loading metrics...

Abstract views

Total abstract views: 168 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st September 2017. This data will be updated every 24 hours.