Skip to main content Accessibility help
Hostname: page-component-747cfc64b6-rxvp8 Total loading time: 0.295 Render date: 2021-06-15T10:03:20.210Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

On the regularity of the Green–Naghdi equations for a rotating shallow fluid layer

Published online by Cambridge University Press:  19 February 2019

David G. Dritschel
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK
Mohammad Reza Jalali
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK


The Green–Naghdi equations are an extension of the shallow-water equations that capture the effects of finite fluid depth at arbitrary order in the characteristic height to width aspect ratio $H/L$ . The shallow-water equations capture these effects to first order only, resulting in a relatively simple two-dimensional fluid-dynamical model for the layer horizontal velocity and depth. The Green–Naghdi equations, like the shallow-water equations, are two-dimensional fluid equations expressing momentum and mass conservation. There are different ‘levels’ of the Green–Naghdi equations of rapidly increasing complexity. In the present paper we focus on the behaviour of the lowest-level Green–Naghdi equations for a rotating shallow fluid layer, paying close attention to the flow structure at small spatial scales. We compare directly with the shallow-water equations and study the differences arising in their solutions. By recasting the equations into a form which both explicitly conserves Rossby–Ertel potential vorticity and represents the leading-order departure from geostrophic–hydrostatic balance, we are able to accurately describe both the ‘slow’ predominantly sub-inertial balanced dynamics and the ‘fast’ residual imbalanced dynamics. This decomposition has proved fruitful in studies of shallow-water dynamics but appears not to have been used before in studies of Green–Naghdi dynamics. Importantly, we find that this decomposition exposes a fundamental inconsistency in the Green–Naghdi equations for horizontal scales less than the mean fluid depth, scales for which the Green–Naghdi equations are supposed to more accurately model. Such scales exhibit pronounced activity compared to the shallow-water equations, and in particular spectra for certain fields like the divergence are flat or rising at high wavenumbers. This indicates a lack of convergence at small scales, and is also consistent with the poor convergence of total energy with resolution compared to the shallow-water equations. We suggest a mathematical reformulation of the Green–Naghdi equations which may improve convergence at small scales.

JFM Papers
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below.


Bonneton, P., Chazel, F., Lannes, D., Marche, F. & Tissier, M. 2011 A Splitting approach for the fully nonlinear and weakly dispersive Green. J. Comput. Phys. 230 (4), 14791498.10.1016/ Scholar
Boussinesq, J. 1871 Théorie de l’intumescence liquide, appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. Acad. Sci. Inst. France. Sci. Math. Phys. 72, 755759.Google Scholar
Boussinesq, J. 1877 Essai sur la théorie des eaux écourantes. Acad. Sci. Inst. France. Sci. Math. Phys. 23, 1680.Google Scholar
Castro-Orgaz, O. & Hager, W. H. 2015 Boussinesq- and Serre-type models with improved linear dispersion characteristics: applications. J. Hydraul. Res. 53 (2), 282284.CrossRefGoogle Scholar
Cosserat, E. & Cosserat, F. 1909 Théorie des corps déformables. Librairie scientifique A. Hermann et Fils.Google Scholar
Demirbilek, Z. & Webster, W. C. 1999 The Green–Naghdi theory of fluid sheets for shallow-water waves. In Developments in Offshore Engineering: Wave Phenomena and Offshore Topics, pp. 154. Gulf Publishing Company.Google Scholar
Deusebio, E., Vallgren, A. & Lindborg, E. 2013 The route to dissipation in strongly stratified and rotating flows. J. Fluid Mech. 720, 66103.10.1017/jfm.2012.611CrossRefGoogle Scholar
Dritschel, D. G. 1989 On the stabilization of a two-dimensional vortex strip by adverse shear. J. Fluid Mech. 206, 193221.10.1017/S0022112089002284CrossRefGoogle Scholar
Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-Lagrangian algorithm for the simulation of fine-scale conservative fields. Q. J. R. Meteorol. Soc. 123, 10971130.CrossRefGoogle Scholar
Dritschel, D. G. & Fontane, J. 2010 The combined Lagrangian advection method. J. Comput. Phys. 229, 54085417.CrossRefGoogle Scholar
Dritschel, D. G., Gottwald, G. A. & Oliver, M. 2017 Comparison of variational balance models for the rotating shallow water equations. J. Fluid Mech. 822, 689716.CrossRefGoogle Scholar
Dritschel, D. G. & McKiver, W. J. 2015 Effect of Prandtl’s ratio on balance in geophysical turbulence. J. Fluid Mech. 777, 569590.CrossRefGoogle Scholar
Dritschel, D. G., Qi, W. & Marston, J. B. 2015 On the late-time behaviour of a bounded, inviscid two-dimensional flow. J. Fluid Mech. 783, 122.CrossRefGoogle Scholar
Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V. 2009 Late time evolution of unforced inviscid two-dimensional turbulence. J. Fluid Mech. 640, 215233.10.1017/S0022112009991121CrossRefGoogle Scholar
Dritschel, D. G. & Tobias, S. M. 2012 Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit. J. Fluid Mech. 703, 8598.CrossRefGoogle Scholar
Dritschel, D. G. & Vanneste, J. 2006 The instability of a potential vorticity front. J. Fluid Mech. 561, 237254.10.1017/S0022112006000644CrossRefGoogle Scholar
Dutykh, D., Clamond, D., Milewski, P. & Mitsotakis, D. 2013 Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations. Eur. J. Appl. Maths 24 (5), 761787.CrossRefGoogle Scholar
Ertekin, R. C.1984 Soliton generation by moving disturbances in shallow water. PhD thesis, University of California, Berkeley, CA, USA.Google Scholar
Ford, R., McIntyre, M. E. & Norton, W. A. 2000 Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57, 12361254.2.0.CO;2>CrossRefGoogle Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic.Google Scholar
Green, A. E., Laws, N. & Naghdi, P. M. 1974 On the theory of water waves. Proc. R. Soc. Lond. A 338 (1612), 4355.10.1098/rspa.1974.0072CrossRefGoogle Scholar
Green, A. E. & Naghdi, P. M. 1976a A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78 (2), 273–246.10.1017/S0022112076002425CrossRefGoogle Scholar
Green, A. E. & Naghdi, P. M. 1976b Directed fluid sheets. Proc. R. Soc. Lond. A 347 (1651), 447473.10.1098/rspa.1976.0011CrossRefGoogle Scholar
Green, A. E., Naghdi, P. M. & Wainwright, W. L. 1965 A general theory of a Cosserat surface. Arch. Rat. Mech. Anal. 20 (4), 287308.10.1007/BF00253138CrossRefGoogle Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential-vorticity maps. Q. J. R. Meteorol. Soc. 111, 877946.10.1002/qj.49711147002CrossRefGoogle Scholar
Jalali, M. R.2016 One-dimensional and two-dimensional Green–Naghdi equation solvers for shallow flow over uniform and non-uniform beds. PhD thesis, University of Edinburgh, Edinburgh, Scotland, UK.Google Scholar
Juckes, M. N. & McIntyre, M. E. 1987 A high-resolution one-layer model of breaking planetary waves in the stratosphere. Nature 328, 590596.10.1038/328590a0CrossRefGoogle Scholar
Kantorovich, L. V. & Krylov, V. I. 1958 Approximate Methods of Higher Analysis. P. Noordhoff Ltd.Google Scholar
Le Métayer, O., Gavrilyuk, S. & Hank, S. 2010 A numerical scheme for the Green–Naghdi model. J. Comput. Phys. 229 (6), 20342045.CrossRefGoogle Scholar
Miles, J. W. & Salmon, R. 1985 Weakly dispersive, nonlinear gravity waves. J. Fluid Mech. 157, 519531.CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2000 On the representation of gravity waves in numerical models of the shallow water equations. Q. J. R. Meteorol. Soc. 126, 669688.CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2001 Hierarchies of balance conditions for the f-plane shallow water equations. J. Atmos. Sci. 58 (16), 24112426.2.0.CO;2>CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2004 Contour-advective semi-Lagrangian algorithms for many-layer primitive equation models. Q. J. R. Meteorol. Soc. 130, 347364.10.1256/qj.03.49CrossRefGoogle Scholar
Mohebalhojeh, A. R. & Dritschel, D. G. 2007 Assessing the numerical accuracy of complex spherical shallow water flows. Mon. Weath. Rev. 135 (11), 38763894.10.1175/2007MWR2036.1CrossRefGoogle Scholar
Naghdi, P. M. 1972 The theory of shells and plates. In Handbuch der Physik (ed. Flügge, S.), vol. V1a/2, p. 425. Springer.Google Scholar
Norbury, J. & Roulstone, I. 2002a Large-scale Atmosphere–Ocean Dynamics: Vol. I: Analytical Methods and Numerical Models. Cambridge University Press.10.1017/CBO9780511549991CrossRefGoogle Scholar
Norbury, J. & Roulstone, I. 2002b Large-scale Atmosphere–Ocean Dynamics: Vol. II: Geometric Methods and Models. Cambridge University Press.CrossRefGoogle Scholar
Pearce, J. D. & Esler, J. G. 2010 A pseudo-spectral algorithm and test cases for the numerical solution of the two-dimensional rotating Green–Naghdi shallow water equations. J. Comput. Phys. 229 (20), 75947608.CrossRefGoogle Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27 (4), 815827.10.1017/S0022112067002605CrossRefGoogle Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximation behind them. In Waves on Beaches and Resulting Sediment Transport (ed. Meyer, R.), pp. 95121. Academic.CrossRefGoogle Scholar
Płotka, H. & Dritschel, D. G. 2014 Simply-connected vortex-patch shallow-water quasi-equilibria. J. Fluid Mech. 743, 481502.10.1017/jfm.2014.48CrossRefGoogle Scholar
Polvani, L. M., McWilliams, J. C., Spall, M. A. & Ford, R. 1994 The coherent structures of shallow-water turbulence: deformation-radius effects, cyclone/anticyclone asymmetry and gravity-wave generation. Chaos 4 (2), 177186.CrossRefGoogle ScholarPubMed
Rayleigh, Lord 1876 On waves. Phil. Mag. 1 (4), 257279.Google Scholar
Read, P. 2011 Dynamics and circulation regimes of terrestrial planets. Planet. Space Sci. 59, 900914.CrossRefGoogle Scholar
Serre, F. 1953 Contibution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche 8 (12), 830887.CrossRefGoogle Scholar
Shields, J. J. & Webster, W. C. 1988 On direct methods in water-wave theory. J. Fluid Mech. 197, 171199.10.1017/S0022112088003222CrossRefGoogle Scholar
Skjelbreia, L. & Hendrickson, J. 1960 Fifth order gravity wave theory. Proc. 7th Conf. Coastal Engng 27 (4), 184196.Google Scholar
Smith, R. K. & Dritschel, D. G. 2006 Revisiting the Rossby–Haurwitz wave test case with Contour Advection. J. Comput. Phys. 217 (2), 473484.10.1016/ Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441455.Google Scholar
Thomson, A. F. 2008 The atmospheric ocean: eddies and jets in the antarctic circumpolar current. Phil. Trans. R. Soc. A 366, 45294541.10.1098/rsta.2008.0196CrossRefGoogle Scholar
Thomson, S. I. & McIntyre, M. E. 2016 Jupiter’s unearthly jets: a new turbulent model exhibiting statistical steadiness without large-scale dissipation. J. Atmos. Sci. 73 (3), 11191141.10.1175/JAS-D-14-0370.1CrossRefGoogle Scholar
Vallis, G. K. 2008 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Viúdez, Á. & Dritschel, D. G. 2004 Optimal potential vorticity balance of geophysical flows. J. Fluid Mech. 521, 343352.10.1017/S0022112004002058CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2006 The transition from geostrophic to stratified turbulence. J. Fluid Mech. 568, 89108.10.1017/S0022112006002060CrossRefGoogle Scholar
Waugh, D. W. & Dritschel, D. G. 1991 The stability of filamentary vorticity in two-dimensional geophysical vortex-dynamics models. J. Fluid Mech. 231, 575598.CrossRefGoogle Scholar
Webster, W. C., Duan, W. Y. & Zhao, B. B. 2011 Green–Naghdi theory, Part A: Green–Naghdi (GN) equations for shallow water waves. J. Mar. Sci. Appl. 10 (3), 253258.10.1007/s11804-011-1066-1CrossRefGoogle Scholar
Whitham, G. B. 1967 Variational methods and applications to water waves. Proc. R. Soc. Lond. A 299 (1456), 625.Google Scholar
Zhao, B. B. & Duan, W. Y. 2010 Fully nonlinear shallow water waves simulation using Green–Naghdi theory. J. Mar. Sci. Appl. 9 (1), 17.CrossRefGoogle Scholar
Zhao, B. B., Duan, W. Y. & Ertekin, R. C. 2014 Application of higher-level GN theory to some wave transformation problems. Coast. Engng 83, 177189.10.1016/j.coastaleng.2013.10.010CrossRefGoogle Scholar
Zhao, B. B., Duan, W. Y., Ertekin, R. C. & Hayatdavoodi, M. 2015 High-level Green–Naghdi wave models for nonlinear wave transformation in three dimensions. J. Ocean Engng Mar. Energy 1, 121132.CrossRefGoogle Scholar
Cited by

Linked content

Please note a has been issued for this article.

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On the regularity of the Green–Naghdi equations for a rotating shallow fluid layer
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On the regularity of the Green–Naghdi equations for a rotating shallow fluid layer
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On the regularity of the Green–Naghdi equations for a rotating shallow fluid layer
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *