# American Institute of Mathematical Sciences

June  2015, 20(4): 961-988. doi: 10.3934/dcdsb.2015.20.961

## An energy-consistent depth-averaged Euler system: Derivation and properties

 1 INRIA Roquencourt, B.P. 105, 78153 Le Chesnay Cedex 2 University Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, Seismology group, 1 rue Jussieu, 75005 Paris, France 3 CEREMA, 134 rue de Beauvais, F-60280 Margny-Lès-Compiègne, France 4 Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  April 2014 Revised  September 2014 Published  February 2015

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.
Citation: Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, Nicolas Seguin. An energy-consistent depth-averaged Euler system: Derivation and properties. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 961-988. doi: 10.3934/dcdsb.2015.20.961
##### References:
 [1] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4. [2] B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57 (2008), 97-131. doi: 10.1512/iumj.2008.57.3200. [3] A.-J.-C. Barré de Saint-Venant, Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit, C. R. Acad. Sci. Paris, 73 (1871), 147-154. [4] J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. [5] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes, European Journal of Mechanics - B/Fluids, 30 (2011), 589-597, URL http://www.sciencedirect.com/science/article/pii/S0997754611000185, Special Issue: Nearshore Hydrodynamics. doi: 10.1016/j.euromechflu.2011.02.005. [6] F. Bouchut, A. Mangeney-Castelnau, B. Perthame and J.-P. Vilotte, A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows, Comptes Rendus Mathematique, 336 (2003), 531-536, URL http://www.sciencedirect.com/science/article/B6X1B-487KH4G-8/2/027c686fc96421e65f60c0171c8c3c12. doi: 10.1016/S1631-073X(03)00117-1. [7] F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography, Comm. in Math. Sci., 2 (2004), 359-389, URL http://projecteuclid.org/euclid.cms/1109868726. doi: 10.4310/CMS.2004.v2.n3.a2. [8] Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12 (1999), 495-512. doi: 10.1088/0951-7715/12/3/004. [9] M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model, Computers & Fluids, 47 (2011), 51-64. doi: 10.1016/j.compfluid.2011.02.013. [10] M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733-759. doi: 10.3934/dcdsb.2008.10.733. [11] R. Camassa, D. D. Holm and C. D. Levermore, Long-time effects of bottom topography in shallow water, Phys. D, 98 (1996), 258-286, Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). doi: 10.1016/0167-2789(96)00117-0. [12] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Math., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [13] F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model, J. Sci. Comput., 48 (2011), 105-116. doi: 10.1007/s10915-010-9395-9. [14] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [15] A. Decoene, L. Bonaventura, E. Miglio and F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for river hydraulics, M3AS, 19 (2009), 387-417. doi: 10.1142/S0218202509003474. [16] D. Dutykh, T. Katsaounis and D. Mitsotakis, Finite volume methods for unidirectional dispersive wave models, Internat. J. Numer. Methods Fluids, 71 (2013), 717-736. doi: 10.1002/fld.3681. [17] S. Ferrari and F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography, M2AN Math. Model. Numer. Anal., 38 (2004), 211-234. doi: 10.1051/m2an:2004010. [18] J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 89-102. doi: 10.3934/dcdsb.2001.1.89. [19] A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Roy. Soc. (London) Ser. A, 338 (1974), 43-55. doi: 10.1098/rspa.1974.0072. [20] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425. [21] E. Grenier, On the derivation of homogeneous hydrostatic equations, ESAIM: M2AN, 33 (1999), 965-970. doi: 10.1051/m2an:1999128. [22] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21 (2009), 016601. doi: 10.1063/1.3053183. [23] O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model, J. Comp. Phys., 229 (2010), 2034-2045. doi: 10.1016/j.jcp.2009.11.021. [24] C. D. Levermore, Entropy-based moment closures for kinetic equations, in Proceedings of the International Conference on Latest Developments and Fundamental Advances in Radiative Transfer (Los Angeles, CA, 1996), 26 (1997), 591-606. doi: 10.1080/00411459708017931. [25] C. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515. doi: 10.1088/0951-7715/14/6/305. [26] Y. A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math., 59 (2006), 1225-1285. doi: 10.1002/cpa.20148. [27] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible models, Oxford University Press, 1996. [28] A. Lucas, A. Mangeney and J. P. Ampuero, Frictional weakening in landslides on earth and on other planetary bodies, Nature Communication, 5 (2014). doi: 10.1038/ncomms4417. [29] N. Makarenko, A second long-wave approximation in the Cauchy-Poisson problem(in russian), Dyn. Contin. Media, 77 (1986), 56-72. [30] A. Mangeney, F. Bouchut, N. Thomas, J. P. Vilotte and M.-O. Bristeau, Numerical modeling of self-channeling granular flows and of their levee-channel deposits, Journal of Geophysical Research - Earth Surface, 112 (2007), 2003-2012, URL http://hal.archives-ouvertes.fr/hal-00311797. doi: 10.1029/2006JF000469. [31] A. Mangeney-Castelnau, F. Bouchut, J. P. Vilotte, E. Lajeunesse, A. Aubertin and M. Pirulli, On the use of Saint-Venant equations to simulate the spreading of a granular mass, Journal of Geophysical Research: Solid Earth, 110 (2005), 1978-2012. doi: 10.1029/2004JB003161. [32] F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects, European Journal of Mechanic /B, 26 (2007), 49-63. doi: 10.1016/j.euromechflu.2006.04.007. [33] N. Masmoudi and T. Wong, On the Hs theory of hydrostatic Euler equations, Archive for Rational Mechanics and Analysis, 204 (2012), 231-271. doi: 10.1007/s00205-011-0485-0. [34] J. Miles and R. Salmon, Weakly dispersive nonlinear gravity waves, J. Fluid Mech., 157 (1985), 519-531. doi: 10.1017/S0022112085002488. [35] B. T. Nadiga, L. G. Margolin and P. K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle, Phys. Fluids, 8 (1996), 2066-2077. doi: 10.1063/1.869009. [36] O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 119 (1993), 618-638. doi: 10.1061/(ASCE)0733-950X(1993)119:6(618). [37] D. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. doi: 10.1017/S0022112067002605. [38] B. Perthame, Kinetic Formulation of Conservation Laws, Oxford University Press, 2002. [39] J. Sainte-Marie, Vertically averaged models for the free surface Euler system. Derivation and kinetic interpretation, Math. Models Methods Appl. Sci. (M3AS), 21 (2011), 459-490. doi: 10.1142/S0218202511005118. [40] C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Mathematical Phys., 10 (1969), 536-539. doi: 10.1063/1.1664873. [41] W. C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid Mech., 107 (1981), 499-508. doi: 10.1017/S0022112081001882.

show all references

##### References:
 [1] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4. [2] B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations, Indiana Univ. Math. J., 57 (2008), 97-131. doi: 10.1512/iumj.2008.57.3200. [3] A.-J.-C. Barré de Saint-Venant, Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit, C. R. Acad. Sci. Paris, 73 (1871), 147-154. [4] J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032. [5] P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes, European Journal of Mechanics - B/Fluids, 30 (2011), 589-597, URL http://www.sciencedirect.com/science/article/pii/S0997754611000185, Special Issue: Nearshore Hydrodynamics. doi: 10.1016/j.euromechflu.2011.02.005. [6] F. Bouchut, A. Mangeney-Castelnau, B. Perthame and J.-P. Vilotte, A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows, Comptes Rendus Mathematique, 336 (2003), 531-536, URL http://www.sciencedirect.com/science/article/B6X1B-487KH4G-8/2/027c686fc96421e65f60c0171c8c3c12. doi: 10.1016/S1631-073X(03)00117-1. [7] F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography, Comm. in Math. Sci., 2 (2004), 359-389, URL http://projecteuclid.org/euclid.cms/1109868726. doi: 10.4310/CMS.2004.v2.n3.a2. [8] Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12 (1999), 495-512. doi: 10.1088/0951-7715/12/3/004. [9] M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model, Computers & Fluids, 47 (2011), 51-64. doi: 10.1016/j.compfluid.2011.02.013. [10] M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733-759. doi: 10.3934/dcdsb.2008.10.733. [11] R. Camassa, D. D. Holm and C. D. Levermore, Long-time effects of bottom topography in shallow water, Phys. D, 98 (1996), 258-286, Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). doi: 10.1016/0167-2789(96)00117-0. [12] R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Math., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [13] F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model, J. Sci. Comput., 48 (2011), 105-116. doi: 10.1007/s10915-010-9395-9. [14] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2. [15] A. Decoene, L. Bonaventura, E. Miglio and F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for river hydraulics, M3AS, 19 (2009), 387-417. doi: 10.1142/S0218202509003474. [16] D. Dutykh, T. Katsaounis and D. Mitsotakis, Finite volume methods for unidirectional dispersive wave models, Internat. J. Numer. Methods Fluids, 71 (2013), 717-736. doi: 10.1002/fld.3681. [17] S. Ferrari and F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography, M2AN Math. Model. Numer. Anal., 38 (2004), 211-234. doi: 10.1051/m2an:2004010. [18] J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 89-102. doi: 10.3934/dcdsb.2001.1.89. [19] A. E. Green, N. Laws and P. M. Naghdi, On the theory of water waves, Proc. Roy. Soc. (London) Ser. A, 338 (1974), 43-55. doi: 10.1098/rspa.1974.0072. [20] A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425. [21] E. Grenier, On the derivation of homogeneous hydrostatic equations, ESAIM: M2AN, 33 (1999), 965-970. doi: 10.1051/m2an:1999128. [22] D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation, Physics of Fluids, 21 (2009), 016601. doi: 10.1063/1.3053183. [23] O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model, J. Comp. Phys., 229 (2010), 2034-2045. doi: 10.1016/j.jcp.2009.11.021. [24] C. D. Levermore, Entropy-based moment closures for kinetic equations, in Proceedings of the International Conference on Latest Developments and Fundamental Advances in Radiative Transfer (Los Angeles, CA, 1996), 26 (1997), 591-606. doi: 10.1080/00411459708017931. [25] C. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515. doi: 10.1088/0951-7715/14/6/305. [26] Y. A. Li, A shallow-water approximation to the full water wave problem, Comm. Pure Appl. Math., 59 (2006), 1225-1285. doi: 10.1002/cpa.20148. [27] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1: Incompressible models, Oxford University Press, 1996. [28] A. Lucas, A. Mangeney and J. P. Ampuero, Frictional weakening in landslides on earth and on other planetary bodies, Nature Communication, 5 (2014). doi: 10.1038/ncomms4417. [29] N. Makarenko, A second long-wave approximation in the Cauchy-Poisson problem(in russian), Dyn. Contin. Media, 77 (1986), 56-72. [30] A. Mangeney, F. Bouchut, N. Thomas, J. P. Vilotte and M.-O. Bristeau, Numerical modeling of self-channeling granular flows and of their levee-channel deposits, Journal of Geophysical Research - Earth Surface, 112 (2007), 2003-2012, URL http://hal.archives-ouvertes.fr/hal-00311797. doi: 10.1029/2006JF000469. [31] A. Mangeney-Castelnau, F. Bouchut, J. P. Vilotte, E. Lajeunesse, A. Aubertin and M. Pirulli, On the use of Saint-Venant equations to simulate the spreading of a granular mass, Journal of Geophysical Research: Solid Earth, 110 (2005), 1978-2012. doi: 10.1029/2004JB003161. [32] F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects, European Journal of Mechanic /B, 26 (2007), 49-63. doi: 10.1016/j.euromechflu.2006.04.007. [33] N. Masmoudi and T. Wong, On the Hs theory of hydrostatic Euler equations, Archive for Rational Mechanics and Analysis, 204 (2012), 231-271. doi: 10.1007/s00205-011-0485-0. [34] J. Miles and R. Salmon, Weakly dispersive nonlinear gravity waves, J. Fluid Mech., 157 (1985), 519-531. doi: 10.1017/S0022112085002488. [35] B. T. Nadiga, L. G. Margolin and P. K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle, Phys. Fluids, 8 (1996), 2066-2077. doi: 10.1063/1.869009. [36] O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 119 (1993), 618-638. doi: 10.1061/(ASCE)0733-950X(1993)119:6(618). [37] D. Peregrine, Long waves on a beach, J. Fluid Mech., 27 (1967), 815-827. doi: 10.1017/S0022112067002605. [38] B. Perthame, Kinetic Formulation of Conservation Laws, Oxford University Press, 2002. [39] J. Sainte-Marie, Vertically averaged models for the free surface Euler system. Derivation and kinetic interpretation, Math. Models Methods Appl. Sci. (M3AS), 21 (2011), 459-490. doi: 10.1142/S0218202511005118. [40] C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Mathematical Phys., 10 (1969), 536-539. doi: 10.1063/1.1664873. [41] W. C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations, J. Fluid Mech., 107 (1981), 499-508. doi: 10.1017/S0022112081001882.
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