# American Institute of Mathematical Sciences

October  2016, 9(5): 1565-1574. doi: 10.3934/dcdss.2016063

## About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows

 1 Univ. Grenoble Alpes, LJK, BP 53, 38041 Grenoble cedex 9, France 2 Inria and Institut Montpelliérain Alexander Grothendieck, Team LEM0N, Bat 5 - CC05 017, 860 rue Saint-Priest, 34095 Montpellier Cedex 5, France

Received  June 2015 Revised  August 2015 Published  October 2016

In this work we are interested in the search of interface conditions to couple hydrostatic and nonhydrostatic ocean models. To this aim, we consider simplified systems and use a time discretization to handle linear equations. We recall the links between the two models (with the particular role of the aspect ratio $\delta=H/L\ll 1$) and introduce an iterative method based on the Schwarz algorithm (widely used in domain decomposition methods).
The convergence of this method depends strongly on the choice of interface conditions: this is why we look for exact absorbing conditions and their approximations in order to provide tractable and efficient coupling algorithms.
Citation: Eric Blayo, Antoine Rousseau. About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1565-1574. doi: 10.3934/dcdss.2016063
##### References:
 [1] E. Audusse, P. Dreyfuss and B. Merlet, Schwarz waveform relaxation for primitive equations of the ocean, SIAM J. Sci. Comput., 32 (2010), 2908-2936. doi: 10.1137/090770059. [2] P. Azerad and F. Guillen, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluids dynamics, SIAM J. Math. Anal., 33 (2001), 847-859. doi: 10.1137/S0036141000375962. [3] E. Blayo, D. Cherel and A. Rousseau, Towards optimized Schwarz methods for the Navier-Stokes equations, J. Sci. Comput., 66 (2016), 275-295. doi: 10.1007/s10915-015-0020-9. [4] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245. [5] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4. [6] O. B. Fringer, M. Gerritsen and R. Street, An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal-ocean simulator, Ocean Modell., 14 (2006), 139-173. doi: 10.1016/j.ocemod.2006.03.006. [7] O. B. Fringer, J. C. McWilliams and R. L. Street, A new hybrid model for coastal simulations, Oceanography, 19 (2006), 46-59. doi: 10.5670/oceanog.2006.91. [8] P. C. Gallacher, D. A. Hebert and M. R. Schaferkotter, Nesting a nonhydrostatic model in a hydrostatic model: The boundary interface, Ocean Modell., 40 (2011), 190-198. doi: 10.1016/j.ocemod.2011.08.006. [9] M. J. Gander, Schwarz methods over the course of time, Elec. Trans. Num. Anal., 31 (2008), 228-255. [10] G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020. [11] J.-L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002. [12] C. Lucas and A. Rousseau, New developments and cosine effect in the viscous shallow water and quasi-geostrophic equations, Multiscale Model. Simul., 7 (2008), 796-813. doi: 10.1137/070705453. [13] J. Marshall, A. Adcroft, C. Hill, L. Perelman and C. Heisey, A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. Geophys. Res. Ocean., 102 (1997), 5753-5766. doi: 10.1029/96JC02775. [14] J. Marshall, C. Hill, L. Perelman and A. Adcroft, Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling, J. Geophys. Res. Ocean., 102 (1997), 5733-5752. doi: 10.1029/96JC02776. [15] F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, Math. Models and Methods in Applied Sciences, 5 (1995), 67-93. doi: 10.1142/S021820259500005X. [16] F. Nataf, F. Rogier and E. de Sturler, Domain decomposition methods for fluid dynamics, in Navier-Stokes Equations and Related Nonlinear Problems (ed. A. Sequeira), Springer US, (1995), 367-376. [17] M. F. Peeters, W. G. Habashi and E. G. Dueck, Finite element streamfunction - vorticity solutions of the incompressible Navier-Stokes equations, Int. J. Num. Methods in Fluids, 7 (2008), 17-27. [18] M. Petcu, R. Temam and M. Ziane, Some mathematical problems in fluid dynamics, in Handbook of Numerical Analysis (eds. P.G. Ciarlet), Elsevier, 14 (2009), 577-750. doi: 10.1016/S1570-8659(08)00212-3. [19] A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domain, in Handbook of Numerical Analysis, (eds. P.G. Ciarlet), Elsevier, 14 (2008), 481-575. doi: 10.1016/S1570-8659(08)00211-1.

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##### References:
 [1] E. Audusse, P. Dreyfuss and B. Merlet, Schwarz waveform relaxation for primitive equations of the ocean, SIAM J. Sci. Comput., 32 (2010), 2908-2936. doi: 10.1137/090770059. [2] P. Azerad and F. Guillen, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluids dynamics, SIAM J. Math. Anal., 33 (2001), 847-859. doi: 10.1137/S0036141000375962. [3] E. Blayo, D. Cherel and A. Rousseau, Towards optimized Schwarz methods for the Navier-Stokes equations, J. Sci. Comput., 66 (2016), 275-295. doi: 10.1007/s10915-015-0020-9. [4] C. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267. doi: 10.4007/annals.2007.166.245. [5] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629-651. doi: 10.1090/S0025-5718-1977-0436612-4. [6] O. B. Fringer, M. Gerritsen and R. Street, An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal-ocean simulator, Ocean Modell., 14 (2006), 139-173. doi: 10.1016/j.ocemod.2006.03.006. [7] O. B. Fringer, J. C. McWilliams and R. L. Street, A new hybrid model for coastal simulations, Oceanography, 19 (2006), 46-59. doi: 10.5670/oceanog.2006.91. [8] P. C. Gallacher, D. A. Hebert and M. R. Schaferkotter, Nesting a nonhydrostatic model in a hydrostatic model: The boundary interface, Ocean Modell., 40 (2011), 190-198. doi: 10.1016/j.ocemod.2011.08.006. [9] M. J. Gander, Schwarz methods over the course of time, Elec. Trans. Num. Anal., 31 (2008), 228-255. [10] G. Kobelkov, Existence of a solution 'in the large' for the 3D large-scale ocean dynamics equations, C. R. Math. Acad. Sci. Paris, 343 (2006), 283-286. doi: 10.1016/j.crma.2006.04.020. [11] J.-L. Lions, R. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053. doi: 10.1088/0951-7715/5/5/002. [12] C. Lucas and A. Rousseau, New developments and cosine effect in the viscous shallow water and quasi-geostrophic equations, Multiscale Model. Simul., 7 (2008), 796-813. doi: 10.1137/070705453. [13] J. Marshall, A. Adcroft, C. Hill, L. Perelman and C. Heisey, A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers, J. Geophys. Res. Ocean., 102 (1997), 5753-5766. doi: 10.1029/96JC02775. [14] J. Marshall, C. Hill, L. Perelman and A. Adcroft, Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling, J. Geophys. Res. Ocean., 102 (1997), 5733-5752. doi: 10.1029/96JC02776. [15] F. Nataf and F. Rogier, Factorization of the convection-diffusion operator and the Schwarz algorithm, Math. Models and Methods in Applied Sciences, 5 (1995), 67-93. doi: 10.1142/S021820259500005X. [16] F. Nataf, F. Rogier and E. de Sturler, Domain decomposition methods for fluid dynamics, in Navier-Stokes Equations and Related Nonlinear Problems (ed. A. Sequeira), Springer US, (1995), 367-376. [17] M. F. Peeters, W. G. Habashi and E. G. Dueck, Finite element streamfunction - vorticity solutions of the incompressible Navier-Stokes equations, Int. J. Num. Methods in Fluids, 7 (2008), 17-27. [18] M. Petcu, R. Temam and M. Ziane, Some mathematical problems in fluid dynamics, in Handbook of Numerical Analysis (eds. P.G. Ciarlet), Elsevier, 14 (2009), 577-750. doi: 10.1016/S1570-8659(08)00212-3. [19] A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domain, in Handbook of Numerical Analysis, (eds. P.G. Ciarlet), Elsevier, 14 (2008), 481-575. doi: 10.1016/S1570-8659(08)00211-1.
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