Etudes et évaluation de processus océaniques par des hiérarchies ...
Etudes et évaluation de processus océaniques par des hiérarchies ... Etudes et évaluation de processus océaniques par des hiérarchies ...
88 CHAPITRE 4. ETUDES DE PROCESSUS OCÉANOGRAPHIQUES 0.006 A. Wirth / Ocean Modelling 9 (2005) 71–87 85 U (m/s) 0.004 0.002 0 8 16 32 64 128 256 256 long 512 -0.002 -0.004 850 900 950 1000 Depth (m) Fig. 9. Horizontal velocity along x ¼ 250 m near the lower boundary. Symbols correspond to the different spatial resolution (as given in the figure). tel-00545911, version 1 - 13 Dec 2010 W (m/s) 0.003 0.002 0.001 0 -0.001 -0.002 850 900 950 1000 Depth (m) Fig. 10. Vertical velocity along x ¼ 500 m near the lower boundary. Symbols correspond to the different spatial resolution (as given in the figure). No spectral convergence is observed in this test-case. This comes at no surprise as the dynamics even in the interior of the domain is completely slaved to the dynamics at the boundaries and thus shows the corresponding convergence properties. It is indeed clearly visible from Fig. 7 that no structures smaller than the basin scale appear in the problem. A problem involving scales smaller than those imposed by the boundaries would cease to be stationary and the here presented analysis could not be performed. With the insight obtained from boundary-layer theory, the velocity components in the buffer zone can be adjusted so that the real dynamics at the boundary is modeled to higher order. The convergence properties of the dynamics in the interior fluid would then follow. This is the subject of future research. 8 16 32 64 128 256 256 long 512
4.3. A NON-HYDROSTATIC FLAT-BOTTOM OCEAN MODEL ENTIRELY BASE ON FOURIER EX 86 A. Wirth / Ocean Modelling 9 (2005) 71–87 LOG(error) -3 -4 -5 -6 -7 -8 -9 -10 -11 Gibbs at bottom u-comp. min. val. u-comp. top u-comp. bottom w-comp. min. val. w-comp top w-comp. -12 2 3 4 5 6 LOG(n) Fig. 11. Log–log plot of the error for different variables: amplitude of the Gibbs oscillation in the horizontal velocity component along x ¼ 250 m at the lower boundary (circles), minimum value of the horizontal velocity component along x ¼ 250 m (squares), horizontal velocity component at x ¼ 250 m, y ¼ 125 m (diamonds), vertical velocity component at x ¼ 500 m, y ¼ 875 m (upward triangles), minimum value of the vertical velocity component along x ¼ 250 m (leftward triangles) vertical velocity component at x ¼ 500 m, y ¼ 175 m (downward triangles). tel-00545911, version 1 - 13 Dec 2010 References Chorin, J.A., 1968. Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762. Davies, A.M., Lawrence, J., 1994. Examining the influence of wind and wind wave turbulence on tidal currents, using a three-dimensional hydrodynamic model including wave current interaction. J. Phys. Oceanogr. 24, 2441–2460. Goldstein, D., Handler, R., Sirovich, L., 1993. Modeling a no-slip flow boundary with an external force field. J. Comp. Phys. 105, 354–366. Gottlieb, D., Orszag, S., 1977. Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia. Iaccarino, G., Verzicco, R., 2003. Immersed boundary technique for LES/RANS simulations. Applied Mech. Rev., ASME, in press. John, F., 1991. Partial Differential Equations, fourth ed. Springer-Verlag, New York. Jones, H., Marshall, J., 1993. Convection with rotation in a neutral ocean: a study of open-ocean deep convection. J. Phys. Oceanogr. 23, 1009–1039. Julien, K., Legg, S., McWilliams, J., Werne, J., 1996. Rapidly rotating Rayleigh–Benard convection. J. Fluid Mech. 322, 243–273. Lamb, K.G., 1994. Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge. J. Geophys. Res. 99, 843–864. Marshall, J., Hill, C., Perelman, L., Adcroft, A., 1997. Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Phys. Oceanogr. 27, 5733–5752. Maxworthy, T., Narimousa, S., 1994. Unsteady, turbulent convection into a homogeneous, rotating fluid, with oceanographic applications. J. Phys. Oceanogr. 24, 865–887. Mohd-Yosuf, J., 1997. Combined immersed boundary/B-spline methods for simulation in complex geometries. Annual Research Briefs, Center for Turbulence Research, pp. 317–328. Padilla-Barbosa, J., Metais, O., 2000. Large-eddy simulations of deep-ocean convection: analysis of the vorticity dynamics. J. Turbul., 1009. Peskin, C.S., 1977. Flow patterns around heart valves: a numerical method. J. Comp. Phys. 25, 220–252. Peyret, R., 2002. Spectral Methods with Application to Incompressible Viscous Flow. Springer-Verlag, New York. 432 pp. Schlichting, H., 1968. Boundary-Layer Theory. McGraw-Hill, New York. 817 pp.
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4.3. A NON-HYDROSTATIC FLAT-BOTTOM OCEAN MODEL ENTIRELY BASE ON FOURIER EX<br />
86 A. Wirth / Ocean Mo<strong>de</strong>lling 9 (2005) 71–87<br />
LOG(error)<br />
-3<br />
-4<br />
-5<br />
-6<br />
-7<br />
-8<br />
-9<br />
-10<br />
-11<br />
Gibbs at bottom u-comp.<br />
min. val. u-comp.<br />
top u-comp.<br />
bottom w-comp.<br />
min. val. w-comp<br />
top w-comp.<br />
-12<br />
2 3 4 5 6<br />
LOG(n)<br />
Fig. 11. Log–log plot of the error for different variables: amplitu<strong>de</strong> of the Gibbs oscillation in the horizontal velocity<br />
component along x ¼ 250 m at the lower boundary (circles), minimum value of the horizontal velocity component<br />
along x ¼ 250 m (squares), horizontal velocity component at x ¼ 250 m, y ¼ 125 m (diamonds), vertical velocity<br />
component at x ¼ 500 m, y ¼ 875 m (upward triangles), minimum value of the vertical velocity component along<br />
x ¼ 250 m (leftward triangles) vertical velocity component at x ¼ 500 m, y ¼ 175 m (downward triangles).<br />
tel-00545911, version 1 - 13 Dec 2010<br />
References<br />
Chorin, J.A., 1968. Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762.<br />
Davies, A.M., Lawrence, J., 1994. Examining the influence of wind and wind wave turbulence on tidal currents, using a<br />
three-dimensional hydrodynamic mo<strong>de</strong>l including wave current interaction. J. Phys. Oceanogr. 24, 2441–2460.<br />
Goldstein, D., Handler, R., Sirovich, L., 1993. Mo<strong>de</strong>ling a no-slip flow boundary with an external force field. J. Comp.<br />
Phys. 105, 354–366.<br />
Gottlieb, D., Orszag, S., 1977. Numerical Analysis of Spectral M<strong>et</strong>hods: Theory and Applications. SIAM, Phila<strong>de</strong>lphia.<br />
Iaccarino, G., Verzicco, R., 2003. Immersed boundary technique for LES/RANS simulations. Applied Mech. Rev.,<br />
ASME, in press.<br />
John, F., 1991. Partial Differential Equations, fourth ed. Springer-Verlag, New York.<br />
Jones, H., Marshall, J., 1993. Convection with rotation in a neutral ocean: a study of open-ocean <strong>de</strong>ep convection.<br />
J. Phys. Oceanogr. 23, 1009–1039.<br />
Julien, K., Legg, S., McWilliams, J., Werne, J., 1996. Rapidly rotating Rayleigh–Benard convection. J. Fluid Mech.<br />
322, 243–273.<br />
Lamb, K.G., 1994. Numerical experiments of internal wave generation by strong tidal flow across a finite amplitu<strong>de</strong><br />
bank edge. J. Geophys. Res. 99, 843–864.<br />
Marshall, J., Hill, C., Perelman, L., Adcroft, A., 1997. Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean<br />
mo<strong>de</strong>ling. J. Phys. Oceanogr. 27, 5733–5752.<br />
Maxworthy, T., Narimousa, S., 1994. Unsteady, turbulent convection into a homogeneous, rotating fluid, with<br />
oceanographic applications. J. Phys. Oceanogr. 24, 865–887.<br />
Mohd-Yosuf, J., 1997. Combined immersed boundary/B-spline m<strong>et</strong>hods for simulation in complex geom<strong>et</strong>ries. Annual<br />
Research Briefs, Center for Turbulence Research, pp. 317–328.<br />
Padilla-Barbosa, J., M<strong>et</strong>ais, O., 2000. Large-eddy simulations of <strong>de</strong>ep-ocean convection: analysis of the vorticity<br />
dynamics. J. Turbul., 1009.<br />
Peskin, C.S., 1977. Flow patterns around heart valves: a numerical m<strong>et</strong>hod. J. Comp. Phys. 25, 220–252.<br />
Peyr<strong>et</strong>, R., 2002. Spectral M<strong>et</strong>hods with Application to Incompressible Viscous Flow. Springer-Verlag, New York. 432<br />
pp.<br />
Schlichting, H., 1968. Boundary-Layer Theory. McGraw-Hill, New York. 817 pp.
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