Finite speed propagation in the relaxation of vortex patches
Authors:
Carole Rosier and Lionel Rosier
Journal:
Quart. Appl. Math. 61 (2003), 213-231
MSC:
Primary 76B47; Secondary 35K65, 35Q30, 76F99
DOI:
https://doi.org/10.1090/qam/1976366
MathSciNet review:
MR1976366
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A degenerate parabolic equation has been proposed by Robert and Sommeria to describe the relaxation towards a statistical equilibrium state for a two-dimensional incompressible perfect fluid with a vortex patch as initial vorticity. In this paper, flows obtained by numerical integration of the Robert-Sommeria equation over a long-time interval are compared with those obtained for the Navier-Stokes equation at high Reynolds number. A finite speed propagation for the extremal values of the vorticity is numerically shown to hold for the Robert-Sommeria equation. A rigorous proof of this (fine) property is also provided.
- S. N. Antontsev and Kh. I. Dias, New results on the localization of solutions of nonlinear elliptic and parabolic equations that are obtained by the energy method, Dokl. Akad. Nauk SSSR 303 (1988), no. 3, 524–529 (Russian); English transl., Soviet Math. Dokl. 38 (1989), no. 3, 535–539. MR 980777
- S. N. Antontsev and J. I. Díaz, Space and time localization in the flow of two immiscible fluids through a porous medium: energy methods applied to systems, Nonlinear Anal. 16 (1991), no. 4, 299–313. MR 1093843, DOI https://doi.org/10.1016/0362-546X%2891%2990032-V
X. J. Carton, G. R. Flierl, L. M. Polvani, The generation of tripoles from unstable axisymmetric isolated vortex structure, Europhys. Lett. 9 (1989), 339–344.
- Xavier Carton and Bernard Legras, The life-cycle of tripoles in two-dimensional incompressible flows, J. Fluid Mech. 267 (1994), 53–82. MR 1279083, DOI https://doi.org/10.1017/S0022112094001114
- J. I. Diaz and G. Galiano, On the Boussinesq system with nonlinear thermal diffusion, Proceedings of the Second World Congress of Nonlinear Analysts, Part 6 (Athens, 1996), 1997, pp. 3255–3263. MR 1602972, DOI https://doi.org/10.1016/S0362-546X%2897%2900330-1
- J. Ildefonso Díaz and Laurent Véron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc. 290 (1985), no. 2, 787–814. MR 792828, DOI https://doi.org/10.1090/S0002-9947-1985-0792828-X
J. B. Flor, Coherent vortex structures in stratified fluids, Thesis, Eindhoven, 1994.
- Gérard Gagneux and Monique Madaune-Tort, Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 22, Springer-Verlag, Berlin, 1996 (French, with French summary). With a preface by Charles-Michel Marle. MR 1616513
- G. Galiano, Spatial and time localization of solutions of the Boussinesq system with nonlinear thermal diffusion, Nonlinear Anal. 42 (2000), no. 3, Ser. A: Theory Methods, 423–438. MR 1774271, DOI https://doi.org/10.1016/S0362-546X%2898%2900355-1
- Andro Mikelić and Raoul Robert, On the equations describing a relaxation toward a statistical equilibrium state in the two-dimensional perfect fluid dynamics, SIAM J. Math. Anal. 29 (1998), no. 5, 1238–1255. MR 1628271, DOI https://doi.org/10.1137/S0036141096306509
- Jonathan Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett. 65 (1990), no. 17, 2137–2140. MR 1074119, DOI https://doi.org/10.1103/PhysRevLett.65.2137
J. Miller, P. Weichman and M. C. Cross, Statistical mechanics, Euler’s equation, and Jupiter’s Red Spot, Phys. Rev. A 45 (1992), 2328–2359.
- Yves G. Morel and Xavier J. Carton, Multipolar vortices in two-dimensional incompressible flows, J. Fluid Mech. 267 (1994), 23–51. MR 1279082, DOI https://doi.org/10.1017/S0022112094001102
- Raoul Robert, A maximum-entropy principle for two-dimensional perfect fluid dynamics, J. Statist. Phys. 65 (1991), no. 3-4, 531–553. MR 1137423, DOI https://doi.org/10.1007/BF01053743
- R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid Mech. 229 (1991), 291–310. MR 1121395, DOI https://doi.org/10.1017/S0022112091003038
- Raoul Robert and Joël Sommeria, Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics, Phys. Rev. Lett. 69 (1992), no. 19, 2776–2779. MR 1190642, DOI https://doi.org/10.1103/PhysRevLett.69.2776
- Raoul Robert and Carole Rosier, The modeling of small scales in two-dimensional turbulent flows: a statistical mechanics approach, J. Statist. Phys. 86 (1997), no. 3-4, 481–515. MR 1438963, DOI https://doi.org/10.1007/BF02199111
- Carole Rosier and Lionel Rosier, Well-posedness of a degenerate parabolic equation issuing from two-dimensional perfect fluid dynamics, Appl. Anal. 75 (2000), no. 3-4, 441–465. MR 1801698, DOI https://doi.org/10.1080/00036810008840859
- J. Sommeria, C. Staquet, and R. Robert, Final equilibrium state of a two-dimensional shear layer, J. Fluid Mech. 233 (1991), 661–689. MR 1140092, DOI https://doi.org/10.1017/S0022112091000642
- Bruce Turkington and Nathaniel Whitaker, Statistical equilibrium computations of coherent structures in turbulent shear layers, SIAM J. Sci. Comput. 17 (1996), no. 6, 1414–1433. MR 1413708, DOI https://doi.org/10.1137/S1064827593251708
- Nathaniel Whitaker and Bruce Turkington, Maximum entropy states for rotating vortex patches, Phys. Fluids 6 (1994), no. 12, 3963–3973. MR 1306232, DOI https://doi.org/10.1063/1.868386
- Hervé Vandeven, Family of spectral filters for discontinuous problems, J. Sci. Comput. 6 (1991), no. 2, 159–192. MR 1140344, DOI https://doi.org/10.1007/BF01062118
G. J. F. Van Heijst, R. C. Kloosterziel and C. W. M. Williams, Laboratory experiments on the tripolar vortex in a rotating fluid, J. Fluid Mech. 225 (1991), 301–331.
S. N. Antontsev and J. I. Diaz, New results on the localization of solutions of nonlinear elliptic and parabolic equations that are obtained by the energy method, Soviet Math. Dokl. 38 (1989), 535–539.
S. N. Antontsev and J. I. Diaz, Space and time localization in the flow of two immiscible fluids through a porous medium: energy methods applied to systems, Nonlinear Anal., TMA 16 (1991), 299–313.
X. J. Carton, G. R. Flierl, L. M. Polvani, The generation of tripoles from unstable axisymmetric isolated vortex structure, Europhys. Lett. 9 (1989), 339–344.
X. J. Carton, B. Legras, The life-cycle of tripoles in two-dimensional incompressible flows, J. Fluid Mech. 267 (1994), 53–82.
J. I. Diaz and G. Galiano, On the Boussinesq system with non linear thermal diffusion, Nonlinear Anal., TMA 30 (1997), 3255–3263.
J. I. Diaz and L. Veron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Am. Math. Soc. 290 (1985), 787–814.
J. B. Flor, Coherent vortex structures in stratified fluids, Thesis, Eindhoven, 1994.
G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, Mathématiques & Applications, 22, Springer, 1996.
G. Galiano, Spatial and time localization of solutions of the Boussinesq system with nonlinear thermal diffusion, Nonlinear Anal., TMA 42 (2000), 423–438.
A. Mikelic and R. Robert, On the equations describing a relaxation towards a statistical equilibrium state in the two-dimensional perfect fluid dynamics, SIAM J. Math. Anal. 29 (1998), 1238–1255.
J. Miller, Statistical mechanics of Euler equations in two dimensions, Phys. Rev. Lett. 65 (1990), p. 2137.
J. Miller, P. Weichman and M. C. Cross, Statistical mechanics, Euler’s equation, and Jupiter’s Red Spot, Phys. Rev. A 45 (1992), 2328–2359.
Y. G. Morel, X. J. Carton, Multipolar vortices in two-dimensional incompressible flows, J. Fluid Mech. 267 (1994), 23–51.
R. Robert, A maximum entropy principle for two-dimensional Euler equations, J. Statist. Phys. 65 (1991), 531–553.
R. Robert and J. Sommeria, Statistical equilibrium states for two-dimensional flows, J. Fluid Mech. 229 (1991), 291–310.
R. Robert and J. Sommeria, Relaxation towards a statistical equilibrium state in two-dimensional perfect fluid dynamics, Phys. Rev. Lett. 69 (1992), 2276–2279.
R. Robert and C. Rosier, The modeling of small scales in two-dimensional turbulent flows: a statistical mechanics approach, J. Stat. Phys. 86 (1997), 481–515.
C. Rosier and L. Rosier, Well-posedness of a degenerate parabolic equation issuing from two-dimensional perfect fluid dynamics, Applic. Anal. 75 (2000), 441–465.
J. Sommeria, C. Staquet and R. Robert, Final equilibrium state of a two-dimensional shear layer, J. Fluid Mech. 233 (1991), 661–689.
B. Turkington and N. Whitaker, Statistical equilibrium computations of coherent structures in turbulent shear layers, SIAM J. Sci. Comput. 17 (1996), 1414–1433.
B. Turkington and N. Whitaker, Maximum entropy states for rotating vortex patches, Phys. Fluids 6 (1994), 3963–3973.
H. Vandeven, Family of spectral filters for discontinuous problems, Ecole polytechnique, Internal report No. 159, 1987.
G. J. F. Van Heijst, R. C. Kloosterziel and C. W. M. Williams, Laboratory experiments on the tripolar vortex in a rotating fluid, J. Fluid Mech. 225 (1991), 301–331.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76B47,
35K65,
35Q30,
76F99
Retrieve articles in all journals
with MSC:
76B47,
35K65,
35Q30,
76F99
Additional Information
Article copyright:
© Copyright 2003
American Mathematical Society