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Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations. (English) Zbl 1189.35234
Summary: We study the asymptotic behavior and the asymptotic stability of the 2D Euler equations and of the 2D linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles $$U(y)$$ and with stationary streamlines $$y=y_{0}$$ (such that $$U^{\prime}(y_{0})=0)$$, a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms.

##### MSC:
 35Q31 Euler equations 76F20 Dynamical systems approach to turbulence 76D05 Navier-Stokes equations for incompressible viscous fluids 76E20 Stability and instability of geophysical and astrophysical flows 76M22 Spectral methods applied to problems in fluid mechanics
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##### References:
 [1] Eyink, G.; Frisch, U.; Moreau, R.; Sobolevski, A., Euler: 250 years on, Physica D, 237, (2008) [2] Miller, J., Statistical mechanics of Euler equations in two dimensions, Phys. rev. lett., 65, 17, 2137-2140, (1990) · Zbl 1050.82553 [3] Robert, R., A maximum-entropy principle for two-dimensional perfect fluid dynamics, J. stat. phys., 65, 531-553, (1991) · Zbl 0935.76530 [4] Marteau, D.; Cardoso, O.; Tabeling, P., Equilibrium states of two-dimensional turbulence: an experimental study, Phys. rev. E, 51, 5124-5127, (1995) [5] Sommeria, J., Experimental study of the two dimensional inverse energy cascade in a square box, J. fluid mech., 170, 139-168, (1986) [6] Schneider, K.; Farge, M., Final states of decaying 2D turbulence in bounded domains: influence of the geometry, Physica D, (2008) · Zbl 1143.76469 [7] Maassen, S.R.; Clercx, H.J.H.; van Heijst, G.J.F., Self-organization of decaying quasi-two-dimensional turbulence in stratified fluid in rectangular containers, J. fluid mech., 495, 19-33, (2003) · Zbl 1069.76002 [8] Schecter, D.A.; Dubin, D.H.E.; Fine, K.S.; Driscoll, C.F., Vortex crystals from 2D Euler flow: experiment and simulation, Phys. fluids, 11, 905-914, (1999) · Zbl 1147.76489 [9] Paret, J.; Tabeling, P., Intermittency in the two-dimensional inverse cascade of energy: experimental observations, Phys. fluids, 10, 3126-3136, (1998) [10] Dubin, D.H.E.; O’Neil, T.M., Two-dimensional guiding-center transport of a pure electron plasma, Phys. rev. lett., 60, 13, 1286-1289, (1988) [11] Chavanis, P.H., Statistical mechanis of two-dimensional vortices and stellar systems, () · Zbl 1096.85012 [12] Farrell, B.F.; Ioannou, P.J., Structural stability of turbulent jets, J. atmospheric sci., 60, 2101-2118, (2003) [13] Rayleigh, L., On the instability of jets, Proc. lond. math. soc., 10, 4-13, (1879) · JFM 11.0685.01 [14] Thomson, W., Rectilinear motion of viscous fluid between two parallel plates, Phil. mag., 24, 188-196, (1887) [15] Orr, W.M.F., The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, Proc. R. ir. acad., 9-69, (1907) [16] A. Sommerfeld, Ein beitrag zur hydrodynamischen erklaerung der turbulenten fluessigkeitsbewegungen, in: Proceedings 4th International Congress of Mathematicians, Rome, 3, 1908, pp. 116-124. [17] Arnold, V.I., On an a-priori estimate in the theory of hydrodynamic stability, Izv. vyssh. uchebn. zaved. mat., 79, 2, 267-269, (1966), Engl. transl.: Am. Math. Soc. Trans [18] Wolansky, G.; Ghil, M., Nonlinear stability for saddle solutions of ideal flows and symmetry breaking, Commun. math. phys., 193, 713-736, (1998) · Zbl 0910.35016 [19] Simonnet, E., On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains, Physica D, 237, 20, 2539-2552, (2008) · Zbl 1157.35460 [20] Ellis, R.S.; Haven, K.; Turkington, B., Nonequivalent statistical equilibrium ensembles and refined stability theorems for most probable flows, Nonlinearity, 15, 239-255, (2002) · Zbl 1068.76041 [21] Caglioti, E.; Pulvirenti, M.; Rousset, F., The 2D constrained Navier Stokes equation and intermediate asymptotics, J. phys. A: math. gen., 41, H4001+, (2008) · Zbl 1143.76022 [22] Bouchet, F., Simpler variational problems for statistical equilibria of the 2D Euler equation and other systems with long range interactions, Physica D, 237, 1976-1981, (2008) · Zbl 1143.76385 [23] Friedlander, S.; Howard, L., Instability in parallel flows revisited, Stud. appl. math., 101, 1, 1-21, (1998) · Zbl 1136.76360 [24] Shvydkoy, Roman; Friedlander, Susan, On recent developments in the spectral problem for the linearized Euler equation, (), 271-295 · Zbl 1221.35303 [25] Grenier, E.; Jones, C.K.R.T.; Rousset, F.; Sandstede, B., Viscous perturbations of marginally stable Euler flow and finite-time Melnikov theory, Nonlinearity, 18, 465-483, (2005) · Zbl 1067.35073 [26] Belenkaya, L.; Friedlander, S.; Yudovich, V., The unstable spectrum of oscillating shear flows, SIAM J. appl. math., 59, 5, 1701-1715, (1999) · Zbl 0949.76025 [27] Drazin, P.G.; Reid, W.H., Hydrodynamic stability, (2004), Cambridge University Press · Zbl 0449.76027 [28] Farrell, B.F.; Ioannou, P.J., Generalized stability theory. part I: autonomous operators, J. atmospheric sci., 53, 2025-2040, (1996) [29] Case, K.M., Stability of inviscid plane Couette flow, Phys. fluids, 3, 143-148, (1960) · Zbl 0213.54306 [30] Dikii, L.A., The stability of plane-parallel flows of an ideal fluid, Sov. phys. dokl., 5, 1179, (1960) · Zbl 0098.40802 [31] Farrell, B., Developing disturbances in shear, J. atmospheric sci., 44, 2191-2199, (1987) [32] Volponi, F., Local algebraic instability of shear-flows in the Rayleigh equation, J. phys. A: math. gen., 38, 4293-4307, (2005) · Zbl 1086.76023 [33] Antkowiak, A.; Brancher, P., Transient energy growth for the lamb – oseen vortex, Phys. fluids, 16, L1-L4, (2004) [34] Schecter, D.A.; Dubin, D.H.E.; Cass, A.C.; Driscoll, C.F.; Lansky, I.M.; O’Neil, T.M., Inviscid damping of asymmetries on a two-dimensional vortex, Phys. fluids, 12, 2397-2412, (2000) · Zbl 1184.76483 [35] Le Dizès, S., Non-axisymmetric vortices in two-dimensional flows, J. fluid mech., 406, 175-198, (2000) · Zbl 0990.76016 [36] Nolan, D.S.; Montgomery, M.T., The algebraic growth of wavenumber one disturbances in hurricane-like vortices, J. atmospheric sci., 57, 3514-3538, (2000) [37] Balmforth, N.J.; Morrison, P.J., A necessary and sufficient instability condition for inviscid shear flow, Stud. appl. math., 102, 309-344, (1999) · Zbl 1136.76356 [38] Balmforth, N.J.; Del-Castillo-Negrete, D.; Young, W.R., Dynamics of vorticity defects in shear, J. fluid mech., 333, 197-230, (1997) · Zbl 0896.76020 [39] Briggs, R.J.; Daugherty, J.D.; Levy, R.H., Role of Landau damping in crossed-field electron beams and inviscid shear flow, Phys. fluids, 13, 421-432, (1970) [40] Yamagata, T., On trajectories of Rossby wave-packets released in a lateral shear flow, J. oceanogr., 32, 162-168, (1976) [41] Tung, K.K., Initial-value problems for Rossby waves in a shear flow with critical level, J. fluid mech., 133, 443-469, (1983) · Zbl 0547.76032 [42] Brunet, G.; Haynes, P.H., The nonlinear evolution of disturbances to a parabolic jet, J. atmospheric sci., 52, 464-477, (1995) [43] Brunet, G.; Montgomery, M.T., Vortex Rossby waves on smooth circular vortices—part I. theory, Dyn. atmos. oceans, 35, 153-177, (2002) [44] Brunet, G.; Warn, T., Rossby wave critical layers on a jet, J. atmospheric sci., 47, 1173-1178, (1990) [45] Brown, S.N.; Stewartson, K., On the algebraic decay of disturbances in a stratified linear shear flow, J. fluid mech., 100, 811-816, (1980) · Zbl 0447.76041 [46] Lundgren, T.S., Strained spiral vortex model for turbulent fine structure, Phys. fluids, 25, 2193-2203, (1982) · Zbl 0536.76034 [47] Rosencrans, S.I.; Sattinger, D.H., On the spectrum of an operator occuring in the theory of hydrodynamic stability, J. math. phys., 45, 289-300, (1966) · Zbl 0144.47006 [48] Isichenko, M.B., Nonlinear Landau damping in collisionless plasma and inviscid fluid, Phys. rev. lett., 78, 2369-2372, (1997) [49] Bassom, A.P.; Gilbert, A.D., The spiral wind-up of vorticity in an inviscid planar vortex, J. fluid mech., 371, 109-140, (1998) · Zbl 0918.76009 [50] Smith, R.A.; Rosenbluth, M.N., Algebraic instability of hollow electron columns and cylindrical vortices, Phys. rev. lett., 64, 649-652, (1990) [51] Chavanis, P.-H., Quasilinear theory of the 2D Euler equation, Phys. rev. lett., 84, 5512-5515, (2000) [52] C. Mouhot, C. Villani, On the Landau damping. arXiv:0904.2760. · Zbl 1239.82017 [53] Pego, R.L.; Weinstein, M.I., Asymptotic stability of solitary waves, Comm. math. phys., 164, 305-349, (1994) · Zbl 0805.35117 [54] Caglioti, E.; Maffei, C., Time asymptotics for solutions of Vlasov Poisson equation in a circle, J. stat. phys., 92, 1, 301-323, (1998) · Zbl 0935.35116 [55] Mishalkin; Sinai, Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid, J. appl. math. mech., 25, 1700-1705, (1961) · Zbl 0108.39501 [56] Nicholson, D., Introduction to plasma theory, (1983), Wiley New York [57] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, (1987), Society for Industrial Mathematic [58] Frisch, U.; Matsumoto, T.; Bec, J., Singularities of Euler flow? not out of the blue!, J. stat. phys., 113, 761-781, (2002) · Zbl 1058.76011 [59] Turner, M.R.; Gilbert, A.D., Linear and nonlinear decay of cat’s eyes in two-dimensional vortices, and the link to Landau poles, J. fluid mech., 593, 255-279, (2007) · Zbl 1128.76015 [60] Bajer, K.; Bassom, A.P.; Gilbert, A.D., Accelerated diffusion in the centre of a vortex, J. fluid mech., 437, 395-411, (2001) · Zbl 0981.76023 [61] Balmforth, N.J.; Smith, S.G.L.; Young, W.R., Disturbing vortices, J. fluid mech., 426, 95-133, (2001) · Zbl 0988.76016 [62] Schecter, D.A.; Montgomery, M.T., On the symmetrization rate of an intense geophysical vortex, Dyn. atmos. oceans, 37, 55-88, (2003) [63] Bouchet, F.; Simonnet, E., Random changes of flow topology in two dimensional and geophysical turbulence, Phys. rev. lett., 102, 9, 094504-+, (2009) [64] Dubrulle, B.; Nazarenko, S., Interaction of turbulence and large-scale vortices in incompressible 2D fluids, Physica D, 110, 123-138, (1997) · Zbl 0925.76266 [65] Laval, J.-P.; Dubrulle, B.; Nazarenko, S.V., Fast numerical simulations of 2D turbulence using a dynamic model for subfilter motions, J. comput. phys., 196, 184-207, (2004) · Zbl 1115.76388 [66] Nazarenko, S.; Laval, J.-P., Non-local two-dimensional turbulence and batchelor’s regime for passive scalars, J. fluid mech., 408, 301-321, (2000) · Zbl 0979.76031 [67] Glassey, R.; Schaeffer, J., Time decay for solutions to the linearized Vlasov equation, Transport theory statist. phys., 23, 411-453, (1994) · Zbl 0819.35114 [68] Glassey, R.; Schaeffer, J., On time decay rates in Landau damping, Comm. partial differential equations, 20, 647-676, (1995) · Zbl 0816.35110 [69] Erdélyi, A., Asymptotic expansions, (1956), Dover New York · Zbl 0070.29002 [70] Bleistein, N.; Handelsman, R., Asymptotic expansions of integrals, (1975), Dover New York · Zbl 0327.41027
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