Abstract
This paper is devoted to the stability analysis of the Lamb–Oseen vortex in the regime of high circulation Reynolds numbers. When strongly localized perturbations are applied, it is shown that the vortex relaxes to axisymmetry in a time proportional to \({Re^{2/3}}\) , which is substantially shorter than the diffusion time scale given by the viscosity. This enhanced dissipation effect is due to the differential rotation inside the vortex core. Our result relies on a recent work by Li et al. (Pseudospectral and spectral bounds for the Oseen vortices operator, 2017, arXiv:1701.06269), where optimal resolvent estimates for the linearized operator at Oseen’s vortex are established. A comparison is made with the predictions that can be found in the physical literature, and with the rigorous results that were obtained for shear flows using different techniques.
Similar content being viewed by others
References
Bajer, K., Bassom, A.P., Gilbert, A.D.: Accelerated diffusion in the centre of a vortex. J. Fluid Mech. 437, 395–411 (2001)
Bassom, A.P., Gilbert, A.D.: The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109–140 (1998)
Bassom, A.P., Gilbert, A.D.: The spiral wind-up and dissipation of vorticity and a passive scalar in a strained planar vortex. J. Fluid Mech. 398, 245–270 (1999)
Bassom, A.P., Gilbert, A.D.: The relaxation of vorticity fluctuations in locally elliptical streamlines. Proc. R. Soc. A 456, 295–314 (2000)
Bedrossian, J., Coti Zelati, M.: Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shearflows. Arch. Ration. Mech. Anal. 224, 1161–1204 (2017)
Bedrossian, J., Masmousi, N.: Inviscid damping and the asymptotic stability of planar shear flows in the 2D Eulerequations. Publ. Math. IHES 122, 195–300 (2015)
Bedrossian, J., Masmousi, N., Vicol, V.: Enhanced dissipation and inviscid damping in the inviscid limit of theNavier-Stokes equations near the two-dimensional Couette flow. Arch. Ration. Mech. Anal. 219, 1087–1159 (2016)
Bedrossian, J., Vicol, V., Wang, Fei: The Sobolev stability threshold for 2D shear flows near Couette, J. Nonlinear Sci., 1–25 2017
Bernoff, A.J., Lingevitch, J.F.: Rapid relaxation of an axisymmetric vortex. Phys. Fluids 6, 3717–3723 (1994)
Bouchet, F., Morita, H.: Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations. Physica D 239, 948–966 (2010)
Brézis, H., Gallouët, Th: Nonlinear Schrödinger evolution equations. Nonlinear Anal. Theory Methods Appl. 4, 677–681 (1980)
Brézis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Commun. Partial Diff. Equ. 5, 773–789 (1980)
Carlen, E., Loss, M.: Optimal smoothing and decay estimates for viscously damped conservation laws, with applicationsto the 2-D Navier-Stokes equation. Duke Math. J. 81, 135–157 (1995)
Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics. Oxford University Press, Oxford, An Introduction toRotating Fluids and the Navier-Stokes Equations (2006)
Deng, W.: Resolvent estimates for a two-dimensional non-self-adjoint operator. Commun. Pure Appl. Anal. 12, 547–596 (2013)
Deng, W.: Pseudospectrum for Oseen vortices operators. Int. Math Res. Not. 2013, 1935–1999 (2013)
Gallagher, I., Gallay, Th, Nier, F.: Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int. Math. Res. Not. 2009, 2147–2199 (2009)
Gallay, Th.: Interaction of vortices in weakly viscous planar flows, Arch. Ration. Mech. Anal. 200, 445–4902011
Gallay, Th.: Stability and interaction of vortices in two-dimensional viscous flows, Discrete Contin. Dyn. Syst. Ser.S 5, 1091–1131 2012
Gallay, Th, Maekawa, Y.: Three-dimensional stability of Burgers vortices. Commun. Math. Phys. 302, 477–511 (2011)
Gallay, Th., Maekawa, Y.: Existence and stability of viscous vortices. In: Handbook of Mathematical Analysis inMechanics of Viscous Fluids (Eds. Giga Y. and Novotny A.), Springer, Berlin, 2017
Gallay, Th, Wayne, C.E.: Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticityequations on \({\mathbb{R}}^2\). Arch. Ration. Mech. Anal. 163, 209–258 (2002)
Gallay, Th, Wayne, C.E.: Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Commun. Math. Phys. 255, 97–129 (2005)
Gallay, Th, Wayne, C.E.: Existence and stability of asymmetric Burgers vortices. J. Math. Fluid Mech. 9, 243–261 (2007)
Gallay, Th, Rodrigues, L.M.: Sur le temps de vie de la turbulence bidimensionnelle (in French). Ann. Fac. Sci. Toulouse 17, 719–733 (2008)
Helffer, B.: Spectral theory and its applications. In: Cambridge Studies in Advanced Mathematics, vol. 139, CambridgeUniversity Press, Cambridge, 2013
Jiménez, J., Moffatt, H.K., Vasco, C.: The structure of the vortices in freely decaying two-dimensional turbulence. J. Fluid Mech. 313, 209–222 (1996)
Kato, T.: Perturbation theory for linear operators. In: Grundlehren der mathematischen Wissenschaften, vol. 132, Springer, New York, 1966
Li, T., Wei, D., Zhang,Z.: Pseudospectral and spectral bounds for the Oseen vortices operator. arXiv:1701.06269 2017
Lieb, E., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (2001)
Lundgren, T.S.: Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193–2203 (1982)
Maekawa, Y.: Spectral properties of the linearization at the Burgers vortex in the high rotation limit. J. Math. FluidsMech. 13, 515–532 (2011)
Maekawa, Y.: On the existence of Burgers vortices for high Reynolds numbers. J. Math. Anal. Appl. 349, 181–200 (2009)
Maekawa, Y.: Existence of asymmetric Burgers vortices and their asymptotic behavior at large circulations. Math. Models Methods Appl. Sci. 19, 669–705 (2009)
Moffatt, H.K., Kida, S., Ohkitani, K.: Stretched vortices–the sinews of turbulence; large-Reynolds-numberasymptotics. J. Fluids Mech. 259, 241–264 (1994)
Ogawa, T., Taniuchi, Y.: The limiting uniqueness criterion by vorticity for Navier-Stokes equations in Besov spaces. Tohoku Math. J. 56, 65–77 (2004)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. In: Applied MathematicalSciences, vol. 44, Springer, New York, 1983
Prochazka, A., Pullin, D.I.: On the two-dimensional stability of the axisymmetric Burgers vortex. Phys. Fluid 7, 1788–1790 (1995)
Prochazka, A., Pullin, D.I.: Structure and stability of non-symmetric Burgers vortices. J. Fluid Mech. 363, 199–228 (1998)
Rhines, P.B., Young, W.R.: How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133–145 (1983)
Robinson, A.C., Saffman, P.G.: Stability and structure of stretched vortices, Stud. Appl. Math. 70, 163–1811984
Ting, L., Klein, R.: Viscous Vortical Flows. Lecture Notes in Physics, vol. 374. Springer, Berlin (1991)
Ting, L., Tung, C.: Motion and decay of a vortex in a nonuniform stream. Phys. Fluids 8, 1039–1051 (1965)
Wei, D., Zhang, Z., Zhao, W.: Linear inviscid damping for a class of monotone shear flow in Sobolev spaces. Commun. Pure Appl, Math (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Šverák
Rights and permissions
About this article
Cite this article
Gallay, T. Enhanced Dissipation and Axisymmetrization of Two-Dimensional Viscous Vortices. Arch Rational Mech Anal 230, 939–975 (2018). https://doi.org/10.1007/s00205-018-1262-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-018-1262-0