On instability of the Ekman spiral

André Fischer; Jürgen Saal

doi:10.3934/dcdss.2013.6.1225

Article Contents

2013, Volume 6, Issue 5: 1225-1236. Doi: 10.3934/dcdss.2013.6.1225

This issue Previous Article Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains Next Article Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane

On instability of the Ekman spiral

André Fischer^1, and
Jürgen Saal^1,

1.
Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstraße 32, 64287 Darmstadt

Received: October 31, 2011

Revised: January 31, 2012

Published: September 2013

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Abstract

The aim of this note is to present a new approach to linear and nonlinear instability of the Ekman spiral, the famous stationary geostrophic solution of the 3D Navier-Stokes equations in a rotating frame. As former approaches to the Ekman boundary layer problem, our result is based on the numerical existence of an unstable wave perturbation for Reynolds numbers large enough derived by Lilly in [15]. By the fact that this unstable wave is tangentially nondecaying at infinity, however, standard approaches (e.g. by cut-off techniques) to instability in standard function spaces (e.g. $L^p$) remain a technical and intricate issue. In spite of this fact, we will present a rather short proof of linear and nonlinear instability of the Ekman spiral in $L^2$. The results are based on a recently developed general approach to rotating boundary layer problems, which relies on Fourier transformed vector Radon measures, cf.[11].

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Mathematics Subject Classification: Primary: 76E20, 76U05; Secondary: 35Q30.

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References

[1]	H. Abels, Bounded imaginary powers and $H^\infty$-calculus of the Stokes operator in unbounded domains, in " Nonlinear Elliptic and Parabolic Problems" (eds. M. Chipot and J. Escher), Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, (2005), 1-15.doi: 10.1007/3-7643-7385-7_1.
[2]	R. A. Adams and J. J. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.
[3]	B. Desjardins, E. Dormy and E. Grenier, Stability of mixed Ekman-Hartmann boundary layers, Nonlinearity, 12 (1999), 181-199.doi: 10.1088/0951-7715/12/2/001.
[4]	B. Desjardins and E. Grenier, Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. I. H. Poincaré Anal. Non Linéaire, 20 (2003), 87-106.doi: 10.1016/S0294-1449(02)00009-4.
[5]	J. Diestel and J. J. Uhl, Jr., "Vector Measures," Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977.
[6]	V. W. Ekman, On the influence of the earth's rotation on ocean currents, Arkiv Matem. Astr. Fysik, 11 (1905), 1-52.
[7]	K.-J. Engel and R. Nagel, "A Short Course on Operator Semigroups," Universitext, Springer, New York, 2006.
[8]	Y. Giga, K. Inui, A. Mahalov and S. Matsui, Uniform local solvability of the Navier-Stokes equations with the Coriolis force, Methods Appl. Anal., 12 (2005), 381-394.
[9]	Y. Giga, K. Inui, A. Mahalov and J. Saal, Global solvability of the Navier-Stokes equations in spaces based on sum-closed frequency sets, Adv. Differ. Equ., 12 (2007), 721-736.
[10]	Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data, Indiana Univ. Math. J., 57 (2008), 2775-2791.doi: 10.1512/iumj.2008.57.3795.
[11]	Y. Giga and J. Saal, An approach to rotating boundary layers based on vector Radon measures, preprint.
[12]	L. Greenberg and M. Marletta, The Ekman flow and related problems: Spectral theory and numerical analysis, Math. Proc. Cambridge Philos. Soc., 136 (2004), 719-764.doi: 10.1017/S030500410300731X.
[13]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations", Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981
[14]	M. Hess, M. Hieber, A. Mahalov and J. Saal, Nonlinear stability of Ekman boundary layers, Bull. London Math. Soc., 42 (2010), 691-706.doi: 10.1112/blms/bdq029.
[15]	D. Lilly, On the instability of Ekman boundary flow, J. Atmospheric Sci., 23 (1966), 481-494.
[16]	M. Marletta and C. Tretter, Essential spectra of coupled systems of differential equations and applications in hydrodynamics, J. Diff. Eq., 243 (2007), 36-69.doi: 10.1016/j.jde.2007.09.002.