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October  2013, 6(5): 1225-1236. doi: 10.3934/dcdss.2013.6.1225

## On instability of the Ekman spiral

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

Received  November 2011 Revised  February 2012 Published  March 2013

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].
Citation: André Fischer, Jürgen Saal. On instability of the Ekman spiral. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1225-1236. doi: 10.3934/dcdss.2013.6.1225
##### 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.

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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.
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