Drobinski, Philippe

Dubos, Thomas

Mkhinini, Nadia

HAL-INSU

Secondary instability of stratified Ekman layer rollvorticesNadia Mkhinini, Thomas Dubos, Philippe DrobinskiTo cite this version:Nadia Mkhinini, Thomas Dubos, Philippe Drobinski. Secondary instability of stratified Ek-man layer roll vortices. EUROMECH Colloquium 525 - Instabilities and transition in three-dimensional flows with rotation, Jun 2011, Ecully, France. <hal-00600394>HAL Id: hal-00600394https://hal.archives-ouvertes.fr/hal-00600394Submitted on 14 Jun 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.EUROMECH Colloquium 525, 2123 June 2011, Écully, FranceInstabilities and transition in three-dimensional flows with rotationSECONDARY INSTABILITY OF STRATIFIED EKMAN LAYER ROLL VORTICESNadia Mkhinini1, Thomas Dubos1, Philippe Drobinski11Laboratoire de MÉtÉorologie Dynamoique / IPSL, École Polytechnique, PalaiseauWe study the transition to turbulence of the Ekman ﬂow in a linear density stratiﬁcation, an exact solutionof the Boussinesq equations occurring near boundaries in rotating ﬂuids. The control parameters are theRichardson number Ri = δ2N2G2 , the Reynolds number Re =Gδν and the Prandtl number Pr =νκ whereδ =√2ν/f is the Ekman length, ν is the kinematic viscosity, f is the Coriolis parameter, G is the geostrophicvelocity far from the lower boundary and κ is the thermal diﬀusivity. We consider Re = 500 and Re = 1000. Inorder to assess separately the eﬀect the diﬀusion of momentum and heat, we consider two values of the Prandtlnumber : Pr = 1 and Pr = 4.1 Nonlinear primary instabilityThe Ekman ﬂow is subject to linear instability for Ri < Ric(Re, Pr) where Ric depends very little on Pr(Brown, 1972). This instability develops into traveling Kelvin-Helmholtz roll vortices which we compute as afunction of stratiﬁcation as in Dubos et al. (2008). Fig. 1 presents the amplitude of the equilibrated vortices asmeasured by their kinetic, potential and total energy. While for Pr = 1 the roll amplitude vanishes at Ri = Ricthis is not the case at Pr = 4. So far only supercritical bifurcations had been identiﬁed in the Ekman layerstability diagram (Haeusser and Leibovich, 2003; Dubos et al., 2008). The subcritical bifurcation observed atPr = 4 allows us to compute equilibrated vortices at Ri > Ric.2 Linear secondary instabilityFig. 2 presents the growth rate σ2 of inﬁntesimal three-dimensional perturbations to the travelling rolls as afunction of their horizontal wavenumber ky. For Ri = 0 we recover at ky > 1 the unstable modes identiﬁed byDubos et al. (2008). New, less unstable modes branches are obtained for 0 < ky < 1. However as Ri increasesthe modes for ky > 1 become less unstable, become dominated by the unstable modes for ky < 1 and eventuallydisappear. For Pr = 1 the overall maximum growth rate decreases as Ri increases, but not for Pr = 4. Thismust be related to the fact that the roll vortices weaken as Ri approaches Ric if Pr = 1, but not if Pr = 4.3 ConclusionWe have studied the inﬂuence of an ambient linear stratiﬁcation on the secondary instability of the Ekman ﬂow.This inﬂuence depends markedly on the Prandtl number. Our results reveal that the bifurcation at Ri = Ric issubcritical Pr = 4. In this regime, ﬁnite amplitude vortices exist beyond Ri = Ric and the growth rate of thesecondary instability increases as the ambient stratiﬁcation increases, instead of decreasing as occurs for Pr = 1and as intuition suggests. A complementary study of the nonlinear development of the secondary instabilty isunder way.ReferencesBrown, R. A. (1972). On the inﬂection point instability of a stratiﬁed Ekman boundary layer. J. Atmos. Sci,29:850859.Dubos, T., Barthlott, C., and Drobinski, P. (2008). Emergence and secondary instability of ekman layer rolls.J. Atmos. Sci, 65(7):23262342.Haeusser, T. M. and Leibovich, S. (2003). Pattern formation in the marginally unstable Ekman layer. J. FluidMech., 479:125144.EUROMECH Colloquium 525, 2123 June 2011, Écully, FranceInstabilities and transition in three-dimensional flows with rotation0 0.005 0.01 0.015 0.02 0.025 0.0300.0050.010.0150.020.0250.03RiEnergyRe= 500 et Pr= 1  TKETPETotal0 0.01 0.02 0.0300.050.10.150.2RiEnergyRe= 500 et Pr= 4  0 0.005 0.01 0.015 0.02 0.02500.010.020.030.04RiEnergyRe= 1000 et Pr= 1  0 0.01 0.02 0.03 0.0400.050.10.150.20.25RiEnergy Re= 1000 et Pr= 4  TKETPETotalTKETPETotalTKETPETotalFigure 1: Energy of roll vortices as a function of ambient stratiﬁcation. Arrows point to Ri = Ric0 1 2 3 4−0.0100.010.020.030.04kyσ2Re=1000, Pr=1  0 0.1 0.2 0.3 0.4 0.500.0050.010.0150.02kyσ2Re=1000, Pr=10 1 2 3 400.010.020.030.040.05kyσ2Re=1000, Pr=40 0.1 0.2 0.3 0.4 0.500.0050.010.0150.020.025kyσ2Re=1000, Pr=4  Ri=0Ri=0.005Ri=0.010Ri=0.015Ri=0Ri=0.005Ri=0.010Ri=0.015Figure 2: Growth rate σ2 of secondary instability of Ekman roll vortices as a function of wave vector ky (right: zoom)

Secondary instability of stratified Ekman layer roll vortices

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Secondary instability of stratified Ekman layer roll vortices

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