Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence         Generation of large scales by modulational instability                    Fee...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
Introduction: Rossby wave turbulence       Generation of large scales by modulational instability                  Feedbac...
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Feedback of zonal flows on Rossby-wave turbulence driven by small scale instability, Wave-Flow Interactions, University of Cambridge April 13-15 2011

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Feedback of zonal flows on Rossby-wave turbulence driven by small scale instability, Wave-Flow Interactions, University of Cambridge April 13-15 2011

  1. 1. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scales Feedback of zonal flows on Rossby-wave turbulence driven by small scale instability Colm Connaughton Centre for Complexity Science and Mathematics Institute University of Warwick Joint work with S. Nazarenko (Warwick), B. Quinn (Warwick). Wave-Flow Interactions: Fourth Meeting Cambridge, 13-15 Apr 2011Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  2. 2. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesOutline 1 Introduction: Rossby wave turbulence 2 Generation of large scales by modulational instability 3 Feedback of large scales on small scalesColm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  3. 3. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesCharney-Hasegawa-Mima equation driven by smallscale "instability" (∂t − L) (∆ψ − F ψ) + β ∂x ψ + J[ψ, ∆ψ] = 0. Operator L has Fourier-space representation Lk = γk − νk 2m which mimics a small scale instability (and hyperviscosity). γk peaked around (kf , 0). Choose kf2 ∼ F . Forcing excites meridionally propagating Rossby waves with wavelength comparable to deformation radius.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  4. 4. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesLarge scale – small scale feedback loop Initialise vorticity field with white noise of very low amplitude. What are the dynamics? The following scenario was proposed by Zakharov and coworkers (1990’s): Waves having γk > 0 initially grow exponentially. When amplitudes get large enough, nonlinearity initiates cascades. Inverse cascade transfers energy to large scales leading to zonal jets. Jets shear small scale waves generating negative feedback. "Switches off" the forcing. This talk: is this what really happens?Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  5. 5. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesWeak and strong turbulence regimes Nonlinearity measured by dimensionless amplitude: Ψ0 k 3 M= β where k typical scale, Ψ0 is typical amplitude. Two limits: M 1 : Euler limit. M 1 : Wave turbulence limit. Wave turbulence limit is analytically tractable: Evolution of turbulence spectrum described by a closed kinetic equation. Kinetic equation has exact stationary solutions describing cascades.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  6. 6. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesWave turbulence kinetic equation ∂nk 2 k = 4π Vq r δ(k − q − r)δ(ωk − ωq − ωr )× ∂t [nq nr − nk nq sgn(ωk ωr ) − nk nr sgn(ωk ωq )] dqdr + γk nk . where β kx ωk = − k2+F k i qy ry ky Vq r = β |kx qx rx | + 2 − 2 . 2 q2 +F r +F k +F Problem: stationary Kolmogorov solution describing the inverse cascade is non-local.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  7. 7. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesModulational instability of Rossby waves Small scale Rossby waves are unstable to large scale modulations (Lorenz 1972, Gill 1973) In wave turbulence limit, M 1, unstable perturbations concentrate on resonant manifold: p = q+r ω(p) = ω(q) + ω(p− ). Perturbations with fastest growth rate become close to zonal p = (1, 0). for M 1.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  8. 8. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesGeneration of jets by modulational instability t=0 t=6 t=8 t=9 t=12 t=18 t=24 t=42Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  9. 9. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesDistortion of small scales by large scales in weaklynonlinear regime Modulational instability generates large scales directly from small scale waves (non-locality). Subsequent evolution of the small scales can be described by a nonlocal turbulence theory: Assume major main contribution to collision integral for small scales comes from modes q having q k (scale separation). Taylor expand in q. Leading order equation for small scales is an anisotropic diffusion equation in k-space: ∂nk ∂ ∂nk = Sij (k1 , k2 ) . ∂t ∂ki ∂kj Diffusion tensor, Sij , depends on structure of the large scales: more intense zonal flows give faster diffusion.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  10. 10. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesMore on the spectral diffusion approximation Diffusion tensor, Sij turns out to be degenerate A change of variables shows that diffusion is along 1-d curves: β k 2 kx Ω= = const k2 + F Wave-action, nk , of a small scale wavepackets is conserved by this motion. Energy, ωk nk , of small scale wavepackets is not conserved.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  11. 11. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesNumerical view of spectral transport Wave-action diffuses along the Ω curves from the forcing region until it is dissipated at large wave-numbers.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  12. 12. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesMechanism for saturation of the large scales Energy lost by the small scales is transferred to the large scale zonal flows which grow more intense. More intense large scales results in a larger diffusion coefficient. Increases the rate of dissipation of the small scale wave-action → negative feedback. Prediction: Growth of the large scales should suppress the small scale instability which turns off the energy source for the large scales leading to a saturation of the large scales without large scale dissipation.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  13. 13. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesNumerical observations of feedback loop Feedback loop is very evident in weakly nonlinear regime. Scaling arguments allow one to predict the saturation level of the large scales in terms of the small scale grow rate: [U ( U)]LS ∼ γ β M 1 [U ( U)]LS ∼ γ U M 1 These estimates are reasonably supported by numerics - especially the cross-over from the weak to strong turbulence regimes (see inset).Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
  14. 14. Introduction: Rossby wave turbulence Generation of large scales by modulational instability Feedback of large scales on small scalesConclusions The CHM equation driven by “instability” forcing behaves differently than one might expect. There is no inverse cascade but rather direct generation of large scales by modulational instability (this could change with broadband forcing). Evolution of subsequent turbulence is non-local in scale and partially tractable analytically in the wave turbulence limit. A negative feedback loop is set up whereby the growth of the small scales is suppressed by the large scales leading to a saturation of the large scales without any large scale dissipation mechanism.Colm Connaughton: http://www.slideshare.net/connaughtonc arXiv:1008.3338 nlin.CD
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