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ICNES 2007 Linear Coupling of Modes in Shear Flows A. G. Tevzadze E.Kharadze National Astrophysical Observatory. Shear Flow Analysis Linear Mode coupling Resonant mode conversion Non-resonant mode conversion DNS results Linear mode coupling Nonlinear developments Turbulence models Summary.

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  1. ICNES 2007Linear Coupling of Modes in Shear FlowsA. G. TevzadzeE.Kharadze National Astrophysical Observatory

  2. Shear Flow AnalysisLinear Mode couplingResonant mode conversion Non-resonant mode conversionDNS results Linear mode coupling Nonlinear developments Turbulence modelsSummary ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  3. Shear Flow Analysisshear flows are non-normalNon-self adjoint operators;Eigenfunctions are not ortogonal;(non-Hermitian system)Modal analysis fail(eigenvalue+eigenfunction)- exponential behavior - algebraic behavior ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  4. Shear Flow AnalysisRigorous consideration of non-Hermitian systems- pseudospectral;Threfethen et al. 1993- non-modal analysis;uniform shear: Kelvin modes,differential rotation – local frame: Goldreich Lynden-Bell 1965,kinematically complex shear: Mahajan & Rogava 1999,nonuniform shear: Volponi & Yoshida 2002 ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  5. Shear Flow Analysisshearing sheet transformation;spatial inhomogeneity -> temporal inhomgeneitySpatial Fourier transform;Dynamics of SFH in time;k=k(t)Linear drift of harmonics; (effect of shearing background)w=w(t)modes with variable frequencies modified initial value problem ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  6. Shear Flow Analysistransient growth(algebraic behavior, flow stability)Two linear channels of energy exchange:background flow ßà perturbations(WKB, adiabatic, non-adiabatic)perturbation ßà perturbations (different modes; adiabatic,non-adiabatic)emphasis on: Linear Coupling ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  7. Linear Mode coupling Temporal dynamics of SFHLinear modes are coupledVelocity shear originates coupling terms in linear equationstwo types of coupling:- resonant (wave-wave)- non-resonant (vortex-wave, wave-wave, spectrally unstable modes) ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  8. Linear Mode coupling: Resonant mode conversion Resonant wave interactions linear drift of SFH + mode coupling ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  9. Linear Mode coupling: Resonant mode conversionMathematical formalism:Resonance conditions: ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  10. Linear Mode coupling: Resonant mode conversion Unbounded 3D ideal compressible MHD shear flow Horizontal shear flow in the uniform magnetic field along the streamlines: V0=(Ay,0,0) , B0=(B0,0,0) • þ Fast magnetosonic and Alfven waves • wf2»wA2 (b <1, kz / kx << 1) • þ Alfven and slow magnetosonic waves • wA2»ws2(b>1) • þ Fast, slow magnetosonic and Alfven waves • wf2»wA2»ws2 (b =1, kz / kx << 1) ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  11. Transformation of the Alfven into the fast magnetosonic wave ky(0)/kx =2, kz/kx = 0.25, A/(VAkx) = 0.025 TripleResonance ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007 • The case of double transformation: Alfvenic perturbations generates fast and slow magnetosonic waves simultaneously. • = 1, ky(0)/kx = 5, kz/kx = 0.05, A/(Cskx) = 0.1

  12. Linear Mode coupling: Resonant mode conversion- More complex magnetic configurations;- Stratification;- Rotation;- analytic form of the transformation coefficients; (asymptotic cases) Direct resonance:Energy exchange between the linear modesResonance conditions: increase of the shear parameter may decrease of the transformation rate ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  13. Linear Mode coupling: non-resonant mode conversionCoupling formalism2D unbounded compressible parallel shear flow: V = (Ay,0)zero shear limit: (Vortex + Wave)velocity shear induced coupling: (Vortex + Wave)vortex is able to excite wave: non-resonant interaction ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  14. Linear Mode coupling: non-resonant mode conversion Evolution of vortex SFH in compressible shear flows: Upper panels: A / cskx = 0.2 Lower panels: A / cskx = 0.4 Generated wave amplitude ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  15. Linear Mode coupling: non-resonant mode conversionHD Keplerian disksInterplay of the transient growth and mode conversionHD turbulence transition: bypass model ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  16. Linear Mode coupling: non-resonant mode conversioncomplex systemsvortex generation: baroclinic production;Differentially rotating disk: Entropy gradient S=S(r)Multi mode conversionCoupling scheme: ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  17. Linear Mode coupling: non-resonant mode conversionLinear interaction of spectrally unstable modes and waves: wave excitationBuoyancy -> Waves ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  18. Linear Mode coupling: non-resonant mode conversionExcitation asymmetry: Vortex -> wavePV conservation prevents generation of the vorticesExcitation rate growth with shear parameterWave excitation is quite abruptwaves are excited when ky(t) = 0Excited waves are fed by the mean shear flow energyGenerated waves can have more energy then the source vortex: vortical perturbations only trigger the wave excitation while the energy is supported by the mean shear flowGenerated waves are spatially correlated with sources ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  19. DNS results- Linear dynamics of vortices in plane shear flows; mode conversion;- Dynamics of vortices in Keplerian disks: global HD simulationsRiemann, Godunov DNS Linear amplitudes; Nonlinear consequences;Transition to turbulence - bypass modelHD nonlinearitiesl pseudospectral code;transverse cascade ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  20. Localized vortex packet with linear geometry DNS results Enhencement of the vortex packet amplitude is followed by the wave excitation ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007 Excited waves propagate in the opposite directions

  21. DNS results Ring-type vortex ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  22. DNS resultsKeperian disk flow:Nonlinear self-sustained vortices;mode conversiondevelopment of shock waves ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  23. DNS resultsKeperian disk flow: ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  24. DNS results Nonlinear interactions in shear flows: transverse cascade ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  25. DNS results- Plane shear flowsvortices with different configuration generate waves linearly- Keplerian Disk flowslinear wave excitationnonlinear excitationPlanet formation modelsShock development- Nonlinear interaction of modes in shear flowstransverse cascade – bypass modelL-H transition in tokamaks ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

  26. Summarymultiple brunches in linear spectrum – shear flow couples all of them(some restrictions due to the nonlinear invariants, e.g. conservation of PV)modes can fall in resonanse(subject to resonance conditions)modes interact nonadiabatically at higher shear rates(trigger excitation, energy comes from background)new energy channels between intrinsically different modes(vortices, waves, unstable branches)mode coupling is efficient even at nonlinear amplitudesnumber of applications can play a central role and define the flow structure/stability itself(vortex stability, HD turbulence, increased energy supply) ___________________________________________________________________ A. G. Tevzadze, Linear coupling of modes in shear flowsICNES2007

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