—— Some physics of shear flows The A personal view (no review) Volker Naulin O.E. Garcia, A.H. Nielsen, J. Juul Rasmussen... volker.naulin@risoe.dk Association EURATOM-Risø National Laboratory OPL-128, Risø, DK-4000 Roskilde, Denmark 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 1/36
Contents Flows HM experiment/numerics Turbulence and flows Stabilisation Energy Transfer Generation of flows Flows in Fusion 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 2/36
Turbulence and flows Definitions: G. Falkovich: Turbulence is a state of a nonlinear physical system that has energy distribution over many degrees of freedom strongly deviated from equilibrium. Turbulence is irregular both in time and in space. Turbulence can be maintained by some ..... influ- ence or it can decay on the way to relaxation to equilibrium. The term first appeared in fluid mechanics and was later generalized to include far-from-equilibrium states in solids and plasmas. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 3/36
Turbulence and flows Laminar flow with low level of mixing. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 3/36
Turbulence and flows Turbulence mixes fast (Chimneys, milk in the coffee). 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 3/36
Turbulence and flows Turbulence in a working fusion reactor nearby 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 3/36
Turbulence and flows Turbulence in soapfilms 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 3/36
Turbulence and flows Plasma turbulence is mainly 2D turbulence, perpendicular to magnetic field. Structures evolve: vortices, eddies and flows 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 3/36
Turbulence and flows H-mode (Wagner, Asdex 1992) is essential to modern tokamak (stellarator) operation and connected to edge shear flows. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 3/36
What do shear flows do? Consider the standard advection-diffusion equation ∂ t Θ + v 0 ( x , t ) ∂ y Θ = µ ∇ 2 ⊥ Θ For a uniformly sheared flow, v 0 ( x ) = v 0 x , a formal spectral transformation yields ∂ t ˆ Θ k + V ′ ⊥ ˆ 0 k y ∂ k x = − µk 2 Θ k indicating a spectral expulsion of the scalar field fluctuations towards large absolute value radial wave numbers. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 4/36
What do shear flows do? The sheared plane wave for µ = 0 . The evolution of the initial plane wave exp ( ik x x + ik y y ) is exp [ i ( k x − v 0 k y t ) x + ik y y ] = exp [ i ( 1 − t / t ) k x x + ik y y ] with the tilting time T = k x / k y v 0 . Two cases may take place: T > 0 : plane wave is tilted against the flow T < 0 : plane wave is tilted with the flow The radial wave number is k x ( 1 − t / T ) = 0 at the tilting time t = T . 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 5/36
Sheared wave 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 6/36
Sheared wave Adding diffusion, the solution for any spectral component is � � � � T + t 2 1 − t ˆ − µk 2 − µk 2 θ k ( t ) = exp x t y t 3 T 2 There are three phases of evolution: exponential decay on the diffusive time scale τ µ = 1 / µk 2 x For t > 0 the decay rate is transiently halted at the T (structure aligned). for long times increase of damping rate exp ( − µt 3 3 T 2 ) These effects are known as: reduced radial correlation length turbulence decorrelation time 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 7/36
Sheared wave 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 8/36
Basic concepts Prototype Equation: Charney-Obukov-Hasegawa-Mima Equation: � � 1 − ∇ 2 φ + J ( φ , ∇ 2 ∂ t ⊥ φ )+ κ n ∂ y φ = 0 ⊥ Scaling: Crossover from linear to non-linear regime: ω turb = k 4 φ / ( 1 + k 2 ) ω wave = κ n k y / ( 1 + k 2 ) Use average Velocity U = k φ to equate the isotropic Rhines length (1975) a � k R = κ n / U a Rhines, J. Fluid Mech. 69 , 691, 1975 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 9/36
Basic concepts Anisotropic Rhines length: � � k Rx = κ n / U sin ( θ ) cos ( θ ) � � k Ry = κ n / U sin ( θ ) sin ( θ ) Crossover Turbulence/Waves 1 0.5 k_y/k_Rhines 0 -0.5 -1 -1 -0.5 0 0.5 1 k_x/k_Rhines Formation of elongated structures with k y → 0 is favoured. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 10/36
Basic concepts Numerical simulation of decaying turbulence: Driver in this case: density inhomogeneity. Note: Density and Potential ( e.g. momentum) transport are coupled!!! Source: Nonlinearity: Polarization drift. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 11/36
ZF: rotating tank Experimental setup, rotating tank with a rigid lid. R = 19.4 cm, D = 20 cm, η = 5 cm, rotation rate 12 rpm. Π = ω + β r ( expansion H ( r ) = 1 − β r ) Mixing: periodically pumping water in and out of two holes (diameter 2 cm ). Forcing period: T F ( T F = 6 . 6 s ) Diagnostics: particle tracking: instantaneous velocity field 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 12/36
Vorticity field Velocity field shown by arrows and vorticity contours averaged over 10 forcing periods. An anticyclonic circulation is observed 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 13/36
Vorticity Vorticity field averaged over 20 forcing periods. Red designates negative vorticity and blue positive 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 14/36
Azimuthal velocity The azimuthal velocity component averaged over 20 forcing periods. Blue designates negative velocity, i.e. anti-cyclonic motion and red designates positive velocity 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 15/36
Averaged flow 0.20 Cone.�Averaged�over�20�periodes.�T�=�6.6s 0.10 Azimuthalvelocity[cm/s] 0.00 -0.10 Forcing -0.20 0.00 4.00 8.00 12.00 16.00 20.00 Radius�[cm] Cone 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 16/36
Averaged flow 0.20 Cone.�Averaged�over�20�periodes.�T�=�6.6s 0.10 Azimuthalvelocity[cm/s] 0.00 -0.10 Forcing -0.20 0.00 4.00 8.00 12.00 16.00 20.00 Radius�[cm] Cone 0.20 Flat.�Averaged�over�20�periodes.�T�=�6.6s 0.10 Azimuthalvelocity[cm/s] 0.00 -0.10 Forcing -0.20 0.00 4.00 8.00 12.00 16.00 20.00 Radius�[cm] Flat 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 16/36
Numerical results The forced quasi-geostrophic vorticity equation on a disk with no-slip boundary conditions at the walls. ∂ω r [ φ , ω ] − β ∂φ ∂ t + 1 ∂θ = − νω + 1 Re ∇ 2 ω + F , (1) r √ Length is scaled as R , time as f − 1 , and β by f / R . ν = E , Ekman number E = µ / D 2 Ω with a spin down time τ E ≈ 90 s . The forcing is modeled by localized vorticity sources with alternating positive and negative vorticity: F = A 0 [ G ( x , y ; r 1 ) sin ( σ F t )+ G ( x , y ; r 2 ) sin ( σ F t + π )] , G ( x , y , r 1 , 2 ) localized at the positions of the two holes. For the experimental condition the scaled values of β = 0 . 256 and E = 4 . 55 × 10 − 4 . While Re ≈ 80 . 000 and volume viscosity is negligible. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 17/36
Vorticity field Numerical solution for the same parameters as in the experiment. Vorticity field averaged over 20 forcing periods for the case of a conical bottom. Red: negative vorticity and blue positive. 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 18/36
Zonal bands Finite Rossby radius ρ s = 1 The number of bands and their width depends on many parameters: β , strength of forcing Is this a case for turbulence spreading?? 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 19/36
ZF: rotating fluid Homogenization of potential vorticity (PV) in quasi 2-D flows (geophysical flows) P . Rhines The Sea (1977); (1979) Ann. Rev. Fluid Mech. 11 , 401 (1979) � ω + f � D Π Dt = D = 0 H ( r ) Dt D / Dt ≡ ∂ / ∂ t + v · ∇ v , ω is the relative vorticity of a fluid element, f is background vorticity, H ( r ) is the depth of the fluid layer. Movement towards deeper regions stretch the vortices and enhance ω ; towards shallower regions compress the vortices and decrease ω . Mixing of Π → low relative vorticity over shallow regions and higher relative vorticity over deeper regions. Plasma case: Ion vorticity equation (cold ions): � ω + ω ci � D Π i Dt = D = 0 n ( r ) Dt 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 20/36
Flows: Reynolds stress Momentum equation/vorticity equation: ∂ω ∂ t + { φ , ω } = µ ∇ 2 ω . 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 21/36
Flows: Reynolds stress ∂ω ∂ t + { φ , ω } = µ ∇ 2 ω . Reynolds decomposition (Reynolds (1894)): ω , φ = Φ + � ω = Ω + � φ , v = V + � v Z L y Ω = � ω � ≡ 1 ω dy L y 0 Zonal velocity V = � v � ; U = 0 2005 Festival du Theorie, Aix en Provence, France Back Next – p. 21/36
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