Collaborators: A. Almagri, C. Forest, M. Nornberg, K. Rahbarnia, J. Sarff UW Madison S. Prager, Y. Ren PPPL D. Hatch, F. Jenko IPP G.

Transcription

1 Dissipa&on Range Turbulent Cascades in Plasmas P.W. Terry Center for Magne,c Self Organiza,on in Laboratory and Astrophysical Plasmas University of Wisconsin Madison Collaborators: A. Almagri, C. Forest, M. Nornberg, K. Rahbarnia, J. Sarff UW Madison S. Prager, Y. Ren PPPL D. Hatch, F. Jenko IPP G. Fiksel Rochester

2 High frequency spectra in laboratory and astrophysical plasmas show breaks in power law behavior FAST, Chaston et al., PRL 2008 f Interpreta&on of new behavior at high wavenumber: New iner&al physics? Dissipa&on effects? Kine&c processes? Sensible to interpret as sequence of power laws?

3 MST: Toroidal wavenumber spectrum bever fit by an exponen&al decay than a power law Hydrodynamic dissipa&on range turbulence has exponen&al decay Is there a dissipa&on range spectrum for plasma turbulence like hydrodynamics? What does it look like? Is there scaling behavior? Can we accommodate behaviors of confined plasmas? Kine&c effects Dissipa&on over all scales?

4 To understand dissipa&on range cascade consider iner&al range Hydrodynamic iner&al range cascade is self similar across scales Nonlinearity has no intrinsic scale Dissipa&on is negligible on dynamical &me scales => no energy loss Energy transfer rate T k at every scale equal to constant energy input rate ε Since T k = v k3 k ε = T k => E(k) = v k2 /k = ε 2/3 k -5/3 Kolmogorov spectrum

5 In reality, dissipa&on not zero in iner&al range magnitude scales con&nuously with k R 1 = Dissipation rate Transfer rate = υk 2 E(k) = υk 4 / 3 T k ε 1/ 3 R = 1 for k = k υ (Kolmogorov wavenumber) Nominal division between iner&al and dissipa&on ranges, but dissipa&on con&nuous over all scales

6 Accoun&ng for dissipa&on rate scaling introduces exponen&al decay at all scales Spectrum: E(k) = aε 2 / 3 k 5 / 3 exp b k α Smith and Reynolds, 1991 k υ Power law and exponen&al envelopes apply over all scales Power law dominates for k < k υ, exponen&al dominates for k > k υ Exponen&al index α between 1 and 2, depending on theory Spectrum obtained from transfer rate avenua&on balance υk 2 E(k) = dt k dk Tennekes and Lumley, 1971 T k transfers in all scales, but it is avenuated by dissipa&on Iner&al balance with energy input ε enters as boundary condi&on Can be adapted to diverse plasma effects use to formulate dissipa&on range spectra for plasmas

7 Plasma physics affects υk 2 E = dt k /dk, leading to changes in power law, exponen&al decay Different dissipa&on rates, nonlocality MHD, η > υ => k η < k υ Nonlocal triad reaching into iner&al scales of B allows power law for k η < k < k υ Scale dependent modifica&on of transfer strength scaling Reduced T k => shallower power law => steeper exponen&al decay Reduced T k : more dissipa&on in nonlinear &me Dissipa&on rate scaling Damped modes and kine&c damping puts damping in largest scales can dissipa&on range s&ll be iden&fied?

8 Remainder of talk: Present illustra&ons and comparisons 1. MHD turbulence Comparison: magne&c turbulence in MST 2. Low magne&c Prandtl number MHD turbulence Comparison: magne&c turbulence in liquid sodium Madison Dynamo Expt 3. Tokamak microturbulence Comparison: High wavenumber gyrokine&c simula&ons of ion temperature gradient (ITG) turbulence

9 Illustra&on 1: Pm = 1 MHD turbulence shows how vector field alignment affects T k, power law and exponen&al behavior Two avenua&on balances for magne&c energy and kine&c energy: 2ηk 2 E B (k) = dt B (k)/dk 2υk 2 E V (k) = dt V (k)/dk Transfer rates from nonlineari&es of MHD: T B (k) = [ B (v )B B (B )v] exp(ik x)d 3 x T V (k) = [ v (v )v v (B )B] exp(ik x)d 3 x Write in terms of E B and E V (closure problem) Unaligned turbulence (v B) : T B = E B ε 1/3 k 5/3 E B = E V = aε 2 / 3 k 5 / 3 exp 3 2 k k η un Aligned turbulence (v B ~ v k B k k -1/4 ) : T B = E B ε 1/4 V A -3/2 k 3/4 4 / 3 k η un = ε1/ 4 η 3 / 4 Terry, Tangri, PoP 2009 E B = E V = a < ε 1/ 2 1/ V 2 A k 3 / 2 exp 4 k 3 k η al E B = E V = a < ε 1/ 2 1/ V 2 A k 5 / 3 1/ k 6 η al e 1/ 6 exp / 2 k k η al (k < k η al ) 4 / 3 (k k η al ) k η al = ε 1/ 3 V A 1/ 3 η 2 / 3

10 Comparison: magne&c spectrum in MST is well fit MHD dissipa&on range theory B spectrum is anisotropic Most power in electron direc&on k spectrum appears exponen&al Fit to k -α exp[-b(k/k d ) β ] Fit: α = 1.79±0.23 (95% confidence) β = 1.64±0.52 For theory values of α = 1.77, β = 1.33, spectrum matches theore&cal form Inferred Kolmogorov scale: k d = 0.8 cm -1 From parameters: k d = 3.0 cm -1 dissipa&on present in spectrum that is stronger than resis&vity Possible source: cyclotron damping Ren et al., PRL 2011

11 Illustra&on 2: Pm < 1 MHD turbulence shows how nonlocal coupling allows power laws in intermediate range between Kolmogorov scales Pm < 1 => η > υ => magne&c fluctua&ons dissipate at lower k In range k η < k < k υ B 2 avoids exponen&al decay by nonlocal coupling B k : above resis&ve dissipa&on scale B k : In magne&c iner&al range v k-k : In flow iner&al range Closure: T B = B k B k k 3/2 [E v (k)] 1/2 Dissipa&ve balance: T B = -ηk 2 B k 2 E B (k) k 11/ 3 exp[ (k /k υ ) 4 / 3 ] E v (k) decays exponen&ally above k υ Terry, Tangri, PoP 2009

12 Comparison: Madison Dynamo Experiment consistent k 11/3 range Liquid sodium experiment Pm 10-5 Provides best test of MHD theories (No plasma effects like kine&c damping) Observa&on: B 2 spectrum (in frequency) steepens to something like k 11/3 Consistent with theory Diagnos&c refinements for k spectra are being pursued

13 Illustra&on 3: Ion temperature gradient turbulence has novel dissipa&on range that s&ll fits in theore&cal framework Kine&c physics affects damping Damping occurs in wavenumber range of instability From nonlinear excita&on of damped modes (other zeros of plasma dielectric) Large numbers excited Most instability energy damped at low k Damping rate largest at low k, smaller at high k Depends of turbulence level Calculate from energy evolu&on E k t! = Qk + Ck N.C. For gyrokine&cs: Qk = # dv dµdzj(z)"n 0T0 B0 (v 2 + µb0 )$T g*iky % Ck is collisional damping Hatch et al, PoP 2011

14 A spectrum valid in dissipa&on and iner&al ranges can be derived from damping measured in gyrokine&c ITG simula&ons Q k is negligible beyond k = 1 C k /E k is smallest at k = 1, slowly grows for larger k To obtain dissipa&on range spectrum fit damping rate for k > 1 to γ k = -γ 0 (k/k 0 ) δ Fit yields δ = 0.17 Use in γ k E(k) = dt k /dk with T k = v k3 k Spectrum: E(k) = Cε 2 / 3 k 5 / 3 exp 1 k 2 k d 1/ 2 ~ Cε2 / 3 k 5 / 3 (k ) Exponen&al dominates at low k (dissipa&on range) Power law dominates at high k (iner&al range) kρ

15 Comparison: high resolu&on, high k GENE simula&ons are consistent with asympto&c behavior of theore&cal spectrum Gyrokine&c simula&ons saturate without need for high resolu&on Reason: large frac&on of energy damped at low k by damped modes Higher resolu&on does not change transport rates, but reveals cascade of residual energy to high k Spectrum is a power law From E v (k) ~ Cε 2 / 3 k 11/ 3 exp 1 2 k d k with E v = v 2 /k; v = kφ; n = φ 1/ 2 E n (k) = E v(k) k 2 E n (k) ~ cε 2 / 3 k 11/ 3 exp 1 k 2 k d 1/ 2 ~ k 11/ 3 for k > k d Compare with red ITG spectrum for E n Görler, Jenko, PoP 2008

16 Conclusions Cascades with exponen&al energy decay occur in plasmas Spectra formed from product of power law and exponen&al, valid for all wavenumbers: E k (k) = k -α exp[-b(k/k d ) β ] Power law dominates in iner&al scales, exponen&al dominates in dissipa&ve scales Spectra obtained when dissipa&on and energy transfer have scaling Stronger nonlinear energy transfer => steeper power law, more shallow exponen&al decay Stronger damping => steeper exponen&al decay Spectrum applies to diverse situa&ons, including MHD and tokamak microturbulence Qualita&ve agreement with experiments, simula&on Certain cases access ranges where exponen&al is dominant

17 Future Look for Pm > 1 dissipa&on range in Madison Plasma Dynamo Experiment Probe connec&on between exponen&al spectrum and ion hea&ng in MST Understand energy distribu&on in damped mode space for plasma microturbulence