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
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?
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?
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
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
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
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?
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
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 3 2 3 / 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
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
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
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
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
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
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
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
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