On the Comparison of Magnetofluid Turbulence in Laboratory and Astrophysical Plasmas

1. On the Comparison of Magnetofluid Turbulence in Laboratory and Astrophysical Plasmas P.W. Terry University of Wisconsin-Madison Ackn: Weixing Ding, Lionello Marelli, John Sarff, and MST group There are issues which experiments could help clarify Relating present measurements to astrophysical plasmas difficult Relevant measurements can be done Improvements in diagnostic sensitivity Specialized analysis techniques Appropriate experimental design (scans, parameters, diagnostic)

2. Can lab experiments tell us about astrophysical b-turbulence? MST ISM • MHD: equilibrium, global scale fluctuations • MHD: model of choice • Evidence for inertial range (high freq) • Evidence for inertial range • Easier to probe • Harder to probe • Knobs available • What you see is what you get Low k driving source B0 strength

3. Laboratory and astrophysical plasmas can have very different parameter values   astrophysical plasmas laboratory plasmas 

4. Nature of plasma confinement affects fluctuation properties Laboratory plasmas: Plasma confined by external magnetic field • • Low b • B, J strong • Fusion plasmas: n, T, P strong • • Instabilities driven by inhomogeneities • Global scale fluctuation properties governed by instabilities • Sources, sinks on multiple scales Example: electrostatic potential fluctuationspectrum in tokamak So, what is possible basis for comparison? • RFP: one instability dominates  Inertial range can develop at small scales Small fluctuations reflect NL inertial force, not instability • Shear Alfvén waves as paradigm for interstellar turbulence • Isolate, study nonlinear forces (common to all types of mag turb)

5. Outline 1. Basic issues and controversies in astrophysical turbulence • Turbulent decorrelation • Turbulent spectrum • Fluctuation anisotropy • Not covered: transport, alignment, dissipation, driving 2. Current laboratory plasma results (drawn from MST experiment) • Decorrelation • Spectrum • Anisotropy • Driving Source 3. Proposed laboratory plasma turbulence studies • Studies for better understanding of existing results • Anisotropy • Decorrelation

6. Turbulent Decorrelation Controversy: Does mean or large scale B field affect decorrelation in magnetic turbulence? Basic Issues in Astrophysical Turbulence Turbulent decorrelation is fundamentally important • Mediates rate of spectral transfer  affects spectrum shape • Responsible for introducing wave-induced anistropy in cascade dynamics • Mediates cascade direction changes associated with symmetry breaking • Quantity where wave physics and turbulent motions interface • Directly affects transport rates Given its importance, it is noteworthy that it is not understood

7. Two views on turbulent decorrelation in magnetic turbulence 1. Alfvénic motions (along large scale B) decorrelate turbulence • Small scale fluctuations propagate as Alfvén waves along large scale B • Large scale B is big  fast propagation  decorrelation set by propagation speed along large scale B  wt = VAk||~ Bk|| 2. Eddy motions (perpendicular to B) decorrelate turbulence • Eddy turnover rate independent of B • Proportional to smaller flow vk at small scale k • Smaller than Alfvénic decorrelation rate, unless anisotropy develops with k|| reduced until eddy turnover governs decorrelation  wt = vkk k||-1

8. Conventional wisdom on turbulent decorrelation has problems CW: Isotropic turbulence  Alfvénic decorrelation Anisotropic turbulence  Fluid straining decorrelation Probs: 1) Equipartition of v and b requires Alfvénic motion Equipartition and no Alfvénic decorrelation are inconsistent 2) Geostrophic turbulence: Development of anisotropy requires dominance of wave rate over fluid straining rate, not reverse 3) Reduced MHD turbulence with maximal anisotropy (k|| = 0): Alfvénic decorrelation still dominates Origin of effect:  small scale turb field   zero under anisotropy Large scale turbulent field Small scale fluct prop along it not eliminated by anisotropy because it has components  to B0 Fernandez and Terry, PoP ‘97

9. Turbulent decorrelation governs spectrum falloff Balance of energy transfer rate and energy input rate: If turbulent decorrelation governed by fluid straining (wt = vkk = bkk) • No dependence on large scale b-field • Kolmogorov spectrum • nk2/k ~ k-5/3 (advected electron density) If turbulence decorrelation governed by Alfvénic time • Turb level depends on large scale field • Iroshnikov-Kraichnan Spectrum • nk2/k ~ k-7/4 • gentler slope  faster decorrelation Both indices reported in simulation literature Turbulent decorrelation rate  Energy input rate 

10. MHD turbulence is anisotropic, but what is its nature? •Universal criterion (many systems with anisotropic wave physics): Anisotropy set by balance of isotropic nonlinearity and anisotropic wave term B0k| | = bk (parallel scales coarsen until balance achieved) •Conventional interpretation: balance sets k| |, eddy aspect ratio (using Kolmogorov spectrum bk2/k = e2/3k-5/3)  Cascade from large k| | stops where B0k| | = bk , spectrum peaks at that k| |

11. Turbulence occupies available scales  conventional interpretation is too simple MHD similar to quasigeostrophic (Rossby-wave) turbulence Balance of wave term with nonlinearity defines k-space boundary (Rhines) • Separates regions where wave term important, unimportant • Turbulence populates scales on both sides of boundary Only seen in very long time numerical integration • Spectrum is modified to maintain balance Strong excitation of zonal modes (k| |=0) by anisotropic transfer Correct interpretation: Eddy aspect ratio set by where spectrum is sampled in k-space Eddy probability, mean wavenumbers set by spectrum shape Must know Em(k| |, k) in all regions of k-space Computation limited by resolution

12. Current Laboratory Plasma Turbulence Results

13. Mean field dependence in spectrum may indicate mean field dependence in decorrelation rate Decorrelation rate inferred from correlation time, spectrum • Single mode time history indicates correlation time Scan mean current to see mean field dependence • Dependence of spectrum on mean current Reminiscent of IK spectrum: Em ~ B01/2k-3/2 • Problem: What part of dependence from decorrelation, what from tearing mode drive? Br Time [ms]

14. Small scale spectrum has two decay subranges Measured by probes at edge and FIR polarimetry (Faraday rot) in core Large scales dominated by tearing mode drive Intermediate scales have power law consistent with k-3/2 or k-5/3 (higher J) Smallest scale subrange may have exponential falloff If this range has power law, steeper slope is not understood (e– dynamics at k~r-1, diamagnetic freq. in decorr., alignment, etc.?) Intermediate scales probably inertial, but carry imprint of tearing instability

15. Spectrum may have multiple driving sources • Large scale drive by trearing instability is well established • Small scales excited by cascade from large scales or by small scale instability ? • To probe, modify tearing drive with current gradient control (PPCD) • Decreased tearing drive  flatter spectrum in high frequencies -Above noise level -Slope consistent with ultraviolet catastrophe  independent small scale source • Nature of small scale source not understood • b-fluctslikely related to measured small scale electrostatic fluctuations

16. Large scaleanisotropy is dominated by geometry and tearing instability • k| | is fixed by B0, geometry, and fluctuation extent For RFP, B0 lies on torus; k: n=kfR, m=–kqr • On resonant torus (m=nBfr/BqR), k| | = 0 • Shear in B0: k| | increases from resonant surface • k| | limited by finite extent of fluctuation m, n • Magnetic fluctuation spectrum dominated by global scale tearing fluctuations  anisotropy set by shear and geometry Can RFP yield any useful information on anisotropy in astrophysical magnetic turbulence? q r R f

17. Ideas for laboratory studies Need to understand more about laboratory turbulence Knobs: Driving strength (PPCD to reduce tearing instability drive) Mean magnetic field strength (discharge current) Dissipation strength (plasma temperature) Q: 1) Is there an inertial range? (Key for validating comparisons) Scale transition of (NL force/linear force) under drive variation 2) What is origin of dual spectrum ranges? Vary dissipation - track transition wavenumber, falloff rate Vary ri - track transition wavenumber Measure partitions (v, b, n ) as function of wavenumber 3) What is origin of br bf  bq? Track changes through transition to inertial range Relate to spatial anisotropy Determine role of plasma boundary 4) What is origin of fluctuation differences between core and edge?

18. Anisotropy measurements of relevance to astrophysics • Determine if experiment has range in which anisotropy is independent of tearing instability Measure anisotropy for k in driving range, power law decay range • If transition observed, relate k| | to k and compare to critical balance hypothesis k| | ~ k2/3 • To measure k| |: Measure bras function of n and m for various radii Calculate k| |(m,n,r) from equilibrium field profiles Construct

19. Making turbulent decorrelationmeasurements of relevance to astrophysics Small scale decorrelation in time histories, spectrum affected by tearing Certain analysis techniques yield pure decorrelation rate: 1) From bispectrum , if statistics close to Gaussian; form appropriate for v b 2) Turbulent response function – Perturb plasma with source localized to small scale – Measure relaxation of b to steady state level – From ensemble, extract wt as exponent of decay – Method used in simulations Both techniques must be applied to inertial scales Both extract decorrelation rate free of driving and other effects

20. Conclusions There are issues which experiments could help clarify Relating current measurements to astrophysical plasmas difficult Relevant measurements could be done Improvements in diagnostic sensitivity Specialized analysis techniques Appropriate experimental design (scans, parameters, diagnostic)