On the Comparison of Magnetofluid Turbulence in Laboratory and Astrophysical Plasmas

P.W. TerryUniversity of Wisconsin-Madison

Ackn: Weixing Ding, Lionello Marelli, John Sarff, and MST group

There are issues which experiments could help clarifyRelating present measurements to astrophysical plasmas difficultRelevant measurements can be done Improvements in diagnostic sensitivity Specialized analysis techniques Appropriate experimental design (scans, parameters, diagnostic)

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 sourceB0 strength

Laboratory and astrophysical plasmas can have very different parameter values

ICM ISMwm ion

Acrtn

Disk

Solar

Corona

Solar

WindMST MRX SSPX

1 - 20 ~ 1 1 -

102

10-4 -

102

~ 1 0.1 0.1

S 1027 < 109 1012 -

1015

105 -

106

102 -

103

104

> 3 ~ 3 ~ 1 10-2 ~ 5 10-2

few 10-2

% Ion-

ization99% Vari-

able

Vari-

able100% 100% 100% 100% 100%

€

˜ b B0

astrophysical plasmas laboratory plasmas

Nature of plasma confinement affects fluctuation properties

Laboratory plasmas: Plasma confined by external magnetic field • Low

• 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 potentialfluctuation spectrum 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)

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

Turbulent Decorrelation Controversy: Does mean or large scale B field affect decorrelation in magnetic 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

Basic Issues in Astrophysical Turbulence

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

t = 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

t = vkk

k||-1

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:

€

∂∂t

+ B0k|| + bk0

k0⊥ + bkk⊥

zero under anisotropy

Large scale turbulent field

Small scale fluct prop along it not eliminated by anisotropy because it has components to B0

small scale turb field

Fernandez and Terry, PoP ‘97

Turbulent decorrelation governs spectrum falloff

Balance of energy transfer rate and energy input rate:

If turbulent decorrelation governed by fluid straining (t = vkk = bkk)

• No dependence on large scale b-field• Kolmogorov spectrum• nk

2/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

Energy input rate Turbulent decorrelation rate

€

Em(k) ≡ bk

2

k =

B01 2ε1 2

k 3 2

€

ε = k2bk

4

ωt

€

Em(k) = bk

2

k =

ε1 3

k5 3

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 = ε2/3k-5/3)

Cascade from large k| | stops where B0k| | = bk , spectrum peaks at that k| |€

k|| = bk⊥

B0

= ε1 3k2 3

B0

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 boundaryOnly 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

Current Laboratory Plasma Turbulence Results

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 ~ B0

1/2k-3/2

• Problem: What part of dependence from decorrelation, what from tearing mode drive?

Time [ms]

Br

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~-1, diamagnetic freq. in decorr., alignment, etc.?)

Intermediate scales probably inertial, but carry imprint of tearing instability

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-flucts likely related to measured small scale electrostatic fluctuations

80

60

40

20

0

P(f) [Gs

2

/kHz]

806040200f [kHz]

standard 400ka ppcd 400ka

magnetic turbulence

Tearing Modes

Large scale anisotropy is dominated by geometry and tearing instability

• k| | is fixed by B0, geometry, and fluctuation extent

For RFP, B0 lies on torus; k: n=kR, m=–kr

• On resonant torus (m=nBr/BR), 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 anisotropyin astrophysical magnetic turbulence?

–10–50510024681012k||, m–1×10−6=0m=1m=15n=6n

˜ B r2(k||)Bo

2

€

k|| = Bφ (r)kφ + Bθ (r)kθ

B2

Rr

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 i - track transition wavenumber Measure partitions (v, b, n ) as function of wavenumber

3) What is origin of br b b? 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?

Ideas for laboratory studies

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 br as function of n and m for various radii

Calculate k| |(m,n,r) from equilibrium field profiles

Construct

€

k|| = dr k||(m,n,r) ˜ b r∫

dr ˜ b r∫

Making turbulent decorrelation measurements 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 t 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

€

s(n1,n2 ,n3) = b*(n1)b*(n2 )b(n3)

b(n1)2

b(n2 )2

b(n3)2

[ ]1 2

€

t (n1) = −n2

R s(n1,n2 ,n3)

b(n2 )2

b(n3)

b(n1)

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

1 2

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)