Spectrum of MHD turbulence

Stanislav Boldyrev

University of Chicago

Ref: astro-ph/0503053; ApJ 626, L37, 2005

(June 20, 2005)

2

Introduction: Kolmogorov turbulence

3/221

2

||21 ~ xxxVxV

213

||21 ~ xxxVxV

1xLv Re=Lv/η>>1

Reynolds number:

Random flow of incompressible fluid

η-viscosity

2x

If there is no intermittency, then:

3/522~~ kkVE kk

Kolmogorov spectrum [Kolmogorov 1941]

and

3

Kolmogorov energy cascade

kE

k kEnergy of an eddy of size is ;

it is transferred to a smaller-size eddy during time: k/1~

2/51~/~ kVV k

322 ~~ kVVE k

The energy flux, 3/5~/~ kEEJ k

3/522~~ kkVE kk

, is constant for the

Kolmogorov spectrum!

local energy flux

- “eddy turn-over” time.

fk

3/5k

4

MHD turbulence

dxBVE 22

V/~

No exact Kolmogorov relation. Phenomenology:

Energyis conserved, and cascades toward small scales.

•Is energy transfer time ?No, since dimensional arguments do not work!

•Need to investigate interaction of “eddies” in detail!

Non-dimensional parameter AVV / can enter the answer.

This is also the main problem in the theory of weak (wave) turbulence. (waves is plasmas, water, solid states, liquid helium, etc…)

[Kadomtsev, Zakharov, ... 1960’s]

5

Iroshnikov-Kraichnan spectrum

After interaction, shape of each packet changes, but energy does not.

wz

w z

6

Iroshnikov-Kraichnan spectrum

λ

λ

during one collision:

number of collisions required to deform packet considerably:

Constant energy flux:

[Iroshnikov (1963); Kraichnan (1965)]

7

Goldreich-Sridhar theoryAnisotropy of “eddies”

Lλ

Shear Alfvén wavesdominate the cascade:

B

┴ BCritical Balance

λL>>

[Goldreich & Sridhar (1995)]

8

Spectrum of MHD Turbulence in Numerics

[Müller & Biskamp, PRL 84 (2000) 475]

9

Goldreich-Sridhar Spectrum in Numerics

Cho & Vishniac, ApJ, 539, 273, 2000Cho, Lazarian & Vishniac, ApJ, 564, 291, 2002

10

Strong Magnetic Filed, NumericsContradictions with Goldreich-Sridhar model

[Maron & Goldreich, ApJ 554, 1175, 2001]

Iroshnikov-Kraichnan scaling

11

Strong Magnetic Filed, NumericsContradictions with Goldreich-Sridhar model

Scaling of field-parallel and field-perpendicular structure functions for different large-scale magnetic fields.

[Müller, Biskamp, GrappinPRE, 67, 066302, 2003]

Weak field, B→0: Goldreich-Sridhar (Kolmogorov)scaling

2VB 2VB

B-parallel scaling

B-perp scaling

Strong field, B>>ρV : Iroshnikov-Kraichnan scaling

2

12

New Model for MHD Turbulence

Depletion of nonlinear interaction:

Nonlinear interaction is depleted

Interaction time is increased

A

1 2

This balances terms and in the MHD equations, as in the Goldreich-Sridhar picture, however, the geometric meaning is different.

1 2

[S.B., ApJ, 626, L37,2005]

For perturbation cannot propagate along the B-line faster than V , therefore, correlation length along the line is

Analytic Introduction

13

New Model for MHD TurbulenceAnalytic Introduction

Nonlinear interaction is depleted

Interaction time is increased

Constant energy flux,

3/2~ N 3/2~ l

Goldreich-Sridhar scaling corresponds to α=0:

2/1~ N

“Iroshnikov-Kraichnan” scaling is reproduced for α=1:

2/1~ l

Explainsnumerically observed scalings for strong B-field !

[Maron & Goldreich, ApJ 554, 1175, 2001]

[Müller, Biskamp, Grappin PRE, 67, 066302, 2003]

[S.B., ApJ, 626, L37,2005]

14

New Model for MHD TurbulenceGeometric Meaning

2/1~ l

S.B. (2005) “eddy”:

4/3~ line displacement:

3/2~ l

Goldreich-Sridhar 1995 “eddy”:

~line displacement:

As the scale decreases, λ→0,

turns into filament

turns into current sheet

agrees with numerics!

15

New Model for MHD Turbulence

4/3~

4/1~

Depletion of nonlinearity

S.B. (2005) “eddy”:

line displacement:

Remarkably, we reproduced the reduction factor in the original formula:

θ

In our “eddy”, w and z are aligned within small angle . One can check that: θ

In our theory, this angle is:

2/1~ l

The theory is self-consistent.

λ

16

Summary and Discussions

3. The spectrum of MHD turbulence may be non-universal.Alternatively, it may always be E~K , but in case 1, resolution of numerical simulations is not large enough to observe it.

┴-3/2

2VB 1. Weak large-scale field:

~dissipative structures: filaments

energy spectrum: E(K)~K ┴-5/3

3/2~ l[Goldreich & Sridhar’ 95]

2/1~ l4/3~

2VB 2. Strong large-scale field:

dissipative structures: current sheets

energy spectrum: E(K)~K ┴-3/2

4/1~ scale-dependentdynamic alignment

17

Conclusions

• Theory is proposed that explains contradiction between Goldreich-Sridhar theory and numerical findings.

• In contrast with GS theory, we predict that turbulent eddies are three-dimensionally anisotropic, and that dissipative structures are current sheets.

• For strong large-scale magnetic field, the energy spectrum is E~K . It is quite possible that spectrum is always E~K , but for weak large-scale field, the resolution of numerical simulations is not large enough to observe it.

┴-3/2 -3/2

┴