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Physics of Alfvenic MHD Turbulence. Jungyeon Cho. Chungnam National Univ., Korea. 들어 가기에 앞서 …. 1. Heisenberg 가 죽기 전, 신을 만나면 다음 두가지 질문을 하겠다고 했다 한다:. Why relativity and why turbulence?. 2. Feynman said …. “turbulence is the last great unsolved problem of classical physics.”. +. = -.

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Physics of Alfvenic MHD Turbulence

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Physics of Alfvenic MHD Turbulence

Jungyeon Cho

Chungnam National Univ., Korea


들어 가기에 앞서…

1.Heisenberg가 죽기 전, 신을 만나면 다음 두가지 질문을 하겠다고 했다 한다:

Why relativity and why turbulence?

2. Feynman said …

“turbulence is the last great unsolved problem of classical physics.”


+

= -

What is turbulence?

  • Reynolds number: Re=VL/n (V2/L) / (nV/L2)

  • V2/L nV/L2

  • When Re << Recritical, flow = laminar

    When Re >> Recritical, flow = turbulent


Example: wake behind a sphere

critical Re = 40~50

Re~15,000


Example: cylinder in water

nwater ~ 0.01

(cgs)

If v=10cm/sec & D=1 cm, Re~1000.

==> turbulence!


Measured spectrum

E(k) ~ k-5/3


Interstellar gas

Orion nebula

Astrophysical fluids are turbulent and magnetized

(Re > 1010)


Solar Surface

Doppler image

SOHO/MDI


Magnetic fluctuations in the solar wind


Plan

Ordinary MHD turbulence

Electron MHD turbulence

= small scale turbulence

3. Relativistic force-free MHD turbulence

<= extremely strong B field


Topic 1: ordinary MHD turbulence

To study magnetized turbulence,

we use the ordinary MHD equations:

Orion nebula


Topic 1. Ordinary MHD turbulence

magnetic field

Suppose that we perturb magnetic field lines.

We will only consider Alfvenic perturbations.

(restoring force=tension)

We can make the wave packet move to one direction.

(We need to specify velocity)


Dynamics of one wave packet

Suppose that this packet is moving to the right. What will happen?

VA: Alfven speed


One wave packet

FFMHD

643

Nothing happens.


Dynamics of two opposite-traveling wave packets

Now we have two colliding wave packets. What will happen?


Two wave packets

This is something we call turbulence


What happens?

What happens when two Alfvenic wave packets collide?

l||

l^

B0

VA

VA B0


What happens?

What happens when two Alfvenic wave packets collide?

energy~b2/2

l||

l^

B0

VA

VA =B0

*From now on, B = actually B/(4pr)1/2


NOTE:

=> db/dt ~ b2/l^

=> dE/dt ~ b3/l^

  • When they collide, a packet loses energy of

  • DE~(dE/dt)Dt~ (b3/ l^ )tcoll ~ (b3/ l^ )(l||/VA).

  • Therefore DE /E ~ (b3/ l^ )(l||/VA) / b2

  • = (b l|| / l^B0)

  • = (l||/B0)/( l^ /b )

  • = tw/teddy = c


  • c ~tw/teddy ~ (b l|| / l^B0) ~ DE /E

  • Suppose that c ~1 .

    e.g.) When B0~bl and l|| ~ l^ , we have c ~1.

=>1 collision is enough to complete cascade!


c~1

E(k)

c~1

k

  • c ~tw/teddy ~ (b l|| / l^B0) ~ DE /E

  • Goldreich & Sridhar (1995) found that, when c ~1 on a scale, c ~1 on all smaller scales.

    * c ~1 is called critical balance

    *This regime is called strong turbulence regime


l

b

Energy Cascade

b2/tcas = constant


b^l2

= const

(l^/b^l)

b^l2

= const

l^l||

tcas

=

b^lB0

Goldreich-Sridhar model (1995)

  • Critical balance

  • Constancy of energy cascade rate

b^~l^1/3

Or, E(k)~k-5/3

l|| ~l^2/3

back


Numerical test: Cho & Vishniac (2000)

B

-pseudo-spectral method

-2563


|B|

B0

Spectra: Cho & Vishniac (2000)

See also Muller & Biskamp (2000); Maron & Goldreich (2001)


B

Anisotropy

Smaller eddies are more elongated

=> Relation between parallel size and perp size?


Anisotropy:Cho & Vishniac (2000)

* Maron & Goldreich (2001) also obtained a similar result


Summary for ordinary MHD

  • Spectrum: E(k)~k-5/3

  • Anisotropy: l|| ~l^2/3

  • Theory: Goldreich-Sridhar (1995)

    Numerical test: Cho-Vishniac (2000)


So far, we have considered ordinary MHD

B0

Example of incompressible MHD simulations

What about small scales?

(earth magnetosphere, crust of neutron stars,

ADAFs, or any small scales)


B

What do I mean by small scales?

Scales smaller than rg

crust of neutron star

Protons=background; only electrons move


Topic 2: EMHD - Introduction

B

B

Protons => smooth background

Electrons carry current

=> J  v


J  v

+

0

Electron MHD eq

v B


Ordinary MHD vs. EMHD turbulence

incompressible

-Studied since 1960’s

-spectrum: Kolmogorov

-scale-dependent anisotropy

  • Studied since 1990’s

  • Energy spectrum: known

  • Biskamp-Drake group:

  • E(k)  k-7/3

  • -Anisotropy: not known


Scaling of EMHD turbulence

Consider two EMHD wave packets:

l||

l^

B0

Vw

Vw  kB0


b^l2

= const

(l^2/b^l)

b^l2

= const

l^2l^l||

tcas

=

b^lB0

Cho & Lazarian (2004)

  • Critical balance (teddy =tW)

  • Constancy of energy cascade rate

b^~l^2/3

Or, E(k)~k-7/3

l|| ~l^1/3

Cf. Ordinary MHD


Numerical Results: spectrum

2883

Biskamp & Drake’s group obtained a k-7/3 spectrum in late 90’s.


Illustration of anisotropy

This is only for illustration.


Numerical Results: anisotropy


Numerical Results: critical balance


Summary for EMHD

  • Spectrum: E(k)~k-7/3

  • Anisotropy: l|| ~l^1/3

  • Critical balance: c ~ 1

  • Theory & test: Cho & Lazarian (2004)


Topic3: Relativistic Force Free MHD

  • Force-free ( B2 >> rc2 => reE+B x J=0 )

    e.g) magnetospheres of NS, BH, …

  • Theory: Thompson & Blaes (1998)

* c=1, flat space-time

Conserved form!


Scaling of Relativistic FF-MHD turbulence

Consider two wave packets:

l||

l^

B0

Vw

Vw =c =1


Simulation

-5123

-MUSCL type scheme with HLL flux

-Constrained transport scheme for div B=0 (Toth 2000)


E(k)

k

4

6

t=0

t > 0


spectrum

anisotropy

Results:

Relativistic MHD ~ classical MHD !

Cho (2005)


Results: eddy shapes

Scale-dependent anisotropy


Results: critical balance

  • ~ DE /E

    ~ tw/teddy


Summary for Relativistic FFMHD

  • Kolmogorov spectrum: E(k) ~ k-5/3

  • Scale-dependent anisotropy: l|| ~ l^2/3

  • Theory: Thompson & Blaes (1998)

    Numerical test: Cho (2005)


Summary

  • We have considered 3 types of Alfvenic turbulence:

  • - ordinary MHD turbulence

  • - electron MHD turbulence

  • - relativistic force-free MHD turbulence

  • They all show anisotropy and critical balance


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