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Physics of Alfvenic MHD Turbulence

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Physics of Alfvenic MHD Turbulence

Jungyeon Cho

Chungnam National Univ., Korea

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.”

+

= -

- Reynolds number: Re=VL/n (V2/L) / (nV/L2)
- V2/L nV/L2
- When Re << Recritical, flow = laminar
When Re >> Recritical, flow = turbulent

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!

E(k) ~ k-5/3

Interstellar gas

Orion nebula

Astrophysical fluids are turbulent and magnetized

(Re > 1010)

Doppler image

SOHO/MDI

Magnetic fluctuations in the solar wind

Ordinary MHD turbulence

Electron MHD turbulence

= small scale turbulence

3. Relativistic force-free MHD turbulence

<= extremely strong B field

To study magnetized turbulence,

we use the ordinary MHD equations:

Orion nebula

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)

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

VA: Alfven speed

FFMHD

643

Nothing happens.

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

This is something we call turbulence

What happens when two Alfvenic wave packets collide?

l||

l^

B0

VA

VA B0

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

b2/tcas = constant

b^l2

= const

(l^/b^l)

b^l2

= const

l^l||

tcas

=

b^lB0

- 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

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

B

Smaller eddies are more elongated

=> Relation between parallel size and perp size?

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

- 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

Scales smaller than rg

crust of neutron star

Protons=background; only electrons move

B

B

Protons => smooth background

Electrons carry current

=> J v

J v

+

0

v B

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

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

- 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

2883

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

This is only for illustration.

- Spectrum: E(k)~k-7/3
- Anisotropy: l|| ~l^1/3
- Critical balance: c ~ 1
- Theory & test: Cho & Lazarian (2004)

- 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!

Consider two wave packets:

l||

l^

B0

Vw

Vw =c =1

-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

Relativistic MHD ~ classical MHD !

Cho (2005)

Scale-dependent anisotropy

- ~ DE /E
~ tw/teddy

- Kolmogorov spectrum: E(k) ~ k-5/3
- Scale-dependent anisotropy: l|| ~ l^2/3
- Theory: Thompson & Blaes (1998)
Numerical test: Cho (2005)

- 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