Weak and Strong regimes of imbalanced

MHD Turbulence

Giga Gogoberidze

Centre for Plasma Astophysics, K. U. Leuven

Ilia State Univrsity, Georgia

Why we study MHD turbulence Why we study MHD turbulence

Various astrophysical applications Solar wind (proton temperature at 1AU; multi

spacecraft measurements); ISM (DM of Pulsar radio emission); Accretion disks (magneto-rotational

instability); Cosmology – background gravity waves

(Gogoberidze et al, PRD (2008), Kahniashvili, Gogoberidze & Ratra PRL, (2008)).

Theoretical Physics: the simplest form of plasma turbulence.

Rich linear theory; Propagation effects; Kinetic effects; Anisotropy.

From Denskat et al. (1983)

Timescales of the turbulenceTimescales of the turbulence

Turbulent fields are random and vary in space and time. Consequently, any model of turbulence should describe spatial and temporal correlations.

Two point two time correlation function Rij(r,)=<Vi(x+r,t+) Vj(x,t)>

Qij(k,)=exp(-k +ik)ĒkPij

Autocorrelation timescale ac=1/k - “Duration of unit act of nonlinear

interaction”.

Cascade timescale cas - “Lifetime of a wave packet”.

For stationaty turbulence cas is

determined by spectrum ~ Ēk k3/cas.

For Kolmogorov turbulence

ac ~ cas ~ 1/kvl

l=ka

c

Alfven Effect and IK Model of MHDTAlfven Effect and IK Model of MHDT

Nonlinear interaction is possible only between oppositely propagating Alfven waves – Alfven effect.Assumptions of IK (Iroshnikov 1963; Kraichnan 1965) model:

(i) Strong external field VA >> v;(ii) Isotropy - k||~ k~k;(iii) Locality of nonlinear interactions.

Then the autocorrelation timescale becomes ac ~ 1/kVADistortion of a wave packet during ac is v(ac )/ v ~ ac vk=v/ VA<<1Because these distortions are summed up with random phases, N=(v/ VA)2 collisions are necessary to achieve the distortion of order

unity. Then cas ~ N ac, and

• Due to the conditions v/ VA<<1 and N >>1 it is sometimes stated that IK model describes weak turbulence

B0

~ 1/k||

~ 1/kv

VA

VA

B0

EkIK=(VA)1/2k-3/2

Weak Turbulence TheoryWeak Turbulence Theory

•WTT considers nonlinear interactions among waves in perturbation manner. The formalism is the same as in non-stationary perturbation theory of quantum mechanics;

•The closure is provided by random phase approximation which is almost equivalent to Gaussianity

WTT leads to kinetic equation for waves

Zakharov (1966) found stationary solutions of KE

WTT – resonant conditionsWTT – resonant conditionsNonlinear interaction is possible only among waves satisfying resonant conditions

If these conditions can not be satisfied then nonlinear interactions are dominated by 4 wave interactions

WTT advantage – sound and clear formalism;WTT disadvantage – rarely Realizable in the nature. Only Example where WTT

spectrum is Observed - surface gravity waves In the ocean

k = k1 + k2, k = k1 + k2

Solution of resonant conditions k||2=0, (k||1=k||)

and consequently in the WTT Cascade is purely Perpendicular. Zakharov transformation method yields

EkWTT=[f(k||)VA/k||]1/2k

-2

Galtier et al (2000).

WTT for MHD turbulenceWTT for MHD turbulenceWe rederived WTT equations in the framework Of the Weak Coupling Approximation

(Kadomtsev 1964). The method in mainly equivalent to DIA (Kraichnan 1959).

k

k = k1 + k2

k1

k2 General validity criteria for WTT

1. k/k= 1/ac k <<1

2. kĒ(k||,k)/kĒ(0,k)<<1

Similar condition was derived by Ottaviani & Krommes (1996) for drift wave turbulence.

Gogoberidze, Mahajan & Poedts, PoP (2009)

cas ~ ac

In WTT k~ (k/k)k<<k and L~ (k /k)>> . Although the interactions are weak, the wave packets are destroyed completely before they pass through each other.

In WTT

Consequently, any model of the turbulence which suggests big number of collisions among wave packets for effective energy transfer is not related to the WTT

Imbalanced MHD TurbulenceImbalanced MHD TurbulenceRecently several phenomenological models of MHD turbulence with nonzero cross helicity has been developed

Lithwick, Goldreich & Sridhar 2007; Beresnyak & Lazarian 2008; Chandran 2008; Perez & Boldyrev 2009; Podesta & Bhattacharjee 2009.

Trying to explain IK like spectrum observed in numerical simulation of MHD turbulence with strong background magnetic fieldBoldyrev (2006) proposed ‘scale dependent dynamicAlignment’ effect. Alternatively, Gogoberidze, PoP (2007)Proposed that observed weakening of the cascadeIs related to the decorrelation caused by low frequencyModes. Perez & Boldyrev (2009) model predicts

2/3~

kEk

B0

W-

W+

From Mueller et al (2003)

Imbalanced MHD TurbulenceImbalanced MHD TurbulenceLithwick, Goldreich & Sridhar (2007) assume

coherent straining imposed by subdominant waves on a dominant wave packet

Beresnyak & Lazarian (2008) and Chandran (2008) assume incoherent straining

Due to this degeneracy determination of the spectral slopes need extra assumption – Pinning effect (Grappin et al. 1983).

B0

W-

W+

W

cas

3/5~

kEk

kWW

2~

E+

E-

Ek

k

Energy imbalance vs flux imbalanceEnergy imbalance vs flux imbalanceSpectral indices are diffucult to determine from DNS Nonlocal interactions are much more strong in MHD then in hydro turbulence; In imbalanced case cascade of dominant component weakens and the time

necessary to reach stationary state increases. Currently available supercomputers can reach only .

2/1

0

0

W

W

32~/ Beresnyak & Lazarian (2010) have noticed that the ratio of energyinjection/dissipation rates is much more robust quantity thenspectral indices, and therefore can be used to differentiate amongvarious models of strong imbalanced MHD turbulence.

PB09 LGS07 C08 & BL08 Dynamic Alignment Coherent Incoherent

0

0

W

W ?

ResultsResultsFor ‘incoherent’ models we derived

‘Incoherent’ models predict weaker cascade compared to the DNS. Coherent model fits well the numerical results for weak imbalance, but overestimates cascade rate in strongly imbalanced case.

Gogoberidze, Poedts and Akhalkatsi, ApJL, submitted (2010)

20

2

00

03/1

03/4

3/20

3/4

20

40

4

,,1

1

)ln(

)ln(

W

WE

k

kK

EK

EK

EK

EK d

ConclusionsConclusionsWeak MHD turbulence In WTT cas ~ ac and therefore number of collisions between wave packets

before nonlinear destroy is always of order unity.

Strong imbalanced MHD turbulence None of the existing phenomenological models of imbalanced MHD

turbulence fits well the data of recent DNS.

Thank You!