Weak and Strong regimes of imbalanced MHD Turbulence

1. Weak and Strong regimes of imbalanced MHD Turbulence Giga Gogoberidze Centre for Plasma Astophysics, K.U.Leuven Ilia State Univrsity, Georgia

2. 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)

3. l=kac Timescales 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 +ik)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  ~ Ēkk3/cas. • For Kolmogorov turbulence • ac ~ cas ~ 1/kvl

4. VA v ~ 1/k B0 B0 VA  ~ 1/k|| Alfven 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 nonlinearinteractions. Then the autocorrelation timescale becomes ac ~1/kVA Distortion of a wave packet during ac is v(ac)/ v ~ ac vk=v/ VA<<1 Because 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<<1and N >>1 it is sometimes stated that IK model describes weak turbulence EkIK=(VA)1/2k-3/2

5. Weak 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

6. WTT – resonant conditions • Nonlinear 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).

7. WTT for MHD turbulence We rederived WTT equations in the framework Of the Weak Coupling Approximation (Kadomtsev 1964). The method in mainly equivalent to DIA (Kraichnan 1959). k2 General validity criteria for WTT k = k1 + k2 1. k/k= 1/ack <<1 k Similar condition was derived by Ottaviani & Krommes (1996) for drift wave turbulence. 2. kĒ(k||,k)/kĒ(0,k)<<1 cas ~ac In WTT k1 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 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. Gogoberidze, Mahajan & Poedts, PoP (2009)

8. W- B0 Imbalanced MHD Turbulence W+ Recently 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 field Boldyrev (2006) proposed ‘scale dependent dynamic Alignment’ effect. Alternatively, Gogoberidze, PoP (2007) Proposed that observed weakening of the cascade Is related to the decorrelation caused by low frequency Modes. Perez & Boldyrev (2009) model predicts From Mueller et al (2003)

9. Ek E+ W- E- B0 k Imbalanced MHD Turbulence W+ Lithwick, 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).

10. Energy imbalance vs flux imbalance Spectral 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 . Beresnyak & Lazarian (2010) have noticed that the ratio of energy injection/dissipation rates is much more robust quantity then spectral indices, and therefore can be used to differentiate among various models of strong imbalanced MHD turbulence. PB09 LGS07 C08 & BL08 Dynamic Alignment Coherent Incoherent

11. Results For ‘incoherent’ models we derived Gogoberidze, Poedts and Akhalkatsi, ApJL, submitted (2010) ‘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.

12. Conclusions • Weak 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.

13. Thank You!