Introduction to Magnetized Turbulence in Astrophysical Fluids Jungyeon

Introduction to Magnetized Turbulence in Astrophysical Fluids Jungyeon Cho (Chungnam National Univ. , Korea)

Plan Weak B 0 case MHD Strong B 0 case *Small-scale turbulence?

What is turbulence? Reynolds number: Re=VL/n § + =V 2/L § (V 2/L) / (n. V/L 2) n. V/L 2 When Re << Recritical, flow = laminar When Re >> Recritical, flow = turbulent

Example of turbulence: Turbulence created by an alligator Photo taken during the boat trip nwater ~ 0. 01 (cgs) V=50 cm/s & L=100 cm Re~5 x 105!

Onset of turbulence Re < 1 Re ~ 40 Re ~ 104 www-pgss. mcs. cmu. edu

Energy cascade -Da Vinci’s view Turbulence = S eddies! -Richardson (1920’s): concept of eddy and energy cascade Big whorls have little whorls / That feed on their velocity And little whorls have lesser whorls / And so on to viscosity. . . a Flea Hath smaller Fleas that on him prey, And these have smaller Fleas to bite 'em, And so proceed ad infinitum. [1733 Swift]

Kolmogorov theory: incompressible hydrodynamic turbulence Vl 2 = const Vl 3 tcas = const, Vl ~ l 1/3 l tcas= l/V Or, E(k)~k-5/3 l v

Measured spectrum (on the Earth) Energy injection dissipation Inertial range E(k) ~ k-5/3

Turbulence is everywhere! ( Re is huge!) Intracluster medium Interstellar medium The solar wind Turbulence! 3 C 465 -- Abell 2634

We also observe power-law spectra: e. g. ) Magnetic spectrum in the solar wind e. g. ) electron density spectrum in the ISM Slope ~ -5/3 Slope = -5/3 Spacecraft-frame frequency (Hz) Leamon+ (1999) pc AU Armstrong, Rickett & Spangler (1995)

Topic 1. Amplification of B fields in turbulence - How can MHD turbulence amplify B fields? Weak seed field (B 0)

Stretching of field lines B 0 t=0: Fluid elements and field lines move together *Back reactions are negligible if Emag<Ekin

Small-scale structures change faster

Expectations: Stretching on the dissipation scale will occur first because eddy turnover time is shortest there B E(k) k Exponential growth (Batchelor 50; Kazantsev 67; Zel’dovich+84; …)

Earlier simulations confirmed this Meneguzzi et al. 1981 (Resolution = 643)

Eturb(k) Expectations: E(k) What will happen when Eturb ~ Emag on the dissipation scale? Exponential growth stage will end! k Stretching scale gradually moves to larger scales. (see, for example, Cho & Vishniac 2000)

Efficiency of stretching Magnetic spectra Cho & Vishniac (2000 a) Dissipation scale

Cho & Vishniac (2000 a) Saturation is reached when B 2 ~ V 2 Schekochihin+(2007) later showed that the growth rate of the 2 nd stage is linear. B 2

Results of simulations linear exponential Cho, Vishniac, Beresnyak, Lazarian, Ryu (2009); * See also Cho & Vishniac (2000)

linear growth exponential growth Cho et al. (2009)

Conclusions for Topic 1 -Turbulence can amplify weak seed B fields -Two stages of amplification: exp. and linear E(k) B 2 time k

Plan -Weak B 0 case -Strong B 0 case *Small-scale turbulence?

Topic 2: Strong B 0 case Alfven wave Suppose that we perturb magnetic field lines. We will only consider Alfvenic perturbations. (restoring force=tension) We can make the wave packet move in 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 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^ B 0 VA VA =B 0

Goldreich & Sridhar (1995): In strong turbulence, 1 collision is enough to complete cascade!

1 collision is enough to complete cascade! -Distortion time scale ~ l^/vl -Duration of collision ~ l|| /B 0 tw/teddy ~ (l|| /B 0) /(l^/v) ~(b l|| / l^B 0) ~1

Energy Cascade l bl bl 2/tcas = constant

Goldreich-Sridhar model (1995) • Critical balance l^ bl l|| = B 0 bl 2 (l^/bl ) = const • Constancy of energy cascade rate bl 2 tcas = const bl ~ l^1/3 Or, E(k)~k-5/3 l|| ~l^2/3

Numerical test: Cho & Vishniac (2000 b) B -pseudo-spectral method -2563

Spectra: |B| B 0 See also Muller & Biskamp (2000); Maron & Goldreich (2001)

Anisotropy Smaller eddies are more elongated B => Relation between parallel size and perp size?

Anisotropy: Cho & Vishniac (2000) * Maron & Goldreich (2001) also obtained a similar result

Summary for strong B 0 case (i. e. Scaling relations for Alfvenic MHD turbulence) § § Theory: Goldreich & Sridhar (1995) Numerical test: Cho & Vishniac (2000) Maron & Goldreich (2001) Spectrum = Kolmogorov But, structures are anisotropic. *Recent issues: 1. Spectrum: Mueller+03; Boldyrev 05; Beresnyak & Lazarian 06; Mason+ 06; Gogoberidze 07; Matthaeus+08; Cho 10, … 2. Imbalance: Lithwick+ 07; Beresnyak & Lazarian 08; Chandran 08; Perez & Boldyrev 09; Podesta & Bhattacharjee 09, …

Actually strong turbulence is very common… (For simplicity, let’s suppose that driving is isotropic. ) b>>B 0 b<<B 0 (b l|| / l^B 0) =(bk^/k||B 0)~1

Critical balance may be a very common state in strongly magnetized plasmas… -Relativistic force-free MHD turbulence (in magnetospheres of BHs or NSs) * Thompson & Bleas (1999): theory Cho (2005): numerical test -EMHD model for small-scale MHD turbulence crust * Cho & Lazarian (2004, 2009) Neutron star

Conclusion for MHD turbulence (i. e. large-scale magnetized turbulence) -Turbulence can efficiently amplify weak seed fields -Alfvenic MHD turbulence : Kolmogorov spectrum + anisotropy

Small-scale turbulence: spectrum=? Leamon et al (1999) Spectrum of magnetic fluctuations in the solar wind

How can we describe small-scale physics? EMHD B B Protons smooth background Electrons carry current J v

Electron MHD eq J v + 0 v B

incompressible Ordinary MHD vs. EMHD turbulence -Studied since 1960’s -Goldreich & Sridhar 1995 E(k) k-5/3 k|| k^2/3 -Numerical test: Cho & Vishniac 2000 -Studied since 1990’s -Energy spectrum: E(k) k-7/3 (Vainshtein 1973; Biskamp-Drake 1990’s) -Anisotropy: k|| k^1/3 (Cho & Lazarian 2004)

Conclusion for small-scale turbulence -Small scale turbulence k-7/3 spectrum + stronger anisotropy

Small-scale turbulence: spectrum? E(k) k-5/3 k-7/3 Alfvenic turbulence ~ lmfp No more turbulence ~ri ~re k Biskamp’s group (1990’s), Cho & Lazarian (04, 09)