Turbulent Dynamo Stanislav Boldyrev (Wisconsin-Madison) Fausto Cattaneo (Chicago) Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas

Questions 1. Can turbulence amplify a weak magnetic field? 2.What are the scales and growth rates of magnetic fields? Kinematic Theory

MHD Equations   v + (v¢r)v = -rp + (r£B)£B +  v + f   B = r£(v£B) +  B P m = /  - magnetic Prandtl number R m =VL/  - magnetic Reynolds number V1V1 V2V2 B B Kinematic dynamo Re=VL/ - Reynolds number

Kinematic Turbulent Dynamo: Phenomenology   B = r£(v£B) +  B V(x, t) is given. Consider turbulent velocity field V(x,t) with the spectrum: K EKEK K0K0 K K -5/3  V / 1/3  » /  V / -2/3 smaller eddies rotate faster Magnetic field is most efficiently amplified by the smallest eddies in which it is frozen. The size of such eddies is defined by resistivity. » 1/K B

Kinematic Turbulent Dynamo: Phenomenology K EKEK K0K0 K K -5/3 KK KK E M (K) Role of resistivity  Small Prandtl number,Large Prandtl number, Dynamo growth rate: “smooth” velocity  V / “rough” velocity  V /   1/3  » 1/    » 1/  P M ´ /  ¿ 1 P M À 1

Phenomenology: Large Prandtl Number Dynamo Large Prandtl number P M = /  À 1 Interstellar and intergalactic media K EKEK K0K0 K K -5/3 KK E M (K) Magnetic lines are folded Schekochihin et al (2004)  B Cattaneo (1996)

Phenomenology: Small Prandtl Number Dynamo Small Prandtl number P M = /  ¿ 1 Stars, planets, accretion disks, liquid metal experiments KK0K0 K KK EKEK K -5/3 E M (K)  V » 1/3  / (R M ) -3/4  » /  V / 2/3  » 1/   / (R M ) 1/2 Dynamo growth rate: [S.B. & F. Cattaneo (2004)] Numerics: Haugen et al (2004)  RMRM

Kinematic Turbulent Dynamo: Theory   B = r£(v£B) +  B V(x, t) is a given turbulent field Two Major Questions: 1. What is the dynamo threshold, i.e., the critical magnetic Reynolds number R M, crit ? 2. What is the spatial structure of the growing magnetic field (characteristic scale, spectrum)? These questions cannot be answered from dimensional estimates!

Kinematic Turbulent Dynamo: Theory Dynamo is a net effect of magnetic line stretching and resistive reconnection. R M > R M, crit : stretching wins, dynamo R M < R M, crit : reconnection wins, no dynamo R M =R M, crit : stretching balances reconnection: B  When R M exceeds R M, crit only slightly, it takes many turnover times to amplify the field

Kinematic Turbulent Dynamo: Kazantsev Model No Dynamo Dynamo incompressibility homogeneity and isotropy

Kazantsev Model: Large Prandtl Number Large Prandtl number: P M = /  À 1 Kazantsev model predicts: 1. Dynamo is possible; 2. E M (K)/ K 3/2 Schekochihin et al (2004) Numerics agree with both results: If we know  (r, t), we know growth rate and spectrum of magnetic filed KEKEK K0K0 K K -5/3 KK E M (K)

Kazantsev Model: Small Prandtl Number P M = /  ¿ 1 KK0K0 K KK EKEK K -5/3 E M (K) Is turbulent dynamo possible? Batchelor (1950): “analogy of magnetic field and vorticity.” No Kraichnan & Nagarajan (1967): “analogy with vorticity does not work.” ? Vainshtein & Kichatinov (1986): Yes Direct numerical simulations: (2004): No  (2007): Yes resolution

Small Prandtl Number: Dynamo Is Possible K EKEK K0K0 K K -5/3 KK E M (K) KK0K0 K KK EKEK K -5/3 E M (K) Keep R M constant. Add small-scale eddies (increase Re). P M = /  À 1 P M = /  ¿ 1 Kazantsev model: dynamo action is always possible, but for rough velocity ( P M ¿1) the critical magnetic Reynolds number ( R M =LV/  ) is very large.

Kazantsev Model: Small Prandtl Number Theory S. B. & F. Cattaneo (2004) Simulations P. Mininni et al (2004) A. Iskakov et al (2007) R M, crit L/  May be crucial for laboratory dynamo, P M ¿ 1 Re P M À 1 P M ¿ 1 smooth velocity Kolmogorov (rough) velocity

Kinematic Dynamo with Helicity K EKEK K0K0 K K -5/3 KK It is natural to expect that turbulence can amplify magnetic field at K ¸ K 0 Can turbulence amplify magnetic field at K ¿ K 0 ? Yes, if velocity field has nonzero helicity “large-scale dynamo”

Dynamo with Helicity: Kazantsev Model energyhelicity magnetic energymagnetic helicity given need to find Equations for M(r, t) and F(r, t) were derived by Vainshtein and Kichatinov (1986) h=s v¢ (r £ v)d 3 x  0

Dynamo with Helicity: Kazantsev Model Two equations for magnetic energy and magnetic helicity can be written in the quantum-mechanical “spinor” form: h=s v¢ (r £ v)d 3 x  0 r r h=s v¢ (r £ v)d 3 x  0 S. B., F. Cattaneo & R. Rosner (2004)

Kazantsev model and  -model r  - model Kazantsev model, NO scale separation The  - model approaches the exact solution only at r  1 assumes scale separation

Conclusions 1.Main aspects of kinematic turbulent dynamo is relatively well understood both phenomenologically and analytically. 2. Dynamo always exists, but 3. Separation of small- and large-scale may be not a correct procedure. PMPM R M, crit 1