The Effects of Magnetic Prandtl Number On MHD Turbulence Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France

(Accretion) Flows May Be Classified into Three Regimes: rgy << Lglobal << mfp : Collisionless Regime. rgy << mfp << Lglobal : Dilute mfp << rgy << Lglobal : Collisional The collisionless regime requires a kinetic approach; the dilute regime requires transport to follow B; the collisional regime is the standard for stars and disks.

Two collisional subregimes of interest: Ratio of kinematic viscosity to resistivity is called “Magnetic Prandtl Number.” Pm = /. Pm = (T/4.2 X 104)4 (1014/n) (Spitzer value.) Pm>>1: ISM (1014), ICM (1029), Solar Wind (1021) (all dilute!) Pm <<1: Liquid Metals (10-6), Stars (10-3), Accretion Disks (10-4)

WHY SHOULD WE CARE? Because MHD turbulence seems to care a lot. The Kolmogorov picture of hydrodynamical turbulence (large scales insensitive to small scale dissipation) … Re=1011 Re=104 …appears not to hold for MHD turbulence.

Iskakov et al., PRL, 98, 208501 (2007) 5123, white noise, nonhelical forcing in a box

Magnetic Field Structure (Iskakov et al.): Pr = 1, Re=Rm=440 Pr = 0.07, Re=430, Rm=6200

MRI SIMULATIONS w/ VARYING Pm: (Fromang et al. arXiv 0705.3622v1 24/5/07) with no accretion, is perfectly OK. Pm regimes of sustained MHD turbulence in shearing box.

16 8 4 1 2  evolutionary history of <B>=0 runs, Rm=12500, Pm as shown. (Fromang et al. 2007).

Pm Effect for <B> .ne. 0: (Lesur & Longaretti 2007 arXive 0704.29431v1)

Schematic Behavior of Fluctuations with Pm 2  B + - Pm

Schematic Behavior of Fluctuations with Pm 2  B + - computational regime Pm

In Brief: MHD turbulence is sustained more easily, at higher levels, and with greater field coherence as Pm increases at fixed Re, for values of Pm ~1. Three independent groups have found this trend. Why should it be so?

B fields in the process of reconnection (Balbus & Hawley 1998)

Associated velocity fields:

Associated velocity fields: Viscous stress in the resistive layer is large.

Are there astrophysical flows that have Pm << 1, Pm ~ 1, Pm >> 1 ?

Are there astrophysical flows that have Pm << 1, Pm ~ 1, Pm >> 1 ? YES.

Are there astrophysical flows that have Pm << 1, Pm ~ 1, Pm >> 1 ? YES. Compact X-ray sources.

Behavior of Pm in  models: We are motivated to find Pm dependence in alpha models. Balbus & Henri 2007 based on Frank, King, & Raine:

Behavior of Pm in  models: We are motivated to find Pm dependence in alpha models. Balbus & Henri 2007 based on Frank, King, & Raine:

Behavior of Pm in  models: We are motivated to find Pm dependence in alpha models. Balbus & Henri 2007 based on Frank, King, & Raine: where Mdot = fEdd X Mdot (Eddington).

M=10 Msol Mdot=.01 Edd Rcr =22 RS 50 Pm=10 Pm=1

Pm transition at M=10Msolar Mdot =0.1 Edd R=60RS

M=108 Msol Mdot=.01 Edd Rcr =10 RS Pm=1

Pm transition at M=108 Msolar Mdot =0.1 Edd R=34RS

Stability of Pm=1 Transition MRI Dispersion Relation: Stability of Pm=1 Transition At the Pm=1 transition, a little extra heating goes a long way: Pm~T5 at constant pressure. A little heating causes a lot of Pm. Growing Pm causes higher turbulence fluctuation levels, yet more heating . . . Possible that the transition is rapid, even eruptive.

This evidence is rather circumstantial, MRI Dispersion Relation: This evidence is rather circumstantial, and circumstantial evidence can be, well, misleading…

Can matters be examined more carefully?

An analogue nonlinear system: 1. Linear growth independent of temperature. Non-linear saturation A(Pm) dependent on T. 3. Non-linear heating ~y2, cooling unspecified function of T. What are the stability properties of the saturated states?

Steady State: Linearize about (y0, T0), seek solutions of the form est . Then, a necessary condition for stability is: (Balbus &Lesaffre, 2007) C(T) is normally an increasing function of T. But A is a steeply decreasing function of T (Pm~T5) near the Pm=1 critical point. The transition need not be smooth and stable.

Schematic Behavior of Fluctuations with Pm 2  B + stable unstable - stable Pm

ASTROPHYSICAL IMPLICATIONS Pm transition changes accretion from resistive to viscous dissipation. a.) Preferential ion heating. b.) Little direct dissipation of electrical current. Critical to determine the different radiative properties of Pm >1 and Pm < 1 flows; relative dominance. Pm >1 transition flow poorly described by alpha disk theory. (Large thermal energy flux.) Related to state changes in compact X-ray sources?

SUMMARY Character of MHD turbulence is sensitive to Pm, at least in the regime Pm ~ 1. Larger Pm lead to higher turbulence levels. 2. Classical BH and NS accretion disks appear to have a radius at which Pm passes through unity (10-100 RS). Larger stars do not. Inner zone (Pm>1) and outer zone (Pm<1) likely to have different dynamical and thermal properties. 4. Nonlinear “dynamical systems” model suggests Pm transition is unstable. Regime accessible by numerical simulation. Relative dominance of Pm <1, Pm>1 zones and observational states?