Steven A. Balbus

Ecole Normale SupérieurePhysics Department

Paris, France

The Effects of Magnetic Prandtl NumberOn MHD Turbulence

(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; thecollisional regime is the standard for stars and disks.

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)

Two collisional subregimes of interest:

• Because MHD turbulence seems to care a lot. The Kolmogorov picture of hydrodynamical turbulence (largescales insensitive to small scale dissipation) …

WHY SHOULD WE CARE?

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

Pr = 1, Re=Rm=440 Pr = 0.07, Re=430, Rm=6200

Magnetic Field Structure (Iskakov et al.):

with no accretion,is perfectly OK.

MRI SIMULATIONS w/ VARYING Pm:(Fromang et al. arXiv 0705.3622v1 24/5/07)

Pm regimes of sustained MHD turbulence in shearing box.

16

84

21

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)

B2

Pm

Schematic Behavior of Fluctuations with Pm

+

-

B2

Pm

Schematic Behavior of Fluctuations with Pm

+

-

computational regime

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?

In Brief:

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.

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:

Behavior of Pm in models:

where Mdot = fEdd X Mdot (Eddington).

M=10 Msol

Mdot=.01 EddRcr =22 RS

Pm=10 Pm=1

500

Pm transition at

M=10Msolar

Mdot =0.1 EddR=60RS

M=108 Msol

Mdot=.01 EddRcr =10 RS

Pm=1

Pm transition atM=108 Msolar

Mdot =0.1 EddR=34RS

MRI Dispersion Relation:

Stability of Pm=1 Transition

1. At the Pm=1 transition, a little extra heating goes a long way: Pm~T5 at constant pressure.

2. A little heating causes a lot of Pm. Growing Pm causes higher turbulence fluctuation levels, yet more heating . . .

3. Possible that the transition is rapid, even eruptive.

MRI Dispersion Relation:

This evidence is rather circumstantial,

and circumstantial evidence can be,

well, misleading…

Can matters be examined more carefully?

1. Linear growth independent of temperature.

2. 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?

An analogue nonlinear system:

Steady State:

Linearize about (y0, T0), seek solutions of the form est .Then, a necessary condition for stability is:

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 notbe smooth and stable.

(Balbus &Lesaffre, 2007)

B2

Pm

Schematic Behavior of Fluctuations with Pm

+

-

stable

stableunstable

ASTROPHYSICAL IMPLICATIONS

1. Pm transition changes accretion from resistive to viscousdissipation. a.) Preferential ion heating. b.) Little direct dissipation of electrical current.

2. Critical to determine the different radiative properties ofPm >1 and Pm < 1 flows; relative dominance.

3. Pm >1 transition flow poorly described by alpha disk theory. (Large thermal energy flux.)

4. Related to state changes in compact X-ray sources?

SUMMARY

1. Character of MHD turbulence is sensitive to Pm, at least in theregime 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). Largerstars do not. 3. Inner zone (Pm>1) and outer zone (Pm<1) likely to havedifferent dynamical and thermal properties.

4. Nonlinear “dynamical systems” model suggests Pm transitionis unstable.

5. Regime accessible by numerical simulation. Relativedominance of Pm <1, Pm>1 zones and observational states?