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

2. (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.

3. 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)

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= Re=104
…appears not to hold for MHD turbulence.

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

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

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

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

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

10. Schematic Behavior of Fluctuations with Pm2  B + - Pm

11. Schematic Behavior of Fluctuations with Pm2  B + -
computational regime Pm

12. 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?

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

14. Associated velocity fields:

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

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

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

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

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

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

21. 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).

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

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

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

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

26. 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.

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

28. Can matters be examined more carefully?

29. 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?

30. 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.

31. Schematic Behavior of Fluctuations with Pm2  B +
stable
unstable -
stable Pm

32. 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?

33. 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 ( 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?