Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France
Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France
description
Transcript of Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France
![Page 1: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/1.jpg)
Steven A. Balbus
Ecole Normale SupérieurePhysics Department
Paris, France
The Effects of Magnetic Prandtl NumberOn MHD Turbulence
![Page 2: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/2.jpg)
(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.
![Page 3: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/3.jpg)
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:
![Page 4: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/4.jpg)
• 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.
![Page 5: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/5.jpg)
Iskakov et al., PRL, 98, 208501 (2007)
5123, white noise, nonhelical forcing in a box
![Page 6: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/6.jpg)
Pr = 1, Re=Rm=440 Pr = 0.07, Re=430, Rm=6200
Magnetic Field Structure (Iskakov et al.):
![Page 7: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/7.jpg)
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.
![Page 8: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/8.jpg)
16
84
21
evolutionary history of <B>=0 runs, Rm=12500, Pm as shown. (Fromang et al. 2007).
![Page 9: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/9.jpg)
Pm Effect for <B> .ne. 0:
(Lesur & Longaretti 2007 arXive 0704.29431v1)
![Page 10: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/10.jpg)
B2
Pm
Schematic Behavior of Fluctuations with Pm
+
-
![Page 11: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/11.jpg)
B2
Pm
Schematic Behavior of Fluctuations with Pm
+
-
computational regime
![Page 12: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/12.jpg)
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:
![Page 13: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/13.jpg)
B fields in the process of reconnection(Balbus & Hawley 1998)
![Page 14: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/14.jpg)
Associated velocity fields:
![Page 15: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/15.jpg)
Associated velocity fields:
Viscous stress in the resistive layer is large.
![Page 16: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/16.jpg)
Are there astrophysical flows that have
Pm << 1, Pm ~ 1, Pm >> 1 ?
![Page 17: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/17.jpg)
Are there astrophysical flows that have
Pm << 1, Pm ~ 1, Pm >> 1 ?
YES.
![Page 18: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/18.jpg)
Are there astrophysical flows that have
Pm << 1, Pm ~ 1, Pm >> 1 ?
YES.Compact X-ray sources.
![Page 19: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/19.jpg)
We are motivated to find Pm dependence in alpha models.
Balbus & Henri 2007 based on Frank, King, & Raine:
Behavior of Pm in models:
![Page 20: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/20.jpg)
We are motivated to find Pm dependence in alpha models.
Balbus & Henri 2007 based on Frank, King, & Raine:
Behavior of Pm in models:
![Page 21: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/21.jpg)
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).
![Page 22: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/22.jpg)
M=10 Msol
Mdot=.01 EddRcr =22 RS
Pm=10 Pm=1
500
![Page 23: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/23.jpg)
Pm transition at
M=10Msolar
Mdot =0.1 EddR=60RS
![Page 24: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/24.jpg)
M=108 Msol
Mdot=.01 EddRcr =10 RS
Pm=1
![Page 25: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/25.jpg)
Pm transition atM=108 Msolar
Mdot =0.1 EddR=34RS
![Page 26: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/26.jpg)
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.
![Page 27: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/27.jpg)
MRI Dispersion Relation:
This evidence is rather circumstantial,
and circumstantial evidence can be,
well, misleading…
![Page 28: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/28.jpg)
Can matters be examined more carefully?
![Page 29: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/29.jpg)
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:
![Page 30: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/30.jpg)
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)
![Page 31: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/31.jpg)
B2
Pm
Schematic Behavior of Fluctuations with Pm
+
-
stable
stableunstable
![Page 32: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/32.jpg)
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?
![Page 33: Steven A. Balbus Ecole Normale Supérieure Physics Department Paris, France](https://reader033.fdocuments.in/reader033/viewer/2022052401/56814748550346895db486ac/html5/thumbnails/33.jpg)
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?