Models of Turbulent Angular Momentum
Transport Beyond the  Parameterization
Martin Pessah
Institute for Advanced Study
C.K. Chan - ITC, Harvard
D. Psaltis - University. of Arizona
Workshop on Saturation and Transport Properties of MRI-Driven Turbulence - IAS- June 16, 2008
Timescales and Lengthscales in Accretion Disks
Hubble Time
1Gyr
Mean Field
Models
Inside
Horizon
1yr
1sec
Disk Spectrum
107Mo
BH
Growth
BLR
BH
Variability
Numerical
Simulations
0.1AU 1AU
1000AU
Standard Accretion Disk Model
Angular momentum transport in turbulent media
l
1  2
   (lv)  
(r T r )
t
r r
Shakura & Sunyaev (‘70s)
Angular momentum transport:
Turbulence and magnetic fields

This prescription dictates the
structure of the accretion disk

T r
 d ln  
 P

 d ln r 
, P,T, H, vr , ...
MRI Disk
Standard Disk
MRI

Tr
T
Bz
H
Pmag
H
P 
Tr
P

T
The need to go beyond -viscosity
Issues…
*Dynamics: is not constant
*Causality: Information propagates across sonic points
(Stress reacts instantly to changes in the flow)
*Stability criterion: Standard model works always
(Stresses only for MRI-unstable disks)
*”Viscosity”: MRI does not behave like Newtonian viscosity
Long-Term Evolution of MRI
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
large
scales
small
scales
MRI drives MHD turbulence at a well defined scale
which depends only on the magnetic field and the shear
MRI Accretion Disks in Fourier Space
Mode interactions
“Stress
Power”
Parasitic Instabilities
small
scales
large
scales
Parker
MRI
Dissipation
k
(Talk by C.K. Chan later today)
The Closure Problem in MHD Turbulence
We can derive an equation for angular momentum conservation
BrB
l
l
1 1 2 22
 (lv)
   r  
vrr) 
(rv(R
 (lv)
[r
 M r )]
rT
t
r rr r
4

t





which involves 2nd order correlations…
We can derive equations for 2nd order correlations…
… but they involve 3rd order correlations…
We can derive equations for 3nd order correlations…
Need for a CLOSURE model beyond alpha!!
(Closure models with no mean fields; papers by Kato et al 90’s; Ogilvie 2003)
Stress Modeling with No Mean Fields
Kato & Yoshizawa, 1995
Pressure-strain tensor,
tends to isotropize the
turbulence
Stress Modeling with No Mean Fields
Ogilvie 2003
C1, C5 energy dissip.
C2 return to isotropy
in decaying turbulence
C3, C4 energy transfer
A Model for Turbulent MRI-driven Stresses
New correlation drives the growth
of Reynolds & Maxwell stresses
d ln 
q
d ln r

k ij   kmax
W ik  vijk
M 0  0 H0 v Az
W in the Turbulent State
W tensor
dynamically
important!
-Maxwell
-Wr
Reynolds
Calibration of MRI Saturation with Simulations
Pessah, Chan, & Psaltis, 2006a
The parameter 
controls the ratio
between Reynolds
& Maxwell stresses
=0.3
k   kmax
The model produces initial exponential growth and leads
to stresses with ratios in agreement with the saturated regime
Calibration of MRI Saturation with Simulations
Pessah, Chan, & Psaltis, 2006b
The parameter 
controls the level
at which the magnetic
energy saturates
M 0  0 H0 v Az
=11.3
Hawley et al ‘95
A physically motivated model for angular momentum transport
Keplerian disks that incorporates the MRI
A Local Model for Angular Momentum Transport in
Turbulent Magnetized Disks
Pessah, Chan, & Psaltis, 2006b
Pessah, Chan, & Psaltis, 2006b
Pessah, Chan, & Psaltis, 2008
Shakura & Sunyaev, 1973
R r  M r
Simulations by Hawley et al. 1995
Pessah, Chan, & Psaltis, 2006b
Simulations by Hawley et al. 1995
 d ln  
 P

 d ln r 

A model for MRI-driven angular
momentum transport in agreement
with numerical simulations
Scaling in MRI Simulations
Pessah, Chan, & Psaltis, 2007; Sano et al. 2004
Deeper
understanding
of scalings?
Non-zero Bz
(few % Beq)
Magnetic pressure
must be important
in setting H
 ~ Bz
Spectral hardening
(Blaes et al. 2006)
(Talk by S. Fromang later today)
Pm>>1
Pm>>1
Pm<<1
kmax
Pm<<1



 max
Pessah & Chan, 2008
Viscous, Resistive MRI
4 2

2


kmax
1 4 

 4 2
 max
1 4 

 4 2
(Various limits studied by Sano et al., Lesaffre &Balbus, Lesur & Longaretti, … )
Viscous, Resistive MRI: Stresses & Energy
Pm>>1
Pm<<1
Pm>>1
Pm<<1
* Lowest ratio magnetic-to-kinetic: Ideal MHD
* Linear MRI effective at low/high Pm as long as Re large
* Need higher Re to drive MHD turbulence at low Pm
(Lesur & Longaretti, 2007; Fromang et al., 2007)
Viscous, Resistive MRI Modes
Pm=1
v,B  v,B ( ,)
v(z,t) 
t
v 0e (cos  v sin kz,sin v sin kz)
t
b(z,t)  b 0e (cos  B cos kz,sin  B cos kz)
Viscous, Resistive MRI Modes
Ideal MHD
kh
Parasitic Instabilities sensitive
to viscosity and resistivity
(Goodman & Xu, 1994)
Viscous, Resistive MRI Modes
Pm>>1
Pessah & Chan, 2008
Pm<<1
Viscous, Resistive Parasitic Instabilities
1ow Pm
high Pm
Parasitic Instabilities seem to be weaker at high Pm
Hint for difficulties to drive MRI at low Pm?
Can the destruction of primary MRI modes by parasitic
instabilities explain (Re, Pm) dependencies?