The Source of Turbulence in
Astrophysical Disks:
An Ill-posed Problem.
Denis Richard, 
NASA Ames Research Center
UCSC-Ames Planet and Star Formation Meeting
March 29, 2007
Astrophysical Disks
Disks are ubiquitous in Astrophysics : 
* Planetary disks
* Circumstellar Disks (around young stars)
* Binary systems
* Active Galactic Nuclei (around black holes)
Therefore, understanding disks is fundamental to understand
planetary and stellar formation and evolution. 
Wide range of sizes :
From Saturn's rings ( ~ 107 km) to AGN disks (~ parsec = 3.1 10 13 km)
Wide variety of complex physical processes, 
one of them being the transport of angular momentum. 
Astrophysical Disks (Artist view)
Astrophysical Disks (HST)
Turbulence in Disks
* Accretion Disk : gas and “dust” falling inward, toward the central object.
* Thin disk = H/R << 1    →   Keplerian rotation : Ω ~ r -3/2 (no radial or 
vertical velocity in first approximation)
* To maintain stationary rotation, angular momentum needs to be 
transported outward.
* Need for an adequate transport mechanism.
* Molecular viscosity is too small. 
*  Turbulence :  instability mechanism ? Transport properties ?
* Early models : Shakura & Sunyaev 1973 :
* Turbulence most likely generated by differential rotation (shear flow).
* Ad hoc model for transport : turbulent viscosity  based on smallest constraints : 
 = α . Cs . H 
A Short History of Turbulence Models for Accretion Disks
* 1973 – 1991 : Shakura & Sunyaev era : 
- Source of turbulence unknown (shear flow assumption ?)
- Work on turbulence, in order to improve transport model.
- meanwhile, transport model : ν
 = α . Cs . H 
* 1991  :  Magneto Rotational Instability rediscovered
  Weak magnetic field coupled to shear flow gives rise to a linear instability.
  (Chandrasekhar 1960, Balbus & Hawley 1991)
* 1991 – present : MRI era :  
- Source of turbulence : MRI
- Turbulent model : ν
 = α . Cs . H 
Turbulent Viscosity Model
What is it ?
* Analytically : A description of turbulent transport as a diffusive mechanism.
* For numerical simulation : a type of basic subgrid model.
Where does it come from ?
* Built ad hoc (alpha-viscosity) : relevant length scale x relevant velocity.
* Measured experimentally (lab or numerical) : Reynold stress.
When should it be used ?
* When studying anything BUT turbulence as a fundamental physical process.
* When simulations do not have adequate resolution to describe the  whole range of 
scales of the flow, in which case the viscosity model has to be chosen as to describe 
only the subgrid scales. 
 From Instabilities to Turbulence
Two types of Instabilities :
* Linear : flow unstable to infinetesimal perturbations (super-critical transitions)
ex : thermal convection, Rayleigh/centrifugal instability in rotating flows.
* Non-Linear : flow unstable to finite amplitudes (sub-critical transitions)  
ex : plan shear flow, Differential rotation (?)
Analytical : Linear can be well treated (transition) / no general model for Non-Linear.
Numerical : Linear : generally low Reynolds, large scale flows = “lower” resolution
      Non-linear : generally high Reynolds, small scale flows = “high” resolution
Laboratory : In theory can study both types equally well.
In an Astrophysical context : Both types are equally difficult to study,
because what ultimately matters is the turbulent state, which will generally be at very 
high Reynolds, thus difficult to describe. 
Thus, while MRI has been around for more than 15 years in the disk community, there is 
no associated description for turbulent transport. An instability easy to describe 
does not mean that the induced turbulence is equally easy to quantify.  
Polemic ? What Polemic ?
* Little to no doubt that MRI is at work in most accretion disks.
* But : Is MRI the only turbulent mechanism relevant to Astrophysical disks ?
“Schools of thoughts” :
MRI  school
* MRI is necessary to power accretion disks
   (Only MRI can provide adequate angular momentum transport.) 
* MRI is sufficient to power all accretion disks 
                       (giving birth to such things as “Dead zone models”)
Instability X school
* Instability X is also relevant to Astrophysical Disks dynamics.
  (Differential Rotation, Plane Shear, Strato-rotational, Baroclinic,...)
No-school school
* Just tell me how turbulence acts in my disk, so that I can react, 
  coagulate, form a planet, evolve, etc... 
Arguments against Differential Rotation
Analytical : radial and azimuthal fluctuations can
                    not grow at the same time.
                    (Balbus, Hawley & Stone, 1996) 
Problem : this set of equations is linear,
                       thus irrelevant to non-linear instabilities
                       (~ Rayleigh criterion)
                        
Numerical : No-show in numerical simulations
                    (Balbus, Hawley & Stone, 1996) 
Problem : Reynolds numbers are “low”
                       (a 1024 3 grid can simulate a maximum  
                        Re of order 10,000)
                        
Experimental : H.Ji, M.Burin, E.Schartman & 
J.Goodman, 2006 (accompanied by comments by
S.Balbus) 
Can Differential Rotation lead to Turbulence in Disks ?
* High Reynolds Shear flow : Astrophysical Disks Re ~ 106 to 1020
* Early (1930's) laboratory experiments : 
         Couette-Taylor flow unstable at high Re.
  (Richard & Zahn, 1999) 
Data from Wendt (1933) 
and Taylor (1936)
Can Differential Rotation lead to Turbulence in Disks ?
* New experimental setup (Richard, 2001).
Conclusion : In a laboratory
experiment (a not-so-close analog to 
a disk), differential rotation can give 
rise to turbulence despite published
arguments.
Differential Rotation may lead 
to turbulence in Keplerian disks.
Princeton Experiment
(H.Ji, M.Burin, E.Schartman & J.Goodman, 2006)
(Comments  requested by meeting organizers...)
* Couette-Taylor flow.
* No flow-visualization, due to technical difficulties (private 
communication, M.Burin). Turbulent flows can not be positively 
differenciated from laminar flows. (As stated in the paper itself.)
This is why Ji et al never claim that the flows are stable. They 
merely discuss fluctuation levels.
* Boundary conditions : 
* Stubby aspect ratio with independently rotating rings to 
         compensate and avoid large scale circulation.
* clever setup but : 
- experimental calibration does not agree with 
                       numerical simulations of this setup 
                       (see Burin et al, 2006).
- calibration done “blind” (no visualization) and only close by the Rayleigh 
  boundary where the flow could be turbulent (priv. comm. M.Burin). Should 
  have been done in a  regime where there are no doubts that the flow is 
         laminar. Could have been returning the flow to a laminar regime after a sub-
  critical transition.
Princeton Experiment
(H.Ji, M.Burin, E.Schartman & J.Goodman, 2006)
(Comments  requested by meeting organizers...)
* Main argument for Astrophysical flows : 
* measured β < 6.2 10-6
*  then compares 
          α and β parameters 
          numerically, by deriving 
   a formulation of the α viscosity for the laboratory setup. Concludes that α and β 
   should have a similar value, thus that α ~ β < 6.2 10-6  is too small (α ~ 10 -3).
* BUT : this formulation for the α viscosity has meaning only in a thin keplerian disk 
  where Cs = Ω.H. It makes absolutely no sense for this experimental setup.
* What makes sense is to compare α and β value in an Astrophysical context.
  For α.Ω.H2 = β.Ω.R2, then β / α = (H/R)2, therefore for α ~ 10 -3, and 0.001< H < 0.1, 
β ~ 10 -9 - 10 -5, still provides adequate transport of angular momentum.
Conclusion
* The issue of differential rotation is still very much open.
* The debate about the origin of turbulence in disks should be a search to characterize 
the turbulent state of disks and to model transport properties. Today, it sometimes 
seems to be more an effort to eliminate non-fashionable instabilities. 
* Considering the complexity of disks, it would be naive to think that one process only 
participate in angular momentum process.
* Discouraging work on various instabilities is damaging for disk understanding. Not 
only angular momentum is transported. Chemistry, planet formation, etc. are affected by 
turbulence that may not ultimately be relevant for angular momentum transport.
* Should be acknowledged :
- The limitations of our tools : numerical, analytical, and experimental. 
   Being able to describe a process better/more easily does not make it more 
   relevant or important. 
- The lack of observational constraints on disks physics.
Real object
Analytical model
LES / DNS models
Laboratory model
...but quite possibly...Hopefully... ...while most often
         presented as :Tools

  An critical overview of the current state of research in turbulence in
astrophysical disks.

The Source of Turbulence in
Astrophysical Disks:
An Ill-posed Problem.
Denis Richard, 
NASA Ames Research Center
UCSC-Ames Planet and Star Formation Meeting
March 29, 2007
Astrophysical Disks
Disks are ubiquitous in Astrophysics : 
* Planetary disks
* Circumstellar Disks (around young stars)
* Binary systems
* Active Galactic Nuclei (around black holes)
Therefore, understanding disks is fundamental to understand
planetary and stellar formation and evolution. 
Wide range of sizes :
From Saturn's rings ( ~ 107 km) to AGN disks (~ parsec = 3.1 10 13 km)
Wide variety of complex physical processes, 
one of them being the transport of angular momentum. 
Astrophysical Disks (Artist view)
Astrophysical Disks (HST)
Turbulence in Disks
* Accretion Disk : gas and “dust” falling inward, toward the central object.
* Thin disk = H/R << 1    →   Keplerian rotation : Ω ~ r -3/2 (no radial or 
vertical velocity in first approximation)
* To maintain stationary rotation, angular momentum needs to be 
transported outward.
* Need for an adequate transport mechanism.
* Molecular viscosity is too small. 
*  Turbulence :  instability mechanism ? Transport properties ?
* Early models : Shakura & Sunyaev 1973 :
* Turbulence most likely generated by differential rotation (shear flow).
* Ad hoc model for transport : turbulent viscosity  based on smallest constraints : 
 = α . Cs . H 
A Short History of Turbulence Models for Accretion Disks
* 1973 – 1991 : Shakura & Sunyaev era : 
- Source of turbulence unknown (shear flow assumption ?)
- Work on turbulence, in order to improve transport model.
- meanwhile, transport model : ν
 = α . Cs . H 
* 1991  :  Magneto Rotational Instability rediscovered
  Weak magnetic field coupled to shear flow gives rise to a linear instability.
  (Chandrasekhar 1960, Balbus & Hawley 1991)
* 1991 – present : MRI era :  
- Source of turbulence : MRI
- Turbulent model : ν
 = α . Cs . H 
Turbulent Viscosity Model
What is it ?
* Analytically : A description of turbulent transport as a diffusive mechanism.
* For numerical simulation : a type of basic subgrid model.
Where does it come from ?
* Built ad hoc (alpha-viscosity) : relevant length scale x relevant velocity.
* Measured experimentally (lab or numerical) : Reynold stress.
When should it be used ?
* When studying anything BUT turbulence as a fundamental physical process.
* When simulations do not have adequate resolution to describe the  whole range of 
scales of the flow, in which case the viscosity model has to be chosen as to describe 
only the subgrid scales. 
 From Instabilities to Turbulence
Two types of Instabilities :
* Linear : flow unstable to infinetesimal perturbations (super-critical transitions)
ex : thermal convection, Rayleigh/centrifugal instability in rotating flows.
* Non-Linear : flow unstable to finite amplitudes (sub-critical transitions)  
ex : plan shear flow, Differential rotation (?)
Analytical : Linear can be well treated (transition) / no general model for Non-Linear.
Numerical : Linear : generally low Reynolds, large scale flows = “lower” resolution
      Non-linear : generally high Reynolds, small scale flows = “high” resolution
Laboratory : In theory can study both types equally well.
In an Astrophysical context : Both types are equally difficult to study,
because what ultimately matters is the turbulent state, which will generally be at very 
high Reynolds, thus difficult to describe. 
Thus, while MRI has been around for more than 15 years in the disk community, there is 
no associated description for turbulent transport. An instability easy to describe 
does not mean that the induced turbulence is equally easy to quantify.  
Polemic ? What Polemic ?
* Little to no doubt that MRI is at work in most accretion disks.
* But : Is MRI the only turbulent mechanism relevant to Astrophysical disks ?
“Schools of thoughts” :
MRI  school
* MRI is necessary to power accretion disks
   (Only MRI can provide adequate angular momentum transport.) 
* MRI is sufficient to power all accretion disks 
                       (giving birth to such things as “Dead zone models”)
Instability X school
* Instability X is also relevant to Astrophysical Disks dynamics.
  (Differential Rotation, Plane Shear, Strato-rotational, Baroclinic,...)
No-school school
* Just tell me how turbulence acts in my disk, so that I can react, 
  coagulate, form a planet, evolve, etc... 
Arguments against Differential Rotation
Analytical : radial and azimuthal fluctuations can
                    not grow at the same time.
                    (Balbus, Hawley & Stone, 1996) 
Problem : this set of equations is linear,
                       thus irrelevant to non-linear instabilities
                       (~ Rayleigh criterion)
                        
Numerical : No-show in numerical simulations
                    (Balbus, Hawley & Stone, 1996) 
Problem : Reynolds numbers are “low”
                       (a 1024 3 grid can simulate a maximum  
                        Re of order 10,000)
                        
Experimental : H.Ji, M.Burin, E.Schartman & 
J.Goodman, 2006 (accompanied by comments by
S.Balbus) 
Can Differential Rotation lead to Turbulence in Disks ?
* High Reynolds Shear flow : Astrophysical Disks Re ~ 106 to 1020
* Early (1930's) laboratory experiments : 
         Couette-Taylor flow unstable at high Re.
  (Richard & Zahn, 1999) 
Data from Wendt (1933) 
and Taylor (1936)
Can Differential Rotation lead to Turbulence in Disks ?
* New experimental setup (Richard, 2001).
Conclusion : In a laboratory
experiment (a not-so-close analog to 
a disk), differential rotation can give 
rise to turbulence despite published
arguments.
Differential Rotation may lead 
to turbulence in Keplerian disks.
Princeton Experiment
(H.Ji, M.Burin, E.Schartman & J.Goodman, 2006)
(Comments  requested by meeting organizers...)
* Couette-Taylor flow.
* No flow-visualization, due to technical difficulties (private 
communication, M.Burin). Turbulent flows can not be positively 
differenciated from laminar flows. (As stated in the paper itself.)
This is why Ji et al never claim that the flows are stable. They 
merely discuss fluctuation levels.
* Boundary conditions : 
* Stubby aspect ratio with independently rotating rings to 
         compensate and avoid large scale circulation.
* clever setup but : 
- experimental calibration does not agree with 
                       numerical simulations of this setup 
                       (see Burin et al, 2006).
- calibration done “blind” (no visualization) and only close by the Rayleigh 
  boundary where the flow could be turbulent (priv. comm. M.Burin). Should 
  have been done in a  regime where there are no doubts that the flow is 
         laminar. Could have been returning the flow to a laminar regime after a sub-
  critical transition.
Princeton Experiment
(H.Ji, M.Burin, E.Schartman & J.Goodman, 2006)
(Comments  requested by meeting organizers...)
* Main argument for Astrophysical flows : 
* measured β < 6.2 10-6
*  then compares 
          α and β parameters 
          numerically, by deriving 
   a formulation of the α viscosity for the laboratory setup. Concludes that α and β 
   should have a similar value, thus that α ~ β < 6.2 10-6  is too small (α ~ 10 -3).
* BUT : this formulation for the α viscosity has meaning only in a thin keplerian disk 
  where Cs = Ω.H. It makes absolutely no sense for this experimental setup.
* What makes sense is to compare α and β value in an Astrophysical context.
  For α.Ω.H2 = β.Ω.R2, then β / α = (H/R)2, therefore for α ~ 10 -3, and 0.001< H < 0.1, 
β ~ 10 -9 - 10 -5, still provides adequate transport of angular momentum.
Conclusion
* The issue of differential rotation is still very much open.
* The debate about the origin of turbulence in disks should be a search to characterize 
the turbulent state of disks and to model transport properties. Today, it sometimes 
seems to be more an effort to eliminate non-fashionable instabilities. 
* Considering the complexity of disks, it would be naive to think that one process only 
participate in angular momentum process.
* Discouraging work on various instabilities is damaging for disk understanding. Not 
only angular momentum is transported. Chemistry, planet formation, etc. are affected by 
turbulence that may not ultimately be relevant for angular momentum transport.
* Should be acknowledged :
- The limitations of our tools : numerical, analytical, and experimental. 
   Being able to describe a process better/more easily does not make it more 
   relevant or important. 
- The lack of observational constraints on disks physics.
Real object
Analytical model
LES / DNS models
Laboratory model
...but quite possibly...Hopefully... ...while most often
         presented as :Tools
