1. Modeling Astrophysical Turbulence
9/14/ :37 PM
Modeling Astrophysical Turbulence
Jonathan Carroll-Nellenback
Center for Integrated Research Computing
University of Rochester
Turbulence Workshop
August 4th 2015
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries.
The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

2. 9/14/ :37 PM
Talk Outline
Introduction to Turbulence in the context of gaseous flows
Supersonic Turbulence
Outflow driven turbulence
Turbulent support of Molecular Clouds
Modeling Molecular Clouds with AstroBEAR
© 2007 Microsoft Corporation. All rights reserved. Microsoft, Windows, Windows Vista and other product names are or may be registered trademarks and/or trademarks in the U.S. and/or other countries.
The information herein is for informational purposes only and represents the current view of Microsoft Corporation as of the date of this presentation. Because Microsoft must respond to changing market conditions, it should not be interpreted to be a commitment on the part of Microsoft, and Microsoft cannot guarantee the accuracy of any information provided after the date of this presentation. MICROSOFT MAKES NO WARRANTIES, EXPRESS, IMPLIED OR STATUTORY, AS TO THE INFORMATION IN THIS PRESENTATION.

3. Introduction to Turbulence
The Navier-Stokes equation defines the motion of fluids
Viscosity
(Friction/Dissipation)
Convective Acceleration (Non-linear)
Pressure
(Resistance to Compression)
The relative size of various terms in the equation can be used to characterize the fluid motion.
The lower the viscosity, the larger the Reynolds number and the more turbulent the flow becomes.

4. Transition to turbulence occurs around Re=103
Astrophysical flows generally have very large Reynolds numbers,

5. Introduction to Turbulence
The Navier-Stokes equation defines the motion of fluids
Viscosity
(Friction/Dissipation)
Convective Acceleration (Non-linear)
Pressure
(Resistance to Compression)
The relative size of various terms in the equation can be used to characterize the fluid motion.
The lower the viscosity, the larger the Reynolds number and the more turbulent the flow becomes.
The higher the mach number, the less effective pressure forces are at resisting compression resulting in strong shocks and large jumps in density and temperature.

6. Astrophysical flows often have large Mach numbers.
Incompressible
Compressible
Mach number = 10
Astrophysical flows often have large Mach numbers.

7. Introduction to Turbulence
For incompresible flows, the conservation of mass implies that the velocity field is solenoidal.
This permits a simplified evolution equation for the vorticity
Taking the curl of the Navier Stokes equation (and dividing by the density) gives the vorticity equation:
Viscosity
(Friction/Dissipation)
Advection of vorticity
Vortex stretching

8. Introduction to Turbulence
As you stretch a vortex, you must have compression in the plane of the vortex so the vortex shrinks in diameter.
As the diameter of the vortex shrinks, the vorticity must increase to conserve angular momentum.
Note in 2D, there is no vortex stretching!
“Big whorls have little whorls, which feed on their velocity, And little whorls have lesser whorls, and so on to viscosity.”
- Lewis Richardson -

9. Introduction to Turbulence1 2 3
Energy Spectrum
Large Scales Small Scales
Size of initial eddies
Energy
Turbulence involves motions on a range of scales.
One way to characterize turbulence is by looking at the velocity in Fourier space.
We can then compute the energy as a function of scale.
Note that k

10. Introduction to Turbulence1 2 3
Energy Spectrum
Large Scales Small Scales
Driving Scale
Energy
In 3D, vortex stretching leads to a cascade in energy from large scales to small scales
Simulation of mach 6 turbulence computed on a grid

11. Introduction to Turbulence
Between the driving scale and the dissipation scale, lies the inertial scale where the cascade becomes scale free and the spectrum obeys a power law. For incompressible turbulence, the spectral index is -5/3 (Kolmogorov 1941)
Eddy turn-over time
Inertial Range
Dissipation Scale
Driving Scale
Small k Large k

12. Introduction to Turbulence
Between the driving scale and the dissipation scale lies the inertial scale where the cascade becomes scale free and the spectrum obeys a power law.
For incompressible turbulence, the spectral index is experimentally -5/3
However for compressible turbulence, the velocity spectrum steepens to a spectral index of -2.
Inertial Range
Dissipation Scale
Driving Scale
Small k Large k

13. Compressive vs. Solenoidal Turbulence
Astrophysical turbulence is often highly supersonic and compressible.
Studies of the velocity spectra for supersonic turbulence indicate a steeper spectral index of -2, however the meaningfulness of this is complicated by the fact that a network of overlapping shocks is expected to give the same spectral index.
Energy cascade arguments have focused on the spectra of ρ1/3v which seem to follow the Kolmogorov scaling (at least for solenoidal forcing).
Federrath 2013 demonstrated that the spectral index for ρ1/3v depends on the degree of compressive motions, which depends on the nature of the driving (solenoidal vs. compressive).
For purely compressive driving, the spectral index for ρ1/3v is closer to -2.1 consistent with Galtier & Banerjee’s model that predicts a spectral index for for ρ1/3v of -19/9 for highly compressive turbulence.

14. Feedback Stars do not usually form in isolation
Dense regions (like NGC 1333) are the primary sight of star formation.
As these new stars form, some of the gravitational energy is released back into the surrounding material in the form of powerful outflows, jets, and winds.

15. Outflow Driven Turbulenceρ0 P
What will the velocity spectra look like?
Matzner 2007

16. Velocity Spectra
Outflow Scale

17. Outflow Driven Turbulence
Energy is present at large scales , but why does this energy not cascade as in isotropically forced case?
Turbulence energy deposited at outflow scale should cascade in eddy turnover time:
Eddy turnover time is the same as the outflow time scale which is the length of time outflows will grow unimpeded. It is also the length of time a fixed point in space will experience before being overrun by the next outflow
Eddies are swept up before they have time to break up.
Outflows suppress cascade of energy from larger scales!

18. Feedback within an isotropic cascade
What about outflows/feedback within an isotropically driven turbulent cascade?
Cascade
Both
Outflows

19. Feedback within an isotropic cascade
Above the outflow scale the energy spectra is dominated by the cascade.
Below the outflow scale the energy spectra is dominated by outflow driving.
The density spectra is much flatter when outflows are present.
Outflows disrupt large scale density structures.

20. The Formation of Molecular Clouds via colliding flows
D = 40 pc V = 8.25 km/s Res = 20483
It is thought that molecular clouds may form as the result of colliding streams of gas in the ISM.
This can lead to rapid cooling and collapse and explain the apparent simultaneity of cloud formation and star formation
This also provides an explanation for the low star formation efficiency of molecular clouds as a whole. Clouds need not be gravitationally bound.

21. Colliding Flows

22. Smooth
Clumpy

25. Magnetic Fields
&
Shear

26. AstroBEAR
Modelling astrophysical flows often requires capturing physics simultaenously over a wide range of scales. This requires an Adaptive Mesh Refinement (AMR)
AstroBEAR is a parallel AMR code that scales well out to 10’s of 1000’s of cores.

27. Questions?