1. Radial and Poloidal Dynamics of Turbulent Transport in a SOC Model with Sheared Flow
Ryan Woodard (Univ. of Alaska - Fairbanks)
David Newman (Univ. of Alaska - Fairbanks)
Raul Sánchez (Universidad Carlos III, Madrid, Spain) B. A. Carreras (Oak Ridge National Laboratory)
APS Division of Plasma Physics Meeting
Québec City, Canada
October

2. Motivation
SOC seems to be a useful paradigm to try to understand plasma anomalous transport.
PDFs of the local duration of turbulent transport events are experimentally measurable quantities.
Understanding PDF behavior in simpler SOC sandpile models can help reveal signatures of complex behavior in real experimental data.

3. PDF distortion due to shear flows. PDF distortion due to rotation.
Goals
To understand:
PDF distortion due to shear flows.
PDF distortion due to rotation.
PDF dependence on driving rate.
Effects caused by the presence of poloidal diffusion or parallel equilibration dynamics.

4. Motivation for SOC
Universality in transport dynamics--experiments show
Bohm scaling (scaling with machine size) while the fluctuation correlation lengths are much smaller.
Gradients often marginal or sub-marginal for modes thought to be important.
Transport scaling seems to change from Bohm to gyro-Bohm in enhanced confinement modes.
Self organized criticality (SOC) model
Transport involves system scale lengths without relying on the underlying mechanisms.
SOC profiles are linearly stable.
Simple paradigm allows investigation of the effect of sheared flow on scaling.
Cellular automaton sandpile model can be used to study SOC dynamics and normal transport with addition of shear

5. Sandpile
BTW proposed a cellular automaton model describing the flow of sand in an open box as the simplest model for self-organized criticality.
Under the action of simple dynamical rules, the model evolves into a stationary state that lacks characteristic time and space scales.
Such a system is in a critical state.
The dynamics satisfy a conservation law. r

6. Basic sandpile SOC model
Cellular automaton model for critical gradient transport dynamics
random rain of grains hn Zn Nf
Simple model gives rich dynamics
(model similar to P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev Lett., (1987) and T. Hwa and M. Kardar, Phys. Rev A, (1992))

7. Marginal vs. SOC Avalanches
For Nf > 1 unstable sites (avalanches) propagate up and down the slope
Slope is subcritical on average
For Nf=1 unstable site propagates down the slope
Slope remains marginal

8. SOC profile sub-marginal
SOC system leads to sub-marginal profile while maintaining active transport
SOC profile sub-marginal
Marginal
SOC r
All parameters the same except amount transported when site is unstable (Nf)
Sub-marginal profile is robust for all Nf > 1 

9. SOC avalanches
Propagation of individual avalanches in SOC case r
time

10. Analogy between sandpile dynamics and turbulent transport

11. System Geometry System is a periodic slab
periodic in  direction
open at r = L, closed at r = 0 r 
System simulates a poloidal cross section

12. What is an event?
Individual relaxations occur at single unstable cell.
Relaxations can propagate up and/or down the sandpile, creating more unstable sites and individual relaxations.
These subsequent relaxations propagate back through original cell.
An event consists of total number of consecutive relaxations at a single site.

13. Shear flows reduces large transport events
The inclusion of a shear zone in the model has a major effect on the PDFs of avalanche events--increased shear reduces maximum event size.
Shear region
slope ~ -2.5

14. Time histories of avalanches with and without sheared flow
Transport events are decorrelated by sheared flow
No Sheared Flow
Sheared Flow r
time r
time

15. Moving and fixed observation points:
Sheared sandpile has poloidal flow of avalanche events--events of all sizes are measured but:
Measurements in reference frames fixed with relation to the flow obtain only small values.
Measurements in reference frames moving with flow obtain small and large values.
Measurements in both frames in zero-flow location can obtain all values.

16. Moving and fixed observation points:
uniform flow region, below shear region

17. Moving and fixed observation points:
non-uniform flow region, within shear region

18. Moving and fixed observation points:
zero-flow region, within shear region

19. Effect of driving rate:
Shear creates avalanche size barrier--this can be crossed by increased driving rate that creates overlapping avalanches.

20. Parallel equilibration dynamics
Different sand relaxation schemes to emulate poloidal and parallel transport:
‘Triangular’ spreading--poloidal diffusion
‘Random’ spreading--parallel equilibration   r r r  

21. Parallel equilibration dynamics
Poloidal cross-correlation and correlation length of shear-free sandpile
one cell
separation

22. Parallel equilibration dynamics
Effect of shear on poloidal correlation length
(with triangular spreading)

23. Experiments with this model show:
Conclusions
Experiments with this model show:
Increased shear flow decreases frequency of large avalanche events
Fixed observation points do not measure large events inside flowing region
Increased driving can cause large events to occur beyond shear-induced barrier
Poloidal diffusion causes increased poloidal correlation length for radial flux events.
Increased shear decreases poloidal correlation length