2. Self-organization to Stable magnetic Plasmas by
T. R. Jarboe, D. A. Sutherland, A. C. Hossack, B. A. Nelson, K. D. Morgan, C. J. Hansen, T. K. Benedett, C. J. Everson, and J. M. Penna, University of Washington, Seattle WA To
PSI-Center
July 2018

3. Outline
Taylor minimum energy principle: Magnetic plasma relaxes toward a state of minimum energy while conserving helicity. The MECH state is stable. (well established)
Imposed dynamo current drive: Helecity injection and Imposed perturbations may allow the sustainment of stable two step equilibria. (observed, topic of research)
Velocity gradient stabilization: Electron velocity shear may locally stabilizes and symmetrizes most unstable modes of an MHD kink-stable equilibrium. (observed, topic of research)
A. C. asymmetric helicity injection may have benefits: imposed perturbations may sustain kink-stable closed-flux equilibria and stabilize slow growing instabilities. (topic of future research)

4. Helicity is the best conserved quantity
Helicity dissipation is proportional to E  B
Only the resistive term in the generalized Ohm’s law gives helicity dissipation. (grad p is mostly perp. to B)
Helicity decays on the resistive time scale (which is the longest) and the MECH state is stable.
With magnetic flux B.C.s the MECH state is the Taylor state and λ = oj/B is uniform.

5. HIT-SI uses imposed fluctuations and an externally driven edge current to sustain the stable equilibrium
Current amplification of 3.9, a new spheromak record
90 kA of toroidal current
Stable sustained equilibria ohmically heat to the beta limit,*achieving the current drive goal
*B. S. Victor et al., Physics of Plasmas, 21, (2014)

6. Conditions needed to allow IDCD with good confinement
The perturbation must be applied for a time much shorter then the time it take resistive effects to change the magnetic topology from nested closed flux surfaces and longer than it takes to setup IDCD:
Equilibrium must be magnetically stable before during and after the perturbation is applied.
Perturbation is allowed to lower the beta-limit for ELM-pacing effect. (Plasma come out in bursts.)

7. Imposed dynamo current drive may allow the sustainment of stable two step equilibria
Sustained MECH states can have λ as a B.C. and a current separatrix can form between the λ from the B.C. and the λ due to self-organization.
Two-step λ-profile, which fits data well, may stabilize and symmetrize separatrix.

8. An explanation of the effect of perturbations on current starts with symmetric current
Electron fluid is frozen to magnetic fields. (Two-fluid ideal MHD)
Current flow is also magnetic field flow.
For a stable equilibrium the magnetic flux surface is resilient. It resists deformation.
Toroidal current is from electrons frozen in nested resilient shells that are rotating at different speeds.
Symmetric flux surfaces allow free differential current flow. (Free means unobstructed by magnetic interference.)
Toroidal current in a torus with a hollow current profile

9. Sheared electron flow plus perturbations give current drive across closed flux surfaces
Now add a magnetic perturbation and differential flow is no longer free.
If perturbation is large enough the flow locks across flux surfaces (inner flux surfaces). This is call “zonal flow” that is caused by turbulence.
If the perturbations are small enough the differential flow can symmetrize the perturbation and differential flow continues.
A viscosity-like drag force will drive the current inside the symmetrized closed flux surface.
Steep flow shear give transport barrier
Both effects are current self-organizing across closed flux surfaces towards uniform λ.
HIT-SI data indicate that the viscosity-like force per unit area needed to symmetrize is:
Externally driving the edge results in Imposed Dynamo Current Drive (IDCD).*
*T. R. Jarboe et al., Nucl. Fusion, 52, (2012)

10. IDCD effect on tokamaks can be estimated
Viscous force on the flux surface area = Force require to drive the current throughout that flux surface volume.

11. IDCD equation agrees with radiative disruption* perturbations
Disruption created by Argon injection.
Cold edge peaks the current until it is unstable. Instability cools plasma.
The low edge current is maintained by the Argon and drags down the plasma current:
1.5 MA profile is flattened in 1.2 ms
IDCD requires δB of 190G (at s)
A 10 ms current quench.
IDCD requires δB of 60 G. (at s)
*P. L. Taylor et al., Phys. Rev. Lett, 76, (1996)

12. On a reactor and ITER the perturbation levels required to drive the current are a little higher than considered acceptable (confirming the effect). They are small.
Parameter
Present tokamaks
ARIES-AT
ITER
Itor (MA)
4.5
12.8
15.
Temp. (keV) 2 18
8.1
a (m) 1
1.3
L/R (s) 15
605
454
Brms /B
0.0004
0.0001
Current driving perturbations can also flatten the λ-profile which flattens the q-profile leading to poor performance.
In the face of such a powerful flattening effect it is not surprising how difficult it is to maintain a high-performance profile often ending in disruption.

13. Perturbations crossing flux surfaces give rotation
In the externally driven regions the drag force (blue) brakes electrons so the force is in the direction of the current giving plasma velocity in that direction.
In the dynamo driven regions the force is with the electron flow resulting in plasma flow against the current.
Thus, the core plasma rotates with current in normal tokamak because the core is externally driven and against the current when LHCD is used in the edge because the core is dynamo driven (seen on C-mod).*
*J. E. Rice et al., Nucl. Fusion (8pp) (2009)

14. This mechanism provides an explanation for:
Summary of evidence that magnetic perturbations make j/B uniform in a stable symmetric equilibrium
This mechanism provides an explanation for:
The level of field error (δB/B = 10-4) that spoils tokamak performance*.
The rate of poloidal flux loss in Argon-induced disruptions in DIII-D*.
The change in plasma rotation direction with LHCD in the edge on C-mod*.
The toroidal current versus time, the injector impedance, and the current profile of HIT-SI***.
Perturbations cause zonal flows and transport barries
*T. R. Jarboe, B. A. Nelson, and D. A. Sutherland, Phys. Plasmas 22, (2015)
**M. C. ArchMiller et al, Physics of Plasmas 21, (2014)
***T. R. Jarboe et al., Nucl. Fusion, 52, (2012)

15. Gradients in vetor inhibit instability
Imposed perturbations cause a force that inhibits differential motion of resilient flux surfaces.
Conversely, differential motion of resilient flux surfaces cause a force that inhibits perturbations, including eigenmodes of instabilities, preventing instability growth.
Transport barriers depend on the E × B shearing rate.*
In ideal two-fluid MHD ve = E × B/B2, and assuming B2 has a small gradient, the shearing velocity is the electron velocity shear.
The divertor may help in forming a pedestal, in part, because it causes electron velocity shear.
*K H Burrell, M E Austin, C M Greenfield, L L Lao, B W Rice, G M Staebler and B W Stallard, “Effects of E × B velocity shear and magnetic shear in the formation of core transport barriers in the DIII-D tokamak”, Plasma Phys. Control. Fusion, 40 (1998) 1585–1596.

16. IDCD perturbations may dynamically stabilizes slow growing instabilities.
Instabilities with growth rates are less than the injector frequency might be stabilized by IDCD.
Pressure driven instability meet this test and its stabilization might explain the observed higher beta at higher frequency.
As beta is increased above the threshold the growth rate increases.
Higher growth rates are stabilized by higher frequencies, allowing higher beta.

17. Summary
Taylor minimum energy principle: Magnetic plasma relaxes toward a state of minimum energy while conserving helicity. The MECH state is stable.
Imposed dynamo current drive: Helecity injection and Imposed perturbations may allow the sustainment of stable two step equilibria.
Velocity gradient stabilization: Electron velocity shear may locally stabilizes and symmetrizes most unstable modes of an MHD kink-stable equilibrium.
A. C. asymmetric helicity injection may have benefits: imposed perturbations may sustain kink-stable closed-flux equilibria and stabilize slow growing instabilities.
Next slide

18. Accurate electron skin depth with real electron mass.
Quantitative simulations of interfering resilient electron shells requires accurate modeling of electrons. Probably need to capture:
Accurate electron skin depth with real electron mass.
Electron viscosity
Non-local transport
Most of the entire generalized Ohms law
Two fluid might be better (physics can be the same)
Not surprising, current drive is a force on the electrons
Self-organization probably can be captured in fluid models (better electron physics led to better understanding of transport barriers)
We have work to do

19. Possible re-organization
Group
leader
code
Dynamic Neutrals and Boundary Conditions
Uri
NIMROD
Transport and Kinetic Effects
Carl
Boundary Conditions, Sustainment and Confinement with 3D Geometry
Chris
PSI-Tet
Plasma Self-Organization and Sustainment
John
Interfacing
Brian

20. Forced rotation of resilient shells scraps out asymmetries
Symmetric perturbation gives no net drag.
Resilient flux surface motion slacks one crossing and stretches the other.
Gives net viscous-like force between flux surfaces.
Maximum force is related to the amount of perpendicular flux.

21. B-field tension is used to analysis HIT-SI.
Differential flow turns all δBperp in to current driving.
Think of B forces as:
Isotropic pressure
Tension
Cos from tangential component
Sin from projected area
Isotropic pressure makes no tangential force
Maxwell stress on mean flux surface = current driving force inside flux surface
“IDCD equation”
Assuming gross distortion the δBll depends on original δB┴.

22. Self-organization make optimizing a normal tokamak difficult
A uniform j/B profile in a normal tokamak is nearly uniform q and has a very low β-limit.
Taylor relaxation must be disobeyed by using the inherit stability of the tokamak.
Perturbation flattening is defeated by keeping field errors and perturbations extremely low.
Shear stabilization is used to create pedestals and internal transport barriers for high performance.
While locally stabilizing, E x B shear can be globally destabilizing → perturbations → uniform λ → loss of pressure →thermal quench (start of disruption).

23. Solutions
Drive the edge current high and impose a perturbation profile that sustains the desired reversed-shear current profile.
Select a high performance equilibrium that has a uniform j/B (low aspect ratio)
Rigorously sustain the rock-stable profile by edge current drive and repeated application of non-resonant perturbations of IDCD.
The method has been demonstrated on HIT-SI.
Solves the sustainment problem. (IDCD is over two orders of magnitude more power efficient on a reactor than rf)
High edge current prevent the edge from using perturbations to drag down the current in disruptions.