1. Kinetic Effects on Slowly Rotating Magnetic Islands in Tokamaks
Max-Planck-Insititut für Plasmaphysik
Kinetic Effects on Slowly Rotating Magnetic Islands in Tokamaks
Mattia Siccinio, Emanuele Poli
Ringberg Schloss Seminar, November 2008
Manuel García-Muñoz

2. OUTLINE THE NEOCLASSICAL TEARING MODE MOTIVATION AND GOALS
The Polarization Current
MOTIVATION AND GOALS
APPROACHING THE PROBLEM
Magnetic Geometry
The Drift-Kinetic Equation
The HAGIS Code
THE w ~ w* REGIME
The w > k//vth Assumption -The Polarization Current
The w ~ k//vth Assumption
THE w ~ wtp REGIME
The Non Collisional Solution
Physical Interpretation
The Effect of Electric Precession
The Collisional Solution
SUMMARY
Manuel García-Muñoz

3. THE NEOCLASSICAL TEARING MODE
The Neoclassical Tearing Mode (NTM) is a non-ideal MHD instability which changes the topology of the magnetic surfaces allowing the appearance of rotating magnetic islands.
In the inner region of the island, both particle and energy confinement are strongly deteriorated. This instability represents a serious limit for the performances of the ITER machine.
Manuel García-Muñoz

4. THE NEOCLASSICAL TEARING MODE
THE POLARIZATION CURRENT
The polarization current is caused by the time dependent electric field (generated by the island itself) a trapped particle experiences while moving along the island.
It is not divergence free, and the parallel closure current which appears by consequence can influence the island stability according to the well- known Rutherford equation.
Manuel García-Muñoz

5. MOTIVATION AND GOALS
The polarization current in absence of gradients has been foreseen to scale as 2 [Smolyakov et al., Wilson et al., 1996].
This scaling holds for the perpendicular current if >> k//vth. Otherwise, it has been shown to break down, so other contributions are expected to play a role. [Poli et al., 2005].
N.B. A THEORY WHICH PREDICTS THE VALUE OF THE ISLAND PROPAGATION FREQUENCY  IS STILL MISSING!! THE RANGE OF FREQUENCIES WE ARE INTERESTED IN CAN BE RELEVANT FOR REALISTIC SCENARIOS.
Manuel García-Muñoz

6. MOTIVATION AND GOALS
Studying the current generated by the island rotation by means of analytical and numerical techniques. The rotation frequency is treated as a free input parameter. The equilibrium gradients are set to zero to concentrate on the island's rotation effects.
Two regimes are in particular identified and discussed separately:
w ~ w* which implies  ~ k//vth
 ~  tp
These regimes correspond to order the island rotation frequency as the passing / trapped particle moving along the island, respectively.
This work consists in a generalization of existing theoretical studies. It represents an investigation of frequency ranges which have not been explored yet, although they could be experimentally meaningful.
Manuel García-Muñoz

7. APPROACHING THE PROBLEM
MAGNETIC GEOMETRY
Toroidal Geometry, Circular Section, Large-Aspect-Ratio Approximation.
Nested surfaces, Positive Magnetic Shear.
Coordinates
 Poloidal Flux
 Poloidal Angle
 Toroidal Angle   
Manuel García-Muñoz

8. APPROACHING THE PROBLEM
MAGNETIC GEOMETRY
It is worthwhile to define the island flux coordinates. We introduce the helical angle:
And the perturbed normalized flux label:
Separatrix
Manuel García-Muñoz

9. APPROACHING THE PROBLEM
THE DRIFT-KINETIC EQUATION
It describes the time evolution of the distribution of the guiding centers (Finite-Larmor Radius effects are not considered). Solved both analytically and numerically.
In the standard neoclassical theory
Analytically solved with the help of a f method, and with a double parameter series expansion [Wilson et al., 1996 ], [Carrera et al., 1986]
Manuel García-Muñoz

10. APPROACHING THE PROBLEM
THE HAGIS CODE
The HAGIS code (Hamiltonian GuIding center System) is the numerical tool adopted in our work.
It solves the drift-kinetic equation in toroidal geometry calculating the time evolution of “markers” which span the phase space.
The time evolution of these “markers” is calculated by means of the Hamiltonian equations of motion.
The f method previously described is also adopted, assuming an isotropic homogeneus Maxwellian as F0.
[Pinches S.D. et al.,1998]
CAVEAT: In both analytical and numerical calculations the electrostatic potential has not been calculated self-consistently, but an analytical expression is given.
Manuel García-Muñoz

11. automatically implies w ~ k//vth.
THE w ~ w* REGIME
With the help of the df technique previously described, we get the following expression for the drift-kinetic equation:
N.B. This ordering
automatically implies
w ~ k//vth.
w* is defined supposing the equilibrium pressure gradient being of the order of 1/a. However, going on with the calculation, equilibrium gradients are neglected.
which is ordered:
Manuel García-Muñoz

12. THE w > k//vth ASSUMPTION - THE POLARIZATION CURRENT
THE w ~ w* REGIME
THE w > k//vth ASSUMPTION - THE POLARIZATION CURRENT
To isolate the contribution of the polarization current, we assume [Wilson et al., 1996]
This perturbed distribution will be used in the quasi-neutrality equation to calculate the polarization current.
The lowest-order perturbed
distribution turns out to be:
Manuel García-Muñoz

13. THE w ~ k//vth ASSUMPTION Discarding the assumption
THE w ~ w* REGIME
THE w ~ k//vth ASSUMPTION
Discarding the assumption
brings to the following result for the perturbed distribution:
N.B. Collisions have been neglected in this first approach
There is a resonance for
critical values of the parallel velocity
Manuel García-Muñoz

14. THE w ~ k//vth ASSUMPTION
THE w ~ w* REGIME
THE w ~ k//vth ASSUMPTION
It is possible to show that the pertubed current is proportional respectively to:
w > k//vth Assumption
w ~ k//vth Assumption
The resonance for a critical value of the parallel velocity still exists, but the corresponding term has been shown not to contribute to the lowest order current perturbation. The existence of the resonating integral is expected to be preserved by collisions.
Manuel García-Muñoz

15. THE w ~ wtp REGIME
We now repeat the previous calculation following a different scaling for the island precession frequency, so that in this case the island propagation frequency is comparable to the trapped particle drift discussed before.
We obtain
Manuel García-Muñoz

16. THE TRAPPED PARTICLE DRIFT
The same thing happens in presence of a radial electric field because of the poloidal component of the ExB drift.
Trapped particles experience also other toroidal precessional equilibrium effects, linked to magnetic shear.
We define
Manuel García-Muñoz

17. is very different between positive and negative values of .
THE w ~ wtp REGIME
THE NON COLLISIONAL SOLUTION
As the trapped particle precession points always in the positive -direction, the behaviour of this
Precessional Current
is very different between positive and negative values of .
w ~wtp+wE
Manuel García-Muñoz

18. CASE  0 CASE > 0 THE w ~ wtp REGIME PHYSICAL INTERPRETATION
(Particles and island rotate in the same direction)
CASE
> 0
(Particles and island rotate in opposite directions)
f NEGATIVE - Fast
E> 0
f POSITIVE - Slow
All trapped particles are accelerated. The local density decreases.
Particles are accelerated. Slow particles reduce their relative speed with respect to the island.
f NEGATIVE
E< 0
All trapped particles are decelerated. The local density increases.
Particles are decelerated. Slow particles increase their relative speed with respect to the island.
f POSITIVE
f POSITIVE - Fast
f NEGATIVE - Slow
Manuel García-Muñoz

19. THE EFFECT OF ELECTRIC PRECESSION
THE w ~ wtp REGIME
THE EFFECT OF ELECTRIC PRECESSION
The electric precession frequency changes locally the number of „fast“ and „slow“ particles. The sign of the overall current can change in a very complicated way along the separatrix of the island for negative frequencies.
Yellow = X-Point cell
Green = Intermediate cell
Blue = O-Point cell
Red = Total
Manuel García-Muñoz

20. THE COLLISIONAL SOLUTION
THE w ~ wtp REGIME
THE COLLISIONAL SOLUTION
In the non collisional case, the integral in Rutherford’s equation can not be carried out because of the resonant denominator.
We include a very simple Krook collision operator in the last steps of the solution.
This solution allows us to calculate the equivalent ’.
The sign of this contribution depends only on K1().
Manuel García-Muñoz

21. SUMMARY D’Pr < 0 D’Pr > 0 POLARIZATION CURRENT PREVALES IF
For higher absolute values of  the standard polarization current prevales.
At low absolute value frequencies the precessional current prevales.
POLARIZATION CURRENT
PREVALES IF
D’Pr < 0
At low negative frequency quite all particles are faster than the island, but the potential reverses its sign. For not too high negative frequencies, the contribution of slow particles dominates.
D’Pr > 0
Manuel García-Muñoz

22. SUMMARY
The standard model for the polarization current turns out to be not satisfactory when the island propagation frequency starts to be comparable with particles' streaming along the island.
For frequencies close to the passing particles' streaming, a resonance has been outlined, although it has been shown not to contribute in a significant way to the perturbed current.
For frequencies close to the trapped particles' precession, the contribution is important, and it could be such to modify the sign of the overall current, which relates to its stabilizing power.
This approach underlines once more the importance of toroidicity and of kinetic effects in the dynamics of Neoclassical Tearing Modes.
Manuel García-Muñoz