1. APS DPP 2006 October 31 2006 1 Dependence of NTM Stabilization on Location of Current Drive Relative to Island J. Woodby 1 L. Luo 1, E. Schuster 1 F. D. Halpern 2,G. Bateman 2, A. H. Kritz 2 1 Department of Mechanical Engineering 2 Department of Physics Lehigh University, Bethlehem, PA 18015 48 th Annual Meeting of the Division of Plasma Physics American Physical Society 30 October – 3 November 2006 Philadelphia, Pennsylvania

2. APS DPP 2006 October 31 2006 2 Abstract High plasma pressure can cause ideally nested magnetic flux surfaces to tear and reconnect, leading to the formation of magnetic islands. The neoclassical tearing mode (NTM) instability drives the islands to grow to their saturated widths, at which they can persist stably in the plasma. The presence of magnetic islands leads to a local flattening of the pressure and current profiles, which is undesirable as it degrades plasma confinement. One common method of replacing lost current within islands is direct current injection via electron cyclotron current drive (ECCD). The greatest effect of current drive on the island width is achieved when current is driven near the island center. In preparation for feedback stabilization of NTMs, the effect of off-center current drive on the saturated magnetic island widths is considered here. The off-center drive is modeled mathematically by a parabolic current density peak (the center of which need not coincide with the island center), which is then transformed into Hamada coordinates. The results are implemented in the ISLAND module, which is used in BALDUR.* * JP1.00142

3. APS DPP 2006 October 31 2006 3 Objectives It has been shown that increasing current within magnetic islands shrinks the island Localized current drive (e.g. ECCD) can be used to inject current into islands, with the strongest shrinking effect when current is injected into island center Difficulty lies in determining the position of the island flux surface in real time Past work has lead to an expression in the ISLAND module which allows for centered current drive only This work seeks to model current drive with the possibility of shifting the drive center off the island This is in preparation for feedback stabilization, for which it is necessary to consider off-center current drive

4. APS DPP 2006 October 31 2006 4 References ISLAND module from NTCC module library: http://w3.pppl.gov/ntcchttp://w3.pppl.gov/ntcc Background, finding saturated magnetic island widths, ISLAND: - F. Halpern, Physics of Plasmas 13 (2006) 062510 - C.N. Nguyen and G. Bateman, Physics of Plasmas 11 (7) (2004) - G. Bateman and R.N. Morris, Phys. Fluids 29 (3) (1986) Similar work expressing current drive in Hamada coordinates: - G. Giruzzi et al., Nuclear Fusion 39 (1999) 107-125 - C. Hegna and J. Callen, Physics of Plasmas 4 (1997) 2940 Computing elliptic integrals: www.netlib.orgwww.netlib.org JP1.00142 “Adaptive Extremum Seeking Control of Current Drive for NTM Stabilization”

5. APS DPP 2006 October 31 2006 5 Using Hamada-like coordinate system ( V is any quantity which is constant over a flux surface, such as volume, are anglelike variables in the poloidal and toroidal directions, respectively) Get set of coupled ODEs which describe change in background current and pressure profiles due to presence of island Implemented in ISLAND module (used in BALDUR) which computes saturated magnetic island widths NTM=neoclassical tearing mode, magnetic “islands” result from tearing and reconnection of ideally nested magnetic flux surfaces Starting from force-balance eqs Background

6. APS DPP 2006 October 31 2006 6 (1) (2) where J is the Jacobian, the elements B 0 and J 0 are the background magnetic field and current density, and the perturbations on these, B 1 and J 1, are expressed in terms of both contravariant and covariant components in Hamada coordinates ODEs to find saturated magnetic island widths

7. APS DPP 2006 October 31 2006 7 Previous work: Toroidal current density is proportional to Current density is flattened by magnetic island, centered current peaking is controlled by Current profile for centered peak = island halfwidth

8. APS DPP 2006 October 31 2006 8 The ISLAND module uses, = averaged current To find integrate (at fixed V ) over the angle Result without current drive: where is the incomplete elliptic integral of the first kind Derivative of averaged current

9. APS DPP 2006 October 31 2006 9 Preliminary code modification Implemented exact elliptic integral in term in ISLAND to replace analytic approximations Result: Similar saturated island widths Factor proportional to current drive u = radial coordinate centered on island

10. APS DPP 2006 October 31 2006 10 with different portions of the current drive both inside and outside the island, depending on the current drive shape Generalize current drive term to include drive offset from island center Modified current profile

11. APS DPP 2006 October 31 2006 11 Assume that the current drive has the following form Off-center current drive

12. APS DPP 2006 October 31 2006 12 Current applied in u-coordinates gets spread over magnetic flux surfaces, given by Relation between u-coordinates and flux surfaces (Taylor expansion of near mode-rational surfaces): Finding average driven current

13. APS DPP 2006 October 31 2006 13 Current applied in u- coordinates onto a differential area dA gets spread over flux surface it hits, over dS, the corresponding differential area in flux surface coordinates Or, inserting the Taylor expansion for Now integrate over and to get the averaged current drive density Derivation of average driven current - 1

14. APS DPP 2006 October 31 2006 14 The average current density is given by Note that the average must be taken separately within the island ( is the separatrix, or island edge) and outside, since the area ratio dA / dS will vary Problem: even if dA / dS is assumed to be constant, this integral cannot be done analytically Derivation of average driven current - 2

15. APS DPP 2006 October 31 2006 15 1: Assume all the current is driven near the widest part of the island, at 2: Assume that dA / dS is constant and the same constant both inside and outside the island Simplifying assumptions

16. APS DPP 2006 October 31 2006 16 Result for averaged current drive (setting dA / dS = 1): Taking the derivative where is the complete elliptic integral of the first kind and Result of current drive derivation where is the complete elliptic integral of the second kind and

17. APS DPP 2006 October 31 2006 17 In this derivation, since the integral was taken separately for positive and negative u, the result is not symmetric within the island, even though flux surfaces form closed loops there. In effect, we assumed that any current applied in positive u will get spread over the flux surfaces it hits in the positive u region only For implementation in ISLAND, we have symmetrized the result by assuming that when current is driven at a positive offset ( +a ), an equivalent current is driven on the other side, at ( -a ). We then normalized the result so that increasing the offset would not result in a larger total current drive. Additional assumption - Symmetry

18. APS DPP 2006 October 31 2006 18 Symmetrized current drive density “Typical” case Results not normalized

19. APS DPP 2006 October 31 2006 19 Symmetrized current drive density Centered drive Results not normalized

20. APS DPP 2006 October 31 2006 20 Symmetrized current drive density Increased power Results not normalized

21. APS DPP 2006 October 31 2006 21 Symmetrized current drive density Narrow drive Results not normalized

22. APS DPP 2006 October 31 2006 22 Symmetrized current drive density Completely off island Results not normalized

23. APS DPP 2006 October 31 2006 23 Summary Implemented exact elliptic integrals in ISLAND to replace analytic approximations; similar saturated island widths After making several simplifying assumptions, found analytical expression for averaged current drive in Hamada coordinates, which was then symmetrized, normalized and implemented in ISLAND Future work Generalize result to include current driven at all toroidal angles simultaneously, not only near island center Symmetrize results only within island, where flux surfaces form closed loops Include effect of dA / dS, which is not constant