Nonlinear interactions between micro-turbulence and macro-scale MHD

1. Nonlinear interactions between micro-turbulence and macro-scale MHD A. Ishizawa, N. Nakajima, M. Okamoto, J. Ramos* National Institute for Fusion Science *Massachusetts Institute of Technology US/Japan JIFT Workshop “Issues in the theoretical analysis of three dimensional configuration” Princeton, March 14-16, 2006

2. Our goal • Effects of MHD instabilities and micro-turbulence on plasma confinement have been investigated separately. • But these instabilities usually appear in the plasma at the same time. • Our goal is to understand multi-scale-nonlinear interactions among micro-instabilities, macro-scale-MHD instabilities and zonal flows.

3. Introduction MHD activities are observed in reversed shear plasmas with a transport barrier related to zonal flows and micro-turbulence. Time evolution of local electron temperature Takeji, et.al., Nuclear Fusion (2002)

4. Motivation and results • The appearance of the macro-scale MHD instability, which leads to the disruption, can be affected by micro-turbulence and zonal flows. • We investigate multi-scale-nonlinear interactions among micro-instabilities, macro-scale tearing instabilities and zonal flows, by solving reduced two-fluid equations numerically. We find that the nonlinear interactions of these instabilities lead to an alteration of macro-equilibrium magnetic field, then this alteration spreads the micro-turbulence over the plasma.

5. Reduced two-fluid equations • We carry out three-dimensional simulations with a reduced set of two-fluid equations that extends the standard four-field model, by including temperature gradient effects. • Basic assumptions • Flute approximation • Large aspect ratio • High-beta ordering • By solving this set of equations, we can describe the nonlinear evolution of tearing modes, interchange modes, ballooning modes and ion-temperature gradient modes. Magnetic surfaces tearing m=2 kink m=1 Electric potential KBM m>10 ITG m>10

6. Basic equations

7. Double tearing mode Micro-instabilities Growth rate Toroidal mode number: n Initial equilibrium and linear growth rate We examine the multi-scale nonlinear interaction among instabilities in a reversed shear plasma.

8. Linear instabilities:micro-instabilities and double-tearing mode n=9 n=1 n=14 Macro-scale MHD Double tearing mode Micro-instability Micro-instability • In the linear phase, a ballooning structure of micro-instability appears in the bad curvature region. The structure is twisted by ion and electron diamagnetic effects. • A double-tearing mode is also unstable, but its growth rate is small compared to that of the micro-instability.

9. Zonal flow Zonal flow t=63 total t=63 n=1 Twisted micro-instability Twisted double-tearing mode linear The zonal flow induced by the micro-instability has a stabilizing effect on the micro-instability and on the tearing mode by twisting their radial structure.

10. Nonlinear evolution of electric potential

11. Details of nonlinear evolution I Zonal flow Magnetic energy 1 total Toroidal mode number: n n=1 t=63 t=63 The nonlinear mode coupling is so strong that it overcomes the stabilizing effect due to the zonal flow, and the tearing mode growth rate is enhanced by the nonlinear-mode-coupling. Thus an m=3 double tearing mode appears. 1 2 3 Nonlinear growth of tearing mode Tearing mode dominates Alteration of equilibrium magnetic field

12. Details of nonlinear evolution II The tearing mode affects the micro-turbulence by breaking the magnetic surfaces 2 Magnetic energy total n=1 Toroidal mode number: n t=99 t=99 Since the tearing mode breaks the magnetic surfaces through magnetic reconnections, the dominance of n=1 tearing mode results in an alteration of the equilibrium magnetic field. 1 2 3 Nonlinear growth of tearing mode Tearing mode dominates Alteration of equilibrium magnetic field

13. Details of nonlinear evolution III 3 total n=1 Magnetic energy Toroidal mode number: n t=108 t=108 The alteration spreads the micro-turbulence over the plasma after , and it also increases the energy of the turbulence as indicated by the traces with n>1. 1 2 3 Nonlinear growth of tearing mode Tearing mode dominates Alteration of equilibrium magnetic field

14. Double tearing mode Micro-instabilities Growth rate Toroidal mode number: n Initial equilibrium and linear growth ratebeta=1.5%

15. Linear analysis: beta=1.5%Double-tearing modes and micro-instabilities n=10 n=14 n=1 Macro-scale MHD Double tearing mode Micro-instability Micro-instability

16. Nonlinear evolution of electric potential Beta=1.5%, eta3

17. Summary • We have found that the multi-scale nonlinear interactions among micro-turbulence, tearing modes, and zonal flows lead to an alteration of the macro-magnetic field, then this alteration spreads the turbulence over the plasma. • The mechanism of the spreading is as follows. The micro instability induces zonal flows which attempt to suppress the tearing mode. However, the nonlinear-mode-coupling due to the micro-instability overcomes this suppression and accelerates the growth of the tearing mode. This tearing mode alters the macro-equilibrium magnetic field by breaking the magnetic surfaces, and thus the tearing mode spreads the micro-turbulence over the plasma. Macro-MHD n=1 (tearing modes) Micro-turbulence n>>1 Linear instabilities 3. destabilize 2.stabilize 1.destabilize 2. stabilize n : toroidal mode number Zonal flow n=0 4. destabilize

18. Future plan • Include the effects of a radial electric field in the initial equilibrium • The present simulation adopted an initial static equilibrium without radial electric field. • The importance of the choice of initial perturbation • The present simulation is based on a linear mode initial condition. • Solve a set of reduced two-fluid equations derived by Prof. Ramos numerically

19. Zonal flow Kinetic energy

20. Electro-static (beta=0) and back ground profiles are fixed Time evolution of zonal flow

21. Single helicity: double tearing: background profile relax t=96 n=1 /home/ishizawa/fivefieldVer8/reversed/singleHelicity