Neoclassical Effects in the Theory of Magnetic Islands: Neoclassical Tearing Modes and more A. Smolyakov* University of Saskatchewan, Saskatoon, Canada,

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Neoclassical Effects in the Theory of Magnetic Islands: Neoclassical Tearing Modes and more A. Smolyakov* University of Saskatchewan, Saskatoon, Canada, *Presently at CEA Cadarache, France IAEA Technical Meeting on Theory of Plasmas Instabilities: Transport, Stability and their Interaction, 2-4 Mar, 2005, Trieste, Italy

Acknowledgements/Contributors: J.D. Callen, U Wisconsin J. Connor, UKAEA R. Fitzpatrick, IFS, UT X. Garbet, CAE Cadarache E. Lazzaro, IFP, CNR A.B. Mikhailovskii, Kurchatov Institute M. Ottaviani, CAE Cadarache P.H. Rebut, JET A. Samain, CAE Cadarache B. Scott, IPP K.C. Shaing, U Wisconsin F. Waelbroeck, IFS,UT H. Wilson, UKAEA …………………..

Additional to the usual current drive?

Outline Basic island evolution -- extended Rutherford equation Finite pressure drive: Bootstrap current Stabilization mechanisms: Removal of pressure flattening due to finite heat conductivity Polarization current Other neoclassical effects Neoclassical coupling of transverse and longitudinal flows Enhanced polarization current due to neoclassical flow damping New stabilization mechanism due to parallel dynamics and neoclassical coupling Ion sound effects Island rotation frequency

Ideal region Resistive layer

Current driven vs pressure gradient driven tearing modes Ideal region: Solved with proper boundary conditions to determine Current drive Nonlinear/resistive layer: Full MHD equations (including neoclassical terms/bootstrap current) are solved Bootstrap current drive

Diamagnetic banana current +friction effects Loss of the bootstrap current around the island Diamagnetic banana current +friction effects Driving mechanism Bootstrap current Constant on magnetic surface

Qu, Callen 1985 Qu, Callen 1985

Beta dependence signatures are critical for NTM identification Saturation for Rutherford growth Beta dependence signatures are critical for NTM identification Bootstrap growth

A problem:

Competition between the parallel (pressure flattening) and Fitzpatrick, 1995 Gorelenkov, Zakharov, 1996 Competition between the parallel (pressure flattening) and transverse (restoring the gradient) heat conductivity ->restores finite pressure gradient

Other stabilizing mechanisms? Bootstrap current is divergent free: Diamagnetic current Glasser-Green Johson Neoclassical viscosity, enhanced polarization Inertia, polarization current

Note the dependence on the frequency of island rotation! Bootstrap current drive Slab polarization current, Smolyakov 1989 Note the dependence on the frequency of island rotation!

Also finite banana width, Fitzpatrick, 1995 Gorelenkov, Zakharov, 1996 Smolyakov, 1989; Zabiego, Callen 1995; Wilson et al, 1996 Also finite banana width, Poli et al., 2002

Parallel ion dynamics effects Neoclassical viscous current Parallel ion dynamics effects Enhanced inertia, replaces the standard polarization current

Neoclassical inertia enhancement

Neoclassical polarization

Neoclassically enhanced inertia standard inertia Neoclassically enhanced inertia depends on collisionality regime and may have further dependence on frequency, Mikhailovskii et al PPCF 2001

“Ion-sound” effects on the island stability Parallel “ion-sound” dynamics Why Finite ion –sound Larmor radius/banana width Finite effect (near the separatrix)

For finite Inertial drift off the magnetic surface Finite orbit effect provides threshold of the same order as the polarization current !

bootstrap drive is reduced, Ware pinch contributes to stabilization Fitzpatrick PoP 2, 895 (1995) Ware pinch contributes to stabilization

but

Ion sound is stabilizing, but ~

Additional stabilization due to “ion-sound” dynamics Pressure variations within the magnetic surface, provide additional stabilization of magnetic islands - finite orbits/banana - finite These effects are amplified by the neoclassical inertia enhancement Caveat: Island rotation frequency? - Useless without the knowledge of the rotation frequency

Island Rotation Frequency Island rotation is determined by dissipation - minimum dissipation principle Dissipation: Classical collisions: resistivity and heat conductivity Collisionless (Landau damping) Perpendicular diffusion density/energy: classical/anomalous Perpendicular anomalous viscosity Neoclassical flow damping/symmetry breaking

Classical dissipation: parallel resistivity and heat conductivity is due to the coupling to external perturbations/wall; otherwise =0 Smolyakov, Sov J Pl Phys 1989 Connor et al; PoP, 2001

Collisional dissipation in toroidal plasma: mainly collisions at the passing/trapped boundary Weakly collisional regime, electron dissipation, Wilson et al, 1996 Mikhailovski, Kuvshinov, PPR, 1998 Ion dissipation is important for larger collisionality

These effects are shown to affect the island rotation: Only preliminary work has been done, no expressions for W are available with few exceptions Neoclassical magnetic damping: Mikhailovskii, 2003 Drift waves emission: Connor et al., 2001; Waelbroeck 2004 Anomalous viscosity/diffusion: Fitzpatrick, 2004 Symmetry breaking, neoclassical losses in 3D: K C Shaing -helical magnetic perturbation + toroidicity locally creates 3D (stellarator-like) configuration->neoclassical like fluxes-> local modification of the plasma profiles->Er is uniquely determined Local plasma rotation frequency=island rotation

Beyond the Rutherford equation? Single mode perturbations are well identified in the experiment: m/n=2/1, 3/2, 4/3,…. - single harmonic approximation seems to be justified However importance of the resonant coupling has been shown in the experiment, e.g. 3/2+1/1=4/3, Frequently Interrupted NTM: 3/2 NTM is stabilized by 4/3 mode, Gunter et al., 2000, 2004 NTM mode stabilization via the resonant coupling, Yu et al, PRL 2000 - separatrix stochastization -> enhanced radial transport -> radial plasma pressure gradient is restored -> bootstrap current is restored -> island drive is reduced Will also affect the radial fluxes -> island rotation frequency

Variety of mechanisms affect the island stability: Summary Variety of mechanisms affect the island stability: neoclassical/bootstrap, polarization/inertial drifts, magnetic field curvature/plasma pressure, parallel heat conductivity, banana orbits, ion-sound effects, … Each of these has to be carefully evaluated Critical issues: Island rotation frequency? Nonlinear trigger/excitation mechanism "Cooperative effects" of the error field and neoclassical/bootstrap drive?