INTERACTION OF DRIFT WAVE TURBULENCE AND MAGNETIC ISLANDS A. Ishizawa and N. Nakajima National Institute for Fusion Science F. L. Waelbroeck, R. Fitzpatrick, W. Horton Institute for Fusion Studies, University of Texas Sherwood conference, University of Texas at Austin, May 2, 2011
OUTLINE 1. Introduction 1. Motivation 2. Neo-classical tearing modes (NTMs) 2. Effect of turbulence on seed island formation 1. 3D electromagnetic simulation in toroidal plasma 2. 2D electromagnetic simulation in slab plasma 3. Effect of turbulence on island propagation o and NTM growth 1. 3D electromagnetic simulation in toroidal plasma 2. 2D electromagnetic simulation in slab plasma 3. 2D electrostatic simulation with a fixed magnetic island 2
MOTIVATION Turbulence is able to influence the formation of coherent structures in nature. Are macro-mhd activities influenced by drift-wave turbulence in magnetically confined plasmas? Jeju island Kyushu Reynolds number of Karman vortices flow 100 Reynolds number of this flow 10^10 3
STUDY OF MULTI-SCALE INTERACTION BETWEEN MACRO-MHD MHD AND MICRO-TURBULENCE Micro-turbulence Zonal flow P.H. Diamond et.al., Phys. Fluids (1984) D. Biskamp, Plasma Phys. Controlled Fusion (1984) W. Horton, Journal of Statistical ti ti Physics (1985) S.-I. Itoh et. al., Phys. Rev. Lett. (2003) CJ C.J. McDevitt et.al., Phys. Plasmas (2006) A. Sen et al, Nuclear Fusion (2009) M. Muraglia, et.al., Nucl. Fusion (2009) A. Ishizawa et.al, Phys. Plasmas (2010) M. Yagi, et al., Nuclear Fusion (2005) A. Ishizawa et.al, Phys. Plasmas (2007) A. Ishizawa et.al, Nuclear Fusion (2007) A. Ishizawa et.al, AIP Conf. Proc. (2008) J. Li, et al., Nuclear Fusion (2009) A. Ishizawa et.al, Nuclear Fusion (2009) Macro-MHD (tearing mode) F. Militello, et.al, Phys. Plasmas (2008) F. Waelbroeck, et.al, Plasma Phys. Controlled Fusion (2009) Z.X. Wang et.al, Phys. Plasmas (2009) A. Ishizawa et.al, Phys. Plasmas (2010) W. A. Hornsby, Phys. Plasmas (2010) 4
MACRO-MHD CHANGES THE BALANCE BETWEEN TURBULENCE AND ZONAL FLOW t=84 log( E M ( n)) t=180 KBM Time evolution of zonal flow tv ti / a Helical flux m/n=2 q profile Helical flux m/n=2 q profile r / a tv ti / a A. Ishizawa et.al, Phys. Plasmas (2007) 5
MOTIVATION: EXCITATION OF NTM Neo-classical tearing mode is a nonlinear instability. '< 0 R.J. LA HAYE and O. SAUTER, NUCLEAR FUSION, Vol. 38, (1998) R. J. La Haye, et al., Phys. Plasmas (2000) 6
REDUCED TWO-FLUID EQUATIONS 7
THE SEED ISLAND PROBLEM: EFFECT OF TURBULENCE ON ISLAND FORMATION 8
MAGNETIC ISLANDS ARE PRODUCED BY TURBULENCE IN 3D ' 2 Electrostatic potential Kinetic ballooning turbulence m/n=3/2 helical flux Initial equilibrium β = 0.02 Electrostatic potential r 9
ISLAND FORMATION ALSO OCCURS IN 2D The ion-temperature gradient turbulence produces long-wavelength magnetic islands, even if the equilibrium is stable against tearing instabilities. stable against tearing mode '= 0. 0 76 Energy spectrum Electrostatic potential y ρ i B B y ρ i wave number Magnetic flux x / ρi
ISLAND WIDTH IS INDEPENDENT OF FOR <0 η i = 3.5 Island width Threshold expected by conventional theory Stability parameter Thick current layer Thin current layer Turbulence modifies the threshold of magnetic island formation predicted by the conventional stability parameter of tearing instability. The width of island is several times ion Larmor radius. 11
STRONGER TURBULENCE CAUSES FASTER GROWTH AND WIDER MAGNETIC ISLANDS. '= 3.4 β = 0.02 tv Ti / a 12
TURBULENT MIXING OF MAGNETIC FLUX Ohm s law of k=1 mode on the resonant surface of tearing mode ~ ψ β β + t ~ ~ ~ ~ ~ 1 = β [ ~ ψ, Φ ] ~ ~ ~ 1 [ ψ, p ] 1 [, 00 ] 1 a [, p ] e + β ψ Φ β ψ eeq 1 η J1 neq Turbulent mixing terms Rotation of island lue of ea ach term Abs solute va ηj 1 ~ ~ [ ψ, pe] 1 ~ [ ~ ψ, Φ ] 1 tv Ti / a Turbulent flow mixing terms may play important role in magnetic island formation. 13
THE ISLAND GROWS THROUGH MERGING Magnetic flux on the neutral sheet and X and O points Blue y Red ρ i tv Ti / a It seems that merging of small magnetic islands produces long- wavelength magnetic islands The propagation direction reverses as the islands grow Black points: O-points, Blue points: X-points Zonal flow ' = 2.4 = 3.5 η i 14
NTM EVOLUTION: ROLE OF ISLAND PROPAGATION VELOCITY 15
NTM EVOLUTION DEPENDS ON ISLAND PROPAGATION The sign of the polarization current depends on the island propagation. Modified Rutherford equation Temporal evolution of magnetic il island width bootstrap Polarization term It is important to determine the direction of the island propagation in the control of NTM. w Island width Island frequency ω Connor et al. Phys. Plasmas (2001) 16
PROPAGATION OF ISLANDS (3D) The rotation direction of (m,n)=(3,2) is unclear. tvt Ti / a 17
TO UNDERSTAND PROPAGATION, WE EXAMINE THE FORCE ON A DRAGGED ISLAND IN 2D We consider the force acting on the island when we are in the island fixed frame. Electrostatic η J = // Φ // pe Flow Boundary velocity u=-0.2 Magnetic flux Electrostatic potential R. Fitzpatrick, et.al, Phys. Plasmas (2006) F. Waelbroeck, et.al, Plasma Phys. Controlled Fusion (2009) 18
TURBULENCE DRAMATICALLY CHANGES THE FORCE ACTING ON THE ISLAND Electromagnetic force (drag u=-0.5 05 force) acting on the island is strong in the presence of ITG turbulence. Electromagnetic force Fy Electrostatic potential w/o turbulence with turbulence Magnetic flux Boundary velocity u 19
SUMMARY We have found that turbulence produces long-wavelength magnetic islands in a current sheet, even if the current sheet is so thick that there is no spontaneous magnetic reconnection. A. Ishizawa and N. Nakajima, Phys. Plasmas (2010) 072308 Thus, turbulence modifies the threshold of magnetic island formation predicted by the conventional stability parameter of tearing instability. This suggests that turbulence can trigger NTM. Mechanism of the island formation The long-wavelength magnetic islands are formed by merging of small-scale l magnetic islands. Turbulent mixing of magnetic flux plays a role in producing longwavelength magnetic islands. 3D: strong 2D: weak Drag force acting on magnetic island The force is strong in the presence of turbulence. 20