Dynamics of ITG driven turbulence in the presence of a large spatial scale vortex flow Zheng-Xiong Wang, 1 J. Q. Li, 1 J. Q. Dong, 2 and Y. Kishimoto 1 1 Graduate School of Energy Science, Kyoto University, Japan 2 Southwestern Institute of Physics and Zhejiang University, China

outline Introduction: Background Model and equations Simulation results:  Linear stabilization of ITG mode by vortex flows (VFs)  Nonlinear dynamics of ITG turbulence w/o zonal flows  Characteristics of Zonal flows in the presence of VFs  Transport structure Summary Effects of magnetic island on ITG evolution

Background Interactions in multiple T-S scales Zonal structures Interaction between a large spatial scale vortex flow and ion-scale micro-turbulence Turbulent transport due to ITG instability is crucially important for improved operation!

Radial profile of pressure Background Internal transport barrier (ITB) Radial profile of conductivity A inner region of reduced anomalous transport A key factor for ITBs EXB shear flows often lead to ITB formation

Background Mean flows (MFs)  toroidally and poloidally symmetric  suppress turbulence by radial flow shear effects  lead to internal transport barriers MFs  driven by external momentum source or by neoclassic effect x y x y (a) (b) eddy eddies Flow shearing Mode coupling Spectral scattering

Background Vortex flows (large scale VFs)  Varying in both radial and poloidal directions VFs  Often observed in natural and laboratory plasmas  Inevitable due to the inherent poloidal asymmetry of equilibrium pressure or magnetic field anisotropy  Induced by numerous plasma instabilities, such as Kelvin–Helmholtz (KH) or tearing mode, Moderate scale VFsLarge scale VFs Observation in experiments M.G. Shats et al., PRL, 2007 KH induced vortex in simulations Pegoraro et al., JoP, conference, 2008

Background purpose In comparison with MFs What are the roles of VFs in ITG evolution?  Linear stabilization of ITG mode by VFs  Direct interaction between VFs and ITG turbulence  Multiple interactions among turbulence, zonal flows, and VFs Understanding of these fundamental processes is crucially desirable but seldom

Model and equations A stationary large scale VF may be represented Streamer-like flows(SF) The VFs may then be understood as a combination of MF and SF structures. ---poloidal wave number ---flow strength

Model and equations: ITG with imposed VFs MF effect, SF effect Satisfying wave matching condition Multiplied Effect!!! Global ITG !!!

Model and equations: ITG with imposed VFs An initial value code with simulation box and In equations, for ZF component and for ITG fluctuations. Finite difference method in x direction with periodic boundary condition Fourier decomposition in y direction. ParametersNormalization

Numerical results Linear growth rate of ITG modes versus amplitude of flows 2-dimensional linear calculations VFs show strong stabilization effect in comparison with MFs and SFs

Linear growth rate of ITG modes versus amplitude of flows growth rate versus MF MFs Wang, Diamond, Rosenbluth, 1992 growth rate of Hamiltonian number L Mechanism MFs couple the modes with different radial scale The results of MFs agree with the previous works Hamaguchi and Horton, PoF B, (1992) Wang, Diamond, Rosenbluth, PoF B, (1992) 2-dimensional linear calculations Numerical results

2-dimensional linear calculations Linear growth rate of ITG modes versus amplitude of flows SF Linear evolution of each poloidal mode Growth rate of each SFs couple together all the poloidal modes and lead to a global ITG mode with an identical, greatly reduced growth rate Li and Kishimoto PoP (2008) Numerical results usual ITGWith SF

Linear growth rate of ITG modes versus amplitude of flows VF Growth rate of each Normalized spectra in linear ITG modes Multiplied effect of MF and SF Growth rate versus VF wave number Numerical results

Effects of magnetic shear s Numerical results Decreasing s reduces the stabilizing roles of MS, SF, and VF

Effects of magnetic shear s on role of MFs Numerical results ITG structures s=0.4 s=0.1 Decreasing s enlarges the width of ITG linear structure beyond the effective shear region of MFs MFs does not change mode width With MFs W/o MFs s=0.1

Numerical results Effects of magnetic shear s on role of SFs Streamer-like flows(SF) W/o SFsWith SFs s=0.1 ITG structures Ｌ ｎｕｍｂｅｒ for each poloidal mode

Numerical results When s becomes small, the increasing of L number is a destabilizing Effect, counteracting the stabilizing effect of poloidal mode coupling induced by SF Effects of magnetic shear s on role of SFs Growth rate of L with magnetic shear s Hamaguchi Horton, 1990 Terry, et al 1988 Growth rate of L with different s Small s large s

Numerical results Effects of magnetic shear s on role of VFs s=0.1 W/o VFs With VFs ITG structures VFs structure

3-dimensional nonlinear simulations w/o ZFs Contours of electric potential (a) w/o VF and (b) w VF (a) (b) Ion heat conductivity: Radial profile of conductivity in the quasisteady state. (a) For and (b) for <> the average in y and z directions and in time VFs can suppress turbulent transport more effectively than MFs with same amplitude Numerical results

3-dimensional nonlinear simulations w/o ZFs Numerical results Turbulent heat flux Profiles of electric potential fluctuations (EPF), pressure fluctuations (PF), and cross phase factor (CPF) for VFs Suppression by VFs is due mainly to synergetic decreases of EPF and CPF rather than PF This trend is similar to the results of MFs in toroidal simulations by Benkadda and Garbet X. Garbet, et al., PoP (2002). C.F. Figarella, et al., PRL (2003).

3-dimensional nonlinear simulations with ZFs Numerical results ZF equation Reynolds stress Coupling of VF and ITG Viscosity term low frequency finite frequency Three wave resonance condition Assumption of zero frequency VF 

3-dimensional nonlinear simulations with ZFs Numerical results Conductivity profile Time history of ZF profile Region (I) suppression by VFs Region (II) suppression by VFs is reduced due to finite frequency w/o ZF Region (III) suppression by ZFs Frequency spectra of ZF and ITB

Summary  VFs can stabilize linear ITG mode more effectively than MFs due to multiplied effect.  VFs can suppress turbulent transport more strongly than MFs with same amplitude.  An oscillatory ZF is found due to the interaction between VFs and ITG turbulence.  The resultant transport structure is consistent with that of ITBs observed experimentally.

Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ ITG mode structure Island structure

Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ Equations:

Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ Poloidal wave number of island Coupling mechanism: parallel compressibility or acoustic wave

Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ 2D linear calculation Time evolution of poloidal modes w/o islandwith island

Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ 2D linear calculation ITG growth rate versus island width Linear spectra of ITG mode in different island width Island can stabilize linear ITG mode

Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ 2D nonlinear simulation without ZFs Time evolution of heat conductivity in the whole system The presence of island will increase heat transport

Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ 2D nonlinear simulation without ZFs Magnetic field Contour of conductivity Contour of potential Fluctuation Contour of pressure Fluctuation O X Conductivity is enhanced around X point but O point of island

 Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ Dynamics of ZFs in the presence of islands (2d/3d simulation)  Combined effects of vortex flows and islands Further work