The Residual Zonal Flow in Toroidally Rotating Tokamak Plasmas ASIPP The Residual Zonal Flow in Toroidally Rotating Tokamak Plasmas 周登 中科院等离子体所 中科院磁约束等离子体理论中心 2013.7.4 天堂寨

Background introduction Gyrokinetic equation in rotating plasmas ASIPP Background introduction Gyrokinetic equation in rotating plasmas Solution to GK equation Summary

ASIPP Introduction ZFs are electrostatic modes with mode number . They are driven by turbulence, like ITG Turbulence The long term evolution of the initial perturbation in a collisionless plasma was investigated by Rosenbluth and Hintin (PRL98)

ASIPP Introduction(2) Plasma flows ( Poloidal or Toroidal) seems ubiquitous in tokamaks. So we ---- need to investigate ZFs in tokamaks with flow. Zonal Flow as an eigen-mode has been investigated by some authors (Wang PRL06 and Wahlberg PRL08) in toroidally rotating tokamaks, and ( Zhou POP10) in poloidally rotating plasmas 4

Introduction(3) What is the residual ZF? ASIPP Introduction(3) What is the residual ZF? Nonlinear interaction of DW induces an initial axisymmetric distribution, then classic polarization causes an axisymmetric electrostatic potential in a few cyclotron time, while in long term the potential is determined by neoclassic polarization, this is the so called Residual Zfs What is the effect of plasma equilibrium flow on Residual Zfs? = This work 5

Basic equation The gyro-kinetic equation in toroidally rotating plasma ASIPP Basic equation The gyro-kinetic equation in toroidally rotating plasma               6

Basic equation(2) To satisfy equilibrium quasi-neutrality ASIPP Basic equation(2) To satisfy equilibrium quasi-neutrality Total perturbation distribution Equilibrium distribution Eikonal form The evolution of ZF is determined using quasi-neutrality condition 7

How to Solve this GK Equation ASIPP How to Solve this GK Equation Explicit form 8

How to Solve this GK Equation(2) ASIPP How to Solve this GK Equation(2) The analytical solution obtained in two limit: 9

Solve GK Equation at low speed ASIPP Solve GK Equation at low speed Analytical solution in low rotation Expanding by ordering in O(ω/ωt) The first order term 10

Solve GK Equation at low speed(2) ASIPP Solve GK Equation at low speed(2) The 2nd order term Taking an orbit averaging Taking Laplace transformation for ions 11

Solve GK Equation at low speed(3) ASIPP Solve GK Equation at low speed(3) The source term taking the form The Laplace transformation of quasi-neutrality equation 12

Solve GK Equation at high speed ASIPP Solve GK Equation at high speed Analytical solution in high rotation Expanding by ordering in O(ω/ωt) The first order term 13

Solve GK Equation at high speed(2) ASIPP Solve GK Equation at high speed(2) The 2nd order term 14

The final form of Residual ZFs ASIPP The final form of Residual ZFs Exactly the same form as RH, but different definition of and bounce average. 15

ASIPP Approximate form large aspect ratio tokamak with concentric circular magnetic surface. 16

ASIPP Approximate form(2) Evaluation of orbit average 17

ASIPP Approximate form(3) 18

Summary The general form is the same as RH form. ASIPP Summary The gyrokinetic equation is solved as an initial value problem to study the ZF dynamic in rotating plasma. The general form is the same as RH form. The residual ZF decrease with increasing rotation, for rotation at sound speed it value is almost 1/3 of static plasmas 19