1. The Residual Zonal Flow in Toroidally Rotating Tokamak Plasmas
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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
天堂寨

2. Background introduction Gyrokinetic equation in rotating plasmas
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Background introduction
Gyrokinetic equation in rotating plasmas
Solution to GK equation
Summary

3. 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)

4. 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

5. Introduction(3) What is the residual ZF?
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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

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

7. Basic equation(2) To satisfy equilibrium quasi-neutrality
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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

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

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

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

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

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

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

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

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

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

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Approximate form(2)
Evaluation of orbit average 17

18. ASIPP
Approximate form(3) 18

19. Summary The general form is the same as RH form.
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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