Numerical Simulation on Flow Generated Resistive Wall Mode

1. Numerical Simulation on Flow Generated Resistive Wall Mode Shaoyan Cui (1,2), Xiaogang Wang (1), Yue Liu (1), Bo Yu (2) 1.State Key Laboratory of Materials Modification by Laser, Ion and Electron Beams, Department of Physics, and College of Advanced Science & Technology, Dalian University of Technology 2. Department of Applied Mathematics, Dalian University of Technology

2. Outline • Introduction • Slab geometry • The model • Plasma region • Inner and outer vacuum region • The Wall • Linear expansion • Initial and boundary conditions • Numerical results • Summary

3. Introduction • J. A. Wesson, Phys. Plasma 5, 3816(1998) Magnetohydrodynamic flow instability in the presence of resistive wall • C. N. Lashmore-Davies, J. A. Wesson, and C. G. Gimblett, Phys. Plasma 6, 3990(1999) The effect of plasma flow ,compressibility, and Landau damping on resistive wall modes • B. M. Veeresha, S. N. Bhattacharyya, and K. Avinash, Phys. Plasma 6, 4479(1999) Flow driven resistive wall instability • C.N. Lashmore-Davies, Phys. Plasmas 8,151 (2001) The resistive wall instability and critical flow velocity

4. Introduction • The uniform flow of an ideal magnetohydrodynamic fluid along a uniform magnetic field is stable for all velocities, unless there is a boundary. • For incompressible plasma, if v0 > sqr(2)VA the instability is then generated. • For compressible plasma, the resistive wall instability occurred if v0 > cs, for the finite layer width and compressibility of the plasma significantly lowers the flow velocity required for instability to set in. • With various parameters, the linear instabilities are studied in many kinds of instances in our work.

5. Slab geometry J. A. Wesson, Phys. Plasma 5,3816(1998) Magnetohydrodynamic flow instability in the presence of resistive wall

6. Slab geometry

7. Physical Models (1) Plasma region • The linearized MHD equations:

8. Physical Models（2）In the inner and outer vacuum • The linearized equation for magnetic flux is Laplace’s equation:

9. Physical Models （3）In the resistive wall • Magnetic fluxsatisfies diffusion equation • Assuming that the wall is thin compared to the skin depth, so that the current can be assumed to be approximately constant across the wall, by integrating it across the wall, where

10. Linear Expansion • Perturbed variables

11. Initial conditions • Plasma region • Vacuum and wall

12. Boundary conditions (1) • The lower boundary • The upper boundary

13. Boundary conditions (2) • At the plasma-vacuum interface the boundary condition satisfies the force balance across the surface:

14. Numerical Results

15. Wave number kL/2p = 1,the growth rates versus the initial plasma flow velocities for cases with ,

16. Wave number kL/2p = 2, the growth rates versus the initial plasma flow velocities for cases with

17. Wave number kL/2p = 3,the growth rates versus the initial plasma flow velocities for cases with

18. (1)Unstable critical velocities vs. the wave numbers for cases with , and

19. (2)Linear growth rates vs. the wave numbers for cases with .

20. (3)Unstable critical velocities vs. viscosity for cases with

21. (4)Linear growth rates vs. viscosity for cases with

22. (5)Linear growth rates vs. plasma beta for the cases with

23. Linear growth rates vs. Cw for the cases with

24. Summary • Fixed on some parameters, the critical velocities are calculated for the different wave numbers; • With the increase of wave numbers, the mode is stabilized ; • The unstable regions vs. wave number and viscosity are presented. It is shown that the system tends to stable when the wave number and viscosity increase; • For the parameter beta, it is found that the linear growth rates increase as beta increases in realistic beta regime; • Cw has little effects on the linear growth rate

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