1. Chernoshtanov I.S., Tsidulko Yu.A.
Alfvén ion-cyclotron instability in a mirror trap with highly-anisotropic plasma
Chernoshtanov I.S., Tsidulko Yu.A.

2. Outline Motivation Simple estimations
Specifics of wave propagation and ion motion
Integral equation for perturbations in non-uniform plasma
Analytical solution for asymptotic case
Summary picture
Conclusions

3. Motivation GDT end-cell:
The purpose of this work: AIC instability in the highly-anisotropic case.
Traditional stability scaling:

4. Estimation unstable when
R.C. Davidson, J.M. Ogden. Phys. Fluids, 1975
unstable when
( : resonant ions move along isolines of distribution function)
For plasma with finite scale
stability

5. Specific of wave propagation
Watson’s case:
Our case:
Reflection from turning points ( )
Reflection from plasma non-uniformity
WKB

6. Specific of ion motion Bounce frequency:
Unstable perturbation frequency:
Local dispersion relation
Resonances:
Non-locality

7. Non-uniform plasma. Eigenmode equation.
The equations for the circularly polarized Fourier components:
Here
For , and

8. Analytical solution at
The equation is
Here ,
Wave vanishes at if
Minimal asymptotic stability
criterion

9. The AIC instability threshold ( )
GDT end-cell: the margin density:

10. Conclusions
A linear theory of AIC instability for highly anisotropic mirror-confined bi-Maxwellian plasmas is presented.
The asymptotic stability threshold and spatial distribution of the eigenmodes are found analytically in the limit of infinite anisotropy.
The wave energy localization length as well as the unstable mode wavelength are of the order of anisotropic plasma scale length.
Numerical results of the theory are in approximate agreement with preliminary results of GDT end-cell experiment.
The mirror-confined highly anisotropic plasma can be much more stable than it follows from the traditional scaling.

11. Thank you for your attention.

12. References
M.N. Rosenbluth, Bull. Am. Phys. Soc., Ser. II, 4 (197) (unpublished)
R.Z. Sagdeev, V.D. Shafranov. JETP. 12, 1960.
R.C. Davidson, J.M. Ogden. Phys. Fluids. 18(8), 1975.
D.C. Watson. Phys. Fluids, 23(12), 1980
T.A. Casper, G.R. Smith. Physical Review Letters 48(15), 1982
R.F. Post. Nuclear Fusion 27(10), 1987

13. Bi-Maxwellian plasma in the non-uniform magnetic field
Axisymmetric magnetic field
Distribution function:
plasma density:

14. The absolute instability in the uniform plasma
The criterion for absolute instability in the collisionless bi-maxwellian plasma with is
The perturbation frequency and wave number are

15. Numerical results Eigenvalues of the equation at fixed
Eigenfunctions in z representation:

16. Simulation of non-linear saturation in uniform plasma
R.C.Davidson, J.M.Ogeden, Phys.Fluids, 1975
P.Hellinger et al, Geophysical Research Letters, 2003
Plasma compression in the magnetosphere

17. Nonlinear saturation in highly-anisotropic case
Possible scenario: the instability modifies ion distribution function only near resonant orbits.
Nonlinear saturation does not lead to substantial anisotropy reducing

18. Estimations for highly-anisotropic uniform plasma?
Цидулко, Черноштанов, препринт ИЯФ
Черноштанов, Цидулко, Вестник НГУ, 2010
- shape of resonant ions

19. Addition of cold plasma

20. Instability growth rate

21. Dielectric permeability for the non-uniform plasmas

22. Qualitative consideration
Resonant condition
Existence of inverse population on resonant orbits
instability
Для неустойчивости на резонансных траекториях частиц с большей энергией должно быть больше.

23. TMX, 2XIIB
TMX:
2XIIB:
R TMx=10, R 2XIIB= 7

24. The stability margin ( )