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

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

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

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

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

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

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

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

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

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.

Thank you for your attention.

References M.N. Rosenbluth, Bull. Am. Phys. Soc., Ser. II, 4 (197) 1959. (unpublished) R.Z. Sagdeev, V.D. Shafranov.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

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

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

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

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

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

Estimations for highly-anisotropic uniform plasma ? Цидулко, Черноштанов, препринт ИЯФ 2009-3 Черноштанов, Цидулко, Вестник НГУ, 2010 - shape of resonant ions

Addition of cold plasma

Instability growth rate

Dielectric permeability for the non-uniform plasmas

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

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

The stability margin ( )