Alfvén ion-cyclotron instability in a mirror trap with highly-

anisotropic plasmaChernoshtanov 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:

• Traditional stability scaling:

•The purpose of this work: AIC instability in the highly-

anisotropic case.

Estimation• 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

Reflection from turning points ( )

Specific of wave propagation

Reflection from plasma non-uniformity

Watson’s case:

kk

Our case:

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 isHere , • 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 (unpublished)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 criterion for absolute instability in the collisionless bi-maxwellian plasma with is

• The perturbation frequency and wave number are

The absolute instability in the uniform plasma

Numerical resultsEigenvalues 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

Possible scenario: the instability modifies ion distribution function only near resonant orbits.

Nonlinear saturation in highly-anisotropic case

Nonlinear saturation does not lead to substantial anisotropy reducing

Estimations for highly-anisotropic uniform plasma

Цидулко, Черноштанов, препринт ИЯФ 2009-3Черноштанов, 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

TMX, 2XIIB• TMX:

• 2XIIB:

The stability margin ( )