Influence of Striction Nonlinearity and Parametric Ion Cyclotron Turbulence on the Structure of Alfven Resonance in a Helical Confining Magnetic Field

Girka I.O.1 and Lapshin V.I.2
1Kharkiv V.N. Karazin National University, Svobody sq., 4, 61077, Kharkiv, Ukraine
2National Science Center “Kharkiv Institute of Physics and Technology”,
Akademichna str., 1, 61108, Kharkiv, Ukraine
Abstract. Helical inhomogeneity of the steady magnetic field causes coupling of separate spatial modes of
electromagnetic fields. Condition is derived under which striction nonlinearity governs spatial distribution of wave fields
within the Alfven resonance region. This condition is valid for the larger amplitudes of pumping wave than in axial
magnetic field, since in a helical magnetic field the Alfven resonance region is wider and the characteristic value of
amplitude of the fundamental harmonic is less than in axial magnetic field. Reverse effect of kinetic parametric ion
cyclotron instabilities of plasma on the pumping wave is studied. Account for the amplitudes of the first sidebands results
in noticeable increase of effective collision frequency as compared with its value in the case of axial steady magnetic
field.
Keywords: Stellarator, Helical Magnetic Field, Alfven Resonance, Striction Nonlinearity, Kinetic Parametric Ion
Cyclotron Instabilities, Effective Scattering Frequency.
PACS: 52.35.Bj, 52.50.Qt (PACS numbers choose from this list: http://www.aip.org/pacs/index.html)
FINE STRUCTURE OF AR
Helical inhomogeneity of the steady magnetic field is one of the main specific features of stellarators. We restrict
our consideration to the following representation of the magnetic field: B0 ( r, ,z ) erB0r e B0 ezB0z r r r r
ϑ = + ϑ ϑ + in
cylindrical coordinates,
B0r =δ sinlθ B0 , ( ) h 0
( l )
B0ϑ =∈ αr cos lθ B , B B (1 h cos lθ )
( l )
0 z = 0 −∈ . (1)
here θ = ϑ – α z, α=2π/L, L is the pitch length of the helical winding, ∈
h=lblIl(ksr), ( )( ) 1
l s
2
bl 8Jaks K k a lc = ′ − , a is the
radius of the cylindrical surface carrying a thin helical winding with the current J, Kl(ξ ) is McDonald function,
Il(ξ ) is modified Bessel function, the prime denotes the derivative with respect to the argument, l is the polarity of
helical coils, ks= lα, δ=(1 k )d h dr
( l )
s ∈ .
Unlike in the case of an axial magnetic field, in which each spatial mode of electromagnetic wave excited by an
external source propagates independently, helical inhomogeneity ∝exp (±ilθ) of the confining magnetic field 0 B
r
(1)
causes coupling of separate spatial modes of electromagnetic fields. That is why electromagnetic waves propagate in
helical magnetic field in the form of an envelope. Sidebands ∝exp{i[(kz m jks)z+(m±jl)θ – ωt]}, j=1,2,3… are present
in the envelope side by side with the main mode ∝exp{i[kzz+mθ – ωt]}. Below we restrict our consideration by the
following expansion for the radial electric field of the wave:

This condition (15) is valid (under the same other conditions) for the larger amplitudes of pumping wave than in a
straight magnetic field, since in a helical magnetic field the AR is wider and the characteristic value of amplitude of
a fundamental harmonic of a radial electrical field is less than in a straight magnetic field. The distribution of RF
fields within AR in the case the condition (15) nevertheless is valid, is studied numerically in the paper [7].
Let us consider reverse influence of kinetic parametric ion cyclotron instabilities of plasma on a pumping wave
which excites them. At non-linear stage of growth of cyclotron oscillations the ions are scattered by turbulent
pulsations, so that one can speak of the effective scattering frequency, i.e., the effective collision frequency [8]. The
account for turbulent absorption of MHD waves can be carried out by replacement of a collision frequency of
plasma particles by an effective collision frequency with turbulent fluctuations,
( )
 


 


+  


 

 

 

→ = 2
pi
2 2
z
2
Ti
3
Ti 0
2
0
i
1eff
c
1
c 1 k c
B
c E
B
16 nT
υ υ ω
π
ε ε . (18)
As the effective scattering frequency of ions on turbulent fluctuations is determined by amplitude of an electrical
field of the pumping wave and the amplitudes of sidebands are small as compared with the amplitude of the
fundamental harmonic, the availability of the helical non-uniformity of the confining magnetic field can not cause
the determining influence on turbulent heating of plasma of a stellarator. Nevertheless, account for the amplitudes
(9) Er
(±1) of the first sidebands results in noticeable increase of ε1eff within the AR as compared with its value in the
case of straight confining magnetic field, by the order of magnitude ε1eff→ε1
(0)
eff+Δε1eff, where
Δε1eff≈(δkz
2/(ks
3a*))2/5 ε
1
(0)
eff.
Power WT, absorbed per unit of plasma cylinder within AR region due to effective dissipation – ion cyclotron
turbulence, equals
WT~(ω/4) rA ΔrNL ε1eff |Er|2∝|Er|7. (19)
Here ΔrNL is the width of of the region in which the striction is sufficient. Thus the account for the amplitudes Er
(±1)
of the first sidebands (caused by the helical inhomogeneity of the confining magnetic field in stellarators) results in
the enhancement of the Alfven plasma heating: WT→ WT(0)+ ΔWT, where ΔWT ≈(δkz
2/(ks
3a*))2/5 WT(0).
ACKNOWLEDGMENTS
The research was supported by Science and Technology Center in Ukraine, Project # 2313.
REFERENCES
1. I. O. Girka, Physica Scripta 73, (2006). Accepted for publication.
2. I. O. Girka, V. I. Lapshin and R. Schneider, Plasma Physics and Controlled Fusion 45, 121–132 (2003).
3. V. V. Dolgopolov and K. N. Stepanov, Nuclear Fusion 5, № 4, 276 – 278 (1965).
4. I. O. Girka and P. K. Kovtun, Plasma Physics Reports 26, 33-40 (2000).
5. T. A. Davydova and V. M. Lashkin, Soviet Journal of Plasma Physics 16, 907 – 915 (1990).
6. R. Klima, Czechoslovak Journal of Physics B18, 1280 -1291 (1968).
7. V. I. Lapshin, K. N. Stepanov and V. O. Shtrasser, Soviet Journal of Plasma Physics 18, 344-347 (1992).
8. V. P. Silin, Parametric Effect of Powerful Radiation on Plasma, Moscow: Nauka, 1973.

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