STUDYOFALFVENWAVESINSTELLARPLASMAS
pB2
ANDINPLASMASOFMAGNETICCONFINMENT(TOKAMAK)
Amel BENAHMEDetAbdelazizSID
PRIMALaboratory,PhysicsDepartment,FacultyofSciences,UniversityofBatna
E-mail:
ABSRACT:Alfvènwavesarelowfrequencymagnetohydrodynamicsplasma wavesoroscillationswhich propagateinthedirectionofthemagneticfield.Alfvèn wavesare of fundamental importanceinthe behaviorof manylaboratoryandspaceplasmas.Theyplayimportantrolesintheheating, stabilityandtransportofplasmas. Solar plasmasarestructured andstratifiedboth vertically and horizontally.Thestability is discussed of the drift- Alfvènwavewhichisdrivenbytheequilibriumdensitygradient,collisionalsolarplasma,includingtheeffects ofboth hot ionsandafinite ionLarmorradius.Ananalyticalmodeanalysis isusedforthedescriptionofthe wavesinspatially unlimitedplasma.Intheanalysis ofmodes,theexchange ofidentity betweentheelectrostatic and electromagneticmodes isdemonstrated.Theresults areapplied tocoronal andchromosphericplasmas.
KEYWORDS: Alfvèn waves, instabilities, magnetized plasmas, stellar plasmas
1. Model and derivations:
The momentumequations for ions and electrons are:
⎡∂vi
⎤ ⎛ ∂AZ ⎞
mini⎢ ∂
+(vi.∇)vi⎥=eni⎜−∇φ−
ez +vi∧B⎟−kTi∇ni −∇.Πi −miniνivi
(1)
⎣ t ⎦ ⎝ ∂t ⎠
⎡∂ve
⎤ ⎛ ∂AZ ⎞
mene ⎢ ∂
+(ve.∇)ve⎥=eni⎜−∇φ−
ez +ve∧B⎟−kTe∇ne −∇.Πe −mene(νeve −veivi)
(2)
⎣ t ⎦ ⎝ ∂t ⎠
The parallel electron dynamics, in the limit when ion motion is predominantly polarized in the perpendicular plain, is described by:
⎛ ∂
⎞ ∂φ1
kTe ∂ne1
meve 2
⎜ +ve0∇⊥⎟Az1+ −
∂t ∂z n e ∂z
−
μe2n
∇⊥Az1=0
(3)
⎝ ⎠ e0
0 e0
and by usingthe Ampere law, the electron continuity becomes:
∂ne1 + 1 (e
∧∇φ).∇n
+ 1 ∂ ∇2A =0
(4)
∂t B0
⊥ 1 ⊥ e0
μ0e∂z
⊥ z1
By the same method, the ion continuity equation is finding and combined by (4) using the quasi-neutrality to obtain:
⎛ ∂ ⎞
∂ kT⎛ ∂ ⎞
∂ ⎛ kT ⎞
⎜ +v⎟∇2φ
+c2 ∇2A
+ i ⎜
+v⎟∇2n
−ρ2 ∇4⎜φ
+ i n⎟=
(5)
⎝∂t
i ⊥ 1
⎠
a ∂z
⊥ Z1
en0⎝∂t
i ⊥ 1
⎠
i ∂t ⊥⎝ 1
0
en0 ⎠
The given set ofEqs (3), (4) and (5) will be usedin the description of“The drift-Alfvèn waves in solar plasma”.
2.Waves in unlimited plasma:
In Cartesian geometry, for perturbations≈exp(−iωt+ikyy+ikzz). Eqs (3), (4) and (5) yield
⎡ ⎛ ⎞⎤
ω3−ω2⎢ω
+ω −i⎜δ+ vi ⎟⎥+i vi [ωω
−k2k2c2(ρ2+ ρ2)]=0
(6)
∗e ∗i ⎜
1+k2ρ2⎟
1+k2ρ2
∗e ∗i
y z a s i
⎝ y i ⎠ y i
2.1. Coronal plasma:
To discuss the roots and increments/decrements, we solve eq (6)numerically bytaking parameters values that are typical for the solar atmosphere (Fig1, 2 and 3)
Figure1:Frequenciesωrand
incrementωiintermsofthecouplingtermskyρs.
Figure2:Thedriftwave Figure3:Thefrequencyof frequency(fullline),anditsincrement thekineticAlfvènmodes (dashedline)intermsofthedensityscalelengh. correspondingtothedrift
modeinFig.2.
2. 2. Application to the chromosphere:
Equation (6)is solved also for the quiet sun parameters of “The chromospheric plasma”.
3. Confinement and heating of plasmas by Alfvèn waves in the Tokamak:
What’s Tokamak?
The word Tokamak is an acronymfor the Russian wordsToroidal’nayaKamera Magnitnoi
Katushki, meaning toroidal chamber and magnetic coil.
3.1. Plasma heating:
The most efficient way to heat a tokamakplasma is by passing through it a current induced by the primary coil (as seen fromFig. 4) This coil is the primary circuit ofatransformer in which the plasma ring constitutes the secondary circuit. It works like an electric heater, the amount of heat generated depending on thecurrent and the resistance of the plasma.
Unfortunately, the plasma resistivity decreasesasthetemperature rises and the heating process becomes less effective. The maximum temperature that can beachieved intokamaks by the resistance heating (or homic heating) method is about 3*107 k, twice the temperature
in the center of the sun but less thanneeded to startup a reactor, about 108K. In tokamak experiments auxiliaryheating is used to reach temperaturescurrentlyashighas5*108k (more than 30 times the temperature at the sun-center).The two main methods of additional heating is by the injection of high-energy neutral particle beams and radiofrequency waves of various
types.
Figure1:Maincomponentsofthetokamak
Conclusion:
The frequency of Alfvèn waves modes dependson the density gradients (as seen from 1, 2 and 3), and on the coupling with thecorrespondingdrift mode which is driven by the density gradient. The change in the frequency of the Alfvèn modes should be taken into account in the analysis of observed modes in the solar corona. In fact, thisintroduces acertain freedomin
the fitting ofobservations into the theorical modeling.We note in the particular that the widely used one-fluid (MHD) model is intrinsicallyunabletodescribethesephenomena.
References:
[1]Bostick,Winston H and MortonA; Experimental Demonstration in the laboratory of the Existence of Magneto-HydrodynamicWaves in Ionized Helium
87 671-677 (1952).
[2] Berthod, Harris and Hope H; World-Wide Effects of Hydro magnetic waves Due toArgus
65 2233(1960).
[3] Alfven H;Cosmic Plasma, Holland (1981).