Bachmann, P.

Suender, D.

English

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REACTION-DIFFUSION PROCESSES IN IMPURITY SEEDEDRADIATIVE PLASMASP. Bachmann and D. SünderMax-Planck-Institut für Plasmaphysik, EURATOM AssociationBereich Plasmadiagnostik, D-10117 Berlin, Germany1. IntroductionThe aim of this paper is to describe impurity seeded radiative plasmas by a single reactiondiffusion equation (RDE) resulting from multi-fluid equations (cp. [1]). The starting point of [1]are the one-dimensional, time-dependent hydrodynamic equations describing the self-consistentdynamics of the plasma hydrogen ions (i), the impurity ions (j) with the charge state Zj , andthe electrons (e) along the magnetic field lines. We start here with the currentless plasma modelequations of [1] which result in the RDE when the Lagrangian mass variable and the equationof state are included. The influence of the impuries carbon and beryllium on the RDE is treatedby solving steady and time-dependent problems.2. Reaction Diffusion EquationApplying the average ion approximation with the average charge< Zj >, assuming all velocitiesto be equal, ve = vi = vj ≡ v, introducing the mass density ρm = mini + mjnj , the totaldensity N = ne + ni + nj = 2ni + (1+ < Zj >)nj , the pressure p = N · T (T - temperature)we obtain [1]:dNdt+N∂v∂x= 0, (1)ρmdvdt+∂p∂x= 0, (2)32dpdt+52p∂v∂x− ∂∂xκe(T )∂T∂x= H(x)− (ni+ < Zj > nj)njLrad(T ). (3)ddt=∂∂t+ v∂∂x, ni =12[N − (1+ < Zj >)nj]; (4)ne,i,j - densities, me,i,j - masses, H - external heat source. To close the system of equations (1)- (3) one needs the impurity density nj in dependence of the other model functions.Introducing Lagrangian coordinates τ, y (y - mass variable):τ = t, y(x, t) =∫ xx1(t)dx′N(x′, t),ddt=∂∂τ,∂∂x= N(x, t)∂∂y, (5)x1(t) = {x|v(x, t) = 0}, and the equation of state, p = NT = p(T ), p(T ) - given function,leads to the Lagrangian RDE for the temperature:∂T∂τ− 25ξp∂∂yp(T )κeT∂T∂y=25Tp(T )ξp{H(y)− 12[N + (< Zj > −1)nj]njLrad(T )}, (6)ξ−1p = 1−(2/5)∂ ln p(T )/∂ lnT . nj is expressed either by (i)nj = ξjN, ξj = const (simplifiedapproximation - SA) or (ii) as a function of T : nj = nj(T ).The impurity affects (i) the radiation loss term QR = 12 [N + (< Zj > −1)nj]njLrad(T )(Lrad - radiation loss function) and (ii) the electron heat conduction coefficientκe = C3(Zeff )neT/(meνee) = C3(Zeff ) · κ0T δ, δ = 5/2 (νee - ee collision frequency, κ0 = const); C3(x) =3.9(1 + 1.7x)/[(1 + 2.65x)(1 + 0, 28x)], Zeff = (ni/ne)(1 + Z0), Z0 = (nj/ni)< Z2j >. nj is determined by the differential equationddTln nj+ < Zj >ddTln[p(T )T+ (< Zj > −1)nj]=Gj − 1T, (7)derived from the impurity balance equation (cp. [1]); Gj = α− β− < Zj >, α = [C2(Zeff ) +C2(Z0)] < Z2j >, β = C2(Zeff )Zeff < Zj >, C2(x) = 2.2(1 + 0.52x)/[(1 + 2.65x)(1 +0.28x)], < Z2j >= (1/nj)∑Zj nZjZ2j . Solutions of Eq. (7) under the condition of an isobaricchange p = p0 will be included.3. EquilibriumThe possible equilibrium states both in Lagrangian and Eulerian coordinates are determinedby very similar equations, i.e. the Lagrangian representation has no advantage in investigatingsteady states. Therefore we will consider equilibrium solutions in Eulerian coordinates:∂∂xκ∂T∂x+H −QR = 0, QR =1 + (< Zj > −1)ξj2ξj(p0T)2Lrad, ξj =njN, (8)x ∈ X = [xmin, xmax]. ξj is either (i) calculated or (ii) estimated by introducing averaged values(SA): < ξj >= a, < C3 >= b; a, b = const.We apply sheath boundary conditions:κ(T )∂T∂x∣∣∣∣∣x=xmin= α0T (0, t)β0, κ(T )∂T∂x∣∣∣∣∣x=xmax= −αnT (1, t)βn, α0,n ∝ p0. (9)Parameters: X = [0, L], L = 60 m - connection length; neτ = 1016 m−3s; κ0 = 1.5 1022(eV)−5/2/(ms); H(x) = H0 = const = 6 1025 eV/(m3s) - symmetrical task; p0 = 1021 eV/m3,nj0 = nj(T = 1 eV) = 1018cm−3;α0 = αn = 2 1026 (eV)1/2/(m2s), β0,n = 0.5 ; ForLrad, < Zj >the ADPAK data are used.Impurity data for C: Fig. 1. ⇒ aC=0.1, bC=1.7. Impurity data for Be: Fig. 3. ⇒ aBe=0.15,bBe=1.85.We compute possible solutions for the symmetrical task, i.e. we solve initial valueproblems at the maximum temperaturesTmax = T (L/2) 80 to 120 eV with vanishing derivativesand compare these with SA results with the estimated constants aC, bC, aBe, bBe. The exactsolutions are displayed in Figs. 2 (a), 4 (a) as profiles T (x), and as phase space portraits in thephase plane (T, Tx) where the ”boundary value curves” (9) as dashed lines are also displayed.Solutions to our boundary value problem are curves whose beginning and ending points are onthe curves (9) in the phase plane. There exists only one solution, i.e. no bifurcation occursfor the parameters used. SA results: Figs. 2 (b), 4 (b). Comparison: Both the phase spaceportraits and the boundary value curves are changed. The resulting solution to our boundaryvalue problem (dashed line) shows that the SA gives only a qualitative agreement with the exactsolution, but it cannot be used for higher or lower temperatures.4. Time-Dependent SolutionConsidering the SA equilibrium solution for carbon (dashed line in Fig. 2 (b)) in Euleriancoordinates, transforming it to Lagrangian coordinates (18) with x1 = L/2 ⇒ T (y) (full linein Fig. 5); y(0) = −y(L)=-0.334 p0 m−2; modulating this state, and solving the LagrangianRDE (6) for this initial temperature distribution with Dirichlet’s boundary condition T (y(0)) =T (y(L)) = 80 eV, proves this state to be the only existing steady state which is stable (Fig. 5).5. SummaryThe impurities affect, with respect to their nj dependence, the reaction diffusion process viathe radiation loss term and the electron heat conductivity. This is demonstrated for carbon andberyllium: For both impurities the density behaves non-monotonically for temperatures lowerthan 10 eV. Mean quantities are estimated that can be used in a simplified model (SA) to solve theRDE. Equilibrium: We consider the symmetrical task to compute possible solutions to sheathboundary value problems (phase space portraits and temperature profiles) in the laboratoryframe. Transformation to Lagrangian coordinates: The boundary value problems in Euleriancoordinates lead to Dirichlet problems in Lagrangian coordinates. The solution (for carbon) ofthe RDE shows the time evolution to the above mentioned steady state which thus is proved tobe stable.References[1] P. Bachmann and D. Sünder: “1D Multi-Fluid Plasma Models.”Report IPP 8/13 (January 1998);P. Bachmann, D. Sünder: Contrib. Plasma Phys. 38 (1998) 290.a    = 0.1Cb    = 1.7CFigure 1. nj, nj/N, njneLrad, C3(Zeff ) asfunctions of T for C.a) b)Figure 2. Profiles T (x) and phase planeportraits (T, Tx) for C; (a) - exact solution, (b) -SA with aC=0.1, bC=1.7.a     = 0.1Beb     = 0.1Be Figure 3. nj, nj/N, njneLrad, C3(Zeff ) asfunctions of T for Be.a) b)Figure 4. Profiles T (x) and phase planeportraits (T, Tx) for Be; (a) - exact solution, (b)- SA with aBe=0.15, bBe=1.85.Figure 5. Time evolution of the temperature profile in Lagrangian coordinates to the steady state (fullline) which corresponds the equlibrium solution in Fig. 6 (b) in Eulerian coordinates.

Reaction-Diffusion Processes in Impurity Seeded Radiative Plasmas

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Reaction-Diffusion Processes in Impurity Seeded Radiative Plasmas

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