ASPECTS OF THE DIFFUSION OF ELECTRONS AND IONS IN TOKAMAK PLASMA

Transcription

1 Dedicated to Professor Oliviu Gherman s 8 th Anniversary ASPECTS OF THE DIFFUSION OF ELECTRONS AND IONS IN TOKAMAK PLASMA M. NEGREA 1,3,, I. PETRISOR 1,3, DANA CONSTANTINESCU,3 1 Department of Physics, University of Craiova, 13 A.I. Cuza Street, RO-585, Craiova, Romania Department of Applied Mathematics, University of Craiova, 13 A.I. Cuza Street, RO-585 Craiova, Romania 3 Association Euratom-MECI, Romania Received November 1, 9 Two distinct problems concerning the anomalous transport in tokamak plasma were analysed. The first one is related to the diffusion of ions in a stochastic magnetic field with curvature starting from Langevin equations of the guiding centre approximation. We analysed the influence of the drift Kubo number, the magnetic Kubo number and of the anisotropy on the diffusion of ions. The second problem is related to the diffusion of electrons in a combination of an electrostatic stochastic field with an unperturbed sheared magnetic field in slab geometry. The global effects of the parameters on the diffusion tensor components are exhibited. 1. INTRODUCTION A central issue for fusion consists in the study of turbulence phenomena in a plasma at high temperature. The magnetic and electrostatic turbulence appear as plausible candidates for determining the anomalous transport properties of a hot magnetized plasma. We have treated in our paper two distinct problems concerning the anomalous transport in tokamak plasma. The first one is related to some aspects of the diffusion of ions in a stochastic magnetic field with curvature starting from Langevin equations of the guiding centre and applying specific approximations. We analysed the influence of the stochastic anisotropy [1]-[4] of the magnetic field on the diffusion of ions when the magnetic Kubo number and the drift Kubo number are fixed and we have shown that the stochastic drifts provide a decorrelation mechanism of the particles from the magnetic lines. Because the Corrsin approximation ignores the trapping effect which necessarily exists in a relatively strongly turbulent plasma, the method of the decorrelation trajectories was applied [5]. The trapping effect is more pronounced, the larger anisotropy parameter is. The diagonal coefficients start with a linear part, defining a ballistic regime followed by a trapping regime before reaching the satura- mnegrea@yahoo.com Rom. Journ. Phys., Vol. 55, Nos. 9 1, P , Bucharest, 1

2 114 M. Negrea, I. Petrisor, Dana Constantinescu tion asymptotic value. Thus the global trapping effect is enhanced at larger stochastic magnetic drift has practically the same influence on ion s diffusion as the magnetic shear on the intrinsic diffusion of magnetic field lines [6]. The second problem is related to some aspects of the diffusion of electrons in a combination of an electrostatic stochastic field with an unperturbed sheared magnetic field in slab geometry. The global effects of electrostatic Kubo number and the shear parameter on the running and asymptotic diagonal diffusion tensor components are exhibited. The running diffusion coefficients start with a linear part characteristic to a ballistic regime. In all of them a trapping effect appears for large enough values of electrostatic Kubo number and/or the shear parameter. The trapping regime does not appear at the same time interval for the two diagonal coefficients and does not appear with the same strength in the two running diffusion coefficients. This behaviour is expected because of the existence of the shear term. The paper is organized as follows. In section 1 we presented the Langevin equations for the ions and the framework of the decorrelation trajectory method. In section the magnetic Langevin equations for ions were established. In section 3 the diffusion coefficients for the ions were analysed for different values of the stochastic anisotropy parameter and fixed values for the magnetic and drift Kubo numbers. In section 4 we presented the model for the electron s diffusion and in the section 5 we analysed the diffusion coefficients. The conclusions were presented in the last section.. MAGNETIC LANGEVIN EQUATIONS FOR THE IONS We consider the model with a shearless slab geometry for the confining magnetic field with a strong component B along the z axis and a fluctuating perpendicular component in the (X,Y ) plane: B(X;Z) = B {e Z + βb X (X;Z)e X + βb Y (X;Z)e Y } (1) where X =(X,Y ) and β is the dimensionless parameter measuring the amplitude of the magnetic field fluctuations, which are described by the dimensionless functions b i (X;Z), i = (X,Y ), taken to be Gaussian processes. We assume that the unperturbed field is strong enough so that the motion of the particles can be described in the drift approximation. We are interested in evaluating the influence of the magnetic drifts on particle behaviour. The time variation is slow so that the induced electric field is negligible. Thus, the guiding centre equations of motion are the following: dx dt = βb X (X;Z)V + V drx ()

3 3 Aspects of the diffusion of electrons and ions in tokamak plasma 115 dy dt = βb Y (X;Z)V + V dry (3) dz dt = V (4) where V is the component of the particle velocity parallel to the reference magnetic field and V dr (V drx, V dry ) is the drift velocity which, for our stationary magnetic field, is given as: V dr 1 [ ] Ω n µ B + V (n )n (5) where Ω = eb/mc is the Larmor frequency, n = B/B is the unit vector along the field line, µ = V /B is the magnetic moment and V is the component of the particle velocity perpendicular to the reference magnetic field. Since the amplitude of the magnetic field fluctuations is very small, we can neglect the terms in [b i (X;Z)] k, k >, in the drift velocity and retain only the dominant, first order, terms [7]: V dr X V Ω V dr Y V Ω b Y [X(Z);Z] Z V Ω b ZY [X(Z);Z] (6) b X [X(Z);Z] Z Secondly, the parallel velocity V appearing in eq. (4) is the particle velocity along magnetic lines and is considered to be constant and equal to the thermal velocity V th. Ψ is taken as a Gaussian random field, spatially homogeneous, isotropic in the (X,Y ) plane, and stationary. The Eulerean autocorrelation function is taken as [6]: where and V Ω b ZX [X(Z);Z] (7) M(X;Z) = Ψ(;)Ψ(X;Z) = M 1 (X)M (Z) (8) M 1 (X) = β Bλ x exp( X λ )exp( Y x λ ) (9) y M (Z) = exp( Z ) (1) Here, three characteristic correlation lengths are defined: the parallel correlation length λ z and the perpendicular correlation lengths λ x and λ y. Dimensionless coordinates {x = (x,y),z} may then be defined by x = X/λ x, y = Y/λ y and z = Z/λ z ; we define also the magnetic dimensionless potential as ψ = B 1 β 1 λ 1 x Ψ. Using these dimensionless quantities and the assumptions which we have made above, the system given in eqs. () - (4) becomes: [ ] ψ(x(z);z) dx(z) dz =ΛK m ψ(x(z);z) y + K dr x=x(z) λ z z v x [x(z);z] = K m b x (x;z) K dr b zy (x;z) x x=x(z) (11)

4 116 M. Negrea, I. Petrisor, Dana Constantinescu 4 dy(z) dz = ΛK m ψ(x(z);z) x + Λ K dr x=x(z) z v y [x(z);z] = ΛK m b y (x;z) + ΛK dr b zx (x;z) [ ] ψ(x(z);z) y x=x(z) The stochastic components βb b are represented by a vector potential in order to have the condition of zero divergence automatically fulfilled: (1) A(X;Z) = B λ x β ψ(x;z) e z (13) The Kubo numbers that appear in eqs. (11) - (1) are [6] K m = β λ z λ x, Λ = λ x λ y, K dr = β V th Ωλ x (14) β is the dimensionless amplitude of magnetic field fluctuations (considered here as relatively strong, i.e. β 1 ), V th is the thermal velocity of the ions and Ω their Larmor frequency. Usually V th Ω 1 1 m (Larmor radius for ions), λ x λ y 1 m and λ z 1 m. 3. DIFFUSION COEFFICIENTS FOR IONS In the following we apply the DCT method [5] where general expressions for the running (and consequently asymptotic) diffusion coefficients can be derived for different Kubo number regimes. An extensive explanation of the DCT method can be found in the excellent book of Radu Balescu [8]. The main idea of this method concerns in the study of the Langevin system (11-1) not in the whole space of the realizations of the potential fluctuations; the whole space is subdivided into subensembles S, characterized by given values of the potential and of the different fluctuating field components at the starting point of the trajectories. The exact expression of the Lagrangian correlation can be written in the form of a superposition of Lagrangian correlations in various subensembles. We define a set of subensembles S of the realizations of the stochastic sheared magnetic field that are defined by given values of the potential ψ and characteristic magnetic field fluctuation b and b z in the point x = at the moment z = : ψ(;) = ψ, b(;) = b, b z (;) = b z (15) We will not expose here the decorrelation trajectory method and the reader can use for further details in [5] and [8]. We only write the expressions of the Lagrangian correlations and the diffusion coefficients. The global Lagrangian correlations are: L ij (z) = dψ db db zp (ψ,b,b z) v i (;)v j [x(z);z] S (16)

5 5 Aspects of the diffusion of electrons and ions in tokamak plasma 117 where the components of v [x(z);z] are defined as: v x (x;z) = K m b x (x;z) K dr b zy (x;z) v y (x;z) = ΛK m b y (x;z) + ΛK dr b zx (x;z) and the components of the Lagrangian correlation tensor in a subensemble are L S ij (ψ,b,b z,z) v i (;)v j [x(z);z] S. The diffusion components are calculated as usually as: D ij (z) = z (17) dζl ij (ζ) (18) In Fig.1 we represented the radial Lagrangian correlation, the magnitude of the total velocity and the running diagonal diffusion coefficients for different values of the anisotropy parameter, for a fixed value of the magnetic turbulence, i.e. for the magnetic Kubo number K m = 3 and a drift Kubo number fixed for all the subplots at K dr =.. The influence of the stochastic anisotropy is exhibited in the presence of a trapping (when compared to cases Λ.5) for both radial and poloidal diffusion coefficients. The asymptotic regime is practically achieved for z for all values of the anisotropy parameter. The radial Lagrangian correlation presents a negative minimum for Λ >.5. Thus the corresponding diffusion coefficient D xx (z) presents a maximum followed by a decay, which is a signature of the trapping effect that is present also for D yy (z) with a different value of the maximum. It is obvious that the smaller anisotropy parameter the smaller value of the maximum of the D yy (z) is. The running radial diffusion coefficient has practically a constant maximum value if the anisotropy parameter vary. The position z m corresponding to the maximum value increases when the anisotropy parameter decreases for the radial diffusion coefficient and decreases when the Λ increases for the poloidal one. 4. ELECTROMAGNETIC LANGEVIN EQUATIONS FOR ELECTRONS The electrostatic turbulence that determines the electron transport regime is assumed to be related to low frequency and long wavelength drift modes. As a result, the electron motion is, in practice, given by the guiding centre equations of motion. In our model, a reduced set of equations in sheared magnetic field is used. Here, a two-dimensional sheared-slab system is considered with the main magnetic field B oriented in the Oz-direction and the sheared term in the Oy-direction. The unperturbed sheared magnetic field defines the slab geometry which mimics a portion of a toroidal chamber. Thus, with e z as unit vector along the toroidal magnetic axis, and e y as poloidal unit vector, we have: [ B(X)=B ez + XL 1 ] s e y (19)

6 118 M. Negrea, I. Petrisor, Dana Constantinescu 6 D xx D yy Λ=. Λ=.35 Λ=.5 Λ=.75 Λ=1 Λ=1.5 Λ= 1 3 z 1 3 z L xx Λ=. Λ=.35 Λ=.5 Λ=.75 Λ=1 Λ=1.5 Λ= v z 1 3 z Fig. 1 Running diffusion coefficients D xx, D yy, radial Lagrangian correlation L xx and the total velocity V for K m = 3, K dr =. and for different values of the anisotropy parameter. In tokamak experiments, the order of magnitude of XL 1 s is 1 3 for a typical length X 1 m and a shear length L s 1 m. The particular choice of the magnetic field, (19), implies that all magnetic field gradients ((b )b, B etc.) reduce to zero and that the guiding centre position (the coordinates of the test electron) X (X, Y, Z) is given by [9]: X = U b + c B E B () where b = B/B is the unit vector along the (total) magnetic field and U is the parallel guiding centre velocity. Dimensionless quantities x (x, y), z, τ and ϕ are defined in terms of the dimensional variables by the following expressions: z = Z λ ; τ t τ c ; x = X λ ; Φ(X,Z,t) εϕ( X λ, Z λ, t τ c ) (1)

7 7 Aspects of the diffusion of electrons and ions in tokamak plasma 119 As a result, the dimensionless Langevin electromagnetic system of equations that we consider is: dx(τ) dτ dy (τ) dτ = K ϕ(x(τ),τ) y = K s x(τ) + K ϕ(x(τ),τ) x V x (x(τ),τ) () K s x(τ) + V y (x(τ),τ) (3) where the electrostatic Kubo number K and the shear Kubo number K s are defined respectively as: K = εcτ c B λ, K s = V el th τ c L s (4) Here, λ is the perpendicular correlation length, λ is the parallel correlation length along the main magnetic field, τ c is the correlation time of the fluctuating electrostatic field, Vth el is the thermal electron velocity and ε is a dimensional quantity measuring the amplitude of the electrostatic field fluctuation. The correlation time τ c is the maximum time interval over which the field (the electrostatic potential in our case) maintains a given structure. In the framework of the decorrelation trajectory method [1] and [5] the system (-3) becomes dy S (τ) dτ dx S (τ) dτ = Kvx S ( x S (τ) ) B (τ) wx S ( x S (τ),τ ) (5) = x S (τ)k s + Kvy S ( x S (τ) ) B (τ) wy S ( x S (τ),τ ) (6) and permit to obtain, in a subensemble S, a deterministic trajectory with the initial condition x S ()= (7) In deriving the system (5-7), which determines the motion of the fictitious quasiparticle characteristic to the DCT method, an important approximation, specific to the DCT method has been made: it has been assumed that the contribution of the subensemble averaged shear term x(τ)k s S in each subensemble S, is equal to its value along the deterministic trajectory, x S (τ)k s. The deterministic DCT solution of the system (5-7) is used to obtain the approximate expression of the Lagrangian correlation where, instead of the Lagrangian average of the velocity, the Eulerean average of the velocity calculated along the solution of the system (5-7) is used. The global Lagrangian correlation tensor is: { L ij (τ) = (π) 3 dϕ dvy exp 1 [ (v ) ( ) x + v ( y + ϕ ) ]} L S ij (τ) (8) dv x

8 1 M. Negrea, I. Petrisor, Dana Constantinescu 8 where L S ij (τ) is the Lagrangian correlation in a subensemble while the running diffusion tensor is given by: τ D ij (τ) = L ij (θ)dθ (9) The behaviour of the diffusion tensor components are analysed in the next section. 5. DIFFUSION COEFFICIENTS FOR ELECTRONS The Lagrangian correlation tensor, the running and the asymptotic diffusion tensor components were numerically calculated. A code based on the Runge-Kutta- Fehlberg 45 (RKF45) method has been developed [1] and [11]. A severe selection of the obtained results was done in order to display two relevant cases. Fig. shows the diagonal running diffusion tensor components for different values of the shear parameter: K s [, 1] in the case of the weak electrostatic turbulence with an electrostatic Kubo number fixed at K =.1 or for the high electrostatic turbulence K = 1, respectively. Only the diagonal diffusion tensor components are considered because the nondiagonal diffusion tensor components (not displayed) are smaller at least by an order of magnitude than the diagonal ones. In the weak turbulence case K =.1 (see Fig. (top)) and small shear Kubo number K s < 1 the shapes of D xx (τ) are practically the same i.e. a continuous increase up to a saturation value is obvious; the saturation value decreases as K s is increased. A different behaviour appears at relatively high electrostatic turbulence (K 3) for D xx (τ). The running diffusion increases up to a maximum value, then decreases before reaching a final saturated regime. This behaviour is another manifestation of the trapping effect. We note that the asymptotic value D as xx of the diffusion coefficient is decreasing when the shear Kubo number is increasing. This specific signature of the trapping appears even for small values of the electrostatic Kubo number but then combined with relatively high values of the shear Kubo number (K s = 3, 6, 1). In general, the saturation values (the asymptotic ones) of the diffusion coefficients are obtained for τ τ cr 3. The maximum asymptotic value for the radial diffusion coefficient is obtained for K s = (the shearless case) and for K s = 1 we note the minimum value. In Fig. (bottom) the coefficient D xx (τ) is represented for K = 1 and for the same values of the shear Kubo number K s as in the case of weak turbulence. For K s < 1 the trapping effect is present and after an increase followed by a decrease, D xx (τ) reaches its asymptotic value, which is larger than for the corresponding weak turbulence case (e.g. K =.1).

9 9 Aspects of the diffusion of electrons and ions in tokamak plasma D xx.5 D yy K s τ K s τ D xx 4 D yy τ 1 5 K s τ 5 K s 1 Fig. Running diffusion coefficients for two fixed values of Kubo number and different values of the shear Kubo number (K s = [,1]): (upper side) D xx(τ) (left) respectively D yy(τ) (right) for K =.1; (lower side) D xx(τ) and D yy(τ) for K = 1. [Note the reverse orientation (just for a better show) of the τ-axis when compared to those from the above subplots]. For 1 > K s > 1, the trapping effect is also present and after an increase followed by a decrease, D xx (τ) increases again to reach its asymptotic value. We can see that the order of magnitude of the radial diffusion coefficient is increasing with the increase of the level of electrostatic turbulence. For the running poloidal diffusion coefficient D yy (τ), in the case of the weak turbulence, there is no trapping effect for any value of the shear Kubo number (see Fig. on top right). Practically for all values of the shear Kubo number, the maximum values (the saturation values) of the poloidal diffusion coefficient are the same, i.e. D as yy 1. For the relatively high electrostatic turbulence the asymptotic values of the poloidal diffusion coefficient increases with the increasing of the shear Kubo number. The maximum value D as yy is obtained for K s = 1. As a conclusion, the trapping is more pronounced for the relatively high electrostatic regime (K = 1) for both diffusion coefficients in comparison with the case of weak electro-

10 1 M. Negrea, I. Petrisor, Dana Constantinescu 1 static turbulence (K =.1); in the last case, only for the radial diffusion coefficient a small trapping is present for relatively high shear Kubo numbers. It is clear that the magnetic shear influences the asymptotic diffusion coefficients: the latter increases when K s increases. 6. CONCLUSIONS Two distinct problems concerning the anomalous transport in tokamak plasma were analysed. In the first one we studied the diffusion of ions in a stochastic magnetic field with curvature using the Langevin equations of the guiding centre approximation. We analysed the influence of the magnetic Kubo number and the drift Kubo number on the diffusion of ions and we have shown that the magnetic stochastic drifts provide a decorrelation mechanism of the particles from the magnetic lines. Because the Corrsin approximation ignores the trapping effect that necessarily exists in relatively strongly turbulent plasma, the method of the decorrelation trajectories was applied. It was shown that the trapping effect is more pronounced, the larger drift Kubo number is. The diagonal coefficients start with a linear part, defining a ballistic regime followed by a trapping regime before reaching the saturation asymptotic value. Thus the global trapping effect is enhanced at larger drift Kubo number; the stochastic magnetic drift has practically the same influence on ion s diffusion as the magnetic shear on the intrinsic diffusion of magnetic field lines. The second problem deals with the diffusion of electrons in a combination of an electrostatic stochastic field with an unperturbed sheared magnetic field in slab geometry. The global effects of the electrostatic Kubo number and the shear parameter on the running and asymptotic diagonal diffusion tensor components are exhibited. The trapping effect appears for large enough values of electrostatic Kubo number and/or the shear parameter but not at the same time interval for the two diagonal coefficients and not with the same strength in the two running diffusion coefficients. This behaviour is expected because of the existence of the shear term. The behaviour of the diffusion coefficients has practically the same feature: it starts with a linear part (signature of the ballistic regime) and if the trapping is present it continues with a decrease before reaching the saturation regime. The trapping is present for a relatively high turbulence level K = 1 for all magnetic shear values and only for the radial diffusion coefficient for a relatively high shear value K s = 1. The case of the electron s diffusion in a stochastic anisotropic electrostatic turbulence combined with a sheared magnetic field is left for a future work. Acknowledgements. This work was supported by the European Communities under the contract of Association between the EURATOM-MEdC, Romania, and the views and opinions expressed herein do not necessarily reflect those of the European Commission. We want to acknowledge the warm hos-

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