Dedicated to Professor Oliviu Gherman’s 80th Anniversary
ASPECTS OF THE DIFFUSION OF ELECTRONS AND IONS
IN TOKAMAK PLASMA
M. NEGREA1,3,∗ , I. PETRISOR1,3 , DANA CONSTANTINESCU2,3
1
Department of Physics, University of Craiova,
13 A.I. Cuza Street, RO-200585, Craiova, Romania
2
Department of Applied Mathematics, University of Craiova,
13 A.I. Cuza Street, RO-200585 Craiova, Romania
3
Association Euratom-MECI, Romania
Received November 10, 2009
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∗
E-mail: [email protected]
Rom. Journ. Phys., Vol. 55, Nos. 9–10, P. 1013–1023, Bucharest, 2010
1014
M. Negrea, I. Petrisor, Dana Constantinescu
2
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 2 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.
2. MAGNETIC LANGEVIN EQUATIONS FOR THE IONS
We consider the model with a shearless slab geometry for the confining magnetic field with a strong component B0 along the z axis and a fluctuating perpendicular component in the (X, Y ) plane:
B(X; Z) = B0 {eZ + βbX (X; Z) eX + βbY (X; Z) eY }
(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
bi (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
= βbX (X; Z) Vk + VdrX
dt
(2)
3
Aspects of the diffusion of electrons and ions in tokamak plasma
1015
dY
= βbY (X; Z) Vk + VdrY
(3)
dt
dZ
= Vk
(4)
dt
where Vk is the component of the particle velocity parallel to the reference magnetic
field and Vdr (VdrX , VdrY ) is the drift velocity which, for our stationary magnetic
field, is given as:
i
h
1
(5)
Vdr ' n × µ∇B + Vk2 (n · ∇) n
Ω
where Ω = eB/mc is the Larmor frequency, n = B/B is the unit vector along the
field line, µ = V⊥2 /2B 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 [bi (X; Z)]k ,
k > 2, in the drift velocity and retain only the dominant, first order, terms [7]:
Vk2 ∂bY [X(Z); Z]
Vk2
≡ − bZY [X(Z); Z]
(6)
Ω
∂Z
Ω
Vk2 ∂bX [X(Z); Z] Vk2
Vdr Y '
≡
bZX [X(Z); Z]
(7)
Ω
∂Z
Ω
Secondly, the parallel velocity Vk appearing in eq. (4) is the particle velocity along
magnetic lines and is considered to be constant and equal to the thermal velocity
Vth . 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]:
Vdr X
' −
M (X; Z) = hΨ(0; 0)Ψ(X; Z)i = M1 (X) M2 (Z)
where
M1 (X) = β 2 B02 λ2x exp(−
X2
Y2
)
exp(−
)
2λ2x
2λ2y
and
(8)
(9)
Z2
)
(10)
2λ2z
Here, three characteristic correlation lengths are defined: the parallel correlation
length λz and the perpendicular correlation lengths λx and λy. Dimensionless coordinates {x = (x, y) , z} may then be defined by x = X/λx , y = Y /λy and z = Z/λz ;
we define also the magnetic dimensionless potential as ψ = B0−1 β −1 λ−1
x Ψ. Using
these dimensionless quantities and the assumptions which we have made above, the
system given in eqs. (2) - (4) becomes:
dx(z)
∂ψ(x(z); z) ∂ ∂ψ(x(z); z) =ΛKm
+
K
≡
dr
dz
∂y
∂z
∂x
(11)
x=x(z)
x=x(z)
≡vx [x(z); z] = Km bx (x; z) − Kdr bzy (x; z)
M2 (Z) = exp(−
1016
M. Negrea, I. Petrisor, Dana Constantinescu
4
dy(z)
∂ψ(x(z); z) ∂ ∂ψ(x(z); z) 2
+ Λ Kdr
≡
= − ΛKm
dz
∂x
∂z
∂y
(12)
x=x(z)
x=x(z)
≡vy [x(z); z] = ΛKm by (x; z) + ΛKdr bzx (x; z)
The stochastic components βB0 b are represented by a vector potential in order to
have the condition of zero divergence automatically fulfilled:
A(X; Z) = B0 λx β ψ(x; z) ez
(13)
The Kubo numbers that appear in eqs. (11) - (12) are [6]
Km = β
λz
λx
Vth
, Λ=
, Kdr = β
λx
λy
Ωλx
(14)
β is the dimensionless amplitude of magnetic field fluctuations (considered here as
relatively strong, i.e. β ' 10−2 ), Vth is the thermal velocity of the ions and Ω their
Larmor frequency. Usually VΩth ' 10−1 m (Larmor radius for ions), λx ' λy '
10−2 m and λz ' 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 (11-12) 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 bz in the point
x = 0 at the ”moment” z = 0:
ψ(0; 0) = ψ 0 , b(0; 0) = b0 , bz (0; 0) = b0z
(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:
Z
Lij (z) = dψ 0 db0 db0z P (ψ 0 , b0 , b0z ) hvi (0; 0)vj [x(z); z]iS
(16)
5
Aspects of the diffusion of electrons and ions in tokamak plasma
1017
where the components of v [x(z); z] are defined as:
vx (x; z) = Km bx (x; z) − Kdr bzy (x; z)
vy (x; z) = ΛKm by (x; z) + ΛKdr bzx (x; z)
(17)
and the components of the Lagrangian correlation tensor in a subensemble are
LSij (ψ 0 , b0 , b0z , z) ≡ hvi (0; 0)vj [x(z); z]iS . The diffusion components are calculated
as usually as:
Zz
Dij (z) = dζLij (ζ)
(18)
0
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 Km = 3 and a drift Kubo number fixed for all the subplots at
Kdr = 0.2. The influence of the stochastic anisotropy is exhibited in the presence of
a trapping (when compared to cases Λ ≤ 0.5) for both radial and poloidal diffusion
coefficients. The asymptotic regime is practically achieved for z > 2 for all values
of the anisotropy parameter. The radial Lagrangian correlation presents a negative
minimum for Λ > 0.5. Thus the corresponding diffusion coefficient Dxx (z) presents
a maximum followed by a decay, which is a signature of the trapping effect that is
present also for Dyy (z) with a different value of the maximum. It is obvious that the
smaller anisotropy parameter the smaller value of the maximum of the Dyy (z) is.
The running radial diffusion coefficient has practically a constant maximum value if
the anisotropy parameter vary. The position zm 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 B0
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 ez as unit vector along the ”toroidal”
magnetic axis, and ey as ”poloidal” unit vector, we have:
B (X) =B0 ez + XL−1
(19)
s ey
1018
M. Negrea, I. Petrisor, Dana Constantinescu
1.4
3
1.2
2.5
0.6
1.5
1
0.4
0.5
0.2
0
Λ=0.2
Λ=0.35
Λ=0.5
Λ=0.75
Λ=1
Λ=1.5
Λ=2
2
0.8
Dyy
Dxx
1
0
1
2
0
3
0
1
z
3
2
3
0.7
Λ=0.2
Λ=0.35
Λ=0.5
Λ=0.75
Λ=1
Λ=1.5
Λ=2
6
4
0.6
0.5
0.4
v
8
Lxx
2
z
10
0.3
2
0.2
0
−2
6
0.1
0
0.5
1
1.5
2
0
0
z
1
z
Fig. 1 – Running diffusion coefficients Dxx , Dyy , radial Lagrangian correlation Lxx and the total
velocity V for Km = 3, Kdr = 0.2 and for different values of the anisotropy parameter.
−3 for a typical length
In tokamak experiments, the order of magnitude of XL−1
s is 10
X ∼ 10−2 m and a shear length Ls ∼ 10 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
E×B
B2
(20)
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
t
X
X Z t
; τ ≡ ; x=
; Φ(X, Z, t) → εϕ( , , )
λk
τc
λ⊥
λ⊥ λk τc
(21)
7
Aspects of the diffusion of electrons and ions in tokamak plasma
1019
As a result, the dimensionless Langevin electromagnetic system of equations that we
consider is:
dx (τ )
∂ϕ (x (τ ) , τ )
= −K
≡ Vx (x (τ ) , τ )
dτ
∂y
dy (τ )
∂ϕ (x (τ ) , τ )
= Ks x (τ ) + K
≡ Ks x (τ ) + Vy (x (τ ) , τ )
dτ
∂x
(22)
(23)
where the electrostatic Kubo number K and the shear Kubo number Ks are defined
respectively as:
el τ
Vth
ε c τc
c
K=
,
K
=
(24)
s
Ls
B0 λ2⊥
Here, λ⊥ is the perpendicular correlation length, λk is the parallel correlation length
along the main magnetic field, τc is the correlation time of the fluctuating electrostatic
el is the thermal electron velocity and ε is a dimensional quantity measuring
field, Vth
the amplitude of the electrostatic field fluctuation. The correlation time τc 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
[10] and [5] the system (22-23) becomes
dxS (τ )
= KvxS xS (τ ) B (τ ) ≡ wxS xS (τ ) , τ
dτ
(25)
dy S (τ )
= xS (τ ) Ks + KvyS xS (τ ) B (τ ) ≡ wyS xS (τ ) , τ
(26)
dτ
and permit to obtain, in a subensemble S, a deterministic trajectory with the initial
condition
xS (0) = 0
(27)
In deriving the system (25-27), 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 hx (τ ) Ks iS in each subensemble S, is equal to its
value along the deterministic trajectory, xS (τ ) Ks . The deterministic DCT solution
of the system (25-27) 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 (25-27) is used.
The global Lagrangian correlation tensor is:
Z Z Z
2
2 i
3
1 h 0 2
Lij (τ ) = (2π)− 2 dϕ0 dvx0 dvy0 exp −
vx + vy0 + ϕ0
LSij (τ ) (28)
2
1020
M. Negrea, I. Petrisor, Dana Constantinescu
8
where LSij (τ ) is the Lagrangian correlation in a subensemble while the running diffusion tensor is given by:
Zτ
Dij (τ ) = Lij (θ) dθ
(29)
0
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-KuttaFehlberg 45 (RKF45) method has been developed [10] and [11]. A severe selection
of the obtained results was done in order to display two relevant cases.
Fig. 2 shows the diagonal running diffusion tensor components for different
values of the shear parameter: Ks ∈ [0, 10] in the case of the weak electrostatic
turbulence with an electrostatic Kubo number fixed at K = 0.1 or for the high electrostatic turbulence K = 10, 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 = 0.1 (see Fig. 2(top)) and small shear Kubo
number Ks < 1 the shapes of Dxx (τ ) are practically the same i.e. a continuous
increase up to a saturation value is obvious; the saturation value decreases as Ks
is increased. A different behaviour appears at relatively high electrostatic turbulence (K ≥ 3) for Dxx (τ ). The running diffusion increases up to a maximum value,
then decreases before reaching a final saturated regime. This behaviour is another
as of the
manifestation of the trapping effect. We note that the asymptotic value Dxx
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 (Ks = 3, 6, 10). In general, the saturation values (the asymptotic ones) of the
diffusion coefficients are obtained for τ ≥ τcr ' 3. The maximum asymptotic value
for the radial diffusion coefficient is obtained for Ks = 0 (the shearless case) and for
Ks = 10 we note the minimum value.
In Fig. 2 (bottom) the coefficient Dxx (τ ) is represented for K = 10 and for the
same values of the shear Kubo number Ks as in the case of weak turbulence.
For Ks < 1 the trapping effect is present and after an increase followed by a
decrease, Dxx (τ ) reaches its asymptotic value, which is larger than for the corresponding weak turbulence case (e.g. K = 0.1).
9
Aspects of the diffusion of electrons and ions in tokamak plasma
0.01
Dyy
Dxx
0.01
0.005
0.005
0
0
0
0
5
Ks
10
0
τ
5
10
5
Ks
8
0
τ
10
5
15
Dyy
Dxx
10
20
6
4
10
5
2
10
0
6
1021
5
4
2
τ
0
0
Ks
10
0
6
5
4
τ
2
0
0
Ks
Fig. 2 – Running diffusion coefficients for two fixed values of Kubo number and different values of the
shear Kubo number (Ks = [0, 10]): (upper side) Dxx (τ ) (left) respectively Dyy (τ ) (right) for
K = 0.1; (lower side) Dxx (τ ) and Dyy (τ ) for K = 10. [Note the reverse orientation (just for a better
show) of the τ -axis when compared to those from the above subplots].
For 10 > Ks > 1, the trapping effect is also present and after an increase followed by a decrease, Dxx (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 Dyy (τ ), in the case of the weak
turbulence, there is no trapping effect for any value of the shear Kubo number (see
Fig. 2 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,
as ' 10−2 . For the relatively high electrostatic turbulence the asymptotic vai.e. Dyy
lues of the poloidal diffusion coefficient increases with the increasing of the shear
as ' 20 is obtained for K = 10. As a conKubo number. The maximum value Dyy
s
clusion, the trapping is more pronounced for the relatively high electrostatic regime
(K = 10) for both diffusion coefficients in comparison with the case of weak electro-
1022
M. Negrea, I. Petrisor, Dana Constantinescu
10
static turbulence (K = 0.1); 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 Ks 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 = 10 for all magnetic shear values and only for the
radial diffusion coefficient for a relatively high shear value Ks = 10. 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-
11
Aspects of the diffusion of electrons and ions in tokamak plasma
1023
pitality of the members of Statistical Physics and Plasmas from Université Libre de Bruxelles, Belgium.
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