Contribution of electrostatic and magnetic turbulence to anomalous transport in tokamak

P ^HYSICAL J ^OURNAL

A PPLIED P HYSICS c EDP Sciences 2001

Contribution of electrostatic and magnetic turbulence to anomalous transport in tokamak

F. Miskane ^1,2 , A. Dezairi ^2,1 , D. Saifaoui ^1,a , H. Imzi ¹ , H. Imrane ¹ , and M. Benharraf ¹

1

Laboratoire de Physique Th´ eorique, Facult´ e des Sciences Ain Chok, Casablanca, Morocco

2

Laboratoire de Physique de la Mati` ere Condens´ ee, Facult´ e des Sciences Ben M’sik, Casablanca, Morocco

Received: 11 August 2000 / Revised: 17 November 2000 / Accepted: 24 November 2000

Abstract. In this paper we have studied the contribution of electrostatic and magnetic turbulence on particles anomalous transport in the tokamak. The diffusion coeffficient is evaluated. Comparison between diffusion magnetic field lines and particles is carried out. We also treat the effects of the reversed shear barrier in reducing the particles anomalous transport in the tokamak.

PACS. 52.65.-y Plasma simulation

1 Introduction

Anomalous transport in a magnetically confined plasma in a region where the magnetic surfaces are destroyed is a problem of a great importance for fusion. Anomalous dif- fusion results from fluctuating electrostatic and magnetic field [1–3]. It is very useful to evaluate the relation ship be- tween diffusion coefficients and the fluctuating fields. By solving the equations of motion ^dx _dt = v(x, t), where the v is the electric drift ^{B ∧∇} _B

2

^ϕ , it is possible to relate the diffu- sion coefficient to the amplitude of electromagnetic field.

The motion of a guiding centre is then described by a 4D system, where the dimensions correspond to the radial di- rection, the poloidal and toroidal angles, and the velocity along the field line. We will see that the main difference between the magnetic and electrostatic cases is that the diffusion increases with the parallel velocity in the mag- netic case, whereas it decreases with the parallel velocity in the electrostatic case.

In this paper, we present briefly the equations for particle guiding center and the quasi-linear calculations in Section II. Section III is devoted to study of electro- static turbulence. In section IV, we focus on the magnetic turbulence. A comparison between field lines diffusion and particles diffusion across magnetic surfaces is treated in Section V. The numerical results are compared to the quasi-linear prediction. In Section VI, we have studied the effect of reversed shear on particles diffusion through mag- netics surfaces. To see this, we consider two cases for the q profile of safety factor: the normal and the reversed shear.

In the first analysis we consider the Poincare cross sec- tion for reversed shear case. When drift wave turbulence

a

e-mail: saifaoui@facsc-achok.ac.ma

is present, a strong transport barrier exist near minimum q surface. This barrier reduces the transport and the dif- fusion of the particles. Taking the normal profile case the KAM theory is applicable for weak perturbations of the system. However as the perturbation is in increased. There is a transition to a global stochasticity with causes an in- crease in diffusion of particles. In the following, we de- scribe the motion of charged particles in the presence of drift waves. We write the guiding-center orbits equation for a given model of the drift wave [13] and replace theses equations with a local map which has the same Hamilto- nian structure. Individual orbits obtained from the map can differ qualitatively from those obtained from differen- tial equations, but statistically maps tend to give correct quantitative descriptions. At the shearless point, this map has the form of the standard nontwist map [15, 16]. Finally we treat the global drift wave map taking account on the reversed shear with radial electric field.

2 The equation of motion and Hamiltonian description

2.1 The equations of motion

Let us consider a plasma in a magnetic field of a toka- mak.We use here a simplified equilibrium field

B eq = B θ e θ + B 0 e ϕ (1) where B θ = _q(r)R ^r

0

B 0 is the poloidal magnetic field, B 0 is

the toroidal field, r the minor radius, θ and ϕ the poloidal

and toroidal angles, and q(r) the safety factor. In the fol- lowing, we will neglect the field curvature, and the trap- ping of particles in the local magnetic mirrors.

Since the cyclotron frequency is much larger than the transit and drift frequencies, the motion can be separated in a fast cyclotron gyration and a slow guiding-center mo- tion. The equations of the guiding center motion are:

dx dt = v _k B

B + B ∧ ∇ φ B ² m i

dv _k dt = − e i

B

B ∇ φ (2)

where m i , e i , v _k are particle mass, charge and parallel ve- locity. We suppose that we have a toro¨ıdal magnetic field with safety factor equal to q(r) = _{R B} ^{r B}

^ϕ

θ

with r is minus radius.

In toroidal coordinates we have x ≡ (r, θ, ϕ). At or- der 1 in _R ^r

0

, this system can be written as E = −∇ φ where φ is the electrostatic potential. At order 1 in _R ^r

0

, this system can be written as:

dr dt = − 1

Br

∂φ

∂θ dθ

dt = v _k q(r)R 0

+ 1 Br

∂φ

∂r dϕ

dt = v _k R 0

dv _k

dt = − e i

m i R 0

∂φ

∂ϕ + 1 q(r)

∂φ

∂θ

(3) where φ, the electric potential perturbation is given by:

φ(r, θ, ϕ, t) = X

mnω

φ ˜ mnω (r) cos(mθ − nϕ − ωt + α mnω ) (4) and α mnω is the random phase. We shall assume that all perturbations rotate with the same phase velocity

ω

m = ω 0 . In this case, there exists a frame where the per- turbation is static. The main advantage of this procedure is that the energy H = ^mv

^k

2

2 + eφ remains invariant in this frame. In order to further simplify our notations, we introduce a reference magnetic surface r = r 0 such that q 0 = ^m _n

₀⁰

, where m 0 and n 0 are integers. The summation will be restricted to only one poloidal wave number m 0

and the amplitude are assumed to be radially constant.

We also introduce the helical angle α = m 0 θ − n 0 ϕ instead of the poloidal angle θ. In this case the electric potential perturbation becomes

φ(r, θ, ϕ) = X L p= − L

φ ˜ p cos(α − pϕ + α p ). (5) The inverse of the safety factor is expanded as:

1 q(r) = 1

q 0 − s 0

q 0

r − r 0

r 0

(6)

where s 0 = ^s _q

⁰

0

dq

dr | ^r=r

0

is the local magnetic shear and r − r 0

is the distance from the resonant rational surface.

We introduce the further normalisation τ = v T

s

R 0

t, ξ = v _k v T

_s

, x = r − r 0

w , φ p = e i φ ˜ p

2T s

(7) consequently, the system then becomes:

dx dτ =

X L

p= − L

Φ p sin(α − pϕ + α p ) dα

dτ = xξ − Ω dϕ

dτ = ξ dξ dτ =

X L

p= − L

(x − p)φ p sin(α − pϕ + α p ) (8) where we have chosen ^m _r

⁰₀

^ρ

^s

^R _w

⁰

= 1 (ρ s is the Larmor ra- dius), ρ s = ^m _eB

ⁱ

^v

^Ts

0

and Ω = m 0 ω

₀

R

₀

v

_Ts

. T s is a temperature, v T

_s

=

2T

_s

m

_i

1/2

a thermal velocity and ω = − m ^q

⁰₀

^r s

⁰₀

is a distance between resonant surfaces.

2.2 Hamiltonian description

The Hamiltonian description of charged particle dynamics provides a convenient theoretical framework. The charged particles trajectories are described by an action-angle vari- ables system. The question of particle diffusion in a toka- mak may then be treated with techniques developed for the study of Hamiltonian chaos. In fact, the charged par- ticle motion is Hamiltonian.

It is then convenient to introduce the action variables:

I = ξ − x ²

2 and J = x. (9)

The above system is Hamiltonian and can be written as dα

dτ = ∂H

∂J

dJ

dτ = − ∂H

∂α dϕ

dτ = ∂H

∂I

dI

dτ = − ∂H

∂ϕ (10)

where H = H eq (I, J) + P

p φ p cos(α − pϕ + α p ) and H eq (I, J) = ξ ²

2 − Ωx = 1 2

I + J ²

2 2

− ΩJ. (11) H eq is the unperturbed Hamiltonian. For a passing parti- cle, the angle variable are the poloidal and toroidal angle θ and ϕ.

The associated Hamiltonian is H = ^m

ⁱ

^v

^k

2

2 + e i φ.

It can be checked out that the system (2) is equivalent to

Hamiltonian equations with this set of variables. It can be

also verified that by replacing the angle θ with the angle α, the action M is changed into I, with the appropriate normalisation. The frequencies associated with the equi- librium Hamiltonian are:

ω I = ∂H eq

∂I

J

= I + J ²

2 = ξ(I, J) ω J = ∂H eq

∂J

I

=

I + J ² 2

J − Ω = ξ(I, J)x − Ω (12) with x = J.

Therefore, the resonance condition for one perturba- tion is

ω J − pω 1 = ξ(x − p) − Ω = 0. (13) This condition is strictly equivalent to the Landau reso- nance condition ω − k _k v _k = 0. In the limit of large veloc- ities or for zero frequency, the resonance are localized on resonant surfaces x = p.

We consider resonant initial conditions which sat- isfy the condition (13) x 0 ξ 0 = Ω. Developing the Hamiltonian (11), one finds the trajectories describe an island topology in the phase space.

K = 1

2 C J ˜ ² + φ 0 cos α (14) where we have chosen α 0 = 0, and

C = ∂ ² H eq

∂J ²

I

₀

,J

₀

= I 0 + 3

2 J ₀ ² = ξ 0 + x ² ₀ . (15) Using the resonance condition (13) for p = 0, the curva- ture C can also be written as C = _x ^Ω

0

+ x ² ₀ . The motion in (α, J) is described by the equation

∂α

∂τ = ∂K

∂J

∂ J ˜

∂τ = − ∂K

∂α (16)

whereas the angle ϕ is a solution of the equation.

dϕ

dτ = ξ ∼ = x 0 J . ˜ (17) For small perturbations, the width of this island is W J = 2

φ

₀

C

1/2

. We will call it an electrostatic island by anal- ogy with a magnetic island. The island shape is then es- sentially in the real space i.e.

K = 1

2 ξ 0 x ˜ ² + φ 0 cos α. (18) In the opposite case where the initial velocity is close to zero, ξ 0 ≺≺ x ² ₀ , then the position of the particle is al- most constant and the particle is trapped along field lines.

By trapping we mean that the velocity ξ(t) reverses it sign periodically, similar to the bounce motion of parti- cles trapped in the minimum of a tokamak magnetic field.

For small deviations ˜ x, the bounce motion is described by the island equation

K = 1

2 x ² ₀ x ˜ ² + φ 0 cos α. (19) The minimum of potential correspond to α = π. The con- dition for trapping along the field line is that the veloc- ity ξ 0 at α 0 = π verifies the constraint ξ 0 ≺ 2[φ 0 ] ^1/2 . The bounce angle is then determined by the equation 2 cos(α 0 /2) = _2[φ ^ξ

⁰

0

]

^1/2

·

2.3 Particle diffusion

The diffusion coefficient is evaluated in studying the parti- cle response to perturbations. We can study this response by investigation the evolution of the particle distribution function F (J, t), where J is the vector (I, J).

F(J, t) fulfills the Vlasov equation

∂F

∂t + [F, H] = 0 (20)

with F and H represented by the Fourier decompositions H = H eq (J ) + X

nω

h nω exp i(nΦ − ωt) F = F eq (J ) + X

nω

f nω exp i(nΦ − ωt) (21) where Φ is the vector (α, ϕ).

The mean on angular variables of the Vlasov equation gives the flux term

Γ = X

n

h ₋ n (inf n ). (22) The linearisation of the Vlasov equation yields the rela- tions

∂F eq

∂t = ∂

∂J k

D ql

∂

∂J l

F eq

(23) with

Dql = π X

nω

h n ² n l n k δ X

l

n l

∂H eq

∂J k − ω

!

(24) in the radial direction D ql becomes

D ql = π 2

(φ 0 ) ²

| ξ 0 | · (25) The island width is close to

W J = 2 φ 0

ξ 0

1/2

· (26)

3 Electrostatic turbulence

3.1 Particles trajectories

The trajectories described by the system (8) are computed with a fourth order Runge-Kutta numerical scheme. The time step is determined in order to maintain a constant energy for each particle with a good accuracy. The island topology as described by equation (14) has been verified for the case of a single perturbation, as well as the time invariance of the action variable I = ξ − ^x 2

²

·

In the following are Ω = 0, and all the phases α p = 0. In this particular case, the potential is given by the relation

φ = φ 0 cos(α) sin

(2L + 1) ^ϕ ₂

sin ^ϕ ₂ · (27)

Particle trajectories are followed with a Poincar´ e map cor- responding to the “poloidal” plane, i.e. by plotting the position (r, α) of each particle when it crosses the plane ϕ = 0 [mod 2π]. Figure 1 shows Poincar´ e maps of N = 40 particles, which a initial energy E 0 = 4 eV, for 6 values of potential φ 0 . The number of perturbations is 3 (L = 1).

The first case is still integrable: 3 islands appear, local- ized on the resonant surfaces x = − 1, 0 and 1. Localized chaotic regions appear for φ 0 = 0.1 and spread up to a fully stochastic situations (φ 0 = 0.8). The critical thresh- old is determined in the following manner: particles are launched at x 0 = 0, ϕ 0 = 0 with initial α 0 equally dis- tributed over the interval [0, π] (this restriction is allowed since with zero phases α p , the system is symmetric when changing α in − α). When one particle reaches the edge of the box, it is considered that the last KAM surface is broken. We found that the onset of global stochasticity agrees rather well with the S = 2/3 criterion.

The case with many perturbations is a problem of Hamiltonian chaos [4]. For evaluate the transition to chaos, we apply the 2/3 rule, i.e. when we have the condi- tion ^φ _ξ

⁰

0

¹ 9 which is in good agreement with our results.

3.2 Diffusion coefficient

In the case where the Chirikov parameter is above the critical value, the particles are expected to exhibit a ran- dom motion. We will compute here two effective “diffu- sion” coefficient. The first one is determined from the ratio (x − h x i ) ² /2t, whereas the second is determined by com- puting an average exit time. It is indeed well known that for particles launched in the interval [ − δd, δd] in the radial (x), the diffusion coefficient is given by the relation [9]

D J = 1

β (2) + β(4)δ ² + β (6)δ ⁴ d ²

4 lim

N →∞

1 N

X N i=1

1 τ i

(28)

where β(2) =

+ ∞

X

k=0

( − 1) ^k

(1 + 2k) ^2p and β(4) = β(6) = 0.

We find a good agreement between the two methods. Fig- ure 2 shows quasi-linear and numerical diffusion coefficient as function of the perturbation amplitude φ 0 for several parallel initial velocities ξ 0 . This figure demonstrates that the computed diffusion coefficient agrees with the quasi- linear prediction. We can observe that at low velocity, the diffusion coefficient is lower than the quasi-linear value, this can be explained by the fact that many particles at low velocity are trapped in the potential minima and leave the domain in a very short time scale, with a non diffusive motion. Figure 3 displays the same data as function of ξ 0

for different values of the potential. In this figure, we can observe that the diffusion coefficient decreases with the initial velocities. The case at large velocity is close to the classical problem of particle trajectories in the presence of magnetic islands [1, 5–9]. At low initial velocities, our re- sults show that the behavior with velocity is less divergent than the 1/ξ 0 dependence predicted by the quasi-linear theory. In Figure 4, we represent the D/D ql ratio as func- tion of the perturbation amplitude φ 0 for several initial velocities ξ 0 according to Mendon¸ca model. The D/D ql

ratio converge to one for the case where the numerical diffusion coefficient matches the quasi-linear diffusion co- efficient.

4 Magnetic turbulence

4.1 Equations of motion

If we consider a plasma in a magnetic field represented in the form

B = B eq + ˜ B (29) where ˜ B = ∇ ∧ A ˜ is the fluctuating part and ˜ A is the fluctuating vector potential.

A ˜ can be represented by a Fourier decomposition

A(r, θ, ϕ) = ˜ X L p= − L

A ˜ p cos(α − pϕ + α p ) (30)

where we impose the amplitude ˜ A p constant for all per- turbations.

We have obtained nearly the same results, but the dif-

ference with the electrostatic case is that the parallel ve-

locity still remains constant. The differential system for

Fig. 1. Poincar´ e maps for 40 particles, with a initial energy E

0

= 4, for 6 values of potential φ

0

. (a) φ

0

= 0.02, (b) φ

0

= 0.06, (c) φ

0

= 0.1, (d) φ

0

= 0.3, (e) φ

0

= 0.4, and (f) φ

0

= 0.8. The number of perturbations is 3 (L = 1).

this case is then dx dt =

X L

p= − L

− v T

_s

ξA p sin(α − pϕ + α p ) dα

dt = xξ − Ω dϕ

dt = ξ dξ

dt = 0 (31)

where A p = ^e _2T

ⁱ

^{A ˜}

^p

s

·

This result has been already obtained by Sabot [10].

The Hamiltonian for magnetic perturbation is then, where

H = H eq (I, J) + X

p

( − v T

_s

ξ eq A p ) cos(α − pϕ + α p ) (32)

Fig. 2. Electrostatic diffusion coefficient DES calculated from the particle exit times as function of the perturbation amplitude φ

0

for six initial velocities ξ

0

: ξ

0

= 0.2, ξ

0

= 1.2, ξ

0

= 1.4, ξ

0

= 1.6, ξ

0

= 1.8 and ξ

0

= 5. - - - indicate the quasi-linear prediction,

∆∆∆∆ indicate the computed diffusion coefficients.

SDUDOOHOYHORFLW\

'(6

I

I

I

Fig. 3. Electrostatic diffusion coefficient DES calculated as function of the initial velocities ξ

0

for three values of the per- turbation amplitude φ

0

: φ

0

= 1.8, φ

0

= 1, φ

0

= 0.5.

with H eq (I, J) = ξ eq

2 − Ωx = 1 2

I + J ²

2 ²

− ΩJ the unperturbed Hamiltonian

ξ = ξ eq + ˜ ξ and

ξ ˜ = X

p

v T

_s

A p cos(α − pϕ + α p ).

For one perturbation, the calculation of the Hamilto- nian involves trajectories which describe a magnetic island topology in the phase space.

K = 1

2 C J ˜ ² + h 0 cos α where C = ∂ ² H eq

∂J ²

I

0

,J

0

= I 0 + 3

2 J ₀ ² = ξ eq,0 + x ² ₀ h 0 = − v T ξ eq,0 A 0 . (33) The width of magnetic island is then

W J = 4

− v T

s

ξ eq,0 A 0

ξ eq,0 + x ² ₀ 1/2

· (34)

4.2 Diffusion coefficient

For the calculation of the diffusion coefficient, we pro- ceeded in the same way as for the electrostatic case. The

usual assumption about transport due to magnetic turbu- lence was that many modes are present and that the tur- bulence is well developed, so that the quasi-linear theory is applicable [1]. We compute the diffusion coefficient for dif- ferent values of magnetic perturbation amplitude with 100 initial conditions and we have considered three interact- ing islands. The evolution is the same for the electrostatic case, i.e., the diffusion increases with the perturbation.

This is shown in Figure 5 for six values of numerical diffu- sion coefficient compared to D ql . As expected, the agree- ment is fairly good. The diffusion increases with parallel velocities (see Fig. 6) in contradiction to the electrostatic case where the diffusion decreases with parallel velocities.

This result is in agreement with the quasi-linear theory.

The ratio D/D ql is plotted for several perturbation am- plitudes (see Fig. 7).

5 Comparison between field lines diffusion and particles diffusion across magnetic surfaces

The evolution of stochastic magnetic field lines has been already studied numerically [11]. We have developed a nu- merical technique in order to study the transition from partial stochasticity to global stochasticity of magnetic field lines. We also introduced a model of diffusion coeffi- cient in order to study the diffusion of lines through mag- netic surfaces. The non-Gaussian dynamics of lines has been analyzed using the Kurtosis parameter. Laval [12]

has given the particles diffusion coefficient D p and field lines diffusion coefficient D L according to relation

D L = vD p

where v is the particle constant velocity along the field lines. So the field lines diffuse across the plasma and the particle motion along these field lines gives rise to the anomalous diffusion.

5.1 Equation of field lines and discrete mapping The magnetic field lines may be transformed in mapping structure. Their general form is found in [8]:

I k+1 = I k + X

m

K m sin(mθ k )

θ k+1 = θ k + I k+1 (35) where

I k + 2πΨ k , K m = (2π) ² mf m . (36) For m = 1, we find the standard mapping.

The generalized mapping that we have used in this work correspond to m = 2 writes as

I k+1 = I k + K 1 sin(θ k ) + K 2 sin(2θ k )

θ k+1 = θ k + I k+1 . (37)

Fig. 4. The ratio D/D

ql

calculated numericallty as function of the electrostatic perturbation amplitude φ

0

for six initial

velocities ξ

0

: ξ

0

= 0.2, ξ

0

= 1.2, ξ

0

= 1.4, ξ

0

= 1.6, ξ

0

= 1.8 and ξ

0

= 5.

Fig. 5. Magnetic diffusion coefficient calculated from the particle exit times as function of the perturbation amplitude φ

0

for

six initial velocities ξ

0

: ξ

0

= 0.2, ξ

0

= 1.2, ξ

0

= 1.4, ξ

0

= 1.6, ξ

0

= 1.8 and ξ

0

= 5. - - - indicate the quasi-linear prediction,

4 4 4 4 indicate the computed diffusion coefficient.

SDUDOOHOYHORFLW\

'P

I

I

I

Fig. 6. Magnetic diffusion coefficient D

m

calculated as func- tion of the initial velocities ξ

0

for three values of the perturba- tion amplitude φ

0

: φ

0

= 1.8, φ

0

= 1, φ

0

= 0.5.

5.2 The Poincar´ e section of magnetic field lines – Simulations

To study the stochastic magnetic field lines we have repre- sented the Poincar´ e section described by the generalized mapping equations (35) in the (Ψ, θ) and (x, y) for dif- ferent values of (K 1 , K 2 ). In Figures (8–12) the Poincar´ e section is represented in (Ψ, θ) plan, whereas Figures (13–

16) are in the (x, y) plan with x = Ψ cos θ and y = Ψ sin θ.

Figures 8, 9, 13 and 14 correspond to the case of weak islands overlaping (partial stochasticity). There exist many integrable trajectory (KAM tori) separating differ- ent stochastic regions. On the other hand, in plots V 10,11 and 14,15 the islands are large enough to overlap (global stochasticity). The stochastic regions cover all space (there are no KAM tori). We note also that in this case (K 2 6 = 0) the last KAM tori is destroyed for values of K 1 different to K c (K c = 0.97 is the Chirikov constant). The transition points (K 1 , K 2 ) from partial to global stochasticity have been calculated numerically in [4].

6 Contribution of reversed shear in reducing of particles diffusion

Anomalous transport observed in tokamaks is known as the result of the electrostatic and magnetic turbulence.

Thus, in the presence of electric perturbation and for the normal profile of the safety factor q, the stochasticity of the trajectories increases and this is the principal cause

of diffusion of particles through magnetic surfaces. How- ever for the reversed shear case, the main result proves that there exists a strong transport barrier near the sur- face of minimum value, and in this case, the diffusion is considerably reduced.

We have studied the guiding-center orbits in tokamak when one perturbed electric field is present. To see the effect of electrostatic turbulence on the confined plasma, we consider tow cases for the q profile of safety factor: the normal and the reversed shear.

The first analysis consist on studying of the Poincare cross section for the normal profile under a weak pertur- bation, the KAM theory is sufficiently applicable. How- ever if we increase the level of the perturbation, there is a transition to global stochasticity and this causes a high diffusion of particles. On the other hand, by taking the re- versed shear case, when drift wave turbulence is present, a strong transport barrier exists near the minimum q sur- face. This barrier reduces the transport and the diffusion of the particles.

In the following, we describe the motion of charged particles in the presence of drift waves, we write the guiding-center orbits equation in a given model of drift wave [13], and we replace these equations with a local map that has the same Hamiltonian structure. Individ- ual orbits obtained from the map can differ qualitatively from those obtained from differential equations, but sta- tistically maps tend to give correct quantitative descrip- tions. At the shearless point, this map has the form of the standard nontwist map. In our simulation, we consider the global drift wave map taking account on the reversed shear with radial electric field.

In order to develop the reversed magnetic shear ob- served in experiment on tokamak TFTR, where the par- ticle transport behaves as though a barrier to transport exists near a minimum of the safety factor q min . We con- sider the guiding center drift theory described by many authors as the motion of charged particles in presence of electrostatic drift wave fluctuations.

6.1 The guiding centre drift orbit

The guiding centre drift orbit theory has been described by many authors. In this section, we consider the motion of charged particle in presence of electrostatic drift wave fluc- tuations. The simplest expression for the guiding-center velocity in Gaussian units is:

dx

dt = v _k B

| B | + c E ∧ B

B ² · (38)

E and B are electric and magnetic field, v _k is the parallel velocity, and the second term of the right side is the drift velocity.

For the fluctuating term of electric field, we take the model of drift wave spectrum which were employed by Horton [13, 17, 18], and we have:

φ(x, t) = ˜ X

m,l,n

φ m,l,n cos(mθ − lϕ − nw 0 t) (39)

Fig. 7. The ratio D/D

ql

calculated numerically as function of the magnetic perturbation amplitude φ

0

for six initial velocities

ξ

0

: ξ

0

= 0.2, ξ

0

= 1.2, ξ

0

= 1.4, ξ

0

= 1.6, ξ

0

= 1.8 and ξ

0

= 5.

Fig. 8. Poincar´ e section of the magnetic field lines for (K

1

, K

2

) = (0.08, 0.04).

Fig. 9. Poincar´ e section of the magnetic field lines for (K

1

, K

2

) = (0.12, 0.08).

Fig. 10. Poincar´ e section of the magnetic field lines for (K

1

, K

2

) = (0.9, 0.9).

Fig. 11. Poincar´ e section of the magnetic field lines for (K

1

, K

2

) = (1.8, 1.6).

Fig. 12. The evolution of stochastic magnetic field lines in the plane (x, y) for values of (K

1

, K

2

) = (0.08, 0.04).

Fig. 13. The evolution of stochastic magnetic field lines in the plane (x, y) for values of (K

1

, K

2

) = (0.12, 0.08).

Fig. 14. The evolution of stochastic magnetic field lines in the plane (x, y) for values of (K

1

, K

2

) = (0.9, 0.9).

Fig. 15. The evolution of stochastic magnetic field lines in the plane (x, y) for values of (K

1

, K

2

) = (1.8, 1.6).

where ˜ φ is the electrostatic potential such that:

E ˜ = −∇ φ ˜ (40)

and ˜ E is the perturbed electric field.

In a toroidal magnetically confined plasma, B ≈ B ϕ ii B θ and B r = 0, which are related to the toroidal

coordinate (r, θ, ϕ), the equation of motion becomes:

dr dt = − c

B 1 r

∂ φ ˜

∂θ r dθ

dt = v _k B θ

B + c B

∂ φ ˜

∂r − c E ˜ r

B R dϕ

dt = v _k (41)

where ˜ E r is the equilibrium radial electric.

To replace these equations with a local map, we assume that one mode (M, L) of perturbation dominate and in- troduce the news variables: I =

r a

2

and Ψ = M θ − Lϕ.

In the first approximation we neglect the radial field, we can obtain this structure of mapping:

I N+1 = I N + 4πc a ² B 0

M φ w 0

sin(M θ N − Lϕ N ) (42) X N+1 = X N + R1(I N+1 ) + R2(I N+1 ) (43) R1(I) = v _k (I)

w 0 q R (M − Lq(I)) (44) R2(I) = cM

w 0 aB 0

E r (I)

√ I (45)

v _k (I) = r 2

m (ζ t − eΦ 0 (I)) (1 − λB 0 ) . (46) Here ζ t is the initial total energy, m and e the mass and charge of particle, µ = ^{m v} _2B

^2⊥

is the magnetic moment and E r (I) = − ^∂Φ ∂r

⁰

_r=a ^√ _I is the equilibrium radial electric field and Φ 0 is the equilibrium radial potential .

In the next section, we neglect the electric field radial component and investigate the map phase structure by calculating 1000 massive ⁺ D particles trajectories with various initial conditions in configuration spaces for the reversed and normal shear case.

In the calculations below, we have used the Texas Experimental Tokamak (TEXT) system parameters, with major radius R 0 = 10 cm, minor radius a = 26 cm, and center-line field B = 3 tesla. We take w 0 = 1.93 × 10 ⁵ , we choose M = 12; L = 6 and λ = ^µ _ζ with ζ = 167 eV.

6.2 Numerical results and comments

In our analysis, we study the Poincare cross section in the phase space for two cases of profile of safety factor: the normal profile q(r) = 1.99 + 1.94(r/a) ² and the reserved profile q(r) = 1.99 + 7.76(r/a − 0.5) ² are used. This can be seen in Figure 16 where we have represented the variation of profile of the dimensionless safety factor function r/a.

We observe the effects on trajectories with the action of electric perturbation. Figures (18–23) and (24–29) show the surfaces of section for the structure in the global map for the reversed and normal profile respectively. We rep- resent the trajectories for diverse values of the drift wave potential amplitude in the (r/a, X) plane.

In the reversed shear case and for small perturbation (pot = 0.02 eV), the surface with the rotation number is located near the minimum q surface at r/a = 0.54 (Figs. 18, 19), with the increase in the level of perturbation until the value pot = 1 eV (Fig. 20), this surface play an important role in producing barrier transport. This is the case for large value of the potential where we see clearly this phenomenon.

In Figures 21, 22 and 23 we show that this barrier transport is a principal source of the reduction of trans- port observed in the experiment. The particle diffusion is reduced and there is an improvement in confinement.

0 .0 0 0 .2 0 0 .4 0

r / a 0 .0 0

2 .0 0 4 .0 0

q nor m al s hear

Fig. 16. Profile of safety factor versus on r/a the normal and the reversed shear. We start by plotting the trajectories with- out perturbation, this is seen in Figure 17 and is the ideal case.

Fig. 17. Trajectories without perturbation in the (X, r/a)

plane.

Fig. 18. Poincar´ e map of 1000 particles in the (X, r/a) plane for the value 0.02 eV of the perturbation – reversed shear.

Fig. 19. Poincar´ e map of 1000 particles in the plane (X, r/a) for the value pot = 0.01 eV of the perturbation – reversed shear.

Fig. 20. Poincar´ e map of 1000 particles in the (X, r/a) plane for the value 1 eV of the perturbation – reversed shear.

Fig. 21. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 1.5 eV of the perturbation – reversed shear.

Fig. 22. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 1.5 eV of the perturbation – reversed shear.

Fig. 23. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 2 eV of the perturbation – reversed shear.

(g)

Fig. 24. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 0.02 eV of the perturbation – normal shear.

(h)

Fig. 25. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 0.1 eV of the perturbation – normal shear.

(i)

Fig. 26. Poincar´ e map of 1000 particles in the ( ¯ X, ¯ r/a) plane for the 0.5 eV of the perturbation – normal shear.

(j)

Fig. 27. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 1 eV of the perturbation – normal shear.

(k)

Fig. 28. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 1.25 eV of the perturbation – normal shear.

For the normal shear case, when a weak perturbation is applicable (pot = 0.02 eV), there is apparition of magnetic islands near the resonate surface, where we can apply the KAM theory; the trajectory is integrable and diffusion is not important (see Fig. 24). However when we increase the amplitude of perturbation until the value pot = 1 eV of the potential (Fig. 27), KAM torus disappear and there is transition to global stochasticity. This is the case for Figure 28. Consequently the trajectory is no longer inte- grable and diffusion become large. The last Figure 29 for the value (pot = 2.5 eV) show the transition to a global stochasticity. Diffusion of particles increases and this is a principal source of anomalous transport of particles.

7 Conclusion

It has been possible to study the anomalous transport

induced by electrostatic and magnetic fluctuation. The

trajectories of particle guiding centers in presence of per-

turbed electrostatic and magnetic field in a tokamak is

shown to be described by four equations of motion. This

system can be treated with Hamilton equations, using

a set of action-angle variables. With only one perturba-

tion, the trajectories are associated with an island chain

in phase space. The transition to stochasticity occurs for

increasing perturbation strength between regions of the

phase space. The diffusion coefficient of particles is com-

puted and compared to the quasi-linear theory in each

Fig. 29. Poincar´ e map of 1000 particles in the (X, r/a) plane for the 2.5 eV of the perturbation – normal shear.

case. It is shown that the diffusion due to magnetic tur- bulence increases with the parallel velocity whereas the diffusion due to electrostatic turbulence decreases with the parallel velocity. This behavior is well described by the quasi-linear expression. A comparison between field lines diffusion and particles diffusion across magnetic sur- faces shows that the field lines diffuse across the plasma and the particle motion along these field lines give rise to the anomalous diffusion.

However anomalous transport of particles and energy observed in the tokamak is reduced in the presence of the electrostatic turbulence for the reversed shear case. The the main result prove that there exists a strong transport barrier near the surface of minimum q value. This surface reduce the diffusion of particles through magnetic surfaces and this causes an improvement in the confinement [19].

F. Miskane have pleasure to thank Dr. X. Garbet for many helpful discussions during her stay in research center Cadarache.

References

1. A.B. Rechester, M.N. Rosenbluth, Phys. Rev. Lett. 40, 38 (1978).

2. H.B. Park, E.G. Heo, W. Horton, D.-I. Choi, Phys. Plas- mas 4, 3273 (1997).

3. W. Horton, H.B. Park, J.-M. Kwon, D. Strozzi, P.J.

Morrison, D.-I. Choi, Phys. Plasmas 5, 3910 (1998).

4. A.J. Lichtenberg, M.A. Liberman, Regular and Stochastic Motion (Springer Verlag, New York, 1983).

5. R. Balescu, M. Vlad, F. Spineanu, in Chaos, Kinetics and Non linear Dynamics in Fluid and Plasmas, edited by S.

Benkadda, G. Zaslavsky (Springer, Berlin, 1998).

6. A.B. Rechester, M.N. Rosenbluth, R.B. White, Phys. Rev.

Lett. 42, 1247 (1979).

7. P. Ghendrih, A. Grosman, H. Capes, Plasma Phys. Cont.

Fusion 38, 1653 (1996).

8. R. Tabet, D. Saifaoui, A. Dezairi, A. Rouak, Eur. Phys.

Lett. 4, 329 (1998).

9. R. Sabot, M.A. Dubois, Phys. Lett. A 212, 201 (1996).

10. R. Sabot, importance de la structure magn´ etique fine dans un tokamak pour le transport anormal et les disruptions internes, th` ese, ´ Ecole Centrale de Paris, 1996.

11. A. Oualyoudine, D. Saifaoui, A. Dezairi, A. Rouak, J.

Phys. 7, 1045 (1997).

12. G. Laval, Phys. Fluids B 5, 711 (1993).

13. Horton et al., Drift test transport in reserved shear profil, Phys. Plasmas 5, 11 (1998).

14. O. Fisher, W.A. Cooper, L. Villard, Magnetic topology and guiding-center orbits in a reserved shear Tokamak, LRP 636/99, Juin 1999.

15. D. del-Castillo-Negrete, P.J. Morrison, Phys. Fluids A 5, 948 (1993).

16. D. del-Castillo-Negrete, J.M. Greene, P.J. Morrison, Phys- ica D 100, 311 (1997).

17. D.L. Brower, W.A. Peebles, N.C. Luhmann Jr., R.L. Sav- age Jr., Phys. Rev. Lett. 54, 689 (1985).

18. E. Mazzucato, R. Nazikian, Phys. Rev. Lett. 71, 1840 (1983).

19. H. Takenga et al., Plasma Phys. Controlled Fusion 40, 183 (1998).

20. H. Imzi, D. Saifaoui, A. Dezairi, F. Miskane, Contribu-

tion of reversed shear in reducing to anomalous transport,

submited to publication.