Contribution of reversed shear in reducing the anomalous transport

DOI: 10.1051/epjap:2001004

P ^HYSICAL J ^OURNAL

A

PPLIED

P

HYSICS c EDP Sciences 2002

Contribution of reversed shear in reducing the anomalous transport

H. Imzi

¹

, D. Saifaoui

^1,a

, A. Dezairi

^1,2

, F. Miskane

^1,2

, and M. Benharraf

¹

1 Laboratoire de Physique Th´eorique, Facult´e des Sciences Ain Chok, BP 5666, Maarif, Casablanca, Morocco

2 Laboratoire de Physique Th´eorique et de la Mati`ere Condens´ee, Facult´e des Sciences Ben M’Sike, Casablanca, Morocco

Received: 26 January 2001 / Revised: 12 April 2001 / Accepted: 3 September 2001

Abstract. In this paper we have studied the contribution of reversed shear in reducing the anomalous transport in a Tokamak using both an analytical approach and simulations. We have used a special model for the drift wave fields. Comparison between particles trajectories for normal and negative shear is carried out. The diffusion coefficient of particles for the two cases; normal and reversed shear is evaluated.

PACS. 52.65.-y Plasma simulation

1 Introduction

One of the major challenges in the research towards a fusion power plant is the understanding and control of plasma turbulence which may lead to anomalous trans- port of particles and energy. Experimentally obtained im- proved scenarios such as H-mode confinement regimes show drastically reduced radial transport. The generation of H-mode confinement [2] regimes seem to be closely re- lated to poloidal shear flows in the edge region of the plasma. Generally, it is observed experimentally [6] and in numerical models [3] that the shear flows in plasmas suppress turbulence and transport. The generation mech- anism of these flows in thus of some interest [1].

It is known that shear in the V

E

=

EXB/B²

velocity caused by the rapid variation of radial electric field E

r

can suppress turbulence and anomalous transport from plasma instabilities [2].

The shear in the radial electric field E

r

, is believed to be important for the suppression of edge turbulence which is a dominant feature of the transition to H-mode confinement with an edge transport barrier [2].

Anomalous transport observed within a Tokamak is known to be the result of the electrostatic and magnetic turbulence. In the presence of electric perturbations, and for the normal profile of the safety factor q, the stochas- ticity of the trajectories increases and this is the princi- pal cause of diffusion of particles through magnetic sur- faces [8]. However for the reversed shear case, the main result proves that there exists a strong transport barrier near the surface at the minimum q-value. Such a barrier plays an important role in the reduction of the transport

a e-mail:[email protected]

and diffusion of particles. This leads to an improvement in the confinement of the plasma and the values are lower than those calculated for the first case.

We have studied the guiding center drift orbits in TEXTOR Tokamak, when a drift wave turbulence is present. The study has been performed with two different safety factor profiles. In the first case, we impose a mono- tonic profile while in the second case, a reversed shear configuration is assumed.

The first analysis consists of the study of the Poincare cross section for the normal profile, with a weak pertur- bation applied to the system. KAM theory can be suf- ficiently applied, and with the increasing perturbation, there is a transition to global stochasticity. This leads to a high diffusion of the particles. In the second case, we take a reversed configuration. When drift wave turbulence is present, a strong transport barrier occurs near the reso- nant surface. The physics is now totally different with the barrier playing an important role in the reduction of the transport and diffusion of particles. This leads to an im- provement in the confinement of the plasma and the values of diffusion coefficient are lower than those calculated for the first case.

This work is organized as follows: in the first section,

we reviewe the theory of local magnetic shear using the

Grad-Shafranov equation [7] and deduce the analytical ex-

pression of the safety factor q. In Section 2 we recall re-

cent experimental data in the formation of barrier trans-

port in improved regimes of confinement. In Section 3,

we describe the motion of charged particles in the pres-

ence of drift waves. We write down the guiding-center or-

bits equation for a given model drift wave and replace

theses equations with a local map which has the same

Hamiltonian structure [10]. Individual orbits obtained

from the map can differ qualitatively from those obtained from the differential equations, but statistical maps tend to give correct quantitative descriptions.

At the shearlesspoint, this map has the form of the standard nontwist map [12, 13]. In Section 4, we present a numerical simulation of the transport barrier and it’s contribution in reducing anomalous transport in the Toka- mak.

The study has been carried out with two different safety factor profiles; in the first case, we impose a mono- tonic profile while in the second case, a reversed shear con- figuration is assumed. We are interested in investigating the differences between the two configurations, to com- pare the trajectories and finally to evaluate the ratios of the diffusion coefficients for the reversed and normal pro- files.

2 The local and global magnetic shear

In this section, we will review the solution of the Grad- Shafranov equation and give analytical expressions for the local and magnetic shear [7].

For an equilibrium with toroidal symmetry, the equi- librium magnetic field can be written in the form:

B

= f

1

(ψ , θ

∗

)

∇

ψ + f

2

(ψ , θ

∗

)(

∇

ψΛ

∇

θ

∗

)

∇

ψ (1) where ψ is the magnetic poloidal flux, and θ

∗

is a general poloidal angle. The unknown functions f

1

and f

2

represent the two degrees of freedom in the representation.

The first degree of freedom in eliminated by choosing the Jacobian of the form J = F(ψ)/B

²

, where F(ψ) = I(ψ)+qg(ψ) is constant for a given magnetic surface. Here 2πg is the poloidal current flowing outside the flux surface.

2πI is the toroidal current flowing inside the flux surface and q is the safety factor. The second degree of freedom is eliminated by choosing the function λ so that the general toroidal angle ξ = ϕ

−

λ(ψ , θ

∗

) is such that the magnetic field lines appear to be straight in the (ξ, θ

∗

) plane. ϕ is the toroidal angle.

The safety factor can be written as:

B

· ∇

ψ

B

· ∇

θ

∗ ≡

q(ψ). (2) The contravariant form of the magnetic field can be writ- ten in the Clebsch form:

B

=

∇

α

∧ ∇

ψ (3) where α

≡

ξ

−

q(ψ)θ

∗

is the field line label.

The covariant representation of

B

can be written in the form:

B

= g(ψ)

∇

ξ + I(ψ)

∇

θ

∗

+ δ(ψ , θ

∗

)

∇

ψ. (4) Plasma equilibria in axisymmetric systems are governed by the Grad- Shafranov equation:

∇ · ∇

ψ X

²

q + q dp

dψ + gq X

²

dg

dψ = 0. (5)

Here X is the distance from the axis of revolution to a point on a magnetic surface.

In general equation (5) has be solved numerically, but in the low β limit, solutions can be found order by order by expanding in the inverse aspect ratio ε =

_R^a

, where a and R are the minor and major radius of the plasma.

To second order in our expansions, assuming that β = o(ε

²

), and we work in cylindrical coordinates:

X = R

0

+ r cos(θ)

−

∆(r) Z = r sin(θ)

where ∆(r) is the Shafranov shift and R

0

is the major radius of the magnetic axis. To second order, equation (5) decouples into a radial and a poloidal part. The radial equation provides an expression for the Shafranov shift:

∆(r) = 1 R

0

q

²

r

^,3

r

^,,3

q

²

1

−

2 R

²₀

q

²

B

₀²

r

^,,

dp dr

^,,

dr

^,,

dr

^,

(6) where B

0

is the magnetic field strength on the magnetic axis. For a given pressure and q profiles, the Shafranov shift is determined and the poloidal current can be deter- mined. The poloidal part of (6) provides an expression for λ as follows

∂λ

∂θ = qη X

1

−

X

η

(7) where η = R

0

(1

−

∆ ˙ cos θ).

Substituting equation (7) into the contravariant repre- sentation (3) we get to second order:

B

θ

= B

0

r qR

0

1

−

r

R

0

cos θ + ˙ ∆ cos θ

(8) and

B

φ

= B

0

1

−

r

R

0

cos θ + ∆(r) R

0

·

(9) The local magnetic shear (LMS) is a structural parameter of the magnetic field lines, crucial to plasma instability.

The LMS is conveniently divided into two parts:

S = ˆ S + R (10)

where ˆ S the global shear is the surface average of LMS and R is known as the residual shear.

Related to equation the LMS can be written as:

S

≡ −s·

(

∇ ∧s)

(11) where

s

=

B· ∇

Ψ

∇

Ψ

∇

Ψ (12)

is a vector lying in the magnetic surface along the direction of the binormal vector.

By using the normalized poloidal flux ψ(r) = B

0

Z

r

q(r) dr (13)

and the gradient operator

∇

=

∇

r ∂

∂r +

∇

θ ∂

∂θ +

∇

ϕ

−

∂λ

∂θ

∇

θ

∂

∂θ (14) both accurate to second order in ε, we obtain the local magnetic shear

S = S

⁽⁰⁾

+ S

⁽¹⁾

+ S

⁽²⁾

(15) where

S

⁽⁰⁾

= s ˆ R

³₀

S

⁽¹⁾

=

−

r

R

₀⁴

cos θ 1 + R

0

∆ ¨ + R

0

∆ ˙

r (1

−

2ˆ s)

!

S

⁽²⁾

=

−

r

R

₀⁴

cos

²

(θ)

2 ˙ ∆

−

r R

0

+ R

0

∆ ¨

3 ˙ ∆

−

r R

0

+R

0

ˆ s ∆ ˙

²

r

!

−

∆

²

1 + cos(θ)

²

1

−

r R

0

∆

·

The LMS scales likes R

₀⁻³·

s ˆ =

^r_q^dq_dr

and is the lowest-order global shear.

The surface average

h

S

i

=

R2π

0

J S(r, θ)dθ

R2π

0

J dθ

where J = qXη is the Jacobian of the transformation.

For the following, the plasma pressure is p(r) = p(0)

1

−

r

a

2

(16) and except for the case with negative shear, the q profile is

q(r) = q(0)

"

1 + r

a

2

q(a) q(0)

2

−

1

!#

·

(17)

3 Drift wave maps

3.1 Equation of motion

We use here a simplified equilibrium field in circular ge- ometry (i.e. a large aspect ratio) we have:

B

= B

θ

(r)e

θ

+ B

ϕeϕ

(18)

where B

θ

(r) =

_q(r)R^r

0

B

ϕ

is the poloidal magnetic field.

B

ϕ

is the toroidal magnetic field, r is the minor radius, θ and ϕ the poloidal and toroidal field and q(r) is the safety factor. This equilibrium corresponds to the usual development at order one in the small parameter r/R

0

, neglecting the Shafranov shift, and the field curvature.

In Gaussian units, the guiding centre equations of mo- tion are:

dx

dt = ν

_k B

|B|

+ c

EΛB

B

²

(19)

where

x

= (r, θ, ϕ), ν

_k

is the parallel velocity,

E

and

B

are the electric and magnetic fields, and the second term of the right side is the drift velocity.

The electric field satisfies the relation:

E

=

−∇

φ.

The potential φ can be written as the sum of a radial and a fluctuating part [10]. For the fluctuating term φ, we use

e

the model drift wave spectrum and we have:

φ(x, t) =

e X

m,l,n

φ

m,l,n

cos(mθ

−

lϕ

−

nw

0

t) (20) where w

0

is the lowest angular frequency in the drift wave spectrum.

The perturbed electric field E

e

is then E

e

=

−∇

φ.

e

In the following, we suppose B

≈

B

ϕ

B

θ

and B

r

= 0, using the toroidal coordinates (r, θ, ϕ) and we introduce E ¯

r

as the equilibrium radial electric field, the equation of motion becomes:

dr dt =

−

c

B 1 r

∂ φ

e

∂θ r dθ

dt = ν

_k

B

θ

B + c B

∂ φ

e

∂r

−

c E

er

B R dϕ

dt = ν

_k

. (21)

Substituting (20) into (21) we have:

dr dt =

−

c

Br

∂

∂θ

X

m,l,n

φ

m,l

[cos(mθ

−

lϕ) cos(nw

0

t) + sin(mθ

−

lϕ) sin(nw

0

t)] . (22) Using the fact that:

+∞

X

n=−∞

cos(nw

0

t) = 2π

X

δ(w

0

t

−

2πn)

and

+∞

X

n=−∞

sin(nw

0

t) = 0

we obtain:

dr dt = 2πc

Br

X

m,l

mφ

m,l

sin(mθ

−

lϕ)δ(w

0

t

−

2πn). (23) Hence this model spectrum, gives impulsive jumps in r at time tn =

^2πn_ω0 ·

3.2 Transformation in a mapping equations

It’s useful to replace the actual guiding-centre equations with those of a simple map. It’s convenient for this to introduce a new canonical variables.

We define the angle-action variables (χ, J) such that J = r

a

2

and

χ = M θ

−

Lϕ

where a is the minor radius of the torus. If we assume one mode (M, L) dominate in equation (22). From equa- tion (23) we have:

dJ dt = 2r

a

²

dr dt = 4πc

a

²

B M φ

m,l

sin(M θ

−

Lϕ)

×X

n

δ(w

0

t

−

2πn). (24)

By integration over one jump at time

tn =

^2nπ_ω0

, equa- tions (21) give us in terms of χ:

dχ

dt = M B

θ

rB

ν

_k−

c E ¯

r

B

θ

−

L ν

_k

R

·

(25) Ignoring in the first approximation ¯ E

r

, the integration on equations (24) and (25) over one jump give us the struc- ture of a simplectic mapping:

J

N+1

= J

N

+ 4πc a

²

B

M φ ω

0

sin (M θ

N−

L

ϕN

) (26) χ

N+1

= χ

N

+ 2π

ω

0

ν

_k

qR (M

−

Lq) . (27)

3.3 The standard Nontwist mapping [11]

Suppose we have a toro¨ıdal magnetic field with safety fac- tor equal to

q(r) = rB

ϕ

RB

θ

with a minimum local at r

i.e.

q

m

= q(r

m

); q

⁰

(r

m

) = 0 since

^dq_dJ

r=rm

=

^dq_dr

r=rm

dr dJ

r=rm

= 0.

Then q possess a minimum at J

m

= J(r

m

).

We consider the motion of particle near r and Taylor expanding q about J

m

, we can write:

q(J ) = q(J

m

) + q

_m⁰⁰

2 (J

−

J

m

)

²

. Substituting this into equation (25) gives us:

dχ dt = ν

_k

Rq

m

M

−

Lq

m−

M q

m⁰⁰

2q

m

(J

−

J

m

)

²

!

·

(28)

By integrating equation (28) over time step

_∆w¹

we obtain:

χ

_N+1

= χ

_N

+ 2π w

0

ν

_k

Rq

m

δ

−

M q

_m⁰⁰

2q

m

(J

N+1−

J

m

)

²

(29) where δ = M

−

Lq

m

.

We introduce the dimensionless variables K and T such that:

K = χ

2π ; T = M q

⁰⁰_m

2q

m

δ

!1/2

(J

−

J

m

) = k (J

−

J

m

) .

Hence, we can transform equations (26) and (27) in the form of the SNM:

K

N+1

= K

N

+ ν

_k

δ Rq

m

w

0

1

−

T

_N+1²

= X

N

+ α 1

−

T

_N+1²

T

N+1

= T

N

+

2πcM φ a

²

Bw

0

2M q

⁰⁰m

q

m

δ

!1/2

sin (2πK

N

)

T

N+1

= T

N−

β sin (2πK

N

) (30)

where α =

_Rq^νkδ

mw₀

; β =

−2πcM φ a²Bw₀

2M q_m⁰⁰ q_mδ

1/2

·

3.4 Global map

To introduce the effects of both a reversed shear and radial electric field, we have to work with a global map which has the following form:

J

N+1

= J

N

+ 4πc a

²

B

0

M φ ω

0

sin (M θ

N−

Lϕ

N

) (31) K

N+1

= K

N

+ RK 1 (J

N+1

) + RK2 (J

N+1

) (32) RK1(J) = ν

_k

(J)

ω

0

qR (M

−

Lq(J )) (33) RK2(J) =

−

cM

ω

0

aB

0

E ¯

r

(J )

√

J (34)

ν

_k

(J ) =

r

2

m (ζ

t−

eΦ

0

(J )) (1

−

λB

0

). (35)

Here ζ

t

is the initial total energy, e the charge of particle, λ = µ/ζ

t

where µ is the magnetic momentum and Φ

0

is the equilibrium potential such that

E ¯

r

(J ) =

−

∂Φ

0

∂r

_r=a√

J

.

In the following we will used two q profiles.

The normal profile q(r) = 1.99 + 1.94(r/a)

²

and the reversed profile q(r) = 1.99 + 7.76(r/a

−

0.5)

²

.

The choice of the potential Φ

0

depend on the nature of the profile q, therefore in the normal case we take Φ

0

(r) =

−

Φ

0

(1

−

(r/a)

²

) and for the reversed profile we have Φ

0

(r) = Φ

0

(1

−

(1

−

2r/a)

²

).

4 Experimental data on the formation of transport barrier

Observation of spontaneous bifurcation to reduced trans- port states has opened up a new era of confinement physics. Such bifurcation produces transport barriers. The edge transport barrier (ETB) is commonly observed as H-mode and the internal transport barrier (ITB) is now observed in a number of devices using a variety of heating and fuelling regimes [3, 6].

Many of the high-confinement modes are characterized by transport barriers such as the central pellet injection mode, the high-β

p

mode, the reversed shear mode and the H-mode with the ETB. These internal and external transport barriers are widely observed in a number of de- vices with various control schemes such as ion heating, electron heating, induced E

r

by loss of charged particles or by biased electrodes. Also many of the enhanced con- finement modes are characterized by a flat or a reversed safety factor profile, and this is the reason why the weak or negative shear modes have been studied extensively in recent Tokamaks.

Furthermore, the reaction plasma operation region has a high density which means that the equipartition be- tween ions and electrons is strong and for current drive, higher electron temperature is more beneficial. In H-mode, diffusivities of both ions and electrons are reduced, but for ITBs, some modes show strong reduction of ion ther- mal, electron thermal, particle and momentum diffusivi- ties, while others show less improvement in the electron transport.

In addition to the possible effects of magnetic shear on electron transport, a change of the safety factor profile affects the radial shapes of density and temperature pro- files. Also locations of q

min

or q = integer surfaces seem o affect the barrier formation and propagation [4, 5]. The

Fig. 1. Difference of profiles of diffusivities and safety factor.

(a) The reversed shear H-mode and (b) the weak positive shear in JT-60U.

radial profiles of diffusivities and the produced profiles of density and temperature are different for strongly negative shear case and weak monotonic shear.

The following figure compares the reversed shear H-mode and the high-β

p

mode in JT-60U.

In the reversed shear mode (Fig. 1a, χ

i

and χ

e

decrease quickly down to the ion neoclassical level in a narrow layer of ITB. In the positive shear case Figure 1b diffusitivi- ties are decreased largely at the ITB foot, but the level is still much higher than the ion neoclassical level. Such a gradual decay of χ

i

down to the neoclassical level at the center is also observed in JET and ASDEX U with the weak shear configuration. These results suggest that, in the weak shear mode, turbulent transport is not de- termined by a single instability, and if the dominant one is suppressed the residual second instability governs the anomalous transport. In the strong reversed shear mode, in turn, the second instability is stable and if the first in- stability is suppressed the diffusivity jumps easily down to the neoclassical level.

In the reversed shear mode, χ decreases quickly down to the neoclassical level in a narrow layer of ITB (Fig. 1a).

In the positive shear case, diffusivities are decreased at

the ITB foot, but the level is still much higher than the

neoclassical level (Fig. 1b) [6].

Fig. 2.Profile of safety factorversusr/afor normal and reversed shear.

5 Numerical results and comments

In the next section, we neglect the electric field radial com- ponent and investigate the map phase structure by calcu- lating 1000 massive D

⁺

particle trajectories with various initial conditions in configuration spaces for the reversed and normal shear cases.

In the calculations below, we have used the Texas Ex- perimental 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.

In our analysis, we study the Poincare cross section in the phase space for two cases of safety factor profile: 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 the following Figure 2 where we have represented the variation of the profile of the dimensionless safety fac- tor function r/a.

We start by plotting the trajectories without pertur- bation, the variable X used here replace the variable K.

First, we have seen in Figure 3 the ideal case without per- turbation; trajectories are regular.

In the following, we observe the effects on trajectories with the action of electric perturbation. Figures 4 and 5 show the Poincare section respectively for the reversed and normal shear case and for the value 1 eV of the drift wave potential amplitude. The representation was carried out in the (X , r/a) plane.

For the reversed shear profile, the surface located near the minimum value of the safety factor q play an

Fig. 3. Trajectories without perturbation in the (X, r/a) plane.

important role in producing a transport barrier. Such a formation has been observed experimentally and defined a high confinement mode, in which transport of the par- ticle was reduced considerably with a simultaneous im- provement in the confinement. On the other hand, for the normal shear profile, magnetic islands appear near the resonant surface, where we can apply the KAM theorem;

the trajectories are integrable for low value of the per-

turbation, but for higher values, KAM torus disappear

and there is a transition to global stochasticity. Transport

and diffusion of particles increase and this is the princi-

pal source of the anomalous transport observed in this

case [9].

Fig. 4.Poincare map of 1000 particles in the (X,r/a) plane for the value 1 eV of the perturbation - reversed shear-.

6 Particles diffusion

In order to study the particles diffusion through magnetic surfaces in Tokamak, we have evaluated in two cases (nor- mal and reversed shear) the diffusion coefficient. In our simulations of diffusion we have used the following model for the coefficient of diffusion

D = lim

N→∞

D

(r

N−

r

0

)

²E

2t

N ·

(36)

In Figure 6, we have represented the time evolution of the ratio of the diffusion coefficient in the reversed shear (D

r

) to the diffusion coefficient in the normal case (D

n

) for a perturbation value of 8 eV. Here t

n

is a time step. We observe a reduction of particles diffusion in the reversed case, due to transport barriers which have a tendency to suppress anomalous transport.

We represented this ratio of diffusion coefficients as a function of the perturbation see Figure 7. We can see that with the increase in the level of perturbation the diffu- sion in the reversed case becomes lower than for the nor- mal case. The formation of a transport barrier which sup- presses turbulent transport may explain this reduction [4].

7 Conclusion

We have studied the guiding-centre orbit in a TFTR Toka- mak using a mapping formulation and a specifical model

Fig. 5. Poincare map of 1000 particles in the (X,r/a) plane for the value 1 eV of the perturbation - normal shear -.

for the drift wave. Two safety factor profiles are assumed in order to describe the trajectories; the normal and the reversed profiles. The main results show that by taking a reversed magnetic shear (i.e. safety factor q decreasing with minor radius in the central region and increasing in the outer region).

An internal transport barrier is formed near the lo-

cation of the minimum value of q. Such ITB formation

is caused by a turbulent transport suppression caused by

sheared rotation [4, 5]. We have also evaluated the ratio

of diffusion coefficient for the reversed shear D

r

and the

normal shear D

n

as a function of time and of the pertur-

bation. The results show a reduction of particle diffusion

in the reversed configuration. Our analytical and numeri-

cal approach are qualitatively in good agreement with the

theoretical and experimental results [6, 10].

Fig. 6. Ratio of diffusion coefficient in the reversed and normal shear as function of time steptn.

Fig. 7.Ratio of diffusion coefficient in the reversed and normal shear as function of the amplitude of perturbation.

References

1. S.B. Korsholmet al., Renold stress and shear flow genera- tion,27 EPS Conference on Plasma Phys. and Controlled Fusion, Budapest, 16–18, 2000.

2. T.D. Rogulienet al., Plasma Phys. Cont. Fusion42, A271 (2000).

3. V. Antoni et al., Plasma Phys. Cont. Fusion 42, A271 (2000).

4. H. Takenaga et al., Plasma Phys. Cont. Fusion 40, 183 (1998).

5. B. Jhowry, Plasma Phys. Cont. Fusion42, 1259 (2000).

6. Y. Kamada, Plasma Phys. Cont. Fusion42, A65 (2000).

7. J.L.V. Lewandowski, M. Persson, Plasma Phys. Cont. Fu- sion37, 1199 (1995).

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

AP4, 329 (1998).

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

Phys.7, 1045 (1997).

10. W. Horton, H. Binpark, J.M. Kwon, D. Strozzi, P.J. Mor- risson, Duk-In Choi, Phys. Plasma5, 3910 (1998).

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

12. D. del-Castillo-Negrete, P.J. Morrison, Phys. Fluids A5, 948 (1993).

13. D. del-Castillo-Negrete, J.M. Greene, P.J. Morrison, Phys- ica D100, 311 (1997).