MAGNETIC TURBULENCE IN TOKAMAKS

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MAGNETIC TURBULENCE IN TOKAMAKS

A. Samain

To cite this version:

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MAGNETIC TURBULENCE IN

TOKAMAKS

A. SAMAIN

Association Euratom-C. E. A. sur la Fusion

DCpartement de Physique du Plasma et de la Fusion Contr8lrk Centre d7Etudes NuclCaires

Boite Postale no 6, 92260 Fontenay aux Roses, France,

R6sum6. - D'une discussion du processus de disruption, il ressort que ce processus consiste probablement en l'apparition d'une turbulence ti grain fin. Cette turbulence doit dtre capable de

1 aP

produire les valeurs Clevees du champ Clectrique inductif

-

,

, assocites a la rkorganisation c of

du flux poloidal Y(r)et de la densite de courant Z(r) sur la surface magnttique de rayon r. I1 est alors plausible que la turbulence consiste en des modes rippling en presence desquels la loi #Ohm prend la forme

1 a l y

- - - _=qZ--

a

_{r K -}

az

.

c at r a r ( ar)

a

az 1 a~

Le terme anormal

-

r K - peut expliquer les valeurs constatkes de

-

si la perturba-

a

ar) c at

tion magnttique realise une substantielle ergodicitk radiale des lignes de flux. I1 est montre que les modes sont instables en presence d'une telle ergodicitk meme en regime non collisionnel, I'ergo- dicite jouant un rBle analogue A celui de la rtsistivitk pour attknuer la contrainte M. H. D.

Abstract. -From a discussion of the disruption process, it is concluded that this process plausibly consists of the onset of a fine grain turbulence. This turbulence must be able to produce the large

1 a l y

values of the inductive electric field - - - which are associated with the reorganization of the c at

poloidal flux Y(r) and the current density Z(r) on the magnetic surfaces of radius r. It is then plau- sible that the turbulence belongs to a class of rippling modes in the presence of which the Ohm law takes the form

1 a l y

- - - _{= q Z - -}

a

_{r K -}

az

_.

c at r a r ( ar)

a

az

The anomalous term - - rK - may explain the experimental values of

-

-1.

for magnetic

a r ( ar) c at

perturbations corresponding to a substantial radial ergodicity of the flux lines. The stability of the modes in the presence of such an ergodicity is accordingly considered. It is found that the modes may be unstable even in collisionless regime, the ergodicity playing a role similar to the resistivity to partially remove the M. H. D. constraint.

1. Discussion of the disruptive process. - The disruptions associated with the surface q = 1 and the soft disruptions associated with the surface q = 2, as they are detected through soft X ray emission

11, 2, 31, consist of a sudden (50-200 ps) partial flatte- ning of the temperature profile in a large domain on both sides of the resonant magnetic surface where the safety factor q = 1 or 2. They are generally preceded by the relatively slow onset of an oscillating structure which has been identified to magnetic islands in the case q = 2 [2,4]. Owing to the strong analogy of the X ray signal (in space and time) in the case q = 1

and q = 2, this interpretation is also plausible for the disruptions q = 1. Eventually the oscillating structure

persists after the disruption. Near the end of the disruptions q = 2 a very strong and short negative voltage pulse around the major axis appears [ 5 ] .

Such a negative spike is absent in the case q = 1. This is not surprising owing to the fact that the effect of the disruptions q = 1, while significant relatively far from the resonant magnetic surface, does not reach the plasma edge. The presence of the voltage spike in the case q = 2 means a sudden variation of the poloidal flux embrassed by the magnetic surface at the plasma edge. It is likely that the poloidal flux

2 n R Y ( r ) embrassed by the surface r (see Fig. 1)

varies by a quantity of the same order in the domain affected by the disruptions. Assuming that this varia-

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C6- 1 04 A. SAMAIN

FIG. 1. - Tokamak Geometry.

tion takes place progressively during the disruption process, this means the existence of an inductive

i ay

toroidal electric field

-

significantly larger than

c at

the normal value of the resistive effect ?I. This con- clusion is plausible also in the case of the disruptions

q = 1 [6]. Assuming that the variation of Y(r) during

the regeneration periods between two disruptions (caused by the rotational part of the electric field y(r) I(r)) is cancelled out by the variation of Y(r)

i ay caused by the disruptions, the value of

-

- - during

c at the latter is

2

3 qI. It must be noted that disruptions q = 1 may exist without magnetic activity on the

surface q = 2 and vice versa, and therefore that the coupling between the two types of perturbation, however interesting, do not play a major role in the disruption mechanism.

The simplest idea to interpret the disruptions is [7, 81 that they consist of the development of the magnetic islands initially present near the resonant surfaces q = 1 or 2 which would invade in particular the domain inside the surface q = 1 or the domain outside the surface q = 2, while keeping roughly

their topology. In the case of the disruption q = 1, this idea has been encouraged by the fact that the tearing mode which is at the origin of the magnetic island has a relatively large linear growth rate. In fact there are serious objections to this interpretation. For instance it is difficult to understand on this basis why the action of the disruptions decreases progressively with the distance from the resonant surface, or why the oscillating structure may persist after the disruption, or why the disruptions q = 2 induce a large voltage spike during a time which is short compared to duration of the disruption. The major objection is however that the assumed topology change of the flux lines seems impossible in the very short time of the disruptions. Let us consider for instance the case of a disruption q = 1, depicted on figure 2. Let @ be the flux between the closed flux lines (3 (initially the helical neutral line of the magnetic island) and 33 (initially the magnetic

axis). We have

FIG. 2.

-

Hypothetical disposition of magnetic island at the beginning (1) and the end (2) of a disruption q = 1.

1 d@

- - - =

1%

E,, dl

-

Je

E,, dl c dt

where EL, and 1 are the component of the electric field and the abscissa along the lines 33 or

C,

and qs, qe, 13, Ie are the values of the resistivity and the current density on 33 and C . The line 33, which is

reminiscent of the plasma center is likely to be hotter than the line (3, and therefore q s < qe. It then results from (1) that

Before the disruption the value of Qi is equal to the difference @,,,

-

Qp,, of the toroidal flux @,,, and

the poloidal flux @,,, embrassed by the surface

q = 1 of radius r,. After the disruption, assumed to have a duration at, the value of @ is 0. We then obtain from (2), assuming a parabolic profile for the initial density current

where q, is the initial value of q on the magnetic axis and 33 is the static field. On the other hand, the

magnetic energy which is initially disponible for the process has the form

3

L(Io

-

I,)', where I, and I, are the initial current density on the axis and the magnetic surface q = 1, and L is a coefficient of self induction. During the disruption, a magnetic energy

3

L'(13

-

Ie)' will appear, with the coefficient L' obviously of the same order as L. The energy conservation imposes that

3

L(Io - IJ2 >

3

Lt(I% -

Id2

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Typically, in the T. F. R. case, the L. H. S. term of this inequation arises to 200 V, to be compared to a value of 2 nRyI

-

2 V. The process is therefore

incompatible with a value L/L1

-

1.

We may then consider as plausible that the disruptions are due to a fine grain turbulence. This turbulence must induce a reorganization of the poloidal flux 2 nRY(r) and the current density I(r) (now taken of course in the average at the scale of the turbulence), corresponding to an inductive electric

i aY

field - - - larger than the normal values of yI.

c at

A first possibility is that the turbulence induces anomalous values of the resistivity y. However the disruptions take place for values of the temperature T, density n and current density which are by no means critical for the onset of such an anomaly. 1 aY is It is more likely that the anomalous field -

-

c at

produced by an electromagnetic effect, namely an

"

associated with a magnetic

effect of the type -

C

perturbation 6B and a velocity perturbation 6u both transverse to the static field. The turbulence may then be specified by a potential vector 6A parallel to the static field B and an electrostatic potential S$ so that

The electric field

Ell

along the perturbed flux lines, averaged at the scale of the turbulence (symbol -) is then given by

1 aul = - - - -

c at div

(g

6$)

.

(5a) As the vector 6B 6$ may vary only in the radial direction along which inhomogeneity exists and as, in the average at the scale of the turbulence, we must haveq, = y z we obtain from (5a)

1

aul

-

- --

= qI

+

c at (5b)

an equation which may be considered as the new Ohm law in the presence of the turbulence. Unstable electromagnetic modes of the form (4) which exhibit a non vanishing value of 6B, 6$ are somewhat similar to the rippling modes which have been investigated by Furth, Kileen and Rosenbluth [9]. For conve- nience we will also call them rippling modes. Actually it is plausible that the turbulence we have in view consists of modes of this class. We will consider them in some detail in the next sections. We note simply

here that if such a turbulence exists, the angular deviation of the magnetic field produces a transverse

-- dB

electron current 61

-

I - which builds up a charge

B

density

The latter is likely to-- be neutralized by a charge density

corresponding to the presence of the potential 6$, the quantity a being a proper constant. We then have

1 87 dB, S$-

a dr B and the Ohm law (5b) becomes

The turbulence liberates a fraction of the magnetic energy W associated with the poloidal field B,. We have in fact

The second term represents the power which is liberated by the turbulence and which is disponible to maintain its amplitude.

2. Rippling modes in the linear regime. - It is convenient to introduce, for a given applied electromagnetic perturbation

6A = a(x) exp iwt

+

Comp. Conj. (7) 6$ = $(x) exp iot

+

Comp. Conj.

the following set of quantities.

10 The first order current and charge densities 61 = j(x) exp iwt

+

Comp. Conj.

6p = p(x) exp iwt

+

Comp. Conj. (8) as they appear from the action of the field 6E, 6B on each plasma species. The field j(x), p(x) results from the field a(x), $(x) by a linear transformation which depends analytically on the frequency o.

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C6-106 A. SAMAIN

which again depends analytically on o . It is readily verified that if the field a, $ and the frequency o

represent a self consistent mode, the Maxwell equations (neglecting the displacement currents) which must be verified by a, $ and o is equivalent to stating that the form T. is an extremum with respect to all variations of the field a*, $*. This implies that C = 0, an equation which gives the frequency w when the geometrical structure a(x), $(x) is known. Also, it may be verified that a field 6E, 6 B of the form (7), with o real, assumed to be created by charge and currents independent of the plasma, provides a power w to the system consisting of the plasma and the magnetic field inside the boundary surface, given by

w = - 2 w Im (C(o ; a , $; a*,$*)). (10) Let us first consider the case of a collisional plasma with negligible temperature, and a mode (see Fig. 1)

a(x) = a(r) exp(il6

+

imq)

,

a

//

B

$(x) = $(r) exp(il6

+

imq) (1 1) exhibiting along unperturbed flux lines the wave number

The ion contribution to j is then given by jll = 0 ; jl = - nec VL* x B

B~ 4 7c

where E = c2/c; and c , is the Alven velocity asso-

ciated with the static field. On the other hand the electron contribution to j i s readily found to be

V x a

jl = I-

+ nec

VL* x B

B B~

Using the charge continuity equation imp

+

div j = 0 to calculate p, and noting that

I

V x a

I

z

I

V,a

I,

we obtain

By putting I)= w$' for w real we transform

T:

as

C(o ; a, $ ; a*, $*) = A(o ; a, $' ; a*,

$'*I

.

The bilinear form A in a, $', a*, $'* may be analytically continued in the plane o. The system of equations which determine the field a, $ and the frequency w is equivalent to stating that C is extremum for all variations of a*, $*. For o real this system is equivalent to state that A is extremum for all variations of a*, $'*.

This equivalence is valid in the complex plane o if A has been analitically continued. For io = y real we obtain

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The tearing modes corresponds to a tnal field

a, $' satisfying the M. H. D. constraint

outside a thin singular layer including the resonant magnetic surface r = r, where K,, = 0. Inside this layer we may take a = constant and interpolate the values of $'. The true rippling modes of Furth

et al. correspond to a = 0 and $' localized in a thin layer on one side of the resonant surface. The rippling modes we have in view correspond to a field a,

$' = i$/y symetric with respect to the surface r = r,,

with the field a and $' localized in radial intervals 6 and 6' respectively. Such a field is depicted on the figure 3. We then obtain from (15), assuming that

l/r

-

1 16 and 6' < 6

FIG. 3.

-

Trial field a, y' for a rippling mode in the collisionnal case.

where

We will assume that

We then find > 0 if

where

and c,, is the Alfven velocity associated with the field Be. The new Ohm law (6a) is easily obtained from (5b) in the form

A value of

significantly larger than ylimplies that

This condition means that the transverse magnetic perturbation 6B is large compared to the transverse component of the equilibrium field which is created by the shear in a radial interval equal to the scale 6 of the magnetic turbulence.

Let us now assume that the electron have a finite temperature T and the collision frequency v, is small. Except in a relatively small layer near the resonance surface we have in that case

The electron contribution to j results from the integration of the Vlasov Equation. Neglecting the gradients of the temperature and of the density n, we obtain instead of (13)

V x a jL = I -

B

-

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C6-108 A. SAMAIN

This expression of C, valid in the domain where (18) similar to (18) is satisfied and that is satisfied, does not contain the terms proportionnal

to rcil 9

I

K I I

I

if

I

r

-

r,

1

5

6 (21a)

which appear in the expression (14) and which are responsible for our rippling instability. In fact, in the domain where KII, is too small for the condition (18) to be satisfied, the expression of C changes progressively from (20) to join (14) in the small layer where

I

KII Vth

1

<

1

o v, Ill2. The thickness of this layer is larger than the scale 6' given by (16) if

The condition (21b) may be verified aposteriori

We first consider the electron contribution to I: in the absence of a gradient df/dr of the averaged current density

Z

The electron response to the field a, $ in that case essentially consists of reaching the new thermodynamical equilibrium along flux lines, in which, because of the condition (21b), the electrostatic potential plays the major role. This response then essentially consists of the charge density

ne2

It may be shown that our rippling modes then persist p = - - $ .

as in the collisional case depicted above.

T

Taking into account the definition (9), the correspond-

3. Stability of the rippling modes in the presence ing value of I: is found equal to

of the turbulence. - We have seen in the preceding section that the rippling modes are able to produce

the large inductive electric fields { ~ { $ I $ ~ I ~ ~ x .

characteristic of the disruptions if the magnetic perturbation satisfies the condition (17). It is then natural to ask the question : are our rippling modes still unstable in these conditions ? We may answer this question by the same technique as in the linear regime by considering a perturbation SE, SB of the form (7), calculating the response in current and charge density of the form (8) in the presence of the turbulence, then calculating the bilinear form C(o ; a,

\CI

; a* $*) specified by (9) which may be used in fact to determine the modes as in the linear case. To simplify the problem we will assume that the linear response of ions is still valid in the presence of the turbulence, focusing our attention on the electron response and neglecting again the density and temperature gradients. We will assume also that the condition (17) is largely satisfied so that the field a, $ exibits a wave number

along perturbed flux lines which is large (the quantity rcll must be of course distinguished from the linear value

We assume in particular that the condition

We now calculate the term of I: which is proportion-

a l

nal to - a. This needs the calculation of that part dr

j' of the electron current response which is proportion-

-

ar

nal to - a. The current j' may be obtained by dr

integration of the Vlasov equation, and in fact is transparent on (19). We have

.

I

a7

jil =

1

- z--ad1

flux line i" B ar

Vxa -

j;

The presence of the turbulence makes the integrand in

5

a stationnary randon function with a small

flux line

coherence length I/IC,~. Therefore j; is the sum of a small value

coherent with the field a, $, and of a series of balistic terms. These terms are incoherent with the field a, $ and have not to be known to calculate the expression of C. Finally we retain only the current

<

j;

>

estimated above and the current j;. Calculating through the charge continuity equation the corresponding charge density

1

- _(iKll_jil

+

_{div j;)} ₌_-

-

(8)

and using (21a), we obtain from the definition (9) the takes a positive value. Unstable rippling modes with

a?

a and $' localized symmetrically on both sides of

desired part of I: proportionnal to - a in the form

dr the resonant surface in the same interval 6 exist by taking

jjjt

f

:a$* d 3 x .

c A ~ Pthi

y - - - .

_,

A ; = , T

To calculate the term of I: which is proportionnal to qRr r T A ne

a?

- $, we may use the fact that the power which is

ar pthi = (c/eB)

TI/'

(ion mass)'/'

.

added to the plasma by a perturbation a, $, w, The Ohmlaw then becomes with w real, must tend to 0 when w tends to 0, i. e.,

for a static perturbation. Taking into account (lo), 1 a F

_-

-

a

q1 - - ( r K

$1

this means that w Im I: tends to 0. This determines c at r dr

the desired vart of I: in the form -

'

"a* d 3 x . r wB ar

Tt finally results from these arguments that the expres- The poloidal magnetic energy liberated by unit time sion of I: in the presence of a strong turbulence is by the turbulence

-

dW/dt is given by (6b). The

given by scale 6 must be large enough for the power - dwldt

to be larger than the joule losses 1

C = j J [ d 3 x ( $ ~ $ ~ 2 - G ~ v L a ~ 2

+

-

111

w 4 c 2 d 3 x .

E I 1

a?

6'

I

4 n

I

+

* + (22)

This condition imposes a lower limit to 6, namely The essential difference between the expressions (22)

and (20) of C is that we have recovered in (22) the terms proportionnal to

which appear in the collisional expression (14), and which are responsible for our rippling instability. It must be noted that the origin of this situation is the fact that the current j; considered above has a small component coherent with the field a, $. This cancellation is fundamentally linked to the existence of balistic terms of j; incoherent with the field a,

$. In fact these terms necessarily generate torsionnal Alven type modes which represent a damping mechanism for the mode a, $. This damping effect is auto- matically taken into account by the variationnal technique of the form C. In the present collisionless case, it partially removes the M. H. D. constraint and allows the onset of unstable rippling modes. Indeed, by putting $ = o$' and transforming the form I:(@ ; a, $ ; a* $*) into A(o ; a, $' ; a*, $'*)

as explained in the

8

2, it is readily shown that unstable modes with a growth rate y exist if there exists a trial field a, $' = i$/y such that the hermitic form

The turbulence produces an anomalous inductive field

larger than qTif we have

4. Concluding remarks. - The turbulence could exist in the disruptive form, with

and accordingly produce anomalous inductive field

It could be triggered by the singularity of the slope daar at the separatrix of the magnetic islands initially present on the resonant surface, and propagate explosively in the plasma as explained in [lo].

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C6-110 A. SAMAIN

and then induces a value the singular current layer to appear. A critical density,

beyond which this stabilization is not effective enough, 1

a@

-

- - - -

_fll and soft or hard disruptions are triggered, could

c at correspond to a critical ratio of the growth rate of

the tearing mode on the surface q = 2 to the inverse which is only a fraction of yx In that case the diffusion scale time y of the turbulence. The new Ohm Law

2

coefficient should roughly have a value

-

6,y

-

yc2, should also result in anomalous skin effects and which is not inconsistent with experimental data. anomalous acceleration of Runaways. Such effects, if The new Ohm law should stabilize the tearing modes present, could be a proof of the presence of the in the linear range on the surface q = 2 by preventing turbulence.

References

[I] VON GOELER, S., STODIEK, W. and SAUTHOFF, N., Phys. Rev. Lett. 33 (1974) 1201.

[2] VON GOELER, S., VIIth European Conference on Con- trolled Fusion and Plasma Physics, Lausanne (1975), Vol. 1, p. 1.

[3] T. F. R. GROUP, in Plasma Physics and Controlled Nuclear Fusion Research (Proc. VIth Int. Conf. Bertchesgaden

(1976)) I. A. E. A., Vienna (1977), Vol. 1, p. 279.

[4] JOBES, F. C., HOSEA, J. C., Proc. 111th International Symp. on Toroidal Confinement, Garching (1973), B 14. [5] HOSEA, J. C., BOBELDIGK, C., GROVE, D. J., in Plasma Physics and Controlled Nuclear Fusion Research

(Proc. IVth Int. Conf. Madison (1971) I. A. E. A. Vienna (1971), Vol. 2, p. 425.

[6] HUTCHINSON, L. H., Phys. Rev. Lett. 37 (1976) 338.

[7] KADOMTSEV, B. B., in Plasma Physics and Controlled Nuclear Fusion Research, I. A. E. A. Vienna (1977), Vol. 1, p. 555.

[8] WHITE, R. B., MONTICELLO, D. A., ROSENBLUTH, M. N., WADDEL, B. V., in Plasma Physics and Controlled Nuclear Fusion Research, I. A. E. A. Vienna (1977), Vol. 1, p. 579.

[9] FURTH, H. P., KILLEEN, J., ROSENBLUTH, M. N., Phys. Fluids 6 (1963) 459.