J. Plasma Fusion Res. SERIES, Vol.l (1998) 88-91

Physics of Collapses in Toroidal Helical Plasmas
ITOH Kimitaka, ITOH Sanae-I.1, FUKUYAMA Atsushi2 and YAGI Masatoshir
National Institute for Fusion Science, Toki 509-5292, JapanlRIAM, Kyushu University, Kasuga 816-8580, Japan
2Faculty of Engineering, Okayama University, Okayama 700-8530, Japan(Received: 30 September 1997 / Accepted: 22 October 1997)

Abstract
Theoretical model for the collapse events in toroidal helical plasmas with magnetic hill is presented.
There exists a turbulent-turbulent transition at a critical pressure gradient, leading to a sudden increaseof the anomalous transport. When the magnetic shear is low, the nonlinear excitation of the global
mode is possible. This model explains an abrupt growth of the perturbations, i.e., the trigger phenomena. Achievable limit of the plasma beta value is discussed.

Keywords:
collapse, transition of turbulence, high beta plasma

1. lntroduction

Here we present the theoretical model of catastrophic events in toroidal helical plasmas. The turbulenceturbulence transition has been predicted (M-mode transition) [6, 7]. The relation between the pressure gradient and the heat flux has the hysteresis characteristics.
At the critical pressure gradient, the turbulent transportcoefficient suddenly increases. The rapid decay of thepressure gradient is predicted to occur. The abrupt destabilization of the global mode is also discussed. The
sudden excitation of the perturbations and the rapid
decay of the pressure gradient are predicted to occur
via two mechanisms. Achievable limit of the plasma

The collapse phenomena in toroidal plasmas arean essential subject. Reports have been made on thebeta-limiting phenomena in Heliotron-E [1,2]. Possible
mechanisms for the onset of the collapses have been
considered to be the MHD instabilities [3].
The pressure driven instabilities have been analyzed in conjunction with the beta{imiting phenomena.
The usual analysis on the condition of the onset isbased on the linear stability. Although there have beenreports on coincidence between the linear stability
boundary and the experimental condition for the onsetof perturbations, there is a fundamental problem: Thecollapses are caused by such the activities of which
growth rate changes abruptly. (See, e.9., [2].) The abrupt jump cannot be explained by the conventional approach to the MHD instabilities. Such a sudden changeof the growth rate of the symmetry-breaking perturbations has also been observed on tokamaks [4], and theunderstanding on the occurrence of the collapse eventswould provide a fundamental elements in the physics oftoroidal plasmas [5].

beta value is discussed.

2. ModelEquation
The change of the growth rate of the symmetrybreaking perturbations (either global or microscopicones) has been known very fast in experiments: It
usually ranges from 1.0 ps to 100 ps [5]. This fastchange of the growth rate shows that the mode growth

is not described by the linear theory, in which thechange of the growth rate is predicted within a time
O1998 by The Japan Society of Plasma
Science and Nuclear Fusion Research88

ItohK. et al., Physics of Collapses in Toroidal Helical plasmas

scale of the global transport. The processes that havethe short correlation time (10 ps to 100 ps, or muchthe turbulent transport, and we havefaster) should play an essential role. In order to analyzel",N - li,Nthe problem, the picture of the self-sustained turbulence is applied to the beta-limiting phenomena.
The method of dressed test mode is applied to thebeta-limiting phenomena. The reduced set of equations
is employed. The interaction with the back-ground tur-If the gradient is large, and magnetic braiding takes
place, then the selective loss of electrons is induced asfm;r"te,r - Ibulence is renormalized as nonlinear transfer rates,
which have the form of diffusion operator. (See[g] forthe detail of the theoretical procedure.) In the systemwith the magnetic shear and magnetic hill, the eigenvalue equation for the stationary state (i.e., the nonlinear critical condition) is given as,,2^z dt6'"

tk rT1dik 0 &)^,4-- dl (dpo",,/dr),(s)sivity is the dominant instability mechanism [10]. On
the contrary, the global mode is destabilized by the resistivity.

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3. M-mode Transition
When shear is strong, tt , Oo, the short wave
length mode is excited [10]. If the pressure gradientreaches the critical gradient

where, ftl : kA + k2, Go denotes the equilibrium pressure gradient coupled to the magnetic curvature,
Go*i :^j,ru,.^

where the coefficient M is determined by the competition between the perpendicular decorrelation and theparallel decorrelation.
For the short wave length mode, the current diffu-+ pt4 Q (k)-#(*.*) ouq:o(4')

Go"* Go, ) G" = s(6)(2\the overlapping of the micro-island takes place[6]. Theturbulent transport coefficient becomes enhanced

which is the origin of interchange instability, s the shear
parameter r(dq/dr)q-2, .f/ the average curvature of themagnetic field,, p the viscosity, l the thermal diffusivity,

4 the resistivity and ,1, the current diffusivity. We employed the normalization for resistive MHD modes as:(euo/a)t - t, r/a - r, z/R - z, f/(eauoBi - /,J(apo/eB) - J, p(2po/eB&) - p, q(rno/poa2\ - q,p(ror/a2) - p, 1(roo/p6oo) - 1, y(roo/a2) * X, where

where 1'- is the turbulent transport coefficient in the
L-mode Xr = (Goi* Gn")3/2s'2(c/aar)2, and the factor
M is approximately given as M = {GoBr. The backtransition occurs at the lower gradient G,.

e is the inverse aspect ratio, a/R, us is the Alfuen veloc-ity, B0 is the main magnetic field, and zoo : R/uo. It isnoted that the finite electron mass and convective non-(^ , o,"ETt(o,,* ^ ,l;i ), ., - nB,

linearlity on the current were kept in the Ohm's law, inorder to discuss properly the electron dynamics.
The transport coefficients (p, ),, X,, 4) are the sumof the collisional transport and turbulent transport. One
has an approximate estimate for high beta plasmas as

The gradient-flux relation has the cusp-type catastrophe, as is illustrated in Fig.1. When the gradientreaches the criterion, the transport suddenly becomeslarge. (The time scale of the growth is given as Gi]/2.)
Due to the enhanced transport, the pressure gradientstarts to decay. Owing to the hysteresis nature of thegradient-flux relation, the crash continues until the gradient reaches to the lower critical point. This transitionis called as M-mode transition [6]. When the transporttelli,N - Xr,N - ?x,l",N - /",N and1",. - (c/aoa)2p.,*,(s)(3)

where suffix N indicates the turbulent transport . Whenthe gradient is weak, the E x B convection dominates
89ItohK. et al., Physics of Collapses in Toroidal Helical Plasmas

catastrophe occurs at a certain magnetic surface, thereappears' an avalanche of the crashes across the magnetic surfaces [11]. The propagation velocity of theavalanche was obtained. Owing to the avalanche, theTNxe

prbssure gradient in the wide region is rapidly depleted.
M-mode4. Destabilization of Low'm Mode

/The low-rn mode is destabilized by the anomalousresistivity. In the weak shear case, sz 1 Go, the destabilization of the global mode (through the nonlinear interactions with the back ground turbulence) takes place.

/Neglecting the 7k2t term in comparison with 4 inEq.(1), the criteria is obtained as [12]rys'x -:x" xi

L- and H-modeoG,c"qC, = #l4lnl2(Go1.r;lytm'4)t/6ll-2 (9)Fig. 1 Turbulent transport coefficient as a function of thegradient. Transition is predicted at critical gradient.

The coefficient Q is weakly varying, but shows that thelowest-m mode is the least stable for the anomalousresistivity driven modes. The critical gradient is determined by the turbulence asx"
,zG^:
'?nt
" c.xNt

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4.

4

Xp

X/T',

nonlinear acceleration of the mode growth is possible.
When the low-rn mode aPpears, the cascade to shorter
wave-length modes induces the increment of the transport coefficients.

0or

c-ls2c-r - -0

3Fig.2 Stability criterion of the global mode in thepresence of back-ground turbulence.

5. Summary and Discussion
The theory of the self-sustained turbulence is applied to investigate the collapse phenomena in toroidal
helical plasmas. The theoretical model for the sudden

The criteria for the onset of the collapse, which arepresented here, have a similarity to the one which has
been obtained in the linear and ideal MHD stability

onset of collapse is made. T$o processes are proposed
as mechanisms of the crash events. One is the transportcatastrophe in the high shear case, and the other is theexcitation of the global mode by the turbulence in thelow shear case. The criterion for the onset of crash istheory. This finding also explains the coincidence thatsome linear theories have shown.

Acknowledgements
This work is partly supported by the Grant-in- Aidfor Scientific Research of the Ministry of Education,
Science, Sports and Culture Japan, by the collaboration
programme of the Advanced Fusion Research Centre

qiven aslRp'l> (1/Rs2)s fors ) L

UnstableTFigure 2 illustrates the stability boundary. The
crossing of two lines, Xt(Go, ' s ' ' ) and that inFig.2,
determines the condition where the excitation of theglobal mode takes place. It should be noted that the(11)of Kyushu University, and by the Collaboration pro-andlRBl> (C/R9'\s2 for s ( L.

sramme of National Institute for Fusion Science.

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90

Itoh K. et al., Physics of Collapses in Toroidal Helical plasmas

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