numerical simulations of sawteeth in tokamaks · simulations of sawteeth in tokamaks bt = 6 t, a =...

NUMERICAL SIMULATIONSOF SAWTEETH IN TOKAMAKS

G. VLADCentro Ricerche Energia Frascati,Associazione Euratom-ENEA sulla Fusione,Frascati, Rome, Italy

A. BONDESONCentre de recherches en physique des plasmas,Association Euratom-Confe'de'ration Suisse,Ecole poly technique fe'de'rale de Lausanne,Lausanne, Switzerland

ABSTRACT. Numerical simulations of sawteeth in tokamaks have been carried out using reduced magneto-hydrodynamics and a simple transport model. The electron temperature is evolved self-consistently, including Ohmicheating and a highly anisotropic thermal diffusivity. The sawtooth period and collapse time found in the simulations forLundquist numbers S below 107 compare favourably with experimental results from small and medium size ohmicallyheated tokamaks. The sawtoothing is found to be sensitive to the values of the transport coefficients. In particular, theperpendicular viscosity must be comparable to, or larger than, the perpendicular thermal diffusivity for distinct relaxa-tion oscillations to occur. To study the scaling with S, the viscosity and perpendicular thermal conductivity have beenscaled as 1/S, and S has been varied. Modifications of the equilibrium, caused by the sawteeth, play an important rolein the scaling of the collapse time and period with the Lundquist numnber. The self-consistently computed q-profilesare very flat in the central region where q « 1. Outside the low-shear region, the shear rises sharply. This q-profileallows a resistive mode to be turned on quickly with a high growth rate. The deviation of the central q from unity overthe sawtooth cycle decreases with increasing S, and the collapse time shows a weaker dependence than the Sm scalingof Sweet and Parker.

1. INTRODUCTION

Sawtooth oscillations, in which the central tempera-ture periodically shows a sudden drop followed by aslow increase until the next drop [1], occur in practi-cally all tokamaks under a variety of experimentalconditions. In 1975, Kadomtsev proposed a theoreticalmodel for sawteeth [2]. In this model, the drop istriggered by the m = l / n = l resistive kink mode, whichbecomes unstable when the safety factor in the centrefalls below unity. Via resistive reconnection of thehelical flux inside the original q = l surface, the defor-mation relaxes to a symmetrical state with q aboveunity everywhere and a flattened temperature profile inthe central region of the plasma. After such an internaldisruption, the temperature and current profiles peakagain under the influence of Ohmic heating and resis-tive diffusion, making the central q fall below unity,and the cycle is repeated. Recent experimental dataindicate that the Kadomtsev model is not alwaysapplicable and that the central q may be significantlyless than unity while the discharge is still sawtoothing[3]. Furthermore, measurements on JET indicate that

the growth time of the instability leading to the drop incentral temperature is too short to be connected with aresistive mode [4, 5].

T&e purpose of the present study has been toexplore, by means of self-consistent numerical simula-tion and a systematic parameter study, the predictionsof the reduced magnetohydrodynamic (MHD) modelfor the sawteeth and to compare the simulation resultswith experimental observations. Waddell et al. [6] firstsimulated a single sawtooth crash, using reducedMHD. Sykes and Wesson [7] studied repeated oscilla-tions, assuming Spitzer resistivity, r\ oc T"3/2, andusing an equation for the temperature evolution thatincluded Ohmic heating and perpendicular thermaldiffusion. In this model, the oscillations were found todecay in time. Denton et al. [8, 9] and Bondeson [10]reproduced periodic oscillations by introducing a largethermal conductivity along the field lines.

All computations carried out so far have been madewith plasma parameters far from those characteristic ofcurrent experiments. In the present paper, we reportresults of self-consistent simulations performed withparameters representative of small and medium size

NUCLEAR FUSION, Vol.29, No.7 (1989) 1139

J

VLAD and BONDESON

tokamaks and compare the results with experimentaldata. Our principal finding is that the reduced MHDsimulations reproduce surprisingly well the behaviourof Ohmic discharges with Lundquist numbers S (ratioof resistive diffusion time to Alfve'n time) up to about107, for example with respect to the sawtooth periodand collapse time. In particular, the collapse timeshows a weaker dependence on resistivity than the S1/2

dependence expected from the Sweet-Parker scaling[11, 12], mainly because the change in central q overthe sawtooth cycle decreases with increasing S. Conse-quently, the amount of helical flux to be reconnecteddecreases with increasing conductivity and this partlycompensates for the decrease in reconnection rate. Inour self-consistent simulations, the maximum growthrate of the resistive kink mode scales weakly with S,roughly as S"l/3. The weak dependence of the growthrate on S and the rapid turn-on of the precursor oscil-lation result from the modification of the equilibriumprofiles by the sawteeth themselves. In the self-consistently computed equilibria, the shear changesfrom nearly zero inside a central region to order unityover a distance comparable to a resistive layer width.The sawtooth period shows a weaker than lineardependence on S, in agreement with experimentalresults.

In carrying out these simulations, we have foundthat the sawtoothing is sensitive to the values of thetransport coefficients, in particular the ratio of perpen-dicular viscosity v to perpendicular heat diffusivity Xi-If v/xx is too small, the characteristic relaxation oscil-lations of the sawteeth are replaced by more or lesscontinuous mode activity and the equilibrium neverdeparts significantly from marginal stability. However,if v > Xx» distinct relaxation oscillations occur. Inthis respect, the primary effect of viscosity is to dampthe successor oscillations. If the damping is too weak,the successor oscillations do not decay sufficientlybefore the next crash is triggered, and continuousmode activity results. Recent experimental results con-cerning momentum confinement in Doublet III [13],TFTR [14] and ASDEX [15] all indicate that the diffu-sivity of momentum is of the same order as that ofthermal energy, although both are anomalous.

2. THE MODEL

Our simulations are based on the standard, low beta,resistive, reduced MHD equations, derived in the largeaspect ratio, cylindrical limit with j3 = O(e2) [16],The code [10] evolves the electron temperature self-

consistently with a highly anisotropic thermal diffusiv-ity and Ohmic heating. In normalized units, the modelequations are:

( d \ -

— + v- Vx J w = B- Vj

+

at(1)

+ V . r X i . V / r + (B-V)xi (B-V)T

Here, B = V\J/ x t + BTf, \j/ is the magnetic fluxfunction, v = V<f> x 2, and <\> is the stream function,CJ = - V j 0 is the vorticity, j = -V 2 .^ is the plasmacurrent, T is the electron temperature, 17 is the Spitzerresistivity and Vx = V - z d/dz. The unit length isthe minor radius a, the unit time is the Alfve'n transittime, TA = R/vA, where R is the major radius and vA

is the Alfve'n speed in the toroidal magnetic field BT.v is the perpendicular viscosity and Xx is the perpen-dicular thermal diffusivity, both multiplied by rA/a2,and xi is the parallel thermal diffusivity multiplied byTA/R2. In Eqs (1), the mass density, viscosity andparallel thermal conductivity are taken to be constantin space and time.

The normalized temperature T corresponds tothe poloidal beta of the electrons divided by q2;T = (R/a)2 0/2. In the following, we refer to the cen-tral value of T as fi^. It is proportional to the ratio ofenergy confinement time, TE = 3nT/2i7J2, to resistivediffusion time, TR = a2/i0/i?. To specify the resistivityof a given simulation, we quote the Lundquist number,S = TR/TA, where TR is evaluated by using the centralresistivity.

The code uses finite differences in the radial direc-tion and Fourier expansion in the azimuthal and toroi-dal directions. In the present study, we have used200 non-equidistant radial points and, in most cases,considered single-helicity perturbations with m/n = 1.Mode numbers up to m = 4 (in some cases m = 8)have been retained. No assumption is made regardingthe symmetry of the perturbations, i.e. \p, <£ and T areexpanded with both cosine and sine components. Theinductive toroidal electric field Ez(t) is adjusted in timeto keep the q-value at the edge fixed.

We have chosen as a reference case a typicalOhmic discharge in the Frascati Tokamak (FT), with

1140 NUCLEAR FUSION, Vol.29, No.7 (1989)

SIMULATIONS OF SAWTEETH IN TOKAMAKS

BT = 6 T, a = 0.20 m, R = 0.83 m, qa = 2.6,central density n(0) * 1.8 X 1020 nr3 and centraltemperature between 500 and 1000 eV. With theseparameters, the AlfVe"n time is TA ~ 0.085 /xs, theLundquist number is S » 107, the normalized parallelthermal conductivity is xi * 20 and the normalizedtemperature is T » 0.02-0.04. We assume a radialdependence for the perpendicular thermal diffusioncoefficient, Xx(r) = X±(0)/[l - (r/r0)2]2, withro/a = 1.1, to represent results from power balancestudies on FT.

Although Eqs (1) neglect toroidal and finite pressureeffects for the MHD perturbations, we have, in most

1.55U01.25-!

1.21.1i.oj0.9

-15-2CH

(c)

10 15 20 25 JO

FIG. 1. Time evolution of the central temperature To, the safetyfactor in the centre q0 and the integrated energies W in the 111 and4/4 Fourier components, for three different values of the poloidalbeta: (a) fa * 0.022, (b) fa - 0.15, (c) fa - 1.35. S = 104

and v = 10'4. The unit time is the Alfvin transit time rA.

cases, kept two neoclassical effects: the bootstrapcurrent j b s and trapped particle corrections to theresistivity [17]. The neoclassical corrections to theresistivity have significant effects on the current pro-file, while the bootstrap current is small for the betavalues achieved in Ohmic tokamaks. The neoclassicalresistivity at the q = 1 radius ( « 0.45a) is typicallyabout twice the central value in our simulations.

In the present discussion of the sawteeth, wheretoroidal and finite pressure effects are looked for toexplain experimental results, as in Refs [3, 4], theuse of a cylindrical zero-beta approximation such asEq. (1) may be considered questionable. We share thispoint of view. On the other hand, there are nonethelessgood reasons for carrying out reduced MHD simula-tions. First of all, self-consistent simulations of saw-toothing have previously not been performed withparameters close to experimental values and, conse-quently, the predictions of the cylindrical approxima-tion for actual tokamak experiments are not reallyknown. Since, in addition, measurements of funda-mental quantities, such as central q, have givensignificantly different results in different experiments[3, 18, 19], it is, ironically, not well understood atpresent to what extent the cylindrical approximationagrees or disagrees with experimental results. The aimof our investigation has been to establish more firmlythe degree to which the cylindrical model reproducesexperimental results concerning the sawteeth, using asrealistic a model as possible for the transport coeffi-cients. Furthermore, by running a numerically efficientreduced MHD code [10], we have been able to under-take an extensive parameter study. This study revealsseveral unexpected and non-trivial dependences of thesawteeth on the transport coefficients. It is, however,evident that the results of a reduced MHD simulationmust be interpreted with due caution.

3. DEPENDENCE ONTHERMAL DIFFUSIVITY AND VISCOSITY

In abstract terms, sawtoothing can be thought ofas a driven, non-linear, dissipative system in whichrelaxation oscillations occur. The behaviour of suchnon-linear dynamic systems is often sensitive to theparameter values, in particular to the dissipation coeffi-cients. We have observed that this is indeed the casefor sawteeth: exploratory simulations, performed withsomewhat randomly chosen parameters, i), v, \x a n ^Xi, would produce very different results, ranging fromdistinct sawteeth to weak, continuous mode activity.


VLAD and BONDESON

0.026

b) (c)0.022

4 6 8 10 12 14xKPt

3 4 5 6 7 8 9X103

t

0.016

0.95

d)

3 4 5 6 7 8 9xK)3

3 4 5 6 7 8 9x103

FIG. 2. Time evolution of the central temperature To, the safety factor in the centre q0 and the integrated energies W in the 111 and

4/4 Fourier components, for four different values of the viscosity: (a) v = 10'3, (b) v = 10'4, (c) v = 10~5, (d) v = 10'6. S = 5 X 10s and

0poi * 0.02 and the time unit is the Alfven transit time rA.

To obtain some understanding of the influence of theparameters on the character of sawteeth, we firststudied the dependence on v and XJ. a t moderate S.We emphasize that the scaling studies presented hereare self-consistent, taking full account of modificationsof the equilibrium profiles resulting from the action ofthe sawteeth themselves. Such modifications lead toimportant and unexpected results for the scaling withthe Lundquist number.

Figure 1 shows the time evolution of the centraltemperature To and the safety factor q0, together withenergies in the Fourier components 1/1 and 4/4(integrated over the cross-section) for three simula-tions at different values of jGpo,: 0.022, 0.15 and 1.35.For all three cases, S = 104 and v = 10~4. (For thetwo cases at high beta in Fig. 1, j3po, = 0.15 and(Spoi = 1.35, we have discarded the neoclassical cor-rections to the resistivity and bootstrap current so that

a direct comparison with simulations published in theliterature [8] is possible.) Figure 1 shows clearly thatas jSpo, is reduced by increasing \± (i-e. by decreasingthe energy confinement time), the period of the saw-teeth decreases and the characteristic relaxation oscilla-tions are lost. Pronounced relaxation oscillations occurin the high beta case, as can be seen from the 1/1mode energy in Fig. l(c). For lower beta, the saw-tooth period is shorter and the successor oscillations donot have sufficient time to decay before the next crashoccurs. In the three cases of Fig. 1, x±(0) takes thevalues 1 X 10"3, 7 X 10"5 and 7 X 10~6, which maybe compared with the viscosity, v = 10"4. The simula-tion with j3poi = 1.35 corresponds closely to that ofDenton et al. [8]. In this case, the variation in q0 overthe sawtooth cycle is large, Aq0 « 0.25.

If we try to represent the sawtooth period of thethree cases as a power law, rsaw oc figfi, the depen-



10" i

t 11saw A

103 .

10'

o o

S = 5 x10

1 0 ' 7 1 0 6 1 0 " 5 1 0 * 1 0 J 1 0 '

FIG. 3. Sawtooth period TSOJTA versus viscosity vforS = 5x 10s and 0^, * 0.02.

dence is rather strong, a0 « 0.8. Thus the energyconfinement time plays an important role in determin-ing the period of sawteeth.

In FT, jSpo, » 0. 02-0.04 and the correspondingcase (a) in Fig. 1 does not show regular sawteeth.However, the behaviour of the sawteeth is alsoaffected by viscosity. Figure 2 shows the time evolu-tion of To and q0, and the mode energies for fourdifferent values of viscosity: 10"3, 10"4, 10"5 and 10~6.The resistivity and the perpendicular heat conductivityare a factor of 20 larger than the reference case:S = 5 x 105, Xi(0) = 2 x 10"5. Distinct relaxationoscillations occur for v = 10"3 and 10"4. However, forv < 10"5, the activity becomes irregular, which is bestseen from the plot of qo(t). The period of the sawteethincreases with viscosity, as shown in Fig. 3, from

Tsaw ® 1 X 10 3 TA at V = 10"6 tO T ^ ~ 3 .3 X 103 TA

at v = 10"3. For v £ 10"5, we find r ^ <x va' withOLV « 0.3.

The reason for the irregular behaviour whenv <, 10"5 (which is similar to the low beta case inFig. 1) can be understood from the time evolution ofthe mode energies. The variation in the 1/1 modeenergy is close to four orders of magnitude whenv = 10'3 and less than one order of magnitude whenv = 10"6. Apparently, an important role of the vis-cosity is to control the decay of the successor oscilla-tions by changing the damping rate. The damping rateof the energy in the 1/1 mode is about 10xl0"3 forv = 10"3 and 5 X 10'3 for v = 10"6. Thus, the damp-ing rate depends weakly on v, but this has a strong(exponential) influence on the sawtoothing, in particu-lar as the sawtooth period also increases with viscosity.

When the viscosity is sufficiently large, it reduces thegrowth rate of the resistive kink mode; asymptoticallyfor v > i), y * 7,=0 1.53 (TJ/I/)"3 [20]. This reductionof the growth rate also tends to make the periodincrease with viscosity.

We conclude that the three transport coefficients Xx»v and rj affect the sawteeth in very different ways.Viscosity increases the damping of the successor oscil-lations and slows down the growth of the precursoroscillation. As a consequence, increasing v gives alonger saawtooth period and more pronounced relaxa-tion oscillations. Increasing 0^ at fixed TJ gives alonger sawtooth period as the Ohmic heating timeincreases. Since XJ. does not significantly influence thedamping of the successor oscillations, a larger j3po,(smaller Xi) also gives more pronounced relaxationoscillations. We note that the experimental relationv ® XJ. lies within, but not far from, the boundary ofthe region that produces sawtooth-like oscillations inFigs 1 and 2.

The influence of resistivity is more complex. Atfixed jSpo! and u, the sawtooth period increases with S,but this is accompanied by a decrease in the dampingand growth rates. The increase in sawtooth period isstronger and the net result is the formation of clearerrelaxation oscillations at large S. This can be seen bycomparing Figs l(a) and 2(b), where S changes from1 X 104 to 5 X 105, while 0^ and v are fixed. Onthe other hand, if v is scaled in proportion to x±> assuggested by experiments, the character of the saw-teeth at fixed /3poi oc TJ/XJ. shows a very weak netdependence on S, as discussed in Section 4.

In addition to the dependence on 0^ and v dis-cussed here, the sawteeth are sensitive to the parallelthermal conductivity. Too small a value of xi willcompletely eliminate the sawteeth and give rise to asteady m = l / n = l convection pattern [9, 10], whichmay be thought of as a non-linear form of the ripplingmode. From our parameter study at low S, it appearsthat the threshold value of xi for regular sawteeth tooccur is approximately that which leads to a decayof the temperature perturbation in the q « 1 region byone order of magnitude between two successivecrashes. In simulations, at large S (> 107), therippling mode slightly influences the sawteeth byspeeding up the linear growth when the central q isvery close to unity. For the range of parameters thatwe have explored, the rippling modes have an insig-nificant effect non-linearly and no major change inthe behaviour of the sawteeth occurs when the (m, n)^ (0,0) components of the resistivity are turned off.The normalized parallel conductivity for FT from

NUCLEAR FUSION, Vol.29. No.7 (1989) 1143

VLAD and BONDESON

10

in od °-

CO.<

FIG. 4. Normalized sawtooth period, TS(IW/TA &£?, versusLundquist number S. The black circles are the simulation resultsobtained for 0^ » 0.02, varying S while keeping vS and Xj.5fixed (vS « 24 and xx(0) S « 10). The other symbols refer totypical shots for different tokamaks.

classical transport theory is about 20, and we presentresults only for a fixed value of xi = 27.

4. DEPENDENCE ON LUNDQUIST NUMBER

As shown in Section 3, the sawteeth are sensitive tothe transport coefficients Xx> v and t\. It would bedesirable to make a complete parameter study (at leastthree-dimensional), but this would be very costly incomputer time. Instead, we have taken the point ofview that in comparing ohmically heated tokamaks ofdifferent size, the primary variations are those in S,while vS and XxS (<* 1//3) stay relatively constant.We therefore consider the reference parameters for FTquoted in Section 2 and scale these by a commonenhancement factor E for all small transport coeffi-cients: Xx» " a nd v- Our self-consistent simulationsshow that the growth rate of the resistive kink modeand the collapse time are strongly influenced by theparticular form of equilibrium profiles that occur as aresult of sawteeth and that the modifications of theequilibrium profiles depend on S.

4.1. Simulation results

Figure 4 shows the sawtooth period as a functionof S (with vS and XxS held constant) for our reducedMHD simulations, together with experimental datapoints for Ohmic discharges [4, 21-26]. To accountfor the dependence on /S^,, we have applied the

approximate scaling of Section 3, rsaw oc j3£,8 j / 0 3 , with" a Xx a tt- This implies that, for fixed S, the netdependence on j3po, is rsaw oc /3j0^. Therefore, theexperimental points have been plotted as rsaw/rA/3^f.(In comparing data for Ohmic discharges, the exponentfor (3poi is not very sensitive, since j3poi does not varymuch between different machines.) The agreement inFig. 4 is striking; the simulation results fall withinthe scatter of the experimental data for S < 106; atS = 107, the period is somewhat too short in compa-rison with FT, for example. We conclude from Fig. 4that with TR/TE fixed, T^JT^ OC Sas, with as » 0.7.

We now discuss in more detail the results forvarious enhancement factors, E = 25, 5 and 1.Figure 5 shows the time histories of the central tem-perature and safety factor together with the modeenergies in the three cases. It is clear that the varia-tions in q0 and To diminish as S increases. Forexample, at S = 4 x 105, we have Aq0 ® 0.05 andAT0/T0 « 40%, whereas, at S = 107, we findAq0 « 0.009 and AT0/T0 « 15%. This is consistentwith the fact that rsaw shows a weaker than lineardependence on S.

It has been noted [4] that the collapse time in JET isshort ( -200 pis), seemingly at variance with the resis-tive reconnection rates and also with the assumptionthat a resistive mode triggers the internal disruption[5]. Instead, ideal instability has been suggested as alikely candidate for the trigger [27, 28]. While thisconclusion may be correct for a machine of the size ofJET, where S is between 108 and 109, our simulationsshow that sawteeth caused by the resistive kink modeare in excellent agreement with experimental data forS up to about 107.

Figure 6 shows the collapse time (i.e. the time-scaleover which To drops) and the growth time of the linearinstability from the non-linear simulations, for thethree values of E. Notably, both of these times scaleweakly with S, and at S = 107 they are in good agree-ment with observations on FT. Figure 6 indicates thatthe maximum linear growth rate scales approximatelyas S"1/3. This would be in line with the prediction ofconventional linear theory, if the shear at the q = lsurface were independent of S. Given that dqo/dtscales as 1/S and that the resistive kink mode becomesunstable as soon as q < 1 anywhere, the S~l/3 scalingindicates that the shear, s = rq'/q, is not a fixed(i.e. S-independent) function of r-rq=1. We find thatthis is indeed the case by comparing the q-profiles inthe three cases of E = 25, 5 and 1. Detailed examina-tion of the q-profiles also shows that the shear variesconsiderably within the resistive layer. At the time of


0.024

0.022

To

0.020

0.018

0.016

E=25

1 2 3 4 5 6xK>3

0.022

OJ020

To

0.018"

0.016

E=5


E=1

f /I10 14 18 22X103

X) K 18 22x1)3

10 14 W 22X103

log(W)

-127OX1O3

FIG. 5. Time evolution of the central temperature To, the safety factor q0, and the integrated energies W in the 111 and 4/4 Fouriercomponents, for three different values of the Lundquist number S = 107/E, with E = 25, E = 5, E = 1. 'the other parameters are the sameas in Fig. 4.

maximum growth rate for the case with S = 2 X 106,-s is about 0.2 at q = 1, and goes from 0.1 one-layerwidth inside the resonant surface to 0.3 one-layerwidth outside. At S = 107, the profiles havesteepened, and the variation of the shear across thelayer is about the same, even though the width of theresistive layer shrinks with increasing S. Thus, the cur-rent gradients become increasingly steep at the edge ofthe low-shear region as S increases so that the shear inthe resistive layer is almost independent of the Lund-quist number.

Figure 7 shows the q-profile together with them=n=0 current density for E = 5 (S = 2 x 106) atthe time of maximum growth rate, for the m = 1component (a) and at the end of the reconnectionphase (c)). The sawteeth maintain the steep currentprofile by generating a sharp current dip just outsidethe low-shear region during the non-linear reconnec-

10" i

i l l

10 10' 10'

FIG. 6. Growth time of the linear instability (open circles) andcollapse time of the central temperature (crosses; for each valueof S, the minimum and maximum values as observed duringseveral sawteeth are shown). The other parameters are the sameas in Fig. 5.


VLAD and BONDESON

0.98

1.0

FIG. 7. (a) Current density profile j(r) (m-n=0) and (b) safety factor profile q(r) (onlyin the central region of the plasma) for the case of E = 5 and S = 2 x 106, at the timeof maximum growth rate (t/rA = 20 720); (c) current density profile and (d) safety factorprofile during the non-linear phase (thA = 22 020).

tion phase (see Fig. 7(c)) near the end of the crash.Figure 7(d) shows the q-profile at the same time,where a slight jump in q can be observed together withthe abrupt change in dq/dr at the edge of the q « 1region. The negative spike in the current profile at theboundary of the reconnection region was predictedtheoretically by Kadomtsev [2], who noted that thepoloidal field will be discontinuous after reconnection.The q-profile shown in Fig. 7(b) is slightly non-monotonic but nevertheless does not produce doublesawteeth [8, 29]. We find that the tendency to form anon-monotonic q-profile is more pronounced at high S.This appears to be in agreement with experimentalobservations; double sawteeth occur much more fre-quently in large machines with S > 107, and onlyvery rarely in machines of the size of TCA [26] whereS is a few times 106.

4.2. Linear stability —comparison with toroidal results

To study the linear evolution of the precursor oscil-lation, we have computed the linear m = 1 growth

rate as a function of qmin for the current density profilein Fig. 7(a) by simply rescaling the current. This is ofinterest when we wish to see how quickly the instabil-ity can be turned on and enables a comparison withtoroidal studies, including the effects of finite pressure[28]. The result for S = 2 x 106 is shown in Fig. 8.Note that a change of less than 2 x 10'3 in qmin

produces a growth rate 7 of 10~3 TA\ i.e. the turn-onof instability is very rapid. We have also computedthe m = 1 linear growth rates for the sequence ofequilibria obtained in the simulations, from the time ofmarginal stability to the time 1000 TA later, whenthe growth rate is about 2 x 10"3 TA'. The dependenceof 7 on qmin is practically identical with that shownin Fig. 8. However, at the marginal point for thissequence of equilibria, q is flatter in the central region:1.002 < q(r) < 1.003 for r < 0.42.

One feature is immediately apparent in Fig. 8:the marginal q^n is slightly above unity, by about2 x 10'3. The mode that is unstable when qmin > 1 isevidently destabilized by rippling, i.e. by perturbationsin the resistivity. With the resistivity perturbationsremoved, the marginal q^n is almost exactly 1, and the



0.001 -

0.0000.98 0.99 1.00 1.01

Pmin

FIG. 8. Linear growth rate (in T'A' unit) as the minimum of thesafety factor qmin is varied by simple reseating of the current profileshown in Fig. 7(a).

growth rates are somewhat lower than those shown inFig. 8.

The sensitivity of the MHD stability properties tothe equilibrium profiles in the q = 1 region makes aprecise comparison with the toroidal results given inRef. [28] somewhat difficult. Among the q-profilesstudied in Ref. [28], the one which comes closest tothat shown in Fig. 7(b) is the 'ultra-flat' profile,q(r) = q0 [l+(r/ro)2X]1/x, with X = 6. Figure 15 ofRef. [28] gives growth rates for the ultra-flat profile asfunctions of q0 for different values of j8 and at tightaspect ratio, A = 2.5. We first compare the case ofj3 = 0 in Ref. [28] with our results presented inFig. 8. The marginal point is qmin « 1 in both cases,but the growth rate shown in Ref. [28] is lower thanthat in our Fig. 8. (In the numerical examples ofRef. [28], S is defined with respect to the minor radius[30] and therefore the resistivity at the q = 1 surfaceis about the same in the two cases.) As already dis-cussed, the resistive kink mode dispersion relation(which is somewhat oversimplifying for the types ofprofile under consideration) predicts a growth rateproportional to sq

2=3j and we see that this varies veryquickly with qmin for the types of profile occurring inthe self-consistent simulations. For example, for theprofile in Fig. 7(b), s = 0.1 is reached already with1 - Qmin = 0.003. The ultra-flat profile, for whichsq=i = 2 (1 - q£), needs 1 - qmin « 0.0085 to reachthe same shear. It is clear that the strong variation ofthe equilibrium over a region comparable to the resis-tive layer permits a rapid turn-on of the resistive modeat high S. Thus, even local changes in the equilibriumprofiles are important for the scaling of the lineargrowth rate with S.

Figure 8 refers to the equilibrium computed atS = 2 x 106. For larger S, the equilibrium profilebecomes even steeper near the minimum q. In the non-linear simulations, where the equilibrium depends on Sin a self-consistent manner, and with resistivity pertur-bations included, the net dependence of the maximum1/1 growth rate over the sawtooth cycle in the non-linear simulations is rather weak, approximatelya S"l/3, as shown in Fig. 6.

It should be stressed that although the linear growthis sped up by the resistivity perturbations, these havelittle effect on the non-linear evolution, for which wefind that the resistive kink mode plays the dominantrole. Almost no change occurs in the non-linear simu-lations at S = 2 x 106 when all the resistivity compo-nents except m=n=0 are turned off; however, atS = 107, the sawtooth period is increased by about20% (which actually brings the simulations closer tothe experimental result, see Fig. 4).

In comparison with toroidal calculations, it is clearthat a significant omission in our model is the neglectof the plasma pressure. For the profiles used inFigs 14 and 15 of Ref. [28], a beta of 1% gives rise toan ideal instability at q0 slightly above 1, with growthrates somewhat larger than those of the resistive modeshown in Fig. 8. However, it should be noted that theincrease in q0 at marginal stability is only about 0.01and that the volume average beta in most ohmicallyheated tokamaks is only a few tenths of a per cent. Inaddition, the pressure profiles are often less peakedthan the one used in Ref. [28], which further reducesthe destabilizing pressure gradient in the centralregion. Thus, Fig. 15 of Ref. [28] refers to a case ofsignificantly higher beta than in Ohmic tokamaks. Thisindicates that the finite beta effects are small in ohmi-cally heated tokamaks at S = 106, although, for qmin

slightly above unity, triggering can occur when finitepressure is accounted for. It is noteworthy that in oursimulations the resistivity perturbations have a similareffect.

The successful reproduction of experimental resultsfor crash times and sawtooth periods by our reducedMHD simulations together with the comparisonbetween the cylindrical and toroidal results for thistype of q-profile gives strong evidence that the resis-tive kink mode plays a dominant role as a trigger ofthe sawteeth in Ohmic discharges with moderate S(S < 107). For theoretical reasons, at higher S, itmust be expected that toroidal modes are involved.This is not in contradiction to the results of our simu-lation study, which fails to reproduce the experimentalbehaviour for S > 107, for example, by giving too


VLAD and BONDESON

short a period. It therefore seems probable that there isa continuous transition between sawteeth triggered byan essentially cylindrical resistive kink mode at low Sand sawteeth triggered by a toroidal (and possiblyideal) mode at larger S, with the transition occurring atLundquist numbers of the order 107. Needless to say,our cylindrical simulations cannot describe sawteeth inwhich the central q remains below unity over the entirecycle.

4.3. Non-linear evolution

Similar to the linear growth time, the collapse timealso shows a weak dependence on S (see Fig. 6) —much weaker than the S1/2 dependence expected fromthe Sweet-Parker scaling [11, 12]. This can be under-stood because the variation in q0 decreases withincreasing S and, therefore, also the helical flux to bereconnected decreases (see Fig. 5). Since the proposalof the Sweet-Parker scaling in 1957/1958, it has beenrealized that higher reconnection rates are possible,depending on global conditions [31]. The physical

situation envisaged by Sweet and Parker was a symme-trical current sheet. This is quite different from thecurrent sheet relevant to the present model of the saw-tooth crash, where the conditions are entirely differenton the two sides of the q = 1 surface; inside, the heli-cal field to be reconnected is small (and decreses whenS increases), but, on the outside, the shear takes offvery sharply. The helical AlfvSn frequency inside thelow shear region, « lq0 - 11, need not be determiningfor the reconnection rate, since the rigid m = 1 dis-placement of the interior is not an Alfve'n wave butrather a fast magnetosonic wave. (In the reduced MHDtheory, the fast wave exists as a 'surface wave', withco = - V i 0 = 0, analogous to pressure perturbationsin an incompressible fluid. For m = 1, the 'surfacewave' is a rigid displacement of the core, due to'forces' B- Vj acting at its boundary.) The effectiveideal time for non-linear reconnection appears to bedetermined by the shear just outside the low-shearregion rather than the Alfve'n time in the centre. Thisis in analogy with linear theory. The weak variation ofthe reconnection time with S seen in Fig. 6 is in sharp

0.022

18 19 20 21 22x103

0.01619 20 21 22x103

t

18 19 20 21 22x103

22xK)3

FIG. 9. Time evolution of the central temperature To and the mode energies Wfor the last crash of Fig. 5 (S = 2 x 106).(a) and (b) Simulation with Fourier components up to mln = 4/4.(c) and (d) Simulation with Fourier components up to mln = 8/8.



t - 2 1 2 2 0 ; -0.13971

= -0.13718

1-21520; y ^ - - 0 . 1 3 9 8 6

V,,^,--0.13812

1-21620; - 0.13987

• -0.13812

t-21720: V m i n - - 0.13991

V -- 0.13813

t -22020:

FIG. 10. Contour plots of the helical flux function for the eight-mode simulation in Fig. 9 (S = 2 x JO6). The contours are plotted only in

the region of low shear.

disagreement with the Sweet-Parker scaling when theeffective ideal time is taken as rA lq0 - 11"'. We planto address this issue in more detail in the future.

Figure 5 shows that during the non-linear part of theinternal disruption the m=4/n=4 mode is excited to arather high amplitude, and the question arises whetherour simulations with m ^ 4 have sufficient angularresolution. To check this, we have rerun the case withS = 2 x 106, keeping Fourier components up tom=8/n=8. The result for T0(t) is shown in Fig. 9.When examining the gross features, such as T0(t), thefour Fourier components give adequate resolution.However, when we consider details of the evolution,certain differences become apparent, notably the for-mation of small secondary islands along the q = 1separatrix [32] which are seen in the simulation witheight modes but not in that with four modes. Thesedetails appear to have only very slight influence on theglobal evolution. Figure 10 presents contour plots of

the helical flux function, \f/» = \p — (1 - r2)/2, atvarious stages of the internal disruption for the eight-mode simulation at S = 2 x 106. The formation ofthe small secondary island is followed by a coalescencewith the primary island. It is easily seen that compo-nents with m > 2 are excited at rather high amplitude,as is evident also in the contour plots of the tempera-ture (Fig. 11). We note from the temperature plots thatthe shape of the hot core changes during the internaldisruption from a circle to a crescent shape, as wasobserved in JET [33] and TFTR [34].

Another issue of resolution concerns the influence ofmultiple helicity interactions. For example, the verysteep current profile in Fig. 7(c) might make tearingmodes with nearby rational surfaces unstable. Thiswas predicted theoretically by Kadomtsev [2] and wasobserved in a multiple helicity simulation at moderateS [10] where the m/n = 4/3, 5/4, 6/5, etc. modeswere periodically destabilized by the sawteeth.


VLAD and BONDESON

t « 21220; Tmj| i-2.48l).l<r3

T - 2.001 x!0-?

t - 2 1 5 2 0 ; Ta i i»2.503»10-3

T =2.O34xlO"2

21620:

T = Z046xl(T2

21720: Tmg)=2.521»IO"3

T = 2.O62*IO"2

1 = 22020: T = 2.530 xlO"3

FIG. 11. Contour plots of the temperature for the eight-mode simulation of Fig. 9 (S = 2 X 10 ) .

We have investigated this by including modes withm = n + 1 in addition to the m = n modes with0 < n < 8. Although the m = n + 1 modes areperiodically destabilized, they do not reach a sufficientamplitude, even at S = 107, to have a significanteffect on sawtoothing. (This result may be quite differ-ent in toroidal geometry because of toroidal coupling.)

5. SCALING OF THE SAWTOOTH PERIOD

Our simulations predict that the sawtooth periodscales as TA(}£\ S07, which, as shown in Fig. 4, givesa good fit for different machines. For a comparisonwith scalings observable in a single machine, it isuseful to display the dependence on the plasmaparameters: the density n, the temperature T, Zeff,the ratio between the ion mass and the proton mass A,the toroidal field BT and the linear dimensions. FromTsaw <* T^P^ S07, we obtain (for fixed qa):

rsaw a (R/BT)'-3 A015 a04 f(Zeff)-°-7 n065 T155 (2a)

Here,

f(Zeff) = Zeff [0.29 + 0.457/(1.077 + Zeff)] (2b)

is the Zeff dependence of the resistivity. When the den-sity is changed at fixed current, the temperature willchange, and Eq. (2), when combined with the relationsT = T(n) and Zeff = Zcff(n), gives a prediction for thesawtooth period as a function of the electron density.The scaling (2) is in reasonable agreement with resultsfrom FT [35], where rsaw ex n05 is observed togetherwith a weak inverse dependence of the temperatureon the density. We note that FT has an unusuallysmall fraction of impurities, and Zeff « 1. In othermachines, the sawtooth period has a stronger depen-dence on density; in TFR [36] and TCA [26], forexample, rsaw increases approximately linearly with n.It is probable that part of this dependence is due to



changes in Zeff; an inverse dependence, ZefT oc 1/n,would be more than sufficient to bring the scaling (2)into agreement with the TFR and TCA results.

6. DISCUSSION

We have simulated the sawtooth activity of ohmi-cally heated tokamaks using a reduced MHD code withtransport. The sawteeth have been found to be sensi-tive to the transport coefficients. In particular, theperpendicular viscosity must be of the same order as,or larger than, the perpendicular heat diffusivity forregular sawtoothing to occur. In the reduced MHDmodel, the sawtooth collapse occurs via a resistivekink mode and complete resistive reconnection. Ourmain conclusion is that, despite its simplicity, themodel is in excellent agreement with experimentalresults as regards the sawtooth period and crash timesfor S-values up to about 107.

The self-consistent simulations produce q-profilesthat are very flat in the central region where q ~ 1.At the edge of this region, the current density dropsand the shear increases sharply. We note that q-profiles of this type have been measured recently forsawtoothing discharges in ASDEX [18] and TCA [19].Such profiles allow a resistive mode to be turned onquickly, even at high S. The drop in the current pro-file outside the q « 1 region becomes increasinglysteep at high S, and this partly compensates for thedecrease in resistive growth rates with S. A secondchange connected with the equilibrium is that the vari-ation of the central q over the sawtooth cycle decreaseswith increasing S. Thus, the magnetic flux to be recon-nected decreases, which has the effect of reducing thedependence of the crash time on S.

To summarize, reduced MHD simulations, in whichthe equilibria are self-consistently computed, give saw-tooth periods, precursor growth rates and crash timesthat are in agreement with those of ohmically heatedtokamaks for S up to about 107. This good agreementgives considerable, although admittedly indirect, evi-dence that the sawteeth in small and medium size toka-maks are caused primarily by the resistive kink mode,similar to the mechanism originally proposed byKadomtsev [2]. However, the q-profiles found in ourstudy differ from the parabolic profile in Kadomtsev'sanalytic calculation by having a flat region withq « 1, outside which q increases sharply, somewhatsimilar to the quasi-interchange model proposed byWesson [5]. These properties are essential to obtainthe rapid turn-on and crash observed experimentally.

Our simulations do not accurately reproduce saw-teeth in large tokamaks, S > 107. This is not surpris-ing, since our reduced MHD model neglects toroidaland finite-beta effects [28, 37]. From a theoreticalpoint of view, it is clear that for high values of S,toroidal and finite pressure effects will influence thebehaviour of the sawteeth. It appears probable thatthere is a transition between sawteeth governed by anessentially cylindrical resistive kink mode at low S andsawteeth governed by toroidal, and possibly ideal,instabilities at higher S. On the basis of the presentstudy, we estimate that the transition occurs for S inthe range of 107 for ohmically heated tokamaks.Experimental evidence for the assumption that idealmodes are involved in triggering of sawteeth in largetokamaks has come from JET [4, 5, 38]. It is alsoevident that the simplest reduced MHD model withoutany toroidal effects cannot reproduce sawteeth inwhich the central q is below unity over the wholesawtooth cycle, as reported from TEXTOR [3].

Finally, since, generally speaking, the ideal MHDstability at q = 1 is close to marginal, many non-MHD effects may be able to significantly modify thebehaviour of the sawteeth. Among these, we mentiontemperature and resistivity perturbations (Section 4.2),drift effects, the influence of hot particles expected incertain RF heating scenarios and, for future experi-ments, alpha particles [39]. In view of several unex-pected results found in the present reduced MHDsimulations and the variety of sawteeth observed underdifferent experimental conditions, we are well awarethat new surprises may be in store when self-consistentsimulations are carried out, using more completemodels at Lundquist numbers characteristic of largetokamaks.

ACKNOWLEDGEMENT

One of the authors (G. Vlad) thanks Euratom forproviding mobility funds to support his stay at theCRPP, where most of this work was done on theCRAY-IS computer at the EPFL.

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VLAD and BONDESON

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(Manuscript received 8 November 1988Final manuscript received 9 March 1989)


numerical simulations of sawteeth in tokamaks · simulations of sawteeth in tokamaks bt = 6 t, a =...

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