P H Y S I C A L R E V I E W L E T T E R S week ending11 JULY 2003VOLUME 91, NUMBER 2

Continuum Self-Organized-Criticality Model of Turbulent Heat Transport in Tokamaks

Varun Tangri, Amita Das, Predhiman Kaw, and Raghvendra SinghInstitute for Plasma Research, Bhat, Gandhinagar 382428, India

(Received 20 July 2002; published 7 July 2003)

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A simple generic one-dimensional continuum model of driven dissipative systems is proposed toexplain self-organized bursty heat transport in tokamaks. Extensive numerical simulations of thismodel reproduce many features of present day tokamaks such as submarginal temperature profiles,intermittent transport events, 1=f scaling of the frequency spectra, propagating fronts, etc. This modelutilizes a minimal set of phenomenological parameters, which may be determined from experimentsand/or simulations. Analytical and physical understanding of the observed features has also beenattempted.

DOI: 10.1103/PhysRevLett.91.025001 PACS numbers: 52.25.Fi, 52.25.Gj, 52.55.Fa

of a coupled set of two equations: The first equation is a1D radial transport equation with sources for the tem-

tion of SOC characteristics; hysteresis is related to thephysical fact that once the turbulence is excited it may be

Recent experimental work on turbulence driven heattransport in tokamaks reveals many features consistentwith a self-organized criticality (SOC) [1] model oftransport [2]. Electron heat transport, in particular, hasmany of these features, notably (i) a threshold gradientlength [3]; (ii) submarginal temperature profiles that areinsensitive to the radial distribution of heat source [4];(iii) large scale intermittent transport events exhibitinglong time autocorrelations [5]; (iv) characteristic fre-quency spectra showing scaling behavior, f�� with ��1 [6]; (v) observation of nondiffusive radial propagation offronts associated with avalanche events with speedsof the order of a few hundred meters=second [7], etc.Most of the features discussed above are generic to tur-bulent transport in toroidal devices and have also beenobserved in experimental studies of core ion transport,edge heat transport, flux driven scrape-off layer transport[8–13], etc.

Full scale three-dimensional gyrokinetic and gyrofluidsimulations [14] of the relevant instabilities have alsobeen carried out and reproduce many of the features ofthe observed transport. However, they tend to be so com-plex that the essential underlying physics often gets ob-scured. This has led to highly oversimplified cellularautomata based models which continue to be popularfor a description of SOC behavior [15,16]. In this Letter,we propose the application of a one-dimensional contin-uum model of driven dissipative systems, to the electronthermal transport problem in tokamaks. Continuum mod-els have the advantage that, in contrast to cellular autom-aton models, variables and algorithms used are based onempirical observations. Recent work by Garcia et al. [17]is an advancement in this direction. The model, however,uses random sources in the equations and there is noindication as to which part of the results depend uponthe choice of such a source. We employ a model (intro-duced by Lu [18] for the study of the problem of solarflares) which, in its adapted form, studies the dynamics

0031-9007=03=91(2)=025001(4)$20.00

perature field and the second equation is a nonlinearrelaxation equation for the turbulence driven transportcoefficient �, which is switched from a low value to ahigh value, when the gradient of temperature locallyexceeds a critical value. For the electron heat transportproblem, such a switch of � is amply supported byexperimental data [19]. Key ingredients of the modelare a finite relaxation time for switching and a hysteresisin the � vs rT behavior, so that the switch of � upwardsand downwards takes place at two different thresholdvalues for rT. This simple model set of equations isable to reproduce all the observed features of the trans-port discussed above.

The model described above is expressible in terms ofthe two equations:

@T@t

�@@x

��@T@x

��S�x; t�; (1)

@�@t

� � � Q�rT�x; t�; (2)

where we have used normalized variables T � �TT=T0, x ��xx=x0, t � �tt=�, � � ����=x20, and S � P�=�3nT0=2�. P is theinput power density which determines the source S. n isthe plasma density, x0, T0 are normalizing variables, and� is the natural nonlinear relaxation time of the � equa-tion. The source function Q for � is a double valuedfunction switching between two values �max and �min;the switch is dictated by the magnitude of the localgradient of temperature and has hysteresis. Thus, Qchanges from �min to �max when jrTj exceeds a criticalvalue K but switches back from �max to �min only ifjrTj<�K; � is the hysteresis parameter and takes val-ues less than 1. �min is the normalized neoclassical trans-port coefficient and �max is the transport coefficientassociated with saturated electron temperature gradientmode (ETG) turbulence. The presence of hysteresis (i.e.,� � 1) in the source function Q is crucial for the depic-

2003 The American Physical Society 025001-1

0 10 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x 10−3

(a)

S(x)

X0 10 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8(b)

T(x)

X

0 0.050

100200

FIG. 1. (a) Two different source profiles indicated by solidand dot-dashed lines. (b) The corresponding relaxed tempera-ture profiles. The straight (dashed) line indicates the criticalgradient profile. The inset shows the probability distributionfunction of the temperature gradient @T=@x.

0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

∆ E

1−β2

(b)

0 200 400 600 800 1000 12002.8

33.23.43.63.8

E

t

(a)

0 200 400 600 800 1000 12000

0.05

0.1

Γ

t

(c)

FIG. 2. Loading/unloading cycle: (a) Plot of E vs time.(b) Mean cycle amplitude �E vs 1� �2. (c) Temporal evolu-tion of flux �@T=@x.

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possible to sustain it even when rT goes below the linearinstability threshold.

We have numerically simulated Eqs. (1) and (2) usingroutines from the Numerical Algorithms Group library.The routines employ finite differencing in space, and thetime advancement is carried out by the gear method(method of lines). We present results for the case whenthe source function is of the form S�x� � S0 sin��x=2L�.We have fixed the box size at L � 20 and the value of thecritical slope parameter and the hysteresis parameter,respectively, are K � 0:04 and � � 0:9. The boundariesat x � 0 and x � L, respectively, satisfy the boundaryconditions T � 0 and @T=@x � 0 for all times, the formercorresponding to the plasma edge and the latter to theplasma core. �max and �min are chosen to be 2 and 0.2,respectively, and S0 is chosen in the range 10�3 to 10�2

and the number of grid points are 200. As explained later,these dimensionless values correspond to typical numberscharacteristic of tokamaks such as JET, TORE-SUPRA,and D-IIID.

Our first observation is that bursty transport is ob-served when the source amplitude lies between two criti-cal values, Slcr < S0 < Sucr. For S0 < Slcr � �minK�=2L,one typically acquires a stationary state with jrTj<Keverywhere and � � �min at all local points. For S0 >Sucr � �max�K�=2L, on the other hand, the stationarystate is such that � � �max and jrTj > �K at all points(except in the vicinity of core boundary). The possibilityof stochastic states with a statistically stationary charac-ter arises only when Slcr < S0 < Sucr, where the local �may switch from �min to �max or vice versa depending onthe values taken by the local jrTj. Such a switching of �back and forth leads to the possibility of bursty transport,propagation of avalanche fronts, etc.

The plot in Fig. 1 obtained numerically shows clearlythat, irrespective of the choice of initial T profile, thetime averaged spatial profile of T after an initial transi-ent) very rapidly approaches a state where rT is below thecritical slope at all positions. Simulations with differentspatial distributions of the source function with its mag-nitude lying between Slcr < S0 < Sucr give similar resultsshowing resilience of the temperature profile. This resultis in agreement with the experimental observation ofprofile consistency and submarginal thresholds. Furtherdetailed results exhibited by this model can be summar-ized as follows: (i) The total energy of the system definedby E�t� �

RdxT2�x; t� [which is like the thermal energy

nT if n�x� and T�x� are taken to be identical functions forsimplicity] shows quasiperiodic behavior with a steadylinear rise with time (loading) and a sudden crash (un-loading) displaying a sawtooth form [Fig, 2(a)]. (ii) Theamplitude �E � Emax � Emin and frequency � of suchsawtooth events are in general of statistical nature.However, their mean values are observed to depend onthe parameters �min and the source strength S0. (iii) Thevalue of �E and � also depend on the hysteresis parame-ter �. The mean value h�Ei scales linearly with �1� �2�

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[Fig. 2(b)]. (iv) The simulations show [Fig. 3(a)] propa-gating frontlike structures in the gradient of T field (aswell as in the diffusivity �). These fronts propagate with aconstant velocity �UU. (v) The velocity �UU is found to scaleas

������������������max=�

p[Figs. 3(b) and 3(c)]. We note that the model

makes predictions which are in conformity with specialfeatures of electron thermal transport in tokamaks.Observations (i), (ii), and (iii) show the intermittentbursty nature of the transport [Fig. 2(c)] [which inciden-tally shows the scaling behavior in power spectrumviz. P�f� � f�� with �� 1] and observations (iv)–(v)are related to the radial propagation of avalanche frontsacross a discharge. While the observations (i), (ii), and(iv) above are similar to those obtained earlier byKlimas et al. [20], who have investigated this model fora description of the magnetospheric substorms, (iii) and(v) are new results and have been obtained by us after

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FIG. 3. (a) Spatiotemporal plots of � showing propagatingfront structures; white regions indicate the supercritical sites.The variation of velocity of fronts [Eq. (4)] with (b) varying�max and (c) varying 1=�1=2.

P H Y S I C A L R E V I E W L E T T E R S week ending11 JULY 2003VOLUME 91, NUMBER 2

extensive numerical simulations.We now provide a simpletheoretical interpretation of the above results; such aninterpretation has not been attempted in earlier work.

The quasiperiodic oscillations of energy having a saw-tooth character in time essentially signify a slow buildingup of the temperature field from �Kx towards Kx by thesource function. The crash is triggered at a certain sitewhich exceeds the critical threshold condition for sloperesulting in high diffusion at that site. When the system iscritically poised, enhanced diffusion at one site makesthe neighboring sites also overcritical leading to an ava-lanche event whereupon there is a sudden release of en-ergy. Clearly, the maximum amplitude of the avalanche isgoverned by the maximum amount of the energy that canbe released by the system. It can be estimated by thefollowing argument. When the typical slope everywhereis of the order of �K, � � �min over the entire range andthe diffusion losses are negligible. Thus, in this state, thedynamics is predominantly governed by source whichtries to build up the energy. The state T � �Kx thusdefines the minimum energy that a system can haveand its magnitude is given by Emin �

RT2dx �

�2K2Rx2dx � �2K2L3=3, where L is the system size.

On the other hand, the maximum energy that can beretained by the system occurs when the slope in the entirerange has a value K, so that T � Kx. A slight increase inthe value of T field at any location by the source leads tothe switching of Q to �max resulting in large diffusivedamping. Thus, any addition in energy over and above thisis necessarily thrown out of the system. So the maximumvalue of energy that such a system can retain is Emax �RT2dx � K2

Rx2dx � K2L3=3. The difference

�E � Emax � Emin � �1� �2�Emax � �1� �2�K2 L3

3(3)

gives the maximum amplitude of �E in the growth and

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decay cycle that the system can exhibit. The observedamplitudes, in general, are typically lower than the aboveestimate. This is reasonable, and shows that in general theavalanche gets triggered before the entire system is fullyloaded. Hence, L, the box size, should be replaced by lrepresenting the avalanche size in Eq. (3). In general, lwill have a distribution, the average of hli � �ll, represent-ing a typical avalanche size. Thus, the average energyrelease in the avalanches is h�Ei � �1� �2�K2 �ll3=3, con-firming the observed scaling with �, which can be seenfrom the plot of Fig. 2(b) and mentioned earlier in point(v) of the summary of results. This agreement also sug-gests that the typical avalanche size �ll is independent of �.

For high amplitude of avalanches, it is necessary tohave good spatial correlation of T during the growthphase so that the transition to �max is triggered in tandemover a large spatial domain. The parameter �min essen-tially determines the spatial correlation length lc for T[note that, for �min � 0 during the growth phase, Eq. (1)turns into an ordinary differential, i.e., lc � 0]. As �min isincreased, it correlates T over disparate spatial regions bydiffusion. Thus, a high value of �min might seem desir-able. However, the dissipation due to �min should notexceed input via source term leading to a dampedfinal state. Hence, an optimum �min can be estimatedby the critical balance condition �min@2T=@x2 ��S0 sin��x=2L� which implies that S0 � Slcr. The abovediscussion establishes the importance of both �min and S0in determining the avalanche sizes and, hence, the energyrelease. For example, for our parameters when Slcr <S0 < 1:5 10�3, the quasiperiodic oscillations have sig-nificant amplitude and are distinct; on the other hand, for1:5 10�3 < S0 < Sucr, the frequency of oscillations be-comes rapid, the amplitude diminishes, and eventuallythe oscillations acquire a stochastic character.

Our simulations show both inward and outward prop-agating spatiotemporal front structures in @T=@x (and soalso in �). Well within the boundaries of x space, thesefronts typically move with a constant velocity [seeFig. 2(a)]. We observe that the propagation speed dependson the parameters �max, �, and � with the scaling of �UU /������������������max=�

pas shown in Fig. 3. Such a scaling can be under-

stood from the following simplified analysis. The frontstructure in the diffusion coefficient arises due to theswitching of Q from �min to �max, at locations wherethe local slope exceeds the critical value K. The subse-quent evolution of the diffusivity with time (so long asQ remains at �max) as governed by Eq. (2) is given bythe following expression: ��t� � �maxf1� exp��t�g ��min exp��t�. For �min � �max (as indeed is the case)and for a time t � 1, we can approximate the expressionfor � as � � �maxt. We may now write the diffusionequation as @T=@� � @2T=@x2, where � � �maxt

2=2, ex-act solution shows diffusion in x� � variables, x2 � � �

�maxt2=2 giving a front propagation speed in dimensional

variables as

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P H Y S I C A L R E V I E W L E T T E R S week ending11 JULY 2003VOLUME 91, NUMBER 2

�UU ��xx�tt�

�������������max

2�

r: (4)

Figures 3(b) and 3(c) give plots illustrating the abovescaling as observed in the numerical simulations. A simi-lar scaling has been derived by Diamond et al. [21] but inthe context of L-H transition fronts.

We now discuss in somewhat more detail the implica-tions of the results of this model for the electron thermaltransport in tokamaks. First, we look at some numbers. Ifwe take x0 � 2:5 cm, T0 � 10 KeV, � � 10�4 s, ���max �

105 cm2=s, n � 1013 5 cm�3, and P � 1 MW=m3, weare considering a 50 cm radius plasma with a peaktemperature of about 8 keV with � 10 MW of input powerwhich is similar to the plasma in the Tore-Supra experi-ment; this choice gives us dimensionless parametersL � 20, S � 10�3, and �max � 2 as shown in our samplesimulation. Experiments such as JETand D-IIID also givea similar parameter range for the simulations. We make achoice for the relaxation time � to interpret D-IIID re-sults. This data is largely related to electron cyclotronemission measurements of local electron temperaturefluctuations. � is a nonlinear relaxation time for theETG turbulence which describes the time taken by � tostabilize at �max after rT crosses the critical value K.This involves saturation of the ETG turbulence, growth ofstreamerlike modulational instabilities, and their satura-tion by Kelvin-Helmholtz secondary instabilities. Anestimate of � can be made from large scale simulationsand/or from experiments [14]; they give a value of theorder of �� 103Lt=Vth � 100 !s, where Vth=LT is thetypical ETG growth rate. ���max is the upper limit to which��� can rise; it typically never reaches there because ava-lanches strike. Thus, ���max � 3 ���meas and is taken as�105 cm2=s. With this choice of parameters, we notethat the radial propagation of avalanchelike fronts takesplace with a speed of the order of � ���max=2��

1=2 �200 m=s, a value which is in excellent agreement withthe measurements of Politzer [6,7] The choice of hyste-resis parameter � � 0:9 is somewhat ad hoc; its maineffect, however, is to limit the size of the fluctuations inenergy.

In conclusion, we have introduced a 1D continuummodel with a few phenomenological parameters such as�max, �min, �, �, K, etc. which explains many features ofthe observed electron thermal transport in tokamaks. Wehave presented numerical results on the model and someanalytical calculations explaining the key features ob-served. We have also described the implications of themodel for the thermal transport problem in fusion de-vices. It would be of interest to verify the physics de-scribed here and to determine the phenomenologicalparameters introduced. For example, active experimentswith localized heat sources such as electron-cyclotron-resonance heating could be carried out to measure �max,�min, the relaxation time �, the hysteresis parameter �,etc. There is already significant experimentation and

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data on the critical threshold gradient rTc � K.Similarly, simulations and analytical theory could beused to understand the magnitudes of these phenomeno-logical parameters and would thus elucidate the physics ofthe phenomena a little better.

Finally, it is worth emphasizing that the paradigmintroduced here for electron thermal transport is muchmore general and with appropriate modifications may beapplicable to a number of observations in magneticallyconfined plasmas such as ELMS, flux driven transport inscrape-off layers, ion thermal transport and particletransport in core regions, etc. It may also be worthwhileto extend these 1D models to situations involvingcoupled transport equations in density, temperatures,currents, etc.

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