Transport barriers and edge localized modes-like bursts in a plasma model with turbulent equipartition profiles

PHYSICS OF PLASMAS VOLUME 10, NUMBER 4 APRIL 2003

Transport barriers and edge localized modes-like bursts in a plasma modelwith turbulent equipartition profiles

V. Naulina) and J. Juul RasmussenAssociation EURATOM-Risø National Laboratory, Optics and Fluid Dynamics Department, OFD-128 Risø,4000 Roskilde, Denmark

J. NycanderDepartment of Meteorology, University of Stockholm, 106 91 Stockholm, Sweden

~Received 13 November 2002; accepted 20 January 2003!

Self-consistent development of transport barriers is investigated analytically and numerically in fluxdriven interchange turbulence with highly intermittent turbulent fluxes. Numerical simulations on abounded domain show turbulence leading to a homogenization of Lagrangian invariants by mixing,resulting in quasisteady pressure profiles predicted by turbulent equipartition. Below a critical aspectratio a5Ly /Lx—for our parameters'3.8—which is related to the rotational transform, large scalepoloidal flows develop. They reduce the energy in the turbulence, prevent mixing, and constitutetransport barriers for the turbulent fluxes, but are intermittently disrupted by strong bursts in thetransport, which may be related to the strong edge localized modes observed in toroidal devices.© 2003 American Institute of Physics.@DOI: 10.1063/1.1559993#

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I. INTRODUCTION

The generation of large scale flows by the rectificationsmall scale turbulent fluctuations is of great importancefluids, e.g., geophysical flows1 and thermal convectiveflows,2 as well as in magnetically confined plasmas.3 Theflows are important as they act back on the turbulenceshearing apart and thereby suppressing fluctuations ontransport scales, thus setting up transport barriers.

More specific in hot magnetized plasmas the main crofield transport is anomalous, i.e., not diffusive in the classsense, and ascribed to low frequency electrostfluctuations.4 It is generally recognized that self-consistendeveloping large scale poloidal flows strongly reduce thedial turbulent transport by ‘‘quenching’’ the turbulence~seeRefs. 3, 5, and references therein!. This mechanism may beresponsible for the rapid transition to an enhanced confiment in magnetically confined plasmas, e.g., the celebrH-mode regime first observed in the ASDEX tokamak.6 TheH-mode7 is often found to be accompanied by bursts in ttransport, related to edge localized modes~ELM!,8,9 whichgo along with magnetohydrodynamic~MHD! activity, theso-called ELMy H-mode. If no such intermittent transpobehavior is observed, rising pressure gradients oftenlently terminate the H-mode plasma by disruption.

Since neither turbulence nor the associated bursty trport can or should be avoided, it is essential to understthe interplay between poloidal flows resp. transport barron the one hand and turbulence as well as transport onother hand. The ultimate goal will be to use this understaing to develop strategies for active control and regulationthe turbulent transport.

In the plasma context, efforts to describe self-consist

a!Electronic mail: [email protected]

1071070-664X/2003/10(4)/1075/8/$20.00

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poloidal flow generation by anisotropic turbulence were itiated by Diamond and Kim.10 The flow is driven by theturbulence via the off-diagonal elements in the Reynostress tensor, i.e.,Ruv5^uv&, whereu and v are the radialand poloidal components of the fluctuating velocity. Preously Biglari et al.11 had shown that a shear flow may astabilizing on turbulent fluctuations via a decorrelatimechanism.

In the present work we consider the evolution and dnamics of transport barriers, in the form of poloidal flowand the interplay with the turbulent transport, which revean intermittent behavior with very strong burst events. Whave employed a self-consistent model for pressure drielectrostatic turbulence of a plasma in an inhomogenemagnetic field. This is a rough model of the outboard sidea toroidal confinement device but includes the effects offavorable curvature in an energy preserving manner andscribes the evolution of profiles as well as fluctuations.

The turbulence is sustained by a heat flux enteringsystem from the hot inside and leaving at the cold outsidesimilar model was used in Ref. 12 to explore the formatiof turbulent equipartition states, the so-called TEP-stat13

achieved by the equipartition of Lagrangian invariants: Tstate determines the mean global behavior of the systethe invariants are sufficiently robust, that is they ‘‘survivefor times much longer than the system mixing time. Taverage profiles of density and temperature in the TEP sare close to the linear stability threshold. Note, however, tany turbulent mixing will arrange that these marginastable profiles are reached, even from below and in thesence of sources. Theories including critical gradients nmally rely on the system relaxing from a state with a gradisteeper than the critical one. TEP predicts thus up-gradtransport while it does not give nor rely on any informatioabout the spatial character of the transport, as f.x. self o

5 © 2003 American Institute of Physics

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1076 Phys. Plasmas, Vol. 10, No. 4, April 2003 Naulin, Rasmussen, and Nycander

nized critical ~SOC! transport models14 do. The latter arecharacterized by a power law behavior of the probabidistribution function of the transport and self-similar charater of the turbulence. Systems showing TEP behavior mthus as well fulfill the characterizations for SOC. Here whowever, want to concentrate on the interplay between sflow and the TEP state.

The TEP state only exists, as mentioned above, ifturbulent mixing is stronger than the viscous diffusion. Thif the turbulence is quenched in some way, the profiles wrelax towards the state controlled by the diffusion terms athe time averaged profile in the driven case will be steethan the marginal stable one.

For a specific parameter regime strong, sheared, poloflows are observed to be generated self-consistently byturbulence. The aspect ratio,a, of the domain seems to bthe most important parameter and the poloidal flows aresistent fora,ac . The flows quench the turbulence. Thatthey transform the energy in the turbulent motion intodered poloidal flow, which provides no cross field transpoMoreover the developing flows may stabilize the instabiliThus, the TEP profiles will no longer be sustained, ansteep pressure gradient builds up due to viscous diffusThis gradient, which is usually steeper than the critical gdient for instability, appears to be stabilized by the poloidflow. However, also the flow decays due to the lack of insbility drive and can at some point no longer stabilize tsteep gradients. This results in a burst of turbulencetransport on a time scale much shorter than the viscousscale. The appearance of the bursts is quasiperiodic witime separation related to the viscous time scale andmay be related to the ELM-like structures discussed aboWe suggest that they may explain at least the class ofstrong type I ELM’s.8,9 A similar behavior with strong inter-mittent bursts is commonly observed in convectiproblems.2 See also Takayamaet al.,15 who employed aBoussinesq model to simulate interchange mode turbule

II. MODEL EQUATIONS

Our model is based on the fluid equations and descrthe fluid drifts accurately in the presence of an inhomoneous, curved magnetic field. Included are the adiabcompression and heating of fluid parcels being displaceda region of higher magnetic field. Density and temperatprofiles are allowed to evolve self-consistently under thefluence of eventual external heating, and so is the baground potential profile, corresponding to a mean flow,

]n

]t1$f,n%1K~n1T2f!5n¹2n, ~1!

]T

]t1$f,T%1

2

3KS n1

7

2T2f D5k¹2T, ~2!

]¹2f

]t1$f,¹2f%1K~n1T!5m¹4f. ~3!

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Here $ f ,g%5(] f /]x)(]g/]y)2(]g/]x)(] f /]y) denotes thePoisson bracket. The potentialf is normalized byT/e, thetime by vci

215mi /(eB0), and the length by r5(T/mi)

1/2/vci . The curvature operator,

K52¹•

B3¹

B2.

The coefficientsn, k, andm model the particle diffusion, theheat diffusion, and the viscosity, respectively. In the derivtion of Eqs.~1!–~3! it is assumed that the fields deviate onslightly from reference levelsN, T, with, e.g., n5N(11n(x,y,t)) or in other words the variation of the profilessmall compared to the absolute levels of the backgroquantities. Modeling the outboard midplane of a tokamaka simplified nonsheared magnetic field configuration, the cvature operator reduces toK( f )5vB] f /]y, where vB

52r/R0 . This corresponds to the large aspect ratio appromation for the magnetic fieldB5(B0R0 /R)b, whereb is theunit vector in the toroidal direction, locally along thz-coordinate.R is the distance from the torus axis andB0 isthe magnetic field atR5R0 . In the coordinate system of thconsidered slab,x corresponds to the radial direction andy tothe poloidal direction.

In the inviscid limit the equations possess the Lagraian invariants,12

l 656A5/2~n1vBx!13T/22n, ~4!

advected by the pseudovelocities

v65 z3¹@f2n2~16~5/2!1/2!T#.

If the turbulence can mix these invariants effectively overavailable volume on a time scale much shorter than thecous time scale, the Lagrangian invariants will on averageuniformly distributed. This leads to TEP profiles

^n&1vBx'const and^T&1 23vBx'const, ~5!

where^•& denotes an ensemble average that we assumereplaceable with an average over time and the periodicloidal coordinate, i.e., flux surface average. Also in the invcid limit Eqs. ~1!–~3! conserve the energy-like integral,

E5E F1

2~¹f!22~n1T!vBxGdxdy. ~6!

The first term is the kinetic energy, while the second hasform of potential energy. It represents that part of the thermenergy which can be converted to kinetic energy when flparcels are displaced to a region with weaker magnetic fiwhich gives rise to instabilities. The approximations behithe derivation of Eqs.~1!–~3! are quite analogous to the clasical Boussinesq approximation.16 In usual fluid convection,thermal energy is converted to kinetic energy, but in tBoussinesq equations commonly used to describe suchvection the energy equation only contains kinetic and pottial energy, while heat is conserved separately, just likeEqs.~1!–~3!.

We should emphasize that in contrast to models, whcurvature is modeled by an effective gravity—the resistg-paradigm—the magnetic field inhomogeneity~correspond-

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1077Phys. Plasmas, Vol. 10, No. 4, April 2003 Transport barriers and edge localized modes-like bursts . . .

ing to curvature! here just couples the equations, i.e., itlows the potential energy in the pressure gradient to be cverted to kinetic energy of the plasma. Thus, the fiinhomogeneity does not act as a source of free energy it

To investigate the linear stability we linearize Eqs.~1!–~3! around the background profilesn0(x), T0(x) assuming awaveform for the potential perturbationck(x)exp(iky2ivt)respecting the boundary conditions in thex-direction, andsimilarly for n and T, wherek is the wave number in they-direction. The equation for the wave amplitudeck reads

d2ck

dx21@2k21D~c,x!#ck5O, ~7!

where

D5cN2~5/3!~vB!2~n081vB!

c~c22~10/3!cvB1~5/3!vB2 !

. ~8!

We have introducedc5v/k, and the ‘‘buoyancy’’ frequency

N[vB~n081T081 53vB!.

Here the ‘‘prime’’ denotes derivative with respect tox.Equation~7! may be solved as an eigenvalue proble

for given profiles and with the boundary conditions,

cku~x50!50 and cku~x5Lx!50,

resulting in an expression for the complex phase velocityc,the dispersion relation. HereLx is the width of the slab in theradial direction. In order to illustrate the features of thestability we consider two simplified cases.

Within a local approximation and considering the lowave limit, i.e.,D'N/(c22 10

3 cvB1 53vB

2), we obtain

c55

3vBS 12

1

2

n081vB

n081T0815

3vB

6AN

K2 D , ~9!

whereK25kx21k2. Thus, we have instability forN,0. This

is the ‘‘standard’’ Rayleigh–Taylor instability with thegrowth rate

g5kAuNu

K2. ~10!

Furthermore with Re(c)5 53vB we have propagation in they

direction.For the special case, wheren0 andT0 are linear inx, N

becomes independent ofx, and we can find the general solution of Eq. ~7!, that satisfies the boundary conditions,

ck~x!5( An sinnpx

Lx

. ~11!

The dispersion relation for these modes is similar to thegiven by Eq.~9! with K25k21(np/Lx)

2, and it is readilyseen that the ‘‘fundamental mode,’’n51, has the largesmaximum growth rate,

g5kA uNu

k21~p/Lx!2

. ~12!

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As was already emphasized, the TEP profile, as given by~5!, is marginally stable.

III. NUMERICAL RESULTS

The model equations are solved numerically using anite difference code, based on the energy and enstrophyserving third order discretization of the Poisson bracke17

which is also used to evaluate the curvature terms.propagation in time we use a third order stiffly stable scheas described in Ref. 18. We consider a two-dimensionalmain which is bounded inx ~the radial direction! with lengthLx and periodic iny ~the poloidal direction! with lengthLy .The poloidal periodicity length may be interpreted as trecurrence length of a magnetic field line: Assuming an innite correlation along magnetic field lines for the safety fator q53 rational surface we would findLy52pr /3 and forq54 the poloidal periodicity length would beLy52pr /4.The aspect ratio of our slab-domaina[Ly /Lx will then bedirectly related to the safety factorq, i.e., a}1/q. The exactvalue naturally depends on the radial length scale. Our splified 2D model does, at the present state, not reflectdetails of magnetic configuration, but it is known from lineanalysis that ballooning modes—being made responsiblegiant ELMs—are rather wide and have an extend of ab20% of the minor radius of the torus.19,20 We will thus takethe radial extend of these linear modes as an estimate onradial scale lengthLx . With a typicalq-value at the edge oq54, we then estimate an aspect ratio for our systemequations. We find an effective aspect ratio ofa5(2p/4)3(1/0.2)'8, which is of the same order of magnitude as wconsider.

We would also like to point out that opposed to turbulefluctuation simulations, we resolve both the fast time scaof the turbulence and the slow profile evolution time scaComputational resources thus restrict our spatial resolutnecessitating artificially high values for viscosity and diffsivities as compared to experimental values, and finally dtate, for the time being, the use of a 2D model.

The diffusivities and dissipation coefficients,n, k, mwere chosen to be equal (51023 in most of the runs!. Thesource of energy to drive the turbulent fluctuations is anposed temperature or pressure difference between the wusing the boundary conditionsTux505T0 and Tux5Lx

50.The diffusive particle flux at the walls was set to zeroprescribing]xnux50,Lx

50, and the potentialf was kept con-stant at the walls, so that the velocity component perpendlar to the walls vanished.

We performed numerical runs for various values of tdifferent parameters of the system: the imposed temperadifferenceT0 , the aspect ratioa[Ly /Lx , the size of thesystemLx , and the diffusivities and the dissipation coefcients. WhenT0 is sufficiently large to drive the instabilitywe observe the following general scenarios, dependingaspect ratio as visualized in Fig. 1:

~a! For sufficiently large a>ac'3.8 ~for n5k5m51023) the system develops into the TEP state dscribed in Eq.~5! regardless of the value ofT0 , dem-

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1078 Phys. Plasmas, Vol. 10, No. 4, April 2003 Naulin, Rasmussen, and Nycander

onstrating profile consistency and resilience.12 There isa radial, turbulent heat flux, which is persistent, bintermittent.

~b! For smallera a different behavior is observed. In ainitial phase the turbulence develops and establisthe TEP profiles with a high flux-level. Later on, thflux is interrupted, that is, an H-mode-like state wisteeper averaged gradients and lower effective dision coefficients develops. For long periods of time tsystem is very quiescent, as seen in Fig. 1 and detain Figs. 2~a! and 2~b! for a51. However, sporadic fluxbursts of high amplitude—analog to ELMs—are oserved to occur at somewhat random intervals. Ttime scale of the quiescent periods between the buis, however, related to the viscous time scale. Increing the viscosity we observe that the average timetervals between the bursts decreases almost proporally to m21, as illustrated in Fig. 2~b!. Here theevolution of the heat flux is shown form50.01, andthe number of bursts is about 10 times larger thanFig. 2~a!. Increasing the viscosity even further wreach a value where the instability becomes too wto drive a sufficient fluctuation level for establishingTEP profile.

The quiet periods are associated with the establishmof a strong poloidal mean flow which characterizes the tra

FIG. 1. Poloidally averaged heat flux vs time for different aspect ratios.an aspect ratio larger than 3.8, no transport suppression is observed.

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port barrier~see Fig. 3!. This flow, which is strongly sheareand often only develops in a part of the domain, quencthe turbulence and acts as an effective barrier for transand mixing. As there is no longer sufficient mixing by thturbulence to maintain the TEP profiles and the nonturbuflux is much smaller than the turbulent one the profiles sto steepen via the diffusive inflow of heat from the hboundary. We estimate the competition between the turbumixing and the diffusion by averaging Eq.~2! over y,

]^T&]t

52]

]x^uT&1k

]2

]x2^T&. ~13!

Here u is the radial velocity component and f &5(1/Ly)*0

Lyf dy is the flux surface average. The turbuleheat fluxGT5^uT& is proportional to the turbulence level;quasilinear estimate using the local dispersion relation giGT}2(T0812vB/3)ufu2. It is directly related to the instabil-ity mechanism, and by using Eq.~6! it is readily observedthat GT is responsible for transferring the free potential eergy stored in the gradient into kinetic energy of the fluctutions. WhenGT is stronger than the diffusive fluxk(]/]x)3^T& the mixing will effectively be sufficient to maintain thTEP profiles. On the other hand, the viscous flux will domnate if the turbulence is suppressed.

The generation of the poloidal flow is described by tReynolds stress. This is seen by averaging Eq.~3! over y toobtain the equation for the evolution of the poloidal veloci

]V

]t52

]

]x^uv&1m

]2

]x2V, ~14!

wherev is the poloidal velocity component andV5^v&. TheReynolds-stress,Ruv5^uv&, which is the flux of momen-tum, is related to the turbulence level. However, a sim

r

FIG. 2. Poloidally averaged heat fluxGT5^uT& vs (x,t) for an aspect ratioa5Ly /Lx51 and dissipation coefficientsn5k5m51023 ~left frame! and1022 ~right frame!. The gray-scale is from20.2 ~light! to 0.5 ~dark!.

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1079Phys. Plasmas, Vol. 10, No. 4, April 2003 Transport barriers and edge localized modes-like bursts . . .

FIG. 3. ~a! Contours of the poloidal velocity vs time and radial coordinawith contour levels spaced at intervals 0.2.~b! and ~c! are enlarged withrespect to time at the occurrence of the flux burst:~b! GT(t) plotted vs (x,t)with dz50.5 and~c! the temperature profile vs time with contour spacingdz51. Same parameters were used for the run as in Fig. 2.

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quasilinear estimate using the linear local dispersion reladoes not give a contribution toRuv , a seed poloidal flow orhigher order nonlinear couplings are essential~see Appen-dix!. Thus, when the turbulence is quenched the lower leturbulence is increasingly unable to transport momentinto the flow, and the poloidal flow will decay due to thviscosity. After the flow has decayed sufficiently and tpressure gradient have built up, the basic instability mayvelop again, resulting in a strong burst of turbulence and hflux. We should emphasize that the heat flux in the fudeveloped turbulent case is always stronger than the momtum flux. Thus, the TEP state is established before thementum flux has build up the poloidal flow.

This behavior is further detailed in Fig. 3, for the samparameters as in Fig. 2~a!. Here we depict the evolution othe poloidal flow by showing the contours ofV in thex– t-plane in Fig. 3~a!. After the initial phase where the turbulence develops and establishes the TEP profiles a poloflow is seen to form att'100. The flow is strongest at thheated boundary where it is in the positivey-direction. Nearthe other wall it is reversed, resulting in a strong shear. Tflow dominates the evolution for a long period, during whithe turbulence is suppressed and the heat flux is negligcompare Fig. 2~a!. The poloidal flow is slowly decaying andat aroundt55200 it has become weak enough for allowinthe onset of the instability and turbulence due to the builttemperature profile. If the viscosity is larger, the poloidflow decays faster, and the turbulent bursts are theremore frequent, as in Fig. 2~b!. Note that the linear growthrate of the Rayleigh–Taylor instability Eq.~10! for thepresent parameters is on the order of 0.1; which explainsfast growth of the turbulence. This leads to a burst in the hflux and an associated flattening of the temperature profilobserved in Figs. 3~b! and 3~c!. The flux is first establishednear the heated wall and the onset of the flux propagoutwards as a sharp front accompanied by the change intemperature profile. Again a poloidal flow builds uparoundt55600. This quenches the turbulence and the fland the previous scenario repeats.

IV. DISCUSSION OF SHEAR FLOW GENERATION

It remains to be explained why the poloidal flow is ondominating the evolution for aspect ratios below a criticvalue ac , which for the low viscosity case consideredFigs. 1–3 is approximately 3.8.ac depends on the viscositywith a tendency to decrease with increasing viscosity.

In turbulent convection flows in neutral fluids governeby the Boussinesq equations Howard and Krishnamurti21 de-scribed the generation of large scale flows by expandingfields in only few fundamental modes and demonstratedthe poloidal flow mode (k50) could grow due to a ‘‘tilting’’instability of the fundamental vortex-like modes. Howevethey did not address the influence of the aspect ratio. Bcally their result shows that the growth of the shear flowstrongly dependent on the wave number (k52p/Ly in ournotations! of the fundamental mode, which was consideras the ‘‘pump’’ wave. The growth rate increases withk2 forsmaller k-values (k!2p/Lx) and with k for larger values.

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1080 Phys. Plasmas, Vol. 10, No. 4, April 2003 Naulin, Rasmussen, and Nycander

Drakeet al.22 applied a similar approach, to explain the fomation of a global shear flow in the two-dimensionNavier–Stokes equations, they termed the instability:‘‘peeling’’ instability. However, they considered a differenset of modes, ensuring that the total circulation (*Vdxdy,whereV is the vorticity! is conserved. The instability wafound to appear for an aspect ratio belowA5 in the case ofweak dissipation. A similar result was recently obtainedthe ‘‘resistive-g paradigm,’’23 i.e., RTI modes driven by onlythe ‘‘gravitational’’ coupling, by using a low-dimensionaGalerkin approach. However, the appearance of a criticapect ratioac appears to be an artifact of the truncation inonly few modes, as shown by Rosenbluth and Shapiro.24 It isinteresting to note that the results of Rosenbluth and Sharecovers the scaling of the growth rate withk2 for the smallk-values, while for higher values~basically k@1/Lx) theyfind a periodic instability with comparatively much lowegrowth rate.

As stated above, we find in our simulations a clear crcal aspect ratio. In the Appendix we have considered a splified model for the shear flow generation again showingvery strong increase withk for small k’s. Expressing thegrowth rate,gV , in terms of a ~A6! we observe thatgV

decrease witha, which agrees with the observations in Fi1, the time scale for the growth of the shear flow to supprthe turbulence is increasing witha. For a sufficiently highvalueac the growth rate can no longer overcome the viscodamping, and we believe that this is the course of the critaspect ratio in our simulations. We also note that even ifhave not yet performed a detailed parameter study ofdependence ofac on m it appears thatac decreases withincreasingm.

We illustrate the mechanism of the flow generationlow aspect ratios by comparing the evolution of the vorticfield for the casesa52 and a54. For the first case thequasistationary state, after the initial growth phase, is donated by two counter rotating convective cells with superposed short scale fluctuations, see Fig. 4. At the ‘‘bordebetween the cells exist very concentrated outward streamwarm plasma and inward streams of cold plasma, i.e.,heat flux is spatially localized in thin channels, which acommonly observed in convection problems in neutral fluiThe borders are strongly wiggling with the largest amptudes near the boundaries. The wiggling finally results insqueezing of one of the convective cells at the expense oother, and a global shear flow is established, manifesting‘‘tilting’’ instability.

In the casea54 the initial evolution is similar, but nowwith four cells. This pattern also gets unstable and leavescounter rotating cells, which are superimposed by relaweak poloidal flows forming and decaying roughly periodcally. These flows are not sufficiently strong to shear apthe cell of the opposite vorticity, and the two-cell pattesurvives. The periodically formed poloidal flows are oserved to change direction between each period, andflows ultimately vanish. Finally a very robust TEP staforms, where not only the temperature and density profifollow the predicted TEP profiles, but also the potential vticity profile ^)&5^¹2f2vBx2n& approaches a constan

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value, implying that ¹2f&5^n1vBx&1const5const. Notethat there is a direct connection between the poloidal flgeneration described by Eq.~14! and the equipartition of thepotential vorticity ) . The flux of the vorticityV5¹2f isgiven by ^uV&5]xRuv , see also Ref. 25.

Once the poloidal flow is formed, it is observed that thflow appears to stabilize the steep supercritical~with respecto the RTI! pressure profile that builds up due to viscodiffusion in the quite periods, see description of Fig. 3. Thmay be explained by the recent analysis of Benilovet al.26

~see also Ref. 27!, who demonstrates that an induced sheflow, irrespective of its actual profile, will tend to stabiliz

FIG. 4. The vorticity~gray scale between@22:2#! ~top frame! and the tem-perature@210:10# ~bottom frame! before the transition to shear-flow fora52 andm51023. The system is heated at the left boundary withT0510.

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1081Phys. Plasmas, Vol. 10, No. 4, April 2003 Transport barriers and edge localized modes-like bursts . . .

the larger wave numbers of the RTI instability of an iversely stratified fluid. This implies that for a given geomeof the domain, i.e., a given aspect ratio, the shear flow mstabilize all the modes allowed by the geometryk.2p/Ly . Thus, even if the shear flow is not stabilizingthe global sense28 it may be stabilizing in a finite systemThis may explain the long quiescent periods—the H-modwith the steep gradients that we observe.

V. CONCLUSION

In conclusion, we have shown that the nonlinear evotion of pressure driven turbulence in an inhomogenemagnetic field depends strongly on the aspect ratio. We csider this as a crude but self-consistent and fully nonlinmodel for turbulence on the outboard side of toroidal mnetic confinement devices. The aspect ratio in our moreflects the fact that when moving poloidally one will crothe same magnetic field line after some distanceL. This setsan effective poloidal periodicity length assuming perfect crelation along magnetic field lines. For low aspect ratwhich would correspond to either high irrationalq or q val-ues close to rational ones, the evolution is characterizedlong lasting quiescent H-mode periods with the turbultransport suppressed by a poloidal shear flow that actstransport barrier. These periods are separated by short vioflux bursts ~ELM! during which the poloidal flow breakdown. A similar behavior was also observed in toroidal grokinetic ion-temperature-gradient turbulence simulatiowhere the poloidal flows was damped by ion collisions.29,30

For large aspect ratio we find a continuous strong turbuflux, which, however, is temporally intermittent and spatialocalized in narrow channels between the dominating cvective rolls.

We have only considered specific typical cases in twork with the aim of bringing out the characteristic behaiors. We expect a very rich dynamical behavior when eploiting the full parameter diagram, in particular the rolethe initial pressure gradient and the viscosity and diffusivitdeserves a much more detailed analysis.

Although the model is simplified and lacks some gemetrical effects, it contributes to the fundamental undstanding of the role of poloidal flows in controlling turbulence and confinement and shows their fundamerelationship to the giant ELM type behavior observed inH-mode. It reproduces the spatial and temporal intermittevolution of the turbulent fluxes generally observed andfers a consistent nonlinear picture of ELM behavior, alloing some prediction of transport barrier formation in relatito the q profile. However, unlessq is changed significantlythis dependence might be weak. Additionally one shonote that our values chosen for viscosity are rather higcompared with the edge region of a tokamak as discusseSec. III. We did find a tendency for a shift to higher criticaspect ratiosac for lower values of viscosities and diffusivities. Thus, at experimentally realistic viscosities/diffusivitiwe would expectac to approach values more consistent wthose estimated from experiment.

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ACKNOWLEDGMENTS

We thank Odd Erik Garcia for fruitful discussions on thshear flow generation.

This work was partly supported by the Danish NatuSciences Foundation~SNF Grant No. 9903273!.

APPENDIX: QUASILINEAR ESTIMATION OF THEREYNOLDS STRESS

The poloidal flow generation is governed by Eq.~14!.We attempt to estimate the Reynolds stress from a ‘‘qulinear’’ approximation. When the wave field is representedthe form above Eq.~7! we find by direct calculation thecontribution to poloidal flow acceleration,]x^uv&, from thekth wave-component,31

]x^uv&522k]x~ ucku2]xuk!, ~A1!

whereuk is the phase ofck . We then use Eq.~7! and obtain,by multiplying by ck* and subtracting the complex conjugated equation multiplied byck ,

2]x~ ucku2]xuk!12DImucku250, ~A2!

and

]x^uv&52kDImucku2. ~A3!

The requirement for a nonvanishing gradient in the Reynostress is thatD ImÞ0. However, it is readily observed that ithe local approximationD must be real. This also applies fothe global modes, Eq.~11!. Thus, the interaction of twowaves that are governed by the dispersion relation willcontribute to a finite gradient in the Reynolds stress. Hoever, if we assume that we have a small seed poloidal fli.e., we replacec by c–V in the expression forD, we get inthe long wave limit,

D Im'2UNci

ci4

. ~A4!

Considering the fundamental mode structure in the radirection @n51 in Eq. ~11!#, which has the largest lineagrowth rate, we obtainci'AuNu/Ak21p2 as the long wavesolution of the linear dispersion relation forN,0. InsertingEq. ~A4! into Eq. ~A3!, and using Eq.~14! in the inviscidlimit, we obtain the growth rate ofV,

gV54k~k21~p/Lx!

2!3/2

AuNuucku2. ~A5!

We now assume that the mode withk52p/Ly is dominatingprior to the shear flow generation, which seems reasonsince the flow generation appears as the ultimate outcomthe inverse cascade. This is further equivalent to the analbased on the few mode truncation, where the fundamemode is considered as the ‘‘pump’’ wave~see the discussionin Sec. III!. We introduce the aspect ratioa5Ly /Lx in Eq.~A5! and obtain

gV58p4~41a2!3/2

Lx4a4AuNu

ucku2. ~A6!

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1082 Phys. Plasmas, Vol. 10, No. 4, April 2003 Naulin, Rasmussen, and Nycander

Thus, for the case of a fixed radial width the growth of tpoloidal flow is decreasing witha4 for a!2, while it de-creases witha for a@2. If we now account for the viscoudissipation ofV, which gives an estimated damping ratem(p/Lx)

2, then it is seen that for a certain value ofa theshear flow growth rate becomes too weak to overcomedamping, which may provide an explanation for the obsercritical a, and indeedac is observed to decrease with increasingm.

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