—journal of December 2003
physics pp. 1163–1169
Dynamical interplay between fluctuations, electric fields and transport in fusion plasmas
C HIDALGO, B GONC¸ ALVES1and M A PEDROSA
Laboratorio Nacional de Fusion, Euratom-Ciemat, 28040 Madrid, Spain
1Associac¸ao EURATOM/IST, Centro de Fus˜ao Nuclear, 1049-001 Lisbon, Portugal MS received 1 April 2003; accepted 31 July 2003
Abstract. A view of recent experimental results and progress in the characterization of the statis- tical properties of electrostatic turbulence in magnetically confined devices is given. An empirical similarity in the scaling properties of the probability distribution function (PDF) of turbulent transport has been observed in the plasma edge region in fusion plasmas. The investigation of the dynamical interplay between fluctuation in gradients, turbulent transport and radial electric fields has shown that these parameters are strongly coupled both in tokamak and stellarator plasmas. The bursty behaviour of turbulent transport is linked with a departure from the most probable radial gradient. The dynam- ical relation between fluctuations in gradients and transport is strongly affected by the presence of sheared poloidal flows which organized themselves near marginal stability. These results emphasize the importance of the statistical description of transport processes in fusion plasmas as an alterna- tive approach to the traditional way to characterize transport based on the computation of effective transport coefficients.
Keywords. Turbulence; edge transport.
PACS Nos 52.55.Fa; 52.55.Hc; 52.35.Ra
1. Introduction
Comparative studies of the structure of plasma turbulence carried out in different magnetic confinement devices have led to insights furthering the understanding of turbulent transport in fusion plasmas [1]. Recently, an empirical similarity in the scaling properties of the probability distribution function (PDF) of turbulent transport has been observed in the plasma edge region [2]. The overall similarity in the structure in the statistical properties of fluctuations has led to the conclusion that plasma turbulence in magnetically confined plasmas, as many other dynamical systems, display universal characteristics [3].
Article presented at the International Conference on the Frontiers of Plasma Physics and Techno- logy, 9–14 December 2002, Bangalore, India.
Recently a new approach to study the relation between gradients and transport, based on the investigation of the dynamical coupling between transport and gradients, has been pro- posed [4,5] emphasizing the importance of the statistical description of turbulent transport in terms of PDF. Based on this approach it has been shown that transport events propagating with radial velocities up to 1000 m/s are particularly significant when the radial gradient increases above its average value in the scrape-off-layer (SOL) region. The dynamical rela- tion between fluctuations in gradients and transport is strongly affected by the presence of sheared poloidal flows, heating power and the proximity to instability thresholds [6]. The statistical properties of the radial propagation of edge localized modes (ELMs) have also been investigated in the SOL region of the JET tokamak [7]. Experimental results suggest a link between the radial velocity and the size of transport events. This paper reports recent results in the characterization of the statistical properties of turbulence and the physics of sheared flows in fusion plasmas.
2. Experimental set-up and analysis tools
Plasma profiles and turbulence have been investigated in the JET tokamak and TJ-II stel- larator plasma boundary region using a fast reciprocating Langmuir probe system. The experimental set-up consists of arrays of Langmuir probes, which allow the simultane- ous investigation of the radial structure of fluctuations and electrostatic-driven turbulent transport. Plasmas studied in this paper were produced in X-point plasma configurations (ohmic and L-mode plasmas) in the JET tokamak and in electron cyclotron resonance heated (ECRH) plasmas in the TJ-II stellarator.
The mean velocity of fluctuations perpendicular to BThas been computed as vphase ΣSkωkωΣSkω, from the wave number and frequency spectra Skω, computed from the two-point correlation technique using floating probes poloidally separated∆θ 05 cm) [8].
Turbulent particle transport has been calculated from the correlation between poloidal electric fields and density fluctuations, ΓEBtn˜tE˜θtB. The poloidal electric field has been estimated from floating potential signals measured by poloidally sepa- rated probes, using the equation Eθ ∆Φ˜f∆θ. Fluctuations in the radial component of ion saturation current gradients have been computed using the equation ∇I˜St
I˜SinnertI˜Soutertwith∆I˜S0, where ˜ISinnerand ˜ISouter are the ion saturation current fluctuations simultaneously measured at two different plasma locations radially separated by 0.5 cm.
In order to study the coupling between probability distribution functions of transport and gradients, we have computed the joint probability Pi j of the two variables X and Y . The probability that at a given instant X and Y occur simultaneously, is given by Pi j PXiYjNi jN where Ni jis the number of events that occur in the intervalXiXi∆X andYiYi∆Yand N the time series dimension.∆X and∆Y are the bin dimensions of X and Y time series, respectively, where the indices stand for ith (or jth) bin average value.
The expected value of X at a given value of Yjis defined as EXYj
∑
i
Pi jXi
∑
i
Pi j
and represents the average value of the probability distribution of X at a given value of Y . An effective radial velocity has been defined as the normalized EB turbulent parti- cle transport to the local density,
veffI˜SE˜θISBT
where ISis the ion saturation current of the inner probe. As this coefficient is not affected by uncertainties in the effective probe area, it provides a convenient way to compare ex- perimental results with edge code simulations.
3. Statistical properties of turbulent transport
In the TJ-II stellarator, as in other devices [9,10], the PDF of the turbulent transport shows significant non-Gaussian features. The PDF of EB turbulent fluxes can be rescaled using a finite-size scaling law [11],
PDFΓEBL 1gΓEBL
whereΓEB is the turbulent EB flux and L is a scaling factor (figure 1). In order to identify the relation of the scaling parameter with plasma parameters, it is important to keep plasmas with similar properties (magnetic topology, collisionality, etc.) but with different magnetic well (i.e. different level of fluctuations). This study has shown that the scaling parameter L is directly related with the level of fluctuations [2]. Similar dependence has been recently observed in the JET tokamak [2].
These results are consistent with previous findings which have shown an empirical similarity of frequency spectra of edge plasma fluctuations in different toroidal mag- netic confinement devices [12]. Frequency spectra can be re-scaled using the expres- sion, PωP0gλω, whereλ and P0are parameters to be determined for each device.
The empirical similarity in turbulent fluxes suggests that edge plasma turbulent transport evolves into a state in which the PDFs of transport exhibit the same behaviour over the entire amplitude range of transport events.
It would be very interesting to look for the properties of PDF in non-fusion plasmas (like linear devices and toroidal devices without rotational transform), where we do not expect to have self-similarity on the structure of edge fluctuations.
4. The naturally occurring velocity shear layer in fusion plasmas
A velocity shear layer has been observed near the location of the last closed flux surface (LCFS) in the TJ-II stellarator and in the JET tokamak which is in agreement with the previous experiments in fusion plasmas [7]. Both in TJ-II and JET devices, radial gradient in vphaseis in the range of 105s 1, which turns out to be comparable to the inverse of the correlation time of fluctuations, in the range of 10µs (figure 2). It should be noted that the present results are consistent with previous observations in tokamaks, stellarators and reversed field pinches [13].
Even though this property is consistent with turbulent-driven radial electric fields, it is difficult to understand in which way other mechanisms, like those based on ion orbit
Figure 1. PDFs of edge turbulence in plasma configurations and the re-scaled PDFs of turbulent transport using the functional form PDFΓEBL 1gΓEBL[2].
losses mechanisms, can allow sheared flows and fluctuations to reach marginal stability.
Numerical simulations have shown that turbulent-driven fluctuating radial electric field via Reynolds stress has the property to get shearing rate critical [14]. Recent experiments carried out in the TJ-II stellarator and JET tokamak have shown that sheared poloidal and parallel flows are linked [15]. The parallel flow is affected by the degree of instability in
Figure 2. Radial gradient of the phase velocity of fluctuations and auto-correlation times in the plasma edge of the TJ-II stellarator [13].
Figure 3. Radial transport vs. fluctuations in gradients in the JET boundary region [5].
the plasma edge and its radial gradient is close to the ratio of the sound speed to the density scale length (e.g. close to the threshold of the Kelvin–Helmholtz instability). These results are also consistent with the concept of sheared poloidal and parallel flows self-organized near marginal stability.
5. Dynamical coupling between fluctuations in gradients and transport
Figure 3 shows the PDF for fluctuations in gradients, and the expected value of the EB flux for a given density gradientEΓEB∇rISin L-mode plasmas in JET tokamak. The results show that most of the time the plasma is at its average gradient and the size of the transport events has minimum amplitude (ΓEBΓEB05). Large amplitude transport events (ΓEBΓEB3–8) take place when the plasma displaces from the most probable gradient average value. The expected value of EB turbulent transport events increases strongly as the gradient increases above its most probable value (i.e.∇I˜Sσ0).
The present experimental results show that the bursty and strongly non-Gaussian be- haviour of the turbulent transport is strongly coupled with fluctuations in gradients. As the density gradient increases above the most probable gradient, the EB turbulent driven- transport increases and the system performs a relaxation which tends to drive the plasma back to the marginal stable situation which minimized the size of transport events. The
increase in the size of transport events as gradient increases is consistent with the self- regulation of turbulent transport and gradients near marginal stability in the plasma bound- ary region.
Transport events, related with small departures from the most probable local gradient, propagates radially with an effective velocity of about 20 m/s, which is consistent with sim- plified simulations of diffusive transport in the SOL region. On the contrary, large transport events, related to significant departures from the most probable gradient, propagates radi- ally with an effective velocity up to 500 m/s. These results suggest a link between the size of transport events and the nature of transport in the plasma boundary region [5].
The dynamical relation between fluctuations in gradients and transport is strongly affected by the presence of sheared poloidal flows, heating power and the proximity to instability thresholds: the size of the large transport events decreases in the proximity of sheared flows and increases with heating power and in the proximity of instability thresh- olds [6]. Furthermore, the radial velocity increases linearly with the size of transport events. This conclusion is consistent with a recent investigation of the radial propaga- tion of ELM events which also suggest an increase in the radial velocity of ELM events with their amplitudes [7].
6. Conclusions
Significant improvement in our understanding of the statistical properties of turbulent transport and the physics of sheared flows has been recently achieved in fusion plasmas.
The main conclusions can be summarized as follows:
An empirical similarity in the PDFs of turbulent transport has been observed in the plasma edge region in fusion plasmas.
The investigation of the dynamical interplay between fluctuations in gradients and turbulent transport has shown that their PDFs are strongly coupled. The bursty be- haviour of turbulent transport is linked with a departure from the most probable radial gradient.
Sheared poloidal/ parallel flows and fluctuations organize themselves near marginal stability. Sheared flows, heating power and the proximity to instability thresholds have a direct impact in the dynamical coupling between gradients and transport.
These results are consistent with the concept of turbulent transport self-regulated via fluctuations near marginal stability and clearly show the importance of the multi- field statistical approach to describe transport in non-equilibrium systems.
Acknowledgements
This work has been performed under the European Fusion Development Agreement.
One of the authors (B Gonc¸alves) acknowledges the support by the Fundac¸˜ao para a Ciˆencia e Tecnologia (Lisbon) under Grant No. PRAXIS XXI/BD/15814/98. This research was sponsored in part by DGICYT (Direcci´on General de Investigaciones Cient´ıficas y Tecnol´ogicas) of Spain under Project No. FTN2000-0924-C03-02.
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