PD/P6-3-1 Estimation of Dynamic and Diffusive Components in Edge Turbulent Particle Fluxes in the L-2M Stellarator and the FT-2 Tokamak N.N. Skvortsova 1), G.M. Batanov 1), D.V. Malakhov 1), A.E. Petrov 1), V.V. Saenko 1), K.A. Sarksyan 1), N.K. Kharchev 1), Yu.V. Kholnov 1), V.Yu. Korolev 2), Yu.V. Zhukov 2), M. Rey 2), A.S. Merkulov 2), S.V. Shatalin 3), S.I. Lashkul 3), E.O. Vekshina 3), A.Yu. Popov 3). 1) A.M. Prokhorov General Physics Institute, Moscow, 119991 Russia 2) Moscow State University, Moscow, 119899 Russia 3) A.F.Ioffe Physico-Technical Institute, St. Petersburg, 194021, Russia e-mail:
[email protected] Abstract. Results are presented from experimental studies of the diffusive and dynamic components of turbulent fluxes measured in the edge plasma in the L-2M stellarator and the FT-2 tokamak. In this report, a new method, which is based on the analysis of time samples of the increments of turbulent particle fluxes, is used to estimate the dynamic and diffusive components in the edge turbulent flux. The flux components are investigated using the socalled "sliding separation of mixtures" (SSM). In this method, volatility of a flux is interpreted as a multidimensional (vector) function of time, which provides a way of separating the dynamic and diffusive components. The dynamic component corresponds to a drift (which can occur by a variety of processes), for example, to ballistic transport. With the use of special mathematical statistical procedures, the diffusive component can be represented as a sum of subcomponents, each being related to the development of a particular type of stochastic structures, their interaction, etc. The turbulent particle flux in the edge plasma of the FT-2 tokamak was investigated in discharges in which L–H transitions occurred. The measurements were performed only in steady-state phases of the discharge. Transient processes will not be discussed here. The intensity of diffusive components after the transition to the H-mode is reduced, and the scatter of values of these components is also reduced. The ratio between the diffusive and dynamic components (calculated over all components) was measured in the edge turbulent flux during the steady-state phase of the L-2M discharge. Turbulent particle transport in L-2M occurs by both the ballistic and diffusive mechanisms. I.
Introduction
Recent investigations of low-frequency (LF) fluctuations in toroidal magnetic confinement systems revealed the presence of strong structural low-frequency turbulence in magnetoactive plasma [1-3]. This state of turbulence occurs in a steady-state plasma in an open thermodynamic system which has a source and sink of energy as a result of the development of transient processes: a growth and saturation of instabilities (by linear or nonlinear mechanisms) and the formation of stochastic plasma structures. An inherent feature of the states of strong structural turbulence observed in different devices is that the probability density functions (PDFs) differ from the normal distributions. The strong structural plasma turbulence can be described by a mathematical model of inhomogeneous random walk with continuous time - a doubly stochastic Poisson process also known as the generalized Cox process [4, 5]. The assumption of an inhomogeneous walk fits well with the idea that the rate
PD/P6-3-2 of variations of the coordinate of a particle during its Brownian motion in a turbulent medium is alterable. The model of inhomogeneous random walk allows us to represent the PDFs of the increments of amplitudes of turbulent particle fluxes as shift-scale mixtures of normal distributions [3, 5] and to estimate turbulent transport coefficients in the tokamak and the stellarator. In this report, a new method, which is based on the analysis of time samples of the increments of turbulent particle fluxes, is used to estimate the dynamic and diffusive components in the edge turbulent flux. II.
Probability-Theoretical Definition of Volatility of a Random Plasma Process
Below, we will be concerned with volatility (the degree of unpredictable change over time) of edge turbulent fluxes. The probabilistic concept of volatility has long been in use in financial statistics; however, to our knowledge, this characteristic has never been employed in papers devoted to plasma turbulence. Volatility of a random process is determined by at least two types of factors. The first type can be characterized as a dynamic factor. The influence of the dynamic factor makes itself evident in the fact that the process varies because of the presence of a certain trend or a combination and, hence, interaction of trends that indicates the presence of dynamic structures in a plasma. The dynamic component corresponds to a drift (which can occur by a variety of processes), for example, to ballistic transport. The second type of factors is termed stochastic or diffusive. With the use of special mathematical statistical procedures, the diffusive component can be represented as a sum of subcomponents, each being related to a particular type of stochastic structures, their interaction, etc. In the basic model of nonhomogeneous random walk described by a subordinate Cox process, volatility is naturally represented as a sum of the dynamic and diffusive components. In previous studies of low-frequency turbulence, it was shown that processes of turbulent particle transport are described by subordinate Cox processes and is described, to a high degree of accuracy, by shift-scale mixtures of normal distributions [5]. If X is an increment of a random process under consideration, then its distribution is a shift-scale mixture of normal distributions if
P ( X < x) = ∫∫ Φ (
x−v x −V )dP(U < u,V < v) = ΕΦ ( ), x ∈ ℜ, u U
(1)
where U ≥ 0 and V are finite random variables. In this case, volatility is characterized by a vector containing components of three types: a weight (positive numbers, which add up 1), drift factors (averages), and diffusion coefficients (standard deviations) of the corresponding components. Each of these mixture components corresponds to a particular mechanism and a particular structure in plasma turbulence, which is characterized by parameters αj и σj, whereas the weight pj defines the proportion of structures of the jth type in the whole process of plasma turbulence. In [6], it is shown that a finite shift-scale mixture of normal distributions (1) is represented in the form k
∑ p Φ( j =1
j
x − aj
σj
) = ΕΦ (
x −V ), U
(2)
PD/P6-3-3 where the pair of random variables U and V has a discrete distribution
P ((U ,V ) = (σ j , a j ),
j = 1, ... , k,
so that k
DV = ∑ p j (a j − a ) 2 , j =1
k
ΕU 2 = ∑ p j σ 2j ,
(3)
j =1
k
where a = ∑ p j a j . j =1
The first expression in (3) defines that volatility component which is associated with the presence of local trends; it will be referred below as a "dynamic component" of volatility. The second expression defines a "diffusive component" of volatility. Total (multidimensional) volatility of the process is the square root of the sum of two components, one of which represents a spread of local trends, whereas the other characterizes diffusion. III.
Analysis of dynamic and diffusive components turbulent fluxes
The L-2M stellarator has two helical windings (l= 2), a major radius R= 100, and a mean plasma radius
= 11.5 cm [7]. The plasma was created and heated by one or two 75GHz gyrotrons under conditions of electron cyclotron resonance heating (ECRH) at the second harmonic of the electron gyrofrequency. The magnetic field at the plasma center was Bt = 1.3–1.4 T. The gyrotron power was P0 = 150–300 kW, and the microwave pulse duration was up to 15 ms. Measurements were carried out in a hydrogen plasma with an average density of = (0.8–2.0) ⋅1013 cm–3 and central temperature of Te = 0.6–1.0 keV. In the edge plasma at a radius r/a = 0.9, the density was at a level of n (r) = (1–2) 1012 cm–3 and the electron temperature was Te (r) = 30–40 eV. The duration of the steady-state phase of the discharge was 10 ms. The FT-2 tokamak has the following basic parameters: R = 0.55 m, a = 0.079 m. Experiments were performed at Bt = 2.2 T and at a plasma current of Ipl = 22 kA. These conditions corresponded to plasma under q = 6, where the transition to improved confinement was realized. The duration of the steady-state phase of the discharge was 40 ms, the plasma current pulse duration being 60 ms. The auxiliary lower hybrid heating (LHH) pulse (frequency F = 920 MHz, power PLHH = 90–100 kW, duration ΔtLHH = 5 ms, N⎢⎢≅ 2.≅ 3) was applied at the middle of plasma-current plateau. Measurements were carried out in a hydrogen plasma with an initial (resonance for LHH) central density of ne(r=0) = 3.3⋅1013 cm–3 and central ion and electron temperatures of Ti = 0.09 – 0.1 keV and Te = 0.3-0.35 keV, respectively. Effective ion and electron heating was observed during the LHH pulse (where Ti rose up to 0.2 – 0.25 keV and Te rose up to 0.5 keV [8,9,]) followed by the L-H transition with ETB [9]. According to simulation with the ASTRA code and diamagnetic measurements, the energy confinement time increased from τOHЕ = (0.8 - 0.9) ms up to τEpost-LHH = (2 – 3.5) ms, and the H-factor changed as Н = τEpost-LHH/τH (y,2) = 2÷3 [8]. Local particle fluxes were measured in the edge plasma of L-2M by probe systems consisting of three single cylindrical probes; a multielectrode probe system was used in FT-2. Measurements in L-2M were performed in several sections along the torus [10]; measurements in FT-2 were performed in one section at different poloidal angles [11].
PD/P6-3-4 The values of the local particle flux measured in successive instants constitute a time sample of a stochastic diffusion process. Hence, all of the conventional methods of spectral, correlation, and probability analysis are applicable for studying the features of this process. The PDF of the local flux in all the experiments in L-2M stellarator and FT-2 tokamak was leptokurtic and had heavier tails as compared to a Gaussian distribution. Time samples of the magnitudes of local fluxes in L-2M and FT-2 are not homogeneous and independent. In contrast, time samples of the increments of local fluxes are independent. For this reason, when analyzing local fluxes, it is reasonable to use time samples of the increments of local fluxes rather than time samples of their magnitudes. The probability distributions of the increments of turbulent fluxes in both devices are well fitted with the finite mixtures of normal distributions [3]. The flux components are investigated using the so-called "sliding separation of mixtures" (SSM). In this method, volatility of a flux is interpreted as a multidimensional (vector) function of time, which provides a way of separating the dynamic and diffusive components. The SSM technique for analyzing the structural plasma turbulence is described in the review [6].
Fig. 1. Diffusion portrait (diffusive components of volatility) of the process of increments turbulent particle flux in the edge plasma of the L-2M stellarator. Calculation by the SSM method with time windows consisting of 200 (upper plots) and 300 (lower plots) points. Figure 1 shows diffusion portraits of turbulent particle transport in the edge plasma of the L-2M stellarator. Here, the successive points in time are plotted on the horizontal axis.
PD/P6-3-5 The vertical axis is the diffusion coefficient of components of a mixture of normal distributions that is fitted to the increments of the process of one-point turbulent flux. The turbulent flux in the stellarator is well modeled by a shift-scale mixture of normal distributions with three components, to accuracy above 90-95% (according to the Kolmogorov--Smirnov criterion). This plot shows the temporal behavior of the diffusive components of volatility calculated using the SSM algorithm with time ("sliding") windows consisting of 200 and 300 points, in two successive tome intervals. The intensity of the diffusive component of each mixture component is shown by color - blue to red. It can be seen from Fig. 1 that the number of components remains constant, whereas the intensity of diffusive components varies with time. The upper picture is more volatile; however, the lower, smoother picture provides more information about the behavior of the diffusive components of volatility of turbulent particle transport. The turbulent particle flux in the edge plasma of the FT-2 tokamak was investigated in discharges in which L–H transitions occurred [3]. Figure 2 shows the temporal behavior of the subcomponents of the diffusive component of the turbulent flux before and after the L–H transition. The measurements were performed only in steady-state phases of the discharge. Transient processes will not be discussed here. Note that particle transport in the FT-2 plasma is governed by only 2-3 stochastic processes of structural turbulence, as was also observed previously. The intensity of diffusive components after the transition to the H-mode is reduced, and the scatter of values of these components is also reduced. The temporal dependence of each of the diffusive subcomponents after the transition to the H-mode becomes smoother. As the number of processes in the mixture after the transition increases (from two to four), the diffusive components of volatility, on the average, decrease resulting in a decrease in the total stochastic component of volatility.
Fig.2. Diffusive components of volatility of the turbulent flux in the FT-2 tokamak in L-mode and H-mode plasmas. Window length is 300 points. Figure 3 shows the ratio between the diffusive and dynamic components (calculated over all components) in the edge turbulent flux during the steady-state phase of the L-2M discharge. The diffusive component, on the whole, seems to be predominant during the discharge, but the dynamic component of volatility exceeds the diffusive component at some instants of time. The dynamic component can be associated with convective transport, whereas the diffusive component characterizes the stochastic component of volatility. Thus, turbulent particle transport in L-2M occurs by both the ballistic and diffusive mechanisms.
PD/P6-3-6
Fig.3. Dynamic (red) and diffusive (blue) components of volatility of the turbulent flux in the L-2M plasma. IV. Conclusions
A new method, which is based on the analysis of time samples of the increments of turbulent particle fluxes, is used to estimate the dynamic and diffusive components in the edge turbulent flux. The flux components are investigated using the so-called "sliding separation of mixtures" (SSM). In this method, volatility of a flux is interpreted as a multidimensional (vector) function of time, which provides a way of separating the dynamic and diffusive components. The probability distributions of the increments of turbulent fluxes in the edge plasma of L-2M stellarator and FT-2 tokamak are well fitted with finite mixtures of normal distributions. The diffusive and dynamic components of turbulent fluxes in the edge plasma were estimated in the L-2M stellarator and the FT-2 tokamak. The dynamic component corresponds to a drift (which can occur by a variety of processes). With the use of special mathematical statistical procedures, the diffusive component can be represented as a sum of subcomponents, each being related to the development of a particular type of stochastic structures, their interaction, etc. The turbulent particle flux in the edge plasma of the FT-2 tokamak was investigated in discharges in which L–H transitions occurred. The intensity of diffusive components after the transition to the H-mode is reduced, and the scatter of values of these components is also reduced.
PD/P6-3-7 The temporal evolution of all diffusive components of volatility was measured for turbulent particle fluxes in L-2M. The ratio between the diffusive and dynamic components (calculated over all subcomponents) was measured in the edge turbulent flux during the steady-state phase of the L-2M discharge. Turbulent particle transport in L-2M occurs by both the ballistic and diffusive mechanisms, and their contributions are of the same order of magnitude. Acknowledgements
This work was supported by the Federal Agency for Atomic Energy of the Russian Federation (contract no. 6.05.19.19.06.900) and the Presidential Program for Support of Leading Scientific Schools (project no. NSh-5382.2006.2), Dynasty Foundation, Basic Research and Education program of found Civilian Research Development Foundation and Department of Education (project no. DSP.2.2.2.3.3560) References
[1] G. M. Batanov, V. E. Bening, V. Yu. Korolev, et al., JETP Lett. 78, 2003. P. 502. [2] V. Yu. Korolev and N.N. Skvortsova. (Eds) “Stochastic Models of Structural Plasma Turbulence”. VSP, Leiden - Boston, The Netherlands. 2006. [3] N.N. Skvortsova, V.Yu. Korolev, G.M. Batanov et al. Plasma Phys. Contr. Fusion. 2006. 48(5A). P. A393. [4] V. Bening and V. Korolev. Generalized Poisson Models and their Applications in insurance and finance. VSP, Utrecht, The Netherlands, 2002. [5] N.N. Skvortsova, V.Yu. Korolev, T.A. Maravina, et al. Plasma Physics Reports. 31. 2005. P. 57. [6] V.Yu. Korolev, N.N. Skvortsova. A new method of the probabilistic and statistical analysis in case of plasma turbulence processes. In “The systems and means of informatics.” Ed. I.A. Sokolov. Moscow. The Institute of Informatics Problems (IPI RAN). 2005. P. 126-179. (In Russian). [7] V.V. Abrakov, D. K. Akulina, E. D. Andryukhina, et al., Nucl. Fusion 37. 1997. P. 233. [8] S.I. Lashkul, V.N. Budnikov, S.V. Shatalin, et al. Czechoslovak J. of Physics. 52. 2002. P. 1149. [9] S.I. Lashkul, S.V. Shatalin, et al. 32th EPS Conf .on Contr. Fusion and Pl. Phys. 2005. P-4.046 [10] G.M. Batanov, O.I. Fedyanin, N.K. Kharchev, et al., Plasma Phys. Controlled Fusion 40.1998. P. 1241. [11] S.V. Shatalin, E.O. Vekshina, S.I. Lashkul, et al. Plasma Phys. Reports. 30. 2004. P. 363. 
