VOLUME 73, NUMBER 25 PHYSICAL REVIEW LETTERS 19 DECEMBER 1994 Characterization of Coherent Structures in Tokamak Edge Turbulence S. Benkadda, ' T. Dudok de Wit, A. Verga, ' A. Sen, ASDEX team, and X. Garbet 'Turbulence Plasma, URA 773 Centre National de la Recherche Scientifique Un-iversite de Provence, Institut Mediterraneen de Technologie, F-13451 Marseille Cedex 20, France Cadarache, F-13108 Saint-Paul-lez-Durance Cedex, France 'Institute for Plasma Research, Bhat, Gandhinagar 382424, India 4Max Pla-nck Inst-itut fii r Plasmaphysik, D 8574-8 Garching, Germany (Received 14 June 1994) A statistical test for extracting and identifying coherent structures in the scrape-off layer turbulence is used to analyze Langmuir probe data from the ADITYA and ASDEX tokamaks. This method, the biorthogonal decomposition, allows one to characterize large-scale coherent structures in plasma turbulence in an unambiguous manner. It is shown that such structures effectively contribute to radial transport and to intermittency. Results from numerical simulations of turbulence driven by the resistive interchange instability in the scrape-off layer are compared to the observed statistical properties. PACS numbers: 52.35. Ra, 02.50. Sk, 52.55. Fa The role played by localized and long-lived structures in plasma turbulence has become an important research issue in the last decades. Both experimental results [1, 2] and numerical simulations [3] support the idea of self- organized turbulence in the scrape-off layer (SOL) of tokamak plasmas. The identification of such coherent structures from spatio-temporal turbulence measurements is fundamental for understanding their dynamics and test- ing their contribution to radial transport. Their extraction, however, has been a challenge problem so far, due to the lack of adequate analysis techniques [4]. It is thus impor- tant to provide appropriate statistical tools for extracting the relevant information from experimental data. and to give an accurate description of turbulence in a relatively low-dimensional space. In this Letter we report a simul- taneous analysis of the space and time dependences of fluctuation data, using a multivariate technique called the biorthogonal decomposition (BD). The analysis of edge plasma turbulence has tradition- ally been based on correlation and spectral techniques [5]. Recently, other techniques such as conditional sampling [6] and bispectral analysis [7] have been used to search for self-organized behavior. Although these methods can be extremely powerful, they do not readily provide a sta- tistical test capable of identifying and extracting coherent structures; they either suffer from an arbitrariness in the definition of coherent structures or lack in adequate spa- tial or temporal resolution. As a result, the experimental evidence for the existence of such structures has remained inconclusive so far. Another shortcoming of these meth- ods is their inability to properly deal with spatio-temporal signals. Indeed, most of them merely proceed with one- dimensional projections of the data and no full spatio- temporal analysis has been reported. The BD provides an objective test for identifying and extracting coherent structures without an a priori specification of their shape or localization. This statistical method has the interesting property to concentrate most of the pertinent dynamics into a few components, thereby allowing a strong reduction in the number of degrees of freedom necessary to describe the data. Its ability to reveal coherent structures will be used to assess the effect of the latter on radial transport and intermittency. The experimental data analyzed in this letter originate from the SOL of ADITYA [8] and ASDEX [9] tokamaks and the simulations data from a model of SOL turbulence driven by the resistive interchange instability [10]. The BD belongs to a class of methods (proper orthogo- nal decomposition, Karhunen-Loeve expansion, and singu- lar value decomposition) that were originally introduced by Lumley in fluid turbulence [11] and has since been used in fiuid mechanics [12] and in magnetohydrodynam- ics (MHD) activity studies [13]. To illustrate the method, we consider a scalar spatio-temporal signal y(x, t) (e. g. , ion saturation current or floating potential) whose tempo- ral evolution is measured simultaneously at M different locations. The signal is subsequently sampled and the data are assembled into an N X M matrix Y, in which the columns are time series. The BD consists in expand- ing the discrete data Y;i = y(xj, t;) into a unique set of modes that are orthonormal in time and in space, i. e. , Y;, = g„, A„v „(t;) u„(x, ), where K = min(N, M) is the fi- nite global dimension of the data set. The base functions u„(x, ) and v„(t;) are, respectively, eigensolutions of the two point temporal and spatial cross-correlation matrices of Y. The weights A„areeither positive or equal to zero, and it is conventional to sort the series in decreasing weight order. The BD, like other proper orthogonal decomposition methods, is based on a diagonalization of the data cross- correlation matrices. Note that we have a one-to-one correspondence between the spatial and temporal modes, Yu„= A„v„, which corresponds to a dispersion relation. A physical interpretation which stems from these defini- 0031-9007/94/73(25)/3403(4)$06. 00 1994 The American Physical Society 3403