- Tex files containing the test Lecture to be held on Wednesday the 22, titled *Examples of experimental data analysis methods for Tokamak and RFP” (examples chosen by the applicants*
- Graphics included in the presentation are in pdf_box
- Programs used for the creation of the graphics are both in idl/ and python/ directory
- This lecture will be devoted to the description of analysis technique suitable for turbulence investigation in magnetized plasmas
- The techniques describes are completely general and can be applied to all types of plasmas, indipendently from the configuration choosen.
- Very basic examples will be provided to ensure clarity and demonstrate practical applications
- All the analysis will use data provided by diagnostics
- Measurements will be available in a variety of different modes:
- Single point local measurements (as Langmuir probes)
- Spatially distributed arrays of measurements (which can cover portion of the plasma with different sizes)
- Line integrated measurements as are the cases for spectroscopic diagnostics
- Array of LoS
- We focuses on techniques which are suitables for the first two cases indicated
- Interpretation of experimental data requires comparison with theoretical prediction, but generally theories are most easily developed in the frequency/k vector space rather than in time/physical space
- On of tehe reason for that is that in the omega/k space mathematical operator like time or spatial derivatives are represented by simple moltiplication for appropriate frequency or k
- Thus before proceeding with the lecture we would like to make some remarks on Fourier trasform giving detailes on the applicability to physical data
- Direct and inverse Fourier transform are defined as a function of frequency f in the following way
- In real case, we will deal with discrete signals with N points in time and a sampling frequency of 1/Delta t. In the case of discrete signals, the corresponding transform is called the Discrete Fourier transform, direct and inverse
- Actually, in the case of discrete signals a theorem exhists which ensure that, if a function has Fourier coefficient equal to zero for frequency greater than a cut-off, than it is fully specified by a discrete series with equispaced values with intervals not exceeding one half of the inverse of the cut-off
- We define consequently the Nyquist frequency, as one half of the inverse of the sampling rate. It represents the last properly resolved frequency in a signal.
- Different theories may be applied to the FT and we cite here only two of them
- The convolution theorem, which ensure that the convolution operator of two functions is transformed in a moltiplication operator between the corresponding fourier components. This theorem is used for example in the treatment of the non-linearities in the Na-St equations or, as we will see later is at the basis of the wavelet transform
- The Rayleigh’s theorem which ensure that the integral of the square modules of the function in the real and fourier domain are equal one to the other. It substantially states an energy conservation principle between real and fourier representation
- The discreteness of the signals requires some trick in order to be properly treated
- For example the presence of frequency higher than the Nyquist frequency may lead to spurious frequency, as the case of this 9 kHz cosine, which if sampled at 10 kHz, manifests itself as a spurious 1 kHz oscillations
- Furthermore actual signals are acquired for a limited period T. From the mathematical point of view this is equivalent to the convolution with a boxcar-like function with support [0,T]
- Thus the Fourier transform of the function is mupltiplied with the corresponding Fourier trasnform of the boxcar which is a function of form sin(x) over x which is not well localized in frequency but cause some leaking from one frequency bin to the adjacents ones.
- A solution is provided by multipling the function with some window function which reduces the lobes. And here we have the typical example of an Hanning window
- It should be known, from basic knowledge of statistical mechanics, that a random process x(t) is completely determined by the knowledge of its moments, i.e. averages over the distribution functions
- On this basis we define the Auto-correlation function, which is the second order momentum of the distribution, and the corresponding quantity for the fluctuation of the signal named Autocovariance function
- Another useful definition is the Auto-correlation coefficient, where normalization is done on the value of the autocorrelation at zero lag.
- We can also write the corresponding formula for digitized signals for the autocovariance
What we can learn from an autocorrelation function. We can measure the degree of correlation of a turbulence field, by properly defining the autocorrelation time as the lag where the autocorrelation is reduced to 1/e….. This is a typical quantities, which is easy to measure which can be compared with turbulence theories
- Once we define the autocorrelation we can also compute its Fourier transform, named Power spectrum
- It describe how the power of the signal is distributed in the frequency spectrum
- Dealing with finite records of signals, it can be shown that the power spectrum is the limit of the periodogram of the signals, which is the average of the square of the FT
- Numerically the spectral estimator of the power spectrum is computed dividing the signals into M slices, treated as indipendent realization, computing the DFT and averaging, frequency by frequency the square of the DFT.
- What physically can be learned from a power spectrum is the presence of modes in the plasma, which results in peaks of the power spectrum
- This is an example taken from doppler reflectometer in a tokamak plasma, where a clear peaks emerge from the spectrum
- To disentangle instabilities which create this peak, we have to
complement the information. And here we have a typical
example. In order to state that this peak, is induced by the
presence of a Geodesic Acoustic Mode. This modes arise as a
consequence of the linear coupling of an
$(m,n)=(0,0)$ electrostatic potential with an$(m,n)=(1,0)$ pressure perturbation. It is found to scale with the ions sound velocity normalized to the major radius. The observation of the scales allow the identification of the origin of the mode
- Another example arises from the computation of the spectrogram. This is based on the Short-Time-Fourier-Transform (i.e. we divide the signals in slices of small extent and compute the corresponding power spectrum). It describe how the power spectral density distributes in time and frequency
- Again the peaks which are observed at higher frequency, in order to be properly interpreted have to be compared with other quantities. In this case we observe a dependence of the value of the frequency from the density, suggesting these are some kind of Alfvenic modes
- The availability of distributed measurements allow acces to the spatial structure of the fluctuations
- In analogy with the definition for single point, given two functions we can define the Cross-correlation function, the Cross-covariance function and the Cross-correlation coefficient function
- Still with analogy to the 1 point case we can define the corresponding quantities in the discrete case
- In analogy to one point measurement, we define the Cross-power spectrum as the Fourier transform of the cross-correlation function
- Dealing with real function, for the studies of cross-power spectrum we can limit to positive frequencies
- The cross-power spectrum is complex-valued and we define the Phase spectrum as the imaginary part of the Cross-power spectrum
- We can define the coherence, which assumes values between [0,1]
- Again for finite discrete series, the cross-power spectrum is built up as average over different slices, or different realizations
- The cross-power analysi may be applied also to measurements of different quantities in the same location
- This is an example using langmuir probes which gives information on density and potential in the same nominal position and these are the derived phase spectrum and coherence
- First we must state that the signals have to be well correlated in order to give a meaningful phase spectrum information
- We have highlighted two different regions where electron density and electron pressure have different phase relation moving from pi half to 0
- The comparison with theories (specifically linear theory) tell us that the two frequency ranges are dominated by different instabilities: in particular we have an interchange dominated region the interchange turbulence results as a combination effect of pressure gradient and bad curvature of the magnetic field and another region which is drift-dominated, i.e. dominated by turbulence whose free energy source is the gradient of the density and characterized by finite parallel wavelenght
- If a reasonably dispersion relation exhists between k and f, the phase spectrum may be used to derive the k spectrum for separated measurements
- This is due that the Fourier transform of the function g(r,t) reduced to a function of the frequency only
- From a practical point of view, having two measurements, the phase spectrum at a given frequency which define the phase shift between the Fourier components at the considered frequency is a function of k moltiplied by the distance d
- If the measurements are azimuthally distributed we can also derived the mode number
- A note should be given: the separation of the probe should not be greater than a wavelenght (equivalently the maximum resolvable k is pi/Delta x), less than a correlation length, to ensure a good coherence between the signals but large enought to give measurable phase difference
- An example of the application of the cross-power spectrum is the determination of the particle flux
- The particle flux is defined as the ensemble average of the v_r and n, where as fluctuating velocities we consider the fluctuating electric drift velocity
- This can be written as the cross-correlation at 0 lag and the corresponding Fourier transformation is the following where we have used the fact that the real part of the cross spectrum is odd in frequency whereas the imaginary is even
- In the case of quasi-static approximation, and assuming an almost deterministic phase dispersion relation the Flux can be computed as the Imaginary part of the cross-spectrum between density and potential moltiplied by the k vector
- This can also be computed in the discrete domain of realistic signals
- This an example of the computation where we can observe which are the frequency which dominates the particle transport. If an almost deterministic phase relation may be deriven as in this case, we can also determine which are the fluctuation responsible for the loss of particles
- An interesting comparison arises whenever we compute a different quantities, still using the cross-power spectrum
- We are going to calculate the reynolds stress, which enters in the equation of momentum flux generation
- Experimentally this can be done for example using probes
- The comparison between the frequency resolved reynolds stress and the frequency resolved particle flux reveal how different scales are involved in the momentum flux and in the particle flux dynamics. The deterministic phase dispersion relation allow also to disentangle the different spatial scales involved in the two processes
- For turbulent medium, with different wave vectors corresponding to the same frequency we need to compute thw Wave number-frequency spectrum
- It is defined in analogies with previous observation as the Fourier transform of the spatial-time cross-correlation
- There is no need to have a full spatial information, with just to points we can relie on the statistics and on different realization and compute the k-omega spectrum according to this formula. This mean that we create a sort of histogram of the cross-power spectrum taking into account that for differen realization different k are determined for different frequencies
- An example of the applicatin is shown in this figure where S(k,omega) is compared in L and H mode discharges in NSTX. We observe that all the fluctuations moves approximately with the same phase velocities, we observe a change in the spatial spectral content between the L an H mode.
- Integrating in frequency we can thus obtain a full spectrum in k of the fluctuations which for examples shows an increase in the relative power of larger scales in H-mode and a faster decays at smaller scales
- The limit of the Fourier decomposition is that it uses trigonometric function as orthogonal basis which are not localized in time but oscillates forever
- If we want a method for maintaining the locality of the information we can use the Wavelet trasnform. Here an example of a typical wavelet, which is a zero-mean function localized both in frequency and in time
- the continuous wavelet transform generates wavelet coefficients as the convolution of the studied function with atoms generated by translating and stretching the mother wavelet
- From the wavelet transform we can build the wavelet cross-power spectrum, and consequently the wavelet phase spectrum
- Here for example we observe an abrupt change of phase between density and potential fluctuation
- This change of phase, which has consequences on transport as we know from the aforementioned calculation is associated to an induced increasing of the ExB shear obtained by a biasing experiment, where an electrode polarized with respect to the vessel of the machine has been used to generate a JxB force which could spin-up the plasma
- Wavelet coefficient exhibits the same scaling properties of the fluctuation of the signals
- They can be used in turbulence analysis for multri-fractal approach, i.e. for the studies of the scaling properties of the fluctuation. For a pure self-similar system, where the energey is equally distributed in the process of energy cascade, the PDF of the normalized coefficients should collapse to a single PDF
- The increasing of the tails of these distributins at smaller time scales is typical of the intermittency process, which cause localization in time and spatial scale of the energy of the system. i.e. it is due to sporadic intense fluctuation
- We can use methods based on the wavelet which allows the identification in time and frequency of those intense fluctuation pertaining to the tails of the PDF. Then we can derive the typical shape of these fluctuations using the CA technique, i.e. averaging time windows centered around the occurrence of these sporadic events
- This is a result, where we have used a combination of probes measuring the potential using as trigger the appearence of an intermittent structures in the more inserted probe. The velocity pattern is then reconstructed, revealing as these blobs are vortices
Presentation on current activities (both scientific and pedagocal) and plans for the future both scientific and pedagogical
- Personal research interest [1 Slides]
- Turbulence & Flows in magnetized plasma
- Statistical characterization of electromagnetic turbulence and of emerging coherent structures
- Spontaneously developed Helical plasmas
- Issue:Turbulence & Flows [2 Slides]
- Role of electrostatic fluctuations in driving perpendicular flow at the edge of an RFP (PRL/NF/PPCF)
- Transport reduction induced by active modification of sheared flow (PPCF)
- Issue: Turbulence/Intermittency & lack of Self-similarity [2 Slides ]
- Evidence of intermittency observed as lack-of-self-similarity + memory processes through laminar times
- Inconsistency with SOC processes (2 PRL)
- Issue: Coherent structures & filaments [2 Slides]
- Transport induced by coherent structures
- Filaments in 3 machines plus planned experiment
- Issue: Spontaneous Helical Plasmas [2 Slides]
- Helical flow & dynamo associated (mappa del flow & histogramma)
- Role of the flow in the high density collapse of an RFP discharges (pattern di velocita ad alta densita)
- Issue: pedagogical [1 Slides]
- 2 Batchelor Thesis plus 1 Master thesis:
- Activities on the Filaments (ASDEX & RFX)
- Activities on the role of m=0 island in modyfing the properties of edge region
- Activites on the phase relation between velocity and
magnetic perturbation during high density discharges
- Issue: Future activities [1 Slides]
- Fast ions & turbulence redistribution of fast particles
- Reconnection (sinergy with plasma science)
- Momentum (also considering the effect of magnetic perturbation on the momentum)
- Issue: Future activities [1 Slides]
- i dati per il plot dell’elettrodo devono essere letti cosi
Flux dFlux k dk flux El dFlux El k El dk El