Level crossings and excess times due to a super-position of uncorrelated exponential pulses

A well-known stochastic model for intermittent fluctuations in physical systems is investigated. The model is given by a super-position of uncorrelated exponential pulses, and the degree of pulse overlap is interpreted as an intermittency parameter. Expressions for excess time statistics, that is, the rate of level crossings above a given threshold and the average time spent above the threshold, are derived from the joint distribution of the process and its derivative. Limits of both high and low intermittency are investigated and compared to previously known results. In the case of a strongly intermittent process, the distribution of times spent above threshold is obtained analytically. This expression is verified numerically, and the distribution of times above threshold is explored for other intermittency regimes. The numerical results compare favorably to known results for the distribution of times above the mean threshold for an Ornstein-Uhlenbeck process. This contribution generalizes the excess time statistics for the stochastic model which find applications in a wide diversity of natural and technological systems.Comment: 40 pages, 26 figures. Longer version of arXiv:1604.0406

Statistical properties of intermittent fluctuations in the boundary of fusion plasmas

Paper I, II, III and V are not available in Munin. Paper I: Theodorsen, A., Garcia, O.E., Horacek, J, Kube, R & Pitts, R.A. (2016). Scrape-off layer turbulence in TCV: evidence in support of stochastic modelling. Available in Plasma Physics and Controlled Fusion, 58(4), 044006 (12pp). Paper II: Theodorsen, A., Garica, O.E. & Rypdal, M. (2017). Statistical properties of a filtered Poisson process with additive random noise: distributions, correlations and moment estimation. Available in Physica Scripta, 92(5), 054002. Paper III: Theodorsen, A., Garica, O.E., Kube, R., LaBombard, B & Terry, J.L. (2017). Relationship between frequency power spectra and intermittent, large-amplitude bursts in the Alcator C-Mod scrape-off Layer. Available in Nuclear Fusion, 57, 114004 (7pp). Paper V: Theodorsen, A. & Garcia, O.E. (2018). Probability distribution functions for intermittent scrape-off layer plasma fluctuations. Available in Plasma Physics and Controlled Fusion, 034006 (14pp).Fluctuation-induced plasma–wall interactions is a major concern for the next generation, high duty-cycle magnetic confinement fusion devices. The turbulence is generated in the outboard midplane transition region between the confined core plasma and the scrape-off layer where magnetic field lines intersect material walls. Here, filaments of hot and dense plasma, elongated in the field direction, detach from the main plasma and move radially outwards, driven by interchange motion. These filaments cause enhanced plasma–wall interactions compared to the level estimated by only considering time-averaged plasma parameters, reduce the efficiency of radio frequency wave heating and is likely related to the empirical discharge density limit. When measured as a time series from a stationary point (either as ion saturation current from electrical probes probes or as emitted light intensity from gas puff imaging), the statistical properties of the turbulent fluctuations in the scrape-off layer are robust across devices, confinement modes and plasma parameters. The highly intermittent fluctuations exhibit skewed and flattened probability density functions and power spectra that are flat for low frequencies and have a power-law tail for high frequencies. Conditional averaging reveals that large-amplitude structures have a sharp, exponential rise and a slower, exponential decay. Both the peak amplitudes of these structures and the waiting time between them are exponentially distributed. In this thesis, a stochastic model describing the time series as a superposition of uncorrelated, two-sided exponential pulses with exponentially distributed amplitudes arriving according to a Poisson process is analysed and its assumptions and predictions are compared with measurement data. This model is consistent with all the above statistical properties. The predictive capabilities of the model are improved by deriving expressions for the rate of threshold crossings and the time the signal spends above a given threshold level. The effects of additive noise and different amplitude distributions are also considered. Parameter estimation from moments, probability density functions and characteristic functions is examined using Monte-Carlo simulations. The model predictions are favorably compared to measurement data from experiments on the TCV and Alcator C-Mod devices

Statistical properties of a filtered Poisson process with additive random noise: Distributions, correlations and moment estimation

Filtered Poisson processes are often used as reference models for intermittent fluc- tuations in physical systems. Such a process is here extended by adding a noise term, either as a purely additive term to the process or as a dynamical term in a stochastic differential equation. The lowest order moments, probability density function, auto-correlation function and power spectral density are derived and used to identify and compare the effects of the two different noise terms. Monte-Carlo studies of synthetic time series are used to investigate the accuracy of model pa- rameter estimation and to identify methods for distinguishing the noise types. It is shown that the probability density function and the three lowest order moments provide accurate estimations of the parameters, but are unable to separate the noise types. The auto-correlation function and the power spectral density also provide methods for estimating the model parameters, as well as being capable of identifying the noise type. The number of times the signal crosses a prescribed threshold level in the positive direction also promises to be able to differentiate the noise type.Comment: 34 pages, 25 figure

A deconvolution method for reconstruction of data time series from intermittent systems

In this manuscript, we will investigate the deconvolution method for recovering pulse arrival times and amplitudes using synthetic data. For the deconvolution procedure to have hope of recovering amplitudes and arrivals, the average waiting time between events must be at least 10 times the time step.Comment: This manuscript currently only contains a few results required for arXiv:1802.0505

Auto-correlation function and frequency spectrum due to a super-position of uncorrelated exponential pulses

Link to publishers version: 10.1063/1.4978955The auto-correlation function and the frequency power spectral density due to a super-position of uncorrelated exponential pulses are considered. These are shown to be independent of the degree of pulse overlap and thereby the intermittency of the stochastic process. For constant pulse duration and a one-sided exponential pulse shape, the power spectral density has a Lorentzian shape which is flat for low frequencies and a power law at high frequencies. The algebraic tail is demonstrated to result from the discontinuity in the pulse function. For a strongly asymmetric two-sided expo- nential pulse shape, the frequency spectrum is a broken power law with two scaling regions. In the case of a symmetric pulse shape, the power spectral density is the square of a Lorentzian function. The steep algebraic tail at high frequencies in these cases is demonstrated to follow from the discontinuity in the derivative of the pulse function. A random distribution of pulse durations is shown to result in apparently longer correlation times but has no influence on the asymptotic power law tail of the frequency spectrum. The effect of additional random noise is also discussed, leading to a flat spectrum for high frequencies. The probability density function for the fluctuations is shown to be independent of the distribution of pulse durations. The predictions of this model describe the variety of auto-correlation functions and power spectral densities reported from experimental measurements in the scrape-off layer of magnetically confined plasmas

Dirac comb and exponential frequency spectra in nonlinear dynamics

An exponential frequency power spectral density is a well known property ofmany continuous time chaotic systems and has been attributed to the presence of Lorentzian-shaped pulses in the time series of the dynamical variables. Here a stochastic modelling of such fluctuations are presented, describing these as a super-position of pulses with fixed shape and constant duration. Closed form expressions are derived for the lowest order moments, auto-correlation function and frequency power spectral density in the case of periodic pulse arrivals and a random distribution of pulse amplitudes. In general, the spectrum is a Dirac comb located at multiple sof the periodicity time and modulated by the pulse spectrum. Of the effects considered, only deviations from periodicity remove the Dirac Comb, and do so rapidly.Randomness in the pulse arrival times is investigated by numerical realizations of the process and the results are discussed in the context of the Lorenz system.Comment: 30 pages, 9 figure

Intermittent electron density and temperature fluctuations and associated fluxes in the Alcator C-Mod scrape-off layer

The Alcator C-Mod mirror Langmuir probe system has been used to sample data time series of fluctuating plasma parameters in the outboard mid-plane far scrape-off layer. We present a statistical analysis of one second long time series of electron density, temperature, radial electric drift velocity and the corresponding particle and electron heat fluxes. These are sampled during stationary plasma conditions in an ohmically heated, lower single null diverted discharge. The electron density and temperature are strongly correlated and feature fluctuation statistics similar to the ion saturation current. Both electron density and temperature time series are dominated by intermittent, large-amplitude burst with an exponential distribution of both burst amplitudes and waiting times between them. The characteristic time scale of the large-amplitude bursts is approximately 15{\mu}s. Large-amplitude velocity fluctuations feature a slightly faster characteristic time scale and appear at a faster rate than electron density and temperature fluctuations. Describing these time series as a superposition of uncorrelated exponential pulses, we find that probability distribution functions, power spectral densities as well as auto-correlation functions of the data time series agree well with predictions from the stochastic model. The electron particle and heat fluxes present large-amplitude fluctuations. For this low-density plasma, the radial electron heat flux is dominated by convection, that is, correlations of fluctuations in the electron density and radial velocity. Hot and dense blobs contribute approximately 6% of the total fluctuation driven heat flux

Scrape-off layer turbulence in TCV: Evidence in support of stochastic modelling.

Manuscript. Published version available in Plasma Physics and Controlled Fusion, vol. 58, no. 4Intermittent fluctuations in the TCV scrape-off layer have been investigated by analysing long Langmuir probe data time series under stationary conditions, allowing calculation of fluctuation statistics with high accuracy. The ion saturation current signal is dominated by the frequent occurrence of large-amplitude bursts attributed to filament structures moving through the scrape-off layer. The average burst shape is well described by a double-exponential wave-form with constant duration, while the waiting times and peak amplitudes of the bursts both have an exponential distribution. Associated with bursts in the ion saturation current is a dipole-shaped floating potential structure and radially outwards directed electric drift velocity and particle flux, with average peak values increasing with the saturation current burst amplitude. The floating potential fluctuations have a normal probability density function while the distributions for the ion saturation current and estimated radial velocity have exponential tails for large fluctuations. These findings are discussed in the light of prevailing theories for filament motion and a stochastic model for intermittent scrape-off layer plasma fluctuations

Fluctuation statistics in the scrape-off layer of Alcator C-Mod

We study long time series of the ion saturation current and floating potential, sampled by Langmuir probes dwelled in the outboard mid-plane scrape off layer and embedded in the lower divertor baffle of Alcator C-Mod. A series of ohmically heated L-mode plasma discharges is investigated with line-averaged plasma density ranging from n_e/n_G = 0.15 to 0.42, where n_G is the Greenwald density. All ion saturation current time series that are sampled in the far scrape-off layer are characterized by large-amplitude burst events. Coefficients of skewness and excess kurtosis of the time series obey a quadratic relationship and their histograms coincide partially upon proper normalization. Histograms of the ion saturation current time series are found to agree well with a prediction of a stochastic model for the particle density fluctuations in scrape-off layer plasmas. The distribution of the waiting times between successive large-amplitude burst events and of the burst amplitudes are approximately described by exponential distributions. The average waiting time and burst amplitude are found to vary weakly with the line-averaged plasma density. Conditional averaging reveals that the radial blob velocity, estimated from floating potential measurements, increases with the normalized burst amplitude in the outboard mid-plane scrape-off layer. For low density discharges, the conditionally averaged waveform of the floating potential associated with large amplitude bursts at the divertor probes has a dipolar shape. In detached divertor conditions the average waveform is random, indicating electrical disconnection of blobs from the sheaths at the divertor targets.Comment: 45 pages, 20 figure