Control system-plasma synchronization and naturally occurring edge localized modes in a tokamak

Abstract

Edge Localised Modes (ELMs) naturally occur in tokamak plasmas in high confinement mode. We find in ASDEX Upgrade that the plasma can transition into a state in which the control system field coil currents, required to continually stabilize the plasma, continually oscillate with the plasma edge position and total MHD energy. These synchronous oscillations are one-to-one correlated with the occurrence of natural ELMs; the ELMs all occur when the control system coil current is around a specific phase. This suggests a phase synchronous state in which nonlinear feedback between plasma and control system is intrinsic to natural ELMing, and in which the occurrence time of a natural ELM is conditional on the phase of the control system field coil current.

Timing Terminators: Forecasting Sunspot Cycle 25 Onset

“…In signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H[u(t)] (Bracewell, 2000;Pikovsky et al, 2002). This linear operator is given by convolution with the function 1/(πt):…”

“…Noting that some authors choose to differ on the choice of ± sign in equation ( 1), we will adopt the −1 convention in defining the transform, such that φ decreases with time, a feature that more easily permits straightforward visual comparison with the decaying sunspot number timeseries; in either case, H [H[u(t)]] = −u(t). It also follows from equation (3) that ω = −dφ/dt, so slope of the changing phase with time has significance as a "localized" or "instantaneous" frequency of the fluctuating quantity (Bracewell, 2000). A second useful feature of the Hilbert phase is in the phase coherence of two time series: if edges/events in one time series occur at constant phase in another, the two are "phase locked" or "synchronized" (Pikovsky et al, 2002;Rial, Oh, and Reischmann, 2013;Chapman et al, 2018a,b).…”

Quantifying the Solar Cycle Modulation of Extreme Space Weather

“…While defined for an arbitrary time series, the analytic signal will only give a physically meaningful decomposition of the original time series if the instantaneous frequency ω(t)=dφ(t)/dt remains positive (Boashash, 1992). We therefore need to remove fast fluctuations and, for a positive definite signal such as the daily sunspot number, a background trend (see, e.g., Boashash, 1992; Chapman, Lang et al 2018). Before performing the Hilbert transform we performed a 180 day moving average and obtained a slowly varying trend by performing a robust local linear regression which down weights outliers (“rlowess”) using a T B =40 year window.…”

“…We express this time series S ( t ) in terms of a time‐varying amplitude A ( t ) and phase φ ( t ) by obtaining its analytic signal (Boashash, 1992; Gabor, 1946) such that the real part of this signal is S ( t ) and the imaginary part is obtained such that where H ( t ) is the Hilbert transform of S ( t ). This is a standard approach that is used to test for synchronization (e.g., Chapman, Lang, et al 2018) and for amplitude‐frequency relationships (Palŭs & Notovná, 1999). Here it is used to provide a mapping between time and signal phase that converts the (variable) duration of each solar cycle into a corresponding uniform phase interval, from 0 to 2 π .…”

Overlapping Magnetic Activity Cycles and the Sunspot Number: Forecasting Sunspot Cycle 25 Amplitude

“…While defined for an arbitrary time series, the analytic signal will only give a physically meaningful decomposition of the original time series if the instantaneous frequency ω(t) = dφ(t)/dt remains positive (Boashash, 1992). For a positive-definite signal such as the monthly sunspot number we therefore need to remove a background trend (see Chapman et al (2018) for an example, and further discussions in Pikovsky et al (2002), Boashash (1992) and Huang et al (1998)). We obtained a slowly-varying trend by performing a robust local linear regression which down-weights outliers ("rlowess") using Matlab's smooth function with a 40 year window.…”