Zonal flows and transient dynamics of the L-H transition

Stochastic Dynamics of Fusion Low-to-High Confinement Mode (L-H) Transition: Correlation and Causal Analyses Using Information Geometry

We investigate the stochastic dynamics of the prey-predator model of the Low-to-High confinement mode (L-H) transition in magnetically confined fusion plasmas. By considering stochastic noise in the turbulence and zonal flows as well as constant and time-varying input power Q, we perform multiple stochastic simulations of over a million trajectories using GPU computing. Due to stochastic noise, some trajectories undergo the L-H transition while others do not, leading to a mixture of H-mode and dithering at a given time and/or input power. One of the consequences of this is that H-mode characteristics appear at a smaller input power QQc as a second peak. The coexisting H-mode and dithering near Q=Qc leads to a prominent bimodal PDF with a gradual L-H transition rather than a sudden transition at Q=Qc and uncertainty in the input power. Also, a time-dependent input power leads to increased variability (dispersion) in stochastic trajectories and a more prominent bimodal PDF. We provide an interpretation of the results using information geometry to elucidate self-regulation between zonal flows, turbulence, and information causality rate to unravel causal relations involved in the L-H transition.

How the birth and death of shear layers determine confinement evolution: from the L → H transition to the density limit

Electric field profile structure-especially its shear-is a natural order parameter for the edge plasma, and characterizes confinement regimes ranging from the H-mode (Wagner et al. 1982 Phys. Rev. Lett. 49, 1408-1412 (doi:10.1103/PhysRevLett.49.1408)) to the density limit (DL) (Greenwald et al. 1988 Nucl. Fusion 28, 2199-2207 (doi:10.1088/0029-5515/28/12/009)). The theoretical developments and lessons learned during 40 years of H-mode studies (Connor & Wilson 1999 Plasma Phys. Control. Fusion 42, R1-R74 (doi:10.1088/0741-3335/42/1/201); Wagner 2007 Plasma Phys. Control. Fusion 49, B1-B33 (doi:10.1088/0741-3335/49/12b/s01)) are applied to the shear layer collapse paradigm (Hong et al. 2017 Nucl. Fusion 58, 016041 (doi:10.1088/1741-4326/aa9626)) for the onset of DL phenomena. Results from recent experiments on edge shear layers and DL phenomenology are summarized and discussed in the light of L [Formula: see text] H transition physics. The theory of shear layer collapse is then developed. We demonstrate that shear layer physics captures both the well known current (Greenwald) scaling of the DL (Greenwald 2002 Plasma Phys. Control. Fusion 44, R27-R53 (doi:10.1088/0741-3335/44/8/201); Greenwald et al. 2014 Phys. Plasmas 21, 110501 (doi:10.1063/1.4901920)), as well as the emerging power scaling (Zanca, Sattin, JET Contributors 2019 Nucl. Fusion 59, 126011 (doi:10.1088/1741-4326/ab3b31)). The derivation of the power scaling theory exploits an existing model, originally developed for the L [Formula: see text] H transition (Diamond, Liang, Carreras, Terry 1994 Phys. Rev. Lett. 72, 2565-2568 (doi:10.1103/PhysRevLett.72.2565); Kim & Diamond 2003 Phys. Rev. Lett. 90, 185006 (doi:10.1103/PhysRevLett.90.185006)). We describe the enhanced particle transport events that occur following shear layer collapse. Open problems and future directions are discussed. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

An overview of dynamical methods for studying transitions between states in sheared plasma flows

The self-organization of structures in a tokamak plasma as it undergoes an [Formula: see text]-mode transition shows properties similar to simpler shear flow configurations. We will describe recent dynamical studies of plasma shear flows, including the idea of tracking the edge of chaos that separates two bistable states, computing the nonlinear minimal seed that can lead to turbulence, finding the attractor solution on the edge and seeing how starting from this solution we can understand the stability of relative period orbits that permeate the turbulent basin of attraction. We present a modus operandi developed for these simple configurations that can be adapted to understand the [Formula: see text]-mode transition. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

Turbulence and E × B flow correlations across the L-H transition in DIII-D deuterium and hydrogen plasmas

The isotope dependence of the low- to high confinement-mode (L- to H-mode) transition power threshold P_LH presents significant challenges for the initial (non-nuclear) hydrogen operations phase of the international thermonuclear experimental reactor (ITER). Here, we examine the isotope dependence of turbulence and E × B flow correlation properties in the L-mode edge plasma, leading up to the L-H transition. At marginal auxiliary power (near P_LH), turbulence is initially suppressed periodically during limit cycle oscillations (LCO) that precede the transition to sustained H-mode confinement. We present evidence that the long-range (toroidal) correlation of the E × B edge plasma flow across the LCO phase is much weaker in hydrogen than in deuterium in the DIII-D tokamak, congruent with the higher threshold power P_LH required to access LCO and H-mode in hydrogen. Concomitantly, the time required to initially quench edge turbulence via localized edge E × B flow shear is significantly longer in hydrogen (1-1.5 ms) than in deuterium (approx. 100 µs). No toroidal long-range correlation of the turbulence amplitude is observed, in agreement with expectations based on the relatively short poloidal turbulence correlation length. Radial edge turbulence and flow correlation lengths are longer in deuterium than in hydrogen plasmas as one would expect from 'naïve' gyro-Bohm isotope transport scaling, despite the substantially higher thermal flux across the last closed flux surface in hydrogen before the L-H transition. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

A stochastic model of edge-localized modes in magnetically confined plasmas

Magnetically confined plasmas are far from equilibrium and pose considerable challenges in statistical analysis. We discuss a non-perturbative statistical method, namely a time-dependent probability density function (PDF) approach that is potentially useful for analysing time-varying, large, or non-Gaussian fluctuations and bursty events associated with instabilities in the low-to-high confinement transition and the H-mode. Specifically, we present a stochastic Langevin model of edge-localized modes (ELMs) by including stochastic noise terms in a previous ODE ELM model. We calculate exact time-dependent PDFs by numerically solving the Fokker-Planck equation and characterize time-varying statistical properties of ELMs for different energy fluxes and noise amplitudes. The stochastic noise is shown to introduce phase-mixing and plays a significant role in mitigating extreme bursts of large ELMs. Furthermore, based on time-dependent PDFs, we provide a path-dependent information geometric theory of the ELM dynamics and demonstrate its utility in capturing self-regulatory relaxation oscillations, bursts and a sudden change in the system. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

H-mode dithering phase studies on ST40

The dithering H-mode phase, characterized by oscillations, is generally observed at input power values close to the L-H transition power threshold and low plasma collisionalities (low electron density and/or high plasma temperature). Measurements to characterize the dithering phase are presented for the low aspect ratio, high magnetic field tokamak, ST40. The dithering phase oscillation frequency is observed between 400 and 800 Hz and demonstrates an inverse relationship with core plasma density. Dithering phase H-modes are documented across a nonlinear, low-density power threshold operational space, with signature low- and high-density branches. The minimum power threshold for dithering H-mode access is measured at a core, line average electron density of 4.7(±0.5) × 10¹⁹ m^-3, close to a predicted value of 4.1(±0.4) × 10¹⁹ m^-3 from multi-machine studies. ASTRA calculated values of power coupled to the ion species, at the dithering H-mode transition, exhibit a similar nonlinear dependence on density. This analysis points to the important contribution of the ion thermal channel to the L-H phase transition. The low-frequency plasma density and D-alpha dithers appear to be accompanied by sudden bursts of magnetohydrodynamic (MHD) activity. A simple model is tested to demonstrate a possible scenario of self-regulation among turbulence, zonal flows, pressure (density) gradient and MHD activities. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

H-mode transition in the TJ-II stellarator plasmas

Since the first H-mode transitions were observed in TJ-II plasmas in 2008, an extensive experimental effort has been done aiming at better physics understanding of confinement transitions. In this article, an overview of the main findings related to the L-H transition in TJ-II is presented including how the radial electric field is driven, what are the possible mechanisms for turbulence suppression and what are the related temporal and spatial scales that can impact the transition. The trigger of the L-H transition in TJ-II plasmas is found to be more correlated with the development of fluctuating [Formula: see text] flows than with steady-state [Formula: see text] effects, pointing to the role played by zonal flows in mediating the transition. Experimental evidence supporting the predator-prey relationship between turbulence and flows as the basis for the L-H transition, found for the first time in TJ-II, reinforces this conclusion. Besides, the reduction in the turbulent transport at the transition is detected at the barrier region but also in a wider radial range with weak or even zero [Formula: see text] flow shear, which points to other mechanisms beyond the turbulence suppression by local sheared flows. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

Self-Sustained Divertor Oscillation Driven by Magnetic Island Dynamics in Torus Plasma

A new type of self-sustained divertor oscillation is discovered in the Large Helical Device stellarator, where the peripheral plasma is detached from material diverters by means of externally applied perturbation fields. The divertor oscillation is found to be a self-regulation of an isolated magnetic field structure (the magnetic island) width induced by a drastic change in a poloidal inhomogeneity of the plasma radiation across the detachment-attachment transitions. A predator-prey model between the magnetic island width and a self-generated local plasma current (the bootstrap current) is introduced to describe the divertor oscillation, which successfully reproduces the experimental observations.

Information Geometry, Fluctuations, Non-Equilibrium Thermodynamics, and Geodesics in Complex Systems

Information theory provides an interdisciplinary method to understand important phenomena in many research fields ranging from astrophysical and laboratory fluids/plasmas to biological systems. In particular, information geometric theory enables us to envision the evolution of non-equilibrium processes in terms of a (dimensionless) distance by quantifying how information unfolds over time as a probability density function (PDF) evolves in time. Here, we discuss some recent developments in information geometric theory focusing on time-dependent dynamic aspects of non-equilibrium processes (e.g., time-varying mean value, time-varying variance, or temperature, etc.) and their thermodynamic and physical/biological implications. We compare different distances between two given PDFs and highlight the importance of a path-dependent distance for a time-dependent PDF. We then discuss the role of the information rate Γ=dLdt and relative entropy in non-equilibrium thermodynamic relations (entropy production rate, heat flux, dissipated work, non-equilibrium free energy, etc.), and various inequalities among them. Here, L is the information length representing the total number of statistically distinguishable states a PDF evolves through over time. We explore the implications of a geodesic solution in information geometry for self-organization and control.

Transport Barrier Triggered by Resonant Three-Wave Processes Between Trapped-Particle-Modes and Zonal Flow

We address the mechanisms underlying low-frequency zonal flow generation in a turbulent system through the parametric decay of collisionless trapped particle modes and its feedback on the stabilization of the system. This model is in connection with the observation of barrier transport in reduced gyrokinetic simulations (A. Ghizzo et al., Euro. Phys. Lett. 119(1), 15003 (2017)). Here the analysis is extended with a detailed description of the resonant mechanism. A key role is also played by an initial polarisation source that allows the emergence of strong initial shear flow. The parametric decay leads to the growth of a zonal flow which differs from the standard zero frequency zonal flow usually triggered by the Reynolds stress in fluid drift-wave turbulence. The resulting zonal flow can oscillate at low frequency close to the ion precession frequency, making it sensitive to strong amplification by resonant kinetic processes. The system becomes then intermittent. These new findings, obtained from numerical experiments based on reduced semi-Lagrangian gyrokinetic simulations, shed light on the underlying physics coming from resonant wave-particle interactions for the formation of transport barriers. Numerical simulations are based on a Hamiltonian reduction technique, including magnetic curvature and interchange turbulence, where both fastest scales (cyclotron and bounce motions) are gyro-averaged.

Development of a Gyrokinetic Particle-in-Cell Code for Whole-Volume Modeling of Stellarators

We present initial results in the development of a gyrokinetic particle-in-cell code for the whole-volume modeling of stellarators. This is achieved through two modifications to the X-point Gyrokinetic Code (XGC), originally developed for tokamaks. One is an extension to three-dimensional geometries with an interface to Variational Moments Equilibrium Code (VMEC) data. The other is a connection between core and edge regions that have quite different field-line structures. The VMEC equilibrium is smoothly extended to the edge region by using a virtual casing method. Non-axisymmetric triangular meshes in which triangle nodes follow magnetic field lines in the toroidal direction are generated for field calculation using a finite-element method in the entire region of the extended VMEC equilibrium. These schemes are validated by basic benchmark tests relevant to each part of the calculation cycle, that is, particle push, particle-mesh interpolation, and field solver in a magnetic field equilibrium of Large Helical Device including the edge region. The developed code also demonstrates collisionless damping of geodesic acoustic modes and steady states with residual zonal flow in the core region.

Direct Measurement of a Toroidally Directed Zonal Flow in a Toroidal Plasma

Zonal flow appears in toroidal, magnetically confined plasmas as part of the self-regulated interaction of turbulence and transport processes. For toroidal plasmas having a strong toroidal magnetic field, the zonal flow is predominately poloidally directed. This Letter reports the first observation of a zonal flow that is toroidally directed. The measurements are made just inside the last closed flux surface of reversed field pinch plasmas that have a dominant poloidal magnetic field. A limit cycle oscillation between the strength of the zonal flow and the amplitude of plasma potential fluctuations is observed, which provides evidence for the self-regulation characteristic of drift-wave-type plasma turbulence. The measurements help advance understanding and gyrokinetic modeling of toroidal plasmas in the pursuit of fusion energy.

Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows

We report the time-evolution of Probability Density Functions (PDFs) in a toy model of self-organised shear flows, where the formation of shear flows is induced by a finite memory time of a stochastic forcing, manifested by the emergence of a bimodal PDF with the two peaks representing non-zero mean values of a shear flow. Using theoretical analyses of limiting cases, as well as numerical solutions of the full Fokker–Planck equation, we present a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing. From time-dependent PDFs, we calculate the information length (L), which is the total number of statistically different states that a system passes through in time and utilise it to understand the information geometry associated with the formation of bimodal or unimodal PDFs. We identify the difference between the relaxation and build-up of the shear gradient in view of information change and discuss the total information length (L∞=L(t→∞)) which maps out the underlying attractor structures, highlighting a unique property of L∞ which depends on the trajectory/history of a PDF’s evolution.

Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation

Inhomogeneous drift-wave turbulence can be modeled as an effective plasma where drift waves act as quantumlike particles and the zonal-flow velocity serves as a collective field through which they interact. This effective plasma can be described by a Wigner-Moyal equation (WME), which generalizes the quasilinear wave-kinetic equation (WKE) to the full-wave regime, i.e., resolves the wavelength scale. Unlike waves governed by manifestly quantumlike equations, whose WMEs can be borrowed from quantum mechanics and are commonly known, drift waves have Hamiltonians very different from those of conventional quantum particles. This causes unusual phase-space dynamics that is typically not captured by the WKE. We demonstrate how to correctly model this dynamics with the WME instead. Specifically, we report full-wave phase-space simulations of the zonal-flow formation (zonostrophic instability), deterioration (tertiary instability), and the so-called predator-prey oscillations. We also show how the WME facilitates analysis of these phenomena, namely, (i) we show that full-wave effects critically affect the zonostrophic instability, particularly its nonlinear stage and saturation; (ii) we derive the tertiary-instability growth rate; and (iii) we demonstrate that, with full-wave effects retained, the predator-prey oscillations do not require zonal-flow collisional damping, contrary to previous studies. We also show how the famous Rayleigh-Kuo criterion, which has been missing in wave-kinetic theories of drift-wave turbulence, emerges from the WME.

Geometric structure and information change in phase transitions

We propose a toy model for a cyclic order-disorder transition and introduce a geometric methodology to understand stochastic processes involved in transitions. Specifically, our model consists of a pair of forward and backward processes (FPs and BPs) for the emergence and disappearance of a structure in a stochastic environment. We calculate time-dependent probability density functions (PDFs) and the information length L, which is the total number of different states that a system undergoes during the transition. Time-dependent PDFs during transient relaxation exhibit strikingly different behavior in FPs and BPs. In particular, FPs driven by instability undergo the broadening of the PDF with a large increase in fluctuations before the transition to the ordered state accompanied by narrowing the PDF width. During this stage, we identify an interesting geodesic solution accompanied by the self-regulation between the growth and nonlinear damping where the time scale τ of information change is constant in time, independent of the strength of the stochastic noise. In comparison, BPs are mainly driven by the macroscopic motion due to the movement of the PDF peak. The total information length L between initial and final states is much larger in BPs than in FPs, increasing linearly with the deviation γ of a control parameter from the critical state in BPs while increasing logarithmically with γ in FPs. L scales as |lnD| and D^{-1/2} in FPs and BPs, respectively, where D measures the strength of the stochastic forcing. These differing scalings with γ and D suggest a great utility of L in capturing different underlying processes, specifically, diffusion vs advection in phase transition by geometry. We discuss physical origins of these scalings and comment on implications of our results for bistable systems undergoing repeated order-disorder transitions (e.g., fitness).

Fast Low-to-High Confinement Mode Bifurcation Dynamics in a Tokamak Edge Plasma Gyrokinetic Simulation

Transport barrier formation and its relation to sheared flows in fluids and plasmas are of fundamental interest in various natural and laboratory observations and of critical importance in achieving an economical energy production in a magnetic fusion device. Here we report the first observation of an edge transport barrier formation event in an electrostatic gyrokinetic simulation carried out in a realistic diverted tokamak edge geometry under strong forcing by a high rate of heat deposition. The results show that turbulent Reynolds-stress-driven sheared E×B flows act in concert with neoclassical orbit loss to quench turbulent transport and form a transport barrier just inside the last closed magnetic flux surface.

Signature of nonlinear damping in geometric structure of a nonequilibrium process

We investigate the effect of nonlinear interaction on the geometric structure of a nonequilibrium process. Specifically, by considering a driven-dissipative system where a stochastic variable x is damped either linearly (∝x) or nonlinearly (∝x^{3}) while driven by a white noise, we compute the time-dependent probability density functions (PDFs) during the relaxation towards equilibrium from an initial nonequilibrium state. From these PDFs, we quantify the information change by the information length L, which is the total number of statistically distinguishable states which the system passes through from the initial state to the final state. By exploiting different initial PDFs and the strength D of the white-noise forcing, we show that for a linear system, L increases essentially linearly with an initial mean value y_{0} of x as L∝y_{0}, demonstrating the preservation of a linear geometry. In comparison, in the case of a cubic damping, L has a power-law scaling as L∝y_{0}^{m}, with the exponent m depending on D and the width of the initial PDF. The rate at which information changes also exhibits a robust power-law scaling with time for the cubic damping.

New Paradigm for Turbulent Transport Across a Steep Gradient in Toroidal Plasmas

First principles gyrokinetic simulation of the edge turbulent transport in toroidal plasmas finds a reverse trend in the turbulent transport coefficients under strong gradients. It is found that there exist both linear and nonlinear critical gradients for the nonmonotonicity of transport characteristics. The discontinuity of the transport flux slope around the turning gradient shows features of a second order phase transition. Under a strong gradient the most unstable modes are in nonground eigenstates with unconventional mode structures, which significantly reduces the effective correlation length and thus reverse the transport trend. Our results suggest a completely new mechanism for the low to high confinement mode transition without invoking shear flow or zonal flow.

Collisional Scaling of the Energy Transfer in Drift-Wave Zonal Flow Turbulence

The collisionality scaling of density and potential coupling together with zonal flow energy transfer and spectral power is investigated at the stellarator experiment TJ-K. With a poloidal probe array, consisting of 128 Langmuir probes, density and potential fluctuations are measured on four neighboring flux surfaces simultaneously over the complete poloidal circumference. By analyzing Reynolds stress and pseudo-Reynolds stress, it is found that, for increasing collisionality, the coupling between density and potential decreases which hinders the zonal flow drive. Also, as a consequence, the nonlinear energy transfer, as well as the zonal flow contribution to the complete turbulent spectrum, decreases the same way. This is in line with theoretical expectations and is a first experimental verification of the importance of collisionality for large-scale structure formation in magnetically confined toroidal plasmas.

Bifurcation and hysteresis of plasma edge transport in a flux-driven system

Transition dynamics and mean shear flow generation in plasma interchange turbulence are explored in a flux-driven system that resembles the plasma edge region. The nonlinear evolution of the interchange mode shows two confinement regimes with different transport levels. Large amplitude oscillations in the phase space of turbulence intensity and mean flow energy are observed and investigated. Both clockwise and counterclockwise oscillations occur during the transition between the two regimes. The Reynolds stress gradients are shown to play a critical role in the generation of mean sheared flows in the edge region. Both the forward and back transitions are simulated self-consistently and a significant hysteresis is found.

Eddy, drift wave and zonal flow dynamics in a linear magnetized plasma

Turbulence and its structure formation are universal in neutral fluids and in plasmas. Turbulence annihilates global structures but can organize flows and eddies. The mutual-interactions between flow and the eddy give basic insights into the understanding of non-equilibrium and nonlinear interaction by turbulence. In fusion plasma, clarifying structure formation by Drift-wave turbulence, driven by density gradients in magnetized plasma, is an important issue. Here, a new mutual-interaction among eddy, drift wave and flow in magnetized plasma is discovered. A two-dimensional solitary eddy, which is a perturbation with circumnavigating motion localized radially and azimuthally, is transiently organized in a drift wave – zonal flow (azimuthally symmetric band-like shear flows) system. The excitation of the eddy is synchronized with zonal perturbation. The organization of the eddy has substantial impact on the acceleration of zonal flow.

Experimental Identification of Electric Field Excitation Mechanisms in a Structural Transition of Tokamak Plasmas

Self-regulation between structure and turbulence, which is a fundamental process in the complex system, has been widely regarded as one of the central issues in modern physics. A typical example of that in magnetically confined plasmas is the Low confinement mode to High confinement mode (L-H) transition, which is intensely studied for more than thirty years since it provides a confinement improvement necessary for the realization of the fusion reactor. An essential issue in the L-H transition physics is the mechanism of the abrupt "radial" electric field generation in toroidal plasmas. To date, several models for the L-H transition have been proposed but the systematic experimental validation is still challenging. Here we report the systematic and quantitative model validations of the radial electric field excitation mechanism for the first time, using a data set of the turbulence and the radial electric field having a high spatiotemporal resolution. Examining time derivative of Poisson's equation, the sum of the loss-cone loss current and the neoclassical bulk viscosity current is found to behave as the experimentally observed radial current that excites the radial electric field within a few factors of magnitude.

Geometric structure and geodesic in a solvable model of nonequilibrium process

We investigate the geometric structure of a nonequilibrium process and its geodesic solutions. By employing an exactly solvable model of a driven dissipative system (generalized nonautonomous Ornstein-Uhlenbeck process), we compute the time-dependent probability density functions (PDFs) and investigate the evolution of this system in a statistical metric space where the distance between two points (the so-called information length) quantifies the change in information along a trajectory of the PDFs. In this metric space, we find a geodesic for which the information propagates at constant speed, and demonstrate its utility as an optimal path to reduce the total time and total dissipated energy. In particular, through examples of physical realizations of such geodesic solutions satisfying boundary conditions, we present a resonance phenomenon in the geodesic solution and the discretization into cyclic geodesic solutions. Implications for controlling population growth are further discussed in a stochastic logistic model, where a periodic modulation of the diffusion coefficient and the deterministic force by a small amount is shown to have a significant controlling effect.

Low-to-High Confinement Transition Mediated by Turbulence Radial Wave Number Spectral Shift in a Fusion Plasma

A new model for the low-to-high (L-H) confinement transition has been developed based on a new paradigm for turbulence suppression by velocity shear [G. M. Staebler et al., Phys. Rev. Lett. 110, 055003 (2013)]. The model indicates that the L-H transition can be mediated by a shift in the radial wave number spectrum of turbulence, as evidenced here, for the first time, by the direct observation of a turbulence radial wave number spectral shift and turbulence structure tilting prior to the L-H transition at tokamak edge by direct probing. This new mechanism does not require a pretransition overshoot in the turbulent Reynolds stress, shunting turbulence energy to zonal flows for turbulence suppression as demonstrated in the experiment.

Variability and degradation of homeostasis in self-sustained oscillators

Homeostasis is known to be absolutely critical to the sustainability of living organisms. At the heart of homeostasis are various feedback loops, which can control and regulate a system to stay in a most favourable stable state upon the influence of various disturbance. While variability has emerged as a key factor in sustainability, too much variability could however be detrimental. It is thus absolutely crucial to understand the effect of fluctuation in different feedback loops. While modelling technique has achieved a great advancement to understand this issue, too a complicated model however often prevents us from disentangling different many processes. Here, we propose a novel model to gain a key insight into the effect of variability in feedback on self-sustained oscillation. Specifically, by taking into account variation in model parameters for self-excitation and nonlinear damping, corresponding to positive and negative feedback, respectively, we show how fluctuation in positive or negative feedback weakens the efficiency of feedback and affects self-sustained oscillations, possibly leading to a complete breakdown of self-regulation. While results are generic and could be applied to different self-regulating systems (e.g. self-regulation of neuron activity, normal cell growth, etc.), we present a specific application to heart dynamics. In particular, we show that fluctuation in positive feedback can lead to slow heart by either amplitude death or oscillation death pathway while fluctuation in negative feedback can lead to fast heart beat.

Experimental confirmation of self-regulating turbulence paradigm in two-dimensional spectral condensation

Turbulent transport in magnetic fusion plasmas can be significantly suppressed by Reynolds-stress-induced zonal flows, allowing effective plasma confinement. We present experimental evidence of spatiotemporal correlation between small-scale turbulence-induced Reynolds stress and large-scale zonal flow production in the E×B driven hydrodynamic spectral condensation. We show that Reynolds stress is generated effectively by anisotropic vorticity structures possessing collective tilt angle. The maximum amplitude of the tilt, the Reynolds stress, and the mean zonal flow production coincide with the transition time of the velocity field, indicating a key role of turbulence-induced Reynolds stress in the condensation of the flow. The analysis of the energy transfer between turbulence and zonal flow shows coherent oscillations with π/2 phase delay, thus indicating a predator-prey-like interaction between zonal flow and turbulence.

Observation of the L-H confinement bifurcation triggered by a turbulence-driven shear flow in a tokamak plasma

Comprehensive 2D turbulence and eddy flow velocity measurements on DIII-D demonstrate a rapidly increasing turbulence-driven shear flow that develops ∼100  μs prior to the low-confinement (L mode) to high-confinement (H mode) transition and appears to trigger it. These changes are localized to a narrow layer 1-2 cm inside the magnetic boundary. Increasing heating power increases the Reynolds stress, the energy transfer from turbulence to the poloidal flow, and the edge flow shearing rate that then exceeds the decorrelation rate, suppressing turbulence and triggering the transition.

Unraveling quasiperiodic relaxations of transport barriers with gyrokinetic simulations of tokamak plasmas

The generation and dynamics of transport barriers governed by sheared poloidal flows are analyzed in flux-driven 5D gyrokinetic simulations of ion temperature gradient driven turbulence in tokamak plasmas. The transport barrier is triggered by a vorticity source that polarizes the system. The chosen source captures characteristic features of some experimental scenarios, namely, the generation of a sheared electric field coupled to anisotropic heating. For sufficiently large shearing rates, turbulent transport is suppressed and a transport barrier builds up, in agreement with the common understanding of transport barriers. The vorticity source also governs a secondary instability--driven by the temperature anisotropy (T(∥)≠T(⊥)). Turbulence and its associated zonal flows are generated in the vicinity of the barrier, destroying the latter due to the screening of the polarization source by the zonal flows. These barrier relaxations occur quasiperiodically, and generically result from the decoupling between the dynamics of the barrier generation, triggered by the source driven sheared flow, and that of the crash, triggered by the secondary instability. This result underlines that barriers triggered by sheared flows are prone to relaxations whenever secondary instabilities come into play.

Spatiotemporal structures of edge limit-cycle oscillation before L-to-H transition in the JFT-2M tokamak

In this Letter, we report analyses of spatiotemporal dynamics of turbulence and structure in the limit-cycle oscillation (LCO) that precedes an L-to-H transition. Zonal flows are not observed during LCO, and the oscillation is the periodic generations or decays of barrier with edge-localized mean flow. Oscillatory Reynolds stress is found to be too small to accelerate the LCO flow, by considering the dielectric constant in magnetized toroidal plasmas. Propagation of changes of the density gradient and turbulence amplitude into the core is also observed.

Dynamics of low-intermediate-high-confinement transitions in toroidal plasmas

The dynamic features of the low-intermediate-high-(L-I-H) confinement transitions on HL-2A tokamak are presented. Here we report the discovery of two types of limit cycles (dubbed type-Y and type-J), which show opposite temporal ordering between the radial electric field and turbulence intensity. In type-Y, which appears first after an L-I transition, the turbulence grows first, followed by the localized electric field. In contrast, the electric field leads type-J. The turbulence-induced zonal flow and pressure-gradient-induced drift play essential roles in the two types of limit cycles, respectively. The condition of transition between types-Y and -J is studied in terms of the normalized radial electric field. An I-H transition is demonstrated to occur only from type-J.

Experimental evidence of turbulent transport regulation by zonal flows

The regulation of turbulent transport by zonal flows is studied experimentally on a flux surface of the stellarator experiment TJ-K. Data of 128 Langmuir probes at different toroidal and poloidal positions on a single flux surface enable us to measure simultaneously the zonal flow activity and the turbulent transport in great detail. A reduction of turbulent transport by 30% during the zonal flow phase is found. It is shown that the reduction process is initiated by a modification in the cross phase between density and electric field followed by a reduction in the fluctuation levels, which sustain low transport levels on larger time scales than the zonal flow lifetime.

Role of zonal flow predator-prey oscillations in triggering the transition to H-mode confinement

Direct evidence of zonal flow (ZF) predator-prey oscillations and the synergistic roles of ZF- and equilibrium E×B flow shear in triggering the low- to high-confinement (L- to H-mode) transition in the DIII-D tokamak is presented. Periodic turbulence suppression is first observed in a narrow layer at and just inside the separatrix when the shearing rate transiently exceeds the turbulence decorrelation rate. The final transition to H mode with sustained turbulence and transport reduction is controlled by equilibrium E×B shear due to the increasing ion pressure gradient.

First evidence of the role of zonal flows for the L-H transition at marginal input power in the EAST tokamak

A quasiperiodic Er oscillation at a frequency of <4 kHz, much lower than the geodesic-acoustic-mode frequency, with a modulation in edge turbulence preceding and following the low-to-high (L-H) confinement mode transition, has been observed for the first time in the EAST tokamak, using two toroidally separated reciprocating probes. Just prior to the L-H transition, the Er oscillation often evolves into intermittent negative Er spikes. The low-frequency Er oscillation, as well as the Er spikes, is strongly correlated with the turbulence-driven Reynolds stress, thus providing first evidence of the role of the zonal flows in the L-H transition at marginal input power. These new findings not only shed light on the underlying physics mechanism for the L-H transition, but also have significant implications for ITER operations close to the L-H transition threshold power.

Mean and oscillating plasma flows and turbulence interactions across the L-H confinement transition

A complex interaction between turbulence driven E × B zonal flow oscillations, i.e., geodesic acoustic modes (GAMs), the turbulence, and mean equilibrium flows is observed during the low to high (L-H) plasma confinement mode transition in the ASDEX Upgrade tokamak. Below the L-H threshold at low densities a limit-cycle oscillation forms with competition between the turbulence level and the GAM flow shearing. At higher densities the cycle is diminished, while in the H mode the cycle duration becomes too short to sustain the GAM, which is replaced by large amplitude broadband flow perturbations. Initially GAM amplitude increases as the H-mode transition is approached, but is then suppressed in the H mode by enhanced mean flow shear.

Formation and structure of transport barriers during confinement transitions in toroidal plasma

Density pedestal formation is studied experimentally during spontaneous low-to-high confinement transitions in the H-1 heliac. Poloidally extended potential structures, or zonal flows, seem to play the major role both in the spatial structure and in the temporal evolution of the pedestal formation. Zonal flows transiently generate radially localized maxima in the radial electric-field shear in L mode which coincides with the radial location of the pedestal in H mode.

Effect of collisional zonal-flow damping on flux-driven turbulent transport

The effect of collisional damping of zonal flows (ZFs) on ion-temperature gradient (ITG) driven turbulence in a toroidal plasma is investigated by means of a 3D global fluid model with flux boundary conditions. Results from simulations show an increase of the energy confinement time and a stabilization of turbulence with the inverse of the collisionality nu(*). The stabilization mechanism is identified as an effect of the increased shearing rate of ZFs, which shift upwards the ITG turbulence effective threshold. The shearing rate of ZFs is also seen to depend on the injected power. As a consequence, the effective heat conductivity depends parametrically on the input power.