Self-Regulating Shear Flow Turbulence: A Paradigm for the L to H Transition

Nonperturbative theory of the low-to-high confinement transition through stochastic simulations and information geometry: Correlation and causal analyses

The low-to-high confinement (L-H) transition signifies one of the important plasma bifurcations occurring in magnetic confinement plasmas, with vital implications for exploring high-performance regimes in future fusion reactors. In particular, the accurate turbulence statistical description of self-regulation and causal relation among turbulence and shear flows is essential for accessing enhanced plasma performance and advanced operation scenarios. To address this, we provide a nonperturbative theory of the L-H transition by stochastic simulations of a reduced L-H transition model and detailed statistical analysis. By calculating time-dependent probability density functions (PDFs) of turbulence, zonal flows, and the mean pressure gradient, we elucidate how statistical properties change over time with the help of the information geometry theory (information rate, causal information rate), highlighting its utility in capturing self-regulation and causal relation among turbulence, zonal flow shears, and the mean flow shears. Furthermore, stochastic noises in turbulence, zonal flows, and/or input power are shown to induce uncertainty in the power threshold Q_{c} above which the L-H transition occurs while leading to a rather gradual L-H transition. A time-dependent PDF of power loss over the L-H transition is presented.

Self-Organized Evolution of the Internal Transport Barrier in Ion-Temperature-Gradient Driven Gyrokinetic Turbulence

Understanding the self-organization of the most promising internal transport barrier in fusion plasmas needs a long-time nonlinear gyrokinetic global simulation. The neighboring equilibrium update method is proposed, which solves the secularity problem in a perturbative simulation and speeds up the numerical computation by more than 10 times. It is found that the internal transport barrier emerges at the magnetic axis due to inward propagated turbulence avalanche, and its outward expansion is the catastrophe of self-organized structure induced by outward propagated avalanche.

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'.

Nonlinear excitation of zonal flows by turbulent energy flux

The nonlinear excitation of zonal flows (ZFs) generated by the ion-temperature-gradient turbulence in a tokamak plasma is investigated by using the global gyrokinetic code nlt. It is found that ZFs are initially driven by the nonlinear self-interaction of the eigenmode. In the nonlinear saturation, the modulational instability becomes important, and its contribution to ZFs can finally be comparable to that of the self-interaction mechanism. More importantly, both types of nonlinear wave-wave interactions can be well described by the turbulent energy flux model.

Stochastic Model for Quasi-One-Dimensional Transitional Turbulence with Streamwise Shear Interactions

The transition to turbulence in wall-bounded shear flows is typically subcritical, with a poorly understood interplay between spatial fluctuations, pattern formation, and stochasticity near the critical Reynolds number. Here, we present a spatially extended stochastic minimal model for the energy budget in transitional pipe flow, which successfully recapitulates the way localized patches of turbulence (puffs) decay, split, and grow, respectively, as the Reynolds number increases through the laminar-turbulent transition. Our approach takes into account the flow geometry, as we demonstrate by extending the model to quasi-one-dimensional Taylor-Couette flow, reproducing the observed directed percolation pattern of turbulent patches in space and time.

Review of plasma turbulence experiments

Understandings of turbulent plasma have been developed along with nuclear fusion research for more than a half century. Long international research has produced discoveries concerning turbulent plasma that allow us to notice the hidden nature and physics questions that could contribute to other scientific fields and the development of technologies. Guiding concepts have been established up to now that stimulate investigations on turbulent plasma. Research based on concepts concerning symmetry breaking and global linkage requires observing the entire field of plasma turbulence for an ultimate understanding of plasma. This article reviews the achievements as well as contemporary problems regarding turbulence experiments associated with strongly magnetized plasmas in the last and present century, and introduces forthcoming experimental issues, including new diagnostics and physics-oriented devices related to plasma turbulence.

The Radial Propagation of Heat in Strongly Driven Non-Equilibrium Fusion Plasmas

Heat transport is studied in strongly heated fusion plasmas, far from thermodynamic equilibrium. The radial propagation of perturbations is studied using a technique based on the transfer entropy. Three different magnetic confinement devices are studied, and similar results are obtained. “Minor transport barriers” are detected that tend to form near rational magnetic surfaces, thought to be associated with zonal flows. Occasionally, heat transport “jumps” over these barriers, and this “jumping” behavior seems to increase in intensity when the heating power is raised, suggesting an explanation for the ubiquitous phenomenon of “power degradation” observed in magnetically confined plasmas. Reinterpreting the analysis results in terms of a continuous time random walk, “fast” and “slow” transport channels can be discerned. The cited results can partially be understood in the framework of a resistive Magneto-HydroDynamic model. The picture that emerges shows that plasma self-organization and competing transport mechanisms are essential ingredients for a fuller understanding of heat transport in fusion plasmas.

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.

Tracing the Pathway from Drift-Wave Turbulence with Broken Symmetry to the Production of Sheared Axial Mean Flow

This study traces the emergence of sheared axial flow from collisional drift-wave turbulence with broken symmetry in a linear plasma device-the controlled shear decorrelation experiment. As the density profile steepens, the axial Reynolds stress develops and drives a radially sheared axial flow that is parallel to the magnetic field. Results show that the nondiffusive piece of the Reynolds stress is driven by the density gradient, results from spectral asymmetry of the turbulence, and, thus, is dynamical in origin. Taken together, these findings constitute the first simultaneous demonstration of the causal link between the density gradient, turbulence, and stress with broken spectral symmetry and the mean axial flow.

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.

Predator-prey model for the self-organization of stochastic oscillators in dual populations

A predator-prey model of dual populations with stochastic oscillators is presented. A linear cross-coupling between the two populations is introduced following the coupling between the motions of a Wilberforce pendulum in two dimensions: one in the longitudinal and the other in torsional plain. Within each population a Kuramoto-type competition between the phases is assumed. Thus, the synchronization state of the whole system is controlled by these two types of competitions. The results of the numerical simulations show that by adding the linear cross-coupling interactions predator-prey oscillations between the two populations appear, which results in self-regulation of the system by a transfer of synchrony between the two populations. The model represents several important features of the dynamical interplay between the drift wave and zonal flow turbulence in magnetically confined plasmas, and a novel interpretation of the coupled dynamics of drift wave-zonal flow turbulence using synchronization of stochastic oscillator is discussed.

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.

Effect of sheared flow on the growth rate and turbulence decorrelation

The effect of a large scale flow shear on a linearly unstable turbulent system is considered. A cubic equation describing the effective growth rate is obtained, which is shown to reduce to well-known forms in weak and strong shear limits. A shear suppression rule is derived which corresponds to the point where the effective growth rate becomes negative. The effect of flow shear on nonlinear mode coupling of drift or Rossby waves is also considered, and it is shown that the resonance manifold shrinks and weakens as the vortices are sheared. This leads to a reduction of the efficiency of three-wave interactions. Tilted eddies can then only couple to the large scale sheared flows, because the resonance condition for that interaction is trivially satisfied. It is argued that this leads to absorbtion of the sheared vortices by large scale flow structures. Studying the form of the effective growth rate for weak shear, it was shown that in addition to reducing the overall growth rate, a weak flow shear also reduces the wave number where the fluctuations are most unstable.

Spatiotemporal structure of the interaction between turbulence and flows at the L-H transition in a toroidal plasma

The spatiotemporal behavior of the interaction between turbulence and flows has been studied close to the L-H transition threshold conditions in the edge region (ρ≥0.7) of TJ-II plasmas. The temporal dynamics of the interaction displays an oscillatory behavior with a characteristic predator-prey relationship. The spatial evolution of this turbulence-flow oscillation pattern has been measured, showing both radial outward and inward propagation velocities of the turbulence-flow front. The results indicate that the edge shear flow linked to the L-H transition can behave either as a slowing-down, damping mechanism of outward propagating turbulent-flow oscillating structures, or as a source of inward propagating turbulence-flow events.

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.

Multichannel Langmuir probe for turbulence study in Heliotron J

New multichannel Langmuir probe system was developed and installed to Heliotron J. The objective of the new probe is to characterize basic turbulence property and the resulting transport in advanced helical configuration. The probe developed here consists of four sets of triple probe and one pin for floating potential measurement. Initial experiments in neutral beam heating plasma were conducted and fluctuation profile of radial and poloidal electric fields and Reynolds stress were estimated. For precise evaluation of the electric fields and Reynolds stress, a technique to compensate radial change of tilt angle between probe array and magnetic surface was proposed and applied to the initial results obtained in edge region of Heliotron J where the complicated magnetic structure exists.

Interplay between gyrokinetic turbulence, flows, and collisions: perspectives on transport and poloidal rotation

The impact of ion-ion collisions on confinement is investigated with the full-f and global gyrokinetic Gysela code through a series of nonlinear turbulence simulations for tokamak parameters. A twofold scan in the turbulence drive and in collisionality is performed, highlighting (i) a heat transport expressed in terms of critical quantities-threshold and exponent, (ii) a first evidence of turbulent generation of poloidal momentum, and (iii) the dominance of mean flow shear, mediated through the turbulent corrugation of the mean profiles, with regard to the oft-invoked zonal flow shear.

Evidence of long-distance correlation of fluctuations during edge transitions to improved-confinement regimes in the TJ-II stellarator

Long-distance coupling between edge parameters' fluctuations has been investigated in the TJ-II stellarator. Results show long-range correlations in potential fluctuations, which are amplified by the development of radial electric fields during transitions to improved-confinement regimes, whereas there is no correlation between ion saturation current signals. These experimental findings suggest the importance of long-range correlations as a new fingerprint of the plasma behavior during the development of edge shear flows and the key role of electric fields to amplify them.

Slow transition of energy transport in high-temperature plasmas

A new slow transition process for energy transport in magnetically confined plasmas is reported. The slow transition is characterized by the change between two metastable transport conditions characterized by a weak and a strong electron temperature (Te) dependence of normalized heat flux. These two branches are found to merge at the critical gradient. In metastable transport, the derivative of normalized heat flux to the Te gradient, [EQUATION: SEE TEXT], is positive, while it becomes negative during the transition phase. The time for the transition increases as the normalized Te gradient is increased and exceeds the transport time scale characterized by the global energy confinement time.

Analysis of bifurcation phenomena in the electron internal transport barrier in the large helical device

The electron internal transport barrier (eITB) formation in the Large Helical Device (LHD) is studied with the transport code TOTAL and a GyroBohm-like model. The reduction of anomalous transport by the E x B shear has been introduced by means of the factor [1 + (tauomega(ExB))(gamma)](-1). Simulation results show a clear critical transition between plasma regimes with rather flat electron temperature profiles (non-) to a steeped one (with eITB) when average density is low enough. With the aim of studying the eITB formation as a phase transition phenomenon, the electron average density is taken as the control parameter and the E x B shearing rate as the order parameter. Results show how the eITB formation in LHD is compatible with a continuum phase transition with critical exponent beta = 0.40.

Spectral condensation of turbulence in plasmas and fluids and its role in low-to-high phase transitions in toroidal plasma

Transitions from turbulence to order are studied experimentally in thin fluid layers and in magnetically confined toroidal plasma. It is shown that turbulence self-organizes through the mechanism of spectral condensation in both systems. The spectral redistribution of the turbulent energy leads to the reduction in the turbulence level, generation of coherent flow, reduction in the particle diffusion, and increase in the system's energy. The higher-order state in the plasma is sustained via the non-local spectral coupling of the linearly unstable spectral range to the large-scale mean flow. Spectral condensation of turbulence is discussed in terms of its role in the low-to-high confinement transitions in toroidal plasma which show similarity with phase transitions.

Experimental evidence of coupling between sheared-flow development and an increase in the level of turbulence in the TJ-II stellarator

The link between the development of sheared flows and the structure of turbulence has been investigated in the plasma boundary region of the TJ-II stellarator. The development of the naturally occurring velocity shear layer requires a minimum plasma density. Near this critical density, the level of edge turbulent transport and the turbulent kinetic energy significantly increases in the plasma edge. The resulting shearing rate in the phase velocity of fluctuations is comparable to the one required to trigger a transition to improved confinement regimes with reduction of edge turbulence, suggesting that spontaneous sheared flows and fluctuations keep themselves near marginal stability. These findings provide the experimental evidence of coupling between sheared flows development and increasing in the level of edge turbulence. The experimental results are consistent with the expectations of second-order transition models of turbulence-driven sheared flows.

Bursting and large-scale intermittency in turbulent convection with differential rotation

The tilting mechanism, which generates differential rotation in two-dimensional turbulent convection, is shown to produce relaxation oscillations in the mean flow energy integral and bursts in the global fluctuation level, akin to Lotka-Volterra oscillations. The basic reason for such behavior is the unidirectional and conservative transfer of kinetic energy from the fluctuating motions to the mean component of the flows, and its dissipation at large scales. Results from numerical simulations further demonstrate the intimate relation between these low-frequency modulations and the large-scale intermittency of convective turbulence, as manifested by exponential tails in single-point probability distribution functions. Moreover, the spatio-temporal evolution of convective structures illustrates the mechanism triggering avalanche events in the transport process. The latter involves the overlap of delocalized mixing regions when the barrier to transport, produced by the mean component of the flow, transiently disappears.

Direct measurement of poloidal long-wavelength E x B flows in the HT-7 tokamak

The poloidal long-wavelength E x B time-varying flows were directly measured using a forked Langmuir probe in the HT-7 tokamak. Low-frequency (<10 kHz) E x B flows were observed at the plasma edge, which possess many of the characteristics of zonal flows, including a poloidal long-wavelength (k(theta)rho(i) approximately 0) and narrow radial extent (k(r)rho(i) approximately 0.1). The cross bicoherence of turbulent Reynolds stress indicates the existence of nonlinear three-wave coupling processes and the generation of low-frequency E x B flows. The estimated flow-shearing rate is of the same order of magnitude as the turbulence decorrelation rate and may thus regulate the fluctuation level and thereby the turbulence-driven transport.

Zonal flows and transient dynamics of the L-H transition

We elucidate the role of zonal flows in transient phenomena observed during L-H transition by studying a simple L-H transition model which contains the evolution of zonal flows, mean ExB flows, and the ion pressure gradient. Zonal flows are shown to trigger the L-H transition and cause time-transient behavior through the self-regulation of turbulence before a mean shearing, due to a steep pressure profile, secures a quiescent H mode. Surprisingly, this self-regulation lowers the power threshold for the ultimate transition to a quiescent H-mode state.

Metamorphosis of plasma turbulence-shear-flow dynamics through a transcritical bifurcation

The structural properties of an economical model for a confined plasma turbulence governor are investigated through bifurcation and stability analyses. A close relationship is demonstrated between the underlying bifurcation framework of the model and typical behavior associated with low- to high-confinement transitions such as shear-flow stabilization of turbulence and oscillatory collective action. In particular, the analysis evinces two types of discontinuous transition that are qualitatively distinct. One involves classical hysteresis, governed by viscous dissipation. The other is intrinsically oscillatory and nonhysteretic, and thus provides a model for the so-called dithering transitions that are frequently observed. This metamorphosis, or transformation, of the system dynamics is an important late side-effect of symmetry breaking, which manifests as an unusual nonsymmetric transcritical bifurcation induced by a significant shear-flow drive.

Slow L-H transitions in DIII-D plasmas

The transition from the low to the high mode of plasma confinement ( L-H transition) is studied in the DIII-D by an experimental technique which allows an arbitrarily slow transition. During an initial transition, periodic turbulent instability bursts are observed near the separatrix which inhibit the full transition. These bursts are damped by self-generated shear flows, and a predator-prey-type relationship is shown to give a good description of the data. As the neutral-beam power is raised, the oscillations change to type III edge localized modes. Another transition then leads to a quiet H mode.

Bifurcations and transport barriers in the resistive-g paradigm

The so-called resistive-g (resistive pressure-gradient-driven turbulence) paradigm is a widely accepted and frequently investigated model for nonlinear plasma dynamics. The parameter dependences of the generated transport barriers as well as third order bifurcations will be discussed numerically and analytically in the present paper. First, using a Galerkin representation, bifurcating states (from the conductive states in a rectangular cell) are investigated for the cases when only one unstable mode dominates. The dependence of the bifurcation properties on the aspect ratio of the domain is discussed, leading to the conclusion that for vanishing (or small) magnetic shear the so-called low, high, and edge localized mode transitions do not occur for small aspect ratios of the domain. Including reasonable magnetic shear, the small-aspect-ratio cutoff disappears, and transport barriers may exist in a broad parameter range. Second, for small aspect ratios, interesting codimension-2 bifurcations occur. When unfolding the dynamics up to third order, e.g., a weakly nonlinear interaction of convection cells is observed. The analytical results are confirmed by numerical simulations.

Role of reynolds stress-induced poloidal flow in triggering the transition to improved ohmic confinement on the HT-6M tokamak

Time and space resolved measurements of electrostatic Reynolds stress, radial electric field E(r), and plasma rotations have been performed across the transition to improved Ohmic confinement in the Hefei Tokamak-6M (HT-6M). The first experimental evidence of the correlation between the enhanced Reynolds stress gradient and the poloidal flow acceleration in the edge plasma is presented. The results indicate that the turbulence-induced Reynolds stress might be the dominant mechanism to create the sheared poloidal flow and E(r), which may further trigger the transition.

Neoclassical radial current balance in tokamaks and transition to the H mode

Monte Carlo ion simulation based on neoclassical radial current balance in a divertor tokamak gives a stationary sheared E-->xB--> flow. The neoclassical radial electric field E(r) shows no bifurcation in contrast with earlier orbit loss models, but the shear in E(r) reaches values at which a transition to enhanced confinement has been observed. Also, MHD turbulence analysis shows that a smooth transition can occur through the neoclassical E-->xB--> flow shear suppression. The parameter scaling of threshold temperature for strong turbulence shear suppression agrees with the H-mode threshold scaling in ASDEX Upgrade.

Proper orthogonal decomposition and galerkin projection for a three-dimensional plasma dynamical system

A general method by which to investigate nonlinear dynamical systems close to a stability threshold is presented. This method combines a proper orthogonal decomposition and a subsequent Galerkin projection. This technique is applied to three-dimensional resistive ballooning plasma fluctuations in a tokamak. The corresponding dynamical system belongs to a large family of convective fluid systems including Rayleigh-Benard convection. A proper orthogonal decomposition of the fluctuating signal obtained by numerical simulation shows that the relevant modes are close to the linear (global) modes. The Galerkin projection provides a low-dimensional system that allows the study of shear flow generation, its subsequent fluctuation reduction, and the evolution to oscillating states.