We show that zonal flow can be preferentially excited by intermediate-scale toroidal electron temperature gradient (ETG) turbulence in tokamak plasmas. Previous theoretical studies that yielded an opposite conclusion assumed a fluid approximation for ETG modes. Here, we carry out a gyrokinetic analysis which ultimately yields a nonlinear Schrödinger equation for the ETG dynamics with a Navier-Stokes type nonlinearity. For typical tokamak parameters, it is found that zonal flow generation plays an important role in the intermediate-scale ETG turbulence. This finding offers an explanation for recent multi-scale gyrokinetic simulations.
A new, frequency modulation mechanism for zonal flow pattern formation is presented. The model predicts the probability distribution function of the flow strength as well as the evolution of the characteristic spatial scale. Magnetic toroidicity-induced global phase dynamics is shown to determine the spatial structure of the flow. A key result is the observation that global phase patterning can lead to zonal flow formation in the absence of turbulence inhomogeneity.
The set of equations describing nonlinear evolution of a single toroidal Alfven eigenmode are derived, including both zero frequency zonal structure (ZFZS) generation and wave-particle phase space nonlinearities. The simplified case of neglecting wave-particle phase space nonlinearity is then investigated to focus on different roles of energetic particles and bulk plasmas on ZFZS generation. It is shown that energetic particles and bulk plasma play dominant roles in ZFZS generation in different nonlinear stages, and the corresponding processes are qualitatively different. Several properties of…
Generation of zonal ow (ZF) by energetic particle (EP) driven toroidal Alfven eigenmode (TAE) is investigated using nonlinear gyrokinetic theory. It is found that, nonlinear resonant EP contri- bution dominates over the usual Reynolds and Maxwell stresses due to thermal plasma nonlinear response. ZF can be forced driven in the linear growth stage of TAE, with the growth rate being twice the TAE growth rate. The ZF generation mechanism is shown to be related to polarization induced by resonant EP nonlinearity. The generated ZF has both the usual meso-scale and micro- scale radial structures. Po…
Turbulence driven zonal flows play an important role in fusion devices since they improve plasma confinement by limiting the level of anomalous transport. Current theories mostly focus on flow excitation but do not self-consistently describe the nearly stationary zonal flow turbulence equilibrium state. First-principles two-fluid turbulence studies are used to construct a Reynolds stress response functional from observations in turbulent states. This permits, for the first time, a reliable charting of zonal flow turbulence equilibria.
Kinetic ballooning modes (KBM) are widely believed to play a critical role in disruptive dynamics as well as turbulent transport in tokamaks. While the nonlinear evolution of ballooning modes has been proposed as a mechanism for detonation in tokamak plasmas, the role of kinetic effects in such nonlinear dynamics remains largely unexplored. In this work global gyrokinetic simulation results of KBM nonlinear behavior are presented. Instead of the finite-time singularity predicted by ideal MHD theory, the kinetic instability is shown to develop into an intermediate nonlinear regime of exponentia…
The linear instability of the electron temperature gradient (ETG) driven modes in the presence of zonal flows is investigated. Random and deterministic $cos$ - like profiles of the zonal flow are considered. It is shown that the presence of shearing by zonal flows can stabilize the linear instability of ETG drift modes.
There is strong evidence in favor for zonal flow suppression of the Ion-Temperature-Gradient (ITG) mode turbulence, specifically close to the linear stability threshold. The present letter attempts to analytically calculate the effects of zonal flow suppression of the ITG turbulence through deriving a modified dispersion relation including the back-reaction of the zonal flows on the ITG turbulence based on the quasilinear theory. The results are manifested in a reduction of the linear growth rate and an increase in the effective linear ITG threshold.
The probability distribution functions (PDFs) of momentum flux and zonal flow formation in ion-temperature-gradient (ITG) turbulence are investigated, including the effect of the shear flow on the PDFs. While ITG turbulence maintains high level of transport, this may be suppressed by shear flow. Zonal flows are also shown here to have an enhanced likelihood of generation further from marginal stability which will then regulate the ITG turbulence, which is more prominent with increased shear flow, leading to a self-regulating system.
The generation of large-scale zonal flows by small-scale electrostatic drift waves in electron temperature gradient(ETG) driven turbulence model is considered. The generation mechanism is based on the modulational instability of a finite amplitude monochromatic drift wave. The threshold and growth rate of the instability as well as the optimal spatial scale of zonal flow are obtained.
The role of self-generated zonal flows in the collisionless trapped-electron mode (CTEM) turbulence is a long-standing open issue in tokamak plasmas. Here we show, for the first time, that the zonal flow excitation in the CTEM turbulence is formally isomorphic to that in the ion temperature gradient turbulence. Trapped electrons contribute implicitly only via linear dynamics. Theoretical analyses further suggest that, for short wavelength CTEMs, the zonal flow excitation is weak and, more importantly, not an effective saturation mechanism. Corresponding controlling parameters are also identifi…
Nonlinear entropy transfer via zonal flows in gyrokinetic plasma turbulence
The Graduate University for Advanced Studies（高级研究大学院大学）；Faculty of Arts and Sciences, Komazawa University（驹泽大学文理学部）；National Institute for Fusion Science（核聚变科学研究所）
Detailed computations of tokamak edge turbulence in three dimensional, globally consistent flux tube geometry show an inhibition of the standard scenario in which zonal ExB flows generated by the turbulence should lead to transport barrier formation. It is found by comparison to slab geometry and by analysis of the energetics that the zonal flow energy is depleted by toroidal coupling to the pressure through the geodesic curvature. Edge transport barriers would then depend on the physics of the neoclassical equilibrium.
The role of parallel ion motion for zonal flow generation in ion-temperature-gradient (ITG) mode turbulence is investigated with focus on the effects of acoustic modes and toroidicity on the zonal flow. One possible reason for the weak suppression of ITG turbulence by zonal flows found in experiments in the Columbia Linear Machine (CLM) [Phys. Plasmas {\bf 13} 055905 (2006)] might be due to the small toroidicity ($ε_n = 2L_n/R$) in the experiment. The zonal flow is often directly dependent on the ITG mode and the coupling of zonal flow to acoustic modes and hence is directly affected by any ch…
We propose a quantitative model of ion temperature gradient driven turbulence in toroidal magnetized plasmas. In this model, the turbulence is regulated by zonal flows, i.e. mode saturation occurs by a zonal-flow-mediated energy cascade ("shearing"), and zonal flow amplitude is controlled by nonlinear decay. Our model is tested in detail against numerical simulations to confirm that both its assumptions and predictions are satisfied. Key results include (1) a sensitivity of the nonlinear zonal flow response to the energy content of the linear instability, (2) a persistence of zonal-flow-regula…
Saturation of magnetised plasma turbulence by propagating zonal flows
Princeton University（普林斯顿大学）；Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·派尔斯理论物理中心）；University of Maryland, College Park（马里兰大学帕克分校）
On nonlinear saturation of toroidal Alfvén eigenmode due to thermal plasma nonlinearities
Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Center for Nonlinear Plasma Science and C.R. ENEA Frascati（非线性等离子体科学中心及恩瑞亚弗拉斯卡蒂研究中心）；Institute for Fusion Theory and Simulation, School of Physics, Zhejiang University（浙江大学物理学院聚变理论与模拟研究所）；Key Laboratory of Frontier Physics in Controlled Nuclear Fusion and Institute of Plasma Physics, Chinese Academy of Sciences（中国科学院等离子体物理研究所及受控核聚变前沿物理重点实验室）；De Vinci Higher Education, De Vinci Research Center（德文奇高等教育机构及德文奇研究中心）
Isotope Effects on TEM-driven Turbulence and Zonal Flows in Helical and Tokamak Plasmas
National Institute for Fusion Science, National Institutes of Natural Sciences（国立研究開発法人 国立統合物理学研究所 核融合科学研究所）；The Graduate University for Advanced Studies（総合研究大学院大学）；Faculty of Arts and Sciences, Komazawa University（駒澤大学 文学部・理学部）；Nagoya University（名古屋大学）
In this paper a conservation equation is derived for the radially dependent entropy in toroidal geometry using the local approximation of the gyro-kinetic framework. This equation naturally leads to an operative definition for the turbulence intensity. It is shown that the conservation equation can be split in two separate conservation equations, one describing the dynamics of the zonal modes and one for the non-zonal modes. In essence the paper provides an operative tool for both analytic as well as numeric studies of the radial propagation of turbulence in tokamak plasmas.
The present work investigates the direct interaction of sheared mean flow with zonal flows (ZF) and the effect of parallel ion motion on ZF generation in ion-temperature-gradient (ITG) background turbulence. An analytical model for the direct interaction of sheared mean flows with zonal flows is constructed. The model used for the toroidal ITG driven mode is based on the equations for ion continuity, ion temperature and parallel ion motion whereas the ZF evolution is described by the vorticity equation. The behavior of the ZF growth rate and real frequency is examined for typical tokamak param…
During the past years the understanding of the multi-scale interaction problems have increased significantly. However, at present there exists a range of different analytical models for investigating multi-scale interactions and hardly any specific comparisons have been performed among these models. In this work, two different models for the generation of zonal flows from ion-temperature-gradient (ITG) background turbulence are discussed and compared. The methods used is the coherent mode coupling model and the wave kinetic equation model (WKE). It is shown that the two models give qualitative…
The self-consistent spatiotemporal evolution of drift wave (DW) radial envelope and zonal flow (ZF) amplitude is investigated in a slab model [1]. Stationary solution of the coupled partial differential equations in a simple limit yields formation of DW-ZF soliton structures, which propagate at group velocity depending on the envelope peak amplitude. Additional interesting physics, e.g. generation, destruction, collision and reflection of solitons, as well as turbulence bursting can also be observed due to effects of linear growth/damping, dissipation, equilibrium nonuniformities and soliton d…
The derivation of an intermediate-scale gyrokinetic-electron theory in nonuniform tokamak plasmas [Chen H. et al 2021 Nucl. Fusion 61 066017] has shown that a Navier-Stokes type nonlinearity couples electron-temperature-gradient (ETG) modes and zonal flow (ZF) modes with wavelengths much shorter than the ion gyroradius but much longer than the electron gyroradius. This intermediate-scale ETG-ZF coupling is typically stronger than the Hasegawa-Mima type nonlinearity characteristic of the fluid approximation and is predicted to lead to relevant zonal flow generation and ETG mode regulation. Elec…
The role of ion polarisation and finite Larmor radius on the isotope effect on turbulent tokamak edge transport and flows is investigated by means of local electromagnetic multi-species gyro-fluid computations. Transport is found to be reduced with the effective plasma mass for protium, deuterium and tritium mixtures. This isotope effect is found for both cold and warm ion models, but significant influence of finite Larmor radius and polarisation effects are identified. Sheared flow reduction of transport through self generated turbulent zonal flows and geodesic acoustic modes in the present m…
The Hasegawa-Wakatani equations, coupling plasma density and electrostatic potential through an approximation to the physics of parallel electron motions, are a simple model that describes resistive drift wave turbulence. We present numerical analyses of bifurcation phenomena in the model that provide new insights into the interactions between turbulence and zonal flows in the tokamak plasma edge region. The simulation results show a regime where, after an initial transient, drift wave turbulence is suppressed through zonal flow generation. As a parameter controlling the strength of the turbul…
In a two-dimensional version of the modified Hasegawa-Wakatani (HW) model, which describes electrostatic resistive drift wave turbulence, the resistive coupling between vorticity and density does not act on the zonal components ($k_{y}=0$). It is therefore necessary to modify the HW model to treat the zonal components properly. The modified equations are solved numerically, and visualization and analysis of the solutions show generation of stable zonal flows, through conversion of turbulent kinetic energy, and the consequent turbulence and transport suppression. It is demonstrated by compariso…
Plasma turbulence described by the Hasegawa-Wakatani equations has been simulated numerically for different models and values of the adiabaticity parameter C. It is found that for low values of C turbulence remains isotropic, zonal flows are not generated and there is no suppression of the meridional drift waves and of the particle transport. For high values of C, turbulence evolves toward highly anisotropic states with a dominant contribution of the zonal sector to the kinetic energy. This anisotropic flow leads to a decrease of a turbulence production in the meridional sector and limits the …
In the present work the generation of zonal flows in collisionless trapped electron mode (TEM) turbulence is studied analytically. A reduced model for TEM turbulence is utilized based on an advanced fluid model for reactive drift waves. An analytical expression for the zonal flow growth rate is derived and compared with the linear TEM growth, and its scaling with plasma parameters is examined for typical tokamak parameter values.
The impact of impurities on the generation of zonal flow (ZF) driven by collisonless trapped electron mode (CTEM) turbulence in deuterium (D)-tritium (T) plasmas is investigated. The expression for ZF growth rate with impurities is derived by balancing the ZF potential shielded by polarization effects and the ZF modulated radial turbulent current. Then, it is shown that the maximum normalized ZF growth rate is reduced by the presence of the fully ionized non-trace light impurities with relatively flat density profile, and slightly reduced by highly ionized trace tungsten (W). While, the maximu…
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 conv…
The role of magnetic shear for zonal flow generation by ion-temperature-gradient (ITG-) and trapped electron (TE-) mode turbulence is studied analytically using fluid descriptions. The scaling of the zonal flow (ZF) growth rate with magnetic shear is examined and compared with linear growth rates for typical tokamak parameter values. The results indicate that large levels of ZF are obtained in regions of negative magnetic shear, in particular for ZF driven by TE mode turbulence. The strong magnetic shear scaling obtained for TE mode driven zonal flows originates from the bounce average of the …
Long global gyrokinetic turbulence simulations are particularly challenging in situations where the system deviates strongly from its initial state and when fluctuation level are high e.g. in strong gradient regions. For Particle-in-Cell simulations, statistical sampling noise accumulation from large marker weights due to large deviations from the control variate of a delta-f scheme make such simulations often impractical. An adaptive control variate in the form of a flux-surface-averaged Maxwellian with a time-dependent temperature profile is introduced in an attempt to alleviate the former p…
On Phase Transition of ITG Turbulence in the Dimits shift
Laboratoire de Physique des Plasmas, CNRS, Ecole Polytechnique, Sorbonne Université, Université Paris-Saclay , Observatoire de Paris, F-91120 Palaiseau, France
The possibility of fast dynamo action by collisionless kinetic Alfven Wave turbulence is demonstrated. The irreversibility necessary to lock in the generated field is provided by electron Landau damping, so the induced electric field does not vanish with resistivity. Mechanisms for self-regulation of the system and the relation of these results to the theory of alpha quenching are discussed. The dynamo-generated fields have symmetry like to that of zonal flows, and thus are termed zonal fields.
A new strategy is presented to explain the creation and persistence of zonal flows widely observed in plasma edge turbulence. The core physics in the edge regime of the magnetic-fusion tokamaks can be described qualitatively by the one-state modified Hasegawa-Mima (MHM) model, which creates enhanced zonal flows and more physically relevant features in comparison with the familiar Charney-Hasegawa-Mima (CHM) model for both plasma and geophysical flows. The generation mechanism of zonal jets is displayed from the secondary instability analysis via nonlinear interactions with a background base st…
3D Braginskii turbulence simulations show that the energy flux in the core/edge transition region of a tokamak is strongly modulated - locally and on average - by radially propagating, nearly coherent sinusoidal or solitary zonal flows. The flows are geodesic acoustic modes (GAM), which are primarily driven by the Stringer-Winsor term. The flow amplitude together with the average anomalous transport sensitively depend on the GAM frequency and on the magnetic curvature acting on the flows, which could be influenced in a real tokamak, e.g., by shaping the plasma cross section. The local modulati…
Anisotropy and intermittency in drift-wave turbulence with zonal flows: a two-dimensional continuous wavelet analysis
Institute of Materials and Systems for Sustainability, Nagoya University（名古屋大学材料系统可持续研究所）；Institut de Mathématiques de Marseille (I2M), Aix-Marseille Université, CNRS（马赛数学研究所（I2M），艾克斯-马赛大学，法国国家科学研究中心）；National Institute for Fusion Science（国立核聚变科学研究所）
The saturated state of turbulence driven by the ion-temperature-gradient instability is investigated using a two-dimensional long-wavelength fluid model that describes the perturbed electrostatic potential and perturbed ion temperature in a magnetic field with constant curvature (a $Z$-pinch) and an equilibrium temperature gradient. Numerical simulations reveal a well-defined transition between a finite-amplitude saturated state dominated by strong zonal-flow and zonal-temperature perturbations, and a blow-up state that fails to saturate on a box-independent scale. We argue that this transitio…
In the complex 3D magnetic fields of stellarators, ion-temperature-gradient turbulence is shown to have two distinct saturation regimes, as revealed by petascale numerical simulations, and explained by a simple turbulence theory. The first regime is marked by strong zonal flows, and matches previous observations in tokamaks. The newly observed second regime, in contrast, exhibits small- scale quasi-two-dimensional turbulence, negligible zonal flows, and, surprisingly, a weaker heat flux scaling. Our findings suggest that key details of the magnetic geometry control turbulence in stellarators.
In electrostatic simulations of MAST plasma at electron-gyroradius scales, using the local flux-tube gyrokinetic code GS2 with adiabatic ions, we find that the long-time saturated electron heat flux (the level most relevant to energy transport) decreases as the electron collisionality decreases. At early simulation times, the heat flux "quasi-saturates" without any strong dependence on collisionality, and with the turbulence dominated by streamer-like radially elongated structures. However, the zonal fluctuation component continues to grow slowly until much later times, eventually leading to a…
The prevailing paradigm for plasma turbulence associates a unique stationary state to given equilibrium parameters. We report the discovery of bistable turbulence in a strongly magnetised plasma. Two distinct states, obtained with identical equilibrium parameters in first-principle gyrokinetic simulations, have turbulent fluxes of particles, momentum and energy that differ by an order of magnitude, with the low-transport state agreeing with experimental observations. Occurrences of the two states are regulated by the competition between an externally imposed mean flow shear and "zonal" flows g…
In the present work the zonal flow (ZF) growth rate in toroidal ion-temperature-gradient (ITG) mode turbulence including the effects of elongation is studied analytically. The scaling of the ZF growth with plasma parameters is examined for typical tokamak parameter values. The physical model used for the toroidal ITG driven mode is based on the ion continuity and ion temperature equations whereas the ZF evolution is described by the vorticity equation. The results indicate that a large ZF growth is found close to marginal stability and for peaked density profiles and these effects may be enhan…
We demonstrate analytically that, in toroidal plasmas, scattering by drift wave turbulence could lead to appreciable damping of toroidal Alfven eigenmodes via generation of short-wavelength electron Landau damped kinetic Alfven waves. A corresponding analytic expression of the damping rate is derived, and found to be, typically, comparable to the linear drive by energetic particles. The implications of this novel mechanism on the transport and heating processes in burning plasmas are also discussed.
We report a non-perturbative study of the effect of different type of shear flows on the evolution of vorticity and particle density fluctuations in interchange turbulence. For the same shear strength, the transport of density is less reduced by streamers than by zonal flows, zonal flows leading to oscillation death. In the inviscid limit, vorticity (density) grows (decays) as a power law due to streamer or zonal flow, and exponentially due to the combined effect of zonal flow and streamer with the same sign of shear. Zonal flow and streamer with the opposite sign of shear lead to oscillation …
