An analytical model for zonal flow generation by toroidal ion-temperature-gradient (ITG) modes, including finite $β$ electromagnetic effects, is derived. The derivation is based on a fluid model for ions and electrons and takes into account both linear and nonlinear $β$ effects. The influence of finite plasma $β$ on the zonal flow growth rate ($γ_{ZF}$) scaling is investigated for typical tokamak plasma parameters. The results show the importance of the zonal flows close to marginal stability where $γ_{ZF}/γ_{ITG}>>1$ is obtained. In this region the parameter $γ_{ZF}/γ_{ITG}$ increases with $β$, indicating that the ITG turbulence and associated transport would decrease with $β$ at a faster rate than expected from a purely linear or quasi-linear analysis.
We present a theory of the nonlinear growth of zonal flows in magnetized plasma turbulence, by the mechanism of secondary instability. The theory is derived for general magnetic geometry, and is thus applicable to both tokamaks and stellarators. The predicted growth rate is shown to compare favorably with nonlinear gyrokinetic simulations, with the error scaling as expected with the small parameter of the theory.
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.
Excitation of zonal flow by the modulational instability in the presence of mean shear flow is considered. It is shown that the small amplitude mean flow favours the modulational instability, increasing the growth rate, whereas sufficiently strong mean shear significantly reduces the instability growth rate.
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.
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…
The dynamics of the radial envelope of a weak coherent drift wave is approximately governed by a nonlinear Schrödinger equation, which emerges as a limit of the modified Hasegawa-Mima equation. The nonlinear Schrödinger equation has well-known soliton solutions, and its modulational instability can naturally generate solitary structures. In this paper, we demonstrate that this simple model can adequately describe the formation of solitary zonal structures in the modified Hasegawa-Mima equation, but only when the amplitude of the coherent drift wave is relatively small. At larger amplitudes, th…
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）
The Dimits shift is the shift between the threshold of the drift-wave primary instability and the actual onset of turbulent transport in magnetized plasma. It is generally attributed to the suppression of turbulence by zonal flows, but developing a more detailed understanding calls for consideration of specific reduced models. The modified Terry--Horton system has been proposed by St-Onge [J. Plasma Phys. $\boldsymbol{\rm 83}$, 905830504 (2017)] as a minimal model capturing the Dimits shift. Here, we use this model to develop an analytic theory of the Dimits shift and a related theory of the t…
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…
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）
We demonstrate theoretically and numerically the zonal-flow/drift-wave feedback mechanism for the LH transition in an idealised model of plasma turbulence driven by a small scale instability. Zonal flows are generated by a secondary modulational instability of the modes which are directly driven by the primary instability. The zonal flows then suppress the small scales thereby arresting the energy injection into the system, a process which can be described using nonlocal wave turbulence theory. Finally, the arrest of the energy input results in saturation of the zonal flows at a level which ca…
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…
We propose a new mechanism for the generation of zonal flows in magnetically confined plasmas, complementing previous theories based on a modulational instability. We derive a new conservation law that operates in the regime of weakly nonlinear dynamics, and show that it serves to focus the inverse cascade of turbulent drift wave energy into zonal flows. This mechanism continues to operate in the absence of the separation of dynamical scales typically assumed in instability calculations.
The tertiary instability is believed to be important for governing magnetised plasma turbulence under conditions of strong zonal flow generation, near marginal stability. In this work, we investigate its role for a collisionless strongly driven fluid model, self-consistently derived as a limit of gyrokinetics. It is found that a region of absolute stability above the linear threshold exists, beyond which significant nonlinear transport rapidly develops. While within this range a complex pattern of transient zonal evolution is observed before a stable profile is found, the Dimits transition its…
In geophysical and plasma contexts, zonal flows are well known to arise out of turbulence. We elucidate the transition from statistically homogeneous turbulence without zonal flows to statistically inhomogeneous turbulence with steady zonal flows. Starting from the Hasegawa--Mima equation, we employ both the quasilinear approximation and a statistical average, which retains a great deal of the qualitative behavior of the full system. Within the resulting framework known as CE2, we extend recent understanding of the symmetry-breaking `zonostrophic instability'. Zonostrophic instability can be u…
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…
The consequences of discrete particle noise for a system possessing a possibly unstable collective mode are discussed. It is argued that a zonostrophic instability (of homogeneous turbulence to the formation of zonal flows) occurs just below the threshold for linear instability. The scenario provides a new interpretation of the random forcing that is ubiquitously invoked in stochastic models such as the second-order cumulant expansion (CE2) or stochastic structural instability theory (SSST); neither intrinsic turbulence nor coupling to extrinsic turbulence is required. A representative calcula…
This paper gives a pedagogic review of the envelope formalism for excitation of zonal flows by nonlinear interactions of plasma drift waves or Rossby waves, described equivalently by the Hasegawa-Mima (HM) equation or the quasigeostrophic barotropic potential vorticity equation, respectively. In the plasma case a modified form of the HM equation, which takes into account suppression of the magnetic-surface-averaged electron density response by a small amount of rotational transform, is also analyzed. Excitation of zonal mean flow by a modulated wave train is particularly strong in the modified…
Solitary zonal structures have recently been identified in gyrokinetic simulations of subcritical drift-wave (DW) turbulence with background shear flows. However, the nature of these structures has not been fully understood yet. Here, we show that similar structures can be obtained within a reduced model, which complements the modified Hasegawa-Mima equation with a generic primary instability and a background shear flow. We also find that these structures can be qualitatively reproduced in the modified Hasegawa-Wakatani equation, which subsumes the reduced model as a limit. In particular, we i…
In this section, we examine the transition from statistically homogeneous turbulence to inhomogeneous turbulence with zonal flows. Statistical equations of motion can be derived from the quasilinear approximation to the Hasegawa-Mima equation. We review recent work that finds a bifurcation of these equations and shows that the emergence of zonal flows mathematically follows a standard type of pattern formation. We also show that the dispersion relation of modulational instability can be extracted from the statistical equations of motion in a certain limit. The statistical formulation can thus …
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…
This paper reports the stability conditions for intense zonal flows (ZFs) and the growth rate $γ_{\rm TI}$ of the corresponding "tertiary" instability (TI) within the generalized Hasegawa--Mima plasma model. The analytic calculation extends and revises Kuo's analysis of the mathematically similar barotropic vorticity equation for incompressible neutral fluids on a rotating sphere [H.-L. Kuo, J. Meteor. $\textbf{6}$, 105 (1949)]; then, the results are applied to the plasma case. An error in Kuo's original result is pointed out. An explicit analytic formula for TI is derived and compared with nu…
Tertiary modes in electrostatic drift-wave turbulence are localized near extrema of the zonal velocity $U(x)$ with respect to the radial coordinate $x$. We argue that these modes can be described as quantum harmonic oscillators with complex frequencies, so their spectrum can be readily calculated. The corresponding growth rate $γ_{\rm TI}$ is derived within the modified Hasegawa--Wakatani model. We show that $γ_{\rm TI}$ equals the primary-instability growth rate plus a term that depends on the local $U''$; hence, the instability threshold is shifted compared to that in homogeneous turbulence.…
The Dimits shift, an upshift in the onset of turbulence from the linear instability threshold, caused by self-generated zonal flows, can greatly enhance the performance of magnetic confinement plasma devices. Except in simple cases, using fluid approximations and model magnetic geometries, this phenomenon has proved difficult to understand and quantitatively predict. To bridge the large gap in complexity between simple models and realistic treatment in toroidal magnetic geometries (e.g. tokamaks or stellarators), the present work uses fully gyrokinetic simulations in Z-pinch geometry to invest…
This paper studies the instability of two-dimensional magnetohydrodynamic (MHD) systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the `clamshell instability', which features field lines on the two hemispheres tilting in opposite directions (Cally 2001, Sol. Phys. vol. 199, pp. 231--249). An asymptotic solutio…
An impact of neutrals on anomalous edge plasma transport and zonal flow (ZF) is considered. As an example, it is assumed that edge plasma turbulence is driven by the resistive drift wave (RDW) instability. It is found that the actual effect of neutrals is not related to a suppression of the instability \textit{per se}, but due to an impact on the ZF. Particularly, it is shown that, whereas the neutrals make very little impact on the linear growth rate of the RDW instability, they can largely reduce the zonal flow generation in the nonlinear stage, which results in an enhancement of the overall…
The phase space of driftons (drift-wave quanta) is studied within the generalized Hasegawa--Mima collisionless-plasma model in the presence of zonal flows. This phase space is made intricate by the corrections to the drifton ray equations that were recently proposed by Parker [J. Plasma Phys. $\textbf{82}$, 95820602 (2016)] and Ruiz $\textit{et al.}$ [Phys. Plasmas $\textbf{23}$, 122304 (2016)]. Contrary to the traditional geometrical-optics (GO) model of the drifton dynamics, it is found that driftons can be not only trapped or passing, but they can also accumulate spatially while experiencin…
In this work we consider magnetic reconnection in thin current sheets with both resistive and electron inertia effects. When the current sheet is produced by a primary instability of the internal kink type, the analysis of secondary instabilities indicates that reconnection proceeds on a time scale much shorter than the primary instability characteristic time. In the case of a sawtooth crash, non-collisional physics becomes important above a value of the Lundquist number which scales like S ~ (R/d_e)^{12/5}, in terms of the tokamak major radius R and of the electron skin depth d_e. This value …
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…
Magnetic reconnection provides the primary source for explosive energy release, plasma heating and particle acceleration in many astrophysical environments. The last years witnessed a revival of interest in the MHD tearing instability as a driver for efficient reconnection. It has been established that, provided the current sheet aspect ratio becomes small enough ($a/L \sim S^{-1/3}$ for a given Lundquist number $S\gg 1$), reconnection occurs on ideal Alfvén timescales and becomes independent on $S$. Here we investigate, by means of two-dimensional simulations, the \emph{ideal} tearing instabi…
The collisional drift wave instability in a straight magnetic field configuration is studied within a full-F gyro-fluid model, which relaxes the Oberbeck-Boussinesq (OB) approximation. Accordingly, we focus our study on steep background density gradients. In this regime we report on corrections by factors of order one to the eigenvalue analysis of former OB approximated approaches as well as on spatially localised eigenfunctions, that contrast strongly with their OB approximated equivalent. Remarkably, non-modal phenomena arise for large density inhomogeneities and for all collisionalities. As…
Consideration of wave--flow resonance addresses the long-standing problem of how zonal flows (ZF) saturate in the limit of weak or zero frictional drag, and also determines the ZF scale. For relevant magnetic geometries, the frequently quoted tertiary instability requires unphysical enhancement of ZF shear and thus is irrelevant to the near-marginal, frictionless regime. We show that resonant vorticity mixing, which conserves potential enstrophy, enables ZF saturation in the absence of drag, and so is effective in the Dimits up-shift regime. Vorticity mixing is incorporated as a nonlinear, sel…
Evolution of the single-mode ablative Rayleigh-Taylor instability in the presence of preheating is investigated numerically. It is found that the instability evolution depends strongly on the preheating length $L$ in front of the ablation-front interface. For weak preheating ($L/L_{\rm SH}\leq f_0$, where $L_{\rm SH}$ is the Spitzer-Harm length and $f_0\sim 20$ is a critical value), the linear growth rate has a sharp maximum and a secondary spike-bubble structure is generated. Interaction of this growing structure with the master spike-bubble leads to rupture of the latter. However, for strong…
Drift instabilities contribute to the formation of edge turbulence and zonal flows, and thus the anomalous transport in tokamaks. Experiments often found micro-scale tur- bulence strongly coupled with large-scale magnetohydrodynamic (MHD) processes, whereas a general framework has been lacking that can cover both regimes, in partic- ular, their coupling. In this paper, the linear resistive drift wave instability (DWI) is investigated using a full 2-fluid MHD model, as well as its numerical implementation in NIMROD code. Both analytical and numerical analyses reveal a macro-scale global drift w…
Planetary and magnetohydrodynamic drift-wave turbulence is observed to self-organize into large scale structures such as zonal jets and coherent vortices. In this Letter we present a non-equilibrium statistical theory, the Stochastic Structural Stability theory (SSST), that can make predictions for the formation and finite amplitude equilibration of non-zonal and zonal structures (lattice and stripe patterns) in homogeneous turbulence. This theory reveals that the emergence of large scale structure is the result of an instability of the interaction between the coherent flow and the associated …
A geometric wedge model of the near-earth nightside plasma sheet is used to derive a wave equation for low frequency shear flow-interchange waves which transmit $\vec{E} \times \vec{B}$ sheared zonal flows along magnetic flux tubes towards the ionosphere. Discrepancies with the wave equation result used in Kalmoni et al. (2015) for shear flow-ballooning instability are discussed. The shear flow-interchange instability appears to be responsible for substorm onset. The wedge wave equation is used to compute rough expressions for dispersion relations and local growth rates in the midnight region …
The kinetic features of secondary magnetic reconnection in a single flux rope undergoing internal kink instability are studied by means of three-dimensional Particle-in-Cell simulations. Several signatures of secondary magnetic reconnection are identified in the plane perpendicular to the flux rope: a quadrupolar electron and ion density structure and a bipolar Hall magnetic field develop in proximity of the reconnection region. The most intense electric fields form perpendicularly to the local magnetic field, and a reconnection electric field is identified in the plane perpendicular to the fl…
The role of different nonlinearities and potential vorticity conservation in two-dimensional fluid ITG models
Laboratoire de Physique des Plasmas, CNRS, Ecole Polytechnique, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris（等离子体物理实验室，法国国家科学研究中心，巴黎综合理工学院，索邦大学，萨克雷大学，巴黎天文台）
Thin current sheets in systems of large size that exceed a critical value of the Lundquist number are unstable to a super-Alfvenic tearing instability. The scaling of the growth rate of the fastest growing instability with respect to the Lundquist number is shown to follow from the classical dispersion relation for tearing modes. As a result of this instability, the system realizes a nonlinear reconnection rate that appears to be weakly dependent on the Lundquist number, and larger than the Sweet-Parker rate by an order of magnitude (for the range of Lundquist numbers considered). This regime …
Hall-effect thrusters (HETs) operated with xenon are one of the most commonly used electric propulsion technologies for a wide range of space missions, including drag compensation in low Earth orbit, station-keeping, and orbital insertion, as access to space becomes more affordable. Although anomalous electron transport, the electron drift instability (EDI), and secondary electron emission (SEE) have been studied experimentally and numerically in xenon-based HETs, the impact of alternative propellants is still poorly characterized. In this work, a two-dimensional particle-in-cell/Monte Carlo c…
Turbulence induced by the ion temperature gradient (ITG) is investigated in the helical and axisymmetric plasma states of a reversed field pinch device by means of gyrokinetic calculations. The two magnetic configurations are systematically compared, both linearly and nonlinearly, in order to evaluate the impact of the geometry on the instability and its ensuing transport, as well as on the production of zonal flows. Despite its enhanced confinement, the high-current helical state demonstrates a lower ITG stability threshold compared to the axisymmetric state, and ITG turbulence is expected to…
We present large scale 3D particle-in-cell (PIC) simulations to examine particle energization in magnetic reconnection of relativistic electron-positron (pair) plasmas. The initial configuration is set up as a relativistic Harris equilibrium without a guide field. These simulations are large enough to accommodate a sufficient number of tearing and kink modes. Contrary to the non-relativistic limit, the linear tearing instability is faster than the linear kink instability, at least in our specific parameters. We find that the magnetic energy dissipation is first facilitated by the tearing insta…
We propose a new reduced fluid model for the study of the drift wave -- zonal flow dynamics in magnetically confined plasmas. Our model can be viewed as an extension of the classic Hasegawa-Wakatani (HW) model, and is based on an improved treatment of the electron dynamics parallel to the field lines, to guarantee a balanced electron flux on the magnetic surfaces. Our flux-balanced HW (bHW) model contains the same drift-wave instability as previous HW models, but unlike these models, it converges exactly to the modified Hasegawa-Mima model in the collisionless limit. We rely on direct numerica…
Recently, secondary island formation due to the tearing instability of the Sweet-Parker current sheet was identified as a possible mechanism that can lead to fast reconnection (less sensitive dependence on Lundquist number $S$) both in numerical simulations using Particle-in-Cell (PIC) method [Daughton et al. 2009], as well as using resistive magnetohydrodynamics (MHD) [Lapenta 2008; Bhattacharjee et al. 2009]. This instability is thought to appear when $S$ is greater than a certain threshold. These recent results prompt us to perform more resistive MHD simulations of a basic reconnection conf…
We study the effect of a tangled sub-fG level intergalactic magnetic field (IGMF) on the electrostatic instability of a blazar-induced pair beam. Sufficiently strong IGMF may significantly deflect the TeV pair beams, which would reduce the flux of secondary cascade emission below the observational limits. A similar flux reduction may result from the electrostatic beam-plasma instability, which operates the best in the absence of IGMF. Considering IGMF with correlation lengths smaller than a kpc, we find that weak magnetic fields increase the transverse momentum of the pair beam particles, whic…
In homogeneous drift-wave (DW) turbulence, zonal flows (ZFs) can be generated via a modulational instability (MI) that either saturates monotonically or leads to oscillations of the ZF energy at the nonlinear stage. This dynamics is often attributed as the predator--prey oscillations induced by ZF collisional damping; however, similar dynamics is also observed in collisionless ZFs, in which case a different mechanism must be involved. Here, we propose a semi-analytic theory that explains the transition between the oscillations and saturation of collisionless ZFs within the quasilinear Hasegawa…
Fast magnetic reconnection plays a fundamental role in driving explosive dynamics and heating in the solar chromosphere. The reconnection time scale of traditional models is shortened at the onset of the coalescence instability, which forms a turbulent reconnecting current sheet through plasmoid interaction. In this work we aim to investigate the role of partial ionisation on the development of fast reconnection through the study of the coalescence instability of plasmoids. Unlike the processes occurring in fully ionised coronal plasmas, relatively little is known about how fast reconnection d…
Using a full kinetic, implicit particle-in-cell code, iPiC3D, we studied the properties of plasma kinetic turbulence, such as would be found at the interface between the solar wind and the Earth magnetosphere at low latitude during northwards periods. In this case, in the presence of a magnetic field B oriented mostly perpendicular to the velocity shear, turbulence is fed by the disruption of a Kelvin-Helmholtz vortex chain via secondary instabilities, vortex pairing and non-linear interactions. We found that the magnetic energy spectral cascade between ion and electron inertial scales, $d_i$ …
Magnetic reconnection mediated by the hyper-resistive plasmoid instability is studied with both linear analysis and nonlinear simulations. The linear growth rate is found to scale as $S_{H}^{1/6}$ with respect to the hyper-resistive Lundquist number $S_{H}\equiv L^{3}V_{A}/η_{H}$, where $L$ is the system size, $V_{A}$ is the Alfvén velocity, and $η_{H}$ is the hyper-resistivity. In the nonlinear regime, reconnection rate becomes nearly independent of $S_{H}$, the number of plasmoids scales as $S_{H}^{1/2}$, and the secondary current sheet length and width both scale as $S_{H}^{-1/2}$. These sc…
