The linear instability and nonlinear dynamics of collisional (resistive) and collisionless (due to electron inertia) double tearing modes (DTMs) are compared with the use of a reduced cylindrical model of a tokamak plasma. We focus on cases where two q = 2 resonant surfaces are located a small distance apart. It is found that regardless of the magnetic reconnection mechanism, resistivity or electron inertia, the fastest growing linear eigenmodes may have high poloidal mode numbers m ~ 10. The spectrum of unstable modes tends to be broader in the collisionless case. In the nonlinear regime, it is shown that in both cases fast growing high-m DTMs lead to an annular collapse involving small magnetic island structures. In addition, collisionless DTMs exhibit multiple reconnection cycles due to reversibility of collisionless reconnection and strong ExB flows. Collisionless reconnection leads to a saturated stable state, while in the collisional case resistive decay keeps the system weakly dynamic by driving it back towards the unstable equilibrium maintained by a source term.
A unified linear tearing-mode formulation is given incorporating both resistivity and Hall effects. A variational method is used that appears to be best suited to deal with the difficulties peculiar to the {\it triple-deck} structure associated with the Hall resistive tearing mode but also to lead to a convenient analytical dispersion relation for the Hall resisitive tearing mode. This analytical dispersion relation - * recovers the Furth-Killeen-Rosenbluth[15] result for the resistive branch; * gives a growth rate for the Hall branch which appears to be consistent with the growth rate of the …
Microtearing modes have been widely reported as a tearing parity electron temperature gradient driven plasma instability, which leads to fine scale tearing of the magnetic flux surfaces thereby resulting in reconnection of magnetic field lines and formation of magnetic islands. In slab geometry it has previously been shown that the drive mechanism requires a finite collision frequency. However, we find in linear gyrokinetic simulations that a collisionless fine-scale tearing parity instability exists even at low and zero collision frequency. Detailed studies reveal that these slab modes are al…
The present paper provides comprehensive description of various regimes involved in the two-fluid model of the resistive tearing instability. These include two novel regimes of this instability, which correspond to the long-wave modes that can develop in a highly-elongated current sheet. This issue is relevant to the study of fast magnetic reconnection and magnetic turbulence in magnetohydrodynamic (MHD) objects with a large value of the Lundquist number.
Recent MHD simulations find that the reversal of the local resistive interchange parameter $D_R$ from negative to positive due to impurity radiation cooling is able to trigger the resistive tearing mode growth in a tokamak above a threshold in impurity level. A layer of perturbed Pfirsch-Schlüter current density and resistivity are also induced by the impurity radiation, which further govern the tearing mode growth and saturation in the nonlinear stage. The impurity threshold and the tearing mode growth strongly depend on the parallel thermal conductivity, and such a dependence derives from th…
Tearing modes with real frequencies in the plasma frame (i.e. in addition to the Doppler shift due to $E\times B$ rotation) are of potential importance because of their effect on the locking process. In particular, it has recently been shown [J. M. Finn, A. J. Cole, and D. P. Brennan, Phys. Plasmas (Letters), 22:120701, 2015] that the Maxwell torque on the plasma in the presence of an applied error field is modified significantly for tearing modes having real frequencies near marginal stability. In addition, it is known [J. M. Finn and R. A. Gerwin, Phys. Plasmas, 3:2344, 1996] that resistive …
A model system of equations has been derived to describe a toroidally rotating tokamak plasma, unstable to Resistive Wall Modes (RWMs) and metastable to Neoclassical Tearing Modes (NTMs), using a linear RWM model and a nonlinear NTM model. If no wall is present, the NTM growth shows the typical threshold/saturation island widths, whereas a linearly unstable kink mode grows exponentially in this model plasma system. When a resistive wall is present, the growth of the linearly unstable RWM is accelerated by an unstable island: a form of coupled RWM-NTM mode. Crucially, this coupled system has no…
The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for a Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle ($θ$) with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes ($θ\neq 0$). We find th…
In this study, the evolution of a highly unstable m = 1 resistive tearing mode, leading to plasmoid formation in a Harris sheet is studied in the framework of full MHD model using the NIMROD simulation. Following the initial nonlinear growth of the primary m = 1 island, the X-point develops into a secondary elongated current sheet that eventually breaks into plasmoids. Two distinctive viscous regimes are found for the plasmoid formation and saturation. In the low viscosity regime (i.e. P r . 1), the plasmoid width increases sharply with viscosity, whereas in the viscosity dominant regime (i.e.…
The effects of line-tying on resistive tearing instability in slab geometry is studied within the framework of reduced magnetohydrodynamics (RMHD).\citep{KadomtsevP1974,Strauss1976} It is found that line-tying has a stabilizing effect. The tearing mode is stabilized when the system length $L$ is shorter than a critical length $L_{c}$, which is independent of the resistivity $η$. When $L$ is not too much longer than $L_{c}$, the growthrate $γ$ is proportional to $η$ . When $L$ is sufficiently long, the tearing mode scaling $γ\simη^{3/5}$ is recovered. The transition from $γ\simη$ to $γ\simη^{3/…
The nonlinear evolution of resistive double tearing modes (DTMs) with safety factor values q=1 and q=3 is studied in a reduced cylindrical model of a tokamak plasma. We focus on cases where the resonant surfaces are a small distance apart. Recent numerical studies have shown that in such configurations high-m modes are strongly unstable. In this paper, it is first demonstrated that linear DTM theory predicts the dominance of high-m DTMs. A semi-empirical formula for estimating the poloidal mode number of the fastest growing mode, m_peak, is obtained from the existing linear theory. Second, usi…
In magnetic configurations with two or three q=1 (with q being the safety factor) resonant surfaces in a tokamak plasma, resistive magnetohydrodynamic modes with poloidal mode numbers m much larger than 1 are found to be linearly unstable. It is found that these high-m double or triple tearing modes significantly enhance through nonlinear interactions, the growth of the m=1 mode. This may account for the sudden onset of the internal resistive kink, i.e., the fast sawtooth trigger. Based on the subsequent reconnection dynamics that can proceed without formation of the m=1 islands, it is propose…
Linear gyrokinetic simulations covering the collisional -- collisionless transitional regime of the tearing instability are performed. It is shown that the growth rate scaling with collisionality agrees well with that predicted by a two-fluid theory for a low plasma beta case in which ion kinetic dynamics are negligible. Electron wave-particle interactions (Landau damping), finite Larmor radius, and other kinetic effects invalidate the fluid theory in the collisionless regime, in which a general non-polytropic equation of state for pressure (temperature) perturbations should be considered. We …
Kinetic simulations have demonstrated that three-dimensional reconnection in collisionless regimes proceeds through the formation and interaction of magnetic flux ropes, which are generated due to the growth of tearing instabilities at multiple resonance surfaces. Since kinetic simulations are intrinsically expensive, it is desirable to explore the feasibility of reduced two-fluid models to capture this complex evolution, particularly, in the strong guide field regime, where two-fluid models are better justified. With this goal in mind, this paper compares the evolution of the collisionless te…
We examine the effect of an external guide field and current sheet thickness on the growth rates and nature of three dimensional unstable modes of an electron current sheet driven by electron shear flow. The growth rate of the fastest growing mode drops rapidly with current sheet thickness but increases slowly with the strength of the guide field. The fastest growing mode is tearing type only for thin current sheets (half thickness $\approx d_e$, where $d_e=c/ω_{pe}$ is electron inertial length) and zero guide field. For finite guide field or thicker current sheets, fastest growing mode is non…
Effect of externally applied resonant magnetic perturbations on resistive tearing modes
State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology（华中科技大学电磁工程与技术国家重点实验室）；Max-Planck-Institut für Plasmaphysik, EURATOM association（德国马普等离子体物理研究所，欧洲原子能共同体联合机构）
A mechanism for explosive magnetic reconnection is investigated by analyzing the nonlinear evolution of a collisionless tearing mode in a two-fluid model that includes the effects of electron inertia and temperature. These effects cooperatively enable a fast reconnection by forming an X-shaped current-vortex layer centered at the reconnection point. A high-resolution simulation of this model for an unprecedentedly small electron skin depth $d_e$ and ion-sound gyroradius $ρ_s$, satisfying $d_e=ρ_s$, shows an explosive tendency for nonlinear growth of the tearing mode, where it is newly found th…
The role of anisotropic thermal diffusivity on tearing mode stability is analysed in general toroidal geometry. A dispersion relation linking the growth rate to the tearing mode stability parameter, Delta, is derived. By using a resistive MHD code, modified to include such thermal transport, to calculate tearing mode growth rates, the dispersion relation is employed to determine Delta in situations with finite plasma pressure that are stabilised by favourable average curvature in a simple resistive MHD model. We also demonstrate that the same code can also be used to calculate the basis-functi…
The effect of shear flow on the resistive tearing instability
University of California, Berkeley（加州大学伯克利分校）；University of Colorado, Boulder（科罗拉多大学博尔德分校）；University of Maryland, College Park（马里兰大学帕克学院）；Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）
Calculations of tearing mode stability in tokamaks split conveniently into an external region, where marginally stable ideal MHD is applicable, and a resonant layer around the rational surface where sophisticated kinetic physics is needed. These two regions are coupled by the stability parameter. Pressure and current perturbations localized around the rational surface alter the stability of tearing modes. Equations governing the changes in the external solution and - are derived for arbitrary perturbations in axisymmetric toroidal geometry. The relationship of - with and without pressure flatt…
Tearing Instability in Gyrotropic MHD: Effects of Equilibrium Pressure Anisotropy
Escola de Artes, Ciências e Humanidades, University of São Paulo（圣保罗大学艺术、科学与人文学院）；Astrophysics Division, National Institute for Space Research（国家空间研究所天体物理部）
In this paper, we study the effects of viscosity on the evolution of double tearing mode (DTM) in a pair of adjacent Harris sheets based on the resistive MHD model in the NIMROD code. Similar to the tearing mode in the conventional single Harris sheet, a transition is observed in the generation of both normal and monster plasmoids at Pr = 1. In the Pr < 1 regime of DTM, normal plasmoids (small plasmoids) are generated along with monster plasmoid, whereas in the single tearing mode (STM) cases such a generation is not observed. When Pr is above the critical value, the generation of monster plas…
The potential formation of the quasi-single-helicity (QSH) state in the Keda Torus eXperiment (KTX) is investigated in resistive MHD simulations using the NIMROD code. We focus on the effects of finite resistivity on the mode structure and characteristics of the dominant linear and nonlinear resistive tearing-mode in a finite $β$, cylindrical configuration of reversed field pinch model for KTX. In the typical resistive regimes of KTX where Lundquist number $S=5 \times 10^4$, the plasma transitions to a steady QSH state after evolving through an initial transient phase with multiple helicities.…
We study a resistive tearing instability developing in a system evolving through the combined effect of Hall drift in the Electron-MHD limit and Ohmic dissipation. We explore first the exponential growth of the instability in the linear case and we find the fastest growing mode, the corresponding eigenvalues and dispersion relation. The instability growth rate scales as $γ\propto B^{2/3} σ^{-1/3}$ where $B$ is the magnetic field and $σ$ the electrical conductivity. We confirm the development of the tearing resistive instability in the fully non-linear case, in a plane parallel configuration wh…
The linear and nonlinear evolutions of the tearing instability in a collisionless plasma with a strong guide field are analyzed on the basis of a two-field Hamiltonian gyrofluid model. The model is valid for a low ion temperature and a finite $β_e$. The finite $β_e$ effect implies a magnetic perturbation along the guide field direction and electron finite Larmor radius effects. A Hamiltonian derivation of the model is presented. A new dispersion relation of the tearing instability is derived for the case $β_e=0$ and tested against numerical simulations. For $β_e \ll 1$ the equilibrium electron…
This paper discusses the transition to fast growth of the tearing instability in thin current sheets in the collisionless limit where electron inertia drives the reconnection process. It has been previously suggested that in resistive MHD there is a natural maximum aspect ratio (ratio of sheet length and breadth to thickness) which may be reached for current sheets with a macroscopic length L, the limit being provided by the fact that the tearing mode growth time becomes of the same order as the Alfvèn time calculated on the macroscopic scale (Pucci and Velli (2014)). For current sheets with a…
The dynamics of a global reconnection in the presence of a poloidal shear flow which is located in between magnetic islands is investigated. Different linear regimes are identified according to the value of the resistivity and the distance between the low-order resonant surfaces. It is found that the presence of a small shear flow affects and significantly delays the global reconnection processes. It is shown that this delay is linked to a breaking of symmetry imposed by the existence of the shear flow and the generation of a mean poloidal flow in the resistive layers.
Dynamics of the magnetospheric plasma configuration intrinsically features intermittent slow and fast phases. The fast transition is a nonlinear process, loss of equilibrium which ends up the slow quasi-static evolution. The process is analysed as a dynamical bifurcation. It appears when marginal stability state is reached in the course of that evolution, either for tearing mode or for ballooning mode disturbances. The resulting force imbalance leads to spontaneous formation of nonlinear kinetic thin current structures. Those are either a pair of slow collisionless shocks or a specific anisotr…
We study, by means of MHD simulations, the onset and evolution of fast reconnection via the "ideal" tearing mode within a collapsing current sheet at high Lundquist numbers ($S\gg10^4$). We first confirm that as the collapse proceeds, fast reconnection is triggered well before a Sweet-Parker type configuration can form: during the linear stage plasmoids rapidly grow in a few Alfvén times when the predicted "ideal" tearing threshold $S^{-1/3}$ is approached from above; after the linear phase of the initial instability, X-points collapse and reform nonlinearly. We show that these give rise to a …
Magnetic reconnection can explosively release magnetic energy when opposing magnetic fields merge and annihilate through a current sheet, driving plasma jets and accelerating non-thermal particle populations to high energy, in plasmas ranging from space and astrophysical to laboratory scales. Through laboratory experiments and spacecraft observations, significant experimental progress has been made in demonstrating how fast dissipation and reconnection occurs in narrow, kinetic-scale current sheets. However, a challenge has been to demonstrate what triggers reconnection and how it proceeds rap…
Turbulence in a conducting plasma can amplify seed magnetic fields in what is known as the turbulent, or small-scale, dynamo. The associated growth rate and emergent magnetic-field geometry depend sensitively on the material properties of the plasma, in particular on the Reynolds number ${\rm Re}$, the magnetic Reynolds number ${\rm Rm}$, and their ratio ${\rm Pm}\equiv{\rm Rm}/{\rm Re}$. For ${\rm Pm} > 1$, the amplified magnetic field is gradually arranged into a folded structure, with direction reversals at the resistive scale and field lines curved at the larger scale of the flow. As the m…
The nonlinear dynamo effect of tearing modes is derived with the resistive MHD equations. The dynamo effect is divided into two parts, parallel and perpendicular to the magnetic field. Firstly, the force-free plasma is considered. It is found that the parallel dynamo effect drives opposite current densities at the different sides of the rational surface, making the $λ=\boldsymbol{j}\cdot\boldsymbol{B}/|\boldsymbol{B}|^2$ profile completely flattened near the rational surface. There are many rational surfaces for the turbulent plasma, which means the plasma is tending to relax into the Taylor s…
The effects that energetic trapped ions have on linear resistive magnetohydrodynamic (MHD) instabilities are studied in a reduced model that captures the essential physics driving or damping the modes through variations in the magnetic shear. The drift-kinetic orbital interaction of a slowing down distribution of trapped energetic ions with a resistive MHD instability is integrated to a scalar contribution to the perturbed pressure, and entered into an asymptotic matching formalism for the resistive MHD dispersion relation. Toroidal magnetic field line curvature is included to model trapping i…
In a magnetized, collisionless plasma, the magnetic moment of the constituent particles is an adiabatic invariant. An increase in the magnetic-field strength in such a plasma thus leads to an increase in the thermal pressure perpendicular to the field lines. Above a $β$-dependent threshold (where $β$ is the ratio of thermal to magnetic pressure), this pressure anisotropy drives the mirror instability, producing strong distortions in the field lines on ion-Larmor scales. The impact of this instability on magnetic reconnection is investigated using a simple analytical model for the formation of …
The neoclassical polarization current, generated when a magnetic island propagates through a tokamak plasma, is believed to influence the initial stage of the neoclassical tearing mode evolution. Understanding the strength of its contribution in the relevant plasma collision frequency regimes for future tokamaks such as ITER is crucial for the successful control and/or avoidance of the neoclassical tearing mode. A nonlinear drift kinetic theory is employed to determine the full collision frequency dependence of the neoclassical polarization current in the small island limit, comparable to the …
A mechanism for fast magnetic reconnection in collisionless plasma is studied for understanding sawtooth collapse in tokamak discharges. Nonlinear growth of the tearing mode driven by electron inertia is analytically estimated by invoking the energy principle for the first time. Decrease of potential energy in the nonlinear regime (where the island width exceeds the electron skin depth) is found to be steeper than in the linear regime, resulting in acceleration of the reconnection. Release of free energy by such ideal fluid motion leads to unsteady and strong convective flow, which theoretical…
Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro kinetic turbulence code, GKW . The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate and frequency of the mode were investigated by varying the current profile, collisionality and the pressure gradients. Both collision-less and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absense of a pres…
The first detailed comparison between gyrokinetic and gyrofluid simulations of collisionless magnetic reconnection has been carried out. Both the linear and nonlinear evolution of the collisionless tearing mode have been analyzed. In the linear regime, we have found a good agreement between the two approaches over the whole spectrum of linearly unstable wave numbers, both in the drift kinetic limit and for finite ion temperature. Nonlinearly, focusing on the small-$Δ'$ regime, with $Δ'$ indicating the standard tearing stability parameter, we have compared relevant observables such as the evolu…
A mechanism for fast magnetic reconnection in collisionless plasma is studied for understanding sawtooth collapse in tokamak discharges by using a two-fluid model for cold ions and electrons. Explosive growth of the tearing mode enabled by electron inertia is analytically estimated by using an energy principle with a nonlinear displacement map. Decrease of the potential energy in the nonlinear regime (where the island width exceeds the electron skin depth) is found to be steeper than in the linear regime, resulting in accelerated reconnection. Release of potential energy by such a fluid displa…
We present experimental results of magnetic feedback control on the m=2, n=1 tearing mode in RFX-mod operated as a circular ohmically heated tokamak. The feedback suppression of the non-resonant m=2, n=1 Resistive Wall Mode (RWM) in q(a)<2 plasmas is a well-established result of RFX-mod. The control of the tearing counterpart, which develops in q(a)>2 equilibrium, is instead a more difficult issue. In fact, the disruption induced by a growing amplitude m=2, n=1 tearing mode can be prevented by feedback only when the resonant surface q=2 is close to the plasma edge, namely 2<q(a)<2.5, and the e…
The energy spectrum of tearing mode turbulence in a sheared background magnetic field is studied in this work. We consider the scenario where the nonlinear interaction of overlapping large-scale modes stirs up a broad spectrum of small-scale modes, generating tearing mode turbulence. The spectrum of such turbulence is of interest since it is relevant to the small-scale back-reaction to the large-scale field. The turbulence we discuss here differs from traditional MHD turbulence mainly in two aspects, one being the existence of many linearly stable small-scale modes which cause an effective dam…
Resistive instabilities of current sheets in stratified plasmas with a gravitational field
Institute for Fusion Studies, Department of Physics, The University of Texas at Austin（德克萨斯大学奥斯汀分校物理系聚变研究所）；Department of Physics, Faculty of Science, Assiut University（阿西尤特大学理学院物理系）
We investigate the effects of runaway electron current on the dispersion relation of resistive magnetohydrodynamic modes in tokamaks. We present a new theoretical model to derive the dispersion relation, which is based on the asymptotic analysis of the resistive layer structure of the modes. It is found that in addition to the conventional resistive layer, a new runaway current layer can emerge whose properties depend on the ratio of the Alfvén velocity to the runaway electron convection speed. Due to the contribution from this layer, both the tearing mode and kink mode will have a real freque…
