A nonlinear gyrokinetic formalism for low‐frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong‐turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magnetic field geometries. The specific case of axisymmetric tokamaks is then considered and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfven wave heating scheme, it is found that nonlinear ion Landau damping of kinetic shear‐Alfven waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency.

Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria

Energetic Particle Physics on the HL-2A Tokamak: A Review

The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a $\mathbb{Z}_2$ quantum spin liquid, and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the universal conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. Moreover, a dynamically self-dual optical conductivity emerges at low temperatures above the transition point, indicating the presence of the elusive vison particles. Our study opens the door for the experimental detection of anyons in a broader regime, and has ramifications in the study of quantum materials, programmable quantum simulators, and ultra-cold atomic gases. In the latter case, we discuss the feasibility of measurements in optical lattices using current techniques.

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Fractionalized conductivity and emergent self-duality near topological phase transitions

The experimental discovery of the fractional Hall conductivity in two-dimensional electron gases revealed new types of quantum particles, called anyons, which are beyond bosons and fermions as they possess fractionalized exchange statistics. These anyons are usually studied deep inside an insulating topological phase. It is natural to ask whether such fractionalization can be detected more broadly, say near a phase transition from a conventional to a topological phase. To answer this question, we study a strongly correlated quantum phase transition between a topological state, called a $${{\mathbb{Z}}}_{2}$$
 quantum spin liquid, and a conventional superfluid using large-scale quantum Monte Carlo simulations. Our results show that the universal conductivity at the quantum critical point becomes a simple fraction of its value at the conventional insulator-to-superfluid transition. Moreover, a dynamically self-dual optical conductivity emerges at low temperatures above the transition point, indicating the presence of the elusive vison particles. Our study opens the door for the experimental detection of anyons in a broader regime, and has ramifications in the study of quantum materials, programmable quantum simulators, and ultra-cold atomic gases. In the latter case, we discuss the feasibility of measurements in optical lattices using current techniques. Conventional quantum particles can break up into fractionalized excitations under the right conditions; however, their direct experimental observation is challenging. Here, the authors predict strong optical conductivity signatures of such excitations in the vicinity of a topological phase transition.

/pdf/fractionalized-conductivity-and-emergent-self-duality-near-53torwelbw.pdf

Fractionalized conductivity and emergent self-duality near topological phase transitions.

The two-field equations governing fully nonlinear dynamics of the drift wave (DW) and geodesic acoustic mode (GAM) in the toroidal geometry are derived in nonlinear gyrokinetic framework. Two stages with distinctive features are identified and analyzed. In the linear growth stage, the set of nonlinear equations can be reduced to the intensively studied parametric decay instability (PDI), accounting for the spontaneous resonant excitation of GAM by DW. The main results of previous works on spontaneous GAM excitation, e.g., the much enhanced GAM group velocity and the nonlinear growth rate of GAM, are reproduced from numerical solution of the two-field equations. In the fully nonlinear stage, soliton structures are observed to form due to the balancing of the self-trapping effect by the spontaneously excited GAM and kinetic dispersiveness of DW. The soliton structures enhance turbulence spreading from DW linearly unstable to stable region, exhibiting convective propagation instead of typical linear dispersive process, and is thus, expected to induce core-edge interaction and nonlocal transport.

Soliton generation and drift wave turbulence spreading via geodesic acoustic mode excitation

Two novel nonlinear mode coupling processes for reversed shear Alfven eigenmode (RSAE) nonlinear saturation are proposed and investigated. In the first process, RSAE nonlinearly couples to a co-propagating toroidal Alfven eigenmode (TAE) with the same toroidal and poloidal mode numbers, and generates a geodesic acoustic mode (GAM). In the second process, RSAE couples to a counter-propagating TAE and generates an ion acoustic wave quasi-mode (IAW). The condition for the two processes to occur is favored during current ramp. Both processes contribute to effectively saturate the Alfvenic instabilities, as well as nonlinearly transfer of energy from energetic fusion alpha particles to fuel ions in burning plasmas.

Nonlinear Coupling of Reversed Shear Alfvén Eigenmode and Toroidal Alfvén Eigenmode during Current Ramp

General nonlinear equations describing reversed shear Alfven eigenmode (RSAE) self-modulation via zero-frequency zonal structure (ZFZS) generation are derived using nonlinear gyrokinetic theory, which are then applied to study the spontaneous ZFZS excitation as well as RSAE nonlinear saturation. It is found that both electrostatic zonal flow and electromagnetic zonal current can be preferentially excited by finite-amplitude RSAE, depending on specific plasma parameters. The modification to local shear Alfven wave continuum is evaluated using the derived saturation level of zonal current, which is shown to play a comparable role in saturating RSAE with the ZFZS scattering.

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Nonlinear reversed shear Alfvén eigenmode saturation due to spontaneous zonal current generation

The local stability of ion-temperature gradient driven mode (ITG) in the presence of a given radial electric field is investigated using gyrokinetic theory and ballooning mode formalism with toroidal effect accounted for. It is found that zero frequency radial electric field-induced poloidal rotation can significantly stabilize ITG, while the associated density perturbation has little effect on ITG stability due to the modification of finite-orbit-width effect. However, the parallel mode structure is slightly affected due to the evenly symmetric density modulation associated with the zero-frequency radial electric field.

Effects of radial electric field on ion temperature gradient driven mode stability

A novel channel for fuel ions heating in tokamak core plasma is proposed and analyzed using nonlinear gyrokinetic theory. The channel is achieved via spontaneous decay of reversed shear Alfvén eigenmode (RSAE) into low frequency Alfvén modes, which then heat fuel ions via collisionless ion Landau damping. The conditions for RSAE spontaneous decay are investigated, and the saturation level and the consequent fuel ion heating rate are also derived. The channel is expected to be crucial for future reactors operating under reversed shear configurations, where fusion alpha particles are generated in the tokamak core with the magnetic shear being, typically, reversed, and there is a dense RSAE spectrum due to the small alpha particle characteristic dimensionless orbits.

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Shizhao Wei

Papers

Nonlinear Coupling of Reversed Shear Alfvén Eigenmode and Toroidal Alfvén Eigenmode during Current Ramp

Soliton generation and drift wave turbulence spreading via geodesic acoustic mode excitation

Nonlinear reversed shear Alfvén eigenmode saturation due to spontaneous zonal current generation

Effects of radial electric field on ion temperature gradient driven mode stability

Core localized alpha-channeling via low frequency Alfvén mode generation in reversed shear scenarios