Collisional Limit Research Articles

Going Beyond the MHD Approximation: Physics-based Numerical Solution of the CGL Equations

We present a new numerical model for solving the Chew–Goldberger–Low system of equations describing a bi-Maxwellian plasma in a magnetic field. Heliospheric and geospace environments are often observed to be in an anisotropic state with distinctly different parallel and perpendicular pressure components. The Chew–Goldberger–Low (CGL) system represents the simplest leading order correction to the common isotropic MHD model that still allows the incorporation of the latter’s most desirable features. However, the CGL system presents several numerical challenges: the system is not in conservation form, the source terms are stiff, and unlike MHD, it is prone to a loss of hyperbolicity if the parallel and perpendicular pressures become too different. The usual cure is to bring the parallel and perpendicular pressures closer to one another, but that has usually been done in an ad hoc manner. We present a physics-informed method of pressure relaxation based on the idea of pitch-angle scattering that keeps the numerical system hyperbolic and naturally leads to zero anisotropy in the limit of very large plasma beta. Numerical codes based on the CGL equations can, therefore, be made to function robustly for any magnetic field strength, including the limit where the magnetic field approaches zero. The capabilities of our new algorithm are demonstrated using several stringent test problems that provide a comparison of the CGL equations in the weakly and strongly collisional limits. This includes a test problem that mimics the interaction of a shock with a magnetospheric environment in 2D.

Taylor dispersion and phase mixing in the non‐cutoff Boltzmann equation on the whole space

AbstractIn this paper we describe the long‐time behavior of the non‐cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (that is, infinite Knudsen number ). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator and its interplay with the singular collision operator. For ‐wavenumbers with , one sees an enhanced dissipation effect wherein the characteristic decay time‐scale is accelerated to , where is the singularity of the kernel ( being the Landau collision operator, which is also included in our analysis); for , one sees Taylor dispersion, wherein the decay time‐scale is accelerated to . Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density , these bounds imply almost uniform‐in‐ decay of in due to phase mixing and dispersive decay.

Microwave plasma interaction in a printed transmission line for a power limiting application: from surface-wave-sustained to leaky-wave-sustained discharge

The coupling between a microwave signal and a plasma discharge in a suspended microstrip transmission line is analytically studied. Maxwell’s equations are solved in a 2D approximation to get the expressions of the electromagnetic field. The wave propagation in the guiding structure is first explored without plasma, and for several modes and frequencies. A unified characterization of the three different modes that can propagate at the interface between two dielectric media, namely the leaky waves, the pseudo-surface wave and the pure surface wave, is given in terms of of both wave vectors and electromagnetic field magnitude distribution. This analyze allow to conclude that the fundamental mode in this case is a pseudo-surface wave. Thereafter, we focus on the microwave propagation with a uniform plasma inside the guiding structure. In the non collisional limit, it appears that the plasma discharge is sustained by the so-called pure surface wave, whereas in the collisional limit, a leaky wave propagates along the plasma column. Finally, a non-uniform density profile is taken into account in the calculation. The numerical results obtained from the self-consistent simulation of the microwave-plasma coupling, in a previous work, are thus analyzed with the aid of the analytical formulas to identify the microwave coupling involved in our plasma-based microwave power limiter. The computed propagation constant from numerical data confirmed the type of coupling exhibited for a uniform electron density. Furthermore, we highlight the role of the dielectric slab, from which electromagnetic power transfer occurs into the plasma discharge.

Numerical minimization of neoclassical poloidal viscosity for supersonic equilibria in tokamak geometry

Extensive experimental evidence has shown that the presence of poloidal flow in tokamaks can dramatically improve transport properties. However, theory indicates that poloidal flows are damped by poloidal viscosity, thus necessitating external drivers, such as neutral beam injection or radio frequency heating. In this work, ideal magnetohydrodynamic equilibria are calculated via the FORTRAN code FLOW [Guazzotto et al., Phys. Plasmas 11, 604 (2004)] and a postprocessor is used to estimate the neoclassical poloidal viscosity. The equilibrium inputs, which correspond to intuitive physical quantities, are then numerically optimized to reduce a viscosity figure of merit. We present supersonic equilibria in tokamak geometry with minimized neoclassical poloidal viscosities for various velocity free function inputs, plasma aspect ratios, and collisionality regimes. Benchmarks are made against an analytic theory as well as a classical expression of poloidal viscosity. Numerical confirmation of the analytic theory is obtained in the high aspect ratio and high collisionality limit. Good agreement is also seen near the plasma core and edge, with discrepancies arising in the intermediate region. Outside of these limits, rotation input function profiles are found that provide ∼order of magnitude improvements over the analytic theory, with additional progress being made toward predictions for tokamak-relevant equilibria.

Drift kinetic theory of the NTM magnetic islands in a finite beta general geometry tokamak plasma

In (Imada et al 2019 Nucl. Fusion 59 046016 and references therein) a new 4D drift kinetic nonlinear theory, valid in the limit of a low beta, small inverse aspect ratio, circular cross section, toroidal geometry, to describe the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is derived. In (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) this theory is reduced in a low collisionality limit, which allows a dimensionality reduction to a 3D problem to efficiently resolve the collisional dissipation layer in the vicinity of the trapped-passing boundary. (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) adopts an improved model for the magnetic drift frequency, which reduces the threshold magnetic island half-width from , where is the trapped ion banana orbit width, to , making it in closer agreement with experimental observations for the large aspect ratio tokamak equilibrium. In the present paper, the theory is extended to a high beta, arbitrary tokamak geometry to capture the plasma shaping effects on the NTM threshold physics with the focus on the non-zero triangularity discharges that are known to have a strong impact on the plasma MHD stability. First, it is found that the higher triangularity plasma is more prone to NTMs which is in agreement with the tearing mode onset relative frequency measurements in DIII-D. Second, the NTM threshold dependence on the tokamak inverse aspect ratio obtained in (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) is refined and extended to a finite aspect ratio limit. Third, the NTM threshold dependence on poloidal beta is obtained and benchmarked against the EAST threshold island width measurements.

Shock structure in collisional positive ion-negative ion plasmas

The weakly nonlinear dynamics of low-frequency sound waves are studied in a positive ion-negative ion (PINI) plasma with the aim of investigating the strong collisional effect on their propagation characteristics. The nonlinear wave dynamics is shown to be governed by a Kakutani-Kawahara equation in the strong collisional limit. In contrast to the electron-ion plasmas, the temperature and inertial effects of both the ions significantly modify the nonlinearity and the dispersion of the wave. Analytical solution together with numerical simulations show that a soliton (localized pulse) first collapses into a weak noise shelf and then forms shock like structures. The relevance and potential application of the results in future numerical simulation and experimental studies of laboratory and space PINI plasmas are also discussed.

Kinetic interpretation of the classical Rayleigh-Taylor instability.

Rayleigh-Taylor (RT) instabilities are prevalent in many physical regimes ranging from astrophysical to laboratory plasmas and have primarily been studied using fluid models, the majority of which have been ideal fluid models. This work presents a five-dimensional (two spatial dimensions, three velocity space dimensions) simulation using the continuum-kinetic model to study the effect of the collisional mean free path and transport on the instability growth. The continuum-kinetic model provides noise-free access to the full particle distribution function permitting a detailed investigation of the role of kinetic physics in hydrodynamic phenomena such as the RT instability. For long mean free path, there is no RT instability growth, but as collisionality increases, particles relax towards the Maxwellian velocity distribution, and the kinetic simulations reproduce the fluid simulation results. An important and novel contribution of this work is in the intermediate collisional cases that are not accessible with traditional fluid models and require kinetic modeling. Simulations of intermediate collisional cases show that the RT instability evolution is significantly altered compared to the highly collisional fluidlike cases. Specifically, the growth rate of the intermediate collisionality RT instability is lower than the high collisionality case while also producing a significantly more diffused interface. The higher moments of the distribution function play a more significant role relative to inertial terms for intermediate collisionality during the evolution of the RT instability interface. Particle energy flux is calculated from moments of the distribution and shows that transport is significantly altered in the intermediate collisional case and deviates much more so from the high collisionality limit of the fluid regime.

Kinetic Landau-fluid closures of non-Maxwellian distributions

New kinetic Landau-fluid closures, based on the cutoff Maxwellian distribution, are derived. A special static case is considered (the frequency ω=0). In the strongly collisional regime, our model reduces to Braginskii's heat flux model, and the transport is local. In the weak collisional regime, our model indicates that the heat flux is non-local and recovers the Hammett–Perkins model while the value of the cutoff velocity approaches to infinity. We compare the thermal transport coefficient χ of Maxwellian, cutoff Maxwellian and super-Gaussian distribution. The results show that the reduction of the high-speed tail particles leads to the corresponding reduction of the thermal transport coefficient χ across the entire range of collisionality, more reduction of the free streaming transport toward the weak collisional regime. In the collisionless limit, χ approaches to zero for the cutoff Maxwellian and the super-Gaussian distribution but remains finite for Maxwellian distribution. χ is complex if the cutoff Maxwellian distribution is asymmetric. The Im(χ) approaches to different convergent values in both collisionless and strongly collisional limit, respectively. It yields an additional streaming heat flux in comparison with the symmetric cutoff Maxwellian distribution. Furthermore, due to the asymmetric distribution, there is a background heat flux q0 though there is no perturbation. The derived Landau-fluid closures are general for fluid moment models, and applicable for the cutoff Maxwellian distribution in an open magnetic field line region, such as the scape-off-layer of Tokamak plasmas, in the thermal quench plasmas during a tokamak disruption, and the super-Gaussian electron distribution function due to inverse bremsstrahlung heating in laser-plasma studies.

Extended electron tails in electrostatic microinstabilities and the nonadiabatic response of passing electrons

Ion-gyroradius-scale microinstabilities typically have a frequency comparable to the ion transit frequency. Due to the small electron-to-ion mass ratio and the large electron transit frequency, it is conventionally assumed that passing electrons respond adiabatically in ion-gyroradius-scale modes. However, in gyrokinetic simulations of ion-gyroradius-scale modes in axisymmetric toroidal magnetic fields, the nonadiabatic response of passing electrons can drive the mode, and generate fluctuations in narrow radial layers, which may have consequences for turbulent transport in a variety of circumstances. In flux tube simulations, in the ballooning representation, these instabilities reveal themselves as modes with extended tails. The small electron-to-ion mass ratio limit of linear gyrokinetics for electrostatic instabilities is presented, in axisymmetric toroidal magnetic geometry, including the nonadiabatic response of passing electrons and associated narrow radial layers. This theory reveals the existence of ion-gyroradius-scale modes driven solely by the nonadiabatic passing electron response, and recovers the usual ion-gyroradius-scale modes driven by the response of ions and trapped electrons, where the nonadiabatic response of passing electrons is small. The collisionless and collisional limits of the theory are considered, demonstrating parallels in structure and physical processes to neoclassical transport theory. By examining initial-value simulations of the fastest-growing eigenmodes, the predictions for mass-ratio scaling are tested and verified numerically for a range of collision frequencies. Insight from the small electron-to-ion mass ratio theory may lead to a computationally efficient treatment of extended modes.

A Snowball in Hell: The Potential Steam Atmosphere of TOI-1266c

Abstract TOI-1266c is a recently discovered super-Venus in the radius valley orbiting an early M dwarf. However, its notional bulk density (∼2.2 g cm−3) is consistent with a large volatile fraction, suggesting that it might have volatile reservoirs that have survived billions of years at more than twice Earth’s insolation. On the other hand, the upper mass limit paints a picture of a cool super-Mercury dominated by >50% iron core (∼9.2 g cm−3) that has tiptoed up to the collisional stripping limit and into the radius gap. Here we examine several hypothetical states for TOI-1266c using a combination of new and updated open-source atmospheric escape, radiative−convective, and photochemical models. We find that water-rich atmospheres with trace amounts of H2 and CO2 are potentially detectable (S/N > ∼ 5) in less than 20 hr of James Webb Space Telescope (JWST) observing time. We also find that water vapor spectral features are not substantially impacted by the presence of high-altitude water or ice clouds owing to the presence of a significant amount of water above the cloud deck, although further work with self-consistent cloud models is needed. Regardless of its mass, however, TOI-1266c represents a unique proving ground for several hypotheses related to the evolution of sub-Neptunes and Venus-like worlds, particularly those near the radius valley.

Dissipative magnetohydrodynamics for nonresistive relativistic plasmas: An implicit second-order flux-conservative formulation with stiff relaxation

Based on a 14-moment closure for nonresistive (general-) relativistic viscous plasmas, we describe a new numerical scheme that is able to handle all first-order dissipative effects (heat conduction, bulk and shear viscosities), as well the anisotropies induced by the presence of magnetic fields. The latter is parametrized in terms of a thermal gyrofrequency or, equivalently, a thermal Larmor radius and allows to correctly capture the thermal Hall effect. By solving an extended Israel-Stewart-like system for the dissipative quantities that enforces algebraic constraints via stiff-relaxation, we are able to cast all first-order dissipative terms in flux-divergence form. This allows us to apply traditional high-resolution shock capturing methods to the equations, making the system suitable for the numerical study of highly turbulent flows. We present several numerical tests to assess the robustness of our numerical scheme in flat spacetime. The 14-moment closure can seamlessly interpolate between the highly collisional limit found in neutron star mergers, and the highly anisotropic limit of relativistic Braginskii magnetohydrodynamics appropriate for weakly collisional plasmas in black-hole accretion problems. We believe that this new formulation and numerical scheme will be useful for a broad class of relativistic magnetized flows.

Semi-implicit Hybrid Discrete $$\left( \text {H}^T_N\right) $$ Approximation of Thermal Radiative Transfer

The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing numerical solutions of the TRT equations accurately and efficiently is challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the P\(_N\) (spherical harmonics) or the S\(_N\) (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H\(^T_N\)) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both P\(_N\) and S\(_N\), and indeed reduces to each of these approximations in various limits: H\(^1_N\) \(\equiv \) P\(_N\) and H\(^T_0\) \(\equiv \) S\(_T\). We prove that H\(^T_N\) results in a system of hyperbolic partial differential equations for all \(T\ge 1\) and \(N\ge 0\). Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop an asymptotic-preserving numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge–Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the H\(^T_N\) ansatz and the numerical discretizations.

Ion temperature anisotropy in the tokamak scrape-off layer

Understanding tokamak exhaust-power heat loads on divertor plates depends critically on having a realistic model of the scrape-off layer (SOL) plasma. The Braginskii fluid model is often solved to understand the SOL plasma behavior. This model is based on the collisional limit for transport along the magnetic field . The ions and electron gyrofrequencies are assumed to be much larger than the Coulomb collision frequencies, which are nonetheless, sufficiently large to yield common parallel and perpendicular temperatures for each species, i.e. the temperatures are assumed to be isotropic. In certain circumstances such as encountered for the tokamak H-mode, the ion temperature can be quite anisotropic. In this work, the anisotropy effects are implemented in the two-dimensional (2D) transport code UEDGE. Various geometries (1D slab, 2D slab and a toroidal tokamak geometry) are used to study the 2D structure of ion temperature anisotropy and its effects on plasma transport in detail. Results show that the effects of ion temperature anisotropy on the plasma parallel transport are substantial near the magnetic X-point, which leads to different steady state density profiles in the divertor regions. The extra mirror force introduced by ion temperature anisotropy can be one of the main forces contributing to the plasma flow in the SOL.

Fast magnetic reconnection induced by resistivity gradients in 2D magnetohydrodynamics

Using two-dimensional (2D) magnetohydrodynamics simulations, we show that Petschek-type magnetic reconnection can be induced using a simple resistivity gradient in the reconnection outflow direction, revealing the key ingredient of steady fast reconnection in the collisional limit. We find that the diffusion region self-adjusts its half-length to fit the given gradient scale of resistivity. The induced reconnection x-line and flow stagnation point always reside within the resistivity transition region closer to the higher resistivity end. The opening of one exhaust by this resistivity gradient will lead to the opening of the other exhaust located on the other side of the x-line, within the region of uniform resistivity. Potential applications of this setup to reconnection-based thrusters and solar spicules are discussed. In a separate set of numerical experiments, we explore the maximum plausible reconnection rate using a large and spatially localized resistivity right at the x-line. Interestingly, the resulting current density at the x-line drops significantly so that the normalized reconnection rate remains bounded by the value ≃0.2, consistent with the theoretical prediction.

New Particle Formation from the Vapor Phase: From Barrier-Controlled Nucleation to the Collisional Limit

Studies of vapor phase nucleation have largely been restricted to one of two limiting cases—nucleation controlled by a substantial free energy barrier or the collisional limit where the barrier is negligible. For weakly bound systems, exploring the transition between these regimes has been an experimental challenge, and how nucleation evolves in this transition remains an open question. We overcome these limitations by combining complementary Laval expansion experiments, providing new particle formation data for carbon dioxide over a uniquely broad range of conditions. Our experimental data together with a kinetic model using rate constants from high-level quantum chemical calculations provide a comprehensive picture of new particle formation as nucleation transitions from a barrier-dominated process to the collisional limit.

Ion attachment rates and collection forces on dust particles in a plasma sheath with finite ion inertia and mobility.

Ion attachment and ion drag to dust particles near the edge of a nonthermal plasma sheath are of interest to better understand how particles become trapped in such sheath regions. While electron-particle collisions in plasmas and sheaths can often be described by orbital motion limited theory, quantification of ion transport about dust particles in collisional sheath regions requires a distinct modeling approach. In this work, the dimensionless ion attachment coefficients and dimensionless collection forces on negatively charged particles are calculated using ion trajectory models accounting for an external electric field in a collisional sheath, ion inertia, and finite ion mobility. By considering both ion inertia and finite ion mobility, results apply for ion transport from the fully collisional regime into a regime of intermediate collisionality. Ion collection forces are calculated in two related limits; first, the nondissipative limit, wherein the dimensionless collection force function coincides with the dimensionless attachment coefficient (anticipated in the collisionless regime), and second, a dissipative limit, wherein neutral gas collisions dissipate ion momentum, which strongly affects the resulting collection force (anticipated in the fully collisional regime). We show that ion motion about a charged particle can be parametrized by the ion Stokes number, which is the ratio of ion inertia to gas resistance to motion and dimensionless electric field strength (the external field strength normalized by the electric field at the particle surface). At intermediate Stokes numbers (10^{1}-10^{2}), ions adopt trajectories that are extremely sensitive to the initial ion-particle impact parameter. Plots of the resulting collision angle at fixed Stokes number and dimensionless field strength as a function of impact parameter contain multiple discontinuities. Nonetheless, we obtain smooth curves for the ion attachment rates and collection forces in both the nondissipative and fully dissipative limits. Increasing the ion Stokes number is found to significantly decrease the dimensionless ion attachment coefficients and ion collection forces in comparison to coefficients evaluated with expressions derived in the fully collisional limit. In all instances, including the dissipative limit, we find the ion collection force acts in the direction of ion migration. Neural network fits are utilized to parametrize the resulting attachment coefficients and ion collection forces, and we apply these fits to examine the charge levels on 1-μm radius particles in external fields in the 3×10^{2}-3×10^{3}Vm^{-1} range and pressures in the 5×10^{-1}-5×10^{1} Torr (66.7-6667 Pa) range. We find the charge level is strongly sensitive to both field strength and pressure in the plasma sheath, ranging from 6 × 10^{3} to 1.8 × 10^{4} over the conditions examined. Calculations are also used to demonstrate that the ion collection force can be sufficiently strong to trap particles not only close to the bottom electrode of a parallel-plate reactor, but also close to the top electrode, with a critical ion density required for trapping.

Magnetic cusp confinement in low-β plasmas revisited

Magnetic cusps have been used for more than 50 years to limit charged particle losses to the walls and confine the plasma in a large variety of plasma sources or ion sources. Quantification of the effective loss area has been the subject of many experimental as well as theoretical investigations in the 1970s–1990s. In spite of this effort, there is no fully reliable expression of the effective wall loss as a function of cusp magnetic field, electron temperature, ion mass, gas pressure, etc. We describe in this paper a first attempt at obtaining scaling laws for the effective loss width of magnetic cusps based on two-dimensional particle-in-cell Monte Carlo collision simulations. The results show that the calculated leak width follows a 1/B scaling in the collisionless low B limit, is approximately proportional to the hybrid gyroradius with an ion velocity equal to the Bohm velocity, and is proportional to the square root of gas pressure in the collisional limit.

Two-dimensional partially ionized magnetohydrodynamic turbulence

Abstract

Reduction of the Coulomb logarithm due to electron-neutral collisions.

The frictional force (stopping power) acting on a test electron moving through an ideal electron gas is calculated by taking into account electron-neutral atom collisions using the linear plasma response formalism. This allows us to elucidate how the effective Coulomb logarithm is affected by electron-neutral collisions. In agreement with a recent investigation by Hagelaar, Donko, and Dyatko [Phys. Rev. Lett. 123, 025004 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.025004], we observe that the effective Coulomb logarithm decreases considerably due to electron-neutral collisions and becomes inversely proportional to the collision frequency in the highly collisional limit.

Catalytic Oxidation of CO by N2O Enabled by Al2O2/3+: Temperature Dependent Kinetics and Statistical Modeling.

The reactions of Al2O2+ + N2O and Al2O3+ + CO, forming a catalytic cycle oxidizing CO by N2O, have been investigated from 300 to 600 K in a variable ion source, temperature adjustable, selected-ion flow tube (VISTA-SIFT). Reaction coordinates have been calculated using density functional theory and statistical modeling of those surfaces compared to experimental kinetics data for mechanistic insight. The reaction of Al2O2+ + N2O proceeds at the Su-Chesnavich collisional limit at all temperatures studied, yielding only Al2O3+, with the exception of a small (<5%) amount of association product, Al2O2(N2O)+ at 300 K. The reaction of Al2O3+ with CO produces Al2O2+ with a rate constant of 4.7 ± 1.2 × 10-10 cm3 s-1 at 300 K, decreasing with temperature as T-0.5±0.2. In addition, a significant amount of association product, Al2O3(CO)+, was observed with rate constants for formation ranging from 10-11 to 10-10 cm3 s-1 dependent upon He buffer gas concentration and temperature. Implications of these kinetic measurements with regard to the reactive surface are discussed.