arXiv subjects

Ammar Hakim

Publications and source records attributed to Ammar Hakim.

At least 37 records · Page 2Linked to original sources

arXiv2023

Convolutional layers are equivariant to discrete shifts but not continuous translations

The purpose of this short and simple note is to clarify a common misconception about convolutional neural networks (CNNs). CNNs are made up of convolutional layers which are shift equivariant due to weight sharing. However, convolutional layers are not translation equivariant, even when boundary effects are ignored and when pooling and subsampling are absent. This is because shift equivariance is a discrete symmetry while translation equivariance is a continuous symmetry. This fact is well known among researchers in equivariant machine learning, but is usually overlooked among non-experts. To minimize confusion, we suggest using the term `shift equivariance' to refer to discrete shifts in pixels and `translation equivariance' to refer to continuous translations.

cs.CV↗

arXiv2023

Towards continuum gyrokinetic study of high-field mirrors

High-temperature superconducting (HTS) magnetic mirrors under development exploit strong fields with high mirror ratio to compress loss cones and enhance confinement, and may offer cheaper, more compact fusion power plant candidates. This new class of devices could exhibit largely unexplored interchange and gradient-driven modes. Such instabilities, and methods to stabilize them, can be studied with gyrokinetics given the strong magnetization and prevalence of kinetic effects. Our focus here is to: a) determine if oft-used gyrokinetic models for open field lines produce the electron-confining (Pastukhov) electrostatic potential; b) examine and address challenges faced by gyrokinetic codes in studying HTS mirrors. We show that a one-dimensional limit of said models self-consistently develops a potential qualitatively reaching the analytical Pastukhov level. Additionally, we describe the computational challenges of studying high mirror ratios with open field line gyrokinetic solvers, and offer a force softening method to mitigate small time steps needed for time integration in colossal magnetic field gradients produced by HTS coils, providing a 19X speedup.

physics.plasm-ph↗

arXiv2023

Invariant preservation in machine learned PDE solvers via error correction

Machine learned partial differential equation (PDE) solvers trade the reliability of standard numerical methods for potential gains in accuracy and/or speed. The only way for a solver to guarantee that it outputs the exact solution is to use a convergent method in the limit that the grid spacing $Δx$ and timestep $Δt$ approach zero. Machine learned solvers, which learn to update the solution at large $Δx$ and/or $Δt$, can never guarantee perfect accuracy. Some amount of error is inevitable, so the question becomes: how do we constrain machine learned solvers to give us the sorts of errors that we are willing to tolerate? In this paper, we design more reliable machine learned PDE solvers by preserving discrete analogues of the continuous invariants of the underlying PDE. Examples of such invariants include conservation of mass, conservation of energy, the second law of thermodynamics, and/or non-negative density. Our key insight is simple: to preserve invariants, at each timestep apply an error-correcting algorithm to the update rule. Though this strategy is different from how standard solvers preserve invariants, it is necessary to retain the flexibility that allows machine learned solvers to be accurate at large $Δx$ and/or $Δt$. This strategy can be applied to any autoregressive solver for any time-dependent PDE in arbitrary geometries with arbitrary boundary conditions. Although this strategy is very general, the specific error-correcting algorithms need to be tailored to the invariants of the underlying equations as well as to the solution representation and time-stepping scheme of the solver. The error-correcting algorithms we introduce have two key properties. First, by preserving the right invariants they guarantee numerical stability. Second, in closed or periodic systems they do so without degrading the accuracy of an already-accurate solver.

math.NA↗

arXiv2023

An investigation of shock formation versus shock mitigation of colliding plasma jets

This work studies the interaction between colliding plasma jets to understand regimes in which jet merging results in shock formation versus regimes in which the shock formation is mitigated due to the collisionless interpenetration of the jets. A kinetic model is required for this study because fluid models will always produce a shock upon the collision of plasma jets. The continuum-kinetic, Vlasov-Maxwell-Dougherty model with one velocity dimension is used to accurately capture shock heating, along with a novel coupling with a moment equation to evolve perpendicular temperature for computational efficiency. As a result, this relatively inexpensive simulation can be used for detailed scans of the parameter space towards predictions of shocked versus shock-mitigated regimes, which is of interest for several fusion concepts such as plasma-jet-driven magneto-inertial fusion (PJMIF), high-energy-density plasmas, astrophysical phenomena, and other laboratory plasmas. The initial results obtained using this approach are in agreement with the preliminary outcomes of the Plasma Liner Experiment (PLX).

physics.plasm-ph↗

arXiv2022

Electron cyclotron drift instability and anomalous transport: two-fluid moment theory and modeling

In the presence of a strong electric field perpendicular to the magnetic field, the electron cross-field (E$\times$B) flow relative to the unmagnetized ions can cause the Electron Cyclotron Drift Instability (ECDI) due to resonances of the ion acoustic mode and the electron cyclotron harmonics. This occurs in collisionless shock ramps in space, and in $\rm{E \times B}$ discharge devices such as Hall thrusters. ECDI can induce an electron flow parallel to the background E field at a speed greatly exceeding predictions by classical collision theory. Such anomalous transport may lead to particle thermalization at space shocks, and may cause unfavorable plasma flows towards the walls of E$\times$B devices. The development of ECDI and anomalous transport is often considered fully-kinetic. In this work, however, we demonstrate that a reduced variant of this instability, and more importantly, the associated anomalous transport, can be treated self-consistently in a two-fluid framework without any collision. By treating electrons and ions on an equal footing, the free energy allows the growth of an anomalous electron flow parallel to the background E field. We first present linear analyses of the instability in the two-fluid 5- and 10-moment models, and compare them against the fully-kinetic theory. At lower temperatures, the two-fluid fastest-growing mode is in good agreement with the kinetic result. Also, by including more ($>=10$) moments, secondary (and possibly higher) unstable branches can be recovered. The dependence of the instability on various parameters is also explored. We then carry out direct numerical simulations of the cross-field setup using the 5-moment model. The growth of the instability and the anomalous transport is confirmed. Finally, 5-moment and Vlasov simulations using identical parameters in the lower-temperature regime are performed, showing reasonable agreement.

physics.plasm-ph↗

arXiv2022

A 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 is the first of its kind to present a 5-dimensional (2 spatial dimensions, 3 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 fluid-like 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.

physics.plasm-ph↗

arXiv2022

Improved multispecies Dougherty collisions

The Dougherty model Fokker-Planck operator is extended to describe nonlinear full-f collisions between multiple species in plasmas. Simple relations for cross-species interactions are developed which obey conservation laws, and reproduce familiar velocity and temperature relaxation rates. This treatment of multispecies Dougherty collisions, valid for arbitrary mass ratios, satisfies the H-Theorem unlike analogous Bhatnagar-Gross-Krook operators.

physics.plasm-ph↗

arXiv2021

Turbulent field fluctuations in gyrokinetic and fluid plasmas

A key uncertainty in the design and development of magnetic confinement fusion energy reactors is predicting edge plasma turbulence. An essential step in overcoming this uncertainty is the validation in accuracy of reduced turbulent transport models. Drift-reduced Braginskii two-fluid theory is one such set of reduced equations that has for decades simulated boundary plasmas in experiment, but significant questions exist regarding its predictive ability. To this end, using a novel physics-informed deep learning framework, we demonstrate the first ever direct quantitative comparisons of turbulent field fluctuations between electrostatic two-fluid theory and electromagnetic gyrokinetic modelling with good overall agreement found in magnetized helical plasmas at low normalized pressure. This framework is readily adaptable to experimental and astrophysical environments, and presents a new technique for the numerical validation and discovery of reduced global plasma turbulence models.

physics.plasm-ph↗

arXiv2021

Mapped discontinuous Galerkin interpolations and sheared boundary conditions

Translations or, more generally, coordinate transformations of scalar fields arise in several applications, such as weather, accretion disk and magnetized plasma turbulence modeling. In local studies of accretion disks and magnetized plasmas these coordinate transformations consist of an analytical mapping and enter via sheared-shift boundary conditions. This work introduces a discontinuous Galerkin algorithm to compute these coordinate transformations or boundary conditions based on projections and quadrature-free integrals. The procedure is high-order accurate, preserves certain moments exactly and works in multiple dimensions. Tests of the proposed approach with increasing complexity are presented, beginning with translations of one and two dimensional fields, followed by 3D and 5D simulations with sheared (twist-shift) boundary conditions. The results show that the algorithm is (p+1)-order accurate in the DG representation and (p+2)-order accurate in the cell averages, with p being the order of the polynomial basis functions. Quantification of the algorithm's diffusion and, for shearing boundary conditions, discussion of aliasing errors are provided.

physics.plasm-ph↗

arXiv2021

A Field-Particle Correlation Analysis of a Perpendicular Magnetized Collisionless Shock

Using the field-particle correlation technique, we examine the particle energization in a 1D-2V continuum Vlasov--Maxwell simulation of a perpendicular magnetized collisionless shock. The combination of the field-particle correlation technique with the high fidelity representation of the particle distribution function provided by a direct discretization of the Vlasov equation allows us to ascertain the details of the exchange of energy between the electromagnetic fields and the particles in phase space. We identify the velocity-space signatures of shock-drift acceleration of the ions and adiabatic heating of the electrons due to the perpendicular collisionless shock by constructing a simplified model with the minimum ingredients necessary to produce the observed energization signatures in the self-consistent Vlasov-Maxwell simulation. We are thus able to completely characterize the energy transfer in the perpendicular collisionless shock considered here and provide predictions for the application of the field-particle correlation technique to spacecraft measurements of collisionless shocks.

physics.plasm-ph↗

arXiv2020

A boundary value "reservoir problem" and boundary conditions for multi-moment multifluid simulations of sheaths

Multifluid simulations of plasma sheaths are increasingly used to model a wide variety of problems in plasma physics ranging from global magnetospheric flows around celestial bodies to plasma-wall interactions in thrusters and fusion devices. For multifluid problems, accurate boundary conditions to model an absorbing wall that resolves a classical sheath remains an open research area. This work justifies the use of vacuum boundary conditions for absorbing walls to show comparable accuracy between a multifluid sheath and lower moments of a continuum-kinetic sheath.

physics.plasm-ph↗

arXiv2020

Conservative discontinuous Galerkin scheme of a gyro-averaged Dougherty collision operator

A conservative discontinuous Galerkin scheme for a nonlinear Dougherty collision operator in full-f long-wavelength gyrokinetics is presented. Analytically this model operator has the advective-diffusive form of Fokker-Planck operators, it has a non-decreasing entropy functional, and conserves particles, momentum and energy. Discretely these conservative properties are maintained exactly as well, independent of numerical resolution. In this work the phase space discretization is performed using a novel version of the discontinuous Galerkin scheme, carefully constructed using concepts of weak equality and recovery. Discrete time advancement is carried out with an explicit time-stepping algorithm, whose stability limits we explore. The formulation and implementation within the long-wavelength gyrokinetic solver of Gkeyll are validated with relaxation tests, collisional Landau-damping benchmarks and the study of 5D gyrokinetic turbulence on helical, open field lines.

physics.plasm-ph↗

arXiv2020

Alias-free, matrix-free, and quadrature-free discontinuous Galerkin algorithms for (plasma) kinetic equations

Understanding fundamental kinetic processes is important for many problems, from plasma physics to gas dynamics. A first-principles approach to these problems requires a statistical description via the Boltzmann equation, coupled to appropriate field equations. In this paper we present a novel version of the discontinuous Galerkin (DG) algorithm to solve such kinetic equations. Unlike Monte-Carlo methods we use a continuum scheme in which we directly discretize the 6D phase-space using discontinuous basis functions. Our DG scheme eliminates counting noise and aliasing errors that would otherwise contaminate the delicate field-particle interactions. We use modal basis functions with reduced degrees of freedom to improve efficiency while retaining a high formal order of convergence. Our implementation incorporates a number of software innovations: use of JIT compiled top-level language, automatically generated computational kernels and a sophisticated shared-memory MPI implementation to handle velocity space parallelization.

physics.comp-ph↗

arXiv2020

Fluid & Gyrokinetic turbulence in open field-line, helical plasmas

Two-fluid Braginskii codes have simulated open-field line turbulence for over a decade, and only recently has it become possible to study these systems with continuum gyrokinetic codes. This work presents a first-of-its-kind comparison between fluid and (long-wavelength) gyrokinetic models in open field-lines, using the GDB and Gkeyll codes to simulate interchange turbulence in the Helimak device at the University of Texas (T. N. Bernard, et. al., Phys. of Plasmas 26, 042301 (2019)). Partial agreement is attained in a number of diagnostic channels when the GDB sources and sheath boundary conditions (BCs) are selected carefully, especially the heat-flux BCs which can drastically alter the temperature. The radial profile of the fluctuation levels is qualitatively similar and quantitatively comparable on the low-field side, although statistics such as moments of the probability density function and the high-frequency spectrum show greater differences. This comparison indicates areas for future improvement in both simulations, such as sheath BCs, as well as improvements in GDB like particle conservation and spatially varying thermal conductivity, in order to achieve better fluid-gyrokinetic agreement and increase fidelity when simulating experiments.

physics.plasm-ph↗

arXiv2020

Dynamo in weakly collisional nonmagnetized plasmas impeded by Landau damping of magnetic fields

We perform fully kinetic simulations of flows known to produce dynamo in magnetohydrodynamics (MHD), considering scenarios with low Reynolds number and high magnetic Prandtl number, relevant for galaxy cluster scale fluctuation dynamos. We find that Landau damping on the electrons leads to a rapid decay of magnetic perturbations, impeding the dynamo. This collisionless damping process operates on spatial scales where electrons are nonmagnetized, reducing the range of scales where the magnetic field grows in high magnetic Prandtl number fluctuation dynamos. When electrons are not magnetized down to the resistive scale, the magnetic energy spectrum is expected to be limited by the scale corresponding to magnetic Landau damping or, if smaller, the electron gyroradius scale, instead of the resistive scale. In simulations we thus observe decaying magnetic fields where resistive MHD would predict a dynamo.

physics.plasm-ph↗

arXiv2020

Investigating shear flow through continuum gyrokinetic simulations of limiter biasing in the Texas Helimak

Previous limiter-biasing experiments on the Texas Helimak, a simple magnetized torus, have been inconclusive on the effect of flow shear on turbulence levels. To investigate this, the first gyrokinetic simulations of limiter biasing in the Helimak using the plasma physics code Gkeyll have been carried out, and results are presented here. For the scenarios considered, turbulence is mostly driven by the interchange instability, which depends on gradients of equilibrium density profiles. An analysis of both experimental and simulation data demonstrates that shear rates are mostly less than than local linear growth rates, and not all requirements for shear stabilization are met. Rather, the mostly vertical shear flow has an important effect on bulk transport and experimental equilibrium density profiles, and changes to the gradients correspond to changes in turbulence levels.

physics.plasm-ph↗

arXiv2020

Measurement of temperature of a dusty plasma from configuration

A new method called `Configurational Temperature' is introduced in the context of dusty plasma, where the temperature of the dust particles, submerged in the plasma, can be measured directly from the positional information of the individual dust particles and the interaction potential between the dust grains. This method does not require the velocity information of individual particles which is a key parameter to measure the dust temperature in the conventional method. The technique is initially tested using two dimensional OpenMP parallel Molecular Dynamics and Monte-Carlo simulation and then compared with the temperature evaluating from the experimental data. The experiments have been carried out in Dusty plasma experimental (DPEx) device where a two dimensional stationary plasma crystal of melamine formaldehyde particles is formed in the cathode sheath of a DC glow discharge argon plasma. The dust kinetic temperature is calculated using standard PIV technique at different pressures. The simulation results matches well with the experimental data at relatively higher pressures where the dust particles arranged into crystalline state or in a strongly coupled fluid state. An extended simulation results for three dimensional case is also presented which can be employed for the temperature measurement of three dimensional dust crystal in laboratory devices.

physics.plasm-ph↗

arXiv2019

Exact and Locally Implicit Source Term Solvers for Multifluid-Maxwell Systems

Recently, a family of models that couple multifluid systems to the full Maxwell equations draw a lot of attention in laboratory, space, and astrophysical plasma modeling. These models are more complete descriptions of the plasma than reduced models like magnetohydrodynamic (MHD) since they naturally retain non-ideal effects like electron inertia, Hall term, pressure anisotropy/nongyrotropy, etc. One obstacle to broader application of these model is that an explicit treatment of their source terms leads to the need to resolve rapid kinetic processes like plasma oscillation and electron cyclotron motion, even when they are not important. In this paper, we suggest two ways to address this issue. First, we derive the analytic forms solutions to the source update equations, which can be implemented as a practical, but less generic solver. We then develop a time-centered, locally implicit algorithm to update the source terms, allowing stepping over the fast kinetic time-scales. For a plasma with $S$ species, the locally implict algorithm involves inverting a local $3S+3$ matrix only, thus is very efficient. The performance can be further elevated by using the direct update formulas to skip null calculations. Benchmarks illustrated the exact energy-conservation of the locally implicit solver, as well as its efficiency and robustness for both small-scale, idealized problems and large-scale, complex systems. The locally implicit algorithm can be also easily extended to include other local sources, like collisions and ionization, which are difficult to solve analytically.

physics.comp-ph↗