arXiv subjects

P. J. Morrison

Publications and source records attributed to P. J. Morrison.

At least 19 recordsLinked to original sources

arXiv2026

Thermodynamics of driven systems via the Kuramoto-Sivashinsky equation

We examine the differences between the driven turbulence described by the Kuramoto-Sivashinsky (KS) equation and the second law of thermodynamics. A general velocity and entropy density system is analyzed with the unified thermodynamic algorithm of metriplectic dynamics, and we show that the positive spectra of the KS equation due to an external energy source prevent its metriplectic description. A variant of the KS equation is produced that monotonically generates an entropy, but the only equilibria of this variant system are spatially constant. Numerical experiments are performed comparing the evolution of the KS equation and its thermodynamic variant. The entropy of this thermodynamic system is increased further by the driving effects of the KS equation, reconciling the generation of entropy with the energy source of the KS equation. Further numerical experiments restrict the positive spectra in the KS equation to determine the effect on the system time evolution. While rescaling the growth rates of instabilities reproduces similar behavior on a slower time scale, introduction of individual positive spectra reproduces the formation of equilibria, relative equilibria, and a transition to chaos.

nlin.CD↗

arXiv2025

Imposing quasineutrality on electrostatic plasmas via the Dirac theory of constraints

We present a method for imposing quasineutrality and, more generally, charge density conservation in the Vlasov-Poisson (VP) and Vlasov-Amp\`ere (VA) systems, which describe electrostatic plasma dynamics, by applying the Dirac theory of constraints. Leveraging the Hamiltonian field formulations of the VP and VA models, we construct generalized Dirac brackets using the Dirac algorithm. The resulting constrained systems enforce charge density conservation, and consequently quasineutrality, given that the initial charge density is zero, through new advection terms in the Vlasov equations involving generalized-force terms, while the electric field is eliminated from the constrained Vlasov dynamics. To verify charge density conservation we conduct one-dimensional numerical experiments using a semi-Lagrangian method, demonstrating that the enforcement of the quasineutrality constraint significantly modifies the dynamics. This approach enables us to identify the forces required to enforce quasineutrality, offering a systematic way to assess the validity of the quasineutral approximation across different kinetic scales.

physics.plasm-ph↗

arXiv2025

Hamiltonian formulation of the quasineutral Vlasov-Poisson system

Slow manifold reduction and the theory of Poisson-Dirac submanifolds are used to deduce a Hamiltonian formulation for a quasineutral limit of the planar, collisionless, magnetized Vlasov-Poisson system. Motion on the slow manifold models plasma dynamics free of fast Langmuir oscillations. Preservation of quasineutrality requires the bulk plasma flow is incompressible. The electric field is determined by counterbalancing plasma stresses that would otherwise produce compression. The Hamiltonian structure for the quasineutral model synthesizes well-known Poisson brackets for incompressible fluids and collisionless kinetic equations.

math-ph↗

arXiv2025

Metriplectic relaxation to equilibria

Metriplectic dynamical systems consist of a special combination of a Hamiltonian and a (generalized) entropy-gradient flow, such that the Hamiltonian is conserved and entropy is dissipated/produced (depending on a sign convention). It is natural to expect that, in the long-time limit, the orbit of a metriplectic system should converge to an extremum of entropy restricted to a constant-Hamiltonian surface. In this paper, we discuss sufficient conditions for this to occur. Then, we construct a class of metriplectic systems inspired by the Landau operator for Coulomb collisions in plasmas, which is included as special case. For this class of brackets, checking the conditions for convergence reduces to checking two usually simpler conditions, and we discuss examples in detail. We apply these results to the construction of relaxation methods for the solution of equilibrium problems in fluid dynamics and plasma physics.

math-ph↗

arXiv2024

Simulated annealing of reduced magnetohydrodynamic systems

Theory of simulated annealing (SA), a method for equilibrium and stability analyses for Hamiltonian systems, is reviewed. The SA explained in this review is based on a double bracket formulation that derives from Hamiltonian structure. In addition to general theoretical aspects, the explicit formulation as well as numerical applications are presented. Both finite and infinite degree-of-freedom systems are treated, in particular, the heavy top, a toy model mimicking low-beta reduced magnetohydrodynamics (MHD) and low- and high-beta reduced MHD. Numerical results successfully demonstrate the usefulness of SA for equilibrium and stability analyses. At the same time, the results raise some future issues that are discussed in the paper.

physics.plasm-ph↗

arXiv2024

Shearless effective barriers to chaotic transport induced by even twin islands in nontwist systems

For several decades now it has been known that systems with shearless invariant tori, nontwist Hamiltonian systems, possess barriers to chaotic transport. These barriers are resilient to breakage under perturbation and therefore regions where they occur are natural places to look for barriers to transport. We describe a novel kind of effective barrier that persists after the shearless torus is broken. Because phenomena are generic, for convenience we study the Standard Nontwist Map (SNM), an area-preserving map that violates the twist condition locally in the phase space. The novel barrier occurs in nontwist systems when twin even period islands are present, which happens for a broad range of parameter values in the SNM. With a phase space composed of regular and irregular orbits, the movement of chaotic trajectories is hampered by the existence of both shearless curves, total barriers, and a network of partial barriers formed by the stable and unstable manifolds of the hyperbolic points. Being a degenerate system, the SNM has twin islands and, consequently, twin hyperbolic points. We show that the structures formed by the manifolds intrinsically depend on period parity of the twin islands. For this even scenario the novel structure, named a torus free barrier, occurs because the manifolds of different hyperbolic points form an intricate chain atop a dipole configuration and the transport of chaotic trajectories through the chain becomes a rare event. This structure impacts the emergence of transport, the escape basin for chaotic trajectories, the transport mechanism and the chaotic saddle. The case of odd periodic orbits is different: we find for this case the emergence of transport immediately after the breakup of the last invariant curve, and this leads to a scenario of higher transport, with intricate escape basin boundary and a chaotic saddle with non-uniformly distributed points.

nlin.CD↗

arXiv2024

Axisymmetric hybrid Vlasov equilibria with applications to tokamak plasmas

We derive axisymmetric equilibrium equations in the context of the hybrid Vlasov model with kinetic ions and massless fluid electrons, assuming isothermal electrons and deformed Maxwellian distribution functions for the kinetic ions. The equilibrium system comprises a Grad-Shafranov partial differential equation and an integral equation. These equations can be utilized to calculate the equilibrium magnetic field and ion distribution function, respectively, for given particle density or given ion and electron toroidal current density profiles. The resulting solutions describe states characterized by toroidal plasma rotation and toroidal electric current density. Additionally, due to the presence of fluid electrons, these equilibria also exhibit a poloidal current density component. This is in contrast to the fully kinetic Vlasov model, where axisymmetric Jeans equilibria can only accommodate toroidal currents and flows, given the absence of a third integral of the microscopic motion.

physics.plasm-ph↗

arXiv2022

Mean field theory for intense light-matter interactions in high energy density plasmas

We present a generalization of Vlasov-Maxwell kinetic theory that accounts for intense electromagnetic fields. A strongly-radiating, possibly optically-thick plasma is decomposed into fragments, each comprising a charged particle together with its self-generated electromagnetic field. Assuming weak inter-fragment correlations, but strong intra-fragment correlations, a mean-field evolution equation for the single-fragment distribution functional is derived. We also identify the equation's Hamiltonian formulation. By incorporating strong correlations between a charged particle and the field it generates, the new model captures the effects of strong radiation reaction non-perturbatively. The fragment kinetic formalism offers an attractive approach to modeling exotic light-matter interactions such as nonlinear and multiple Compton scattering.

physics.plasm-ph↗

arXiv2022

Notes on a 1-dimensional electrostatic plasma model

A starting point for deriving the Vlasov equation is the BBGKY hierarchy that describes the dynamics of coupled marginal distribution functions. With a large value of the plasma parameter one can justify eliminating 2-point correlations in terms of the 1-point function in order to derive the Vlasov Landau Lenard Balescu (VLLB) theory. Because of the high dimensionality of the problem, numerically testing the assumptions of the VLLB theory is prohibitive. In these notes we propose a physically reasonable interaction model that lowers the dimensionality of the problem and may bring such computations within reach. We introduce a 1-dimensional (1-D) electrostatic plasma model formulated in terms of the interaction of parallelly-aligned charged disks. This model combines 1-dimensional features at short distances and 3-dimensional features at large distances.

physics.plasm-ph↗

arXiv2022

Linear stability analysis via simulated annealing and accelerated relaxation

Simulated annealing (SA) is a kind of relaxation method for finding equilibria of Hamiltonian systems. A set of evolution equations is solved with SA, which is derived from the original Hamiltonian system so that the energy of the system changes monotonically while preserving Casimir invariants inherent to noncanonical Hamiltonian systems. The energy extremum reached by SA is an equilibrium. Since SA searches for an energy extremum, it can also be used for stability analysis when initiated from a state where a perturbation is added to an equilibrium. The procedure of the stability analysis is explained, and some examples are shown. Because the time evolution is computationally time consuming, efficient relaxation is necessary for SA to be practically useful. An acceleration method is developed by introducing time dependence in the symmetric kernel used in the double bracket, which is part of the SA formulation described here. An explicit formulation for low-beta reduced magnetohydrodynamics (MHD) in cylindrical geometry is presented. Since SA for low-beta reduced MHD has two advection fields that relax, it is important to balance the orders of magnitude of these advection fields.

physics.plasm-ph↗

arXiv2021

Hamiltonian kinetic-Hall Magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes

We present two generalized hybrid kinetic-Hall magnetohydrodynamics (MHD) models describing the interaction of a two-fluid bulk plasma, which consists of thermal ions and electrons, with energetic, suprathermal ion populations described by Vlasov dynamics. The dynamics of the thermal components are governed by standard fluid equations in the Hall MHD limit with the electron momentum equation providing an Ohm's law with Hall and electron pressure terms involving a gyrotropic electron pressure tensor. The coupling of the bulk, low-energy plasma with the energetic particle dynamics is accomplished through the current density (current coupling scheme; CCS) and the ion pressure tensor appearing in the momentum equation (pressure coupling scheme; PCS) in the first and the second model, respectively. The CCS is a generalization of two well-known models, because in the limit of vanishing energetic and thermal ion densities we recover the standard Hall MHD and the hybrid kinetic-ions/fluid-electron model, respectively. This provides us with the capability to study in a continuous manner the global impact of the energetic particles in a regime extending from vanishing to dominant energetic particle densities. The noncanonical Hamiltonian structures of the CCS and PCS, which can be exploited to study equilibrium and stability properties through the energy-Casimir variational principle, are identified. As a first application here, we derive a generalized Hall MHD Grad--Shafranov--Bernoulli system for translationally symmetric equilibria with anisotropic electron pressure and kinetic effects owing to the presence of energetic particles using the PCS.

physics.plasm-ph↗

arXiv2021

Transport barriers in symplectic maps

Chaotic transport is a subject of paramount importance in a variety of problems in plasma physics, specially those related to anomalous transport and turbulence. On the other hand, a great deal of information on chaotic transport can be obtained from simple dynamical systems like two-dimensional area-preserving (symplectic) maps, where powerful mathematical results like KAM theory are available. In this work we review recent works on transport barriers in area-preserving maps, focusing on systems which do not obey the so-called twist property. For such systems KAM theory no longer holds everywhere and novel dynamical features show up as non-resistive reconnection, shearless curves and shearless bifurcations. After presenting some general features using a standard nontwist mapping, we consider magnetic field line maps for magnetically confined plasmas in tokamaks.

physics.plasm-ph↗

arXiv2020

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac's constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac's method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

physics.plasm-ph↗

arXiv2020

Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria

The formal stability analysis of Eulerian extended magnetohydrodynamics (XMHD) equilibria is considered within the noncanonical Hamiltonian framework by means of the energy-Casimir variational principle and the dynamically accessible stability method. Specifically, we find explicit sufficient stability conditions for axisymmetric XMHD and Hall MHD (HMHD) equilibria with toroidal flow and for equilibria with arbitrary flow under constrained perturbations. The dynamically accessible, second-order variation of the Hamiltonian, which can potentially provide explicit stability criteria for generic equilibria, is also obtained. Moreover, we examine the Lagrangian stability of the general quasineutral two-fluid model written in terms of MHD-like variables, by finding the action and the Hamiltonian functionals of the linearized dynamics, working within a mixed Lagrangian-Eulerian framework. Upon neglecting electron mass, we derive a HMHD energy principle, and in addition, the perturbed induction equation arises from Hamilton's equations of motion in view of a consistency condition for the relation between the perturbed magnetic potential and the canonical variables.

physics.plasm-ph↗

arXiv2019

Hamiltonian Nature of Monopole Dynamics

Classical electromagnetism with magnetic monopoles is not a Hamiltonian field theory because the Jacobi identity for the Poisson bracket fails. The Jacobi identity is recovered only if all of the species have the same ratio of electric to magnetic charge or if an electron and a monopole can never collide. Without the Jacobi identity, there are no local canonical coordinates or Lagrangian action principle. To build a quantum theory of magnetic monopoles, we either must explain why the positions of electrons and monopoles can never coincide or we must resort to new quantization techniques.

math-ph↗

arXiv2019

Computing Casimir invariants from Pfaffian systems

We describe a method for computing Casimir invariants that is applicable to both finite and infinite-dimensional Poisson brackets. We apply the method to various finite and infinite-dimensional examples, including a Poisson bracket embodying both finite and infinite-dimensional structure.

math-ph↗

arXiv2019

Ellipticity conditions for the extended MHD Grad-Shafranov-Bernoulli equilibrium equations

In this study, we find the points of transition between elliptic and hyperbolic regimes for the axisymmetric extended magnetohydrodynamic (MHD) equilibrium equations. The ellipticity condition is expressed via a single inequality but is more involved than the corresponding two-fluid ones due to the imposition of the quasineutrality condition and is also more complicated than the Hall MHD ellipticity condition, due to electron inertia. In fact, the inclusion of electron inertia is responsible for peculiar results; namely, even the static equilibrium equations can become hyperbolic.

physics.plasm-ph↗

arXiv2018

Relaxation to magnetohydrodynamics equilibria via collision brackets

Metriplectic dynamics is applied to compute equilibria of fluid dynamical systems. The result is a relaxation method in which Hamiltonian dynamics (symplectic structure) is combined with dissipative mechanisms (metric structure) that relaxes the system to the desired equilibrium point. The specific metric operator, which is considered in this work, is formally analogous to the Landau collision operator. These ideas are illustrated by means of case studies. The considered physical models are the Euler equations in vorticity form, the Grad-Shafranov equation, and force-free MHD equilibria.

physics.plasm-ph↗