arXiv subjects

Philip J. Morrison

Publications and source records attributed to Philip J. Morrison.

At least 37 records · Page 2Linked to original sources

arXiv2023

A Hamiltonian description of finite-time singularity in Euler's fluid equations

The recently proposed low degree-of-freedom model of Moffat and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes. The Hamiltonian structure of the inviscid limit of the model is obtained. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant, are discovered. It is shown that the system is integrable with a solution that lies on the intersection for the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffat and Kimura model accurately represents Euler's ideal fluid equations of motion, we have shown the existence of finite-time singularity.

physics.flu-dyn↗

arXiv2023

A Hamiltonian and geometric formulation of general Vlasov-Maxwell-type models

Three geometric formulations of the Hamiltonian structure of the macroscopic Maxwell equations are given: one in terms of the double de Rham complex, one in terms of L2 duality, and one utilizing an abstract notion of duality. The final of these is used to express the geometric and Hamiltonian structure of kinetic theories in general media. The Poisson bracket so stated is explicitly metric free. Finally, as a special case, the Lorentz covariance of such kinetic theories is investigated. We obtain a Lorentz covariant kinetic theory coupled to nonlinear electrodynamics such as Born-Infeld or Euler-Heisenberg electrodynamics.

math-ph↗

arXiv2022

Dual PIC: a structure preserving method for discretizing Lie-Poisson brackets

We consider a general discretization strategy for Hamiltonian field theories generated by Lie-Poisson brackets which we call dual PIC (DPIC). This method involves prescribing two different discrete representations of the dynamical variable which are constrained as a Casimir invariant of the flow to coincide with one another via an L2 projection throughout the entire simulation. This allows one to leverage the relative advantages of each discrete representation. We begin by describing DPIC as applied to a general Lie-Poisson system and then provide illustrative examples: the discretization of the two-dimensional vorticity equations and the Vlasov-Poisson equation.

physics.comp-ph↗

arXiv2022

A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus

A Hamiltonian field theory for the macroscopic Maxwell equations with fully general polarization and magnetization is stated in the language of differential forms. The precise procedure for translating the vector calculus formulation into differential forms is discussed in detail. We choose to distinguish between straight and twisted differential forms so that all integrals be taken over densities (i.e. twisted top forms). This ensures that the duality pairings, which are stated as integrals over densities, are orientation independent. The relationship between functional differentiation with respect to vector fields and with respect to differential forms is established using the chain rule. The theory is developed such that the Poisson bracket is metric and orientation independent with all metric dependence contained in the Hamiltonian. As is typically seen in the exterior calculus formulation of Maxwell's equations, the Hodge star operator plays a key role in modeling the constitutive relations. As a demonstration of the kind of constitutive models this theory accommodates, the paper concludes with several examples.

math-ph↗

arXiv2022

A mimetic discretization of the macroscopic Maxwell equations in Hamiltonian form

A mimetic spectral element discretization, utilizing a novel Galerkin projection Hodge star operator, of the macroscopic Maxwell equations in Hamiltonian form is presented. The idea of splitting purely topological and metric dependent quantities is natural in the Hamiltonian modeling framework as the Poisson bracket is metric free with the Hamiltonian containing all metric information. This idea may be incorporated into the mimetic spectral element method by directly discretizing the Poincaré duality structure. This "split exterior calculus mimetic spectral element method" yields spatially discretized Maxwell's equations which are Hamiltonian and exactly and strongly conserve Gauss's laws. Moreover, the new discrete Hodge star operator is itself of interest as a partition of the purely topological and metric dependent portions of the Hodge star operator. As a simple test case, the numerical results of applying this method to a one-dimensional version of Maxwell's equations are given.

physics.comp-ph↗

arXiv2022

Clebsch Canonization of Lie-Poisson Systems

We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie-Poisson equation via a momentum map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. We also find another momentum map so that the pair of momentum maps constitute a dual pair under a certain condition. The dual pair gives a concrete realization of what is commonly referred to as collectivization of Lie-Poisson systems. It also implies that solving the canonized system by symplectic Runge-Kutta methods yields so-called collective Lie-Poisson integrators that preserve the coadjoint orbits and hence the Casimirs exactly. We give a couple of examples, including the Kida vortex and the heavy top on a movable base with controls, which are Lie-Poisson systems on $\mathfrak{so}(2,1)^{*}$ and $(\mathfrak{se}(3) \ltimes \mathbb{R}^{3})^{*}$, respectively.

math-ph↗

arXiv2022

Fluid Flow Induced Deformation of a Boundary Hair

The deformation of a dense carpet of hair due to Stokes flow in a channel can be described by a nonlinear integro-differential equation for the shape of a single hair, which possesses several solutions for a given choice of parameters. While being posed in a previous study and bearing resemblance to the pendulum problem from mechanics, this equation has not been analytically solved until now. Despite the presence on an integral with a nonlinear functional dependence on the dependent variable, the system is integrable. We compare the analytically obtained solution to a finite-difference numerical approach, identify the physically realizable solution branch, and briefly study the solution structure through a conserved energy-like quantity. Time-dependent fluid-structure interactions are a rich and complex subject to investigate and we argue that the solution discussed herein can be used as a basis for understanding these systems.

physics.flu-dyn↗

arXiv2022

The kinetic origin of the fluid helicity -- a symmetry in the kinetic phase space

Helicity, a topological degree that measures the winding and linking of vortex lines, is preserved by ideal (barotropic) fluid dynamics. In the context of the Hamiltonian description, the helicity is a Casimir invariant characterizing a foliation of the associated Poisson manifold. Casimir invariants are special invariants that depend on the Poisson bracket, not on the particular choice of the Hamiltonian. The total mass (or particle number) is another Casimir invariant, whose invariance guarantees the mass (particle) conservation (independent of any specific choice of the Hamiltonian). In a kinetic description (e.g. that of the Vlasov equation), the helicity is no longer an invariant (although the total mass remains a Casimir of the Vlasov's Poisson algebra). The implication is that some "kinetic effect" can violate the constancy of the helicity. To elucidate how the helicity constraint emerges or submerges, we examine the fluid reduction of the Vlasov system; the fluid (macroscopic) system is a "sub-algebra" of the kinetic (microscopic) Vlasov system. In the Vlasov system, the helicity can be conserved, if a special helicity symmetry condition holds. To put it another way, breaking helicity symmetry induces a change in the helicity. We delineate the geometrical meaning of helicity symmetry, and show that, for a special class of flows (so-called epi-2 dimensional flows), the helicity symmetry is written as $\partial_γ=0$ for a coordinate $γ$ of the configuration space.

physics.flu-dyn↗

arXiv2020

A class of three-dimensional gyroviscous magnetohydrodynamic models

A Hamiltonian and action principle formalism for deriving three-dimensional gyroviscous magnetohydrodynamic models is presented. The uniqueness of the approach in constructing the gyroviscous tensor from first principles and its ability to explain the origin of the gyromap and the gyroviscous terms are highlighted. The procedure allows for the specification of free functions, which can be used to generate a wide range of gyroviscous models. Through the process of reduction, the noncanonical Hamiltonian bracket is obtained and briefly analysed.

physics.plasm-ph↗

arXiv2020

Deformation of Lie-Poisson algebras and chirality

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian-symmetric spectra: If $λ$ is an eigenvalue of the linearized generator, $-λ$ and $\barλ$ (hence, $-\barλ$) are also eigenvalues -- the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Phys. Lett. A 381 (2017), 2772--2777]. To elucidate how chiral spectra are generated, we study the 3-dimensional Lie-Poisson systems, and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi's list of all 3-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra so(3), yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.

math-ph↗

arXiv2020

A General Metriplectic Framework with Application to Dissipative Extended Magnetohydrodynamics

General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model have already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view for deriving dissipative brackets, a means of derivation that is natural for understanding and creating dissipative brackets. Finally the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, nonlocal dissipative brackets naturally emerge.

physics.flu-dyn↗

arXiv2018

Jovian vortices and jets

We explore the conditions required for isolated vortices to exist in sheared zonal flows and the stability of the underlying zonal winds. This is done using the standard 2-layer quasigeostrophic model with the lower layer depth becoming infinite; however, this model differs from the usual layer model because the lower layer is not assumed to be motionless but has a steady configuration of alternating zonal flows [1]. Steady state vortices are obtained by a simulated annealing computational method introduced in [2], generalized and applied in [3] in fluid flow, and used in the context of magnetohydrodynamics in [4-6]. Various cases of vortices with a constant potential vorticity anomaly atop zonal winds and the stability of the underlying winds are considered using a mix of computational and analytical techniques.

physics.ao-ph↗

arXiv2018

An integral transform technique for kinetic systems with collisions

The linearized Vlasov-Poisson system can be exactly solved using the $G$-transform, an integral transform introduced in Refs. 1-3 that removes the electric field term, leaving a simple advection equation. We investigate how this integral transform interacts with the Fokker-Planck collision operator. The commutator of this collision operator with the $G$-transform (the "shielding term") is shown to be negligible. We exactly solve the advection-diffusion equation without the shielding term. This solution determines when collisions dominate and when advection (i.e. Landau damping) dominates. This integral transform can also be used to simplify gyro-/drift-kinetic equations. We present new gyrofluid equations formed by taking moments of the $G$-transformed equation. Since many gyro-/drift-kinetic codes use Hermite polynomials as basis elements, we include an explicit calculation of their $G$-transform.

physics.plasm-ph↗

arXiv2018

Direction of cascades in a magnetofluid model with electron skin depth and ion sound Larmor radius scales

The direction of cascades in a two-dimensional model that takes electron inertia and ion sound Larmor radius into account is studied, resulting in analytical expressions for the absolute equilibrium states of the energy and helicities. It is found that typically both the energy and magnetic helicity at scales shorter than electron skin depth have direct cascade, while at large scales the helicity has an inverse cascade as established earlier for reduced magnetohydrodynamics (MHD). It is also found that the introduction of gyro-effects allows for the existence of negative temperature (conjugate to energy) states and the condensation of energy to the large scales. Comparisons between two- and three-dimensional extended MHD models (MHD with two-fluid effects) show qualitative agreement between the two.

physics.plasm-ph↗

arXiv2018

Metriplectic torque for rotation control of a rigid body

Metriplectic dynamics couple a Poisson bracket of the Hamiltonian description with a kind of metric bracket, for describing systems with both Hamiltonian and dissipative components. The construction builds in asymptotic convergence to a preselected equilibrium state. Phenomena such as friction, electric resistivity, thermal conductivity and collisions in kinetic theories all fit within this framework. In this paper an application of metriplectic dynamics is presented that is of interest for the theory of control: a suitably chosen torque, expressed through a metriplectic extension of its "natural" Poisson algebra, an algebra obtained by reduction of a canonical Hamiltonian system, is applied to a free rigid body. On a practical ground, the effect is to drive the body to align its angular velocity to rotation about a stable principal axis of inertia, while conserving its kinetic energy in the process. On theoretical grounds, this example provides a class of non-Hamiltonian torques that can be added to the canonical Hamiltonian description of the free rigid body and reduce to metriplectic dissipation. In the canonical description these torques provide convergence to a higher dimensional attractor. The method of construction of such torques can be extended to other dynamical systems describing "machines" with non-Hamiltonian motion having attractors.

physics.class-ph↗

arXiv2017

Internal wave energy flux from density perturbations in nonlinear stratifications

Internal gravity wave energy contributes significantly to the energy budget of the oceans, affecting mixing and the thermohaline circulation. Hence it is important to determine the internal wave energy flux $\mathbf{J} = p \, \mathbf{v}$, where $p$ is the pressure perturbation field and $\mathbf{v}$ is the velocity perturbation field. However, the pressure perturbation field is not directly accessible in laboratory or field observations. Previously, a Green's function based method was developed to calculate the instantaneous energy flux field from a measured density perturbation field $ρ(x,z,t)$, given a $constant$ buoyancy frequency $N$. Here we present methods for computing the instantaneous energy flux $\mathbf{J}(x,z,t)$ for a spatially varying $N(z)$, as in the oceans where $N(z)$ typically decreases by two orders of magnitude from shallow water to the deep ocean. Analytic methods are presented for computing $\mathbf{J}(x,z,t)$ from a density perturbation field for $N(z)$ varying linearly with $z$ and for $N^2(z)$ varying as $\tanh (z)$. To generalize this approach to $arbitrary$ $N(z)$, we present a computational method for obtaining $\mathbf{J}(x,z,t)$. The results for $\mathbf{J}(x,z,t)$ for the different cases agree well with results from direct numerical simulations of the Navier-Stokes equations. Our computational method can be applied to any density perturbation data using the MATLAB graphical user interface EnergyFlux.

physics.flu-dyn↗

arXiv2017

Symplectic Maps for Diverted Plasmas

Nowadays, divertors are used in the main tokamaks to control the magnetic field and to improve the plasma confinement. In this article, we present analytical symplectic maps describing Poincaré maps of the magnetic field lines in confined plasmas with a single null poloidal divertor. Initially, we present a divertor map and the tokamap for a diverted configuration. We also introduce the Ullmann map for a diverted plasma, whose control parameters are determined from tokamak experiments. Finally, an explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent toroidal equilibrium magnetic surfaces. In this method, we also give examples of onset of chaotic field lines at the plasma edge due to resonant perturbations.

physics.plasm-ph↗

arXiv2017

Unsteady Stokes flow near boundaries: the point-particle approximation and the method of reflections

Problems of particle dynamics involving unsteady Stokes flows in confined geometries are typically harder to solve than their steady counterparts. Approximation techniques are often the only resort. Felderhof (see e.g. 2005, 2009b) has developed a point-particle approximation framework to solve such problems, especially in the context of Brownian motion. Despite excellent agreement with past experiments, this framework has an inconsistency which we address in this work. Upon implementing our modifications, the framework passes consistency checks that it previously failed. Further, it is not obvious that such an approximation should work for short time-scale motion. We investigate its validity by deriving it from a general formalism based on integral equations through a series of systematic approximations. We also compare results from the point-particle framework against a calculation performed using the method of reflections, for the specific case of a sphere near a full-slip plane boundary. We find from our analysis that the reasons for the success of the point-particle approximation are subtle and have to do with the nature of the unsteady Oseen tensor. Finally, we provide numerical predictions for Brownian motion near a full-slip and a no-slip plane wall based on the point-particle approximation as used by Felderhof, our modified point-particle approximation, and the method of reflections. We show that our modifications to Felderhof's framework would become significant for systems of metallic nanoparticles in liquids.

physics.flu-dyn↗