arXiv subjects

Naoki Sato

Publications and source records attributed to Naoki Sato.

53 records · Page 3Linked to original sources

arXiv2020

Realization of Incompressible Navier-Stokes Flow as Superposition of Transport Processes for Clebsch Potentials

In ideal fluids, Clebsch potentials occur as paired canonical variables associated with the Hamiltonian description of the Euler equations. This paper explores the properties of the incompressible Navier-Stokes equations when the velocity field is expressed through a complete set of paired Clebsch potentials. First, it is shown that the incompressible Navier-Stokes equations can be cast as a system of transport (convection-diffusion) equations where each Clebsch potential plays the role of a generalized distribution function. The diffusion operator associated with each Clebsch potential departs from the standard Laplacian due to a term depending on the Lie-bracket of the corresponding Clebsch pair. It is further shown that the Clebsch potentials can be used to define a Shannon-type entropy measure, i.e. a functional, different from energy and enstrophy, whose growth rate is non-negative. As a consequence, the flow must vanish at equilibrium. This functional can be interpreted as a measure of the topological complexity of the velocity field. In addition, the Clebsch parametrization enables the identification of a class of flows, larger than the class of two dimensional flows, possessing the property that the vortex stretching term identically vanishes and the growth rate of entrophy is non-positive.

physics.flu-dyn↗

arXiv2020

Symmetric Ideal Magnetofluidostatic Equilibria with Non-Vanishing Pressure Gradients in Asymmetric Confinement Vessels

We study the possibility of constructing steady magnetic fields satisfying the force balance equation of ideal magnetohydrodynamics with tangential boundary conditions in asymmetric confinement vessels, i.e. bounded regions that are not invariant under continuous Euclidean isometries (translations, rotations, or their combination). This problem is often encountered in the design of next-generation fusion reactors. We show that such configurations are possible if one relaxes the standard assumption that the vessel boundary corresponds to a pressure isosurface. We exhibit a smooth solution that possesses an Euclidean symmetry and yet solves the boundary value problem in an asymmetric ellipsoidal domain while sustaining a non-vanishing pressure gradient. This result provides a definitive answer to the problem of existence of regular ideal magnetofluidostatic equilibria in asymmetric bounded domains. The question remains open whether regular asymmetric solutions of the boundary value problem exist.

physics.plasm-ph↗

arXiv2019

Dissipative Brackets for the Fokker-Planck Equation in Hamiltonian Systems and Characterization of Metriplectic Manifolds

It is shown that the Fokker-Planck equation describing diffusion processes in noncanonical Hamiltonian systems exhibits a metriplectic structure, i.e. an algebraic bracket formalism that generates the equation in consistency with the thermodynamic principles of energy conservation and entropy growth. First, a microscopic metriplectic bracket is derived for the stochastic equations of motion that characterize the random walk of the elements constituting the statistical ensemble. Such bracket is fully determined by the Poisson operator generating the Hamiltonian dynamics of an isolated (unperturbed) particle. Then, the macroscopic metriplectic bracket associated with the evolution of the distribution function of the ensemble is induced from the microscopic metriplectic bracket. Similarly, the macroscopic Casimir invariants are inherited from microscopic dynamics. The theory is applied to construct the Fokker-Planck equation of an infinite dimensional Hamiltonian system, the Charney-Hasegawa-Mima equation. Finally, the canonical form of the symmetric (dissipative) part of the metriplectic bracket is identified in terms of a `canonical metric tensor' corresponding to an Euclidean metric tensor on the symplectic leaves foliated by the Casimir invariants.

math-ph↗

arXiv2019

Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries

We study the existence of steady solutions of ideal magnetofluid systems (ideal MHD and ideal Euler equations) without continuous Euclidean symmetries. It is shown that all nontrivial magnetofluidostatic solutions are locally symmetric, although the symmetry is not necessarily an Euclidean isometry. Furthermore, magnetofluidostatic equations admit both force-free (Beltrami type) and non-force-free (with finite pressure gradients) solutions that do not exhibit invariance under translations, rotations, or their combination. Examples of smooth solutions without continuous Euclidean symmetries in bounded domains are given. Finally, the existence of square integrable solutions of the tangential boundary value problem without continuous Euclidean symmetries is proved.

math-ph↗

arXiv2019

Local representation and construction of Beltrami fields II. Solenoidal Beltrami fields and ideal MHD equilibria

Object of the present paper is the local theory of solution for steady ideal Euler flows and ideal MHD equilibria. The present analysis relies on the Lie-Darboux theorem of differential geometry and the local theory of representation and construction of Beltrami fields developed in [1]. A theorem for the construction of harmonic orthogonal coordinates is proved. Using such coordinates families of solenoidal Beltrami fields with different topologies are obtained in analytic form. Existence of global solenoidal Beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered. It is shown that only singular solutions are admissible, an explicit example is given in a spherical domain, and a theorem on existence of singular solutions is proven. Local conditions for existence of solutions, and local representation theorems are derived for generalized Beltrami fields, ideal MHD equilibria, and general steady ideal Euler flows. The theory is applied to construct analytic examples.

math-ph↗

arXiv2018

Local Representation and Construction of Beltrami Fields

A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie-Darboux theorem of differential geoemtry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold-Beltrami-Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and non-solenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors.

math-ph↗

arXiv2018

Degenerate Laplacian: A Classification by Helicity

We study a class of generalized Laplacian operators by violating the ellipticity with degenerate metric tensors. The theory is motivated by the statistical mechanics of topologically constrained particles. In the context of diffusion models, the metric tensor is given by $g=-\mathcal{J}^2$ with a generalized Poisson matrix $\mathcal{J}$ that dictates particle dynamics. The standard Euclidean metric corresponds to the symplectic matrix of canonical Hamiltonian systems. However, topological constraints bring about nullity to $\mathcal{J}$, resulting in degeneracy in the corresponding diffusion operator; we call such an operator an orthogonal Laplacian (since the ellipticity is broken in the direction parallel to the nullity), and denote it by $Δ_{\perp}$. Although all nice properties pertinent to the ellipticity are generally lost for $Δ_{\perp}$, a finite helicity of $\mathcal{J}$ helps to recover some of them by preventing foliation of space. We show that $-\left(Δ_{\perp}u,v\right)$ defines an inner product of a Sobolev-like Hilbert space, and satisfies a Poincaré-like inequality $-\left(Δ_{\perp}u,u\right)\geq C\lvert\lvert{u}\rvert\rvert^{2}_{L^2}$. Applying Riesz's representation theorem, we obtain a unique weak solution of the orthogonal Poisson equation.

math.AP↗

arXiv2018

Beltrami operators

Beltrami fields occur as stationary solutions of the Euler equations of fluid flow and as force free magnetic fields in magnetohydrodynamics. In this paper we discuss the role of Beltrami fields when considered as operators acting on a Hamiltonian function to generate particle dynamics. Beltrami operators, which include Poisson operators as a special subclass, arise in the description of topologically constrained diffusion in non-Hamiltonian systems. Extending previous results, we show that random motion generated by a Beltrami operator satisfies an H-theorem, leading to a generalized Boltzmann distribution on the coordinate system where the Beltrami condition holds. When the Beltrami condition is violated, random fluctuations do not work anymore to homogenize the particle distribution in the coordinate system where they are applied. The resulting distribution becomes heterogeneous. The heterogeneity depends on the `field charge' measuring the departure of the operator from a Beltrami field. Examples of both Beltrami and non-Beltrami operators in three real dimensions together with the corresponding equilibrium distribution functions are given.

math-ph↗

arXiv2018

The galactic rotation curve of a magnetized plasma cloud

The rotation curve of a magnetized plasma cloud orbiting in the gravitational potential of a galaxy is calculated by statistical arguments. The working assumption is that a mild magnetic field, decreasing with distance from the galactic center, permeates space. It is shown that the resulting probability density is compatible with a flat rotation curve.

astro-ph.GA↗

arXiv2018

Poissonization of Three Dimensional Nonholonomic Dynamics with the Method of Extension

In this study we develop a systematic procedure to construct a Poisson operator that describes the dynamics of a three dimensional nonholonomic system. Instead of reducing by symmetry the antisymmetric operator that links the energy gradient to the velocity on the tangent bundle, the system is embedded in a larger space. Here, the extended antisymmetric operator, which preserves the original equations of motion, satisfies the Jacobi identity in a conformal fashion. Thus, a Poisson operator can be obtained by a further time reparametrization. Such Poissonization does not rely on the specific form of the Hamiltonian function. The theory is applied to calculate the equilibrium distribution function of a non-Hamiltonian ensemble.

math-ph↗

arXiv2017

Diffusion with finite-helicity field tensor: a new mechanism of generating heterogeneity

Topological constraints on a dynamical system often manifest themselves as breaking of the Hamiltonian structure; well-known examples are non-holonomic constraints on Lagrangian mechanics. The statistical mechanics under such topological constraints is the subject of the present study. Conventional arguments based on phase spaces, Jacobi identity, invariant measure, or the H theorem are no longer applicable, since all these notions stem from the symplectic geometry underlying canonical Hamiltonian systems. Remembering that Hamiltonian systems are endowed with field tensors (canonical 2-forms) that have zero helicity, our mission is to extend the scope toward the class of systems governed by finite-helicity field tensors. Here we introduce a new class of field tensors that are characterized by Beltrami vectors. We prove an H theorem for this Beltrami class. The most general class of energy-conserving systems are non-Beltrami, for which we identify the "field charge" that prevents the entropy to maximize, resulting in creation of heterogeneous distributions. The essence of the theory can be delineated by classifying three-dimensional dynamics. We then generalize to arbitrary (finite) dimensions.

math-ph↗

arXiv2017

Charged Particle Diffusion in a Magnetic Dipole Trap

When particles are magnetized, a diffusion process is influenced by the ambient magnetic field. While the entropy increases, the constancy of the magnetic moment puts a constraint. Here, we compare the E-cross-B diffusion caused by random fluctuations of the electric field in two different systems, the Penning-Malmberg trap and the magnetic dipole trap. A Fokker-Planck equation is derived by applying the ergodic ansatz on the invariant measure of the system. In the dipole magnetic field particles diffuse inward and accumulate in the higher magnetic field region, while, in a homogeneous magnetic field, particles diffuse out from the confinement region. The properties of analogous transport in a more general class of magnetic fields are also briefly discussed.

physics.plasm-ph↗

arXiv2017

Relaxation of Compressible Euler Flow in a Toroidal Domain

It is shown that the universal steady Euler flow field, independent of boundary shape or symmetry, in a toroidal domain with fixed boundary obeys a nonlinear Beltrami equation, with the nonlinearity arising from a Boltzmann-like, velocity-dependent factor. Moreover, this is a relaxed velocity field, in the sense that it extremizes the total kinetic energy in the domain under free variations of the velocity field, constrained only by tangential velocity and vorticity boundary conditions and conservation of total fluid helicity and entropy. This is analogous to Woltjer-Taylor relaxation of plasma magnetic field to a stationary state. However, unlike the magnetic field case, attempting to derive slow, quasi-relaxed dynamics from Hamilton's action principle, with constant total fluid helicity as a constraint, fails to agree, in the static limit, with the nonlinear Beltrami solution of the Euler equations. Nevertheless, an action principle that gives a quasi-relaxed dynamics that does agree can be formulated, by introducing a potential representation of the velocity field and defining an analogue of the magnetic helicity as a new constraint. A Hamiltonian form of quasi-relaxed fluid dynamics is also given.

physics.flu-dyn↗

arXiv2016

Up-Hill Diffusion Creating Density Gradient - What is the Proper Entropy?

It is always some constraint that yields any nontrivial structure from statistical averages. As epitomized by the Boltzmann distribution, the energy conservation is often the principal constraint acting on mechanical systems. Here, we investigate a different type: the topological constraint imposed on `space'. Such constraint emerges from the null space of the Poisson operator linking energy gradient to phase space velocity, and appears as an adiabatic invariant altering the preserved phase space volume at the core of statistical mechanics. The correct measure of entropy, built on the distorted invariant measure, behaves consistently with the second law of thermodynamics. The opposite behavior (decreasing entropy and negative entropy production) arises in arbitrary coordinates. An ensamble of rotating rigid bodies is worked out. The theory is then applied to up-hill diffusion in a magnetosphere.

math.ST↗

arXiv2015

Observation of particle acceleration in laboratory magnetosphere

The self-organization of magnetospheric plasma is brought about by inward diffusion of magnetized particles. Not only creating a density gradient toward the center of a dipole magnetic field, the inward diffusion also accelerates particles and provides a planetary radiation belt with high energy particles. Here, we report the first experimental observation of a 'laboratory radiation belt' created in the Ring Trap 1 (RT-1) device. By spectroscopic measurement, we found an appreciable anisotropy in the ion temperature, proving the betatron acceleration mechanism which heats particles in the perpendicular direction with respect to the magnetic field when particles move inward. The energy balance model including the heating mechanism explains the observed ion temperature profile.

physics.plasm-ph↗

arXiv2015

Thermal equilibrium of non-neutral plasma in dipole magnetic field

Self-organization of a long-lived structure is one of the remarkable characteristics of macroscopic systems governed by long-range interactions. In a homogeneous magnetic field, a non-neutral plasma creates a "thermal equilibrium" which is a Boltzmann distribution on a rigidly rotating frame. Here, we study how a non-neutral plasma self-organizes in inhomogeneous magnetic field; as a typical system we consider a dipole magnetic field. In this generalized setting, the plasma exhibits its fundamental mechanism that determines the relaxed state. The scale hierarchy of adiabatic invariants is the determinant; the Boltzmann distribution under the topological constraint by the robust adiabatic invariants (hence, the homogeneous distribution with respect to the fragile invariant) is the relevant relaxed state, which turns out to be a rigidly rotating clump of particles (just same as in a homogeneous magnetic field), while the density is no longer homogeneous.

physics.plasm-ph↗

arXiv2014

A Stochastic Model of Inward Diffusion in Magnetospheric Plasmas

The inward diffusion of particles, often observed in magnetospheric plasmas (either naturally created stellar ones or laboratory devices) creates a spontaneous density gradient, which seemingly contradicts the entropy principle. We construct a theoretical model of diffusion that can explain the inward diffusion in a dipole magnetic field. The key is the identification of the proper coordinates on which an appropriate diffusion operator can be formulated. The effective phase space is foliated by the adiabatic invariants; on the symplectic leaf, the invariant measure (by which the entropy must be calculated) is distorted, by the inhomogeneous magnetic field, with respect to the conventional Lebesgue measure of the natural phase space. The collision operator is formulated to be consistent to the ergodic hypothesis on the symplectic leaf, i.e., the resultant diffusion must diminish gradients on the proper coordinates. The non-orthogonality of the cotangent vectors of the configuration space causes a coupling between the perpendicular and parallel diffusions, which is derived by applying Ito's formula of changing variables. The model has been examined by numerical simulations. We observe the creation of a peaked density profile that mimics radiation belts in planetary magnetospheres as well as laboratory experiments.

physics.plasm-ph↗