arXiv subjects

Zensho Yoshida

Publications and source records attributed to Zensho Yoshida.

At least 19 recordsLinked to original sources

arXiv2026

A Unified theory of transport barriers (TBs) in magnetically confined systems

A thermodynamic model of a plasma boundary layer, characterized by enhanced temperature contrasts is proposed. The theory is constructed to determine the inner boundary temperature $T_1$ for a specified outer (colder) boundary temperature $T_0$, the heat flux $F$ entering the inner boundary, and the parameters defining the layer. The system shows bifurcation and switches to a stable high gradient state if the heat flux $F$ entering through the inner boundary exceeds a critical value $F_c$. However there is an additional stringent condition for the transition to occur; the edge temperature $T_0$ must exceed a critical value $T_c$- no transition is possible if $T_0<T_c$ even for arbitrary large $F$. Equally important is the finding that $F_c$ is not a monotonic function of $T_0$ but has a minimum at $T_{optimum}$ (= $4T_c$ )in the model calculation. The confinement peaks at $T_{optimum}$. The basic conceptual physics is obviously simple: The high contrast state becomes the preferred state when the incoming power into the layer is preferentially converted into coherent motions like the fluid flows and currents (undermining the standard diffusive processes that keep the lower temperature contrast). The purely macroscopic thermodynamic model bears excellent comparison with experimental and detailed microscopic investigations of the H-mode. Deeper plausibility reasons for the workability of this heat engine, creating the simultaneous existence of an ordered state and large entropy production, are suggested.

physics.plasm-ph↗

arXiv2023

Numerical simulation of two-dimensional incompressible Navier-Stokes turbulence by Clebsch potentials

The Clebsch representation of a velocity field represents an effective tool for the analysis of physical properties of fluid flows. Indeed, a suitable choice of Clebsch potentials can be used to extract structural features that would otherwise be hidden within the complexity of fluid patterns and their evolution. In this work, we report the solution of the two-dimensional incompressible Navier-Stokes equations via Clebsch potentials. The results are in agreement with the solution of the vorticity equation for the stream function. Furthermore, we numerically demonstrate that the Shannon information entropy associated with each Clebsch potential is a growing function of time, and that it evolves at a slower rate than the rate of change in energy and enstrophy, as predicted by theory. These results pave the way for an alternative approach in the numerical study of fluid flows.

physics.flu-dyn↗

arXiv2022

Nambu mechanics viewed as a Clebsch parameterized Poisson algebra -- toward canonicalization and quantization

In his pioneering paper [Phys. Rev. E 7, 2405 (1973)], Nambu proposed the idea of multiple Hamiltonian systems. The explicit example examined there is equivalent to the so(3) Lie-Poisson system, which represents noncanonical Hamiltonian dynamics with a Casimir; the Casimir corresponds to the second Hamiltonian of Nambu's formulation. The vortex dynamics of ideal fluid, while it is infinite dimensional, has a similar structure, in which the Casimir is the helicity. These noncanonical Poisson algebras are derived by the reduction, i.e., restricting the phase space to some submanifold embedded in the canonical phase space. We may reverse the reduction to canonicalize some Nambu dynamics, i.e., view the Nambu dynamics as the subalgebra of a larger canonical Poisson algebra. Then, we can invoke the standard corresponding principle for quantizing the canonicalized system. The inverse of the reduction, i.e., representing the noncanonical variables by some canonical variables may be said "Clebsch parameterization" following the fluid mechanical example.

math-ph↗

arXiv2022

Clebsch representation of relativistic plasma and generalized enstrophy

The dynamics of plasma can be formulated as a subalgebra of the Poisson manifold of the Clebsch fields. In this work, we extend this formulation to a Lorentz covariant form. We show that the generalized enstrophy, which means the "charge" of the Clebsch field, is no longer conserved by the relativistic effect, implying that the conservation of circulation is broken. Instead, we formulate a relativistically modified generalized enstrophy that is conserved in the relativistic model.

math-ph↗

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

Partition of enstrophy between zonal and turbulent components

The partition of enstrophy between zonal (ordered) and wavy (turbulent) components of vorticity has been studied for the beta-plane model of two-dimensional barotropic flow. An analytic estimate of the minimum value for the zonal component has been derived. The energy, angular momentum, circulation, as well as the total enstrophy are invoked as constraints for the minimization of the zonal enstrophy. The corresponding variational principle has an unusual mathematical structure (primarily because the target functional is not a coercive form), by which the constraints work out in an interesting way. A discrete set of zonal enstrophy levels is generated by the energy constraint; each level is specified by an eigenvalue that represents the lamination period of zonal flow. However, the value itself of the zonal enstrophy level is a function of only angular momentum and circulation, being independent of the energy (and total enstrophy). Instead, the energy works in selecting the "level" (eigenvalue) of the relaxed state. The relaxation occurs by emitting small-scale wavy enstrophy, and continues as far as the nonlinear effect, scaled by the energy, can create wavy enstrophy. Comparison with numerical simulations shows that the theory gives a proper estimate of the zonal enstrophy in the relaxed state.

physics.flu-dyn↗

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↗

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↗

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

Rattleback: a model of how geometric singularity induces dynamic chirality

The rattleback is a boat-shaped top with an asymmetric preference in spin. Its dynamics can be described by nonlinearly coupled pitching, rolling, and spinning modes. The chirality, designed into the body as a skewed mass distribution, manifests itself in the quicker transition of $+$spin $\rightarrow$ pitch $\rightarrow$ $-$spin than that of $-$spin $\rightarrow$ roll $\rightarrow$ $+$spin. The curious guiding idea of this work is that we can formulate the dynamics as if a symmetric body were moving in a chiral space. By elucidating the duality of matter and space in the Hamiltonian formalism, we attribute asymmetry to space. The rattleback is shown to live in the space dictated by the Bianchi type ${\rm VI}_{h < -1}$ (belonging to class B) algebra; this particular algebra is used here for the first time in a mechanical example. The class B algebra has a singularity that separates the space (Poisson manifold) into mirror-asymmetric subspaces, breaking the time-reversal symmetry of nearby orbits.

math-ph↗

arXiv2017

Multi-region relaxed magnetohydrodynamics in plasmas with slowly changing boundaries --- resonant response of a plasma slab

The adiabatic limit of a recently proposed dynamical extension of Taylor relaxation, \emph{multi-region relaxed magnetohydrodynamics} (MRxMHD) is summarized, with special attention to the appropriate definition of relative magnetic helicity. The formalism is illustrated using a simple two-region, sheared-magnetic-field model similar to the Hahm--Kulsrud--Taylor (HKT) rippled-boundary slab model. In MRxMHD a linear Grad--Shafranov equation applies, even at finite ripple amplitude. The adiabatic switching on of boundary ripple excites a shielding current sheet opposing reconnection at a resonant surface. The perturbed magnetic field as a function of ripple amplitude is calculated by invoking conservation of magnetic helicity in the two regions separated by the current sheet. At low ripple amplitude "half islands" appear on each side of the current sheet, locking the rotational transform at the resonant value. Beyond a critical amplitude these islands disappear and the rotational transform develops a discontinuity across the current sheet.

physics.plasm-ph↗

arXiv2016

Enhanced entropy production in heat-flux-driven plasma sheath

The plasma sheath sets a stage for a strongly nonlinear coupling of the thermal, kinetic, and electric energies of plasma in a non-equilibrium, open environment. The pressure, velocity, and electrostatic potential profiles depend strongly on the boundary condition given on the internal side (pre-sheath). By controlling the boundary values of the heat flux and the ion Mach number, we solve a set of equations for the ion temperature and the electrostatic potential. The boundary values of the ion velocity and the electrostatic potential vary due to the change of the boundary ion temperature. When the heat flux exceeds a threshold value (determined by the ion Mach number), the temperature contrast is enhanced, resulting in a large entropy production.

physics.plasm-ph↗

arXiv2016

Nonlinear Helicons ---an analytical solution elucidating multi-scale structure

The helicon waves exhibit varying characters depending on plasma parameters, geometry, and wave numbers. Here we elucidate an intrinsic multi-scale property embodied by the combination of dispersive effect and nonlinearity. The extended magnetohydrodynamics model (exMHD) is capable of describing wide range of parameter space. By using the underlying Hamiltonian structure of exMHD, we construct an exact nonlinear solution which turns out to be a combination of two distinct modes, the helicon and Trivelpiece-Gould (TG) waves. In the regime of relatively low frequency or high density, however, the combination is made of the TG mode and an ion cyclotron wave (slow wave). The energy partition between these modes is determined by the helicities carried by the wave fields.

physics.plasm-ph↗

arXiv2016

Epi-two-dimensional flow and generalized enstrophy

The conservation of the enstrophy ($L^2$ norm of the vorticity $ω$) plays an essential role in the physics and mathematics of two-dimensional (2D) Euler fluids. Generalizing to compressible ideal (inviscid and barotropic) fluids, the generalized enstrophy $\int_{Σ(t)} f(ω/ρ)ρ\, d^2 x$, ($f$ an arbitrary smooth function, $ρ$ the density, and $Σ(t)$ an arbitrary 2D domain co-moving with the fluid) is a constant of motion, and plays the same role. On the other hand, for the three-dimensional (3D) ideal fluid, the helicity $\int_{M} {V}\cdotω\,d^3x$, ($V$ the flow velocity, $ω=\nabla\times V$, and ${M}$ the three-dimensional domain containing the fluid) is conserved. Evidently, the helicity degenerates in a 2D system, and the (generalized) enstrophy emerges as a compensating constant. This transition of the constants of motion is a reflection of an essential difference between 2D and 3D systems, because the conservation of the (generalized) enstrophy imposes stronger constraints, than the helicity, on the flow. In this paper, we make a deeper inquiry into the helicity-enstrophy interplay: the ideal fluid mechanics is cast into a Hamiltonian form in the phase space of Clebsch parameters, generalizing 2D to a wider category of epi-2D flows (2D embedded in 3D has zero helicity, while the converse is not true -- our epi-2D category encompasses a wider class of zero-helicity flows); how helicity degenerates and is substituted by a new constant is delineated; and how a further generalized enstrophy is introduced as a constant of motion applying to epi-2D flow is described.

physics.flu-dyn↗

arXiv2016

Nonlinear ion acoustic waves scattered by vortexes

The Kadomtsev--Petviashvili (KP) hierarchy is the archetype of infinite-dimensional integrable systems, which describes nonlinear ion acoustic waves in two-dimensional space. This remarkably ordered system resides on a singular submanifold (leaf) embedded in a larger phase space of more general ion acoustic waves (low-frequency electrostatic perturbations). The KP hierarchy is characterized not only by small amplitudes but also by irrotational (zero-vorticity) velocity fields. In fact, the KP equation is derived by eliminating vorticity at every order of the reductive perturbation. Here we modify the scaling of the velocity field so as to introduce a vortex term. The newly derived system of equations consists of a generalized three-dimensional KP equation and a two-dimensional vortex equation. The former describes `scattering' of vortex-free waves by ambient vortexes that are determined by the latter. We say that the vortexes are `ambient' because they do not receive reciprocal reactions from the waves (i.e., the vortex equation is independent of the wave fields). This model describes a minimal departure from the integrable KP system. By the Painlevé test, we delineate how the vorticity term violates integrability, bringing about an essential three-dimensionality to the solutions. By numerical simulation, we show how the solitons are scattered by vortexes and become chaotic.

physics.plasm-ph↗

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↗

arXiv2016

Anisotropy in broad component of H$α$ line in the magnetospheric device RT-1

Temperature anisotropy in broad component of H$α$ line was found in the ring trap 1 (RT-1) device by Doppler spectroscopy. Since hot hydrogen neutrals emitting a broad component are mainly produced by charge exchange between neutrals and protons, the anisotropy in the broad component is the evidence of proton temperature anisotropy generated by betatron acceleration.

physics.plasm-ph↗