arXiv subjects

G. M. Webb

Publications and source records attributed to G. M. Webb.

At least 19 recordsLinked to original sources

arXiv2023

Group classification of the two-dimensional magnetogasdynamics equations in Lagrangian coordinates

The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and nonthermal polytropic gas with infinite electrical conductivity. The equations are considered in mass Lagrangian coordinates. Use of Lagrangian coordinates allows reducing number of dependent variables. The analysis presented in this article gives complete group classification of the studied equations. This analysis is necessary for constructing invariant solutions and conservation laws on the base of Noether's theorem.

physics.class-ph↗

arXiv2022

Action Principles and Conservation Laws for Chew-Goldberger-Low Anisotropic Plasmas

The ideal CGL plasma equations, including the double adiabatic conservation laws for the parallel ($p_\parallel$) and perpendicular pressure ($p_\perp$), are investigated using a Lagrangian variational principle. An Euler-Poincaré variational principle is developed and the non-canonical Poisson bracket is obtained, in which the non-canonical variables consist of the mass flux ${\bf M}$, the density $ρ$, three entropy variables, $σ=ρS$, $σ_\parallel=ρS_\parallel$, $σ_\perp=ρS_\perp$ ($S_\parallel$ and $S_\perp$ are the two scalar entropy invariants), and the magnetic induction ${\bf B}$. Conservation laws of the CGL plasma equations are derived via Noether's theorem. The Galilean group leads to conservation of energy, momentum, center of mass, and angular momentum. Cross helicity conservation arises from a fluid relabeling symmetry, and is local or nonlocal depending on whether the entropy gradients of $S_\parallel$, $S_\perp$ and $S$ are perpendicular to ${\bf B}$ or otherwise. The point Lie symmetries of the CGL system are shown to comprise the Galilean transformations and scalings.

physics.plasm-ph↗

arXiv2022

Generalized Fluid Models of the Braginskii Type

Several generalizations of the well-known fluid model of Braginskii (Rev. of Plasma Phys., 1965) are considered. We use the Landau collisional operator and the moment method of Grad. We focus on the 21-moment model that is analogous to the Braginskii model, and we also consider a 22-moment model. Both models are formulated for general multi-species plasmas with arbitrary masses and temperatures, where all the fluid moments are described by their evolution equations. The 21-moment model contains two "heat flux vectors" (3rd and 5th-order moments) and two "viscosity-tensors" (2nd and 4th-order moments). The Braginskii model is then obtained as a particular case of a one ion-electron plasma with similar temperatures, with de-coupled heat fluxes and viscosity-tensors expressed in a quasi-static approximation. We provide all the numerical values of the Braginskii model in a fully analytic form (together with the 4th and 5th-order moments). For multi-species plasmas, the model makes calculation of transport coefficients straightforward. Formulation in fluid moments (instead of Hermite moments) is also suitable for implementation into existing numerical codes. It is emphasized that it is the quasi-static approximation which makes some Braginskii coefficients divergent in a weakly-collisional regime. Importantly, we show that the heat fluxes and viscosity-tensors are coupled even in the linear approximation, and that the fully contracted (scalar) perturbations of the 4th-order moment, which are accounted for in the 22-moment model, modify the energy exchange rates. We also provide several Appendices, which can be useful as a guide for deriving the Braginskii model with the moment method of Grad.

physics.plasm-ph↗

arXiv2019

An introductory guide to fluid models with anisotropic temperatures Part 2 -- Kinetic theory, Padé approximants and Landau fluid closures

In Part 2 of our guide to collisionless fluid models, we concentrate on Landau fluid closures. These closures were pioneered by Hammett and Perkins and allow for the rigorous incorporation of collisionless Landau damping into a fluid framework. It is Landau damping that sharply separates traditional fluid models and collisionless kinetic theory, and is the main reason why the usual fluid models do not converge to the kinetic description, even in the long-wavelength low-frequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function $Z(ζ)$, and the associated plasma response function $R(ζ)=1+ζZ(ζ)=-Z'(ζ)/2$. We then consider a 1D (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the 4th-order moment level. These closures for parallel moments have general validity from the largest astrophysical scales down to the Debye length, and we verify their validity by considering examples of the (proton and electron) Landau damping of the ion-acoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1D closures at higher-order moments than the 4th-order, and as was concluded in Part 1, this is not possible without Landau fluid closures. We show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing the convergence of the fluid and collisionless kinetic descriptions. We then consider a 3D (electromagnetic) geometry in the gyrotropic (long-wavelength low-frequency) limit and map all closures that are available at the 4th-order moment level. In the Appendix A, we provide comprehensive tables with Padé approximants of $R(ζ)$ up to the 8th-pole order, with many given in an analytic form.

physics.plasm-ph↗

arXiv2019

An introductory guide to fluid models with anisotropic temperatures Part 1 -- CGL description and collisionless fluid hierarchy

We present a detailed guide to advanced collisionless fluid models that incorporate kinetic effects into the fluid framework, and that are much closer to the collisionless kinetic description than traditional magnetohydrodynamics. Such fluid models are directly applicable to modeling turbulent evolution of a vast array of astrophysical plasmas, such as the solar corona and the solar wind, the interstellar medium, as well as accretion disks and galaxy clusters. The text can be viewed as a detailed guide to Landau fluid models and it is divided into two parts. Part 1 is dedicated to fluid models that are obtained by closing the fluid hierarchy with simple (non Landau fluid) closures. Part 2 is dedicated to Landau fluid closures. Here in Part 1, we discuss the CGL fluid model in great detail, together with fluid models that contain dispersive effects introduced by the Hall term and by the finite Larmor radius (FLR) corrections to the pressure tensor. We consider dispersive effects introduced by the non-gyrotropic heat flux vectors. We investigate the parallel and oblique firehose instability, and show that the non-gyrotropic heat flux strongly influences the maximum growth rate of these instabilities. Furthermore, we discuss fluid models that contain evolution equations for the gyrotropic heat flux fluctuations and that are closed at the 4th-moment level by prescribing a specific form for the distribution function. For the bi-Maxwellian distribution, such a closure is known as the "normal" closure. We also discuss a fluid closure for the bi-kappa distribution. Finally, by considering one-dimensional Maxwellian fluid closures at higher-order moments, we show that such fluid models are always unstable. The last possible non Landau fluid closure is therefore the "normal" closure, and beyond the 4th-order moment, Landau fluid closures are required.

physics.plasm-ph↗

arXiv2019

Godbillon-Vey Helicity and Magnetic Helicity in Magnetohydrodynamics

The Godbillon-Vey invariant occurs in homology theory, and algebraic topology, when conditions for a co-dimension 1, foliation of a 3D manifold are satisfied. The magnetic Godbillon-Vey helicity invariant in magnetohydrodynamics (MHD) is a higher order helicity invariant that occurs for flows, in which the magnetic helicity density $h_m={\bf A}{\bf\cdot}{\bf B}={\bf A}{\bf\cdot}(\nabla\times{\bf A})=0$, where ${\bf A}$ is the magnetic vector potential and ${\bf B}$ is the magnetic induction. This paper obtains evolution equations for the magnetic Godbillon-Vey field $\boldsymbolη={\bf A}\times{\bf B}/|{\bf A}|^2$ and the Godbillon-Vey helicity density $h_{gv}=\boldsymbolη{\bf\cdot}(\nabla\times{\boldsymbolη})$ in general MHD flows in which either $h_m=0$ or $h_m\neq 0$. A conservation law for $h_{gv}$ occurs in flows for which $h_m=0$. For $h_m\neq 0$ the evolution equation for $h_{gv}$ contains a source term in which $h_m$ is coupled to $h_{gv}$ via the shear tensor of the background flow. The transport equation for $h_{gv}$ also depends on the electric field potential $ψ$, which is related to the gauge for ${\bf A}$, which takes its simplest form for the advected ${\bf A}$ gauge in which $ψ={\bf A\cdot u}$ where ${\bf u}$ is the fluid velocity. An application of the Godbillon-Vey magnetic helicity to nonlinear force-free magnetic fields used in solar physics is investigated. The possible uses of the Godbillon-Vey helicity in zero helicity flows in ideal fluid mechanics, and in zero helicity Lagrangian kinematics of three-dimensional advection are discussed.

astro-ph.SR↗

arXiv2018

New closures for more precise modeling of Landau damping in the fluid framework

Incorporation of kinetic effects such as Landau damping into a fluid framework was pioneered by Hammett and Perkins PRL 1990, by obtaining closures of the fluid hierarchy, where the gyrotropic heat flux fluctuations or the deviation of the 4th-order gyrotropic fluid moment, are expressed through lower-order fluid moments. To obtain a closure of a fluid model expanded around a bi-Maxwellian distribution function, the usual plasma dispersion function $Z(ζ)$ that appears in kinetic theory or the associated plasma response function $R(ζ)=1 + ζZ(ζ)$, have to be approximated with a suitable Padé approximant in such a way, that the closure is valid for all $ζ$ values. Such closures are rare, and the original closures of Hammett and Perkins are often employed. Here we present a complete mapping of all plausible Landau fluid closures that can be constructed at the level of 4th-order moments in the gyrotropic limit and we identify the most precise closures. Furthermore, by considering 1D closures at higher-order moments, we show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing convergence of the fluid and collisionless kinetic descriptions.

physics.plasm-ph↗

arXiv2017

On Magnetohydrodynamic Gauge Field Theory

Clebsch potential gauge field theory for magnetohydrodynamics is developed based in part on the theory of Calkin (1963). It is shown how the polarization vector ${\bf P}$ in Calkin's approach, naturally arises from the Lagrange multiplier constraint equation for Faraday's equation for the magnetic induction ${\bf B}$, or alternatively from the magnetic vector potential form of Faraday's equation. Gauss's equation, (divergence of ${\bf B}$ is zero), is incorporated in the variational principle by means of a Lagrange multiplier constraint. Noether's theorem, coupled with the gauge symmetries is used to derive the conservation laws for (a)\ magnetic helicity (b)\ cross helicity, (c) fluid helicity for non-magnetized fluids, and (d) a class of conservation laws associated with curl and divergence equations, which applies to Faraday's equation and Gauss's equation. The magnetic helicity conservation law is due to a gauge symmetry in MHD and not due to a fluid relabelling symmetry. The analysis is carried out for the general case of a non-barotropic gas, in which the gas pressure and internal energy density depend on both the entropy $S$ and the gas density $ρ$. The cross helicity and fluid helicity conservation laws in the non-barotropic case, are nonlocal conservation laws, that reduce to local conservation laws for the case of a barotropic gas. The connections between gauge symmetries, Clebsch potentials and Casimirs are developed. It is shown that the gauge symmetry functionals in the work of Henyey (1982) satisfy the Casimir determining equations.

physics.plasm-ph↗

arXiv2016

Vorticity and Symplecticity in Multi-Symplectic Lagrangian Gas Dynamics

The Lagrangian, multi-dimensional, ideal, compressible gasdynamic equations are written in a multi-symplectic form, in which the Lagrangian fluid labels, $m^i$ (the Lagrangian mass coordinates) and time $t$ are the independent variables, and in which the Eulerian position of the fluid element ${\bf x}={\bf x}({\bf m},t)$ and the entropy $S=S({\bf m},t)$ are the dependent variables. Constraints in the variational principle are incorporated by means of Lagrange multipliers. The constraints are: the entropy advection equation $S_t=0$, the Lagrangian map equation ${\bf x}_t={\bf u}$ where ${\bf u}$ is the fluid velocity, and the mass continuity equation which has the form $J=τ$ where $J=\det(x_{ij})$ is the Jacobian of the Lagrangian map in which $x_{ij}=\partial x^i/\partial m^j$ and $τ=1/ρ$ is the specific volume of the gas. The internal energy per unit volume of the gas $\varepsilon=\varepsilon(ρ,S)$ corresponds to a non-barotropic gas. The Lagrangian is used to define multi-momenta, and to develop de-Donder Weyl Hamiltonian equations. The de Donder Weyl equations are cast in a multi-symplectic form. The pullback conservation laws and the symplecticity conservation laws are obtained. One class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws, and another class of symplecticity laws are related to derivatives of the Lagrangian energy conservation law with respect to the Lagrangian mass coordinates $m^i$. We show that the vorticity-symplecticity laws can be derived by a Lie dragging method, and also by using Noether's second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action. We obtain the Cartan-Poincaré form describing the equations and we discuss a set of differential forms representing the equation system.

math-ph↗

arXiv2015

Rossby Wave Green's Functions in an Azimuthal Wind

Green's functions for Rossby waves in an azimuthal wind are obtained, in which the stream-function $ψ$ depends on $r$, $ϕ$ and $t$, where $r$ is cylindrical radius and $ϕ$ is the azimuthal angle in the $β$-plane relative to the easterly direction, in which the $x$-axis points east and the $y$-axis points north. The Rossby wave Green's function with no wind is obtained using Fourier transform methods, and is related to the previously known Green's function obtained for this case, which has a different but equivalent form to the Green's function obtained in the present paper. We emphasize the role of the wave eikonal solution, which plays an important role in the form of the solution. The corresponding Green's function for a rotating wind with azimuthal wind velocity ${\bf u}=Ωr{\bf e}_ϕ$ ($Ω=$const.) is also obtained by Fourier methods, in which the advective rotation operator in position space is transformed to a rotation operator in ${\bf k}$ transform space. The finite Rossby deformation radius is included in the analysis. The physical characteristics of the Green's functions are delineated and applications are discussed. In the limit as $Ω\to 0$, the rotating wind Green's function reduces to the Rossby wave Green function with no wind.

physics.ao-ph↗

arXiv2015

Small-scale magnetic islands in the solar wind and their role in particle acceleration. Part 1: Dynamics of magnetic islands near the heliospheric current sheet

Increases of ion fluxes in the keV-MeV range are sometimes observed near the heliospheric current sheet (HCS) during periods when other sources are absent. These resemble solar energetic particle (SEP) events, but the events are weaker and apparently local. Conventional explanations based on either shock acceleration of charged particles or particle acceleration due to magnetic reconnection at interplanetary current sheets are not persuasive. We suggest instead that recurrent magnetic reconnection occurs at the HCS and smaller current sheets in the solar wind (Zharkova & Khabarova 2012), of which a consequence is particle energization by the dynamically evolving secondary current sheets and magnetic islands (Zank et al. 2014; Drake et al. 2006a). The effectiveness of the trapping and acceleration process associated with magnetic islands depends in part on the topology of the HCS. We show that the HCS possesses ripples superimposed on the large-scale flat or wavy structure. We conjecture that the ripples can efficiently confine plasma and provide tokamak-like conditions that are favorable for the appearance of small-scale magnetic islands that merge and/or contract. Particles trapped in the vicinity of merging islands and experiencing multiple small-scale reconnection events are accelerated by the induced electric field, and experience first-order Fermi acceleration in contracting magnetic islands (Zank et al. 2014). We present multi-spacecraft observations of magnetic island merging and particle energization in the absence of other sources, providing support for theory and simulations that show particle energization by reconnection related processes of magnetic island merging and contraction.

physics.plasm-ph↗

arXiv2015

Multi-Symplectic Magnetohydrodynamics: II, Addendum and Erratum

A recent paper arXiv:1312.4890 on multi-symplectic magnetohydrodynamics (MHD) using Clebsch variables in an Eulerian action principle with constraints is further extended. We relate a class of symplecticity conservation laws to a vorticity conservation law, and provide a corrected form of the Poincaré-Cartan differential form formulation of the system. We also correct some typographical errors (omissions) in arXiv:1312.4890. We show that the vorticity-symplecticity conservation law, that arises as a compatibility condition on the system, expressed in terms of the Clebsch variables is equivalent to taking the curl of the conservation form of the MHD momentum equation. We use the Cartan-Poincaré form to obtain a class of differential forms that represent the system using Cartan's geometric theory of partial differential equations.

physics.plasm-ph↗

arXiv2015

Multi-Symplectic Lagrangian, One-Dimensional Gas Dynamics

The equations of Lagrangian, ideal, one-dimensional (1D), compressible gas dynamics are written in a multi-symplectic form using the Lagrangian mass coordinate $m$ and time $t$ as independent variables, and in which the Eulerian position of the fluid element $x=x(m,t)$ is one of the dependent variables. This approach differs from the Eulerian, multi-symplectic approach using Clebsch variables. Lagrangian constraints are used to specify equations for $x_m$, $x_t$ and $S_t$ consistent with the Lagrangian map, where $S$ is the entropy of the gas. We require $S_t=0$ corresponding to advection of the entropy $S$ with the flow. We show that the Lagrangian Hamiltonian equations are related to the de Donder-Weyl multi-momentum formulation. The pullback conservation laws and the symplecticity conservation laws are discussed. The pullback conservation laws correspond to invariance of the action with respect to translations in time (energy conservation) and translations in $m$ in Noether's theorem. The conservation law due to $m$-translation invariance gives rise to a novel nonlocal conservation law involving the Clebsch variable $r$ used to impose $\partial S(m,t)/\partial t=0$. Translation invariance with respect to $x$ in Noether's theorem is associated with momentum conservation. We obtain the Cartan-Poincaré form for the system, and use it to obtain a closed ideal of two-forms representing the equation system.

math-ph↗

arXiv2014

Hydromagnetic Waves in a Compressed Dipole Field via Field-Aligned Klein-Gordon Equations

Hydromagnetic waves, especially those of frequencies in the range of a few milli-Hz to a few Hz observed in the Earth's magnetosphere, are categorized as Ultra Low Frequency (ULF) waves or pulsations. They have been extensively studied due to their importance in the interaction with radiation belt particles and in probing the structures of the magnetosphere. We developed an approach in examining the toroidal standing Aflvén waves in a background magnetic field by recasting the wave equation into a Klein-Gordon (KG) form along individual field lines. The eigenvalue solutions to the system are characteristic of a propagation type when the corresponding eigen-frequency is greater than a cut-off frequency and an evanescent type otherwise. We apply the approach to a compressed dipole magnetic field model of the inner magnetosphere, and obtain the spatial profiles of relevant parameters and the spatial wave forms of harmonic oscillations. We further extend the approach to poloidal mode standing Alfvén waves along field lines. In particular, we present a quantitative comparison with a recent spacecraft observation of a poloidal standing Alfvén wave in the Earth's magnetosphere. Our analysis based on KG equation yields consistent results which agree with the spacecraft measurements of the wave period and the amplitude ratio between the magnetic field and electric field perturbations. We also present computational results of eigenvalue solutions to the compressional poloidal mode waves in the compressed dipole magnetic field geometry.

physics.space-ph↗

arXiv2014

Structures of Interplanetary Magnetic Flux Ropes and Comparison with Their Solar Sources

Magnetic reconnection is essential to release the flux rope during its ejection. The question remains: how does the magnetic reconnection change the flux rope structure? Following the original study of \citet{Qiu2007}, we compare properties of ICME/MC flux ropes measured at 1 AU and properties of associated solar progenitors including flares, filaments, and CMEs. In particular, the magnetic field-line twist distribution within interplanetary magnetic flux ropes is systematically derived and examined. Our analysis shows that for most of these events, the amount of twisted flux per AU in MCs is comparable with the total reconnection flux on the Sun, and the sign of the MC helicity is consistent with the sign of helicity of the solar source region judged from the geometry of post-flare loops. Remarkably, we find that about one half of the 18 magnetic flux ropes, most of them being associated with erupting filaments, have a nearly uniform and relatively low twist distribution from the axis to the edge, and the majority of the other flux ropes exhibit very high twist near the axis, of up to $\gtrsim 5$ turns per AU, which decreases toward the edge. The flux ropes are therefore not linear force free. We also conduct detailed case studies showing the contrast of two events with distinct twist distribution in MCs as well as different flare and dimming characteristics in solar source regions, and discuss how reconnection geometry reflected in flare morphology may be related to the structure of the flux rope formed on the Sun.

astro-ph.SR↗

arXiv2014

Potential Vorticity in Magnetohydrodynamics

A version of Noether's second theorem using Lagrange multipliers is used to investigate fluid relabelling symmetries conservation laws in magnetohydrodynamics (MHD). We obtain a new generalized potential vorticity type conservation equation for MHD which takes into account entropy gradients and the ${\bf J}\times{\bf B}$ force on the plasma due to the current ${\bf J}$ and magnetic induction ${\bf B}$. This new conservation law for MHD is derived by using Noether's second theorem in conjunction with a class of fluid relabelling symmetries in which the symmetry generator for the Lagrange label transformations is non-parallel to the magnetic field induction in Lagrange label space. This is associated with an Abelian Lie pseudo algebra and a foliated phase space in Lagrange label space. It contains as a special case Ertel's theorem in ideal fluid mechanics. An independent derivation shows that the new conservation law is also valid for more general physical situations.

math-ph↗

arXiv2014

Multi-Symplectic Magnetohydrodynamics

A multi-symplectic formulation of ideal magnetohydrodynamics (MHD) is developed based on a Clebsch variable variational principle in which the Lagrangian consists of the kinetic minus the potential energy of the MHD fluid modified by constraints using Lagrange multipliers, that ensure mass conservation, entropy advection with the flow, the Lin constraint and Faraday's equation (i.e the magnetic flux is Lie dragged with the flow). The analysis is also carried out using the magnetic vector potential $\tilde{\bf A}$ where $α=\tilde{\bf A}{\bf\cdot}d{\bf x}$ is Lie dragged with the flow, where ${\bf B}=\nabla\times\tilde{\bf A}$. The symplecticity conservation laws are shown to give rise to the Eulerian momentum and energy conservation laws in MHD. Noether's theorem for the multi-symplectic MHD system is derived, including the case of non-Cartesian space coordinates, where the metric plays a role in the equations.

math-ph↗

arXiv2014

Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether's Theorems and Casimirs

Conservation laws in ideal gas dynamics and magnetohydrodynamics (MHD) associated with fluid relabelling symmetries are derived using Noether's first and second theorems. Lie dragged invariants are discussed in terms of the MHD Casimirs. A nonlocal conservation law for fluid helicity applicable for a non-barotropic fluid involving Clebsch variables is derived using Noether's theorem, in conjunction with a fluid relabelling symmetry and a gauge transformation. A nonlocal cross helicity conservation law involving Clebsch potentials, and the MHD energy conservation law are derived by the same method. An Euler Poincaré variational approach is also used to derive conservation laws associated with fluid relabelling symmetries using Noether's second theorem.

math-ph↗