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Results for “Guiding center motion”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1971

Hamiltonian formulation of guiding center motion

The nonrelativistic guiding center motion of a charged particle in a static magnetic field is derived using the Hamiltonian formalism. By repeated application of first-order canonical perturbation theory, the first two adiabatic invariants and their averaged Hamiltonians are obtained, including the first-order correction terms. Other features of guiding center theory are also given, including lowest order drifts and the flux invariant.

Stern, D. P.↗
NASA NTRS1978

Extensions of guiding center motion to higher order

In a static magnetic field, some well-known guiding-center equations maintain their form when extended to next order in gyroradius. In these cases, it is only necessary to include the next order term in the magnetic moment series. The differential equation for guiding-center motion which describes both the parallel and perpendicular velocities correctly through first order in gyroradius is given. The question of how to define the guiding center position through second order arises and is discussed, and second order drifts are derived for one usual definition. The toroidal canonical angular momentum, P-phi, of the guiding center in an axisymmetric field is shown to be conserved using the guiding center velocity correct through first order. When second-order motion is included, P-phi is no longer a constant. The above extensions of guiding-center theory help to resolve the different tokamak orbits obtained either by using the guiding-center equations of motion or by using conservation of P-phi.

Northrop, T. G.↗
DOE OSTI2024

Relativistic Guiding-Center Motion: Action Principle, Kinetic Theory and Hydrodynamics

We treat the guiding-center dynamics in a varying external Maxwell field using a relativistically covariant action principle which reproduces the known Vandervoort expression for the drift velocity and extends it to curved spacetime. We derive the corresponding kinetic theory and ideal hydrodynamic theory. In contrast to conventional five-equation hydrodynamics, the guiding-center hydrodynamics needs only three equations due to a constraint on the motion across magnetic field. Furthermore, we argue that such a hydrodynamics is applicable to strongly coupled plasmas where kinetic theory fails.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS1977

Adiabatic particle motion in a nearly drift-free magnetic field: Application to the geomagnetic tail

The guiding center motion of particles in a nearly drift free magnetic field is analyzed in order to investigate the dependence of mean drift velocity on equatorial pitch angle, the variation of local drift velocity along the trajectory, and other properties. The mean drift for adiabatic particles is expressed by means of elliptic integrals. Approximations to the twice-averaged Hamiltonian W near z = O are derived, permitting simple representation of drift paths if an electric potential also exists. In addition, the use of W or of expressions for the longitudinal invariant allows the derivation of the twice averaged Liouville equation and of the corresponding Vlasov equation. Bounce times are calculated (using the drift-free approximation), as are instantaneous guiding center drift velocities, which are then used to provide a numerical check on the formulas for the mean drift.

Stern, D. P.↗
DOE OSTI2023

Modification of Lie's transform perturbation theory for charged particle motion in a magnetic field

It is pointed out that the conventional Lie transform perturbation theory for the guiding center motion of charged particles in a magnetic field needs to be modified for ordering inconsistency. There are two reasons. First, the ordering difference between the temporal variation of gyrophase and that of the other phase space coordinates needs to be taken into account. Second, it is also important to note that the parametric limit of the derivative of a function is not equivalent to the derivative of the limit function. When these facts are taken into account, the near identity transformation rule for one form related to the Lagrangian is modified. With the modified near identity transformation rule, the drift motion of charged particles can be described in the first order, instead of the second order and beyond through a tedious expansion process as in the conventional formulation. Here, this resolves the discrepancy between the direct and Lie transform treatments in the Lagrangian perturbation theory for charged particle motion in a magnetic field.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1980

Motion of trapped electrons and protons in Saturn's inner magnetosphere

A summary is given of basic formulas for the guiding center motion of energetic charged particles trapped in a dipolar magnetic field. These formulas for longitudinal drift rates, latitudinal bounce periods, equatorial gyroradii, and equatorial gyroperiods are then stated in convenient numerical form for electrons and protons as functions of kinetic energy E, magnetic shell parameter L, and equatorial pitch angle alpha 0 for a slightly simplified model of the observed magnetic field of Saturn. To aid in the study of the interaction of charged particles with the rings and inner satellites of Saturn, additional formulas are given for the time interval between successive encounters of charged particles with a satellite in a circular prograde orbit and for the energy of electrons whose longitudinal angular velocity is resonant, or synchronous, with the Keplerian angular velocity of such a satellite.

Thomsen, M. F.↗
NASA NTRS1994

Kinetic and thermodynamic properties of a convecting plasma in a two-dimensional dipole field

Charged particle guiding center motion is considered in the magnetic field of a two-dimensional ('line') dipole on which is superimposed a small, static, perpendicular electric field. The parallel equation of motion is that of a simple harmonic oscillator for cos theta, the cosine of magnetic colatitude theta. Equations for the perpendicular electric and magnetic drifts are derived as well as their bounce-averaged forms. The latter are solved to yield a bounce-averaged guiding center trajectory, which is the same as that obtained from conversation of magnetic moment mu, longitudinal invariant J, and total (kinetic plus electrostatic) energy K. The algebraic simplicity of the trajectory equations is also manifest in the forms of the invariants. An interesting result is that guiding centers drift in such a way that they preserve the values of their equatorial pitch angles and (equivalently) mirror latitudes. The most general Maxwellian form of the equilibrium one-particle distribution function f is constructed from the invariants, and spatially varying density and pressure moments, parallel and perpendicular to the magnetic field, are identified. Much of the paper deals with the more restricted problem in which f is specified as a bi-Maxwellian over a straight line of finite length in the equatorial plane of the dipole and perpendicular to field lines. This might be thought of as specifying a cross-tail ion injection source; our formalism then describes the subsequent spatial development. The distribution away from the source is a scaled bi-Maxwellian but one that is cut off at large and small kinetic energies, which depend on position. Density and pressure components are reduced from the values they would have if the total content of individual flux tubes convected intact. The equatorial and meridional variations of density and pressure components are examined and compared systematically for the isotropic and highly anisotropic situations. There appears to be little qualitative difference due to anisotropy. An anisotropy measure is defined, and its spatial variation determined as a signature of possible MHD instability. Extreme values are found, larger than at the source, but the plasma beta in such regions is probably so low as to render the effect inconsequential energetically. Finally, the possible consequence of 'nonadia- batic' pressure profiles on electrostatic interchanges is considered, and a boundary delineating stabilizing and destabilizing regions determined.

Huang, T. S.↗
DOE OSTI2023

Response to “Comment on ‘Modification of Lie's transform perturbation theory for charged particle motion in a magnetic field’” [Phys. Plasmas 30 , 104701 (2023)]

Dr. Brizard's comment on my work is based on a conceived procedure that does not come from my work. The defense of his claim that the modification of the so-called standard Lie's transform theory is unnecessary is also unsupported. This response reveals in detail the inconsistency issues in the so-called standard Lie's transform theory by analyzing both its results and root causes. The problem in the so-called standard Lie's transform theory is beyond the issue to take into account the ordering difference between the guiding center motion and gyromotion. Here, the inconsistent commutation of derivative and limit causes another issue. In addition, the so-called standard Lie's transform formulation leads to an unnecessarily lengthy and tedious derivation process for a one or two page task under the singular (or renormalized) formalism described in my paper.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Some mathematics for quasi-symmetry

Quasi-symmetry of a steady magnetic field means integrability of first-order guiding-center motion by a spatial symmetry. Here, we derive many restrictions on the possibilities for a quasi-symmetry. We also derive an analog of the Grad–Shafranov equation for the flux function in a quasi-symmetric magnetohydrostatic field.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1979

A search for the interplanetary quantity controlling the development of geomagnetic storms

An historical account is presented concerning the evolution of our present concept of geomagnetic storms. The present concept was formulated by Chapman (1927) in his magnetic data statistical studies of 'the initial rise' (now termed the initial phase) 'and subsequent larger decrease' (now termed the main phase) in H, followed by 'slow recovery'. The concept introduced by Alfven in 1940 of guiding center motions of a charged particle in a nonuniform magnetic field (ring currents) is also discussed. By 1963 it became quite certain that the ring current, namely a storm-time Van Allen belt, is formed in the magnetosphere during the storm's main phase. The search then began for the solar wind quantity controlling the development of the main phase. The author then gives a personal account of how our concept of geomagnetic storms has advanced and how new findings based on satellite and ground-based observations have made it possible to arrive at a first-approximation expression for the interplanetary quantity controlling the development of geomagnetic storms. Since a geomagnetic storm is a magnetic manifestation of a magnetospheric storm, which is a nonlinear superposition of intense magnetospheric substorms, the main emphasis is shifted toward the understanding of magnetospheric substorms in order to arrive at the parameters controlling the development of geomagnetic storms.

Akasofu, S.-I.↗
NASA NTRS1982

Evidence for specularly reflected ions upstream from the quasi-parallel bow shock

Ion velocity distributions in the form of bunches of gyrating particles traveling along helical paths have been observed moving sunward immediately upstream from quasi-parallel parts of the earth's bow shock using Los Alamos/Garching instruments on ISEE-1 and -2. These distributions have characteristics which indicate that they are produced by the nearly specular reflection at the shock of a portion of the incident solar wind ions. In particular, the guiding center motion and the gyrospeeds of the gyrating ions are quantitatively consistent with simple geometrical considerations for specular reflection. These considerations reveal that specularly reflected ions can escape upstream when the angle between the upstream magnetic field and the local shock normal is less than 45 deg but not when the angle is greater than 45 deg. These upstream gyrating ions are an important signature of one of the processes by which solar wind streaming energy is dissipated into other forms of energy at the shock.

Gosling, J. T.↗
NASA NTRS1983

Ions upstream of the earth's bow shock - A theoretical comparison of alternative source populations

The trajectories of ions reflected or leaked upstream from the earth's bow shock and subject solely to the Lorentz force in a steady interplanetary magnetic field B and the V x B electric field are studied theoretically. Expressions are obtained for the guiding center motion and gyromotion in a frame (the Hoffman-Teller frame) moving parallel to the shock surface with sufficient speed to transform the incident solar wind velocity into motion entirely along the interplanetary magnetic field. Equations are derived which transform these motions back to the observer's frame. The predicted upstream motions for four different source models for upstream ions are compared using these expressions: magnetic moment-conserving reflection of solar wind ions, specular reflection of solar wind ions, magnetic moment-conserving leakage of magnetosheath ions, and leakage of magnetosheath ions parallel to the shock normal.

Schwartz, S. J.↗
NASA NTRS1986

Thermal equilibrium of a cryogenic magnetized pure electron plasma

The thermal equilibrium correlation properties of a magnetically confined pure electron plasma (McPEP) are related to those of a one-component plasma (OCP). The N-particle spatial distribution rho sub s and the Helmholtz free energy F are evaluated for the McPEP to O(lambda sub d-squared/a-squared), where lambda sub d is the thermal de Broglie wavelength and is an interparticle spacing. The electron gyromotion is allowed to be fully quantized while the guiding center motion is quasi-classical. The distribution rho sub s is shown to be identical to that of a classical OCP with a slightly modified potential. To O(lambda sub d-squared/a-squared) this modification does not affect that part of F that is caused by correlations, as long as certain requirements concerning the size of the plasma are met. This theory is motivated by a current series of experiments that involve the cooling of a magnetically confined pure electron plasma to the cryogenic temperature range.

Dubin, D. H. E.↗
NASA NTRS1993

Stormtime transport of ring current and radiation belt ions

A dynamical guiding-center simulation model is used to study the stormtime ion transport which leads to the formation of the ring current and diffusion in the radiation belts. Representative ions guiding-center motion in response to model storm-associated impulses in the convection electric field is traced for a range of ion mu values. The present numerical results are compared with previously formulated limiting idealization of particle transport in order to assess the limits of validity of these approximations. For ions having drift periods that exceed the duration of the main phase of the storm, their inward transport to form the stormtime ring current is appropriately described as direct convective access. For ions having drift periods comparable to the duration of the main phase of the storm, there is a transition between direct convective access and transport that resembles radial diffusion. Lower-energy ring-current ions at L of about 3 are freshly injected there from open adiabatic trajectories, whereas the higher-energy ring-current population consists of a mixture of freshly injected and previously trapped ions.

Chen, Margaret W.↗
NASA NTRS1993

Stormtime transport of ring current and radiation belt ions

This is an investigation of stormtime particle transport that leads to formation of the ring current. Our method is to trace the guiding-center motion of representative ions (having selected first adiabatic invariants mu) in response to model substorm-associated impulses in the convection electric field. We compare our simulation results qualitatively with existing analytically tractable idealizations of particle transport (direct convective access and radial diffusion) in order to assess the limits of validity of these approximations. For mu approximately less than 10 MeV/G (E approximately less than 10 keV at L equivalent to 3) the ion drift period on the final (ring-current) drift shell of interest (L equivalent to 3) exceeds the duration of the main phase of our model storm, and we find that the transport of ions to this drift shell is appropriately idealized as direct convective access, typically from open drift paths. Ion transport to a final closed drift path from an open (plasma-sheet) drift trajectory is possible for those portions of that drift path that lie outside the mean stormtime separatrix between closed and open drift trajectories, For mu approximately 10-25 MeV/G (110 keV approximately less than E approximately less than 280 keV at L equivalent to 3) the drift period at L equivalent to 3 is comparable to the postulated 3-hr duration of the storm, and the mode of transport is transitional between direct convective access and transport that resembles radial diffusion. (This particle population is transitional between the ring current and radiation belt). For mu approximately greater than 25 MeV/G (radiation-belt ions having E approximately greater than 280 keV at L equivalent to 3) the ion drift period is considerably shorter than the main phase of a typical storm, and ions gain access to the ring-current region essentially via radial diffusion. By computing the mean and mean-square cumulative changes in 1/L among (in this case) 12 representative ions equally spaced in drift time around the steady-state drift shell of interest (L equivalent to 3), we have estimated (from both our forward and our time-reversed simulations) the time-integrated radial-diffusion coefficients D(sup sim)(sub LL) for particles having selected values of mu approximately greater than 15 MeV/G. The results agree surprisingly well with the predictions (D(sup ql)(sub LL)) of quasilinear radial diffusion theory, despite the rather brief duration (approximately 3 hrs) of our model storm and despite the extreme variability (with frequency) of the spectral-density function that characterizes the applied electric field during our model storm. As expected, the values of D(sup sim)(sub LL) deduced (respectively) from our forward and time-reversed simulations agree even better with each other and with D(sup sim)(sub LL) when the impulse amplitudes which characterize the individual substorms of our model storm are systematically reduced.

Chen, Margaret W.↗
NASA NTRS2002

Wave-Particle Interactions Associated with Nongyrotropic Distribution Functions: A Hybrid Simulation Study

Nongyrotropic plasma distribution functions can be formed in regions of space where guiding center motion breaks down as a result of strongly curved and weak ambient magnetic fields. Such are the conditions near the current sheet in the Earth's middle and distant magnetotail, where observations of nongyrotropic ion distributions have been made. Here a systematic parameter study of nongyrotropic proton distributions using electromagnetic hybrid simulations is made. We model the observed nongyrotropic distributions by removing a number of arc length segments from a cold ring distribution and find significant differences with the results of simulations that initially have a gyrotropic ring distribution. Model nongyrotropic distributions with initially small perpendicular thermalization produce growing fluctuations that diffuse the ions into a stable Maxwellian-like distribution within a few proton gyro periods. The growing waves produced by nongyrotropic distributions are similar to the electromagnetic proton cyclotron waves produced by a gyrotropic proton ring distribution in that they propagate parallel to the background magnetic field and occur at frequencies on the order of the proton gyrofrequency, The maximum energy of the fluctuating magnetic field increases as the initial proton distribution is made more nongyrotropic, that is, more highly bunched in perpendicular velocity space. This increase can be as much as twice the energy produced in the gyrotropic case.

Convery, P. D.↗