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Z Pinch Kinetics I -- A Particle Perspective: Transitional Magnetization and Cyclotron and Betatron Orbits
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Abstract
Single-particle orbits are investigated as the basis of a coherent kinetic theory. Analytic solutions including azimuthal circulation are obtained in elliptic functions, of which ideal cyclotron and betatron motions are limiting solutions. The elliptic modulus depends on a trapping parameter and an axis-encircling to axial-drift kinetic energy ratio. Large trapping parameters limit to guiding-center cyclotron motion, while small trapping parameters describe ideal betatron motion. The analytic solutions inform a separation of phase space into trapped and passing trajectories, with which the Bennett solution is decomposed into cyclotron and betatron distributions. The two partial densities are computed to reveal a transitional magnetization layer the extent of which depends only on an ensemble-averaged trapping parameter, equivalent to: the Budker parameter, the pinch-to-Alfv\'{e}n current ratio, the Larmor radius parameter squared, the Hall parameter squared, the drift parameter squared, and the Hartmann number squared. The interplay of diamagnetic current and axial betatron flux is clarified through partial current densities, and the radial electric fields of shear flows are observed to influence ion orbits particularly around unity Budker parameter.
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Continuum kinetic simulations are increasingly capable of resolving high-dimensional phase space with advances in computing. These capabilities can be more fully explored by using linear kinetic theory to initialize the self-consistent field and phase space perturbations of kinetic instabilities. The phase space perturbation of a kinetic eigenfunction in unmagnetized plasma has a simple analytic form, and in magnetized plasma may be well approximated by truncation of a cyclotron-harmonic expansion. We catalogue the most common use cases with a historical discussion of kinetic eigenfunctions and by conducting nonlinear Vlasov–Poisson and Vlasov–Maxwell simulations of singlemode and multimode two-stream, loss-cone and Weibel instabilities in unmagnetized and magnetized plasmas with one- and two-dimensional geometries. Applications to quasilinear kinetic theory are discussed and applied to the bump-on-tail instability. In order to compute eigenvalues we present novel representations of the dielectric function for ring distributions in magnetized plasmas with power series, hypergeometric and trigonometric integral forms. Eigenfunction phase space fluctuations are visualized for prototypical cases such as the Bernstein modes to build intuition. In addition, phase portraits are presented for the magnetic well associated with nonlinear saturation of the Weibel instability, distinguishing current-density-generating trapping structures from charge-density-generating ones.
The FuZE sheared-flow-stabilized Z pinch at Zap Energy is simulated using whole-device modeling employing an axisymmetric resistive magnetohydrodynamic formulation implemented within the discontinuous Galerkin WARPXM framework. Simulations show formation of Z pinches with densities of approximately 10²² m⁻³ and total DD fusion neutron rate of 10⁷ per µs for approximately 2 µs. Simulation-derived synthetic diagnostics show peak currents and voltages within 10% and total yield within approximately 30% of experiment for similar plasma mass. The simulations provide insight into the plasma dynamics in the experiment and enable a predictive capability for exploring design changes on devices built at Zap Energy.
The sheared-flow-stabilized Z pinch concept has been studied extensively and is able to produce fusion-relevant plasma parameters along with neutron production over several microseconds. We present here elevated electron temperature results spatially and temporally coincident with the plasma neutron source. An optical Thomson scattering apparatus designed for the FuZE device measures temperatures in the range of 1–3 keV on the axis of the device, 20 cm downstream of the nose cone. The 17-fiber system measures the radial profiles of the electron temperature. Scanning the laser time with respect to the neutron pulse time over a series of discharges allows the reconstruction of the Te temporal response, confirming that the electron temperature peaks simultaneously with the neutron output, as well as the pinch current and inductive voltage generated within the plasma. Comparison to spectroscopic ion temperature measurements suggests a plasma in thermal equilibrium. The elevated Te confirms the presence of a plasma assembled on axis, and indicates limited radiative losses, demonstrating a basis for scaling this device toward net gain fusion conditions.
When a particle crosses a region of space where the curvature radius of the magnetic field line shrinks below its gyroradius , it experiences a non-adiabatic (magnetic moment violating) change in pitch angle. The present paper carries that observation into magnetohydrodynamic (MHD) turbulence to examine the influence of intermittent, sharp bends of the magnetic field lines on particle transport. On the basis of dedicated measurements in a simulation of incompressible turbulence, it is argued that regions of sufficiently large curvature exist in sufficient numbers on all scales to promote scattering. The parallel mean free path predicted by the power-law statistics of the curvature strength scales in proportion to ( is the coherence scale of the turbulence), which is of direct interest for cosmic-ray phenomenology. Particle tracking in that numerical simulation confirms that the magnetic moment diffuses through localized, violent interactions, in agreement with the above picture. Correspondingly, the overall transport process is non-Brownian up to length scales .
This Rapid Communication identifies the physical mechanism for the quench of turbulent resistivity in two-dimensional magnetohydrodynamics. Without an imposed, ordered magnetic field, a multiscale, blob-and-barrier structure of magnetic potential forms spontaneously. Magnetic energy is concentrated in thin, linear barriers, located at the interstices between blobs. The barriers quench the transport and kinematic decay of magnetic energy. The local transport bifurcation underlying barrier formation is linked to the inverse cascade of 〈A2〉 and negative resistivity, which induce local bistability. For small-scale forcing, spontaneous layering of the magnetic potential occurs, with barriers located at the interstices between layers. This structure is effectively a magnetic staircase.
The sheared-flow stabilized Z pinch has demonstrated long-lived plasmas with fusion-relevant parameters. We present the first experimental results demonstrating sustained, quasi-steady-state neutron production from the fusion Z-pinch experiment, operated with a mixture of 20% deuterium/80% hydrogen by pressure. Neutron emissions lasting approximately 5 μs are reproducibly observed with pinch currents of approximately 200 kA during an approximately 16 μs period of plasma quiescence. The average neutron yield is estimated to be (1.25±0.45)×105 neutrons/pulse and scales with the square of the deuterium concentration. Coincident with the neutron signal, plasma temperatures of 1–2 keV and densities of approximately 1017 cm−3 with 0.3 cm pinch radii are measured with fully integrated diagnostics.
To explain many natural magnetized plasma phenomena, it is crucial to understand how rates of collisionless magnetic reconnection scale in large magnetohydrodynamic (MHD) scale systems. Simulations of isolated current sheets conclude such rates are independent of system size and can be reproduced by the Hall-MHD model, but neglect sheet formation and coupling to MHD scales. Here, it is shown for the problem of flux-rope merging, which includes this formation and coupling, that the Hall-MHD model fails to reproduce the kinetic results. The minimum sufficient model must retain ion kinetic effects, which set the ion diffusion region geometry and give time-averaged rates that reduce significantly with system size, leading to different global evolution in large systems.
The observation of the motion of particles and light near a gravitating object is until now the only way to explore and to measure the gravitational field. In the case of exact black hole
solutions of the Einstein equations the gravitational field is characterized by a small number of parameters which can be read off from the observables related to the orbits of test particles and light rays. Here we review the state of the art of analytical solutions of geodesic
equations in various space–times. In particular we consider the four dimensional black hole space–times of Plebański-Demiański type as far as the geodesic
equation separates, as well as solutions in higher dimensions, and also solutions with cosmic strings. The mathematical tools used are elliptic and hyperelliptic functions. We present a list of analytic solutions which can be found in the literature.
The kinetic theory of the m = 1 kink instability of a Z-pinch with Bennett profile is presented. The dominant particle trajectories in the equilibrium field are the large excursion betatron orbits. To deal with these orbits, an integral formulation of the stability analysis is adopted. In this case, the dispersion relation is expressed as the determinant of a matrix. In the limit where the electrostatic perturbation is neglected, the eigenmodes are computed numerically from this dispersion relation. Two methods are used to obtain the growth rates, and the computed values agree very well. The Landau damping of the mode is found to be strong enough to stabilize the mode at shorter wavelengths. This may explain the stability of the kink mode in some experiments.
Theoretical studies have predicted that the Z-pinch can be stabilized with a sufficiently sheared axial flow [U. Shumlak and C. W. Hartman, Phys. Rev. Lett. 75, 3285 (1995)]. A Z-pinch experiment is designed to generate a plasma which contains a large axial flow. Magnetic fluctuations and velocity profiles in the plasma pinch are measured. Experimental results show a stable period which is over 700 times the expected instability growth time in a static Z-pinch. The experimentally measured axial velocity shear is greater than the theoretical threshold during the stable period and approximately zero afterwards when the magnetic mode fluctuations are high.
The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi), the motion of a force-free asymmetric top (Jacobi), the motion of a spherical pendulum (Weierstrass), and the motion of a heavy symmetric top with one fixed point (Weierstrass). The problem of the planar pendulum, in fact, can be used to construct the general connection between the Jacobi and Weierstrass elliptic functions. The easy access to mathematical software by physics students suggests that they might reappear as useful tools in the undergraduate curriculum.
A complete quasilinear model is derived for the electrostatic acceleration-driven lower hybrid drift instability in a uniform two-species low-beta plasma in which current is perpendicular to the background magnetic field. The model consists of coupled nonlinear velocity space diffusion equations for the volume-averaged ion and electron distribution functions. Each species' diffusion coefficient depends on a time-evolving spectral density of the electric-field energy per unit volume and a time-evolving dispersion relation. The dispersion relation is expressed analytically in integral form without the use of asymptotic limits and applies to arbitrary distribution functions, so long as they can be expressed as a function of one velocity coordinate, e.g., f(vy) or f(v⊥). The quasilinear model conserves energy and is complete in that it fully describes the evolution of the distribution functions, including resonant and nonresonant particle-wave interactions, while accounting for distribution-function-dependent mixed-complex frequencies. The quasilinear diffusion model is solved numerically and self-consistently using a Crank-Nicolson temporal discretization and a second-order finite-volume velocity-space discretization. Numerical solutions are compared to nonlinear fourth-order accurate continuum kinetic Vlasov-Poisson simulations. Evolution of electric-field energy, growth rates, distribution functions, and diffusion coefficients are shown to be in agreement with Vlasov simulations. The quasilinear model is shown to predict anomalous transport terms, like resistivity and heating, to within a factor of order unity. Discrepancies between the quasilinear model and Vlasov simulations are assessed and attributed primarily to lack of damping in the quasilinear description and to the use of unperturbed-orbit susceptibilities in the linear theory dispersion relation. The results illuminate the predictive accuracy of the quasilinear model, place approximate bounds on its validity, and provide much needed vetting of quasilinear theory's ability to predict the nonlinear state of a microturbulent plasma.
The field of elliptic functions, apart from its own mathematical beauty, has many applications in physics in a variety of topics, such as string theory or integrable systems. This book, which focuses on the Weierstrass theory of elliptic functions, aims at senior undergraduate and junior graduate students in physics or applied mathematics. Supplemented by problems and solutions, it provides a fast, but thorough introduction to the mathematical theory and presents some important applications in classical and quantum mechanics. Elementary applications, such as the simple pendulum, help the readers develop physical intuition on the behavior of the Weierstrass elliptic and related functions, whereas more Interesting and advanced examples, like the n=1 Lamé problem-a periodic potential with an exactly solvable band structure, are also presented.
This mini-tutorial summarizes the plasma characteristics important for the Z -pinch research, with an emphasis on high-density collisional plasmas. It begins with the discussion of the most basic plasma properties related to collisionality and magnetization and then proceeds to more complex phenomena associated with magnetic field evolution in a highly dynamical plasma. Plasma transport properties are discussed mostly in conjunction with the Magnetized Liner Inertial Fusion concept. Issues of interplay of the classical and anomalous transport in a plasma whose pressure is higher than the magnetic pressure are elucidated. Differences in magnetic reconnection in weakly versus highly collisional plasmas are discussed. Kinetic effects and the role of microturbulence are mentioned in conjunction with the formation of high-energy tails in the particle distribution functions and the generation of particle beams. The discussion is based on the order-of-magnitude estimates suitable for initial orientation in the problem. The two appendixes contain some auxiliary material.
A kinetic description of intense nonneutral beam propagation through a periodic solenoi-dal focusing field B so1 (x) = Bz(z)e z -(1/2)B;(z)(xe x + yey) is developed, where Bz(z + S) = Biz), and S = const. is the axial periodicity length. The analysis is carried out for a thin beam with characteristic beam radius rb« S, and directed axial momentum 'Ybm!3bc (in the z-direction) large compared with the transverse momentum and axial momentum spread of the beam particles. Making use of the nonlinear Vlasov-Maxwell equations for general distribution functionfb(x,p, t) and self-consistent electrostatic field ES(x, t) = -\7q/(x, t) consistent with the thin-beam approximation, the kinetic model is used to investigate detailed beam equilibrium properties for a variety of distributions functions. Examples are presented both for the case of a uniform solenoidal focusing field Biz) = B o = const. and for the case of a periodic solenoidal focusing field Bz(z + S) = Bz(z). The nonlinear Vlasov-Maxwell equations are simplified in the thin-beam approximation, and an alternative Hamiltonian formulation is developed that is particularly well-suited to intense beam propagation in periodic focusing systems. For the case of a uniform focusing field, the nonlinear Vlasov-Maxwell equations are used to investigate a wide variety of azimuthally symmetric (B/BB = 0) intense beam equilibria characterized by B/Bt = 0 = B/Bz, ranging from distributions that are isotropic in momen-tum dependence in the frame of the beam, to anisotropic distributions in which the momentum spreads are different in the axial and transverse directions. As a general remark, for a uniform focusing field, it is found that there is enormous latitude in the choice of equilibrium distribution function f b o, and the corresponding properties of the Vlasov-Maxwell equations to investigate intense beam propagation in a periodic sole-noidal field Bz(s + S) = BzCs), in which case the properties of the beam are modulated as a . function of s by the focusing field. The nonlinear Vlasov equation is transformed to a frame of reference rotating at the Larmor frequency nL(s) = -ZieBz(s)/2,bmc, and the description is further simplified by assuming azimuthal symmetry, in which case the canonical angular momentum is an exact single-particle constant of the motion. As an application of this general formalism, the specific example is considered of a periodically focused, rigid-rotor Vlasov equilibrium with step-function radial density profile and average azimuthal motion of the beam corresponding to a rigid rotation (in the Larmor frame) about the axis of symmetry. A wide range of beam properties are calculated, such as the average flow velocity, transverse temperature profile, and transverse thermal emit-tance. This example represents an important generalization of the familiar Kapchinskij-Vladimirskij beam distribution to allow for average beam rotation in the Larmor frame. Based on the present analysis, the Vlasov-Maxwell description of intense nonneutral beam propagation through a periodic solenoidal focusing field Bso1(x) is found to be remarkably tractable and rich in physics content. The Vlasov-Maxwell formalism devel-oped here can be extended in a straightforward manner to investigate detailed stability behavior for perturbations about specific choices of beam equilibria.
This Resource Letter surveys the literature on momentum in
electromagnetic fields, including the general theory, the relation
between electromagnetic momentum and vector potential, ``hidden''
momentum, the 4/3 problem for electromagnetic mass, and the
Abraham-Minkowski controversy regarding the field momentum in
polarizable and magnetizable media.
Heat and particle transports which are associated with the helicity transport are obtained in a current-carrying plasma. The helicity flux represents the transport of parallel current density and is produced by an electric field with a circularly polarized component. Fluctuations with circularly polarized components induce finite average in cross-field nonlinear parallel current, which leads to generation of frictional electron heat flux as well as ion nonlinear polarization current that produces particle flux. The circular polarization of the perturbed electric field thus relates the helicity flux, electron heat flux, and the particle flux in such a manner that the heat and the particle transports in the direction opposite to the helicity flux. This result applies whether the helicity is injected externally by oscillating fields or it is generated internally in the plasma.
The plasma is treated as an assembly of classical particles having masses m+ and m− and charges ±e. It is shown that if such a plasma is in thermodynamic equilibrium it is unaffected by a magnetostatic field. In particular, it cannot be confined by a magnetostatic field and at the same time be in equilibrium. In absence of collisions there exist stationary solutions for the magnetic field and the particle distribution in phase space such that the gas is confined. The solutions are self-consistent, meaning that the particle motions generate the field, which in turn maintains the particle distribution. However, this distribution may not be Maxwellian. As an example the linear pinch is treated. The fields, the particle number densities, and the current density are calculated for one particular case.
Streams of fast electrons which can accumulate positive ions in sufficient quantity to have a linear density of positives about equal to the linear density of electrons, along the stream, become magnetically self-focussing when the current exceeds a value which can be calculated from the initial stream conditions. Focussing conditions obtain when breakdown occurs in cold emission. The characteristic features of breakdown are explained by the theory. Failure of high voltage tubes is also discussed.
The dissipation mechanism of guide field magnetic reconnection remains a subject of intense scientific interest On one hand one set of recent studies have shown that particle inertia-based processes which include thermal and bulk inertial effects provide the reconnection electric field in the diffusion region On the other hand a second set of studies emphasizes the role of wave-particle interactions in providing anomalous resistivity in the diffusion region In this presentation we present analytical theory results as well as 2 5 and three-dimensional PIC simulations of guide-field magnetic reconnection We will show that diffusion region scale sizes in moderate and large guide field cases are determined by electron Larmor radii and that analytical estimates of diffusion region dimensions need to include description of the heat flux tensor The dominant electron dissipation process appears to be based on thermal electron inertia expressed through nongyrotropic electron pressure tensors We will argue that this process remains viable in three dimensions by means of detailed comparisons with high resolution particle-in-cell simulations and perform detailed analyses of electron distributions The latter will demonstrate how electron gyroscales set up local pressure nongyrotropies near the maximum parallel electric field We will discuss in detail the measurement requirements that emerge from the present understanding of magnetic reconnection physics
Fiber-initiated High Density Z-Pinches at Los Alamos, NRL, and Karlsruhe have shown anomalously good stability. Kink modes are never seen, and sausage modes are at least delayed until late in the discharge. The success of these devices in reaching fusion conditions may depend on maintaining and understanding this anomalous stability. We have developed two numerical methods to study the stability in the regime where fluid theory is valid. While our methods are applicable to all modes, we will describe them only for the m=0 sausage mode. The appearance of sausage modes late in the discharge and the total absence of kink modes suggest that an understanding of sausage modes is more urgent, and it is also simpler.
This paper considers all resistive instabilities of a self-pinched cylindrically symmetric beam of charged particles in a finite or an infinite Ohmic plasma channel. The problem is reduced to an ordinary second-order linear differential equation for the longitudinal component of the perturbed electric field. The equation can be solved for a uniform beam shape, yielding an implicit transcendental equation whose roots define the various modes. We find that for each azimuthal ``quantum number'' m there are two infinite sequences of modes and two exceptional modes, except that some of these modes are missing for m = 0, 1, and 2. In all modes we find stable oscillation at very low and very high frequencies, and instability at intermediate frequencies, the growth rates generally reaching maxima somewhat less than the betatron frequency ωβ. The largest maximum growth rate is in the ``hose'' mode (the only exceptional mode for m = 1), where it is approximately 0.29 ωβ. For a general smooth beam shape, the catalog of modes is similar to that for a uniform beam, except that there also appears a continuous spectrum. It is also proved for general beam shape that at low frequencies the ``hose'' dispersion relation becomes the same as that derived earlier under the assumption of rigid beam displacement; this is not the case at higher frequencies.
Trapped nonadiabatic charged particle motion in two-dimensional taillike magnetic field reversals with arbitrary curvature radii is studied. It is shown that the type of trajectory is dependent on a curvature parameter representing the ratio between the minimum curvature radius and the maximum particle Larmor radius. Consideration is given to the breakdown of regularity and charged particle motion in a sharply curved field reversal. The basic equations for describing particle motion in sharply curved fields are derived. The applications of the research for magnetotail physics are discussed.
The adiabatic theory is presented for charged particles moving in a
region where the main component of the magnetic field reverses its
direction. The unperturbed field configuration is one-dimensional and
involves two regions of uniform and opposite magnetic fields separated
by a reversal region of small but finite extent. The adiabatic theory is
capable of describing particle orbits when gradual spatial and/or
temporal field variations are added to this configuration. Several
applications are given. These include orbits during a null-sheet
compression, orbits in the sheets studied by Alfvén and Speiser,
and orbits in the vicinity of the magnetic null line of the so-called
field-reconnection geometry.
We explore the mechanism for ion acceleration during magnetic reconnection to understand the energetic particle spectra produced during flares and in the solar wind. Reconnection driven ion acceleration is initiated as particles move from upstream into the Alfvenic exhaust. In the case of a weak guide field, protons and higher mass particles behave like pickup particles in that they abruptly cross a narrow boundary layer and find themselves in a region of Alfvenic outflow. Their motion then mimics that of a classic pickup ion, gaining an Alfvenic ExB flow in the jet and a thermal speed close to the Alfven speed. In the case of a strong guide field particle acceleration is strongly enhanced for ions with high mass-to-charge (m/q) ratio since these particles act as pick-up particles while small m/q particles are adiabatic. Once ions become super-Aflvenic their acceleration is, like electrons, dominated by Fermi reflection during island contraction and their energy increases until it is limited by firehose marginal stability. The ion distribution function for super-Alfvenic ions then takes the form of a v-5 distribution. During reconnection in a multi-island environment, as in flares, strong enhancements in high m/q ions are expected. This picture is consistent with several observations related to flare and local solar wind ion acceleration: (1) the ubiquitous observations of energy proportional to mass; (2) strong enhancements in high m/q ions during impulsive flares; and (3) the temperature increments of solar wind exhausts.
The physical principles underlying the formation of a relativistic stabilized electron beam are presented and questions connected with the stability of such a configuration are discussed.
The macroscopic dynamical equations of a tenuous ionized gas in a magnetic field are developed by averaging over the individual ion and electron motions, which do not necessarily possess an isotropic distribution. It is shown that the principal motion of the gas is related to the magnetic field by the usual hydromagnetic equations, as developed for conducting liquids and dense gases; the anisotropy of the individual particle motions shows up primarily as a coefficient multiplying the pondermotive force exerted by the magnetic field on the plasma. The results reduce properly to the earlier work of Schl\"uter, Cowling, and Spitzer for isotropic pressure, and are in agreement with the recent developments from the Boltzmann equation. It is pointed out that the magnetic lines of force are permanently connected and move in the frame of reference of the electric drift. It is shown that near static equilibrium, when the principal motions vanish, there remain small macroscopic drift motions of the gas in the field inhomogeneities.
The limitation of a current consisting of charged high energy particles (cosmic rays) passing through interstellar space is discussed. As interstellar matter is ionized, interstellar space is considered as a good conductor so that the electric field always equals zero. The motion of the particles is then governed by the magnetic fields produced by themselves. This sets a rather low upper limit to the currents through space so that a difference in intensity of cosmic rays in different points is smoothed out very slowly. This means that the intensity may vary considerably even within the galactic system. An explanation of the excess of positive particles in cosmic radiation is tentatively suggested. Arguments are given for the view that most of the rays we receive on the earth are generated within less than 1000 light years from us.
The Bennett equilibrium is shown to be a particular case of a class of Vlasov equilibria characterized by a constant axial macroscopic velocity for the ions and the electrons, and by a constant ratio of the densities. It is established that any collisionless plasma column in radial equilibrium, carrying a very high current, may be considered as the superposition, on the thermal background of ions, of two counterstreaming electron beams, the more intense of which carries most energetic electrons in the vicinity of the axis. This property could be interpreted as a partial 'magnetic neutralization', and thus explain the absence of any current limitation.
A two-dimensional magnetohydrodynamic code that includes resistivity and viscosity has been used to study the growth of m = 0 perturbations of a Z pinch in which the current increases with time. Stability for all wave numbers was found when the product SR was less than about 1000, where S is the Lundquist number and R is the Alfven-Reynolds number. Since SR depends only on the line density N, the stability criterion translates as N < 2 X 10(16) ions cm-1 (deuterium). This is two to three orders of magnitude smaller than the line density of fiber pinch experiments that have shown anomalous stability, and it is concluded that viscoresistive effects are inadequate to account for these observations.
It is well known both in early Z-pinch experiments and more recently in plasma focus experiments that highly accelerated ion beams are formed, this being inferred from the axial centre-of-mass motion of the neutrons produced by nuclear reactions. Two acceleration mechanisms are discussed here. The first, proposed in ref. 1, concerns the acceleration of ions across an electrostatic sheath adjacent to the anode, this sheath having a thickness less than an ion Larmor radius but greater than an electron Larmor radius. The electrons in the sheath, being magnetised, have a cE × B/B2 drift through which the JrBθ force is provided to satisfy momentum conservation. The second is a new mechanism arising from the occurrence of a violent m = 0 instability, and relies on large ion Larmor radius effects. It is shown that the inclusion of the Hall terms and finite ion Larmor radius (FLR) terms in the MHD equations removes the mirror symmetry each side of an m = 0 neck. A detailed consideration of particle orbits during the formation of the highly compressed neck shows that on the cathode side of the neck an energetic ion beam is formed close to the axis, whilst on the anode side an equal and opposite momentum is distributed among a larger number of ions throughout the cross-section.
The spectacular progress made during the last few years in reaching high energy densities in fast implosions of annular current sheaths (fast Z pinches) opens new possibilities for a broad spectrum of experiments, from x-ray generation to controlled thermonuclear fusion and astrophysics. Presently Z pinches are the most intense laboratory X ray sources (1.8 MJ in 5 ns from a volume 2 mm in diameter and 2 cm tall). Powers in excess of 200 TW have been obtained. This warrants summarizing the present knowledge of physics that governs the behavior of radiating current-carrying plasma in fast Z pinches. This survey covers essentially all aspects of the physics of fast Z pinches: initiation, instabilities of the early stage, magnetic Rayleigh-Taylor instability in the implosion phase, formation of a transient quasi-equilibrium near the stagnation point, and rebound. Considerable attention is paid to the analysis of hydrodynamic instabilities governing the implosion symmetry. Possible ways of mitigating these instabilities are discussed. Non-magnetohydrodynamic effects (anomalous resistivity, generation of particle beams, etc.) are summarized. Various applications of fast Z pinches are briefly described. Scaling laws governing development of more powerful Z pinches are presented. The survey contains 36 figures and more than 300 references.
New computational results are presented which advance the understanding of the stability properties of the field-reversed configuration (FRC). The results of hybrid and two-fluid [Hall-MHD (magnetohydrodynamic)] simulations of prolate FRCs are presented. The n = 1 tilt instability mechanism and growth rate reduction mechanisms are investigated in detail, including resonant particle effects, finite Larmor radius and Hall stabilization, and profile effects. It is shown that the Hall effect determines the mode rotation and the change in the linear mode structure in the kinetic regime; however, the reduction in the growth rate is mostly due to finite Larmor radius effects. Resonant wave–particle interactions are studied as a function of (a) elongation, (b) the kinetic parameter S∗, which is proportional to the ratio of the separatrix radius to the thermal ion Larmor radius, and (c) the separatrix shape. It is demonstrated that, contrary to the usually assumed stochasticity of the ion orbits in the FRC, a large fraction of the orbits are regular in long configurations when S∗ is small. A stochasticity condition is found, and a scaling with the S∗ parameter is presented. Resonant particle effects are shown to maintain the instability in the large gyroradius regime regardless of the separatrix shape. © 2003 American Institute of Physics.
The equations governing the guiding center motion of a charged particle in an electromagnetic field are obtained simultaneously and deductively, without considering individually the special geometric situations in which one effect or another occurs alone. The general expression is derived for the guiding center velocity at right angles to the magnetic field BbdB. This expression contains five terms arising in the presence of an electric field. They are in addition to the usual “E × B” drift. Because these terms are unfamiliar objects in the literature on plasmas, they are illustrated by simple examples. Three of the five drifts occur in rotating plasma machines of the Ixion type. One of these three is also shown to be responsible for the Helmholtz instability of a plasma. A fourth one gives the (low frequency) dielectric constant, while the fifth arises if the direction of B is time dependent. A detailed geometric picture of the fifth drift is given.The equation governing the guiding center motion parallel to B is also derived for the general time-dependent field. The conditions are discussed under which it can be integrated into the form of an energy integral.Finally the component of current density perpendicular to B in a collisionless plasma is shown to be the current due to the guiding center drift plus the perpendicular component of the curl of the magnetic moment per unit volume. Proofs of this have been given in the past for special cases, such as static fields, ∇ × B = 0, etc. This proof holds in general, provided conditions for adiabaticity are met. It is also true, but not proven in this paper, that the component of the current density parallel to B is the current due to the guiding center velocity parallel to B plus the parallel component of the curl of the magnetic moment per unit volume. A proper proof of the parallel component is quite lengthy.
From the Vlasov-fluid model a set of approximate stability equations describing the stability of the pure z–pinch is derived. The equations are valid for equilibria with small gyroradius compared with the pinch radius, but the perturbation wavenumber k may be of the order of the gyroradius ρi, δ = kρi = 0(1) - so-called gyrokinetic ordering. The equations are used to study the stability of the m = 0 and m = 1 internal modes of the z–pinch. In the limit of zero gyroradius δ → 0 we recover previously obtained results. For δ ≠ 0 we find that increasing δ at first gives a rapidly decreasing growth rate, and for δ ≈ l the growth rate compared with perpendicular MHD is γ/γMHD ≈ 0·09. For larger δ however, the growth rate increases to a quite large value. For the m = O mode we find, provided that drift resonances can be neglected, a stability criterion for δ ≥ 1, which is fulfilled both for the Bennett equilibrium and the constant-current-density equilibrium.
Both dynamic and equilibrium z pinches are mostly
described, whether in theory or in simulation, by conventional
magnetohydrodynamic (MHD) fluid equations. However there are
several key phases in z-pinch behavior when kinetic
effects are important. Runaway electrons can occur close to
the axis especially in an m = 0 neck or during the
subsequent disruption. Large ion-Larmor orbits can, if sufficiently
collisionless, lead to stabilizing effects. During a disruption,
ion beams can be produced. For deuterium discharges, the
interaction of an ion beam with the ambient plasma can lead
to a significant neutron yield. If the drift velocity of the
current-carrying electrons exceeds a threshold for generating
microinstabilities (lower-hybrid or ion-acoustic instabilities),
this leads to anomalous resistivity. This can occur not only
during a disruption, but also in the low-density (usually outer)
plasma boundary of an equilibrium or dynamic pinch. Related
to this is the open question of whether current reconnection
can occur in fully developed magneto-Rayleigh–Taylor
instabilities in the low-density coronal plasma. Two other kinetic
effects are new to z pinches. First, there are the
collisions of low-density energetic ions from a wire array as
they pass close to the axis to form a precursor plasma. Second,
there is the possible erosion and ablation of wire cores in
the necks of m = 0 coronal current-carrying plasma
by flux-limited heat flow with its attendant deviation from
a Maxwellian of the isotropic part of the distribution function.
The relationship of the singular current which flows within one Larmor radius of the axis of a Z-pinch to the current due to guiding centre and diamagnetic motion in the magnetised region is found. In particular for zero local ion centre-of-mass velocity it is shown that ions flow on-axis from the anode to the cathode and there is a return flow by guiding centre drift in the off-axis magnetised region. The analogous electron flow indicates that in a steady pinch held under pressure balance the heat losses to the electrodes will be dominant and indeed must be balanced by Ohmic heating for an equilibrium to hold.
Hall magnetohydrodynamics (MHD) near a two-dimensional (2D) X-type magnetic neutral line is considered. The Hall effects are shown to be able to sustain the hyperbolicity of the magnetic field (and hence a more open X-point configuration) near the neutral line in the steady state. This result is predicated on considering the steady Hall MHD state as the temporal asymptotic limit of the corresponding time-dependent problem. For the time-dependent Hall MHD problem, the Hall effects are shown to have a negligible impact on the current-sheet formation process near the X-type magnetic neutral line at short times but, subsequently, to quench the finite-time singularity exhibited in ideal MHD and, hence to prevent the plasma collapse, in consistency with the sustenance of the hyperbolicity of the magnetic field in the corresponding steady problem.
The linear stability problem of a current flowing perpendicularly to a magnetic field is analysed. With the restriction that the growth rate should be larger than the ion cyclotron frequency, all the fast growing electrostatic modes are computed and compared. It is argued that the most significant modes of the problem (relevant to perpendicular collisionless shocks, pinches and the plasma focus) can be obtained without inclusion of effects due to gradients of the equilibrium quantities. The computations are divided into three broad categories: Te Ti, Te = Ti and Ti = 10 Te. With the aid of computation and analysis an attempt has been made to cover the whole range of parameter space for this problem.
The Z-pinch, perhaps the oldest subject in plasma physics, has achieved a remarkable renaissance in recent years, following a few decades of neglect due to its basically unstable MHD character. Using wire arrays, a significant transition at high wire number led to a great improvement in both compression and uniformity of the Z-pinch. Resulting from this the Z-accelerator at Sandia at 20 MA in 100 ns has produced a powerful, short pulse, soft x-ray source >230 TW for 4.5 ns) at a high efficiency of ~15%. This has applications to inertial confinement fusion. Several hohlraum designs have been tested. The vacuum hohlraum has demonstrated the control of symmetry of irradiation on a capsule, while the dynamic hohlraum at a higher radiation temperature of 230 eV has compressed a capsule from 2 mm to 0.8 mm diameter with a neutron yield >3 × 1011 thermal DD neutrons, a record for any capsule implosion. World record ion temperatures of >200 keV have recently been measured in a stainless-steel plasma designed for Kα emission at stagnation, due, it was predicted, to ion-viscous heating associated with the dissipation of fast-growing short wavelength nonlinear MHD instabilities. Direct fusion experiments using deuterium gas-puffs have yielded 3.9 × 1013 neutrons with only 5% asymmetry, suggesting for the first time a mainly thermal source. The physics of wire-array implosions is a dominant theme. It is concerned with the transformation of wires to liquid-vapour expanding cores; then the generation of a surrounding plasma corona which carries most of the current, with inward flowing low magnetic Reynolds number jets correlated with axial instabilities on each wire; later an almost constant velocity, snowplough-like implosion occurs during which gaps appear in the cores, leading to stagnation on the axis, and the production of the main soft-x-ray pulse. These studies have been pursued also with smaller facilities in other laboratories around the world. At Imperial College, conical and radial wire arrays have led to highly collimated tungsten plasma jets with a Mach number of >20, allowing laboratory astrophysics experiments to be undertaken. These highlights will be underpinned in this review with the basic physics of Z-pinches including stability, kinetic effects, and finally its applications.
At an elliptic magnetic stagnation line, the combined effect of viscosity ν and resistivity η stabilizes the ideally unstable kinks with short axial wavelengths λ. For small ν and η the critical wavelength below which all modes are stable is O(ν1/4 η1/4). If only one of the two parameters ν and η is nonzero (purely resistive or purely viscous plasma), the modes remain unstable, but the maximum growth rate is O(λ2/η) or O(λ2/ν), as opposed to the ideal growth rate (ν = η = 0) which is O(1). This result provides one possible mechanism to explain the absence of these instabilities in Z-pinch experiments such as EXTRAP. It is derived for the simple but generic case of the incompressible motion about circularly cylindrical equilibria with purely azimuthal magnetic fields and arbitrary axial current density profiles. For the equilibrium with constant current density the entire short wavelength spectrum is explicitly given. The eigenvalues (but not the eigenfunctions) are invariant upon interchanging ν and η. The eigenvalues pertaining to different radial mode numbers have spacing O(λ) and are on curves in the complex frequency plane which depend only on the azimuthal mode number m and on the two parameters λ2/ν and λ2/η.
A linear initial-value code is used to study the combined effect of resistivity and viscosity on the stability of a static z-pinch. Resistivity alone is seen to have only a minor effect, even at Lundquist parameters for which neglecting the evolution of the equilibrium is unjustified. However, the introduction of a scalar viscosity leads to strong damping of m=0 and m=1 modes, and can completely stabilize short-wavelength perturbations. In agreement with Spies (1988), the stability threshold is given in terms of the product of the Lundquist parameter and the viscous Reynolds number, so that the mode stability is due to the combined effect of resistivity and viscosity. When the scalar viscosity is replaced by the full ion stress tensor, the damping is seen to become less effective as the viscosity becomes more anisotropic. However, linear growth-rates are still significantly reduced for typical Z-pinch parameters.
Recent advances in both experimental and theoretical studies on neutron generation in various Z-pinch facilities are reviewed.
The main methods for enhancing neutron emission from the Z-pinch plasma are described, and the problems of igniting a thermonuclear
burn wave in this plasma are discussed.
We observe linear and nonlinear features of a strong plasma/magnetic field interchange Rayleigh-Taylor instability in the limit of large ion Larmor radius. The instability undergoes rapid linear growth culminating in free-streaming flute tips.
The regimes for the applicability of various theoretical models for the stability of a {ital Z} pinch under pressure balance are shown to be clearly delineated in a diagram of ln({ital I}{sup 4}{ital a}) vs ln{ital N}, where {ital I}, {ital a}, and {ital N} are the current, pinch radius, and line density, respectively. In particular, the most unstable regime where ideal magnetohydrodynamics applies is shown to be restricted to a small-wedge-shaped region bounded by resistive, viscous, anisotropic, and finite Larmor radius effects. Recent experimental results of anomalous stability can be interpreted in terms of resistive or large-ion Larmor-radius effects.
A scaling approach to the simplest viscoresistive MHD model reveals that the Prandtl number acts only through the inertia term. When this term is negligible the dynamics is ruled by the Hartmann number H only. This occurs for the reversed field pinch dynamics as seen by numerical simulation of the model. When H is large the system is in a multiple helicity state. In the vicinity of H = 2500 the system displays temporal intermittency with laminar phases of quasi-single-helicity (SH) type. For lower H's two basins of SH are shown to coexist. SH regimes are of interest because of their nonchaotic magnetic field.