Small Larmor Radius Research Articles

Small-signal analysis of the induced-resonance electron-cyclotron maser.

The induced-resonance electron-cyclotron maser operates at Doppler-upshifted frequencies \ensuremath{\omega}\ensuremath{\simeq}${\ensuremath{\gamma}}^{2}$${\ensuremath{\Omega}}_{0}$, with \ensuremath{\gamma} the relativistic factor and ${\ensuremath{\Omega}}_{0}$=${\mathrm{eB}}_{0}$/\ensuremath{\gamma}mc. In addition, the mechanism is insensitive to the beam thermal spreads when the index of refraction n=${k}_{z}$c/\ensuremath{\omega} is properly adjusted. A set of simplified equations, describing the electron dynamics near the axis of the configuration, is derived in the limit of small Larmor radius. The linear efficiency is derived for radiation having a Gaussian profile and propagating at an angle \ensuremath{\alpha} relative to a constant external magnetic field. The start-up current for operation in the submillimeter regime is well within the capabilities of today's long-pulse relativistic electron beams.

Inadequacies of finite Larmor radius treatments of the internal tilting instability in field-reversed configurations

A finite Larmor radius treatment of the tilting instability for field-reversed configurations is presented. It is shown on two accounts that such a treatment is inappropriate. First, the finite Larmor radius terms become infinite at the field null of the magnetic axis, because of the breakdown of the small Larmor radius assumption. Second, numerical computations using realistic parameters show that the magnetic moments of the ions are very poorly conserved at the tips of the flux surfaces in field-reversed configuration equilibria. Therefore finite Larmor radius theories cannot be used reliably to explain the observed stability to tilting of field-reversed configuration experiments.

Electromagnetic theory of curvature-driven trapped-electron instabilities

An analytical theory of the curvature-driven trapped-electron instability including effects of finite $\ensuremath{\beta}$, toroidal coupling, and realistic aspect ratio is presented. The mode number of the purely growing mode is modified by the effect of finite $\ensuremath{\beta}$ and inverse aspect ratio, but the ubiquitous mode still remains. A destabilizing effect of finite $\ensuremath{\beta}$ for small Larmor radius is obtained.

Theory of ion-cyclotron resonances and diffusion at high energy in the 2XIIB magnetic mirror experiment

It is shown that a monochromatic ion-cyclotron wave can cause stochastic ion motion at high energy in a mirror plasma, as observed in the 2XIIB Magnetic Mirror Experiment. In earlier studies Rosenbluth, Timofeev, and Smith predicted regular (superadiabatic) motion at energies above 10 or 20 keV, in a slab (or small Larmor radius) model. Particle motion is described here by a Hamiltonian valid for arbitrary Larmor and guiding-center radii, in cylindrical geometry, and in either axisymmetric or quadrupole magnetic fields. The weakness of the self-dug magnetic well at moderate plasma pressure is used as an expansion parameter. Analytical and numerical calculations show large, stochastic changes in the particles’ kinetic energy, in the range of parameters observed in the experiment.

Comment on a paper by Virtamo (1979) (plasma theory)

The author stresses that the integral equation (which has already been derived for the slab geometry by Croci and Saison, 1973) has a more far-reaching theoretical interest than merely that of being a possible starting point for a numerical investigation. In fact, an asymptotic method developed by Croci and Saison (1973) made it possible to derive from the integral equation a dispersion relation correct for small density gradients irrespective of the value of kperpendicular to rho , whereas the usual 'small Larmor radius' approach is only valid for kperpendicular to rho <1. The presence of this limit is evident from the form of the differential equation for the electric potential in the x-space, which is obtained with the 'small Larmor radius' approach.

Stability of general plasma equilibria. III

For pt.II see ibid., vol.23, p.265 (1971). A general method for investigating stability of low-frequency electrostatic oscillations of magnetically confined plasma, which was developed in Part I (1968) for equilibria with closed field lines (e.g. multipoles), is extended to axisymmetric toroidal equilibria with finite magnetic shear (e.g. Tokamaks). The analysis encompasses all perturbations whose parallel wavelengths are comparable to equilibrium scale lengths and whose perpendicular wavelengths are comparable to ion Larmor radii. Once the problem of reconciling these characteristics with toroidal periodicity has been overcome, the investigation of any axisymmetric toroidal equilibrium becomes very similar to that of closed line equilibria and the ion and electron charge densities resulting from an arbitrary potential perturbation are calculated by a small Larmor radius expansion as in Part I. Using these expressions the determination of stability is reduced to a single one dimensional integro-differential equation-which must be solved numerically for each given equilibrium. In the most general case this requires considerable computation, but in many circumstances one can use simpler approximate forms of this equation which are also derived.

Flute modes in a plasma in the presence of nonuniform electric fields

The effects of the presence of a nonuniform electric field on the stability of flute modes in a plasma is studied in the approximation of small Larmor radius. Most of the work is concerned with discussing and applying to the corresponding eigenvalue problem a step approximation for the profiles of the electric and diamagnetic drifts. Furthermore, a new sufficient stability criterion for flute modes is formulated. The influence of the drifts on the stability properties of the flute modes is discussed, and, in particular, the stabilizing effect of a density gradient is displayed.

Numerical Study of Strong Plasma Shock Waves Produced in an Electromagnetic Shock Tube

A set of two-fluid Navier-Stokes equations with classical physical transport coefficients is used to compute the evolution and structure of collisional plasma shock waves in an electromagnetic shock tube. Shock speeds up to 200 cm/μsec and shocked-plasma temperatures of the order of kilovolts are studied. A strong transverse bias magnetic field is employed, which significantly alters the size and shape of the shock profiles, when compared with the zero bias field case. The wave structure is different from that of a two-fluid gas-dynamic shock without transverse magnetic field. Especially significant is the effect of the small ion Larmor radius in reducing the gasdynamic shock thickness by at least an order of magnitude in the transverse shock case, which permits collisional shocks to have thicknesses much smaller than the post-shock mean free path. These collisional shock waves produce very hot ions, the ion temperature increasing with shock speed, although the formation time and distances also increase substantially with shock speed. Generally, the results compare favorably with electromagnetic shock tube experimental data.

Radiation from charged particles in weakly inhomogeneous magnetic fields

The spectral distribution of radiation emitted by a charged particle moving nonrelativistically in a prescribed electromagnetic field, consisting of a strong uniform magnetic induction on which is superimposed a weak inhomogeneity, is determined. The calculation is confined to trajectories characterized by small Larmor radii, and to the azimuthal average of the full angular distribution of the radiation, the azimuthal angle being measured in a co-ordinate system for which the uniform component of the field defines the polar axis. If the particle gyrates for a time long compared to the gyro-period, the spectrum separates into several independent lines. Quantitative expressions for the distribution of energy within each line are presented, and then evaluated for selected inhomogeneities. Qualitatively, it is found that inhomogeneities (i) may distort the distribution of energy within the spectral line at the gyro-frequency by broadening, displacing, destroying the symmetry of, and/or splitting that line, and (ii) may also generate spectral lines at aipproximate harmonics of the gyro-frequency. These harmonics are second order in the inhomogeneity, are of successively higher order in the Larmor radius, involve successively higher-order spatial derivatives of the inhomogeneities, may occur at frequencies whose ratios are not precisely ratios of integers, and may be split into two or more closely spaced components.

Stability of a plasma stream in a magnetic field

The system studied is a plasma streaming parallel to a uniform magnetic field with a velocity which varies in a transverse direction. The flow is bounded at y =± a by perfectly conducting planes.The Poisson-Vlasov equations are used to derive an integro-differential equation for ø the Fourier transform of the electrostatic potential. The kernel of this equation is expanded using a small Larmor radius expansion for ø and for the equilibrium distribution function f0.

A Flute-like Plasma Instability Driven by Field Inhomogeneities

We investigate the stability of ring-like plasma configurations with small Larmor radii, precessing axi-symmetrically in mirror machines. If the precession frequency increases with radius, such equilibria may be subject to a low frequency flute-like instability which is driven by field inhomogeneities rather than by the density gradient. The phase velocity of the unstable waves is twice that occurring in the case of the ordinary flute-instability. Though the physical situation is quite different from that of circular beams of charged particles, the driving mechanism shows a certain analogy to that of the negative mass instability. It is noteworthy, however, that in our case equilibria with a finite spread in the transverse particle energies may be less stable than those with zero energy spread.

Electromagnetic Electron-Ion Streaming Instability

Instabilities produced by a current across a magnetic field and propagating obliquely to the field are studied in connection with the description of hydromagnetic shock waves. It is shown that there exist unstable waves of mixed polarization which can be described as coupled ion beam and whistler modes with frequencies given by ω≈k·u where u is the streaming velocity of the ions relative to the electrons. These waves are investigated in the small electron Larmor radius approximation and in the frequency range Ωi ≪ ω ≪ Ωe where Ωi and Ωe are the cyclotron frequencies of the ions and electrons, respectively. The maximum growth rate is limited by γ &amp;lt; 123 (meu/2miυ¯e)1/3Ωe where υ¯e is the electron thermal velocity. The effect of collisions is considered, and the possibility of observing the instability in oblique shock wave experiments is discussed.

Trajectory of Charged Particles in Weakly Inhomogeneous Magnetic Fields

The detailed time dependence of the trajectory, and particularly of the acceleration, of a charged particle moving nonrelativistically in a weakly inhomogeneous, slowly varying electromagnetic field is calculated by a method that involves superimposing a conventional perturbation expansion in the weak field on the expansion of Berkowitz and Gardner and of Kruskal, the latter expansion being asymptotically valid in the limit of small Larmor radius and slowly varying fields. Fourier transforms are introduced to facilitate isolation of rapidly varying terms from the most general solutions for the coefficients in the expansion. The calculations are carried to first order in the weak inhomogeneity and (at least formally) to all orders in Larmor radius. The resulting velocity reproduces the familiar transverse drift velocities of first-order orbit theory, emphasizes that in the presence of fields varying slowly in time the polarization and curvature drifts are to be calculated from the time derivative of the fields as seen from a frame of reference fixed with respect to the guiding center of the particle's orbit, and permits calculation of higher-order contributions to the drift velocities, contributions of second order in Larmor radius being explicitly evaluated; the resulting acceleration prepares the way for a study of the effect of weak inhomogeneities on the radiative output of single particles.

Resonance interaction of flute oscillations with ion Larmor drift

It is found that, in a regime in which flute oscillations are usually assumed to be stabilized effects of the finite Larmor radius, they are characterized by resonances with the ion Larmor drift. It is shown that a correct study of these resonances requires direct use of an integral equation, without expansion in the small Larmor radius. The initial value problem is solved by the method of Laplace transformation, and it is shown that the flute instabilities are indeed stable.

Focusing properties of some static non-uniform electric and magnetic fields

The motion of a charged particle in the superposition of the electric field of a cylindrical capacitor on the axial homogeneous magnetic field and the magnetic field of line current flowing along the axis of the capacitor, is considered. The equations of trajectories are written in the form of integrals, which are then evaluated approximately, under the assumption of small Larmor radius. Application of the results for coaxial plasma accelerators and rotating plasma devices is briefly considered. The focusing properties of these fields are analysed in some detail. It is shown particularly that second order focusing can be accomplished providing that the field intensities are properly chosen.

Coupling of Transverse and Longitudinal Waves at the Upper Hybrid Frequency

Propagation of an electromagnetic wave into an increasingly dense plasma perpendicularly to a constant magnetic field is considered. The reflection coefficient is calculated in the limit of small electron Larmor radius and high frequency.

Stability of Sheet Pinch

The growth rate of the sheet pinch instability is calculated in the small Larmor radius limit using the method of integration along unperturbed orbits. It is found that the instability is driven by a small group of ``resonant'' electrons which cross and drift about the neutral plane having zero velocity along the field lines. The microinstability is absolute and develops fastest for a critical perturbation wavelength, of the order of several times the thickness of the plasma sheet.

Quasi-Linear Theory for Small Larmor Radius Universal Instabilities

A class of universal instabilities is discussed. The basic physical mechanism of these instabilities is illustrated with a simple physical picture, and the large amount of energy available to drive these instabilities is pointed out. A small Larmor radius quasi-linear theory is developed for a class of unstable ion waves whose phase velocities along the magnetic field lie between the thermal velocities of ions and electrons. In the approximation of localized perturbations, the final state quantities of the plasma are given without specifying the detailed instability mechanism. An essential feature is that the quasi-linear theory predicts an anomalous ``displacement'' of a group of resonant particles rather than any anomalous diffusion. The theory is applied to a special type of the current-driven universal instability. Its quasi-linear consequences are explored.

Some Stable Plasma Equilibria in Combined Mirror-Cusp Fields

The problem of equilibrium and stability of plasma confined in certain magnetic fields of combined mirror-cusp form is discussed. These fields have the properties that they are nowhere zero and everywhere increase toward the periphery. Attention is drawn to the importance of the existence of closed surfaces of constant |B|—the magnetic isobars. The conditions for plasma equilibrium are derived and interpreted; then by exploiting the existence of closed magnetic isobars certain low-β confined equilibria are constructed. These equilibria are shown to be stable according to the fluid (double adiabatic) energy principle and according to the small Larmor radius limit theory. A direct proof of stability against motions which preserve the magnetic moment is given. These equilibria have the property that there is no current along lines of force so that they are also immune to several drift instabilities.

Theoretical Structure of Plasma Equations

In high-temperature plasmas, collisions are very infrequent. Thus the charged particles travel on independent orbits determined by the electromagnetic field. At first sight this would seem completely different from a conventional fluid where particles are closely hemmed in by their neighbors. However, there can exist collective modes of motion in which the particles interact with each other by altering the fields. In this paper a new method is developed for the solution of the linearized transport equation. By facilitating direct use of the properties of particle orbits, a considerable simplification is achieved. In particular, a variational expression is derived for determining stability which is rigorous in the limit of small Larmor radius.