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Volume 5, Issue 7, July 1993

The fluctuation‐induced Hall effect
View Description Hide DescriptionThe fluctuation induced Hall term, <J̃×B̃≳, has been measured in the Madison Symmetric Torus (MST) [R. N. Dexter et al., Fusion Technol. 19, 131 (1991)], reversed‐field pinch. The term is of interest as a possible source of current self‐generation (dynamo). It is found to be non‐negligible, but small in that it can account for less than 25% of the dynamo driven current.

Observations of frequency upshift in a pulsed free‐electron laser amplifier
View Description Hide DescriptionA new phenomenon, the frequency upshift in a pulsed free‐electron laser (FEL) amplifier is reported. The measurement is made in an FEL that uses a mildly relativistic electron beam (750 keV, 90 A, 30 nsec) and is driven by a copropagating electromagnetic wave generated by a 33.39 GHz magnetron source. The frequency upshift is ascribed to the rapid temporal change in the effective dielectric coefficient that can be associated with the FEL interaction.

The magnetoacoustic cyclotron instability of an extended shell distribution of energetic ions
View Description Hide DescriptionThe magnetoacoustic cyclotron instability is a mechanism by which waves on the perpendicular fast Alfvén‐ion Bernstein branch can be excited through cyclotron resonance with an energetic ion population. It is a candidate emission mechanism for the superthermal ion cyclotron radiation, apparently associated with the products of fusion reactions, that has been observed from tokamak plasmas. In the present paper, an extended shell model is adopted for the energetic ion distribution function, f _{α}(v)∼n _{α} exp[−(v−v _{0})^{2}/v _{ T } ^{2}]. An analytical formulation of the dispersion relation is obtained, whose numerical solution yields quantitative information on the role of v _{ T } in stabilizing wave growth at ion cyclotron harmonics. The results show that, for typical plasma parameters of interest, the degree of instability is significantly depressed, relative to its level for v _{ T }=0, once v _{ T }/v _{0}≂0.1. Gaps appear in typical multiple cyclotron harmonic excitation patterns for 0.1≤v _{ T }/v _{0}≤0.2, and most harmonics are stable for v _{ T }/v _{0}≥0.25. Thus the energetic ion shell‐driven magnetoacoustic cyclotron instability typically occurs only when the shell is relatively narrow in velocity space.

Quasioptical treatment of electromagnetic Gaussian beams in inhomogeneous and anisotropic plasmas
View Description Hide DescriptionThe main effect of the Gaussian behavior of an electromagnetic beam consists in a waist formation in focal regions, where the ordinary geometric optics would collapse. The characteristic features of Gaussian beams, both when they interact with the components of transmission lines and when they propagate through inhomogeneous and anisotropic media (as happens, for instance, in the case of diagnostic or heating experiments in magnetoactive plasmas of fusion interest), are of crucial relevance for many technical and scientific purposes. The present paper is devoted to the analysis of the propagation of Gaussian beams, showing in particular that a properly formulated eikonal equation contains all the elements required by a correct ray tracing procedure, basically amounting to a first‐order description of the beam diffraction. Simple methods are proposed, apt to follow numerically the beam evolution for a quite general choice of refractive media and of wave launching conditions. Numerical results are presented for Gaussian beam propagation in vacuum, in isotropic and anisotropic media, and compared (evidencing significant deviations) to the corresponding ones in the optical case.

Conservation of canonical circulation and its relation to finite Hall term magnetohydrodynamics
View Description Hide DescriptionThe axisymmetric, compressible visco‐resistive two‐fluid plasma equations are examined under the constraint that the current is purely poloidal and the pressure is a function of density only (‘‘barotropic’’). For ideal plasmas (zero resistivity and zero viscosity) the Kelvin circulation theorem of fluid mechanics and the concept of frozen‐in field lines turn out to be limiting cases of a more general concept, namely, that the canonical circulation S _{σ}=∮ (m _{σ} u _{σ}+q _{σ} A) ⋅d l of a toroidal fluid element, is exactly conserved as the toroidal element convects and/or is compressed. Appropriate linear combinations of the electron and ion fluid equations give a magnetohydrodynamic vorticity transport equation and an induction equation with a nonlinear Hall term. The finite Hall term is identical to the source term in the vorticity transport equation [P. M. Bellan, Phys. Rev. Lett. 69, 3515 (1992)], except for a constant factor related to the ion collisionless skin depth.

Two‐fluid equations for low‐n singular modes in the low‐frequency regime
View Description Hide DescriptionBraginskii’s two‐fluid equations are employed to obtain the eigenmode equations governing low‐n singular modes in toroidal geometry. Collisional effects are neglected. The equilibrium electric field is taken into account. The frequency of the modes is assumed to be lower than that of the ion acoustic wave along the magnetic field lines. In this low‐frequency regime it is shown that the pressure of the ion fluid tends to be constant along the magnetic field lines, as also does that of the electron fluid, and the parallel electric field tends to vanish. The finite‐gyroradius (FGR) effect is recovered by taking into consideration the diamagnetic drift velocity and the gyroviscosity of the ion fluid. The results show that the toroidal effect gives rise not only to the so‐called apparent mass effect, but also to an enhancement of the FGR and equilibrium electric field effects. The toroidicity‐induced FGR and equilibrium electric field effects are shown to be even more important than those exhibited in the cylindrical equilibrium model.

Effect of finite ion gyroradius on the fire‐hose instability in a high beta plasma
View Description Hide DescriptionIn this paper, a generalized kinetic dispersion equation that supports various hydromagnetic waves and instabilities is derived. The general dispersion equation is derived under the usual assumption of hydromagnetic perturbations [i.e., ‖ω‖^{2}≪Ω_{ i } ^{2}, and (k _{ z }ν_{A}/Ω_{ i })^{2}≪β_{∥i }, where Ω_{ i } and ν_{A} are the ion gyrofrequency and Alfvén speed, respectively, and β_{∥i } is the parallel ion beta], but for arbitrary values of the quantity λ_{ i }=(k _{⊥}ρ_{⊥i })^{2}/2=(k _{⊥}ν_{A}/Ω_{ i })^{2} β_{⊥i }/2 that appears in the dielectric tensor. Here, ρ_{⊥i } refers to the mean ion gyroradius, and β_{⊥i } is the perpendicular ion beta. Otherwise, the dispersion equation is fairly general with no additional approximation, such as ignoring certain off‐diagonal dielectric tensor elements (which is usually done in the literature) employed. In the subsequent numerical analysis, special attention is paid to the fire‐hose instability in a high beta plasma. The numerical results reveal that the conventional treatment of the fire‐hose instability (i.e., taking zero ion gyroradius limit at the outset) is not adequate, and that the effect of finite ion gyroradius results in a significant enhancement of the growth rate over a large range of wave numbers.

Hamiltonian description and stability of magnetic electron vortices
View Description Hide DescriptionA Hamiltonian description and integral invariants of two‐dimensional magnetic electron modes are derived. Vortex street solutions are given. Using the integrals, linear stability for long wavelength perturbations is established. It is shown that nonlinear stability cannot be proven using Arnol’d’s method.

Dynamics of ponderomotive self‐focusing and periodic bursts of stimulated Brillouin backscattering in plasmas
View Description Hide DescriptionThe space–time evolution of ponderomotive self‐focusing of electromagnetic beams in a plasma is investigated. The quasineutral, hydrodynamic plasma response to the ponderomotive force is considered. The set of coupled quasioptic and acoustic equations is solved both analytically and numerically for slab and cylindrical beams. It is shown that the transient process of self‐focusing has the form of a nonlinear wave propagating along the beam axis from boundary into the interior of a plasma with velocity considerably higher than the ion‐sound velocity. Mutual dynamics of self‐focusing and stimulated Brillouin backscattering (SBBS) is computed. It is shown that self‐focusing results in the high intensity periodical bursts of SBBS. However, the time average level of scattered radiation is quite low.

Correlation theory of a two‐dimensional plasma turbulence with shear flow
View Description Hide DescriptionWhen the ion sound effect is neglected, a wide class of electrostatic plasma turbulence can be modeled by a two‐dimensional equation for the generalized enstrophy Ψ, an inviscid constant of motion along the turbulent orbits. Under the assumption of a Gaussian stochastic electrostatic potential, an averaged Green’s function method is used to rigorously derive equations for the N‐particle correlation functions for a dissipative and sheared flow. This approach is equivalent to the cumulant expansion method [T. H. Dupree, Phys. Fluids 15, 334 (1972); 21, 783 (1978)] used to study the Vlasov–Poisson system. For various cases of interest, appropriate equations are solved to obtain the absolute level as well as the detailed structure of the two‐point correlation function C(r), and its Fourier transform, the enstrophy spectral function I(k). Uniformly valid analytical expressions are derived for the dissipative but shearless case resulting in a ‘‘fluctuation–dissipation’’ theorem relating the total spectral intensity to classical viscosity. These self‐consistent results show a strong logarithmic modification of the mixing length estimates for the turbulence levels. For the extremely important and interesting problem of a sheared flow, the suppression of turbulence is demonstrated by using asymptotic analytical techniques in the inviscid range, and uniformly valid numerical methods for the dissipative system. The current asymptotic methods reproduce the results obtained in the orbit picture [Y. Z. Zhang and S. M. Mahajan, Phys. Fluids B 4, 1385 (1992)], but provide much clearer physical perspective and a better definition of crucial parameters like the decorrelation time. The uniformly valid numerical approach allows the determination of the change in spectral shape and intensity due to the presence of shear. It is found that the suppression is more effective for longer wavelengths as compared to the shorter wavelengths. This and other relevant issues, concerning the role of flows with shear (including its radial variation) in the understanding of the L–H transitions in tokamaks, are discussed.

Strongly nonlinear evolution of low‐frequency wave packets in a dispersive plasma
View Description Hide DescriptionThe evolution of strongly nonlinear, strongly modulated wave packets is investigated in a dispersive plasma using a hybrid numerical code. These wave packets have amplitudes exceeding the strength of the external magnetic field, along which they propagate. Alfvén (left helicity) wave packets show strong steepening for β<1, while fast (right helicity) wave packets hardly steepen for any β. Substantial regions of opposite helicity form on the leading side of steepened Alfvén wave packets. This behavior differs qualitatively from that exhibited by the solutions to the derivative nonlinear Schrödinger (DNLS) equation.

Behavior of a continuous uranium plasma stream expanding across a weak magnetic field
View Description Hide DescriptionWhen an electron beam generates a uranium vapor in the electron beam evaporator, a uranium plasma is formed on the evaporating surface and expands. It was confirmed experimentally that this plasma could expand with increasing its drift velocity and that the plasma possessed a Boltzmann electron distribution along the streamline, even when it propagated across a magnetic field of 40–50 G. This behavior can be understood by the cross‐field plasma expansion theory at 1/ε→0 limit, where ε is the dielectric constant of plasma [Phys. Fluids 25, 730(1982)]. The ion drift energy calculated from the measured polarization electric field is in good agreement with that estimated from the potential distribution and vapor drift energy, which confirms the ion acceleration along the plasma stream.

Transport due to ion‐temperature‐gradient‐driven magnetic drift modes
View Description Hide DescriptionAn advanced reactive fluid model is used to study the transport due to the toroidal η_{ i } mode in the flat density limit. In this limit the density gradient scale length scales out from both the linear dispersion relation and the quasilinear transport coefficients. Simplified transport coefficients are derived for the flat density limit including also influence of electron trapping and are discussed in relation to the density modification experiments on the Tokamak Fusion Test Reactor (TFTR) [Proceedings of the 11th International Conference on Plasma Physics and Controlled Fusion Research (International Atomic Energy Agency, Vienna 1987), Vol. 1, p. 51]. The scaling properties with respect to various dimensionless parameters are discussed. In particular the scaling X _{ i } ∼ η_{ i } − η_{ ith}, which has also been derived by other authors for somewhat different situations, is found to be the relevant scaling in typical situations. A stabilizing and transport reducing effect of a high degree of electron trapping is pointed out.

Nonlinear ideal magnetohydrodynamics instabilities
View Description Hide DescriptionExplosive phenomena such as internal disruptions in toroidal discharges and solar flares are difficult to explain in terms of linear instabilities. A plasma approaching a linear stability limit can, however, become nonlinearly and explosively unstable, with noninfinitesimal perturbations even before the marginal state is reached. For such investigations, a nonlinear extension of the usual MHD (magnetohydrodynamic) energy principle is helpful. (This was obtained by Merkel and Schlüter, Sitzungsberichted. Bayer. Akad. Wiss., Munich, 1976, No. 7, for Cartesian coordinate systems.) A coordinate system independent Eulerian formulation for the Lagrangian allowing for equilibria with flow and with built‐in conservation laws for mass, magnetic flux, and entropy is developed in this paper which is similar to Newcomb’s Lagrangian method of 1962 [Nucl. Fusion, Suppl., Pt. II, 452 (1962)]. For static equilibria nonlinear stability is completely determined by the potential energy. For a potential energy which contains second‐ and nth order or some more general contributions only, it is shown in full generality that linearly unstable and marginally stable systems are explosively unstable even for infinitesimal perturbations; linearly absolutely stable systems require finite initial perturbations. For equilibria with Abelian symmetries symmetry breaking initial perturbations are needed, which should be observed in numerical simulations. Nonlinear stability is proved for two simple examples, m=0 perturbations of a Bennet Z‐pinch and z‐independent perturbations of a θ pinch. The algebra for treating these cases reduces considerably if symmetries are taken into account from the outset, as suggested by M. N. Rosenbluth (private communication, 1992).

Nonlinear waves in two‐fluid hydrodynamics
View Description Hide DescriptionTwo assumptions, one‐dimensionality and quasineutrality, in the framework of the two‐fluid hydrodynamics for hot plasmas, allow a close set of three equations (for inverse density and two components of the transverse magnetic field) to be obtained. These equations describe nonlinear waves in a wide range of wave vectors (up to the inverse electron inertial length) and frequencies (up to the low‐hybrid frequency or in some cases electron gyrofrequency). The obtained set of equations is valid for arbitrary plasma temperatures. Linear dispersion relations are easily recovered from the obtained nonlinear equations. Nonlinear wave equations for different modes, which include new terms due to finite pressure, are derived using methods of the reductive perturbation theory. Stationary solutions are analyzed by the pseudopotential method. Conditions for the existence of solutions with homogeneous asymptotics are found.

The spectral signatures of a Langmuir soliton instability
View Description Hide DescriptionTwo‐dimensional simulation of a soliton instability in a weakly magnetized plasma is performed in the Zakharov model of strong Langmuir turbulence. Growth rate spectra are calculated showing the suppression of a magnetic‐field effect with a growing soliton strength. Time resolved spectra of the Langmuir energy and electric‐field components are exposed with important nonlinear effects of a nonlinear frequency shift and spectral broadening observed already in an early stage.

Nonconservative and reverse spectral transfer in Hasegawa–Mima turbulence
View Description Hide DescriptionThe dual cascade is generally represented as a conservative cascade of enstrophy to short wavelengths through an enstrophy similarity range and an inverse cascade of energy to long wavelengths through an energy similarity range. This picture, based on a proof due to Kraichnan [Phys. Fluids 10, 1417 (1967)], is found to be significantly modified for spectra of finite extent. Dimensional arguments and direct measurement of spectral flow in Hasegawa–Mima turbulence indicate that for both the energy and enstrophy cascades, transfer of the conserved quantity is accompanied by a nonconservative transfer of the other quantity. The decrease of a given invariant (energy or enstrophy) in the nonconservative transfer in one similarity range is balanced by the increase of that quantity in the other similarity range, thus maintaining net invariance. The increase or decrease of a given invariant quantity in one similarity range depends on the injection scale and is consistent with that quantity being carried in a self‐similar transfer of the other invariant quantity. This leads, in an inertial range of finite size, to some energy being carried to small scales and some enstrophy being carried to large scales.

Nonlinear hydromagnetic waves in a thermally stratified cylindrical fluid: Exact helically symmetric solutions
View Description Hide DescriptionThe propagation of nonlinear hydromagnetic waves in a highly conducting, self‐gravitating fluid in a cylindrical geometry, subject to the convective forces produced by a radial temperature gradient, is treated in a Boussinesq approximation. Exact wave solutions of the nonlinear magnetohydrodynamic equations (in the Boussinesq approximation) in the presence of convective forces are obtained for the case when the physical quantities are helically symmetric in cylindrical coordinates. The solutions represent waves propagating helically on the cylindrical surfaces, under the influence of the helical magnetic and velocity fields and the convective forces. The solutions may be applicable to the hydromagnetic waves in the Earth’s fluid core and the solar convection zone with suitable modifications to account for spherical geometry.

Kadomtsev–Petviashivili equation for an ion‐acoustic soliton in a collisionless weakly relativistic plasma with finite ion temperature
View Description Hide DescriptionA Kadomtsev–Petviashivili equation is derived for a two‐dimensional ion‐acoustic soliton propagating in a collisionless weakly relativistic plasma containing the finite temperature ions. This equation is solved in a stationary frame to obtain expressions for the phase velocity, amplitude, width, peak ion density, peak pressure, and energy of the soliton. It is shown that both the relativistic effect and the ion temperature greatly influence the phase velocity, amplitude, and the width. The soliton behavior is described in detail, and a comparison is made between the present results and the previous theories.

Probability of orbit transition in asymmetric toroidal plasma
View Description Hide DescriptionTrapped particles in stellarators and tokamaks can be either locally trapped in the ripple wells or toroidally trapped by the 1/R variation of the magnetic field strength. Transitioning particles make transitions back and forth between these two states. The transition probability gives the relative rate of toroidally trapped particles to become locally trapped. In this paper the transition probably is uniquely defined, and an improved analytical calculation of the transition probability is given. It is then calculated numerically by following the full guiding‐center dynamics. Comparison with the numerical results shows that the present calculation is more accurate than those previously presented.