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Volume 3, Issue 8, August 1991

Theory of hydrodynamic trapped‐ion‐temperature‐gradient‐driven turbulence
View Description Hide DescriptionA novel, nonlinearly coupled, two‐dimensional (2‐D), two‐field set of equations is derived to model the turbulent dynamics of trapped‐ion‐temperature‐gradient‐driven convective cells. Statistical mechanical arguments are employed to show that, in spite of the two‐dimensional character of the system, the direction of spectral energy flow is from large to small scales. Steady‐state fluctuation levels are derived from the evolution equation for temperature spectra, and the ensuing particle and heat transport coefficients are shown to be comparable to conventional estimates associated with electron drift wave turbulence.

Magnetic neutral point stretching and coalescence in tearing‐generated magnetohydrodynamic structures
View Description Hide DescriptionThe time evolution of the instability of a sheet pinch is numerically studied using a sufficiently high ratio of system length to width in order to allow the simultaneous growth of several unstable wavelengths. This numerical simulation provides new insights into the nonlinear development of the tearing instability. Before the instability saturates, the nonlinear interactions among the unstable modes produce local coalescence phenomena that destroy the weaker current pinches and reduce the number of magnetic islands. In contrast with the usual picture, this coalescence is not due to the attraction between the current maxima, but is due to the stretching of the X‐neutral points associated with the most intense current pinches. The global perturbation growth rate remains essentially unchanged in time, being of the order of the resistive instability growth rate.

The Bohm criterion for rf discharges
View Description Hide DescriptionThe well‐known dc Bohm criterion is extended to rf discharges. Both low‐ (ω_{rf}≪ω_{ pi }) and high‐(ω_{ pi } ≪ ω_{rf}) frequency regimes are considered. For low frequencies, the dc Bohm criterion holds. This criterion states that the initial energy of the ions entering the sheath must exceed a limit in order to obtain a stable sheath. For high frequencies, a modified limit is derived, which is somewhat lower than that of the dc Bohm criterion. The resulting ion current density in a high‐frequency sheath is only a few percent lower than that for the dc case.

Unified theory of ballooning instabilities and temperature gradient‐driven trapped ion modes
View Description Hide DescriptionA unified theory of temperature gradient‐driven trapped ion modes and ballooning instabilities is developed using kinetic theory in banana regimes. All known results such as electrostatic and purely magnetic trapped particle modes and ideal magnetohydrodynamic ballooning modes (or shear Alfvén waves) are readily derived from the present single general dispersion relation. Several new results from ion–ion collision, finite beta stabilization of ion temperature gradient‐driven trapped particle modes, and trapped particle modification of ballooning modes are derived and discussed. The interrelationship between these modes is established.

Magnetohydrodynamic wave drag
View Description Hide DescriptionA calculation is made of the energy loss by emission of shear‐Alfvén and slow magnetoacoustic waves by a perfectly conducting solid spherical projectile moving with uniform velocity through an ideal magnetoplasma.

Relaxation for one‐dimensional plasma: Test particles versus global distribution behavior
View Description Hide DescriptionThe behavior of a one‐dimensional weakly correlated plasma is studied using an exact algorithm. The simulations deal with a crucial point of the kinetic linearized theory (Lenard–Balescu equation). This equation describes the evolution of different subpopulations, introduced by a ‘‘gedanken experiment’’ easily performed on the computer, which exhibits the different scalings of the relaxation times of ‘‘distinguished’’ particles and overall population.

The modified plasma dispersion function
View Description Hide DescriptionIn the linear theory of waves in a hot plasma if the zeroth‐order velocity distribution function is taken to be Maxwellian, then there arises a special, complex‐valued function of a complex variable ξ=x+i y, namely Z(ξ), known as the plasma dispersion function. In space physics many particle distributions possess a high‐energy tail that can be well modeled by a generalized Lorentzian (or kappa) distribution function containing the spectral index κ. In this paper, as a natural analog to the definition of Z(ξ), a new special function Z ^{*} _{κ}(ξ) is defined based on the kappa distribution function. Here, Z ^{*} _{κ}(ξ) is called the modified plasma dispersion function. For any positive integral value of κ, Z ^{*} _{κ}(ξ) is calculated in closed form as a finite series. General properties, including small‐argument and large‐argument expansions, of Z ^{*} _{κ}(ξ) are given, and simple explicit forms are given for Z ^{*} _{1}(ξ), Z ^{*} _{2}(ξ), ..., Z ^{*} _{6}(ξ). A comprehensive set of graphs of the real and imaginary parts of Z ^{*} _{κ}(ξ) is presented. It is demonstrated how the modified plasma dispersion function approaches the plasma dispersion function in the limit as κ→∞, a result to be expected since the (appropriately defined) kappa distribution function formally approaches the Maxwellian as κ→∞. The function Z ^{*} _{κ}(ξ) is expected to be instrumental in studying microinstabilities in plasmas when the particle distribution function is not only the standard generalized Lorentzian, but also of the Lorentzian type, including i n t e r a l i a, the loss‐cone, bi‐Lorentzian, and product bi‐Lorentzian distributions.

Relaxation processes in a low‐order three‐dimensional magnetohydrodynamics model
View Description Hide DescriptionThe selective decay and dynamic alignment relaxation theories are used to interpret the time asymptotic behavior of a Galerkin model of three‐dimensional (3‐D) magnetohydrodynamics (MHD). A large number of simulations are performed that scan a parameter space defined by the rugged ideal invariants: energy, cross helicity, and magnetic helicity. Ranges of the initial parameters are found where one or both of the relaxation theories are needed to describe the time asymptotic properties of the system, as previously found in analogous studies of two‐dimensional (2‐D) MHD [Ting e t a l., Phys. Fluids 2 9, 3261 (1986)]. In many cases, the time asymptotic state can be interpreted as a relaxation to minimum energy. For certain parameter ranges spectral back transfer of cross helicity can lead to growth in velocity‐magnetic field correlation [Stribling and Matthaeus, Phys. Fluids B 2, 1979 (1990)]. A simple decay model, based on absolute equilibrium theory, predicts a mapping of initial onto time asymptotic states, and accurately describes the long time behavior of the runs when magnetic helicity is present. We also discuss two processes, operating on time scales shorter than selective decay and dynamic alignment, in which the ratio of kinetic to magnetic energy relaxes to values O(1). The faster of the two takes states initially dominant in magnetic energy to a state of near‐equipartition between kinetic and magnetic energy through power law growth of kinetic energy. The other process takes states initially dominant in kinetic energy to the near‐equipartitioned state through exponential growth of magnetic energy.

Ellipticity induced Alfvén eigenmodes
View Description Hide DescriptionIt is shown that noncircularity of tokamak flux surfaces leads to frequency gaps in the magnetohydrodynamic Alfvén continuum. Within these gaps discrete modes having macroscopic structure are shown to exist and have many common features with toroidicity induced Alfvén eigenmodes. The present work focuses on ellipticity. Since κ−1>ε in many tokamaks the ellipticity induced Alfvén eigenmode may indeed be a more robust mode. The most global mode couples the m=1, n=1 and m=3, n=1 ‘‘cylindrical’’ eigenmodes. The region of strong coupling occurs at the q(r)=2 surface and the width of the coupling region is finite and of order (κ−1)a. Furthermore, for typical limiter q(r) profiles satisfying 1≲q≲3, the dominant mode harmonics do not intersect the continuum Alfvén spectrum.

Relativistic hydrodynamics and thermodynamics of anisotropic plasmas
View Description Hide DescriptionState equations for a fully relativistic anisotropic plasma are obtained which generalize the Chew–Goldberger–Low theory. These equations are also a generalization of the isentropic law onto the anisotropic case. The problem of temperature definition in the anisotropic case is discussed.

Particle simulation model of transport in a bounded, Coulomb collisional plasma
View Description Hide DescriptionThe transport of particles and energy in a fully ionized, collisional plasma is studied through the use of a kinetic transport model. A particle‐in‐cell (PIC) code has been coupled to a Monte Carlo, binary particle model of Coulomb collisions, to provide a fully kinetic, self‐consistent description of transport and potential formation in a single spatial dimension and two velocity components (parallel and perpendicular to the spatial coordinate). The dependence of plasma transport on Coulomb collisionality is investigated by varying the normalized collision frequency within the range 10^{−2}≤ν_{*}≡ν_{ c0}/ν_{ b e0}≤5, where ν_{ c0} is the average electron/ion collision frequency and ν_{ b e0} is the frequency at which thermal electrons bounce between the collector sheath potential drops located adjacent to the absorbing plates at each end of the system. Collisions between charged‐plasma and recycled‐neutral particles are omitted in this study. For finite values of ν_{*}, the heat conduction flux is found to be reduced from the value predicted by classical, hydrodynamic transport theory. The electron heat conduction flux is shown to lie between 12% and 21% of the free‐streaming thermal flux q ^{ e } _{ f s }≡n _{ e } v _{∥,t e } k T _{ e }, where n _{ e }, v _{∥,t e }, and k T _{ e } are the steady‐state values of the electron density, parallel thermal velocity, and temperature, respectively. The variation of several transport quantities with collisionality is presented, and the results are compared against those from other collisional plasma transport models.

The expansion of polarization charge layers into a magnetized vacuum: Theory and computer simulations
View Description Hide DescriptionWhen a sufficiently dense plasma stream moves across a magnetic field, the stream will form polarization charge layers and will E×B drift across the field. One charge layer is composed of electrons and the other is composed of ions. The phenomena associated with the expansion of these polarization charge layers along the magnetic field away from the stream is investigated by means of analytic theory and is confirmed by two‐dimensional electrostatic particle‐in‐cell computer simulations. At very early times, the expansion of the electron charge layer is described by single‐particle motion in a dipolelike electric field. Eventually, the electron expansion is halted by the net positive charge left behind at the stream. Then an ambipolar expansion of the electron and ion charge layers forms, with the expansion velocity set by the stream voltage rather than by plasma temperatures. Steady decreases in the cross‐field‐propagation velocities of the streams are observed, owing to the steady losses of charge from the stream edges.

Penetration and expulsion of magnetic fields in plasmas due to the Hall field
View Description Hide DescriptionThe axial penetration of an azimuthal magnetic field into a short‐duration hollow cylindrical plasma is studied. When the process is so fast that the ion motion is small and the plasma dissipative resistivity, electron inertia, and pressure are small, the evolution of the magnetic field is governed by the Hall field. When the radial current flows inward, the magnetic field penetrates in the form of a Hall‐induced shock wave with a narrow current channel. When outward, the magnetic field does not penetrate the plasma. Moreover, in the latter case the magnetic field is expelled from an initially magnetized plasma. The increase and decrease of the magnetic field intensity in the cylindrical plasma are shown to result naturally from the frozen‐in law.

Toroidal ion temperature gradient‐driven weak turbulence
View Description Hide DescriptionIn this paper, a theory of toroidal ion temperature gradient‐driven weak turbulence near the threshold is presented. The model considers gyrokinetic ions and adiabatic electrons in toroidal geometry. The linear theory considers modes with k _{θ}ρ_{ i }∼O(1) (generally not considered in toroidal theories), giving a toroidal threshold regime dominated by transit resonances (as opposed to the more usually considered drift resonances), and with a stability threshold of η^{tor} _{thresh}≂1+2ε_{ n }[2/τ+1/(1+ŝ)]. It is shown that when 0<η_{ i }−η^{tor} _{thresh}<2(1+1/τ)^{1/2} ×L _{ n }/(R L _{ T })^{1/2} then 0<γ<ω_{ r }, and a weak turbulence expansion can be used to treat the nonlinearity. The instability is saturated via nonlinear ion resonance, and it is shown that this nonlinear process transfers energy directly from the waves to the ion distribution function, and does not conserve wave energy. The saturated spectrum is calculated, and the resulting ion thermal conductivity is found to be χ_{ i }=(1+1/τ)^{1/2}[(η_{ i }−η^{tor} _{th}/η_{ i }] ((L _{ T })^{1/2}/R ^{3/2})ρ^{3} _{ i }Ω_{ i }, which is smaller than typical mixing length estimates by a factor of about L _{ T }/R, but in the range of tokamak observations. The diffusive nature of the transport (requiring a short radial step size) is reconciled with the broad radial extent of the toroidally coupled linear modes by postulating that it is the nonlinear beat wave between linear modes (not the radial width of a single linear mode) that determines the step length appropriate for transport.

Two‐dimensional compressible magnetohydrodynamic simulation of the driven reconnection process
View Description Hide DescriptionA two‐dimensional compressible magnetohydrodynamic code is developed to provide simulations of the driven reconnection process in conducting plasma. For a fixed value of the plasma inflow at the boundary (Alfvénic Mach number M _{A}=0.15 ) and a ratio of 2:1 for the two sides of the simulation box, the dependence of the reconnection effect as a function of time on the magnetic Reynolds number is investigated by examining a range of R _{ m } value (400–2500). The size of the current sheet is generally found to cover the horizontal length of the whole simulation box. It is found that as R _{ m } increases, the plasma system will evolve from a configuration of one single X‐line formation to a multiple X‐line formation. Unlike the former situation, the multiple X‐line formation is characterized by quasiperiodic generation of secondary magnetic islands (or plasmoids) of high temperature and dense mass. These two states are separated by a critical value of R ^{*} _{ m }=2000 for the plasma parameters considered. The occurrence of the flux tube transfer events (FTEs) at the magnetopause might be related to the nonstationary driven reconnection with high R _{ m } value as found here.

Three‐dimensional simulation of ∇ T _{ i }‐driven turbulence and transport
View Description Hide DescriptionThe anomalous transport of ion thermal energy in a heated plasma slab confined by a uniform, straight magnetic field B _{ z } has been investigated in three‐dimensional (3‐D) fluid simulations. Convection flows are driven unstable by ∇T _{ i } and nonlinearly develop into narrow streams which carry cold edge plasma into the hot center. The convective flows undergo a sharp transition from laminar to turbulent behavior as the thermal energy confined in the slab is increased beyond a critical level. This transition is reminiscent of similar behavior in Rayleigh–Bénard convection in heated fluids. The convective thermal transport increases sharply as this turbulence threshold is exceeded. The structure of the flow patterns and associated transport also depend strongly on the physical dimensions of the confined plasma (L _{ x },L _{ y },L _{ z }) =(a,2πa,2πR) compared with the ion Larmor radius ρ_{ i }. For α=a ^{2}/Rρ_{ i }>1, the dominant flows have k _{ y }ρ_{ i } ∼a/R and produce an anomalous cross field thermal transport χ_{⊥i } that scales as v _{ i } a ^{2}/R . In the opposite limit α=a ^{2}/Rρ_{ i } <1, dominant flows have k _{ y } a∼1 and the transport is given by χ_{⊥i }∼ρ_{ i } v _{ i }, the Bohm scaling. Close to marginal stability, the transport is greatly reduced. These simulations imply that ion thermal transport in any straight field system such as a tandem mirror or a stellarator with weak shear and nearly rational fields will be strongly anomalous if the ∇T _{ i } threshold for instability is exceeded.

Renormalization group analysis of reduced magnetohydrodynamics with application to subgrid modeling
View Description Hide DescriptionThe technique for obtaining a subgrid model for Navier–Stokes turbulence introduced by Yakhot and Orszag [J. Sci. Comput. 1, 3 (1986); Phys. Rev. Lett. 5 7, 1772 (1986)], based on renormalization group analysis (RNG), is extended to the reduced magnetohydrodynamic (RMHD) equations. A RNG treatment of the Alfvén turbulence (perpendicular scale k ^{−1} _{⊥}≪k ^{−1} _{∥} parallel scale) supported by the RMHD equations leads to effective values of the viscosity and resistivity at large scales, k→0, dependent on the amplitude of turbulence. When the RNG analysis is augmented by the Kolmogorov argument for energy cascade the effective viscosity and resistivity become independent of the molecular quantities. This leads to a ‘‘universal’’ subgrid model, which all models approach at the largest scales. A self‐contained system of equations is derived for the range of scales, 0<k<K, where K=π/Δ, is the maximum wave number for a grid size Δ. In this system the resistive and viscous dissipation is represented by differential operators, whose coefficients depend upon the amplitudes of the large‐scale quantities being computed.

Effects of a nonuniform open magnetic field on the plasma presheath
View Description Hide DescriptionEffects of a nonuniform magnetic field on the plasma presheath are numerically investigated using the plasma equation for a collisionless plasma with a finite‐temperature particle source. The present calculation confirms that previously published analytical solutions [Phys. Fluids B 1, 725 (1989)] are available over a wide range of mirror ratios. Potential drop in the presheath, which depends considerably on both the magnetic strength profile and the spatial distribution of the particle source, is remarkably increased by applying an expanding magnetic field when plasma particles are generated in the inner part of the plasma. An effect of a nonuniform magnetic field on sheath formation is also discussed by using a calculated ion distribution function. If the plasma equation has no singularity at the sheath edge, its solution satisfies the generalized Bohm criterion with the inequality sign in the expanding magnetic field.

Nonlinear wave scattering and electron beam relaxation
View Description Hide DescriptionThe role played by nonlinear scattering during the relaxation of a warm electron beam is investigated with the help of a numerical code. The code is based on kinetic equations and includes the quasilinear wave–electron interaction as well as wave–wave scattering off ion clouds. Both mechanisms have been observed to play key roles in a recent particle simulation with a large number of modes. It is found that (1) ions with velocity 2v _{ i } (v _{ i } being the ion thermal velocity) are the most efficient to scatter the Langmuir waves off their polarization clouds. As a result, the transfer rate of the spectrum out of resonance with the beam is larger by a factor 3 compared to the usual estimates in the literature, which assume a static ion response. The predicted wave number k of the secondary spectrum differs also substantially. (2) If the beam density n _{ b }, drift U _{ b }, and width v _{ b } satisfy the condition n _{ b }/n _{0}>4.2(v _{ e }/U _{ b })^{2} ×(v _{ b }/U _{ b })^{3}, the changes brought to the dispersion relation by the presence of the beam electrons dramatically alter the characteristics of the secondary spectrum.
Forward propagating waves may grow where the conventional picture expects backward propagating waves. Most strikingly, in a late phase the classic condensate about k∼0 is depleted with the formation of a new condensate in resonance with the flat‐topped beam distribution. This contradicts the commonly assumed cascade in wave numbers, but follows simply from the fact that the mere presence of the beam electrons creates a minimum in the frequency–wave‐number relation. There is no contradiction with a cascade toward lower frequencies driven by an isotropic ion distribution. For strong and slow beams (n _{ b }/n _{0}∼10^{−2}, U _{ b }∼10v _{ e }) the predictions of this code can be compared with the results obtained in the particle simulation. The agreement is excellent if one uses a dispersion relation that includes the beam. Complete plateau formation by resonant diffusion and late formation of a secondary spectrum are observed. Time scales and spectral characteristics compare well. For faster and weaker beams, it is demonstrated that the nonlinear wave scattering may intervene before complete quasilinear relaxation. Once the beam top has been erased by diffusion, a wave condensate forms, which inhibits further relaxation toward lower velocities. Modes in resonance with the positive slope at the low‐velocity front of the flat‐topped beam are stabilized by a fast transfer of their energy into the condensate.

Ion acoustic parametric decay instability in laser‐produced plasma with varying ionic charge
View Description Hide DescriptionStudies of the ion acoustic decay instability (IADI) are reported in which the target material of the laser‐produced plasma was varied. The IADI was monitored by observing the Stokes peaks of the second‐harmonic spectrum. Its threshold was quite low (≊2×10^{13} W/cm^{2} ) even in high‐Z plasma. The threshold increased only weakly with Z. On the other hand, the instability intensity decreased strongly with Z, which is attributed to the decrease of the growth rate. A simple theory explains these experimental results reasonably well.