Ronald C. Davidson

Effects of finite plasma beta on the lower-hybrid-drift instability

The local dispersion relation for the lower‐hybrid‐drift isntability is derived in a fully self‐consistent manner including the finite‐beta effects associated with (a) transverse electromagnetic perturbations (δB≠0), and (b) resonant and nonresonant h/B0 electron orbit modifications. Moreover, the analysis is carried out for arbitrary values of local β=8πn (Te+Ti)/B02, Te/Ti, ω2pe/ω2ce, and VE/vi. (Here, VE is the cross‐field E×B velocity, and vi is the ion thermal speed.) For all parameter regimes studied, the net effect of finite plasma beta is to reduce the maximum growth rate γm of the lower‐hybrid‐drift instability. The details, however, vary, depending on plasma parameters. For example, if Te≪Ti and VE<vi, then the maximum growth rate is reduced by a factor (1+βi/2)−1/2, relative to the value obtained when βi=8πnTi/B20→0. On the other hand, for Te≈Ti, there exists a critical value of plasma beta (βcr) such that the lower‐hybrid‐drift instability is completely stabilized (γ<0) for β≳βcr.

Nonlinear Development of Electromagnetic Instabilities in Anisotropic Plasmas

Theory and simulation experiment are presented for a wide variety of transverse electromagnetic instabilities in plasmas with different sources and degrees of anisotropy. In each of the electron bi‐Maxwellian, electron‐pinch, and ion‐pinch experiments, the bulk response of the system during the initial stages of instability is in good agreement with the predictions of quasilinear theory. Furthermore, the two independent energy constants which derive from the fully nonlinear Vlasov‐Maxwell equations are found to remain constant to very good accuracy, even when the magnetic field energy reaches a substantial fraction of the total system energy. In each simulation experiment it is found that the magnetic energy saturates once the magnetic bounce frequency has increased to a value comparable to the linear growth rate prior to saturation, i.e., when ω¯B∼γ¯k. It is concluded that amplitude limitation for Weibel instabilities is a result of magnetic trapping for a broad range of system parameters. In many experi...

Anomalous transport in high-temperature plasmas with applications to solenoidal fusion systems

The linear, non-linear, and anomalous transport properties associated with various microinstabilities driven by cross-field currents in high-temperature plasmas are reviewed. Particular emphasis is placed on instabilities pertinent to the implosion and post-implosion phases of theta-pinch plasmas, e.g. Buneman (electron-ion two-stream), ion acoustic, lower-hybrid-drift, electromagnetic ion cyclotron, and ion-ion cross-field instabilities. Analytic studies of the non-linear and quasi-linear evolution of these instabilities are presented, together with a detailed comparison with computer simulation experiments to test the validity of the various theoretical models and non-linear saturation mechanisms. A general theoretical formalism is presented which describes, in a self-consistent manner, the macroscopic transport produced by the (shortwave-length) turbulence associated with the microinstabilities enumerated above. The experimental evidence that such a self-consistent anomalous transport model is required for describing the implosion behaviour (characterized by diffuse current sheaths) in rapidly pulsed theta pinches is reviewed, together with the early attempts at modelling these implosions numerically with a one-fluid (MHD) model including artificial viscosity. It is shown that fluid-numerical simulations that include (at each space and time step) the effects of anomalous transport in a fully self-consistent manner, explain several features of the experimental observations. The relevance of reflected ions to sheath structure and implosion dynamics is also discussed, and state-of-the-art hybrid-numerical studies (Vlasov ions and fluid electrons) of pinch implosions are presented, which include reflected ion dynamics as well as the anomalous transport associated with cross-field instabilities. Finally, instability mechanisms for producing long-time interpenetration of plasma and magnetic field in post-implosion theta pinches are discussed, together with estimates of the anomalous resistivity.

Vlasov Equilibria and Stability of an Electron Gas

Self‐consistent Vlasov equilibria with space charge are constructed for a pure electron gas radially confined in a uniform external magnetic field. A sufficient condition is obtained for the stability of these nonuniform equilibria to electrostatic perturbations. For the case in which the electron density is constant in the column interior, the dispersion relations for small‐amplitude interior perturbations, electrostatic and electromagnetic, are shown to be similar in structure to the corresponding results for a neutral plasma. Algorithms are given for obtaining electron gas stability information from the neutral plasma literature.

Macroscopic Quasilinear Theory of the Garden‐Hose Instability

The quasilinear stabilization of the garden‐hose instability is discussed from a macroscopic point of view with closure in the fluid model obtained by neglecting the effects of heat flow. In order to keep the problem well‐posed mathematically finite Larmor radius corrections to the conventional growth rate are retained, which leads to a natural cutoff in the growth rate for sufficiently large wavenumber.

Vlasov equilibrium and nonlocal stability properties of an inhomogeneous plasma column

A fully kinetic, nonlocal, matrix dispersion equation is derived for electrostatic perturbations about a spatially nonuniform cylindrical plasma equilibrium. The analysis is carried out for the class of radially confined rigid‐rotor equilibria described by f0j(x,v) = (njmj/2πTj) F (H⊥/Tj− ωjPϑ/Tj,vz), where Pϑ is the canonical angular momentum, vz is the axial velocity, H⊥ is the perpendicular energy, and nj, Tj, and ωj are constants. Assuming equilibrium charge neutrality and negligible spatial variation in the axial magnetic field B0ez, it is shown that the particle trajectories (in the equilibrium electric and magnetic fields) and the orbit integrals required in the stability analysis can be evaluated in closed form. Expanding the perturbed electrostatic potential in terms of the vacuum eigenfunctions {Jl(λnr) } for the conducting cylinder leads to a matrix dispersion equation of the form det[δn′,n+ Σjχjn′,n(ω)]=0, where the susceptibility χjn′,n(ω) is expressed as a phase‐space integral over f0j(x,v...

Ordinary‐Mode Electromagnetic Instability in High‐β Plasmas

An absolute instability associated with electromagnetic waves propagating perpendicular to an external magnetic field is discussed. A lower bound on the growth rate is found and it is shown that growth rates may be of the order of the electron cyclotron frequency.

Self‐consistent theory of cyclotron maser instability for intense hollow electron beams

A self‐consistent theory of the cyclotron maser instability, assuming azimuthally symmetric perturbations about a slowly rotating hollow electron beam propagating parallel to a uniform axial magnetic field B0ez is developed. The stability analysis is carried out within the framework of the linearize Vlasov–Maxwell equations. It is assumed that the beam is thin with radial thickness (2a) much smaller than the beam radius (R0), and that ωp2/ω2c≪1, where ωp and ωc are the electron plasma frequency and electron cyclotron frequency, respectively, in a frame of reference moving with the beam axial velocity cβb. The analysis is carried out for the specific choice of equilibrium electron distribution function in which all electrons have the same value of canonical angular momentum and the same value of energy in a frame of reference moving with axial velocity cβb. Stability properties are investigated including the important influence of finite radial geometry, finite beam temperature, and transverse magnetic per...

Sheath broadening by the lower-hybrid-drift instability in post-implosion theta pinches

Theoretical and numerical studies of anomalous transport due to the lower-hybrid-drift instability show that this mode can cause significant diffusion in post-implosion theta pinches where Ti Te and the electron drift velocity is below the ion thermal speed. Numerical results show that the diffusion scales with the simple diffusion velocity VD ~ (c2νan/ω2pe) [∂lnBz/∂x] = − (c2νanβ/2ωpe2) [∂lnn/∂x], where νan is the anomalous collision frequency for the lower-hybrid-drift instability, estimated from quasilinear theory. (Here, Bz is the magnetic field, n is the density, and β = 8πnTi/Bz2 is the local value of beta.) Moreover, the time evolution of the sheath width in the numerical calculation agrees well with the simple theoretical estimate [Ln(t)/Ln(t=0)]4 = 1 + 4[VD(t=0)/Ln(t = 0)]t, where Ln(t) is the density sheath width [Ln −(∂lnn/∂x)−1]. This simple formula for lower-hybrid-drift sheath broadening is valid for 0.5 rLi < Ln < 0.5 (mi/me)1/2rLi, where rLi is the ion Larmor radius.

A hybrid‐kinetic model for high‐β plasmas

A hybrid‐kinetic model (Vlasov ions and drift‐kinetic electrons) is developed to describe the equilibrium and stability properties of collisionless high‐beta plasmas in arbitrary magnetic field configurations. No ordering of ion properties (such as ω/Ωi and rLi/L⊥) is assumed, so the ions are described by the exact (unexpanded) Vlasov equation. The electrons are assumed to be strongly magnetized with ω/Ωe ∼rLe/L⊥=e≪1, and the electron Vlasov equation is expanded in e to give a drift‐kinetic equation for fe. Examples of linear screw‐pinch equilibria are constructed to demonstrate the ease with which equilibrium solutions are found within this formalism. Also, it is shown that the hybrid‐kinetic model correct to O (e) includes physical effects which are important during the implosion and immediate post‐implosion phases of high‐density pinch experiments, where stability behavior is characterized by the fast time scale ω−1LH= (ΩeΩi)−1/2. In particular, a local dispersion relation is derived for the modified‐t...