The range of ▿ ∧ (u ∧ B) is determined for an axisymmetric poloidal incompressible B and incompressible u, by solving ▿ ∧ (u ∧ B) = ▿ ∧ v for u, assumed to be confined to within a surface of revolution. Two constraints on V are shown to be necessary for the existence of solutions, viz that the integral of rç/|B| must vanish on each flux surface, and that the integral of v1 around any dosed field line must vanish. The construction of the general solution proves that the constraints are sufficient conditions, providing also that the second derivative, with respect to the poloidal flux function, of the volume contained by flux surfaces does not vanish. The general solution is stated for the homogeneous case, v = 0. For the particular case of poloidal v1 an integral property of the solution u is established.

The motion of a particle driven by a slow RF wave inside crossed electrostatic and magnetostatic fields is examined. A time-scale separation exists in the synchronous frame, moving at the slow-wave phase velocity, where all fields appear static in time and the cyclotron motion is much faster than the guidingcentre drift. The averaging of the periodic gyromotion is performed systematically with canonical transformations up to second order in the smallwave amplitude. The first-order guiding-centre drift follows the equipotential surfaces of the transformed electric field. The second-order effects, due to the field-line curvature, and/or the nonlinear variation in the wave phase velocity, cause departures from the parapotential flow. The topology of the trajectories depends on the departure of the drift velocity from synchronism with the wave. Various flow topologies corresponding to different values of the control parameters are examined.

In this paper, we use the hydrodynamic approach to study the stimulated scattering of high-frequency electromagnetic waves by a low-frequency electrostatic perturbation that is either an upper- or lower-hybrid wave in a two-electron-temperature plasma. Considering the four-wave interaction between a strong high-frequency pump and the low-frequency electrostatic perturbation (LHW or UHW), we obtain the dispersion relation for the scattered wave, which is then solved to obtain an explicit expression for the growth rate of the coupled modes. For a typical Q-machine plasma, results show that in both cases the growth rate increases with noh/noc. This is in contrast with the results of Guha & Asthana (1989), who predicted that, for scattering by a UHW perturbation, the growth rate should decrease with increasing noh/noc.

The steady currents driven in a spherical plasma by a rotating magnetic field via the Hall effect are studied analytically. The total field is shown to be symmetric across the origin. Integral relationships are obtained between Ohmic dissipation, angular momentum and the oscillating axial current density. The topology of the sum of a Hill's vortex field and a rotating field is documented. Analytical solutions for the driven current are obtained by expansion for the limits corresponding to small rotation frequency, to small number density, to large rotating-field magnitude, to small resistivity, and to small rotating-field magnitude combined with very small resistivity. The latter solution, relevant to the reactor limit, indicates that, with control of the vertical field magnitude, an MHD equilibrium can be generated with total current any fraction of the currentcorresponding to synchronous rotation of the electrons. Oscillating currents sufficient to drive the synchronous current are determined.

The parametric excitation of surface waves in a warm inhomogeneous plasma is investigated. It is demonstrated that the coupling between the surface wave and plasmons that can be excited in the transition layer must be taken into account. The growth rate and the threshold value for the instability process are calculated. It is also shown that thermal effects are of importance even for very low temperatures, since the plasmons can now propagate out of the resonance region.

We investigate the parametric decay and modulational instabilities of a large-amplitude circularly polarized dispersive Alfvén wave. Our treatment is more general than that of previous derivations based on the two-fluid equations in that we allow for propagation of the unstable daughter waves at arbitrary angles to the background magnetic field, although our main concern in this paper is the exploration of new aspects of propagation parallel to the DC magnetic field. In addition to the well-known coupling of pump waves to electrostatic daughter waves, we find a new parametric channel where the pump wave couples directly to electromagnetic daughter waves. Excitation of the electromagnetic instability occurs only for modulation (k/k0 ≤ 1) and not for decay (k/k0 < 1). In contrast with the modulational instability excited by the electrostatic coupling, the electromagnetic modulational instability exists for both left-hand (K > 0) and right-hand (K < 0) polarization. For large k/k0, the electromagnetic channel dominates, while at lower values the electrostatic channel has a larger growth rate for modest values of dispersion, pump-wave amplitude and plasma β. Unlike the electrostatic modulational instability, the growth rate of the electromagnetic instability increases monotonically with increasing pump wave amplitude. This analysis confirms that, for decay, the dominant process is coupling to electrostatic daughter waves, at least for parallel propagation. For modulation, the coupling to electromagnetic daughter waves usually dominates, suggesting that the parametric modulational instability is really an electromagnetic phenomenon.

An exact solution for the thermal (Xe) or particle (D) diffusivities is derived (up to a constant factor) for the diffusive model in terms of the Fourier coefficients of the perturbed Te (or ne) signals. The common assumptions of constant ne and Xe (or D) generally used in the standard analytical or semi-analytical solutions are not required, since their radial dependences are taken explicitly into account. The effect of a damping (sink) term in the heat equation can also be implemented easily.

The image approach, used extensively to treat bounded unmagnetized plasmas, is extended to the case of an arbitrary homogeneous and non-magnetic medium. A general dispersion relation for electromagnetic surface waves on a plane plasma-vacuum interface is thus obtained, subject only to the suitability of the chosen boundary conditions. The boundary conditions used here are those of Barr and Boyd. It is emphasized that this dispersion relation is applicable to magnetized plasmas. The general dispersion relation is applied to the special case of an isotropie medium, and the dispersion relation of Barr and Boyd for an unmagnetized plasma is reproduced. A major assumption in the image approach is that the semi-infinite bounded medium can be described by the infinite-medium response. The validity of this assumption and of the boundary conditions is discussed. Two conditions are deduced that must be satisfied for the image theory to be self-consistent. It is argued that these can be satisfied in all situations for which the assumed boundary conditions are appropriate.

The Kelvin-Helmholtz instability of a tangential velocity discontinuity in a plasma flow when cosmic-ray pressure is taken into account is studied. It is shown that a new unstable mode can exist. This appears when an effective Mach number exceeds some critical value. In the case of strong flow mediation by cosmic rays the wave vectors of the most-unstable waves are directed along the plasma velocity vector for both subsonic and supersonic flows.

We investigate the parametric instabilities of a large-amplitude circularly polarized dispersive parallel-propagating Alfvén wave. Our treatment is more general than that of previous derivations based on the two-fluid equations in that we allow for propagation of the unstable daughter and side-band waves at arbitrary angles to the background (DC) magnetic field. We present the characteristics of the decay and modulational instabilities as functions of propagation angle. We find, in addition to the well-known decay and modulational instabilities, that at oblique and perpendicular propagation there is another parametric instability, namely the filamentation instability, which is characterized by a broad band-width in wavenumber and which satisfies the condition Re (ω) ≪ γ. A second parametric process at oblique and perpendicular angles of propagation, which has not been reported before is also investigated, namely the parametric magneto-acoustic instability. This instability is distinct from the filamentation instability in that it is characterized by density perturbations with large real frequencies that satisfy the condition Re (ω) ≫ γ. Unlike the filamentation instability, the magneto-acoustic instability extends over a broad angular range, but has a very narrow band-width in wavenumber. We report the dispersive characteristics of the filamentation and magneto-acoustic instabilities as functions of plasma β dispersion η and pump amplitude η for arbitrary propagation angles.

We study the nonlinear stability of a one-dimensional hydromagnetic cavity into which Alfvén waves are fed by harmonic shear motions of its boundaries and where they interact with slow magnetosonic waves. We use characteristic conditions for the outgoing and ingoing Alfven waves at the boundaries where the magnetosonic oscillations are required to vanish. Forcing of Alfven waves takes place at a frequency close to the eigenfrequency of the lowest-order mode of the cavity. We let the frequency detuning δω vary as a free parameter together with the amplitude of the forcing, the plasma β and the compressive Reynolds number Re0. Given these last three parameters and varying δω, we calculate the amplitude of the nonlinear equilibrium state of the cavity as the stationary solution of a simple forced, dissipative dynamical system that governs the evolution of the cavity over a slow time scale and to which we are led by multiple-scale and Galerkin analyses of the one-dimensional MHD equations. This amplitude is a multi-valued function of δω (bistability), and we discuss the possibility of nonlinear stabilization of the Alfven wave by locking it in one of the bistable states. This amplitude undergoes saddle-node bifurcations: we calculate the two values of δω at which this occurs and the lowest value of the Reynolds number (27/2) for this to happen. We show that the magnetic energy density released during a bistable transition scales as (Re0)2; it has a maximum at β = 1 - (⅔)½ and it may amount to a substantial part of the energy originally stored in the unperturbed cavity. The magnetic power density released scales as (Re0)3 and has a maximum at β = 1 ± (⅓)½5. We conclude that the cavity is a good site for plasma heating such as that of the solar corona.

Using first-order smoothing theory, Fourier analysis and perturbation methods, we obtain the evolution equation of the wave spectrum as well as the nonlinear forces generated by Alfvénic turbulence in a finite-β plasma with dispersion and phenomenological Landau-damping effects. The stability of Alfvénic turbulence is then analysed by solving the derived mean-field equations. It is shown that parallel-propagating turbulent Alfvén waves with dispersion can be modulationally unstable, leading to amplification of large-scale Alfvén waves and acoustic waves when Landau-damping effects are strong and weak respectively.

The emission of extraordinary mode radiation in a plasma with Langmuir turbulence driven by an electron beam is considered. The process of emission considered in this paper is the plasma maser effect, which is essentially an energy up-conversion process. The energy necessary for the growth of the extraordinary mode is derived from the Langmuir turbulence. The nonlinear dispersion relation of the extraordinary mode in the presence of Langmuir turbulence is obtained and its growth rate calculated. The scope of application of the results to space-plasma observation is then stressed.

The dissipation of relative magnetic helicity due to the presence of a resistive reconnection region is considered. We show that when the reconnection region has a vanishing cross-section, helicity is conserved, in agreement with previous studies. It is also shown that in two-dimensional systems reconnection can produce highly twisted reconnected flux tubes. Reconnection at a high magnetic Reynolds number generally conserves helicity to a good approximation. However, reconnection with a small Reynolds number can produce significant dissipation of helicity. We prove that helicity dissipation in two-dimensional configurations is associated with the retention of some of the inflowing magnetic flux by the reconnection region, vr. When the reconnection site is a simple Ohmic conductor, all of the magnetic field parallel to the reconnection line that is swept into vr is retained. (In contrast, the inflowing magnetic field perpendicular to the line is annihilated.) We are able to relate the amount of helicity dissipation to the retained flux. A physical interpretation of helicity dissipation is developed by considering the diffusion of magnetic field lines through vr. When compared with helicity-conserving reconnection, the two halves of a reconnected flux sheet appear to have slipped relative to each other parallel to the reconnection line. This provides a useful method by which the reconnected field geometry can be constructed: the incoming flux sheets are ‘cut’ where they encounter vr, allowed to slip relative to each other, and then ‘pasted’ together to form the reconnected flux sheets. This simple model yields estimates for helicity dissipation and the flux retained by vr in terms of the amount of slippage. These estimates are in agreement with those expected from the governing laws.

The quasi-linear diffusion coefficients for the inherently space-dependent wave amplitudes in lower-hybrid heating and current-drive experiments are derived and compared with the usual space-independent approximation. The distribution function is derived and the dependence on the radio-frequency field amplitude gradient of the ‘plateau’, which is an essential quantity in thetheoretical description of LH experiments, is stressed.