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We describe the design and operation of the SPHEX spheromak device and present an overview of its behaviour. The plasma is formed by ejection from a magnetized Marshall gun, and can be sustained as long as the gun is energized. The plasma is divided into the annulus comprising the closed toroidal flux, linked with the open flux forming the central column. The column current is driven directly by the central gun electrode, and the toroidal current in the annulus is driven indirectly by a mechanism associated with a coherent n = 1 oscillation of the column. The configuration exemplifies the operation of the process of relaxation to a state of minimum magnetic energy, which leads to magnetic configurations similar to those observed; to sustain these configurations requires some mechanism of toroidal current drive. Associated with this is the amplification of the poloidal flux, which is typically a factor of about five larger than the flux generated by the gun solenoid; the constancy (to a first approximation) of this factor plays a controlling role in spheromak behaviour. In standard operating conditions there is a `hard' limit, set by the solenoid flux, on the current carried by the column; any current driven by the external circuit above this apparently does not emerge from the gun. Evidence is presented that the column current is carried largely (> 50%) by accelerated ions with energy up to the gun voltage ( for a typical gun current of 60 kA). These ions are poorly magnetized and can escape across the magnetic field to the wall, a likely mechanism for the observed `loss' of current. Hydrogen is the normal operating gas: other gases ( and He) have been used, but the current drive is found to be less effective than in , with lower toroidal current maintained in the annulus.

The structure of the n = 1 mode in the SPHEX spheromak, which plays a central role in relaxation during sustainment, is investigated by analysing the measured voltage fluctuations in the central plasma column. By combining these results with a suitably defined helical magnetic flux function, the mode is found to be due to a rotating helical distortion of the open linked flux. We propose that the distortion is due to a saturated current-driven kink mode of the open flux tube. The prolongation of this `helical column' on its return around the outside of the closed flux is found to be strongly asymmetric. Previously published measurements of the Poynting flux and -profile are re-analysed in the light of these results, and implications for the mechanism of relaxation and non-inductive current drive are discussed.

We present a theoretical and experimental study of the launching impedance , which describes the effectiveness of drift wave launching into the plasma in the quadrupole GOLUX. The theory, valid only for damped modes, incorporates the plasma response function (the inverse of the dielectric function) and measurements of the impedance can, therefore, lead to an experimental test of theoretical forms for this function. Using a lock-in amplifier technique, has been measured for four different branches of the drift wave dispersion curve, two expected to be linearly stable and damped and two believed to be linearly unstable but in practice apparently damped. For the stable modes theory and experiment agree within a factor of two, as good as can be expected given experimental error and approximations in the theory. For the unstable mode, a well defined value of is obtained, but it exceeds the theoretical estimate by two orders of magnitude. Using a different technique we have been able to detect growth of the unstable mode after launching, provided the launched amplitude is sufficiently small; this explains the failure of the theory. It appears, however, that in most cases the wave launched into the plasma is already saturated and does not grow further. We propose that the launched wave amplitude is limited by the power available from the external circuit, and on this basis we have derived an upper limit for the impedance.

We describe experiments on the launching of drift waves in the UMIST Quadrupole GOLUX, presenting the general principles of successful launching and showing how these are applied to launch both the unstable drift waves which occur spontaneously in the device, and other stable modes representing different branches of the drift wave dispersion relation. In each case the measured dispersion curves are presented and compared with theory. All the waves appear to be spatially damped, including those known to be unstable; we propose that this is due to systematic error in the detection system. The results suggest that only if the measured decay constant exceeds about can we be certain that the waves are actually damped. This is the case at low frequencies, where the dissipation is due to ion Landau damping, and again at high wavenumbers where no suitable dissipative process has been discovered. We conjecture that the effect is due to radiative damping, associated with the breakdown of the one-dimensional propagation assumed in the theory.

We develop the linear theory of drift waves in a sheared quadrupole magnetic field configuration, with a view to applying the theory to the UMIST quadrupole GOLUX; shear can be introduced into this system by imposing a uniform longitudinal field. An eigenvalue equation is obtained, and appropriate sets of boundary conditions are proposed. The basic instability is due to the 'dissipative trapped electron' mechanism in both the simple and sheared quadrupole configurations, but the mode structure changes with shear; at sufficiently large values the mode adopts the Pearlstein-Berk (1969) form and may then be stabilized by shear damping. A novel prediction of the theory is that the 'private flux' region of the quadrupole, which without shear is completely stable, is destabilized at very small values of longitudinal field. We propose that the sheared quadrupole will form an excellent laboratory system for testing theories of drift waves developed for tokamak configurations.

Earlier studies indicate that shear in the parallel flow (V∥’(x)) is destabilizing for drift‐like modes. Here, it is shown that taking the flow curvature (V∥‘(x)) effect into account can change the picture altogether. Depending on its sign, parallel flow curvature can be both stabilizing and destabilizing. For a negative value in the curvature [usually found during the neutral beam injection (NBI) heating and the high (H)‐mode edges] parallel flow has a stabilizing effect on drift waves and the mode width gets elongated. Velocity shear, on the other hand, plays an insignificant role both in determining the mode stability and its radial width. As a result, parallel flow must be considered as a viable agent in drift‐like wave stabilization for achieving high confinement operation of fusion devices.

A neutral particle analyser (NPA) and high-resolution spectrometer have been used to make ion energy and rotation measurements during the driven and decaying phases of SPHEX, the UMIST spheromak. During the driven phase NPA spectra reveal the presence of a small (

The behaviour of drift waves localized in a narrow layer in a plasma with a density gradient is discussed. In zero order the plasma is assumed to be in a steady state in which ionization is balanced by a plasma transport process independent of the drift waves, which leads to a flux of plasma through the drift wave layer. It is shown that this plasma flux interacts with the drift wave to increase the damping rate by an amount inversely proportional to the density gradient. Thus as the drift wave amplitude increases, the density gradient flattens until the growth rate is reduced to zero. The theory leads to specific predictions of the saturated amplitude and the drift wave transport: the amplitude rises until the plasma excursion is equal to Delta , the half-thickness of the drift wave layer, and the additional diffusivity is 1/2 gamma Delta 2 where gamma is the linear growth rate at zero amplitude. This theory is shown to agree well with experimental results from the SHEILA heliac, while for comparison a fully nonlinear theory is shown to predict amplitudes much larger than those observed.

Force-free equilibrium magnetic fields in tight-aspect-ratio toroidal configurations are investigated. The study is mainly directed to modelling field configurations in the 'rodomak', a modification to the SPHEX gun-injected spheromak in which a current-carrying rod is inserted along the geometric axis. A family of analytical relaxed states ( Del *B= mu B, mu constant) is presented for a torus of rectangular cross section, with boundary conditions allowing for flux embedded in the walls, representing the gun. Numerically calculated fields in SPHEX geometry, with mu profiles relevant to the driven phase of operation, are also given. The dependence of the field configurations and global quantities such as energy, helicity and toroidal current on the controlling parameters (gun flux, gun current and rod current) and geometry is discussed.

The authors give a more extended discussion of the method of 'amplitude correlations' introduced by Crossley et al. (1992) for the study of nonlinear interactions in plasma turbulence, and in particular for determining the sense of energy flow through a turbulent spectrum. For a particular case, the 'frequency doubling' interaction drift waves in the magnetic quadrupole GOLUX, which transfer energy from driving to driven modes, the authors present a rigorous justification of the method: the driven mode amplitude lags that of the driving modes, as the intuitive interpretation would lead one to expect. The authors compare the theory with experimental results, and show that the method can, in principle, lead to estimates of the damping rate of the driven mode and the nonlinear energy throughput rate. A full determination of these quantities, however, requires further information on the dispersion curve of the driven mode, which is obtainable in principle from wave propagation experiments. Finally the authors show that the method requires many fewer data and less computational effort than the conventional method using the bispectrum and bicoherence.

We present measurements of the magnetohydrodynamic ``dynamo'' due to correlated fluctuations of velocity and magnetic field in the SPHEX spheromak. We show that there are both single-mode and turbulent dynamo processes present, although the single-mode process is in this case an ``antidynamo'' opposing the externally applied electric field. The size of the turbulent dynamo at the magnetic axis is close to that required to drive the toroidal current there.

The authors have used a probe containing both a Rogowski coil and a magnetic pick-up coil to measure both the current density and the mu value in the plasma of the SPHEX spheromak. The results are broadly in agreement with previous measurements but give much better spatial resolution; they reveal a discontinuity at the interface between the 'central column' and 'annulus' regions of the plasma. In the annulus, the range of values of mu spans the spheromak eigenvalue as expected from theory. When the spheromak current drive is terminated by shorting the Marshall gun, the current density at the magnetic axis initially rises to a maximum before decaying, and mu continues to rise towards the eigenvalue. These results are interpreted as showing that the relaxation process continues to operate in the decaying plasma.

The authors describe measurements of nonlinear interactions between drift waves and low-frequency flute-like modes in the UMIST quadrupole. They present experimental measurements of the bispectrum, and also results from a technique of 'amplitude correlation' which they believe provides a more direct way of determining the direction of the energy flow. On the basis of the results they propose a model of energy flow in the drift-wave spectrum: the observed spectrum contains regions of growth at high frequency and damping at low frequency, with flow down the spectrum via a cascade mediated by the flute-like modes.

It is shown that these two models are closely related. The 'Tangled Discharge Model' (TDM) necessarily involves a non-vanishing , the so-called 'dynamo' effect, while the effective functioning of this term in the 'Dynamo' models is shown to require a stochastic magnetic field structure. There is no inconsistency between this conclusion and the fact that the fully-relaxed state may be described by analytical forms such as the 'Bessel Function Model' which apparently contain good flux surfaces. The TDM is not a satisfactory description of the behaviour of present relaxed-state systems, but might be appropriate in the limit of large Lundquist number S.

The spheromak device SPHEX has been modified by adding a current-carrying rod along the geometric axis, providing a preexisting toroidal field. We show that plasma can be succesfully injected into such a field from a helicity source; the field assists plasma ejection from the gun and improves the coupling between gun and plasma, so that T e , T i , and the toroidal current all increase with rod current.

We extend the analysis of the motion of the sprung pendulum to show the full effects of the nonlinear coupling between horizontal and vertical oscillations. We show that even at modest amplitudes a theory involving second‐order terms only is inadequate to explain experimental results, and suggest that the discrepancy is due to nonlinear detuning from the exact resonance condition, an effect which appears only in higher order. Finally, we investigate the effect of damping on the patterns of motion.

Calculations of changes in magnetic moment of a particle passing near an X-type neutral line are discussed. The trapping of particles by these changes, in the low field region about the line, is considered, along with the effects of this trapping around magnetically guarded supports for internal ring machines. Previous calculations concerned particles confined to a symmetrical zone around the separatrices of the field or whose canonical momentum along the neutral line vanished. The present calculation is intended to remove this type of restriction.

Drift waves have been successfully launched from flag probes in a steady-state magnetized plasma, and the launching mechanism has been identified. Nonlinear interactions are observed between launched and intrinsic waves. A wide range of further experimental studies is thus made possible, of fundamental relevance to plasma confinement.

Drift waves in a plasma with a density gradient are expected to behave like standing waves in the direction parallel to the density gradient, confined to a region around the point of maximum gradient and evanescent outside this region. Because of finite Larmor radius effects the higher radial harmonics are predicted to have lower frequencies. The authors show that in the quadrupole, confinement at the inner boundary is probably due to a layer of rapid velocity shear and with this assumption the observed radial eigenfunctions are in reasonable agreement with those predicted; the lower frequency drift wave observed in earlier work is identified as the first radial harmonic. There is a significant departure from the theory, however, in that the drift wave is observed to become a radially propagating wave in the outer regions of the plasma. The energy outflow associated with this propagation is evaluated and shown to be small compared with the input rate due to the drift wave instability mechanism.

The authors describe a method of analysis of random signals containing non-Gaussian features such as pulses or oscillatory bursts; the method involves detecting a pulse locating it in time and classifying it in height and then averaging the signal over a fixed time interval centred on the pulse and over many occurrences of pulses in each class. They show that the technique retains information which is lost in the usual two-point correlation function measurements: this information is that contained in the non-random phase relationships between different Fourier modes, which are necessarily entailed by the non-Gaussian distribution. They also show that in suitable cases the technique allows the direct detection of some kinds of non-linear effect. They illustrate the method with examples from fluctuation measurements in magnetically confined plasmas.

A feature of the plasma in the UMIST quadrupole has been the occurrence of isolated pulses of plasma outside the critical surface, where the plasma is expected to be unstable to interchange modes; isolated plasma structures should then 'fall' outwards and disappear rapidly. The observed pulses propagate along the quadrupole and show no outward motion. The authors propose that the pulses in fact represent vortices, and that their rotation can stabilise them against falling, much as a gyroscope is stabilised by spin. On this basis the authors predict their velocity from first principles and obtain agreement with experiment to an order of magnitude or better, depending on assumptions about the ion temperature; they also show that the vortices can maintain themselves against collisional dissipation by the energy released when plasma is transported outwards in their circulation, and on this basis predict an average potential drop of 0.3-1.5 V between centre and edge of the vortex: the observed value is in the range 0-5 V.

It is shown that random reconnections of magnetic field lines in a diffuse pinch can lead to the appearance of reversed axial field near the discharge tube wall. It is suggested that the magnitude of the effect will depend on the degree to which current can be exchanged between adjacent field lines at a point of reconnection and on this basis a mathematical model is developed which describes magnetic field configurations of the diffuse pinch which are intermediate between two extremes: the force-free paramagnetic model with no field reversal in the limit where the current does not follow the field at all at a recombination point, and the Bessel function model in the limit where the current rigorously follows the field lines. An iterative procedure for the generation of intermediate configurations is described, and numerical results are presented and compared with experimental results from ZETA.

Spontaneous oscillations observed in the UMIST steady-state quadrupole are identified as drift waves with one full wave-length around a closed line of force, and with the potential perturbation anti-symmetric about the maximum field point. Existing theories do not properly describe this mode, for which trapped particles are important. The authors develop the theory of the mode and solve the resulting eigenvalue equation to obtain the dispersion curve; the result agrees with experimental evidence within experimental error. The excitation is identified as due to the 'dissipative trapped particle' mechanism, and the corresponding theoretical growth rate also agrees closely with experiment. In their conditions the excitation is due to electron-neutral rather than electron-ion collisions; as a result the temperature gradient is destabilising rather than stabilising, and they show that this also agrees with experiment.

The authors describe a system suitable for the measurement of electron temperatures in the range 0.1-10 eV and densities of the order of 1015 m-3, consisting of a computer-controlled data acquisition system with off-line data analysis. For the latter stage the authors have developed an algorithm based on curve-fitting techniques to extract the required information in the presence of significant departures from the expected hyperbolic tangent form for the probe characteristic, including in particular the non-zero slope and curvature of the 'saturation' parts of the curve which occur when the probe size is comparable to the Debye length. The authors have tested the system by comparing the results with temperatures obtained by other methods, and have used it in a survey of the temperature and density distributions in the UMIST quadrupole.

The study of plasma fluctuations and confinement in idealized systems such as octupoles and levitrons has contributed to the understanding of cross-field transport processes. The linear theory of plasma instabilities that cause fluctuations is well developed and can predict growth rates γ and wavelengths θ x around lines of force. However, the theoretical prediction of cross-field transport coefficient D ± is restricted to quasilinear estimates of upper bounds (for example, D = 1 2 γ λ x 2 ) because of the complexity of the full nonlinear calculation. Such quasilinear estimates usually far exceed the measured values and are of limited worth. A general view of the results from octupole and levitron experiments shows that under collisional conditions ( λ ei / L < 0 ) the diffusion coefficient, D , scales in the same way as classical collisional diffusion ( D α n / T e 1 2 B 2 ). Agreement is closely approached in many cases, sometimes even in the presence of fluctuations. Under collisionless conditions ( D α n / T e 1 2 B 2 ), Bohm diffusion scaling ( D α T e / B ) is found in the few cases where the scaling law has been determined. This behaviour is consistent with the general scaling laws of Connor & Taylor (1977) but is not understood in detail. In addition there is evidence, both experimental and theoretical, that long-wavelength low-frequency electric fields (convection cells) can be generated nonlinearly from high-frequency fluctuations and can contribute to cross-field transport

For pt.I see ibid., vol.21, no.11, p.907 (1979). It has been suggested that an electric field applied along a neutral line in a plasma should drive an Ohmic current along the line even in a collisionless plasma. The authors have attempted to verify this by studying single particle motion in a quadrupole magnetic field with an applied longitudinal electric field. The results support the existence of the effect, though not conclusively. Expressions are derived for the total current flow along the neutral line and a possible application is discussed briefly.

It has been suggested that charged particle motion near a neutral line, where the adiabatic approximation is inapplicable, can be described approximately as a diffusion process in one dimension. This suggestion is confirmed by following particle trajectories both computationally in an idealised magnetic field and experimentally in the magnetic field of a quadrupole. Measurements made with a beam of electrons give estimates of the diffusion coefficient numerically close to those obtained computationally, and also reproduce the predicted variation with particle energy and magnetic field gradient.

The authors extend the discussion of the bispectrum and bicoherence to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension. They evaluate the bispectral density in terms of the dielectric function of the driven mode and the dimensionless coupling constant, and point out that if the latter can be assumed to be real, the phase of the bispectrum can be used to distinguish resonant from non-resonant modes; the authors then evaluate the power in the driven mode separately for the resonant and non-resonant cases, to give results in a form suitable for comparison with experiment. They then consider the bicoherence, and show that this statistic is of limited value for continuous spectra, since its maximum value is depressed in an ill-defined way. On the other hand, the normalisation inherent in the definition does enhance weak regions of the spectrum and can show up interactions too weak to be readily detected in the bispectrum itself.

An outstanding problem in the behaviour of the Zeta plasma is the origin of the fluctuations of density; it has been shown experimentally that they do not arise directly from the observed turbulent motions of the plasma, and there is no known instability which can produce them directly. It is proposed that they arise as a by-product of resistive instabilities through their action in cross-connecting lines of force belonging to magnetic surfaces at different radii. This idea is developed quantitatively and it is shown that it predicts fluctuation levels in reasonable agreement with experiment. This result is interpreted as giving support to the tangled discharge model of the Zeta plasma.

Plasma from a duoplasmatron is injected along the axis of a linear quadrupole. Peak number densities of about 5*109 cm-3 and temperatures of about 10 eV are achieved. The lifetime is about 100 mu s; losses occur mainly at the ends, transverse losses being negligible. This hot plasma is embedded in a cold plasma (0.5-1 eV) part of which drifts upstream towards the source. It is suggested that this arises from the action of a positive sheath at the far end of the device, maintained by an excess electron current from the source. Density and potential fluctuations are observed throughout the plasma but are large only in a zone bounded approximately by the critical surface and the surface of minimum field line length.

The UMIST quadrupole magnetic confinement system shows a well-developed spectrum of drift waves, which are anti-symmetric between opposite sides of the machine (they have one full wavelength around a closed line of force). The authors consider non-linear interactions of these modes, concluding that these should include a nearly resonant interaction of two drift-waves to give a symmetric drift-wave (with two full wavelengths around a closed line of force) at double the frequency. They show that the observed power-spectrum does show a frequency-doubling peak, and from observations of the bispectrum that it is linked by non-linear interactions with the drift-waves. The authors use the bispectrum as a probe of the eigenfunction at the frequency-doubled peak, and show from this that the resonant interaction described above can contribute only a small part of the total power, and that it must be suppressed relative to non-resonant interactions with other modes. They discuss possible reasons for this, and conclude that considerable caution must be exercised in identifying non-linear interactions, however plausible they may appear.

The authors describe a correlator designed for measurements in a steady state plasma device, including the features of heterodyne filtering to obtain extremely narrow bandwidths and signal normalisation by automatic gain control to give an output proportional to the true correlation coefficient. The possible extension of the device to measure triple correlations is discussed. Typical results are presented to show that with a sufficiently narrow bandwidth the device can be used to determine the intrinsic spread in wave number of modes of oscillation of the plasma.

Regular oscillations have been detected under a limited range of conditions in a 12-in. bore torus. The electric and magnetic fields associated with these have been investigated and indicate a helical notch of reduced density travelling on the pinched current channel with a uniform velocity. A possible explanation of the phenomenon is given. The oscillation is considered to be growing as a sound wave in a partially-ionized medium. The mode is cited as an example of a pinched discharge with poor confinement.

The diffusion of the current sheath in ZETA is shown to be incompatible with classical resistive diffusion at high rates of energy input or low pressures. In the anomalous region the diffusion time is approximately proportional to mass density and inversely proportional to the applied electric field. There are considerable asymmetries in the toroidal discharge and complicated convective motions during the diffusion phase, but these appear not to be the primary cause of the enhanced diffusion. Evidence has been obtained that the enhanced diffusion is not due to the presence of a probe in the plasma. It is therefore probable that there must be an enhanced resistivity, depending on the local values of mass density and electric field, which is presumably due to some form of turbulence, but none of the simple models of such turbulence considered account for the results.

It is known that the adiabatic invariance of the magnetic moment of a charged particle breaks down when the particle passes near a field zero. The extreme case of this breakdown, arising when the particle orbit encloses the field zero, is investigated by computation of particle orbits. It is found that the magnetic moment changes by a factor with a maximum value of about 5.75; the exact value depends on the phase of the particle in its orbit, and the details of the dependence are explained on a simple model.

A theory is developed which leads to an expression for the partition of energy between mechanical and magnetic modes in a turbulent magnetohydrodynamic system in the presence of a strong external field. Modifications to this expression due to specific plasma effects are obtained and are not significant. Results are also obtained for the frequency spectra and diffusion coefficient. Measurements of the turbulent fluctuations in the Zeta device are described, and from these, values for the partition are obtained. It is found that there is only complete agreement with the theory if an anomalous resistivity is assumed at low operating pressures.

The Zeta discharge in the low density regime is characterized by strong turbulence, rapid loss of plasma, a resistance much larger than that predicted from the electron temperature, and a high ion temperature. A model for the energy flow in this regime is proposed, based on the convective transport of the plasma to the walls. This model leads to coupled equations for the ion and electron temperatures and for the turbulent energy. From these equations, and from their integrated forms which govern the energy balance of the discharge as a whole, predictions of the ion and electron temperatures are obtained which are in gross disagreement with observation. It is shown that agreement can be obtained by introducing an additional energy loss process, thermal conduction along the lines of force which are assumed to follow a random path to the wall. Two other possible energy loss mechanisms are considered and shown to be inadequate alone, although some contribution from either in conjunction with thermal conduction cannot be ruled out.

Flute fluctuations have been studied in an afterglow discharge in a linear quadrupole device. They are observed outside the critical surface; inside, their amplitude is unobservably small. The results support the stability criterion including adiabatic effects. The phase velocity and growth rate of the fluctuations outside the critical surface have been measured and compared with a simple linear theory; significant differences from the theory are found, some of which can be explained qualitatively by a more elaborate linear theory. It is not clear, however, whether this theory would account for the differences or whether nonlinear effects are also important. The radial transport of plasma due to these fluctuations is estimated and compared with the observed loss rate, but it is shown that some other mechanisms must exist for transporting plasma from the stable region across the critical surface.