M G Rusbridge

The interpretation of the bispectrum and bicoherence for non-linear interactions of continuous spectra

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.

Observations of the interaction of a plasma stream with neutral gas: evidence of plasma loss through molecular-activated recombination

We describe an experiment, the UMIST Linear System (ULS), in which a hydrogen plasma stream, guided by a longitudinal magnetic field, is injected through a diaphragm containing an orifice into a separately-pumped target chamber in which the neutral hydrogen pressure can be raised to a maximum of 8 mTorr. The stream is about 6 mm in diameter, has an electron temperature of up to 15 eV and an ion flux of 3×1018 s-1; it is supersonic with Mach number up to M≈3. We have studied both the passage of the stream through the orifice and the interaction of the supersonic plasma with neutral hydrogen in the target chamber. We find that transmission is incomplete even when the orifice diameter is five times that of the plasma; we attribute this to the presence of ion trajectories which extend well outside the visible plasma and are intercepted by the diaphragm. In the target chamber, the stream does not broaden, but the ion flux decreases approximately exponentially with distance, with a scale length of the order of the mean free path for momentum transfer in ion-neutral collisions, and much less than that expected for other processes, such as charge exchange or electron-ion recombination. Elastic collisions alone cannot decrease the flux, but would lead to a large accumulation of slow ions in thermal equilibrium with the neutral gas, which must be limited by some other loss process: collisional diffusion and electron-ion recombination are too slow, leading to a density approaching 1020 m-3. The observed density is of the order of 1018 m-3, requiring a process with a rate of 10-100 times faster. Calculated rates for molecular-activated recombination (MAR) of the slow ions are of this order, and the predicted density agrees with our observations to order of magnitude.

The 'boxcar' method for the analysis of non-Gaussian random signals

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.

Experiments on drift wave launching: I. Dispersion curves and damping rates

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.

Measurement of the impedance to current flow in a steady-state magnetically confined plasma

The authors study the flow of current in the plasma of the UMIST quadrupole between two Langmuir probes which served as electrodes. For low frequencies (<10 kHz) the impedance to current flow transverse to the magnetic field behaves as a resistance and capacitance in parallel. Experimental results are in good agreement with the theory and allow a determination of the ion-neutral collision frequency and of the ion density, both of which agree well with values determined by independent methods. They suggest that this may be applicable as a new diagnostic method in suitable cases. There is a definite resistive component of the impedance for measurements along the field lines and the channel for current flow is considerably larger than the tube of flux joining the two probe sheaths. Theory suggest that the actual value of this resistance will be determined almost entirely by the transverse flow of current associated with the current channel expansion, and experiment confirms this.

Non-linear interactions of drift-waves in quadrupole geometry

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.

Experiments on drift wave launching: II. The launching process

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.

Correlation broadening and the study of turbulence in reversed-field pinch plasmas

The use of the spatial correlation function to describe the structure of turbulence in a plasma is well established. The authors consider the extent to which further information about the turbulence can be obtained from the study of the broadening of the correlation function when a time delay is introduced between the correlated signals. They define an experimentally accessible dimensionless measure of the broadening rate and illustrate its use with data obtained from the ZETA and HBTX reversed-field pinch (RFP) devices. They discuss the interpretation of these results in terms of the random motion of turbulent elements or of the broadening due to resistive diffusion, and show that the true value of the fine-scale correlation length in ZETA was about 2.5 cm, half the value given by the magnetic field fluctuations. The results are compatible with an overall picture of the ZETA discharge as dominated by parallel motion along stochastic field lines near the magnetic axis but by radial convective motions near the edge; it remains to be seen whether other RFP devices show a similar pattern.

Quasilinear saturation of drift waves in the presence of a background transport process

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.

Drift wave launching in a linear quadrupole

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.