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A brief history of the concept of negative energy waves in fluid and plasma media is given. The theory of microwave tubes is highlighted, where the small signal power theorem was first derived, which led to the interpretation of these devices in terms of the interaction of positive and negative energy waves. Three examples are described to illustrate the use of negative waves in stability problems. The first is concerned with the Kelvin-Helmholtz instability in neutral fluids. The second comes from a kinetic theory of instability in a collisionless shock wave in a magnetized plasma, and the third is concerned with the effect of a resistive wall and plasma flow on the magnetohydrodynamic stability of a magnetically confined plasma. The paper ends with some general conclusions.

Integrated modelling of important plasma physics issues related to the design of a steady-state spherical tokamak (ST) fusion power plant is described. The key is a steady-state current drive, and 92% of this is provided by a combination of bootstrap and diamagnetic currents, both of which have a substantial toroidal component in a ST. The remaining current is to be provided by either neutral beam injection or radio-frequency waves, and various schemes for providing this are discussed and quantified. The desire to achieve a high bootstrap current drives the design to high plasma pressure, ? (normalized to the magnetic field pressure), and high elongation. Both these requirements have implications for ideal magneto-hydrodynamic instability which are discussed. Confinement is addressed both through comparison with the recent scaling laws developed from the conventional tokamak database and self-consistent one-dimensional modelling of the transport processes. This modelling shows that the power required for the current drive (~50?MW) is sufficient to heat the plasma to a regime where more than 3?GW of fusion power is produced, taking into account the dilution due to He ash and prompt ?-particle losses, which are small. A preliminary study of the micro-instabilities, which may be responsible for the turbulent transport is provided. Given assumptions about the particle confinement, we make estimates of the fuelling requirements to maintain the steady state. Finally, the power loading due to the exhaust is derived using theory-based scalings for the scrape-off layer width.

In previous publications [A. K. Ram and S. D. Schultz, Phys. Plasmas 7, 4084 (2000); A. Bers, A. K. Ram, and S. D. Schultz, in Proceedings of the Second Europhysics Topical Conference on RF Heating and Current Drive of Fusion Devices, edited by J. Jacquinot, G. Van Oost, and R. R. Weynants (European Physical Society, Petit-Lancy, 1998), Vol. 22A, pp. 237–240] it has been shown that, in overdense plasmas of the type encountered in spherical tori, electron Bernstein waves can be excited in a plasma by mode conversion of either an externally launched X mode or an O mode. The electron Bernstein waves are strongly absorbed by electrons in the region where the wave frequency matches the Doppler broadened electron cyclotron resonance frequency or its harmonics. The strong absorption also implies that electron Bernstein waves are emitted by a thermal plasma. These waves can then mode convert to the X mode and to the O mode and be observed external to the plasma. In this paper an approximate kinetic model describing the coupling between the X mode, the O mode, and the electron Bernstein waves is derived. This model is used to study the mode conversion properties of electron Bernstein wave emission from the plasma interior. It is shown, analytically and numerically, that the energy flow conversion efficiencies of the electron Bernstein wave to the X mode and to the O mode are the same as the energy flow conversion efficiency of the X mode to electron Bernstein waves and of the O mode to the electron Bernstein waves, respectively. This has important experimental consequences when designing experiments to heat overdense plasmas by electron Bernstein waves.

The linear and nonlinear dynamics of zonal flows and their interactions with drift wave turbulence is considered in the simple but illuminating generalized Charney–Hasegawa–Mima model due to Smolyakov et al. [Phys. Plasmas 7, 1349 (2000)]. Two positive definite, exact, integral invariants associated with the full generalized Charney–Hasegawa–Mima system are derived. For an initial monochromatic drift wave pump with small but finite amplitude, a modulational instability can occur, characterized by growing zonal flow and sideband perturbations (i.e., a four-wave interaction). The pump threshold for instability is readily satisfied, depending on the zonal flow wave number. The fully nonlinear Charney–Hasegawa–Mima equations are solved with a numerical scheme which is validated by demonstrating the conservation of the two exact invariants. The simulations show that the validity of the four-wave model is limited to approximately three instability growth times. The radial structure of the zonal flow can be “jet...

A cylindrical model with an equilibrium surface current and a uniform equilibrium plasma flow velocity parallel to the axis of the cylinder is used to investigate resistive wall instability. This system can be unstable to the ideal, external kink mode, which can be stabilized by the presence of a perfectly conducting wall. This is the classic condition for the resistive wall instability and the effect of a plasma flow velocity on this mode is explored. It is noted that a stable kink mode, Doppler shifted by the flow velocity, can pass through zero frequency for a velocity which depends on the marginal condition for the external kink instability. The passage through zero frequency is the condition for the kink mode to carry negative energy. It is shown how this mode implies a critical flow speed at which the resistive wall mode is further destabilized, with a growth rate inversely proportional to the square root of the wall time. Under these circumstances, the resistive wall mode behaves more like an ideal instability. All flow velocities are shown to be potentially destabilizing and the flow velocity can produce a resistive wall instability even when the plasma is stable to the external kink mode in the absence of a wall. At velocities well above the critical flow speed, the resistive wall growth rate is much reduced (inversely proportional to the wall time and to the flow speed).

For a fusion power plant to be economically and technologically attractive, it should be as compact as possible and capable of `steady state' operation. One approach is based upon the spherical tokamak (ST) concept. This configuration features many of the qualities of the conventional aspect ratio tokamak, such as good confinement and MHD stability, together with a number of highly promising features for the realization of a cost effective, steady state fusion power core. In particular, it allows high β capability due to the high I/B achievable at low A and strong, natural shaping of the configuration, together with the possibility of approaching 100% pressure driven currents through high natural elongation and access to second stability to ideal ballooning modes. The physics insights gained from experiments on START are discussed and the theoretical modelling work summarized, describing how the beneficial properties of STs can be combined in steady state ST power plant designs such as the Culham STPP concept. Where appropriate, we shall refer to the two new mega-amp machines MAST and NSTX, both designed for a plasma duration approaching the plasma current relaxation time and also other new, smaller, STs, designed to expand the knowledge gained so far.

A uniform compressible plasma with a uniform flow along the magnetic field in the presence of a resistive wall is shown to be subject to two instabilities. For the first instability, the flow velocity is required to exceed the Alfven speed, whereas for the second it need only exceed the sound speed. For a sufficiently high ion temperature, ion Landau damping is shown to stabilize the second instability associated with the sound speed.

Experiments with simultaneous lower hybrid (LH) current drive and ion cyclotron resonance frequency heating of minority protons in JET deuterium plasmas showed efficient coupling of LH power to the MeV energy anisotropic protons. Such an interaction represents parasitic loss of LH power, intended for electron current drive, to the proton population, with important implications for the efficiency of LH current drive schemes in fusion devices containing high-energy charged fusion products. We solve the hot-plasma dispersion relation in the LH range of frequencies for plasmas containing high-energy minority anisotropic protons and electrons, and find that multiple solutions of the dispersion relation with perpendicular refractive index exist in the plasma centre due to the presence of the protons. These hot-plasma modes give damping of LH input power onto the protons by perpendicular Landau damping. Combinations of background plasma and LH antenna parameters exist for which no solutions of the LH dispersion relation are found with , thus reducing damping of LH power to the protons to . This suggests ways of reducing the parasitic loss of LH power to high-energy charged fusion products.

A theory of fast wave absorption in a high-beta plasma is given. A reduced, second-order, ordinary differential equation has been used which includes all collisionless electron dissipation mechanisms and ion cyclotron damping over many harmonics. This is relevant to the high-harmonic fast wave heating scheme proposed by Ono [Phys. Plasmas 2, 4075 (1995)]. The parameters appropriate to the National Spherical Tokamak Experiment NSTX [J. Spitzer et al., Fusion Technol. 30, 1337 (1996)] have been used to investigate the absorption characteristics of electrons and ions under high-beta, high-harmonic conditions.

Detailed ray tracing, of wave propagation in a plasma near electron cyclotron resonances, suggests that refraction can lead to reduced absorption, in some cases. By studying the full wave equation for the ordinary wave near the fundamental, in a slab model, it is shown that such refraction does indeed reduce absorption, for this particular case, but that wave energy tunnelling can significantly modify the result.

A cylindrical model of a tokamak is used to describe the nonlinear coupling between Alfven waves and sound waves. The model is applied to the case of a toroidal Alfven eigenmode (TAE) driven unstable by fusion alpha particles to explore its relevance to the nonlinear saturation of such modes, which can produce a loss of alpha particles in a burning plasma. The mechanism is the modulational instability of a finite-amplitude wave, in this case a TAE mode, in which a density fluctuation and Alfven sidebands are spontaneously excited when the finite-amplitude wave exceeds a threshold amplitude. In general, such a nonlinear calculation in a bounded inhomogeneous plasma would require a numerical solution. However, in the present case, an approximate analytic solution is derived by making use of the fact that the sound wave is logarithmically singular on a particular magnetic surface. The calculation is therefore carried out on the magnetic surface on which the TAE instability is centered and a nonlinear dispersion relation is derived. The threshold for the modulational instability yields a value for the ratio of the perturbed radial component of the magnetic field to the equilibrium magnetic field, which is comparable to the saturated amplitude of a TAE instability obtained by other workers. In addition, the nonlinear dispersion relation also includes a resonant case for which the threshold can be significantly lower. The present, heuristic calculation suggests that the modulational instability may indeed be relevant to the evolution and eventual saturation of a TAE instability.

An analysis of ion cyclotron resonance heating in a plasma containing two ion species is given. Particular emphasis is placed on the important case of comparable concentrations of the two species. The properties of ion-Bernstein waves in such a plasma are discussed. A summary of the cutoff–resonance (Budden) and cutoff–resonance–cutoff (triplet) configurations is given and an analytic expression obtained for the Budden transmission coefficient (relevant also to the triplet case), which is valid for any concentration ratio of the two-ion species. A formula for the damping coefficient of the ion hybrid wave is obtained and a ray tracing code is used to illustrate the absorption of this mode by electrons close to the two-ion hybrid resonance. The reduced, second-order fast wave equation is solved numerically, illustrating the transition from strong electron heating through mode conversion at the two-ion hybrid resonance to direct electron damping of the fast wave in a high beta plasma.

The stability of the ion hybrid wave in a plasma containing two thermal ion species present in comparable proportions and a low density population of superthermal alpha-particles is analyzed. A simple, model distribution function consisting of a ring distribution in the perpendicular velocity and a Maxwellian in the parallel velocity is used for the superthermal alpha-particles. This distribution function is relevant to the core plasma of a tokamak in the immediate post-birth phase before the alpha-particles have had time to relax collisionally and is therefore of interest to the alpha-channelling question. It has also been used to interpret ion cyclotron emission from fusion products in the edge plasma of large tokamaks. An approximate dispersion relation is derived which allows the conditions for instability to be explored and an analytic expression for the growth rate to be obtained. It is found that the ion hybrid wave can be unstable for v⊥0/cA≪1 where v⊥0 is the alpha-particle ring speed and cA is A...

Ion cyclotron emission (ICE) has been observed during neutral beam‐heated supershots in the Tokamak Fusion Test Reactor (TFTR) [Phys. Rev. Lett. 72, 3526 (1994)] deuterium–tritium campaign at fusion product cyclotron harmonics. The emission originates from the outer midplane edge plasma, where fusion products initially have an anisotropic velocity distribution, sharply peaked at a sub‐Alfvenic speed. It is shown that the magnetoacoustic cyclotron instability, resulting in the generation of obliquely propagating fast Alfven waves at fusion product cyclotron harmonics, can occur under such conditions. The time evolution of the growth rate closely follows that of the observed ICE amplitude. Instability is suppressed if the fusion products undergo a moderate degree of thermalization, or are isotropic. In contrast, the super‐Alfvenic fusion products present in the outer midplane of the Joint European Torus (JET) [Nucl. Fusion 33, 1365 (1993)] can drive the instability if they are isotropic or have a broad speed distribution. This may help to account for the observation that fusion product‐driven ICE in JET persists for longer than fusion product‐driven ICE in TFTR supershots.

Ion cyclotron emission (ICE) has been observed from neutral beam heated TFTR, and JET tritium experiments at sequential cyclotron harmonics of both fusion products and beam ions. The emission originates from the outer midplane plasma, where fusion products and beam ions are likely to have a drifting ring-type velocity-space distribution that is anisotropic and sharply peaked. Fusion product driven ICE in both TFTR and JET can be attributed to the magnetoacoustic cyclotron instability, which involves the excitation of obliquely propagating waves on the fast Alfven/ion Bernstein branch at cyclotron harmonics of the fusion products. Differences between ICE observations in JET and TFTR appear to reflect the sensitivity of the instability growth rate to the ratio vbirth/cA where vbirth is the fusion product birth speed and cA is the local Alfven speed for fusion products in the outer midplane edge of TFTR supershots, vbirth < cA for alpha particles in the outer midplane edge of JET, the opposite inequality applies. If sub-Alfvenic fusion products are isotropic or have undergone even a moderate degree of thermalization, the magnetoacoustic instability cannot occur. In contrast, the super-Alfvenic alpha particles that are present in the outer midplane of JET can drive the magnetoacoustic cyclotron instability even if they are isotropic or have a relatively broad distribution of speeds. These conclusions may account for the observation that fusion product driven ICE in JET persists for longer than fusion product driven ICE in TFTR. Moreover, the time evolution of the maximum growth rate, obtained using the Sigmar model for the alpha particle distribution and TFTR data for the fusion product source rate, closely follows the observed time evolution of the ICE amplitude in TFTR supershot discharges. Other observed features of fusion product driven ICE that match the linear instability include the scaling with fusion product density, doublet splitting of spectral peaks, the relative strength of certain harmonics and source localization. A separate mechanism is proposed for the excitation of beam driven ICE in TFTR: electrostatic ion cyclotron harmonic waves, supported by strongly sub-Alfvenic beam ions, can be destabilized by a low concentration of such ions with a very narrow spread of velocities in the parallel direction. Sufficiently narrow distributions are likely to exist in the edge plasma, close to the point of beam injection.

Mode conversion of the fast Alfven wave (FAW) at the ion‐hybrid frequency in the ion cyclotron range of frequencies (ICRF) is studied in the presence of ion cyclotron absorption and direct electron damping in a tokamak plasma. The usual Budden model is extended to include the effect of electron damping and of the high‐field‐side cutoff, and is solved analytically and numerically. The mode‐conversion efficiency is given as a function of the Budden transmission coefficient and of a phase integral, which describes interference between the incoming and outgoing waves. In incidence from the low‐field side, a discrete spectrum of phases exists for which complete absorption (i.e., combined mode conversion and direct electron damping) of the FAW for a single transit of the resonance region can be achieved. This permits efficient electron heating and/or current drive via mode conversion of FAWs.

A mechanism is proposed for the excitation of waves at harmonics of the injected ion cyclotron frequencies in neutral beam‐heated discharges in the Tokamak Fusion Test Reactor (TFTR) [Proceedings of the 17th European Conference on Controlled Fusion and Plasma Heating (European Physical Society, Petit‐Lancy, Switzerland, 1990), p. 1540]. Such waves are observed to originate from the outer midplane edge of the plasma. It is shown that ion cyclotron harmonic waves can be destabilized by a low concentration of sub‐Alfvenic deuterium or tritium beam ions, provided these ions have a narrow distribution of speeds parallel to the magnetic field. Such a distribution is likely to occur in the edge plasma, close to the point of beam injection. The predicted instability gives rise to wave emission at propagation angles lying almost perpendicular to the field. In contrast to the magnetoacoustic cyclotron instability proposed as an excitation mechanism for fusion‐product‐driven ion cyclotron emission in the Joint Europ...

The theory of the magnetoacoustic cyclotron instability, which has been proposed as a mechanism for suprathermal ion cyclotron harmonic emission observed in large tokamaks, is generalized to include finite parallel wave number k∥. This extension introduces significant new physics: the obliquely propagating fast Alfven wave can undergo cyclotron resonant interactions with thermal and fusion ions, which affects the instability driving and damping mechanisms. The velocity–space distribution of the fusion ions is modeled by a drifting ring, which approximates the distribution calculated for the emitting region in tritium experiments on the Joint European Torus (JET) [Cottrell et al., Nucl. Fusion 33, 1365 (1993)]. Linear instability can occur simultaneously at the fusion ion cyclotron frequency and all its harmonics when the fusion ion concentration is extremely low, because the finite k∥ gives rise to a Doppler shift, which decouples cyclotron damping due to thermal ions from wave growth associated with fusi...

In a recent article Cairns et al. [Phys. Fluids B 3, 2953 (1991)] gave a method for the derivation of full wave equations describing propagation through a cyclotron resonance in an inhomogeneous plasma. The simplicity of this method compares favorably with previous derivations, and the damping resulting from the variation in the magnetic field across a Larmor orbit, described by Lashmore‐Davies and Dendy [Phys. Fluids B 1, 1565 (1989)], is included. The effect of the relativistic mass shift on the cyclotron frequency, which plays an important role in the electron cyclotron range of frequencies, was not taken into account, however, and the object of the present work is to remedy this omission. It is shown how equations, valid in the weakly relativistic regime, may be obtained in a rather straightforward way. Results obtained by a number of earlier workers are recovered and can be extended.

A common mechanism for the observed excitation of waves at sequential multiple ion cyclotron harmonics in the JET tokamak and in various space plasmas is proposed. It is suggested that, in each case, waves on the fast Alfven-ion Bernstein branch are excited by the magnetoacoustic cyclotron instability, which is driven by an energetic ion distribution with a local maximum at v⊥ ≠ 0, where v⊥ is the velocity component perpendicular to the magnetic field.

It can be shown that whistler waves with low parallel phase velocity can be subject to strong cyclotron damping, even when the wave frequency is well below the cyclotron frequency. This resonance arises as a result of the Doppler effect which is proportional to the plasma density and increases the frequency experienced by electrons moving in the opposite direction to the wave. In this paper the dispersion relation of whistler waves in a cylindrically bounded plasma is obtained and then solved. This geometry is desirable for its relevance to experiment and also for comparison with theoretical results by other authors. In addition to cyclotron damping, Landau damping as well as electron–ion, electron–electron, and electron–neutral collisions are all included thus enabling the relative importance of these damping mechanisms to be evaluated. The dispersion relation that is obtained is used to explore the transition from Landau dominated damping to cyclotron dominated damping.

Nearly all kinetic treatments of fast wave minority heating of inhomogeneous plasma in the cyclotron range of frequencies assume the magnetic field varies in the direction perpendicular to the magnetic field. However, the toroidal magnetic field of a tokamak varies along a field line due to the rotational transform and causes a small number of trapped particles to turn in the region of cyclotron resonance. In order to include the effects of rotational transform and, hence, trapped particles in the kinetic plasma response, a simplified, concentric circle flux surface model of a tokamak is employed. The most important result of this work is the derivation of response functions for Maxwellian and bi‐Maxwellian minority ions which generalize and extend previous replacement Z function forms obtained from a slab approximation of a tokamak (which also retains the variation of the strength of the magnetic field along a field line). The plasma response functions obtained include both passing and trapped ions, off‐...

The magnetoacoustic cyclotron instability is a mechanism by which waves on the perpendicular fast Alfven‐ion Bernstein branch can be excited through cyclotron resonance with an energetic ion population. It is a candidate emission mechanism for the superthermal ion cyclotron radiation, apparently associated with the products of fusion reactions, that has been observed from tokamak plasmas. In the present paper, an extended shell model is adopted for the energetic ion distribution function, fα(v)∼nα exp[−(v−v0)2/vT2]. An analytical formulation of the dispersion relation is obtained, whose numerical solution yields quantitative information on the role of vT in stabilizing wave growth at ion cyclotron harmonics. The results show that, for typical plasma parameters of interest, the degree of instability is significantly depressed, relative to its level for vT=0, once vT/v0≂0.1. Gaps appear in typical multiple cyclotron harmonic excitation patterns for 0.1≤vT/v0≤0.2, and most harmonics are stable for vT/v0≥0.25...

A full‐wave equation has been obtained from the gyrokinetic theory for the fast wave traversing a minority cyclotron resonance [Phys. Fluids B 4, 493 (1992)] with the aid of the fast wave approximation [Phys. Fluids 31, 1614 (1988)]. This theory describes the transmission, reflection, and absorption of the fast wave for arbitrary values of the parallel wave number. For oblique propagation the absorption is due to both ion cyclotron damping by minority ions and mode conversion to the ion Bernstein wave. The results for a 3He minority in a D plasma indicate that for perpendicular propagation and minority temperatures of a few keV the power lost by the fast wave is all mode converted whereas for minority temperatures ∼100 keV∼30% of the incident power is dissipated by the minority ions due to the gyrokinetic correction. The gyrokinetic correction also results in a significant reduction in the reflection coefficient for low field side incidence when kzLB≲1 and the minority and hybrid resonances overlap.

Intense superthermal ion cyclotron emission (ICE) has been observed from tokamak plasmas. The power spectrum displays narrow peaks at multiple harmonics of the background ion cyclotron frequency [Cottrell and Dendy, Phys. Rev. Lett. 60, 33 (1988)] in Ohmic deuterium plasmas, and the radiation appears to be driven by the fusion ion population in the edge plasma. Theoretical investigation of this phenomenon may be rewarding, in terms of the information about the behavior of energetic ions in tokamaks that can be extracted from ICE observations. The interpretation presented here is based on the resonant excitation of fast Alfven waves with ion Bernstein waves supported by an energetic ion species (α), in the presence of a more numerous thermal ion species (i). Because the ion cyclotron frequencies may be commensurate (lΩα=sΩi for some low integers l,s), and observations indicate that ω is comparable in magnitude to Ωi, the standard theory [Belikov and Kolesnichenko, Sov. Phys. Tech. Phys. 20, 1146 (1976)] which assumes ω≫Ωi is not immediately applicable, and is accordingly extended here to the low‐frequency regime. The results show that excitation of the fast Alfven wave at proton cyclotron harmonics can occur for fusion proton concentrations nα/ni as low as 10−7, and that multiple cyclotron harmonics can be simultaneously unstable. Furthermore, while fusion protons born at 3.0 MeV are above the energy threshold required to drive the instability, the other primary fusion products in deuterium—1.0 MeV tritons and 0.82 MeV helium‐3 nuclei—fall below it, consistent with the observation that radiation at cyclotron harmonics of the latter is not detected. However, the mechanism considered here cannot easily explain the generation of radiation which is observed at deuterium cyclotron harmonics that are degenerate with proton half‐harmonics.

The thin beam model of an axis‐encircling relativistic electron beam in a uniform, constant, external magnetic field in a cylindrical waveguide is used to develop a unified theory of the various interactions of such a beam with the electromagnetic field. The model includes the gyrotron interaction (longitudinal displacement of the beam) and the peniotron interaction (transverse displacement of the beam). Space‐charge effects are included. The resulting self‐fields influence the boundary conditions of the perturbed electromagnetic field across the electron beam. For a beam with only azimuthal equilibrium flow and for propagation perpendicular to the equilibrium external magnetic field the well‐known gyrotron instabilities are recovered. In addition, it is shown that the transverse beam modes (slow peniotron) are unstable for resonant interaction with the transverse electric (TE) waveguide mode. A general dispersion relation for propagation at any angle is derived for a beam with arbitrary energy and arbitrary pitch angle. This dispersion relation includes gyrotron, cyclotron autoresonance maser (CARM), peniotron, and harmonic autoresonant peniotron (HARP) interactions and the coupling between TE and transverse magnetic (TM) waveguide modes. For weak space charge, analytic expressions for the growth rates of the Doppler‐shifted cyclotron resonance interaction of TE and TM modes with the gyrotron and slow peniotron modes are obtained.

The gyrokinetic theory of ion cyclotron resonance is extended to include propagation at arbitrary angles to a straight equilibrium magnetic field with a linear perpendicular gradient in strength. The case of the compressional Alfven wave propagating in a D(3He) plasma is analyzed in detail. A self‐consistent local dispersion relation is obtained using a single mode description; this approach enables three‐dimensional effects to be included and permits efficient calculation of the transmission coefficient. The dependence of this quantity on the species density ratio, minority temperature, plasma density, magnetic field, and equilibrium scale length is obtained. A self‐consistent treatment of the variation of the field polarization across the resonant region is included. Families of transmission curves are given as a function of the normalized parallel wave number for parameters relevant to JET [see, for example, J. Jacquinot et al., Plasma Phys. 30, 1467 (1988)]. Perpendicular absorption by the minority io...

The inclusion of the variation of the equilibrium magnetic field across the Larmor orbits of the resonant particles is crucial for a self‐consistent treatment of cyclotron resonance in plasmas. Two contrasting nonrelativistic self‐consistent calculations [T. M. Antonsen and W. M. Manheimer, Phys. Fluids 21, 2295 (1978); C. N. Lashmore‐Davies and R. O. Dendy, Phys. Fluids B 1, 1565 (1989)] which analyze perpendicular propagation in the same nonuniform magnetic field are compared. It is shown that the first of these, which is a full wave calculation, makes an approximation that eliminates the damping found in the second, which calculates optical depth via a Wentzel–Kramers–Brillouin (WKB) approximation. A new expansion of the exact integral equation describing the problem is given, producing full wave equations which incorporate the perpendicular damping. The equations are of the correct form to ensure energy conservation and can easily be obtained to any order in an expansion in terms of the ratio of Larmo...

A simple MHD model of a plasma rotating through the periodic ripple of the toroidal magnetic field of a Tokamak is considered. It is shown that the plasma becomes unstable when the equilibrium toroidal rotation velocity exceeds the sound speed. The rippled field is equivalent to the magnetic 'wiggler' of a free electron laser. In the plasma case, long wavelength Alfven waves and short wavelength ion acoustic waves are excited.

Local extrema in the equilibrium magnetic field strength can occur in tokamak plasmas when the plasma beta is increased. Furthermore, the magnetic field strength experienced by an electromagnetic wave propagating along a vertical chord in lower‐beta tokamak plasmas may also pass through an extremum. For both these reasons, it is of interest to calculate the absorption of electron cyclotron waves in configurations where a harmonic of the cyclotron resonance lies near an extremum of the equilibrium magnetic field strength. It is shown that strong absorption can occur for both the O mode at second harmonic resonance and the X mode at third harmonic resonance, whereas absorption at these resonances is relatively weak when the gradient in magnetic field strength is linear. The optical depth in a magnetic field with parabolic scale length L is typically enhanced with respect to that in a field with linear scale length R by a factor cL/5vTR, where vT is the electron thermal velocity and c the velocity of light. ...

The conversion of electron neutrinos into either a tau or muon neutrino in a dense plasma, such as in the solar interior, is shown to be described by the Weber equation. We show that the problem is related to propagation through a potential barrier and is identical to classical mode conversion problems in inhomogeneour fluids and plasmas.

The generalized Charney-Hasegawa-Mima equation is unstable to a four wave modulational instability whereby a coherent, monochromatic drift wave can drive a band of modes and associated zonal flows unstable. Although initially the fastest growing modes dominate, a secondary nonlinear instability later drives the longest wavelength zonal flow and its associated sidebands at twice the growth rate of the fastest growing modulationally unstable modes. This results in a direct transfer from strongly unstable short wavelength modes to the weakly unstable long wavelength modes, which drains the short wavelength pump energy. A related but less efficient direct enstrophy cascade generates very short wavelength modes lying outside the band of modulationally unstable modes.

The high‐β magnetically confined plasmas in spherical tori (ST), like NSTX and MAST, provide a unique opportunity for a wide variety of applications of electron Bernstein waves (EBW). These applications range from heating of the ST plasma to modifying and controlling its current profile. Using the fully relativistic dielectric tensor for a Maxwellian distribution function, this paper presents initial results illustrating the effect of relativity on the dispersion characteristics of EBWs. It is found that, even at temperatures relevant to present STs, the relativistic dispersion properties of EBWs are significantly different from their non‐relativistic counterpart.

Gill [JET-P(93)28(1993)] has noted the existence of a population of 35 MeV electrons in JET post disruption discharges. A preliminary consideration of the effect of such a population of relativistic electrons on electron cyclotron emission is described. It is shown that a collective synchrotron instability is possible. The presence of the background plasma causes the low harmonics to be stable but a large number of high harmonics could be unstable.

An approximation to the relativistic broadening of an electron cyclotron resonance is obtained by including only the effect of the momentum parallel to the magnetic field. This approximation is then applied to the calculation of the absorption of the electromagnetic X-mode and compared to the full relativistic analysis with which it agrees very well. The propagation and damping of electron Bernstein waves is calculated using the approximate electrostatic dispersion relation and compared with the result obtained from the full electromagnetic dispersion relation.

The conversion of the O-mode to the X-mode is calculated for a sheared magnetic field. The damping of electron Bernstein waves by Doppler shifted cyclotron resonance is shown to be very strong. The energy of the Bernstein wave is totally absorbed well before the cold resonance.

Electron cyclotron absorption (ECA) of electromagnetic radiation at the second harmonic is of interest in Tokamak physics both for plasma heating and current drive, as well as for some diagnostics. Modelling of the linear wave propagation for such problems is usually done by WKB ray tracing, despite the fact that the WKB theory is not strictly valid in the neighbourhood of a resonance, where reflection and mode conversion can occur. Here, we present a full wave analysis of second harmonic ECA and compare it with various ray tracing models. The full wave equation used differs from previous ones in that it is valid even in regimes where significant absorption occurs on scale lengths comparable to the incident wavelength. The resulting analysis allows us to assess the accuracy of ray tracing in describing such problems.

We apply a fast wave approximation to the propagation of the fast wave across the magnetic field. This approximation involves replacing the hot plasma conductivity tensor by an approximation which reduces the problem to a second order differential equation. Such a procedure is particularly useful in the high harmonic regime, where many terms with the full Bessel function dependence must be retained in the dielectric tensor. However for conditions anticipated in low aspect ratio tokamaks, there are regions of the plasma in which a substantial fraction of the power is carried by the particle kinetic flux, and this must be accounted for if correct results are to be obtained.

A theory of the ICR heating is worked out and applied to the minority heating in a D‐He fusion plasma with the small Larmor radius regime. Computation can be greatly simplified by using the WKB and fast wave approximations.

The fast wave approximation, extended to include the effects of electron dissipation, is used to calculate the power mode converted to the ion hybrid (Bernstein) wave in the vicinity of the ion hybrid resonance. The power absorbed from the fast wave by ion cyclotron damping and by electron Landau and transit time damping (including cross terms) is also calculated. The fast wave equation is solved for either the Budden configuration of a cut‐off‐resonance pair or the triplet configuration of cut‐off‐resonance‐cut‐off. The fraction mode converted is compared for the triplet case and the Budden multi‐pass situation. The electron damping rate of the ion hybrid wave is obtained from the local dispersion relation and a ray tracing code is used to calculate the damping of the mode converted ion hybrid wave by the electrons as it propagates away from the resonance. Quantitative results for a range of conditions relevant to JET, TFTR and ITER are given.

When a wave passes through an electron gyroresonance, in a plasma in the presence of a magnetic field gradient, there is a small spread in the resonance due to the electron’s Larmor radius. Mathematically this is represented by the inclusion of the so called gyrokinetic term in the resonance condition, Lashmore‐Davies and Dendy. The smallness of this term, compared with other effects such as relativistic broadening, suggests that it should be negligible. However, we shall show here, by extending the method of Cairns et al., into the relativistic regime, that its inclusion is vital for producing self consistent full‐wave equations which describe electron gyroresonance. The method is considerably simpler than those used previously by Maroli et al., Petrillo et al., and Lampis et al., for obtaining similar equations. As an example we include a calculation for the O‐Mode passing perpendicularly through the fundamental.

A simulation lower‐hybrid current‐drive code is applied with an electron distribution function in which the perpendicular temperature is calculated analytically based on the extent of the plateau region. The relations are outlined here for both a non‐relativistic and a relativistic analysis. Using JET parameters, we obtain 0.9 MA of rf current using the relativistic relations as compared to 0.64 MA using the non‐relativistic relation, for an input LH power of 0.84 MW.

A generalized Poynting theorem for a system of uniform electron beams is obtained. Two examples of the two-stream instability with beams of equal density are used to discuss the relation between negative wave energy and negative potential energy, which arises in the energy principle of ideal magnetohydrodynamics. In the first example, v10>v20, while in the second example, v20=−v10, where v10,20 are the equilibrium beam velocities. Both cases can be interpreted in terms of the energy density arising from the generalized Poynting theorem. The first instability is due to the coupling of negative and positive energy waves at a frequency k(v10+v20)∕2. The second instability is due to the coupling of the same two perturbations, but at zero frequency. In this case, there is no oscillatory (wave) energy, but the beam electrons still make a negative contribution to the total energy.

The mechanism of four nonlinearly interacting drift or Rossby waves is used as the basic process underlying the turbulent evolution of both the Charney-Hasegawa-Mima-equation (CHME) and its generalized modification (GCHME). Hasegawa and Kodama’s concept of equivalent action (or quanta) is applied to the four-wave system and shown to control the distribution of energy and enstrophy between the modes. A numerical study of the GCHME is described in which the initial state contains a single finite-amplitude drift wave (the pump wave), and all the modulationally unstable modes are present at the same low level (10−6 times the pump amplitude). The simulation shows that at first the fastest-growing modulationally unstable modes dominate but reveals that at a later time, before pump depletion occurs, long- and short-wavelength modes, driven by pairs of fast-growing modes, grow at 2γmax. The numerical simulation illustrates the development of a spectrum of turbulent modes from a finite-amplitude pump wave.

In Shoucri’s Comment, he suggests that the off diagonal terms in the dielectric tensor, Σxz, σyz can be as important as the elements included. We provide an answer to this question by considering a whistler wave propagating in an infinite, uniform plasma with wave number, κ=(κ∥, 0, δz). (AIP)

Electron Bernstein waves are analyzed as possible candidates for heating spherical tokamaks. An inhomogeneous plane slab model of the plasma with a sheared magnetic field is used to calculate the linear conversion of the ordinary mode (O-mode) to the extraordinary mode (X-mode). A formula for the fraction of the incident O-mode energy which is converted to the X-mode at the O-mode cutoff is derived. This fraction is then able to propagate to the upper hybrid resonance where it is converted to the electron Bernstein mode. The damping of electron Bernstein waves at the fourth harmonic resonance, corresponding to a 60 GHz source on the Mega Amp Spherical Tokamak MAST [A. C. Darke et al., Proceedings of the 16th Symposium on Fusion Energy, Champaign-Urbana, Illinois (IEEE, Piscataway, NJ, 1995, Vol. 2, p. 1456)], is computed. For comparison, results are also presented for a lower frequency source, close to the fundamental electron cyclotron resonance. Both the fundamental and the fourth harmonic are shown to be so strongly absorbing that the electron Bernstein wave would be totally absorbed in the outer regions of the resonance. This feature implies that electron Bernstein wave current drive (on- or off-axis) could be very efficient.

We consider a hot plasma, with two ion species, in which the minority ion species has an anisotropic temperature with Tperpendicular to >T//. The conditions for the occurrence of the electromagnetic ion cyclotron instability are obtained within a uniform plasma model. Two regimes of instability are identified, the first for frequencies close to the minority ion cyclotron frequency and the second for frequencies below the majority ion cyclotron frequency. The second instability occurs for a lower value of Tperpendicular to /T// but the first instability has a much faster growth rate. We quantify how parallel propagating left circularly polarized electromagnetic waves can be driven linearly unstable, for parameter regimes which are relevant to minority ion cyclotron heating experiments in tokamak plasmas.

Several spacecraft, notably OGO 3, IMP 6, Hawkeye 1, and GEOS 1 and 2, have detected harmonically structured wave emissions in the ULF range at low magnetic latitudes between L values of 4 and 8. These waves propagate at large angles with respect to the magnetic field, and, in the case of GEOS 1, appear to be associated with ringlike proton distributions. It is demonstrated that the excitation of obliquely propagating fast Alfven waves by such distributions can account for many of the observed features of harmonically structured ULF emissions. Linear instability can occur at, or slightly below, the proton cyclotron frequency and all its harmonics. The maximum growth rates of some harmonics, notably the fundamental, occur at propagation angles which are oblique rather than strictly perpendicular. Because of magnetic field gradients, waves which propagate approximately azimuthally and within a few degrees of the perpendicular direction are likely to be amplified to the highest levels. Certain features of the ULF data, such as the finite bandwidth of the harmonics, can be explained only by allowing the wave vector to have a finite parallel component.

The authors have obtained a solution to the nonlinear MHD equations, which describes a helical magnetic field of arbitrarily large amplitude. The frequency and wave number of this field are not restricted by any particular relation. The stability properties of this large amplitude wave have been considered. This approach enables us to link together parametric instability theory with conventional MHD analysis of spatially varying magnetic fields.

The existence and instability of extended suprathermal tails in the electron velocity distribution are fundamental to RF heating and current drive schemes. The authors contrast the linear stability against emission of longitudinal plasma waves of flat tails with that of tails which have a bump. Tails with a bump support additional wave modes, whose wave-wave resonance instabilities have already been pointed out. Previous authors have ignored the role of these instabilities in the evolution of the tail distribution. The authors examine this topic, and relate it to negative-energy wave considerations. The wave-wave resonance instabilities are found to have higher growth rates than the wave-particle resonance instabilities associated with a flat tail. They occur for waves of different frequency, and in different regions of wavenumber space. The role of wave-wave resonance instabilities in the evolution of tail distributions with a bump may therefore be significant, and they offer in addition an experimental indicator of tail shape.