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The present analysis concerns a plasma confined in a magnetic field, which consists of one part due to the plasma current density and one part due to steady currents in fixed external conductors. Unrestricted free-boundary displacements are studied that lead to an unbalanced pressure force at the plasma boundary and to a corresponding change in potential energy. Thus the total change in energy is due to two factors: (i) the work done against the plasma volume forces, and (ii) the work done against the unbalanced pressure forces at the plasma boundary. As a consequence, the conventional extended energy principle has to be reconsidered. This reconsideration applies in particular to its final formulation, where unrestricted displacements are made to replace restricted ones in an expression for the potential energy change deduced from the work of the plasma volume force only. The conventional MHD analysis on perturbed plasmas thus has to be revised, by taking induced surface currents properly into account. This applies to all plasmas in inhomogeneous externally applied magnetic fields.

An extended analysis is presented of the earlier investigated induced surface current effects which arise in a magnetized plasma when the confining field has an imposed inhomogeneous part generated by currents in external conductors. The electromagnetic induction law requires a corresponding contribution from these effects to be added to the formulation of the energy principle for plasma MHD stability. For a plasma with a free boundary, conventional theory therefore holds in a strict sense only when there is a homogeneous external magnetic field. An illustration is given by two-dimensional straight Extrap geometry with a peaked current distribution, subject to a combined translational and ballooning-like displacement. In this case there are strong restoring forces, leading to stable oscillations around the plasma equilibrium position.

Anisotropic resistivity causes paramagnetic effects (Bz'(r) 1 are unattainable. Diamagnetism caused by radial particle diffusion and the Nernst effect is also discussed. In a tokamak or reactor plasma, diamagnetism is shown to be negligible, whereas it may contribute in present ultra-low q, Extrap and RFP experiments. A basic relation is derived for the coupling between the poloidal and axial magnetic field components with the above effects included. Of specific importance to the Extrap programme is the result that plasma current limitation can arise due to the lack of equilibrium when the plasma radius tends to exceed its upper limit, which is defined by a magnetic or material limiter. Comparisons are made with stability limits depending on the safety factor, the number of contained ion Larmor radii and the line density.

A solution is presented for electron plasma oscillations in a thermalized plasma, at arbitrary ratios of the Debye length AλD and the perturbation wavelength λ. The limit λD≪λ corresponds to the conventional fluid-like theory of small particle excursions, whereas λD≫λ corresponds to the free-streaming limit of strong kinetic phase mixing due to large particle excursions. A strong large-Debye-distance (LDD) effect already appears when λD ≳ λ. The initial amplitude of the fluid-like contribution to the macroscopic density perturbation then becomes small compared with the contribution from the free-streaming part. As a consequence, only a small fraction of the density perturbation remains after a limited number of kinetic damping times of the free-streaming part. The present analysis can be considered as a first exercise in an attempt to tackle the far more difficult problem of large-Larmor-radius (LLR) effects in a magnetized plasma. The analysis further shows that a representation in terms of normal modes of the form exp (— iωt) leads to amplitude factors of these modes that are related to each other and that depend on the combined free-streaming and fluid behaviour of the plasma. Consequently, these modes are coupled and cannot be treated as independent of each other.

Electrostatic perturbations are considered in a model where a plasma is situated in a homogeneous magnetic field and is subject to an equivalent gravitation force representing the magnetic inhomogeneity effects. The perturbations are strongly affected by large Larmor radius (LLR) effects when their wavelength becomes comparable to the ion Larmor radius. In the case of a dilute plasma, two time-dependent constants of he motion are found from which a solution can be deduced for the perturbations. The LLR effects are show to 'chop up' the time development of the macroscopic density and electric field perturbations into a pulse-shaped periodic patterns. The latter is repeated at every half of a gyro period and becomes strongly damped between a set of arising narrow peaks. The LR effects have been demonstrated in a simple special case, but similar results are likely to be obtained in more complicated situations, such as in denser plasmas where the electric field from the space charges affects particle dynamics. Consequently, the fine structure of a number of plasma perturbation is expected to become strongly distorted by LLR effects, and this becomes important to plasma stability.

This paper draws attention to some mechanisms which can contribute to the understanding of particle transport in Tokamaks and similar systems. First, it is demonstrated that, even when there are highly anomalous heat losses due to electron heat conduction, the particle losses do not necessarily have to become strongly anomalous but can remain nearly neoclassical. Second, plasma-neutral gas interaction is shown to have a number of important effects on plasma equilibrium, transport and profile shaping, even in Tokamaks where the boundary layer only contains a small fraction of neutral particles. Thus, under rather general conditions of strong recycling, the characteristic plasma pressure gradient in the boundary layer becomes independent of the rate of particle transport. On the other hand, the ratio between the average electron density and the neutral density at the plasma edge provides a measure of the plasma particle loss rate. Finally, in Tokamaks near the density limit, the heat losses due to plasma-neutral gas interaction affect the energy confinement time. The present analysis appears to be consistent with experiments, as far as orders of magnitude are concerned.

The equilibrium and stability of a fully ionized, dense, rapidly rotating, magnetized plasma is analysed in terms of a simple model: (i) An equivalent beta value βc is defined, being related to the centrifugal force and the corresponding azimuthal plasma current, as well as the associated radial "expansion" of the frozen-in magnetic field lines. (ii) A steady-state plasma balance exists only at beta values below a certain maximum, being either due to an equilibrium limit at βc = 2/3 or to a stability limit at βc < 2/3. (iii) The stability limit is due to the onset of a "centrifugal expansion instability" being similar to the mirror mode, but with the centrifugal force replacing the perpendicular pressure as driving mechanism. (iv) Plasma structures which are strongly elongated in the magnetic field direction depend critically on the beta limitation. Such structures exist both in cosmos and in the laboratory. (v) The theory gives a possible explanation of earlier experiments on voltage limitation with the devices FI and FIII, in terms of a velocity limitation of the former and a beta limitation of the latter.

The particle and energy balance is considered of plasmas which are permeable to neutral gas: (i) There exist two subclasses of permeable plasmas. The first includes 'dilute' plasmas being penetrated by slow neutrals of wall temperature. The second includes 'non-dilute' plasmas being penetrated only by fast neutrals having a temperature comparable to that of the ions. (ii) The energy and particle balance of dilute plasmas are strongly affected by plasma-neutral gas and plasma-wall interaction, as well as by the vacuum conditions, especially at long particle and energy containment times such as in the Spherator. (iii) Earlier experiments with the Spherator can be explained by classical theory including plasma-neutral gas interaction, and need not be interpreted in terms of anomalous diffusion. Thus, the possibilities of the Spherator as a magnetic bottle have to be reconsidered.

A one-dimensional model is being considered where a fully ionized plasma is separated from a neutral gas by a homogeneous magnetic field directed along the plasma boundary. The plasma and the neutral gas consist of two different types of ions and neutral particles. In a stationary state the outflux of plasma by diffusion across the magnetic field is compensated by an influx of neutrals which are ionized in a partially ionized boundary region. It is found that the ratio between the ion densities in the fully ionized region will in general differ from the density ratio of the two types of neutrals being present in the gas region. This provides a separation mechanism with applications both to cosmical and laboratory plasmas, such as in the following cases

It is likely that the velocity limitation observed in some types of rotating plasma devices is caused by an enhanced plasma-neutral gas interaction at the end walls bounding the plasma in the magnetic field direction. If the mechanism underlying the interaction is due to some process which forms anomalous azimuthal electric fields, a short-circuit of the latter would eliminate the velocity limitation If these predictions are true, new possibilities will open up in the use of rotating plasmas. In particular, the power losses due to ions and electrons escaping from a hot isotropic plasma through the magnetic mirror ends can then be kept small enough to balance the thermonuclear power at technically realizable beta values, plasma densities, and magnetic field strengths. At the same time such a strong confinement can only be achieved by very large centrifugal forces, rotational velocities and applied voltages, the technical feasibility of which has to be further examined.

In a hot magnetized plasma large ion excursions can take place along the magnetic field lines, thereby introducing kinetic effects which cannot be treated in terms of a macroscopic fluid model. These effects are studied for transverse wave motions of ions and electrons in a homogeneous thermal plasma which is immersed in a homogeneous magnetic field. Strong kinetic wave damping is found to occur when the product kui of the wave number k and the ion thermal velocity ui exceeds the ion gyro frequency omega i. The physical mechanism of this damping originates from the ion excursions and differs from that of Landau and cyclotron damping. For plasmas having a comparatively low beta value, such as in Tokamak experiments, the present results indicate that kinetic damping becomes important only at small wave lengths, as compared to the dimensions of the plasma body. However, in high-beta systems, such as the Z-pinch, wave lengths in a large range become affected by kinetic damping, in some cases even those which are comparable to the characteristic macroscopic dimensions. This also affects the conditions of plasma stability and plasma high-frequency heating.

The research program conducted in Stockholm is reviewed. It is characterized by a combination of the following approaches: (1) Steady-state confinement is produced by a purely poloidal main magnetic field, generated by internal ring-shaped conductors suspended by magnetically shielded supports. Investigations of open rotating plasma systems are included as a supplementary line. (2) The ion density is chosen in a range in which the plasma body becomes impermeable to neutral gas particles. (3) Stabilization against flute-type disturbances is achieved by the joint action of a number of intrinsic plasma effects, in combination with special boundary conditions and distributions of plasma and neutral gas being established in the case of impermeable steady-state plasma system. (4) Plasma heating is provided by imposed electric fields of high and low frequencies. (MOW)

The plasma-neutral gas balance is considered in the partially ionized transition region situated between a fully ionized magnetically confined plasma and an adjacent neutral gas cloud. A well-defined boundary layer can develop between the plasma and the neutral gas, but only within certain ranges of the relevant parameters such as the plasma density and the magnetic field strength. Consequently, such layers are expected to appear in the atmospheres of normal and magnetic variable stars and in other more dense cosmic clouds of matter, as well as in impermeable laboratory plasmas. On the other hand, sharply defined boundary layers are unlikely to appear in the low-density regions of interstellar and interplanetary space where the plasma becomes permeable to neutral gas and the partially ionized transition region has a minimum depth of the order of 5×1018/nb m, where nb is the corresponding characteristic plasma density.

It may become possible to create powerful neutron sources from a number of magnetic plasma confinement schemes being investigated in fusion research. Among the one- and two- energy-component plasmas discussed so far, special attention is drawn in this paper to quasi-static systems such as the Spherator with magnetically shielded supports, as well as to rotating plasma systems such as the closed Tornado bottle and open bottles. Provided that sufficiently high ion and electron temperatures can be reached, these systems may be operated at higher plasma densities and smaller dimensions than those of mirror and Tokamak neutron sources.

The classical phenomenon of electron plasma oscillations has been investigated from new aspects. The applicability of standard normal-mode analysis of plasma perturbations has been judged from comparisons with exact numerical solutions to the linearized initial-value problem. We consider both Maxwellian and non-Maxwellian velocity distributions. Emphasis is on perturbations for which αλD is of order unity, where α is the wavenumber and λD the Debye distance. The corresponding large-Debye-distance (LDD) damping is found to substantially dominate over Landau damping. This limits the applicability of normal-mode analysis of non-Maxwellian distributions. The physics of LDD damping and its close connection to large-Larmor-radius (LLR) damping is discussed. A major discovery concerns perturbations of plasmas with non-Maxwellian, bump-in-tail, velocity distribution functions f0(ω). For sufficiently large αλD (of order unity) the plasma responds by damping perturbations that are initially unstable in the Landau sense, i.e. with phase velocities initially in the interval where df0/dw > 0. It is found that the plasma responds through shifting the phase velocity above the upper velocity limit of this interval. This is shown to be due to a resonance with the drifting electrons of the bump, and explains the Penrose criterion.

Under the conditions of the quasi-steady tokamak experiments near the density limit performed so far, the plasma pressure gradient in the outer layers of the plasma body becomes determined mainly by the plasma-neutral gas balance. A combination of this balance with the results of MHD stability theory leads to a corresponding restriction in plasma density, being closely related with the empirical Murakami limit.

The problem is considered of a cloud of neutral dust moving into a cloud of static plasma which is confined in a magnetic field. Earlier experiments with rotating plasma devices and plasma guns on critical velocity limitation suggest that such limitation could also arise in the case of plasma-neutral dust interaction in cosmos.

The “Extrap” (External Ring Trap) scheme consists of a toroidal Z-pinch immersed in a transverse (poloidal) magnetic field B v which is produced by currents in a set of external ring-shaped conductors1–4. This scheme has two characteristic features. First, a purely transverse confinement field.B = B p + B v is obtained by generating a plasma current j = curlB p/µo along the zero-field region of the externally imposed transverse “vacuum” field B v. Second, the field B v is strongly inhomogeneous which leads to a non-circular plasma cross section and to a stabilizing effect on a large class of free-boundary and internal modes.

Rotating plasma operation in spiral coil “Tornado” traps provides a unique combination of confinement and heating properties. Such a system consists of a closed and compact magnetic bottle to which the crossed-field technique can be applied, in absence of end insulators and their critical velocity limitation effect. This is expected to lead to the generation and heating of fully ionized plasmas within a large range of ion densities, temperatures, and rotational velocities. Provided that stable operation becomes possible at high temperatures and the effects due to the asymmetries of the spiral coil structure can be neglected, it is likely that Tornado traps can be used as strong neutron sources of moderately large dimensions and technically realizable parameter values. Some detailed data and operation ranges are outlined for the “Tornado 6 650” device in Leningrad, and for a somewhat larger device which may be operated as a neutron source.

The potentialities and properties are discussed of a recently suggested axially symmetric, quasi-steady, closed, high-beta magnetic confinement system. A toroidal pinch of non-circular cross section is immersed in the strong, inhomogeneous polodial vacuum field from a set of ring-shaped conductors, all carrying currents in the opposite direction of the plasma current. This "EXTRAP" concept is expected to have some important advantages, as compared to magnetic bottles based on a main torodial field component. Concerning plasma equilibrium, there are closed local and average guiding centre orbits, beta values of order unity should become available, there is no magnetic surface splitting limit, cyclotron radiation should be kept at a comparatively low level, and bootstrap operation may become possible. Concerning plasma stability, the system should not become sensitive to magnetic island formation and asymmetries, and the short magnetic connection lengths as well as the strong vacuum field gradients should favour stabilization of flute-type, kink, and other magnetohydrodynamic modes. Concerning reactor technology, high power densities are expected to be reached, superconducting coils should not become necessary, the coil stresses are moderate, and repair and replacement are facilitated by the coil geometry. Before a final judgement can be made on this concept, however, extensive experiments and further theoretical analysis are needed.

The influence of plasma-neutral gas interaction on cosmical phenomena is discussed. with applications to steady and non-steady phenomena in cool interstellar and interplanetary clouds, in the photospheres, chromospheres, prominences and filaments of the Sun and other stars, in the planetary ionospheres, as well as in certain cosmogonical theories Special attention is given to filamentary structures of hot plasmas embedded in cooler regions, and to the internal properties of cool partially ionized clouds.

An experimental and theoretical analysis is performed of a fully ionized, magnetized, quasi-steady plasma being separated from a surrounding neutral gas blanket by partially ionized boundary regions: (i) For rotating as well as high-frequency heated plasmas there exists a sharply defined minimum heating power Pm and temperature T0m below which a steady fully ionized state cannot be maintained. At laboratory dimensions and ion densities of about 1021 m-3, these parameters become Pm 0.5 MW and T0m 2 × 104 K. This "minimum-P effect" is important both to the processes of creating and maintaining laboratory plasmas in fusion research and to the studies of certain cosmical plasmas. (ii) The theoretically predicted behaviour of the plasma is roughly in agreement with the experiments in the power range close to the minimum value.

Neutral gas which penetrates into a hot plasma of characteristic dimension L b, average density n̄, and temperature T, consists mainly of a slow and fast component of densities nns, nnf and temperatures T ns≪T, T nf ≃ T, respectively. These components have the penetration lengths L ns = 1/σcsn̄ and L nf = 1σcfn̄ defining the critical densities ncs = 1/σcs L b and n cf = 1σcf L b where 1/σcf ≃ 100/σcs ≃ 5×1018m-2 for hydrogen in the range 106 < T < 107 K. Thus, hot plasmas can be divided into the classes of permeable dilute, permeable non-dilute, and impermeable systems defined by n̄ ≲ n cs≪ncf,ncs≪n̄ ≲ ncf and ncs≪ncf≪n̄ respectively: (i) These classes have different equilibrium and stability properties, depending on the relative concentrations, spatial distributions, and transport processes of the neutral gas. (ii) The density of the neutral gas immersed in a quasi-steady permeable plasma becomes a measure of the plasma particle loss rate. (iii) The ratio between the neutral gas "blanket" density and the average plasma density increases when transitions are made from the permeable dilute to the permeable non-dilute state, and further to the impermeable state. (iv) The spatial distributions and relaxation times of the plasma. The pressure gradient, which acts as a driving force of instabilities, reaches a maximum value when the transition is made from a permeable to an impermeable state.

The pressure gradient dp/dr of a steady-state magnetized thermonuclear plasma column is determined by the Nernst effect, the emitted neutron flux, the ionization rate in regions containing neutral gas, externally imposed plasma sources and sinks, and by the pinch effect of an axial current. When the temperature gradient dT/dr

Experiments on rotating plasmas provide methods for studying part of the important problems of plasma equilibrium and stability at high beta values. This report describes investigations on high-density plasmas being performed by means of probe and Rogowski coil recordings: (i) Appreciable forces can be generated in the plasma, corresponding to equivalent beta values as high as about 20%. The magnetic field lines are in some cases "pushed out" radially by a considerable fraction of the distance from the axis of symmetry. (ii) The observed plasma behaviour agrees roughly with that predicted by a simple classical model. (iii) The plasma is at least macroscopically stable in respect to flute and ballooning modes, and has an outer boundary being sharply defined by the conditions imposed by a limiter and a surrounding neutral gas blanket.

During recent investigations on plasma confinement by internal ring systems with magnetically shielded supports some authors have arrived at contradictory conclusions. This paper reconsiders support shielding in terms of the problems of plasma cut-off at the support surfaces, the plasma equilibrium in presence of the support field, non-adiabatic effects on transit and trapped particles, and magnetic field misalignment in the weak-field regions.

The heat balance is considered of a rotating plasma which is permeable to neutral gas. The neutral particles which enter the plasma and fill its confinement volume are assumed to have small velocities before interacting with ions and electrons. Plasma-neutral gas interaction then gives rise to a type of frictional heating of the plasma. The conditions are analysed under which this effect becomes the dominant plasma heating mechanism. It is found that thermonuclear ion temperatures become available at rotational velocities of the order of 106m/s. The result rests on two crucial questions. First, it requires the densities of the neutral gas and the plasma as well as the geometry of the magnetic configuration and its boundary conditions to be chosen such as to avoid critical velocity limitation by plasma-neutral gas interaction. Such a possibility seems to be supported by recent experiments, but requires further and more detailed investigations. Second, an excessively large power input is required to reach these temperatures, unless the plasma and neutral gas densities are chosen to be small. Thus, the necessary power input by far exceeds the thermonuclear energy production, but the present system could still become useful as a neutron source.

Neutral gas which penetrates into a hot plasma body, of characteristic dimension Lb, average density and temperature T, consists mainly of a slow and a fast component, of densities nns, nnf and temperatures TnsT, TnfT, respectively. These components have the penetration lengths Lns = l/σcs and Lnf= 1/σcf, corresponding to the critical densities ncs=1/σcsLb and ncf=l/σcfLb where l/σcflOO/σcs5 × 1018 m−2 for hydrogen in the range 106 < T < 107 K. Thus, hot plasmas can be divided into the classes of permeable dilute, permeable non-dilute, and impermeable systems defined by ncsncf, ncsncf, and ncs ncf , respectively.Concerning tokamaks, an analysis of the plasma-neutral gas interaction leads to the following conclusions: So far, all reported tokamak experiments have been conducted in the permeable density range and have been limited by instabilities in the transition region close to = ncf. Full-scale tokamak reactors should, on the other hand, operate far inside the impermeable density range. The driving force of instabilities due to the pressure gradient is expected to reach its maximum value when approachesthe density limit ncf from below, i.e. at the transition to the impermeable state. The experimentally observed parameter values at the instability limit agree with those calculated from the theory on collisionless ballooning modes driven by the pressure gradient.

A review is given on the features of magnetic traps with a spherical separatrix, with special emphasis on Tornado spiral coil configurations. The confinement and heating of static plasmas in Tornado traps is treated, including the topology of the magnetic field structure, the magneto-mechanical properties of the magnetic coil system, as well as the particle orbits and plasma behaviour in these traps. In addition, the mode of rotating plasma operation by crossed electric and magnetic fields is described. The results of experiments on static and rotating plasmas are summarized, and conclusions are drawn about future possibilities of Tornado traps in the creation and containment of hot plasmas.

The behavior of an ionized gas in cosmical observations and laboratory experiments is examined by means of dimensionless parameters. The deviations of experimental from theoretical calculations on electric neutrality and "creep diffusion" are estimated. The coupling between plasma and neutral gas is investigated by the basic equations for a partially ionized gas, This was found to give a dispersion relation for plane disturbances of small amplitudes. The effective electrical conductivity can be deduced from the corresponding damplng distance. Agreement was obtained with the conclusions of Piddington and Cowling regarding the increased dissipation of magneto-hydrodynamic waves in the solar chromosphere and of magnetic fields in cool interstellar clouds. It was determined that magnetohydrodynamic oscillations with wave lengths comparable to the thickness of the ionospheric F/sub 1/ region and with periods comparable to those of "giant pulsations" are damped moderately only if the ionization degree is about a hundred times greater inside than outside of the auroral zones. As an experimental illustration of the theory, torsional oscillations in a cylindrical "magnetohydrodynamic wave guide" are discussed. It was found that Alfvon waves will exist In an ionized gas in the laboratory only in a high-current discharge of relatively large dimensions. The dlffusion processesmore » which take place in a plasma column situated in a longitudinal magnetic field are discussed, Experimental data on the electric potential drop were found to agree well with the collision diffusion theory in a range of magnetic fields up to a centain critical value. Above this value a marked change occurs which can be interpreted as a sudden and rapid increase in the diffusion coefficient with a simultaneous rise in the noise level, These results support the "drain" diffusion mechanism proposed by Bohm, Burhop, Massey, and Williams-particle dlffusion across the magnetic field is accelerated by the action of transverse electric fields.« less

A review is given of the experimental evidence and evaluation of macro- and micro-instabilities in a gaseous plasma. The paper briefly describes the physical mechanisms of the instabilities and the experimental methods used in their detection. The present state of research is summarized and discussed in connexion with thermonuclear and cosmic problems, with special emphasis on recently obtained results. An extensive list of references on the experimental work is included.

The Second European Conference on Controlled Fusion and Plasma Physics was held at the Royal Institute of Technology in Stockholm, Sweden, from the 14th to the 18th of August 1967. A total number of 249 participants from 21 countries attended. The present report summarizes the conference proceedings and includes the abstracts of papers presented.

The motion and equilibrium is investigated of a plasma confined by an arbitrary magnetic field where the lines of force are either closed inside the confinement volume or form a bottle between two magnetic mirrors, (a) Stationary states are treated for which macroscopic fluid motions are allowed to take place in the plasma. Such states represent a larger class of equilibria than the static ones where the plasma is assumed to be at rest all over space, (b) For a hot plasma confined i n a closed magnetic bottl e the fluid motions decay at essentially the same rate as the pressure gradients and the electric currents, (c) A modified deduction is presented of the average guiding centre drifts from field line to field line in the αβ space defined by the magnetic field. The drifts include contributions from the electric field and the magnetic gradient, (d) The contribution to the macroscopic particle flux from the gyro motion is averaged along the field lines to form a corresponding flux in αβ space. An average macroscopic fluid velocity is further obtained by superimposing the contributions from the guiding centre drifts and the gyro motion, (e) The mechanisms producing density changes and charge separation phenomena are treated in terms of the average macroscopic fluid velocity. They are found to be due to the average guiding centre drifts only, (f) An analysis is performed in terms of the velocities in αβ space of equilibria for which local fluid velocities are allowed to exist in the plasma. For a particular class of such states an electric field can be set up which, on the average, keeps the plasma at rest on each field line, (g ) The results indicate that plasma confinement should becom e possible in ring-current configurations with magnetically screened supports, provided that the plasma remains stable and that there are no excessive losses in the small weak field regions close to the leads where this analysis is inapplicable.

The partially ionized plasma centrifuge contains a neutral gas mixture of isotopes or elements, the motion of which is driven by a small amount of rotating plasma. The plasma motion is, in its turn, controlled by externally imposed electric and magnetic fields. In this report earlier investigations of the centrifuge are extended by an analysis of its operation conditions. It is shown that high rotational velocities, large separation degrees, and stable velocity profiles can be achieved at comparatively small power losses, within certain ranges of the parameters involved.

In magneto-hydrodynamics conservation of energy leads to the adiabatic relation, du = pd?/? 2 , between the internal energy u per unit mass, the pressure p , and the density ?. The relation is valid for arbitrary amplitudes in a non-dissipative medium. The Hall current modifies the properties of magneto-hydrodynamic waves in an ionized gas. The compressive modes and the Alfven mode earlier discussed by van de Hulst are shown to be coupled by means of the Hall current. The dissipation of slightly damped waves is expressed in terms of the effective conductivity s? 3 = s? 1 + s? 3 2 /s? 1 earlier introduced by Cowling. The theory is applied to magneto-hydrodynamic oscillations in the layers of the ionosphere. The possible periods of the waves are consistent with the periods of “giant pulsations” which have been observed as a type of perturbation in the earth's magnetic field in the auroral zone. The results indicate that magneto-hydrodynamic waves in the E -, F 1 -, and F 2 -layers may explain the existence of giant pulsations as well as the occurrence of rapid vibrations in the terrestrial magnetic field. DOI: 10.1111/j.2153-3490.1956.tb01217.x

The balance of forces is investigated for a rotating plasma contained in a long cylindrical vessel and placed in an axial magnetic field. A relation is deduced between the voltage which arises in a radial direction in the plasma and the electric charge being transmitted through the system from an external source. It is found that the equivalent capacity, which describes the electrical behaviour of the plasma, depends upon the transmitted charge and upon the constitution of the vessel wall. An explanation is suggested for the voltage limiting effect earlier observed in the Ixion device.

Previous investigations on the confinement of a plasma in the magnetic field from one or several ring-shaped magnetic coils are extended in this paper. The energy losses are analysed for a number of magnetically screened leads which support the coils situated inside the plasma volume. Under thermonuclear conditions it is found that the losses due to the leads become negligible compared to the thermonuclear power of the DD and DT reactions when the main magnetic field is purely poloidal. In a sheared field the same losses are larger, but still negligible for the DT reaction, and it should in any case be possible to make them less than the thermonuclear power of the DD reaction. In connexion with the present type of configuration some further comments are made on the problems of stability, plasma generation and heating, the experimental analysis of the confinement, and on some of the engineering problems.

Under certain conditions, plasma may be contained by a transversal magnetic field. If the plasma pressure is low with respect to the energy density of the magnetic field, the field is the same in the presence of plasma and in vacuum. A transversal motion of the plasma is theoretically impossible but other types of motion, not accompanied by a change of the magnetic field, may ultimately result in the escape of the plasma. Such cases, involving flute and screw instabilities were more closely examined; mechanisms involved in the formation of spatial charges, such as the gravitation-induced drift, centrifugal forces, inertia forces generated by magnetic pinch, and the effect of the Larmor radius. The results of the calculations indicated a strong interaction between the mechanisms corresponding to the flute and screw instabilities; these mechanisms are able to create instabilities in both partially and completely ionized gases. The theory developed is based on the equations of the macroscopic motion of the gas, as defined by the Boltzmann equation. (TTT)

The accessible voltage levels have been determined for a rotating plasma device in which the confinement region surrounds a ring-shaped coil that is suspended by a single rod. (i) There exists a sharply defined critical voltage and a corresponding velocity limit which cannot be exceeded by a moderately large power input. This is clearly shown with the present apparatus where the plasma voltage slowly approaches the critical level from below and where the insulator consists only of a narrow rod. (ii) The critical voltage increases with the mean radial ratio r 0/r w in the plasma. This ratio is defined by a magnetic field line which intersects the midplane of the device at a radial distance r 0 from the axis of symmetry, and cuts the end insulator at the corresponding distance r w. (iii) The results can be explained by an interaction between the plasma and a neutral gas layer at the insulator surface that limits the velocity to the critical value v c earlier introduced by Alfvén. As a consequence, the velocity limit becomes (r 0/r w)v c in the equatorial plane. Deductions of the corresponding voltage limit agree with the experiments, (iv) The plasma is heavily disturbed even by a very small obstacle which is introduced in the equatorial plane. This indicates that the losses of the system are small under normal operation, (v) So far the stability, is not influenced by the fact that most of the field lines pass freely around the confinement region and do not end upon an insulator surface.

Experimental studies have been performed of a fully ionized magnetized plasma generated in a poloidal magnetic field by a transverse electric field applied between two electrodes and being surrounded by a neutral gas blanket: (i)The rotating plasma which has been generated in the crossed fields is released from its energy source and its rotation is being stopped. The plasma decay is then studied in the absence of macroscopic motions, with the exception of those from a slow diffusion process. The state of ionization is determined by suddenly giving the plasma a small pulse of angular momentum which, after a short time is recovered as a current pulse being extracted by means of a short-circuit between the electodes. The pulse becomes a measure of the state of ionization and shows that the rate of penetration of neutral gas into the plasma is strongly delayed at high plasma densities, in agreement with theory. (ii)The velocity field of a quasi-stationary, rotating plasma has been investigated spectroscopically by observing the Doppler shifts of various spectral lines. The results agree with theoretical perdictions of the velocity and temperature distributions within the plasma region. The results also support the prediction that the interior of the plasma should be screened from the surrounding neutral gas.

Under laboratory conditions particle losses from a magnetized plasma will produce a neutral particle flux reentering from the surrounding vessel walls into the plasma region. Even at very high temperatures, where the plasma is almost fully ionized, this gives rise to interactions between charged and neutral particles which cannot always be neglected. Three types of interactions are discussed: ionization, charge exchange, and elastic collisions with neutrals. The conservation laws of mass, charge, momentum, and energy are reconsidered for a two-fluid system with ionization, charge exchange, and elastic scattering of neutral particles taken into account. It is found not necessary to derive the energy equation in terms of the joint action of electromagnetic and mechanical changes of state. It has earlier been concluded from the motion of a single particle that a very strong confinement can be obtained in a rotating plasma situated in the magnetic field of a current loop (Bonnevier and Lehneat). These results are verified in a macroscopic theory including the energy balance of stationary motion. It is found that the main part of the heating of the plasma and the scattering of ions are not provided by coulomb collisions, but by collisions with reentering neutral particles. Twomore » special cases are treated in detail, one with vanishing heat conductivity and one with infinite heat conductivity in the direction along the magnetic field. For the latter, which is more likely to occur in a high temperature plasma, the energy loss by escaping particles is deduced. The theory predicts that this loss should be several orders of magnitude less in the current loop configuration than in earlier described homopolar machines. Finally, some comments are given on the nonexistence of interchange instabilities in systems with volume currents.« less

The axially symmetric equilibrium of a magnetically confined plasma is reconsidered, with the special purpose of studying high-beta schemes with a purely poloidal magnetic field. A number of special solutions of the pressure and magnetic flux functions are shown to exist.

A new scheme for steady-state magnetic plasma confinement is suggested, consisting of a superposition of two poloidal magnetic field components. The first arises from the unidirectional currents flowing in a number of ring-shaped external conductors. The second is produced by an oppositely directed azimuthal current in the plasma. Whenever being necessary, an additional toroidal magnetic field component can be introduced. The scheme differs from those having passive guard conductors surrounding the plasma, in the sense that given currents are fed into the rings from external sources. This scheme provides a closed axisymmetric magnetic bottle with closed guiding-centre drift orbits of the plasma particles, as well as possibilities of "bootstrap" operation. The problems are considered of stabilizing electrostatic modes by means of minimum-average-Bgeometry, and of stabilizing electromagnetic modes by means of the mutual forces between the currents in the plasma and those in the external conductors.

Flute-type disturbances are investigated by normal mode analysis in the case of a low-beta plasma being confined in the field from a straight line current. A dispersion relation and stability criterion are derived including the joint effects of the plasma pressure, the finite ion Larmor radius and frequency, and of an imposed radial gravitation field. The results, which may also illustrate the general plasma behaviour in more complicated field geometries, indicate that the stability domain is enlarged substantially by finite Larmor radius effects, especially for small-scale disturbances. Also the centrifugal force in rotating plasmas should have a stabilizing effect under certain conditions. In combination with the stabilizing mechanisms in the boundary regions of impermeable plasmas, the present results provide an alternative to the method of minimum-average-B stabilization.

The confinement and energy balance of rotating plasmas are studied by means of a device whose configurations may be changed to resemble the homopolar, Ixion, and current loop types of plasma generators. The effects of particle losses along the magnetic field lines to a nonconducting wall are examined. For the three configurations studied, the voltage and current characteristics, light intensities, gas production, and wall deterioration effects are determined using H and N gas. The plasma contains a considerable amount of energy for as long as 0.7 msec. (T.F.H.)

Electric current is sent through a hollow jet of mercury which is flowing over the surface of a glass tube. The jet is broken up by the action of the electrodynamic force. However, if at the same time a strong current is sent through a central conductor placed inside the glass tube, the jet can be stabilized. This occurs when the currents are parallel. Sausage instabilities have not been observed in the present experiment.

A model has earlier been proposed for a boundary layer which separates a cloud of matter from one of antimatter in a magnetized ambiplasma. In this model steady pressure equilibrium ceases to exist when a certain beta limit is exceeded. The latter is defined as the ratio between the ambiplasma and magnetic field pressures which balance each other in the boundary layer. Thus, at an increasing density, the high-energy particles created by annihilation within the layer are ‘pumped up’ to a pressure which cannot be balanced by a given magnetic field. The boundary layer then ‘disrupts’. The critical beta limit thus obtained falls within the observed parameter ranges of galaxies and other large cosmical objects. Provided that the considered matter-antimatter balance holds true, this limit is thus expected to impose certain existence conditions on matter-antimatter boundary layers. Such a limitation may apply to certain cosmical objects and cosmological models. The maximum time-scale for the corresponding disruption development has been estimated to be in a range from about 10−4 to 102s for boundary layers at ambiplasma particle densities in the range from 104 to 10−2 m−3, respectively.

In a perfect magnetic bottle, both the field lines and the drift orbits of charged particles have to be closed within the intended confinement region. The field from a circular current loop fulfills these requirements, also in presence of a toroidal field component1,2. To make practical use of this field geometry, it has been proposed to suspend a circular internal coil by magnetically shielded supports, in a way to prevent the confined plasma from reaching the supports and surrounding wall surfaces1.

The possibility is being discussed of creating and heating the plasma in "Tornado" magnetic traps by means of the rotating plasma technique. On account of the particular field geometry of these traps, it is likely that the critical velocity phenomenon of rotating plasmas can be avoided.