D. Pfirsch

On the Relation Between Mixing Length and Direct Interaction Approximation Theories of Turbulence

A systematic procedure is outlined for deriving a spectral equation from the renormalized theory of turbulence based on the so‐called direct interaction approximation. This spectral equation should be valid for a large class of plasma and fluid systems.

New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: basic theory

In this paper we describe a relatively simple and transparent method of obtaining collisionless drift-kinetic, gyrokinetic and more general theories, including local charge, energy and momentum conservation laws. An important feature of the new formalism is, contrary to present-day theories, the exact gauge invariance, thus avoiding certain inconsistencies. The present paper starts with the introduction and proof of the correctness of a Lagrangian for combined Maxwell-kinetic theories in general coordinates as concerns the particle motion. The kinetic part of it is formulated in Eulerian form by means of the equations of motion in the form of Hamilton–Jacobis equation, used only as a tool, and Diracs constraint theory. Charge and current densities automatically distinguish between ‘particle-like’ (guiding-centre), polarization and magnetization contributions. This formalism is applied to averaging coordinates derived by a method similar to Kruskals. Certain properties of the averaging coordinates, according to the basic requirements imposed on them, can be used to obtain a gyroangle independent Lagrangian, from which one can obtain a Lagrangian for the combined Maxwell-kinetic theories in a reduced phase space that is applicable to situations in which one is not interested in the dependence on some kind of gyroangle describing the gyromotion, whose treatment can, however, easily be added. The basic perturbation theory, which aims at obtaining averaging phase-space coordinates, is done solely within the Kruskal formalism in which only the electric and magnetic fields appear, but not the corresponding potentials. This formalism provides, in particular, information about the allowed amplitudes of fluctuations. The results are later used to obtain approximate expressions for the Lagrange functions in the drift-kinetic and the gyrokinetic ordering. For the definition of certain approximations to the exact Lagrangian the central principle is the exact gauge invariance. The terms of the zeroth and first orders needed for such an approximation are given. For the drift-kinetic ordering Littlejohns Lagrangian is readily rederived and hence the drift-kinetic theory as obtained and investigated by the present authors in some previous work. Conservation laws and their mathematical structures corresponding to the gyroangle-independent Lagrangian are obtained in a following paper. There, in particular, a detailed description of how to obtain variations of gyroangle averaged quantities in the reduced phase space is given, and it is explained how these variations are used in a modified form of Noethers theory which observes exact gauge invariance.

Conditions for the Existence of Shock‐Like Solutions of Korteweg‐De Vries Equation with Dissipation

Necessary conditions on dissipative operators for the existence of shock‐like solutions of the Korteweg‐de Vries equation, which limit the damping tolerable at long wavelengths are derived.

New method of deriving local energy and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: conservation laws and their structures

This paper gives a first application of the reduced-phase-space Lagrangian for kinetic theories obtained in a sister paper in this issue by means of Kruskals averaging coordinates. The approximations made within Kiuskals formalism are of first order in the smallness paranieter ∈, given by the ratio of the gyroperiod to the macroscopic time scale, which is the same as the ratio of the gyroradius to the macroscopic scale length in the drift-kinetic case, or as the ratio of the amplitudes of the fluctuations to the background fields in the gyrokinetic case. This paper presents methods and results concerning local conservation laws for the density of gyrocentres and the charge, energy, momentum and angular momentum. A very important feature of our treatment is that throughout the theory is gauge invariant. The methods consist of a modified Noether formalism with gauge-invariant shift variations which in a very straightforward way lead to, in particular, the symmetric energy momentum tensor instead of the canonical tensor. The shift variations are defined both within the reduced phase space, which does not contain the gyroangle, and also for gyroangle-dependent quantities which subsequently have to be averaged. A clear definition of the Lagrange density needed for the derivation of the local conservation laws for energy, momentum and angular momentum is given. The discovery of combinations of terms such as the polarization and the magnetization allows the conservation laws to be cast in a very clear form affording insight into their structure.

On the economic prospects of nuclear fusion with Tokamaks

A method to estimate the cost and construction energy of tokamak fusion power stations based on the present, early stage of fusion development is described. The method is based on first-wall heat load constraints rather than beta limitations, which could eventually be the more critical of the two. The economic efficiency of pure fusion is discussed, with particular reference to a European study. It is shown that the claims made therein for the economic prospects of pure fusion with tokamaks, when discussed on the basis of present-day technology do not stand up to critical examination. A fusion-fission hybrid, however, could afford more positive prospects.

Nonperiodicity in space of the magnetic moment series

The first term in the magnetic moment adiabatic invariant series which is explicitly not periodic in space is calculated. The implications of the existence of such a term are discussed.

Collisional drift fluid equations and implications for drift waves

The usual theoretical description of drift-wave turbulence (considered to be one possible cause of anomalous transport in a plasma), e.g. the Hasegawa - Wakatani theory, makes use of various approximations, the effects of which are extremely difficult to assess. This concerns in particular the conservation laws for energy and momentum. The latter law is important in relation to charge separation and the resulting electric fields, which are possibly related to the L - H transition. Energy conservation is crucial to the stability behaviour; it will be discussed by means of an example. New collisional multi-species drift-fluid equations were derived by a new method which yields, in a transparent way, conservation of energy and total angular momentum and the law for energy dissipation. Both electrostatic and electromagnetic field variations are considered. The only restriction involved is the validity of the drift approximation; in particular, there are no assumptions restricting the geometry of the system. The method is based primarily on a Lagrangian for dissipationless fluids in the drift approximation with isotropic pressures. The dissipative terms are introduced by adding corresponding terms to the ideal equations of motion and of the pressures. The equations of motion, of course, no longer result from a Lagrangian via Hamiltons principle. However, their relation to the ideal equations also implies a relation to the ideal Lagrangian, which can be used to advantage. Instead of introducing heat conduction one can also assume isothermal behaviour, e.g. = constant. Assumptions of this kind are often made in the literature. The new method of introducing dissipation is not restricted to the present kind of theory; it can equally well be applied to theories such as multi-fluid theories without using the drift approximation of the present paper. Linear instability is investigated by means of energy considerations and the implications of taking ohmic resistivity into account are discussed. A feature of the results is that for purely electrostatic perturbations the second spatial derivative of the density profile plays a role, in contrast to the usual approximations. For a class of systems with , it is shown that linear instability can only occur when the resistivity is sufficiently large, while the Hasegawa - Wakatani theory predicts instability for arbitrarily small non-vanishing resistivity. It is shown that, for essentially electrostatic instabilities, magnetic perturbations in resistive systems may not be negligible even for . An example which will be treated in a future paper indicates in addition that, in systems with vanishing ion temperature, electron temperature profiles should strongly influence the stability via resistive effects. This is in addition to effects leading to -modes. It also demonstrates that, in general, it is not possible to perform an expansion with respect to the resistivity near . The new formalism is interesting not only from a theoretical point of view, but also, in particular, as a useful tool for numerical calculations.

Poisson brackets for guiding-centre and gyrocentre theories

Poisson brackets in the phase space of averaging Kruskal coordinates are obtained in a clear and straightforward way. The derivation makes use of the equations of motion of guiding centres and gyrocentres derived from a gyroangle-independent Lagrangian, and from generally valid relations of Hamiltonian mechanics. The usual procedure of matrix inversion to obtain the Poisson tensor from the Lagrange tensor is not required.

Lagrangians for plasmas in the drift-fluid approximation

For drift waves and related instabilities, conservation laws can play a crucial role. In an ideal theory these conservation laws are guaranteed when a Lagrangian can be found from which the equations for the various quantities result by Hamiltons principle. Such a Lagrangian for plasmas in the drift-fluid approximation was obtained by a heuristic method in a recent paper by Pfirsch and Correa-Restrepo. In the present paper the same Lagrangian is derived from the exact multifluid Lagrangian via an iterative approximation procedure which resembles the standard method usually applied to the equations of motion. That method, however, does not guarantee that all the conservation laws hold.

Small scale magnetic flux‐averaged magnetohydrodynamics

By relaxing exact magnetic flux conservation below a scale λ a system of flux‐averaged magnetohydrodynamic equations are derived from Hamilton’s principle with modified constraints. An energy principle can be derived from the linearized averaged system because the total system energy is conserved. This energy principle is employed to treat the resistive tearing instability and the exact growth rate is recovered when λ is identified with the resistive skin depth. A necessary and sufficient stability criteria of the tearing instability with line tying at the ends for solar coronal loops is also obtained. The method is extended to both spatial and temporal averaging in Hamilton’s principle. The resulting system of equations not only allows flux reconnection but introduces irreversibility for appropriate choice of the averaging function. Except for boundary contributions which are modified by the time averaging process total energy and momentum are conserved over times much longer than the averaging time τ bu...