Dario Correa-Restrepo

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.

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.

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.

Localized resistive instabilities in general toroidal configurations, with applications to INTOR equilibria

A useful method for studying localized resistive instabilities in toroidal equilibria at and near the ideal stability limit is presented. In particular, a valuable relation is given explicitly which allows direct calculation of the resistive growth rate at the ideal ballooning limit in terms of line-averaged equilibrium quantities. The method, which is valid not only for axisymmetric equilibria but also for toroidal, nonsymmetric 3-D configurations, is derived without making any of the usual restricting assumptions such as small inverse aspect ratio, low pressure etc. The actual evaluation of stability criteria and dispersion relations is illustrated by studying the stability of typical INTOR equilibria with respect to localized ideal and resistive modes. Configurations which are stable with respect to ideal ballooning modes (and are therefore also Mercier stable) are shown to be unstable with respect to resistive ballooning instabilities, and the corresponding growth rates are calculated. Considerable growth rates ( gamma -1 approximately 1 ms) are found at values of ( beta ) that are 80-90% of the critical value. Below these values of ( beta ), the growth rates decrease very rapidly. A subject of more general interest is also treated, namely the situation which appears when the Mercier exponent sM exceeds the value 1/2, and a new dispersion relation for a model equation is derived.

The electromagnetic gauge in the variational formulation of kinetic and other theories

This paper investigates the implications and consequences of choosing special gauges or gauge-invariant or non-gauge-invariant approximations in the action integral for the variational formulation of theories involving electromagnetic fields. After some interesting special gauges are considered, it is shown that non-gauge-invariant approximations always lead to inconsistent Euler-Lagrange equations. As a concrete example. Maxwell Vlasov theory is investigated. The special non-gauge-invariant case considered, which is sometimes used in the literature. is obtained by replacing the contribution of the electric field to the Maxwell part of the Lagrangian density by the contribution of tlie gradient of the scalar potential alone. The detailed investigation concerns the local energy conservation law, and it is shown that the law thus derived is incorrect since it contains non-physical, spurious terms. An improvement of this situation can lie obtained by the introduction of a Lagrange multiplier in tile non-gauge-invariant theories in order to avoid inconsistencies in the local charge conservation law. The results are also valid for drift-kinetic and gyrokinetic theories, and for other theories, e.g. two-fluid theories.

Noether formalism with gauge-invariant variations

The energy-momentum tensor is usually obtained by symmetrizing the canonical tensor, which is sometimes difficult. The fact that the energy-momentum tensor must always be gauge invariant, and insight gained by investigating the Maxwell kinetic equations, lead one to conclude that this tensor should be readily obtainable for a certain class of Lagrange densities. The appropriate method to achieve this consists in performing gauge-invariant variations within the framework of Noethers theory. These variations are equivalent to shifts in space and time. and are considered here instead of the usual shift variations. which are not gauge Invariant. These features indicate the existence of a certain underlying structure which is made evident by the proposed method. The Lagrange densities of the class in question are characterized by having a gauge-invariant contribution additional to the Maxwell part. For vanishing potentials. such Lagrange densities lead directly to a symmetric canonical tensor; this implies that there are only scalar fields. Maxwells equations, being the basic part of such theories, are treated first. Then the coupled system of electromagnetic fields and the usual scalar fields describing charged matter is considered. Finally, a different kind of scalar field which occurs in a structural non-standard form in collisionless phase-space theories (Vlasov or collisionless Boltzmann-like equations) is treated.

Resistive ballooning modes near the edge of toroidal configurations

The resistive ballooning mode equations are cast in a new form appropriate for evaluation near the plasma edge of toroidal (axisymmetric as well as three-dimensional) configurations, where the resistive ballooning effects outweigh the diamagnetic effects. Explicit evaluation is carried out for cylindrically symmetric plasmas and for a tokamak model with circular cross sections. Owing to the large electric resistivity of the regions considered, resistive ballooning modes with growth rates comparable to the characteristic growth rate of ideal ballooning modes are possible. A general feature is that modes with large growth rates are localized around the regions of bad curvature and become less unstable with increasing shear, while those with smaller growth rates are extended along the magnetic field lines and are insensitive to shear.

Localized resistive modes in the circular tokamak

Localized resistive modes (resistive interchange and resistive ballooning modes) are investigated for self-consistent Tokamak equilibria which have circular cross-sections near the magnetic axis. The second region of stability (which is stable with respect to ideal ballooning modes at high poloidal beta beta p) proves to be unstable when resistive effects are taken into account as well. In the low beta p stability region, the stability boundaries are shifted toward lower beta p values. The magnitude of the shift depends on the degree of localization of the perturbation around the field line, highly localized perturbations being the most unstable ones.