Equilibrium configurations of a toroidal plasma

Laing, E. W.; Roberts, S.J.; Whipple, R.T.P.

doi:10.1088/0368-3281/1/1/308

Cited by 30 publications

(19 citation statements)

References 2 publications

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“…(2.51) In fact the safety factor is as large as 4 at the maximum radius in present toroidal devices. (Grad and Rubin, 1958;Laing, et al , 1959;Shafranov, 1958) …”

Section: B Synopsismentioning

confidence: 99%

Theory of plasma transport in toroidal confinement systems

1976

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The dissipation induced by coulomb-collisional scattering provides an irreducible minimum, and thus a useful standard for comparison, for transport processes in a hot, magnetically confined plasma. The kinetic description of this dissipation is provided by an equation of the Fokker -Planck form. As in the standard transport theory for a neutral gas, approximate solution of the Fokker -Planck equation permits the calculation of transport coefficients, which linearly relate the fluxes of particles, energy, and electric charge, to the density and temperature gradients, and to the electric field. The transport relations are useful in studying the confinement properties of present and future experimental devices for research in controlled thermonuclear fusion. The transport theory for a magnetized plasma (in which the Larmor radius is much smaller than gradient scale lengths describing the plasma fluid) departs from the theory for a neutral gas in several fundamental ways. Thus, transport coefficients for a magnetized plasma can be calculated even when the collisional mean free path is much longer than the gradient scale length (as would pertain in thermonuclear regimes). Such transport coefficients are generally nonlocal, being defined in terms of averages over surfaces with macroscopic dimensions. Furthermore, when the mean free path is long, the magnetized-plasma transport coefficients depend crucially upon the magnetic field geometry, the effects of which must be treated at the kinetic level of the Fokker -Planck equation. The results display several novel couplings between collisional dissipation and the electromagnetic field. The present review of magnetizedplasma transport theory is intended to be as widely accessible as possible. Thus the relevant features of magnetic confinement in closed (toroidal) systems, and of charged particles in spatially varying fields, are derived, at least in outline, from first principles. Although consideration is given to "classical" transport in which most field geometric effects are omitted, major emphasis is placed on the "neoclassical" theory which has been developed over the last decade, Neoclassical transport coefficients are specifically relevant to a magnetically confined plasma, rather than to just a magnetized plasma; their unusual features, such as nonlocality and geometry dependence, become particularly important in the high temperature regime of proposed thermonuclear reactors, The area of neoclassical theory which seems most complete -its application to axisymmetric tokamak-type confinement systems -is correspondingly stressed. CONTENTS

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“…(2.51) In fact the safety factor is as large as 4 at the maximum radius in present toroidal devices. (Grad and Rubin, 1958;Laing, et al , 1959;Shafranov, 1958) …”

Section: B Synopsismentioning

confidence: 99%

Theory of plasma transport in toroidal confinement systems

1976

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“…Because of the intrinsic nonlinearity of the EGSS equation and strong coupling to the Bernoulli law (which is also nonlinear), approximate or numerical techniques need to be invoked to solve the governing equations except for some special examples of static equilibrium or unidirectional (in the toroidal direction) flow (Laing, Roberts & Whipple 1959;Infeld 1971;Maschke & Perrin 1980;Agim & Tataronis 1985). We adopt the moment method of Lao, Hirshmann & Wieland (1981) generalized to the case of flow equilibrium by Zelazny et al (1991).…”

Section: Methodsmentioning

confidence: 99%

Toroidal equilibrium of rotating plasma with adiabatic constraints

Gałkowski

Zelazny

1994

J. Plasma Phys.

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A numerical technique, alternative to Grad's well-known ADM method has been proposed to deal with the slow adiabatic evolution of a toroidal plasma with flow. The equilibrium problem with prescribed adiabatic constraints may be solved by simultaneous calculations of flux surface geometry and original profile functions. Implications for the problem of bifurcation due to nonlinearity of the governing equations are discussed. In the case of field-aligned sub-Alfvénic flow the system is in the second elliptic regime if β <A2/(1 – A2) at the magnetic axis, where A is the Mach Alfvén number of the flow. Super-Alfvénic flows do not satisfy the local firehose stability criterion.

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“…In Ref. [3] the form assumed for the source functions in flux space is such that the terms they contribute to the Grad-Shafranov equation are proportional to the flux function. A perturbative solution is obtained by developing an expansion of the poloidal flux function in powers of the reciprocal of the aspect ratio ( ) of the torus about the corresponding cylindrical configuration (that is, the one with vanishing ), which satisfies the requirement that the normal component of the magnetic field vanishes at the wall of the plasma container, supposed to be made of a perfectly conducting material.…”

Section: Introductionmentioning

confidence: 99%

Solution to the Grad-Shafranov Boundary Value Problem for a Thermonuclear Plasma contained in a Toroidal Vase With a Conducting Wall under the Assumption of Constant Sources to the Equilibrium in Flux Space

Ferreira

2018

Preprint

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A method is proposed to solve the Grad-Shafranov partial differential equation for the poloidal flux function associated with the equilibrium of a plasma magnetically confined in an axisymmetric torus under the assumption that the sources to the equilibrium (the gradient of the plasma pressure and the gradient of half the squared toroidal field function in flux space) are constant, and subjected to the condition of constant value of the poloidal flux at the toroidal boundary. The solution to the equation is written as the sum of the particular solution and a combination of N multipole solutions of the lowest orders in the variables of the toroidal-polar coordinate system having the pole at the centre of the torus cross section. The term of the multipole solution of the zeroth order does not bring about any magnetic field and its inclusion in the combination fulfills the purpose of providing it with a constant term whose value can be adjusted so as to make equal to zero the value taken by the poloidal flux at the boundary; the term of the multipole solution of the first order is associated with a (constant) vertical field, such as required for the attainment of the balance of forces in a confinement system of toroidal geometry, and has the effect of introducing a shift (the Shafranov shift) to the line of null magnetic poloidal field inside the plasma with regard to the centre line of the torus; the terms containing the multipole solutions of higher orders tailor the contours of the flux surfaces, modifying the shape they would take from the sole presence of the particular solution. Imposition of the boundary condition on the assumed solution leads to a

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Equilibrium configurations of a toroidal plasma

Cited by 30 publications

References 2 publications

Theory of plasma transport in toroidal confinement systems

Theory of plasma transport in toroidal confinement systems

Toroidal equilibrium of rotating plasma with adiabatic constraints

Solution to the Grad-Shafranov Boundary Value Problem for a Thermonuclear Plasma contained in a Toroidal Vase With a Conducting Wall under the Assumption of Constant Sources to the Equilibrium in Flux Space

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