On resistive magnetohydrodynamic equilibria of an axisymmetric toroidal plasma with flow

Abstract

It is shown that the magnetohydrodynamic equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field are governed by a second-order partial differential equation for the poloidal magnetic flux function ψ coupled with a Bernoulli type equation for the plasma density (which are identical in form to the corresponding ideal MHD equilibrium equations) along with the relation ∆ ⋆ ψ = V c σ. (Here, ∆ ⋆ is the Grad-Schlüter-Shafranov operator, σ is the conductivity and V c is the constant toroidal-loop voltage divided by 2π). In particular, for incompressible flows the above mentioned partial differential equation becomes elliptic and decouples from the Bernoulli equation [H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas 5, 2378 (1998)]. For a conductivity of the form σ = σ(R, ψ) (R is the distance from the axis of symmetry) several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible σ profiles, i.e. profiles with σ taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For σ = σ(ψ) consideration of the relation ∆ ⋆ ψ = V c σ(ψ) in the vicinity of the magnetic axis leads therein to a proof of the non-existence of either compressible or incompressible equilibria. This result can be extended to the more general case of non-parallel flows lying within the magnetic surfaces.

Energetically resolved multiple-fluid equilibria of tokamak plasmas

“…This is equivalent to equation (28) in the single fluid limit under the replacement B 2 = |∇ | 2 + (B φ R) 2 using equation 9, substitution for v using equation (10) and v φ using equation (12). Correcting for transcription errors in Guazzotto et al 2 , the transformed Grad-Shafranov equation of Guazzotto et al (equation (19a) in [14]) reads…”

“…To help solve the multi-fluid model in realistic configurations, we draw on the extensive body of literature for solving tokamak MHD equilibria with flow (see [6][7][8][9][10][11][12] and references therein). Two recent examples which largely shape our work are McClements and Thyagaraja [13] and Guazzotto et al [14].…”

Self-consistent stationary MHD shear flows in the solar atmosphere as electric field generators

“…To enable quasi-steady and sustainable electric fields with field-aligned flows, it is necessary to simultaneously solve the resistive Ohm's law and the momentum equation. This was first investigated by Grad & Hogan 1970 and subsequently by Throumoulopoulos (1998) and Throumoulopoulos & Tasso (2000) for the case of axisymmetric fields. These authors found several classes of analytical equilibria with physically plausible σ = 1/η profiles, stating clearly that in general, fields with constant resistivity do not exist by proving that only the assumption of a spatially varying σ makes the equilibrium problem well posed.…”

Topological Structures of Velocity and Electric Field in the Vicinity of a Cusp-type Magnetic Null Point

“…In 2D, the search of the Poincaré class of the total electric field is linked to the Poincaré class of the resistivity. It is well known that for exact and analytical reconnection solutions, it is not sufficient to assume any non-ideal term or non-idealness or for example a constant resistivity (Priest et al 1994;Craig & Henton 1995;Watson & Craig 1998;Throumoulopoulos & Tasso 2000;Titov & Hornig 2000;Nickeler et al 2012Nickeler et al , 2014. For the classical role of reconnection in 2D it is inevitable that the plasma flow can cross both magnetic separatrix branches, as this scenario constitutes the reconnection solution.…”