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It is shown that the magnetohydrodynamic (MHD) 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 Δ*ψ = Vcσ (here Δ* is the Grad–Schlüter–Shafranov operator, σ is the conductivity and Vc 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. Plasma 5, 2378 (1998)]. For a conductivity of the form σ = σ(R, ψ) (where 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 Δ*ψ = Vc σ(ψ) in the vicinity of the magnetic axis leads then 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.

A recent study on axisymmetric ideal magnetohydrodynamic equilibria with incompressible flows [H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas 5, 2378 (1998)] is extended to the generic case of helically symmetric equilibria with incompressible flows. It is shown that the equilibrium states of the system under consideration are governed by an elliptic partial differential equation for the helical magnetic flux function containing five surface quantities along with a relation for the pressure. The above-mentioned equation can be transformed to one possessing a differential part identical in form to the corresponding static equilibrium equation, which is amenable to several classes of analytical solutions. In particular, equilibria with electric fields perpendicular to the magnetic surfaces and non-constant-Mach-number flows are constructed. Unlike the case in axisymmetric equilibria with isothermal magnetic surfaces, helically symmetric T = T(ψ) equilibria are overdetermined, i.e. in this case the equilibrium equations reduce to a set of eight ordinary differential equations with seven surface quantities. In addition, the non-existence is proved of incompressible helically symmetric equilibria with (a) purely helical flows and (b) non-parallel flows with isothermal magnetic surfaces and with the magnetic field modulus a surface quantity (omnigenous equilibria).