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We present several numerical solutions to a generalized Grad-Shafranov equation (GGSE), which governs axisymmetric plasma equilibria with incompressible flows of arbitrary direction, using fully connected, feed-forward, deep neural networks, also known as multi-layer perceptrons. Such artificial neural network (ANNs) are trained to approximate tokamak-relevant equilibria upon minimizing the GGSE mean squared residual in the plasma volume and the poloidal flux function on the plasma boundary. Solutions for the Solovev and the general linearizing ansatz for the free functions involved in the GGSE are obtained and benchmarked against known analytic solutions. We also construct a non-linear equilibrium incorporating characteristics relevant to the high confinement mode. In our numerical experiments it was observed that changing the radial distribution of the training points has a surprisingly small effect on the accuracy of the trained solution. In particular it is shown that localizing the training points at the plasma edge results in ANN solutions that describe quite accurately the entire magnetic configuration, thus demonstrating the interpolation capabilities of the ANNs.

We derive a sufficient condition for the linear stability of plasma equilibria with incompressible flow parallel to the magnetic field, $\boldsymbol{B}$ , constant mass density and anisotropic pressure such that the quantity $\unicode[STIX]{x1D70E}_{d}=\unicode[STIX]{x1D707}_{0}(P_{\Vert }-P_{\bot })/B^{2}$ , where $P_{\Vert }$ ( $P_{\bot }$ ) is the pressure tensor element parallel (perpendicular) to $\boldsymbol{B}$ , remains constant. This condition is applicable to any steady state without geometrical restriction. The condition, generalising the respective condition for magnetohydrodynamic equilibria with isotropic pressure and constant density derived in Throumoulopoulos & Tasso (Phys. Plasmas, vol. 14, 2007, 122104), involves physically interpretable terms related to the magnetic shear, the flow shear and the variation of total pressure perpendicular to the magnetic surfaces. On the basis of this condition we prove that, if a given equilibrium is linearly stable, then the ones resulting from the application of Bogoyavlenskij symmetry transformations are linearly stable too, provided that a parameter involved in those transformations is positive. In addition, we examine the impact of pressure anisotropy, flow and torsion of a helical magnetic axis, for a specific class of analytic equilibria. In this case, we find that the pressure anisotropy and the flow may have either stabilising or destabilising effects. Also, helical configurations with small torsion and large pitch seem to have more favourable stability properties.

In this note we propose a trilinear bracket formulation for the Hamiltonian extended Magnetohydrodynamics (XMHD) model with homogeneous mass density. The corresponding two-dimensional representation is derived by performing spatial reduction on the three-dimensional bracket, upon introducing a symmetric representation for the field variables. Subsequently, the trilinear bracket of the resulting two-dimensional, four-field model is discretized using a finite difference scheme, which results in semi-discrete dynamics that involve the Arakawa Jacobian. Simulations of planar dynamics show that this scheme respects the desired conservation properties to high precision.

In this study, we find the points of transition between elliptic and hyperbolic regimes for the axisymmetric extended magnetohydrodynamic (MHD) equilibrium equations. The ellipticity condition is expressed via a single inequality but is more involved than the corresponding two-fluid ones due to the imposition of the quasineutrality condition and is also more complicated than the Hall MHD ellipticity condition, due to electron inertia. In fact, the inclusion of electron inertia is responsible for peculiar results; namely, even the static equilibrium equations can become hyperbolic.

Being motivated by the paper [O. I. Bogoyavlenskij, Phys. Rev. 66, 056410 (2002)] we generalise the symmetry transformations for MHD equilibria with isotropic pressure and incompressible flow parallel to the magnetic field introduced therein in the case of respective CGL equilibria with anisotropic pressure. We find that the geometrical symmetry of the field-aligned equilibria can break by those transformations only when the magnetic field is purely poloidal. In this situation we derive three-dimensional CGL equilibria from given axisymmetric ones. Also, we examine the generic symmetry transformations for MHD and CGL equilibria with incompressible flow of arbitrary direction, introduced in a number of papers, and find that they cannot break the geometrical symmetries of the original equilibria, unless the velocity and magnetic field are collinear and purely poloidal., Comment: 23 pages

Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for the three-dimensional dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (Kaltsas et al., Phys. Plasmas, vol. 24, 2017, 092504). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy–Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad–Shafranov–Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helically symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis.

Motivated by a geometric method employed for the derivation of the Nambu bracket for ideal two-dimensional incompressible hydrodynamics, we reconstruct the reduced magnetohydrodynamic (RMHD) model by a priori imposition of its conservation laws. It turns out that there exists a more general model that conserves the same quantities with RMHD. The noncanonical Hamiltonian and Nambu description of this generic system are derived and in addition a canonical description is formed by Clebsch-parameterizing the vorticity and the magnetic flux function. The method for the construction of the dynamical equations is based on the imposition of the conservation laws as orthogonality conditions. Furthermore, this approach enabled us to construct three families of models that respect any combination of two out of the three conservation laws. Some of these models can serve as conservative regularizations of RMHD since under certain conditions they keep the enstrophy bounded without the need of introducing viscosity and may be candidates for incorporating small length scale physics into the RMHD framework.

An analytic solution to a generalized Grad-Shafranov equation with flow of arbitrary direction is obtained upon adopting the generic linearizing ansatz for the free functions related to the poloidal current, the static pressure, and the electric field. Subsequently, a D-shaped tokamak pertinent equilibrium with sheared flow is constructed using the aforementioned solution.An analytic solution to a generalized Grad-Shafranov equation with flow of arbitrary direction is obtained upon adopting the generic linearizing ansatz for the free functions related to the poloidal current, the static pressure, and the electric field. Subsequently, a D-shaped tokamak pertinent equilibrium with sheared flow is constructed using the aforementioned solution.

Nonlinear tranlational symmetric equilibria with up to quartic flux terms in the free functions, reversed magnetic shear and sheared flow are constructed in two ways: i) quasianalytically by an ansatz which reduces the pertinent generalized Grad-Shafranov equation to a set of ordinary differential equations and algebraic constraints which is then solved numerically, and ii) completely numerically by prescribing analytically a boundary having an X-point. The equilibrium characteristics are then examined by means of the pressure, safety factor, current density and electric field. For flows parallel to the magnetic field the stability of the equilibria constructed is also examined by applying a sufficient condition. It turns out that the equilibrium nonlinearity has a stabilizing impact which is slightly enhanced by the sheared flow. In addition, the results indicate that the stability is affected by the up-down asymmetry., Comment: 16 pages, 12 figures

We employ a conformal mapping transformation to solve a generalized Grad-Shafranov equation with incompressible plasma flow of arbitrary direction and construct particular up-down asymmetric D-shaped and diverted tokamak equilibria. The proposed method can also be employed as an alternative quasi-analytic method to solving two dimensional elliptic partial differential equations., Comment: 9 pages, 4 figures, accepted in Physics of Plasmas

The aim of the present work is to investigate tokamak equilibria with reversed magnetic shear and sheared flow, which may play a role in the formation of internal transport barriers (ITBs), within the framework of two-fluid model. The study is based on exact self-consistent solutions in cylindrical geometry by means of which the impact of the magnetic shear, s, and the "toroidal" (axial) and "poloidal" (azimuthal) ion velocity components on the radial electric field, its shear and the shear of the ExB velocity is examined. For a wide parametric regime of experimental concern it turns out that the contributions of the toroidal and poloidal velocity and pressure gradient terms to the electric field, its shear and ExB velocity shear are of the same order of magnitude. The impact of s on ExB velocity shear through the pressure gradient term is stronger than that through the velocity terms. The results indicate that, alike MHD, the magnetic shear and the sheared toroidal and poloidal velocities act synergetically in producing electric fields and therefore ExB velocity shear profiles compatible with ones observed in discharges with ITBs; owing to the pressure gadient term, however, the impact of s on the electic field, its shear and the shear of ExB velocity is stronger than that in MHD., 25 pages, 21 figures

We identify and study new nonlinear axisymmetric equilibria with incompressible flow of arbitrary direction satisfying a generalized Grad Shafranov equation by extending the symmetry analysis presented in [G. Cicogna and F. Pegoraro, Phys. Plasmas Vol. 22, 022520 (2015)]. In particular, we construct a typical tokamak D-shaped equilibrium with peaked toroidal current density, monotonically varying safety factor and sheared electric field., 4 pages, 6 figures

Analytical solutions of the magnetohydrodynamic equilibrium equations for a cylindrically symmetric magnetically confined plasma with reversed magnetic shear, s < 0, and sheared flow are constructed by prescribing the safety factor, poloidal velocity and axial velocity profiles consistently with experimental ones. On the basis of the solutions obtained, in most of the cases considered, it turns out that an increase of \s\ and of the velocity components result in larger absolute values for the radial electric field, E-r its shear, \dE(r)/dr\ drop \E'(r)\, and the E x B velocity shear, (omega(ExB) = \(d/dr)(E x B/B-2)\, which may play a role in the formation of internal transport barriers in tokamaks. In particular for a constant axial magnetic field, omega(ExB) at the point where E'(r) = 0 is proportional to 1 - s. Also, \E'(r)\ and omega(ExB) increase as the velocity shear takes larger values. The results clearly indicate that s < 0 and sheared flow acts synergetically in an increase of omega(ExB) with the impact of the flow, in particular the poloidal one being stronger than that of s < 0. Plasma Physics and Controlled Fusion

By choosing appropriate deformed Maxwellian ion and electron distribution functions depending on the two particle constants of motion, i.e. the energy and toroidal angular momentum, we reduce the Vlasov axisymmetric equilibrium problem for quasineutral plasmas to a transcendental Grad-Shafranov-like equation. This equation is then solved numerically under the Dirichlet boundary condition for an analytically prescribed boundary possessing a lower X-point to construct tokamak equilibria with toroidal sheared ion flow and anisotropic pressure. Depending on the deformation of the distribution functions these steady states can have toroidal current densities either peaked on the magnetic axis or hollow. These two kinds of equilibria may be regarded as a bifurcation in connection with symmetry properties of the distribution functions on the magnetic axis., Comment: 9 pages, 9 figures

A new constant of motion for helically symmetric equilibria in the vicinity of the magnetic axis is obtained in the framework of Vlasov theory. In view of this constant of motion the Vlasov theory is compared with drift kinetic and gyrokinetic theories near axis. It turns out that as in the case of axisymmetric equilibria [H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas 18, 064507 (2011)] the Vlasov current density thereon can differ appreciably from the drift kinetic and gyrokinetic current densities. This indicates some limitation on the implications of reduced kinetic theories, in particular as concerns the physics of energetic particles in the central region of magnetically confined plasmas., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1104.0427

Nonlinear z-independent solutions to a generalized Grad-Shafranov equation (GSE) with up to quartic flux terms in the free functions and incompressible plasma flow non parallel to the magnetic field are constructed quasi-analytically. Through an ansatz the GSE is transformed to a set of three ordinary differential equations and a constraint for three functions of the coordinate x, in cartesian coordinates (x,y), which then are solved numerically. Equilibrium configurations for certain values of the integration constants are displayed. Examination of their characteristics in connection with the impact of nonlinearity and sheared flow indicates that these equilibria are consistent with the L-H transition phenomenology. For flows parallel to the magnetic field one equilibrium corresponding to the H-state is potentially stable in the sense that a sufficient condition for linear stability is satisfied in an appreciable part of the plasma while another solution corresponding to the L-state does not satisfy the condition. The results indicate that the sheared flow in conjunction with the equilibrium nonlinearity play a stabilizing role., Comment: 26 pages, 16 figures

A generalised Grad-Shafranov equation that governs the equilibrium of an axisymmetric toroidal plasma with anisotropic pressure and incompressible flow of arbitrary direction is derived. This equation includes six free surface functions and recovers known Grad-Shafranov-like equations in the literature as well as the usual static, isotropic one. The form of the generalised equation indicates that pressure anisotropy and flow act additively on equilibrium. In addition, two sets of analytical solutions, an extended Solovev one with a plasma reaching the separatrix and an extended Hernegger-Maschke one for a plasma surrounded by a fixed boundary possessing an X-point, are constructed, particularly in relevance to the ITER and NSTX tokamaks. Furthermore, the impacts both of pressure anisotropy and plasma flow on these equilibria are examined. It turns out that depending on the maximum value and the shape of an anisotropy function, the anisotropy can act either paramagnetically or diamagnetically. Also, in most of the cases considered both the anisotropy and the flow have stronger effects on NSTX equilibria than on ITER ones., 27 pages, 14 figures

The equilibrium of an axisymmetric magnetically confined plasma with anisotropic resistivity and toroidal flow is investigated in the framework of magnetohydrodynamics (MHD). The stationary states are determined by an elliptic differential equation for the poloidal magnetic flux function, a Bernoulli equation for the pressure and two relations for the resistivity components parallel and perpendicular to the magnetic field. The equilibrium characteristics of a tokamak with rectangular cross-section are studied by means of eigenfunctions in connection with exact solutions for the cases of "compressible" flows with constant temperature, but varying density on magnetic surfaces and incompressible ones with constant density, but varying temperature thereon. Those eigenfunctions can describe either single or multiple toroidal configurations. In the former case the equilibrium has the following characteristics: (i) the profiles of the resistivity's components on the poloidal cross-section are roughly collisional (ii) the electric field perpendicular to the magnetic surfaces possesses two local extrema within the plasma and vanishes on the boundary and (iii) the toroidal current density is peaked close to the magnetic axis and vanishes on the boundary. The impact of the flow and the aspect ratio on the aforementioned quantities is evaluated for both "compressible" and incompressible flows., 18 pages, 8 figures

It is proved that irrespective of compressibility tokamak steady states with purely poloidal mass flow can not exist in the framework of either magnetohydrodynamics (MHD) or Hall MHD models. Non-existence persists within single fluid plasma models with pressure anisotropy and incompressible flows., Comment: The conclusion reported in the last sentence of the first paragraph of Sec. V in the version of the paper published in Physics of Plasmas is incorrect. The correct conclusion is given here (15 pages)

In the above entitled recent publication by Giovanni Lapenta [Phys. Rev. Lett. Vol 90, 135005 (2003) ] it is claimed construction of a new class of solitonlike solutions for the Grad-Shafranov equation in plane geometry. It is proved here that, because of the mathematically erroneous choice \nabla p= |\Psi|^2 \Psi \nabla \Psi for an analytic continuation of the poloidal magnetic flux-function \Psi in the complex plane (p is the pressure), the cubic Schr\"odinger equation considered by the author is irrelevant to the equilibrium problem and the Grad-Shafranov equation., Comment: 3 pages, no figures

Exact solutions of the equation governing the equilibrium magetohydrodynamic states of an axisymmetric plasma with incompressible flows of arbitrary direction [H. Tasso and G.N.Throumoulopoulos, Phys. Pasmas {\bf 5}, 2378 (1998)] are constructed for toroidal current density profiles peaked on the magnetic axis in connection with the ansatz $S=-ku$, where $S=d/d u [\varrho (d\Phi/du)^2]$ ($k$ is a parameter, $u$ labels the magnetic surfaces; $\varrho(u)$ and $\Phi(u)$ are the density and the electrostatic potential, respectively). They pertain to either unbounded plasmas of astrophysical concern or bounded plasmas of arbitrary aspect ratio. For $k=0$, a case which includes flows parallel to the magnetic field, the solutions are expressed in terms of Kummer functions while for $k\neq 0$ in terms of Airy functions. On the basis of a tokamak solution with $k\neq 0$ describing a plasma surrounded by a perfectly conducted boundary of rectangular cross-section it turns out that the Shafranov shift is a decreasing function which can vanish for a positive value of $k$. This value is larger the smaller the aspect ratio of the configuration., Comment: 13 pages, 2 figures. v2:Eq (3) has been corrected. A new figure (Fig. 2) has been added in order to illustrate u-contours in connection with solution (24) and the Shafranov shift. Also, a sentence referring to Fig. 2 has been added after Eq. (25)

It is found that the ideal magnetohydrodynamic equilibrium of an axisymmetric gravitating magnetically confined plasma with incompressible flows is governed by a second-order elliptic differential equation for the poloidal magnetic flux function containing five flux functions coupled with a Poisson equation for the gravitation potential, and an algebraic relation for the pressure. This set of equations is amenable to analytic solutions. As an application, the magnetic-dipole static axisymmetric equilibria with vanishing poloidal plasma currents derived recently by Krasheninnikov, Catto, and Hazeltine [Phys. Rev. Lett. {\bf 82}, 2689 (1999)] are extended to plasmas with finite poloidal currents, subject to gravitating forces from a massive body (a star or black hole) and inertial forces due to incompressible sheared flows. Explicit solutions are obtained in two regimes: (a) in the low-energy regime $\beta_0\approx \gamma_0\approx \delta_0 \approx\epsilon_0\ll 1$, where $\beta_0$, $\gamma_0$, $\delta_0$, and $\epsilon_0$ are related to the thermal, poloidal-current, flow and gravitating energies normalized to the poloidal-magnetic-field energy, respectively, and (b) in the high-energy regime $\beta_0\approx \gamma_0\approx \delta_0 \approx\epsilon_0\gg 1$. It turns out that in the high-energy regime all four forces, pressure-gradient, toroidal-magnetic-field, inertial, and gravitating contribute equally to the formation of magnetic surfaces very extended and localized about the symmetry plane such that the resulting equilibria resemble the accretion disks in astrophysics., Comment: 12 pages, latex, to be published in Geophys. Astrophys. Fluid Dynamics

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 $\psi$ 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 $\Delta^\star \psi=V_c \sigma$. (Here, $\Delta^\star$ is the Grad-Schl\"{u}ter-Shafranov operator, $\sigma$ is the conductivity and $V_c$ is the constant toroidal-loop voltage divided by $2 \pi $). 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 {\bf 5}, 2378 (1998)]. For a conductivity of the form $\sigma=\sigma(R, \psi)$ ($R$ is the distance from the axis of symmetry) several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible $\sigma$ profiles, i.e. profiles with $\sigma$ taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For $\sigma=\sigma(\psi)$ consideration of the relation $\Delta^\star\psi = V_c \sigma(\psi)$ 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., Comment: Post script file, 17 pages, no figures, to be published in J. Plasma Physics

A recent study on axisymmetric ideal magnetohydrodynamic equilibria with incompressible flows [H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas {\bf 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 $\psi$ containing five surface quantities along with a relation for the pressure. The above mentioned equation can be transformed to one possessing differential part identical in form to the corresponding static equilibrium equation, which is amenable to several classes of analytic 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 $T=T(\psi)$ equilibria are over-determined, i.e., in this case the equilibrium equations reduce to a set of eight ordinary differential equations with seven surface quantities. In addition, it is proved the non-existence of incompressible helically symmetric equilibria with (a) purely helical flows (b) non-parallel flows with isothermal magnetic surfaces and the magnetic field modulus being a surface quantity (omnigenous equilibria)., Comment: Latex file, 13 pages, accepted in J. Plasma Phys

The impact of an equilibrium radial electric field $E $ on negative-energy perturbations (NEPs) (which are potentially dangerous because they can lead to either linear or nonlinear explosive instabilities) in cylindrical equilibria of magnetically confined plasmas is investigated within the framework of Maxwell-drift kinetic theory. It turns out that for wave vectors with a non-vanishing component parallel to the magnetic field the conditions for the existence of NEPs in equilibria with E=0 [G. N. Throumoulopoulos and D. Pfirsch, Phys. Rev. E 53, 2767 (1996)] remain valid, while the condition for the existence of perpendicular NEPs, which are found to be the most important perturbations, is modified. For $|e_i\phi|\approx T_i$ ($\phi$ is the electrostatic potential) and $T_i/T_e > \beta_c\approx P/(B^2/8\pi)$ ($P$ is the total plasma pressure), a case which is of operational interest in magnetic confinement systems, the existence of perpendicular NEPs depends on $e_\nu E$, where $e_\nu$ is the charge of the particle species $\nu$. In this case the electric field can reduce the NEPs activity in the edge region of tokamaklike and stellaratorlike equilibria with identical parabolic pressure profiles, the reduction of electron NEPs being more pronounced than that of ion NEPs., Comment: 30 pages, latex

The equilibrium of a resistive axisymmetric plasma with purely toroidal flow surrounded by a conductor is investigated within the framework of the nonlinear magnetohydrodynamic theory. It is proved that a) the poloidal current density vanishes and b) apart from an idealized case the pressure profile should vanish on the plasma boundary. For the cases of isothermal magnetic surfaces, isentropic magnetic surfaces and magnetic surfaces with constant density, the equilibrium states obey to an elliptic partial differential equation for the poloidal magnetic flux function, which is identical in form to the corresponding equation governing ideal equilibria. The conductivity, which can not be either uniform or a surface quantity, results however in a restriction of the possible classes of equilibrium solutions, e.g., for the cases considered the only possible equilibria with Spitzer conductivity are of cylindrical shape., Comment: latex, 16 pages, 1 figure available on request to the author (gthroum@cc.uoi.gr), to be published in J. Plasma Physics

It is proved that (a) the solutions of the ideal magnetohydrodynamic equation, which describe the equlibrium states of a cylindrical plasma with purely poloidal flow and arbitrary cross sectional shape [G. N. Throumoulopoulos and G. Pantis, Plasma Phys. and Contr. Fusion 38, 1817 (1996)] are also valid for incompressible equlibrium flows with the axial velocity component being a free surface quantity and (b) for the case of isothermal incompressible equilibria the magnetic surfaces have necessarily circular cross section., Comment: 7 pages, latex, no figures

The conditions for the existence of negative-energy perturbations (which could be nonlinearly unstable and cause anomalous transport) are investigated in the framework of linearized collisionless Maxwell-drift kinetic theory for the case of equilibria of magnetically confined, circularly cylindrical plasmas and vanishing initial field perturbations. For wave vectors with a non-vanishing component parallel to the magnetic field, the plane equilibrium conditions (derived by Throumoulopoulos and Pfirsch [Phys Rev. E {\bf 49}, 3290 (1994)]) are shown to remain valid, while the condition for perpendicular perturbations (which are found to be the most important modes) is modified. Consequently, besides the tokamak equilibrium regime in which the existence of negative-energy perturbations is related to the threshold value of 2/3 of the quantity $\eta_\nu = \frac {\partial \ln T_\nu} {\partial \ln N_\nu}$, a new regime appears, not present in plane equilibria, in which negative-energy perturbations exist for {\em any} value of $\eta_\nu$. For various analytic cold-ion tokamak equilibria a substantial fraction of thermal electrons are associated with negative-energy perturbations (active particles). In particular, for linearly stable equilibria of a paramagnetic plasma with flat electron temperature profile ($\eta_e=0$), the entire velocity space is occupied by active electrons. The part of the velocity space occupied by active particles increases from the center to the plasma edge and is larger in a paramagnetic plasma than in a diamagnetic plasma with the same pressure profile. It is also shown that, unlike in plane equilibria, negative-energy perturbations exist in force-free reversed-field pinch equilibria with a substantial fraction of active particles., Comment: 31 pages, latex

The equilibrium of a cylindrical plasma with purely poloidal mass flow and cross section of arbitrary shape is investigated within the framework of the ideal MHD theory. For the system under consideration it is shown that only incompressible flows are possible and, conscequently, the general two dimensional flow equilibrium equations reduce to a single second-order quasilinear partial differential equation for the poloidal magnetic flux function $\psi$, in which four profile functionals of $\psi$ appear. Apart from a singularity occuring when the modulus of Mach number associated with the Alfv\'en velocity for the poloidal magnetic field is unity, this equation is always elliptic and permits the construction of several classes of analytic solutions. Specific exact equlibria for a plasma confined within a perfectly conducting circular cylindrical boundary and having i) a flat current density and ii) a peaked current density are obtained and studied., Comment: Accepted to Plasma Physics & Controlled Fusion, 14 pages, revtex

A kinetic consideration of an axisymmetric equilibrium with vanishing electric field near the magnetic axis shows that del f should not vanish on axis within the framework of Vlasov theory while it can either vanish or not in the framework of both a drift kinetic and a gyrokinetic theories (f is either the pertinent particle or the guiding center distribution function). This different behavior, relating to the reduction of phase space which leads to the loss of a Vlasov constant of motion, may result in the construction of different currents in the reduced phase space than the Vlasov ones. This conclusion is indicative of some limitation on the implications of reduced kinetic theories in particular as concerns the physics of energetic particles in the central region of magnetically confined plasmas., 9 pages

The cat-eyes steady state solution in the framework of hydrodynamics describing an infinite row of identical vortices is extended to the magnetohydrodynamic equilibrium equation with incompressible flow of arbitrary direction. The extended solution covers a variety of equilibria including one- and two-dimensional generalized force-free and Harris-sheet configurations which are preferable from those usually employed as initial states in reconnection studies. Although the vortex shape is not affected by the magnetic field, the flow in conjunction with the equilibrium nonlinearity has a strong impact on isobaric surfaces by forming pressure islands located within the cat-eyes vortices. More importantly, a magnetic-field-aligned flow of experimental fusion relevance and the flow shear have significant stabilizing effects in the region of pressure islands. The stable region is enhanced by an external axial ("toroidal") magnetic field., 17 pages, 8 figures

We derive exact solutions of a linear form of the Grad-Shafranov (GS) equation, including incompressible equilibrium flow, using ansatz-based similarity reduction methods. The linearity of the equilibrium equation allows linear combinations of solutions in order to obtain axisymmetric MHD equilibria with closed and nested magnetic surfaces which are favorable for the effective confinement of laboratory plasmas. In addition, employing the same reduction methods we obtain analytical solutions for several non-linear forms of the GS equation. In this context analytic force-free solutions in both linear and nonlinear regimes are also derived., 5 pages, 3 figures

Vlasov equilibria of axisymmetric plasmas with vacuum toroidal magnetic field can be reduced, up to a selection of ions and electrons distributions functions, to a Grad-Shafranov-like equation. Quasineutrality narrow the choice of the distributions functions. In contrast to two-dimensional translationally symmetric equilibria whose electron distribution function consists of a displaced Maxwellian, the toroidal equilibria need deformed Maxwellians. In order to be able to carry through the calculations, this deformation is produced by means of either a Heaviside step function or an exponential function. The resulting Grad-Shafranov-like equations are established explicitly., 11 pages

The Hamiltonian structure of ideal translationally symmetric extended MHD (XMHD) is obtained by employing a method of Hamiltonian reduction on the three-dimensional noncanonical Poisson bracket of XMHD. The existence of the continuous spatial translation symmetry allows the introduction of Clebsch-like forms for the magnetic and velocity fields. Upon employing the chain rule for functional derivatives, the 3D Poisson bracket is reduced to its symmetric counterpart. The sets of symmetric Hall, Inertial, and extended MHD Casimir invariants are identified, and used to obtain energy-Casimir variational principles for generalized XMHD equilibrium equations with arbitrary macroscopic flows. The obtained set of generalized equations is cast into Grad-Shafranov-Bernoulli (GSB) type, and special cases are investigated: static plasmas, equilibria with longitudinal flows only, and Hall MHD equilibria, where the electron inertia is neglected. The barotropic Hall MHD equilibrium equations are derived as a limiting case of the XMHD GSB system, and a numerically computed equilibrium configuration is presented that shows the separation of ion-flow from electromagnetic surfaces.

The formal stability analysis of Eulerian extended magnetohydrodynamics (XMHD) equilibria is considered within the noncanonical Hamiltonian framework by means of the energy-Casimir variational principle and the dynamically accessible stability method. Specifically, we find explicit sufficient stability conditions for axisymmetric XMHD and Hall MHD (HMHD) equilibria with toroidal flow and for equilibria with arbitrary flow under constrained perturbations. The dynamically accessible, second-order variation of the Hamiltonian, which can potentially provide explicit stability criteria for generic equilibria, is also obtained. Moreover, we examine the Lagrangian stability of the general quasineutral two-fluid model written in terms of MHD-like variables, by finding the action and the Hamiltonian functionals of the linearized dynamics, working within a mixed Lagrangian-Eulerian framework. Upon neglecting electron mass, we derive a HMHD energy principle, and in addition, the perturbed induction equation arises from Hamilton's equations of motion in view of a consistency condition for the relation between the perturbed magnetic potential and the canonical variables.