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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 present two generalized hybrid kinetic-Hall magnetohydrodynamics (MHD) models describing the interaction of a two-fluid bulk plasma, which consists of thermal ions and electrons, with energetic, suprathermal ion populations described by Vlasov dynamics. The dynamics of the thermal components are governed by standard fluid equations in the Hall MHD limit with the electron momentum equation providing an Ohm's law with Hall and electron pressure terms involving a gyrotropic electron pressure tensor. The coupling of the bulk, low-energy plasma with the energetic particle dynamics is accomplished through the current density (current coupling scheme; CCS) and the ion pressure tensor appearing in the momentum equation (pressure coupling scheme; PCS) in the first and the second model, respectively. The CCS is a generalization of two well-known models, because in the limit of vanishing energetic and thermal ion densities, we recover the standard Hall MHD and the hybrid kinetic-ions/fluid-electron model, respectively. This provides us with the capability to study in a continuous manner, the global impact of the energetic particles in a regime extending from vanishing to dominant energetic particle densities. The noncanonical Hamiltonian structures of the CCS and PCS, which can be exploited to study equilibrium and stability properties through the energy-Casimir variational principle, are identified. As a first application here, we derive a generalized Hall MHD Grad–Shafranov–Bernoulli system for translationally symmetric equilibria with anisotropic electron pressure and kinetic effects owing to the presence of energetic particles using the PCS.

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

We present recent results on steady states of translational symmetric and axisymmetric ITER relevant plasmas with incompressible sheared flow in connection with a generalized Grad-Shafranov (GGS) equation and on their stability. On the basis of a vari- ety of ITER pertinent solutions of the GGS equation constructed it turns out that the flow affects noticeably the equilibrium quantities in qualitative agreement with the advanced confinement regimes phenomenology. Also, application of a sufficient condition for linear stability indicates that stabilization is mainly caused by the variation of the magnetic field in the direction perpendicular to the magnetic surfaces, related to the magnetic shear, depends on the plasma shaping and is sensitive to the up-down asymmetry. In certain cases the sheared flow has a weaker stabilizing effect enhanced by the equilibrium nonlinearity.

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.

We consider a generalized Grad–Shafranov equation (GGSE) in a triangularity-deformed axisymmetric toroidal coordinate system and solve it numerically for the generic case of ITER-like and JET-like equilibria with non-parallel flow. It turns out that increase of the triangularity improves confinement by leading to larger values of the toroidal beta and the safety factor. This result is supported by the application of a criterion for linear stability valid for equilibria with flow parallel to the magnetic field. Also, the parallel flow has a weaker stabilizing effect.

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

The equilibrium of an axisymmetric magnetically confined plasma with anisotropic resistivity and incompressible flows parallel to the magnetic field is investigated within the framework of the magnetohydrodynamic (MHD) theory by keeping the convective flow term in the momentum equation. It turns out that the stationary states are determined by a second-order elliptic partial differential equation for the poloidal magnetic flux function psi along with a decoupled Bernoulli equation for the pressure identical in form with the respective ideal MHD equations; equilibrium consistent expressions for the resistivities eta(parallel to) and eta(perpendicular to) parallel and perpendicular to the magnetic field are also derived from Ohm's and Faraday's laws. Unlike in the case of stationary states with isotropic resistivity and parallel flows [G.N. Throumoulopoulos and H. Tasso, J. Plasma Phys. 64, 601 (2000)] the equilibrium is compatible with nonvanishing poloidal current densities. Also, although exactly Spitzer resistivities either eta(parallel to)(psi) or eta(perpendicular to)(psi) are not allowed, exact solutions with vanishing poloidal electric fields can be constructed with 771, and y, profiles compatible with roughly collisional resistivity profiles, i.e., profiles having a minimum close to the magnetic axis, taking very large values on the boundary and such that eta(perpendicular to)>eta(parallel to). For equilibria with vanishing flows satisfying the relation (dP/dpsi)(dI(2)/dpsi)>0, where P and I are the pressure and the poloidal current functions, the difference eta(perpendicular to)-eta(parallel to) for the reversed-field pinch scaling, B(p)approximate toB(t), is nearly two times larger than that for the tokamak scaling, B(p)approximate to0.1B(t) (B-p and B-t are the poloidal and toroidal magnetic-field components). The particular resistive equilibrium solutions obtained in the present work, inherently free of-but not inconsistent with-Pfirsch-Schluter diffusion, indicate that parallel flows might result in a reduction of the diffusion observed in magnetically confined plasmas. (C) 2003 American Institute of Physics. Physics of Plasmas

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

Analytic solutions of the magnetohydrodynamic equilibrium equations for a symmetric magnetically confined plasma with sheared incompressible flows associated with electric fields similar to those observed in the transition from the low- to the high-confinement mode in tokamaks are constructed in cylindrical and toroidal geometries. In particular, an exact toroidal solution is obtained which for vanishing flows reduces to the Solovev equilibrium which has been extensively employed in tokamak confinement studies. Owing to the flow, several toroidal configurations having either one or two stagnation points are possible in addition to the usual ones with a single magnetic axis. For flows pertaining to tokamak operational regime the extremum of the electric field becomes larger as flow and its shear increase, the location of the extremum being, however, nearly independent of these variations. In addition, the flow affects the safety factor profile and the shape of the magnetic surfaces and results in an increase of the magnetic shear and a decrease of the toroidal beta. The impact of plasma elongation on the above-mentioned confinement figures of merit is also evaluated. (C) 2001 American Institute of Physics. Physics of Plasmas

A necessary condition for the existence of general dissipative magneto-hydrodynamic equilibria is derived. The derivation comprises Ohm's law and the existence of magnetic surfaces, only in the sense of the Kolmogorov–Arnold–Moser (KAM) theorem. All other equations describing the system are only required for evaluating the condition for a specific case.

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

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

A Solovev-like equilibrium solution is extended to plasmas with incompressible flow parallel to the magnetic field. ITER-like configurations are constructed for Alfven Mach functions peaked either on or off the magnetic axis. The linear stability of the equilibrium is also examined by applying a sufficient condition. (C) 2012 American Institute of Physics. [doi:10.1063/1.3672509] Physics of Plasmas

A previous stability condition (see Throumoulopoulos and Tasso, Phys. Plasmas 14, 122104 (2007)) for incompressible plasmas with field-aligned flows is extended to gravitating plasmas, including self-gravitation. It turns out that the stability condition is affected by gravitation through the equilibrium values only.

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

The general expression for the second-order perturbation energy of a Maxwell-drift kinetic system derived by Pfirsch and Morrison [Phys. Fluids B 3, 271 (1991)] is evaluated for the case of a magnetically confined plasma for which the equilibrium quantities depend on one Cartesian coordinate y. The conditions for the existence of negative-energy modes with vanishing initial field perturbations are also obtained. If the equilibrium guiding center distribution function ${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$ of any particle species \ensuremath{\nu} has locally the property ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$)g0, where ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$ is the guiding center velocity parallel to the magnetic field, and if this holds in the minimum-energy reference frame, parallel and oblique negative-energy modes exist with no essential restriction on either the orientation or magnitude of the wave vector. This condition also holds for the equilibria of a homogeneous magnetized plasma and an inhomogeneous force-free plasma with sheared magnetic field. If ${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}${\mathit{v}}_{\mathrm{\ensuremath{\parallel}}}$)0, the oblique negative-energy modes possible in a magnetically confined plasma are nearly perpendicular. The condition for purely perpendicular negative-energy modes reads as (${\mathit{dP}}^{(0)}$/dy)(\ensuremath{\partial}${\mathit{f}}_{\mathit{g}\ensuremath{\nu}}^{(0)}$/\ensuremath{\partial}y)0, where ${\mathit{P}}^{(0)}$ is the plasma pressure.

A class of exact axisymmetric tokamak equilibria with sheared flows parallel to the magnetic field is constructed, generalizing previous work on the subject (Kuiroukidis 2010 Plasma Phys. Control. Fusion 52 015002). The additional free parameters associated with new terms in the solution make it possible to construct up-down asymmetric configurations with a divertor X-point and desirable values of confinement figures of merit as the safety factor on the magnetic axis and plasma betas; in particular, we construct a number of ITER-pertinent equilibria. Their stability with respect to linear MHD perturbations is also examined by applying a sufficient condition. Plasma Physics and Controlled Fusion

A nonlinear two-dimensional steady state solution in the framework of hydrodynamics describing a row of periodic counter-rotating vortices is extended to the magnetohydrodynamic (MHD) equilibrium equation with incompressible flow of arbitrary direction. The extended solution covers a variety of equilibria because four surface quantities remain free. Similar to the case of the MHD cat-eyes equilibrium [Throumoulopoulos et al., J. Phys. A: Math. Theor. 42, 335501 (2009)] and unlike linear equilibria, the flow has a strong impact on isobaric surfaces by forming pressure islands located within the counter-rotating vortices even for values of beta (defined as the ratio of the thermal pressure over the external axial magnetic-field pressure) on the order of 0.01. Also, the axial current density is appreciably modified by the flow. Furthermore, a magnetic-field-aligned flow of experimental fusion relevance, i.e., for Alfven Mach numbers of the order of 0.01, and the flow shear in combination with the variation of the magnetic field perpendicular to the magnetic surfaces have significant stabilizing effects potentially related to the equilibrium nonlinearity. The stable region is enhanced by an external axial magnetic field. (C) 2010 American Institute of Physics. [doi:10.1063/1.3353077] Physics of Plasmas

Axisymmetric equilibria with incompressible flows of arbitrary direction are studied in the framework of magnetohydrodynamics under a variety of physically relevant side conditions consisting, for example, in that the plasma temperature or the magnetic field modulus are uniform on magnetic surfaces. To this end a set of pertinent nonlinear ordinary differential equations (ODEs) are transformed to quasilinear ODEs and the respective initial value problem is solved numerically with appropriately determined initial values near the magnetic axis. Several equilibrium configurations are then constructed surface by surface. It turns out that in addition to the usual configurations with a magnetic axis, the non-field aligned flow results to novel toroidal shell equilibria in which the plasma is confined within a couple of magnetic surfaces. In addition, the flow affects the elongation and triangularity of the magnetic surfaces and opens up the possibility of changing the magnetic field topology by creating double toroidal shell-like configurations.

A previous proof of non- existence of tokamak equilibria with purely poloidal flow within macroscopic theory ( Throumoulopoulos et al 2006 Phys. Plasmas 13 122501) motivated this microscopic analysis near the magnetic axis for toroidal and 'straight' tokamak plasmas. Despite the new exact solutions of Vlasov's equation on the magnetic axis found here, the structure of macroscopic flows remains elusive. However, the treatment within a toroidal system of orthogonal coordinates and comparison with the straight case, which reveals the importance of toroidicity, is encouraging. Journal of Physics a-Mathematical and Theoretical

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)

Steady states are studied in the framework of the ideal Hall magnetohydrodynamics (HMHD) model in arbitrary and axisymmetric geometries. In arbitrary geometry, conditions are found under which certain magnetohydrodynamics (MHD) equilibrium solutions can also satisfy the HMHD equations. For axisymmetric plasmas reduced equations are derived for uniform electron temperatures on magnetic surfaces and either barotropic ions or incompressible ion flows. The Hall and electron pressure gradient terms result in a deviation of the magnetic from the ion velocity surfaces, and consequently, the axisymmetric equilibria obey a set of coupled partial differential equations: one for the poloidal magnetic flux function and the other for a flux function labeling the ion velocity surfaces. Furthermore, the characteristics of certain classes of axisymmetric steady states with side conditions, as flows parallel to the magnetic field or purely poloidal incompressible flows, are identified and compared with respective MHD equilibria. Unlike in the frame of MHD, steady states with parallel axisymetric flows must be incompressible and equilibria with purely poloidal incompressible flows are possible. Certain analytic axisymmetric solutions are also constructed. (c) 2006 American Institute of Physics. Physics of Plasmas

Equilibrium eigenstates of an axisymmetric magnetically confined plasma with toroidal flow are investigated by means of exact solutions of the ideal magnetohydrodynamic equations. The study includes. "compressible" flows with constant temperature T(psi), but varying density on magnetic surfaces and incompressible ones with constant density rho(psi), but varying temperature thereon [the function psi(R, z) is the poloidal magnetic flux function with (R, z, phi) cylindrical coordinates]. These variations are nedessary for the existence of tokamak steady states with flow. The "compressible" and incompressible solutions are associated with the ansatz to omega(2)/T proportional to M-0 = const and d(rho omega(2))/d psi = A psi, respectively, where omega(psi) is the rotation frequency. In both cases eigenfunctions of the form psi(ln) = Z(l)(z)R-n(R) (l, n = 1, 2,...) describe configurations with f X n magnetic axes. A connection between multitoroidal configurations and reversed magnetic shear is also identified. Owing to the flow, the respective eigenvalues can be considered in two alternative ways: either as flow eigenvalues, M-0(l) and A(l), which depend on a pressure parameter P-0 or as pressure eigenvalues P-0(l), which depend on the flow parameters, M-0 or A. In the context of the latter consideration when the flow parameters are varied continuously there are transition points, (M-0)(m) and A(m) (m = 1,2,...), at which an additional magnetic axis is formed. This flow-caused* change in magnetic topology is possible solely in the presence of toroidicity because in the limit of infinite aspect ratio the axial flow does not appear in the equilibrium equation. Also, the lower the aspect ratio the smaller (M-0)(m) and A(m). In addition, the effects of the flow and the aspect ratio on the Shafranov shift are evaluated along with the variations of density and temperature on magnetic surfaces. (c) 2005 American Institute of Physics. Physics of Plasmas

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

A previous theorem by Pfirsch and Tasso [Nucl. Fusion 11, 259 (1971)] concerning "resistive wall modes" is extended in a "weak sense" to the case of time-dependent wall resistivity. Instability growth behavior remains, however, an open problem. (C) 2004 American Institute of Physics. Physics of Plasmas

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)

The Floquet stability of systems of differential equations with piecewise constant periodic coefficients is considered. In the "two-step" case the monodromy matrix is the product of two matrix exponentials. It can be evaluated either by reducing the matrix exponentials to polynomials on the eigenvalues of the corresponding matrices or, without knowledge of the eigenvalues, by using the Baker-Campbell-Hausdorff formula. The two methods are discussed and applied to several examples including the "two-step" dissipative Hill's equation introduced recently H. Tasso, G.N. Throumoulopoulos [Phys. Plasmas 9 (2002) 2662] to stabilize the "resistive wall mode" in magnetically confined plasmas. Application to large systems and matrices is not only possible but needed for the full understanding of the dynamic stabilization of the resistive wall mode. Its practical implementation is discussed. (C) 2002 Elsevier Science B.V. All rights reserved. Physics Letters A

In a previous note [Tasso, Phys. Plasmas 2, 1789 (1995)], dissipative magnetized Trkal flows were proved to be Lyapunov stable nonlinearly under the conditions of sufficiently small Reynolds number and of an upper bound of the "force-free" current. An alternative Lyapunov functional containing the cross helicity and introduced recently for conservative magnetized Beltrami flows [Yoshida , J. Math. Phys. 44, 2168 (2003)] is extended here to the Trkal flows mentioned above. Stability is now proven under limitations on maximal velocity and current without introducing an upper bound on the Reynolds number. (C) 2003 American Institute of Physics. Physics of Plasmas

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

The impact of reversed magnetic shear (s < 0) along with a radial electric field E-r on negative-energy perturbations (NEP's) is investigated for the case of cylindrical tokamak-like equilibria. NEP's can imply instabilities leading to anomalous transport [G. N. Throumoulopoulos and D. Pfirsch, Phys. Rev. E 56, 5979 (1997) and references therein]. For equilibria with E-r corresponding to \e(nu)phi\much less than beta T-nu and eta(nu)< 4/3 (phi is the electrostatic potential, beta equivalent to 8 pi P/B-2, eta(nu)equivalent to partial derivative lnT(nu)/partial derivative lnN(nu) and nu denotes the particle species) s < 0 results in a reduction of the fraction of particles associated with NEP's (active particles) as compared with equilibria with s > 0, by making the pressure profile steeper. The reduction is stronger the closer the minimum of the safety factor (q(min)) to the plasma center and the lower the negative value of s. For equilibria with \e(nu)phi\much less than beta T-nu and eta(nu)> 4/3 the fraction of active particles decreases as the magnetic tension B-theta(2)/r becomes larger. For E-r< 0 corresponding to \e(nu)phi\approximate to beta T-nu the electric field leads to a reduction of the fraction of active ions for eta(i)< 4/3 and of the fraction of active electrons for eta(e)> 4/3 in addition to that caused by s < 0 and B-theta(2)/r, respectively. For T(nu)greater than or equal to\e(nu)phi\much greater than beta T-nu, which corresponds to experimental observations, the reduction of active particles caused by E-r< 0 is insensitive to the sign and the value of s, the position of q(min), and the value of B-theta(2)/r. In addition, it is shown that the experimentally evidenced facts that (a) s < 0 is associated with a steepness of the pressure profile and (b) the position of q(min) is displaced towards the plasma edge as the toroidal current is increased are consistent with equilibrium considerations. It is emphasized that these results cannot yet give a complete picture, since the present paper still neglects toroidal effects. (C) 1999 American Institute of Physics. [S1070-664X(99)03208-5]. Physics of Plasmas

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

It is shown that the ideal magnetohydrodynamic (MHD) equilibrium states of an axisymmetric plasma with incompressible flows are governed by an elliptic partial differential equation for the poloidal magnetic flux function psi containing five surface quantities along with a relation for the pressure. Exact equilibria are constructed including those with nonvanishing poloidal and toroidal flows and differentially varying radial electric fields. Unlike the case in cylindrical incompressible equilibria with isothermal magnetic surfaces which should have necessarily circular cross sections [G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 4, 1492 (1997)], no restriction appears on the shapes of the magnetic surfaces in the corresponding axisymmetric equilibria. The latter equilibria satisfy a set of six ordinary differential equations which for flows parallel to the magnetic field B can be solved semianalytically. In addition, it is proved the nonexistence of incompressible axisymmetric equilibria with (a) purely poloidal flows and (b) nonparallel flows with isothermal magnetic surfaces and \B\ = \B\(psi) (omnigenous equilibria). (C) 1998 American Institute of Physics. [S1070-664X(98)02806-7] Physics of Plasmas

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