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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.

A Solovev-like equilibrium with arbitrary incompressible plasma flow nonparallel to the magnetic field is identified. The equilibrium configuration having a corner on the axis of symmetry resembles the nonlinear Palumbo one. The nonparallel flow associated with the quantity λ ∝ d / d u [ ρ ( d Φ / d u ) 2 ] (where ρ is the density, Φ the electrostatic potential and u labels the magnetic surfaces) affects the plasma shape, the Shafranov shift and the safety factor. For λ 0 the equilibrium has an x -point additional to the magnetic axis and increase of | λ | results in a variety of configurations with changing magnetic topology.

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

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

In a recent publication [H. Tasso and G. N. Throumoulopoulos, Phys. Lett. A 271, 413 (2000)] on Lyapunov stability of general mechanical systems, it is shown that “parametric excitations” can be stabilized by dissipation for positive potential energies. Specializing on the damped Mathieu equation permits one to establish its full stability chart. It is then seen that dissipation broadens the regions of stability to the extent that not only the response to parametric excitations is damped, but even “negative-energy” modes are stabilized by the combined action of the parametric excitation and the damping coefficient. The extension of this analysis to the “two-step” Hill’s equation shows that the stability regions become many times larger than those of the Mathieu equation. By analogy, these findings are a strong indication that the “resistive wall mode” could be stabilized by the joint action of a properly tailored time-dependent wall resistivity and a sufficient viscous dissipation in the plasma. Note that within this scheme neither the wall nor the plasma need to be in motion. An extension of this work to include more realistic models is in progress.

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 sufficient condition for the linear stability of nonautonomous dissipative mechanical systems with circulatory forces is derived. It is applied to autonomous systems transformed to nonautonomous ones by a time-dependent orthogonal transformation. This allows to obtain sufficient stability conditions in case the perturbed potential energy can have negative values, which was not possible up to now. This is, in particular, of importance for plasma equilibria with sheared flows. The nearness of the sufficient condition to necessity is discussed for a particular example.

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 non-linear drift wave equation is derived for a hot plasma confined by a strong magnetic field, the temperature gradient being perpendicular to the magnetic field and unidirectional. This equation, which is obtained through a long wave-length approximation, is the two-dimensional Korteweg-de Vries equation. It can be transformed approximately in the two-dimensional modified Korteweg-de Vries equation via the Miura transformation. A bi-Hamiltonian formulation of the drift wave equation is given, which implies its complete integrability. This derivation of the non-linear drift wave equation gives a strong support to k -spectrum calculations published previously by the author.

The derivation of the odd Burgers hierarchy is revisited. A first Hamiltonian formulation of the basic equation of the hierarchy is presented in relation with its linear counterpart. The generalized Poisson bracket is given explicitly. It contains exponentials of integrals of the dynamic variable. It verifies Jacobi identity by construction and through direct calculations. A second Hamiltonian formulation is also presented. It means that the equation, as expected, is `bi-Hamiltonian'. This property permits, as usual, the construction of all the hierarchy. Extension to matrix Burgers systems is suggested.

In the first part nonlinear stability of dissipative magnetized parallel flows is studied using Lyapunov methods. For two-dimensional perturbations perpendicular to the direction of the flow, a Lyapunov functional is constructed explicitly by two slightly different methods. This insures the nonlinear unconditional stability of the system. Though the extension of this nonlinear work to three-dimensional perturbations seems impossible at present, a conjecture concerning linear stability of magnetized Couette flows is stated, whose proof may become a mathematical challenge. In the second part the addition of parallel flows to ideal static equilibria is investigated. It turns out that Palumbo's “isodynamic” equilibrium plays a special role in this problem.

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 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.

For a system of van der Pol-like oscillators, Lyapunov functions valid in the greater part of phase space are given. They allow a finite region of attraction to be defined. Any attractor has to be within the rigorously estimated bounds. Under a special choice of the interaction matrices the attractive region can be squeezed to zero. In this case the asymptotic behaviour is given by a conservative system of nonlinear oscillators which acts as attractor. Though this systems does not posesses, in general, a Hamiltonian formulation, Gibbs statistics is possible due to the proof of a Liouville theorem and the existence of a positive invariant or “shell” condition. The “canonical” distribution on the attractor is remarkably simple despite nonlinearities. Finally the connection of the van der Pol-like system and of the attractive region with turbulence and fluctuation spectra in fluids and plasmas is discussed.

Non-linear stability in hydrodynamics and magnetohydrodynamics is discussed especially in connection with «unconditional» stability and with severe limitations on the Reynolds number. Two examples in magnetohydrodynamics show that the limitations on the Reynolds numbers can be removed but unconditional stability is preserved. Practical stability needs to be treated for limited levels of perturbations or for conditional stability. This implies some knowledge of the basin of attraction of the unperturbed solution, which is a very difficult problem.

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

A general sufficient condition for nonlinear stability of steady and unsteady flows in hydrodynamics and magnetohydrodynamics is derived. It leads to strong limitations in the Reynolds and magnetic Reynolds numbers. It is applied to the stability of generalized time-dependent planar Couette flows in magneto-hydrodynamics. Reynolds and magnetic Reynolds numbers have to be typically less than 2π2 for stability.

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.

Exact equal-field axisymmetric equipotentials are found for the Poisson equation in terms of elliptic integrals of the first and second kind. The method used has been introduced in 1967 by D. Palumbo in the context of magnetohydrodynamics.

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

Two general problems related to resistive magnetohydrodynamic stability are addressed in this paper. First, a general stability condition previously derived by the author for a class of real systems, occurring especially in plasma physics, is proved to persist to second order, despite the addition of several antisymmetric operators of first order in the linearized stability equation. Second, for a special but representative choice of the stability operators, a nonperturbative analysis demonstrates the existence of a critical density for the appearance of an overstability and the connected Hopf bifurcation, as suggested in a previous paper [Phys. Lett. A 180, 257 (1993)].

Equilibrium statistics of model Hamiltonians for 1D and 2D nonlinear drift wave equations are considered. While 1D calculations of correlation functions, lead to a Lorentz spectrum, the 2D Fourier transforms of expected 2D space correlation functions deliverk−3-dependence at largerk and a maximum atk near to zero, in agreement with recent measurements on large tokamaks.

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

In a general stability condition obtained by the author in a previous work physically motivated test functions are introduced. This leads to simplified versions of the stability functional, which makes its evaluation and minimization more tractable. In the case of special force-free fields the simplified functional reduces to a good approximation of the exact stability functional derived by other means. It turns out that in this case the condition is sufficient for nonlinear stability also.

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 general sufficient condition obtained by the author in a previous work is analysed with respect to its “nearness” to necessity. It is found that for physically reasonable approximations the condition is in some sense necessary and sufficient for stability against all modes. This together with Hermiticity makes its analytical and numerical evaluation worth doing for the optimization of magnetic configurations.

A sufficient stability condition with respect to purely growing modes is derived for resistive MHD. Though it may be, in general, violated, its ability to reduce in the appropriate limits to known necessary and sufficient stability conditions makes it instructive and conceptually useful.

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

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

Starting from a simple, marginally stable model considered for Lyapunov based boundary control of flexible mechanical systems, we add a term driving an instability and prove that for an appropriate control condition the system can become Lyapunov stable. A similar approximate extension is found for the general energy principle of linearized magnetohydrodynamics. The implementation of such external instantaneous actions may, however, impose challenging constraints for fusion plasmas.

A general stability condition for plasma-vacuum systems with resistive walls is derived by using the Frieman Rotenberg Lagrangian stability formulation [Rev. Mod. Phys. 32, 898 ( 1960)]. It is shown that the Lyapunov stability limit for external modes does not depend upon the gyroscopic term but upon the sign of the perturbed potential energy only. In the absence of dissipation in the plasma such as viscosity, it is expected that the flow cannot stabilize the system. (C) 2011 American Institute of Physics. [doi:10.1063/1.3606470] Physics of Plasmas

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

Force free fields can be generalized to Trkal flows in magnetohydrodynamics if a special velocity field parallel to the magnetic field is introduced. Such flows decay exponentially in time in case a constant viscosity and a constant resistivity are added. The stability of generalized Trkal flows is investigated and a sufficient condition for nonlinear stability is derived. For low velocities, this condition reduces essentially to the nonlinear sufficient stability condition for force free fields, previously found by the author.

A special Hopf bifurcation in resistive magnetohydrodynamics can easily be identified by starting from a particularly suitable form of the linearized equations which was previously introduced by the author. As usual, the bifurcation can lead to nonlinear periodic oscillations. These oscillations can eventually be destroyed, which raises the question of the nature of the ultimate attractors.

An 'energy principle' derived previously by Tasso (1978) is applied to the stability of shaped large aspect ratio Tokamaks. Most of the previous results on MHD, resistive and universal-type instabilities can be recovered in a simple way that demonstrates the power of this formation. This allows 3-dimensional tearing modes for general cross-sections to be investigated. An exact proof of the dissipative universal instability is given at least within the assumption of adiabatic motion. The relation of this to current work on collisionless drift is discussed.

An energy principle for 'helical' incompressible perturbations in shaped cross-section plasmas is derived in the Tokamak scaling ( epsilon identical to ka approximately=Bperpendicular to /Bz<