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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

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 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.

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.

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)

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.

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.

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 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<

An energy principle for 2-d resistive instabilities has been found. It leads to a necessary and sufficient condition for stability allowing the use of test functions. One simple consequence is that the current density in a plasma with arbitrary cross section should not increase to the outside. Otherwise the plasma would be unstable against resistive instabilities.

An energy principle has been found for all resistive perturbations of a circular plasma cylinder in the Tokamak scaling (kr approximately=Btheta /Bz approximately= epsilon ). It allows stability to be determined independently of the resistive scaling because resistivity is taken to be finite. The inclusion of FLR effects and viscosity would only reduce the growth rates. Using simple test functions near the resonances always leads to instability. Estimates of growth rates can be made, these being only rough if FLR effects dominate.

The linear tearing mode stability is evaluated for 1D equilibria in cylindrical geometry with arbitrary current profiles. This treatment combines the energy method and the Ritz-Galerkin procedure using finite elements. Particularly for hollow currents with two singular q surfaces extremely accurate results showing the coupling influence of both singularities are obtained.

For the stationary one-dimensional Vlasov equation quasi-neutrality is used instead of the Poisson equation. This allows one to obtain explicit solutions for the density as a function of the potential and to prove the positiveness of the trapped particle distribution function in a potential interval. This method has obvious application to oscillatory waves and shock waves.

The existence of limit cycles of two-dimensional autonomous integrable systems periodically perturbed by delta-peaks is discussed. For that purpose, Poincare maps are introduced considering analytical solutions between successive pulses.

The k-spectrum of Korteweg-de Vries turbulence is computed exactly. It consists of a superposition of lorentzians in k. It is determined by both the steepening and the dispersive term. Application to drift waves in a tokamak shows qualitative agreement with measurements.

The nonlinear drift wave equation is modelled with the Korteweg-de Vries equation. This equation is transformed into canonical form with a hamiltonian interpretable as an energy. This allows exact calculation of the partition function according to previous results in solid-state physics. The procedure seems to afford some hope for investigating general systems by means of Clebsch potentials.

For a class of axially symmetric equilibria, the terms in Mercier's criterion can be reduced to complete elliptic integrals with certain arguments and coefficients. For one particular choice of the equilibrium parameters the latter quantities can be evaluated explicitly without using any expansion. This case corresponds to “vertical” elliptic magnetic surfaces near the magnetica axis.

It is proved that for dispersive evolution equations the equilibrium statistics and in particular the k -spectrum can be calculated exactly. The general shape of the spectrum seems to be insensitive to changes in the steepening term.

The influence of gyroviscosity on the fluctuations of an MHD plasma is investigated by the method reported in a previous note. The main result is that gyroviscosity does not help to remove ultraviolet divergences. For a sub-class of observables it does not even show up. The full nonlinear problem may be needed.

A general sufficient condition for stability against purely growing modes is derived. It is a quadratic symmetric form very suitable to numerical treatment and minimization. It has potential applications in plasma stability with non-ideal equations and real geometry.

A general formula for the fluctuations spectrum of inhomogeneous plasmas and fluids at statistical equilibrium is found. It is valid for linearized equations of conservative systems and uses a rigorous treatment within Gibbs statistics. It opens the way for quantitative calculations of two-dimensional systems.

The theory of the drift-dissipative instability is done in the frame of the macroscopic theory, taking into account resistivity and viscosity and making use of the W.K.B. method. A linear term in k x ( k x is the component of the wave vector parallel to the density gradient) is found which has been neglected in previous papers. This term changes the stability condition noticeably and raises doubt on the existence of stationary waves in the x directions.

It is shown that a simple macroscopic plasma model allows for nonlinear shock-like drift waves satisfying Burger's equation.