Conservation of generalized vorticity and collisionless relaxation and reconnection in two-fluid MHD
V. Yankov, N. Petviashvili
The Canonical Hamiltonian formulation of fluid particles equation of motion leads to the frozen-in law for generalized vorticity for electrons and ions. In one-fluid magnetohydrodynamics (MHD) the frozen-in law for generalized ion vorticity disappears when the Hall term in Ohn's law is neglected. If the field lines of ion and electron generalized vorticities have different topology, the conservation of both generalized vorticities strongly affects plasma relaxation even for large-scale configurations. For example, electron inertia and finite ion Larmor radius does not lead to collisionless reconnection, when magnetic field lines lie on nested tori, and the field lines of generalized ion vorticity are chaotic. The topology of the generalized vorticity field lines breakdown only if the thermal drifts of electrons and ions are both taken into account.
Concept of statistical attractor for solitions in non-integrable Hamiltonian wave systems
Soltion, or solitary wave, is a particular solution even among stationary solutions of nonlinear wave equations. The special cases of solutions represent a nontypical behavior. Particular exact solutions cannot be taken as representative solutions unless they are attractors. The new unusual feature of solitons in nonintegrable systems is that they are the Hamiltonian attractors. How can it be seen? First, and most important, is taht a soliton realizes min and max of energy under the conservation of others invariants. Energy is an invariant and can not relax to an extremum. The energy relaxes to a min only in macroscopic degress of freedom. This means that we must consider the therodynamics of nonlinear waves. The classical thermodynamics are science for ordinary Hamitonian differential equations. We attempt to extend this approach for partial differential equations. The fundamental problem here is the absence of measure in a functional space. Fortunately, the solutions of wave equations can be presented as a sum of solitons and weakly interacting linear waves. This model of the two thermodyamic phases leads to the concept of the statistical attractor for Hamiltonian nonintegrable pertial Hamiltonian equations. Stable solitons are attractors for wave equations. Thermodynamics give an adequate language for the purpose. Equilibrium between solitons and free waves is calculated analytically. There is a numerical evidence confirming the theoretical predictions.
Ion mobility and transport barriers in the tokamak plasmas
H. Xiao, R.D. Hazeltine, Y.Z. Zhang, P.M. Valanju
The character of charged particle motion in an axisymmetric toroidal system with a constant radial electric field is investigated both analytically and numerically. Ion radial mobility caused by the combined effects of the radial electric field and charge exchange is found. A simple moment argument in the banana regime matches the simulation results well. Relation of present work and high confinement (H-mode) experiment is also discussed.
Profile stabilization of tilt mode in a field reversed configuration
J.W. Cobb, T. Tajima, D.C. Barnes
The possibility of stabilizing the tilt mode in field-reversed configurations without resorting to explicit kinetic effects such as large ion orbits is investigated. Various pressure profiles, P(Ψ), are chosen, including ``hollow'' profiles, where current is strongly peaked near the separatrix. Numerical equilibria are used as input for an initial value simulation, which uses an extended magnetohydrodynamic (MHD) model that includes viscous and Hall terms. Tilt stability is found for specific hollow profiles when accompanied by high values of separatrix beta, βsep. The stable profiles also have moderate to large elongation, racetrack separatrix shape, and lower values of s-bar, average ratio of Larmor radius to device radius. The stability is unaffected by changes in viscosity, but the neglect of the Hall term does cause stable results to become marginal or unstable. Implications for interpretation of recent experiments are discussed.
Shear flow generation by reynolds stress and suppression of resistive g modes
H. Sugama, W. Horton
Suppression of resistive g-mode turbulence by background shear flow generated from a small external flow source and amplified by the fluctuation-induced Reynolds stress is demonstrated and analyzed. The model leads to a paradigm for the low-to-high (L–H) confinement mode transition. To demonstrate the L–H transition model, single-helicity nonlinear fluid simulations using the vorticity equation for the electrostatic potential, the pressure fluctuation equation, and the background poloidal flow equation are used in the sheared slab configuration. The relative efficiency of the external flow and the Reynolds stress for producing shear flow depends on the poloidal flow damping parameter ν, which is given by neoclassical theory. For large ν, the external flow is a dominant contribution to the total background poloidal shear flow and its strength predicted by the neoclassical theory is not enough to suppress the turbulence significantly. In contrast, for small ν, it is shown that the fluctuations drive a Reynolds stress that becomes large and suddenly, at some critical point in time, shear flow much larger than the external flow is generated and leads to an abrupt, order unity reduction of the turbulent transport just like that of the L–H transition in tokamak experiments. It is also found that, even in the case of no external flow, the shear flow generation due to the Reynolds stress occurs through the nonlinear interaction of the resistive g modes and reduces the transport. To supplement the numerical solutions, the Landau equation for the mode amplitude of the resistive g mode is derived, taking into account the fluctuation-induced shear flow and the opposite action of the Reynolds stress in the resistive g turbulence compared with the classical shear flow Kelvin–Helmholtz (KH) driven turbulence is analyzed.
Filamentary magnetohydrodynamic plasmas
R. Kinney, T. Tajima, J.C. McWilliams, N. Petaviashvili
A filamentary construct of magnetohydrodynamical plasma dynamics based on the Elsässer variables is developed. This approach is modeled after discrete vortex models of hydrodynamical turbulence, which cannot be expected in general to produce results identical to those based on a Fourier decomposition of the fields. In a highly intermittent plasma, the induction force is small compared to the convective motion, and when this force is neglected, the plasma vortex system is described by a Hamiltonian. A statistical treatment of a collection of discrete current-vorticity concentrations is given. Canonical and microcanonical statistical calculations show that both the vorticity and the current spectra are peaked at long wavelengths, and the expected states revert to known hydrodynamical states as the magnetic field vanishes. These results differ from previous Fourier-based statistical theories, but it is found that when the filament calculation is expanded to include the inductive force, the results approach the Fourier equilibria in the low-temperature limit, and the previous Hamiltonian plasma vortex results in the high-temperature limit. Numerical simulations of a large number of filaments are carried out and support the theory. A three-dimensional vortex model is presented as well, which is also Hamiltonian when the inductive force is neglected. A statistical calculation in the canonical ensemble and numerical simulations show that a nonzero large-scale magnetic field is statistically favored, and that the preferred shape of this field is a long, thin tube of flux. Possible applications to a variety of physical phenomena are suggested.
Structure and damping of toroidal drift waves (and their implications for anomalous transport)
J.B. Taylor, J. Connor, H.R. Wilson
The conventional theory of high-n toroidal drift waves, based on the ballooning representation, indicates that shear-damping is generally reduced in a torus compared to its plane-slab value. It therefore describes the most unstable class of toroidal drift waves. However, modes of this type occur only if the diamagnetic frequency ω *(r) has a maximum in r, and they affect only a small fraction, O(1/n1/2), of the plasma radius around this maximum. Consequently they may produce little anomalous transport. Within the ballooning description, there is another class of toroidal drift waves with very different properties to the conventional ones. The new modes have greater shear-damping (closer to that in a plane-slab) than the conventional ones and so have a higher instability threshold. However, they occur for any plasma profile and at all radii, and they have larger radial extent. Consequently they may produce much greater anomalous transport than the possibly benign conventional modes. This suggests a picture of anomalous transport in which the plasma profile is determined by marginal stability, but marginal to the new class of modes not to the conventional ones.
Diffusion in laminar Rayleigh-Benard convection: Boundary layers versus boundary tubes
A.M. Dykhne, M.B. Isichenko, W. Horton
New results on the advection–diffusion of a passive tracer in a periodic system of hexagonal Rayleigh–Bénard convection cells at high Péclet number P=Lv/D0 >> 1 are presented, where L is the characteristic length scale of the flow, v is the velocity amplitude, and D0 is the molecular diffusivity. It is shown that the transport properties of this three-dimensional (3-D) laminar flow are drastically different from those of the well-studied two-dimensional convection rolls. The 3-D topology of the streamlines in the hexagonal convection leads to the formation of boundary tubes near the axes and the edges of the hexagons, in addition to the standard boundary layers found near the faces and the bases. A scaling theory is given and confirmed by test-particle simulations that show that the transport enhancement due to the hexagonal cells is controlled by the boundary tubes and scales only logarithmically with P. On the other hand, it is found that the subdiffusive regimes of transport in hexagons are similar to those found in other flows with constrained streamlines. The described effects can be used for the experimental investigation of structures in thermal convection.
Non-radiative collisions of Langmuir solitons
B. Breizman, N. Petviashvili, K. Jungwirth
Collisions of Langmuir solitons are described in terms of equations of motion for equivalent point particles. The description is valid in the limit when the eigenfrequencies of bound plasmons are much higher than the inverse interaction time, and the velocities of the solitons are much less than the ion-acoustic velocity. It is shown that the velocities of the solitons do not change due to binary collisions but they generally change when more than two solitons collide simultaneously.
A unified Monte Carlo interpretation of particle simulations and applications to nonneutral plasmas
Using a "Monte Carlo interpretation" of particle simulations, a general description of low-noise techniques, such as the δf method, is developed in terms well-known Monte Carlo variance reduction methods. Some of these techniques then are applied to linear and nonlinear studies of pure electron plasmas in cylindrical geometry, with emphasis on the generation and nonlinear evolution of electron vortices. Long-lived l=1 and l=2 vortices, and others produced by unstable diocotron modes in hollow profiles, are studied. It is shown that low-noise techniques make it possible to follow the linear evolution and saturation of even the very weakly unstable resonant diocotron modes.
The δf algorithm for beam dynamics
J. Koga, T, Tajima
An algorithm is developed to study particle dynamics of beams including collective interaction with high accuracy and low noise. Particle dynamics with collective interactions is treated through particle simulation, where the main or average distribution f0 and the deviation away from it, δf , are separately followed. The main distribution f0 is handled by an analytic equilibrium solution and the perturbation away from it, δf, is followed by the method of characteristics. We call this the δf algorithm. We specifically model a synchrotron collider which includes the collision section where collective effects of collisions are simulated by this δf algorithm and the rest of the collider where single particle dynamics are treated by simple harmonic transport. The most important target of this simulation is to understand and predict the long-time (108-109 rotations) behavior of the beam luminosity and lifetime. The δf method allows us to study the effect of small perturbations over long timescales of beam lifetime by eliminating the numerical noise problem that is inherent in particle-in-cell techniques. In the δf code using the reference parameters of the SSC (Superconducting Super Collider), beam blow-up near resonances and oscillations in the tune shift, Δv, far from resonances are observed. In studying long timescale particle diffusion in the phase space of the beams away from resonances, the δf code performance is compared with a tracking code which does not incorporate collective interaction.
Iso-topological relaxation, coherent structures, and Gaussian turbulence in two-dimensional magnetohydrodynamics
M.B. Isichenko, A.V. Gruzinov
The long-time relaxation of ideal two dimensional magnetohydrodynamic turbulence subject to the conservation of two infinite families of constants of motion - the magnetic and the "cross" topology invariants - is examined. The analysis of the Gibbs ensemble, where all integrals of motion are respected, predicts the initial state to evolve into an equilibrium, stable coherent structure (the most probable state) and decaying Gaussian turbulence (fluctuations) with a vanishing, but always positive temperature. The non-dissipative turbulence decay is accompanied by decrease in both the amplitude and the length scale of the fluctuations, so that the fluctuation energy remains finite. The coherent structure represents a set of singular magnetic islands with plasma flow whose magnetic topology is identical to that of the initial state, while the energy and the cross topology invariants are shared between the coherent structure and the Gaussian turbulence. These conservation laws suggest the variational principle of iso-topological relaxation which allows us to predict the appearance of the final state from a given initial state. For a generic initial condition having X points in the magnetic field, the coherent structure has universal types of singularities: current sheets terminating at Y points. These structures, which are similar to those resulting from the 2D relaxation of magnetic field frozen into an ideally conducting viscous fluid, are observed in the numerical experiment of Biskamp and Welter. The Gibbs ensemble method developed in this work admits extension to other Hamiltonian systems with invariants not higher than quadratic in the highest-order-derivative variables. The turbulence in two dimensional Euler fluid is of a different nature: there the coherent structures are also formed, but the fluctuations about these structures are non-Gaussian.
Gyrosheath near the tokamak edge
R.D. Hazeltine, H. Xiao, P.M. Valanju
A new model for the structure of the radial electric-field profile in the edge during the high confinement (H mode) is proposed. Charge separation caused by the difference between electron and ion gyromotion, or more importantly in the tokamak, the banana motion (halo effect) can self-consistently produce an electric dipole moment that causes the sheared radial electric field. The calculated results based on the model are consistent with DIII-D [Fusion Technol. 8, 441 (1985)] and Tokamak Experiment for Technology Oriented Research (TEXTOR) [J. Nucl. Mater. 122, 123, 1124 (1984)] experimental results.
Electromagnetic effect on the toroidal ion temperature gradient mode
J.Y. Kim, W. Horton, J.Q. Dong
A systematic study of the electromagnetic effects on the toroidal ion temperature gradient mode is presented using the local and nonlocal theories with the full kinetic terms. For the nonlocal study, a numerical code is developed to solve the electromagnetic gyrokinetic equation in the ballooning space. The electromagnetic coupling to the shear Alfvén mode is shown to give a stabilization of the toroidal temperature gradient mode at almost the same plasma pressure as that at which the kinetically modified magnetohydrodynamic (MHD) ballooning mode becomes destabilized. The transitional β value is shown to be lower in the full kinetic description than in the fluid theory. Possible correlations of these stability results with experimental observations are discussed.
Linear stability of stationary solutions of the Vlasov-poisson system in three dimensions
J. Batt, P.J. Morrison, G. Rein
Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in the contexts of both plasma physics and stellar dynamics. It is proved that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i.e., particle, energy. The main tool in the analysis is the free energy, a conserved quantity of the linearized system. In addition, an appropriate global existence result is proved for the linearized Vlasov-Poisson system and the existence of stationary solutions which satisfy the above stability condition is established.
Percolation and turbulent diffusion: Transport on and around percolation clusters
A brief review of percolation theory is given and some of its novel applications are described. There are several continuum generalizations of the lattice percolation problem, of which the main emphasis is given to the "potential model," describing the statistics of the contour lines of a random potential, objects studied in the framework of "statistical topography." The variety of new physical applications come from the idea that many transport processes in turbulent and random media (fluids, plasmas, semiconductors, 2D electron gas, etc.) occur along the contours of some random potential. The simplest and the most findamental example of this sort is the diffusion of a passive tracer in an incompressible two-dimensional flow, whose streamlines are the isolines of a random stream function. In the limit of a large Peclet number P, the effective diffusion in such a flow scales with P as the universal power of 10/13, expressable through the percolation critical exponents. More elaborate examples, where the transport scalings can be also obtained analytically, involve the plasma heat conduction in a stochastic magnetic field and/or electrostatic turbulence, the average conductivity of randomly inhomogeneous conductors, and the quantum Hall effect. A special section is devoted to reviewing the Dykhne reciprocity technique, applicable to two-dimensional, two-phase systems at the percolation threshold, where the effective transport coefficients can be calculated exactly.
Arbitrary mode number boundary layer theory for non-ideal toroidal Alfven modes
H.L. Berk, R.R. Mett, D.M. Lindberg
The theory of toroidicity-induced Alfven eigenmodes (TAE) and kinetic TAE (KTAE) is generalized to arbitrary mode numbers for a large aspect ratio low-beta circular tokamak. The interaction between nearest neighbors is described by a three-term recursion relation that combines elements from an outer region, described by the ideal magnetohydrodynamic equations of a cylinder, and an inner region, which includes the toroidicity and the nonideal effects of finite ion Larmor radius, electron inertia, and collisions. By the use of quadratic forms, it is proven that the roots of the recursion relation are stable and it is shown how perturbation theory can be applied to include frequency shifts due to other kinetic effects. Analytic forms are derived which display the competition between the resistive and radiative damping, where the radiation is carried by kinetic Alfven waves. When the nonideal parameter is small, the KTAE modes appear in pairs. When this parameter is large, previously found scaling for the single gap case is reproduced analytically.
Vortex filament evolution in electron magnetohydrodynamics
V.V. Yankov, A.V. Grechikha
Three-dimensional vortex filament motion in the framework of local approximation is reduced to the 1D nonlinear Schrodinger equation. The essential feature of electron MHD model is that the skinning of the magnetic field of the vortex at the London scale leads to an algebraic accuracy of the local approximation, contrary to logarithmic one as in the well-known Hasimoto vortex in ideal liquid. Filament behavior in an inhomogeneous medium provides a new model of the pinning attraction of the filament to the minima of density.
Hamiltonian dynamics in tokamak configuration
The exact Hamiltonian with guiding-center variables is obtained through canonical transformation for the particles in tokamaks. The canonical variables are simply related to cylindrical coordinates, R, φ, Z. The drift Hamiltonian is calculated to the second order. The gyrokinetic equation is derived in a very simple way. The simplicity and accuracy of canonical transformation make the formalism much easier in the kinetic and stochastic theorties in tokamaks.
Discrete vortex representation of magnetohydrodynamics
R. Kinney, T. Tajima, N. Petviashvili, J.C. McWilliams
We present an alternative approach to statistical analysis of an intermittent ideal magnetohydrodynamics fluid in two dimensions, based on the hydrodynamic discrete vortex model applied to the Elsasser variables. The model contains negative temperature states which predict the formation of magnetic islands, but also includes a natural limit under which the equilibrium states revert to the familiar twin-vortex states predicted by hydrodynamic turbulence theories. Numerical dynamical calculations yield equilibrium spectra in agreement with the theoretical predictions.
Collective transport of alpha particles due to Alfven wave instability
B.N. Breizman, H.L. Berk, H. Ye
Recently a new point of view has been developed for describing saturation of discrete modes excited by weak sources. The method applies to the evolution of energetic particles in the beam plasma instability as well as to the description of how alpha particles evolve when they destabilize Alfvén waves under reactor conditions. Over a wide range of parameters the system produces pulsations, where there are relatively brief bursts of wave energy separated by longer intervals of quiescence. There are two types of pulsations: benign and explosive. In the benign phase, valid when particle motion is not stochastic, the distribution function is close to that predicted by classical transport theory, and the instability saturates when the wave trapping frequency equals the expected linear growth rate. If the field amplitude in a burst reaches the level where orbit stochasticity occurs, the quasilinear diffusion causes rapid transfer of particle energy to wave energy and rapid flattening of the particle distribution function. The bursting phase is followed by a relatively long quiescent time interval, where the source provides the necessary free energy to regenerate the cycle. The critical issue is whether the instability develops to a high enough level to produce stochastic diffusion. In general, this question can be assessed by using mapping methods to obtain criteria of overlapping of orbit resonances. If overlap occurs, then the modes will saturate at a high level, which will result in significant anomalous transport effects. This picture is consistent with recent observations of energetic beam losses in tokamak experiments due to Alfvén mode excitation.
A Fokker-Planck operator for the emission and absorption of electron plasma waves in a magnetized plasma
For slab geometry the perturbation of the electrostatic wake of a superthermal test electron in a magnetized plasma (ωce >> ωpe) due to moderate magnetic shear is determined. Allowing for the spherical symmetry of the surfaces of constant phase to the rear of the test electron, the "resonant'' field electrons causing the damping of the wave in a magnetic surface at a distance x from the test electron are those with parallel velocity v|| = v||cosβ/cos(β + γ). Here β is the angle between the emitted ray and B(0), γ is the angle between B(0) and B(x) and v|| is the velocity of the test electron. As a result the damping in the WKB approximation for the wave emission is a function of both the angle of emission and γ. A Fokker–Planck equation is derived for the rate of change of the electron distribution function (f) due to the emission and absorption of the waves under these conditions; f is assumed approximately Maxwellian for v|| ≤ vT but with an arbitrary tail for v|| ≥ vT.
Self-organized profile relaxation by ion temperature gradient instability in toroidal plasmas
Y. Kishimoto, T. Tajima, M.J. Lebrun, M.G. Gray, J.-Y. Kim, W. Horton
Toroidal effects on the ion-temperature gradient mode are found to dictate the temperature evolution and the subsequent relaxed profile realization according to our toroidal particle simulation. Both in the strongly unstable fluid regime as well as in the near-marginal kinetic regime we observe that the plasma maintains an exponential temperature profile and forces the heat flux to be radially independent. The self-organized critical relaxed state is sustained slightly above the marginal stability, where the weak wave growth balances the wave decorrelation.
Nonlinear instability and chaos in plasma wave-wave interactions
Conventional linear stability analyses may fail for fluid systems with an indefinite free-energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes, or nonlinearly via resonant wave–wave coupling, leading to explosive growth. In the dissipationless case, it is conjectured that intrinsic chaotic behavior may allow initially nonresonant systems to reach resonance by diffusion in phase space. In this and a companion paper (submitted to Phys. Plasmas), this phenomenon is demonstrated for a simple equilibrium involving cold counterstreaming ions. The system is described in the fluid approximation by a Hamiltonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamiltonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, and to either decay or explosive instability via three-wave interactions. These instabilities are described for various integrable systems of waves interacting via single nonlinear terms. This discussion provides the foundation for the treatment of nonintegrable systems in the companion paper.
Self-consistent chaos in the beam-plasma instability
J.L. Tennyson, J.D. Meiss, P.J. Morrison
The effect of self-consistency on Hamiltonian systems with a large number of degrees-of-freedom is investigated for the beam-plasma instability using the single-wave model of O'Neil, Winfrey, and Malmberg. The single-wave model is reviewed and then rederived within the Hamiltonian context, which leads naturally to canonical action-angle variables. Simulations are performed with a large (104) number of beam particles interacting with the single wave. It is observed that the system relaxes into a time asymptotic periodic state where only a few collective degrees are active; namely, a clump of trapped particles oscillating in a modulated wave, within a uniform chaotic sea with oscillating phase space boundaries. Thus self-consistency is seen to effectively reduce the number of degrees-of-freedom. A simple low degree-of-freedom model is derived that treats the clump as a single macroparticle, interacting with the wave and chaotic sea. The uniform chaotic sea is modeled by a fluid waterbag, where the waterbag boundaries correspond approximately to invariant tori. This low degree-of-freedom model is seen to compare well with the simulation.