Spontaneous symmetry breaking and neutral stability in the noncanonical Hamiltonian formalism

P.J. Morrison, S. Eliezer


The noncanonical Hamiltonian formalism is based upon a generalization of the Poisson bracket, a particular form of which is possessed by continuous media fields. Associated with this generalization are special constants of motion called Casimir invariants. These are constants that can be viewed as being built into the phase space, for they are invariant for all Hamiltonians. Casimir invariants are important because when added to the Hamiltonian they yield an effective Hamiltonian that produces equilibrium states upon variation. The stability of these states can be ascertained by a second variation. Goldstone’s theorem, in its usual context, determines zero eigenvalues of the mass matrix for a given vacuum state, the equilibrium with minimum energy. Here, since for fluids and plasmas the vacuum state is uninteresting, we examine symmetry breaking for general equilibria. Broken symmetries imply directions of neutral stability. Two examples are presented: the nonlinear Alfvén wave of plasma physics and the Korteweg–de Vries soliton.


Healing of magnetic stochasticity and islands by self-consistent plasma currents

M. Kotschenreuther


When resonant magnetic perturbations are imposed on a toroidally confined plasma, magnetic stochasticity can arise. It is shown that the transport of electrons in the stochasticity can lead to self-consistent magnetic fields which can greatly reduce, or heal, the stochasticity in steady state. These currents occur if there is another plasma transport process operating in which the radial fluxes of electrons and ions are not automatically equal. The healing effect can be quite strong in stellarator plasma confinement devices for fusion applications.


The linear instability and nonlinear motion of rotating plasma

J. Liu


Two coupled nonlinear equations describing the flute dynamics of the magnetically confined low-β collisionless rotating plasma are derived. The linear instability and nonlinear dynamics of the rotating column are analyzed theoretically. In the linear stability analysis, a new sufficient condition of stability is obtained. From the exact solution of eigenvalue equation for Gaussian density profile and uniform rotation of the plasma, the stability of the system strongly depends on the direction of plasma rotation, FLR effect and the location of the conducting wall. An analytic expression showing the finite wall effect on different normal modes is obtained and it explains the different behavior of (1,0) normal mode from other modes. The sheared rotation driven instability is investigated by using three model equilibrium profiles, and the analytic expressions of eigenvalues which includes the wall effect are obtained. The analogy between shear rotation driven instability and the instability driven by sheared plane parallel flow in the inviscid fluid is analyzed. Applying the linear analysis to the central cell of tandem mirror system, the trapped particle instability with only passing electronics is analyzed. For uniform rotation and Gaussian density profile, an analytic expression that determines the stability boundary is found. The nonlinear analysis shows that the nonlinear equations have a solitary vortex solution which is very similar to the vortex solution of nonlinear Rossby wave equation.


Theory of ion temperature gradient driven turbulence in tokamaks

G.S. Lee, P.H. Diamond


An analytic theory of ion-temperature-gradient-driven turbulence in tokamaks is presented. Energy-conserving, renormalized spectrum equations are derived and solved in order to obtain the spectra of stationary ion-temperature-gradient-driven turbulence. Corrections to mixing-length estimates are calculated explicitly. The resulting anomalous ion thermal diffusivity χi=0.4[(π/2)ln(1+ηi)]2[(1+ηi)/τ]2 ρ2pcs/Ls is derived and is found to be consistent with experimentally deduced thermal diffusivities. The associated electron thermal and particle diffusivity, and particle and heat-pinch velocities are also calculated. The effect of impurity gradients on saturated ion-temperature-gradient-driven turbulence is discussed and a related explanation of density profile steepening during Z-mode operation is proposed.


Study of the magnetic compressional mode in a hot particle plasma

D. Stotler, H.L. Berk, M. Engquist


The integral equation for the magnetic compressional mode, accounting for geometrical effects along the field line and using the eikonal approximation across the field line, is solved numerically for the eigenvalues and eigenfunctions. These results reproduce the analytic estimates when there is strong drift reversal. The representation of the eigenfunction of the form B|| = (C(ψ)/B) x (dP⊥h/dψ) is found to give accurate growth rates over a large range of parameter values. For typical EBT-S (Plasma Phys. 25, 597 (1983)) parameters, instability is predicted for all pressure scale lengths just below those needed for drift reversal, i.e., |R ∂(Pc) +P⊥h)/2B2 ∂r|>1 (where P is the article pressure, c and h refer to cold and hot components, B is the midplane magnetic field, and R is the midplane radius of curvature). If larger core densities are present, a wave--particle resonance arises when the particle drifts are not reversed, causing instability up to much larger pressure scale lengths. Stability for all values of the ratio of hot electron density to core density is obtained with |R ∂Pc/B2 ∂r|>1+P||h/P⊥h.


Three-dimensional toroidal magnetohydrodynamic particle codes

F. Brunel, J.N. Leboeuf, D. Stotler, H.L. Berk, S.M. Mahajan


The magnetohydrodynamic particle code has been developed to three dimensions in a cylindrical coordinate system in order to describe the plasma in a torus. To keep the noise level down, the finite differences are defined halfway between grid points and the magentic force is defined in a non-conservative manner. Two practical examples of using such a code for physics applications are reported: simulations of high amplitude Global Alfven Eigenmodes and stabilization of flute modes by a hot electron ring.


Two and three dimensional particle simulation models for study of plasma microinstabilities

R.D. Sydora, J.N. Lebeouf, T. Tajima


Two and three dimensional particle simulation models suitabe for the study of low frequency instabilities in inhomogeneous magnetized plasmas are described. Using the guiding center approximation for electrons transverse to the magentic field and exact electron dynamics parallel, as well as full ion dynamics, the necessary physics is included to study a class of microinstabilities known as drift waves (the universal mode). Applications of the model to studies of drift wave stability in sheared fields with single and multiple rational surfaces are discussed.


Analytic theory of the nonlinear M=1 tearing mode

R.D. Hazeltine, J.D. Meiss, P.J. Morrison


Numerical studies show that the m = 1 tearing mode continues to grow exponentially well into the nonlinear regime, in contrast with the slow, Rutherford, growth of m > 1 modes. We present a single helicity calculation which generalizes that of Rutherford to the case when the constant-psi approximation is invalid. As in that theory, the parallel current becomes an approximate flux function when the island size, W, exceeds the linear tearing layer width. However for the m = 1 mode, W becomes proportional to δB, rather than (δB)1/2 above this critical amplitude. This implies that the convective nonlinearity in Ohm's law, which couples the m = 0 component to the m = 1 component, dominates the resistive diffusion term. The balance between the inductive electric field and this convective nonlinearity results in exponential growth. Assuming the form of the perturbed fields to be like that of the linear mode, we find that the growth occurs at 71% of the linear rate.


Markov tree model of transport in area preserving maps

J.D. Meiss, E. Ott


A particle in a chaotic region of phase space can spend a long time near the boundary of a regular region since transport there is slow. This "stickiness" of regular regions is thought to be responsible for previous observations in numerical experiments of a long-time algebraic decay of the particle survivial probability, i.e., survival probability ∼t-z for large t. This paper presents a global model for transport in such systems and demonstrates the essential role of the infinite hierarchy of small islands interspersed in the chaotic region. Results for z are discussed.


Self-consistency constraints on turbulent magnetic relaxation and transport in collisionless plasmas

P. Terry, P.H. Diamond


Novel constraints on collisionless relaxation and transport in drift-Alfvén turbulence are reported. These constraints arise as a result of the effects of mode coupling and incoherent fluctuations as manifested by the proper application of self-consistency conditions. The result that electrostatic fluctuations alone regulate transport in drift-Alfvén turbulence follows directly. Quasilinear transport predictions are discussed in light of these constraints.


Effect of toroidicity during lower hybrid mode conversion

S. Riyopoulous, S.M. Mahajan


The effect of toroidicity during lower hybrid mode conversion is examined by treating the wave propagation in an inhomogeneous medium as an eigenvalue problem for ω(2) = ω(2)(m,n),m,n poloidal and toroidal wave numbers. Since the frequency regime near ω(2) = ω(2)LH is an accumulation point for the eigenvalue spectrum, the degenerate perturbation technique must be applied. The toroidal eigenmodes are constructed by a zeroth order superposition of monochromatic solutions with different poloidal dependence m, thus they generically exhibit a wide spectrum in k|| for given fixed omega even for small inverse aspect ratio ε. In case that the average k|| is in the neighborhood of kmin, the minimum wave number for accessibility of the mode conversion regime, it is expected that excitation of toroidal modes rather than geometric optics will determine the wave coupling to the plasma.


Three dimensional particle simulation of drift wave fluctuations in a sheared magnetic field

R.D. Sydora, J.N. Leboeuf, D. Thayer, P.H. Diamond, T. Tajima


Three-dimensional particle simulations of collisionless drift waves in sheared magnetic fields were performed in order to determine the nonlinear behavior of inverse–electron-resonance dynamics in the presence of thermal fluctuations. It is found that stochastic electron diffuision in the electron-resonance overlap region can destabilize the drift-wave eigenmodes. Numerical evaluations based on a nonlinear electron-resonance broadening theory give predictions in accord with the frequency and growth rates found in the simulation of short-wavelength modes (kyρs ≥ 1).


Two-dimensional consideration of large magnetic and electric fields in laser produced plasmas

S. Eliezer, A. Loeb


A simple model in two dimensions is developed and solved analytically taking into account the electric and magnetic fields in laser procuded plasmas. The electric potential in this model is described by a nonlinear differential equation. The stationary solution of this model is consistent for -0.1 ≤ ψ < 1, with electron temperature in the KeV region and a ratio of the electric (E) to magnetic (B) fields of (E/106 v/cm)/(B/MGauss) ≈ 1.


Diamagnetic stability limit for hot particle plasmas

H.L. Berk, Y.Z. Zhang


The ''layer'' precessional mode is analyzed at arbitrary drift reversal by using a cylindrical model with embedded currents which simulate the role of field line curvature in actual mirror experiments. New effects are included by taking into account the Landau damping arising from the spread of hot particle drift orbits. It is found that the mode is destabilized slightly below the value that drift reversal is achieved. Its growth rate is proportional to N-4 where N-2 is a coupling constant. In addition we describe the unstable interaction of the precessional layer mode with surface waves which can propagate below and above the ion cyclotron frequency. Below the ion cyclotron frequency the surface wave resembles the Alfven wave, while above the ion cyclotron frequency the surface wave resembles a whistler wave.


Renormalized turbulence theory of ion pressure gradient driven drift modes

B.G. Hong, D.I. Choi, W. Horton


From the nonlinear gyrokinetic equation we formulate the renormalized turbulence equation for the ηi -mode drift wave instability. The study shows that the dominant nonlinear damping mechanism is from the E x B convection of the pressure fluctuation and that the kinetic modifications to the fluid E x B mode coupling, studied earlier, shift the spectrum toward the shorter wavelengths. Balancing the linear growth rate with the nonlinear damping rate at the linearly most unstable region, we calculate the anomalous ion thermal conductivity, which exceeds the neoclassical plateau formula and gives a value of the same order as that previously computed by Horton, Choi, and Tang (Phys. Fluids 26, 1077 (1981)), but with a kinetic enhancement factor. Also, the thermal conductivity formula remains finite for vanishing density gradient.


Explosive coalescence of magnetic islands and explosive particle acceleration

T. Tajima, J.I. Sakai


An explosive reconnection process associated with the nonlinear evolution of the coalescence instability is found through studies of the electromagnetic particle simulation and the magnetohydrodynamic particle simulation. The explosive coalescence is a process of magnetic collapse, in which we find the magnetic and electrostatic field energies and temperatures (ion temperature in the coalescing direction, in particular) explode toward the explosion time t0 as (t0 - t)-8/3, (t0 - t)-4, and (t0 - t)-8/3, respectively for a canonical case. Single-peak, double-peak, and triple-peak structures of magnetic energy, temperature, and electrostatic energy, respectively, are observed on the simulation as overshoot amplitude oscillations and are theoretically explained. The heuristic model of Brunel and Tajima is extended to this explosive coalescence in order to extract the basic process. Since the explosive coalescence exhibits self-similarity, a temporal universality, we theoretically search for a self-similar solution to the two-fluid plasma equations.


Energy confinement in a high-current reversed field pinch

Z.G. An, G.S. Lee, P.H. Diamond


The ion temperature gradient driven (ηi) mode is proposed as a candidate for the cause of anomalous transport in high current reversed field pinches. A four-field fluid model is derived to describe the coupled nonlinear evolution of resistive interchange and ηi modes. A renormalized theory is discussed, and the saturation level of the fluctuations is analytically estimated. Transport scalings are obtained, and their implications discussed. In particular, these results indicate that pullet injection is a potentially viable mechanism for improving energy confinement in a high temperature reverse field pinch (RFP).


Drift modes with differential rotation and passing electrons

J. Liu, W. Horton, J.E. Sedlak


The stability of drift modes in a radially bounded cylindrical model of the tandem mirror central cell is analyzed as a function of plasma rotation and passing electron density. Analytic solutions for solid-body rotation and variable wall-to-plasma radius are used to derive formulas for the threshold and cutoff values of the passing-to-trapped electron density ratio. For certain density ratios all low m modes are unstable for all solid body rotational speeds. The effect of shear in the rotational profiles is shown to be a significant stabilizing effect on the rotationally-driven interchange instability. For typical profiles with the rotational velocity Ω(r) monotonically decreasing with radius, we show that for some parameters the rotational modes are stabilized by shear well before the onset of the shear-driven instabilities.


Plasma kinetic theory in action-angle variables

S.M. Mahajan, C.Y. Chen


An appropriate canonical perturbation theory to correctly deal with general electromagnetic field perturbation is developed, and is used to set up plasma kinetic theory in action-angle variables. A variety of test problems are solved to show the unifying power of the method. Basic linear, quasilinear, and nonlinear equations are derived which can serve as the starting point for a whole range of plasma problems.


Solitary vortices in a rotating plasma

W. Horton, J. Liu, J.D. Meiss, J.E. Sedlak


Nonlinear equations describing the flute dynamics of rotating plasma are derived and solitary vortex solutions are obtained. The solution takes the form of a shielded dipole vortex, similar to that found for nonlinear Rossby waves. The nonlinear dispersion relation, relating propagation speed to vortex radius, is obtained. Vortex speeds are shown to take values complementary to the phase velocities of the linear modes of the system. The E x B circulation velocity of the plasma trapped in the vortex is comparable to the diamagnetic drift velocity in the equilibrium plasma.


A paradigm for joined Hamiltonian and dissipative systems

P.J. Morrison


A paradigm for describing dynamical systems that have both Hamiltonian and dissipative parts is presented. Features of generalized Hamiltonian systems and metric systems are combined to produce what are called metriplectic systems. The phase space for metriplectic systems is equipped with a bracket operator that has an antisymmetric Poisson bracket part and a symmetric dissipative part. Flows are obtained by means of this bracket together with a quantity called the generalized free energy, which is composed of an energy and a generalized entropy. The generalized entropy is some function of the Casimir invariants of the Poisson bracket. Two examples are considered: (1) a relaxing free rigid body and (2) a plasma collision operator that can be tailored so that the equilibrium state is an arbitrary monotonic function of the energy.


Theoretical studies of magnetized plasma turbulence: Ion-cyclotron turbulence and shear flow turbulence and its applications to tokamak edge phenomena

T. Chiueh


Analytical theories of current-driven ion-cyclotron turbulence, treating incoherent phase-space density granulations (clumps), and strongly magnetized electrostatic MHD turbulence are studied. For the former, the local structure as well as global effects (instability and anomalous transport) of clumps are examined. For the latter, two shear-flow driven systems in the limit of vanishing magnetic shear, two-dimensional flows, are studied. Subsequently, the analysis is extended to a shear flow with finite magnetic shear, in an attempt to explain the turbulence characteristics observed in the tokamak edge. For ion-cyclotron turbulence, the author concentrates on a weak collective dissipation regime, where waves and clumps coexist. Nonlinear instability occurs at a lower threshold current (up to 7%) than the linear one. For two-dimensional flows, mixing-layer and wake (or jet) turbulence are studied. It is predicted that the mixing layer expands linearly in time, using a quasi-linear model. The predicted expansion rate is consistent with those observed in experiments and computer simulations. Finally, the theory is applied to tokamak edge turbulence, where dc electric field induced poloidal flow exists.


High energy plasma accelerators

T. Tajima


Colinear intense laser beams (LC Ω)0, κ0 and (LC Ω)1, κ1 shone on a plasma with frequency separation equal to the electron plasma frequency (LC Ω) pe are capable of creating a coherent large longitudinal electric field E L = mc (LC Ω) pee of the order of 1 GeV/cm for a plasma density of 10 to the 18th/c.c. through the laser beat excitation of plasma oscillations. Accompanying favorable and deleterious physical effects using this process for a high energy beat-wave accelerator are discussed: the longitudinal dephasing, pump depletion, the transverse laser diffraction, plasma turbulence effects, self-steepening, self-focusing, etc. The basic equation, the driven nonlinear Schroedinger equation, is derived to describe this system. Advanced accelerator concepts to overcome some of these problems are proposed, including the plasma fiber accelerator of various variations. An advanced laser architecture suitable for the beat-wave accelerator is suggested. Accelerator physics issues such as the luminosity are discussed. Applications of the present process to the current drive in a plasma and to the excitation of collective oscillations within nuclei are also discussed.


back to top