Introduction to bifurcation theory

J.D. Crawford


The theory of bifurcation from equilibria based on center-manifold reduction and Poincaré-Birkhoff normal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and recent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest generic bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation occurring in a model of three-wave mode coupling and steady-state bifurcations occurring in the real Landau-Ginzburg equation. The former provides an example of the importance of degenerate bifurcations in problems with more than one parameter and the latter illustrates new effects introduced into a bifurcation problem by a continuous symmetry.


Importance of a mirror based neutron source for the controlled fusion program

H.L. Berk, D.D. Ryutov


This comment discusses the importance of a neutron source for the controlled fusion program and the compatibility of mirror machines operating in a beam-target mode, with neutron source requirements. It is pointed out that the 2XIIB experiment has already established most of the feasibility characteristics of the neutron source. However, before a decision can be made to build a mirror machine sources, additional experiments are needed to guarantee that the necessary improvement of plasma parameters can be indeed achieved.


The rapid inward diffusion of cold ions in tokamaks and their effect on ion transport

A.A. Ware


The observed increase with density of the density asymmetry caused by the centrifugal force of toroidal motion in the PDX tokamak ( Plasma Physics and Controlled Nuclear Fusion Research (IAEA, Vienna, 1981), Vol. 1, p. 665), which is contrary to conventional theory, is explained by the presence of an excess of low-energy ions with 10%--15% concentration. The prime source being recycling, it is shown that low-energy ions undergo rapid inward diffusion (too rapid to thermalize with the outward diffusing energetic ions) because of the combined effects of large νPA, electrostatic diffusion, and -Er and ∂Ti/∂r. The presence of the low-energy ions alters dramatically the predictions of neoclassical theory and many hydrogen and impurity ion transport phenomena now have simple explanations.


Stable solitary propagation of optical beams

T. Kurki-Suonio, T. Tajima, P.J. Morrison


The behavior of a short laser pulse with periodically peaked transverse intensify profile is important for the study of laser acceleration of particles. For a specific relation between the amplitudes and the separation of the peaks, this profile should remain undistorted while propagating in plasma. Carrying out numerical particle simulation runs in which a deviation from this relation is present, we have observed the system to exhibit a kind of bistability.


Beam transport in the crystal x-ray accelerator

T. Tajima, B. Newberger, F.R. Huson, W.W. Mackay, B.C. Covington, J. Payne, N.K. Mahale, S. Ohnuma


A Fokker-Planck model of charged particle transport in crystal channels which includes the effect of strong accelerating gradients has been developed for application to the crystal x-ray accelerator and other crystal accelerator schemes. We indicate the implications of the analytic solutions found for a harmonic channeling potential for the accelerating gradient and the multiple scattering which, because we consider only the acceleration of positive particles, is dominated by scattering from the valence electrons. In order to relax the constraints imposed by these, we have been exploring the application of novel materials to this problem. One candidate is porous Si and our investigation into this material which is as yet preliminary is discussed and other possible materials are indicated.


Self-consistent collision code for beam tracking in SSC

T. Tajima, J. Koga


The beam dynamics for an electron (or positron) storage ring and for the Superconducting Super Collider (SSC) at the collision point is crucial in detemining the maximum luminosity and lifetime of the beam. For typical parameters of SSC the beam density is sufficiently high (≈ 1015 cc) at the collision point and the interaction time sufficiently long (Γmaxτcoll ≈ 0.1, where Γmax = √2ωpi/√γ), so that effects of not only rigid body beam-beam interaction but also the softer beam-beam collective interaction should be assessed. A relatively simple model has been introduced to analyze the above problem in a self-consistent manner by the particle simulation method. Relativistic particles interact self-consistently with each other over the collision time τcoll and then their transverse position (or betatron coordinates x,y) and beam angles (or aperature) θx, θy are transformed in a simplectic map of the magnets that lie between the collision point to the next. This is repeated over a necessary number of time steps. Results with parameters for colliders are presented.


The free energy of Maxwell-Vlasov equilibria

P.M. Morrison, D. Pfirsch


A previously derived expression for the energy of arbitrary perturbations about arbitrary Vlasov-Maxwell equilibria is transformed into a very compact form. The new form is also obtained by a canonical transformation method for solving Vlasov's equation, which is based on Lie group theory. This method is simpler than the one used before and provides better physical insight. Finally a procedure is presented for determining the existence of negative-energy modes. In this context the question of why there is an accessibility constraint for the particles, but not for the fields, is discussed.


Inversion of the ballooning transformation

R.D. Hazeltine, W.A. Newcomb


The ballooning formalism can be viewed, not just as an eikonal representation, but as an integral transform, analogous to the Fourier or Laplace transforms, with a uniquely defined inverse. Here, the inversion theorem is proved, and an error in the previous literature is corrected.


Canonical coordinates for guiding center motion

J.D. Meiss, R.D. Hazeltine


We obtain, up to a quadrature, canonical coordinates for the Littlejohn guiding center Hamiltonian. The configuration variables are two angles, and the canonical momenta are angular momenta. These coordinates are valid for arbitrary magnetic fields including non-axisymmetric ones whose field lines may be chaotic. Examples of application of canonical coordinates to tokamak geometry are given.


Current sheets and nonlinear growth of the M=1 kink tearing mode

F.L. Waelbroeck


A calculation is presented that accounts for rapid nonlinear growth of the m=1 kink-tearing instability. The equilibrium analysis contained in the Rutherford theory [Phys. Fluids 16, 1903 (1973)] of nonlinear tearing-mode growth is generalized to islands for which the constant-ψ approximation is not valid. Applying the helicity-conservation assumption introduced by Kadomtsev [Plasma Physics and Controlled Nuclear Fusion Research (IAEA, Vienna, 1977), Vol. I, p. 555], the presence of a current-sheet singularity is shown that gives rise to a narrow tearing layer and rapid reconnection. This rapid reconnection, in turn, justifies the use of the helicity conservation assumption. The existence of a family of self-similar m=1 equilibrium islands is demonstrated. The formalism introduced here is shown to apply both to the case of the m=1 kink-tearing mode and to the case of forced reconnection. These two cases are compared and contrasted.


Monte Carlo simulation study of short-range correlations between itinerant hydrogen in lattice fields: application to cold fusion

S. Ichimaru, S. Ogata, A. Nakano, H. Iyetomi, T. Tajima


We perform Monte Carlo simulation study for short-range correlations between itinerant hydrogen, interacting mutually via electron-screened repulsive forces, in periodic and aperiodic (due to defects) lattice fields of metal hydrides. We find that the screening potentials and the resultant fusion rates depend extremely sensitively on microscopic details in the lattice fields, corroborating qualitatively the varied results in recent “cold fusion” experiments.


Saturation of a single mode driven by an energetic injected beam III: Alfven wave problem

H.L. Berk, B.N. Breizman


The saturation amplitude of Alfven waves, excited by alpha particles produced in an ignited tokamak, is estimated. The formalism that has been developed to describe the saturation of a single mode is generalized to toroidal geometry. The saturation level is estimated for the toroidal Alfven gap mode. The alpha particle radial flux resulting from the finite wave amplitude is found to produce relatively weak energy losses compared to the usual energy drag losses.


Saturation of a single mode driven by an energetic injected beam II: Electrostatic "universal" destabilization mechanism

H.L. Berk, B.N. Breizman


The formalism to describe the saturation of a discrete mode that is destabilized by hot particles fed by neutral beam injection is extended. The destabilization mechanism described in this work arises from the density gradient in the distribution function formed from a spatially inhomogeneous source. Energetic particles are injected at a fixed speed and collisionally relax through drag and pitch-angle scattering with the background plasma. The distribution formed is solved self-consistently in the presence of a finite amplitude wave in a sheared magnetic field. Three regimes of collisionality are found and the expressions for the nonlinear wave--particle power transfer is determined in each regime. With the dissipation processes of the background plasma given, the wave saturation level is then determined. When pitch-angle scattering is sufficiently weak, particles trapped in a wave convect across the magnetic field as they slow down, a phenomenon similar to the Ware pinch.


Saturation of a single mode driven by an energetic injected beam I: Plasma wave problem

H.L. Berk, B.N. Breizman


A formalism is established for calculating the saturation level of a discrete mode that is destabilized by the distribution function formed by a high-energy injected beam. The electrostatic plasma wave interaction is studied here for two problems. In one the distribution function is formed by injection of a source with a velocity spread and a steady-state bump-on-tail instability is established with only particle annihilation taken into account. In the second problem particle drag as well as particle annihilation is accounted for. In both problems the self-consistent distribution function in the presence of a finite amplitude wave needs to be calculated. By calculating the power transfer between particles and a finite amplitude wave, the saturation level of the discrete mode can be predicted. The drag problem with annihilation has the interesting feature that in steady state holes in phase space are formed for a large enough amplitude wave and the power transferred from particles to waves can be greatly enhanced as a result of the drag force on the holes.


Self-similar evolution of nonlinear magnetic buoyancy instability

K. Shibata, T. Tajima, Matsumoto


A program derivation support system called Focus is being constructed. It will formally derive programs using the paradigm of program transformation. The following issues are discussed: (1) the integration of validation and program derivation activities in the Focus system; (2) its tree-based user interface; (3) the control of search spaces in program derivation; and (4) the structure and organization of program derivation records. The inference procedures of the system are based on the integration of functional and logic programming principles. This brings about a synthesis of paradigms that were heretofore considered far apart, such as logical and executable specifications and constructive and transformational approaches to program derivation. A great emphasis has been placed, in the design of Focus, on achieving small search spaces during program derivation. The program manipulation operations such as expansion, simplification and rewriting were designed with this objective. The role of operations that are expensive in search spaces, such as folding, has been reduced. Program derivations are documented in Focus in a way that the high level descriptions of derivations are expressed only using program level information. All the meta-level information, together with dependencies between derivations of program components, is automatically recorded by the system at a lower level of description for its own use in replay.


Resonances - The devil's staircase and transport in area-preserving maps

Q. Chen


Chaotic transport in few degree of freedom Hamiltonian systems is of considerable importance in various fields, including plasma confinement, accelerator physics, intramolecular dynamics, celestial mechanics, and condensed matter physics. In two degree of freedom systems represented by area-preserving maps, we introduce a Markov transport model to describe various statistical properties in the irregular components where there are no invariant tori to prevent global transport. States of the Markov chain are regions delineated by the stable and unstable manifolds of the hyperbolic periodic orbits. We show that resonances give a complete partition of the phase space in the supercritical regime so that almost all points in the phase space are identified with particular resonance states. We then apply the Markov model to a purely chaotic system--the sawtooth map, and derive exact analytic results for transport rates. These are compared with the numerical rates. We conclude that in the chaotic regime, the Markov model gives reasonably good predictions for transport properties in the irregular components. In order to calculate transport rates for real systems in the chaotic regime, we need to locate highly unstable orbits. We develop a numerical method, the orbit extension method, for finding both unstable ordered periodic orbits and the principal heteroclinic orbits between two resonances. This method actually takes advantage of the instability in the chaotic regime and gives both a stable and an efficient prescription for finding unstable orbits.


Resonances and transport in the sawtooth map

Q. Chen, I. Dana, J.D. Meiss, N.W. Murray, I.C. Percival


We study transport in a completely chaotic Hamiltonian system, the hyperbolic sawtooth map. Analytical expressions are obtained for its cantori and resonances. We show that resonances give a complete partition of phase space. The flux leaking out of a resonance is given by its turnstiles, whose form and areas are obtained analytically. When the total flux out of a resonance becomes one third the area of an island, the topology of the turnstiles changes. At the same parameter value, a horseshoe is formed corresponding to the orbits trapped within the resonance. Based on this, a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained. The partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles. We calculate the survival probability within a resonance using the Markov model; the results are compared with results obtained numerically and from periodic-orbit theory.


Reductive perturbation method for quasi one-dimensional nonlinear wave propagation II: Application to magnetosonic waves

T. Taniuti, A. Hasegawa


By means of an application of the generic theory developed in the first paper of this series, the system of two-fluid magnetohydrodynamic equations is reduced to the three-dimensional Kadomtsev-Petviashvili equation. Stability conditions of the fast and slow magnetosonic solitons for transverse slow modulations are established for an isothermal plasma.


Two-stream instability threshold of the migma exyder (Published as: Microinstability threshold of an axially focused neutralized ion beam)

O. Agren, H.L. Berk, H.V. Wong


The response to electrostatic perturbations, φ=cflx φ(r, z)ei(lθ-ωt), is investigated for the Migma Exyder, a proposed disk-shaped device where ions are trapped in the periphery and focused towards the axis (Nucl. Instrum. Methods A 271, 214 (1988)). The ions are modeled as a relativistic monoenergetic distribution with a small spread in the impact parameter (the distance of closest approach to the axis) and with the axial extent ΔZ much shorter than the radial extent R0. It is found that odd l modes do not give rise to a low-frequency response. Forl=0, an energy principle for low frequency ion--ion interactions is formulated, and it is shown that the extremizing perturbation is located in a narrow region surrounding the axis. The threshold is proportional to the total number of stored particles, N = 4ε 0(π-2)γ(γ2-1) x mic2iR0/q2i, where γ is the usual relativistic factor, and this stability condition therefore allows for a high density on axis in highly focused systems. Compared to a cylindrical shape, a disk-shaped plasma gives an improvement in the density by the factor R0/ΔZ. The threshold in particle number for the relativistic ion--ion two-stream instability scales as γ3 and thus with relativistic effects the instability is further suppressed. Inclusion of the electron response increases the threshold even further. Preliminary investigations indicate that finite frequency l=0 modes can give a lower threshold than the zero frequency case, at least for nonrelativistic beam energies. These finite frequency instabilities can arise from coupling between positive and negative energy modes.


Amplitude equations on unstable manifolds: Singular behavior from neutral modes

J.D. Crawford


Perturbation theories that expand in the amplitudes of the unstable modes are an important tool for analyzing the nonlinear behavior of a weak instability which saturates in a final state characterized by small mode amplitudes. If the unstable mode couples to neutrally stable modes, such expansions may be singular because nonlinear effects are very strong even in the regime of weak instability and small amplitudes. Two models are discussed that illustrate this behavior; in each case the unstable mode corresponds to a complex conjugate eigenvalue pair in the spectrum of the linearized dynamics. In the first model, there is only a single neutral mode corresponding to a zero eigenvalue. This example is first solved exactly and then using amplitude expansions. The Vlasov equation for a collisionless plasma is the second model; in this case there are an infinite number of neutral modes corresponding to the van Kampen continuous spectrum. In each of the two examples, the neutral modes sharply reduce the size of the resulting nonlinear oscillation. For the Vlasov instability, the amplitude of the saturated mode is predicted to scale like γ2 where γ is the linear growth rate.


Effect of shear on toroidal ion temperature gradient mode turbulence

B.G. Hong, W. Horton


The effect of magnetic shear on the toroidal ion temperature gradient-driven drift mode is investigated through two-dimensional fluid simulations. Shear reduces the anomalous thermal transport by localizing the turbulence. Mixing length formulas for anomalous ion thermal transport are derived in various regimes with respect to the strength of the magnetic shear.


A study of runaway electron confinement and theory of neoclassical MHD turbulence

O.J. Kwon


Two major studies are presented: a study of runaway electron confinement and a theory of neoclassical MHD turbulence. The aim of the former is to study the structure of internal magnetic turbulence in tokamaks, which is thought by many to be responsible for the heat transport. The aim of the latter is to extend existing theories of MHD turbulence in tokamaks into experimentally relevant low collisionality regimes. This section contains a theory of neoclassical pressure gradient driven turbulence and a theory of neoclassical resistivity gradient driven turbulence.


Theory of high-n toroidicity-induced shear Alfven eigenmodes in tokamaks

G.Y. Fu, C.Z. Cheng


A high-n WKB-ballooning mode equation is employed to study toroidicity-induced shear Alfvén eigenmodes (TAE) in s-α space, where s=(r/q)(dq/dr) is the magnetic shear and α =–(2Rq2/B2)(dp/dr) is the normalized pressure gradient for tokamak plasmas. In the ballooning mode first stability region, TAE modes are found to exist only for alpha less than some critical value αc. It is found that these TAE modes reappear in the ballooning mode second stability region for bands of α values. The global envelope structures of these TAE modes are studied by the Wentzel–Kramers–Brillouin (WKB) method and are found to be bounded radially if the local mode frequency has a maximum in radius.


Nonlinear behavior of the magnetohydrodynamic modes near marginally stable states II: application to the resistive fast interchange mode

N. Nakajima, S. Hamaguchi


With the use of the general formulation developed in an earlier paper, the nonlinear evolution of the resistive fast interchange mode near the marginally stable state is obtained analytically. The nonlinear amplitude equation of the mode is shown to be of the Landau type. It is also shown that there is a stable equilibrium bifurcating from the initial equilibrium. Comparing this analytical result to numerical simulations, it is confirmed that the saturation level and the saturation time are well estimated by this Landau type of nonlinear amplitude equation.


Fluctuation spectrum and transport from ion temperature gradient driven modes in sheared magnetic fields

S. Hamaguchi, W. Horton


The ion temperature gradient driven mode or ηi-mode turbulence is reinvestigated based on two-component compressible fluid equations with the polarization drift velocity and adiabatic electrons. The scaling of the anomalous ion heat conductivity with magnetic shear s=Ln/Ls and the excess of ηi over the critical value ηi,c for marginal stability is found to vary as χi=g(ρs/Ln)(cT/eB) (ηi−ηi,c)exp(−αs), where g≈1 and α≈5.


Chaotic transport in Hamiltonian dynamical systems with several degrees of freedom

H. Kook


The dynamics and transport in the phase space of Hamiltonian systems with N + 1 degrees of freedom are studied. Such systems can be reduced to 2N-dimensional symplectic maps using Poincare's surface of section method. A 2N-dimensional symplectic map which satisfies the twist condition can be derived from a Lagrangian generating function, which enables the action formulation for orbits. Reversible maps of this form have 2rm N+1 invariant symmetry sets. We show that orbits of reversible, symplectic, twist maps can be classified by frequency, symmetry, and Morse index. The properties of orbits in each class are studied, including various bifurcations. The structure of phase space is viewed via resonances and chaotic layers using a four-dimensional example. A fast and stable Newton's algorithm for finding periodic orbits, based on the variational method, is devised for a class of Lagrangian mappings. Its computation time and required storage space are shown to be linear in the period. Partial barriers in the chaotic regions of phase space and Markov models for transport in area preserving maps, as well as the existence of co-dimension one manifolds for higher dimensional maps and their roles in transport are discussed. The characteristic function method is used to obtain the diffusion tensor for 2N-dimensional symplectic maps. At lowest order this method gives the quasilinear result, and a series in higher order correlations is developed. Comparison of the theory to numerical experiments, using a four-dimensional example, shows good agreement for moderately large parameters. A study of Arnol'd diffusion for the thick layer case shows that the short time correlations in one canonical plane affect the diffusion in the other plane even in the limit of zero coupling. Accelerator modes exist for doubly periodic maps, and cause enhancements in the diffusion. The long -time behavior of the correlation function is also discussed.


back to top