**Ion orbit loss and the poloidal electric field in a tokamak **

H. Xiao, R.D. Hazeltine, P.M. Valanju

**Abstract**

Monte Carlo simulation studies for ion orbit loss in limiter tokamaks show a poloidal asymmetry in ion loss arising from differences in ion orbit geometry. Since electron loss to the limiter is uniformly distributed because of its tiny orbit width, the nonuniform ion loss could cause a poloidal electric field that would tend to make the ion loss to the limiter more uniform. A simple analytical derivation of this poloidal electric field and a discussion of its effects on ion movement and transport are also presented.

**Electron-neutrino phase separation instability**

C. Lai, T. Tajima

**Abstract**

Deduced from Weinberg-Salam electroweak theory, a Boltzmann equation and subsequent fluid equations are derived for the primordial electron-positron-neutrino-photon plasma. A collective instability that separates the phases of electrons (and positrons) and neutrinos (and anti-neutrinos) is discussed. Cosmological implications are mentioned.

**Large amplitude localized structures in a relativistic electron-positron ion plasma **

V.I. Berezhiana, S.M. Mahajan

**Abstract**

The nonlinear propagation of circularly polarized electromagnetic waves with relativistically strong amplitude in an unmagnetized cold electron-positron ion plasma is investigated. The possibility of finding soliton solutions in such a plasma is explored. It is shown that the presence of a small fraction of massive ions in the plasma leads to stable localized solutions.

**Helical temperature perturbations associated with tearing modes in tokamak plasmas **

R. Fitzpatrick

**Abstract**

An investigation is made into the electron temperature perturbations associated with tearing modes in tokamak plasmas. It is found that there is a critical magnetic island width below which the conventional picture where the temperature is flattened inside the separatrix is invalid. This effect comes about because of the stagnation of magnetic field lines in the vicinity of the rational surface and the finite parallel thermal conductivity of the plasma. Islands whose widths lie below the critical value are not destabilized by the perturbed bootstrap current, unlike conventional magnetic islands. This effect may provide an explanation for some puzzling experimental results regarding error field-induced magnetic reconnection. The critical island width is found to be fairly substantial in conventional tokamak plasmas, provided that the long mean-free path nature of parallel heat transport and the anomalous nature of perpendicular heat transport are taken into account in the calculation.

**Convective amplification of drift-acoustic waves in sheared flows **

F. Waelbroeck, J.Q. Dong, W. Horton, P. Yushmanov

**Abstract**

The evolution of wave packets is investigated in a cold-ion plasma model with sheared magnetic and velocity fields. Wave packets may be amplified by the drift Kelvin–Helmholtz mechanism even when the velocity shear is such that normal modes are stable. It is shown that the logarithm of the convective amplification can be an order of magnitude greater than the logarithm of the steady-state amplification often taken as the measure of convective instability. For a given wave number, the maximum of either of these amplifications decreases only as the inverse of the perpendicular component of the velocity shear.

**Particle dynamics and collisionless conductivity of the plasma sheet in the geomagnetic tail **

J. Ochoa

**Abstract**

A recurrent theme in magnetospheric research is that of the origin and quantification of the finite, collisionless electrical conductivity of the central plasma sheet in the geomagnetic tail. Outside the quasineutral layer the charged particle orbits are described by the guiding center classical theory and the plasma dynamics is given by ideal MHD theory. In the interior region where the magnetic field is weak and rapidly changing in direction, the particle orbits are complex, nonadiabatic, and receive a net acceleration from the dawn to dusk electric field (*E _{y}*) over the correlation time τ

**Low beta equilibrium and stability for anisotropic pressure closed field line plasma confinement systems **

V.P. Pastukhov, V.I. Ilgisonis, A.A. Subbotin

**Abstract**

A formalism has been developed to analyze the equilibrium and stability of low-beta anisotropic pressure plasmas confined in closed field line magnetic systems. This formalism, based on a joint use of paraxial (long-thin) and low-beta approximations, allows self-consistent equilibrium and stability consideration of rather general magnetic systems with nonuniform axis curvature and longitudinal profiles of toroidal and multipole poloidal field. Strong pressure anisotropy, corresponding to enhanced plasma pressure in mirror cells of the system, may also be considered. Nonconventional features of anisotropic pressure equilibria have been revealed. Application of the above formalism to the recently proposed linked mirror neutron source (LMNS) confirms the basic principles of the LMNS concept, but calculations based on this formalism have appreciably corrected some LMNS parameters. The LMNS longitudinal pressure profile and magnetic field distribution are optimized.

**Proof-of-principle experiments of laser wakefield ****acceleration**

K. Nakajima, T. Tajima

**Abstract**

A principle of the laser wakefield particle acceleration has been tested by the Nd : glass laser system with the peak power of 30 TW and the pulse duration of 1 ps. The particle acceleration up to 18 MeV/c has been demonstrated by injecting 1.0 MeV/c electrons emitted from a solid target by an intense laser impact. The corresponding field gradient achieves 1.7 GeV/m.

**Multifaceted asymmetric radiation from the edge of tokamak plasmas (MARFE): pattern formation under nonlocal constraints **

B. Meerson, N. Petviashvili, T. Tajima

**Abstract**

A simplified two-dimensional nonlinear fluid model is developed to elucidate the basic mechanism of the formation of multifaceted asymmetric radiation from the edge of tokamak plasmas (MARFE). In the framework of a mixed Eulerian–Lagrangian description, the problem is reduced to a reaction-diffusion-type equation with nonlocality, which obeys the constraints of length constancy and mass conservation along the magnetic field. With sufficient radiative cooling, this model predicts formation of MARFE-like plasma condensations from a variety of initial conditions.

**Momentum-energy transport from turbulence driven by parallel flow shear **

J.Q. Dong, W. Horton, R.D. Bengtson, G.X. Li

**Abstract**

The low-frequency E×B turbulence driven by the shear in the mass flow velocity parallel to the magnetic field is studied using the fluid theory in a slab configuration with magnetic shear. Ion temperature gradient effects are taken into account. The eigenfunctions of the linear instability are asymmetric about the mode rational surfaces. Quasilinear Reynolds stress induced by such asymmetric fluctuations produces momentum and energy transport across the magnetic field. Analytic formulas for the parallel and perpendicular Reynolds stress, viscosity, and energy transport coefficients are given. Experimental observations of the parallel and poloidal plasma flows on the Texas Experimental Tokamak Upgrade (TEXT-U) are presented and compared with the theoretical models.

**Electron physics and ambipolarity in the tokamak scrape-off layer **

R.D. Hazeltine, P.J. Catto

**Abstract**

Models of the electron behavior in the scrape-off layer (SOL) of diverted and limited boundary conditions that occurs tokamak plasmas must retain the abrupt change in boundary conditions that occurs across the separatrix or last closed flux surface as well as the electron reflecting Debye sheath established at the limiter or divertor plates. The balance between ion radial diffusion and streaming to the plates sets the SOL width and the electrons must adjust the Debye sheath at the plates to main tain quasineutrality and ambipolarity in the SOL beyond the last closed flux surface. We consider the long mean-free-path limit where a bounce-averaged kinetic electron model results in a steady-state balance in the SOL between radial diffusive feed from the core and velocity space diffusive loss to the plates due to collisional detrapping. In this double diffusion model a velocity space boundary layer occurs about the trapped-passing boundary where strong velocity space gradients must balance the streaming of the newly de-trapped electrons to the plates. The behavior of the electron distribution function in the velocity provides the information needed to evaluate the Debye-sheath-dependent electron loss rate.

**Collective effects of beam-beam interaction in a synchrotron collider **

J.K. Koga, T, Tajima

**Abstract**

The effects of the beam-beam interaction on particle dynamics in a synchrotrom collider are investigated. The main highlight of this work is the investigation of collective effects of the beam-beam interaction in a self-consistent approach that naturally incorporates the correct single particle dynamics. The most important target of this simulation is to understand and predict the long-time (10^{8} - 10^{9} rotations) behavior of the beam luminosity and lifetime. For this task a series of computer codes in one spatial dimension have been developed in increasing order of sophistication. They are: the single particle dynamics tracking code, the strong-strong particle-in-cell (PIC) code, and teh particle code based on the δ*f *algorithm. The later two include the single particle dynamics of the first. The third approach is used to understand beam lifetime by improving the numerical noise problem in the second. Scans in tune ν_{0} and tune shift Δν_{0 }show regions of stability and instability which correspond to the regions predicted by a linear theory. Strong resonance beam blowup is observed just above ν_{0} = 1/2 and ν_{0} = 1/4, where the rate of beam blowup drops witht he order of the resonance. In both the strong-strong code and δ*f* code using the reference parameters of the SSC (Superconducting Super Collider), oscillations in the tune shift, Δν, are observed. The odd moments of the beam are increasing in oscillation amplitude with rotation number, while the amplitudes of the even moments either decrease or remain constant. The "flip-flop" effect is observed in the strong-strong code simulations and is found to be sensitive to the initial conditions. In studying slow particle diffusion in the phase space of the beams away from resonances, the tracking code shows no diffusion of particles from the beam-beam interaction after 10^{5} rotations. The strong-strong code is found too noisy to study particle diffusion from the beam-beam interaction. The much quieter δ*f * code shows all particles diffusive after 10^{5} rotations in contrast to single particle tracking results. The diffusion coefficients are several orders of magnitude higher than the tracking code and increase exponentially with the action. However, this amount of diffusion (D ~ 10^{-10} - 10^{-11} in the normalized unit) is still permissible for the SSC design parameters. This diffusion is caused by the collision induced variation of the second moment of the beams⟨x^{2}⟩.

**Isotope scaling and η _{i} mode with impurities in tokamak plasmas **

J.Q. Dong, W. Horton, W.D. Dorland

**Abstract**

The ion temperature gradient- (ITG) driven instability, or η_{i} mode, is studied for discharges with hydrogen, deuterium, or tritium in a toroidal magnetic configuration. Impurity effects on the mode and the instability (impurity mode) driven by the presence of impurity ions with negative density gradient are studied. It is found that the maximum growth rate of the η_{i} mode scales as M_{i}^{ - 1/2} for pure hydrogenic plasmas, where M_{i} is the mass number of the working gas ion. With the inclusion of impurity ions, the growth rate of the η_{i} mode decreases in all three kinds of plasmas, with a hydrogen plasma still having the highest maximum growth rate, tritium the lowest, and deuterium in between. However, the isotope effects are weaker and scale as M_{eff}^{-1/2} with the presence of impurity ions, where the effective mass number, M_{eff}=(1−f_{z})M_{i}+f_{z}M_{z}, with M_{z} and f_{z}= Zn_{0z}/n_{0e} being the mass number and charge concentration of impurity ions, respectively. For the impurity mode, the scaling is similar to that of the η_{i} mode without impurity ions. The experimental database shows that the plasma energy confinement time scales as τ* _{E}*αM

**Effect of a static external magnetic perturbation on resistive mode stability in tokamaks **

R. Fitzpatrick, T.C. Hender

**Abstract**

The influence of a general static external magnetic perturbation on the stability of resistive modes in a tokamak plasma is examined. There are three main parts to this investigation. First, the vacuum perturbation is expanded as a set of well-behaved toroidal ring functions, and is, thereafter, specified by the coefficients of this expansion. Second, a dispersion relation is derived for resistive plasma instabilities in the presence of a general external perturbation, and finally, this dispersion relation is solved for the amplitudes of the tearing and twisting modes driven in the plasma by a specific perturbation. It is found that the amplitudes of driven tearing and twisting modes are negligible until a certain critical perturbation strength is exceeded. Only tearing modes are driven in low-β plasmas with εβ_{p} << 1. However, twisting modes may also be driven if εβ_{p}1. For error-field perturbations made up of a large number of different poloidal and toroidal harmonics the critical strength to drive locked modes has a ``staircase'' variation with edge-q, characterized by strong discontinuities as coupled rational surfaces enter or leave the plasma. For single harmonic perturbations, the variation with edge-q is far smoother. Both types of behavior have been observed experimentally. The critical perturbation strength is found to decrease strongly close to an ideal external kink stability boundary. This is also in agreement with experimental observations.

**A relativistic solitary wave in electron-positron ion plasma **

V.I. Berezhiana, S.M. Mahajan

**Abstract**

Relativistic solitary-wave propagation is studied in a cold electron-positron plasma embedded in an external arbitrary strong magnetic field. The exact, analytical, solitonlike solution corresponding to a localized, purely electromagnetic pulse with an arbitrarily large amplitude is found.

**Solitary waves and homoclinic orbits **

N.J. Balmforth

**Abstract**

The notion that fluid motion often organizes itself into coherent structures has increasingly permeated modern fluid dynamics. Such localized objects appear in laminar flows and persist in turbulent states; from the water on windows on rainy days, to the circulations in planetary atmospheres. This review concerns solitary waves in fluids. More specifically, it centers around the mathematical description of solitary waves in a single spatial dimension. Moreover, it concentrates on strongly dissipative dynamics, rather than integrable systems like the KdV equation. One-dimensional solitary waves, or pulses and fronts as they are also called, are the simplest kinds of coherent structure (at least from a geometrical point of view). Nevertheless, their dynamics can be rich and complicated. In some circumstances this leads to the formation of spatio-temporal chaos in the systems giving birth to the solitary waves, and understanding that phenomenon is one of the major goals in the theory outlined in this review. Unfortunately, such a goal is far from achieved to date, and the author assesses its current status and incompleteness.

**Comment on chaotic advection in quiet discharges in tokamaks **

S. Tian, W. Horton

**Stability of coupled tearing and twisting modes in tokamaks **

R. Fitzpatrick

**Abstract**

A dispersion relation is derived for resistive modes of arbitrary parity in a tokamak plasma. At low mode amplitude, tearing and twisting modes which have nonideal magnetohydrodynamical (MHD) behavior at only one rational surface at a time in the plasma are decoupled via sheared rotation and diamagnetic flows. At higher amplitude, more unstable "compound'' modes develop which have nonideal behavior simultaneously at many surfaces. Such modes possess tearing parity layers at some of the nonideal surfaces, and twisting parity layers at others, but mixed parity layers are generally disallowed. At low mode number, "compound'' modes are likely to have tearing parity layers at all of the nonideal surfaces in a very low-β plasma, but twisting parity layers become more probable as the plasma β is increased. At high mode number, unstable twisting modes which exceed a critical amplitude drive conventional magnetic island chains on alternate rational surfaces, to form an interlocking structure in which the O points and X points of neighboring chains line up.

**Decay of magnetic helicity producing polarized Alfven waves **

Z. Yoshida, S.M. Mahajan

**Abstract**

When a super-Alfvénic electron beam propagates along an ambient magnetic field, the left-hand circularly polarized Alfvén wave is Cherenkov emitted (two stream instability). This instability results in a spontaneous conversion of the background plasma helicity to the wave helicity. The background helicity induces a frequency (energy) shift in the eigenmodes, which changes the critical velocity for Cherenkov emission, and it becomes possible for a sub-Alfvénic electron beam to excite a nonsingular Alfvén mode.

**Dynamical nature of inviscid power law for 2D turbulences and self-consistent spectrum and transport of plasma filaments **

Y.Z. Zhang, S.M. Mahajan

**Abstract**

On the basis of equal-time correlation theory (a non-perturbative approach) inviscid power laws of 2D isotropic plasma turbulences with one Lagrangian inviscid constant of motion are unambiguously solved by determining the dynamical characteristics. Two distinct types of induced transport, according to the divergence of the inverse correlation length in the inviscid limit are revealed. This analysis also suggests a physically reasonable closure. The self-consistent system (a set of integral equations for plasma filaments is investigated in detail, and is found to be a nonlinear differential eigenvalue problem for the diffusion coefficient D, with the Dyson-like (integral) equation playing the tole of a boundary condition. this new type of transport is non-Bohm-like, and shows quasilinear behavior even in the strong turbulence regime. Physically, this behavior arises from a synchronization of the shrinking squared correlation length with the decorrelation time, for which the "mixing-length" breaks down. The shrinkage of correlation length is a characteristic pertaining to the new type of turbulence; its relationship with the turbulence observed in supershot regime on TFTR is commented on.

**Magnetless magnetic fusion **

A.D. Beklemishev, T. Tajima

**Abstract**

The authors propose a concept of thermonuclear fusion reactor in which the plasma pressure is balanced by direct gas-wall interaction in a high-pressure vessel. The energy confinement is achieved by means of the self-contained toroidal magnetic configuration sustained by an external current drive or charged fusion products. This field structure causes the plasma pressure to decrease toward the inside of the discharge and thus it should be magnetohydrodynamically stable. The maximum size, temperature and density profiles of the reactor are estimated. An important feature of confinement physics is the thin layer of cold gas at the wall and the adjacent transitional region of dense arc-like plasma. The burning condition is determined by the balance between these nonmagnetized layers and the current-carrying plasma. They suggest several questions for future investigation, such as the thermal stability of the transition layer and the possibility of an effective heating and current drive behind the dense edge plasma. The main advantage of this scheme is the absence of strong external magnets and, consequently, potentially cheaper design and lower energy consumption.

**Magnetically constricted intergalactic plasmas **

T. Tajima

**Abstract**

A model of magnetically constricted hot intergalactic plasmas as a source of the cosmic X-ray background radiation satisfies the known observational constraints. The network magnetic fields that weave through clusters of galaxies are strongly constricted by the violent relaxation of the clusters in the supercluster potential. These intercluster fields tend to constrict the trapped plasma, driving them to high densities and high temperatures. These hot (T ≥ 10^{8}K) and dense plasmas are magnetically insulated from colder (T ≤ 10^{4}K) surrounding gases, forming intermittent intercluster medium. The dynamical processes of these fields involve rapid magnetic relaxation toward the nearly force-free state by involving reconnection of field lines and rapid heating of plasmas by being continuously fed energy from the violent gravitational relaxation. The fundamental physical processes of magnetic constriction and subsequent plasma heating by the violent motions of compact objects that trap the magnetic fields are elucidated. Brightening regions of such magnetically constricted plasmas have typical dimensions of order the size of clusters or even less, thus they will be seen as a diffuse X-ray source. This model explains the large amount of necessary thermal energy that results in cosmic X-ray background radiation in a large supercluster spatial scale, the rapid heating, the small amount of deviation of the cosmic microwave background radiation due to the Comptonization, and how to keep colder gases from evaporating. Compatible with this model is the primordial plasma could sustain a large amount of spontaneously generated magnetic fields and thus isothermal density fluctuations with little temperature signatures. We further consider the evolution of such generated magnetic fields by dynamo in the epoch following this and preceding the above X-ray forming epoch. Using the ABC dynamo model, we obtain the cellular morphology of magnetic fields in the expanding Universe.

**Effect of a nonuniform resistive wall on the stability of tokamak **

R. Fitzpatrick

**Abstract**

The effect of a nonuniform resistive wall on the stability of plasma magnetohydrodynamical (MHD) modes is examined. For the case of a tokamak plasma interacting with a wall possessing toroidally nonuniform electrical resistance, the kink mode dispersion relation is found to reduce to a surprisingly simple form, provided that the scale of variation of the resistance is sufficiently large. The influence of a wall with toroidal gaps on tokamak plasma stability is investigated in some detail. Under some circumstances, kink modes are found to ``explode'' through the gaps with ideal growth rates. A similar investigation is made for a modular wall constructed of alternate thick and thin sections.

**Quasi-two-dimensional dynamics of plasmas and fluids **

W. Horton, A. Hasegawa

**Abstract**

In the lowest order of approximation quasi-two-dimensional dynamics of planetary atmospheres and of plasmas in a magnetic field can be described by a common convective vortex equation, the Charney and Hasegawa-Mima (CHM) equation. In contrast to the two-dimensional Navier-Stokes equation, the CHM equation admits "shielded vortex solutions" in a homogeneous limit and linear waves ("Rossby waves" in the planetary atmosphere and "drift waves" in plasmas) in the presence of inhomogeneity. Because of these properties, the nonlinear dynamics described by the CHM equation provide rich solutions which involve turbulent, coherent and wave behaviors. Bringing in nonideal effects such as resistivity makes the plasma equation significantly different from the atmospheric equation with such new effects as instability of the drift wave driven by the resistivity and density gradient. The model equation deviates from the CHM equation and becomes coupled with Maxwell equations. This article reviews the linear and nonlinear dynamics of the quasi-two-dimensional aspect of plasmas and planetary atmosphere starting from the introduction of the ideal model equation (CHM equation) and extending into the most recent progress in plasma turbulence.

**Electron pairing mediated by coulomb repulsion in a periodic potential **

S.M. Mahajan, A. Thyagaraja

**Abstract**

It is shown that the binary Coulomb repusion between two electrons can lead to spatially localized pair states (preformed pairs) in the periodic potential of a crystalline solid. These composite bosonic states, residing in the erstwhile forbidden gaps, could further our understanding of the solid state, in particular, of the high temperature superconducting state.

**The energy of perturbations for Vlasov plasmas **

P.J. Morrison

**Abstract**

The energy content of electrostatic perturbations about homogeneous equilibria is discussed. The calculation leading to the well-known dielectric (or as it is sometimes called, the wave) energy is revisited and interpreted in light of Vlasov theory. It is argued that this quantity is deficient because resonant particles are not correctly handled. A linear integral transform is presented that solves the linear Vlasov–Poisson equation. This solution, together with the Kruskal–Oberman energy [Phys. Fluids 1, 275 (1958)], is used to obtain an energy expression in terms of the electric field [Phys. Fluids B 4, 3038 (1992)]. It is described how the integral transform amounts to a change to normal coordinates in an infinite-dimensional Hamiltonian system.

**Hamiltonian description of the ideal fluid **

P.J. Morrison

**Abstract**

The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable when nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods.

**L-H confinement mode dynamics in three-dimensional state space **

H. Sugama, W. Horton

**Abstract**

A dynamical model for the low-high (L-H) confinement mode transitions consisting of three ordinary differential equations (3-ODE model) for the essential state variables is proposed. The model is derived from the energy balance equations for the resistive pressure-gradient-driven turbulence and describes temporal evolutions of three characteristic variables (u, k, f), the potential energy contained in the pressure gradient, the turbulent kinetic energy and the shear flow energy. The energy input to the peripheral plasma region is included as an external control parameter in the model. The model equations have stationary solutions corresponding to the L- and H-modes. The L to H and H to L transitions are obtained by varying the energy input parameter. The type of L-H transition, whether like a first- or second-order transition, is shown to be determined by the sheer flow damping. At a higher level of the energy input parameter the H-mode stationary solution becomes unstable and bifurcates to a limit cycle which shows periodic oscillations characteristic of the H-localized mode (ELM) confinement state.

**Constructing symplectic maps for application to magnetostatics and Hamiltonian mechanics **

P.M. Abbamonte, P.J. Morrison

**Abstract**

Various methods of constructing two-dimensional, area preserving maps of general Hamiltonian systems are explored. Emphasis is on constructing maps with a given set of fixed points, a given invariant curve, or a given topology, and also on guaranteeing integrability. One method is used to find an integrable Poincare map for the field lines in a tokamak with a single null-divertor where the q-profiles can be arbitrarily chosen.

**Basic principles approach for studying nonlinear Alfven wave-alpha particle dynamics **

H.L. Berk, B.N. Breizman, M. Pekker

**Abstract**

An analytical model and a numerical procedure are presented which give a kinetic nonlinear description of the Alfven-wave instabilities driven by the source of energetic particles in a plasma. The steady-state and bursting nonlinear scenarios predicted by the analytical theory are verified in the test numerical simulation of the bump-on-tail instability. A mathematical similarity between the bump-on-tail problem for plasma waves and the Alfven wave problem gives a guideline for the interpretation of the bursts in the wave energy and fast particle losses observed in the tokamak experiments with neutral beam injection.

**Particle diffusion from the beam-beam interaction in synchrotron colliders **

J.K. Koga, T. Tajima

**Abstract**

We investigate the beam-beam interaction in a synchrotron collider, specifically studying slow particle diffusion in phase space away from tune resonances. Using the tune and tune shift of contemporary large hadron colliders as reference parameters, our computation shows all particles diffusive after 10^{5} rotations in contrast to previous single particle tracking results. The diffusion coefficients are several orders of magnitude higher than the tracking code and increase exponentially with the action, caused by the collision induced variation of the second moment of the beams ⟨x^{2}⟩.