**Large acceleration of electrons by the microstructure of quasi-normal shocks **

Charles Chiu, Wendell Horton, and Yukiharu Ohsawa

**Abstract**

Orbits of high speed test electrons are analyzed for their acceleration by quasi-normal shocks, where the parameters of the electromagnetic fields of the shocks are motivated by recent cluster mission data. Numerical simulations in this regime leads to the following two observations. First, for a given electron initial speed, by adjusting the quasi-normal angle the gyrating electron trajectory may be trapped along the shock front over a long range, and the electron may gain large energy. Second, this energy gain is predominantly associated with the kinetic energy parallel to the magnetic field. Based on a simple analytic model together with the nucleon-to-electron-mass *(M/m)*-scaled models, one may deduce an analytic expression which gives the maximum energy in terms of the optimal quasi-normal angle. The same approach also gives a understanding for the growth of the parallel kinetic energy. Since the ejected electron has a small pitch angle and thus follows the magnetic field line in the foreshock region. Simulation of electrons with uniform angular distributions on a fixed energy shell show that electrons from a substantial part of the shell acquire energies which are over one-half of the maximum energy. This suggests that quasi-normal shocks provide a mechanism to accelerate high-speed electrons to higher energy.

**A Model for Multi-Wave Beam-Plasma Interaction**

Evstati Evstatiev

**Abstract**

A system that describes the interaction of an electron beam, plasma waves, and electromagnetic waves in a cold plasma is presented and studied. A multiwave model is developed that allows for e cient computational and analytical study. The model is based on the slow amplitude and phase change approximations. Using a Lagrangian approach, the continuous system of electron beam, background plasma, and waves is reduced to a nite degree-of-freedom system. This model, describes an e cient energy transfer mechanism between electromagnetic waves and the plasma wave, via the particles trapped in the plasma wave. It is suggested that this energy transfer be used in plasma-based accelerators to further increase the energy of the accelerated particles. Numerical and analytical studies of this mechanism are performed and an experimental test is proposed.

**Self-Consistent Dynamics of Nonlinear Phase Space
Structures **

Denis Eremin

**Abstract**

This thesis investigates the self-consistent dynamics of nonlinear "hole" and "clump" phase space structures and the nonlinear modes supported by the structures in the presence of dissipation due to the background plasma. A system consisting of a single mode driven by a weakly destabilizing distribution function in a dissipative medium close to the threshold of linear instability exhibits explosive instability. This instability results in the formation of the phase space structures and the corresponding modes. The holes and clumps were expected to persist for an appropriate collisional time scale. However, for certain initial conditions Fokker-Planck calculations for the nonlinear system abruptly break down in the course of the calculation. We find that this is because an adiabatic description of phase space structures at zero collisionality does not necessarily lead to continual adiabatic frequency sweeping.

**Nonlinear physics of laser-irradiated micro-clusters**

B. N. Breizman, Alexey V. Arefiev, and Mykhailo V. Fomyts'kyi

**Abstract**

A nonlinear theory has been developed to describe electron response and ion acceleration in dense clusters that are smaller in size than the laser wavelength. This work is motivated by high-intensity laser-cluster interaction experiments. The theory reveals that the breakdown of quasi-neutrality affects the cluster dynamics in a dramatic way: the laser can create a positively charged ion shell that expands due to its own space charge much faster than the central part of the cluster. The developed theory also shows a trend for the electron population to have a two-component distribution function: a cold core that responds to the laser field coherently and a hot halo that undergoes stochastic heating. The hot electrons expand together with the equal number of ions that are accelerated to supersonic velocities in a double layer at the cluster edge. This mechanism produces fast ions with energies much greater than the ponderomotive potential and it suggests that larger deuterium clusters can significantly enhance the neutron yield in future experiments.

**Meanders and Reconnection-Collision Sequences in the Standard Nontwist Map**

A. Wurm, A. Apte, K. Fuchss, and P. J. Morrison

**Abstract**

New global periodic orbit collision/separatrix reconnection scenarios in the standard nontwist map in different regions of parameter space are described in detail, including exact methods for determining reconnection thresholds that are implemented numerically. The results are compared to a break-up diagram of shearless invariant curves. The existence of meanders (invariant tori that are not graphs) is demonstrated numerically for *both* odd and even period reconnection for certain regions in parameter space, and some of the implications on transport are discussed.

**Magnetohydrodynamic equilibrium and stability of rotating plasmas in a mirror geometry**

A. Y. Aydemir

**Abstract**

A systematic study of equilibrium and stability of rotating plasmas in a mirror geometry is presented. At supersonic azimuthal rotation velocities, centrifugal forces lead to equilibria with nearly detached plasmas, where the plasma mass concentrates at the mirror center. However, isolation of the plasma from the ends is not complete due to high parallel thermal conduction. Temperature remains a flux function, connecting the centrifugally confined region to the end points. Rotation shear is found to be strongly stabilizing for the interchange (flute) modes, as demonstrated in earlier works. However, all configurations studied to date with various rotation profiles are found to be linearly unstable to other ideal magnetohydrodynamic (MHD) modes. Thus, it is unlikely that these centrifugally detached states can be accessed within ideal MHD.

**On acoustic wave generation in uniform shear flow**

G. Gogoberidze, P. J. Morrison, and M. Akhalkatsi

**Abstract**

The linear dynamics of acoustic waves and vortices in uniform shear flow is studied. For flows with very low shear rates, the dynamics of perturbations is adiabatic and can be described by the WKB approximation. However, for flows with moderate and high shear rates the WKB approximation is not appropriate, and alternative analysis shows that two important phenomena occur: acoustic wave over-reflection and wave generation in shear flows, a mechanism that is related to the continuous spectrum that arises in linear flow dynamics. A detailed analytical study of these phenomena is performed and the main quantitative and qualitative characteristics of the radiated acoustic field are obtained and analyzed.

**MHD scenario of plasma detachment
in a magnetic nozzle**

Alexey V. Arefiev, B. N. Breizman

**Abstract**

Some plasma propulsion concepts rely on a strong magnetic field to guide the plasma flow through the thruster nozzle. The question then arises of how the magnetically confined plasma can detach from the spacecraft. This work presents an MHD detachment scenario in which the plasma flow stretches the magnetic field lines to infinity. Detachment takes place after the energy density of the expanding magnetic field drops below the kinetic energy density of the plasma. As plasma flows along the magnetic field lines, the originally sub-Alfvénic flow becomes super-Alfvénic; this transition is similar to what occurs in the solar wind. In order to describe the detachment quantitatively, the ideal MHD equations have been solved for a cold plasma flow in a paraxial nozzle. The solution exhibits a well-behaved transition from sub- to super-Alfvénic flow inside the nozzle and a rarefaction wave at the edge of the outgoing flow. It is shown that efficient detachment is feasible if the nozzle is sufficiently long.

**Thermal density fluctuations and correlations in homogeneous plasmas **

R. D. Hazeltine and S. M. Mahajan

**Abstract**

The spatial correlation function for thermal plasma density fluctuations is computed from the plasma entropy. The method is demonstrated by three examples: a Maxwellian plasma, a strongly magnetized plasma, and a plasma dominated by Coulomb collisions. In each case the entropy is computed from the one-particle distribution function and then, following the Einstein method, used to construct the probability distribution for density fluctuation.

**Fluid description of ion dynamics in a toroidally confined plasma **

Abinadab Dieter, R. D. Hazeltine

**Abstract**

Fluid equations describing ion dynamics in a toroidally confined plasma at low collision frequency are derived. The ions are assumed to be magnetized in the sense that relevant scale lengths are much longer than the ion gyroradius, and time scales of interest are assumed long compared to the ion bounce time. These assumptions are consistent with, for example, the evolution of unstable magnetic islands, as well as conventional transport. A special case of the present description is the quasistatic, axisymmetric state with nearly uniform pressure and density on flux surfaces. In that case the equations reproduce the radial ion heat transport predicted by neoclassical transport theory. The essential feature of our derivation is its emphasis on heat flow in the direction of the magnetic field.

**Separatrix Reconnection and Meanders in the Standard
Nontwist Map **

A. Wurm, A. Apte, K. Fuchss, P.J. Morrison

**Abstract**

Contribution to the Proceedings of the 31st European Physical Society Conference on Plasma Physics, 28th June to 2nd July 2004, Imperial College, London.

**An Exact Nonlinear Hall-MHD Waves **

S. M. Mahajan, V. Krishan

**Abstract**

A fully nonlinear three-dimensional propagating solution of Hall MHD is constructed. In the appropriate limcits, the solution reprodues the known nonlinear Alfvénic state and the circularly polarized Whistler states.

**Vorticity probes and the characterization of vortices in the Kelvin-Helmholtz instability in the LArge Plasma Device (LAPD) experiment **

W. Horton, J. Perez, T. Carter, and R. Bengtson

**Abstract**

A new five-pin probe design called the Vorticity Probe is presented that explicitly measures the vorticity in the ** Ε** x

**Renormalization for breakup of invariant tori**

A. Apte, A. Wurm, and P. J. Morrison

**Abstract**

We present renormalization group operators for the breakup of invariant tori with winding numbers that are quadratic irrationals. We find the simple fixed points of these operators and interpret the map pairs with critical invariant tori as critical fixed points. Coordinate transformations on the space of maps relate these fixed points, and also induce conjugacies between the corresponding operators.

**Hamiltonian formulation and coherent structures in electrostatic turbulence**

F. L. Waelbroeck, P. J. Morrison, W. Horton

**Abstract**

A Hamiltonian formulation is constructed for a finite ion Larmor radius fluid model describing ion temperature-gradient driven and drift Kelvin-Hemholtz modes. The Hamiltonian formulation reveals the existence of three invariants obeying detailed conservation properties, corresponding roughly to generalized potential vorticity, internal energy, and ion momentum parallel to the magnetic field. These three invariants are added to the energy to form a variational principle that describes coherent structures, such as monopolar and dipolar vortices or modons. It is suggested that the invariants are responsible for the coherence and longevity of coherent structures and for their robustness during binary collisions.

**Collisionless magnetic reconnection with arbitrary guide-field**

Richard Fitzpatrick and Franco Porcelli

**Abstract**

A new set of reduced equations governing two-dimensional, two-fluid, collisionless magnetic reconnection with arbitrary guide-field is derived. These equations represent a significant advance in magnetic reconnection theory, since the existing reduced equations used to investigate collisionless reconnection are only valid in the large guide-field limit. The new equations are used to calculate the linear growth-rate of a strongly unstable, spontaneously reconnecting, plasma instability, as well as the general linear dispersion relation for such an instability.

**Electron Transport and the Critical Temperature Gradient**

W. Horton, G. T. Hoang, C. Bourdelle, X. Garbet, M. Ottaviani, L. Colas

**Abstract**

Tore Supra electron thermal fluxes, analyzed over a range of heating powers and plasma densities, are
shown to vary parametrically according to the small-scale electron temperature gradient (ETG)
model, rather than the ion inertial scale electrostatic gyro-Bohm model. Steady-state power balance
analysis and time-varying interpretative transport simulations, performed on the Tore Supra Fast
Wave Electron Heating (FWEH) database, validate the ETG thermal flux-versus-gradient relation and
the existence of a critical electron temperature gradient. The critical gradient length R/L_{c} and the
parametric dependence of the thermal flux, q_{e}(n_{e}, T_{e}, ∇T_{e}, q, s), agree well with the ETG model.

**Observation of tearing mode deceleration and locking due to eddy currents induced in a conducting shell**

B. E. Chapman, R. Fitzpatrick, D. Craig, P. Martin, and G. Spizzo

**Abstract**

Growth to the large amplitude of a single core-resonant tearing mode in the Madison Symmetric Torus [R. N. Dexter *et al.*, Fusion Technol. **19**, 131 (1991)] reversed-field pinch is accompanied by braking and eventual cessation of mode rotation. There is also a concurrent deceleration of bulk plasma rotation. The mode deceleration is shown to be well described by a time-dependent version of a magnetohydrodynamical model [R. Fitzpatrick *et al.*, Phys. Plasmas **6**, 3878 (1999)] in which a braking torque originates from eddy currents induced by the rotating mode in the conducting shell surrounding the plasma. According to the model, the electromagnetic braking torque is localized to the plasma in the immediate vicinity of the modeÕs resonant surface, but viscosity transfers the torque to the rest of the plasma. Parametrizing the plasma viscous momentum diffusivity in terms of the global momentum confinement time, the model is used to predict both the momentum confinement time and the time evolution of the decelerating mode velocity. In both respects, the model is quite consistent with experimental data.

**Scaling of forced magnetic reconnection in the Hall-magnetohydrodynamic Taylor problem**

Richard Fitzpatrick

**Abstract**

Two-dimensional, incompressible, zero guide-field, nonlinear Hall-MHD (magnetohydrodynamical)
simulations are used to investigate the scaling of the rate of forced magnetic reconnection in the
so-called Taylor problem. In this problem, a small-amplitude boundary perturbation is suddenly
applied to a tearing stable, slab plasma equilibrium; the perturbation being such as to drive magnetic reconnection within the plasma. This type of reconnection, which is not due to an intrinsic plasma instability, is generally known as ‘‘forced reconnection.’’ The inclusion of the Hall term in the plasma Ohm’s law is found to greatly accelerate the rate of magnetic reconnection. In the linear Hall-MHD regime, the peak instantaneous reconnection rate is found to scale like* d*Ψ/*dt~d*_{i}*η*^{ 1/3}Ξ_{0} where Ψ is the reconnected magnetic flux, *d _{i}* the collisionless ion skin depth

**The effects of presheath dynamics on rf sheaths**

N. Xiang and F. L. Waelbroeck

**Abstract**

A common approach to the study of rf sheaths is to separate the
description of the sheaths from that of the bulk plasma. In order to
solve the resulting set of equations, an appropriate boundary condition for the sheath model has to be specified at the sheath-plasma
boundary. In the existing sheath models, this boundary condition is
assumed to be stationary. In present paper, we investigate the ion
dynamics in both presheath region and sheath region by using numerical as well as analytical methods. It is found that the presheath
introduces an additional time scale ω* _{pre}* =

**Collisional sheath dynamics in the intermediate rf
frequency regime**

N. Xiang and F. L. Waelbroeck

**Abstract**

A sheath model is proposed for the case when the rf frequency ω is
comparable to or larger than the ion plasma frequency of the bulk plasma
ω* _{pi}* and the ion collisionality in the sheath is significant. In this case, the
ion momentum equation can be solved easily. We find that the ion velocity
in the sheath varies with time and the resulting ion energy distribution is
bimodal even though the rf frequency is much larger than the ion plasma
frequency in the sheath. The results of the model are compared with
the numerical solutions of the fluid equations. Both are in very good
agreement.

**Closed fluid description of relativistic, magnetized plasma interacting with radiation field**

R. D. Hazeltine and S. M. Mahajan

**Abstract**

A closed set of averaged fluid equations for a relativistic plasma immersed, simultaneously, in a slowly varying magnetizing field and a sharply varying electromagnetic field (radiation field, for example) of arbitrary intensity are derived. The modifications due to the radiation field on the plasma stress tensor and the Lorentz force are explicitly displayed. The resulting equations include the effects of radiation reaction as well as radiation pressure.

**Scaling of forced magnetic reconnection in the
Hall-magnetohydrodynamical Taylor problem with
arbitrary guide-field**

Richard Fitzpatrick

**Abstract**

Two-dimensional, nonlinear, Hall-magnetohydrodynamical (MHD) numerical simulations are used to investigate the scaling of the rate of forced magnetic reconnection in the so-called Taylor problem. In this problem, a small amplitude boundary perturbation is suddenly applied to a tearing stable, slab plasma equilibrium. The perturbation is such as to drive magnetic reconnection within the plasma. This type of reconnection, which is not due to an intrinsic plasma instability, is generally termed "forced reconnection." Hall effects are found to greatly accelerate the rate of magnetic reconnection, relative to the well-known Sweet–Parker rate. In the nonlinear Hall-MHD regime with arbitrary guide field, the peak reconnection rate is found to be independent of the resistivity, and to scale like dψ/dt~[β/(1 + β)]^{3/4}d_{i}^{3/2}Ξ_{0}^{2}, where beta is the plasma beta calculated using the guide field, di the collisionless ion skin depth, and Ξ_{0} the amplitude of the boundary perturbation. ©2004 *American Institute of Physics.*

**Radiation reaction in fusion plasmas**

R. D. Hazeltine, S. M. Mahajan

**Abstract**

The effects of radiation reaction on thermal electrons in a magnetically confined plasma, with parameters typical of planned burning plasma experiments, are studied. A fully relativistic kinetic equation that includes radiation reaction is derived. The associated rate of phase-space contraction is computed and the relative importance of radiation reaction in phase space is estimated. A consideration of the moments of the radiation reaction force show that its effects are typically small in reactor-grade confined plasmas, but not necessarily insignificant.

**On Reconnection Phenomena in the Standard
Nontwist Map**

A. Wurm, A. Apte, P. J. Morrison

**Abstract**

Separatrix reconnection in the standard nontwist map is described, including exact methods for determining the reconnection threshold in parameter space. These methods are implemented numerically for the case of odd-period orbit reconnection, where meanders (invariant tori that are not graphs) appear. Nested meander structure is numerically demonstrated, and the idea of meander transport is discussed.