**Two and three-dimensional magnetoinductive particle codes with guiding center electron motion **

J.L. Geary, T. Tajima, J.N. Leboeuf, E.G. Zaidman, J.H. Han

**Abstract**

A magnetoinductive (Darwin) particle simulation model developed for examining low frequency plasma behavior with large time steps is presented. Electron motion perpendicular to the magnetic field is treated as massless keeping only the guiding center motion. Electron motion parallel to the magnetic field retains full inertial effects as does the ion motion. This model has been implemented in two and three dimensions. Computational tests of the equilibrium properties of the code are compared with linear theory and the fluctuation dissipation theorem. This code has been applied to the problems of Alfven wave resonance heating and twist-kink modes.

**The effect of energetic particles on stability of mirror and tokamak plasmas **

D. Stotler

**Abstract**

Effects of an energetic particle species on high-mode-number, curvature-driven instabilities in magnetic mirror and tokamak plasmas are studied. The author investigates whether or not these hot particles can stabilize the magnetohydrodynamic (MHD) ballooning mode by having magnetic drift velocities large enough that they do not respond on the usual time scale of the instability and consequently allow thermonuclear fusion devices to operate at higher, more efficient plasma pressures. However, the energetic particles themselves are subject to instabilities that limit the effectiveness of this procedure. Using an MHD particle simulation code, the stabilizing effect of a diamagnetic well formed by the energetic particles is demonstrated in an axisymmetric mirror by treating the hot species as a rigid current ring. The results match those predicted by an analytic theory based on the MHD equations. More general aspects of linear stability in mirrors containing energetic particles are examined through analysis of equations derived from the drift kinetic equation. Numerical techniques are used to show that a magnetic compressional instability can arise if the core plasma density is to high or if the hot particle pressure gradient is too large. A similar set of equations is solved numerically in a tokamak geometry to determine whether or not energetic particles will allow access to the desirable second stability region for MHD ballooning modes.

**Resonances in area-preserving maps **

R.S. Mackay, J.D. Meiss, I.C. Percival

**Abstract**

A resonance for an area-preserving map is a region of phase space delineated by ``partial separatrices'', curves formed from pieces of the stable and unstable manifold of hyperbolic periodic points. Each resonance has a central periodic orbit, which may be elliptic or hyperbolic with reflection. The partial separatrices have turnstiles like the partial barriers formed from cantori. In this paper we show that the areas of the resonances, as well as the turnstile areas, can be obtained from the actions of homoclinic orbits. Numerical results on the scaling of areas of resonances with period and parameter are given. Computations show that the resonances completely fill phase space when there are no invariant circles. Indeed, we prove that the collection of all hyperbolic cantori together with their partial barriers occupies zero area.

**Electron diffusion in tokamaks due to electromagnetic fluctuations **

W. Horton, D.I. Choi, P.N. Yushmanov, V.V. Parail

**Abstract**

Calculations for the stochastic diffusion of electrons in Tokamaks due to a spectrum of electro-magnetic drift fluctuations are presented. The parametric dependence of the diffusion coefficient on the amplitude and phase velocity of the spectrum, and the bounce frequency for the electrons is studied. The wavenumber spectrum is taken to be a low order (5*5) randomly-phased, isotropic, monotonic spectrum extending from k_{⊥min} ≈ ω_{ci}/c_{s} to k_{⊥to max} ≈ 3 ω_{pe}/c with different power laws of decrease phi_{k} ≈ phi_{1}/k^{m}, 1 ≤ m ≤ 3. A nonlinear Ohm's law is derived for the self-consistent relation between the electrostatic and parallel vector potentials. The parallel structure of the fluctuations is taken to be such that k_{||}^{nl}ν_{e} < ω_{k} due to the nonlinear perpendicular motion of the electrons described in the nonlinear Ohm's law. The diffusion coefficient scales approximately as the neo-Alcator and Merezhkin-Mukhovatov empirical formulas for plasma densities below a critical density.

**Theory of hot particle stability **

H.L. Berk, H.V. Wong, K.T. Tsang

**Abstract**

The investigation of stabilization of hot particle drift reversed systems to low frequency modes has been extended to arbitrary hot beta β_{H}, for systems that have unfavorable field-line curvature. Steep profile equilibria are considered where the thickness of the pressure drop Δ, is less than plasma radius r_{p}. The analysis describes layer modes which have mΔ/r_{p} < 1, where m is the mode number, and radial structure larger than Δ. Stabilization is classified as either being ``robust,'' where all excitations are positive energy, or ``fragile,'' where stability criteria exist to magnetohydrodynamic (MHD)-like and drift-compressional instabilities, but positive and negative energy waves are present (with the possibility that negative energy waves are destabilized by dissipative mechanisms). It is shown that in making a continuous transition from the fragile stability regime to the robust stability regime one must go through an unstable region. To bridge the unstable band in a physical manner one must either produce robust stability conditions very rapidly, or use transient stability techniques such as ponderomotive forces or transient minimum-B coils. The positive energy stabilization terms of the layer mode from wall stabilization terms and finite Larmor radius terms are explicitly exhibited.It is shown the robust stability can even be achieved with only wall stabilization for all possible m values of the layer modes if β_{H} > (2)/(3) . When robust stability conditions are fulfilled, the hot particles will have their bounce frequency less than their grad-B drift frequency. This allows for a low bounce frequency expansion to describe the axial dependence of the magnetic compressional response. Such as expansion provides for another negative energy source in the theory, with the system then being susceptible to the drift compressional instability if dβ_{H}/dr > dβ_{w}/dr, where β_{w} is the beta of the background plasma. If robust stabilization conditions are not fulfilled, the regions of fragile stability are extremely small if the MHD-like modes, the diamagnetic compressional and drift-compressional modes are to be simultaneously avoided.

**Turbulence in toroidally confined plasma-ion-temperature-gradient-driven turbulence; Dynamics of magnetic relaxation **

G.S. Lee

**Abstract**

This thesis is devoted to two studies of low-frequency turbulence in toroidally confined plasma. Low-frequency turbulence is believed to play an important role in anomalous transport in toroidal confinement devices. The first study pertains the development of an analytic theory of ion-temperature-gradient-driven turbulence in tokamaks. 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 is derived and is found to be consistent with experimentally-deduced ion thermal diffusivities. The associated electron thermal diffusivity, particle and heat-pinch velocities are also calculated. The second study is devoted to the role of multiple helicity nonlinear interactions of tearing modes and dynamics of magnetic relaxation in a high-temperature current carrying plasma. To extend the resistive MHD theory of magnetic fluctuations and dynamo activity observed in the reversed field pinch, the fluid equations for high temperature regime are derived and basic nonlinear interaction mechanism and the effects of diamagnetic corrections to the MHD turbulence theory are studied for the case of fully developed, densely packed turbulence.

**Particle simulations of current driven drift waves in shearless and sheared magnetic fields **

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

**Abstract**

The nonlinear behavior of the collisionless current-driven drift instability in shearless and sheared magnetic fields is studied by means of particle simulation. Electrostatic models with guiding center electrons and full dynamic ions are used in both two-and-one-half and three dimensions. The electron current (J_{||}), in the direction parallel to the magnetic field but perpendicular to the density gradient in the x direction, is maintained throughout the simulation. Instability thresholds, growth rates, real frequency spectra, and mode structures observed in the simulation are in good agreement with theory. Saturation of the unstable modes occurs by a flattening of the electron distribution function in x-v_{||} space. The measured potential saturation levels and final distribution functions are consistent with a quasilinear plateau theory for both shearless and sheared magnetic field configurations.

**Anchor stabilization of trapped particle modes in mirror machines **

H.L. Berk, G.V. Roslyakov

**Abstract**

It is shown that for trapped particle modes in tandem mirrors, the pressure of the passing particles in the anchor region introduces a stabilizing term proportional to the sum of the anchor's field line curvature and total diamagnetic pressure gradient. The theory is applied to the proposed gas dynamic trap experiment.

**Resistive fluid turbulence in diverted tokamaks and the edge transport barrier in H-mode plasmas **

T.S. Hahm, P.H. Diamond

**Abstract**

The thermal and particle diffusivities driven by resistive fluid turbulence in diverted tokamak edge plasmas are calculated. Diverted tokamak geometry is characterized by increased global shear near the separatrix and the tendency of field lines to linger near the x-point. For resistive fluid turbulence, the dominant effect is increased global shear, which causes a reduction in the effective step-size of the turbulent diffusion process and corresponding improvements in heat and particle confinement close to the separatrix. Stability of resistive kink modes resonant near separatrix is also ensured by the increased global shear. The relevance of these considerations to the L → H transition and to the edge transport barrier in H-mode plasmas is discussed.

**Enhanced electron stochasticity from electrostatic waves in a sheared magnetic field **

J.A. Robertson, W. Horton, D.I. Choi

**Abstract**

Electron motion in a single electrostatic wave in a sheared magnetic field is shown to become stochastic in the presence of a second wave at an amplitude well below that obtained from the overlapping pendulum resonance approximation. The enhanced stochasticity occurs for low parallel velocity electrons for which the parallel trapping motion from eE_{||}/m interacts strongly with the E×B trapping motion due to the presence of magnetic shear. The single-wave particle motion is given by a two-parameter family of one-degree-of-freedom Hamiltonians which bifurcate from a pendulum phase space to a topology with three chains of elliptic and hyperbolic fixed points separated in radius about the mode-rational surface. In the presence of a perturbing wave with a different helicity, electrons in the small parallel velocity regime become stochastic at an amplitude scaling as the fourth root of the wave potential. The results obtained for stochastic motion apply directly to the problem of the diffusion of electrons in drift waves.

**Simulation study of two-ion hybrid resonance heating **

S. Riyopoulos, T. Tajima

**Abstract**

A one-dimensional low-noise, low-frequency electromagnetic particle simulation code that is appropriate for investigation of ion-cyclotron resonance heating (ICRH) is developed. Retaining the hyperbolicity of the electromagnetic waves and exploiting nearly one-dimensional characteristics (perpendicular to the external magnetic field) of the ICRH, the guiding center electron approximation for the transverse electronic current calculation is used. Mode conversion of the incoming magnetosonic wave into the electrostatic ion–ion hybrid mode is observed accompanied by strong ion heating. The dependence of this heating on the different plasma parameters is examined through a series of simulations, focusing mainly on wave incidence from the high field side. Because k_{||}=0 in the runs, the conventional Landau damping cannot explain the ion heating. Nonlinear mechanisms for energy transfer are discussed. The numerical results demonstrate the importance of the nonlinear wave–particle interaction for energy absorption during radio-frequency heating in the ion-cyclotron regime.

**Class renormalization: Islands around islands **

J.S. Meiss

**Abstract**

An orbit of ''class'' is one that rotates about a periodic orbit of one lower class with definite frequency. This contrasts to the ''level'' of a periodic orbit which is the number of elements in its continued fraction expansion. Level renormalization is conventionally used to study the structure of quasi-periodic orbits. The scaling structure of periodic orbits encircling other periodic orbits in area preserving maps is discussed here. Fixed points corresponding to the accumulation of p/q bifurcations are found and scaling exponents determined. Fixed points for q > 2 correspond to self-similar islands. Frequencies of the island boundary circles at the fixed points are obtained. Importance of this scaling for the motion of particles in stochastic regions is emphasized.

**Destabilization of Alfven resonant modes by resistivity and diamagnetic drifts **

R.D. Hazeltine, A. Aydemir, J.D. Meiss, M. Kotschenreuther

**Abstract**

The existence of unstable resistive drift-Alfven modes is shown in a cylindrical geometry. The modes, driven by nonuniformities in the diamagnetic drift frequency, are unstable even when the associated purely resistive mode is stable. The growth rate has a weak, η^{1/4}, dependence on resistivity. Because of their large growth rates, they may play an important role in tokamak confinement.

**Reply to comments of J.A. Krommes on theory of dissipative density-gradient driven turbulence in the tokamak edge **

P. Terry, P.H. Diamond

**Abstract**

We appreciate the interest of Krommes in our recent paper and welcome the opportunity to discuss his comments and other related issues. In our opinion, most of the objections hea has raised follow from a misunderstanding of the physics treated by clump and hole theory. In particular, throughout his critique Krommes attempts to extrapolate results and intuition of homogeneous Navier-Stokes turbulence (HN-ST) to the more complicated case of dissipative drift-wave turbulence (DD-WT). Since these two cases are so dissimilar with regard to their fundamental constituents, drive, characteristic scales and interaction mechanisms, extrapolations from one case to the other are unwarranted and misleading. Moreover, the hypotheses and results of clump and hole theories have fared well in several tests using laboratory and simulation data which is relevant to the theoretical models analyzed.

**Muon catalyzed fusion-fission reactor driven by a recirculating beam **

S. Eliezer, T. Tajima, M.N. Rosenbluth

**Abstract**

The recent experimentally inferred value of multiplicity of fusion of deuterium and tritium catalyzed by muons has rekindled interest in its application to reactors. Since the main energy expended is in pion (and consequent muon) productions, we try to minimize the pion loss by magnetically confining pions where they are created. Although it appears at this moment not possible to achieve energy gain by pure fusion, it is possible to gain energy by combining catalyzed fusion with fission blankets. We present two new ideas that improve the muon fusion reactor concept. The first idea is to combine the target, the converter of pions into muons, and the synthesizer into one (the synergetic concept). This is accomplished by injecting a tritium or deuterium beam of 1 GeV/nucleon into DT fuel contained in a magnetic mirror. The confined pions slow down and decay into muons, which are confined in the fuel causing little muon loss. The necessary quantity of tritium to keep the reactor viable has been derived. The second idea is that the beam passing through the target is collected for reuse and recirculated, while the strongly interacted portion of the beam is directed to electronuclear blankets. The present concepts are based on known technologies and on known physical processes and data.

**The intrinsic electromagnetic solitary vortices in magnetized plasma **

J. Liu, W. Horton

**Abstract**

Several Rossby type vortex solutions constructed for electromagnetic perturbations in magnetized plasma encounter the difficulty that the perturbed magnetic field and the parallel current are not continuous on the boundary between two regions. We find that fourth order differential equations must be solved to remove this discontinuity. Special solutions for two types of boundary value problems for the fourth order partial differential equations are presented. By applying these solutions to different nonlinear equations in magnetized plasma, the intrinsic electromagnetic solitary drift-Alfven vortex (along with solitary Alfven vortex) and the intrinsic electromagnetic solitary electron vortex (along with short-wavelength drift vortex) are constructed. While still keeping a localized dipole structure, these new vortices have more complicated radial structures in the inner and outer regions than the usual Rossby wave vortex. The new type of vortices guarantees the continuity of the perturbed magnetic field ΔB_{⊥} and the parallel current j_{||} on the boundary between inner and outer regions of the vortex. The allowed regions of propagation speeds for these vortices are analyzed, and we find that the complementary relation between the vortex propagating speeds and the corresponding phase velocities of the linear modes no longer exists.

**Flux and differences in action for continuous time Hamiltonian systems **

R.S. Mackay, J.D. Meiss

**Abstract**

For a time-periodic Hamiltonian H(p,q,t) of period T, the area crossing a collection of curves at time 0 spanning two homotopic orbits of common period nT, in a time T, is shown to be the difference between the actions, contour integral pdq-Hdt, of the orbits. Similarly in an autonomous Hamiltonian system of two degrees of freedom the flux of energy surface volume per unit time through a surface spanning two homotopic orbits of the same energy is given by the difference between the actions, contour integral p.dq, of the orbits. Analogous results hold for pairs of orbits which converge together in both directions of time.

**Computer simulation of driven Alfven waves **

J. Geary

**Abstract**

The first particle simulation study of shear Alfven wave resonance heating is presented. Particle simulation codes self-consistently follow the time evolution of the individual and collective aspects of particle dynamics as well as wave dynamics in a fully nonlinear fashion. Alfven wave heating is a possible means of increasing the temperature of magnetized plasmas. A new particle simulation model was developed for this application that incorporates Darwin`s formulation of the electromagnetic fields with a guiding center approximation for electron motion perpendicular to the ambient magnetic field. The implementation of this model and the examination of its theoretical and computational properties are presented. With this model, several cases of Alfven wave heating is examined in both uniform and nonuniform simulation systems in a two dimensional slab. For the inhomogeneous case studies, the kinetic Alfven wave develops in the vicinity of the shear Alfven resonance region.

**Destabilization of MHD modes with finite larmor radius effects in tandem mirrors **

M. Kotschenreuther, H.L. Berk

**Abstract**

Difficulties in the stabilization of ideal magnetohydrodynamic (MHD) ballooning modes by finite Larmor radius (FLR) effects are considered, in tandem mirror geometry for azimuthal mode numbers l > 1. A kinetic formalism is used to obtain corrections to the long thin approximation, when keeping terms of quadratic order in the curvature. If η_{i} = par. δlnT_{i} par. δlnn_{i} ≥ 0, with T_{i}, n_{i}, the ion temperature and density, ion resonance effects eliminate absolute FLR stability, though the residual growth rates are substantially reduced from the MHD values. Even lowest order FLR stability is difficult to achieve with choke coils present, but is possible with a more gradually tapered mirror. However, the residual modes are still important, and mixing length estimates of the confinement degradation from modes with l > 1 indicate they can still severely limit the achievement of reactor-grade operation near and above the threshold beta value predicted from the ideal MHD theory. This is most severe if the ion temperature gradient decreases radial (η_{i} > 0), whereupon significant instabilities can even arise below the ideal threshold. However, if the ion temperature gradient can be made positive, and -η_{i} < 2/3, the lowest order FLR theory suffices to produce stability.

**Dynamics and fluctuation spectra of electrostatic resistive interchange turbulence **

R.D. Sydora, J.N. Leboeuf, Z.G. An, P.H. Diamond, G.S. Lee, T.S. Hahm

**Abstract**

The saturation mechanism for density and potential fluctuation spectra which evolve from linearly unstable electrostatic resistive interchange modes, are investigated using particle simulations. Detailed comparisons of the nonlinear evolution, saturation levels and resultant spectra between two- and three-dimensional sheared magnetic field configurations are made. Significant differences appear. The single rational surface, quasilinear-dominated evolution, fluctuation spectrum is adequately described using a density convection model. For the multiple rational surface case, the potential fluctuations are adequately represented by a balance between the nonlinearly modified source (curvature drive) and linear sink (parallel resistive field line diffusion). An accurate description of the density spectrum requires a mode coupling theory based on the two-point density correlation evolution equation.

**Local effect of equilibrium current on tearing mode stability **

F. Cozzani

**Abstract**

The local effect of the equilibrium current on the linear stability of low poloidal number tearing modes in tokamaks is investigated analytically. The plasma response inside the tearing layer is derived from fluid theory and the local equilibrium current is shown to couple to the mode dynamics through its gradient, which is proportional to the local electron temperature gradient under the approximations used in the analysis. The relevant eigenmode equations, expressing Ampere's law and the plasma quasineutrality condition, respectively, are suitably combined in a single integral equation, from which a variational principle is formulated to derive the mode dispersion relations for several cases of interest. The local equilibrium current is treated as a small perturbation of the known results for the m greater than or equal to 2 and the m = 1 tearing modes in the collisional regime, and the m greater than or equal to 2 tearing mode in the semicollisional regime; its effect is found to enhance stabilization for the m greater than or equal to 2 drift-tearing mode in the collisional regime, whereas the m = 1 growth rate is very slightly increased and the stabilizing effect of the parallel thermal conduction on the m greater than or equal to 2 mode in the semicollisional regime is slightly reduced.

**Existence and calculation of sharp boundary MHD equilibrium in 3-D toroidal geometry **

H.L. Berk, J. Freidberg, X. Llobet, P.J. Morrison, J.A. Tataronis

**Abstract**

The problem of sharp boundary, ideal magnetohydrodynamic equilibria in three dimensional toroidal geometry is addressed. The sharp boundary, which separates a uniform pressure, current-free plasma from a vacuum, is determined by a magnetic surface of a given vacuum magnetic field. The pressure balance equation has the form of a Hamilton-Jacobi equation with a Hamiltonian that is quadratic in the momentum variables, which are the two covariant components of the magnetic field on the outer surface is identical with finding phase space tori in nonlinear dynamics problems and the KAM theorem guarantees that such solutions exist. When tori exist, renormalized perturbation theory is used to calculate the properties of the magnetic field just outside the plasma.

**Island bootstrap current modification of the nonlinear dynamics of the tearing mode **

R. Carrera, R.D. Hazeltine, M. Kotschenreuther

**Abstract**

A kinetic theory for the nonlinear evolution of a magnetic island in a collisionless plasma confined in a toroidal magnetic system is presented. An asymptotic analysis of a Grad-Shafranov equation including neoclassical effects such as island bootstrap current defines an equation for the time dependence of the island width. Initially, the island bootstrap current strongly influences the island evolution. As the island surpasses a certain critical width the effect of the island bootstrap current diminishes and the island grows at the Rutherford rate. For current profiles such that Δ' < 0 the island bootstrap current saturates the island.

**The electromagnetic solitary vortices in rotating plasma **

J. Liu, W. Horton

**Abstract**

The nonlinear equations describing drift-Alfven solitary vortices in a low ..beta.., rotating plasma are derived. Two types of solitary vortex solutions along with their corresponding nonlinear dispersion relations are obtained. Both solutions have the localized coherent dipolar structure. The first type of solution belongs to the family of the usual Rossby or drift wave vortex, while the second type of solution is intrinsic to the electromagnetic perturbation in a magnetized plasma and is a complicated structure. While the first type of vortex is a solution to a second-order differential equation the second one is the solution of a fourth-order differential equation intrinsic to the electromagnetic problem. The fourth-order vortex solution has two intrinsic space scales in contrast to the single space scale of the previous drift vortex solution. With the second short scale length the parallel current density at the vortex interface becomes continuous. As special cases the rotational electron drift vortex and the rotational ballooning vortex also are given.

**Effects of ballooning instability on tokamak confinement **

G. Fu, J.W. Van Dam

**Abstract**

Using the ballooning mode transport model proposed by Connor, Taylor, and Turner (1984), the thermal conductivity induced by ideal ballooning instability is derived and compared to experimental observations from auxiliary-heated tokamaks. It is shown how this model can be improved by means of a finite-beta equilibrium and also applied to obtain a confinement scaling law for high-beta, purely ohmically heated tokamaks. Finally, this transport model is employed to find that tokamaks with supplemental stabilization, for example, due to gyroradius, energetic particle, or shaping effects, can self-consistently access the second stability regime at rather high heating power.