**Parallel processing particle trajectory code for anomalous transport studies **

J. Biswas, L. Leonard, W. Horton

**Abstract**

The particle trajectory code that calculates the guiding trajectories of electrons in a tokamak taking into account a spectrum of low frequency electromagnetic fluctuations has been reconfigured to run on eight parallel processors. The new code is expressed in Task Level Data Flow Language (TDFL) developed for parallel processing. A data flow graph of the problem is given to describe the split-up and recombination of the data vectors at the central points in the fortran code. A speed up factor of six over single processor computation time is obtained.

**Kelvin-Helmholtz instability and vortices in magnetized plasma **

W. Horton, T. Tajima, T. Kamimura

**Abstract**

Analytic theory and implicit particle simulations are used to describe the evolution of the Kelvin–Helmholtz instability and the plasma vortices driven by a nonuniform E×B flow velocity. Formulas for plasma convection arising from the self-consistent plasma electric field give the rate of anomalous momentum transport across the magnetic field. The momentum transport is shown to be controlled by the tilting angle of the elliptical vortex with respect to the direction of the parallel flow. Three stages of evolution are investigated and formulas for the final vortex state are given.

**Finite pressure ballooning mode stability in toroidal equilibria **

X. Llobet, H. Berk, M.N. Rosenbluth

**Abstract**

We examine the effect of finite pressure on the ballooning instability in toroidal MHD equilibria of steep boundary stellarators and tokamaks. Ballooning modes tend to arise near the place where the local shear vanishes and the normal curvature (the curvature component perpendicular to the flux surface, pointing away from the magnetic axis) is negative. We will show how the pressure gradient determines the position of the shearless points, and demonstrate in detail how this effect explains the existence of second stability in tokamaks. For large aspect ratio circular cross-section tokamaks the second stability condition is found to scale as α = const S^{ 1.25}. Stellarators are inherently more stable, due to the negative vacuum shear which at moderate pressure gradients allows the zero shear point to localize on the inner side of the flux surface. However, at high pressure gradients the Pfirsch-Schlueter current produces a positive mean shear when the total toroidal current on a flux surface is zero. This causes the zero shear point to localize on the outer edge, near the vertical extremes of the flux surface. This causes the zero shear point to localize on the outer edge, near the vertical extremes of the flux surface. This effect, together with helical contributions to the helical curvature, allows for ballooning instability to arise. At higher pressure gradients, with zero net toroidal current, an unstable ballooning mode which localizes to within a helical period always arises where the normal curvature is unfavorable.

**Sideband instabilities in free electron lasers **

M. Rosenbluth, H. Wong, B.N. Moore

**Abstract**

The linear stability of sideband modes for a one-dimensional free electron laser is investigated in detail. The dependence on wiggler taper, slippage between optical pulse and electrons, and trapped electron distribution functions are included in the analysis. Nyquist plots are used to delineate the parameter space in which sideband instabilities occur and approximate analytic expressions for the linear growth rate are derived. In special cases a complete analytic solution is given. Essentially all equilibria are unstable to sideband growth. The linear growth rates agree well with numerical simulations.

**Effective diffusion in laminar convection patterns **

M. Rosenbluth, H.L. Berk, W. Horton, I. Doxas

**Abstract**

The effective diffusion coefficient D* of a passive component, such as test particles, dye, temperature, magnetic flux, etc., is derived for motion in periodic two-dimensional incompressible convective flow with characteristic velocity v and size d in the presence of an intrinsic local diffusivity D. Asymptotic solutions for effective diffusivity D*(P) in the large P limit, with P approx. vd/D, is shown to be of the form D* = cDP^{1/2} with c being a coefficient that is determined analytically. The constant c depends on the geometry of the convective cell and on an average of the flow speed along the separatrix. The asymptotic method of evaluation applies to both free boundary and rough boundary flow patterns and it is shown that the method can be extended to more complicated patterns such as the flows generated by rotating cylinders, as in the problem considered by Nadim, Cox, and Brenner (J. Fluid Mech., 164: 185 (1986)). The diffusivity D* is readily calculated for small P, but the evaluation for arbitrary P requires numerical methods. Monte Carlo particle simulation codes are used to evaluate D* at arbitrary P, and thereby describe the transition for D* between the large and small P limits.

**A split-timestep high-resolution simulation of 2D drift-wave turbulence **

B. Scott

**Abstract**

A split-timestep scheme for the simulation of drift-wave turbulence is presented. Large coupling terms in all equations are evaluated at mid-step to avoid severe timestep restrictions. The need for high resolution, not always recognized, is demonstrated. The intermediate scales are kept out of the dissipation range of numerical or artificial diffusion. Coherent structure dynamics at these scales are shown to play an important role in the essential physics of the turbulence. A number of tests of the numerics are presented with the results of a typical run. The numerical methods used here should be useful in most simulations involving large terms which linearly couple separate equations as well as dissipationless intermediate scales.

**The effects of sloshing energetic particles on ballooning modes in tokamaks **

D.P. Stotler, H.L. Berk

**Abstract**

Distributions that give rise to energetic trapped particle pressures peaked in the ``good curvature'' region of a tokamak (sloshing distributions) are examined in an attempt to find stable regimes for both the magnetohydrodynamic (MHD) and precessional modes. It is the precessional drift destabilization of ballooning modes that inhibits bridging the unstable gap to second stability by the use of deeply trapped energetic particles unless the hot particles have an extremely large energy (~0.35 MeV for a tokamak like PDX [Phys. Rev. Lett. 49, 326 (1982)]). Unfortunately, our calculations indicate that the sloshing particles do not have a significant stabilizing effect. An analytic treatment shows that stability for the precessional mode can be found only if the sign of the energetic particle magnetic drift frequency can be reversed from its value in vacuum bad curvature without hot species diamagnetism. This is difficult to do in a tokamak because of the destabilizing contribution of the geodesic curvature to the drift frequency. Furthermore, for each of the two sloshing distributions employed (one contains only trapped particles; the other includes trapped and passing particles), a new ``continuum instability'' (where asymptotically along the field line the mode is a propagating plane wave) is found to be driven by geodesic curvature. These results indicate that energetic sloshing particles are not able to bridge the unstable gap to second stability.

**Minimum scaling laws in tokamaks **

Y.Z. Zhang, S.M. Mahajan

**Abstract**

Scaling laws governing anomalous electron transport in tokamaks with ohmic and/or auxiliary heating are derived using renormalized Vlasov-Ampere equations for low frequency electromagnetic microturbulence. It is also shown that for pure auxiliary heating (or when auxiliary heating power far exceeds the ohmic power), the energy confinement time scales as τ_{E} ≈ P_{inj}^{-1/3}, where P_{inj} is the injected power.

**Implicit particle simulation of electromagnetic plasma phenomena **

T. Kamimura, E. Montalvo, D.C. Barnes, J.N. Leboeuf, T. Tajima

**Abstract**

A direct method for the implicit particle simulation of electromagnetic phenomena in magnetized, multi-dimensional plasmas is developed. The method is second-order accurate for ωδt < one, with ω a characteristic frequency and time step δt. Direct time integration of the implicit equations with simplified space differencing allows the consistent inclusion of finite particle size. Decentered time differencing of the Lorentz force permits the efficient simulation of strongly magnetized plasmas. A Fourier-space iterative technique for solving the implicit field corrector equation, based on the separation of plasma responses perpendicular and parallel to the magnetic field and longitudinal and transverse to the wavevector, is described. Wave propagation properties in a uniform plasma are in excellent agreement with theoretical expectations. Applications to collisionless tearing and coalescence instabilities further demonstrate the usefullness of the algorithm.

**Muonic X-ray laser assisted by catalyzed fusion of deuterium and tritium **

T. Tajima, S. Eliezer

**Abstract**

The possibility of an X-ray laser by irradiation of muon beams on a thin rod of deuterium-tritium mixture is discussed. The excited state of dtmu-mesomolecules (J υ)= (1,0) can be induced to make radiative chain transitions with photon energy of 90eV to the ground state (0,0), evaluated by fusion.

**Confinement of a self-stabilized tokamak under average magnetic well conditions **

V.V. Demchenko, G.Y. Fu, J.W. Van Dam

**Abstract**

It is well known that the average favorable magnetic curvature of a tokamak is stabilizing with respect to pressure-driven magnetohydrodynamic instabilities at low beta and that self-stabilization occurs at finite beta in the so-called second stability regime. Here we self-consistently investigate how these two effects, viz., the mean magnetic well and the self-stabilization, influence the energy confinement time in a tokamak, using the ballooning mode transport model.

**Papers presented by IFS/FRC members at the international workshop on small scale turbulence and anomalous transport in magnetized plasmas **

B. Scott, P. Diamond, P. Terry, R. Bengston

**Stochastic electron dynamics due to drift waves in a sheared magnetic field and other drift motion problems **

J. Robertson

**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/sub parallel//m interacts strongly with the E x B trapping motion due to the presence of magnetic shear. The guiding-center equations for single particle electron orbits in given fields are investigated using both analytical and numerical techniques. The model assumes a slab magnetic field geometry with shear and two electrostatic plane waves propagating at an angle with respect to each other. Collisions and the self-consistent effect of the electron motion upon the fields are ignored. The guiding-center motion in an inertial reference frame moving in phase with the two waves is given by a two degree-of-freedom, autonomous Hamiltonian system. The single wave particle motion may be reduced to 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 electron diffusion in drift waves occurring in toroidal fusion confinement devices. The effect of an adiabatically changing radial electric field upon guiding-center orbits in tokamaks is also investigated. This perturbation causes a radial polarization drift of trapped particle tokamak orbits.

**A mechanism for rapid sawtooth crashes in tokamaks **

A. Aydemir, R. Hazeltine

**Abstract**

The standard picture of Kadomtsev reconnection process predicts sawtooth crash times that are longer than those observed in the present day large tokamaks. Ideal kink modes are investigated as a possible mechanism for these fast crashes, by use of fully toroidal, compressible, full magnetohydrodynamic equations. In systems with low shear, parallel-current and pressure-driven modes are identified well below the previously accepted poloidal beta limits. Linear and nonlinear calculations show good agreement with experiments and indicate that such modes may explain fast collapse times reported in the recent literature.

**Wakeless triple soliton accelerator **

Mima, Ohsuga, Takabe, Nishihara, T. Tajima, E. Zaidman, W. Horton

**Abstract**

We introduce and analyze the concept of a wakeless triple-soliton accelerator in a plasma fiber. Under appropriate conditions the triple soliton with two electromagnetic and one electrostatic waves in the beat-wave resonance propagates with velocity c leaving no plasma wake behind, while the phase velocity of the electrostatic wave is made also c in the fiber.

**Solution of hot particle ballooning mode integral equation in tokamaks **

D. Stotler, H.L. Berk

**Abstract**

The solution for the stability of ballooning mode equations in tokamaks with hot particles is presented. The system of equations requires solving a combined set of differential and integral equations accounting for the detailed hot particle orbits. Special techniques are developed to describe the equilibrium, the linear set of matrix equations and the boundary conditions. A modified WKB technique is developed to treat the boundary conditions. A detailed discussion is given of how boundary conditions are determined. We rigorously show why, in some cases, the eigenfunction may be exponentially growing with an outgoing group velocity. Sample results are presented.

**Theory of resistive pressure-gradient-driven turbulence **

P.H. Diamond, B. Carreras, L. Garcia

**Abstract**

Saturated resistive pressure-gradient-driven turbulence is studied analytically and with numerical calculations. Fluid viscosity and thermal diffusivity are retained in the analysis and calculations. Such dissipation guarantees the existence of a stable, high-m dissipation range, which serves as an energy sink. An accurate saturation criterion is proposed. The resulting predicted pressure diffusivity scales similarly to the mixing length estimate but is significantly larger in magnitude. The predictions of the analytic theory are in good quantitative agreement with the numerical results for fluctuation levels.

**Kinetic simulation of nonlinear kink instabilities **

E. Zaidman

**Abstract**

A particle simulation study of kink and twist -kink instabilities in magnetized plasmas is presented. Plasma 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. The twist-kink mode is a possible mechanism for fast release of magnetic energy into thermal or mechanical in modeling the physics of solar flares, tandem mirrors, and other laboratory devices. A particle simulation model has been developed for these applications that incorporates Darwin's formulation of the electromagnetic fields with a guiding center approximation for electron motion perpendicular to the ambient magnetic field with the inclusion of all three dimensions. This enables us to explore magnetoactive kinetic plasma physical processes at low frequencies without restrictions of lower dimensionality, which have heretofore not been available. The implementation of this model and the examination of its theoretical and computational properties are presented. Using this model, we examine several cases of kink and twist-kink instabilities in a three-dimensional slab as candidates for a fast energy release mechanism in a plasma, three dimensional extensions of two dimensional process such as the coalescence instability. In this dissertation we describe a first-cut attempt of investigations of a qualitive nature of non-linear evolution of kink-related instabilities. For the kink simulations an applied current in the z direction induced a magnetic field in the theta direction which leads to instability in the case of sufficient current magnitude as expected. The plasma twisting in the twist-kink simulations was applied by an external radial electric field and resultant E x B totation. A pinching of plasma near r = 0 and the increase of plasma density are observed. The field aligned current is induced, yielding twisted field lines. When these field lines are twisted strong enough to wrap around (approximately) one revolution in the poloidal direction, the plasma column is observed to kink. Further twist results in a disruption of the plasma column and turbulent flows within the plasma, releasing the magnetic energy stored by the mechanical twist. After this relaxation the system undergoes a similar cycle, exhibiting relaxation oscillations.

**Stability of ballooning modes in tokamaks with energetic particles **

N. Vergara

**Abstract**

The effects of energetic particles are of interest since fast ions are present in neutral beam and rf-heated tokamaks and will occur in ignition devices in the form of alpha particles. Moreover, it may be desirable to create such particles by auxiliary heating in order to exploit their stabilizing properties and thus attain a high beta plasma. Here a range of issues related to the stabilization of MHD ballooning modes in tokamaks by using energetic particles is investigated analytically and numerically. The presence of a highly energetic plasma component can stabilize MHD ballooning modes in tokamaks and may allow direct access to the high-beta second stability regime. Here, an improved estimate of such stability was obtained, in the large-aspect-ratio circular limit, by means of a variational refinement of the lower bound for the energetic-particle potential energy. The effect of various profiles for the hot-particle pressure on stability is investigated and the stability of off-angle modes is explored. Moderately energetic particles, however, can destabilize the plasma through resonant interaction at their curvature drift frequency. Finally, even if the ideal modes and resonantly excited modes can be simultaneously stabilized, resistive ballooning instabilities may persist.

**Limiter effects on scrape-off layer fluctuations and transport **

D. Thayer, P. Diamond, A. Wootton

**Abstract**

An analysis is presented of scrape-off layer (SOL) fluctuations using a rippling mode or resistivity gradient driven turbulence (RGDT) calculation which incorporates the essential limiter boundary condition. The line-tying effects caused by a simple conducting poloidal limiter can lead to an order of magnitude reduction of the growth rate in the SOL region when parallel thermal conductivity is neglected. The basic effect of the limiter boundary condition can be understood by exploiting the conjugate relation between the parallel and radial mode widths, a consequence of magnetic shear. A reduction of the connection length, due to the limiter, implies an increase in the radial mode width. Assuming a conducting poloidal limiter with a large radial extent, the increased radial mode width causes an increase in thermal conduction damping which is sufficiently large enough so that the rippling mode is rendered stable for typical parallel thermal conductivities. We have also included the destabilizing effect of impurity radiative cooling in the RGDT analysis which leads to an increase of the growth rate, the saturation potential, and the diffusion coefficient by approximately 30% for typical parameters. Additionally, dissipative density gradient driven turbulence (DDGDT) is reconsidered as a viable model of SOL turbulence since it exhibits shorter toroidal mode widths than RGDT; thus, it is less sensitive to the limiter line-tying effect. Saturation level estimates for DDGDT with limiter effects are presented.

**Structural stability and chaotic solutions of perturbed Benjamin-ono equations **

B. Birnir, P.J. Morrison

**Abstract**

A method for proving chaos in partial differential equations is discussed and applied to the Benjamin-Ono equation subject to perturbations. The perturbations are of two types: one that corresponds to viscous dissipation, the so-called Burger`s term, and one that involves the Hilbert transform and has been used to model Landau damping. The method proves chaos in the PDE by proving temporal chaos in its pole solutions. The spatial structure of the pole solutions remains intact, but their positions are chaotic in time. Melnikov`s method is invoked to show this temporal chaos. It is discovered that the pole behavior is very sensitive to the Burger`s perturbation, but is quite insensitive to the perturbation involving the Hilbert transform.

**Local effect of equilibrium current of tearing mode stability in the semicollisional regime (Part II) **

F. Cozzani, S. Mahajan

**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.

**Local effect of equilibrium current on tearing mode stability in the collisional regime (Part I) **

F. Cozzani, S. Mahajan

**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.

**Axisymmetric MHD stable sloshing ion distributions **

H.L. Berk, N. Dominguez, G.V. Roslyakov

**Abstract**

The MHD stability of a sloshing ion distribution is investigated in a symmetric mirror cell. Fokker-Planck calculations show that stable configurations are possible for ion injection energies that are at least 150 times greater than the electron temperture. Special axial magnetic field profiles are suggested to optimize the favorable MHD properties.