Session BP1: Poster Session I: MHD and Stability; Stellarator; Basic Plasma Physics; MHD Phenomena in Astrophysics and Laboratory; Beams, Radiation, and Microwaves

Analysis of Options for Resistive Wall Mode Control Coils for ITER

Several fusion devices have found improvement in plasma performance from the application of either static or dynamic magnetic perturbations from a set of coils. DIII-D has found that static fields can prevent formation of locked modes and create ergodic structures in the plasma edge that decrease the size of ELMS. They have also used such coils in a feedback loop to control the growth of resistive wall modes. Similar effects have been observed on NSTX, C-Mod, ASDEX, and JET. In all cases, the coils were placed close to the plasma either inside the vessel or immediately outside a thin vessel. Because ITER is a burning plasma device with a long pulse length, thick nuclear shielding must be placed between the plasma and the vacuum vessel. If ITER is to realize the confinement and operation benefits of resistive wall mode control coils, locations and coil designs must be found where such coils can be deployed. Two generic locations have been identified. The most accessible location is immediately outside the vessel and around the mid-plane ports. An alternative location closer to the plasma is inside the mid-plane ports but behind the port shield module. We have used an electromagnetic modeling code to evaluate both the static and dynamic field perturbations at the plasma edge for both of these coil options for frequencies from 1 Hz to 6kHz. *Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.   

Nonlinear simulation of plasma disruption caused by resistive wall mode in a cylindrical tokamak

Nonlinear behavior of resistive wall mode (RWM) has been studied using nonlinear three-dimensional simulation code based on the reduced MHD equations in a cylindrical tokamak. For the case that the RWM with (m,n)=(3,1) mode and the (m,n)=(5,2) mode are linearly unstable, the magnetic field line stochastization is obtained around the plasma edge, where m and n are poloidal and toroidal mode number, respectively. Even when the (m,n) = (2,1) tearing mode is linearly stable, the (m,n)=(2,1) mode rapidly grows by nonlinear mode coupling of (m,n)=(3,1) and (5,2) modes. Since the stochastization causes the changes of the current and resistivity profile, the (m,n)=(2,1) mode still grows after the saturation of the (m,n)=(3,1) and (5,2) modes. Finally, the energy of the (m,n)=(2,1) mode becomes predominant and the stochastic region covers all plasma region.   

RWM Analyses of JT-60SA and JT-60U Tokamak Plasmas

JT-60SA is a tokamak device, being now designed at JAEA with collaboration of EU. One of the main purposes of JT-60SA is to realize the steady state plasma with high normalized beta values, 3.5$\sim $5.5. Our previous analyses have shown that the critical normalized beta value was 3.8 with the effect of the stabilizing structure with finite resistivity and the active feedback control. The critical beta value is low compared to the critical normalized beta of 5.5 in the case using ideal stabilizing structure, which results in very low C$\beta $ value of 0.37 and poor efficiency of feedback control. To overcome the poor efficiency of feedback control, we consider the new configuration of stabilizing structure and feedback control coil. The analyses are being carried out for new equilibrium including transport analyses of JT-60SA plasma. These critical beta analyses are carried out by VALEN code developed in Columbia University. We also present the results of analyses of experimental data of current driven and pressure driven RWM in JT-60U plasma. These analyses are carried out by AEOLUS-FT code developed in JAEA.   

Does Flow Stabilize or Destabilize MHD Instabilities?

It is well known that a static (i.e.${\rm {\bf v}}=0)$ closed field line configuration, such as a levitated dipole, or hard-core Z-pinch, can be stabilized against ideal MHD interchange modes when the pressure gradient is sufficiently weak. The stabilizing effect is provided by plasma compressibility. However, many laboratory plasmas exhibit a sheared velocity flow, (i.e.${\rm {\bf n}}\cdot \nabla {\rm {\bf v}}\ne 0)$, and this flow may affect the marginal stability boundary. The present work addresses this issue by an analysis of the effect of axially sheared flow on ideal MHD stability in a hard-core Z-pinch, which is a cylindrical model for the levitated dipole configuration. Specifically, the goal is to learn whether sheared flow is favorable, unfavorable, or neutral with respect to MHD stability. Analytic calculations of marginal stability for several idealistic velocity profiles show that all three options are possible depending on the shape of the shear profile. This variability reflects the competition between the destabilizing Kelvin-Hemholtz effect and the fact that shear makes it more difficult for short wavelength interchange perturbations to form. Numerical calculation are also presented for more realistic experimental profiles and compared with the results for the idealized analytic profiles.   

A general MHD stability formulation for plasmas with flow and resistive walls

Toroidal rotation, either induced by means of neutral beams (e.g$.$ in NSTX and DIII-D) or appearing spontaneously (e.g$.$ in Alcator C-Mod, JET and Tore Supra) is routinely observed in modern tokamak experiments. Poloidal rotation is also commonly observed, in particular in the edge region of the plasma. Plasma rotation has a major effect on plasma stability. Flow and flow shear stabilize external modes such as the resistive wall mode (as observed e.g$.$ in DIII-D), suppress turbulence when the flow shear is large enough, and also have a significant influence on the stability and nonlinear evolution of the internal kink and ballooning modes. Flow shear can in particular have both a stabilizing (by breaking up unstable structures) and destabilizing (through the Kelvin-Helmholtz mechanism) effect. A self-consistent analysis of the effect of rotation requires the use of numerical tools. In this work, we present a general eigenvalue formulation based on a variational stability analysis, including arbitrary (both toroidal and poloidal) plasma rotation and a thin resistive wall of arbitrary shape and resistivity. It is shown the problem can always be reduced to a classic eigenvalue formulation of the kind $i \omega \underline{\underline A} \cdot {\boldmath \zeta} = \underline{\underline B} \cdot {\boldmath \zeta}$, where $\boldmath \zeta$ is an unknown eigenvector related to the plasma displacement, and $\omega$ the (complex) evolution frequency of the perturbation. The formulation is well suited for a finite element analysis.   

Numerical studies for the linear growth of resistive wall modes generated by plasma flows in a slab model

The resistive wall mode generated by plasma-wall relative rotations is studied numerically in a slab model with a compressible plasma flow parallel to the magnetic field. The linear growth of the mode is investigated with different parameters in numerical simulations. The critical plasma flow velocities for the instability are calculated as the wavenumber of the mode and other parameters vary. It is found that in the long wavelength regime, the critical velocity is in the range of the sound speed $c_s$, as predicted in theory. In the short wavelength regime however, the critical velocity increases to a level of tens of times of Alfv\'en velocity $V_A$. Also a second stable region is found in the short wavelength regime which eventually merges with the first stable region as the wavenumber increases and stabilizes the mode. The growth rate of the mode is found decreasing with the wavenumber of the mode and the plasma viscosity. The critical wavenumber for the instability is also calculated as the plasma velocity changes. In addition, the linear growth rate versus the plasma $\beta$ is also presented.   

Control of magnetohydrodynamic modes with a resistive wall above the wall stabilization limit

Studies are shown of control of magnetohydrodynamic (MHD) modes in the presence of a resistive wall, below and above the regime for which stabilization is possible with a perfectly conducting wall, i.e.~below and above the ideal wall limit. The results show that resistive plasma (tearing-like) modes can be feedback stabilized for current profiles which are unstable \emph{above} the ideal wall limit, both for tokamak-like and reversed field pinch (RFP)-like profiles. However, above the limit for wall stabilization of ideal plasma modes, resonant or non-resonant, the feedback scheme cannot provide stabilization. The control scheme senses both normal and tangential components of the perturbed magnetic field, and the feedback is proportional to a linear combination of the two. Neither plasma rotation nor complex gain is included. A cylindrical reduced MHD model, in resistive or ideal form, is used, with tokamak-like profiles {[}increasing profile of $q(r)${]} or RFP-like profiles {[}decreasing $q(r)${]}. The possible relevance to RFPs and tokamaks will be discussed.   

Three-dimensional equilibria in axially symmetric tokamaks

The NSTAB computer code was developed to study equilibrium and stability in fusion plasmas with three-dimensional (3D) geometry, and it is now being applied to calculate islands in tokamaks like ITER. The significance of 3D magnetohydrodynamic (MHD) equilibria in axially symmetric tokamaks is that they may contribute to the prompt loss of $\alpha $ particles or to disruptions, since they model neoclassical tearing modes and edge localized modes. The NSTAB code captures islands correctly despite a nested surface hypothesis because the MHD equations are in conservation form. This works well for islands whose widths are comparable to the mesh size. Convergence studies were made to check runs of NSTAB used to design quasiaxially symmetric stellarators. Residuals of $10^4$ discrete equations can be reduced by $10^6$ iterations of an accelerated method of steepest descent to the level $10^{-13}$ of round off error in double precision, showing that the 3D solutions are valid.   

Magnetic island induced bootstrap current on island evolution in tokamaks

In a previous paper [K. C. Shaing, and D. A. Spong, Phys. Plasmas \textbf{13}, 2006], the effects of the even component (relative to the mode rational surface) of the magnetic island induced bootstrap current on the island dynamics in tokamaks are investigated, It is found that island induced bootstrap current density has the stabilizing influence on the island stability. The theory is extended here to including the effects of the odd component of the bootstrap current on the island evolution by taking into account the asymmetric island shape relative to the mode rational surface in the derivation of the island evolution equation. There are two types of the perturbed bootstrap current density that is odd relative to the mode rational surface. One type is the island modified which is independent of the collision frequency and the other is the island induced which is inversely proportional to the collision frequency. Both of these contributions will be included in the island evolution equation. The effects of the island induced bootstrap current on the island stability in the high poloidal beta tokamak plasmas will be discussed.   

The interaction of electrostatic turbulence and magnetic islands

A model describing the multiscale interaction of short wavelength electrostatic fluctuations with a long wavelength nonlinear magnetic island is developed. The magnetic island modifies the magnetic geometry and produces a helically modulated local shear that can modify the stability properties of the electrostatic fluctuations. Additionally, sheared flows associated with the magnetic islands can modify the turbulence. These effects combine to modulate the amplitude of the turbulence in the vicinity of the island as a function of helical angle. This can ultimately produce spatially dependent turbulent diffusion and viscosity coefficients. The turbulence affects magnetic island dynamics principally through a mean field- like force in the momentum equation (Reynolds stress) that is self-consistently modified by the helical nature of the island potential and profiles. This force subsequently affects island dynamics through a perturbed perpendicular current that enters into the island quasineutrality equation; in general, both island growth and rotation are altered. Implications for islands in 3-D equilibria and neoclassical tearing mode dynamics will be addressed.   

Self-consistent evolution of a magnetic island with drift wave turbulence

In this work, we treat the self-consistent evolution of a magnetic island in the presence of drift wave turbulence. As is well known, the evolution of a magnetic island, including neoclassical destabilization, depends strongly on the transport coefficients present within the magnetic island equations. Although these transport coefficients are known to have a turbulent origin, and thus will be strongly effected by the dynamics of the magnetic island, they're typically modelled as constants. Here, wave kinetics and adiabatic theory are used to treat the feedback of the large scale magnetic island on the drift wave turbulence via shearing, radial advection, and temperature profile modification. The stresses exerted by the drift wave turbulence on the magnetic island are calculated by mean field methods, allowing for a fully self-consistent description of the coupled evolution of the system. It is found that nonlocal interactions between the drift wave turbulence and magnetic island lead to the formation of strong shears flows. Discussion of the impact of shear flow excitation on magnetic island evolution, especially perpendicular heat transport, is presented.   

Finite Ion Orbit Effects on Magnetic Islands in Toroidal Plasmas

A kinetic theory for the interaction of an ion population with an isolated magnetic island in a high aspect ratio tokamak plasma is presented. We examine islands whose characteristic widths are larger than the ion gyro radius but comparable to the ion banana width. In this regime, the ion response to the island has a non-local feature due to the banana drifts. When solving the drift kinetic equation for ions, a change in coordinates is used to account for this behavior. A bounce averaging procedure is developed to separate out and solve the lower order distribution function. For small islands relative to banana width, the trapped ions do not respond to the island electrostatic potential and helical magnetic geometry. A two-fluid model is adopted to determine the density of electrons. Quasineutrality leads to a self-consistent calculation for the electrostatic potential. An iteration procedure is introduced to calculate the potential, which is shown to be a combination of functions of the helical flux surfaces and the topologically toroidal flux surfaces. The contribution to the perturbed current is composed of bootstrap current and the perpendicular ion polarization current. The closure parallel current (J$_{//})$ is calculated and compared with some recent numerical results. Using this current in the Rutherford equation, the island width evolution equation is determined and compared with calculations for large island width.   

Kinetic effects on MHD instabilities with extended gyrokinetic theory

In order to recover linear MHD from gyrokinetics and self-consistently obtain FLR effects on MHD modes, we revisit gyrokinetic theory and introduce two key modifications. First, the solution for the equilibrium gyrokinetic distribution function is obtained to higher order in Larmor radius. Second, additional gyrophase-dependent parts of the perturbed distribution function are retained. This extended linear gyrokinetic theory can then be used to study kinetic effects on tokamak MHD instabilities (e.g., resistive wall mode).   

Free boundary simulations of MHD instabilities with a finite volume based full-MHD code

MHFVSP is a finite volume based code for 3D full-MHD simulations in tokamak plasmas using either structured (rectangular) or unstructured (triangular) mesh in the poloidal plane, and fast Fourier transforms in the toroidal direction. The code is being modified to handle issues on the free boundary configurations related to the growth of MHD instabilities. The pseudo-vacuum model is implemented in the code, and then free boundary MHD simulations are carried out for the equilibrium including the vacuum region which is obtained by the MEUDAS code$^{1}$. We describe the algorithm in the MHFVSP code and report on the development and validation of such a modification. The initial results from simulation runs for free boundary configurations are also presented. \\$^{1}$ T. Takeda and S. Tokuda, J. Comp. Phys. 93, 1 (1991).   

Explosive growth and nonlinear dynamics of the forced magnetic island

The whole process of nonlinear dynamics of magnetic island due to the external perturbation in rotating tokamak plasmas is studied numerically. The island formation is a critical issue severely affecting the performance of the tokamak plasmas. The theoretical work has been done to understand the forced magnetic island suppression by the plasma rotation and to evaluate its threshold value, while less attention has been paid to the subsequent break up process and the long term behavior of the forced island. It was found that the magnetic island grows explosively with changing its structure and the localized plasma current is formed around the X-point. Contrary to the standard magnetic reconnection theory, this localized plasma current causes the enhanced magnetic reconnection in the low resistivity regime. As the result, the long term evolution of the forced magnetic island is dominated by the secondary reconnection as the resistivity becomes small.   

MHD Turbulence, div B = 0 and Lattice Boltzmann Simulations

The question of div B = 0 in MHD simulations is a crucial issue. Here we consider lattice Boltzmann simulations for MHD (LB-MHD). One introduces a scalar distribution function for the velocity field and a vector distribution function for the magnetic field. This asymmetry is due to the different symmetries in the tensors arising in the time evolution of these fields. The simple algorithm of streaming and local collisional relaxation is ideally parallelized and vectorized -- leading to the best sustained performance/PE of any code run on the \textit{Earth Simulator}. By reformulating the BGK collision term, a simple implicit algorithm can be immediately transformed into an explicit algorithm that permits simulations at quite low viscosity and resistivity. However the div B is not an imposed constraint. Currently we are examining a new formulations of LB-MHD that impose the div B constraint -- either through an entropic like formulation or by introducing forcing terms into the momentum equations and permitting simpler forms of relaxation distributions.   

Computation of Singular MHD Instabilities with DCON and MATCH

The DCON code is in wide use for computing the ideal MHD stability of axisymmetric toroidal plasmas. It uses an adaptive numerical integrator to solve a system of ordinary differential equations for the radial dependence of complex Fourier coefficients of the normal displacement, a generalization of Newcomb's equation, from the magnetic axis to the plasma-vacuum interface. Fixed-boundary stability is determined by a toroidal generalization of Newcomb's criterion. Free-boundary stability is determined by the sign of the lowest eigenvalue of the sum of plasma and vacuum response matrices. DCON has been extended to compute the outer ideal region matching conditions for singular modes such as resistive and neoclassical tearing modes. A matching matrix is constructed from asymptotic coefficients of resonant and nonresonant solutions on either side of each singular surface and corresponding terms from any singular surface model. A dispersion relation for growth rates and eigenfunctions is obtained from the roots of the determinant of this matrix. Numerous computational improvements will be described which now make the code accurate, robust, and reliable. Benchmarks with the PEST 3 resistive stability code will be presented.   

Edge Stability in ELM-free QH and RMP Plasmas

The peeling-ballooning model proposes that intermediate wavelength MHD instabilities cause edge localized modes (ELMs) and impose constraints on the pedestal height. In typical discharges, the pedestal goes unstable to coupled peeling-ballooning modes shortly before an ELM is observed. However, in ELM-free discharges, such as in the promising Quiescent (QH) and resonant magnetic perturbation (RMP) H-mode regimes, the edge collisionality is low, and the resulting large bootstrap current in the pedestal region drives kink/peeling modes ($n\sim\,$1-10). Both flows and the conducting wall have significant impact in this regime, and an edge localized resistive wall mode can be unstable. We present a theory for the occurrence of QH-mode, in which the observed edge harmonic oscillation (EHO) is a saturated low-n kink/peeling mode, which drives particle transport and allows a steady quiescent pedestal. In RMP discharges, we find that the imposed magnetic perturbation plays the role of the EHO, similarly allowing steady state quiescent discharges.   

ELMs explained as resistive vacuum modes excited by SOL current onset

Recent DIII-D experiments show that SOL current onset occurs slightly before the edge electron temperature collapse associated with ELMs. We propose the excitation of resistive vacuum modes during SOL current onset as an explanation for ELMs. Wall effects are weak for high-n modes; however, for high-n modes, a vacuum region with resistive plasma can play a similar role to a resistive wall. High-n resistive vacuum external modes can be excited above a critical plasma beta limit and then resonantly enhanced by SOL current onset. In turn, enhanced MHD activity causes stronger radial transport and induces a larger SOL current. This process explains the initial bursting nature of ELMs. The radial transport also reduces the pedestal pressure gradient, which stabilizes the resistive vacuum modes in the later ELM stage. The ELM cycle repeats when heating again increases the beta value.   

ELM simulations with M3D

Large scale ELM simulations using the M3D extended MHD code with up to 40 toroidal modes were carried out as part of a DOE milestone. Simulations were done starting from DIIID EFIT equilibria. Nonlinear computations were performed both with and without gyroviscous stabilization. Gyroviscosity with a relevant Hall parameter had little effect on the nonlinear ELM behavior, which is dominated by moderate toroidal modes. Nonlinear ELM simulations have also been carried out in ITER geometry. In the ITER simulations, substantial outflow of density to the divertor was observed. Upwinded numerical methods were introduced in M3D to deal with nonlinear advection of the pedestal density. Mesh generation was improved to deal with a realistic boundary shape. Simulations have also been done in which the bootstrap current, pressure pedestal, and density pedestal were calculated with the XGC kinetic neoclassical edge code. The effects on ELMs of varying the bootstrap current and pressure profiles will be considered.   

Computation of two-fluid, flowing equilibria

Equilibria of flowing two-fluid plasmas are computed for realistic compact-toroid and spherical-tokamak parameters. In these examples the two-fluid parameter \textit{$\varepsilon $} (ratio of ion inertial length to overall plasma size) is small, \textit{$\varepsilon $ }$\sim $ 0.03 -- 0.2, but hardly negligible. The algorithm is based on the nearby-fluids model [1] which avoids a singularity that otherwise occurs for small \textit{$\varepsilon $}. These representative equilibria exhibit significant flows, both toroidal and poloidal. Further, the flow patterns display notable flow shear. The importance of two-fluid effects is demonstrated by comparing with analogous equilibria (e.g. fixed toroidal and poloidal current) for a static plasma (Grad-Shafranov solution) and a flowing single-fluid plasma. Differences between the two-fluid, single-fluid, and static equilibria are highlighted: in particular with respect to safety factor profile, flow patterns, and electrical potential. These equilibria are computed using an iterative algorithm: it employs a successive-over-relaxation procedure for updating the magnetic flux function and a Newton-Raphson procedure for updating the density. The algorithm is coded in Visual Basic in an Excel platform on a personal computer. The computational time is essentially instantaneous (seconds). [1] L.C. Steinhauer and A. Ishida, Phys. Plasmas \textbf{13}, 052513 (2006).   

Resistive ballooning modes in general toroidal geometry

Resistive ballooning modes (RBM) could be unstable and responsible for edge plasma fluctuations and anomalous transport in tokamaks and stellarators. A linear stability theory of RBM is investigated using a two fluid model based on the reduced Braginskii equations for arbitrary three dimensional geometry. RBM eigenvalues and eigenfunctions are calculated for a variety of equilibria including axisymmetric shifted circular geometry ($s-\alpha$ model) and configurations of interest to the Helically Symmetric stellarator (HSX). Attempts to generalize previous analytic calculations of RBM stability using a two scale analysis on $s-\alpha$ equilibria to more general 3-D equilibria will be addressed.   

The role of resistivity on line-tying kink modes

In a recent paper [1], Evstatiev et al. proposed a new method to analyze the linear stage of line-tied kink modes in cylindrical geometry. The method consists of summing up a number of one-dimensional (radial) eigenfunctions to obtain the full two-dimensional solution of the problem and has been successfully applied to both ideal and resistive MHD [1]. The present work investigates the role of resistivity on line-tied kink modes. Resistivity affects the problem in two ways. First, it disallows perfect line-tying at the two end-plates of the cylinder. Second, some of the radial eigenfunctions used to construct the full solution of the problem can be unstable tearing modes instead of stable ideal modes, thus opening the possibility of tearing-like instabilities in line-tied configurations. In order to address these two issues, we will use our new method to study different equilibria where the field line pitch as a function of radius can be monotonically increasing (tokamak-like), monotonically decreasing (RFP-like) or constant. Furthermore, a parametric study will be presented by varying resistivity and the results will be compared with the ideal MHD case. [1] E. G. Evstatiev, G. L. Delzanno, J. M. Finn, Physics of Plasmas 13, 072902 (2006).   

Ellipticity and triangularity effects in tokamak Alfven spectrum

Plasma configurations with ellipticity and triangularity are usual in tokamak experiments. These plasmas can be studied using a new system of coordinates of recent publications. Here this method has been applied to study Alfven spectrum in axisymmetric tokamaks with different values of ellipticity and triangularity [1-3]. Previous authors have developed numerical methods to obtain the Alfven spectrum using the Shafranov-Solove'v equilibrium flux function where the parameter ellipticity is also included [3]. Here more general configurations are treated and compared with the results of these authors, as well as those derived for the geometric optics or WKBJ approximation. The Alfven wave dispersion relation is obtained by the linearization of the MHD equations around a stationary equilibrium and the results are obtained by numerical calculations. [1] P. Martin, M. G. Haines and E. Castro, Phys. Plasma \underline {\textbf{12}}, 082506 (2005) [2] L. L. Lao, S. P. Hishman and R. M. Wieland, Phys. Fluids \underline {\textbf{24}}, 1431 (1981); H. Weitzner's Appendix. [3] G. O. Ludwig, Plasma Phys. Controlled Fusion \underline {\textbf{37}}, 633 (1995) [4] S. Novo, M. N\'{u}\~{n}ez and J. Rojo, Phys. Fluids B\underline { }\underline {\textbf{3}}, 2967 (1991)   

Breaking Quasi-translational Invariance of Toroidal Eigenmodes

Toroidal eigenmode structure in the presence of strong ${\bf E \times B}$ shear is examined to explore a possible spontaneous symmetry breaking in the parallel wave vector of turbulence which can contribute to radial flux of toroidal angular momentum. Even in the weak shear limit, FULL code [1]results have shown a significant departure of mode structure from the usual ballooning-type eigenmode towards the CZC-like [2] mode structure with the amplitude peaking between rational surfaces. In the strong shear limit, Ballooning Mode Formalism is confusing and invalid. At the low B field side, the radial eigenmode equation can be reduced to the Whittaker's form which should obey the Berk outgoing wave condition [3]. However, the magnetic shear induced damping is negligible and the most effective damping is from the perpendicular ${\bf E \times B}$ flow shear. \newline [1] G. Rewoldt et al., Nucl. Fusion, 42, 403 (2002). \newline [2] C.Z. Cheng, L. Chen, and M. Chance, Ann. Phys. 161, 21 (1985). \newline [3] L.D. Pearlstein and H. Berk, Phys. Rev. Lett. 23, 2209 (1982).   

Nonlinear Localized Interchange Mode and Current Sheet Formation

The localized interchange mode is a fundamental instability in inhomogeneous plasmas with finite $\beta$ and magnetic shear. Recent work based on a reduced MHD formulation indicates that, while the violation of the Suydam criterion itself may not impose a practical operational constraint, the nonlinear development of a robust localized interchange instability could lead to the formation of thin current sheets, which are the prelude to reconnection processes in the system [Gupta, Callen, and Hegna, 2004]. The spontaneous formation of thin current sheets from interchange-like processes, such as the magnetic Rayleigh-Taylor instability, has also been found in simulations of solar coronal mass ejection [Isobe {\it et al.}, 2005]. In this work, we explore the relation between the nonlinear development of the localized interchange mode and the formation of thin current sheet with direct ideal and extended MHD simulations using the NIMROD code. Initial NIMROD simulation results confirm the general sequence of the linear growth phase and nonlinear saturating phase for the evolution of the localized interchange mode found earlier by Gupta, Callen, and Hegna (2004). The formation of the Rayleigh-Taylor finger-mushroom pattern is also evident in NIMROD simulations of the nonlinear localized interchange mode. Investigation of current sheet formation and comparison with the earlier reduced MHD simulation will be presented.   

Magnetorotational Instability of Electrically Driven Flow in Circular Channel: Spectral Analysis of Global Modes

The spectral MHD stability of liquid metal differentially rotating in transverse magnetic field is studied numerically by solving the eigenvalue problem with rigid-wall boundary conditions. The equilibrium velocity profile used in calculations corresponds to the electrically driven flow in circular channel with the rotation law $\Omega(r)\propto 1/r^2$. This type of flow is planned to be used in new experimental device developed in RRC ``Kurchstov institute'' to test the magnetorotational instability (MRI) in laboratory. Our analysis includes calculations of the eigen-frequency spectra for both axisymmetric (with azimuthal wave-number $m=0$) and non-axisymmetric ($m\ne0$) modes. The dependence of MRI increments on mode wave-numbers is found numerically, the approximate analytical expression is obtained for marginal stability. It is shown that for the given parameters of the design the flow is unstable due to MRI with the fastest growth rate corresponding to the axisymmetric mode. For other parameters the axisymmetric MRI modes can be suppressed and the instability develops only for modes with $m\ne0$.   

Spontaneous generation of large-scale fields in plasmas under directional noise.

Generation of a large-scale magnetic filed in a plasma has been proposed as a way of stabilizing measure when the plasma is stirred by a parity-nonconserving noise. In this work, reduced MHD equations are used to reconsider the problem. It is shown that, unlike the full MHD equations, it is the renormalized response of the velocity that is unstable to the noise that has a unique direction. Introduction of the mean constant magnetic field is shown to play a stabilizing role and the size of the mean field is computed by balancing the two opposing effects. The role of the mean velocity will be discussed in the presentation.   

Status of NCSX Construction

The National Compact Stellarator Experiment is a new facility being built to study the physics of compact stellarators for fusion and plasma science. Construction is proceeding on schedule toward First Plasma in July, 2009. The major components are well into production. As of July 2006, all 18 modular coil winding forms have been cast in industry. Finished winding forms, with critical surfaces machined to 0.25~mm tolerance, are arriving at PPPL at the rate of one per month, with eight delivered so far. Two of the three vacuum vessel sectors have been delivered, with the last due in September. Three of the 18 modular coils have been wound, and the first of several machine assembly stages has begun. The manufacturing processes were developed through R{\&}D, including the construction of prototypes, prior to going into production. Because of that, the component production efforts are now succeeding in meeting the challenging requirements-- complex geometries and tight tolerances-- that are necessary to achieve desired plasma properties.   

Magnetic Alignment of NCSX.

The National Compact Stellarator Experiment (NCSX), currently under construction, is a modular quasi-axisymmetric stellarator designed to study confinement and stability of high-beta plasmas. It has 18 modular coils, 18 planar weak toroidal field coils, and 6 pairs of poloidal field coils. A novel technique has been developed to magnetically measure the relative deviations in the coils' shapes and positions by measuring differences between mutual or self-inductances (between field-coils) that should be identical due to design symmetries. Non-zero differences indicate deviations in the location, orientation, or shape of one or more coils. For the fully assembled NCSX, there are 1001 linearly independent symmetric inductance differences, which is sufficient to determine, in principle, the 270 relative location and orientation parameters for the coils and $\sim $15 relative shape moments per coil. From the calculated coupling coefficients and estimates of measurement signal to noise, the minimum measurable position or shape deviation amplitude is $\sim $ 0.02 -- 0.1 mm. The technique is applicable to either modular stellarators or tokamaks.   

Equilibrium Flux Surface Calculations for the W7AS and NCSX Stellarators.

Calculations of equilibrium flux surface loss in the W7AS stellarator using the PIES code are found to be consistent with the observed maximum $\beta $ values. A stochastic region is calculated to appear at the plasma edge when the magnitude of $\beta $ is above a threshold value, and the stochastic region is calculated to broaden as $\beta $ increases further. An interesting discovery is that the field line trajectories behave as if the flux surfaces are broken locally near the outer midplane and remain intact elsewhere. This is consistent with the expectation, which has been widespread in the stellarator community, that the 3D surfaces are broken by the strong local compression and distortion produced by the Shafranov shift. An estimate of the heat transport due to the field line stochasticity is consistent with the observed global energy confinement time. Divertor control coil currents are calculated to have a strong effect on the width of the stochastic region, and these calculations are consistent with the observed variation of the maximum $\beta $.   

Comparisons of transport in quasi-axisymmetric stellarators

We compare the confinement characteristics of some related QA stellarator designs, including LI383, the basis for the NCSX stellarator now being constructed. LI383 has very good thermal neoclassical confinement, while its energetic particle confinement is more problematic for a reactor. We find that both thermal and energetic confinement in LI383 can be appreciably improved,\footnote{ H.E. Mynick, A.H. Boozer, L.-P. Ku, {\em Phys. Plasmas} (2006).} by small perturbations which largely eliminate a class of bad drift trajectories not previously recognized. The coil set planned for NCSX should be able to reduce this class of bad trajectories. As part of this assessment, we also consider perturbations which violate stellarator symmetry, including up/down symmetry of the device.\footnote{ P.N. Yushmanov, J.R. Cary, S.G. Shasharina, {\em Nucl.Fusion} {\bf 33}, 1293 (1993).} We find that such perturbations neither improve nor degrade confinement, in general, but their effects depend on details. Work supported by U.S.DOE Contract No.DE-AC02-76-CHO3073.   

Heat Load on Divertors in NCSX

We have continued our study[1-3] of the effect of divertors in NCSX, using magnetic field data generated by both the PIES and VMEC/MFBE equilibrium codes. Results for comparable equilibria from the two codes agree to within statistical uncertainty. We follow field lines from a surface just outside and conformal with the LCMS until they strike a divertor plate or the first wall, or exceed 1000m in length, with effects of particle scattering mimicked by field-line diffusion. Current candidate divertor designs efficiently collect field lines, allowing fewer than 0.1\% to reach the wall. The sensitivity of localized power deposition, assumed to be proportional to the density of field- line strike-points, to adjustments in the divertor configuration is under investigation.\\ 1. T.B. Kaiser, et al, Bull. Am. Phys. Soc., 48, paper RP1-20, 2003.\\ 2. T.B. Kaiser, et al, Bull. Am. Phys. Soc., 49, paper PP1-73, 2004.\\ 3. R. Maingi, et al, EPS Conf. Rome, Italy, paper P5.116, 2006.   

Stellarator Coil Optimization

Coil design is critical to the cost and viability of stellarator fusion reactors. Coil design is a complex inverse problem with many subtleties. The complexity and inefficiency of the coils increases exponentially with the number of magnetic features retained. The physics properties of a stellarator plasma are determined by the shape of the outermost surface. Only a limited set of shape parameters are essential to that design. The optimal coil set for a given plasma equilibrium controls only the essential shape parameters. The study of small perturbations about the essential shape parameters yields the magnetic features that the coils must produce. This study allows the design of a coil set that possesses minimal complexity and a minimal ratio of the magnetic field on the coil surface to that on the plasma surface. Coil set designs for several plasmas are presented.   

Decomposition of Magnetic Field Boundary Conditions into Parts Produced by Internal and External Sources

Given the total magnetic field on a toroidal plasma surface, a method for decomposing the field into a part due to internal currents (often the plasma) and a part due to external currents is presented. The method exploits Laplace theory which is valid in the vacuum region between the plasma surface and the chamber walls. The method is developed for the full three dimensional case which is necessary for studying stellarator plasma configurations. A change in the plasma shape is produced by the total normal field perturbation on the plasma surface. This method allows a separation of the total normal field perturbation into a part produced by external currents and a part produced by the plasma response. There are immediate applications to coil design. The computational procedure is based on Merkel's 1986 work on vacuum field computations. Several test cases are presented for toroidal surfaces which verify the method and computational robustness of the code.   

Profile modifications by magnetic islands in the H-1NF plasma

Magnetic islands in fusion devices have serious impacts on plasma confinement. Islands can in general degrade the confinement by mixing up different regions of the plasma. However there has been experimental evidence of confinement improvement by transport barriers induced by the formation of islands, under favourable conditions. Detailed behaviour of magnetic islands is not fully understood, experimentally. Magnetic islands in the H-1NF heliac have recently been exploited to study this issue in detail. Accurate mapping of vacuum magnetic islands has been carried out using wire tomography. A Langmuir probe has been used to obtain the temporal and spatial profiles of local plasma parameters. Our experimental results indicate that under some conditions, magnetic islands serve as pockets of improved confinement regions in the plasma. This results in significant profile modifications including excitation of large radial electric fields in the range of 5kV/m. Experimental results are discussed.   

Local Observations of Energetic Particle Modes and Fast Ion Responses in the Compact Helical System

MHD instabilities induced by energetic ions are crucial issues in burning plasmas, and loss mechanisms of energetic ions have been strongly studied. In this research, we have applied a directional probe method to plasmas in CHS for local measurements of energetic ion responses to EPMs, and observed two kinds of responses. Moreover, magnetic fluctuations of EPMs were simultaneously measured at almost the same position with the energetic particle measurements. The phase relation between them was experimentally investigated. Temporal change of the spatial structure of EPM was also observed during the frequency chirping-down.   

ECRH and its effects on neoclassical transport in stellarators

The effect of ECRH heating on the neoclassical transport of stellarators is addressed.~ We present a calculation that proceeds by solving for the lowest order electron distribution function using a Fokker-Planck equation including the effects of ECRH.~A lowest order energetic electron population is described by balancing collisions off of a background Maxwellian plasma and ECRH heating as described by a quasilinear diffusion operator.~Finite ECRH beam width and relativistic detuning effects are accounted for in the diffusion operator.~Radial particle fluxes are calculated from the 1st order corrections to the kinetic equation. With the presence of a large energetic trapped electron population, enhanced neoclassical transport is generally expected in low collision frequency plasmas.~However, a self-consistently generated E X B poloidal drift reduces the direct losses of trapped electrons.~ Progress in using ambipolarity constraints to determine the radial electric field and implications for the achievement of electron root and associated enhanced confinement regimes will be addressed.   

Recent Developments in Quasi-Poloidal Stellarator Physics

The Quasi-Poloidal Stellarator (QPS) is a different type of compact stellarator with very low aspect ratio $R$/$a\sim $ 2.7, 1/4--1/2 that of existing stellarators. QPS has little variation of $\vert $B$\vert $ in the poloidal direction and larger variation in the toroidal direction, and is thus more like a linked magnetic mirror than a tokamak or other stellarators. The quasi-poloidal symmetry reduces anomalous transport by decreasing the poloidal viscosity by a large factor, thus strongly promoting self-generation of sheared flows that break up turbulent eddies. The magnitude, direction and variation within a flux surface of plasma flows in QPS that affect transport and stability differ from those in other stellarators (LHD, W 7-X, NCSX). The self-generated flow shearing is sufficient to impact temperature gradient modes. QPS is the only toroidal device stable to drift wave turbulence over a range of temperature and density gradients, which should reduce anomalous transport even in absence of flow shearing. The magnetic field structure has a large fraction of trapped particles in regions of low/favorable field line curvature while all other toroidal devices have a significant fraction of the trapped particles in regions with bad curvature. This strongly reduces the drive for a class of trapped-particle instabilities.   

Innovations in Quasi-Poloidal Stellarator Design

The Quasi-Poloidal Stellarator (QPS) is being developed with very low plasma aspect ratio, 1/2-1/4 that of existing stellarators. Design innovation is driven by both the complex 3-D geometry and the need for reduced cost and risk in fabrication, so QPS differs significantly in design and construction from other toroidal devices. An internally cooled, compacted cable conductor consisting of stranded copper filaments wound around an internal copper cooling tube was developed that can be wound into complex 3-D shapes. This conductor is wound directly onto the complex, highly accurate, stainless steel coil winding forms. Simplified coil winding procedures lead to faster fabrication and reduced technical risk. A full-size prototype of the largest and most complex of the winding forms has been cast using a patternless process (machined sand molds) and a high-temperature pour, which resulted in $<$1/10 the major weld repairs of similar sand castings using conventional patterns, and machined to high precision. A vacuum-tight cover is welded over each coil pack and a high-temperature cyanate ester resin is used for vacuum pressure impregnation of the coils because it has several important advantages over the usual epoxy. The completed coils are then installed in an external vacuum vessel.   

An overview of ideal MHD stability calculations for the Quasi-Poloidal Stellarator

An overview of the status of ideal MHD stability calculations for the Quasi-Poloidal Stellarator (QPS) is presented. The primary focus of these calculations has been infinite-$n$ ballooning modes using the COBRAVMEC code. Previously, it was shown that QPS is potentially susceptible to ballooning instabilities for $\left< \beta \right> > 2\%$ and that regions of second stability exist for $\left< \beta \right> > 4\%$ [A. S. Ware, {\it et al.}, Phys. Plasmas {\bf 11}, (2453)]. Recent calculations have examined the possibility of testing ballooning stability in QPS at lower plasma beta. An optimization is underway that targets degraded ballooning stability at $\left< \beta \right> \sim 1\%$ without degrading the neoclassical confinement propoerties. The results of this optimization will be presented. In addition to the infinite-$n$ calculations, previous work has been done on global, finite-$n$ calculations using the TERPSICHORE code. Here, we present a status report on initial attempts to apply both the TERPSICHORE and CAS3D codes to examine global ideal MHD stability for QPS plasmas.   

Effect of symmetry-breaking on ballooning modes in quasi-symmetric stellarators

The impact of degraded symmetry on global ballooning stability is examined in the quasi-poloidally symmetric QPS, the quasi- helically symmetric HSX, and an axisymmetric configuration. In the ray tracing method, global ballooning mode stability is calculated by following rays in the eigenvalue space determined by the results of local, infinite-$n$ ballooning theory. The eigenvalue is a function of the flux coordinate $q$ (the safety factor), the field line label $alpha$, and the ballooning parameter, $\theta_{k}$. For each of the different magnetic configurations, the impact of breaking the symmetry (or degrading the quasi-symmetry) on ballooning modes is examined. For the HSX configurartion, three cases are examined: the standard quasi-helically symmetric case, a mirror case, and a hill case. The mirror and hill cases represent degraded symmetry configurations for the HSX experiment. The weak global shear in HSX results in modes which only weakly depend on the ballooning parameter. For QPS, the standard quasi-poloidally symmetric case and a degraded symmertry case are examined. In all of the non-axisymmetric cases studied, the unstable modes are localized in the field- line label, $\alpha$. The results for these cases will be presented.   

Shear flow generation in stellarators - configurational variations

Plasma momentum transport within magnetic surfaces is important for a number of toroidal plasma physics issues, such as: turbulence suppression, impurity transport, and bootstrap current generation. Stellarators provide new opportunities for understanding of plasma flow effects because (a) new forms of quasi-symmetry (e.g., helical, poloidal) can be produced that differ significantly from the tokamak; and (b) symmetry breaking effects remove the degeneracy between parallel and cross-field transport characteristic of symmetric systems. A method has been developed to evaluate neoclassical self-generated plasma flows in stellarators; this indicates that flow directionality and shearing rates are significantly influenced by the magnetic structure. The possibility that such effects could be a new hidden variable in the stellarator confinement database is an important focus for our model. For example, a recent analysis of a series of inwardly shifted LHD discharges has indicated that decreases of up to a factor of 10 in the neoclassical viscosity (allowing greater flow shearing) were correlated with the experimentally observed improved confinement times.   

Stability of Super Dense Core plasmas in the Large Helical Device

Recent experiments [N. Ohyabu et al, Phys. Rev. Lett, in press] using pellet injection into reduced-recycling discharges in the Large Helical Device have yielded Super Dense Core (SDC) plasmas with very peaked density profiles, high central density $\sim $ 4.5 x 10$^{20}$ m$^{-3}$, and improved confinement. We have examined ideal MHD stability of these SDC configurations the using the 3-D COBRA stability code [R. Sanchez et al, Comp. Phys. Comm \textbf{141}, 55 (2001)]. These calculations show that the core region inside the zero-shear radius has direct access to second stability, i.e., the stability margin increases with $\beta$. Outside the zero-shear radius, the plasma becomes unstable to ballooning modes at average $\beta \sim $3-4{\%}. Of course, resistive versions of the modes are expected to appear at lower $\beta$. These MHD effects may play a role in improving core confinement, and may also provide a useful mechanism to constrain the plasma pressure in the outer plasma region and thus help maintain the favorable SDC state.   

Three-Dimensional Equilibrium Reconstruction: V3FIT

Axisymmetric equilibrium reconstruction has proven invaluable for equilibrium control, and for comparisons with MHD stability and confinement predictions. The V3FIT code, currently under construction, will perform fast, accurate reconstruction for stellarators. To be most useful for experiments, the V3FIT code will need to a) run rapidly, b) be flexible, and c) be extensible. V3FIT uses the VMEC three-dimensional equilibrium code to solve the forward problem: given parameters such as the current and pressure profile, find the expected signals from diagnostics like magnetic diagnostic loops and microwave interferometers. The reconstruction algorithm is tightly coupled to the VMEC force balance equilibration iterations. Preliminary results from V3FIT reconstructions will be shown, and potential improvements to the reconstruction algorithm will be presented.   

ECRH and Ohmic Plasmas in the Compact Toroidal Hybrid (CTH) Experiment

Plasma is generated in the low-aspect ratio CTH torsatron (R$_{O}$ = 0.75 m, R$_{O}$/a$_{p} \quad \le $ 4, B $\le $ 0.7 T) by electron cyclotron heating at a frequency of 18 GHz. Reliable breakdown at the fundamental frequency is obtained when RF power is launched into a broad, resonant saddle region present on the inboard side of the magnetic axis. Discharges up to 1 second duration have been produced with input RF power $\le $ 10 kW. Ohmic currents are induced in ECRH-formed plasmas by capacitor bank discharge into a dedicated OH transformer coil set. Present experiments have begun with low plasma currents of approximately 5 kA with an ohmic phase lasting 75 msec. These experiments are the first in the planned studies of equilibrium and stability of current-carrying stellarator discharges. Ongoing efforts aim at improving plasma operation and implementing SXR arrays and a full set of magnetic diagnostics.   

Vacuum Field Mapping of Magnetic Equilibria and Islands in the Compact Toroidal Hybrid Experiment

Vacuum field mapping experiments are being performed on the Compact Toroidal Hybrid (CTH) to get an accurate model the magnetic coils and to minimize magnetic islands. Using the electron-beam, phosphor-coated screen/wand techniques, comparisons are made between the designed and experimentally achieved magnetic configurations. A Single Value Decomposition (SVD) technique uses the modeled and experimentally measured magnetic axis position and rotational transform to attempt to give a more accurate model of the CTH coils. Furthermore, magnetic islands are observed at low order rational surfaces. Corrections to reduce the resonant island sizes are computed with the Fix Stellarator code and applied by a set of up to 15 circular error-correction coils. We have successfully minimized the size of the island on the iota = 1/3 rational surface. Efforts are underway to reduce the island width of multiple island chains in the same equilibrium.   

Overview of Recent Results from HSX and the Planned Experimental Program

HSX has previously demonstrated that the quasihelical symmetry (QHS) does indeed improve single-particle confinement and reduce parallel viscous damping over a non-optimized 3-D configuration. New neoclassical differences have been observed under the present operating conditions including reduced thermodiffusion and electron thermal conductivity in the QHS case as compared to the mirror case. Current measurements are consistent with bootstrap calculations and show the current flows in a direction opposite to the tokamak. MHD modes have been observed tied to the presence of energetic electrons in the QHS configuration. A new ECRH transmission line now permits operation at full tube power. Our goals are to increase the density, the magnetic field and heating power to accentuate neoclassical transport relative to anomalous. A CHERS system is being implemented to infer radial electric fields for transport analysis. Preparations are near complete for going to B=1.0T for O-mode heating.   

Higher power ECRH in the Helically Symmetric Experiment

A new transmission line has increased the maximum available ECRH power to HSX from 50 to 200 kW. Using a 28 GHz gyrotron at a magnetic field of 0.5 T, the extraordinary wave is launched at the second harmonic of the electron cyclotron frequency. In this new regime, the plasma stored energy, confinement time and electron temperature are studied as a function of the absorbed power and the plasma density. Comparisons with the international stellarator transport scaling database will be presented. In the configuration with quasihelical symmetry, the central electron temperature is higher ($\ge $1 keV) than in configurations without symmetry. The ECE diagnostic indicates the presence of a non-Maxwellian component in the emission spectrum. Results from the CQL3D Fokker-Planck code used to model the ECRH absorption and Electron Cyclotron Emission spectrum will also be presented.   

The HSX Hybrid Quasioptical Waveguide System

Plasma formation and heating in the HSX stellarator is accomplished via 28 GHz ECRH. The ECRH transmission line has been upgraded recently from an over-sized wave-guide to a hybrid quasioptical line to increase maximum launched power and to reduce the chance of arcing. A Vlasov mode converter transforms the dominant $TE_{02}$ gyrotron output mode to a Gaussian beam. The quasioptical portion of this system refocuses the beam, corrects the astigmatism, and rotates the beam polarization for X-mode or O-mode heating. A circular cross-section, dual-mode waveguide transmits microwave power from the quasioptical unit to the launching port where an ellipsoidal mirror refocuses the beam into the corrugated waveguide. This system has been successfully tested for 50 ms pulses with launched power up to 100 kW. Modal analysis of the microwave beam is accomplished with a thermal imaging technique which utilizes a ceramic target at various points along the beam path for 50 kW, 2 ms microwave pulses.   

Theory Based Transport Modeling of B=0.5 T, ECRH Plasmas in HSX

Theory based models for both neoclassical transport [1] and ITG/TEM anomalous transport [2] are used to predict density and temperature profiles in the HSX stellarator. Although the axisymmetric ITG/TEM model of [2] can not treat 3D geometry effects, recent 3D gyrokinetic linear stability calculations [3] have demonstrated the impact of the local geometry on ITG/CTEM linear growth rates in stellarators. As an approximation to the results of [3], the largest value of local bad curvature in the low field/ballooning region of HSX is used in the evaluation of [2], with no free fit parameters. The model input particle and ECRH power source rates are determined from 3D neutral gas simulations and ray tracing calculations, respectively. The predicted density and temperature profiles are in reasonable quantitative agreement with a number of experimental profiles in the quasi-helically symmetric configuration. The model also predicts profile changes similar to experiment in a configuration with the quasi-symmetry intentionally degraded. [1] J.N. Talmadge et al., Fus. Sci. {\&} Tech. \textbf{46}, 255 (2004) [2] H. Nordman et al., Nucl. Fusion \textbf{30}, 983 (1990) [3] G. Rewoldt et al., Phys. Plasmas \textbf{12}, 102512 (2005)   

Measurement of the Bootstrap Current in HSX

The bootstrap current in the quasihelically symmetric stellarator HSX is in the opposite direction to that in a tokamak, reducing the rotational transform but at the same time reducing particle excursions from flux surfaces. Knowledge of the bootstrap current is critical to predicting operation of advanced stellarators. The current as measured with an external Rogowski coil in HSX rises throughout the discharge on a 10-40 ms timescale and approaches a maximum value between 100-300A. Profiles are measured with a 10-chord Thomson scattering system, showing central electron temperatures up to 700 eV and peak densities of 4 x 10$^{12}$ cm$^{-3}$. Both the equilibration time and maximum value can be adjusted by variation of the electron pressure profile and associated gradients. Reversal of the magnetic field reverses the direction of the toroidal current. Theoretical models used to predict and model the bootstrap current will be presented. Initial estimates predict between 200-400A, consistent with the measured values.   

Studies of Energetic-Electron-Driven Alfv\'{e}nic Modes in HSX

Coherent, global fluctuations in the range of 20-120 kHz are observed for quasi-helically-symmetric 2$^{nd}$ Harmonic X-mode ECRH produced plasmas in HSX. Measurements and theory indicate that the mode is likely the global Alfv\'{e}n eigenmode (GAE). Dependence of mode amplitude and frequency with heating power (electron temperature) will be measured using up to 200 kW of ECRH. Plasma stored energy loss related to onset of GAE mode activity will be explored. Preliminary results indicate that the toroidal precession of energetic electrons might be responsible for driving and resonant to the mode. Measurements indicate the mode has helicity m/n=1/1, and inversion of interferometry data shows the amplitude peaks in the region of steepest density gradient (r/a=0.4). Newly constructed magnetic coil arrays will be used to provide further information.   

HIBP Designs for measurement of the electric field in HSX

Understanding the relative roles of neoclassical and anomalous transport in advanced stellarators is critically dependent on knowledge of the radial electric field. A feasibility study has shown that it is practical to measure the radial electric field in the Helically Symmetric eXperiment, HSX, using ion beams. Two options have been explored, a standard Heavy Ion Beam Probe, HIBP, and a system that measures the deflection of an ion beam due to the plasma electric field. The standard HIBP measures the local space potential at multiple points, allowing a calculation of the radial electric field. Estimated signal levels are similar to some previous systems, most notably the EBT HIBP. It is also capable of measuring fluctuations in potential and density. The second option studied measures the change in a probing ion beam trajectory due to the electric field. HSX vacuum magnetic fields are virtually unchanged by the plasma, therefore changes in the beam trajectory due to plasma would be dominated the plasma electric field. The changes are path integrated and the local electric field is determined by running multiple trajectories and inverting. A beam deflection system is simpler and needs a lower ion accelerator voltage than a HIBP, but it provides less information.   

Statistical mechanics of Vlasov continua

Ideas of statistical mechanics have been applied to describe the turbulence of continuum systems, such as those described by the Vlasov and two-dimensional Euler equations, since the early work of Burgers, Onsager, Lee, Lynden-Bell, Kraichnan, Montgomery, and others. Results using two new approaches for calculating the partition function and obtaining turbulent spectra will be discussed. The first is based on the use of normal coordinates associated with continuum eigenfunctions, which are used in the manner of particle degrees of freedom for finite systems. The second is based on an experimentally verifiable definition of independent subsystems and concomitant independent invariants. Appropriate additive invariants are use in the calculation of the partition function. Regions of validity of the two approaches will be discussed.   

A Critical Analysis of Semi-Classical Potentials

Semi-classical potentials have recently seen a surge in interest because of the possibility of their use in simulations to determine equilibrium and non-equilibrium properties of plasmas under extreme conditions. Various authors have developed potentials for Coulomb interactions as well as to mimic degeneracy effects. However, there exists little guidance regarding the physical regimes in which these semi-classical potentials are applicable. We present a critical analysis, using analytical and numerical techniques, of the success and breakdown of the assumptions behind semi-classical potentials. The effectiveness of pair potentials in reproducing many-body degeneracy effects is determined in comparisons with exact quantum ideal gas results. The effectiveness of semi-classical Coulomb interactions in reproducing equilibrium properties is studied via comparison with the principle Hugoniot, as determined both by other simulation techniques and experiments. The principal Hugoniot provides an especially strong test because of its sensitivity to the relationship between pressure and energy, as well as the role played by strong coupling, degeneracy, and atomic and molecular physics at low temperatures. In determining at what temperatures and densities and for what reasons these potentials succeed and fail, we hope to provide insight into a powerful technique available to the simulation community.   

Numerical and analytical study of nonlinear Landau damping

Nonlinear interaction of a plasma wave with resonant particles is essential classical problem in plasma physics. Studying of its nature has been performed for almost fifty years and resulted with a number of models. Most of these models are complicated and it is hard to apply them to solve particular problems. New simplified model of quasilinear saturation of Landau damping of growing in time plasma wave is proposed. This model describes interaction in terms of fluid equations on the wave amplitude and several low-order coefficients of Taylor expansion of the distribution function within trapped region. These parameters allow determining both nonlinear Landau damping rate and nonlinear frequency shift of the plasma wave due to its interaction with trapped particles. Proposed model is verified with PIC simulations using XES1 code. The results of the numerical experiment are in reasonable agreement with analytical. This work was supported by DOE Contracts No. AC02-76CH0-3073 and DE-FG52-04NA00139.   

Suppression of drift wave instability due to sheared field-aligned flow and negative ions

Sheared field-aligned plasma flow is a significant topic in space/circumterrestrial plasmas. Taking into account negative ions or dust grains will make the space plasma physics more general and accurate. Using the Q$_T$-Upgrade Machine, we have conducted laboratory experiments to examine negative ion effects on shear-modified drift waves. Field-aligned K$^+$ ion flow and its shear strength are controlled with a concentrically segmented W hot plate. Negative ions SF$_6^-$ are produced by introducing SF$_6$ gas in the plasma. The drift wave shows a gradual monotonic decrease in amplitude as the shear strength is increased from zero. However, as the shear strength is decreased from zero to negative values, the amplitude increases up to a certain shear strength and rapidly decreases after the peaking. The negative ion introduction, in general, suppresses this instability while retaining the dependence of the amplitude on the shear. These wave characteristics are interpreted using the theories of current-driven (kinetic) and of D’Angelo (fluid) instabilities.   

Theory of Horseshoe Maser Instability

When a beam of electrons with a thermal spread moves along converging magnetic field lines, the distribution function evolves into a horseshoe shape in (p$_{\parallel}$ , p$_{\perp}$) space. In such a distribution there is a population inversion of particles in perpendicular momentum which results in the plasma being subject to a cyclotron maser instability. There is strong evidence that this type of instability is the source of auroral kilometric radiation. The theory of the instability indicates that the phenomenon can be scaled to laboratory dimensions, with centimetre rather than kilometre wavelengths, and an experiment to do this has been constructed at the University of Strathclyde (see the paper of S D.C. Speirs et al at this conference). Recently we have carried out an analysis of this horseshoe instability in cylindrical geometry and have shown that the predicted growth rate is in line with what is seen in simulations and sufficient to account for growth of the instability within the dimensions of the experiment. Here we extend the analysis in order to understand in more detail the properties of the dispersion relation and attempt to relate these properties to experimental observations in both the laboratory and space. Theory predicts high growth rate of several spatial modes propagating almost perpendicularly to the electrons motion, with quasi-TE mode structure and sharp spectrum with a resonance at the frequency just below the cyclotron frequency.   

Investigation of instability of the shell distribution produced in the process of interaction of whistler waves with a magnetized plasma

A theoretical model of cyclotron interaction of electrons with whistler waves excited in the process of laser pulse propagation in a magnetized plasma, connected with the pitch-angle diffusion of electrons, leads to formation of the shell-like distribution of resonant electrons [1]. This distribution formed due to quasi-linear diffusion of resonant electrons in the presence of whistler waves, produced in the process of propagation of a laser pulse along the external magnetic field, can be responsible for the excitation of electrostatic waves propagating at an angle to the magnetic field. These waves can then cause heating of the cold electrons in the system. In order to better understand the heating mechanisms associated with the instability of the shell distribution, we performed 2D PIC simulations. These provided an estimate of the amount of energy transferred from the laser pulse to the cold plasma particles. [1] V. I. Sotnikov, Y. Sentoku, V.B. Krasovitskiy, Physics of Plasmas, \textbf{12}, 1 (2005).   

Experimental observation of whistler waves arising from mode transitions between pairs of discrete Langmuir cavity modes.

Bursty whistler wave packets, excited spontaneously in the Green Tank at KTH, by a B-aligned electron beam from a hot cathode, appear as wave packets, each with 0.1 - 1 microseconds duration and altogether covering a few percent of time. Wave packets, each dominated by a single frequency, are found in a broad frequency range, 7 -- 40 MHz [Brenning et al., J. Geophys. Res. submitted]. Also seen are electrostatic oscillations, 300-500 MHz, covering numerous standing-wave frequencies, in a narrow HF-spike structure [Gunell, PhD thesis, 1997] located at the high potential side. Here we demonstrate the correlation between electrostatic mode transitions (occurrence, mode-pair frequency difference, and mode frequency-modulation) and wave-packets (occurrence, frequency, and envelope). We conclude that an individual whistler wave packet arises from a cavity-mode transition. Wave packets are observed only when the cavity mode separations are within the whistler range.   

Laboratory Investigation of Whistler and Lower Hybrid Wave Propagation

Many interesting \textit{in situ} and laboratory observations of whistler and lower hybrid waves have been made over the past few decades. Observations such as these have prompted NRL Space Physics Simulation Chamber investigations of nonlinear whistler wave dynamics and lower hybrid solitary structures. For the initial experiments, we have fabricated and tested transmitting and receiving magnetic loop antennas and crossed electric field dipole receiving antennas. Electromagnetic modes launched in the Space Chamber plasma have been identified as whistler waves. Propagation characteristics of both whistler and lower hybrid waves have been investigated in homogeneous plasma conditions. Experiments into the interaction of whistler/lower hybrid waves with pre-existing plasma density structures have begun. In addition, preliminary investigations into the nonlinear properties of whistlers have provided indication of whistler wave ducting. Experimental results related to the propagation characteristics of whistler/lower hybrid waves under these conditions will be presented.   

Experimental Antenna Impedance Measurements in a Magnetized Plasma

The characteristic impedances of two antennas immersed in a magnetized plasma have been measured by monitoring the reflection coefficient $\Gamma$ using a network analyzer. The first antenna is spherical, the second an electric dipole. Previous theoretical and experimental studies have focused almost exclusively on very weakly magnetized ($\omega_c<<\omega_p$) plasmas or short ($L<<\lambda$) antennas. Here we present results which are an extension of this work and of our previous unmagnetized plasma antenna impedance experiments into higher density, strongly magnetized plasmas. Where applicable, experimental impedance curves are compared with analytic results. The experimental work was performed in the large volume Space Physics Simulation Chamber (SPSC) at NRL where the plasma is characterized by electron densities and temperatures of n$_e$ = 10$^6$-10$^{11}$ cm$^{-3}$ and T$_e$ = 0.5 eV.   

Inferring electron sheath density profiles in collisionless plasmas: a non-perturbative technique

We extend a recently published work$^{2}$ which uses a simpler derivation than previous authors of plasma collisionless resistance in spherical geometry. The experimental work is based on measurements of the rf impedance characteristics of a small spherical probe immersed in collisionless laboratory plasma. The plasma resistance is shown to be inversely proportional to the plasma density gradient evaluated at the location where the plasma frequency is equal to the applied frequency. With the assumption that electrons obey the Boltzmann relation, and cold, massive ions, which obey the Bohm condition at the sheath edge, we relate n$_{e}$ and potential in the presheath. This allows an expression for collisionless resistance in the sheath as a function of applied frequency. Using numerical integration of measured resistance, we are able to infer the sheath electron density profile. We will present results of three experimental studies showing how to construct sheath density profiles using spheres of differing radii. $^{2}$Walker, D.N., R.F. Fernsler, D.D. Blackwell, W.A. Amatucci, S.J. Messer, \textit{Phys of Plasmas}, $13$, 032108 (2006).   

Ion collection by a sphere in a weakly magnetized plasma

Ion collection by a sphere in a collisionless flowing magnetoplasma is studied using the kinetic code SCEPTIC[1]. The key features of this 2d3v electrostatic PIC code are a spherical geometry accurately resolving the sheath at the collector's edge, and a Boltzmann treatment of the electrons. We concentrate on the transition between unmagnetized and weakly magnetized regimes: ion thermal Larmor radius $\rho_i > r_p$ (sphere radius). Two different Debye length ($\lambda_D$) regimes will be covered, giving the most relevant effects of the magnetic field in each regime. In the case $\lambda_D\ll r_p$, relevant to mach-probe physics, as the magnetic field increases the angular collection distribution changes particularly strongly on the downstream side. The flow and the magnetic field effect being correlated, we will deduce to what extent previous calibrations based on their independence are still valid. In the long Debye length case, $\lambda_D\ge r_p$, most relevant to dust in plasmas, one interesting effect of the magnetic field is to cancel the flux reversal caused by ion focusing downstream of the plasma flow. [1] I.H. Hutchinson {\bf PPCF 45} (2003) 1477.   

Alfv\'{e}nic phenomena triggered by resonant absorption

A simulation and modeling study is made of the nonlinear interaction of an electromagnetic pulse with a magnetized plasma having a cross-field density gradient. For small amplitudes the pulse propagates up to the cut-off layer where an Airy pattern develops. Beyond a certain power level the ponderomotive force produced by the standing electromagnetic fields carves density cavities. The excess density piled-up on the side of the cavities causes secondary, field- aligned plasma resonances to arise. Due to the short-scale of the secondary resonant fields excited, strong electron acceleration occurs. The fast electrons exiting the new resonant layers induce a return current system in the background plasma. This generates a packet of shear Alfv\'{e}n waves of small transverse scale and increasing frequency. The results provide insight into microscopic processes associated with a recent laboratory investigation in which large amplitude Alfv\'en waves have been generated upon application of high-power microwaves [B. Van Compernolle et al., to appear in {\it{Phys. of Plas}}].   

Low Velocity Ion Stopping in a Strongly magnetized Plasma

We focus attention on low velocity stopping of heavy ions in a strongly magnetized electron-ion plasma target considered in a guiding center approximation with electron Larmor radius much shorter than correponding Debye length. Elaborating on an exact equivalence between low velocity stopping and particle diffusion through the magnetic field B [1],we can proceed to an analytic determination of the given friction coefficient pertaining to a stopping power~projectile velocity. The latter emboddies a combination of hydrodynamical in 1/B and kinetic in 1/B$^2$ contributions. It is appropriate to use effective interactions regularizing quantum-mechanically the short range electron-ion interaction [2] \newline \newline [1] Y. Furutani, Y. Oda, C.Deutsch and M.M. Gombert, Phys Rev 26A,2913 (1982) \newline [2] C Deutsch,Phys Lett A60,317[1977)   

Gyrokinetic Particle Simulation of Mirror Instability

Magnetic mirror instability is the slow magnetosonic wave driven unstable by the anisotropic pressure in collisionless high-beta plasmas. In previous analysis, it is found that the maximum growth rate increases with perpendicular wave number until the perpendicular wavelength becomes comparable to the ion-gyroradius. Therefore, the finite Larmor radius effects are important in determining the threshold and the wavelength of the kinetic mirror instability. We have developed a general dispersion relation of the mirror mode with finite Larmor radius effects. In the short wavelength limit, we find that the finite Larmor radius effect has stabilizing effects. Besides, the coupling to the slow sound wave is also found to be stabilizing. Linear gyrokinetic particle has recovered the linear dispersion relation. Nonlinear simulation will be carried out to study saturation mechanism of mirror instability.   

Self-interactions of electron Bernstein waves in an inhomogeneous plasma

The nonlinear dynamics of the electron Bernstein waves in the X- B mode conversion is simulated via both the $\delta$f and full particle-in-cell (PIC) simulations. It is shown that the nonlinear self-interaction of the electron Bernstein wave (EBW) can give rise to the second harmonic generation at a pump power as low as three orders smaller than the electron thermal energy. The theory describing this nonlinear wave coupling in an inhomogeneous plasma is obtained and is compared to the simulations. It is found that the amplitude of the second harmonic EBW excited can exceed that of the fundamental wave if the field of the fundamental EBW is sufficiently strong. With a high incident power, the generations of the non-propagation third and fourth harmonic modes are also observed.   

Numerical Study of Alfv\'{e}n Eigenmodes in a High-Beta Toroidal Plasma

\def\aTAE{$\alpha$TAE} \def\aTAEs{$\alpha$TAEs} Discrete toroidal Alfv\'{e}n eigenmodes trapped in $\alpha$-induced potential wells (so-called {\aTAE}) and their interaction with trapped energetic ions is studied numerically. Here, $\alpha = -q^2R\beta'$ is a measure for the pressure gradient. Previous investigations using positive magnetic shear ($s \equiv rq'/q > 0$) are extended to the $s<0$ negative shear regime. It is found that {\aTAEs} exist as bound states (radially and along the field line) regardless of the sign of $s$. While for $s>0$ {\aTAEs} tend to be localized in the bad-curvature region ($|\vartheta| < \pi/2$), for $s<0$ {\aTAEs} tend to peak at the top and bottom ($\vartheta \sim \pm \pi/2$), and also have larger amplitudes in the good-curvature region. These quasi-marginally stable modes can be excited by trapped energetic ions through resonance with the precessional drift or bounce-precession resonances, whereby excitation is easier for $s>0$. Extensions to regimes with low magnetic shear and the inclusion of thermal ions are currently underway in order to study the properties of {\aTAEs} near the minimum of the safety factor $q$ in reversed-shear configurations and near the second ballooning stability boundary. Corresponding results will be reported as they become available.   

Experimental study of non line-tied kink instability for different axial boundary conditions in RSX

The typical theoretical description of the kink, which is either toroidal or periodic, cannot describe non toroidal natural systems, e.g. astrophysical jets and erupted solar prominences. Using flux ropes in Reconnection Scaling eXperiment (RSX), a linear plasma device, it has recently been demonstrated that the kink instability is greatly affected by the axial boundary conditions. For example, when one end of the current rope is free to move, i.e., not line tied, the kink stability threshold is reduced to less that half of the Kruskal-Shafranov prediction [1]. We are investigating the kink behavior of finite length flux ropes with a focus on the axial boundary conditions. Different shapes and sizes of end plate will be used to test the dynamical evolution, relaxation response and saturated long lifetime end state of the kink motion. We will show magnetic and electrostatic fluctuation data at different places in the plasma column and boundary. \newline \newline [1] I. Furno, et al, Phy. Rev. Lett. 97, 015002 (2006)   

Onset and saturation of the kink instability in a current carrying, line-tied plasma surrounded by a conducting shell

The MHD stability properties of a line-tied plasma have been studied in a linear screw pinch device. An internal kink instability is observed to grow when the safety factor $q = \frac{4\pi^2 r^2 B_z}{\mu_0 I_p(r) L}$ drops below 1 inside the plasma. This mirrors the stability condition for external kinks that the edge $q$ remain above 1. The growth rate scales with the wall time, as predicted by theory. After a brief growth phase, the mode saturates as a helical equilibrium. The main diagnostics for characterizing the MHD activity is a 2D array of 80 radial magnetic field pickup coils surrounding the plasma column, a segmented anode, which serves to measure current distribution inside the plasma, and an array of 40 poloidal and axial magnetic field coils inside the conducting shell. In addition to the ideal mode, reconnection events are observed to periodically flatten the current profile and alter the magnetic topology. The 2D array indicates a plasma dominated by an m=1 mode, while internal axial magnetic field sensors highlight an n=1 mode. Finally, initial results of a resistive wall with a longer wall time surrounding the plasma will be presented. This work was supported by the DoE.   

Axially Inserted Magnetic/Langmuir Probe

A new probe has been designed to provide internal magnetic and Langmuir diagnostics throughout the rotating wall machine. The machine is a linear screw-pinch built to study the role of different wall boundary conditions on the resistive wall mode (RWM). Individual mode stability depends on the value of the safety factor at the plasma edge ($q_a$). The plasma is produced by an array of nineteen guns, creating a column one meter in length and up to 20 cm in diameter. The central guns in the array can be biased to individually discharge 1 kA of plasma current. Different instabilities are studied by changing the current and density profiles of the guns. But profile measurement has been limited to the top and bottom of the plasma column by radial port access. The new axially-inserted \mbox{Q-Tip} probe can travel along the entire span of the column. The \mbox{Q-Tip} combines ten pickup loops with four triplets of electrodes for simultaneous magnetic and Langmuir measurements. The edge $a$ and safety factor $q$ of the plasma can now be found for different instabilities seen along the column. This poster tracks the change of $n_e, T_e, \Phi_p$, and $q$ throughout the plasma.   

Atomic Coherence Length Spectroscopy of Pulsed Plasma

The movement of atoms at an interface must entail sequences of atomic collisions. Such collisions reduce the coherence of atomic emissions. Since the sequence of such collisions is stochastic, we postulate that coherence lengths associated with plasma emissions from the interfacial atoms must vary in accordance with the correlated movements of the neighboring atoms of the interface. To explore the connection between atomic fluctuations and the onset of instability, we have developed an interferometer coupled to a spectrometer for measurement of the coherence length of plasma emissions at a selected wavelength. The interferometer is capable of utilizing the plasma emissions from a point in the plasma to form interference fringes, or extracting two separate beams from two adjacent but different points of the plasma to construct the interference fringes. The fringes are processed in a spectrometer so that internal excitation of plasma atoms may be examined. The fringe field is captured time resolved by using a gated intensified array detector. We show an interference-fringe field constructed from a spark-gap discharge and also from a laser produced plasma plume that is confined in a dense neutral gas. Visibility of the interference fringe field is used to determine the coherence length as a function of position within the plasma and as a function of angle the line of sight makes with a symmetry axis.   

Confinement of enhance-transmitted light beam by surface plasma induced on sub-wavelength slits.

The enhancement of light transmission through sub-wavelength slit is an interesting and important recent discovery in plasmonic nano-photonics. This is studied with our newly developed two dimensional simulation code of finite-difference time-domain method.~The simulations verify the enhanced transmission that far exceeds the diffraction limit and help to understand in depth the resultant beaming of light that will be important in nano-optical applications. The beaming of light is found due to the interference of the first order scattered light with the zeroth order transmitted light and is divergent with a finite angle. By proposing the interference between the zeroth and zeroth orders, we found the light can be confined as an almost divergence-less beam. The dynamics and interesting physics of our new scheme and founding will be discussed.   

Structure of KEEN Waves in Phase Space: Partitions, Coherence, Spectrum

We will show just where in the ponderomotive driver, w-k plane, KEEN waves [1] can be excited and sustained after the drive is turned off. For k lambda Debye values of 0.1-0.6 and w/wp values of 0.1-1.6, we will examine the morphological differences in sustained KEEN waves based on the stiffness of the background plasma and the self-organization properties of these highly nonlinear coherent structures using Vlasov- Poisson simulations. The partitions of phase space into a series of concentric oscillating traps where on average many more particles are trapped than untrapped will be explained [2]. The extent to which these are entirely different objects than the stationary BGK modes often sought in nonlinear systems will be explained. The impact these waves can have in the evolution of nonlinear electron plasma waves will also be given. \newline [1] B. Afeyan, et al., Proc. Inertial Fusion Sciences and Applications, 213, eds. B. Hammel, D. Meyerhofer, J. Meyer-ter-Vehn and H. Azechi, Amer. Nucl. Soc. 2004 \newline [2] B. Afeyan, V. Savchenko, K. Won, T.W. Johnston ``New Long-Lived Nonstationary Coherent Structures in Vlasov Plasmas: KEEN Waves,'' submitted to Physical Review Letters, 2006.   

White light parametric instabilities in plasmas

Parametric instabilities are pervasive in many fields of science, its importance stemming from their close connection to the onset of nonlinear and collective effects such as solitons, vortices, self-organization, and spontaneous ordering. Developments in light sources and laser technology, continue to reveal novel features of the parametric instabilities in plasmas, and have revived many questions associated with parametric instabilities driven by partially coherent radiation sources. These instabilities are critical in many problems, and have eluded a self-consistent treatment because of the lack of the appropriate theoretical framework. We employ a formalism, directly inspired in the Wigner-Moyal formalism for Quantum Mechanics, to establish the general dispersion relation for parametric instabilities driven by electromagnetic radiation, with arbitrary statistics, coupled to the electron collective dynamics in a unmagnetized plasma. The one-dimensional analysis for Stimulated Raman Scattering reveals a growth rate dependence with the coherence width $\sigma$ of the radiation field scaling as $1/\sigma$ for backscattering, and $1/\sigma^{1/2}$ for forward scattering, and a significant dependence of the instability growth rate on the shape of the power spectrum of the radiation. The results open the way to a full multi-dimensional description of parametric instabilities driven by intense radiation sources, with arbitrary statistics, interacting with plasmas.   

Modulational Instabilities and wave Collapse Driven by Broadband Turbulence

The interaction of broadband turbulence with plasmas has been studied in a number of regimes. Beam instabilities occur in this interaction when the phase velocity of the long- wavelength monochromatic wave is nearly equal to the group velocity of short-wavelength wavepackets, or quasi-particles, associated with the turbulent spectrum. It is shown that quasi-particle Landau-damping can take place, as well as quasi-particle modulational instabilities, thus establishing a direct link between short- and long-wavelength perturbations of the medium. Numerical simulations are reported for three cases (i) intense laser pulse driving a plasma wakefield, for use in plasma-based acceleration. (ii) a broadband distribution of drift modes coupling to zonal flows in a tokamak-like configuration, as well as the magnetopause, and (iii) wave collapse of strong Langmuir turbulence driven by electron beams.   

Ion-acoustic surface modes of a plasma with kappa distribution

Electrostatic surface waves propagating on the interface between a vacuum and a plasma with kappa distribution are kinetically derived by using the Vlasov-Maxwell equations. The plasma is semi-bounded, uniform, isotropic, collisionless, and unmagnetized. A specular reflection condition in which the charged particles undergo a mirror reflection is used for the boundary condition on a small perturbation distribution. It is shown that the ion-acoustic surface modes can be varied with the values of the spectral index $\kappa $, especially in the long wavelength limit.   

Shock Wave Propagation Measurements in Glow Discharge Plasmas

Mach 1.5--2.2 shock waves are produced in argon over a range of pressures 3--15 Torr by a fast capacitor discharge (quarter period $\tau _{1/4}$ = 1.4 $\mu $s). The shock waves are allowed to traverse through a glow discharge plasma inside the shock tube, where the deflections of the laser beams, caused by the density jump at the shock front, are recorded on a fast oscilloscope. An average shock wave velocity in plasma is determined from the time history of the laser deflection signals. Shock wave speeds in plasma are found to be dependent on the plasma discharge current. Shock wave speeds increase by 18{\%} for the plasma at 3.6 Torr over a range of plasma discharge current I = 0--150 mA and by 46{\%} for the plasma at 15 Torr over I = 7--150 mA. In addition, shock wave amplitudes are attenuated in plasma and show linear dependence on the shock wave speed or Mach number.   

Numerical Simulations of Gas Breakdown in Two-Dimensions

Time domain simulations of gas breakdown are performed in two- dimensions with unusual cathode and anode shapes and with various external circuit loads. The simulations determine the voltages for breakdown and model current oscillations known to be associated with the transition between normal and Townsend discharge regimes[1]. The negative differential resistance magnitude in the transition region between the discharge regimes is found. Relationship between the magnitude of radial diffusion, breakdown voltage, and oscillation waveform for various cases is explored.\\ $[1]$ Kolobov, V.I., and A. Fiala, Phys. Rev. E, V. 50, No. 4, (1994), p. 3018.   

Kinetic Theory of Collisional Sheath

Electron dynamics in the collisional sheath of a weakly ionized gas are investigated in the regime in which the electron mean free path is very small compared with the Debye length. The electron distribution function and the Boltzmann collision operator for electron neutral collisions are expanded in spherical harmonics to produce a modified relaxation time approximation. We believe this approximation to be valid even when the distribution function is very far from Maxwellian, as is the case near a particle absorbing boundary. The Boltzmann equation is then formulated for monoenergetic groups of particles and integrated along the characteristics of the collisionless equation to reduce the problem to a solving a single Fredholm integral equation. Results illustrate and quantify the remarkable role played by electron inertia at distances on the order of one mean free path from the boundary in the otherwise collision dominated system. Comparisons with approximate hydrodynamic boundary conditions made in recent literature are given and implications for dust grain charging discussed.   

Simulation study on non-collisional thermalization of fast ions in magnetized plasma

The first non-collisional process causing the perpendicular energy of fast ions to be thermalized toward Maxwellian [K. R. Chen, Phys. Rev. Lett. 72, 3534 (1994)] was done by Particle-In-Cell (PIC) simulation with limited particle numbers. Due to a two-gyro-stream relativistic cyclotron instability driven by fast ions, harmonic ion cyclotron waves are excited and the wave-ion interaction induces this non-collisional thermalization process. This is investigated in detail and further with much more particles to reduce system noise and to achieve high precision in statistics. Thus, the fluctuation at low energy portion and the limitation on the high energy tail can be improved. With more particle numbers, the fluctuation is greatly reduced and the fast ion perpendicular energy distribution becomes a smooth curve and is closer toward Maxwellian.   

Onsager Regression applied to fluctuations of the ion distribution function.

We compare the two-point correlation function of the ion distribution function, resolved in space and in particle velocity along a fixed magnetic field to the linear response function of a cylindrical plasma column at low frequency. Even though the plasma is has unstable fluctuations (convectively limited) the fluctuations nevertheless carry information on the linear response. The experimentally observed particle-velocity averaged fluctuation is symmetric in delay time even for spatially separated points and agrees with dissipative driftwave theory. There is a kinetic component, with a very low degree of symmetry, which appears to be driven by nonlinear interaction of the driftwaves even at low amplitude. We explore what can be learned from these time correlation functions about the kinetic component. The experiments are performed on a CW, singly-ionized argon plasma column immersed in a 0.1T magnetic field. The fluctuation data are collected using laser induced fluorescence and movable light collection periscopes. Progress on a second laser system to extend the measurements to two different ion velocities as well as separate positions and times will be presented.   

High time resolution LIF in pulsed argon plasma

A high time resolution Laser Induced Fluorescence (LIF) method for obtaining the temporal evolution of the ion velocity distribution function in pulsed argon plasma is presented. A single mode tunable ring dye laser pumped by a 6 W argon-ion laser is used to excite a classic three-level Ar II LIF scheme $3d^{'2}G_{9/2} \to 4p^{'2}F_{7/2} \to 4s^{'2}D_{5/2} $. An LIF system used for steady-state plasma is slightly modified by addition of a digital oscilloscope and by replacing the mechanical chopper with a high frequency acousto-optic modulator. Ion velocity distribution measurements as a function of time during the rf pulse are obtained by taking temporal slices of the LIF intensity -- wavelength plane. The parallel ivdf measurements indicate that in the region of high magnetic field gradient two ion groups coexist: a fast moving $^{\mbox{`}}$beam$^{\mbox{'}}$ and a slower moving background population. The formation of a weak double layer below a threshold pressure due to ion acceleration in the expansion region of plasma appears to be a characteristic of such low pressure helicon plasmas. We will present sub-ms time resolution measurements of the temporal evolution of the parallel and perpendicular ion velocity distribution functions (ivdf) in a pulsed, helicon-generated, expanding, argon plasma.   

Physics and Design of the Driven Relaxation Experiment (DRX) at Los Alamos

The Driven Relaxation Experiment (DRX) is under construction at LANL with first plasma scheduled for early 2007. Recent theoretical work by Tang \& Boozer\footnote{X. Z. Tang and A. H. Boozer, PRL~{\bf 94}, 225004 (2005).} has provided new insights into partially relaxed force-free plasmas, namely that nonlinearity removes the resonances of the linear force-free equation, and that new branches of relaxed states, such as the flipped spheromak and other high $\lambda$ states become energetically accessible. The primary research objective of DRX is to create and sustain a partially relaxed driven steady-state above the first linear resonance by ``over-driving'' a coaxial plasma gun with $\lambda_{\rm gun} \approx (2.5$--3)$\lambda_1$. Two other important objectives are to measure the spatial wave-number spectrum of magnetic energy for the relaxed states and to explore the dynamics of the relaxation. This poster will review the project theoretical motivations, experimental design, and construction status.   

Excitation and Detection of Drift Waves in TORPEX toroidal plasma

Low frequency $\omega \ll \omega_{ci}$ electrostatic fluctuations are ubiquitous in laboratory plasmas and play an important role in anomalous cross-field transport. In TORPEX plasmas, Langmuir probes covering the whole plasma cross-section are used to determine the local dispersion relation. The low frequency fluctuations are identified as drift interchange-modes. Coupling to the stable modes using an antenna is a first step in understanding the underlying excitation mechanism. Moreover, this may lead to methods that alter the instability spectrum. The antenna consists of a movable poloidal arrangement of four electrodes. Each electrode is phased independently and is driven with a sinusoidal potential. The wave field is obtained using a synchronous detection technique applied to the probes in the ion saturation regime. Results of the wave field dependency on the launched frequency, on the relative phase shift between electrodes, and on the driving amplitude are presented. On the passive side (with no excitation), an investigation of the energy transfer between distinct frequency bands on the unstable modes is conducted. The status of the implementation of non-pertubative optical diagnostics on TORPEX will also be discussed.   

Experimental studies of particle transport in the TORPEX toroidal plasma.

TORPEX is a toroidal device where plasmas produced by microwaves are embedded in a helical magnetic field. It is mainly dedicated to basic plasma physics studies on instabilities and transport. In this type of devices, the plasma is primarily lost along the open field lines. Nevertheless, particle fluxes across the magnetic field are clearly measurable using different, complementary techniques. The local particle flux can be estimated over most of the cross-section from the plasma response to a modulation of the injected microwave power, and quantified on the basis of a diffusive-convective model. The fraction of the total particle flux caused by plasma instabilities, identified as drift-interchange modes, is also measured and related to the observed spectral features. The results are compared with the transport associated with macroscopic fluctuation structures, reconstructed from a 2D imaging of the spatio-temporal behavior of the density fluctuations.   

Early-out of Equilibrium Beam-Plasma Evolution

We solve analytically the out-of equilibrium initial stage that follows the injection of a radially finite electron beam into a plasma at rest and test it against particle-in-cell simulations. For initial large beam edge gradients and moderate beam radius w.r.t skin depth the electron beam is seen to evolve into a ring structure. For low enough transverse temperatures, filamentation instability proceeds and saturates when transverse isotropy is reached. This analysis agrees with very recent experimental beam transverse observations.   

Overview of the Madison Dynamo Experiment

A spherical dynamo experiment has been constructed at the University of Wisconsin-Madison's liquid-sodium facility. The experiment is designed to self-generate magnetic fields from flows of conducting metal. The apparatus consists of a 1 m diameter, spherical stainless steel vessel filled with liquid sodium. Two 100 Hp motors drive impellers which generate the flow. The motors have been operated up to 1300 RPM (70\% of design specification), achieving a magnetic Reynolds number of 130, based on impeller tip speed. Various polarizations of external magnetic fields have been applied to the sodium, and the induced magnetic field has been measured by both internal and external Hall probe arrays. The voltage induced across the sphere by the turbulent flow has been measured. Techniques for using ultrasound Doppler velocimetry have been explored in the water model of the experiment, including the use of high-pressure bubbles as seed particles.   

Intermittent magnetic field excitations in the Madison Dynamo Experiment

Determining the onset conditions for magnetic field growth in magnetohydrodynamics is fundamental to understanding how astrophysical dynamos such as the Earth, the Sun, and the galaxy self-generate magnetic fields. The role of turbulence in modifying these onset conditions is studied in the Madison Dynamo Experiment. A turbulent flow of liquid sodium, composed primarily of two counter-rotating helical vortices, is generated by impellers. Laser Doppler velocimetry measurements of the flow in an identical-scale water experiment demonstrate that the turbulence is isotropic, though not homogeneous, with particularly long-lived eddies in the shear layer between the two flow cells. The magnetic field induced when an axial field is applied shows intermittent periods of growth and has a spatial structure consistent with the fastest growing magnetic eigenmode predicted by a laminar kinematic dynamo model of the mean flow.\footnote{Nornberg {\em et al.}, Phys.\ Rev.\ Lett., in press (2006), physics/0606239.} Turbulent fluctuations of the velocity field change the flow geometry such that the eigenmode growth rate is temporarily positive, thus generating the magnetic bursts. It is found from ensemble averaging that the bursts gain strength and frequency with increased impeller rotation rate, though they become shorter so that each burst remains a rare, random event.   

Fluctuation-induced Magnetic Fields in the Madison Dynamo Experiment

The Madison Dynamo Experiment is designed to function as a simply-connected, homogeneous dynamo. A turbulent flow of liquid sodium is driven by two counter-rotating impellers in a one-meter-diameter sphere. A model of the mean velocity field is constructed from measurements made in an identical-scale water experiment. This model is used to predict the mean induced magnetic fields when exposed to an external magnetic field. The magnetic field is measured by arrays of both internal and external hall probes. The predicted and measured magnetic fields are then compared to determine the effects of turbulence on the mean field. An external dipole moment is measured which cannot be generated by the mean axisymmetric velocity field. The measured toroidal and poloidal induced magnetic fields within the sphere are significantly weaker than predicted. These effects are attributed to a turbulent electromotive force.   

Ultrasonic Doppler Velocimetry Measurements on the Madison Dynamo Experiment

The Madison Dynamo Experiment is used to study the generation of magnetic fields in a homogeneous fluid. Flows of liquid sodium in the one-meter-diameter spherical vessel are generated by two counter-rotating impellers. Both the shape and magnitude of the velocity field must be well-understood to predict whether the magnetic field will grow or decay. Ultrasonic Doppler Velocimetry (UDV) is used to measure components of the velocity field in a dimensionally-identical water version of the experiment. Several ultrasonic transducers are used to simultaneously measure the flow along several chords of the vessel. Both neutral density polystyrene beads and air bubbles are used to reflect the ultrasonic pulses. UDV measurements, supplemented by two-component Laser Doppler Velocimetry measurements, are used in a fitting routine to model the flow in terms of spherical harmonics.   

Turbulent Dynamos and a non Turblent Dynamo Experiment, NMTech and LANL

The liquid sodium $\alpha \omega$ dynamo experiment is designed to demonstrate how magnetic fields are generated in AGN and stars. Turbulence is strongly constrained by the stability of Couette flow and by the short transient time of driven plumes. Similarly we expect low turbulent large scale $\alpha \omega$ dynamos in MBH accretion disks and in stars where diffusive transport of magnetic flux by turbulence is much less than the advected magnetic flux, thus avoiding a common problem with unconstrained shear flows. Kinematic exponential growth of magnetic field has been predicted to occur in the presence of random or chaotic three dimensional or turbulent motions of a highly conducting fluid. We predict that no dynamo growth can occur in unconstrained shear flows because of the irreversible nature of turbulence and hence the enhanced dissipaton of magnetic flux or reduced magnetic Reynolds number. Analytic formulations suffer from the time reversible invariance of the equations and hence no change in enropy.   

First Experimental Results From the Princeton MagnetoRotational Instability (MRI) Experiment

The inferred rate of angular momentum transport in accretion disks is too large to be explained by a molecular viscosity. Two sources of instability to drive turbulence have been proposed: the MRI and subcritical hydrodynamic instability. In the MRI, a weak magnetic field can use the angular velocity gradient as a source of free energy. In the subcritical case, finite amplitude disturbances are expected to allow access to non-linear instabilities. Recent experimental investigations have claimed to observe both mechanisms in the laboratory, but neither has been conclusively demonstrated. During the first year of operation, the Princeton MRI Experiment has been searching for conclusive evidence of these instabilities. The experiment is a Couette-Taylor apparatus which uses water or liquid Gallium alloy to generate rotating shear flows with linear stability properties analagous to astrophysical disks. In the purely hydrodynamic case we do not find evidence of angular momentum transport great enough to be astrophysically important. We will also present initial results of our search for the MRI using liquid Gallium as our working fluid.   

Magnetorotational instability in magnetized Taylor-Couette flows

We present non-ideal magnetohydrodynamic simulations of MRI in the geometry of the Princeton MRI experiment. MRI saturates in a resistive current-sheet with significant reduction of the mean shear, and with poloidal circulation scaling as the square root of resistivity. Angular momentum transport scales as the reciprocal square root of viscosity but hardly depends on resistivity. Separately, we have studied MRI in the presence of a current-free combination of toroidal and axial magnetic field. The new mode (HMRI) persists to smaller magnetic Reynolds number and Lundquist number than standard MRI, which relies on axial field alone. In vertically infinite or periodic cylinders, resistive HMRI is a weakly destabilized hydrodynamic inertial oscillation propagating axially along the background Poynting flux. Growth rates are small, however, and require large axial currents. Furthermore, highly resistive HMRI is stabilized in finite cylinders with insulating endcaps, and also in keplerian flow profiles regardless of end conditions. Comparison of models and measurements is used to validate our theoretical tools, which we will apply to nonlinear saturation of resistive MRI in astrophysical systems. Theoretical modeling has already played a major role in the design of the MRI experiment, and the physics of these modes may be of interest for fluid dynamics and geophysics as well as astrophysics   

Magneto-rotational instability and turbulent angular momentum transport.

We present numerical simulations of magnetized-Couette flow between concentric rotating cylinders in axisymmetric and fully three-dimensional geometry. This work complements the Princeton liquid gallium experiment by Goodman and Ji to study the Magneto-Rotational Instability (MRI). The simulations are carried out with a spectral element code incorporating realistic hydro boundary conditions at the upper and lower boundaries and consisting of differentially rotating rings aimed at minimization of the effects of Ekman circulation. We have studied changes in the flow structure and in the mechanism for angular momentum transport in the magnetic and non-magnetic cases as well as the impact of the boundary conditions (periodic vs. finite container). The angular momentum transport by Reynolds stresses and comparable viscous and ohmic dissipation were observed in the inner region of the annulus while the flow in the outer region was dominated by Maxwell stresses and exhibit a tendency toward constant AZIMUTHAL velocity with the increase of the external axial magnetic field.   

Complex Behavior of a Fluid Jet within Rotating Annular Flow

2-D simulations of fluid flow between co-rotating cylinders have been performed where the velocities of the inner and outer cylinders are set so that the flow is stable to centrifugal instabilities. For small aspect ratios, the flow is significantly altered by viscous boundary effects that cause a large imbalance between the centrifugal force and pressure gradient near the vertical boundaries. As a result, a secondary circulation appears. This circulation produces a jet flowing radially outward at the mid-plane of the system. Above a critical Reynolds number, this jet is observed to exhibit a regular flapping motion. As the Reynolds number is further increased, the flapping motion becomes more complex. This oscillation is globally reflected in the torque required to rotate the cylinders. We have analyzed time-series of the torque and stream function of the fluid at various Reynolds numbers to look for signatures of a transition to chaos.   

Laboratory Study of MHD Effects on Stability of Free-surface Liquid Metal Flow

The dynamics of free-surface MHD shear flows is potentially important to both astrophysics (e.g. in the mixing of dense plasma accreted upon neutron star surfaces) and fusion reactors (e.g. in liquid metal ‘first walls’). To date however few relevant experiments exist. In order to study the fundamental physics of such flows, a small-scale laboratory experiment is being built using a liquid gallium alloy flowing in an open- channel geometry. The flow dimensions are nominally 10cm wide, 1cm deep, and 70cm long under an imposed magnetic field up to 7kG, leading to maximum Hartman number of 2000 and maximum Reynolds number of $4\times 10^5$. Two basic physics issues will ultimately be addressed: (1) How do MHD effects modify the stability of the free surface? For example, is the flow more stable (through the suppression of cross-field motions), or less stable (through the introduction of new boundary layers)? We also investigate whether internal shear layers and imposed electric currents can control the surface stability. (2) How do MHD effects modify free-surface convection driven by a vertical and/or horizontal temperature gradient? We discuss aspects of both of these issues, along with detailed descriptions of the experimental device. Pertinent theoretical stability analyses and initial hydrodynamic results are presented in companion posters. This work is supported by DoE under contract \#DE-AC02-76-CH03073.   

Linear Stability Analysis of Free Surface Liquid Metal Flow

We study the linear stability of the flow of a liquid metal on a planar surface in the presence of an external magnetic field. The objective is to account for the behavior encountered in a free surface MHD experiment at Princeton, but the model has a range of astrophysical and industrial applications (see companion poster). This class of free surface flow exhibits two mechanisms of linear instability. In the so-called `soft' instability, a downstream propagating surface wave of large wavelength becomes mildly unstable. The second, `hard', instability is of the critical layer type and takes place at shorter wavelengths. Solving the eigenvalue problem posed by the coupled Orr-Sommerfeld and induction equations via a spectral method, we find that in the regime of relevance to the Princeton experiment (Reynolds number, magnetic Reynolds number and Hartmann number up to $ 10 ^ { 5 } $, $ 10 ^ { - 1 } $, and $ 10 ^ 3 $, respectively) MHD effects suppress both types of instability. The soft instability is efficiently suppressed via resistive dissipation if the background magnetic field is normal to the basic flow. In contrast, the hard instability is strongly suppressed irrespective of the details of the background magnetic field configuration, even at moderate Hartmann numbers.   

Irregular singularity of the magneto-rotational instability in a Keplerian disk.

Magneto-rotational instabilities in a Keplerian disk are studied via both the eigenvalue and initial-value approaches on the basis of the incompressible magnetohydrodynamics model. The center of the Keplerian disk is an irregular singularity for the eigenfunctions of the magneto-rotational instabilities. This singularity yields continuous eigenvalues (growth rates in the unstable regime and frequencies in the stable regime). The eigenfunctions belonging to the continuum are square-integrable and are not orthogonal with each other. Physical implications of such a continuum and the eigenfunctions are rather complex, because of the ``non- Hermitian'' nature of the rotating plasma system. Invoking the Laplace transform, as well as numerical simulations, interesting long-term behavior of the instability, or the slow change of the mode structure in the exponentially growing phase, has been found.   

Do Accretion Disks Exist in High Energy Astrophysics?

The familiar concept of an accretion disk is based on its gas dynamic description where, in particular, the vertical equilibrium is maintained by the (weak) vertical component of the gravitational force due to the central object. When a plasma structure differentially rotating around the same kind of object is considered in which the magnetic field diffusion due to finite resistivity is realistically weak, a radially periodic sequence of pairs of opposite current channels is found.\footnote{B. Coppi, \textit {Phys. Plasmas} \textbf{12}, 057301, (2005)}$^,$\footnote{B. Coppi and F. Rousseau, \textit{ Ap. J. }\textbf{641 (1)}, 458 (2006)} Moreover, the vertical confinement of the structure is maintained by the resulting Lorentz force rather than by gravity. Thus, a ``Lorentz compression'' occurs. In addition, sequences of plasma rings$^2 $ rather than disks emerge. (Note that H. Alfv\'en had proposed that planetary rings may be ``fossils'' of pre- existing envisioned plasma rings. Moreover, a large ring is the most prominent feature emerging from the high resolution X- ray image of the Crab). The ``seed'' magnetic field in which the structure is immersed is considerably smaller than that produced by the internal toroidal currents. The magnetic pressure is of the order of the plasma pressure. Thus, ring sequence configurations can be suitable for the emergence of a jet from their center. Two coupled non-linear equations have been solved, representing the vertical and the horizontal equilibrium conditions for the structure.*Sponsored in part by the U.S. D.O.E.   

Self-gravitating Disks and Plasma Structures Immersed in Them*

When considering axisymmetric differentially rotating plasma structures in the prevalent gravity of a central object these are found to be characterized by a sequence of current filaments and to develop a corresponding ring sequence\footnote{B. Coppi and F. Rousseau, \textit{Ap. J.} \textbf{641} (1), 458 (2006).} configuration for the plasma. The same type of structure can be found when the self-gravity of a differentially rotating plasma component is no less important than the gravity of the central object. Then in addition to the vertical and horizontal equilibrium equations to be solved, Poisson's equation for the gravitational potential has to be dealt with. The fact that the vertical equilibrium is ensured by the vertical component of the Lorentz force due to the internal plasma currents (``Lorentz compression'') simplifies the problem considerably. \newline*Sponsored in part by the Universita$'$ di Milano and by the U.S. DOE   

Differentially Rotating Structures and Angular Momentum Transport in the Prevalent Gravity of a Central Object*

The presence of angular momentum transport associated with an accretion process in an axisymmetric differentially rotating structure affects the equilibrium configuration that this can take and can introduce a toroidal Lorentz force with the associated poloidal current densities. All three components (vertical, radial and toroidal) of the total momentum conservation equation are considered. A sequence of ring solutions\footnote{B. Coppi and F. Rousseau\textit{ Ap. J. } \textbf{641} {(1)}, 458 (2006)} can be found by making use of the inequalities $v_{NJ} [Preview Abstract]

MHD collimation and kink instability driven by differential rotation

Recent observations of astrophysical outflows from active galactic nuclei suggest that these jets are permeated with helical magnetic fields. We are conducting simulations that are relevant to these jet systems using the NIMROD MHD code. In these simulations an initial seed field is twisted by a differentially rotating flow boundary condition. The zero beta axisymmetric nonlinear system is considered. The degree of collimation in this system is shown to depend on the ratio of the Alfven velocity in the domain to the driving velocity on the boundary. Non-axisymmetric linear eigenmodes of the collimated axisymmetric results are calculated. It is shown that an external force, provided by an outer boundary in the simulations discussed here, is necessary for growing Eigenmodes. By comparing to simulations of kink modes in a paramagnetic pinch, it is shown that these growing Eigenmode solutions are indicative of the kink mode.   

Ion-Neutral Collisions and the Propagation Distance of Interstellar Turbulence

Plasma turbulence exists throughout the pervasive ``Diffuse Ionized Gas'' (DIG) phase of the interstellar medium. The generators of this turbulence are unknown, but are widely believed to be supernova remnants. The DIG is a partially ionized plasma, since at least half of the helium is neutral, and perhaps a small fraction of the hydrogen. The damping rate of MHD waves on neutral helium in the DIG is of order $(3-6) \times 10^{-12}$ Hz. With a corresponding Alfven speed of 23 kilometers per sec, interstellar turbulence should be strongly damped within a parsec of its sources. This result is strongly discordant with the observation that turbulence in the DIG seems relatively uniformly distributed and is found far from obvious turbulence ``generators'' such as supernova remnants and star formation regions. I discuss possible ways in which turbulence could be generated or propagate through the highly lossy interstellar medium.   

Simulation of the Magnetothermal Instability

In many magnetized, dilute astrophysical plasmas, thermal conduction occurs almost exclusively parallel to magnetic field lines. In this case, the usual stability criterion for convective stability, the Schwarzschild criterion, which depends on entropy gradients, is modified. In the magnetized long mean free path regime, instability occurs for small wavenumbers when $(\partial P/\partial z)(\partial \ln T / \partial z) < 0$, which we refer to as the Balbus criterion. We refer to the convective-type instability that results as the magnetothermal instability (MTI). We use the equations of MHD with anisotropic electron heat conduction to numerically simulate the linear growth and nonlinear saturation of the MTI in plane-parallel atmospheres that are unstable according to the Balbus criterion. The linear growth rates measured from the simulations are in excellent agreement with the weak field dispersion relation. The instability saturates when the atmosphere becomes isothermal as the source of free energy is exhausted. By maintaining a fixed temperature difference between the top and bottom boundaries of the simulation domain, sustained convective turbulence can be driven. This paper presents 2D and 3D studies of the MTI in a variety of geometry and field configurations. In addition, we explore the implications of this instability for a variety of astrophysical systems. The most important application is to the dilute, magnetized plasma that makes up the hot intracluster medium of galaxy clusters.   

Transport of Poloidal Magnetic Flux Along a Flux Rope in Stratified Solar Subphotospheric Regions

A three-dimensional (3D) MHD simulation model of vertical, cylindrical magnetic flux rope embedded in a stratified background plasma has been constructed. The initial flux rope consists of a current channel of finite radius and is in non-force-free equilibrium with pressure and constant downward gravity. The poloidal magnetic field is increased in time at the base of the simulation region. The focus of the study is to understand the resulting plasma dynamics inside and outside the current channel, corresponding to relatively low-$\beta$ and high-$\beta$ regions, respectively. The role of magnetic reconnection in the evolution of the system is discussed. Fluctuations in density are imposed, and a horizontal flow from the side boundary is allowed. The poloidal flux injected from the base propagates outward and upward, developing highly incoherent structures due to both gravitational effects and the imposed noise. For comparison, similar simulations are carried out for the Gold-Hoyle flux rope. Possible observable manifestations of the transport of poloidal flux through the photosphere are discussed.   

Stability and Acceleration of Solar Flux Ropes: Application to Coronal Mass Ejections

The dynamics of solar flux ropes have received much attention in connection with coronal mass ejections (CMEs). A major unanswered question is how initial quasi-equilibrium flux ropes are driven. The Lorentz hoop force, originally derived for toroidal tokamak equilibrium, has been extented to expanding solar flux ropes with stationary footpoints [1]. We discuss the results of extensive comparisons between calculated flux-rope dynamics and recently observed CME dynamics (17 events). The agreement is found to be very good. In particular, the intrinsic spatial and temporal scales produced by the model equations are manifested in observed CME acceleration profiles [2]. More recently, a simplified equation based on the same concept has been proposed to describe CME dynamics [3]. This equation describes a system with no fixed footpoints and yields fundamentally different scales. We discuss how the differences are manifested in observed acceleration and how they can be used as observational discriminators. \newline [1] Chen, J., Astrophy. J., 338, 453, 1989. Garren, D. and Chen, J., Phys. Plasmas, 1, 3425, 1994. Chen, J., J. Geophys. Res., 101, 27499, 1996. \newline [2] Chen, J. and Krall, J., in press, Astrophys. J., 2006. \newline [3] Kliem, B. and Torok, T., Phys. Rev. Lett., 96, 255002, 2006.   

Electrostatic potential drop with no electric field

It is commonly believed that because plasma is a near perfect electrical conductor, one can assume that electron motion will short out any spatial variation in the electrostatic potential $V$ and so the electrostatic electric field $-\nabla V$ can be assumed to be zero. As an example of this sort of argument, see p. 23 of Parker's book, Cosmical Magnetic Fields (Oxford, 1979). However, assuming that a perfectly conducting plasma cannot support an electrostatic potential drop is a serious error, because the property of a perfect conductor is that the net electric field must be zero, not that just the electrostatic component is zero. Since $\mathbf{E=}% -\nabla V-\partial \mathbf{A/}\partial t$ it is quite possible to have $% \mathbf{E}$ vanish while $\nabla V$ is finite, provided $\partial \mathbf{% A/}\partial t$ is such as to cancel $\nabla V.$ This is of special interest in the classic MHD situation of a straight cylindrical flux tube being twisted up. The axial component of the ideal Ohm's law is $% E_{z}+U_{r}B_{\phi }-U_{\phi }B_{r}=0$ and since $U_{r}$ and $B_ {r}$ are both zero for the twisting of a straight flux tube, the axial component of the ideal MHD\ Ohm's law reduces to $E_{z}=-\partial V/\partial z-\partial A_{z}/\partial t=0.$ Twisting up a flux tube, i.e., increasing $B_{\phi },$ corresponds to increasing $A_{z}$ since $B_{\phi }=-\partial A_ {z}/\partial r.$ \textit{Thus, twisting up a flux tube requires that a nontrivial electrostatic potential drop }$\partial V/\partial z=-\partial A_{z}/\partial t$ \textit{develop along the length of the flux tube during the twisting up process.} This electrostatic potential drop is typically neglected in studies of the twisting of solar corona flux tubes, but is known to be essential in laboratory experiments involving the twisting flux tubes. *Supported by DoE.   

Observation of Strong Internal Magnetic Field in Laboratory Simulations of Coronal Loops

Solar coronal loops have been simulated in a Caltech laboratory experiment using a magnetized plasma source. The laboratory plasma loops are formed in three steps; namely, application of an arched vacuum magnetic field spanning both electrodes, injection of neutral gas through orifices on each electrode, and application of high voltage between the electrodes. The foot-points of the loops on the orifices are separated by 8 cm. Visual images of the dynamical evolution of the laboratory loops are recorded using a digital framing camera. A loop probe array measures all three components of the magnetic field at four distinct locations. Our new result is the observation of a strong magnetic field carried by the plasma loop as it propagates away from the electrodes. Within 5 $\mu $s the loop expands $\sim $ 16 cm away from the electrode. Thereafter, the loop magnetic field ($\sim $ 500 G) is measured to be more than two orders of magnitude larger than the pre-existing vacuum magnetic field at the apex of the loop. We also plan to present results showing the deviation of the measured magnetic field from the calculated force-free magnetic field.   

Dual gas laboratory coronal loops

Solar coronal loops are arched plasma structures that protrude from the Sun's photosphere into its corona, some extending hundreds of thousands of kilometers. These structures exhibit a ubiquitous axial uniformity; it has been proposed that this is due to convection of toroidal flux. Their behaviors include filamentation of the current channel, expansion of the arch, formation of helical twists, and eruption. These features have all been observed in laboratory simulations of coronal loops, created with a magnetized plasma gun. Both single loops and pairs of loops have already been studied in detail. Until now, however, each experiment used only one gas. This can make it difficult to resolve the details of processes whereby different portions of the plasma merge. By incorporating a second gas into the laboratory coronal loops, this investigation seeks a clearer understanding of how these plasmas evolve and interact. For a single loop experiment, a different gas can be used at each of the two footpoints, highlighting convection processes and interactions at the apex. Additionally, two adjacent loops can be formed using two distinct gases, elucidating the manner in which they merge.   

Images of 50 km/s laboratory plasma jet colliding with neutral gas cloud

Jets of magnetized plasma are common in astrophysical settings; their interaction with the surrounding medium is of particular interest. We have developed an experimental facility to better understand this interaction. The experiment accelerates a magnetized plasma jet$ ^1$ to velocities of $\sim$10-50 km/s into a target cloud of neutral gas with densities in the range of 10$ ^{16}$ to 10$ ^{20}$ m$ ^{-3}$. We view the interaction using a high speed camera that produces 16 images in 5 $\mu$s with an exposure time of 10 ns. Preliminary experiments show different qualitative behavior in each of three regimes defined by the mass ratio of the target gas to the plasma ions. A plasma jet whose mass is less than the mass of the target gas is observed to “pile up” at the boundary of the target gas cloud, whereas a jet of mass equal to that of the target slows but does not have significant thickening at its front. When a heavy plasma jet hits a light target gas cloud, a front that we believe may be a shock wave is pushed away from the interaction boundary between the target and the jet at a velocity greater than that of the jet. \newline \newline $ ^1$ S. C. Hsu and P. M. Bellan, Mon. Not. R. Astron. Soc. {\bf 334}, 257 (2002).   

Visualization of the collisional process and kinetic energy dissipation of two colliding suprathermal plasma jets

We present an experimental study of an arch-shaped laboratory plasma structure, similar to solar coronal loops, having two counterstreaming internal flows using a high-speed camera and a high-resolution spectroscopic system. The spectroscopic system measures emission spectra from multiple (up to 12) locations of the plasma simultaneously with variable time and space resolution, permitting observation of plasma parameters such as velocity, density and temperature along the jet axis. The camera and spectroscopic diagnostics show that an arched flux tube flared in the middle is initially formed with both of its footpoints being attached to gas feeds and subsequently become collimated by an MHD pumping process$^{\dagger}$, creating a high velocity ($\sim20~km/s$) jet outflowing from each footpoint toward the middle. The two counterstreaming jets collide in the middle, which creates a bright region at the apex. However, the excitation temperature estimated from line intensity ratios shows negligible temperature variation along the jet axis including the bright region. We plan to visualize in detail the bright region, and plan to measure the ion temperature and density gradients along the jet axis to study the dissipation of the kinetic energy of the colliding jets. $\dagger$ P. M. Bellan, \textit{Why current-carrying magnetic flux tubes gobble up plasma and become thin as a result}, Phys. Plasmas 10 Pt 2, 1999 (2003).   

A Low Cost Photo-Electric Detector for a Solar Coronal Loop Experiment

A low cost photo-electric detector was tested for use in a laboratory experiment simulating solar coronal loops. An array of such detectors is desired to image the ultraviolet radiation previously observed in this experiment. The detector had a coaxial design with a magnesium inner disk (chosen for its low work function) and a steel outer cylinder. The detector was placed in a vacuum chamber along with a pulsed mercury flash lamp separated by 50 centimeters. A current-to-voltage amplifier was used to read the detector's output. Numerous tests were conducted, including varying bias voltage, introducing magnetic fields, and occulting and collimating the incoming radiation. Two types of signals, distinguished by the bias voltage polarity, were observed. When the inner disk was negatively biased, the signal appears consistent with a photo-electric signal. With a positively biased inner disk, the signal seemed to be produced by secondary sources: electrons photo-emitted at the chamber walls or released from ionized gas.   

Experimental and Computational Investigations of a High-Power, Long-Pulse Relativistic Klystron Oscillator

A high-power, long-pulse source of high-power microwaves has been investigated experimentally and through a variety of modeling and simulation efforts at the Air Force Research Laboratory. The relativistic klystron oscillator (RKO) is an injection-locked oscillator capable of producing 200ns duration pulses exceeding 1 GW output power at 1270-1275 MHz. Extensive experiments have been closely coupled with computational modeling and simulation to explore a wide range of issues encountered in the operation and diagnostics of the device. The experiment uses coupled three-quarter lambda cavities to modulate the electron beam. Calculations using both HFSS and ICEPIC have been used to reproduce cold test frequency characteristics of the isolated and of the coupled cavities, including the finite conductivity of the RKO walls. Calculations using reduced physics models and ICEPIC have been used to explore the coupling between the beam and the cavities. A highlight of the modeling efforts is a series of calculations, which for the first time predict cavity saturation voltages at sub-virtual-cathode levels. Previous calculations were either restricted to quarter lambda cavities, or showed saturation voltages at the virtual cathode levels. Comparisons between experiment and computation will be presented.   

Experimental study of a high efficiency 1.5 MW, 110 GHz gyrotron with a depressed collector

A 1.5 MW, 110 GHz gyrotron is being developed mainly for plasma heating in the DIII-D tokamak. Research at MIT with a short pulse prototype gyrotron is dedicated to understand the physics and engineering challenges facing megawatt class gyrotrons with high efficiency ($>$50 {\%}). We have successfully tested a low ohmic loss cavity (named V-2005) providing 1.67 MW of output power with a corresponding efficiency of 42 {\%} in an axial configuration (without an internal mode converter and without a depressed collector). This paper will report the most recent experimental study with an internal mode converter and a single-stage depressed collector. The operating mode was the TE22,6 mode and an existing MIG gun was used at 96 kV of beam voltage and 40 A of beam current with 3 microsecond pulses. We have successfully achieved 50 {\%} of gyrotron overall efficiency at 1.5 MW of output power with 25 kV of beam depression using the new cavity, V-2005. Analysis of the experimental results will be discussed.   

Status of Megawatt Gyrotrons at CPI for ECRH Applications

Long-pulse and CW gyrotrons that generate output power levels of around 1 MW have been developed at frequencies of 110 GHz and 140 GHz for use in electron cyclotron heating experiments. A total of six, 110 GHz, 1 MW, 10-s-pulse gyrotrons have been designed and fabricated for General Atomics. Four of the tubes have been delivered to General Atomics and the final two tubes are in the final stages of test. Operating experience with the 110 GHz gyrotrons will be reviewed. A 110 GHz, 1.3 MW gyrotron that employs a depressed collector has been designed and fabricated and will soon undergo final testing at General Atomics. In initial tests at CPI the tube was tested to a power level of 1.25 MW for pulse durations of a few ms and an output power of 500 kW was achieved for 10-s pulses. A 140 GHz gyrotron that also employs a depressed collector was tested to an output power of 900 kW for pulse durations of 30 minutes during final tests at the Max Planck Institute for Plasma Physics in Greifswald, Germany. Design features and updated test results on the two gyrotrons with depressed collectors will be presented.   

Experimental measurements of electron emission uniformity from cold cathodes

We report measurements of electron emission, including spatial uniformity, from cold field emission cathodes. The measurements are taken on a system designed to examine the nanophysics of field emission from advanced Al, W and CsI cathodes that operate in the 1-1000 A/cm$^{2}$ regime. Operation is for UHV vacuum (10$^{-10}$ Torr) with bake-out up to temperatures of 450 C to eliminate residual water vapor or other contaminants. The current-voltage characteristics, field emission, work function, space charge effects and the Fowler-Nordheim coefficients are examined. Negative pulses of 0-20 kV and 1-5000 $\mu $s duration are applied between the cathode and anode to obtain the current-voltage characteristics. A linear translation stage is used to adjust the cathode-anode gap from 0-1.25 cm with a resolution of 0.025 mm. A small $\sim $4 mm$^{2}$ second ``local anode'' maps the spatial uniformity of the emission current density across the cathode surface. Measurements have been carried out on ALF (Ablation Line Focus) and knife edge Si-Ni cathodes. Additional tests will be accomplished on both single and multi-tip sharp tip/knife-edge cathodes to determine the interaction between local neighboring tips during electron emission.   

Analysis of a Simple Gyrotron Model

A simple standard gyrotron model is studied. The model employs the paraxial approximation and involves only the axial coordinate as a space coordinate. The model contains a particle distribution function for the velocity in the perpendicular direction and is coupled to equations for the electromagnetic field. It is shown that there can be no blow-up of, or infinite gradients in, the fields or the distribution function. One can analyze the initial and final states under the assumption the tube has constant radius in those regions. When the final state is a pure plane wave one can characterize all final states dynamically accessible from the initial state. In many cases more than one final state is possible, although their number is not infinite. The description of the final state depends on a Hamiltonian formulation of the problem for the distribution function.   

Modeling of Magnetron Injection Locking Characteristics

A magnetron-specific, phase-locking model has been developed [1] to qualitatively explain the various regimes observed in magnetron injection-locking experiments [2]. The experiments utilize two continuous wave oven magnetrons; one functions as an oscillator and the other as a driver. Both time and frequency domain solutions are developed from the model, allowing investigations into the growth and saturation as well as the frequency response of the output signal. This paper extends this locking theory for various configurations of magnetron coupling. Also studied is the general effect of driving frequency chirp on Adler's classical locking condition. This work was supported by AFOSR. \newline \newline [1] P. Penvanich et al., J. Appl. Phys. 98, 114903 (2005). \newline [2] V. B. Neculaes, Ph.D. Dissertation, University of Michigan, Ann Arbor, MI (2005).   

Experiments on dielectric window breakdown at atmospheric pressure using a 1.5 MW, 110 GHz gyrotron

It is increasingly important to know the power threshold for breakdown of high-power microwave (HPM) transmission through a window as the output powers of HPM devices continue to rise. One of the frequency ranges where breakdown threshold data are scarce is the W-band (75 to 111 GHz). We report preliminary experimental results on window breakdown at atmospheric pressure using the near-Gaussian output beam generated from a 1.5 MW, 110 GHz gyrotron with a pulse length of 3 microseconds. Successful breakdowns were achieved by focusing the beam down to approximately 6-mm radius at a polycarbonate window. The threshold power density and peak electric field were determined to be roughly 2 MW/cm$^{2}$ and 50 kV/cm for 50{\%} breakdown probability. Also, we have observed periodic plasma array structures in the time-integrated photographs of breakdown plasmas, and the origin of this periodicity is currently under investigation. These results will be compared with extensive data taken at Texas Tech Univ. at lower frequency, in S-Band.   

Electromagnetic effect on a discharge generated in the window breakdown on a dielectric

In high-power microwave sources and rf accelerators, the suppression of undesirable dielectric window breakdown is an issue. The vacuum multipactor discharge [1], often considered a candidate which initiates window breakdown on the vacuum side, is an avalanche caused by secondary electron emission from the dielectric window. On the air side, the collision of electrons with the background gas alters the multipactor discharge significantly, and an rf discharge plasma is generated. The additional electron generation mechanism, electron-impact ionization, can lead to high electron density which significantly changes the wave impedance. In that regime, the self-consistent interaction between the wave and plasma can play an important role. Considering the wave equation for a nonuniform electron density distribution, the electromagnetic effect is investigated by using particle-in-cell simulations. [1] H. C. Kim and J. P. Verboncoeur, Phys. Plasmas \textbf{12}, 123504 (2005).   

Evolution of Modes in Magnetically Insulated Crossed Field Diodes

The time-dependent behavior of electron sheaths in a magnetically insulated B$>$B$_{Hull}$ anode-cathode gap with crossed electric and magnetic fields is studied. The crossed-field, space-charge limited diode is modeled for various magnetic fields by means of multidimensional (1d and 2d), self consistent, electromagnetic, particle-in-cell (PIC) simulations in both cylindrical and planar geometries. The transient behavior of the system is examined in detail and is divided into three separate stages: cycloidal flow, collapse of cycloidal flow and sheared (near-Brillouin) flow. Our 2d electromagnetic PIC simulations (both planar and cylindrical) show that cycloidal flow also collapses into a perturbed flow that is dominated by the E cross B drift, but is neither steady nor stable. This observed cycloidal flow instability is a kinetic mode, not a fluid mode such as the magnetron or diocotron instability. The growth of the kinetic mode is faster than that of either of the above mentioned fluid instabilities. After the kinetic mode saturates, the fastest growing fluid mode grows to dominate the system. The SWS is added by three different methods to separate the RF effects from the DC electric field effects created by the SWS. The first method is to add a circuit to the anode that does not effect the DC electric fields, the second is to add the SWS by placing a thin dielectric (with and unphysical large dielectric constant), and last is to add the geometric SWS.   

Effects of ions on gap closure in crossed-field Diodes

The presence of plasmas has long been suspected to be a major cause of gap closure in a high power diode whose electrons are presumably magnetically insulated. In this paper, we report an in-depth study of the effects of ions on magnetic insulation of electrons. We find that, in general, the presence of ions in a crossed-field gap always increases the electrons' excursions toward the anode, regardless of the location of the ions. This effect is much more pronounced with ions situated in the cathode region than ions in the anode region. These properties are shared by single particle orbit models, shear flow models, and particle-in-cell simulations. Thus, the rate at which the electrons migrate toward the anode is related to the rate at which ions are introduced into the crossed-field gap. We should point out that this anode-migration of electrons is unrelated to crossed-field ambipolar diffusion. The implications of these findings are explored, such as pulse shortening in relativistic magnetrons [1] and bipolar flows in pulsed-power systems. [1] Luginsland et al., in this conference.   

Impact of AK gap plasma on magnetron performance

A magnetron is a cross-field source of high power microwaves. While magnetrons consistently produces very high peak power [1], the source suffers from rather poor pulse length. Previous work [2] posited the existence of neutral moving cathode plasma with sufficient density to short out the electric fields in the device, acting as a kind of moving AK gap which de-tunes the device, changed the mode of operation, and/or caused a lack of insulation sufficient to terminate the output microwave power pulse. The challenge with this model is the basis of the plasma with sufficient density to short out the electric field. In this work, we report on recent developments [3] showing that plasma, anywhere in the anode-cathode gap, is sufficient to modify the electron flow hub height. The modification of the hub height is sufficient to break the Buneman-Hartree resonance condition and drastically reduce the efficiency of microwave production. The density of the ion population, while important for the degree of hub modification, need not short out the electric field in the diode. We will show 2D EM ICEPIC calculations of this novel pulse-shortening effect, comparing the simulation data with both theoretical and experimental results. Plausible schemes for the ion source will be discussed. [1] M. R. Lopez et. al., IEEE Trans. Plasma Sci, 30, 3, 947, 2002. [2] D. Price, J.S. Levine, and J.N. Benford. IEEE Trans. Plasma Sci. 26, 348, 1998. [3] Lau et. al. at this conference.   

Enhanced Performance of a Relativistic Magnetron by Magnetic Priming

Magnetic priming [1] was applied to the UM/L-3 Titan relativistic magnetron (6-vane, -300kV, 5-10KA, 0.3-0.5 $\mu $s). Three 4-cm-long Mu-Metal were inserted within the cathode, centered beneath the emission region, and spaced 120 degrees apart. These wires produce magnetic perturbations with N/2 azimuthal symmetry (for pi-mode, N vane magnetron). Experimental results using the non-symmetric waveguide load array showed dramatic reduction in pi-mode starting current. Magnetic priming increased the percentage of pi-mode shots from 35{\%} to 58{\%}. Preliminary data also yielded increases in pi-mode peak power and mean pulse width. Symmetric waveguide load array data showed similar trends in magnetron performance improvement. A second series of experiments using three 6-cm wires within the cathode showed an 11{\%} increase in the probability of pi-mode shots over the baseline case. MAGIC simulations combining magnetic priming on the cathode and anode have shown faster startup than the baseline case without magnetic priming, as well as improvements over magnetic priming applied only on the cathode or anode. [1] V.B. Neculaes, R.M. Gilgenbach, and Y.Y. Lau, US Patents 6,872,929 and 6, 921,890 (2005).   

Vortex structures in planar magnetrons

In recent theoretical work, we proved the existence of vortex flows in the electron beam equilibria of magnetron structures consisting of a central cylindrical cathode and a periodically corrugated cylindrical anode [J. A. Davies and C. Chen, Phys. Plasmas 13, 012310 (2006)]. While the previous treatment was non-relativistic, the present theoretical work focuses on the relativistic regime. An analogous geometry for which a relativistic treatment is relatively simple is that of a planar cathode and a periodically corrugated planar anode. We will present results of vortex analyses in both the non-relativistic regime and the relativistic regime, and discuss the implications of our theoretical predictions.   

Simulation of electron emission from conformal boundaries for applications to high-power microwave sources

The finite difference time domain (FDTD) approach for electromagnetic particle-in-cell (EM-PIC) is a proven method for many problems involving interactions of charged particles with electromagnetic fields. Applying these methods to complex geometries that occur in high-power microwave (HPM) sources requires methods to accurately model fields and deal with particle emission and absorption at complex boundaries. We have recently developed conformal boundaries for the FDTD electromagnetic solver in the VORPAL code that have be shown to be 2nd order accurate in space. VORPAL also has models for the field emission of electrons as well as models for secondary electron emission. In order to use these advances to study particle effects in HPM devices we have begun modifying the particle boundaries in VORPAL so they can be used with the conformal geometry. We will present the current results of this work including the addition of current correction algorithms to prevent the build up of unphysical image charges when particles are removed and simulations involving field emission and secondary emission of electrons from conformal surfaces.   

Resonances in Saturated Stage of a Magnetron

Theoretical studies of the cold-fluid model of crossed-field electron vacuum devices such as magnetrons and crossed-field amplifiers have shown that there are two important stages to their operation [1]. First there is the ``initiation stage'' wherein an instability in the rf fields grows. When this instability saturates, the device enters into the ``saturation stage.'' A drift (diocotron) resonance dominates this stage. The importance of the rf drift resonance to the steady operation of the device is clear from the fact that this resonance can only significantly couple to the usual rf modes of the initiation stage when it is located along the ``edge'' of the sheath. Theory [1] has outlined the major features of the drift resonance in the saturation stage, however a complete understanding requires a study of numerical solutions. The problem here is one of scales. In the initiation stage there are two rf modes with oscillations on the order of unity. In the saturation stage there are five rf modes; the original two modes of the initiation stage and three additional fast modes with fast vertical oscillations on the order of 100-1000 times that of the initiation rf modes. We will present a WKB approach to decompose the rf equations of the saturated modes into individual modal equations, which can be rapidly integrated numerically. Computational results using this method will also be presented. \noindent [1] D.J. Kaup, Phys. Plasmas {\bf 13}, 053113 (2006).