# Plasma Seminar Series

## Upcoming Seminars

## Friday Dec. 11, 2020, 10am (MT) (Zoom)

A coupled particle pusher for multi-scale compact-object magnetosphere simulations

### Fabio Bacchini, CIPS, CU Boulder

## Past Seminars

Turbulent transport in the inner core of the high-JET hybrid discharge.

75225 is investigated extensively through linear and non-linear gyrokinetic simulations using the gyro-kinetic code GKW in the local approximation limit. Compared to previous studies [Citrin et al 2015 Plasma Phys. Control. Fusion 57 014032], the analysis has been extended towards the magnetic axis, rho<0.3 (rho is toroidal flux coordinate), where the turbulence characteristics remain an open question.

Understanding turbulent transport in this region is crucial to predict core profile peaking that in turn will impact the fusion reactions and the tungsten neoclassical transport, in present devices as well as in ITER. At rho= 0.15, a linear stability analysis indicates that Kinetic Ballooning Modes (KBMs) dominate, with an extended mode structure in ballooning space due to the low magnetic shear. The sensitivity of KBM stability to main plasma parameters is investigated. In the non-linear regime, the turbulence induced by these KBM modes drives a significant ion and electron heat flux. Standard quasi-linear models are compared to the non-linear results. The standard reduced quasi-linear models work well for the ExB part of heat flux, but fail to capture magnetic flutter contribution to the electron heat flux induced by the nonlinear excitation of low k_theta*rho_i (poloidal wave vector) micro-tearing modes (MTM) that are linearly stable. An extension of the quasi-linear models is proposed allowing better capturing the magnetic flutter flux.

[1] S. A. Slutz, et al., Pulsed-power-driven cylindrical liner implosions of laser preheated fuel magnetized with an axial field), Physics of Plasmas 17 (2010). URL: http://scitation. aip.org/content/aip/journal/pop/17/5/10.1063/1.3333505. doi:http://dx.doi.org/10.1063/1.3333505.

[2] D. B. Sinars, et al., Measurements of magneto-rayleigh-taylor instability growth during the implosion of initially solid metal liners, Physics of Plasmas 18 (2011) 056301. URL: http://aip.scitation.org/doi/abs/10.1063/1.3560911. doi:10.1063/1.3560911. arXiv:http://aip.scitation.org/doi/pdf/10.1063/1.3560911.

[3] K. J. Peterson, D. B. Sinars, E. P. Yu, M. C. Herrmann, M. E. Cuneo, S. A. Slutz, I. C. Smith, B. W. Atherton, M. D. Knud- son, C. Nakhleh, Electrothermal instability growth in magnetically driven pulsed power liners, Physics of Plasmas 19 (2012) 092701.

**Microwave fruit plasma and the shape of water**, Aaron Slepkov (Trent University). Friday Oct. 2, 2020

**Passive Thin Plasma Lens: Next Generation Electron Beam Optics**, Christopher Doss, CU. April 10, 2020

Finally I will talk about the FLARE experiment and how it can address the onset of reconnection in laboratory plasma, helping to understand fluid vs kinetic modeling of reconnection onset.

**Characterizing magnetic reconnection regions using Gaussian mixture models on particle velocity distributions from simulations:**We present a method based on unsupervised machine learning to identify regions of interest using electron velocity distributions as a signature pattern. An automatic density estimation technique is applied to particle distributions provided by PIC simulations to study magnetic reconnection https://arxiv.org/abs/1910.10012. This method identifies automatically the presence of complex distributions due to beams or other non-Maxwellian features and can be used as a detection algorithm able to identify reconnection regions. The method is also compared to more physical plasma properties: Swisdak's agyrotropy parameter. The approach is demonstrated for specific double Harris sheet simulations but it can in principle be applied to any other type of data in the future, in particular 3D simulations.

**Magnetic Reconnection in Relativistic Plasmas**, Greg Werner, University of Colorado. October 18, 2019

*t*, where

^{-1}*t*is time normalized to the reconnection timescale; and the field correlation length grows as

*t*. Critical balance is shown (by magnetic structure functions) to govern the aspect ratio of the flux ropes in 3D RMHD simulations. This quantitative description of inverse energy transfer could improve our understanding of longstanding problems such as coronal heating, galactic magnetogenesis, and high-energy emission in gamma-ray bursts.

^{1/2}From an arcade of magnetic flux loops on the surface of the sun to tokamak sawteeth, magnetic reconnection is not only fast, but also impulsive -- a slow buildup of magnetic energy is followed by a fast release. But the physics responsible for this is still an open question. Impulsive events observed in recent laboratory experiments [1-2] required 3-D physics to explain and exhibited features in common with events observed in the magnetosphere. Future work will include: 1) Satellite observations aimed at investigating events comparable to the experiment, and 2) Laboratory experiments on the new FLARE user facility at Princeton aimed at exploring a multi-X-line regime comparable to the magnetosphere.

Alfvén waves, a fundamental mode of magnetized plasmas, are ubiquitous in lab and space. The nonlinear behavior of these modes is thought to play a key role in important problems such as the heating of the solar corona, solar wind turbulence, and Alfvén eigenmodes in tokamaks. In particular, theoretical predictions show that these Alfvén waves may be unstable to various decay instabilities, even at very low amplitudes (δB/B<10-3), but this key physics has only recently been demonstrated for the first time in the laboratory [3-5]. My ongoing and future work aims to compare laboratory observations and satellite measurements in order to determine the regions of the heliosphere where Alfvén wave decay instabilities may play an important role. Proposed new projects include investigations of fundamental non-linear interaction physics using Alfvén eigenmodes in tokamak plasmas and whistler waves in a basic laboratory device.

My present and future research interests lie in the astrophysical analogs of the HED systems that we create in the lab using high-power lasers. The fundamental plasma science at the heart of this field is essential to understanding some of the most interesting phenomena in our universe. In this field of research, large-scale physics experiments are often executed at user-facilities in the US and abroad, while much of the diagnostic and technological development is done at the academic institution. My research is motivated by outstanding questions in astrophysics, but the technologies developed for this work are of great importance to the whole HED physics community. I will briefly cover upcoming experiments at the Jupiter Laser Facility that will lay the ground work for future studies of shockprocessed dust-destruction rates in supernova remnants. I will also cover my plans to build a first class facility at CU Boulder for training students in the rapidly growing field of HED Laboratory Astrophysics.

We derive the Hamiltonian formulation of EMHD in a canonical form; we calculate the matrix elements for the three-wave interaction of whistlers and show that (i) harmonic whistlers are exact non-linear solutions; (ii) co-linear whistlers do not interact (including counter- propagating); (iii) whistler modes have a dispersion that allows a three-wave decay, including into a zero frequency mode; (iv) the three-wave interaction effectively couples modes with highly different wave numbers and propagation angles.

We solve numerically the kinetic equation and show that, generally, the EMHD cascade is non-univeral - it depends on the forcing and often fails to reach a steady state. Analytical estimates predict the spectrum of magnetic fluctuations for the quasi-isotropic cascade ~ k^?2. The cascade remains weak (not critically- balanced). The cascade is UV-local, while the infrared locality is weakly (logarithmically) violated.

Spectral properties of coherent waves in an argon plasma column are examined using fluctuation data from fast imaging. Experimental dispersion relation estimates are constructed from imaging data alone using a cross-spectral-density technique. Electron drift waves are identified by comparison with theoretical dispersion curves, and a tentative match of a low-frequency spectral feature to Kelvin-Helmholtz-driven waves is presented.

Transient fluctuations are examined in a local spheromak merging experiment using a multi-channel magnetic probe. A histogram cross-spectral analysis technique allows experimental dispersion relation estimates to be made from magnetic measurements. Hints of waves in the range of ion-cyclotron frequency harmonics are observed in conjunction with merging events.

We have used the Reconnection Scaling Experiment (RSX) to study flux ropes, and have found many new features involving unexpected 3D dynamics, kink instability driven reconnection, non linearly stable but kinking flux ropes, large flows, and shear flow induced magnetic fields. For example the onset threshold for external kink instability depends upon boundary conditions that can be adjusted between line tied and free. These two boundary conditions could correspond to CME eruption flux ropes that are anchored ("line tied") at one end to solar coronal holes while the other end remains "free" to drift as magnetic clouds in the solar system. The dynamics of two flux ropes form a fairly simple 3D system that allowed the first identification of how a plasma instability (in this case the kink) initiated magnetic reconnection. When there is significant guide magnetic field, flux ropes bounce off each other much of the time instead of merging and reconnecting. As we assemble large 3D experimental data sets for density, temperature, pressure, magnetic field, and current density we observe local violations of MHD, and strongly sheared flow and fields. We show data where magnetic field is generated from sheared electron fluid flow. Movies from 3D experimental data also show that MHD forces fail to balance, i.e. JxB - grad P_e does not vanish at the 30% level, and we evaluate some candidates for the missing physics. We intend to model these 3D data with a PIC code (VPIC), 2 fluid code (HiFi) and possibly other hybrid approaches, and solicit collaborators.

*DOE Fusion Energy Sciences DE-AC52-06NA25396, NASA Geospace NNHIOA044I, Basic

[1] P. Snyder, et al., Phys. Plasmas 16 056118 (2009).

[2] Z. Yan, et al., Phys. Plasmas 18 056117 (2011).

[3] J. Callen, et al. Nucl. Fusion 50 064004 (2010).

Computations with multiple modes similar to reversed-field pinch discharges show that both MHD and Hall dynamos contribute to relaxation events. The presence of Hall dynamo implies a fluctuation-induced Maxwell stress, and the simulation results show net transport of parallel momentum. The magnitude of force densities from the Maxwell stress and a competing Reynolds stress, and changes in the parallel flow profile are within a factor of 1.5 of measurements [Kuritsyn et al., Phys. Pl. 2009] during a relaxation event in the Madison Symmetric Torus.

Over the past half-century, solar dynamo theory has proceeded along two parallel tracks: axisymmetric mean-field models, particularly the so-called 'flux-transport' models, and full 3D MHD models. Breakthroughs using mean-field models include explaining the solar cycle period, how the fields reverse, and what features can be predicted. Full 3D MHD models now produce cyclic evolution of the fields. I will review recent developments for both model types.

The biggest remaining challenge is how to include the effects of unresolved small scale processes, such as the rising of magnetic flux-tubes and MHD turbulence. Current full 3D MHD models are themselves mean-field models -- very sophisticated, but very expensive -- that are very hard to use to advance our understanding of the solar dynamo. But axisymmetric kinematic flux-transport models also have limitations. There is a third class of model, intermediate in complexity and expense, that offers the opportunity to extend our understanding of the solar cycle by explaining global departures from axisymmetry, such as 'active longitudes', 'sector boundaries', and 'tilted dipole' structures. I will describe how this model can be built as a generalization of axisymmetric flux transport dynamo models.

- G. Lapenta, J.U. Brackbill, Nonlinear Evolution of the Lower Hybrid Drift Instability: Current Sheet Thinning and Kinking, Physics of Plasmas, 9, 1544-1554, 2002.
- P. Ricci, J.U. Brackbill, W.S. Daughton, G. Lapenta, Influence of the Lower-Hybrid Drift Instability on the onset of Magnetic Reconnection, Physics of Plasmas, 11, 4489-4500, 2004.
- W. Daughton, G. Lapenta, P. Ricci, Nonlinear Evolution of the Lower-hybrid Drift Instability in a Current Sheet, Physical Review Letters, 93, 105004, 2004.
- G. Lapenta, J. King, Study of Current Intensification by Compression in the Earth Magnetotail, Journal of Geophysical Research, 112, A12204, doi:10.1029/2007JA012527, 2007.
- G. Lapenta, Large scale momentum exchange by microinstabilities: a process happening in laboratory and space plasmas, Physica Scripta, 80, 035507, 2009.

[1] J. Egedal, N. Katz, et al., J. Geophys. Res. 113, A12207 (2008).

[2] A. Le, J. Egedal, et al., Phys. Rev. Lett., 102, 085001 (2009).

[3] A. Le, J. Egedal, et al., Geophys. Res. Lett. 37, L03106 (2010).

[4] J. Egedal, A. Le, et al., Geophys. Res. Lett. 37, L10102 (2010).

**Momentum transport and the radial electric field in tokamaks**Peter Catto: MIT. Wednesday May 05 2010

During recent years the Heidelberg dust group designed and built high resolution mass spectrometers of large and intermediate sensitive areas. An important constraint for the optimum design of the spectrometer's field optics is the properties of the plasma produced by the hypervelocity impacts. Furthermore, the reliable interpretation of a mass spectrum of an impact plasma requires a good knowledge of the various processes that produce the impact plasma. To this aim we performed experiments with a mass spectrometer designed to investigate the energy distributions of the ionized species. Furthermore, we reanalyzed the mass spectra of various materials such as silicates obtained from laboratory experiments using the Heidelberg dust accelerator as well as cosmic water ice dust impacts recorded by the Cassini dust detector CDA. After the initial acceleration the plasma energy was found to be surprisingly low, with a non-Maxwellian distribution. We also investigated which impact parameter actually controls the properties of the impact plasma. Our experiments suggest that this parameter is the energy density at the site of impact rather than the impact speed.

What is the analogous structure for volume preserving systems? For the case of incompressible fluids, Bernoulli's theorem can be interpreted to say that an incompressible flow with a symmetry has an integral.This implies that a 3D fluid flow with a symmetry can be effectively reduced to a 2D flow that is Hamiltonian, and therefore integrable. For the case of maps, however, it seems that symmetries and invariants are independent. A 3D volume preserving map with a symmetry can be effectively reduced to an area preserving map. However, this map need not be integrable. To insure integrability requires an additional invariant. The result is similar to a notion of " broad integrability " for general flows defined by Bogoyavlenskij.

In this talk, two new physics ideas are developed and applied to this problem. First idea: electrostatic wells are replaced by centrifugal wells and non-neutral plasma replaced by quasi-neutral plasma. Previously known physics are applied, leading to many arrangements which form a high-efficiency (<90%) heat engine. Simplest of all is the Pastukov problem, in which a single well confines a low-collisionality nearly thermal plasma. It is shown that a proper arrangement of magnetic field (essentially the open field of a field-reversed configuration - FRC) can make this into a heat engine, so that plasma heat becomes rotation. The energy cycle is completed by converting rotation to beam energy. It is shown how to use the high electrical potentials induced by rotation to electrostatically accelerate a beam into the confined plasma.

Second idea: plasma rotation can produce plasma waves from a static magnetic perturbation, using nothing more than the Doppler effect. These waves can also be used to produce a desired beam by resonant absorption. Another use for such waves is to drive currents. As already demonstrated experimentally, such currents can form a FRC.

All of this leads to lowering the fusion threshold. In particular, required temperatures are greatly reduced, leading to very small, very high-power density systems. The non-thermal fusion also means that aneutronic fuel cycles can be used. Some examples are given. Finally, a small experiment to test these physics is being planned and some details of this design are given.

**Magnetohydrodynamic Activity Inside a Sphere**David Montgomery: Dartmouth College. Friday May 12 2006

[1] A. Gailitis et al, Phys. Rev. Lett. 86, 003024 (2001).

[2] P.D. Mininni and D.C. Montgomery, Phys. Rev. E72, 056320 (2005).

[3] P.D. Mininni and D.C. Montgomery, "Magnetohydrodynamic activity inside a sphere," arXiv:physics/0602147 (submitted to Phys. Fluids, 2006).

In this talk, I will show that parallel electric fields can be supported by double layers (DL). I will then show how we solve for such a double layer using methods similar to Bernstein, Green and Kruskal [1957] (the so called BGK method). The distribution functions that we use to construct the DL are modeled from FAST data. Finally, to test whether such a DL is stable, I have initialized a Vlasov simulation with a typical auroral cavity plasma, and have included a double layer to see how it evolves. I will briefly discuss the Vlasov algorithm for evolving distribution functions. One of the new features of the simulation is that we have included two ion species (H+ and O+) in addition to electrons. As a result of having two ion species, I will show how ion phase space holes and other non-linear structures, which are often seen with FAST, form in the simulation.

Collaborators: John R. Cary, Daniel C. Barnes and Johan Carlsson

At low temperatures, the m? = 0, kz = 1, 2, 3, . . . Trivelpiece-Gould modes (standing waves of density fluctuation along the z-axis; i.e., center of mass motion, breathing mode, and higher modes) are weakly damped and dominate, since the random particle component is suppressed by Debye-shielding. As the temperature increases, the broad random particle component increases in between the modes. The thermally excited mode is physically interesting because it exhibits both the individual particle behavior and the collective mode (wave) behavior of equilibrium plasmas. Also, the thermally excited mode leads to an important application, which is a passive temperature diagnostic of electron plasmas.

In the present seminar, I describe my recent work in the field of flow-magnetic field interaction applied to processes typical of the solar and Earth environment. I will describe the specific examples of the genesis of the slow solar wind in the solar corona and of reconnection in the Earth's magnetosphere. I will describe fundamental processes related to the interplay of flows and magnetic fields and I will address how microscopic and macroscopic processes interact to determine the overall evolution. A unique tool to handle such multiple scale problems at within a fully kinetic approach, CELESTE3D, will also be described.

*In collaboration with D.A. D'Ippolito, D.A. Russell [Lodestar], L.A. Berry, E.F. Jaeger, and M.D. Carter [ORNL]

** Work supported by U.S. DOE grant DE-FG02-97ER54392 and the RF-SciDAC project.

Acknowledgements:

This work is being supported by the US DOE, OFES Novel Diagnostics Initiative.

Key contributions to the PCI diagnostic by J. Dorris and C. Rost (at DIII-D), N. Basse, L. Lin, and E. Edlund (at C-Mod) are acknowledged. In addition, key contributions to the C-Mod ICRF physics by P. Bonoli, Y. Lin, J. Wright, and S. Wukitch are noted. Past contributions by S. Coda, A. Mazurenko and E. Nelson-Melby are also noted here.

The talk will be divided into three parts.

First, I will discus the challenges of multiple scale problems in plasma physics. Plasmas host a variety of processes, often some are of more interest than others. Often the processes of interest are on long space and time scales. The implicit approach is an excellent way to handle this situation. It focuses on the long scales of interest, with proportionate resolution, without needing to resolve smaller scales accurately. The method implicitly averages over the smaller and faster scales. I will discuss the general properties of the implicit method.

Second, I will discuss how the implicit moment method is designed and turned into a computer code. I will summarize the actual formulation we currently use in our CELESTE3D code. I will spend a little more time discussing the most recent advances in this area: the formulation of the Maxwell's equations and the boundary conditions for them.

Lastly, I will discuss some benchmark calculations meant to illustrate the performance of CELESTE3D.

These mappings and the circular modes are parametrized similar to the Courant-Snyder forms for the conventional uncoupled, or planar, case. The planar-to-circular and reverse transformers (beam adapters) are introduced; their implementation on the basis of skew quadrupole blocks is described. Applications of the planar-to-circular, circular-to-planar and circular-to-circular transformers are discussed. A range of applications includes round beams at the interaction region of circular colliders, flat beams for linear colliders and relativistic electron cooling.

NIMROD(NonIdeal MHD with Rotation - Open Discussion) is a massively parallel three dimensional magnetohydrodynamic simulation utilizing finite elements (FE) to represent the poloidal plane and a fourier decomposition in the toroidal direction. The use of finite elements allows flexibility in the representation of the simulation domain. The ability to model experimental shots with NIMROD provides a platform to test new ideas of plasma behavior. To expand the physics capabilities of NIMROD, kinetic effects have been added to NIMROD by the addition of delta-f particle-in-cell (PIC) module. The addition of kinetic particle effects captures essential wave-particle interactions important in the saturation of various magentohydrodynamic (MHD) instabilities such as the internal kink mode, sawtooth and fishbone instabilities, and toroidal Alfvén eigenmodes. Particle simulation capabilities in NIMROD can also be extended to simulate various phenomena such as neutral beam injection, ion cyclotron resonance heating, and anomalous los mechanisms. In addition, this hybrid kinetic-MHD technique lays the foundations for a kinetic closure to the MHD equations.

This talk will briefly introduce NIMROD and delta-f PIC in general, then detail the development of PIC in finite elements and their implementation and some preliminary results.

[1] Nickel, Parker, Gould. Phys. Fluids. 7:1489. 1964.

[2] Cooperberg. Phys. Plasmas. 5, No. 4, April 1998.

Work in collaboration with J. R. Myra (Lodestar) and S. I. Krasheninnikov (UCSD)

First, microscopic processes will be considered. Over the last few years, I have developed several simulation techniques implemented in computer codes and I have developed mathematical physics models for studying fundamental processes in dusty plasmas. I will briefly overview these efforts. The main focus of the presentation will be the results obtained in using such techniques to understand the presence of attractive forces in dusty plasmas. Attractive forces are believed to be responsible for some lattice structures observed in dusty plasma crystals as well as for some coagulation processes. Two mechanisms will be presented, discussed and analyzed quantitatively using simulation methods: plasma wakes and dipole moments induced by asymmetric charging in flowing plasmas (found in space and at the edge of sheaths in glow).

Second, global collective effects will be discussed. The simulation techniques used here differ profoundly to those considered above. Monte Carlo and Molecular Dynamics methods will be presented and applied to study macroscopic effects and transitions of state in complex fluids and in complex plasmas.

Here I report on a new method of predicting MeV electron fluxes at geostationary orbit 1-2 days in advance using only solar wind measurements. Using this method we have achieved a prediction efficiency of 0.82 and a linear correlation of 0.84 for the two years 1995 and 1996. Using the same model parameters based on the years 1995-1996, the prediction efficiency and the linear correlation for the five year period 1995-1999 are 0.61 and 0.77, respectively. The model is based on the standard radial diffusion equation, which is solved by setting the phase space density larger at the outer boundary than the inner boundary and by making the diffusion coefficient a function of solar wind velocity and interplanetary magnetic field. This model also provides a physical explanation for some observed features of the correlation between the solar wind and the electron flux.

In physics, we have built an electronic homework system which results in improved test and final exam scores, sometimes adding a full letter grade to the student score. Additionally, the weaker students receive the greatest benefit from this system. In chemistry, Web-based homework program includes thirty-five interactive discovery environments that provide guided inquiry, feedback to student responses and tracking of student performance. Intelligent tutors in chemistry provide levels of customized responses to expose students to the depths of chemical reactions. One tutor enables students to directly manipulate images and work with a palette of tools for placing and moving symbols. Tutors have been tested with over 900 students and show positive improvement in final exams. The electronic homework system provides an open architecture allowing for rapid extensions to new departments.

In addition, engineering tutors provide animated 3D tooling solutions of student designs and advice about relative costs. Evaluation demonstrates that these tutors are as effective as several lectures and homework assignments within a traditional classroom setting. A microbiology tutor provides visual support for an entire undergraduate course in molecular biology with rich 3-D animations depicting production of proteins through the interaction of DNA and RNA. A mathematics tutor uses machine learning to individualize problems and hints.

Each demonstrated project has been evaluated for effectiveness and efficiency. The talk will show how technology's impact on leaning has been quantified and that these systems can be of general use on a national scale.

In practical application, velocity distributions are often determined experimentally, or are numerically generated by computer simulations. These general distribution functions cannot be assumed analytic, nor can they necessarily be well-modeled by a superposition of analytic functions (e.g., Maxwellians).

An alternative perspective on the problem of determining the dispersive properties of damped plasma waves will be presented. This method draws a connection between the plasma susceptibility and the convolution (or deconvolution) of the distribution function with simple Lorentzians. This approach lends itself to the numerical determination of the plasma susceptibility for piecewise-analytic distributions that have been subjected to Gaussian smoothing.

A one-dimensional analytical linear model based on a fluid description of the plasma was initially developed. It was found that the threshold intensity of the absolute instability and the steady-state spatial growth rate of the convective instability are both independent of the scattering angle. However, the saturation time of the convective instability exhibits a strong inverse dependence on the scattering angle.

The basic model was improved by extending the one-dimensional analysis to include two spatial dimensions and time. In order to assess the effects that the finite size of the laser beam has on SBS, wide and narrow laser-beam geometries were considered. Detailed comparison were made between the predictions of a reduced 1D and 2d models, which can be solved and analytically, and the results of 2D numerical simulations.

The influence that nonlinear and kinetic effects have on SBS was investigated by performing particle-in-cell (PIC) simulations. The results of these PIC simulations were compared against fluid simulations, and good agreement was obtained for sufficiently weak laser intensities. When the laser intensity is sufficiently strong for ion trapping to be significant, PIC and fluid simulations differ substantially. The SBS reflectivity is shown to depend sensitively on the frequency mismatch between the light wave used to seed the instability and the incident laser.

A possible solution to this problem is the addition of collective gravitational wave dynamics in the disk. Planetary emryos tend to excite waves in the disk which can strongly damp the velocity dispersion of the larger bodies, leading to larger collision cross sections and more rapid planetary growth. Althought the theory of wave-planet interactions is fairly mature, practical techniques for including wave-planet interactions in N-body simulations remain to be developed. I will discuss a reduced description of wave-planet interactions that is somewhat similar to reduced descriptions of wave-particle interactions that are used in plasma physics simulations. The goal of this talk is to draw useful analogies between the methods of plasma physics and the methods of solar system dynamics and to facilitate the exchange of ideas between these two disciplines of classical physics.

We report on the results of simple, 1D particle based equilibrium models of the current sheet. In particular, we discuss the importance of particle stochasticity in determining the gross current sheet structure. We have found that small numbers of stochastic particles can lead to significant thickening of the current sheet, with potential consequences for current sheet instability theories. In addition, the quasi-trapped nature of these particles can lead to force balance failure, so that in certain parameter regimes (particulary in the near Earth, dipolar region) thin, 1D current sheet solutions do not exist.