Modeling Long Probes in Flowing Plasmas using KiPS-2D, a Novel Steady-State Vlasov Solver

Transcription

1 39th AIAA/ASME/SAE/ASEE Joint Proplsion Conference and Exhibit 2-23 Jly 23, Hntsille, Alabama AIAA Modeling Long Probes in Flowing Plasmas sing KiPS-2D, a Noel Steady-State Vlaso Soler Éric Choinière and Brian E. Gilchrist Uniersity of Michigan, Ann Arbor, Michigan 489 A Consistent steady-state kinetic 2D plasma model and the corresponding comptational soler were deeloped and sed for the modeling of long condctie electron-collecting probes in flowing mesosonic plasmas. Sheath asymmetries, not acconted for in preios treatments of ion-collecting probes in flowing plasmas, are modeled here and shown to consititte an important mechanism for the departre from OML theory for electron crrent collection in the mesosonic regime. The effects of collisions are addressed by diiding the space srronding the probe into a collisionless comptational space and a collisional backgrond plasma. The implementation of the soler consists of sccessie linearizations of the nonlinear Poisson-Vlaso operator, within a Tikhono-reglarized Newton iteratie process. The Finite Element Method is sed for the Poisson soler, while the inside-ot trajectory tracking procedre is sed for the Vlaso soler. The parallel soler allows for the arbitrary elocity distribtions of both species within the comptational domain, proides an adaptie, nstrctred meshing strategy, and allows simlation of ery large comptational domains. Reslts show indication of a small enhancement, with respect to OML theory, of the collected crrent to an electron-attracting probe in a flowing plasma. This enhancement is attribted to the elongation of the pre-sheath into the collisional zone of the plasma, which cases an enhanced density of incoming electrons pstream from the probe, and is seen to dominate the opposing decrease in electron collection de to additional potential barriers created by a wake-side depression of the electric potential. The primary isse is the accmlation of noise in the soltion that sbsides in spite of the employed Tikhono reglarization, limiting the progress of the iteratie scheme. Improements in the Vlaso soler to redce the amont of qadratre noise it generates are planned to improe the consistency of soltions. Introdction LANGMUIR probes and bare space electrodynamic tethers are two important applications inoling long electron-collecting condctie cylinders in flowing plasmas. The interpretation of Langmir probe data in flowing plasma diagnostics, as well as the prediction of crrent collection to bare space electrodynamic tethers, are lacking an accrate model that acconts for the sheath asymmetries that arise in mesosonic flowing plasmas. Space electrodynamic tethers offer the opportnity for propellantless proplsion of near-earth orbiting spacecraft, based on the conersion of the geomagnetic force on an electric crrent along a tether into a proplsie force. One of the key parameters affecting thrst is the crrent leel flowing Gradate Stdent, Radiation Laboratory, EECS Department, AIAA Stdent Member, echoinie@mich.ed Professor, Electrical Engineering and Space Sciences, AIAA Senior Member, gilchrst@mich.ed Copyright c 23 by Éric Choinière. Pblished by the, Inc. with permission. throgh the tether, which in trn is limited by the amont of electron crrent collected from the ionosphere. Bare condctie tethers are belieed to be ery efficient electron collectors when compared to configrations sing an inslated tether combined with an end collector. 2 Howeer, space tethers are moing throgh the ionosphere at orbital elocities, effectiely adding a flow component to the srronding plasma. It is desirable to assess how the electron collection capability of a cylindrical bare tether, immersed in a flowing plasma, departs from that predicted by the Orbital Motion Limit, which is only alid in the case of thin wires 3 in non-flowing plasmas. In both space electrodynamic tether applications and plasma diagnostic deices, a detailed model of the plasma kinetics is reqired in order to accrately predict the crrent collection to cylinders of arios cross-section geometries in flowing plasmas and/or to extract estimates of the plasma parameters from measred oltage-crrent characteristic cres. of Copyright 23 by Eric Choiniere. Pblished by the, Inc., with permission.

2 Existing Models Exact analytical soltions to the symmetric problem of an attracting rond condctie cylinder immersed in a non-flowing plasma only exist in the thin sheath (infinite cylinder radis) and orbital-motion (anishing cylinder radis) limits. The thin-sheath and OML normalized crrents are gien by 4 I ts =, φ > () I th { I oml 2 ( ) } = π φ + exp (φ)erfc φ, φ > (2) I th where the thermal crrent, which normalizes both et eqations, is gien by I th = A p nq 2πm, where A p is the cylinder area, n is the nmber density of attracted particles in m 3, m and q are the particle mass and charge, T is the temperatre of the species in nits of ev, and e =.6 9 J/eV. The normalized bias potential φ is related to the physical bias potential V by φ = V V p T, where V p is the plasma potential. To address cases other than those two limits, a consistent soltion to the problem of the non-flowing ion-attracting rond cylindrical probe was first proided by Bernstein and Rabinoitz, 5 who deeloped a Poisson/Vlaso consistent, -D model based on approximations of mono-energetic ions and negligible electron crrent. The comptations consisted of nmerically integrating an ordinary differential eqation of one ariable. Laframboise 6 later deeloped a consistent iteratie nmerical scheme that allowed for Maxwellian distribtions (far from the probe) for both the attracted and repelled species, and inoled the iteratie resoltion of a nonlinear system of integral eqations, with reslts applicable to all potential ales, temperatres, and probe sizes with respect to the Debye length. Seeral athors hae also addressed, in a firstorder sense, the problem of ion collection by a rond cylindrical probe immersed in a flowing plasma, sing the crcial assmption of a radially symmetric potential profile, naffected by any flow effects. Mott-Smith and Langmir 4 deried an asymptotic formla alid in the limit of large speed ratios (relatie to the ion thermal elocity) for the crrent characteristic in the large sheath limit (orbital motion limit). Kanal 7 deried similar expressions alid in the limit of small speed ratios. Hoegy and Wharton 8 generalized those reslts by proiding expressions alid for all speed ratios, for the limiting cases of thin sheath, large sheath (orbital motion limit) and retarding regimes. Godard and Laframboise 9 went frther by deeloping a nmerical model that allowed for all probe radises to be considered in the flowing case by sing the -D cylindrically symmetric potential profiles obtained by Laframboise 6 as the assmed electric potential. In the case of the mesosonic regime, where the elocity of the flow is mch larger than the ion thermal elocity bt mch smaller than the electron thermal elocity, only ion collection can be addressed by an approximate soltion based on an assmed symmetric potential profile. Sch an approximation wold show irtally no departre from the non-flowing reslts in the case of electron collection, de to their large thermal elocity as compared to the speed of the mesothermal flow. In the mesothermal regime, the effects of the flow on the collection of the light species - the electrons- are only indirect. That is, these effects only occr de to the asymmetries in the potential profile arond the probe that are indced by the heaier ion species. Een for the ion-attracting case, the assmption of a symmetric profile cold proe erroneos at least in cases showing one or more of these two conditions: 9 the probe radis is not small with respect to the Debye length, implying a non negligible and likely asymmetric ion space charge distribtion near the collector; the ratio of flow energy to bias potential is neither ery small (a small flow cold only case small asymmetries) nor ery large (in which case the bias potential cold not significantly affect the flow). Proposed 2D Steady-State Kinetic Model We present the deelopment and software implementation of a consistent, steady-state, electrodynamic plasma model based on kinetic theory. The soler presented here is best classified as a Vlaso code based on a kinetic description of plasmas. It differs from Particle-In-Cell methods in that the Vlaso eqation is soled directly instead of resorting to a set of discrete macro-particles. It is also distinct from Elerian Vlaso implementations, owing primarily to the fact that a steadystate non time-arying soltion is soght directly, withot resorting to incremental time stepping. The steady-state soler presented here bares some similarities with preios work addressing other geometries, 2 4 and may be regarded as a 2-D extension of the -D model deeloped by Laframboise. 6 The aim of this model is to nmerically sole, self-consistently, the Poisson and Vlaso eqations in a steady-state oer a large comptational region 2 of

3 arond an arbitrarily-shaped 2D condctie body in a collisionless, nmagnetized, flowing 2-species plasma. The implementation of the soler, called KiPS-2D (Kinetic Plasma Soler, 2-dimensional) consists of sccessie linearizations of the nonlinear Poisson-Vlaso operator, within a Tikhonoreglarized Newton iteratie process. Following are the main featres of the proposed model and soler: it allows for the representation of the complete, arbitrary elocity distribtion of both plasma species at all sampled locations in the comptational domain; it can model plasma cross flow in all speed regimes with respect to the thermal elocities of the species; it proides a dynamic, adaptie, nstrctred meshing strategy, allowing for the resoltion of sheath asymmetries indced by the plasma flow and the spport of condctie objects with arbitrary 2D geometries; it incldes a Finite-Element based Poisson soler incorporating a backgrond plasma bondary condition based on asymptotic analytical reslts; 5 it allows for the simlation of ery large comptational domains in order to accommodate the prononced pre-sheath elongation (seeral hndred Debye lengths) along the direction of plasma flow. The ariable grid size is tailored locally to efficiently accommodate potential ariations. Nmerical instabilities reslting from large grid sizes are handled sing a Tikhono reglarization process; its steady-state Vlaso soler featres a parallel implementation based on the PVM library, 5 allowing it to rn on either a single host, a parallel architectre, or a scattered network of workstations based on a MPMD (Mltiple Processors, Mltiple Data) scheme. Definition of the Problem The basic probe geometry nder consideration is an arbitrarily-shaped 2-dimensional object, i.e. a cylinder of arbitrary cross-section geometry and infinite length. Probe end effects are neglected in this treatment, which is hardly an approximation in the case of bare space electrodynamic tethers, which are typically seeral kilometers long. This 2D condctie object (referred to as the probe) is immersed in a flowing 2-species nmagnetized plasma with the plasma flow directed along the Assmed Profile: V=V(R, ) R/R A θ A R A Ions: Shifted Maxwellian Electrons: Maxwell-Bolztmann Reg. A & B: Collisionless electrons & ions V=V Consistent charge & pot. profiles A R B B C Region C: Collisionless ions Collision-dominated electrons, E<T e /mfp Fig. Geometry of the plasma model. Regions A and B are part of simlation space, while region C defines a bondary condition on the oter edge of region B. x axis. A bias potential V = V is applied between the probe and a srface located at infinity, r =. In order to accrately represent the elongated sheath and pre-sheath strctres established in a flowing plasma, it is necessary to simlate a ery large region arond the probe. With that objectie in mind, we diide the space arond the object into three regions, as seen on Fig.. Regions A and B, defined by radises R A and R B respectiely, are part of the comptational domain, while region C contains the backgrond plasma. Following are descriptions of the assmptions made in each of the regions: Region A This innermost region extends from r = to r = R A and is the smallest region enclosing the probe, the sheath and part of the pre-sheath. In this region, charge densities are compted throgh a fll kinetic treatment of Vlaso s eqation, assming that the plasma is collisionless. In other words, binary short-range collisions among particles are deemed insignificant as compared to the large electric field forces reslting from the collectie longrange Colomb interactions. Vlaso s eqation is soled by tracking the electron and ion trajectories back in time, all the way to the oter bondary at r = R B throgh regions A and B. No assmptions are made with regards to the potential strctre in region A, bt an oter bondary condition is specified by the assmed potential profile in region B. Region B This intermediate region extends from r = R A to r = R B and coers the blk of the presheath. In this region, the electric field is still strong e 3 of

4 enogh to oerwhelm the effect of binary collision forces. Ths, the particle orbits followed from region A into region B still obey collisionless trajectories. In this region, howeer, the electric potential follows an assmed shape: Φ(r, θ) = Φ M(θ)R A r (3) where Φ M (θ) is the nknown potential profile on the interface between region A and region B at r = R A. This assmed profile follows from the asymptotic reslt obtained for a rond cylinder immersed in a non-flowing plasma (deried by Bernstein 5 and sed by Laframboise 6 ). It assmes that the ariables r and θ are separable for sfficiently large ales of R A. Any rotational asymmetries fond in the potential profile at the edge of region A (r = R A ) are propagated into region B, neglecting the damping of the θ-ariation of the potential in region B. The latter approximation cold eentally be improed, bt a conergence analysis of or reslts as a fnction of the size of region A will be performed to insre that this approximate behaior does not reflect in or conerged reslts. An improement in this approximation wold merely help redcing the size reqired on region A for proper conergence. Region C This region contains the backgrond plasma. It extends beyond the comptational domain, from r = R B to r =. In this region, electric fields are assmed to hae dropped to a low leel, sch that collision forces are dominating electron behaior. De to the dominant collisions beyond r = R B and the small ale of the electric potential at the oter bondary r = R B, we can safely assme a drifting Maxwell-Boltzmann elocity distribtion for the electrons at this location: f e ( x, y ) = n m e 2πeT e exp { Φob T e m e 2eT e ( ( x f ) y) } (4) where n is the backgrond plasma density and f is the speed of the flow, which is negligible for electrons in the mesosonic regime. The electric field leel at which collisional forces are on par with field forces can be defined as the following: an electric field that generates sfficient accelerations to significantly affect the thermal eqilibrim oer the same time interal as the mean time between collisions, that is, the relaxation time oer which elocity distribtions are thermalized to a Maxwellian eqilibrim. Eqating the mean time between collisions with the eqilibrim distrbance time interal, which is roghly eqal to the time reqired to accelerate a still electron to the aerage 2D elocity, we obtain: τ ee/m e (5) λ mfp ee/m e (6) E m e 2 = π m e th 2 eλ mfp 2 eλ mfp (7) E π T e 2 λ mfp (8) where we hae sed the 2D aerage elocity = π 2 th, since the z component of the electric field is assmed to be negligible. The mean free path of the dominating collision mechanism haing a thermalizing effect on electrons is sed here, e.g. the mean free path for electron-electron or electron-netral collisions (or any other electron scattering mechanism). As for the ions, their trajectories are assmed collisionless becase of their high speed as compared to their temperatre, which translates into dominating inertial forces. Ions are assmed to be distribted according to a drifting Maxwellian distribtion at infinity, and to follow straight-line collisionless trajectories within region C. In other words, they are accelerated in the direction of their initial elocity at r = ntil they reach r = R B, after which they follow collisionless trajectories gided by the potential distribtion in regions B and A. We assme for their elocity distribtion at infinity (r = ): f i ( x, y ) = n m i 2πeT { i exp m ( i ( x f ) 2 + 2eT y) } 2 (9) i Finally, the electric potential is assmed to ary piecewise-bilinearly oer the whole comptational domain, i.e. it aries bilinearly within any of the mesh s triangles. This simplifies the analytical tracking of sb-trajectories, since the electric field is conseqently piecewise-constant. Sfficient meshing accracy will ensre an accrate representation of the fields. Formlation of the Poisson-Vlaso Operator Obtaining a simltaneos soltion in two dimensions for the asymmetric steady-state electric potential distribtion and charge density distribtions of both plasma species (ions and electrons) reqires the ability to sole, self-consistently, Vlaso s eqation for each species and Poisson s eqation for the 4 of

5 electric potential and charges, while satisfying appropriate bondary conditions. Both the electric potential and charge densities are sampled on a piecewise-trianglar mesh, which is generated sing the Bidimensional Anisotropic Mesh Generator (BAMG) 6 deeloped at INRIA, France. An adaptie process was implemented whereby the BAMG mesher is sed to periodically refine the mesh based on the approximate net charge density and potential distribtions. At the nodes of this mesh are collocated samples of the continos ion density distribtion n i (x, y), electron density distribtion n e (x, y), net charge density distribtion ρ(x, y) = e (n i (x, y) n e (x, y)) and electric potential distribtion V (x, y). These samples are packed into the ectors n i, n e, ρ and V, respectiely. In the remainder of this paper, these sample ectors will be loosely referred to as distribtions of the corresponding physical qantities. A Poisson soler f P ( ) is deried, based on Poisson s eqation and the bondary conditions, that soles for the electric potential distribtion for a gien net charge density distribtion ρ, i.e.: V = f P ( ρ) () Likewise, a Vlaso soler f V ( ) is deried based on Vlaso s eqation and the bondary conditions, that soles for the density distribtions of both species, for a gien electric potential distribtion: ρ = f V ( V ) () Gien the Poisson and Vlaso operators, the whole problem can be specified either as a fnction of the net charge density distribtion or the potential distribtion. Here we choose to work in terms of the net charge density distribtion ρ, de to a stability adantage inherent to the reglarization process discssed later. When combined together, both solers form a Poisson-Vlaso operator. The soltion lies at the fixed point of this operator, i.e. is the root of the following eqation: ρ = f P V ( ρ), with f P V ( ) = f ( ) fp V ( ) (2) where f V, f P and f P V are the Vlaso, Poisson and PV operators. In order to get to the soltion, we mst ths find the root of (2), a nonlinear ector eqation. Poisson soler based on the FEM The Poisson soler, which comptes a potential distribtion from a gien net charge density distribtion, was implemented sing a formlation based on the well-known Finite-Element Method as applied to Poisson s eqation. The Poisson soler looks for a soltion for the potential profile V (x, y) which satisfies: ɛ 2 V (x, y) = ρ(x, y) (3) V condctie srface = V (4) V = V (R A, θ) (5) r R A R=RA,θ where the last 2 eqations are bondary conditions at the probe and at the oter bondary of region B, respectiely. The details of the deriation are not shown here since this is a standard Finite Element implementation of Poisson s eqation sbject to mixed bondary condition. 7 We hae sed piecewise bilinear charge density and potential distribtions in the formlation. This approach has proed more robst and flexible (in terms of applying mixed bondary conditions) than the point-matching techniqe sed preiosly. 8 Vlaso soler The Vlaso soler described here allows each species to express the fll kinetic natre of its 2D elocity distribtion. This featre allows for the realistic and accrate modeling of non-thermal plasmas, that is, plasmas that are not at thermal eqilibrim de to large electric fields and high elocity flows. It acconts for the largest part of the comptational complexity of the techniqe presented in this paper. Assming a collisionless, qiescent, nmagnetized plasma in a steady state, Vlaso s eqation states that the elocity distribtion fnction f e,i (x, y, x, y ) for any gien plasma species is constant along particle orbits in a gien electric potential distribtion: 9 df e,i dt = x f e,i x + f e,i y y + q e,i V m e,i x + q e,i m e,i V y f e,i x f e,i y = (6) where we sbstitted the electric field components with the gradient of the electric potential, i.e. E = V. The indices e and i refer to electrons and ions, respectiely. f e,i (x, y, x, y ) is the elocity distribtion of a gien species and V (x, y) is the electric potential distribtion fnction. Gien a known potential distribtion V (x, y) and a bondary condition for the incoming particles on the oter bondary, the method of characteristics can be sed to sole (6) for the elocity distribtions f e,i (x, y, x, y ) of both the electrons and ions. The 5 of

6 y (m) x 3 more efficient than sing a fixed time step particle psher, since the amont of comptations necessary for one trajectory depends on the nmber of edges being crossed rather than the nmber of time steps necessary to reach a bondary. Also, gien the assmption of a piecewise-bilinear potential distribtion, it proides exact (nearly exact if we accont for rondoff errors) conseration of energy along orbits, which contribtes to the accracy of the oerall approach. x (m) x 3 Fig. 2 Example of the Semi-analytical Particle Tracking Process Throgh the Potential Mesh. densities of both species are then obtained by integrating oer all elocity space: n e,i = f e,i (x, y, x, y ) d x d y (7) and the total charge density reslts from ρ(x, y) = q i n i q e n e, where q i and q e are the ion and electron particle charge. The se of the method of characteristics for the resoltion of Vlaso s eqation in plasma problems was reported early on, 3 and was referred to as the inside-ot procedre. It consists of tracking a particle s trajectory back in time ntil it intersects the oter bondary of comptational space (interface between regions B and C on Fig. ), where the elocity distribtion fnction is sampled. Any trajectory not originating from the oter shell is deemed npoplated. 6 Sch is the case for trajectories originating from the tether itself, which we assme does not emit charged particles, as well as trapped trajectories which hae no sorces in the collisionless case. Howeer, it shold be pointed ot that the Maxwell-Boltzmann elocity distribtion sed for electrons (see Eq. (4)) allows nonetheless some electrons haing a total energy less than zero into the comptational space; these particle may be regarded as trapped particles as well. Fig. 2 illstrates the particle tracking process. The trajectories are tracked analytically from one edge of the mesh to another, assming a constant electric field within any gien triangle of the mesh. Eery sb-trajectory is ths resoled by compting the intersection of a qadratic parametric cre with a segment on the mesh. This techniqe is mch To obtain a ale for the particle density at a gien point, one needs to integrate the ales obtained for f e,i (x, y, x e, y ) sing Eq. (7). To limit the comptational task, the domain of integration is restricted to a limited region otside of which the elocity distribtion fnction is known to be ery low. Conseration of energy, together with knowledge abot the elocity distribtion fnctions at the oter bondary, is sed to define a annlar domain of integration in two-dimensional elocity space, 8 which corresponds to a rectanglar domain of integration in the two-dimensional space defined by the kinetic energy and the anglar direction. The integration is performed in the latter 2D space, sing an adaptie cbatre of trianglar elements. A modified ersion of rotine TOMS76 2, 2 from the Transactions on Mathematical Software repository was sed to this effect. The main integration sbrotine for compting the integral oer a single trianglar region was sbstitted with Algorithm 7 from the Jornal of Comptational Applied Mathematics, sing a 64-point generalized Gass-Legendre prodct rle. 64 points are ths sed to ealate the integral oer each of the trianglar elements of the domain in this adaptie trianglation of the kinetic-energy/direction-angle space. Typical calclations inole from 5 to a few hndred trianglar elements, therefore reqiring seeral thosand ealations of the elocity distribtion fnction for each sample of the nmber density. In order to obtain an estimate of the error on the nmber density calclation, the integral oer each trianglar element is actally compted twice, once sing the 64-point rle discssed aboe, and once sing a 49-point rle. The error estimate is taken as the difference between the 2 ealations. In addition to proiding an estimate for the error on the density, these error estimates are sed in the adaptie refinement of the trianglation. The error estimate on the density is sed as part of the reglarization of the Newton iteration, as shown in the following section. 6 of

7 M Poisson Soler ')(+*-,/. Vlaso Soler "!$#&% 32/4)57698;:=< FHGJI-K=L?24 Fig. 3 Fixed Point Operator Comprised of Both the Poisson and Vlaso Solers. Tikhono-reglarized Newton Iteratie Method Using the Poisson and Vlaso solers, we seek a soltion for the potential and charge distribtions that simltaneosly satisfies the Poisson and Vlaso eqations. Fig. 3 depicts the general fixed-point operator comprised of both the Poisson and Vlaso solers. The fixed-point operator takes a charge distribtion ρ in at its inpt and generates a new estimate ρ ot : ρ ot = g ( ρ in ) (8) There are known difficlties arising in soling sch a problem. 6 Simple iteration of the fixed point operator does not in general yield conergence, since it is a non-contractie mapping. 22 Instead, we se a procedre based on Newton s method for nonlinear systems of eqations 23 which is depicted in Fig. 4. This necessitates the Jacobian matrix J g of the fixed point operator and is eqialent to iterating the following fixed point fnction: ρ ot = h ( ρ in ) = ρ in + (J g ( ρ in ) I) ( ρ in g ( ρ in )) (9) where the symbol represents a reglarized inersion based on colmn normalization and Tikhono reglarization, 24 both of which are necessary to filter ot the errors introdced by the Vlaso soler. These errors originate from the finite precision with which densities are compted, combined with the fact that like nmbers are sbtracted to obtain the net charge density in the pre-sheath where the ion and electron densities are ery close to each other. J g ( ρ in ) is the Jacobian matrix of operator g ( ) ealated at ρ in. If there are N nknowns (N charge nodes), the Jacobian matrix is a N N matrix and is defined by: J g (i, j) = g i ρ j, i =,..., N, j =,..., N. (2) Using finite differences to compte an approximation for J g wold be prohibitiely expensie comptationally. Instead, direct comptation of J g is performed. This inoles combining the gradients of eery sample of the elocity distribtion fnction that were obtained at that particlar iteration, and Poisson Soler Vlaso Soler WXZY\[Q]_^a`b] V W"XY-cedgfcih ] V W"XYjd V oqp V WkmlNn$c Fig. 4 Tikhono-reglarized Newton Iteratie Poisson/Vlaso Procedre replicating all the operations that were applied to those samples. To obtain the gradients of each elocity distribtion sample, the analytical gradient of the elocity distribtion at the oter bondary is propagated and transformed along the corresponding particle trajectory. To redce the comptation time of each iteration, the Jacobian compted at the first iteration may be sed for sbseqent iterations, a techniqe know as the Chord method. 25 This has the disadantage of increasing the nmber of reqired iterations, bt it stabilizes the Newton iterations by aoiding the noise generated de to the finite precision of each Jacobian comptation. The principal difficlty inoled in the reglarization process is the choice of the reglarization parameter. Both the reglarization parameter and the precision of the Vlaso soler mst be selected sch that the Vlaso noise will penetrate the soltion at a slower rate than the rate of conergence of the process. Details of the implementation of this reglarization procedre will be presented in an pcoming jornal article. Parallel Implementation The iteratie resoltion high-leel algorithm and the Poisson soler, both fairly light comptationally, were implemented in Matlab. The Vlaso soler, being mch heaier comptationally, was implemented in Fortran 9 sing an MPMD parallel processing scheme based on the Parallel Virtal Machine library( home.html). The program is being rn on two different platforms. The primary platform is the AMD Linx Mltiple Program, Mltiple Data 7 of

8 clster at the Uniersity of Michigan s Center for Adanced Compting, accessed throgh the National Partnership for Adanced Comptational Infrastrctre (NPACI). The secondary platform is based on opportnistic pooling of local Linx and Sn workstations. The parallelization is implemented at a fairly high leel: each node is assigned the task of compting one or more samples of the nmber density. Twenty processors are sed in a typical simlation on the Linx clster. Simlation Reslts Earlier alidation rns 8 had confirmed that or code has excellent agreement with OML theory for both electron and ion collection in a non-flowing plasma, in satration as well as in retardation regimes. Here we present recent reslts obtained for a long probe with a one-debye-length radis, immersed in a Xenon plasma. The flow energy is 4 times the electron temperatre, and the applied bias is 4 times the electron temperatre. For this simlation, region A had a radis of Debye lengths, and region B extended to a radis of 54 Debye lengths. Based on Eq. (8), the electric field at the oter edge of region B has a magnitde sch that the forces it creates on the electrons are eqialent to that cased by a collisional process haing a mean free path between 2 and 26 Debye lengths. The extent of region B was actally sized with the intent of achieing an edge electric field eqialent to a 9 Debye length mean free path, which is the theoretical ale for the electron-netral collisions corresponding to the backgrond pressre and plasma parameters of a acm experiment reported in Ref. 26. Fig. 5 shows two different iews of the arios profiles obtained at the 2 th iteration of the simlation. The plasma flows along the x axis (shown here in scaled form as the axis) from negatie to positie x s. The charge distribtion ρ in and potential distribtion correspond to the inpt and otpt of the Poisson soler, respectiely. The ion density and electron density profiles correspond to the otpt of the Vlaso soler, and are combined to prodce the net charge otpt ρ ot of the Poisson-Vlaso operator. The conergence parameter (shown in Fig. 5) is expressed in the form of a relatie difference between the inpt and otpt net charge density distribtions. Here the error was redced down to 22.5%. As discssed earlier, the electron and ion density profiles generated by the Vlaso soler are compted with finite precision. Despite the reglarization, some of the noise present in the net charge ρ ot gets inserted into the soltion ρ in, and bilds p oer the iterations. The present simlation was not iterated any frther than 2 iterations, in order to limit the amont of noise that gets inserted into the soltion. Frther work is needed to redce the amont of noise generated by the qadratre integration process in the Vlaso soler and allow s to redce the error below %. The reslts indicate an enhancement in the collected crrent as compared to the OML-predicted leel of crrent (I = 7.76I the as opposed to I oml = 7.23I the ). Varios nmerical experiments hae indicated that there are two opposing flow-related phenomena that can be associated with the change in the amont of collected crrent as compared to OML theory:. the potential depression seen on the wake side of the probe acts as a potential barrier that preents some electrons from reaching the probe, thereby redcing the amont of collected electron crrent. The ery low electron nmber density seen on the wake side proides confirmation of that phenomenon; 2. pstream from the probe, the electric potential has a small bt finite ale on the edge of comptational space (in the case presented here, the pstream edge potential is. T e ). Still, the electric field at this location is sfficiently weak that collisions dominate the motion of electrons, based on a mean free path as large as 2 to 26 Debye lengths as mentioned aboe. The density of the electrons incoming from the collisional region C(see Fig. ) is gien by a Maxwell-Boltzmann law (see Eq. 4), owing to the collisional natre of the electrons beyond or comptational space. Ths, the density of the incoming electrons from the ram side is proportional to e φ edge Te e. =.2, which increases the amont of collected electron crrent. This phenomenon also allows the electron density to rise aboe the backgrond plasma density, as seen on Fig. 5. The plasma flow creates an elongation of the pre-sheath in the direction of the flow, which cases, on the ram side, the electric field to drop below the collisional inflence limit before the potential as reached a small ale, thereby casing an enhancement in the nmber density of incoming electrons that are transitioning from collisional region C to collisionless region B. It appears that, in the present case, the second ef- 8 of

9 ρ/(e n ) Net charge ρ in 2 2 Ion density V in /T e Potential 2 2 Electron density iteration #2/2 X Y AXES LOG SCALE: =log (r/r )x/r =log (r/r )y/r ρ ot ρ in / ρ in =.225 Normalized Collected Crrent: I/I the =7.76 > I oml /I the =7.225 Net charge ρ ot n i /n n e /n ρ/(e n ) Net charge ρ in 2 2 (a) Potential 2 2 iteration #2/2 ρ/(e n ) V in /T e X Y AXES LOG SCALE: =log (r/r )x/r =log (r/r )y/r ρ ot ρ in / ρ in =.225 Normalized Collected Crrent: I/I the =7.76 > I oml /I the =7.225 Ion density Electron density Net charge ρ ot n i /n n e /n (b) Fig. 5 Simlation reslt shown for a -Debye-length radis probe biased at 4T e and immersed in a Xenon flowing plasma with beam energy of 4T e. To help the isalization of the reslts, the x and y axes were stretched based on the following mappings: = log (r/r )x/r and = log (r/r )y/r; (a) side iew (b) slanted iew. 9 of.5 ρ/(e n )

10 fect dominates the first one, reslting in an oerall crrent enhancement. This obsered enhancement is qalitatiely consistent with recent experimental reslts, althogh not as important (7.4% in these simlation reslts as opposed to 39% in or experimental reslts 26 ). This difference in the magnitde of the enhancement might be explained by a shorter thermalizing collision mean-free-path in or experiment as compared to the one sed in the simlation to determine the extent of region B. One of the reasons for this discrepancy cold be that the backgrond netral pressre within the plasma beam of these experiments was actally higher than the measred ale sed to determine the theoretical mean free path of 9 Debye lengths that was referred to aboe. The backgrond pressre was measred at another location within the 6m 9m acm chamber, ot of the plasma beam. Clearly, a smaller mean-free-path wold increase the collected crrent in or simlation. In a preios commnication, 8 we had shown simlation reslts indicating a collection crrent below OML. Howeer, the model sed then assmed the presence of a localized grond at some distance from the tether and did not allow for the electron density enhancements consistent with a slow increase of the potential pstream from the probe combined with thermalizing collisions. Ot of the two phenomena identified aboe, only the first one (i.e., a potential depression on the wake side) was then simlated, ths reslting in a redced collected crrent de to potential barriers. Present Stats and Conclsions We hae presented a steady-state 2-species plasma model applicable to arbitrary 2D condctie objects in flowing plasmas with ery long electron collision mean free paths and collisionless ions. The effects of collisions were addressed by diiding the space srronding the condctie object (the probe) into a collisionless comptational space (regions A and B) and a collisional backgrond plasma (region C). Reslts were shown pertaining to the case of a circlar cylinder in a flowing plasma, with indication of a small enhancement of the collected crrent with respect to OML theory. This enhancement is attribted to the elongation of the pre-sheath into the collisional zone of the plasma, which cases an enhanced density of incoming electrons pstream from the probe, and is seen to dominate the opposing decrease in electron collection de to additional potential barriers created by a wake-side depression of the electric potential. The primary isse that will need to be addressed is the accmlation of noise in the soltion that sbsides in spite of the employed Tikhono reglarization. Improements mst be made in the Vlaso soler in order to redce the amont of qadratre noise generated and conseqently redce the error leel, thereby improing the consistency of the soltion. Frther deelopments will also address noncirclar cross-sections. Acknowledgments The athors wold like to thank Dr. Dae Cooke, Dr. Ira Katz and Dr. Jan Sanmartín for sefl discssions on the topic. É.C. acknowledges the scholarship spport of the Horace H. Rackham School of Gradate Stdies at the Uniersity of Michigan, the Natral Sciences and Engineering Research Concil of Canada and the Commnications Research Centre(Canada). 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