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Two-Fluid Physics and Field-Reversed Configurations PHYSICS OF PLASMAS 14, 055911 ͑2007͒ Two-fluid physics and field-reversed configurationsa… ͒ A. Hakimb Tech-X Corporation, 5621 Arapahoe Avenue – Suite A, Boulder, Colorado 80303 U. Shumlak Aerospace and Energetics Research Program, University of Washington, Seattle, Washington 98195-2600 ͑Received 6 November 2006; accepted 30 April 2007; published online 24 May 2007͒ In this paper, algorithms for the solution of two-fluid plasma equations are presented and applied to the study of field-reversed configurations ͑FRCs͒. The two-fluid model is more general than the often used magnetohydrodynamic ͑MHD͒ model. The model takes into account electron inertia, charge separation, and the full electromagnetic field equations, and it allows for separate electron and ion motion. The algorithm presented is the high-resolution wave propagation scheme. The wave propagation method is based on solutions to the Riemann problem at cell interfaces. Operator splitting is used to incorporate the Lorentz and electromagnetic source terms. The algorithms are benchmarked against the Geospace Environmental Modeling Reconnection Challenge problem. Equilibrium of FRC is studied. It is shown that starting from a MHD equilibrium produces a relaxed two-fluid equilibrium with strong flows at the FRC edges due to diamagnetic drift. The azimuthal electron flow causes lower-hybrid drift instabilities ͑LHDI͒, which can be captured if the ion gyroradius is well resolved. The LHDI is known to be a possible source of anomalous resistivity in many plasma configurations. LHDI simulations are performed in slab geometries and are compared to recent experimental results. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2742570͔ I. INTRODUCTION electromagnetism. The two-fluid model retains both electron inertia effects and the displacement currents and also allows The dynamical behavior of plasmas is strongly depen- for ion and electron demagnetization. The rest of this paper dent on frequency. At the lowest frequency, the motion of the is organized as follows. First, some aspects of ideal two-fluid electrons and ions are locked together by electrostatic forces physics are described. Length scales at which two-fluid ef- and the plasma behaves like an electrically conducting fluid. fects become important are then derived. Next, a high- ͑ ͒ This is the regime of magnetohydrodynamics MHD .At resolution wave-propagation scheme2 for the solution of somewhat higher frequencies, the electrons and ions can these equations is presented. This scheme, originally devel- move relative to each other, behaving like two separate, in- oped by LeVeque, has been extensively used to study fluid terpenetrating fluids. At still higher frequencies, the distribu- dynamics, elasticity, MHD,3 etc. The algorithms are bench- tion function of the plasma species is driven by anisotropies marked against the Geospace Environmental Modeling in the velocity space. This regime is best described by the ͑GEM͒ magnetic reconnection challenge problem. It is collisionless Boltzmann equation or Vlasov equation of ki- shown that the reconnection flux from the two-fluid model netic theory. In this paper, numerical schemes are developed agrees well with that computed from full particle and hybrid to simulate two-fluid plasma dynamics, i.e., physics in the simulations. Simulations of field-reversed configurations intermediate frequency regime between MHD and full ki- ͑FRCs͒ equilibria and lower-hybrid drift instability ͑LHDI͒ netic theory. Due to the disparate scales on which plasma are presented. dynamics occurs, a complete spectrum of mathematical mod- els of plasmas can be derived. Among the most commonly used fluid models are the magnetohydrodynamics model1 and the Hall MHD model. In MHD, the plasma is treated as II. TWO-FLUID PHYSICS a single electrically conducting fluid. Although in the Hall- The two-fluid plasma equations can be obtained by talk- MHD model a distinction is made between the bulk plasma ing moments of the Boltzmann equation. Assuming no heat velocity and electron velocity, electron inertia and displace- flow and a scalar fluid pressure, the following five-moment ment currents are ignored and electron and ion number den- ideal two-fluid equations listed below are obtained for each sities are assumed to be the same ͑quasineutrality͒. A more species in the plasma: general approach, used in this paper, is to treat the plasma as a mixture of multiple fluid species. In this five-moment ideal ץ nץ two-fluid model, each plasma species is described by a set of + ͑nu ͒ =0, ͑1͒ j ץ ץ fluid equations with electromagnetic body forces. The elec- t xj tromagnetic fields are modeled using Maxwell equations of ץ ץ a͒ Paper GI2 3, Bull. Am. Phys. Soc. 51, 100 ͑2006͒. m ͑nu ͒ + ͑p␦ + mnu u ͒ = nq͑E + ⑀ u B ͒, ͑2͒ kj k j k kij i j ץ k ץ b͒ Invited speaker. t xj 1070-664X/2007/14͑5͒/055911/11/$23.0014, 055911-1 © 2007 American Institute of Physics Downloaded 25 Jun 2007 to 128.95.35.171. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp 055911-2 A. Hakim and U. Shumlak Phys. Plasmas 14, 055911 ͑2007͒ ⍀ץ ץ Eץ ١p/n͒. ͑14͒͑ ϫ ١−= ϫ ͑u ϫ ⍀͒ ١ − u p + u E͒ = qnu E . ͑3͒͑ + ץ j j j j ץ ץ t xj t Here n is the number density, u is the mean fluid velocity, p This equation applies to each species in the plasma, and, for E is the fluid scalar pressure, and is the fluid total energy example for a hydrogen plasma, there are two such equa- given by tions. Equation ͑14͒ can be compared to the ideal MHD re- p 1 sult E = + mnu u , ͑4͒ ␥ −1 2 i i Bץ ϫ ͑v ϫ B͒ =0, ͑15͒ ١ − tץ where ␥=5/3 is the adiabatic index. Further E is the electric field, B is the magnetic flux density, q and m are the charge ⑀ and mass of the plasma species, and kmj is the completely where v is the “bulk” or MHD single-fluid velocity, the Hall- antisymmetric Cevi-Levita pseudotensor, which is defined to MHD result6 be ±1 for even/odd permutations of ͑1,2,3͒ and zero other- Bץ -wise. Summation over repeated indices is assumed. The elec ١p /en͒, ͑16͒͑ ϫ ١−= ϫ ͑u ϫ B͒ ١ − t e eץ tromagnetic fields appearing in the source terms of the fluid equations are determined using Maxwell equations,4 ͑ ͒ B where e is electron charge, and the Euler neutral fluid resultץ ϫ E =− , ͑5͒ ٌ ␻ץ tץ ١p/␳͒, ͑17͒͑ ϫ ١−= ϫ ͑v ϫ ␻͒ ١ − tץ Eץ 1 ϫ B = ␮ J + , ͑6͒ ٌ ץ 2 0 c t where ␳ is the mass density. From these equations, it is clear that the two-fluid equations span the complete range from → E = c , ͑7͒ neutral fluids, to Hall-MHD, to MHD: B 0 corresponds to · ٌ ␧ → 0 the neutral fluid limit, me /mi 0 corresponds to Hall-MHD, while ␻/͑qB/m͒→0, which, as shown below, is the same as ,B =0. ͑8͒ vanishing electron and ion skin depths or ion Larmor radii · ٌ corresponds to the ideal MHD limit. Here ␮ and ␧ are the permeability and permittivity of free 0 0 Examining the generalized vorticity ⍀=m͑␻+qB/m͒ it space, c=͑␮ ␧ ͒−1/2 is the speed of light, and and J are the 0 0 c is clear that for two-fluid effects to be important, charge density and the current density defined by ϵ ͑ ͒ ␻/␻ Ն O͑1͒, ͑18͒ c ͚ qn, 9 c ␻ ϵ where c qB/m is the cyclotron frequency. Using the fluid J ϵ ͚ qnu. ͑10͒ ϵͱ ͑ ͒ thermal velocity uT 2p/ mn as a reference speed and some reference length L, ␻Ϸu /L and hence the condition The summations in Eqs. ͑9͒ and ͑10͒ are over all species T present in the plasma. For a plasma with s species, there are u /͑L␻ ͒ = r /L Ն O͑1͒, ͑19͒ 5s+8 equations in the system. T c L The ideal two-fluid model is more general that the MHD where r ϵu /␻ is defined as the Larmor radius, is ob- or the Hall-MHD models. To derive conditions under which L T c tained. Instead of the fluid thermal velocity, if the typical two-fluid effects, not included in the MHD or Hall MHD speed is assumed to be the Alfven speed, u ϵB/ͱ␮ mn, model, are important, scalar and vector potentials, ␾ and A, A 0 then the condition are introduced. In terms of these, the electric and magnetic fields are expressed as ͑ ␻ ͒ ͑ ␻ ͒ Ն ͑ ͒ ͑ ͒ uA/ L c = c/ p /L O 1 , 20 t, ͑11͒ץ/Aץ + ١␾−= E ␻ ϵͱ 2 ⑀ where p nq / 0m is the plasma frequency, is obtained. It -ϫ A. ͑12͒ should be emphasized that the plasma frequency, Larmor ra ١ = B dii, and cyclotron frequency are each defined separately for ϵ Next, defining a generalized momentum, P mu+qA and a each plasma species. ١ ⍀ϵ١ϫ ␻ ␻ generalized vorticity, P=m +qB, where = In summary, two-fluid effects are important when, for ϫu is the fluid vorticity, the nonconservative form of the Ն ͑ ͒ both ions and electrons, rL /L O 1 and/or when l/L momentum equation is written as Ն ͑ ͒ ϵ ␻ O 1 , where l c/ p is the skin depth. Conversely, in the ١p limit in which the length scales are much larger than the Pץ ١͑mu2/2 + ␾͒, ͑13͒ + −= u ϫ ⍀ − -t n electron skin depth, but smaller than the ion skin depth, Hallץ MHD is an adequate model, while in the limit in which the which is a balance law for the generalized momentum.5 Tak- scale lengths are larger than ion skin depths, the MHD de- ing the curl of Eq.
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