Computer studies on the fast reconnection mechanism under an externally driven boundary condition

The spontaneous fast reconnection model is applied to the earthward fast flow events observed in the near‐Earth plasma sheet. Here, the earthward fast flow events include both of bursty bulk flow events and flow burst events. In order to apply it directly to actual observations, virtual probes are located in the plasma sheet region in the three‐dimensional simulation domain so that we can directly observe the temporal variations of plasma quantities in accordance with the growth and proceeding of the fast reconnection. In this model, magnetic reconnection drastically evolves and Alfvénic fast plasma jet flows in the very restricted narrow channel, and a large‐scale plasmoid is formed ahead of the fast plasma jet. The results of virtual observation of these evolutions are found to be in good agreement with actual satellite observations. At the same time, in the lobe region, travelling compression regions (TCRs) are observed in connection with the fast flow events. The temporal profiles of magnetic fields detected by the virtual probes are also in good agreement with actual satellite observations. It is concluded that the earthward fast flow events and earthward TCR events result from the fast reconnection mechanism.

On the basis of the spontaneous reconnection model, computer simulations study the physical mechanism by which magnetic energy, initially stored in a current sheet system, is released into plasma energies. For the uniform resistivity model, the Sweet–Parker mechanism is eventually set up with the diffusion region becoming longer with time. It is the Ohmic heating ηJ2 in the diffusion region that plays the dominant role in releasing the magnetic energy. Attached to the diffusion region, a long plasmoid is formed and propagates like a large-amplitude Alfvén pulse, where the generator and motor effects are canceled along the plasmoid boundary. For the anomalous resistivity model, the fast reconnection mechanism is eventually set up with the diffusion region remaining to be localized near an X neutral point. It is the powerful motor effect [u⋅(J×B)≳0] along the slow shock layers that drastically releases the stored magnetic energy. A large-scale plasmoid distinctly swells, so that the ambient magnetic fields are compressed (by the generator effect), and the enhanced magnetic energy is then reduced by the strong motor effect in the backward half of the plasmoid. The slow shocks extend with time from near the X point, leading to a drastic catastrophe for the overall magnetic field system.

The present paper studies the basic physics of the spontaneous fast reconnection model in a three-dimensional (3D) situation for different resistivity parameter values, where the threshold for occurrence of current-driven anomalous resistivity is allowed to increase with the thermal velocity (T), and the initial plasma density notably changes in space with the plasma pressure in the current sheet system. For any case, once the anomalous resistivity is ignited, the 3D fast reconnection mechanism explosively evolves as a nonlinear instability by the positive feedback between the anomalous resistivity and the reconnection flow, even if the threshold significantly increases with the thermal velocity; for the larger threshold values, the fast reconnection evolution becomes more drastic and the reconnection rate, finally attained on the nonlinear saturation phase, becomes larger. In the resulting 3D fast reconnection configuration, slow shocks stand and extend outwards in the finite extent; also, ahead of the fast reconnection jet, a large-scale 3D plasmoid swells and propagates in the central current sheet, and a vortex flow is formed near the plasmoid side boundary. In the wide range of parameter values, the basic physics of the 3D spontaneous fast reconnection evolution in the finite extent is found to be, qualitatively, consistent with the well-known two-dimensional one.

The spontaneous evolution of fast reconnection is studied in three dimensions by extending (in the z direction) the previous two-dimensional model that considered only the x-y plane [M. Ugai, Phys. Fluids B 4, 2953 (1992)]. It is demonstrated that the reconnection development strongly depends on three-dimensional effects; only when the central current sheet is sufficiently long in the z direction, say more than a few times larger than the current sheet width, the fast reconnection mechanism fully develops by the self-consistent coupling between the global reconnection flow and the current-driven anomalous resistivity. In this case, the reconnection flow can grow so powerfully as to enhance the current density (the current-driven resistivity) locally near an X line; otherwise, such a vital reconnection flow cannot be caused. The resulting quasisteady fast reconnection mechanism is significantly confined in the z direction, where a strong (Alfvénic) plasma jet results from standing switch-off shocks; accordingly, a large-scale plasmoid is formed and propagates in the middle of the system. It is concluded that the well-known two-dimensional spontaneous fast reconnection model can reasonably be extended to three dimensions.

In the last two decades, a wide variety of plasmoids events have been observed, ranging from space and astrophysical phenomenon to magnetically confined laboratory plasmas, in which there are a lot of evidence of observational plasmoid-like features supported by direct large-scaled computer simulations. A super-Alfvénic instability, named plasmoid instability, occurs in an extended current sheet, when the Lundquist number exceeds a critical value. The large-aspect-ratio current sheet is fragmented by generating, growing, coalescing and ejecting of plasmoids so that this phenomenon has been proposed as a possible mechanism for fast reconnection scenario. This super-Alfvénic plasmoid instability has been usedin the significant new development of reconnection theory, and thus can provide alternative and more convincing mechanism for fast reconnection. In this work, a “driving” kind of shear flow in the out-of-plane direction is imposed on a two-dimensional, three-component magnetohydrodynamic model with a current sheet system to study the dynamic process of the plasmoids in a current sheet system. The effect of the width and strength of the driving flow on the reconnection rate of plasmoids are numerically analyzed in detail. It is found that the plasmoids are easily formed in the case of strong and wide out-of-plane driving flow. The reconnection rate and the number of the plasmoids increase with the driving flow width and/or driving flow strength increasing. In the presence of guiding field, it is found that the symmetry of the plasmoids is broken in the reconnection plane. In addition, for the fixed guiding field, the growth rate of plasmoids increases much faster when the strength of driving flow increases.

Magnetic reconnection is widely believed to be involved in dynamical phenomena such as solar flares or magnetospheric substorms. The Sweet–Parker model of magnetic reconnection in a Y-type geometry predicts a characteristic time scale proportional to S1/2 (where S is the Lundquist number), which is too slow to account for the observed time scales. The Petschek model, in contrast, predicts a time scale proportional to ln S in an X-point geometry. Numerical magnetohydrodynamic (MHD) simulations in the high-S regime generally validate the Sweet–Parker model, unless the resistivity is enhanced in the diffusion region to large values (such that typically S<103). It is demonstrated in this paper that nonlinear reconnection dynamics in a Harris sheet driven by inward boundary flows occurs on a nonlinear time scale that is proportional to S1/5 and thus has a weaker dependence on resistivity than the Sweet–Parker time scale. The current sheet amplitude at the separatrix (spanning Y points) grows algebraically in the linear regime but is suddenly enhanced after it makes a transition to the nonlinear regime. An analytical calculation is given for both the linear and the nonlinear regimes, and supported by two-dimensional resistive MHD simulations. The features of sudden current sheet enhancement and fast reconnection, controlled by boundary flows, are relevant to the phenomena of substorm onset or the impulsive phase of flares.

The self-consistent behavior of an isolated current-sheet system enclosed by free boundaries is numerically studied in the framework of magnetohydrodynamics. Initiated by a disturbance, the middle current sheet drastically collapses into a thin layer in a collisionless plasma. It is argued that an anomalous resistance should limit the extreme current concentration. When the current sheet, initially of width 2L, becomes thin, an anomalous resistivity η is introduced that increases with the relative electron–ion drift velocity ‖VD‖ when ‖VD‖ exceeds a threshold value VC. It is found that, when the reconnection flow has just been coupled to the anomalous resistivity, the reconnection growth rate [normalized by (L/VA)−1 ] becomes extremely large and scales as (∂η/∂‖VD‖)0.6V2.2C (normalization is based on the initial configuration); accordingly, the fast reconnection mechanism is set up on a short time scale and leads to the rapid collapse of the system at large. The initial growth of the fast reconnection is shown to be in good agreement with the underlying physical mechanism of the modified tearing instability.

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)

Time evolutions of the electron diffusion region embedded in the ion-scale diffusion region and the reconnection rate associated with magnetic reconnection are investigated using 2-1/2 dimensional full kinetic simulations in a large system, so that any effects of the downstream boundary conditions are negligible. The simulation code employs the adaptive mesh refinement technique and the particle splitting algorithm to the conventional particle-in-cell code, which enable us to perform large-scale full particle simulations. It is shown that the reconnection rate increases associated with magnetic reconnection and reaches a peak value large enough for fast reconnection, but then it decreases as time goes on, even though the effects of the system boundary are negligible. The key process responsible for slowing the reconnection processes is the extension of the electron diffusion region in association with the enhancement of the polarization electric field directing toward the neutral sheet in the electron inflow region. The polarization electric field is caused by the inertia difference between ions and electrons, and enhanced by the meandering motions of the background ions. In order to confirm the role of the polarization electric field, we compare the simulation results with mi∕me=1 and 100. It is found that (1) a quasi-steady reconnection is achieved in the system where the polarization electric field does not arise, (2) a large reconnection rate is obtained, even in the system without the Hall effects. It is suggested that the anomalous resistivity due to the Buneman-type instability might be required to support a steady-state fast reconnection.

The basic plasma processes associated with magnetic reconnection in the Earth's magnetosphere are studied in our proposed project. In the past year, several problems associated with the 2-D and 3-D nonlinear reconnection processes were investigated. A new theory for nonlinear fast steady-state magnetic reconnection is proposed. The inflow region possesses highly curved magnetic field lines, which differs from that in the classical model of Petschek. A separatrix jet of plasma is ejected from the central diffusion region along the magnetic separatrix. Two types of outflow are studied, the simplest one possessing a potential outflow magnetic field and the other containing weak standing shock waves attached to the ends of the diffusion region. Across the separatrix jet, the outflow can be either slowed down by the fast-mode shocks or speeded up by the slow-mode shocks, depending on the downstream boundary conditions. In the 3-D MHD simulations, frayed magnetic flux ropes are found to be formed during the dayside magnetic reconnection process. Strong flux rope-aligned plasma flows are observed. Simulations of the driven reconnection process also indicate that the focusing of the Poynting flux has an important effect on the onset of reconnection and the location of reconnection sites.

Fast magnetic reconnection mechanism triggered by current sheet ejection is studied in plasma merging laboratory experiment with a strong guide field. The current sheet is hardly compressed in the presence of the guide field and slow steady reconnection evolves in the early phase. It then transits to impulsively fast magnetic reconnection when the current sheet is ejected from the X point region. Behavior of the current sheet is investigated in detail by comparison with the fast reconnection case provided by anomalous resistivity.

Dissipation processes responsible for fast magnetic reconnection in collisionless plasmas are investigated using 3D electromagnetic particle-in-cell simulations. The present study revisits the two simulation runs performed in the previous study (Fujimoto, Phys. Plasmas 16, 042103 (2009)); one with small system size in the current density direction, and the other with larger system size. In the case with small system size, the reconnection processes are almost the same as those in 2D reconnection, while in the other case a kink mode evolves along the current density and deforms the current sheet structure drastically. Although fast reconnection is achieved in both the cases, the dissipation mechanism is very different between them. In the case without kink mode, the electrons transit the electron diffusion region without thermalization, so that the magnetic dissipation is supported by the inertia resistivity alone. On the other hand, in the kinked current sheet, the electrons are not only accelerated in bulk, but they are also partly scattered and thermalized by the kink mode, which results in the anomalous resistivity in addition to the inertia resistivity. It is demonstrated that in 3D reconnection the thickness of the electron current sheet becomes larger than the local electron inertia length, consistent with the theoretical prediction in Fujimoto and Sydora (Phys. Plasmas 16, 112309 (2009)).

A new theory for fast steady-state magnetic reconnection is proposed that includes many features of recent numerical experiments. The inflow region differs from that in the classical model of Petschek (1964) and the unified linear solutions of Priest & Forbes (1986) in possessing highly curved magnetic field lines rather than ones that are almost straight. A separatrix jet of plasma is ejected from the central diffusion region along the magnetic separatrix. Two types of outflow are studied, the simplest possessing an outflow magnetic field that is potential. The other contains weak standing shock waves attached to the ends of the diffusion region and either slowing down the flow (fast-mode shock) after it crosses the separatrix jet or speeding it up (slow-mode), depending on the downstream boundary conditions. A spike of reversed current slows down the plasma that emerges rapidly from the diffusion region into the more slowly moving downstream region, and diverts most of it along the separatrix jets. In the simplest case the outflow possesses no vorticity over most of the downstream region. The models demonstrate that both upstream and downstream boundary conditions are important in determining which regime of reconnection is produced from a wide variety of possibilities.

Standard magnetohydrodynamic (MHD) theory predicts reconnection rate that is far too slow to account for a wide variety of reconnection events observed in space and laboratory plasmas. Therefore, it was commonly accepted that some non-MHD (kinetic) effects play a crucial role in fast reconnection. A recently renewed interest in simple MHD models is associated with the so-called plasmoid instability of reconnecting current sheets. Although it is now evident that this effect can significantly enhance the rate of reconnection, many details of the underlying multiple-plasmoid process still remain controversial. Here, we report results of a high-resolution computer simulation which demonstrate that fast albeit intermittent magnetic reconnection is sustained by numerous small-scale Petschek-type shocks spontaneously formed in the current sheet due to its plasmoid instability.

In two or three dimensions, magnetic reconnection may occur at neutral points and is accompanied by the transport of magnetic field lines across separatrices, the field lines (or flux surfaces in three dimensions) at which the mapping of field lines is discontinuous. Here we show that reconnection may also occur in three dimensions in the absence of neutral points at so‐called “quasi‐separatrix layers,” where there is a steep gradient in field line linkage. Reconnection is a global property, and so, in order to determine where it can occur, the first step is to enclose the volume being considered by a boundary (such as a spherical surface). Then the mapping of field lines from one part of the boundary to another is determined, and quasi‐separatrix layers may be identified as regions where the gradient of the mapping or its inverse is very much larger than normal. The most effective measure of the presence of such layers is the norm of the displacement gradient tensor; their qualitative location is robust and insensitive to the particular surface that is chosen. Reconnection itself occurs when there is a breakdown of ideal MHD and a change of connectivity of plasma elements, where the field line velocity becomes larger than the plasma velocity, so that the field lines slip through the plasma. This breakdown can occur in the quasi‐separatrix layers with an electric field component parallel to the magnetic field. In three dimensions the electric field E (and therefore the field line velocity v⊥) depends partly on the imposed values of E (or v⊥) at the boundary and partly on the gradients of the inverse mapping function. We show that the inverse mapping determines the location of the narrow layers where the breakdown of ideal MHD can occur, while the imposed boundary values of v⊥ determine mainly the detailed flow pattern inside the layers. Thus, in general, E (and therefore v⊥) becomes much larger than its boundary values at locations where the gradients of the inverse mapping function are large. An example is given of a sheared X field, where a slow smooth continuous shear flow imposed on the boundary across one quasi‐separatrix produces a flipping of magnetic field lines as they slip rapidly through the plasma in the other quasi‐separatrix layer. It results in a strong plasma jetting localized in, and parallel to, the separatrix layers.