Resistive MHD Model Research Articles

THE DETECTION OF NUMEROUS MAGNETIC SEPARATORS IN A THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC MODEL OF SOLAR EMERGING FLUX

Magnetic separators in three-dimensional (3D) magnetic fields are believed to be often associated with locations of magnetic reconnection. In this preliminary study, we investigate this relationship using data from a numerical resistive 3D MHD experiment of a solar flux emergence event. For the first time separators are detected in complex magnetic fields resulting from a 3D resistive MHD model of flux emergence. Two snapshots of the model, taken from different stages of its evolution, are analyzed. Numerous separators are found in both snapshots, and their properties, including their geometry, length, relationship to the magnetic null points, and integrated parallel electric field are studied. The separators reside at the junctions between the emerging flux, the overlying field, and two other flux domains that are newly formed by reconnection. The long separators, which connect clusters of nulls that lie either side of the emerging flux, pass through spatially localized regions of high parallel electric field and correspond to local maxima in integrated parallel electric field. These factors indicate that strong magnetic reconnection takes place along many of the separators, and that separators play a key role during the interaction of emerging and overlying flux.

Second Core Formation and High-Speed Jets: Resistive Magnetohydrodynamic Nested Grid Simulations

The stellar core formation and high-speed jets driven by the formed core are studied using three-dimensional resistive MHD nested grid simulations. Starting with a Bonnor-Ebert isothermal cloud rotating in a uniform magnetic field, we calculate the cloud evolution from the molecular cloud core (nc = 106 cm-3) to the stellar core (nc 1023 cm-3), where nc denotes the central density. For comparison, we calculate two models: resistive and ideal MHD models. Both the resistive and ideal models have the same initial condition, but the former includes the dissipation process of magnetic field while the latter does not. The magnetic fluxes in the resistive MHD model are extracted from the first core during 1012 cm-3 < nc < 1016 cm-3 by ohmic dissipation. Magnetic flux density of the formed stellar core (nc 1020 cm-3) in the resistive MHD model is 2 orders of magnitude smaller than that in ideal MHD model. Since magnetic braking is less effective in the resistive MHD model, a rapidly rotating stellar core (the second core) is formed. After stellar core formation, the magnetic field of the core is largely amplified, and high-speed (45 km s-1) jets are driven by the second core, which results in strong mass ejection. A cocoon-like structure around the second core also forms with clear bow shocks.

Coalescence of Magnetic Islands in the Low-Resistivity, Hall-MHD Regime

The coalescence of magnetic islands in the low-resistivity eta, Hall-MHD regime is studied. The interaction between the ion inertial length d(i) and the dynamically evolving current sheet scale length deltaJ is established. Initially, d(i) << deltaJ. If eta is such that deltaJ dynamically thins down to d(i) prior to the well-known sloshing phenomena, then sloshing is avoided. This results in eta independent peak reconnection rates. However, if d(i) is small enough that deltaJ cannot be thinned down to this scale prior to sloshing, then sloshing proceeds as in the resistive MHD model.

Quasi-steady state, low current behaviour of a magnetized coaxial plasma source

The Coaxial Plasma Source-1 facility (Mayo R M et al 1995 Plasma Sources Sci. Technol. 4 47) was modified from a short pulse, high current (SPHC) pulse forming network (PFN) with very low inductance (∼200 nH) to a large inductance ladder circuit. This modification allows for a longer, flat top gun current pulse that eliminates the under-damped, sinusoidal behaviour of the gun current with consequent interruptions in plasma parameters. The new PFN was designed to produce a current waveform for a much longer period (∼1 ms). As a consequence of increasing the pulse length, the magnitude of the gun current was reduced as no additional energy storage was added to the PFN. The characterization of the electrical and plasma behaviour of the experiment operated with the long pulse, low current (LPLC) PFN is presented. The gun currents produced by the LPLC PFN are approximately one-fifth in magnitude of the gun currents produced by the SPHC PFN. Axial plasma parameters were measured near the muzzle of the plasma source, and electron densities were found to range from 1 × 1019 m−3 to 7 × 1019 m−3 depending upon the axial location. These values are approximately 1–2 orders of magnitude less than the electron densities produced by the SPHC PFN at the same locations. Electron temperatures range from 30 to 60 eV at these locations and are very similar to those produced by the SPHC PFN. A resistive MHD model was applied as an order estimate of the plasma resistivity and demonstrates reasonable agreement with measured values of the magnetized coaxial gun resistance.

Effects of Hall electric fields on the saturation of forced antiparallel magnetic field merging

The role of Hall electric fields in flux pile-up antiparallel magnetic field merging is addressed. Analytical solutions of the resistive Hall magnetohydrodynamics (Hall-MHD) equations are obtained, describing stagnation point flows in a thin current sheet. The stagnation point flow solutions explain a number of interesting effects observed in two-dimensional resistive Hall-MHD simulations of forced magnetic reconnection. In particular, when Hall electric fields are important within the current sheet, less pile-up of magnetic energy is required upstream of the current sheet to support a given reconnection electric field. Fast electron flows transport magnetic flux into the diffusion layer without requiring a compensating drop in plasma pressure upstream of the current sheet. The maximum flux pile-up merging rate allowed by the external plasma pressure becomes independent of the Lundquist number, scaling like the square root of the ratio of the ion inertial length to the spatial scale of the stagnation point flow. Thus, Hall electric fields provide a possible solution to the problem of flux pile-up saturation in two-dimensional, resistive MHD models of forced magnetic reconnection.

Evolution of Global‐Scale Coronal Magnetic Field due to Magnetic Reconnection: The Formation of the Observed Blob Motion in the Coronal Streamer Belt

Recent SOHO/LASCO/EIT observations showed that the global corona at the minimum phase of cycle 22 is an organized, simple, and persistent dipolar configuration streamer belt (Michels). But the low-lying, closed-loop, multipolar magnetic structure of the inner corona at low- to mid-latitudes revealed in EIT and LASCO images is not static. It stretches continuously outward, feeding plasma into higher coronal structures and eventually into the solar wind, including plasma blobs (Sheeley et al.; Wang et al.). To understand the physics of this slow dynamic evolution process, we use a two-dimensional, planar, resistive MHD model and observed pre-event characteristics for the model input to simulate the formation and propagation of these observed plasma blobs. It seems that reconnection processes among the low-lying multipolar coronal loops and their overlying streamers cause the slow time evolution. The motion of plasma blobs frequently observed by LASCO C2/C3 is indeed reproduced by this model through reconnection in the multipolar loops. Results presented in this paper are the global field evolution, mass, momentum, and energy transport from the inner corona to the outer corona and up into the streamer belt region where slow solar wind forms. We conclude that these plasma blobs may be the origin of the lumps in the solar wind observed in interplanetary space.

Comment on “Modeling the magnetosphere for northward interplanetary magnetic field: Effects of electrical resistivity” by Joachim Raeder

In a recent article Raeder [1999] presented a series of MHD simulations using a resistive MHD model to describe the interaction of the terrestrial magnetosphere with the solar wind for due northward interplanetary magnetic field (IMF) conditions. Depending on the initial conditions, the simulated magnetotail remained open for at least 6 hours after the IMF turned from south to north, or remained closed at all times when the simulation started with northward IMF. When an increasing amount of uniform resistivity was added, the magnetotail became closed, and its length decreased with increasing resistivity. At the same time, increasing uniform resistivity diffused the bow shock heavily. Raeder [1999, p. 17,357] speculated that models that predict the rapid closure of the magnetosphere and the formation of a steady, finite-length tail are “possibly in error due to numerical resistivity.” The variation of explicit resistivity of Raeder [1999], intended to mimic the artificial resistivity present in MHD codes, is an interesting exercise. However, we disagree with Raeder’s statements about the numerical resistivity being the cause of closed magnetotails for northward IMF. In fact, as it will be described in this comment, there are several reasons to believe that numerical resistivity does not play a role in the formation of closed magnetotails. In this Comment we point out the following: 1. A number of investigators, using a wide variety of MHD codes, have reported closed configurations for the northward IMF condition. Raeder’s [this issue] implication that all these codes are only as accurate as the first-order Rusanov solution, provided by him for comparison, is contradicted both by the marked similarities of some codes (particularly that of Raeder [1999] and that of Fedder and Lyon [1995]) and by published convergence studies [e.g. Powell et al., 1999]. 2. Closed configurations have been obtained with MHD codes with similar underlying schemes as the approach employed by Raeder [1999] and on meshes more highly resolved than those employed by him [cf. Song et al., 1999]. 3. In the simulations presented by Raeder [1999], added uniform resistivity simultaneously resulted in closed magnetotails and substantial diffusion of the bow shock (the upstream plasma pressure in the equatorial plane significantly exceeded the thermal pressure of the ambient solar wind). By interpreting the difference between his results and those of other investigators in terms of numerical resistivity, Raeder [1999] reaches unjustified and incorrect conclusions. In particular, in his simulations, short, closed tails come together with the disappearance of well-defined bow shocks. Since other simulations obtaining closed magnetotails have sharp and well-resolved bow shocks, Raeder’s [1999] simulations imply that these results cannot be due to uniform numerical resistivity.

Impulsive reconnection dynamics in collisionless laboratory and space plasmas

There are some phenomena involving magnetic reconnection in collisionless laboratory and space plasmas that are characterized not only by fast growth but also by an impulsiveness, that is, a sudden increase in the time derivative of the growth rate. Three examples are the sawtooth collapse in tokamaks, the sudden enhancement of the cross‐tail current density just prior to substorm onset in the Earth's magnetotail, and the impulsive phase of a solar flare. Not only is the peak growth rate of these phenomena large (near‐Alfvénic), they are triggered suddenly, with the growth rate increasing from small to large values very rapidly. Traditional steady state reconnection models, such as those due to Sweet‐Parker or Petschek, do not include time dependency, and hence cannot account for the impulsiveness. It is demonstrated that in collisionless plasmas, the effects of electron pressure gradients and Hall currents (referred to collectively as Hall MHD effects) can provide a plausible explanation not only of the fast growth but also of the impulsiveness. The geometrical structure of thin current sheets realized during the dynamical evolution of such collisionless systems changes spontaneously from a Y point in the early nonlinear regime (characteristic of the models of Sweet‐Parker and Syrovatsky) to an X point (characteristic of the Petschek model). However, the underlying physics of reconnection is qualitatively different from that in resistive MHD models. In Hall MHD, the electron and ion dynamics are decoupled from each other in the reconnection layer. Such decoupling is forbidden in resistive MHD. In Hall MHD, while the spatial scale of the parallel electric field (that is, the component of the electric field parallel to the background magnetic field) is the ion skin depth, the spatial scale of the parallel current density is narrower, determined by the Lundquist number (or the electron skin depth), if resistivity (or electron inertia) is the dominant mechanism that breaks field lines. Analytical results as well as high‐resolution numerical simulations are presented, and the predictions of the model are compared with data from laboratory experiments as well as space.

Structure of the dayside reconnection layer in resistive MHD and hybrid models

The structure of reconnection layer at the Earth's dayside magnetopause is studied based on one‐dimensional simulations for the evolution of an initial current sheet. A resistive MHD code and a hybrid code are used in simulations. Initially, the magnetopause current sheet separates two uniform plasma regions: magnetosheath (high plasma density and low magnetic field) and magnetosphere (low plasma density and high magnetic field). In MHD simulations, it is found that steady intermediate shock, time‐dependent intermediate shock (TDIS), Alfvén wave pulse, slow shock, slow expansion wave, or contact discontinuity can be formed as a result of magnetic reconnection. The current density in the magnetopause‐boundary layer region is mainly concentrated at the intermediate shock on the magnetosheath side. High‐speed flow is observed in the boundary layer. The Alfvén wave pulse, which propagates to the magnetosphere, is found to have a large amplitude in plasma flow speed. In hybrid simulations, the reconnection layer has striking differences from that in MHD simulations: (1) Because of the mixing of hot magnetospheric plasma and cold magnetosheath plasma along magnetic field lines, the contact discontinuity does not appear in the reconnection layer. (2) In general cases, the TDIS evolves quickly to a steady rotational discontinuity with a constant width. In the singular case in which the magnetic fields in the magnetosheath and in the magnetosphere are exactly antiparallel, a weak steady intermediate shock is formed. (3) No clear shock front can be identified for the slow shock, and the slow expansion wave is strongly modified as compared with that in the MHD simulations. (4) The boundary layer is a transition region between the magnetosheath and the magnetosphere with a mixing of plasmas from the two sides; the plasma density decreases monotonically from the magnetosheath side to the magnetosphere side, while the total temperature and magnetic field strength increase monotonically toward the magnetosphere side. (5) A decrease in T⊥/T∥ is observed in the boundary layer. Our results indicate that in general cases rotational discontinuities bound the reconnection layer at the dayside magnetopause.

Effect of resonant magnetic perturbations on COMPASS-C tokamak discharges

Experimental results from the COMPASS-C tokamak reveal a sharp threshold in amplitude above which externally applied static resonant magnetic perturbations (RMPs) induce stationary magnetic islands. Such islands (in particular, m=2, n=1 islands) give rise to a significant degradation in energy and particle confinement, suppression of the sawtooth oscillation and a large change in the impurity ion toroidal velocity. The observed threshold for inducing stationary (2,1) islands is consistent with a phenomenological resistive MHD model which takes into account plasma rotation (including poloidal flow damping) and externally applied resonant fields. Broadly similar results are found for applied fields other than m=2, n=1. Other results from RMP experiments are also discussed, such as the stabilization of rotating MHD activity, stimulated disruptions and extensions to the disruptive density limit. Finally, the likely effect of field errors on large tokamaks is briefly examined in the light of the COMPASS-C results

The structure of resistive‐dispersive intermediate shocks

The structure of intermediate shocks is studied on the basis of the resistive, nonviscous two‐fluid equations. Electron inertia effects are neglected so that the generalized Ohm's law contains only the Hall current and the electron pressure terms in addition to the usual resistive term and the electric field. As for the case of purely resistive MHD, reported recently by Hau and Sonnerup (Journal of Geophysical Research, 94, 6539, 1989), fixed‐point analysis is performed to examine the nature of the magnetic structure near the upstream and downstream states of the intermediate shock. The one‐dimensional, steady state, resistive Hall MHD equations are then integrated numerically to generate complete shock structures which are presented in the form of magnetic hodograms. These hodograms describe fast and slow shocks in addition to intermediate shocks. As expected, the calculations show that the main effect of Hall currents is to remove the symmetry between left‐hand and right‐hand polarized shock structures found in the purely resistive case and sometimes to convert the smooth shock transitions obtained from the resistive MHD model into transitions that incorporate oscillatory standing wave train structures at their upstream and/or downstream edge. The magnetic structure in the plane of the shock near the possible upstream and downstream states of the intermediate shock, which in the case of purely resistive MHD is either a node or a saddle, can be either a node, a saddle or a spiral point, the latter corresponding to a standing wave train, when the Hall term is included. As a result, the number of possible types of magnetic hodogram topology increases from 3 in the resistive case, to a total of 20. However, it appears that the constraints provided by the shock jump conditions make certain of these topologies unattainable: only 13 of the 20 cases have been found and are reported in the paper. The relationship between small‐amplitude dispersive waves in the flow upstream or downstream of a shock and the nature of the corresponding fixed point is also discussed.

Simple resistive MHD models for instability revisited

Two MHD models for instability, viz., the Shafranov's [1] resistive free-boundary and resistive fixed-boundary step models have been revisited. In the former, a constant longitudinal current density passes through a cylindrical column of a resistive plasma of radius a separated by vacuum from a perfectly conducting wall at a radius b; in the latter, the current density has the same radial profile but the plasma fills the whole cylinder up to the conducting wall.For the former, the explicit dependence of resistivity η on the maximum normalized growth rate Γmax is exhibited showing a stabilizing effect on kinks. A fact that is overlooked by some authors is also pointed out here in the form of a correction of the original growth rate γmax. For the latter, a special code set up to solve the (stiff) system of differential equations for instability in the region a ⩽ r ⩽ b, together with a set of jump conditions at the current channel radius a, yield no solution satisfying the appropriate boundary conditions at the wall.

Plasmoid‐associated energetic ion bursts in the deep magnetotail: Numerical modeling of the boundary layer

Single particle orbits are numerically followed in a time dependent two‐dimensional resistive MHD model of tail reconnection. In this model, near‐Earth reconnection disconnects the plasma sheet and the tailward departing closed field line region is surrounded by a thickening layer of disconnected lobe field lines. Orbits starting in this layer are traced backward in time, and their interaction with the current sheet close to the X line is studied. The model predicts the spatial distribution of energetic protons in the boundary layer surrounding plasmoids. It is argued that lower energy protons are injected after their acceleration onto lobe field lines closer to the neutral line than higher energy ions. Thus the low‐energy proton boundary layer will be found closer to the separatrix than the higher energy proton layer. This is confirmed by the numerical calculations. Further downtail from the neutral line the energy dispersion due to the dawn‐dusk electric field becomes larger and leads to the usual inward dispersion. Applying Liouville's theorem, we calculate the distribution function of accelerated ions in the boundary layer. Comparison with measurements obtained in the deep tail indicates that proton intensities observed in the layer around plasmoids at energies above ∼30 keV can be obtained by the neutral line acceleration of a suitable lobe ion distribution.

Relaxation Oscillation in an Ohmic-Heated Heliotron E Plasma with a Zero Rotational Transform Surface

Nonlinear evolution of tearing instability with a poloidal mode number m =1 and a toroidal mode number n =0 is studied in a straight helical system with an \(\imath{=}0\) surface, where \(\imath\) denotes a rotational transform. On the basis of a low beta resistive MHD model derived by using the stellarator expansion, it is shown that the m =1/ n =0 magnetic island grows to cover the whole central region; then, plasma current is readjusted and it almost recovers the initial profile. The large magnetic island causes outflow of ions, since the level of poloidal field fluctuations reaches about 10%. This whole process seems to repeat and may explain relaxation oscillations of ion temperature detected by neutral particle analyzer in the Heliotron E experiment.