Normal Component Of Magnetic Field Research Articles

Structure of the reconnection layer and the associated slow shocks: Two‐dimensional simulations of a Riemann problem

The kinetic structure of the reconnection layer in the magnetotail is investigated by two‐dimensional hybrid simulations. As a proxy, the solution of the Riemann problem of the collapse of a current sheet with a normal magnetic field component is considered for two cases of the plasma beta (particle to magnetic field pressure): β = 0.02 and β = 0.002. The collapse results in an expanding layer of compressed and heated plasma, which is accelerated up to the Alfvén speed vA. The boundary layer separating this hot reconnection like layer from the cold lobe plasma is characterized by a beam of back‐streaming ions with a field‐aligned bulk speed of ≃ 2vA relative to the cold lobe ion population at rest. As a consequence, obliquely propagating waves are excited via the electromagnetic ion/ion cyclotron instability, which led to perpendicular heating of the ions in the boundary layer as well as further outside the layer in the lobe. In both regions, waves are found which propagate almost parallel to the magnetic field and which are identified as Alfvén ion cyclotron (AIC) waves. These waves are excited by the temperature anisotropy instability. The temperature anisotropy increases with decreasing plasma beta. Thus the anisotropy threshold of the instability is exceeded even in the case of a rather small beta value. The AIC waves, when convected downstream of what can be defined as the the slow shock, make an important contribution to the ion thermalization process. More detailed information on the dissipation process in the slow shocks is gained by analyzing individual ion trajectories.

A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods

The exact Riemann problem solutions of the usual equations of ideal magnetohydrodynamics (MHD) can have negative pressures, if the initial data has ∇·B≢0. This creates a problem for numerical solving because in a first-order finite-volume conservative Godunov-type method one cannot avoid jumps in the normal magnetic field component even if the magnetic field was divergenceless in the three-dimensional sense. We show that by allowing magnetic monopoles in MHD equations and properly taking into account the magnetostatic contribution to the Lorentz force, an additional source term appears in Faraday's law only. Using the Harten–Lax–vanLeer (HLL) Riemann solver and discretizing the source term in a specific manner, we obtain a method which is positive and conservative. We show positivity by extensive numerical experimentation. This MHD-HLL method is positive and conservative but rather diffusive; thus we show how to hybridize this method with the Roe method to obtain a much higher accuracy while still retaining positivity. The result is a fully robust positive conservative scheme for ideal MHD, whose accuracy and efficiency properties are similar to the first order Roe method and which keeps ∇·B small in the same sense as Powell's method. As a special case, a method with similar characteristics for accuracy and robustness is obtained for the Euler equations as well.

Numerical study of asymmetric driven reconnection at dayside magnetopause

A two-dimensional compressible MHD code has been used to numerically study the asymmetric driven reconnection processes in the vicinity of the magnetopause. The initial magnetic field configuration is assumed to be in a mechanical equilibrium state. The cases with identical temperatures (Tm0/Ts0=1.0) and four different ratios of magnetic field strength (Q=Bm0/Bs0=1.0, 1.5, 2.0, 2.5), and the case withTm0/Ts0=2.0 andQ=1.5 are investigated (Bm0,Tm0 andBs0,Ts0 are the initial magnetic strength and temperature outside the current sheet on the magnetosphere and the magnetosheath, respectively). When the magnetic field on the magnetosheath side is set as southward, a recurrent formation of multiple magnetic bubbles with various scales occurs under the action of the inward plasma flow imposed at the left and right boundaries. In the simulation, some bubbles coalesce into a bigger one and then it is convected out of the simulation domain; the others are convected through the top boundary all alone. Thus, the plasmoid events with different scales and different time intervals take place intermittently and the impulsive features of magnetic reconnection are clearly shown. The multiple magnetic islands are all high-temperature and large-density regions in comparison with the ambient environment. The bipolar signatures or fluctuant variations of normal magnetic field component are generated by the formation of multiple magnetic islands. This result is similar to the FTEs signature.

Small‐amplitude bipolar flows in the near‐Earth tail

Geotail observations in the near‐Earth magnetotail at XGSM=−11RE revealed the small‐amplitude bipolar (first tailward then earthward) ion flow signature. Tailward‐flow part contained spikes of negative GSM Bz magnetic field, while earthward‐flow part included transient Bz increase and subsequent dipolarization. This and similar structures had maximal velocities less than 200 km/s, duration of about one minute and were associated with a stretched magnetotail configuration and substorm activity. We interpret them as tailward‐propagating reconnection events involving only magnetic field lines adjacent to the neutral sheet and therefore having low merging rates. Such structures might also be important because they increase the normal magnetic field component and stability in the neutral sheet.

Collisionless slow shocks in magnetotail reconnection

The kinetic structure of collisionless slow shocks in the magnetotail is studied by solving the Riemann problem of the collapse of a current sheet with a normal magnetic field component using 2‐D hybrid simulations. The collapse results in a current layer with a hot isotropic distribution and backstreaming ions in a boundary layer. The lobe plasma outside and within the boundary layer exhibits a large perpendicular to parallel temperature anisotropy. Waves in both regions propagate parallel to the magnetic field. In a second experiment a spatially limited high density beam is injected into a low beta background plasma and the subsequent wave excitation is studied. A model for slow shocks bounding the reconnection layer in the magnetotail is proposed where backstreaming ions first excite obliquely propagating waves by the electromagnetic ion/ion cyclotron instability, which lead to perpendicular heating. The T⟂/T∥ temperature anisotropy subsequently excites parallel propagating Alfvén ion cyclotron waves, which are convected into the slow shock and are refracted in the downstream region.

On the kinetic structure of the magnetotail reconnection layer

The structure of the reconnection layer in the magnetotail is studied by solving the Riemann problem of the decay of a current sheet with a normal magnetic field component using 2‐D hybrid and Hall MHD simulations. In the first run a segment of the reconnection layer obtained from a hybrid simulation of reconnection is used as initial state in a periodic system. The thin central current sheet becomes instable in the direction parallel to the current sheet: the electrical current becomes filamentary and a warped magnetic field structure develops. In a second run, where the initial state is a Harris sheet with a superposed normal magnetic field component, a similar behavior is found. In both cases the boundary of the layer is not bounded by slow mode switch‐off shocks and no large‐amplitude rotational wave train develops within the layer, in contrast to 1‐D hybrid simulations of the same Riemann problem. The corresponding 2‐D Hall MHD simulation results in a stable layer bounded by slow‐mode switch‐off shocks.

On the origin of the ion‐electron temperature difference in the plasma sheet

The results of a study of proton and electron acceleration in the Earth's magnetotail are presented. By following the trajectories of thousands of charged particles launched from mantle and tail lobe source regions, distribution functions are calculated at different locations in a model magnetotail. The magnetic field is based on the Tsyganenko [1989] model combined with a constant cross‐tail convection electric field. Despite the simplicity of the model and the lack of self‐consistent fields, a qualitative picture of the proton/electron plasma sheet emerges including an ion to electron temperature ratio Ti/Te ranging from about 4 to 6 in the magnetotail, in approximate agreement with plasma sheet observations. To explain this result, an analytic expression for Ti/Te is derived based on the particle motion of ions and electrons in a current sheet configuration with a varying normal magnetic field component. The derived expression depends on the ion to electron mass ratio to the one‐third power (mi/me)1/3 and a factor that takes the local field gradient into account. Using numerical values from the Tsyganenko [1989] field model in the derived equation gives Ti/Te ∼ 5. Another result is that the heated electron distribution functions formed in the plasma sheet are not Maxwellian but instead have power law high‐energy tails much like the so‐called “kappa” distributions reported by Christon et al. [1989]. At the edge of the plasma sheet, the calculations show the electron plasma sheet boundary layer extends further towards the lobe than the ion plasma sheet boundary layer, also in agreement with observations [Takahashi and Hones, 1988]. Nonisotropic distribution functions form at different locations in the plasma sheet, including electron beams streaming along field lines just inside the separatrix in the deep magnetotail. Electron distributions that are highly skewed in velocity space are found very near the magnetic null point. The nonisotropic distributions suggest that plasma instabilities and wave‐particle interactions could occur in those regions. That such a simple model should reproduce many of the features of the observed plasma sheet indicates that adiabatic and nonadiabatic single‐particle motion play important roles in the quiet time magnetotail and suggests that ion and electron plasma sheet formation is a natural consequence of single‐particle motion in an X line type magnetotail geometry.

A study of the effect of pitch angle and spatial diffusion on tearing instability using a new finite element based linear code

Although the collisionless tearing mode has been a leading candidate as the mechanism responsible for substorm expansion phase onset, it has not been established that the near‐Earth plasma sheet can become unstable to ion tearing owing to electron compression stabilization. The observation that the electron orbits become stochastic when the current sheet thins prior to substorm onset has given impetus to a view that pitch angle diffusion might overcome electron compression sufficiently to establish an ion mode. Theoretical attempts to demonstrate that intrinsic stochasticity in the electron orbits can destabilize the tearing mode have not been upheld without controversy. In order to examine the effect of pitch angle scattering and spatial diffusion on the tearing mode, we have developed a new linear code based on the finite element technique. This code incorporates numerical particle orbits and overcomes many of the usual difficulties associated with solving kinetic linear stability problems. We find that neither intrinsic stochasticity in the particle orbits nor externally imposed pitch angle scattering can destabilize the ion mode. The electron mode, however, can be reestablished by pitch angle diffusion, although only for values of the normal magnetic field component that are too small to be of physical significance. We also show that for the observed values of the normal magnetic field, the magnetotail is linearly stable to ion tearing even in the presence of a large spatial diffusion. The possibility of ion tearing due to nonlinear and/or three‐dimensional effects cannot, however, be ruled out at this time.

Singularities and Discontinuities in the Eigenfunction Expansions of the Dyadic Green's Functions for Biisotropic Media

Based on the theory of distributions, this paper presents a simple derivation of the singularities and discontinuities associated with the eigenfunction expansions of the dyadic Green's functions for biisotropic media. The approach deals directly with Maxwell equations cast into dyadic forms prior to explicit determination of the dyadic Green's function formula. For both electric and magnetic dyadic Green's functions, the source point singularities are obtained without any integration and the discontinuities in their tangential and normal components across the source point are determined in compact forms directly from Maxwell dyadic equations. From the discontinuity relations, boundary conditions involving the tangential and normal components of electric and magnetic fields are derived. Due to the availability of different sets of constitutive relations characterizing a biisotropic medium, the singularities and discontinuities are discussed based on the commonly used Post-Jaggard and Drude-Born-Federov relations.

Self-consistent models of incompressible stationary reconnection by weakly two dimensional extension of magnetic annihilation

Weakly two-dimensional self-consistent solutions of the stationary resistive magnetohydrodynamic equations are presented. The solutions are found by applying an asymptotic expansion procedure to the stationary resistive MHD equations with the annihilation solution as lowest order solution. Two different approximation schemes are investigated. Whereas the first scheme allows the introduction of a normal magnetic field component in the lowest order, the second scheme also introduces a weak variation of the current density along the current sheet in the lowest order. This is closely related to the fact that in the first case the inflow velocity does not vary along the current sheet, whereas in the second case the inflow velocity may vary weakly along the current sheet.

Two‐dimensional HaIl‐MHD simulation of current sheet dynamics during substorm growth phase

In this paper we study the dynamical evolution of the current sheet in the near‐Earth geomagnetic tail region during the substorm growth phase by means of HaIl‐MHD simulation. Special attention is paid to the development of cross‐tail electric current components attributed to the electrons and ions. It is found that the induced component of the electric field generated during the current sheet thinning process leads to E×B drift of both the electrons and ions toward the dawnward direction, which augments the electron diamagnetic drift but opposes the ion diamagnetic drift. This effect causes the electrons to carry a higher and higher portion of the cross‐tail current as time progresses, even if the bulk of the initial current may be carried by the ions. However, the,increase of electron current strongly depends on the boundary condition and on the location along the distant‐tail/near‐Earth axis. It is found that close to the near‐Earth simulation boundary, a sizable normal magnetic field component Bz persists even in the late phase of the simulation. This limits the north‐south electric field Ez to a relatively low value, and as a result, the cross‐tail current is largely carried by the ions there. However, toward the distant tail, Bz is much weaker, and thus the electron current is seen to increase quite rapidly.

Distributed Source Estimation Using 3-D Measurements of Magnetoencephalogram

In neuromagnetism research, it is important to accurately estimate internal electrical sources in the human brain from the spatial and temporal distributions of magnetoencephalogram (MEG) measurements on the surface of the head. In this study, we compared the performance of distributed internal source estimation in the human brain under two different MEG measurement conditions: (a) measurements of only the normal components of the external magnetic fields and (b) three-dimensional vector measurements of the external magnetic fields. We applied the sub-optimal least-squares subspace scanning source estimation technique to both measurement conditions. The results indicated that the accuracy of the distributed source estimation could be improved by using three-dimensional vector magnetic field measurements.

The Importance of Photospheric Intense Flux Tubes for Coronal Heating

Dissipation of magnetic energy in the corona requires the creation of very fine scale-lengths because of the high magnetic Reynolds number of the plasma. The formation of current sheets is a natural possible solution to this problem and it is now known that a magnetic field that is stressed by continous photospheric motions through a series of equilibria can easily form such sheets. Furthermore, in a large class of 3D magnetic fields without null points there are locations, called ‘quasi-separatrix layers’ (QSLs), where the field-line linkage changes drastically. They are the relevant generalisation of normal separatrices to configurations without nulls: along them concentrated electric currents are formed by smooth boundary motions and 3D magnetic reconnection takes place when the layers are thin enough. With a homogenous normal magnetic field component at the boundaries, the existence of thin enough QSL to dissipate magnetic energy rapidly requires that the field is formed by flux tubes that are twisted by a few turns. However, the photospheric field is not homogeneous but is fragmented into a large number of thin flux tubes. We show that such thin tubes imply the presence of a large number of very thin QSLs in the corona. The main parameter on which their presence depends is the ratio between the magnetic flux located outside the flux tubes to the flux inside. The thickness of the QSLs is approximately given by the distance between neighbouring flux tubes multiplied by the ratio of fluxes to a power between two and three (depending on the density of flux tubes). Because most of the photospheric magnetic flux is confined in thin flux tubes, very thin QSLs are present in the corona with a thickness much smaller than the flux tube size. We suggest that a turbulent resistivity is triggered in a QSL, which then rapidly evolves into a dynamic current sheet that releases energy by fast reconnection at a rate that we estimate to be sufficient to heat the corona. We conclude that the fragmentation of the photospheric magnetic field stimulates the dissipation of magnetic energy in the corona.

Collisionless reconnection in the magnetotail

Recent developments regarding collisionless reconnection in current sheets with a finite normal magnetic field component ( B z ) are reviewed. In 2-D x, z configurations the ion tearing mode is stabilized by the electron compressibility. When the y dependence is included, cross-field current instabilities can be excited. Of these, the drift kink mode appears to be particularly important. 3-D electromagnetic particle simulations indicate that this mode can act as the precursor to the growth of tearing modes and subsequent reconnection.

Comparison of the performance of distributed source estimation in the human brain using normal and vector magnetic field measurements

In neuromagnetism research, it is important to accurately estimate internal electrical sources in the human brain from the spatial and temporal distributions of magnetoencephalogram (MEG) activities over the head. In this study, we compared the performance of distributed internal source estimation in the human brain under two different MEG measurement conditions: (a) measurements of only the normal components of the external magnetic fields and (b) three-dimensional vector measurements of the external magnetic fields. We applied the sub-optimal least-squares subspace scanning source estimation technique to both measurement conditions. The results showed that with measurement of only the normal components, distributed sources tend to be estimated deeper in the brain than they should be, while three-dimensional vector measurements had the potential to provide a better estimation of the depth of the internal source distribution.

Magnetic field and plasma flow structure near the magnetopause

It has been experimentaly established that there exists a normal component of magnetic field on the magnetopause and that open field lines connect polar cap with interplanetary medium. To explain this information, it is necessary to develop a quantitative theory of an open magnetosphere and to study the structure of the magnetopause. The magnetosheath and the magnetosphere are considered as a unitary system. The dissipative MHD approach is used outside the magnetopause, and the magnetostatic approach is used inside. The frozen‐in condition is valid anywhere except at the magnetopause which is considered to be a thin dissipative boundary layer separating the magnetosheath and the magnetosphere. The magnetopause thickness is proportional to Rm−1/2 (where Rm is the magnetic Reynolds number) and is about 100 times less than stand‐off distance. The magnetic field near the magnetopause is formed due to mutual diffusion of the magnetospheric magnetic field in the magnetosheath and IMF into the magnetosphere. The magnetospheric magnetic field component normal to the magnetopause is proportional to Rm−1/2 and is about 1 nT at the dayside magnetopause. The related potential difference accross the open field lines is about 15 kV, which can be responsible for the background electric field measured in the polar caps. The portion of IMF which penetrates into the magnetosphere is proportional to Rm−1/4 and is about a factor of 10 less than the IMF. The potential difference accross the polar cap is about 100 kV for the southward IMF of −10 nT.

Properties of high-Tc superconducting circular waveguides with Meissner boundary

The propagation properties of the TE-modes in a high-temperature superconducting circular waveguide using the Mei\ner boundary conditions on the wall are presented for the first time. The results show that now the normalized cutoff parameterk c R, (whereR is the radius of the superconducting circular waveguide andk c the cutoff wavenumber,) is dependent on the radius unlike conventional metallic circular waveguide whose normalized cutoff parameterk c R is a constant for a given mode and the filled dielectrics. Instead of TE11-mode now TE01 mode becomes the dominant mode and the normal component of magnetic field for the dominant mode is not equal to zero on the wall. Other unique results of high-T c superconducting circular waveguides are illustrated, too.

Effect of electron dynamics on collisionless reconnection in two‐dimensional magnetotail equilibria

Analytic theory has yielded contradictory predictions regarding the stabilizing role of electron dynamics on the collisionless tearing instability in a field‐reversal region with finite normal magnetic field component Bz. Two‐dimensional (x, z) simulations in which both electrons and ions are treated as particles demonstrate that, for the case of thin current sheets (Ri/ λ ∼ 1, where Ri is the ion Larmor radius based on the lobe field and λ is the current sheet half thickness) and with weakly chaotic electron orbits, the ion tearing mode is indeed stabilized under the mild condition that kρe ≲ 1, where ρe is the electron Larmor radius in the Bz field.

Structure of the tail plasma/current sheet at ∼11 RE and its changes in the course of a substorm

At the end of April 2, 1978, the ISEE 1 and 2 spacecraft moved inbound at ∼11 RE on the nightside (0130 MLT). Due to a flapping motion of the plasma sheet the spacecraft crossed the neutral sheet region (central region of the plasma sheet) more than 10 times in the hour between 2115 and 2215 UT. This provided a unique opportunity to study the structure of the plasma/current region and its evolution during substorm growth and early expansion before the final disruption of the current sheet. Using minimum variance analysis of the magnetic field variations during the crossings as well as finite ion gyroradius diagnostics, we determine the orientation of the current sheet (CS) and then estimate the CS thickness as well as the value of its normal component, Bn. Typically, the current distribution was inferred to be very inhomogeneous with a current concentrated in a very thin CS (only 0.2 to 0.8 RE as thick) embedded inside the thicker plasma sheet. Current sheet crossings could be classified as regular or turbulent. The first type prevailed during the growth phase and at the initial stage of expansion when the spacecraft were well outside (in longitude) of the active region of the substorm and no large plasma flow was detected. The normal field component Bn was typically very small (∼1 nT) in the CS center in comparison to the larger shear magnetic By component. In the course of the growth phase we inferred an increase of the lobe field Bx and a decrease of the CS half thickness h (from h∼3000 km to ∼800 km just before the expansion onset), i.e., a very large increase (up to an order of magnitude) of the current density. At the same time, in disagreement with the usual cartoon picture of magnetic reconfiguration, the magnetic field magnitude in the CS center increased (instead of decreased) at the expense of the shear component. Three turbulent crossings were found during substorm expansion within the longitude range of the substorm current wedge (SCW). The second of them was detected ∼1 min before the main dipolarization and was characterized by a rather small CS thickness (h < 600 km), by strong earthward plasma flow and by a positive normal magnetic field component. That period showed signatures of concentration of both cross‐B and field‐aligned current at the outer edge of CS and may indicate a nearby reconnection region. The main result of this study is that the region of very thin current sheet (thickness of the order of the gyroradius of thermal protons in the field just outside the current sheet), which contained a very small normal component, clearly appeared in the near tail prior to the sudden onset of current disruption as predicted by some quantitative models of quasi‐static evolution of earthward convecting plasma sheet flux tubes. Comparing these observations to theoretical results, we find that the threshold conditions for the growth of the tearing mode instability in sheared magnetic fields were apparently satisfied in this case, but the growth rate was too slow for sudden initiation of substorm expansion.

Observational support for the current sheet catastrophe model of substorm current disruption

It has recently been found that a one‐dimensional current sheet equilibrium with a non‐zero confection electric field, Ey, and a non‐zero normal magnetic field component, Bn, can reach a point of catastrophe through either the reduction of the drift velocity, υD = cEy/Bn or the increase of Bn. This point of catastrophe coincides with a value of κA ≃ 0.7, where κA is the self‐consistent value of κ = (Rmin/ρA max)½ corresponding to ions of average energy. (Here Rmin is the minimum field‐line radius of curvature and ρA max is the maximum gyroradius for ions of average energy.) The point of catastrophe was found to be preceded by a twisting of the current sheet field‐lines into the dawnward direction, i.e. by the development of a y‐component of B, with odd symmetry in z, and a sign opposite Bx, where positive x is earthward, positive y is in the dawn‐to‐dusk direction and positive z is northward. Since the loss of the current sheet would cause the local configuration to become more dipolar, it was suggested that the catastrophic loss of the local current sheet equilibrium could correspond to local current disruption and dipolarization. In this paper, observations of some of the signatures predicted by theory are presented.