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We have constructed a new class of exact, nonlinear solutions to the Grad-Shafranov equation, representing force-free magnetic fields with translational symmetry. These exact solutions are pertinent to the study of magnetic structures in the solar corona that are subjected to photospheric shearing motions.
It is shown that the solution, with one ignorable coordinate, for the Taylor minimum energy state (resulting in a force-free magnetic field) in either a straight cylindrical or a toroidal geometry with arbitrary cross section can be reduced to the solution of either an inhomogeneous Helmholtz equation or a Grad-Shafranov equation with simple boundary conditions. Standard Green's function theory is, therefore, applicable. Detailed solutions are presented for the Taylor state in toroidal and cylindrical domains having a rectangular cross section. The focus is on solutions corresponding to the continuous eigenvalue spectra. Singular behavior at 90 deg corners is explored in detail.
A general adaptive grid algorithm is developed on triangular grids. The adaptivity is provided by a combination of node addition, dynamic node connectivity and a simple node movement strategy. While the local restructuring process and the node addition mechanism take place in the physical plane, the nodes are displaced on a monitor surface, constructed from the salient features of the physical problem. An approximation to mean curvature detects changes in the direction of the monitor surface, and provides the pulling force on the nodes. Solutions to the axisymmetric Grad-Shafranov equation demonstrate the capturing, by triangles, of the plasma-vacuum interface in a free-boundary equilibrium configuration.
This review paper demonstrates how the magnetohydrostatic equilibrium (MHE) theory can be used to describe the large-scale magnetic field configuration of the magnetosphere and its time evolution under the influence of magnetospheric convection. The equilibrium problem is reviewed, and levels of B-field modelling are examined for vacuum models, quasi-static equilibrium models, and MHD models. Results from two-dimensional MHE theory as they apply to the Grad-Shafranov equation, linear equilibria, the asymptotic theory, magnetospheric convection and the substorm mechanism, and plasma anisotropies are addressed. Results from three-dimensional MHE theory are considered as they apply to an intermediate analytical magnetospheric model, magnetotail configurations, and magnetopause boundary conditions and the influence of the IMF.
A model is presented of an axially symmetric pole-on magnetosphere in MHD force balance, in which both plasma thermal pressure gradients and centrifugal force are taken into account. Assuming that planetary rotation leads to differentially rotating magnetotail field lines, the deformation of magnetotail field lines under the influence of both thermal plasma pressure and centrifugal forces was calculated. Analytic solutions to the Grad-Shafranov equation are presented, which include the centrifugal force term. It is shown that the nonrotational magnetosphere with hot thermal plasma leads to a field configuration without a toroidal B(phi) component and without field-aligned Birkeland currents. The other extreme, a rapidly rotating magnetosphere with cold plasma, leads to a configuration in which plasma must be confined within a thin disk in a plane where the radial magnetic field component B(r) vanishes locally.
Two dimensional and three dimensional MHD equilibrium models were begun for Earth's magnetosphere. The original proposal was motivated by realizing that global, purely data based models of Earth's magnetosphere are inadequate for studying the underlying plasma physical principles according to which the magnetosphere evolves on the quasi-static convection time scale. Complex numerical grid generation schemes were established for a 3-D Poisson solver, and a robust Grad-Shafranov solver was coded for high beta MHD equilibria. Thus, the effects were calculated of both the magnetopause geometry and boundary conditions on the magnetotail current distribution.
Previous work on the origin of self-collimated electromagnetic jets is extended to the general case where there is no reflection symmetry of the magnetic field about the equatorial plane of the disk. The axisymmetric field structure inside the disk is obtained by solving for the magnetic flux function, and the toroidal magnetic field, from the generalized thin-disk induction equation, which is derived. The asymptotic (large-z) magnetic field structure outside the disk is obtained by solving the force-free Grad-Shafranov equation semianalytically. Jet solutions are found in which the power flow is carried mainly by the Poynting flux of the electromagnetic field and the angular momentum outflow from the disk is carried by the magnetic field. The ratio of jet luminosities (top/bottom) depends directly on the degree of asymmetry of the field and can easily be much greater than unity. It is argued that the degree of field asymmetry in the disk is determined by the asymmetry of the weak galactic field fed into the disk at large distances over long periods of time.
We explore the generalized Grad-Shafranov (GS) equation (Lovelace et al. 1986), which describes the cross-field balance of the magnetic flux surfaces in a stationary, axisymmetric, cold relativistic MHD wind. We construct a family of self-consistent, jet-type, non-self-similar solutions in which flux surfaces thread the equatorial plane vertically and are eventually collimated into nested cylinders. These solutions conserve the total energy and angular momentum along each flux surface and satisfy the relativistic GS equation in the cross-field direction. We find that the total specific energy in the flow depends primarily on a dimensionless combination of the poloidal magnetic flux, the rotation, and the mass loading rate, which reduces to Michel's (1969) magnetization parameter for radial winds. The final width of the flow and its ratio of kinetic energy flux to Poynting flux depend sensitively on the pressure distribution of the ambient medium, which is required to contain the jet at its outer edge when no jet-confirming external toroidal magnetic field is present. Furthermore, we show that the magnetic pressure in the flow can be much higher near the axis than close to the edge, reflecting the 'pinching' effect of the toroidal magnetic field.
We formulate the dynamical problem of a cool wind centrifugally driven from the magnetic interface of a young star and an adjoining Keplerian disk. We examine the situation for mildly accreting T Tauri stars that rotate slowly as well as rapidly accreting protostars that rotate near break-up. In both cases a wind can be driven from a small X-region just outside the stellar magnetopause, where the field lines assume an open geometry and are rooted to material that rotates at an angular speed equal both to the local Keplerian value and to the stellar angular speed. Assuming axial symmetry for the ideal magnetohydrodynamic flow, which requires us to postpone asking how the (lightly ionized) gas is loaded onto field lines, we can formally integrate all the governing equations analytically except for a partial equation that describes how streamlines spread in the meridional plane. Apart from the difficulty of dealing with PDEs of mixed type, finding the functional forms of the conserved quantities along streamlines - the ratio beta of magnetic field to mass flux, the specific energy H of the fluid in the rotating frame, and the total specific angular momentum J carried in the matter and the field - constitutes a standard difficulty in this kind of (Grad-Shafranov) formalism. Fortunately, because the ratio of the thermal speed of the mass-loss regions to the Keplerian speed of rotation of the interface constitutes a small parameter epsilon, we can attack the overall problem by the method of matched asymptotic expansions. This procedure leads to a natural and systematic technique for obtaining the relevant functional dependences of beta, H, and J. Moreover, we are able to solve analytically for the properties of the flow emergent from the small transsonic region driven by gas pressure without having to specify the detailed form of any of the conserved functions, beta, H, and J. This analytical solution provides inner boundary conditions for the numerical computation in a companion paper by Najita & Shu of the larger region where the main acceleration to terminal speeds occurs.
We determine the photospheric boundary conditions which maximize the magnetic energy released by a loss of ideal-magnetohydrodynamic (MHD) equilibrium in two-dimensional flux-rope models. In these models a loss of equilibrium causes a transition of the flux rope to a lower magnetic energy state at a higher altitude. During the transition a vertical current sheet forms below the flux rope, and reconnection in this current sheet releases additional energy. Here we compute how much energy is released by the loss of equilibrium relative to the total energy release. When the flux-rope radius is small compared to its height, it is possible to obtain general solutions of the Grad-Shafranov equation for a wide range of boundary conditions. Variational principles can then be used to find the particular boundary condition which maximizes the magnetic energy released for a given class of conditions. We apply this procedure to a class of models known as cusp-type catastrophes, and we find that the maximum energy released by the loss of equilibrium is 20.8% of the total energy release for any model in this class. If the additional restriction is imposed that the photospheric magnetic field forms a simple arcade in the absence of coronal currents, then the maximum energy release reduces to 8.6%
We study the conditions which lead to the conversion of most Poynting flux into kinetic energy flux in cold, relativistic hydromagnetic winds. It is shown that plasma acceleration along a precisely radial flow is extremely inefficient due to the near cancellation of the toroidal magnetic pressure and tension forces. However, if the flux tubes in a flow diverge even slightly faster than radially, the fast magnetosonic point moves inward from infinity to a few times the light cylinder radius. Once the flow becomes supermagnetosonic, further divergence of the flux tubes beyond the fast point can accelerate the flow via the 'magnetic nozzle' effect, thereby further converting Poynting flux to kinetic energy flux. We show that the Grad-Shafranov equation admits a generic family of kinetic energy-dominated asymptotic wind solutions with finite total magnetic flux. The Poynting flux in these solutions vanishes logarithmically with distance. The way in which the flux surfaces are nested within the flow depends only on the ratio of angular velocity to poliodal 4-velocity as a function of magnetic flux. Radial variations in flow structure can be expressed in terms of a pressure boundary condition on the outermost flux surface, provided that no external toriodal field surrounds the flow. For a special case, we show explicitly how the flux surfaces merge gradually to their asymptotes. For flows confined by an external medium of pressure decreasing to zero at infinity we show that, depending on how fast the ambient pressure declines, the final flow state could be either a collimated jet or a wind that fills the entire space. We discuss the astrophysical implications of our results for jets from active galactic nuclei and for free pulsar winds such as that believed to power the Crab Nebula.
We have carried out a theoretical analysis of the stability and parallel structure of coupled shear Alfven and slow magnetosonic waves in Earth's inner magnetopause (i.e., at equatorial distances between about five and ten Earth radii) including effects of finite anisotropic Grad-Shafranov equation yields an approximate self-consistent magnetohydrodynamic (MHD) equilibrium. This MHD equilibrium is used in the numerical solution of a set of eigenmode equations which describe the field line eigenfrequency, linear stability, and parallel eigenmode structure. We call these modes anisotropic Alfven-ballooning modes. The main results are: (1) The field line eigenfrequency can be significantly lowered by finite pressure effects. (2) The parallel mode structure of the transverse wave components is fairly insensitive to changes in the plasma pressure, but the compressional magnetic component can become highly peaked near the magnetic equator as a result of increased pressure, especially when P(sub perpendicular to) is greater than P(sub parallel) (here P(sub perpendicular to) and P(sub parallel) are the perpendicular and parallel plasma pressure). (3) For the isotropic (P(sub parallel) = P(sub perpendicular to) = P) case ballooning instability can occur when the ratio of the plasma presure to the magnetic pressure, beta = P/(B squared/8 pi), exceeds a critical value beta(sup B)(sub 0) is approximately equal to 3.5 at the equator. (4) Compared to the isotropic case the critical beta value is lowered by anisotropy, either due to decreased field line bending stabilization when P(sub parallel) is greater than P(sub perpendicular to) or due to increased ballooning-mirror destabilization when P(sub perpendicular to) is greater than P(sub parallel). (5) We use a beta-delta stability diagram to display the regions of instability with respect to the equatorial values of the parameters bar beta and delta, where bar beta = (1/3)(beta(sub parallel) + 2 beta(sub perpendicular to)) is an average beta value and delta = 1 - P(sub parallel)/P(sub perpendicular to) is a measure of the plasma anisotropy. The diagram is divided into regions corresponding to the firehose, mirror and ballooning instabilities. It appears that observed values of the plasma pressure are below the critical value for the isotropic ballooning instability but it may be possible to approach a ballooning-mirror instability when P(sub perpendicular to)/P(sub parallel) is greater than or approximately 2.
We present initial results of strong field amplitude flux ropes observed by Mars Atmosphere and Volatile EvolutioN (MAVEN) mission around Mars during the interplanetary coronal mass ejection (ICME) passage on 8 March 2015. The observed durations were shorter than 5 s and the magnetic field magnitudes peaked above 80 nT, which is a few times stronger than those usually seen in the magnetosheath barrier. These are the first unique observations that MAVEN detected such flux ropes with a strong field at high altitudes (greater than 5000 km). Across these structures, MAVEN coincidentally measured planetary heavy ions with energies higher than a few keV. The spatial properties inferred from the Grad-Shafranov equation suggest that the speed of the structure can be estimated at least an order of magnitude faster than those previously reported quiet-time counterparts. Hence, the space weather event like the ICME passage can be responsible for generating the observed strong field, fast-traveling flux ropes.
We investigate the physics of the X-ray ionized absorbers often identified as warm absorbers (WAs) and ultra-fast outflows (UFOs) in Seyfert AGNs from spectroscopic studies in the context of magnetically-driven accretion-disk wind scenario. Launched and accelerated by the action of a global magnetic field anchored to an underlying accretion disk around a black hole, outflowing plasma is irradiated and ionized by an AGN radiation field characterized by its spectral energy density (SED). By numerically solving the Grad-Shafranov equation in the magnetohydrodynamic (MHD) framework, the physical property of the magnetized disk-wind is determined by a wind parameter set, which is then incorporated into radiative transfer calculations with xstar photoionization code under heating-cooling equilibrium state to compute the absorber's properties such as column density N(sub H), line-of-sight (LoS) velocity v, ionization parameter xi, among others. Assuming that the wind density scales as n varies as r(exp. -1), we calculate theoretical absorption measure distribution (AMD) for various ions seen in AGNs as well as line spectra especially for the Fe K alpha absorption feature by focusing on a bright quasar PG 1211+143 as a case study and show the model's plausibility. In this note we demonstrate that the proposed MHD-driven disk-wind scenario is not only consistent with the observed X-ray data, but also help better constrain the underlying nature of the AGN environment in a close proximity to a central engine.
We discuss Wind observations of a long and slow magnetic cloud (MC) propagating through large-amplitude Alfvén waves (LAAWs). The MC axis has a strong component along GSE X, as also confirmed by a Grad-Shafranov reconstruction. It is overtaking the solar wind at a speed roughly equal to the upstream Alfvén speed, leading to a weak shock wave âLij17 hrs. ahead. We give evidence to show that the nominal sheath region is populated by LAAWs: (i) a well-defined deHoffman-Teller frame in which there is excellent correlation between the field and flow vectors, (ii) constant field and total pressure, and (iii) an Alfvén ratio (i.e. ratio of kinetic-to-magnetic energy of the fluctuations) near unity at frequencies much lower than the ion cyclotron frequency in the spacecraft frame. In the region where the LAAWs approach the MCâAZs front boundary there are field and flow discontinuities. At the first, magnetic reconnection is taking place, as deduced from a stress balance test (Walen test). This severs connection of some field lines to the Sun and the solar wind strahl disappears. There follows a aLij2-hour interval where the magnetic field strength is diminished while pressure balance is maintained. Here the bi-directionality of the suprathermal electron flows is intermittently disrupted. This interval ends with a slow expansion fan downstream of which there is a dropout of halo electrons just inside the front boundary of the MC. This study illustrates an untypical case of a slow MC interacting with LAWWs in the slow solar wind.