Stability of tangential discontinuities

The MHD stability of tangential discontinuities is first considered. We treat these discontinuities as structured forms rather than as sharp breaks in the magnetic field. An unfamiliar form of the MHD energy principle is applied, and stability is proved provided that there is no fluid flow tangent to the discontinuity plane. Perturbations which simply transform the system from one equilibrium to another are neutrally stable. Using comparison theorems we conclude that the observed stability of tangential forms in the solar wind implies near isotropy of the particle pressure in them.     

Magnetosonic shocks in nonlinear dispersive media

When account is taken of the finite resistivity, energy equation, and generalized Ohm's law, it can be shown that the propagation of nonlinear magnetosonic waves in a warm plasma is described by a KdV-Burgers' equation. The stationary shock solutions for the latter are discussed. It is suggested that shocks may be responsible for the observed discontinuities in MHD fluctuations in the solar atmosphere.     

Weak MHD discontinuities in a radiation-induced flow field

The growth of weak MHD discontinuities have been studied in a radiation induced flow field at very high temperature. Growth and decay properties of weak MHD discontinuities have been discussed under the influences of time-dependent gasdynamic field, the radiation field and the magnetic field with finite electrical conductivity. The effects of thermal radiation and conduction of the global behaviour of weak MHD discontinuities have been studied under a quasi-equilibrium and quasi-isotropic hypothesis of the differential approximation to the radiative heat transfer equation. It is shown that the existence of the time-dependent radiation field gives rise to a radiation induced wave which has a negligibly small effect on the non-relativistic flow properties of the gasdynamic field. It is also shown that the radiation stresses resist the steepening tendency of a compressive weak wave and help in stabilizing it whereas the thermal conduction effects counteracts to destabilize it. It is found that under radiation effects the shock formation is either disallowed or delayed. The two cases of diverging waves and converging waves have been studied separately to answer a particular question as to when a shock discontinuity or a coustic will be formed or disallowed under curvature effects.     

A simple unsplit Godunov method for multidimensional MHD

We describe a numerical algorithm based on Godunov methods for integrating the equations of compressible magnetohydrodynamics (MHD) in multidimensions. It combines a simple, dimensionally-unsplit integration method with the constrained transport (CT) discretization of the induction equation to enforce the divergence-free constraint. We present the results of a series of fully three-dimensional tests which indicate the method is second-order accurate for smooth solutions in all MHD wave families, and captures shocks, contact and rotational discontinuities well. However, it is also more diffusive than other more complex unsplit integrators combined with CT. Thus, the primary advantage of the method is its simplicity. It does not require a characteristic tracing step to construct interface values for the Riemann solver, it is straightforward to extend with additional physics, and it is suitable for use with nested and adaptive meshes. The method is implemented as one of two dimensionally unsplit MHD integrators in the Athena code, which is freely available for download from the web.     

Kelvin-Helmholtz instability of the magnetopause boundary: Comparison with observed fluctuations

We present a numerical investigation of the Kelvin-Helmholtz instability problem, in MHD approximation, for transitions involving continuous variations of both magnetic field and plasma parameters. The variations have been chosen in such a way as to reproduce general features of possible magnetopause transitions. The study, which leads to prediction of a wavelength of maximum instability, has been confronted with recent observational results of surface fluctuations of the magnetopause.     

General polytropic magnetofluid under self-gravity: Voids and shocks

We study the self-similar magnetohydrodynamics (MHD) of a quasi-spherical expanding void (viz. cavity or bubble) surrounding the centre of a self-gravitating gas sphere with a general polytropic equation of state. We show various analytic asymptotic solutions near the void boundary in different parameter regimes and obtain the corresponding void solutions by extensive numerical explorations. We find novel void solutions of zero density on the void boundary. These new void solutions exist only in a general polytropic gas and feature shell-type density profiles. These void solutions, if not encountering the magnetosonic critical curve (MCC), generally approach the asymptotic expansion solution far from the central void with a velocity proportional to radial distance. We identify and examine free-expansion solutions, Einstein–de Sitter expansion solutions, and thermal-expansion solutions in three different parameter regimes. Under certain conditions, void solutions may cross the MCC either smoothly or by MHD shocks, and then merge into asymptotic solutions with finite velocity and density far from the centre. Our general polytropic MHD void solutions provide physical insight for void evolution, and may have astrophysical applications such as massive star collapses and explosions, shell-type supernova remnants and hot bubbles in the interstellar and intergalactic media, and planetary nebulae.     

MHD shock interactions in coronal structures

We consider the magnetohydrodynamic (MHD) interactions of solar coronal fast shock waves of flare and/or nonflare origin with the boundaries of coronal streamers and coronal holes. Boundaries are treated as MHD tangential discontinuities (TD). Different parameters of the observed corona are used in the investigation. The general case of the oblique interaction is studied.It is shown that a solar fast shock wave must be refracted usually as a fast shock wave inside the coronal streamer. For the special case of the velocity shear across TD, a slow shock wave is generated. On the contrary, the shock wave refracted inside the coronal hole is indeed a slow shock wave.The significance of different effects due to the interaction of fast and slow shock waves on the coronal magnetic field is noticed, especially at the time of a coronal mass ejection (CME). It is also shown, that an oblique fast MHD coronal shock wave may trigger an instability at the boundary of a streamer considered as a TD. It might have a relation with the observed process of abrupt disappearance of the streamer's boundary in the solar corona.On leave from the Academy of Sciences, Central Astronomical Observatory Pulkovo, 196140, St. Petersburg, Russia.     

Generalized analytical models of Syrovatskii’s current sheet

Two-dimensional stationary magnetic reconnection models that include a thin Syrovatskii-type current sheet and four discontinuous magnetohydrodynamic flows of finite length attached to its endpoints are considered. The flow pattern is not specified but is determined from a self-consistent solution of the problem in the approximation of a strong magnetic field. Generalized analytical solutions that take into account the possibility of a current sheet discontinuity in the region of anomalous plasma resistivity have been found. The global structure of the magnetic field in the reconnection region and its local properties near the current sheet and attached discontinuities are studied. In the reconnection regime in which reverse currents are present in the current sheet, the attached discontinuities are trans-Alfvénic shock waves near the current sheet endpoints. Two types of transitions from nonevolutionary shocks to evolutionary ones along discontinuous flows are shown to be possible, depending on the geometrical model parameters. The relationship between the results obtained and numerical magnetic reconnection experiments is discussed.     

Generation of rotational discontinuities by magnetic reconnection associated with microflares

Magnetic reconnection can take place between two plasma regions with antiparallel magnetic field components. In a time-dependent reconnection event, the plasma outflow region consists of a leading bulge region and a trailing reconnection layer. Magnetohydrodynamic (MHD) discontinuities, including rotational discontinuities, can be formed in both the bulge region and the trailing layer. In this paper, we suggest that the rotational discontinuities observed in the solar wind may be generated by magnetic reconnection associated with microflares in coronal holes. The structure of the reconnection layer is studied by solving the one-dimensional Riemann problem for the evolution of an initial current sheet after the onset of magnetic reconnection as well as carrying out two-dimensional MHD simulations. As the emerging magnetic flux reconnects with ambient open magnetic fields in the coronal hole, rotational discontinuities are generated in the region with open field lines. It is also found that in the solar corona with a low plasma beta ( 0.01), the magnetic energy is converted through magnetic reconnection mostly into the plasma bulk-flow energy. Since more microflares will generate more rotational discontinuities and also supply more energy to the solar wind, it is expected that the number of rotational discontinuities observed in the solar wind would be an increasing function of solar wind speed. The observation rate of rotational discontinuities generated by microflares is estimated to be dN _RD/dt - f/63 000 s (f > 1) at 1 AU. The present mechanism favors the generation of rotational discontinuities with a large shock normal angle.     

wham: a WENO-based general relativistic numerical scheme – I. Hydrodynamics

Active galactic nuclei, X-ray binaries, pulsars and gamma-ray bursts are all believed to be powered by compact objects surrounded by relativistic plasma flows driving phenomena such as accretion, winds and jets. These flows are often accurately modelled by the relativistic magnetohydrodynamic (MHD) approximation. Time-dependent numerical MHD simulations have proven to be especially insightful, but one regime that remains difficult to simulate is when the energy scales (kinetic, thermal, magnetic) within the plasma become disparate. We develop a numerical scheme that significantly improves the accuracy and robustness of the solution in this regime. We use a modified form of the weighted essentially non-oscillatory (WENO) method to construct a finite-volume general relativistic hydrodynamics code called wham that converges at fifth order. We avoid (1) field-by-field decomposition by adaptively reducing down to two-point stencils near discontinuities for a more accurate treatment of shocks and (2) excessive reduction to low-order stencils, as in the standard WENO formalism, by maintaining high-order accuracy in smooth monotonic flows. Our scheme performs the proper surface integral of the fluxes, converts cell-averaged conserved quantities to point-conserved quantities before performing the reconstruction step, and correctly averages all source terms. We demonstrate that the scheme is robust in strong shocks, very accurate in smooth flows and maintains accuracy even when the energy scales in the flow are highly disparate.     

Magnetic field reconnection in a compressible plasma

An extension of Sonnerup's model for the magnetic field-line reconnection for a compressible plasma is given. The plasma is considered to be only slightly compressible so that the leading wave in Sonnerup's model can still be taken to be a thin discontinuity. The flow is assumed to occur under adiabatic conditions, and de Hoffmann-Teller jump conditions are used to connect the state variables across the shocks. The compressibility effects are found to increase the reconnection rate. The signaling problem is finally considered to study the evolution of MHD waves in a compressible, dissipative plasma so as to investigate the conditions under which the assumption of MHD waves in a compressible plasma to be thin discontinuities is valid.     

Non-evolutionarity of a reconnecting current sheet as a cause of its splitting into MHD shocks

Numerical simulations of the magnetic reconnection process in a current sheet show that, in some cases, MHD shocks appear to be attached to edges of the sheet. The appearance of the shocks may be considered to be a result of splitting of the sheet. In the present paper we suppose that this splitting takes place in consequence of non-evolutionarity of the reconnecting current sheet as a discontinuity. The problem of time evolution of small perturbations does not have a unique solution for a non-evolutionary discontinuity, and it splits into other (evolutionary) discontinuities. Such an approach allows us to determine conditions under which the splitting of the-sheet occurs. The main difficulty of this approach is that a current sheet is not reduced to a classified 1D discontinuity, because inhomogeneity of flow velocity inside the sheet is two-dimensional. To formulate the non-evolutionarity problem, we solve the linear MHD equations inside and outside the sheet and deduce linearized 1D boundary conditions at its surface. We show that for large enough conductivity, small perturbations exist which interact with the sheet as with a discontinuity. Then we obtain a non-evolutionarity criterion, with respect to these perturbations, in the form of a restriction on the flow velocity across the surface of the sheet.     

AMR Simulations of Magnetohydrodynamic Problems by the CESE Method in Curvilinear Coordinates

The objective of this paper is to present new extensions of the space – time conservation element and solution element (CESE) method for simulations of magnetohydrodynamic (MHD) problems in general curvilinear coordinates by using an adaptive mesh refinement (AMR) grid system. By transforming the governing MHD equations from the physical space (x,y,z) to the computational space (ξ,η,ζ) while retaining the form of conservation, the CESE method is established for MHD in the curvilinear coordinates. Utilizing the parallel AMR package PARAMESH, we present the first implementation of applying the AMR CESE method for MHD (AMR-CESE-MHD) in both Cartesian and curvilinear coordinates. To show the validity and capabilities of the AMR-CESE-MHD code, a suite of numerical tests in two and three dimensions including ideal MHD and resistive MHD are carried out, with two of them in both Cartesian and curvilinear coordinates. Numerical tests show that our results are highly consistent with those obtained previously by other authors, and the results under both coordinate systems confirm each other very well.     

Self-similar dynamics of a magnetized polytropic gas

In broad astrophysical contexts of large-scale gravitational collapses and outflows and as a basis for various further astrophysical applications, we formulate and investigate a theoretical problem of self-similar magnetohydrodynamics (MHD) for a non-rotating polytropic gas of quasi-spherical symmetry permeated by a completely random magnetic field. Within this framework, we derive two coupled nonlinear MHD ordinary differential equations (ODEs), examine properties of the magnetosonic critical curve, obtain various asymptotic and global semi-complete similarity MHD solutions, and qualify the applicability of our results. Unique to a magnetized gas cloud, a novel asymptotic MHD solution for a collapsing core is established. Physically, the similarity MHD inflow towards the central dense core proceeds in characteristic manners before the gas material eventually encounters a strong radiating MHD shock upon impact onto the central compact object. Sufficiently far away from the central core region enshrouded by such an MHD shock, we derive regular asymptotic behaviours. We study asymptotic solution behaviours in the vicinity of the magnetosonic critical curve and determine smooth MHD eigensolutions across this curve. Numerically, we construct global semi-complete similarity MHD solutions that cross the magnetosonic critical curve zero, one, and two times. For comparison, counterpart solutions in the case of an isothermal unmagnetized and magnetized gas flows are demonstrated in the present MHD framework at nearly isothermal and weakly magnetized conditions. For a polytropic index γ=1.25 or a strong magnetic field, different solution behaviours emerge. With a strong magnetic field, there exist semi-complete similarity solutions crossing the magnetosonic critical curve only once, and the MHD counterpart of expansion-wave collapse solution disappears. Also in the polytropic case of γ=1.25, we no longer observe the trend in the speed-density phase diagram of finding infinitely many matches to establish global MHD solutions that cross the magnetosonic critical curve twice.      

The structure of MHD shock waves in a thermally-radiating gas at high mach numbers

The structure of a strong MHD shock wave which radiates thermally downstream of the shock is studied by asymptotic expansion. The exact integral equation for radiation is adopted for the study. Hence, the optically thick (and thin), the general differential approximate and the exact integral equation solutions may now be compared.     

Damping of Longitudinal Magneto–Acoustic Oscillations in Slowly Varying Coronal Plasma

We investigate the propagation of MHD waves in a magnetised plasma in a weakly stratified atmosphere, representative of hot coronal loops. In most earlier studies, a time-independent equilibrium was considered. Here we abandon this restriction and allow the equilibrium to develop as a function of time. In particular, the background plasma is assumed to be cooling due to thermal conduction. The cooling is assumed to occur on a time scale greater than the characteristic travel times of the perturbations. We investigate the influence of cooling of the background plasma on the properties of magneto–acoustic waves. The MHD equations are reduced to a 1D system modelling magneto–acoustic modes propagating along a dynamically cooling coronal loop. A time-dependent dispersion relation that describes the propagation of the magneto–acoustic waves is derived using the WKB theory. An analytic solution for the time-dependent amplitude of waves is obtained, and the method of characteristics is used to find an approximate analytical solution. Numerical calculations of the analytically derived solutions are obtained to give further insight into the behaviour of the MHD waves in a system with a variable, time-dependent background. The results show that there is a strong damping of MHD waves and the damping also appears to be independent of the position along the loop. Studies of MHD wave behaviour in a time-dependent backgrounds seem to be a fundamental and very important next step in the development of MHD wave theory that is applicable to a wide range of situations in solar physics.     

Magnetic equilibria and instabilities

Solar flares are understood as a process of explosive liberation of magnetic energy, coming after a slow phase of energy build-up. The slow evolution of magnetic equilibria may end up with (a) the termination of an equilibrium sequence, or (b) an instability. The distinction between the two can be made by drawing schematic potential curves. Case (a) has been extensively studied in two-dimensional models. The appearance of multiple solutions, or disappearance of a solution takes place as the system evolves away from the current-free configuration. Case (b) can be discussed in terms of ideal MHD or resistive MHD instabilities. A possible route to explosive energy release is suggested by combining these two cases.     

Dynamic evolution of a quasi-spherical general polytropic magnetofluid with self-gravity

In various astrophysical contexts, we analyze self-similar behaviours of magnetohydrodynamic (MHD) evolution of a quasi-spherical polytropic magnetized gas under self-gravity with the specific entropy conserved along streamlines. In particular, this MHD model analysis frees the scaling parameter n in the conventional polytropic self-similar transformation from the constraint of n+γ=2 with γ being the polytropic index and therefore substantially generalizes earlier analysis results on polytropic gas dynamics that has a constant specific entropy everywhere in space at all time. On the basis of the self-similar nonlinear MHD ordinary differential equations, we examine behaviours of the magnetosonic critical curves, the MHD shock conditions, and various asymptotic solutions. We then construct global semi-complete self-similar MHD solutions using a combination of analytical and numerical means and indicate plausible astrophysical applications of these magnetized flow solutions with or without MHD shocks.     

Eastern Aphrodite Terra on Venus: Characteristics,structure, and mode of origin

Eastern Aphrodite Terra and Western Aphrodite form an altimetrically prominent 14,000 km long part of the equatorial highlands on Venus. Several parallel linear discontinuities striking northwest across the general east-west regional strike of the highlands are mapped in the altimetric and radar image data of Eastern Aphrodite and identified on the basis of abrupt termination of rift-like central chasma, offset and segmentation of the center of the highlands, and radar image discontinuities in the lowlands to the north. These characteristics are similar to those of linear discontinuities previously mapped in Western Aphrodite in terms of length, orientation, and influence on the central highlands and adjacent lowlands.Altimetric profiles in directions parallel to the discontinuities are regionally symmetric, more ridge-like in Eastern Aphrodite compared to the plateau-dominated form of topography in Western Aphrodite, and are characterized by alternating paired ridge-and-trough forms near their crests and on their flanks. By mapping the center of symmetry in multiple profiles, the prominent segmentation of the highland is shown to be imparted by an offset of the regional symmetry along the mapped discontinuities. These characteristics are morphologically similar to several of the large-scale characteristics of divergent plate boundaries of Earth, including mid-ocean rise crests and rifts, offset at fracture zones and transform faults, and symmetric thermal boundary layer topography.The altitude of the surface in profiles parallel to the discontinuities decreases as the square root of distance from the symmetry axes and with a slope similar to that predicted for thermal boundary layer topography associated with rates of divergence on Venus of ~ 1 ± 0.5 cm/yr. In order to test the hypothesis that the linear discontinuities are analogous to fracture zones, the predicted altitude of the surface at great distance from the centers of symmetry of the central highland and in directions across the discontinuities was calculated on the basis of a thermal boundary layer topography model with offset of altimetric symmetry at each discontinuity. Similarity of observed Arecibo high-resolution altimetric profiles across the discontinuities with that calculated for thermal boundary layer topography offset by transform faults reveals that in terms of the sense and magnitude of regional steps in altimetry across discontinuities and the altitude of the surface, Eastern Aphrodite is similar to the known characteristics of crustal spreading at divergent boundaries. The plateau-like form of Western Aphrodite and the ridge-like form of Eastern Aphrodite are analogous respectively to the difference between areas of anomalous (Iceland) and normal crustal production along rise crests on Earth. Estimates of volumetric differences in crustal production in the environment of Venus and as it would be influenced by differences in mantle temperature beneath Western and Eastern Aphrodite imply that Eastern Aphrodite represents normal crustal production. On this basis, Western Aphrodite may be characterized by a mantle temperature that is warmer than the mantle beneath Eastern Aphrodite Terra, perhaps in association with deep convective mantle upwelling.'Geology and Tectonics of Venus', special issue edited by Alexander T. Basilevsky (USSR Acad. of Sci., Moscow), James W. Head (Brown University, Providence), Gordon H. Pettengill (MIT. Cambridge, Massachusetts) and R. S. Saunders (J.P.L., Pasadena).