Search Results - (Author, Cooperation:B. C. Low)

2011-04-15

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1089-7674

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Two-dimensional ideal magnetohydrodynamic (MHD) simulations are presented that demonstrate several novel phenomena in MHD shock formation. The stationary symmetrical flow of a uniform, planar, field-aligned, low-β and superfast magnetized plasma around a perfectly conducting cylinder is calculated. The velocity of the incoming flow is chosen such that the formation of fast switch-on shocks is possible. Using a time marching procedure, a stationary bow shock is obtained, composed of two consecutive interacting shock fronts. The leading shock front has a dimpled shape and is composed of fast, intermediate and hydrodynamic shock parts. A second shock front follows the leading front. Additional intermediate shocks and tangential discontinuities are present in the downstream part of the flow. The intermediate shocks are of the 1–3, 1–4, 2–4 and 1=2–3=4 types. This is a confirmation in two dimensions of recent results on the admissibility of these types of shocks. Recently it has also been shown that the 1=2–3=4 shock, embedded in a double compound wave, is present in the analytical solution of some planar one-dimensional MHD Riemann problems. This MHD flow with interacting shocks may have applications for some observed features of fast solar Coronal Mass Ejections and other phenomena in low-β space plasmas. © 1998 American Institute of Physics.

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1089-7674

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Physics

The magnetized, million-degree solar corona evolves in cycles of about 11 years, in dynamical response to newly generated magnetic fluxes emerging from below to eventually reverse the global magnetic polarity. Over the larger scales, the corona does not erupt violently all the time. Violent events like the flares and episodic ejections of material into interplanetary space occur frequently, several times a day, but they often originate in long-lived magnetic structures that form continually throughout the solar cycle. In this paper, the creation, stability, and eventual eruption of these structures are discussed from basic principles, drawing on recent advances in observation and theory. A global view is offered in which different pieces of observation relate physically, with distinct roles for the conservation of magnetic helicity and the release of magnetic energy in dissipated and ordered forms.

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1089-7674

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Physics

A simple, compact, and systematic derivation is given of the characteristic properties of the magnetohydrodynamic (MHD) equations with two independent variables (time-dependent MHD in the xt plane and steady MHD in the xy plane), based on the symmetrizable Galilean invariant form of the equations and using a matrix approach. A numerically obtained stationary planar field-aligned MHD bow shock flow with interacting shocks is then analyzed in terms of hyperbolic and elliptic regions, steady xy characteristics, limiting lines, and allowed shock transitions. With the help of this analysis, a wave structure present in the bow shock flow can be interpreted as a double steady compound shock. This interpretation is based on the complete analogy demonstrated in our analysis, between the xy characteristic structure of this novel steady compound shock and the xt characteristic structure of the well-known time-dependent MHD compound shock. © 1999 American Institute of Physics.

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1089-7674

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Physics

A physical analysis is presented of two distinct families of two-dimensional (2D) analytical solutions for isothermal periodic magnetostatic atmospheres in uniform gravity, one arrived at by Dungey and the other arrived at by Low and Manchester. It is demonstrated that particular members of the two families of 2D equilibria may be generated from the same planar state by plasma displacements which move the system through continuous sequences of equilibria while ensuring flux freezing. The two families of solutions both possess undulating magnetic field lines but geometrically different flux surfaces. The Dungey solutions can be created from a planar state by an undulating deformation whose spatial variation is along the field lines. By contrast, the 2D plane of variation of the Low-Manchester solutions lies at an angle to the field lines of the planar state. As a result, a mixed mode of undulating, interchange and shearing displacements must be made to the planar state to produce the more complex 2D state. Finally, the physical properties of these topologically equivalent states, including the magnetic and electrical helicities and the hydromagnetic potential energy, as introduced by Mouschovias, are discussed in terms of the variational principles of Kruskal and Kulsrud. © 2000 American Institute of Physics.

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1089-7666

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Physics

A family of time-dependent solutions is derived, which describe the self-similar motion of an inviscid γ=5/3 adiabatic fluid in arbitrary three-dimensional geometry. The derivation identifies a non-Galilean frame of reference in which the self-similar flows are observed to be steady, subject to a (fictitious) inertial force. This theory generalizes a primitive version, previously published, to allow for quite general velocity structures which are topologically no less complex than those belonging to the set of all steady flows. The existence of these self-similar flows stems from a compatibility between the conservation laws governing the internal energy of the γ=5/3 adiabatic gas and its angular momentum. Illustrative analytic solutions are presented for vortices of finite spatial extent isolated in a body of fluid undergoing purely radial expansion. These vortices consist of axisymmetric, closed meridional circulations, with or without swirls. A related set of self-similar, magnetohydrodynamic solutions for the γ=2 adiabatic fluid in the Cartesian plane are derived in the second of two appendices.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract The theoretical force-free magnetic fields in the first paper of this series, modeling magnetic configurations associated with polarity intrusion in active regions, are established to be all stable to linear ideal hydromagnetic perturbations under the boundary condition that anchors the lines of force rigidly to the photosphere. It is shown first that these force-free fields belong to an even larger class found by Chang and Carovillano (1981). A proof by the energy principle is then given to establish that all force-free magnetic fields in the larger class are absolutely stable. The physical implications of this result are discussed.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract Three-dimensional, quasi-static evolutions of coronal magnetic fields driven by photospheric flux emergence are modeled by a class of analytic force-free magnetic fields. Our models relate commonly observed photospheric magnetic phenomena, such as the formation and growth of sunspots, the emergence of an X-type separator, and the collision and merging of sunspots, to the three-dimensional magnetic fields in the corona above. By tracking the evolution in terms of a continuous sequence of force-free states, we show that flux emergence and submergence along magnetic neutral lines in the photosphere are essential processes in all these photospheric phenomena. The analytic solutions we present have a parametric regime within which the magnetic energy attained by an evolving force-free field may be of the order of 1030 ergs to several 1031 ergs, depending on the magnetic environment into which an emerging flux intrudes. The commonly used indicators of magnetic shear in magnetogram interpretation are discussed in terms of field connectivity in our models. It is demonstrated that the crossing angle of the photospheric transverse magnetic field with the neutral line may not be a reliable indicator of the magnetic shear in the coronal field above, due to the complexity of three-dimensionality. The poorly understood constraint of magnetic-helicity conservation on the availability of magnetic free energy for a flare is briefly discussed.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract This is a study of the relationship between a magnetic field and its embedding plasma in static equilibrium in a uniform gravity. The ideal gas law is assumed. A system invariant in a given direction is treated first. We show that an exact integral of the equation for force balance across field lines can be derived in a closed form. Using this integral, exact solutions can be generated freely by integrating directly for the distributions of pressure, density and temperature necessary to keep a given magnetic field in equilibrium. Particular solutions are presented for illustration with the solar atmosphere in mind. Extending the treatment to the general system depending on all three spatial coordinates, we arrive at the general form of a theorem of Parker that a magnetic field in static equilibrium must possess certain symmetries. We derive an equation involving the Euler potentials of the magnetic field stipulating these necessary symmetries. Only those magnetic fields satisfying this equation can be in static equilibrium and for these fields, the endowed symmetries make the construction of exact solutions an essentially two dimensional problem as exemplified by the special case of invariance in a given direction.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract We construct exact, non-linear, solutions for an horizontal, cylindrical, current-carrying, prominence supported against solar gravity by the action of a Lorentz force. The solutions incorporate the photosphere boundary condition, proposed by van Tend and Kuperus (1978), and analyzed by them for line filaments. Our solutions have finite radius for the prominence material and, as well as satisfying the equations of magnetostatic equilibrium, they allow for the continuity of gas pressure, and of the normal and tangential components of magnetic field across the circular prominence boundary. We show that an infinity of solutions is possible and we illustrate the basic behavior by investigation of a special case. We also give a prescription for constructing equilibrium fields for any horizontal prominence with arbitrary cross-section and with an arbitrary external magnetic field. The prescription is ideally suited for numerical codes and we suggest that both the equilibrium of such shapes can easily be accomplished numerically together with their evolutionary history.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract A method is prescribed for generating exact solutions of magnetostatic equilibrium describing a cylindrically symmetric magnetic flux tube oriented vertically in a stratified medium. Given the geometric shape of the field lines, compact formulae are presented for the direct calculation of all the possible distributions of pressure, density, temperature and magnetic field strength compatible with these field lines under the condition of static equlibrium. The plasma satisfies the ideal gas law and gravity is uniform in space. A particular solution is obtained by this method for a medium sized sunspot whose magnetic field obeys the similarity law of Schlüter and Temesváry (1958). With this solution, it is possible for the first time to illustrate explicitly the confinement of the magnetic field of the cool sunspot by the hotter external plasma in an exact relationship involving both magnetic pressure and field tension as well as the support of the weight of the plasma by pressure gradients. It is found that the cool region of the sunspot is not likely to extend much more than a few density scale heights below the photosphere. The sunspot field approaches being potential in the neighbourhood of the photosphere so that the Lorentz force exerting on the photosphere is less than what the magnetic pressure would suggest. This accounts for how the sunspot field can be confined in the photosphere where its magnetic pressure is often observed to even exceed the normal photospheric pressure. The energy mechanism operating in the sunspot and the question of mechanical stability are not treated in this paper.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract This paper considers the structural properties of a sunspot-like magnetic flux tube which lacks perfect axisymmetry. The flux tube is taken to be in static equilibrium with an atmosphere in a uniform gravity. Assuming the departure from axisymmetry to be slight, the equations for the first order non-axisymmetric part of the equilibrium are derived in cylindrical coordinates. These first order equations reduce to a linear second order hyperbolic partial differential equation in the r-z plane. Whereas Cauchy type boundary conditions are appropriate for hyperbolic equations, physical considerations dictate the specification of boundary conditions on a closed surve for our problem of interest. The construction of solutions to this boundary value problem is illustrated with three analytically soluble cases, where the zero-order axisymmetric equilibria are chosen to have magnetic field geometry of different complexity. A physical discussion of the results is given.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract Using a Lagrangian approach to the equations describing small amplitude departures from equilibrium of solar prominences, we derive seven quantities which, by analogy to the concepts of energy, momentum and angular momentum, are conserved under circumstances corresponding to ignorable coordinates of classical mechanics. In a pragmatic sense it is expected that these conservation laws will be useful as criteria of accuracy in obtaining eigen-frequencies for the perturbation equations when numerical techniques are employed. We also demonstrate that: (i) the perturbation equations (which are not self-adjoint) follow from an external variation of a Lagrangian; (ii) the equations adjoint to the perturbation equations also follow from an extremal variation of the same Lagrangian; (iii) the Lagrangian approach automatically gives the conserved quantities, together with a variational principle for computing the eigen-frequencies of the perturbation equations. In view of the paucity of analytic solutions (and their eigen-frequencies) to the perturbation equations (due essentially to the complicated spatial dependence of thermodynamic quantities describing the equilibria) we believe that the technique developed here - which is capable of handling complicated spatial dependences and which has internal criteria for assessing accuracy of any given numerical convergence scheme - is a valuable addition to the arsenal of methods used for discussing the equilibria, and their stability, of models thought to represent the basic physical processes underlying the quiescent solar prominence phenomenon.

Electronic Resource

1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract An elementary analysis based on Ampére's Law is given to separate the general magnetic field above the photosphere into two parts B=B 1+B *. The field B 1 is a potential field due to electric currents below the photosphere. The field B * is produced by electric currents above the photosphere combined with the induced mirror currents. By symmetry, B * has a set of field lines lying entirely in the plane z = 0 taken to be the photosphere. This set of field lines can be constructed from given vector magnetograph measurements and represents all the information on the electric currents above the photosphere that a magnetograph can provide. Theoretical illustrations are given and the implications for data analysis are discussed.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract We present a simple magnetostatic theory of the thin vertical filaments that make up the quiescent prominence plasma as revealed by fine spatial resolution Hα photographs. A class of exact equilibrium solutions is obtained describing a horizontal row of long vertical filaments whose weights are supported by bowed magnetic field lines. A free function is available to generate different assortments of filament sizes and spacings, as well as different density and temperature variations. The classic Kippenhahn-Schlüter solution for a long sheet without filamentary structures is a particular member of this class of solutions. The role of the magnetic field in supporting and thermally shielding the filament plasma is illustrated. It is found that the filament can have a sharp transition perpendicular to the local field, whereas the transition in the direction of the local field is necessarily diffuse. A consequence of the filamentary structure is that its support by the Lorentz force requires the electric current to have a component along the magnetic field. This electric current flowing into the rarefied region around the prominence can contain substantial energy stored in the form of force-free magnetic fields. This novel feature has implications for the heating and the disruption of prominences.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract The free energy associated with current sheets formed by displacing magnetic dipoles in a highly conducting medium is discussed. Specific models are illustrated, based on the idea that the free energy of a magnetic field in the solar atmosphere is the energy librated after the field has relaxed to a potential state that preserves the photospheric flux distribution. Previous calculations by other authors based on the consideration of discrete currents were incorrect because the interaction between the atmospheric currents and the highly conducting photosphere was not accounted for properly. It is shown that when this interaction is included, the consideration of discrete currents leads to the same result based on continuous magnetic field consideration.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract This paper presents a new class of exact solutions describing the non-linear force-free field above a spatially localized photospheric bipolar magnetic region. An essential feature is the variation in all three Cartesian directions and this could not be modelled adequately with previously known symmetric force-free fields. Sequences of force-free fields are constructed and analyzed to simulate the slow growth of a pair of spots on the photosphere. The axis connecting the spots executes rotational motion, distorting the photospheric neutral line separating fluxes of opposite signs. We show directly from the analytic solutions that the resulting reversal of the positions of the spots relative to the background field is associated with (i) the creation of magnetic free energy, (ii) the severe shearing of localized low-lying loops in the vicinity where the photospheric transverse field aligns with the photospheric neutral line, and (iii) the emergence and disappearance of flux from the photosphere at these highly stressed regions. The model relates theoretically for the first time these different magnetic field features that have been suggested by observation and theoretical considerations to be flare precursors. A general formula, based on the virial theorem, is also given for the free energy of a force-free field, strictly in terms of the field value at the photosphere. This formula has obvious practical application.

Electronic Resource

1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract This article describes recent developments in the theoretical investigation of magnetostatic equilibrium in the presence of gravity, nonequilibrium in hydromagnetics, and classical problems in hydromagnetic stability. The construction of magnetostatic equilibria has progressed beyond geometrically idealized systems, such as the axisymmetric system, to fully three-dimensional systems capable of modelling realistic solar structures. Nonequilibrium in a magnetic field with an arbitrary interweaving of lines of force due to random footpoint motion is a novel and subtle property with important implications for the solar atmosphere. Work begun by Parker and subsequent developments are described. To the extent quasi-static solar structures are approximated by stable equilibrium, ideal hydromagnetic stability theory provides a first insight into how stability is achieved in the solar environment. A qualitative physical picture based on recent stability analyses is given. The article places emphasis on understanding basic principles and issues rather than detailed results which can be found in the published literature.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract We present a theoretical model of quiescent prominences in the form of an infinite vertical sheet. Self-consistent solutions are obtained by integrating simultaneously the set of nonlinear equations of magnetostatic equilibrium and thermal balance. The basic features of the models are: (1) The prominence matter is confined to a sheet and supported against gravity by a bowed magnetic field. (2) The thermal flux is channelled along magnetic field lines. (3) The thermal flux is everywhere balanced by Low's (1975b) hypothetical heat sink which is proportional to the local density. (4) A constant component of the magnetic field along the length of the prominence shields the cool plasma from the hot surrounding. We assume that the prominence plasma emits more radiation than it absorbs from the radiation fields of the photosphere, chromosphere and corona, and we interpret the above hypothetical heat sink to represent the amount of radiative loss that must be balanced by a nonradiative energy input. Using a central density and temperature of 1011 particles cm−3 and 5000 K respectively, a magnetic field strength between 2 to 10 gauss and a thermal conductivity that varies linearly with temperature, we discuss the physical properties implied by the model. The analytic treatment can also be carried out for a class of more complex thermal conductivities. These models provide a useful starting point for investigating the combined requirements of magnetostatic equilibrium and thermal balance in the quiescent prominence.

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1573-093X

Springer Online Journal Archives 1860-2000

Physics

Abstract We consider the mechanical equilibrium of a cylinder of plasma suspended horizontally by magnetic fields in uniform gravity. This configuration is what may be expected if a quiescent prominence were to condense in a region initially filled with a uniform magnetic field. A set of exact solutions describing the equilibrium situation is presented. Although the plasma distribution is assumed to be cylindrically symmetric to obtain tractibility of the problem, exact force balance between plasma pressure, the Lorentz force and gravity is achieved everywhere in space. The set of solutions covers a particular case of a uniform temperature as well as cases where the temperature rises from zero at the center of the plasma cylinder to rapidly reach a constant asymptotic value outside the cylinder. The physical properties of these solutions are described. A suggestion is made for future development, based on the present work, to construct a prominence model in which the requirements of both mechanical and radiative equilibrium are satisfied.

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