Non-radial oscillations in an axisymmetric MHD incompressible fluid

It is well known from Helioseismology that the Sun exhibits oscillations on a global scale, most of which are non-radial in nature. These oscillations help us to get a clear picture of the internal structure of the Sun as has been demonstrated by the theoretical and observational (such as GONG) studies. In this study we formulate the linearised equations of motion for non-radial oscillations by perturbing the MHD equilibrium solution for an axisymmetric incompressible fluid. The fluid motion and the magnetic field are expressed as scalarsU, V, P andT, respectively. In deriving the exact solution for the equilibrium state, we neglect the contribution due to meridional circulation. The perturbed quantitiesU *, V *, P *, T * are written in terms of orthogonal polynomials. A special case of the above formulation and its stability is discussed.     

Propagation of MHD body and surface waves in magnetically structured regions of the solar atmosphere

The fact that magnetically structured regions exist in the solar atmosphere has been known for a number of years. It has been suggested that different kinds of magnetohydrodynamic (MHD) waves can be efficiently damped in these regions and that the dissipated wave energy may be responsible for the observed enhancement in radiative losses. From a theoretical point of view, an important task would be to investigate the propagation and dissipation of MHD waves in these highly structured regions of the solar atmosphere. In this paper, we study the behavior of MHD body and surface waves in a medium with either a single or double (slab) magnetic interface by use of a nonlinear, two-dimensional, time-dependent, ideal MHD numerical model constructed on the basis of a Lagrangean grid and semi-implicit scheme. The processes of wave confinement and wave energy leakage are discussed in detail. It is shown that the obtained results depend strongly on the type of perturbations imposed on the interface or slab and on the plasma parameter, . The relevance of the obtained results to the heating problem of the upper parts of the solar atmosphere is also discussed.     

Waves and Oscillations in the Corona – (Invited Review)

It has long been suggested on theoretical grounds that MHD waves must occur in the solar corona, and have important implications for coronal physics. An unequivocal identification of such waves has however proved elusive, though a number of events were consistent with an interpretation in terms of MHD waves. Recent detailed observations of waves in events observed by SOHO and TRACE removes that uncertainty, and raises the importance of MHD waves in the corona to a higher level. Here we review theoretical aspects of how MHD waves and oscillations may occur in a coronal medium. Detailed observations of waves and oscillations in coronal loops, plumes and prominences make feasible the development of coronal seismology, whereby parameters of the coronal plasma (notably the Alfvén speed and through this the magnetic field strength) may be determined from properties of the oscillations. MHD fast waves are refracted by regions of low Alfvén speed and slow waves are closely field-guided, making regions of dense coronal plasma (such as coronal loops and plumes) natural wave guides for MHD waves. There are analogies with sound waves in ocean layers and with elastic waves in the Earth's crust. Recent observations also indicate that coronal oscillations are damped. We consider the various ways this may be brought about, and its implications for coronal heating.     

The effects on the MHD stability of field line tying to the end faces of a cylindrical magnetic loop

We examine the magnetohydrodynamic (MHD) stability of a magnetic loop, taking into account field line tying at its foot points. We use the ideal MHD energy equation to derive a stability equation for a specific class of perturbations.We found that for a loop with large aspect ratio (10) the field line tying effect is negligible to the m = 1 kink mode but important to the localized modes. The stability criterion for high m localized modes is derived and compared with the Suydam criterion. The result shows that for the perturbation of the class studied, there are two effects of field line tying; one is a field line bending effect which is always stabilizing and the other is a shear effect which is stabilizing or destabilizing depending on the sign of the gradient of potential magnetic field. The net effect of field line tying is determined by the sum of these two effects.The result of this work is contrary to the result of Hood and Priest, in which they found that the field line tying effect is significant to the m = 1 mode. We believe that the contradiction comes from their incomplete minimization of the energy equation.     

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.      

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.     

Present and Future Observing Trends in Atmospheric Magnetoseismology

With modern imaging and spectral instruments observing in the visible, EUV, X-ray, and radio wavelengths, the detection of oscillations in the solar outer atmosphere has become a routine event. These oscillations are considered to be the signatures of a wave phenomenon and are generally interpreted in terms of magnetohydrodynamic (MHD) waves. With multiwavelength observations from ground- and space-based instruments, it has been possible to detect waves in a number of different wavelengths simultaneously and, consequently, to study their propagation properties. Observed MHD waves propagating from the lower solar atmosphere into the higher regions of the magnetized corona have the potential to provide excellent insight into the physical processes at work at the coupling point between these different regions of the Sun. High-resolution wave observations combined with forward MHD modeling can give an unprecedented insight into the connectivity of the magnetized solar atmosphere, which further provides us with a realistic chance to reconstruct the structure of the magnetic field in the solar atmosphere. This type of solar exploration has been termed atmospheric magnetoseismology. In this review we will summarize some new trends in the observational study of waves and oscillations, discussing their origin and their propagation through the atmosphere. In particular, we will focus on waves and oscillations in open magnetic structures (e.g., solar plumes) and closed magnetic structures (e.g., loops and prominences), where there have been a number of observational highlights in the past few years. Furthermore, we will address observations of waves in filament fibrils allied with a better characterization of their propagating and damping properties, the detection of prominence oscillations in UV lines, and the renewed interest in large-amplitude, quickly attenuated, prominence oscillations, caused by flare or explosive phenomena.     

Analytical theory of tachocline transport at mid latitudes

We provide a theory of magnetic diffusion, momentum transport, and mixing in the solar tachocline by considering magnetohydrodynamics (MHD) turbulence on a β plane subject to a large scale shear (provided by the latitudinal differential rotation). In the strong magnetic field regime, we find that the turbulent viscosity and diffusivity are reduced by magnetic fields only, similarly to the two-dimensional MHD case (without Rossby waves). In the weak magnetic field regime, we find a crossover scale (L_R) from a Alfvén dominated regime (on small scales) to a Rossby dominated regime (on large scales). For parameter values typical of the tachocline, L_R is larger than the solar radius so that Rossby waves are unlikely to play an important role in the transport of magnetic field and angular momentum. This is mainly due to the enhancement of magnetic back-reaction by shearing which efficiently generates small scales, thus strong currents. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamo theory and liquid metal MHD experiments

High values of magnetic Reynolds number R_m are characteristic not only to astrophysics, but also to other interesting objects, including liquid metal (LM) flows. Since sixties LM experiments have been performed illustrating important predictions of the dynamo theory, for example, about the existence and features of the α effect. Consideration of so called “laminar” dynamos provides a theoretical base for direct experimental realization and examination of the dynamo process. First step results, gathered at subcritical conditions, confirm the statement that self-excitation in LM experiments can be achieved practically today. In such devices as LM (sodium) cooled fast breeders R_m can reach values up to 50 and specific MHD phenomena have been observed in operating fast reactors. Cautions against crisis like processes have been expressed. It is important for the dynamo theory to understand what kind of perturbed motion is able to coexist with the generated magnetic field. Fundamentally new ideas here are issuing from the theory of 2D MHD turbulence. LM MHD served for the first direct proves, confirming, that the predicted surprising features of 2D turbulence can be observed in reality. It is worth incorporating these already not new ideas in the dynamo theory. In such a way a field for new solutions could be established.     

The visco-resistive stability of arcades in the corona of the Sun

The stability of ballooning modes in coronal arcades is studied using linear visco-resistive MHD. Rigid wall conditions are adopted for modelling the photospheric line-tying of the magnetic field. The full Braginskii viscosity stress tensor is used and particular attention is given to the effect of the viscosity coefficient ₃ which was left out of an earlier investigation by Van der Linden, Goossens, and Hood (1987, 1988). The numerical results for shearless arcades show that the coefficient ₃ has a stabilizing effect. However, for realistic values of the equilibrium quantities the stabilizing effect by ₃ can be neglected in comparison with the strong stabilizing effect of the perpendicular viscosity. The effect of magnetic field strength and mode number on stability are determined. In particular it is found that there exists a critical field strength for every mode number such that the mode is stable for weaker fields and unstable for stronger fields.     

Oscillatory cosmic-ray shock structures

A multiple scales analysis is used to derive a mixed Burgers-Korteweg-de Vries (BKdV) equation in the long wavelength regime for a two-fluid MHD model used to describe cosmic-ray acceleration by the first-order Fermi process in astrophysical shocks. The BKdV equation describes the time evolution of weak shocks in the theory of diffusive shock acceleration for all possible cosmic-ray pressures. Previous work on weak shocks in the cosmic-ray MHD model has assumed that dissipation alone is sufficient to balance nonlinearity, but, as cosmic-ray pressures become small, the weak shock becomes discontinous. By including Hall current effects into the MHD model, the low cosmic-ray pressure limit leads smoothly into solitary wave behaviour. For low cosmic-ray pressures, the shock has a downstream oscillatory precursor which is smoothed into the standard Taylor shock profile with increasing cosmic-ray pressure. As a by-product of the perturbation analysis, a dissipative KdV equation is derived. In conclusion, dispersive effects on Alfvén waves are discussed and a modulational stability analysis is presented.     

The heating of the solar corona

The heating of the solar corona has been a fundamental astrophysical issue for over sixty years. Over the last decade in particular, space-based solar observatories (Yohkoh, SOHO and TRACE) have revealed the complex and often subtle magnetic-field and plasma interactions throughout the solar atmosphere in unprecedented detail. It is now established that any energy release mechanism is magnetic in origin - the challenge posed is to determine what specific heat input is dominating in a given coronal feature throughout the solar cycle. This review outlines a range of possible magnetohydrodynamic (MHD) coronal heating theories, including MHD wave dissipation and MHD reconnection as well as the accumulating observational evidence for quasi-periodic oscillations and small-scale energy bursts occurring in the corona. Also, we describe current attempts to interpret plasma temperature, density and velocity diagnostics in the light of specific localised energy release. The progress in these investigations expected from future solar missions (Solar-B, STEREO, SDO and Solar Orbiter) is also assessed.Received: 6 February 2003, Published online: 14 November 2003 Correspondence to: R. W. Walsh     

Rayleigh-Taylor instability of a plasma with finite ion Larmor radius effects

The stability of an infinitely conducting plasma of variable density has been investigated taking into account the finiteness of the ion Larmor radius. The perturbations propagating along the ambient magnetic field are considered. It is established that, in general,n ² is necessarily real, wheren is the growth rate of disturbance, thus ruling out the possibility of overstability or damped oscillations. The solution is shown to be characterized by a variational principle, which provides the basis for obtaining an approximate solution of the problem. Two density distributions are considered: (i) a continuously stratified plasma layer and (ii) two semi-infinitely extending plasmas of constant densities separated by a horizontal interface. In both cases it has been shown that for the said disturbances the stability criterion remains unaffected by the inclusion of finite Larmor radius effects, though the amplified motion is strongly inhibited due to their inclusion.     

A class of semi-analytical solutions for a rotating toroidal plasma

In this paper, the problem of stationary MHD flow for a rotating toroidal plasma is investigated by assuming that the entropy is a surface quantity. Then, the system of ideal MHD equations is reduced to a single second-order elliptic partial differential equation known as the modified Grad-Shafranov (or Maschke-Perrin) equation. Under the assumption that both the function,P _s andf ² are quadratic polynomials of the flux function, a class of semi-analytical solutions is obtained for a plasma contained in a perfectly conducting toroidal boundary with a rectangular cross section. The flux function, poloidal current and the generalized pressure are obtained and discussed for relevant values of the parameters.     

Quasi-periodic Oscillations of Solar Active Regions in Connection with Their Flare Activity – NoRH Observations

The sunspot-associated sources at the frequency of 17 GHz give information on plasma parameters in the regions of magnetic field about B=2000 G at the level of the chromosphere-corona transition region. The observations of short period (from one to ten minutes) oscillations in sunspots reflect propagation of magnetohydrodynamic (MHD) waves in the magnetic flux tubes of the sunspots. We investigate the oscillation parameters in active regions in connection with their flare activity. We confirm the existence of a link between the oscillation spectrum and flare activity. We find differences in the oscillations between pre-flare and post-flare phases. In particular, we demonstrate a case of powerful three-minute oscillations that start just before the burst. This event is similar to the cases of the precursors investigated by Sych et al. (Astron. Astrophys. 505, 791, 2009). We also found well-defined eight-minute oscillations of microwave emission from sunspot. We interpret our observations in terms of a relationship between MHD waves propagating from sunspots and flare processes.     

The thermal structure of the magnetized solar transition region

The detailed thermal structure of the magnetized solar transition region, as measured by itsdifferential emission measure [DEM(T)], is unknown. Proposals have been made that envision a significant lower-temperature contribution to the energy balance from cross-field (ion) heat flux. In this paper, we describe a self-consistent, 2-D, MHD simulation (including the full effects of anisotropic thermal conduction) of a conceptual model due to Athay (1990). We display the detailed, irregular, thermal and magnetic structure of the transition region, and demonstrate that the predicted DEM agrees with observations, particularly in theT < 10⁵ K regime where previous theories had difficulty.     

Structures in a class of magnetized scale-free discs

We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamics (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density  Σ₀∝ r ^−α  , a coplanar azimuthal magnetic field   B ₀∝ r ^−γ  , a consistent self-gravity and a power-law rotation curve   v ₀∝ r ^−β  , where v ₀ is the linear azimuthal gas rotation speed. The barotropic equation of state  Π∝Σ ⁿ   is adopted for both MHD background equilibrium and coplanar MHD perturbations where Π is the vertically integrated pressure and n is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among  α, β, γ  and n such that only one parameter (e.g. β) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For non-axisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e. fast and slow MHD density waves) can be derived once β is specified. To complement the magnetized singular isothermal disc analysis of Lou, we extend the analysis to a wider range of  −1/4 < β < 1/2  . As an illustrative example, we discuss specifically the  β= 1/4  case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.     

Modeling the Resistive MHD by the Cese Method

In this paper, the Space–Time Conservation Element and Solution Element (CESE) method is applied to 2.5-dimensional resistive magnetohydrodynamics (MHD) equations in Cartesian coordinates, with the purpose of modeling the magnetic reconnection study. To show the validity and capacity of its application to MHD reconnection problem, spontaneous fast reconnection and magnetic reconnection in multiple heliospheric current sheets are studied, which show good consistency with those obtained formerly by other authors. In order to assess the ∇ ⋅ B = 0 constraint numerically, the contours and evolution of ∇ ⋅ B are analyzed. The numerical results tell us that the CESE numerical scheme not only has good numerical resolution but also can keep the divergence-free condition for magnetic fields in the reconnection problems during the evolutionary process without any special treatment.     

Jump relations for magnetohydrodynamic shock waves in non-ideal gas flow

The generalized jump relations across the magnetohydrodynamic (MHD) shock front in non-ideal gas are derived considering the equation of state for non-ideal gas as given by Landau and Lifshitz. The jump relations for pressure, density, and particle velocity have been derived, respectively in terms of a compression ratio. Further, the simplified forms of the MHD shock jump relations have been obtained in terms of non-idealness parameter, simultaneously for the two cases viz., (i) when the shock is weak and, (ii) when it is strong. Finally, the cases of strong and weak shocks are explored under two distinct conditions viz., (i) when the applied magnetic field is strong and, (ii) when the field is weak. The aim of this paper is to contribute to the understanding of how shock waves behave in magnetized environment of non-ideal gases.