The continuous spectrum of MHD waves in 2D solar loops and arcades. First results on poloidal mode coupling for poloidal magnetic fields

A first attempt is made to study the continuous spectrum of linear ideal MHD for 2D solar loops and to understand how 2D effects change the continuum eigenfrequencies and continuum eigenfunctions. The continuous spectrum is computed for 2D solar loops with purely poloidal magnetic fields and it is investigated how non-circularity of the cross-sections of the poloidal magnetic surfaces and variations of density along the poloidal magnetic field lines change the continuous spectrum and induce poloidal wave number coupling in the eigenfunctions. Approximate analytical results and numerical results are obtained for the eigenfrequencies and the eigenfunctions and the poloidal wave number coupling is clearly illustrated. It is found that the continuum frequencies are substantially increased, that the ranges of the continuum frequencies are considerably enlarged and that the derivatives of the continuum frequencies normal to the magnetic surfaces are substantially increased. The eigenfunctions are strongly influenced by poloidal wave number coupling. Implications of these findings for the heating mechanisms of resonant absorption and phase mixing are briefly considered.Research Assistant of the Belgian National Fund for Scientific Research.     

Magnetic and boundary effects on thermal instabilities in solar magnetic fields: Localized modes in a slab geometry

The coupling of thermal and ideal MHD effects in a sheared magnetic field is investigated. A slab geometry is considered so that the Alfvén mode can be decoupled from the system. With the total perturbed pressure approximately zero, the fast mode is eliminated and a system of linearized equations describing magnetic effects on the slow mode and thermal mode is derived. These modes evolve independently on individual fieldlines. One of the main features of this approach is that the influence of the dense photosphere can be included. A variety of different conditions that simulate the photospheric boundary are presented and the different results are discussed. A choice of field geometry and boundary conditions is made which removes mode rational surfaces so that there are no regions in which parallel thermal conduction can be neglected. This provides a stabilizing mechanism for the thermal mode. Growth rates are reduced by 30–40% and there is complete stabilization for sufficiently short fieldlines. The influence of dynamic and thermal boundary conditions on the formation of prominences is discussed.     

Coronal heating by selective decay of MHD turbulence

We suggest that the solar corona is stationarily heated by Joule dissipation of magnetohydrodynamic (MHD) turbulence. This turbulence is continuously being fed by subphotospheric convective motions at very low wavenumbers where the plasma can be considered ideal. The cascade process due to the nonlinear interaction of the turbulent modes, effectively transfers energy to even shorter wavenumbers until Joule effect becomes dominant and converts the cascading magnetic energy into heat.Assuming that a stationary spectrum is established, we show that MHD turbulence naturally provides a way of enhancing the Joule dissipation of macroscopic magnetic stresses and a heating rate can be derived.     

The relevance of the ballooning approximation for magnetic,thermal, and coalesced magnetothermal instabilities

Approximate solutions of the linearized non-adiabatic MHD equations, obtained using the ballooning method, are compared with exact numerical solutions of the full equations (including the effects of optically thin plasma radiation). It is shown that the standard ballooning method, developed within the framework of ideal linear MHD, can be generalized to non-ideal linear MHD. The localized (ballooning) spectrum has to be used with caution, but can give valuable (though limited) information on non-ideal stability.The numerical analysis also confirms and quantifies the interesting connection between magnetic and thermal instabilities. The existence of such a coupling is inherent in many qualitative discussions of magnetic disruptions. Finally, the hitherto unrecognized role of the thermal continuum in the unstable part of the magnetothermal spectrum is investigated.Research Assistant of the National Fund for Scientific Research, Belgium.     

Influence of a uniform coronal magnetic field on solar p modes: coupling to slow resonant MHD waves

The influence of a constant coronal magnetic field on solar global oscillations is investigated for a simple planar equilibrium model. The model consists of an atmosphere with a constant horizontal magnetic field and a constant sound speed, on top of an adiabatic interior having a linear temperature profile. The focus is on the possible resonant coupling of global solar oscillation modes to local slow continuum modes of the atmosphere and the consequent damping of the global oscillations. In order to avoid Alfvén resonances, the analysis is restricted to propagation parallel to the coronal magnetic field. Parallel propagating oscillation modes in this equilibrium model have already been studied by Evans and Roberts (1990). However, they avoided the resonant coupling to slow continuum modes by a special choice of the temperature profile. The physical process of resonant absorption of the acoustic modes with frequency in the cusp continuum is mathematically completely described by the ideal MHD differential equations which for this particular equilibrium model reduce to the hypergeometric differential equation. The resonant layer is correctly dealt with in ideal MHD by a proper treatment of the logarithmical branch cut of the hypergeometric function. The result of the resonant coupling with cusp waves is twofold. The eigenfrequencies become complex and the real part of the frequency is shifted. The shift of the real part of the frequency is not negligible and within the limit of observational accuracy. This indicates that resonant interactions should definitely be taken into account when calculating the frequencies of the global solar oscillations.     

Numerical MHD Simulations of Solar Magnetoconvection and Oscillations in Inclined Magnetic Field Regions

The sunspot penumbra is a transition zone between the strong vertical magnetic field area (sunspot umbra) and the quiet Sun. The penumbra has a fine filamentary structure that is characterized by magnetic field lines inclined toward the surface. Numerical simulations of solar convection in inclined magnetic field regions have provided an explanation of the filamentary structure and the Evershed outflow in the penumbra. In this article, we use radiative MHD simulations to investigate the influence of the magnetic field inclination on the power spectrum of vertical velocity oscillations. The results reveal a strong shift of the resonance mode peaks to higher frequencies in the case of a highly inclined magnetic field. The frequency shift for the inclined field is significantly greater than that in vertical-field regions of similar strength. This is consistent with the behavior of fast MHD waves.     

Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity-damped regime and astrophysical implications

We present numerical simulations and explore scalings and anisotropy of compressible magnetohydrodynamic (MHD) turbulence. Our study covers both gas-pressure-dominated (high β) and magnetic-pressure-dominated (low β) plasmas at different Mach numbers. In addition, we present results for super-Alfvénic turbulence and discuss in what way it is similar to sub-Alfvénic turbulence. We describe a technique of separating different magnetohydrodynamic modes (slow, fast and Alfvén) and apply it to our simulations. We show that, for both high- and low-β cases, Alfvén and slow modes reveal a Kolmogorov   k ^−5/3  spectrum and scale-dependent Goldreich–Sridhar anisotropy, while fast modes exhibit a   k ^−3/2  spectrum and isotropy. We discuss the statistics of density fluctuations arising from MHD turbulence in different regimes. Our findings entail numerous astrophysical implications ranging from cosmic ray propagation to gamma ray bursts and star formation. In particular, we show that the rapid decay of turbulence reported by earlier researchers is not related to compressibility and mode coupling in MHD turbulence. In addition, we show that magnetic field enhancements and density enhancements are marginally correlated. Addressing the density structure of partially ionized interstellar gas on astronomical-unit scales, we show that the viscosity-damped regime of MHD turbulence that we reported earlier for incompressible flows persists for compressible turbulence and therefore may provide an explanation for these mysterious structures.     

A toy model for global magnetar oscillation

The presence of a magnetic field in a neutron star interior results in a dynamical coupling between the fluid core and the elastic crust. We consider a simple toy-model where this coupling is taken into account and compute the system’s mode oscillations. Our results suggest that the notion of pure torsional crust modes is not useful for the coupled system, instead all modes excite Alfvén waves in the core. However, we also show that among a rich spectrum of global MHD modes the ones most likely to be excited by a fractured crust are those for which the crust and the core oscillate in concert. For our simple model, the frequencies of these modes are similar to the “pure crustal” frequencies. We advocate the significant implications of these results for the attempted theoretical interpretation of QPOs during magnetar flares in terms of neutron star oscillations.      

Short-Wave Oscillations of Coronal Magnetic Arcades

We investigate MHD waves in potential and force-free magnetic arcades describing bipolar active regions. The eikonal method allows us to study analytically the short waves, which are divided into Alfvén and magnetosonic waves. The eigen-modes of magnetic arcades are formed as a result of their reflection at the photosphere. The Alfvén mode oscillations of a certain frequency take place on magnetic surfaces. The fast-mode oscillations also take place on some surfaces but they are not magnetic surfaces. Both the Alfvén and fast-mode eigen-frequencies change continuously from one such surface to another. Each oscillation surface has a discrete set of eigen-frequencies.     

The Effect of a Twisted Magnetic Field on the Nature of Kink MHD Waves

We consider a pressureless plasma in a thin magnetic-flux tube with a twisted magnetic field. We study the effect of twisted magnetic field on the nature of propagating kink waves. To do this, the restoring forces of oscillations in the linear ideal magnetohydrodynamics (MHD) were obtained. In the presence of a twisted magnetic field, the ratio of the magnetic-tension force to the gradient of the magnetic pressure increases for the mode with negative azimuthal wave number, but it decreases for the mode with positive azimuthal wave number. For the kink mode with positive azimuthal mode number, the ratio of the forces is more affected by the twisted magnetic field in dense loops. For the kink mode with negative azimuthal mode number, the perturbed magnetic pressure is negligible under some conditions. The magnetic twist increases (diminishes) the damping of the kink waves with positive (negative) azimuthal mode number due to resonant absorption. Our conclusion is that introducing a twisted magnetic field breaks the symmetry between the nature of the kink waves with positive and negative azimuthal wave number, and the wave can be a purely Alfvénic wave in the entire loop.     

Linear Analysis of Axial Sheared Flow in Astrophysical Jets

A linear analysis of axial sheared flow in magnetohydrodynamic (MHD) jets with helical magnetic fields is presented. A linearized set of ideal MHD equations allows the investigation of plasmas with both magnetic shear and flow shear included in the equilibrium profile. These equations are integrated numerically by following the linear development in time of an initial seed perturbation. Global instability growth rates are obtained after the numerical solution converges to the fastest growing mode. It is shown that axial sheared flow reduces the growth of current-driven instabilities in plasma jets with constant magnetic pitch P = rB _z /B _θ.     

耀斑大尺度电流片的湍流耗散和能谱分析

磁重联在宇宙的许多动力学现象中都是非常核心的过程.磁流体动力学(MHD)数值模拟是研究磁重联过程以及相应物理图像的一种很有效的手段.通过不同的参数组合,来研究MHD数值模拟中磁雷诺数和空间分辨率对磁重联率、数值耗散和能谱分布的影响.对得到的数据进行分析后,发现磁雷诺数对磁重联率和能谱分布有一定的影响.磁雷诺数越大,磁重联过程进入非线性阶段所需的特征时间越短,磁重联率就越早发生跃升.磁雷诺数R_m对耗散开始发挥作用的Kolmogorov微观尺度l_ko有明显影响:R_m越大,l_ko就越小.研究了磁重联过程中包括数值耗散在内的额外耗散对重联过程的影响.结果表明,撕裂模不稳定性开始之前的额外耗散以纯数值耗散为主,撕裂模不稳定性出现之后,额外耗散出现同步跃升,说明不稳定性导致的湍流明显增强了耗散的效果,相当于在局部湍流区引入了超电阻.能谱分析进一步表明,大尺度电流片的l_ko完全可能出现在宏观的MHD尺度上.     

The effect of non-uniform magnetic field on the slow mode oscillations

In this paper, the slow MHD mode oscillations of the coronal plasma are studied. The aim is to identify the effect of structuring (such as magnetic field, temperature, density, and pressure) on the frequencies of oscillations. We modelled the coronal medium as a low-β plasma with longitudinally density and pressure stratifications and a weakly inhomogeneous magnetic field varied slowly with height and radial directions. The linearized ideal MHD equations reduced to a single Klein–Gordon differential equation for square of oscillatory frequencies. The eigenfunctions and analytical dispersion relations are derived. The dispersion relations were solved numerically. In the case of uniform magnetic field, the previous studies verified. Our numerical results show that, the frequencies and their ratios are very sensitive functions of pressure scale height, and slightly varying functions of inhomogeneity parameter of magnetic field. By changing the magnetic field strength between the apex and footpoints of the loop about 50%, the frequencies ratio are changed about 5%. We concluded that, the pressure scale height and temperature gradient are first order effects and inhomogeneity of magnetic field is a second order effect on the slow mode oscillations.     

The effect of density stratification on the resonant absorption of MHD waves in coronal loops

The standing quasi-modes of the ideal magnetohydrodynamics (MHD) in a zero-β cylindrical magnetic flux tube that undergoes a longitudinal density stratification and radial density structuring are considered. The radial structuring is assumed to be a linearly varying density profile. Using the relevant connection formulae of the resonant absorption, the dispersion relation for the fast MHD body waves is derived and solved numerically to obtain both the frequencies and damping rates of the fundamental and first-overtone,   k = 1, 2  , modes of both the kink  ( m = 1)  and fluting  ( m = 2)  waves, where k and m are the longitudinal and azimuthal mode numbers, respectively.     

Reflection Properties of Gravito-MHD Waves in an Inhomogeneous Horizontal Magnetic Field

We derive the dispersion equation for gravito-magnetohydrodynamical (MHD) waves in an isothermal, gravitationally stratified plasma with a horizontal inhomogeneous magnetic field. Sound and Alfvén speeds are constant. Under these conditions, it is possible to derive analytically the equations for gravito-MHD waves. The high values of the viscous and magnetic Reynolds numbers in the solar atmosphere imply that the dissipative terms in the MHD equations are negligible, except in layers around the positions where the frequency of the MHD wave equals the local Alfvén or slow wave frequency. Outside these layers the MHD waves are accurately described by the equations of ideal MHD. We consider waves that propagate energy upward in the atmosphere. For the plane boundary, z=0, between two isothermal plasma regions with horizontal but different magnetic fields, we discuss the boundary conditions and derive the equations for the reflection and transmission coefficients. In the simpler case of a gravitationally stratified plasma without magnetic field, these coefficients describe the reflection and transmission properties of gravito-acoustic waves.     

Excess heating of corona and chromosphere above magnetic regions by non-linear Alfvén waves

Excess heating of the active region solar atmosphere is interpreted by the decay of MHD slow-mode waves produced in the corona through the non-linear coupling of Alfvén waves supplied from subphotospheric layers. It is stressed that the Alfvén-mode waves may be very efficiently generated directly in the convection layer under the photosphere in magnetic regions, and that such magnetic regions, at the same time, provide the ‘transparent windows’ for Alfvén waves in regard to the Joule and frictional dissipations in the photospheric and subphotospheric layers. Though the Alfvén waves suffer considerable reflection in the chromosphere and in the transition layer, a certain fraction of this large flux is propagated out to the corona, and a large velocity amplitude exceeding the local Alfvén velocity is attained during the propagation along the magnetic tubes of force into a region of lower density and weaker magnetic field. The otherwise divergence-free velocity field in Alfvén waves gets involved in such a case with a compressional component (slow-mode waves) which again is of considerable velocity amplitude relative to the local acoustic velocity when estimated by using the formulation for non-linear coupling between MHD wave modes derived by Kaburaki and Uchida (1971). Therefore, the compressional waves thus produced through the non-linear coupling of Alvén waves will eventually be thermalized to provide a heat source. The introduction of this non-linear coupling process and the subsequent thermalization of thus produced slow-mode waves may provide means of converting the otherwise dissipation-free Alfvén mode energy into heat in the corona. The liberated heat will readily be redistributed by conduction along the magnetic lines of force, with higher density as a consequence of increased scale height, and thus the loop-like structure of the coronal condensations (or probably also the thread-like feature of the general corona) may be explained in a natural fashion.     

Effect of rotation and self-gravity on the propagation of MHD waves

Magnetohydrodynamics waves and instabilities in rotating, self-gravitating, anisotropic and collision-less plasma were investigated. The general dispersion relation was obtained using standard mode analysis by constructing the linearized set of equations. The wave mode solutions and stability properties of the dispersion relations are discussed in the propagations transverse and parallel to the magnetic field. These special cases are discussed considering the axis of rotation to be in transverse and along the magnetic field. In the case of propagation transverse to the magnetic field with axis of rotation parallel to the magnetic field, we derived the dispersion relation modified by rotation and self-gravitation. In the case of propagation parallel to the magnetic field with axis of rotation perpendicular to the magnetic field, we obtained two separate modes affected by rotation and self-gravitation. This indicates that the Slow mode and fire hose instability are not affected by rotation. Numerical analysis was performed for oblique propagation to show the effect of rotation and self-gravitation. It is found that rotation has an effect of reducing the value of the phase speeds on the fast and Alfven wave modes, but self-gravitation affect only on the Slow modes, thereby reducing the phase speed compare to the ideal magneto hydrodynamic (MHD) case.     

Resistive relaxation of a magnetically confined mountain on an accreting neutron star

Three-dimensional numerical magnetohydrodynamic (MHD) simulations are performed to investigate how a magnetically confined mountain on an accreting neutron star relaxes resistively. No evidence is found for non-ideal MHD instabilities on a short time-scale, such as the resistive ballooning mode or the tearing mode. Instead, the mountain relaxes gradually as matter is transported across magnetic surfaces on the diffusion time-scale, which evaluates to  τ_I∼ 10⁵–10⁸ yr  (depending on the conductivity of the neutron star crust) for an accreted mass of   M _a= 1.2 × 10⁻⁴ M_⊙  . The magnetic dipole moment simultaneously re-emerges as the screening currents dissipate over  τ_I  . For non-axisymmetric mountains, ohmic dissipation tends to restore axisymmetry by magnetic reconnection at a filamentary neutral sheet in the equatorial plane. Ideal-MHD oscillations on the Alfvén time-scale, which can be excited by external influences, such as variations in the accretion torque, compress the magnetic field and hence decrease  τ_I  by one order of magnitude relative to its standard value (as computed for the static configuration). The implications of long-lived mountains for gravitational wave emission from low-mass X-ray binaries are briefly explored.     

ANALYTICAL SOLUTIONS FOR CUSP RESONANCE IN DISSIPATIVE MHD

The present paper considers resonant slow waves in 1D non-uniform magnetic flux tubes in dissipative MHD. Analytical solutions are obtained for the Lagrangian displacement and the Eulerian perturbation of the total pressure for both static and stationary equilibrium states. From these analytical solutions we obtain the fundamental conservation law and the jump conditions for resonant slow waves in dissipative MHD. The validity of the ideal conservation law and jump conditions obtained by Sakurai, Goossens, and Hollweg (1991) for static equilibria and Goossens, Hollweg, and Sakurai (1992) for stationary equilibria is justified in dissipative MHD.     

Nonlinear Generation of Fluting Perturbations by Kink Mode

We study the excitation of fluting perturbations in a magnetic tube by an initially imposed kink mode. We use the ideal magnetohydrodynamic (MHD) equations in the cold-plasma approximation. We also use the thin-tube approximation and scale the dependent and independent variables accordingly. Then we assume that the dimensionless amplitude of the kink mode is small and use it as an expansion parameter in the regular perturbation method. We obtain the expression for the tube boundary perturbation in the second-order approximation. This perturbation is a superposition of sausage and fluting perturbations. The amplitude of the fluting perturbation takes its maximum at the middle of the tube, and it monotonically decreases with the distance from the middle of the tube.