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.     

Dissipative MHD solutions for resonant AlfvÉn waves in 1-dimensional magnetic flux tubes

The present paper extends the analysis by Sakurai, Goossens, and Hollweg (1991) on resonant Alfvén waves in nonuniform magnetic flux tubes. It proves that the fundamental conservation law for resonant Alfvén waves found in ideal MHD by Sakurai, Goossens, and Hollweg remains valid in dissipative MHD. This guarantees that the jump conditions of Sakurai, Goossens, and Hollweg, that connect the ideal MHD solutions for _r , andP across the dissipative layer, are correct. In addition, the present paper replaces the complicated dissipative MHD solutions obtained by Sakurai, Goossens, and Hollweg for _r , andP in terms of double integrals of Hankel functions of complex argument of order with compact analytical solutions that allow a straightforward mathematical and physical interpretation. Finally, it presents an analytical dissipative MHD solution for the component of the Lagrangian displacement in the magnetic surfaces perpendicular to the magnetic field lines which enables us to determine the dominant dynamics of resonant Alfvén waves in dissipative MHD.     

Resonant Absorption of Nonlinear Slow MHD Waves in Isotropic Steady Plasmas – II. Application: Resonant Acoustic Waves

Nonlinear theory of driven magnetohydrodynamic (MHD) waves in the slow dissipative layer in isotropic steady plasmas developed by Ballai and Erdélyi (Solar Phys. 180 (1998)) is used to study the nonlinear interaction of sound waves with one-dimensional isotropic steady plasmas. An inhomogeneous magnetic slab with field-aligned plasma flow is sandwiched by a homogeneous static magnetic-free plasma and by a homogeneous steady magnetic plasma. Sound waves launched from the magnetic-free plasma propagate into the inhomogeneous region interacting with the localised slow dissipative layer and are partially reflected, dissipated or transmitted by this region. The nonlinearity parameter, introduced by Ballai and Erdélyi, is assumed to be small and a regular perturbation method is used to obtain analytical wave solutions. Analytical studies of resonant absorption of sound waves show that the efficiency of the process of resonant absorption strongly depends on both the equilibrium parameters and the characteristics of the resonant wave. We also find that a steady equilibrium shear flow can significantly influence the nonlinear resonant absorption in the limits of thin inhomogeneous layer and weak nonlinearity. The presence of an equilibrium flow may therefore be important for the nonlinear resonant MHD wave phenomena. A parametric analysis also shows that the nonlinear part of resonant absorption can be strongly enhanced by the equilibrium flow.     

Viscous computations of resonant absorption of MHD waves in flux tubes by fem

A numerical code is presented for computing the stationary state of resonant absorption of MHD waves in cylindrical flux tubes in linear, compressible, and viscous MHD. The full viscosity stress tensor is included in the code with the five viscosity coefficients as given by Braginskii (1965). Also non-zero plasma pressure effects are taken into account, and the finite elements discretization with the Galerkin method has been used. The implementation of the stress tensor and the numerical accuracy of the tensorial viscous MHD code are scrutinized in test case. The test case involves the absorption of waves in cylindrical flux tubes considered by Lou (1990) and Goossens and Poedts (1992) in the context of absorption of acoustic oscillations. The results for the absorption rates obtained with the tensorial viscous code agree completely with the results obtained by Lou in a scalar viscous MHD and by Goossens and Poedts in resistive MHD. This verifies not only the complicated tensor viscous code but again proves that the absorption rate is independent of the actual dissipation mechanism.     

Nonlinear wave propagation along a magnetic flux tube

The weakly nonlinear wave propagation of a slow sausage surface wave traveling along a magnetized slab with a thin nonuniform boundary layer is considered. The ideal incompressible MHD equations are used and the nonlinearities are assumed to be due to second harmonic generation. A nonlinear dispersion relation and the related nonlinear Schrödinger equation is derived. The existence of a continuous thin interface leads to sharply peaked field amplitudes due to resonant interaction with local Alfvén waves. It is shown that the nonlinear effects from processes within the thin layer are much more important than those from the main slab. Furthermore, the nonlinear interaction with local Alfvén waves yields a nonlinear damping rate of the wave that is much larger than the linear damping rate when the transition layer is sufficiently thin.     

Resonant behaviour of MHD waves on magnetic flux tubes

The resonances that appear in the linear compressible MHD formulation of waves are studied for equilibrium states with flow. The conservation laws and the jump conditions across the resonance point are determined for 1D cylindrical plasmas. For equilibrium states with straight magnetic field lines and flow along the field lines the conserved quantity is the Eulerian perturbation of total pressure. Curvature of the magnetic field lines and/or velocity field lines leads to more complicated conservation laws. Rewritten in terms of the displacement components in the magnetic surfaces parallel and perpendicular to the magnetic field lines, the conservation laws simply state that the waves are dominated by the parallel motions for the modified slow resonance and by the perpendicular motions for the modified Alfvén resonance.The conservation laws and the jump conditions are then used for studying surface waves in cylindrical plasmas. These waves are characterized by resonances and have complex eigenfrequencies when the classic true discontinuity is replaced by a nonuniform layer. A thin non-uniform layer is considered here in an attempt to obtain analytical results. An important result related to earlier work by Hollweg et al. (1990) for incompressible planar plasmas is found for equilibrium states with straight magnetic field lines and straight velocity field lines. For these equilibrium states the incompressible and compressible surface waves have the same frequencies at least in the long wavelength limit and there is an exact correspondence with the planar case. As a consequence, the conclusions formulated by Hollweg et al. still hold for the straight cylindrical case. The effects of curvature are subsequently considered.     

Resonant Damping of Propagating Kink Waves in Time-Dependent Magnetic Flux Tube

We explore the notion of resonant absorption in a dynamic time-dependent magnetised plasma background. Very many works have investigated resonance in the Alfvén and slow MHD continua under both ideal and dissipative MHD regimes. Jump conditions in static and steady systems have been found in previous works, connecting solutions at both sides of the resonant layer. Here, we derive the jump conditions in a temporally dependent, magnetised, inhomogeneous plasma background to leading order in the Wentzel–Kramers–Billouin (WKB) approximation. Next, we exploit the results found in Williamson and Erdélyi (Solar Phys. 289, 899, 2014) to describe the evolution of the jump condition in the dynamic model considered. The jump across the resonant point is shown to increase exponentially in time. We determined the damping as a result of the resonance over the same time period and investigated the temporal evolution of the damping itself. We found that the damping coefficient, as a result of the evolution of the resonance, decreases as the density gradient across the transitional layer decreases. This has the consequence that in such time-dependent systems resonant absorption may not be as efficient as time progresses.     

Nonlinearity effects on resonant absorption of surface Alfvén waves in incompressible plasmas

The resonant absorption of small amplitude surface Alfvén waves is studied in nonlinear incompressible MHD for a viscous and resistive plasma. The reductive perturbation method is used to obtain the equation that governs the spatial and temporal behaviour of small amplitude nonlinear surface Alfvén waves. Numerical solutions to this equation are obtained under the initial condition that att = 0 the spatial variation is purely sinusoidal. The numerical results show that nonlinearity accelerates the wave damping due to resonant absorption. Resonant absorption is a more efficient wave damping mechanism than can be anticipated on the basis of linear theory.     

Resonant Absorption of Alfvén Waves in Steady Coronal Loops

The effect of equilibrium flow on linear Alfvén resonances in coronal loops is studied in the compressible viscous MHD model. By means of a finite element code, the full set of linearised driven MHD equations are solved for a one-dimensional equilibrium model in which the equilibrium quantities depend only on the radial coordinate. Computations of resonant absorption of Alfvén waves for two classes of coronal loop models show that the efficiency of the process of resonant absorption strongly depends on both the equilibrium parameters and the characteristics of the resonant wave. We find that a steady equilibrium shear flow can also significantly influence the resonant absorption of Alfvén waves in coronal magnetic flux tubes. The presence of an equilibrium flow may therefore be important for resonant Alfvén waves and coronal heating. A parametric analysis also shows that the resonant absorption can be strongly enhanced by the equilibrium flow, even up to total dissipation of the incoming wave.     

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.     

Linear and non-linear MHD wave propagation in steady-state magnetic cylinders

The propagation of linear and non-linear magnetohydrodynamic (MHD) waves in a straight homogeneous cylindrical magnetic flux tube embedded in a homogeneous magnetic environment is investigated. Both the tube and its environment are in steady state. Steady flows break the symmetry of forward (field-aligned) and backward (anti-parallel to magnetic field) propagating MHD wave modes because of the induced Doppler shifts. It is shown that strong enough flows change the sense of propagation of MHD waves. The flow also induces shifts in cut-off values and phase-speeds of the waves. Under photospheric conditions, if the flow is strong enough, the slow surface modes may disappear and the fast body modes may become present. The crossing of modes is also observed due to the presence of flows. The effect of steady-state background has to be considered particularly carefully when evaluating observation signatures of MHD waves for diagnostics in the solar atmosphere.     

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.     

Viscous Damping of Non-Adiabatic MHD Waves in an Unbounded Solar Coronal Plasma

The damping of MHD waves in solar coronal magnetic field is studied taking into account thermal conduction and compressive viscosity as dissipative mechanisms. We consider viscous homogeneous unbounded solar coronal plasma permeated by a uniform magnetic field. A general fifth-order dispersion relation for MHD waves has been derived and solved numerically for different solar coronal regimes. The dispersion relation results three wave modes: slow, fast, and thermal modes. Damping time and damping per periods for slow- and fast-mode waves determined from dispersion relation show that the slow-mode waves are heavily damped in comparison with fast-mode waves in prominences, prominence–corona transition regions (PCTR), and corona. In PCTRs and coronal active regions, wave instabilities appear for considered heating mechanisms. For same heating mechanisms in different prominences the behavior of damping time and damping per period changes significantly from small to large wavenumbers. In all PCTRs and corona, damping time always decreases linearly with increase in wavenumber indicate sharp damping of slow- and fast-mode waves.     

Total resonant absorption of acoustic oscillations in sunspots

The question of total resonant absorption of acoustic oscillations in sunspots is studied for cylindrical 1-D flux tubes that are stratified only in the radial direction and surrounded by a uniform, non-magnetic plasma. The numerical investigation of Goossens and Poedts (1992) in linear resistive MHD is taken further by increasing the strength of the azimuthal magnetic field in the equilibrium flux tubes. For relatively strong azimuthal magnetic fields, total absorption is found over a relatively wide range of spot radii.     

On Ponderomotive Effects Induced by Alfvén Waves in Inhomogeneous 2.5D MHD Plasmas

Where spatial gradients in the amplitude of an Alfvén wave are non-zero, a nonlinear magnetic-pressure gradient acts upon the medium (commonly referred to as the ponderomotive force). We investigate the nature of such a force in inhomogeneous 2.5D MHD plasmas by analysing source terms in the nonlinear wave equations for the general case of inhomogeneous B and ρ, and consider supporting nonlinear numerical simulations. Our equations indicate that there are two distinct classes of ponderomotive effect induced by Alfvén waves in general 2.5D MHD, each with both a longitudinal and transverse manifestation. i) Geometric effects: Gradients in the pulse geometry relative to the background magnetic field cause the wave to sustain cospatial disturbances, the longitudinal and transverse daughter disturbances – where we report on the transverse disturbance for the first time. ii) ?(c _A) effects: Where a pulse propagates through an inhomogeneous region (where the gradients in the Alfvén-speed profile c _A are non-zero), the nonlinear magnetic-pressure gradient acts to accelerate the plasma. Transverse gradients (phase mixing regions) excite independently propagating fast magnetoacoustic waves (generalising the result of Nakariakov, Roberts, and Murawski (Solar Phys. 175, 93, 1997)) and longitudinal gradients (longitudinally dispersive regions) perturb along the field (thus creating static disturbances in β=0, and slow waves in β≠0). We additionally demonstrate that mode conversion due the nonlinear Lorentz force is a one-way process, and does not act as a mechanism to nonlinearly generate Alfvén waves due to propagating magnetoacoustic waves. We conclude that these ponderomotive effects are induced by an Alfvén wave propagating in any MHD medium, and have the potential to have significant consequences on the dynamics of energy transport and aspects of dissipation provided the system is sufficiently nonlinear and inhomogeneous.     

Magnetohydrodynamics in superconducting-superfluid neutron stars

Magnetohydrodynamic (MHD) equations are presented for the mixture of superfluid neutrons, superconducting protons and normal electrons believed to exist in the outer cores of neutron stars. The dissipative effects of electron viscosity and mutual friction resulting from electron-vortex scattering are also included. It is shown that Alfvén waves are replaced by cyclotron-vortex waves that have not been previously derived from MHD theory. The cyclotron-vortex waves are analogous to Alfvén waves with the tension arising from the magnetic energy density replaced by the vortex energy density. The equations are then put into a simplified form useful for studying the effect of the interior magnetic field on the dynamics. Of particular interest is the crust–core coupling time, which can be inferred from pulsar glitch observations. The hypothesis that cyclotron-vortex waves play a significant role in the core spin-up during a glitch is used to place limits on the interior magnetic field. The results are compared with those of other studies.     

A Global Model of MHD Wave Support of a Stratified Molecular Cloud

We perform numerical simulations of nonlinear MHD waves in a gravitationally stratified molecular cloud that is bounded by a hot and tenuous external medium, within a 1.5-dimensional approximation. Under the influence of a driving source of Alfvénic disturbances, the cloud is lifted up by the pressure of MHD waves and reaches a steady state characterized by oscillations about a new time-averaged equilibrium state. The nonlinear effect results in the generation of longitudinal motions and many shock waves. Models of an ensemble of clouds show that, for various strengths of the input energy, the velocity dispersion in the cloud σ ∝ Z ^0.5, where Z is a characteristic size of the cloud. Furthermore, σ is always comparable to the mean Alfvén velocity of the cloud, consistent with observational results.     

Numerical simulation of slow shocks in the solar wind

The evolutionary state of slow forward shock waves is examined with the use of two MHD numerical codes. Our study is intended to be exploratory rather than a detailed parametric one. The first code is one-dimensional (with three components of velocity and magnetic field) which is used to follow a slow shock that propagates into a positive gradient of density versus distance. It is found that the slow shock evolves into an extraneous (intermediate) shock wave. The second code has a spherical, one-dimensional, planar geometry (with two velocity and magnetic field components) which is used to follow a spiral interplanetary magnetic field. It is found that a slow shock type perturbation can generate a forward slow shock; a fast forward shock is generated in the front of the slow shock; a contact discontinuity is formed behind the slow shock, and a compound nonlinear MHD wave is formed behind the contact discontinuity with a fast reverse shock formed further behind. Thus, we demonstrate that the evolution of a slow shock into (solely) a fast shock, as suggested by Whang (1987), is much more complicated.