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 - I. Theory

This paper considers driven resonant nonlinear slow magnetohydrodynamic (MHD) waves in dissipative steady plasmas. A theory developed by Ruderman, Hollweg, and Goossens (1997) is used and extended to study the effect of steady flows on the nonlinear resonant behaviour of slow MHD waves in slow dissipative layers. The method of matched asymptotic expansions is used to describe the behaviour of the wave variables in the slow dissipative layer. The nonlinear analogue of the connection formulae for slow MHD waves obtained previously by Goossens, Hollweg, and Sakurai (1992) and Erdélyi (1997) in linear MHD, are derived. The effect of an equilibrium flow results partly in a Doppler shift of the available frequency for slow resonance and partly in the modification of the width of the dissipative layer.     

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.     

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.     

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.     

Non-adiabatic discrete Alfvén waves in coronal loops and prominences

Discrete Alfvén waves in coronal loops and prominences are investigated in non-ideal magnetohydrodynamics. The non-ideal effects included are anisotropic, thermal conduction, and optically thin radiation. The classic ideal Alfvén continuum is not altered by these non-ideal effects, but the discrete Alfvén modes, which exist under certain conditions above or below the Alfvén continuum in ideal MHD, are shown to be influenced by non-adiabatic effects.The existence of discrete, non-adiabatic Alfvén waves, and their damping and overstability are examined for 1D cylindrical equilibrium states with twisted magnetic fields. First, analytic results are obtained for modes of high radial order by means of a WKB-analysis. The subspectrum of discrete Alfvén modes is computed with a numerical code, with particular emphasis on the modes of low radial order. The results show that discrete Alfvén waves are of potential importance for solar applications and also that the information obtained with the WKB-analysis is of limited use in this context.Research Assistant of the Belgian National for Scientific Research.     

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.     

Absorption of acoustic waves in monolithic and fibril sunspot models

It has been hypothesized that the observation of substantial absorption of acoustic power in the vicinity of sunspots may be explained by the transformation of acoustic oscillations into highly damped shear Alfvén waves in thin resonant layers. Analytical estimates of the efficiency of this process (Hollweg, 1988) are compared with direct one-dimensional numerical simulations of absorption by a magnetic barrier in a viscous medium. After slight modification, the estimate is found to give a good approximation to the numerical absorption rate.Further calculations are performed for scattering from a magnetic field of fibril structure. Such models are better able to explain the spatial structure of the absorbing region implied by the observations. It is found that the existence of a multiplicity of surfaces at which resonant absorption occurs can considerably increase the total energy absorption coefficient. Resonant effects involving the multiple reflection of acoustic waves within such structures can also lead to enhanced absorption. Fibril models, therefore, produce significantly increased absorption over a wide range of plausible parameter values, and are a more plausible explanation for the observed p-mode scattered power deficit than resonant absorption in a monolithic structure.     

SUMER Observations of the Evolution and the Disappearance of a Solar Prominence

The mechanisms that lead to the formation and the disappearance of prominences are poorly understood, at present. An arch-shaped prominence was observed with the Solar Ultraviolet Measurements of Emitted Radiation (SUMER) spectrometer on board the Solar and Heliospheric Observatory (SOHO) on 31 March–1 April 1996. The observations were performed at three wave-bands in the Lyman continuum. Ten successive images were obtained at 41-minute time intervals. Based on computed models of Gouttebroze, Heinzel, and Vial (1993), we have determined the temperature distribution of the prominence using the intensity ratio of 876 Å and 907 Å. The observed time sequence shows that parts of the prominence disappear possibly by heating, while other parts exhibit heating and cooling with apparent outward motion. We model the heat input with the linearized MHD equations using a prescribed initial density and a broad-band spectrum of Alfvén waves. We find a good qualitative agreement with observations. In the model the prominence is heated by the resonant absorption of Alfvén waves with frequencies that match the resonant condition for a particular flux tube structure that is determined by the magnetic field topology and plasma density.