Nonstationary driven oscillations of a magnetic cavity

The problem of transition to the steady state of driven oscillations in a magnetic cavity in a cold resistive plasma is addressed. The foot point driving polarized in the inhomogeneous direction is considered, and it is assumed that the cavity length in the direction of the equilibrium magnetic field is much larger than the cavity width in the inhomogeneous direction. The latter assumption enables one to neglect the variation of the magnetic pressure in the inhomogeneous direction, which strongly simplifies the analysis. The explicit solution describing the nonstationary behavior of the magnetic pressure and the velocity is obtained. This solution is used to study the properties of the transition to the steady state of oscillation. The main conclusion is that, in general, there are two different characteristic transitional times. The first time is inversely proportional to the decrement of the global mode. It characterizes the transition to the steady state of the global motion, which is the coherent oscillation of the cavity in the inhomogeneous direction. The second time is the largest of the two times, the first transitional time and the phase-mixing time, which is proportional to the magnetic Reynolds number in 1/3 power. It characterizes the transition to the steady state of the local motion, which is oscillations at the local Alfvén frequencies, and the saturation of the energy damping rate. An example from solar physics shows that, in applications, the second transitional time can be much larger than the first one

Non-axisymmetric oscillations of stratified coronal magnetic loops with elliptical cross-sections

We study non-axisymmetric oscillations of a straight magnetic tube with an elliptic cross-section and density varying along the tube. The governing equations for kink and fluting modes in the thin tube approximation are derived. We found that there are two kink modes, polarised along the large and small axes of the elliptic cross-section. We have shown that the ratio of frequencies of the first overtone and fundamental harmonic is the same for both kink modes and independent of the ratio of the ellipse axes. On the basis of this result we concluded that the estimates of the atmospheric scale height obtained using simultaneous observations of the fundamental harmonic and first overtone of the coronal loop kink oscillations are independent of the ellipticity of the loop cross-section

On the vertical and horizontal transverse oscillations of curved coronal loops

Kink oscillations of curved coronal loops with the density varying along the loop are studied in the thin tube approximation. The equilibrium magnetic field is assumed to be potential, and the field potential and flux function are used as curvilinear coordinates. It is also assumed that the loop expansion is weak, and the solution to the problem is looked for in the form of power series with respect to the tube expansion parameter lambda << 1. The main result of the study is that the eigenfrequencies of the vertical and horizontal tube oscillations are, in general, different, their difference being proportional to lambda. As an example a simple equilibrium with the magnetic field magnitude exponentially decaying with the height is considered. The implication of the obtained results for the interpretation of observational data is discussed

Stability of an MHD shear flow with a piecewise linear velocity profile

In this paper we present the results of the stability analysis of a simple shear flow of an incompressible fluid with a piecewise linear velocity profile in the presence of a magnetic field. In the flow, a finite transitional magnetic-free layer with a linear velocity profile is sandwiched by two semi-infinite regions. One of these regions is magnetic-free and the flow velocity in the region is constant. The other region is magnetic and the fluid in it is quiescent. The magnetic field is constant and parallel to the flow in the transitional layer. The fluid density is constant both in the magnetic as well as the magnetic-free regions, while it has a jump-type discontinuity at the boundary between the transitional layer and the magnetic region. The effect of gravity is included in the model, and it is assumed that the lighter fluid is overlaying the heavier one, thus no Rayleigh-Taylor instability is present. The dispersion equation governing the normal-mode stability of the flow is derived and its properties are analysed. We study stability of two cases: (i) magnetic-free flow in the presence of gravity, and (ii) magnetic flow without gravity. In the first case, the flow stability is controlled by the Rayleigh number, R. In the second case, the control parameter is the inverse squared Alfvénic Mach number, H . Stability of a particular monochromatic perturbation also depends on its dimensionless wavenumber α. We combine the analytical and numerical approaches to obtain the neutral stability curves in the (α,R)-plane in the case of the magnetic-free flow, and in the (α,H)-plane in the case of the magnetic flow. The dependence of the instability increment on R in the first case, and on H in the second case is treated. We apply the results of the analysis to the stability of a strongly subsonic portion of the heliopause. Our main conclusion is as follows: The inclusion of a transitional layer near the heliopause into the model increases by an order of magnitude the strength of the interstellar magnetic field required to stabilize this portion of the heliopause in comparison with the corresponding stabilizing strength of the magnetic field required when modelling the heliopause as a tangential discontinuity

On the validity of nonlinear Alfvén resonance in space plasmas

Aims. In the approximation of linear dissipative magnetohydrodynamics (MHD), it can be shown that driven MHD waves in magnetic plasmas with high Reynolds number exhibit a near resonant behaviour if the frequency of the wave becomes equal to the local Alfvén (or slow) frequency of a magnetic surface. This behaviour is confined to a thin region, known as the dissipative layer, which embraces the resonant magnetic surface. Although driven MHD waves have small dimensionless amplitude far away from the resonant surface, this near-resonant behaviour in the dissipative layer may cause a breakdown of linear theory. Our aim is to study the nonlinear effects in Alfvén dissipative layer Methods. In the present paper, the method of simplified matched asymptotic expansions developed for nonlinear slow resonant waves is used to describe nonlinear effects inside the Alfvén dissipative layer. Results. The nonlinear corrections to resonant waves in the Alfvén dissipative layer are derived, and it is proved that at the Alfvén resonance (with isotropic/anisotropic dissipation) wave dynamics can be described by the linear theory with great accuracy

Resonantly damped oscillations of longitudinally stratified coronal loops

Soon after coronal loop oscillations were observed by TRACE spacecraft for the first time in 1999, various theoretical models have been put forward to explain the rapid damping of the oscillations of these intriguing objects. Coronal loop oscillations are often interpreted as fast kink modes of a straight cylindrical magnetic flux tube with immovable edges modelling dense photospheric plasma at the ends of the loop. Taking this model as a basis we use cold plasma approximation and consider the tube to be thin to simplify the problem and be able to deal with it analytically. In its equilibrium state the tube is permeated by a homogeneous magnetic field directed along the tube axis. We include the effect of stratification in our model supposing that plasma density varies along the tube. There is also density inhomogeneity in the radial direction confined in a layer with thickness much smaller than the radius of the tube. Considering the system of linearized MHD equations we study the dependence of the spectrum of tube oscillations and its damping due to resonant absorption on the parameters of the unperturbed state. The implication of the obtained results on coronal seismology is discussed

The resonant damping of oscillations of coronal loops with elliptic cross-sections

Motivated by recent Transition Region and Coronal Explorer (TRACE) observations of damped oscillations in coronal loops, Ruderman & Roberts (2002), studied resonant damping of kink oscillations of thin straight magnetic tubes in a cold plasma. In their analysis, Ruderman & Roberts considered magnetic tubes with circular cross-sections. We extend their analysis for magnetic tubes with elliptic cross-sections. We find that there are two infinite sequences of the eigenfrequencies of the tube oscillations, {omega(nc)} and {omega(ns)}, n = 1,2,.... The eigenfrequencies {omega(nc)} and {omega(ns)} correspond to modes with 2n nodes at the tube boundary. In particular, omega(1c) and omega(1s) correspond to two kink modes. These modes are linearly polarized in the direction of the large and small axis of the tube elliptic cross-section respectively. The sequence {omega(nc)} is monotonically growing and {omega(ns)} monotonically decreasing, and they both tend to omega(k) as n --> infinity, where omega(k) is the frequency of the kink mode of tubes with circular cross-sections. In particular, omega(1c) < omega(k) < omega(1s). We calculate the decrements of the two kink modes and show that they are of the order of decrement of the kink mode of a tube with a circular cross-section