Relation between Magnetic Fields and Kink Oscillations with Flows

A. Satya Narayanan

Indian Institute of Astrophysics, Bangalore – 560 012, India

*Corresponding Author E-mail: satya@iiap.res.in

ABSTRACT:

In this paper, we discuss a method by which Magnetic Fields which are prevalent in the Sun’s Atmosphere, namely, the Corona, can be calculated with the Kink Oscillations, which is basically an MHD (Magneto Hydro Dynamic) wave. Kink Oscillations are found in structured media, in particular in cylindrical geometries, which are modeled in the Coronal Loops. The MHD equations are linearized and the solution of the resulting wave equation, is written in terms of Bessel Functions. The boundary conditions (continuity of velocity and total pressure ) at the interface of the cylinder are applied. The resulting dispersion relation is a complicated transcendental function. Analytical solutions of this relation are hard to find. However, in the limit of large wave numbers, ka → ꝏ, the dispersion relation can be simplified and an analytical solution of this relation can be found. Using the phase velocity of the Kink Oscillations, and a few parameters such as the loop length, the density ratio and the frequency (or period), one can determine the magnetic field. The dispersion relation with the effect of uniform flows has also been derived. Limiting cases of incompressible flows, curvature effects and no flows are presented briefly.

INTRODUCTION :

The corona is highly structured in the magnetic field, plasma density and temperature. In this study, we model a coronal loop to be made up of a cylindrical tube of constant cross section. The different modes of coronal oscillations has different properties such as dispersion relations, characteristic speeds, excitation conditions and observational manifestation. In this paper we have discussed about kink oscillations and the dispersion relation assuming plasma-β to be small. The phase speed of the kink mode can be used as a diagnostic for determining the magnetic field of corona.

MHD waves and oscillations have been observed for a long time in radio and optical bands. Their presence in the corona is known both from theoretical predictions and observations such as SOHO and TRACE. The magnetic field structures exist on the corona, often called as coronal loop and these coronal oscillations are known as kink and sausage mode oscillations. The effect of flow on kink oscillations is one of the main aims of this study. There have been several studies on MHD waves in the corona in the context of coronal heating and acceleration of the solar wind, both theoretically as well as from an observational point of view. These waves also play an important role in the solar-terrestrial connections. Observations of MHD oscillations and waves have been done in almost all possible bands in the past few decades, in particular radio pulsations (Aschwanden 1987, Aschwanden et al. 1999).

Significant progress in MHD wave theory has been made in the recent past (see reviews of Roberts 2000, Nakariakov 2003). The discussion of other aspects of coronal oscillations and waves can be found, in particular, wave theory (Goossens 1991), observations (Aschwanden et. al 1999) and prominence oscillations (Oliver 1999).

Coronal loops are anchored in the dense plasma of the photosphere, so it is resonable to assume that any motions in the corona are effectively zero at the base of a loop. This condition may be met by specifying the wave number along the loop. Coronal loop is assumed to be a straight cylindrical tubeand also an uniform flow of the plasma inside the tube of radius ‘a’ as shown in Figure 1. The case of an infinite, uniform plasma in a uniform magnetic field provides a guide to the behaviour of MHD waves. We have considered a flux tube having radius much smaller than the wavelength of a disturbance, in which two characterstic speeds of propogation have been identified namely tube speed and Alfven velocity.

Figure 1. The Basic Model of a Flux Tube in the form of a cylinder

The cylindrical flux tube model is presented in Figure 1, which has a uniform magnetic field and flow inside the tube, while the magnetic field is different outside the tube. Also the pressure and the density are assumed to be uniform but different inside and outside the tube. Small amplitude perturbations (linear wave theory) are studied about the basic equilibrium state, given by

Basic Equations of Motion:

The basic equations of motin are the conservation of mass, conservation of linear momentum, magnetic induction, energy equation, which are well known in the literature .

The Dispersion Relation

The dispersion relation for a system is derived by the straight forward method of expanding the pertrubation inside and outside the cylinder in the normal modes and matching these modes across cylindrical vortex sheet. Considering cylindrical coordinate system . For the mode which varies as

f(r,ø,z,t) = f(r)exp[i(kz+lø-z)] (6)

where k is the axial wave number, l is the azimuthal wave number and ω is the angular frequency, we can show that the radial dependence of the flow variables satisfies the Bessel differential equation.

Consider a straight magnetic flux tube or slab of strength with cross- sectional area , filled by a plasma of density and kinetic pressure .

By simple algebraic simplifications, one can show that the dispersion relation for the cylindrical geometry with flows ( Satya Narayanan, 1990, Satya Narayanan, 2013) as

Figure 6. The phase speed as a function of the density ratio for the kink mode

CONCLUSIONS:

In this study, we have discussed the relation between the magnetic field of the Sun and its relationship with the kink oscillations, with flows. The dispersion relation has been solved under the assumption that the wavenumber (normalized) . We have also invoked the low beta plasma approximation, which is very justified, considering that the outer atmosphere of the Sun (corona) has a very low density, so that the inertial forces are small compared to the magnetic forces. The study is far from complete. We hope to continue the study with the effects of thermal and viscous dissipaton, so that the problem of coronal heating will also be addressed. This will be taken up in the near future.