Astron. Astrophys. 362, 1151-1157 (2000)

2. The model and governing equations

As a first step, we consider a semi-circular loop of constant cross-section with the curvature radius . The loop is filled with a magnetized isothermal plasma. In our model, we restrict our attention to strictly longitudinal, with respect to the magnetic field, perturbations. In the following, we neglect 2D effects such as the loop curvature and twisting, and transversal structuring. Consequently, we can consider the loop as a straight cylinder, confined between two planes representing the footpoints. The geometry of the problem is shown in Fig. 1.

Fig. 1. The sketch of the model considered. A coronal loop is considered as a straight magnetic field line with density and gravitational acceleration varying along the axis of the cylinder.

The plasma is supposed to be much smaller than unity. As we consider purely parallelly propagating waves, there are two magnetohydrodynamic wave modes in the model, Alfvén waves propagating with the Alfvén speed , and slow magnetoacoustic waves degenerated to the pure acoustic waves, with the sound speed . The Alfvén waves have to be excluded from the consideration, because (a) their speed in the corona is much higher than the observed speed of the running disturbances and (b) the waves are almost incompressible and are not able to create the emission intensity perturbations observed. On the other hand, the slow magnetoacoustic waves are the primary candidates for the interpretation, because they do perturb the plasma density, creating the emission intensity variations and their speed is about the observed propagation speed.

We consider a slow wave propagating strictly along the magnetic field, in the z-direction. The plasma motions are described by the equations

where is the plasma density, V is the longitudinal speed, p is the plasma pressure, T is plasma temperature, is the adiabatic index, is thermal conductivity along the magnetic field, is the compressive viscosity coefficient, g is the projection of the gravitational acceleration on the z-axis,

with G is the gravitational constant, is the loop radius and and are solar radius and mass, respectively. In Eq. (3) we neglect radiative losses and heating terms. Connection of T with and p can be obtained from the ideal gas law,

where is the gas constant.

The magnetic field guides the waves, but is not explicitly presented in the governing equations. This is because we consider strictly longitudinal waves only. The waves do not perturb the field and their speed is independent of the field strength. Consequently, the magnetic field is absent from the governing equations of our model.

The stationary density and pressure are connected with each other by relations

which follows from Eq. (1) and the state equation

We restrict our attention to consideration of the isothermal loops with the stationary temperature , giving . The density profile along the loop, following from (4), (6) and (7) is

© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000
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