2. The model
We consider a spherically stratified atmosphere permeated by a magnetic field, which models an open magnetic structure of a coronal hole. The magnetic field is assumed to be strictly radial and described by the function
where the coordinate is the solar radius. Of course, we consider the domain only.
The atmosphere is assumed to be isothermal, with constant temperature T and sound speed .
Both the scale height H and sound speed are determined by the temperature T. In the solar corona, one can use estimations and .
There are three independent spatial scales in the problem considered, the radius of the Sun , the wavelength and the density scale height H. The characteristic scale of the Alfvén speed variations which can be defined as , can be expressed through and H as
For typical coronal conditions and, consequently, . Also, in the spatial domain considered, we can take . Under these assumptions, we obtain .
As the governing set of equations, magnetohydrodynamics is used,
where is the kinematic viscosity and other notations are standard. Only shear viscosity, which affects Alfvén waves, is taken into account. Although compressive viscosity can affect the Alfvén waves indirectly, affecting compressive waves generated nonlinearly by the Alfvén waves, the effect is weak and is not considered here. We do not have reliable information about the radial dependence of the viscosity coefficient in coronal holes so here the viscosity is assumed to be constant. Also, the induction equation should contain resistivity. However, viscosity and resistivity affect the steepening of the Alfvén waves in a similar way (Ofman et al. 1994). Consequently, finite resistivity does not introduce new physical effects and can be neglected with respect to the viscosity when the Lundquist number is much larger than the Reynolds number. This may occur in the corona if small scale turbulence is present, and the viscosity is enhanced. In these studies we neglect the effects of finite resistivity on Alfvén waves.
Eq. (5) have to be supplemented by the equation of state, . In this model the effects of kinetic pressure anisotropy are neglected since they are believed to be small for the coronal Alfvén waves (e.g. Nakariakov & Oraevsky 1995).
© European Southern Observatory (ESO) 2000
Online publication: December 17, 1999