2. Physical model
A coronal loop is modelled as a static, straight, gravitationless plasma slab with thickness b, obeying the standard set of ideal linearized MHD equations. In the Cartesian coordinate system we use, the x-coordinate corresponds to the radial direction (the direction of the inhomogeneity in the equilibrium), the y-coordinate is the (ignorable) azimuthal coordinate and the z-coordinate represents the direction along the loop.
At z we impose a given footpoint motion whereas at z we assume the loop to be line-tied. This can be done without any loss of generality because of the principle of superposition for solutions of linear equations. The boundary planes model the sharp transition from the corona to the photosphere (i.e. transition region, chromosphere and photosphere). We refer to these boundary planes as the 'photospheric edges' of the loop and we implicitly assume that a disturbance initiated in the photosphere indeed reaches the corona. In the radial direction we assume for mathematical tractability rigid wall conditions at x and x. The two boundaries x and x correspond to the interior of the loop and the exterior coronal environment respectively.
The plasma is permeated by a uniform magnetic field () and has a uniform pressure p which we neglect in comparison with the magnetic pressure ('zero-beta-approximation'). The inhomogeneity of the plasma is introduced by a continuously varying density
which models the higher density inside the loop.
The plasma is being shaken by small-amplitude perturbations at the footpoints of the magnetic field lines on the z plane. As long as non-linear and non-ideal effects are negligible we can follow the temporal evolution of the excited MHD waves inside the loop by solving the linear ideal MHD equations. These equations reduce for a pressureless plasma to
where is the Lagrangian displacement and the Alfvén speed v is given by
This coupled system of partial differential equations in and describes the coupled fast-Alfvén waves. Slow waves are absent () because the plasma pressure is neglected.
Since the equilibrium quantities are constant in the y-coordinate which runs over an infinite interval, we can Fourier analyse with respect to y. For the Fourier component corresponding to wave number k, the time evolution and the spatial variation in x and z are described by
For k these equations are decoupled: Eq. (4) describes the evolution of the fast waves, whereas Eq. (5) governs the behaviour of the Alfvén waves. In this paper we shall focus on y-independent fast waves excited by stochastic radially polarized footpoint motions. For the rest of the paper, length, speed, magnetic field strength and density are non-dimensionalized with respect to b, v, B and respectively.
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998