3. The linearized MHD equations
In the presence of linear perturbations it is not any longer true that the plasma can be taken to be ideal everywhere. We shall see that dissipation cannot be ignored near the possible resonant points due to the local quasi-singular behaviour of the solutions. In our model we shall include the Ohmic heating as the only dissipation mechanism and exclude the effects of viscosity and thermal conduction.
The standard set of linearized MHD equations for a resistive plasma is
As the equilibrium quantities depend on the x -coordinate only, the perturbed quantities are Fourier analyzed with respect to y, z and t. The amplitudes of the corresponding Fourier components remain dependent:
Because of the very high values of the magnetic Reynolds number, e.g. for the solar coronal conditions, the dissipation due to the finite electrical resistivity can be ignored except in narrow layers of steep gradients (e.g. around resonances). Outside these dissipative layers the Eqs. (6) reduce to the following two coupled first order differential equations for the perturbations of the normal component of the Lagrangian displacement and of the Eulerian perturbation of total pressure P:
can be rewritten as
where and are known as the cut-off frequencies.
The inclusion of the equilibrium flow V causes only a replacement of the frequency by the Doppler shifted frequency in the equations with respect to the static case. (see e.g. Goossens, Sakurai & Hollweg 1992).
The set of ordinary differential Eqs. (7) has mobile regular singularities at the positions and/or where vanishes:
As both and are functions of x, they define two continuous ranges of frequencies referred to as the Alfvén continuum and the slow continuum respectively. These continua are now Doppler shifted due to the presence of the flow as indicated by Eqs. (8).
>From a physical point of view, the conditions (8) mean that the eigenmodes resonantly interact with one of the two continua and also with the flow. The interaction with the continua in the absence of the flow causes damping of the eigenmode due to resonant wave transformation and we talk about quasi-modes with complex eigenfrequency with in this case (Tirry & Goossens 1996 and references therein). If however a flow is present then it not only Doppler shifts the continuum frequencies but it can also affect the energy of the eigenmodes. At the resonance, the flow can namely drain energy away from the surface modes, which additionally increases the wave damping, but the flow can also be an energy source in which case the surface modes gain energy and become unstable, i.e. .
© European Southern Observatory (ESO) 1998
Online publication: March 23, 1998