3. The evolutionary equation for slow waves
For the following analysis we assume that the effects of nonlinearity and dissipation are both weak. Weakly nonlinear and weakly dissipative perturbations of the stationary state are described by the equations
In Eq. (3) and consequent Eqs. (11) and (14), we took into account that the background temperature assumed to be constant and, therefore, the thermal conductivity is independent of the coordinate z. Additional terms, connected with the modification of by perturbations of the temperature, are of higher orders as the thermal conductivity itself is assumed to be small. Consequently, the modification of the thermal conduction by the temperature perturbations is negligible.
Using expressions (17) and (18), Eq. (16) can be re-written as
Wave equation (19) describes two slow magnetoacoustic waves propagating in opposite directions. If the nonlinear and dissipative terms on the right handside of (19) and the "inhomogeneous", two last terms on the left handside are all zero, the waves propagating in the opposite directions are decoupled with each other. We restrict our attention to one of the waves, which propagates in the positive direction of z. In the absence of the nonlinearity, dissipation and inhomogeneity, the wave propagates with a constant amplitude and shape in the running frame of reference. The inhomogeneity, dissipation and nonlinearity affect the wave parameters leading to evolution of the wave. We assume that effects of inhomogeneity, dissipation and nonlinearity are weak. These assumptions are expressed by the non-equalities,
where is the wavelength, is the density scale height and is the density perturbation. The last inequality can also be represented as . Inequalities (20) are supposed to be fulfilled at any time and in any point of the domain considered.
The fourth term on the lefthand side of (19) reaches a maximal value near the footpoints , and decreases to zero near the loop apex. We observe that the ratio of the fourth term to the third term is about and, according to the first inequality from (20), the fourth term can be neglected with respect to the third term.
where is a small parameter of order of the inhomogeneity, dissipation and nonlinearity. Note, that the three factors of evolution are not necessary to be of the same order, but each of them is at least of order of less than the first two terms on the lefthand side of Eq. (19) (the "wave" terms).
In the running frame of reference (21), Eq. (19) is re-written as
were used in the derivation of Eqs. (22)-(25). Note that only linear expressions were applied, because we restrict our attention to quadratically nonlinear processes.
Taking that, according to (20), H and are functions of the "slow" coordinate Z and integrating Eq. (22) with respect to , we obtain the evolutionary equation for the density perturbations
Eq. (27) is the modified Burgers equation, which takes into account nonlinearity, viscosity and thermal conductivity, stratification and structuring of the plasma.
Solutions of Eq. (28) allows us to determine behaviour of other physical values, using expressions (26). The relative perturbations of density, pressure and temperature show the same behaviour as V, but with the different coefficients of proportionality.
© European Southern Observatory (ESO) 2000
Online publication: October 30, 2000