## 3. The evolutionary equation for slow wavesFor 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 where Eq. (5) provides us with the linear expression for perturbations of the temperature: 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
Eqs. (9)-(11) can be combined into the wave equation where is the local density scale height, and 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 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. Under assumptions (20), the wave evolution is slow (with respect to the wave period), which allows us to apply the method of slowly varying amplitude. We change the independent variables 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 In the running frame of reference (21), Eq. (19) is re-written as Perturbations of other physical values, expressed through 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), Eq. (27) is the modified Burgers equation, which takes into account nonlinearity, viscosity and thermal conductivity, stratification and structuring of the plasma. It is convenient to use the normalized variables , , , and . In the normalized variables, Eq. (27) is re-written as is the normalized dissipation coefficient, and . 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 © European Southern Observatory (ESO) 2000 Online publication: October 30, 2000 |