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Astron. Astrophys. 362, 1151-1157 (2000)

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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

[EQUATION]

[EQUATION]

[EQUATION]

where p, [FORMULA] and V are perturbations of pressure, density and velocity. Nonlinear and dissipative terms are gathered on the right handsides of the equations,

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Eq. (5) provides us with the linear expression for perturbations of the temperature:

[EQUATION]

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 [FORMULA] is independent of the coordinate z. Additional terms, connected with the modification of [FORMULA] 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.

Eqs. (9)-(11) can be combined into the wave equation

[EQUATION]

According to (8),

[EQUATION]

where [FORMULA] is the local density scale height, and

[EQUATION]

Using expressions (17) and (18), Eq. (16) can be re-written as

[EQUATION]

[EQUATION]

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,

[EQUATION]

[EQUATION]

where [FORMULA] is the wavelength, [FORMULA] is the density scale height and [FORMULA] is the density perturbation. The last inequality can also be represented as [FORMULA]. 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 [FORMULA], and decreases to zero near the loop apex. We observe that the ratio of the fourth term to the third term is about [FORMULA] 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

[EQUATION]

where [FORMULA] 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 [FORMULA] 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

[EQUATION]

[EQUATION]

with

[EQUATION]

[EQUATION]

[EQUATION]

Perturbations of other physical values, expressed through V,

[EQUATION]

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 [FORMULA] are functions of the "slow" coordinate Z and integrating Eq. (22) with respect to [FORMULA], we obtain the evolutionary equation for the density perturbations

[EQUATION]

[EQUATION]

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 [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. In the normalized variables, Eq. (27) is re-written as

[EQUATION]

where

[EQUATION]

is the normalized dissipation coefficient,

[EQUATION]

where [FORMULA], and

[EQUATION]

and [FORMULA].

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.

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000
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