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Astron. Astrophys. 353, 741-748 (2000)

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3. The evolutionary equation

Linearly polarized Alfvén waves propagating along the radial magnetic field are described by the equations

[EQUATION]

and

[EQUATION]

where [FORMULA] and [FORMULA] are transverse components of the perturbation velocity and magnetic field, respectively, and

[EQUATION]

[EQUATION]

[EQUATION]

Eqs. (6) and (7) have to be supplemented by equations for perturbations of the density and the longitudinal component of the velocity,

[EQUATION]

[EQUATION]

Here we assumed that the perturbations of [FORMULA] and [FORMULA] were initially absent from the system, and were generated nonlinearly by the Alfvén wave. Consequently, only nonlinear terms containing [FORMULA] and [FORMULA] had to be taken into account. Second order nonlinear terms containing [FORMULA] and [FORMULA] are neglected, because they are responsible for fourth and higher order nonlinear effects on the Alfvén waves, which are not considered in these studies. Also, we neglect the effects of viscosity on the forced perturbations of density and longitudinal motions.

Eqs. (6) and (7) can be combined into the wave equation

[EQUATION]

[EQUATION]

The linear terms on the left hand side of this equation coincide with the equation shown in Ofman & Davila (1998) for linear spherical Alfvén waves.

In the following, we restrict our derivation to short wavelength motions, [FORMULA] and, consequently, [FORMULA] and [FORMULA]. This restriction allows us to treat the effect of the stratification as a small perturbation. Nonlinear effects are assumed to be weak too, viz. [FORMULA].

Neglecting terms proportional to [FORMULA] with respect to terms proportional to [FORMULA], we rewrite equations (10) as

[EQUATION]

[EQUATION]

with

[EQUATION]

[EQUATION]

Note that the expressions [FORMULA] and [FORMULA] include appropriate nonlinear terms from [FORMULA] and [FORMULA] together with the linear terms which come from the left hand side of Eq. 10.

In addition, from Eqs. (8) and (9) we obtain

[EQUATION]

[EQUATION]

which allow us to determine the value [FORMULA] and [FORMULA] required for [FORMULA] as functions of [FORMULA] (or [FORMULA]).

As discussed above, the effects of stratification and nonlinearity are taken to be weak and, consequently, the right hand side of Eq. (11) is smaller than the left hand side. This allows us to apply the method of slowly varying amplitudes. Considering a wave propagating in the positive r-direction, we change the independent variables to

[EQUATION]

where [FORMULA] is a small positive parameter of order of the nonlinearity and inhomogeneity,

[EQUATION]

The background variables [FORMULA] and [FORMULA] depend on the "slow" coordinate R.

Using these new variables, Eq. (11) is rewritten as

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

Perturbations of other physical variables are expressed through [FORMULA] as

[EQUATION]

[EQUATION]

Integrating equation (15) with respect to [FORMULA] and taking the integration constant to be zero, we arrive at the evolutionary equation

[EQUATION]

Eq. (19) is an analog of the scalar Cohen-Kulsrud-Burgers equation for the case of spherical geometry.

In the following analysis, it is convenient to use normalized variables,

[EQUATION]

Below, the primes are omitted. In the normalized variables, Eq. (19) is written as

[EQUATION]

where both the Alfvén and sound speeds are measured in units of the Alfvén speed at the base of the corona, [FORMULA], the scale height H is measured in units of the solar radius, and the normalized viscosity [FORMULA] is introduced. The background Alfvén speed is re-written in the dimensionless form as

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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