3. The evolutionary equation
where and are transverse components of the perturbation velocity and magnetic field, respectively, and
Here we assumed that the perturbations of and were initially absent from the system, and were generated nonlinearly by the Alfvén wave. Consequently, only nonlinear terms containing and had to be taken into account. Second order nonlinear terms containing and 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
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, and, consequently, and . This restriction allows us to treat the effect of the stratification as a small perturbation. Nonlinear effects are assumed to be weak too, viz. .
Neglecting terms proportional to with respect to terms proportional to , we rewrite equations (10) as
Note that the expressions and include appropriate nonlinear terms from and together with the linear terms which come from the left hand side of Eq. 10.
which allow us to determine the value and required for as functions of (or ).
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
where is a small positive parameter of order of the nonlinearity and inhomogeneity,
The background variables and depend on the "slow" coordinate R.
Using these new variables, Eq. (11) is rewritten as
Perturbations of other physical variables are expressed through as
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,
where both the Alfvén and sound speeds are measured in units of the Alfvén speed at the base of the corona, , the scale height H is measured in units of the solar radius, and the normalized viscosity is introduced. The background Alfvén speed is re-written in the dimensionless form as
© European Southern Observatory (ESO) 2000
Online publication: December 17, 1999