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Astron. Astrophys. 354, 334-348 (2000)

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2. Equilibrium and linearised MHD equations

We consider an inhomogeneous density with an associated inhomogeneous Alfvén speed. If we assume that the background Alfvén speed (only) has variations in the horizontal direction, then Alfvén waves on neighbouring field lines, driven with the same frequency, will have different wavelengths. This will cause them to become out of phase as they propagate up in height and, therefore, large transverse gradients will build up. In this way, short lengthscales are created which means dissipation eventually becomes important and allows the energy in the wave to dissipate and heat the plasma. To study the effect of a radially diverging background magnetic field on phase mixing of Alfvén waves, spherical coordinates will be the most convenient choice and we therefore set up the basic equilibrium in spherical coordinates.

Assuming a low-[FORMULA]-plasma and an isothermal atmosphere, i.e. [FORMULA] uniform, the equilibrium is expanded in powers of [FORMULA]. Following Del Zanna et al. (1997), the leading order solution is a radially diverging field, [FORMULA], where [FORMULA] is the surface field strength and [FORMULA] is the solar radius. Note that we cannot assume a uniform field because of flux conservation, which shows that [FORMULA] has to be constant. At order [FORMULA] the radial magnetohydrost atic force balance equation reduces to

[EQUATION]

with [FORMULA] and, therefore,

[EQUATION]

and

[EQUATION]

where [FORMULA] is the pressure scale height. The force balance in the [FORMULA]-direction determines the finite [FORMULA] correction to the magnetic field (Del Zanna et al. 1997).

We now analyse Alfvén waves by considering perturbations [FORMULA] and [FORMULA] in the velocity and the magnetic field. Assuming a time dependence of the form [FORMULA] for both the perturbed magnetic field [FORMULA] and the velocity [FORMULA][FORMULA], the linearised MHD equations become:

[EQUATION]

and

[EQUATION]

where the magnetic diffusivity [FORMULA] and the dynamic viscosity [FORMULA] only depend on the temperature (Priest 1982) and in an isothermal atmosphere are assumed constant. The Alfvén speed is given by

[EQUATION]

where [FORMULA]. These equations can be combined to give either one single equation for the perturbed magnetic field b,

[EQUATION]

where we neglect the dynamic viscosity [FORMULA], or to give an equation for the perturbed velocity v,

[EQUATION]

where we neglect the resistivity [FORMULA]. Including both dissipation terms increases the order of the equations and obscures the physical effects of each term (see De Moortel et al. 1999).

As it will be convenient to use dimensionless variables for the numerical solutions to Eqs. (2) and (3), we set [FORMULA], [FORMULA] and

[EQUATION]

where [FORMULA] regulates the magnitude of the equilibrium density variations in the [FORMULA]-direction for given radial disctance r, and m is the number of density inhomogeneities inside the coronal hole. This could be, for example, the number of coronal plumes. The equation for the perturbed magnetic field then becomes

[EQUATION]

where

[EQUATION]

is the basic wavelength, [FORMULA] is the pressure scale height measured in units of the solar radius and [FORMULA]. Similarly we can rewrite the equation for the perturbed velocity as

[EQUATION]

where [FORMULA].

From now on we will drop the barred variables and work in terms of dimensionless variables. We consider either the ohmic heating, and solve Eq. (5) with [FORMULA], or the viscous heating, and solve Eq. (7) with [FORMULA].

The boundary conditions are chosen as

[EQUATION]

[EQUATION]

on the photospheric base and an outward propagating wave on the upper boundary. To obtain this upper boundary condition we assume that the density remains constant and the dissipation negligible outside the computational box. The solution corresponding to an outward propagating wave is

[EQUATION]

and matching v and [FORMULA] onto the solution inside the computational box gives

[EQUATION]

where

[EQUATION]

When dissipation is included and the height of the numerical box, i.e. [FORMULA], is taken sufficiently large, the waves are damped and the actual choice of the upper boundary condition is unimportant.

The spherical geometry allows the effect of flux tube area divergence to be studied. We can retrieve the non-diverging Cartesian case in the following manner. If we assume

[EQUATION]

we see that at low heights, i.e. [FORMULA], and for small initial wavelengths [FORMULA], Eqs. (5) and (7) transform to

[EQUATION]

and

[EQUATION]

with [FORMULA] and [FORMULA] where [FORMULA] and [FORMULA] with [FORMULA] as in Eq. (6). As we expect that the dominant damping terms are the second order derivatives in Eqs. (5) and (7), we dropped the other damping terms. So, at low heights and for small [FORMULA], we recover the standard Heyvaerts and Priest case (Heyvaerts & Priest 1983).

We note here that we are looking at torsional Alfvén waves, i.e. [FORMULA] and [FORMULA], in open field regions. Examples of studies of torsional Alfvén waves in closed field regions such as coronal loops and arcardes, can be found in Ruderman et al. (1997a, 1997b). Furthermore, we remark that the choice of analysing torsional Alfvén waves excludes all possible coupling with slow and fast magneto-acoustic waves. Coupling of the different types of MHD waves and more complicated classes of motions are considered by e.g. Nakariakov et al. (1997, 1998), Berghmans & Tirry 1997; Tirry & Berghmans 1997; Ruderman 1999.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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