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Astron. Astrophys. 318, 957-962 (1997)

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2. Basic equations

The simplest model of phase mixing consists of a vertical magnetic field with the photospheric footpoints being oscillated at frequency [FORMULA]. If the Alfvén speed is non-uniform in the horizontal direction, then the waves generated at the footpoints propagate with different wave speeds and soon become out of phase. Large horizontal gradients are created and dissipation can then damp the waves and release their energy.

Thus, we assume the equilibrium is

[EQUATION]

and consider a perturbation of the velocity and magnetic field in the [FORMULA] direction only so that phase mixing of Alfvén waves is produced. Hence, the linearised MHD equations are

[EQUATION]

and

[EQUATION]

for the perturbed velocity, v, and magnetic field B. These equations can be combined to give (see Heyvaerts & Priest 1983)

[EQUATION]

where [FORMULA] is the Alfvén speed squared. The important terms for phase mixing are the first and second terms on the right-hand side of (12). The first term indicates that the wavelength is different on each field line and this generates the large horizontal gradients that are damped by the second term. The last term in (12) is not important for phase mixing and will be neglected from now on.

A simple self-similar solution is presented in the next section that allows us to investigate the efficiency of phase mixing as a mechanism for heating coronal holes.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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