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

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3. Similarity solution

A self-similar solution can be found for the particular Alfvén speed profile

[EQUATION]

Set

[EQUATION]

where the self-similar variable is

[EQUATION]

Choosing the vertical lengthscale as [FORMULA] and defining

[EQUATION]

the dimensionless form of (12) reduces to

[EQUATION]

which may be rewritten as

[EQUATION]

This may be expressed in terms of a new independent variable w as

[EQUATION]

where w is related to s by

[EQUATION]

The boundary conditions are taken as

[EQUATION]

Thus, we are modelling the generation of waves with no preferred wavelength at the base that are outward propagating and damped at large heights. The similarity variable s can simply be thought of as height, since the equation of the field lines is x a constant so that s is linearly proportional to z.

Before considering the exact solution to (17) we consider a couple of special cases. For [FORMULA] the solution to (17) is approximately

[EQUATION]

whereas for [FORMULA] it is

[EQUATION]

with [FORMULA].

Finally, the height at which the solution switches from (22) to (23) is controlled by the size of [FORMULA] and is approximately given by

[EQUATION]

The exact solution to (17) that satisfies the boundary conditions (21) is

[EQUATION]

The branches of the square roots are defined by taking [FORMULA] and the logarithmic function by [FORMULA]. The real part of the solution is plotted in Fig. 2 as a function of height for the particular field line given by [FORMULA]. For small enough values of [FORMULA] (e.g. [FORMULA]) it is clear that the solution simply oscillates with very little damping evident at these heights. However, [FORMULA] shows strong damping with little oscillation. Finally, [FORMULA] has fast damping with no oscillations. These different types of solution can be understood by examining the real and imaginary parts of the exponent in (25).

[FIGURE] Fig. 2. The real part of the perturbed magnetic field as a function of height [FORMULA] for various values of [FORMULA].

3.1. [FORMULA] and [FORMULA]

Consider the case [FORMULA]. The exponent in (25) may be expanded to give

[EQUATION]

This shows that the basic Alfvén wave is only exponentially damped when [FORMULA] is of order unity. There is minimal damping if [FORMULA] is small. This form of the damping is in agreement with the results of Heyvaerts & Priest 1983. Thus, there is a "window" in height for which the wave is damped before the weaker algebraic damping of (23) takes over. This window is given by

[EQUATION]

Within this region the wave amplitude will decay by an amount [FORMULA]. So the smaller the value of [FORMULA] the more the wave is damped but the further the wave travels before damping sets in.

The way the wave damps can be investigated by studying the real part of the exponent of (25). For [FORMULA] the damping is given by (26), namely,

[EQUATION]

and for [FORMULA] by

[EQUATION]

The real part of the exponent is shown in Fig. 3 a together with the approximations (28) and (29).

[FIGURE] Fig. 3. a The real part of the exponent in (25) (solid line), scaled with respect to [FORMULA], is shown as a function of [FORMULA]. The dashed curve refers to (28) and the dot-dashed curve to (29). b The corresponding phase of the exponent (solid curve), scaled with respect to [FORMULA], is shown as a function of [FORMULA]. The dashed curve refers to the imaginary part of the exponent in (26) and the dot-dashed curve refers to (35)

The variation of the phase is given by the imaginary part of the exponent in Eq. (25) and is shown in Fig. 3 b. Since the phase is not simply the straight line [FORMULA], this means that the wavefronts will turn with height, as discussed in the next section.

We now interpret these results in terms of the assumptions of weak damping and strong phase mixing. The definitions given by Heyvaerts & Priest 1983 and in Sect. 1 are limited by their choice of ansätz and were used to simplify the equations. Weak damping means that the wave propagates for many periods before significant damping occurs. Thus, by expressing the perturbed magnetic field in the form

[EQUATION]

where [FORMULA] and [FORMULA] are the real and imaginary parts of the exponent in (25), weak damping can then be interpreted as

[EQUATION]

Hence, weak damping occurs only when

[EQUATION]

Strong phase mixing means that the horizontal length-scales are much shorter than the vertical length-scales, so that

[EQUATION]

This reduces to

[EQUATION]

so that strong phase mixing occurs above heights corresponding to the horizontal inhomogeneity length.

3.2. Wave turning

As was mentioned in Sect. 1, the inhomogeneity in the background Alfvén speed causes Alfvén waves on neighbouring field lines to move out of phase with each other. Hence, as the wave progresses in z, the wavefront will turn consistent with the background inhomogeneity. One can view the creation of shorter lengthscales in the plasma as a result of wavefront turning. Fig. 4 shows the real part of the self-similar solution to the non-dissipative Eq. in contour form, with initial condition [FORMULA]. As z increases, the number of wavefronts crossing the line [FORMULA] increases. Therefore, cross-sections along the [FORMULA] direction show far more structure due to the appearance of gradients. These gradients can be dissipated, as is shown in Figs. 5 and 6 (plotted using Eq. with [FORMULA] and [FORMULA]).

[FIGURE] Fig. 4. Contour plot of the real part of the Alfvén wave profile for Eq. in the absence of damping, [FORMULA]. Lighter shades of grey correspond to higher values of [FORMULA], darker shades to lower values of [FORMULA].
[FIGURE] Fig. 5. Contour plot of the real part of the Alfvén wave profile for [FORMULA]. Lighter shades of grey correspond to higher values of [FORMULA], darker shades to lower values of [FORMULA].
[FIGURE] Fig. 6. Surface plot of the Alfvén wave profile for [FORMULA].

Fig. 6 shows a surface plot of the Alfvén wave profile. Starting from a disturbance that is uniform in x at [FORMULA], the surface plot clearly shows the rapid generation of short length-scales in the x -direction: the shorter these length-scales are, the faster the wave damps.

As can be seen, Fig. 5 has far less fine structure than Fig. 4, due to dissipation at high values of z, and so the increased wave turning in Fig. 5 is not visible.

The non-dissipative Eq., has self-similar solution [FORMULA], which has the same phase as Eq. , obtained under the assumption [FORMULA]. The equation for the position of the wavefront is simply [FORMULA] for some [FORMULA]. In the limit [FORMULA], the phase is given by

[EQUATION]

Fig. 3 b demonstrates the behaviour of the phase of the solution with increasing [FORMULA]. The phase in this limit is advanced with respect to the non-dissipative solution.

3.3. Ohmic dissipation

The earlier discussion has shown that the wave amplitude is decaying with height. The energy of the wave is transferred into heat through ohmic dissipation. The analytical solution for the perturbed magnetic field allows the current and ohmic dissipation to be calculated. Assuming [FORMULA], so that [FORMULA], the ohmic dissipation is

[EQUATION]

The modulus of B can be estimated using the approximation (26) and [FORMULA] so that

[EQUATION]

The exact ohmic dissipation and the approximation (37) are plotted in Fig. 7. The curves lie almost on top of each other for [FORMULA]. Using (37), the location and the value of the maximum dissipation can be estimated as

[EQUATION]

and

[EQUATION]

For this value of [FORMULA] to be valid we must make sure that [FORMULA], which is ensured by having [FORMULA].

[FIGURE] Fig. 7. The ohmic dissipation, in arbitrary units, as a function of height for [FORMULA].

Obviously there is also the magnitude of the wave amplitude still to be included. However, using the definition of [FORMULA], the maximum ohmic dissipation scales as

[EQUATION]

Thus, the amount of coronal heating by phase mixing depends on the frequency and the phase mixing length as well as the resistivity.

Finally, the total amount of Ohmic heating can be estimated by integrating (37) over height to give

[EQUATION]

Thus, the total Ohmic dissipation is independent of resistivity, as one would expect but it does depend on the phase mixing length and the wave frequency.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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