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

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4. Alfvén wave dynamics

Eq. (21) does not have obvious exact solutions. However, qualitative analysis gives us some important insight into the dynamics of spherical weakly nonlinear and weakly dissipative short wavelenghts Alfvén waves.

When the wave amplitude and viscosity are negligible, the last two terms can be dropped and Eq. (21) has the solution

[EQUATION]

showing that the amplitude of an upwardly propagating linear Alfvén wave grows with height. (We must note that expression (23) was derived under the assumption that [FORMULA] and, consequently, consideration of the limiting case of the non-stratified atmosphere, [FORMULA], is meaningless.) Thus, the second term in evolutionary Eq. (21) leads to amplification of the Alfvén waves, which can be understood by simple geometrical reasoning (see also, Ofman & Davila 1998, Torkelsson & Boynton 1998). In the absence of dissipation and backward waves, the radial component of the Poynting flux [FORMULA] is proportional to [FORMULA]. On the other hand, the radial component is given by the expression [FORMULA], or, taking into account (1) and (18), [FORMULA]. Consequently, the combination [FORMULA] is constant, which gives us

[EQUATION]

which coincides with the expression (23).

The third and fourth terms in the Eq. (21) describe the nonlinear distortion of the wave profile through the generation of higher harmonics and the dissipation. When all three terms are comparable with each other, the dynamics of the Alfvén waves is determined by an interplay between the geometrical amplification, nonlinear generation of higher harmonics and dissipation.

Fig. 1 shows the typical evolution of a wave generated by a harmonic driver on the solar surface, with distance from the Sun. We can distinguish three main stages of the wave dynamics: the geometrical amplification, wave breaking and enhanced dissipation. Initially, the wave is amplified by the stratification, keeping nearly the same harmonic shape. When the amplitude reaches a certain value, nonlinear effects become more pronounced and the wave steepens. The shape of the wave evolves to the typical shape described by the scalar Cohen-Kulsrud equation. When the wave steepens, dissipation comes into play and the wave shows intensive decay. Also, in this stage, the nonlinear wave accelerates, propagating faster than the local Alfvén speed. (The wave maxima and nodes shift to the left).

[FIGURE] Fig. 1. Evolution of the temporal dependences of the initially harmonic nonlinear spherical Alfvén waves with the distance from the Sun. The solid line shows the variation of the velocity at the solar surface, the dotted line at about 2 [FORMULA], the dashed line at 5 [FORMULA], and the dotted-dashed line at 9 [FORMULA]. The initial wave period is 50 s and the amplitude is 25 km s-1. The atmosphere is isothermal with [FORMULA] MK and the Alfvén speed near the base of the corona is 1000 km s-1.

This scenario of the wave evolution is almost independent of the specific value of the viscosity [FORMULA], provided the viscosity is sufficiently weak ([FORMULA] for the parameters considered). Fig. 2 shows the wave shape in the nonlinear dissipation stage for three different values of the viscosity, [FORMULA], [FORMULA] and [FORMULA]. The wave shape remains almost the same for all three values of the viscosity, with only a slight decrease for the case of stronger dissipation. Also, for stronger dissipation the wave is smoother in the vicinity of wave fronts. This independence of the wave evolution on the viscosity allows us to conclude that the wave shows a similar behavior in cases of smaller viscosities, which are difficult to model numerically.

[FIGURE] Fig. 2. The shape of an Alfvén wave at 8 solar radii for three different dimensionless viscosities [FORMULA] (the solid line), [FORMULA] (the dotted line) and [FORMULA] (the dashed line). The initial wave period is 50 s and the amplitude is 25 km s-1. The Alfvén speed is taken to be 1000 km s-1 near the base of the corona.

Figs. 3-5 show the dependence of the wave amplitude on the height for different parameters of the waves and the corona: wave amplitudes (Fig. 3), wave periods (Fig. 4) and atmosphere temperatures (Fig. 5). It is seen in Fig. 3, that, for higher amplitudes, the stage of nonlinear dissipation begins earlier and the following dissipative decrease of the amplitude is steeper than for lower initial wave amplitudes. The transverse velocity reaches its maximum value up to 200 km s-1 at 7-9 solar radii, for various initial amplitudes (15-25 km s-1 for runs performed). The maximum amplitude is also connected with the wave frequency, as seen in Fig. 4. Higher frequency waves reach their amplitude maxima at shorter distances. However, lower frequency waves reach higher overall maximum amplitudes. Fig. 5 shows that the maximum amplitude is quite sensitive to the temperature of the atmosphere. From Eq. (19), it follows that the coefficient of the nonlinear term, which determines the nonlinear steepening, depends on the temperature in two different ways: directly, through the sound speed [FORMULA], and indirectly through the scale height H, which determines the radial dependence of the density [FORMULA] and, consequently, the Alfvén speed [FORMULA]. In the cooler atmosphere, the Alfvén waves reach higher amplitudes. The position of the wave breaking and the nonlinear dissipation do not depend strongly on the temperature.

[FIGURE] Fig. 3. Dependence of the nonlinear spherical Alfvén wave amplitude on the distance from the Sun for three different initial amplitudes A. The solid line corresponds to [FORMULA] km s-1, the dash-dotted line to [FORMULA] km s-1 and the dashed line to [FORMULA] km s-1. The dotted curves show the growth of the waves according to linear solution (23). The initial wave period is 50 s. The atmosphere is isothermal with [FORMULA] MK and the Alfvén speed near the surface is 1000 km s-1.

[FIGURE] Fig. 4. Dependence of the nonlinear spherical Alfvén wave amplitude on the distance from the Sun for three different wave periods P. The solid line corresponds to [FORMULA] s, the dashed line to [FORMULA] s and the dotted-dashed line to [FORMULA] s. The dotted curve correspond to linear solution (23). The initial wave amplitude is 25 km s-1. The atmosphere is isothermal with [FORMULA] MK and the Alfvén speed near the base of the corona is 1000 km s-1.

[FIGURE] Fig. 5. Dependence of the nonlinear spherical Alfvén wave amplitude on the distance from the Sun for three different temperatures T of the isothermal atmosphere. The solid line corresponds to [FORMULA] MK, the dotted line to [FORMULA] MK and the dashed line to [FORMULA] MK. The initial wave period is 50 s and the amplitude is 25 km s-1. The Alfvén speed is 1000 km s-1 near the base of the corona.

Figs. 6 and 7 show driven perturbations of density and longitudinal velocity, respectively, for three different temperatures of the atmosphere. The driven perturbations are connected with the amplitude of Alfvén waves by expressions (18). The evolution of the driven perturbations follows the growth of the Alfvén wave amplitude. For typical parameters, [FORMULA] km s-1, [FORMULA] s, [FORMULA] MK and [FORMULA] km s-1, the driven density perturbations reach 1.5% of the background density, while the longitudinal speed reaches 15-20 km s-1. The amplitudes of the driven perturbations of the density and the longitudinal velocity are sensitive to the temperature of the atmosphere and are strongest for lower temperatures. It can be explained qualitatively: lower temperatures correspond to smaller density scale heights, which leads to stronger geometrical amplification according to Eq. (23); a higher amplitude wave is more nonlinear and drives the density and the longitudinal velocity perturbations more efficiently.

[FIGURE] Fig. 6. Dependence of relative perturbation of density generated by nonlinear Alfvén waves, on the distance from the Sun, for three different temperatures T of the isothermal atmosphere. The solid line corresponds to [FORMULA] MK, the dashed line to [FORMULA] MK and the dash-dotted line to [FORMULA] MK. The initial wave period is 50 s and the amplitude is 25 km s-1. The Alfvén speed is taken 1000 km s-1 near the base of the corona.

[FIGURE] Fig. 7. Longitudinal motions generated by nonlinear Alfvén waves as a function of the distance from the Sun, for three different temperature T of the isothermal atmosphere. The solid line corresponds to [FORMULA] MK, the dashed line to [FORMULA] MK and the dash-dotted line to [FORMULA] MK. The initial wave period is 50 s and the amplitude is 25 km s-1. The Alfvén speed is 1000 km s-1 near the base of the corona.

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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