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Astron. Astrophys. 357, 1073-1085 (2000)

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5. Parametric decay and damping of the phase-mixed AW in the corona

5.1. Dynamic equation and coupling coefficients of the three-wave AW interaction

We shall proceed further in the local approximation and expand all perturbations that appear in the eigenmode Eq. (31) into Fourier series. We use the second-order approximation for all quantities, which are present in the nonlinear right-hand side, expressing all perturbations through the parallel component of the electromagnetic potential [FORMULA]. Then, we average over time and space intervals which are short compared to the intervals over which the amplitudes of the waves vary, but [FORMULA], [FORMULA], and obtain the dynamic equations describing the slow variations of the wave amplitude:

[EQUATION]

where [FORMULA] for [FORMULA] [FORMULA] ([FORMULA] is chosen to be [FORMULA]).

The coefficients [FORMULA] in the Eq. (42) take into account that the phase velocity of [FORMULA]-th wave, [FORMULA], can be positive ([FORMULA] with [FORMULA]) or negative ([FORMULA] with [FORMULA]), and the indices [FORMULA], [FORMULA] and [FORMULA] denote complex conjugate terms [FORMULA] and [FORMULA] (note that only real parts, Re[FORMULA], should be taken of the complex amplitudes in binary combinations).

The coupling coefficient of the three-wave resonant interaction is

[EQUATION]

where [FORMULA] is the kinetic factor, accounting for temperature (FLR and electron pressure) effects. From (43) we see that three-wave resonant interaction among AWs is local in k-space, i.e. the wavenumbers of the effectively interacting AWs are of the same order, [FORMULA]. Note that the non-zero value of [FORMULA] and the non-linear (three-wave) coupling among AW triads becomes possible due to the finite value of the kinetic factor [FORMULA]. In MHD Alfvén waves [FORMULA], making [FORMULA] so that [FORMULA] for resonant triads in agreement with the well-known fact that three-wave resonant interaction is impossible among ideal MHD Alfvén waves.

Since we consider AWs with frequencies [FORMULA] in a low-[FORMULA] coronal plasma, we have dropped the terms of the order [FORMULA] and [FORMULA] in comparison to those retained in the expression for the matrix element (43). In this situation, the most important effect for parametric decay is the vector nonlinearity, [FORMULA], retained in (43 ). The converse situation, where the scalar nonlinearity, [FORMULA], becomes more important, is considered by Voitenko & Goossens (in preparation).

The matrix element [FORMULA] is the key mathematical object in the non-linear wave theory. Once this matrix element is found, we can follow theoretical methods developed earlier in order to study nonlinear properties of a given mode.

Using expressions (42)-(43), we study the parametric decay of AWs in the next section. This nonlinear process is important for the AWs and often plays a crucial role in wave propagation, nonlinear saturation of instabilities, and spectral dynamics (Voitenko 1998a,b).

5.2. Parametric decay instability of the short-scale Alfvén waves

The parametric decay of a pump oblique AW into two daughter oblique AWs in a hot plasma has been first examined by Erokhin et al. (1978), followed by other authors (see e.g. Volokitin & Dubinin 1989; Voitenko 1996c; Yukhimuk & Kucherenko 1993). The common approach so far was to consider weak wave dispersion, [FORMULA], and to assume that all three interacting AWs propagate in the same direction along the background magnetic field. However, it was shown in the framework of kinetic theory by Voitenko (1998a) that the strongest decay of the weakly dispersing AW was missed in the previous investigations. Indeed, the strongest decay occurs when the daughter waves propagate in opposite directions along the background magnetic field. Also, in contrast with previous studies, we consider the influence of resistive dissipation of AWs and the detuning of the perpendicular wavenumbers, caused by the transverse inhomogeneity. The inclusion of these effects is important when investigating the phase mixing of the low-frequency part of the AWs in the corona. Hence, we shall study the KAW decay instability for an arbitrary value of the kinetic factor [FORMULA], based on the dynamic Eq. (42) found from the two-fluid resistive MHD plasma model.

Let us consider a pump KAW with frequency [FORMULA], wave vector [FORMULA], and amplitude [FORMULA], propagating in the positive z-direction, i.e. [FORMULA]. From (42) it follows that the waves [FORMULA] and [FORMULA] can effectively interact with the pump wave, provided the following resonant conditions:

[EQUATION]

are satisfied. The dynamical equations for waves [FORMULA] and [FORMULA], coupled via the parametric pump wave, are:

[EQUATION]

[EQUATION]

Acting by the operator [FORMULA] on the Eq. (45) and eliminating [FORMULA] by use of Eq. (46), we obtain the equation for the amplitude [FORMULA] (the corresponding equation for the amplitude [FORMULA] is obtained in the same way):

[EQUATION]

An exponential solution to (47), [FORMULA], has indices

[EQUATION]

where the rate of non-linear interaction is

[EQUATION]

The waves 1 and 2 grow if

[EQUATION]

For the rate of the non-linear interaction [FORMULA] to be real, i.e. [FORMULA], the dispersion functions [FORMULA] have to satisfy the decay condition that

[EQUATION]

In the case of weak wave-particles interaction, [FORMULA], the growth rate of the waves 1 and 2 is determined by the rate of the non-linear interaction: [FORMULA].

The decay condition (50) implies that at least one dispersion function of the decay product should be smaller than the dispersion function of the pump wave ([FORMULA] for arbitrary [FORMULA] and [FORMULA] if both [FORMULA] and [FORMULA] [FORMULA]). In what follows we choose [FORMULA], so that decay is possible: into (1) parallel-propagating daughter KAWs, [FORMULA] [FORMULA] ([FORMULA]), and into (2) counterstreaming daughter KAWs, [FORMULA], [FORMULA] ([FORMULA], [FORMULA]) (here [FORMULA] is chosen for the pump wave).These decay channels are shown on Fig. 1 as parallelograms in the ([FORMULA])-plane, reflecting matching conditions for corresponding AW triads.

[FIGURE] Fig. 1. Two possible channels for decay of the pump KAW k: 1) decay into two parallel-propagating KAWs [FORMULA] ([FORMULA], [FORMULA]); 2) decay into two counterstreaming KAWs [FORMULA] ([FORMULA], [FORMULA]), where we have taken [FORMULA] for simplicity. The dispersion of the MHD Alfvén wave [FORMULA] is shown for reference.

Let us consider these two cases separately for the weakly dispersed KAWs, [FORMULA] (we shall not consider here strongly dispersed AWs, [FORMULA], which are unlikely for the AW phase-mixing in corona).

5.3. Excitation of the parallel-propagating decay AWs ([FORMULA], [FORMULA])

As follows from (50) with [FORMULA] and [FORMULA], the decay process including parallel-propagating KAWs [FORMULA] [FORMULA] can take place if the dispersion functions of the decay products [FORMULA] and [FORMULA] are smaller and larger respectively than the dispersion function [FORMULA] of the pump KAW:

[EQUATION]

Since the function K is a monotonous function of its argument [FORMULA], we conclude that the wavenumbers of the interacting waves have to satisfy the following sequence of inequalities:

[EQUATION]

The last inequality here follows from a combination of the resonant condition for the wave vectors and the second inequality.

For [FORMULA], we use the asymptotic expression [FORMULA] in (48) so that

[EQUATION]

The dependence of the normalized decay rate,

[EQUATION]

on the wavenumbers of the decay waves is shown on the Fig. 2.

[FIGURE] Fig. 2. The dependence of the normalized rate of parametric decay on the wavenumbers of the parallel-propagating decay waves, [FORMULA] and [FORMULA].

The spectrum of the generated daughter waves has a sharp maximum. To find the wavenumbers of the daughter waves which are most effectively excited, we determine the maximum of [FORMULA] given by (53) with respect to the wavenumbers of the decay products. Thus we obtain the rate of resonant decay instability into parallel-propagating AWs as

[EQUATION]

which is achieved for [FORMULA] (here we take [FORMULA] for simplicity).

Expression (55) shows a strong dependence of the interaction rate on the perpendicular wavenumber of the pump wave, [FORMULA], and this results in a rapid decrease of the interaction rate with decreasing [FORMULA].

5.4. Counterstreaming decay KAWs ([FORMULA], [FORMULA])

Eq. (53) is now replaced with

[EQUATION]

The dependence of the normalized decay rate [FORMULA] on the wavenumbers,

[EQUATION]

is shown on the Fig. 3.

[FIGURE] Fig. 3. The normalised rate of parametric decay depending on the wavenumbers of counterstreaming decay waves, [FORMULA] and [FORMULA].

When we compare the counterstreaming decay waves with the parallel-propagating decay AWs, we find that the spectrum of the counterstreaming decay waves is concentrated around another pair of wavenumbers: [FORMULA]. With these wavenumbers the maximum growth rate is

[EQUATION]

With decreasing [FORMULA], the decrease of the rate of decay into counterstreaming decay waves is not as strong as for parallel-propagating decay waves. This means that the decay into counterstreaming AWs is more effective for the weakly dispersing waves, created by phase mixing. Consequently, (57) gives a good approximation for the damping rate of the pump AW, and for the rate of the three-wave AWs interaction in [FORMULA]-space as well, if the decay into counterstreaming waves is not forbidden.

Summarizing, there are significant differences between the decay into parallel-propagating waves, and decay into counter-streaming (antiparallel-propagating) waves. Namely, the decay of a weakly-dispersing AW into counter-streaming waves:

  1. is much stronger for a weakly dispersed AW in a homogeneous plasma, but

  2. is more sensitive to the field-aligned inhomogeneity;

  3. transfers the wave energy into lower wavenumbers only, [FORMULA], thus reducing collisional dissipation, whereas the decay into parallel-propagating waves excite both shorter- and longer-wavelength waves, [FORMULA];

  4. in part, reverses the direction of wave energy propagation.

Expressions (55) and (57) will be used to estimate the nonlinear damping of the phase-mixed AWs with height in solar corona.

5.5. Collisional threshold of decay instability

Let us estimate the threshold amplitude for the decay instability of the phase-mixed pump in the corona, where the growth of the decay products due to nonlinear coupling via the pump wave may be balanced by their collisional dissipation. In a local approximation, the threshold amplitude for the pump wave to excite waves 1 and 2 is obtained from the marginal condition for decay (49), [FORMULA], where [FORMULA] and [FORMULA] are to be determined from (38).

In the coronal plasma, the collisional damping is

[EQUATION]

and the marginal decay condition is

[EQUATION]

For the strongest counterstreaming decay (57), the threshold appears to be independent of the perpendicular wavenumbers or other wave characteristics, and the threshold value depends only on plasma parameters:

[EQUATION]

The actual value of the threshold with the typical coronal parameters is extremely low:

[EQUATION]

The decay condition (49) is certainly satisfied for the AWs with [FORMULA], able to heat the corona.

Eqs. (55) - (61) clearly demonstrate that the phase-mixed AWs with [FORMULA] can become nonlinearly unstable at very small amplitudes, [FORMULA]. With these amplitudes, the parametric decay becomes stronger than the collisional dissipation of the phase-mixed AW, [FORMULA].

5.6. Nonuniformity thresholds

We consider three effects of nonuniformity on the parametric decay of phase-mixed AW.

1. It should be noted that although the decay of the [FORMULA] pump wave into counterstreaming KAWs is local in [FORMULA]-space, it is non-local in [FORMULA]-space because [FORMULA]. This means that our results on this decay are applicable as long as the large parallel wavelength of the counterstreaming (decay) KAW is shorter than the inhomogeneity length scale, [FORMULA]. Although our results can be used for waves with [FORMULA] as order of magnitude estimates, for larger wavelengths they can be significantly affected by the non-uniformity. The corresponding marginal perpendicular wavenumber of the pump wave and its parallel wavelength are related through [FORMULA]

2. Also, the efficiency of the decay including counterstreaming AWs in a nonuniform plasma may be reduced by the finite correlation length along the field lines, [FORMULA]. This correlation length may be estimated as [FORMULA], where [FORMULA] is the duration of the quasi-periodic pulse that excites the pump wave. We can estimate the corresponding threshold for decay into counterstreaming KAWs from the condition that the waves should overlap during the growth time ([FORMULA]):

[EQUATION]

which can be high for the waves with short correlation lengths.

3. The progressive detuning of the perpendicular wavenumbers of the interacting AWs triplet can limit the decay if the time of the nonlinear interaction is shorter than the time required to change the wavenumbers. The physics of this process is as follows. In the course of time, the initially most effectively interacting triplet of resonant waves is removed from its top position in the wavenumber space, because the different temporal behaviour of the constant and evolving components of the perpendicular wave vectors changes the wavenumber's ratios in the triplet. The corresponding limitation on the pump wave amplitude may be found from the condition [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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