Alfvén wave attenuation by solar wind turbulence scattering

N. A. Barkhatov

Nizhny Novgorod State Pedagogical University, Nizhny Novgorod, Russia

P. A. Bespalov

Institute of Applied Physics, Nizhny Novgorod, Russia

Received December 15, 1999

Contents

Abstract

Introduction

Complicated events with different characteristic scales taking place on the solar surface and in the interplanetary space initiate disturbance of wide spectrum inhomogeneities in the solar wind [Schmidt and Marsch, 1995]. The interplanetary medium parameters' fluctuations usually correspond to weak plasma turbulence and can be described in terms of wave turbulence [Chashey and Shishov, 1981]. Degree law of background wave turbulence spectrum [Denskat and Neubauer, 1982] may be the evidence of the non-linear turbulence scales cascade. Modern observational possibilities do not allow to define exactly which phenomenologic theory is true: Kolmogorov's [Kolmogorov, 1965] or Iroshnikov's theory [Iroshnikov, 1963]. More over, the dependence of turbulence spectrum shape on radial coordinate is attributed to the existence of the solar wind. The average background turbulence characteristics are considered to be well-known and are defined from observation [Readhead et al., 1978; Riley et al., 1996; Sari and Valley, 1976].

Conditions existing in the solar wind cause generation of comparatively stable magnetohydrodynamic (MHD) disturbances of various scales. Parametric decay of the solar wind Alfvén wave was studied thoroughly by Hollweg [1974] for small scale turbulence.

Solar wind Alfvén waves have rather weak collision and collisionless dampings. Therefore, when excited near the Sun, they bear the information on the solar wind far regions, which are not available for direct space observations. It is possible to define the corporal angle of source from which these waves could come in ultra Alfvénic flow. Some scientific schools have developed models of the evolution of the Alfvén wave power spectrum with solar distance [Goldstein et al., 1995; Marsch and Tu, 1996; [Zank et al., 1996]. A relatively recent review of the subject was given by Tu and Marsch, [1995]. To calculate MHD disturbance transport, it is necessary to take into consideration its modification as a result of interaction with the solar wind background turbulence. If the direct influence of MHD background turbulence is ignored, then it is possible to study wave parametric interaction against the background turbulence and to estimate the effective regular MHD wave attenuation caused by this process. However, it is necessary to take into account inhomogeneous medium peculiarities which become relevant within the long distances. They are: spherical solar wind, spiral interplanetary magnetic field and radial distribution of typical turbulence scales. It is also necessary to take into account the peculiarities of basic turbulence parameters' distribution.

This paper studies regular Alfvén wave attenuation caused by parametric interaction with background turbulence according to wave characteristic scales and the position of generation region in the solar wind. Attenuation as a result of such non-linear interaction between the given regular wave and the solar wind turbulence is compared with the findings obtained when studying the given wave dispersion on stationary inhomogeneities [Barkhatov and Belliustin, 1983]. Outcomes of numerical experiments and theoretical calculations [Kadomtsev, 1976] have shown, that Alfvén waves interact weakly. Therefore, of the entire hierarchy of three-wave processes, fast and slow magnetosound disturbances, which are effective sourses of Alfvén wave in the solar wind, are relevant. In this case some circumstances may result from that dispersing characteristics of Alfvén and fast magnetosound waves for small angles propagation to a magnetic field are close. Here it is possible to expect increase of interaction of Alfvén waves with large-scale fast magnetosound waves.

Of special interest to this research are two circumstances. At first, available experimental data do not allow immediately to measure Alfvén wave attenuation. For this purpose synchronous observations along the group trajectory are necessary in view of time delay and doppler of shifts on two spacecrafts. Secondly, the results obtained in the study allow to reveal the value of a large-scale turbulence as the reason of Alfvén wave attenuation. It should be mentioned, when it comes to problem of scattering, small-scale turbulence is considered.

Problem Statement and Initial Equations

We assume that Alfvén wave spectrum density W_k is not only geometrically developing along the beam, but also suffers the influence of the solar wind microstructure described in terms of wave turbulence. Both noted processes can be considered within the kinetic equation for waves [Kadomtsev, 1964]

(1)

When the collisional term considered only three-wave process of wave interaction [Akhiezer, 1974] in random phase approximation, the right part of the equation (1) is the following

Here, W _kt is the spectral density of turbulence energy, index k_t corresponds the background turbulence, H_int is the Hamilton function of wave interaction.

Alfvén wave is regarded to have enough high energy density in comparison with turbulence background, therefore, energy pumping into it can be ignored. Then, in coordinate system connected with beam trajectory, equation (1) for stationary spectral density distribution W _k can be also written down as

(2)

where V_l is the group velocity along the trajectory and n is the effective collision frequency which describe nonlinear interaction of regular Alfvén wave with background turbulence.

The integration of equation (2) is carried out in the Sun ecliptic plane along the beam trajectory for regular Alfvén waves with dispersion equations:

(3)

where signs correspond to Alfvén waves with wave vectors along and against the interplanetary magnetic field (IMF); V_sw is a solar wind velocity; V_Ar and V_{A<FONT FACE='Symbol'>j} are components of Alfvén velocity. The solution of equation (2) has the following form

(4)

Here, r, , j are spherical coordinates; r_s is a source coordinate; b (r) = p - arctan [r sin j/(r_s - r cos j)], R = r² + r_s² - 2 r r_s cos j ; W _k⁰ is the solution of uniform equation (1) calculated in geometric optics approximation for n=0 and / t = 0. Note, that the geometrooptic wave vector evaluation in space is taken into account in the design of n ( r).

According to [Akhiezer, 1974; Barkhatov and Bespalov, 1988], regular Alfvén (index "A") wave nonlinearly interacting with fast magnetosound (index "F") and slow magnetosound (index "S") waves can be examined by the method used for waves' research in plasma physics. The principle of energy and impulse conservation, in case of wave interaction provides the conditions of synchronizing, under which integrals' expressions of collisions for three-waves interactions are written. Investigation of Alfvén wave attenuation take into account background turbulence invariability. This phenomenon can be defined as Alfvén wave parametric interaction. The probabilities of interaction in the system of MHD waves, one of which corresponds to the background turbulence, are included in the expression for effective collision frequency. This expression is written in cylindrical coordinate system with z axis, oriented along IMF B(r) = (B_r² + Bj²)^1/2

(5)

Here, k has components k_z, k y and they are functions of r and , defined by geometric optics

k² = k² + (k_r sin a - kj cos a)²

tan y = k^-1 (k_r sin a - k_j cos a)

cos a (r, ) = B_r (B_r² + Bj²)^-1/2

Component k_tz is evaluated taking into consideration synchronizing conditions; formulas for W / k_{t z} are given below; M is proportional to the probability of selected three-waves process.

In (5), integral over the turbulence is taken in each point r, , j, which is reached by regular Alfvén radiation with dispersion equations (3) and W _k (r, , j) spectrum, obtained in geometric optic approximation. The form of equation (1) characteristics, when is fixed, leads to simplification and, therefore, n = n (r).

The processes which give irrevocable losses of Alfvén waves are the following: A + F_t F, A + S_t F, A F + S_t, A F_t + S, A + S_t F. For example, sinchronising conditions w₁( k₁) w₂( k₂) - w₃( k₃) = 0, ( k₁ k₂ - k₃ = 0 ) for waves with wave vectors along IMF are defined by equations

(6)

in which V_s is the sound velocity. Quite naturally, that weak attenuation of wave disturbances is necessary for realization of sinchronising conditions in form (6). Particulary for slow magnetosound, it occurs when the temperature of electron component largely exceeds the ionic one [Kadomtsev, 1964].

The comparison of levels of Alfvén wave attenuation, which resulted from different processes, is presented below; it showed, that the most significant process is A + F_t F process. Its probability is as follows

[k_z k_tz + k k_t cos F

Here, n is the proton concentration; m is the proton mass; w^A (k), w^F (k^F), w^F (k_t) are dispersion equations for interacting waves.

Model Background Turbulence Spectrum

The results of attenuation coefficient calculation (4) are largely defined by turbulence spectrum in the solar wind. Turbulence energy spectral density W _kT is determined on the basis of direct observation results, which have been summarized in Sari and Valley [1976], and experimental findings of Readhead et al. [1978]. The findings obtained on space craft Ulysses [Riley et al., 1996] should be taken into account. Assuming, that medium inhomogeneities are basically caused by magnetosound waves, magnetic field inhomogeneities' spectrums for fast and slow magnetosound turbulence will take the correspondent form

(7)

where z is the angle between k_t and B ; V_ph is the magnetosound wave phase velocity.

As a background turbulence model, choose the turbulence with isotropical angle spectrum and with one characteristic scale of disturbance a(r). Batchellor turbulence corresponds to these conditions [Batchelor, 1960]. Its spectral energy density takes the form

(8)

Coefficient a is determined as a function of distance to the Sun, taking into consideration the above mentioned experimental data. On this basis, spectrum of concentrational inhomogeneities was obtained

(9)

where d(r) = 2.73 10^-3 (r/1.5 10¹³)^-4, and characteristic turbulence scale increases with a distance from the Sun:

(10)

Here, AU is astronomical unit.

Coefficient value q 2.8 2.9 is calculated on the basis of data about spectral turbulence energy density [Sari and Valley, 1976].

If the turbulence is generated only by fast or slow magnetosound waves, then by order of quantity

(11)

Thus, for ka 1 spectral density for a turbulence of fast and slow magnetosound waves is determined by the formula (8), close to degree relation with parameters (10) and (11).

At ka > 1 according to expression (9) spectral density of energy has an exponential blockage. The selected model of turbulence spectrum does not contradict the recent experimental research [Bavassano, 1994].

The using of these expressions allow us to evaluate the maximum coefficient of relative Alfvén wave attenuation in the solar wind, namely L(r,k) - factor in expression (4).

Effective Attenuation of Regular Alfvén Wave

Relative effective attenuation of regular Alfvén wave is analyzed. Let's consider the Alfvén waves with wave vectors k, which have components along the interplanetary magnetic field.

The expressions of effective collision frequency of the three-waves processes are calculated, taking into account equations (5)-(11). For the cases of small scale turbulence (SST) in comparison with Alfvén wave scale k_t k_o a^-1 k (index ss) and large scale turbulence (LST) k_t k_o a^-1 k (index ls), the following expressions, correct to order of quantity, were obtained

Here, d is a coefficient of concentrate changing, V_A, V_s, V_sw, n, B, d, a, k are the distance functions. The medium and the small scale turbulence conditions within the whole examined interval from r_s till r = 1 AU. are accomplishing for regular wave with wave number k (r = r_s) 4 10^-8 cm ^-1 (for Alfvén wave with frequency w_A < 1.5 s^-1 ) and large scale conditions for k(r = r_s) 10^-7 cm ^-1 (  w_A > 1.5 s ^-1 ).

Estimations of n quantities for average values of the solar wind parameters show the importance of A + F_t F process providing the regular Alfvén wave energy pumping into the fast magnetic sound. In Figure 1 there are given results of numerical calculation of the factor L in expression (4) characterizing Alfvén wave relative dumping as a result of confluence with fast magnetosound turbulence. The factor L is presented as a dependence on k_A wave vector value of regular wave, when r = r_s. The distance being counted from Earth orbit to the region, where Alfvén disturbance was given, is taken as a parameter in astronomic units in Figure 1.

Figure 1 indicates the effectiveness increase of this kind of parametric Alfvén wave scattering with their length reducing. The efficiency of energy pumping is decreasing with the distance from the Sun. This is explained by features of parameters' distribution of interplanetary medium and magnetic field along the trajectory. The results allow to define spheric layers of the near solar space responsible for Alfvén waves of different scales registered in the Earth's vicinity.

Discussions

Analysis of parametrical interaction of the regular Alfvén wave along the trajectory with fast magnetosound turbulence has shown that Alfvén wave package attenuation takes place in the beginning of the way from the solar vicinity. The rate of energy pumping decreases with distance from the Sun. This can be attributed to the distribution peculiarities of interplanetary medium parameters and magnetic field along the trajectory. Alfvén wave attenuation due to energy pumping into fast magnetosound wave is most effective on large scale turbulence (LST) (Figure 1).

The role of small scale turbulence (SST) when compared to regular wave length turns to be significant for parametric interaction of given Alfvén wave with slow magnetosound and Alfvén waves in presence of fast magnetosound and slow magnetosound turbulences, respectively. In spite of turbulence characteristic scales growth, when distance to the Sun increases, the analyzed processes attenuate because of the general changing of plasma parameters and magnetic field.

Comparison of extinction obtained here with that calculated in Dobrowolny and Torricelly-Ciamponi [1985] Alfvén wave dissipation due to linear Landau attenuation in the solar wind shows the predominance of the parametrical effect.

Turbulence model in form of Batchelor turbulence used above can be used only for estimation of the process effectiveness, as it was showed before, the turbulence spectrum in the solar wind is likely to have degree character. This means, that it is not obligatory to investigate the case of middle scale turbulence k k_t a^-1. Such proportions are related to resonance of regular wave scattering [Ishimaru, 1978]. However, in this case resonance scattering contribution to the attenuation is, perhaps, less than from large scale turbulence because of degree spectrum drop.

There is an intimate connection between the problem of non-linear interaction of the waves in plasma and wave transformation as a result of scattering by rather small chaotic inhomogeneities [Ginzburg, 1967]. Indeed, the obtained results of parametric wave interaction correspond to our findings concerning the wave scattering by the stationary inhomogeneities.

For interpretation of MHD waves' scattering mechanism and quality explanation of obtained results it is helpful to use geometric relationships in k space: comparison of space spectrum surfaces of wave vectors of inhomogeneities scattering the waves [Barkhatov and Belliustin, 1983]. The results obtained above can be explained by wave vector's surfaces for MHD waves presented in Figure 2. Wave vector surfaces of slow magnetosound and Alfvén waves are not close. It is attributed to plasma resonances. Reradiation by inhomogeneities as the sources of these waves will be the more considerable, the less their space scales and, consequently, the broader their wave vectors' spectrum are. Reradiation of these inhomogeneities in fast magnetosound wave is too small and not connected with plasma resonance. It doesn't depend on inhomogeneities' dimensions, when they are smaller than the wave length (SST). In Figure 2 arrows show the wave vectors of given regular Alfvén waves. The regions, occupied with wave vectors of scattering turbulence inhomogeneities for cases of SST and LST, respectively, are depicted with dotted and point-circular lines. For cases studied above, when the given Alfvén wave vector is parallel to IMF direction, the total power scattered in slow magnetosound and Alfvén wave turns to be less than the power radiated in fast magnetosound wave by LST inhomogeneity. This takes place in spite of the fact, that space spectrums of SST inhomogeneities cover large regions and have resonance radiation directions. Of cause, the scattering picture is enriched because of participation of the near wave vector surfaces in this process.

Thus, the connection between the problem of nonlinear interaction of waves in plasma and the question of wave transformation as a result of their scattering by stationary chaotic inhomogeneities is confirmed.

Conclusions

In this paper parametric interaction processes of Alfvén waves with different types of waves against the solar wind wave turbulence background were studied within the kinetic equation for wave spectral energy density in inhomogeneous medium. Special attention is paid to Alfvén wave attenuation with wave vectors near interplanetary magnetic field direction. Principal efficiency of three-wave processes for this wave group is explained by closeness of their dispersive characteristic to fast magnetosound wave dispersive characteristic. Here, the parameter distribution of interplanetary medium along wave transport trajectory was taken into account.

1. Alfvén wave attenuation was determined. Wave vector along IMF Alfvén wave attenuation resulting from energy pumping into fast magnetosound waves is most effective on large scale disturbance. Effectiveness of this process decreases with distance from the Sun.

2. Conclusions were made about the growth of Alfvén wave attenuation with increase of their length. This allows to define spherical layers of near solar space, which may be responsible for Alfvén wave of different scales, registred in the Earth vicinity. Deep corona layers are responsible for admitted Alfvén waves of large scales; external layers - for waves of small scales. That is why experimentally registered growth of turbulence scales with distance to the Sun doesn't seem to be strange.

3. It was noted, that the role of small scale turbulence in comparison to the length of regular wave, turns out to be relevant for parametrical interaction of given Alfvén wave with slow magnetosound and Alfvén wave. These processes occur in presence of fast and slow magnetosound turbulence, respectively. The scattering of given Alfvén wave scattering into fast magnetosound wave is small.

4. The obtained results correspond qualitatively to regular Alfvén wave scattering by moving stationary inhomogeneities observed before [Barkhatov and Belliustin, 1983]. This confirms the close connection of the problem of non-linear interaction of waves in plasma with the question of wave transformation as a result of their scattering by stationary chaotic inhomogeneities.

Acknowledgments

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