Fig. 2.23.4. Hemispherical scattering (q = 0.9) for rectilinear (left) and focusing (L = 1, right) geometries. According to Kota (1999).

From Fig. 2.23.4 it can be seen that for the rectilinear case the dispersion relations are quite similar to those of the telegrapher's equation (see Fig. 2.23.1). Moreover, the higher eigenvalues vj (j = 2,3,...) are remarkably large. For instance, the second eigenvalue is already v2 =23, thus the contributions from the higher eigenfunctions vanish quickly and can be neglected. This also reaffirms that the use of the two distinct levels, f- and f+ is a good approximation. For the evolution of f+ and f-, Kota (1999) suggest the coupled equations f V df-~ f-~ f+ (2.23.12)

which, in the focusing term, are somewhat different from the equations of Schwadron (1998) and Isenberg (1997). Combining Eq. 2.23.12 and Eq. 2.23.13 leads to v2-fe + fK02-f) = 0 (2.23.-4)

implying a modification of vo (k) owed to the effect of focusing. The right panels of Fig. 2.23.4 show that, as expected from Eq. 2.23.14, the dispersion d 2v/ dk2 does indeed decrease in the presence of focusing.

2.24. The dynamics of dissipation range fluctuations with application to CR propagation theory

2.24.1. The matter of the problem

According to Leamon et al. (1999a), relatively few studies of the dissipation range of interplanetary magnetic turbulence exist when compared to the inertial range at lower frequencies. Fig. 2.24.1 shows an example of a high-resolution spectrum taken by the WIND spacecraft in near-Earth orbit and its associated reduced magnetic helicity spectrum.

From Fig. 2.24.1 can be seen that the inertial range spectrum terminates at 0.44 Hz in a spectral break to a steeper spectral index. This break marks the onset of the dissipation range. The possible involvement of ion cyclotron activity in the observed onset of steepening has been discussed by Behannon (1976) and Denskat et al. (1983). In all events observed by Leamon et al. (1999a) this steepening of the dissipation range sets in at fsc > Qpj2n, where Q p is the proton cyclotron frequency, but the k • VSW Doppler shift makes it likely that m<Qp . In the spacecraft's frame it may be found that as a reasonable first approximation, the break frequency fsc is about 4 times the gyro-frequency Qpj2n.

Although the dissipation range contains very little energy, it is important because low rigidity particles and all particles at large pitch-angles become resonant with fluctuations at those scales. Magneto-static quasi-linear scattering by the 'slab' geometry which omits consideration of the dissipation range gives too much scattering, especially at low rigidity. To counter this Bieber et al. (1988) and Smith et al. (1990) argue that incorporation of a dissipation range in magneto-static scattering significantly alters the mean-free-paths of energetic particles. Bieber et al. (1994) employ the dissipation range, together with magneto-dynamic effects, to produce mass-dependent mean-free-paths that are distinct from the usual rigidity-dependent forms. This leads to differing mean-free-paths for protons and electrons of equal rigidity, in general agreement with a large class of solar energetic particle observations. Understanding supra thermal particle scattering therefore requires better determination of the turbulence geometry, i.e., the direction of k.

Fig. 2.24.1. Typical interplanetary power spectrum showing the inertial and dissipation ranges: (a) Trace of the spectral matrix with a break at = 0.4 Hz where the dissipation range sets in; (b) The corresponding magnetic helicity spectrum. According to Leamon et al. (1999a).

PSD Spacecraft -Frame Frequency [Hz]

Fig. 2.24.1. Typical interplanetary power spectrum showing the inertial and dissipation ranges: (a) Trace of the spectral matrix with a break at = 0.4 Hz where the dissipation range sets in; (b) The corresponding magnetic helicity spectrum. According to Leamon et al. (1999a).

Traditionally the reported observation of magnetic fluctuations perpendicular to the mean magnetic field (Belcher and Davis, 1971) has been used to motivate k||B. However, the possibility that an energetically significant fraction of the wave vectors could be nearly at k 1B was shown by Matthaeus et al. (1990). Bieber et al. (1996) assumed a composite two-dimensional (2D)/slab model for the magnetic turbulence and determined that in the inertial range there is a dominant (= 85% by energy) 2D component. The 2D component does not contribute to resonant scattering of very energetic particles (CR) and can explain the observed problem of 'too small' CR mean free paths (Bieber et al., 1994). Whereas Bieber et al. (1996) considered solar particle events, we shall extend their methods to the undisturbed solar wind and in frequency to the high-frequency end of the inertial range (~ 0.02 to ~ 0.2 Hz) and the low-frequency end of the dissipation range (~ 0.5 to < 2 Hz).

2.24.2. Magnetic helicity according to WIND spacecraft measurements

The results presented by Leamon et al. (1999a) are based on the analysis of 33 one-hour intervals of quiet solar wind data from the magnetic field and thermal plasma instruments of the WIND spacecraft. This data set and the method of analysis is described in detail by Leamon et al. (1998). For the 33 quiet solar wind intervals the spectral indices of the inertial range were between -1.46 and -1.93, with an average of -1.67. The dissipation range indices range from -1.93 to -4.43, with the average being -3.01. No clear correlation between the fitted indices of the two ranges is observed. The panel b in Fig. 2.24.1 shows the reduced magnetic helicity spectrum for that interval. Note the negative signature at dissipation frequencies, averaging -0.275 over those frequencies used to compute the dissipation range spectral slope. If there is finite magnetic helicity, the sign of the particle's charge can enter into the rigidity dependence of the mean free path as a second order effect (Goldstein and Matthaeus, 1981; Bieber et al., 1987, 1994). This is accomplished by changing the amount of energy available for resonant scattering by adjusting the net polarization of the power spectrum within any given range. perhaps more importantly, nonzero magnetic helicity can lead to resonant scattering dependent on the sign of the particle's charge. The apparent depletion of outward propagating Alfvén waves at frequencies comparable to the proton gyro-frequency naturally suggests resonant cyclotron damping of such Alfven waves as the leading candidate for the formation of the dissipation range.

2.24.3. Anisotropy according to WIND spacecraft measurements

The classic study of inertial range magnetic fluctuations is that of Belcher and Davis (1971). They defined a coordinate system relative to the mean magnetic field direction B , and radial direction R , according to {b x R,B x(B x R),B} and showed that the average variances for these three components are in the ratio 5:4:1. Leamon et al. (1999) note that this implies a ratio for the total variance transverse to aligned with the mean field of 9 : 1. This high level of anisotropy is in accordance with the fluctuations consisting of Alfvén waves. Leamon et al. (1999a) define P// to be the power in fluctuations parallel to B and P± to be the total power in both components perpendicular to the mean field. For the high-frequency end of the inertial range it was find a mean PLjP// ratio of 14 : 1, with a range 3.0 <P±/P// < 53.2. For the dissipation range there was found a mean ratio of 5.4:1 with a range 2.36 <P1/P// < 12.8. The dissipation range ratios P]_¡P// are consistently less than inertial range ratios, implying a decreased importance of transverse fluctuations in the dissipation range and an increase in the compression of the plasma at these scales.

2.24.4. Slab waves and 2D turbulence according to WIND spacecraft measurements

The Belcher and Davis (1971) anisotropy is usually taken as evidence of slab waves, even though it is consistent with 2D turbulence. By 2D turbulence is meant fluctuations which have wave-vectors that are nearly transverse to B. Most people interpret Belcher and Davis (1971) 5:4:1 anisotropy as a P1/P// = 9:1 ratio; the 5:4 part is considered physically unimportant. However, there is physical meaning to the ratio of the power in the two perpendicular directions (i.e., x and y in the mean-field coordinate system outlined in Section 2.24.3), and reason to expect that they should not be equal. Following Bieber et al. (1996), in a test based on the analysis of Oughton (1993) this ratio was used as a direct link to the percentage of slab waves and 2D modes in the fluctuations. Bieber et al. (1996) use a coordinate system that is a 90° right-handed rotation away from Belcher and Davis (1971) around the z or B axis). In the analysis that follows Bieber's conventions were used such that y = B x R . It was assumed that the magnetic fluctuations consist of a mixture of slab and 2D geometries and compute their relative strengths from the ratio of transverse spectral powers: CS and C2D are the amplitudes of the slab and 2D components, respectively; i.e., the slab spectrum in the range of interest is parameterized by CSk~q and the 2D spectrum by C2Dk~q . It was further assumed that the two components obey the same power law (that is, they have the same spectral index -q). This is equivalent to the statement that Pxx and Pyy obey the same power law, which is not strictly obeyed, at least not within data set, but is approximately true. The 'slab fraction'

is the contribution of the slab component to the energy spectrum, relative to the total energy. From Bieber et al. (1996) and the above definition leads to the following formula for the ratio of power between components:

VSW cos 6

VSW sin 6

VSW cos 6

VSW sin 6

The ratio Pyy/P^ (which becomes independent of frequency in the relevant range) and the parameters VSW , 9, the angle between the magnetic field and solar wind's velocity, and q are derivable from observations by a single spacecraft. Thus the only unknown in Eq. 2.24.2 is ratio C2D/CS , which, in turn, gives the slab fraction r determined by Eq. 2.24.1. For the 'middle' of the inertial range, Bieber et al. (1996) conclude that IMF geometry is ~ 85% 2D and only ~ 15% slab waves. Results of Leamon et al. (1999a,b) provide an essentially identical result for the high-frequency end of the inertial range, with ~ 89% of the energy in 2D fluctuations. In the dissipation range, on the other hand, the 2D component falls to ~ 55%, which it may explain by preferential dissipation of 2D structures. In terms of application to scattering theory, the large 2D component reduces the overall scattering rate by the same percentage. Perpendicular wavevectors are inefficient scatterers of particles, essentially making their percentage of the total energy unavailable for particles. In Leamon et al. (1998, 1999a,b) have shown that there is both observational and theoretical evidence to support the claim that the dissipation range forms as the result of dissipating energy associated with wave vectors at large angles to the mean magnetic field. This is consistent with inertial range studies (Matthaeus et al., 1990; Bieber et al., 1996) that indicate the same geometry at these larger scales and CR mean-free-path analyses (Bieber et al., 1994). The results described above are expected to aid in the refinement of ongoing CR propagation analyses. Also important is to examine the possible role of magnetic helicity within the dissipation range in determining CR propagation. Since resonant scattering of large pitch-angle particles by the dissipation range is balanced against magneto-dynamic effects and other considerations, the possible role of helicity at the small scales is unclear.

2.25. A path integral solution to the stochastic differential equation of the Markov process for CR transport

2.25.1. The matter of the problem

CR transport in interplanetary or interstellar magnetic fields is often studied in the framework of diffusion models (e.g., Parker, 1965; Ginzburg and Ptuskin, 1975). For interplanetary transport a Fokker-Planck diffusion equation for the isotropic part of the CR distribution function can be derived from the collisionless Boltzmann equation with the help of observations of interplanetary magnetic fields (Skilling, 1976). The mechanism for motion of CR in the interstellar medium is not yet clear simply because of insufficient information on the interstellar medium and the galactic magnetic field. But on the overall scale size of the galaxy and on the time scale of CR life time (~ 107 years) the diffusion approximation seems to be a suitable approach because it is consistent with the observation of small CR flux anisotropy and large amount of secondly produced nuclei in CR relative to the interstellar medium composition. In addition, acceleration of CR by astrophysical shocks may also be studied in the framework of diffusion models (Drury, 1983). According to Zhang (1999a,b,c), CR transport in interplanetary or interstellar magnetic fields can be viewed as a Markov stochastic process and the transport equation has therefore been reformulated with a set of stochastic differential equations that describe the guiding center and the momentum of individual charged particles. The Fokker-Planck diffusion equation for the CR flux can be derived from these stochastic differential equations. Alternatively, the Fokker-Planck equation, like the Schrodinger equation in quantum mechanics, can be solved with a path integral method. Both new methods enable one to solve modulation, propagation and acceleration problems for CR spectra. In addition, both can reveal insights into the physical processes behind the solutions to these problems since they follow the trajectory and the momentum of individual particles. In papers of Zhang (1999a,b,c) stochastic differential equations were used that describe Markov stochastic processes to replace the diffusion equation as the fundamental transport equation of CR. From the stochastic differential equation, the stochastic process was discretized to obtain a path integral solution for the transition probability, which is consistent with the Green's function of the diffusion equation. A Lagrangian was found which, if minimized, describes the most probable trajectory of particles in diffusion process. The path integral derived from the Markov stochastic process is consistent with the path integral derived from the diffusion equation with quantum mechanics method (Zhang, 1999a). In Zhang (1999a,b,c) was shown that both the stochastic process method and path integral approach give excellent results for CR spectrum calculation (see below, Section 2.25.4).

2.25.2. Diffusion and Markov stochastic processes; using definitions

In diffusion models for CR studies, the distribution function or flux obeys a second-order d-dimensional partial differential equation, which can be in general written as:

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