## 12 The Origin of the High-Frequency Region

How is the high-frequency region of the spectrum generated? This has become the urgent topic which must be addressed. Ghosh et al. (1996) appeals to change of invariants in controlling the flow of spectral energy transfer in the cascade process, and in this picture no dissipation is required to explain the steepening of the magnetic power spectrum. Furthermore it is believed that the high-frequency region is highly anisotropic, with a significant fraction of turbulent energy cascades mostly in the quasi 2D structures, perpendicular to the background magnetic field. How magnetic energy dissipated in the anisotropic energy cascade still remains an open question.

### 12.1 A dissipation range

As we have already said, in their analysis of Wind data Leamon et al. (1998) attribute the presence of the region at frequencies higher than the ion-cyclotron frequency to a kind of dissipative range. Apart for the power spectrum, the authors examined the normalized reduced magnetic helicity , and they found an excess of negative values at high frequencies. Since this quantity is a measure of the spatial handedness of the magnetic field (Moffatt, 1978) and can be related to the polarization in the plasma frame once the direction propagation direction is known (Smith et al., 1983), the above observations should be consistent with the ion-cyclotron damping of Alfvén waves. Using a reference system relative to the mean magnetic field direction and radial direction as , they conclude that transverse fluctuations are less dominant than in the inertial range and the high frequency range is best described by a mixture of 46% slab waves and of 54% 2D geometry. Since in the low-frequency range they found 11% and 89% respectively, the increased slab fraction my be explained by the preferential dissipation of oblique structures. Thermal particles interactions with the 2D slab component may be responsible for the formation of dissipative range, even if the situation seems to be more complicated. In fact they found that kinetic Alfvén waves propagating at large angles to the background magnetic field might be also consistent with the observations and may form some portion of the 2D component.

Recently the question of the increased anisotropy of the high-frequency region has been addressed by Perri et al. (2009) who investigated the scaling behavior of the eigenvalues of the variance matrix of magnetic fluctuations, which give information on the anisotropy due to different polarizations of fluctuations. The authors investigated data coming from Cluster spacecrafts when satellites orbited in front of the Earth’s parallel Bow Shock (Perri et al., 2009). Results indicates that magnetic turbulence in the high-frequency region is strongly anisotropic, the minimum variance direction being almost parallel to the background magnetic field at scales larger than the ion cyclotron scale. A very interesting result is the fact that the eigenvalues of the variance matrix have a strong intermittent behavior, with very high localized fluctuations below the ion cyclotron scale. This behavior, never investigated before, generates a cross-scale effect in magnetic turbulence. Indeed, PDFs of eigenvalues evolve with the scale, namely they are almost Gaussian above the ion cyclotron scale and become power laws at scales smaller than the ion cyclotron scale. As a consequence it is not possible to define a characteristic value (as the average value) for the eigenvalues of the variance matrix at small scales. Since the wave-vector spectrum of magnetic turbulence is related to the characteristic eigenvalues of the variance matrix (Carbone et al., 1995a), the absence of a characteristic value means that a typical power spectrum at small scales cannot be properly defined. This is a feature which received little attention, and represents a further indication for the absence of universal characteristics of turbulence at small scales.

### 12.2 A dispersive range

The presence of a magnetic power spectrum with a slope close to 7/3 (Leamon et al., 1998; Smith et al., 2006), suggests the fact that the high-frequency region above the ion-cyclotron frequency might be interpreted as a kind of different energy cascade due to dispersive effects. Then turbulence in this region can be described through the Hall-MHD models, which is the most simple model to investigate dispersive effects in a fluid-like framework. In fact, at variance with the usual MHD, when the effect of ion inertia is taken into account the generalized Ohm’s law reads

where the second term on the r.h.s. of this equation represents the Hall term ( being the ion mass). This means that MHD equations are enriched by a new term in the equation describing the magnetic field and derived from the induction equation

which is quadratic in the magnetic field. The above equation contains three different physical processes characterized by three different times. By introducing a length scale and characteristic fluctuations , , and , we can define an eddy-turnover time , related to the convective process, an Hall time which characterizes typical processes related to the presence of the Hall term, and a dissipative time . At large scales the first term on the r.h.s. of Equation (75) describes the Alfvénic turbulent cascade, realized in a time . At very small scales, the dissipative time becomes the smallest timescale, and dissipation takes place.^{12}However, one can conjecture that at intermediate scales a cascade is realized in a time which is no more and not yet , rather the cascade is realized in a time . This happens when . Since at these scales density fluctuations becomes important, the mean volume rate of energy transfer can be defined as , where is used as a characteristic time for the cascade. Using the usual Richardson’s cartoon for the energy cascade which is viewed as a hierarchy of eddies at different scales, and following (von Weizsäcker, 1951), the ratio of the mass density at two successive levels of the hierarchy is related to the corresponding scale size by where is a measure of the degree of compression at each level . Using a scaling law for compressive effects and assuming a constant spectrum energy transfer rate, we have , from which the spectral energy density The observed range of scaling exponents observed in solar wind (Leamon et al., 1998; Smith et al., 2006), can then be reproduced by different degree of compression of the solar wind plasma .