The primary process governing the solar wind heating is probably active locally in the wind. However, since collisions are very rare in the solar wind plasma, the usual viscous coefficients have no meaning, say energy must be transferred to very small scales before it can be efficiently dissipated, perhaps by kinetic processes. As a consequence, the presence of a turbulent energy flux is the crucial first step towards the understanding of solar wind heating (Coleman, 1968; Tu and Marsch, 1995a) because, as said in Section 2.4, the turbulent energy cascade represents nothing but the way for energy to be efficiently dissipated in a high-Reynolds number flow.11 In other words, before to face the problem of what actually be the physical mechanisms responsible for energy dissipation, if we conjecture that these processes happens at small scales, the turbulent energy flux towards small scales must be of the same order of the heating rate.
Using the hypothesis that the energy dissipation rate is equal to the heat addition, one can use the omnidirectional power law spectrum derived by Kolmogorov
( is the Kolmogorov constant that can be obtained from measurements) to infer the energy dissipation rate (Leamon et al., 1999)Zhou and Matthaeus, 1989, 1990; Marsch, 1991) MacBride et al., 2010).
A different estimate for the energy dissipation rate in spherical symmetry can be derived from an expression that uses the adiabatic cooling in combination with local heating rate . In a steady state situation the equation for the radial profile of ions temperature can be written as (Verma et al., 1995)–1. ( is the Boltzmann constant). Equation (73) can be solved using the actual radial profile of temperature thus obtaining an expression for the radial profile of the heating rate needed to heat the wind at the actual value (Vasquez et al., 2007) 74) should thus be only seen as a first approximation that could be improved with better models of the heating processes. Using the expected solar wind parameters at 1 AU, the expected heating rate ranges from for cold wind to in hot wind. Cascade rates estimated from the energy-containing scale of turbulence at 1 AU obtained by evaluating triple correlations of fluctuations and the correlation length scale of turbulence give values in this range (Smith et al., 2001a, 2006; Isenberg, 2005; Vasquez et al., 2007)
Rather than estimating the heating rate by typical solar wind fluctuations and the Kolmogorov constant, it is perhaps much more convenient to get a direct estimate of the energy dissipation rate by measurements of the turbulent energy cascade using the Yaglom’s law, say from measurements of the third-order mixed moments of fluctuations. In fact, the roughly constant values of , or alternatively their compressible counterpart will result in an estimate for the pseudo-energy dissipation rates (at least within a constant of order unity), over a range of scales , which by definition is unaffected by intermittency. This has been done both in the ecliptic plane (MacBride et al., 2008, 2010) and in polar wind (Marino et al., 2009; Carbone et al., 2009b). Even preliminary attempts (MacBride et al., 2008) result in an estimate for the energy dissipation rate which is close to the value required for the heating of solar wind. However, refined analysis (MacBride et al., 2010) give results which indicate that at 1 AU in the ecliptic plane the solar wind can be heated by a turbulent energy cascade. As a different approach, Marino et al. (2009) using the data from the Ulysses spacecraft in the polar wind, calculate the values of the pseudo-energies from the relation , and compare these values with the radial profile of the heating rate (74) required to maintain the observed temperature against the adiabatic cooling. The Ulysses database provides two different estimates for the temperature, , indicated as in literature, and , known as . In general, and are known to sometimes give an overestimate and an underestimate of the true temperature, respectively, so that analysis are performed using both temperatures (Marino et al., 2009). The heating rate are estimated at the same positions for which the energy cascade was observed. As shown in Figure 109 results indicate that turbulent transfer rate represents a significant amount of the expected heating, say the MHD turbulent cascade contributes to the in situ heating of the wind from 8% to 50% (for and respectively), up to 100% in some cases. The authors concluded that, although the turbulent cascade in the polar wind must be considered an important ingredient of the heating, the turbulent cascade alone seems unable to provide all the heating needed to explain the observed slowdown of the temperature decrease, in the framework of the model profile given in Equation (74). The situation is completely different as far as compressibility is taken into account. In fact, when the pseudo-energy transfer rates have been calculated through , the radial profile of energy dissipation rate is well described thus indicating that the turbulent energy cascade provides the amount of energy required to locally heat the solar wind to the observed values.
As we saw in Section 8, the energy cascade in turbulence can be recognized by looking at Yaglom’s law. The presence of this law in the solar wind turbulence showed that an energy cascade is at work, thus transferring energy to small scales where it is dissipated by some mechanism. While, as we showed before, the inertial range of turbulence in solar wind can be described more or less in a fluid framework, the small scales dissipative region can be much more (perhaps completely) different. The main motivation for this is the fact that the collision length in the solar wind, as a rough estimate the thermal velocity divided by the collision frequency, results to be of the order of 1 AU. Then the solar wind behaves formally as a collisionless plasma, that is the usual viscous dissipation is negligible. At the same time, in a magnetized plasma there are a number of characteristic scales, then understanding the physics of the generation of the small-scale region of turbulence in solar wind is a challenging topic from the point of view of basic plasma physics. With small-scales we mean scales ranging between the ion-cyclotron frequency (which in the solar wind at 1 AU is about ), or the ion inertial length , and the electron-cyclotron frequency. At these scales the usual MHD approximation could breaks down in favour of a more complex description of plasma where kinetic processes must take place.
Some times ago Leamon et al. (1998) analyzed small-scales magnetic field measurements at 1 AU, by using 33 one-hour intervals of the MFI instrument on board Wind spacecraft. Figure 110 shows the trace of the power spectral density matrix for hour 1300 on day 30 of 1995, which is the typical interplanetary power spectrum measured by Leamon et al. (1998). It is evident that a spectral break exists at about , close to the ion-cyclotron frequency. Below the ion-cyclotron frequency, the spectrum follows the usual power law , where the spectral index is close to the Kolmogorov value . At small-scales, namely at frequencies above , the spectrum steepens significantly, but is still described by a power law with a slope in the range (Leamon et al., 1998; Smith et al., 2006), typically . As a direct analogy to hydrodynamic where the steepening of the inertial range spectrum corresponds to the onset of dissipation, the authors attribute the steepening of the spectrum to the occurrence of a “dissipative” range (Leamon et al., 1998). Statistical analysis by Smith et al. (2006), showed that the distribution of spectral slopes (cf. Figure 2 of Smith et al., 2006), is broader for the high-frequency region, while it is more peaked around the Kolmogorov’s value in the low-frequency region. Moreover, as a matter of fact, the high-frequency region of the spectrum seems to be related to the low-frequency region (Smith et al., 2006). In particular, the steepening of the high-frequency range spectrum is clearly dependent on the rate of the energy cascade obtained as a rough estimate (cf. Figure 4 of Smith et al., 2006).
Further properties of turbulence in the high-frequency region have been evidenced by looking at solar wind observations by the FGM instrument onboard Cluster satellites (Alexandrova et al., 2008) spanning a frequency range. The authors found that the same spectral break by Leamon et al. (1998) exists when different datasets (Helios for large scales and Cluster for small scales) are used. The break (cf. Figure 1 of Alexandrova et al., 2008) has been found at about , near the ion cyclotron frequency , which roughly corresponds to spatial scales of about 1900 km (being the ion-skin-depth). However, as evidenced in Figure 1 of Alexandrova et al. (2008), the compressible magnetic fluctuations, measured by magnetic field parallel spectrum , are enhanced at small scales. This means that, after the break compressible fluctuations become much more important than in the low-frequency part. The parameter in the low-frequency range ( is the total power spectrum density and brackets means averages value over the whole range) while compressible fluctuations are increased to about in the high-frequency part. The increase of the above ratio were already noted in the paper by Leamon et al. (1998). Moreover, Alexandrova et al. (2008) found that, as results in the low-frequency region (cf. Section 7.2), intermittency is a basic property of the high-frequency range. In fact, the authors found that PDFs of normalized magnetic field increments strongly depend on the scale (Alexandrova et al., 2008), a typical signature of intermittency in fully developed turbulence (cf. Section 7.2). More quantitatively, the behavior of the fourth-order moment of magnetic fluctuations at different frequencies is shown in Figure 111
It is evident that this quantity increases as the frequency becomes smaller, thus indicating the presence of intermittency. However the rate at which increases is pronounced above the ion cyclotron frequency, meaning that intermittency in the high-frequency range is much more effective than in the low-frequency region. Recently, by analyzing a different dataset from Cluster spacecraft, Kiyani et al. (2009) using high-order statistics of magnetic differences, showed that the scaling exponents of structure functions, evaluated at small scales, are no more anomalous as the low-frequency range, even if the situation is not so clear (Yordanova et al., 2008, 2009). This is a good example of absence of universality in turbulence, a topic which received renewed attention in the last years (Chapman et al., 2009; Lee et al., 2010; Matthaeus, 2009).