EL ROL DE LA FAMILIA EN EL PROCESO DEL APRENDIZAJE

The application of the basic concepts of turbulent theory (Section 1.8) to the so- lar wind is not a simple task, as solar wind plasma differs from that of an ideal turbulent fluid in a number of different ways (Meyer-Vernet, 2007, and references within). The solar wind has a viscosity which gives a Reynolds number that is not vary large and thus, dissipation is expected to be driven by waves and instabilities rather than by fluid viscosity. In addition, itermittency is observed to be greater than for ordinary fluid turbulence and takes the form of current sheets as opposed to filamentary vortices as in fluid turbulence. Finally, the solar wind is a magne- tised plasma which introduces anisotropy and the presence of a magnetic field also introduces additional forces, so that the turbulent ‘eddies’ in the solar wind take the form of MHD waves (generally Alfv´en waves, as the others are typically damped).

Figure 2.10: Typical interplanetary magnetic field power spectrum at 1 AU (Bruno and Carbone, 2013, and references within). Vertical dashed lines represent the correlation (or inertial) length, the Taylor scale and the Kolmogorov scale. The correlation length is the upper limit for the inertial range or the largest septation distance over which eddies are still correlated, i.e., the largest turbulent eddy size. The Taylor scale is the point at which at which viscous dissipation begins to take effect. The Kolmogorov scale is the point at which the transition from the inertial range to the dissipation range occurs.

2.1. The Solar Wind 84 Solar wind observations have revealed that the turbulent state, and its radial evolution during solar wind expansion, differs significantly between the slow and fast winds. This difference is likely, at least in part, to be related to the differ- ent origins, and thus different different macrostructure, of the slow and fast wind (e.g., Tu and Marsch, 1995a; Goldstein et al., 1995b). One key difference between slow and fast streams is that the fast wind contains more strongly Alfv`enic fluctu- ations i.e., the magnetic field and velocity fluctuations are highly correlated, with δ v ∼ ±δ B/ (µ0ρ )1/2 (Bruno and Carbone, 2013, and references within). Alfv`enic

fluctuations have also been found to be stronger, and present at increasing lower frequencies, for smaller heliocentric radial distances.

Observations within the ecliptic solar wind have also demonstrated that the spectral index of slow and fast streams are different, see Figure 2.11. The slow wind does not display any radial dependence and can be characterized by a single Kolmogorov-type index. Whereas, the fast wind is characterized by two distinct spectral slopes; the first, a ∼ f−1 relation at low frequencies and the second, a Kolmogorov-like spectrum at higher frequencies. These two regimes are separated by a knee (or break) in the spectrum which, as the wind expands, moves to lower and lower frequencies. This implies that larger and larger scales become part of the Kolmogorov-like turbulence spectrum, i.e, the inertial range, and that the spec- trum of solar wind fluctuations is a function of heliocentric radial distance as well as frequency. Thus observed turbulent evolution is likely influenced by velocity shear, parametric decay (in which large amplitude Alfv´en waves are unstable to ran- dom field perturbations and decay into two secondary Alfv´en modes propagating in opposite directions), and interaction of Alfv´enic waves with convected structures (Bruno and Carbone, 2013).

The spectral break has also been observed to move to lower frequencies at greater radial distances within the polar solar wind (e.g., Horbury et al., 1996a). In addition, it was found that the turbulent evolution in the polar wind was slower than that of fast streams in the ecliptic. A key difference between the polar and ecliptic fast wind is that the former has no interactions with slower streams and thus freely flows into interplanetary space. This suggests that, free of co-rotating shear events, the mechanism determining this evolution is an intrinsic property of the turbulence.

2.1. The Solar Wind 85

Figure 2.11: Magnetic field power spectra at different radial distances in the ecliptic plane for slow (right) and fast (left) wind streams (Bruno and Carbone, 2013, and references within). The spectral index of the slow wind does not show any radial dependence. Whereas the spectral breakobserved in the fast wind (marked by blue dots) moves to lower and lower frequency with increasing heliocentric distance.

The solar wind inertial range appears to fit with fluid theory. However, the dissipative behaviour at small scales is very different from that of an ordinary fluid. This is because the solar wind mean free path is so large that the plasma is es- sentially collisionless and thus viscous dissipation is negligible. In Section 1.7 it was shown that at smaller scales, i.e., the ion gyro-frequency and electron gyro- frequency, the single-fluid MHD approximation breaks down and a more complex description of plasma must be used. In the solar wind, a spectral break at the upper frequency limit of the inertial range is observed close to the ion gyro-scale (see, Figure 2.10), beyond this break the spectrum steepens but the exact nature of the fluctuations beyond this point is much debated.

The region of small scale turbulence beyond this second spectral knee is known as the kinetic turbulence range and it is where dissipation and plasma heating is though to occur. There are two main scenarios for the energy cascade within this

2.1. The Solar Wind 86 range: dispersion via Whistler-mode waves and dispersion via kinetic Alfv´en waves (KAWs). A key difference between the expected spectra of these two forms of tur- bulence is that whistler turbulence is expected to be steeper than for KAW turbu- lence (Chen, 2016). A key physical difference between these two forms of turbu- lence is that KAW turbulence is low frequency, ω  k⊥,i, whereas whistler turbu-

lence is high frequency, ω  k⊥,i (Chen, 2016). This means that density fluctua-

tions for KAW turbulence are non-negligible (e.g., Chen et al., 2013b) and it has been shown that, in this case, both the magnetic and density fluctuations within the kinetic range should have the same scaling relation (e.g., Boldyrev et al., 2013). Near-identical scaling relations for magnetic and density fluctuations within the kinetic range have been observed in solar wind turbulence (Chen et al., 2013a). However, the spectral index is also steeper than expected (∼ k−2.8 as opposed to ∼ k−2.3). Modifications to theory, such as including the presence of electron Lan- dau damping (Howes et al., 2012) or the effect of intermittent structures (Boldyrev and Perez, 2012) may explain this difference. However, it should be noted that there have also been observations, , achieved via the very high resolution data provided by the Magnetospheric Multiscale Mission (MMS), of whisltler turbulence at these scales (Narita et al., 2016). Thus the exact nature of turbulence at kinetic scales, and the relative roles of whistler and KAW fluctuations, has yet to be fully resolved. Furthermore, it should be noted that determining the nature of dispersion within this region has significant implications with regard to both solar wind heating (see Section 2.2.2) and the evolution of the strahl (see Section 2.2.6).