"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
1 Introduction
1.1 What does turbulence stand for?
1.2 Dynamics vs. statistics
2 Equations and Phenomenology
2.1 The Navier–Stokes equation and the Reynolds number
2.2 The coupling between a charged fluid and the magnetic field
2.3 Scaling features of the equations
2.4 The non-linear energy cascade
2.5 The inhomogeneous case
2.6 Dynamical system approach to turbulence
2.7 Shell models for turbulence cascade
2.8 The phenomenology of fully developed turbulence: Fluid-like case
2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case
2.10 Some exact relationships
2.11 Yaglom’s law for MHD turbulence
2.12 Density-mediated Elsässer variables and Yaglom’s law
2.13 Yaglom’s law in the shell model for MHD turbulence
3 Early Observations of MHD Turbulence in the Ecliptic
3.1 Turbulence in the ecliptic
3.2 Turbulence studied via Elsässer variables
4 Observations of MHD Turbulence in the Polar Wind
4.1 Evolving turbulence in the polar wind
4.2 Polar turbulence studied via Elsässer variables
5 Numerical Simulations
5.1 Local production of Alfvénic turbulence in the ecliptic
5.2 Local production of Alfvénic turbulence at high latitude
6 Compressive Turbulence
6.1 On the nature of compressive turbulence
6.2 Compressive turbulence in the polar wind
6.3 The effect of compressive phenomena on Alfvénic correlations
7 A Natural Wind Tunnel
7.1 Scaling exponents of structure functions
7.2 Probability distribution functions and self-similarity of fluctuations
7.3 What is intermittent in the solar wind turbulence? The multifractal approach
7.4 Fragmentation models for the energy transfer rate
7.5 A model for the departure from self-similarity
7.6 Intermittency properties recovered via a shell model
8 Observations of Yaglom’s Law in Solar Wind Turbulence
9 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data
9.1 Structure functions
9.2 Probability distribution functions
10 Turbulent Structures
10.1 On the statistics of magnetic field directional fluctuations
10.2 Radial evolution of intermittency in the ecliptic
10.3 Radial evolution of intermittency at high latitude
11 Solar Wind Heating by the Turbulent Energy Cascade
11.1 Dissipative/dispersive range in the solar wind turbulence
12 The Origin of the High-Frequency Region
12.1 A dissipation range
12.2 A dispersive range
13 Two Further Questions About Small-Scale Turbulence
13.1 Whistler modes scenario
13.2 Kinetic Alfvén waves scenario
13.3 Where does the fluid-like behavior break down in solar wind turbulence?
13.4 What physical processes replace “dissipation” in a collisionless plasma?
14 Conclusions and Remarks
A Some Characteristic Solar Wind Parameters
B Tools to Analyze MHD Turbulence in Space Plasmas
B.1 Statistical description of MHD turbulence
B.2 Spectra of the invariants in homogeneous turbulence
B.3 Introducing the Elsässer variables
C Wavelets as a Tool to Study Intermittency
D Reference Systems
D.1 Minimum variance reference system
D.2 The mean field reference system
E On-board Plasma and Magnetic Field Instrumentation
E.1 Plasma instrument: The top-hat
E.2 Measuring the velocity distribution function
E.3 Computing the moments of the velocity distribution function
E.4 Field instrument: The flux-gate magnetometer
F Spacecraft and Datasets

11 Solar Wind Heating by the Turbulent Energy Cascade

The Parker theory of solar wind (Parker, 1964) predicts an adiabatic expansion from the hot corona without further heating. For such a model, the proton temperature T(r) should decrease with the heliocentric distance r as T(r) ∼ r−4∕3. The radial profile of proton temperature have been obtained from measurements by the Helios spacecraft at 0.3 AU (Marsch et al., 1982; Marsch, 1983; Schwenn, 1983; Freeman, 1988; Goldstein, 1996), up to 100 AU or more by Voyager and Pioneer spacecrafts (Gazis, 1984; Gazis et al., 1994; Richardson et al., 1995Jump To The Next Citation Point). These measurements show that the temperature decay is in fact considerably slower than expected. Fits of the radial temperature profile gave an effective decrease T ∼ T0 (r0∕r )ξ in the ecliptic plane, with the exponent ξ ∈ [0.7;1], much smaller than the adiabatic case. Actually ξ ≃ 1 within 1 AU, while ξ flattens to ξ ≃ 0.7 beyond 30 AU, where pickup ions probably contribute significantly (Richardson et al., 1995; Zank et al., 1996; Smith et al., 2001b). These observations imply that some heating mechanism must be at work within the wind plasma to supply the energy required to slow down the decay. The nature of the heating process of solar wind is an open problem.

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, 1995aJump To The Next Citation Point) 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

P (k) = CK 𝜖2P∕3k −5∕3

(CK is the Kolmogorov constant that can be obtained from measurements) to infer the energy dissipation rate (Leamon et al., 1999)

[ ]3∕2 5- −1 5∕2 𝜖P = 3P (k)CK k , (71 )
where k = 2πf ∕V (f is the frequency in the spacecraft frame and V is the solar wind speed). The same conjecture can be made by using Elsässer variables, thus obtaining a generalized Kolmogorov phenomenology for the power spectra P ± (k ) of the Elsässer variables (Zhou and Matthaeus, 1989, 1990; Marsch, 1991)
∘ ------- 𝜖±P = C −k3∕2P ±(k) P ∓(k)k5∕2. (72 )
Even if the above expressions are affected by the presence of intermittency, namely extreme fluctuations of the energy transfer rate, and an estimated value for the Kolmogorov constant is required, the estimated energy dissipation rates roughly agree with the heating rates derived from gradients of the thermal proton distribution (MacBride et al., 2010Jump To The Next Citation Point).

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 𝜖. In a steady state situation the equation for the radial profile of ions temperature can be written as (Verma et al., 1995)

dT (r) 4T (r) mp𝜖 --dr--+ 3--r-- = (3∕2)V---(r)k--, (73 ) SW B
where mp is the proton mass and VSW (r) is the radial profile of the bulk wind speed in km s–1. (kB is the Boltzmann constant). Equation (73View Equation) 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., 2007Jump To The Next Citation Point)
3( 4 ) V (r)k T (r) 𝜖(r) = -- --− ξ -SW-----B-----. (74 ) 2 3 rmp
This relation is obtained by considering a polytropic index γ = 5∕3 for the adiabatic expansion of the solar wind plasma, the protons being the only particles heated in the process. Such assumptions are only partially correct, since the electrons could play a relevant role in the heat exchange. Heating rates obtained using Equation (74View Equation) 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 102 J ∕Kg s for cold wind to 104 J∕Kg s 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, 2006Jump To The Next Citation Point; 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 Yℓ±âˆ•â„“, or alternatively their compressible counterpart W ± ∕ℓ ℓ will result in an estimate for the pseudo-energy dissipation rates ± 𝜖 (at least within a constant of order unity), over a range of scales ℓ, which by definition is unaffected by intermittency. This has been done both in the ecliptic plane (MacBride et al., 2008Jump To The Next Citation Point, 2010Jump To The Next Citation Point) and in polar wind (Marino et al., 2009Jump To The Next Citation Point; Carbone et al., 2009b). Even preliminary attempts (MacBride et al., 2008) result in an estimate for the energy dissipation rate 𝜖E 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. (2009Jump To The Next Citation Point) using the data from the Ulysses spacecraft in the polar wind, calculate the values of the pseudo-energies from the relation ± Y ℓ ∕ ℓ, and compare these values with the radial profile of the heating rate (74View Equation) required to maintain the observed temperature against the adiabatic cooling. The Ulysses database provides two different estimates for the temperature, T1, indicated as Tlarge in literature, and T2, known as Tsmall. In general, T1 and T2 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 109View Image 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 T1 and T2 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 (74View Equation). 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 ± W ℓ ∕ℓ, 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.

View Image

Figure 109: Radial profile of the pseudoenergy transfer rates obtained from the turbulent cascade rate through the Yaglom relation, for both the compressive and the incompressive case. The solid lines represent the radial profiles of the heating rate required to obtain the observed temperature profile.

11.1 Dissipative/dispersive range in the solar wind turbulence

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 fci = eB∕mi (which in the solar wind at 1 AU is about fci ≃ 0.1 Hz), or the ion inertial length λi = c∕ωpi, 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.

View Image

Figure 110: a) Typical interplanetary magnetic field power spectrum obtained from the trace of the spectral matrix. A spectral break at about ∼ 0.4 Hz is clearly visible. b) Corresponding magnetic helicity spectrum. Image reproduced by permission from Leamon et al. (1998Jump To The Next Citation Point), copyright by AGU.

Some times ago Leamon et al. (1998Jump To The Next Citation Point) analyzed small-scales magnetic field measurements at 1 AU, by using 33 one-hour intervals of the MFI instrument on board Wind spacecraft. Figure 110View Image 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. (1998Jump To The Next Citation Point). It is evident that a spectral break exists at about fbr ≃ 0.44 Hz, close to the ion-cyclotron frequency. Below the ion-cyclotron frequency, the spectrum follows the usual power law f−α, where the spectral index is close to the Kolmogorov value α ≃ 5∕3. At small-scales, namely at frequencies above fbr, the spectrum steepens significantly, but is still described by a power law with a slope in the range α ∈ [2 –4] (Leamon et al., 1998Jump To The Next Citation Point; Smith et al., 2006Jump To The Next Citation Point), typically α ≃ 7∕3. 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., 1998Jump To The Next Citation Point). Statistical analysis by Smith et al. (2006Jump To The Next Citation Point), showed that the distribution of spectral slopes (cf. Figure 2 of Smith et al., 2006Jump To The Next Citation Point), 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., 2006Jump To The Next Citation Point). In particular, the steepening of the high-frequency range spectrum is clearly dependent on the rate of the energy cascade 𝜖 obtained as a rough estimate (cf. Figure 4 of Smith et al., 2006Jump To The Next Citation Point).

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., 2008Jump To The Next Citation Point) spanning a 0.02 ÷ 0.5 Hz frequency range. The authors found that the same spectral break by Leamon et al. (1998Jump To The Next Citation Point) exists when different datasets (Helios for large scales and Cluster for small scales) are used. The break (cf. Figure 1 of Alexandrova et al., 2008Jump To The Next Citation Point) has been found at about fbr ≃ 0.3 Hz, near the ion cyclotron frequency f ≃ 0.1 Hz ci, which roughly corresponds to spatial scales of about 1900 km ≃ 15λi (being λi ≃ 130 km the ion-skin-depth). However, as evidenced in Figure 1 of Alexandrova et al. (2008Jump To The Next Citation Point), the compressible magnetic fluctuations, measured by magnetic field parallel spectrum S∥, 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 ⟨S ⟩∕⟨S ⟩∕ ≃ 0.03 ∥ in the low-frequency range (S is the total power spectrum density and brackets means averages value over the whole range) while compressible fluctuations are increased to about ⟨S∥⟩∕⟨S ⟩∕ ≃ 0.26 in the high-frequency part. The increase of the above ratio were already noted in the paper by Leamon et al. (1998Jump To The Next Citation Point). Moreover, Alexandrova et al. (2008Jump To The Next Citation Point) 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., 2008Jump To The Next Citation Point), 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 K (f) is shown in Figure 111View Image

View Image

Figure 111: The fourth-order moment K (f ) of magnetic fluctuations as a function of frequency f is shown. Dashed line refers to data from Helios spacecraft while full line refers to data from Cluster spacecrafts at 1 AU. The inset refers to the number of intermittent structures revealed as da function of frequency. Image reproduced by permission from Alexandrova et al. (2008), copyright by AAS.

It is evident that this quantity increases as the frequency becomes smaller, thus indicating the presence of intermittency. However the rate at which K (f) 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).

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