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"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
Abstract
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
Acknowledgments
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
References
Footnotes
Updates
Figures
Tables

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. (1998Jump To The Next Citation Point) 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 σ (f ), 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 eB and radial direction eR as (eB × eR, eB × (eB × eR ),eB), 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. (2009Jump To The Next Citation Point) 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., 1998Jump To The Next Citation Point; Smith et al., 2006Jump To The Next Citation Point), 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

E = − V × B + mi(∇ × B ) × B, ρe

where the second term on the r.h.s. of this equation represents the Hall term (mi 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

[ ] ∂B-- mi- ∂t = ∇ × V × B − ρe (∇ × B ) × B + η∇ × B , (75 )
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 ℓ and characteristic fluctuations ρ ℓ, B ℓ, and u ℓ, we can define an eddy-turnover time T ∼ ℓ∕u NL ℓ, related to the convective process, an Hall time 2 TH ∼ ρℓℓ ∕B ℓ which characterizes typical processes related to the presence of the Hall term, and a dissipative time 2 TD ∼ ℓ∕η. At large scales the first term on the r.h.s. of Equation (75View Equation) describes the Alfvénic turbulent cascade, realized in a time TNL. 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 TNL and not yet TD, rather the cascade is realized in a time TH. This happens when TH ∼ TNL. Since at these scales density fluctuations becomes important, the mean volume rate of energy transfer can be defined as 𝜖 ∼ B2 ∕T ∼ B3∕ ℓ2ρ V ℓ H ℓ ℓ, where T H 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 ρℓ at two successive levels ℓν > ℓν+1 of the hierarchy is related to the corresponding scale size by
( ) ρν ℓν −3r -----∼ ------ , (76 ) ρν+1 ℓnu+1
where 0 ≤ |r | ≤ 1 is a measure of the degree of compression at each level ℓν. Using a scaling law for compressive effects ρ ℓ ∼ ℓ− 3r and assuming a constant spectrum energy transfer rate, we have (2∕3−2r) B ℓ ∼ ℓ, from which the spectral energy density
− 7∕3+r E (k) ∼ k . (77 )
The observed range of scaling exponents observed in solar wind α ∈ [2,4] (Leamon et al., 1998Jump To The Next Citation Point; Smith et al., 2006), can then be reproduced by different degree of compression of the solar wind plasma − 5∕6 ≤ r ≤ 1∕6.


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