"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

C Wavelets as a Tool to Study Intermittency

Following Farge et al. (1990Jump To The Next Citation Point) and Farge (1992Jump To The Next Citation Point), intermittent events can be viewed as localized zones of fluid where phase correlation exists, in some sense coherent structures. These structures, which dominate the statistics of small scales, occur as isolated events with a typical lifetime which is greater than that of stochastic fluctuations surrounding them. Structures continuously appear and disappear, apparently in a random fashion, at some random location of fluid, and they carry most of the flow energy. In this framework, intermittency can be considered as the result of the occurrence of coherent (non-Gaussian) structures at all scales, within the sea of stochastic Gaussian fluctuations.

It follows that, since these structures are well localized in spatial scale and time, it would be advisable to analyze them using wavelets filter instead of the usual Fourier transform. Unlike the Fourier basis, wavelets allow a decomposition both in time and frequency (or space and scale). The wavelet transform W {f(t)} of a function f (t) consists of the projection of f (t) on a wavelet basis to obtain wavelet coefficients w (τ,t). These coefficients are obtained through a convolution between the analyzed function and a shifted and scaled version of an optional wavelet base

∫ ( ′) w(τ,t) = f(t′)√1--Ψ t −-t- dt′, (126 ) τ τ
where the wavelet function
1 ( t − t′) Ψt ′,τ(t) = √---Ψ ------ τ τ

has zero mean and compact support. Some examples of translated and scaled version of this function for a particular wavelet called “charro”, because its profile resembles the Mexican hat “El Charro”, are given in Figure 116View Image, and the analytical expression for this wavelet is

[ ( ( ′)2) ( ( ′)2 )] -1-- t-−-t- 1- t −-t- Ψt′,τ(t) = √ τ 1 − τ exp − 2 τ .

Since the Parceval’s theorem exists, the square modulus |w(τ,t)|2 represents the energy content of fluctuations f (t + τ) − f(t) at the scale τ at position t.

View Image

Figure 116: Some examples of Mexican Hat wavelet, for different values of the parameters τ and ′ t.

In analyzing intermittent structures it is useful to introduce a measure of local intermittency, as for example the Local Intermittency Measure (LIM) introduced by Farge (see, e.g., Farge et al., 1990; Farge, 1992)

2 LIM = -|w-(τ,t)|-- (127 ) ⟨|w(τ,t)|2⟩t
(averages are made over all positions at a given scale τ). The quantity from Equation (127View Equation) represents the energy content of fluctuations at a given scale with respect to the standard deviation of fluctuations at that scale. The whole set of wavelets coefficients can then be split in two sets: a set which corresponds to “Gaussian” fluctuations wg(τ,t), and a set which corresponds to “structure” fluctuations ws (τ,t), that is, the whole set of coefficients w (τ,t) = wg(τ,t) ⊕ ws(τ,t) (the symbol ⊕ stands here for the union of disjoint sets). A coefficient at a given scale and position will belong to a structure or to the Gaussian background according whether LIM will be respectively greater or lesser than a threshold value. An inverse wavelets transform performed separately on both sets, namely f (t) = W −1{w (τ,t)} g g and f (t) = W −1{w (τ,t)} s s, gives two separate fields: a field fg(t) where the Gaussian background is collected, and the field fs(t) where only the non-Gaussian fluctuations of the original turbulent flow are taken into account. Looking at the field fs(t) one can investigate the spatial behavior of structures generating intermittency. The Haar basis have been applied to time series of thirteen months of velocity and magnetic data from ISEE space experiment for the first time by Veltri and Mangeney (1999).

In our analyses we adopted a recursive method (Bianchini et al., 1999; Bruno et al., 1999a) similar to the one introduced by Onorato et al. (2000) to study experimental turbulent jet flows. The method consists in eliminating, for each scale, those events which cause LIM to exceed a given threshold. Subsequently, the flatness value for each scale is checked and, in case this value exceeds the value of 3 (characteristic of a Gaussian distribution), the threshold is lowered, new events are eliminated and a new flatness is computed. The process is iterated until the flatness is equal to 3, or reaches some constant value, for each scale of the wavelet decomposition. This process is usually accomplished eliminating only a few percent of the wavelet coefficients for each scale, and this percentage reduces moving from small to large scales.

View Image

Figure 117: The black curve indicates the original time series, the red one refers to the LIMed data, and the blue one shows the difference between these two curves.

The black curve in Figure 117View Image shows the original profile of the magnetic field intensity observed by Helios 2 between day 50 and 52 within a highly velocity stream at 0.9 AU. The overlapped red profile refers to the same time series after intermittent events have been removed using the LIM method. Most of the peaks, present in the original time series, are not longer present in the LIMed curve. The intermittent component that has been removed can be observed as the blue curve centered around zero.

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