"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

D Reference Systems

Interplanetary magnetic field and plasma data are provided, usually, in two main reference systems: RTN and SE.

The RTN system (see top part of Figure 118View Image) has the R axis along the radial direction, positive from the Sun to the s/c, the T component perpendicular to the plane formed by the rotation axis of the Sun Ω and the radial direction, i.e., T = Ω × R, and the N component resulting from the vector product N = R × T.

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Figure 118: The top reference system is the RTN while the one at the bottom is the Solar Ecliptic reference system. This last one is shown in the configuration used for Helios magnetic field data, with the X-axis positive towards the Sun.

The Solar Ecliptic reference system SE, is shown (see bottom part of Figure 118View Image) in the configuration used for Helios magnetic field data, i.e., s/c centered, with the X-axis positive towards the Sun, and the Y-axis lying in the ecliptic plane and oriented opposite to the orbital motion. The third component Z is defined as Z = X × Y. However, solar wind velocity is given in the Sun-centered SE system, which is obtained from the previous one after a rotation of ∘ 180 around the Z-axis.

Sometimes, studies are more meaningful if they are performed in particular reference systems which result to be rotated with respect to the usual systems, in which the data are provided in the data centers, for example RTN or SE. Here we will recall just two reference systems commonly used in data analysis.

D.1 Minimum variance reference system

The minimum variance reference system, i.e., a reference system with one of its axes aligned with a direction along whit the field has the smallest fluctuations (Sonnerup and Cahill, 1967). This method provides information on the spatial distribution of the fluctuations of a given vector.

Given a generic field B (x,y, z), the variance of its components is

2 2 2 2 2 2 ⟨B x⟩ − ⟨Bx⟩ ;⟨B y⟩ − ⟨By ⟩ ;⟨Bz⟩ − ⟨Bz⟩ .

Similarly, the variance of B along the direction S would be given by

VS = ⟨B2 ⟩ − ⟨BS ⟩2. S

Let us assume, for sake of simplicity, that all the three components of B fluctuate around zero, then

⟨Bx ⟩ = ⟨By ⟩ = ⟨Bz⟩ = 0 =⇒ ⟨BS ⟩ = x⟨Bx ⟩ + y⟨By⟩ + z⟨Bz ⟩ = 0.

Then, the variance VS can be written as

V = ⟨B2 ⟩ = x2⟨B2 ⟩ + y2⟨B2 ⟩ + z2 ⟨B2 ⟩ + 2xy ⟨B B ⟩ + 2xz⟨B B ⟩ + 2yz ⟨B B ⟩, S S x y z x y x z y z

which can be written (omitting the sign of average ⟨⟩) as

VS = x(xB2x + yBxBy + zBxBz ) + y (yB2y + xBxBy + zByBz ) + z(zB2z + xBxBz + yByBz ).

This expression can be interpreted as a scalar product between a vector S (x, y,z) and another vector whose components are the terms in parentheses. Moreover, these last ones can be expressed as a product between a matrix M built with the terms B2 x, B2 y, B2 z, BxBy, BxBz, ByBz, and a vector S (x,y,z). Thus,

VS = (S,M S),


( ) x S ≡ ( y ) z


( B 2 B B B B ) ( x x 2y x z) M ≡ BxBy By ByBz2 . BxBz ByBz Bz

At this point, M is a symmetric matrix and is the matrix of the quadratic form V S which, in turn, is defined positive since it represents a variance. It is possible to determine a new reference system [x,y,z] such that the quadratic form VS does not contain mix terms, i.e.,

VS = x′2Bx′2 + y′2B′y2 + z′2B ′z2.

Thus, the problem reduces to compute the eigenvalues λi and eigenvectors V&tidle;i of the matrix M. The eigenvectors represent the axes of the new reference system, the eigenvalues indicate the variance along these axes as shown in Figure 119View Image.

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Figure 119: Original reference system [x,y,z] and minimum variance reference system whose axes are V1, V2, and V3 and represent the eigenvectors of M. Moreover, λ1, λ2, and λ3 are the eigenvalues of M.

At this point, since we know the components of unit vectors of the new reference system referred to the old reference system, we can easily rotate any vector, defined in the old reference system, into the new one.

D.2 The mean field reference system

The mean field reference system (see Figure 120View Image) reduces the problem of cross-talking between the components, due to the fact that the interplanetary magnetic field is not oriented like the axes of the reference system in which we perform the measurement. As a consequence, any component will experience a contribution from the other ones.

Let us suppose to have magnetic field data sampled in the RTN reference system. If the large-scale mean magnetic field is oriented in the [x, y,z] direction, we will look for a new reference system within the RTN reference system with the x-axis oriented along the mean field and the other two axes lying on a plane perpendicular to this direction.

Thus, we firstly determine the direction of the unit vector parallel to the mean field, normalizing its components

ex1 = Bx ∕|B |, ex2 = By∕ |B |, ex3 = Bz∕|B |,

so that ˆe′x(ex1,ex2,ex3) is the orientation of the first axis, parallel to the ambient field. As second direction it is convenient to choose the radial direction in RTN, which is roughly the direction of the solar wind flow, ˆeR(1,0, 0). At this point, we compute a new direction perpendicular to the plane ˆeR − ˆex

′ ′ ˆez(ez1,ez2,ez3) = ˆex × ˆeR.

Consequently, the third direction will be

′ ′ ′ ˆey(ey1,ey2,ey3) = ˆez × ˆex.

At this point, we can rotate our data into the new reference system. Data indicated as B (x, y,z) in the old reference system, will become ′ ′ ′ ′ B (x ,y,z ) in the new reference system. The transformation is obtained applying the rotation matrix A

( e e e ) x1 x2 x3 A = ( ey1 ey2 ey3) ez1 ez2 ez3

to the vector B, i.e., B ′ = AB.

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Figure 120: Mean field reference system.

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