"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

E On-board Plasma and Magnetic Field Instrumentation

In this section, we briefly describe the working principle of two popular instruments commonly used on board spacecraft to measure magnetic field and plasma parameters. For sake of brevity, we will only concentrate on one kind of plasma and field instruments, i.e., the top-hat ion analyzer and the flux-gate magnetometer. Ample review on space instrumentation of this kind can be found, for example, in Pfaff et al. (1998a,bJump To The Next Citation Point).

E.1 Plasma instrument: The top-hat

The top-hat electrostatic analyzer is a well known type of ion deflector and has been introduced by Carlson et al. (1982). It can be schematically represented by two concentric hemispheres, set to opposite voltages, with the outer one having a circular aperture centered around the symmetry axis (see Figure 121View Image). This entrance allows charged particles to penetrate the analyzer for being detected at the base of the electrostatic plates by the anodes, which are connected to an electronic chain. To amplify the signal, between the base of the plates and the anodes are located the Micro-Channel Plates (not shown in this picture). The MCP is made of a huge amount of tiny tubes, one close to the next one, able to amplify by a factor up to 106 the electric charge of the incoming particle. The electron avalanche that follows hits the underlying anode connected to the electronic chain. The anode is divided in a certain number of angular sectors depending on the desired angular resolution.

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Figure 121: Outline of a top-hat plasma analyzer.

The electric field E(r) generated between the two plates when an electric potential difference δV is applied to them, is simply obtained applying the Gauss theorem and integrating between the internal (R1) and external (R2) radii of the analyzer

R1R2 1 E (r) = δV ---------2. (128 ) R1 − R2 r

In order to have the particle q to complete the whole trajectory between the two plates and hit the detector located at the bottom of the analyzer, its centripetal force must be equal to the electric force acting on the charge. From this simple consideration we easily obtain the following relation between the kinetic energy of the particle Ek and the electric field E (r):

Ek 1 --- = --E (r )r. (129 ) q 2

Replacing E (r) with its expression from Equation (128View Equation) and differentiating, we get the energy resolution of the analyzer

δEk δr ---- = ---= const., (130 ) Ek r
where δr is the distance between the two plates. Thus, δEkβˆ•Ek depends only on the geometry of the analyzer. However, the field of view of this type of instrument is limited essentially to two dimensions since δΨ is usually rather small (∘ ∼ 5). However, on a spinning s/c, a full coverage of the entire solid angle 4 π is obtained by mounting the deflector on the s/c, keeping its symmetry axis perpendicular to the s/c spin axis. In such a way the entire solid angle is covered during half period of spin.

Such an energy filter would be able to discriminate particles within a narrow energy interval (Ek, Ek + δEk ) and coming from a small element dΩ of the solid angle. Given a certain energy resolution, the 3D particle velocity distribution function would be built sampling the whole solid angle 4π, within the energy interval to be studied.

E.2 Measuring the velocity distribution function

In this section, we will show how to reconstruct the average density of the distribution function starting from the particles detected by the analyzer. Let us consider the flux through a unitary surface of particles coming from a given direction. If f (vx,vy, vz) is the particle distribution function in phase space, f (vx, vy,vz)dvxdvy dvz is the number of particles per unit volume (ppβˆ•cm3 ) with velocity between vx and v + dv ,v x x y and v + dv ,v y y z and v + dv z z, the consequent incident flux Φ i through the unit surface is

∫ ∫ ∫ Φi = vf d3ω, (131 )
where d3ω = v2dv sinπœƒ dπœƒdΟ• is the unit volume in phase space (see Figure 122View Image).
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Figure 122: Unit volume in phase space.

The transmitted flux t C will be less than the incident flux Φi because not all the incident particles will be transmitted and Φi will be multiplied by the effective surface S(< 1), i.e.,

∫ ∫ ∫ ∫ ∫ ∫ Ct = Svf d3ω = Svf v2dv sin πœƒd πœƒdΟ• (132 )

Since for a top-hat Equation 130View Equation is valid, then

dv v2 dv = v3 ---∼ v3. v

We have that the counts recorded within the unit phase space volume would be given by

t 4 dv- 4 C Ο•,πœƒ,v = fΟ•,πœƒ,vSv δ πœƒδΟ• v sinπœƒ = f Ο•,πœƒ,vv G, (133 )
where G is called Geometrical Factor and is a characteristic of the instrument. Then, from the previous expression it follows that the phase space density function fΟ•,πœƒ,v can be directly reconstructed from the counts
t fΟ•,πœƒ,v = C-Ο•,πœƒ,v. (134 ) v4G

E.3 Computing the moments of the velocity distribution function

Once we are able to measure the density particle distribution function fΟ•,πœƒ,v, we can compute the most used moments of the distribution in order to obtain the particle number density, velocity, pressure, temperature, and heat-flux (Paschmann et al., 1998).

If we simply indicate with f(v) the density particle distribution function, we define as moment of order n of the distribution the quantity Mn, i.e.,

∫ 3 Mn = vnf (v)d ω. (135 )
It follows that the first 4 moments of the distribution are the following:
  • the number density
    ∫ n = f(v)d3ω, (136)
  • the number flux density vector
    ∫ nV = f(v)vd3 ω, (137)
  • the momentum flux density tensor
    ∫ 3 Π = m f(v)vvd ω, (138)
  • the energy flux density vector
    ∫ m 2 3 Q = -- f(v)v vd ω. (139) 2

Once we have computed the zero-order moment, we can obtain the velocity vector from Equation (137View Equation). Moreover, we can compute Π and Q in terms of velocity differences with respect to the bulk velocity, and Equations (138View Equation) and (139View Equation) become

∫ P = m f (v)(v − V )(v − V )d3ω, (140 )
∫ m- 2 3 H = 2 f(v)|v − V |(v − V) d ω. (141 )

The new Equations (140View Equation) and (141View Equation) represent the pressure tensor and the heat flux vector, respectively. Moreover, using the relation P = nKT we extract the temperature tensor from Equations (140View Equation) and (136View Equation). Finally, the scalar pressure P and temperature T can be obtained from the trace of the relative tensors

T r(P ) P = ------ij- 3


Tr (T ) T = -----ij-. 3

E.4 Field instrument: The flux-gate magnetometer

There are two classes of instruments to measure the ambient magnetic field: scalar and vector magnetometers. While nuclear precession and optical pumping magnetometers are the most common scalar magnetometers used on board s/c (see Pfaff et al., 1998b, for related material), the flux-gate magnetometer is, with no doubt, the mostly used one to perform vector measurements of the ambient magnetic field. In this section, we will briefly describe only this last instrument just for those who are not familiar at all with this kind of measurements in space.

The working principle of this magnetometer is based on the phenomenon of magnetic hysteresis. The primary element (see Figure 123View Image) is made of two bars of high magnetic permeability material. A magnetizing coil is spooled around the two bars in an opposite sense so that the magnetic field created along the two bars will have opposite polarities but the same intensity. A secondary coil wound around both bars will detect an induced electric potential only in the presence of an external magnetic field.

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Figure 123: Outline of a flux-gate magnetometer. The driving oscillator makes an electric current, at frequency f, circulate along the coil. This coil is such to induce along the two bars a magnetic field with the same intensity but opposite direction so that the resulting magnetic field is zero. The presence of an external magnetic field breaks this symmetry and the resulting field ⁄= 0 will induce an electric potential in the secondary coil, proportional to the intensity of the component of the ambient field along the two bars.

The field amplitude BB produced by the magnetizing field H is such that the material periodically saturates during its hysteresis cycle as shown in Figure 124View Image.

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Figure 124: Left panel: This figure refers to any of the two sensitive elements of the magnetometer. The thick black line indicates the magnetic hysteresis curve, the dotted green line indicates the magnetizing field H, and the thin blue line represents the magnetic field B produced by H in each bar. The thin blue line periodically reaches saturation producing a saturated magnetic field B. The trace of B results to be symmetric around the zero line. Right panel: magnetic fields B1 and B2 produced in the two bars, as a function of time. Since B1 and B2 have the same amplitude but out of phase by 180 ∘, they cancel each other.

In absence of an external magnetic field, the magnetic field B1 and B2 produced in the two bars will be exactly the same but out of phase by 180 ∘ since the two coils are spooled in an opposite sense. As a consequence, the resulting total magnetic field would be 0 as shown in Figure 124View Image. In these conditions no electric potential would be induced on the secondary coil because the magnetic flux Φ through the secondary is zero.

On the contrary, in case of an ambient field HA ⁄= 0, its component parallel to the axis of the bar is such to break the symmetry of the resulting B (see Figure 125View Image). HA represents an offset that would add up to the magnetizing field H, so that the resulting field B would not saturate in a symmetric way with respect to the zero line. Obviously, the other sensitive element would experience a specular effect and the resulting field B = B1 + B2 would not be zero, as shown in Figure 125View Image.

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Figure 125: Left panel: the net effect of an ambient field HA is that of introducing an offset which will break the symmetry of B with respect to the zero line. This figure has to be compared with Figure 124View Image when no ambient field is present. The upper side of the B curve saturates more than the lower side. An opposite situation would be shown by the second element. Right panel: trace of the resulting magnetic field B = B1 + B2. The asymmetry introduced by HA is such that the resulting field B is different from zero.

In these conditions the resulting field B, fluctuating at frequency f, would induce an electric potential V = − dΦ βˆ•dt, where Φ is the magnetic flux of B through the secondary coil.

View Image

Figure 126: Time derivative of the curve B = B1 + B2 shown in Figure 125View Image assuming the magnetic flux is referred to a unitary surface.

At this point, the detector would measure this voltage which would result proportional to the component of the ambient field HA along the axis of the two bars. To have a complete measurement of the vector magnetic field B it will be sufficient to mount three elements on board the spacecraft, like the one shown in Figure 123View Image, mutually orthogonal, in order to measure all the three Cartesian components.

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