"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

B Tools to Analyze MHD Turbulence in Space Plasmas

No matter where we are in the solar wind, short scale data always look rather random as shown in Figure 114View Image.
View Image

Figure 114: BY component of the IMF recorded within a high velocity stream.

This aspect introduces the problem of determining the time stationarity of the dataset. The concept of stationarity is related to ensembled averaged properties of a random process. The random process is the collection of the N samples x(t), it is called ensemble and indicated as {x (t)}.

Properties of a random process {x(t)} can be described by averaging over the collection of all the N possible sample functions x(t) generated by the process. So, chosen a begin time t1, we can define the mean value μx and the autocorrelation function Rx, i.e., the first and the joint moment:

N ∑ μx (t1) = Nli−m→∞ xk(t1), (78 ) k=1
N R (t ,t + τ) = lim ∑ x (t )x (t + τ). (79 ) x 1 1 N −→∞ k 1 k 1 k=1

In case μx(t1) and Rx (t1,t1 + τ) do not vary as time t1 varies, the sample function x (t) is said to be weakly stationary, i.e.,

μx(t1) = μx, (80 )
Rx (t1,t1 + τ) = Rx (τ ). (81 )

Strong stationarity would require all the moments and joint moments to be time independent. However, if x (t) is normally distributed, the concept of weak stationarity naturally extends to strong stationarity.

Generally, it is possible to describe the properties of {x (t)} simply computing time-averages over just one x (t). If the random process is stationary and μx(k) and Rx (τ,k) do not vary when computed over different sample functions, the process is said ergodic. This is a great advantage for data analysts, especially for those who deals with data from s/c, since it means that properties of stationary random phenomena can be properly measured from a single time history. In other words, we can write:

μx (k ) = μx, (82 )
Rx (τ,k) = Rx (τ). (83 )

Thus, the concept of stationarity, which is related to ensembled averaged properties, can now be transferred to single time history records whenever properties computed over a short time interval do not vary from one interval to the next more than the variation expected for normal dispersion.

Fortunately, Matthaeus and Goldstein (1982a) established that interplanetary magnetic field often behaves as a stationary and ergodic function of time, if coherent and organized structures are not included in the dataset. Actually, they proved the weak stationarity of the data, i.e., the stationarity of the average and two-point correlation function. In particular, they found that the average and the autocorrelation function computed within a subinterval would converge to the values estimated from the whole interval after a few correlation times tc. More recent analysis (Perri and Balogh, 2010) extended the above studies to different parameter ranges by using Ulysses data, showing that the stationarity assumption in the inertial range of turbulence on timescales of 10 min to 1 day is reasonably satisfied in fast and uniform solar wind flows, but that in mixed, interacting fast, and slow solar wind streams the assumption is frequently only marginally valid. If our time series approximates a Markov process (a process whose relation to the past does not extend beyond the immediately preceding observation), its autocorrelation function can be shown (Doob, 1953) to approximate a simple exponential:

−-t R (t) = R (0)e tc (84 )
from which we obtain the definition given by Batchelor (1970Jump To The Next Citation Point):
∫ ∞ R (t) tc = -----dt. (85 ) 0 R(0)

Just to have an idea of the correlation time of magnetic field fluctuations, we show in Figure 115View Image magnetic field correlation time computed at 1 AU using Voyager 2’s data.

View Image

Figure 115: Magnetic field auto-correlation function at 1 AU. Image reproduced by permission from Matthaeus and Goldstein (1982bJump To The Next Citation Point), copyright by AGU.

In this case, using the above definition, 3 tc ≃ 3.2 × 10 s.

B.1 Statistical description of MHD turbulence

When an MHD fluid is turbulent, it is impossible to know the detailed behavior of velocity field v(x,t) and magnetic field b(x,t), and the only description available is the statistical one. Very useful is the knowledge of the invariants of the ideal equations of motion for which the dissipative terms μ∇2b and ν∇2v are equal to zero because the magnetic resistivity μ and the viscosity ν are both equal to zero. Following Frisch et al. (1975) there are three quadratic invariants of the ideal system which can be used to describe MHD turbulence: total energy E, cross-helicity Hc, and magnetic helicity Hm. The above quantities are defined as follows:

1 2 2 E = -⟨v + b⟩, (86 ) 2 Hc = ⟨v ⋅ b ⟩, (87 ) Hm = ⟨A ⋅ B ⟩, (88 )
where v and b are the fluctuations of velocity and magnetic field, this last one expressed in Alfvén units b (b − → √4πρ), and A is the vector potential so that B = ∇ × A. The integrals of these quantities over the entire plasma containing regions are the invariants of the ideal MHD equations:
∫ 1- 2 2 3 E = 2 (v + b )d x, (89 ) ∫ Hc = 1- (v ⋅ b )d3x, (90 ) ∫2 3 Hm = (A ⋅ B )d x, (91 )

In particular, in order to describe the degree of correlation between v and b, it is convenient to use the normalized cross-helicity σ c:

2H σc = ---c, (92 ) E
since this quantity simply varies between +1 and − 1.

B.2 Spectra of the invariants in homogeneous turbulence

Statistical information about the state of a turbulent fluid is contained in the n-point correlation function of the fluctuating fields. In homogeneous turbulence these correlations are invariant under arbitrary translation or rotation of the experimental apparatus. We can define the magnetic field auto-correlation matrix

Rb (r) = ⟨b (x)b (x + r)⟩, (93 ) ij i j
the velocity auto-correlation matrix
Rvij(r) = ⟨vi(x)vj(x + r)⟩, (94 )
and the cross-correlation matrix
1 Rvibj(r) = -⟨vi(x)bj(x + r) + bi(x)vj(x + r)⟩. (95 ) 2
At this point, we can construct the spectral matrix in terms of Fourier transform of Rij
∫ b -1- b −ik⋅r 3 Sij(k ) = 2π Rij(r)e d r, (96 ) ∫ Sv (k ) = -1- Rv (r)e−ik⋅rd3r, (97 ) ij 2π ∫ ij vb 1 vb −ik⋅r 3 S ij (k ) = --- Rij(r)e d r. (98 ) 2π

However, in space experiments, especially in the solar wind, data from only a single spacecraft are available. This provides values of Rb ij, Rv ij, and Rvb ij, for separations along a single direction r. In this situation, only reduced (i.e., one-dimensional) spectra can be measured. If r1 is the direction of co-linear separations, we may only determine Rij(r1,0,0) and, as a consequence, the Fourier transform on Rij yields the reduced spectral matrix

1 ∫ ∫ Srij(k1) = --- Rij(r1,0,0)e−ik1⋅r1 dr1 = Sij(k1,k2,k3)dk2 dk3. (99 ) 2 π

Then, we define r H m, r H c, and r r r E = E b + Ev as the reduced spectra of the invariants, depending only on the wave number k1. Complete information about Sij might be lost when computing its reduced version since we integrate over the two transverse k. However, for isotropic symmetry no information is lost performing the transverse wave number integrals (Batchelor, 1970Jump To The Next Citation Point). That is, the same spectral information is obtained along any given direction.

Coming back to the ideal invariants, now we have to deal with the problem of how to extract information about Hm from Rij(r). We know that the Fourier transform of a real, homogeneous matrix Rij(r) is an Hermitian form Sij, i.e., S = S&tidle;∗ − → sij = s∗ji, and that any square matrix A can be decomposed into a symmetric and an antisymmetric part, As and Aa:

s a A = A + A , (100 )
s 1- &tidle; A = 2(A + A ), (101 ) 1 Aa = -(A − &tidle;A ). (102 ) 2

Since the Hermitian form implies that

S = S&tidle;∗ − → sij = s∗ji, (103 )
it follows that
s 1- &tidle; 1- S = 2 (S + S) = 2 (Sij + Sji) = real, (104 )
1 1 Sa = --(S − &tidle;S) = -(Sij − Sji) = imaginary. (105 ) 2 2

It has been shown (Batchelor, 1970; Matthaeus and Goldstein, 1982bJump To The Next Citation Point; Montgomery, 1983) that, while the trace of the symmetric part of the spectral matrix accounts for the magnetic energy, the imaginary part of the spectral matrix accounts for the magnetic helicity. In particular, Matthaeus and Goldstein (1982b) showed that

Hrm (k1 ) = 2Im Sr23(k1)∕k1, (106 )
where Hm has been integrated over the two transverse components
∫ ∫ k1- Im S23(k)dk2 dk3 = 2 Hm (k) dk2dk3. (107 )

In practice, if co-linear measurements are made along the X direction, the reduced magnetic helicity spectrum is given by:

Hr (k ) = 2Im Sr (k )∕k = 2Im (YZ ∗)∕k , (108 ) m 1 23 1 1 1
where Y and Z are the Fourier transforms of By and Bz components, respectively.

Hm can be interpreted as a measure of the correlation between the two transverse components, being one of them shifted by ∘ 90 in phase at frequency f. This parameter gives also an estimate of how magnetic field lines are knotted with each other. Hm can assume positive and negative values depending on the sense of rotation of the correlation between the two transverse components.

However, another parameter, which is a combination of H m and E b, is usually used in place of H m alone. This parameter is the normalized magnetic helicity

σm (k) = kHm (k)∕Eb (k), (109 )
where Eb is the magnetic spectral power density and σm varies between +1 and − 1.

B.2.1 Coherence and phase

Since the cross-correlation function is not necessarily an even function, the cross-spectral density function is generally a complex number:

Wxy (f ) = Cxy(f) + jQxy (f),

where the real part Cxy(f) is the coincident spectral density function, and the imaginary part Qxy(f) is the quadrature spectral density function (Bendat and Piersol, 1971). While Cxy (f) can be thought of as the average value of the product x(t)y (t) within a narrow frequency band (f,f + δf ), Qxy (f) is similarly defined but one of the components is shifted in time sufficiently to produce a phase shift of 90∘ at frequency f.

In polar notation

Wxy (f) = |Wxy (f)|e−j𝜃xy(f).

In particular,

∘ ----------------- 2 2 |Wxy (f)| = C xy(f ) + Q xy(f),

and the phase between C and Q is given by

Qxy(f-) 𝜃xy(f) = arctan C (f ). xy


|W (f )|2 ≤ W (f)W (f ), xy x y

so that the following relation holds

2 γ2 (f ) = -|Wxy-(f)|---≤ 1. xy Wx (f)Wy (f)

This function 2 γxy(f), called coherence, estimates the correlation between x(t) and y(t) for a given frequency f. Just to give an example, for an Alfvén wave at frequency f whose k vector is outwardly oriented as the interplanetary magnetic field, we expect to find 𝜃vb(f) = 180∘ and γ2vb(f ) = 1, where the indexes v and b refer to the magnetic field and velocity field fluctuations.

B.3 Introducing the Elsässer variables

The Alfvénic character of turbulence suggests to use the Elsässer variables to better describe the inward and outward contributions to turbulence. Following Elsässer (1950); Dobrowolny et al. (1980b); Goldstein et al. (1986); Grappin et al. (1989); Marsch and Tu (1989); Tu and Marsch (1990a); and Tu et al. (1989c), Elsässer variables are defined as

± --b--- z = v ± √4-πρ, (110 )
where v and b are the proton velocity and the magnetic field measured in the s/c reference frame, which can be looked at as an inertial reference frame. The sign in front of b, in Equation (110View Equation), is decided by sign [− k ⋅ B ] 0. In other words, for an outward directed mean field B 0, a negative correlation would indicate an outward directed wave vector k and vice-versa. However, it is more convenient to define the Elsässers variables in such a way that + z always refers to waves going outward and − z to waves going inward. In order to do so, the background magnetic field B0 is artificially rotated by 180 ∘ every time it points away from the Sun, in other words, magnetic sectors are rectified (Roberts et al., 1987a,b).

B.3.1 Definitions and conservation laws

If we express b in Alfvén units, that is we normalize it by √ ---- 4π ρ we can use the following handy formulas relative to definitions of fields and second order moments. Fields:

± z = v ± b, (111 ) 1- + − v = 2(z + z ), (112 ) 1 b = -(z+ − z− ). (113 ) 2
Second order moments:
1 z+ and z − energies −→ e± = -⟨(z±)2⟩, (114 ) 2 kinetic energy −→ ev = 1-⟨v2⟩, (115 ) 2 b 1 2 magnetic energy −→ e = 2-⟨b ⟩ , (116 ) v b total energy −→ e = e + e , (117 ) residual energy −→ er = ev − eb, (118 ) 1 cross- helicity −→ ec = --⟨v ⋅ b ⟩. (119 ) 2
Normalized quantities:
+ − c normalized cross-helicity −→ σ = e--−-e--= -2e----, (120 ) c e+ + e− ev + eb ev − eb 2er normalized residual-energy −→ σr = -v---b-= -+----−-, (121 ) ev+ e e + e Alfv´en ratio −→ r = e- = 1-+-σr , (122 ) A eb 1 − σr e− 1 − σc Els¨asser ratio −→ rE = -+-= ------. (123 ) e 1 + σc

We expect an Alfvèn wave to satisfy the following relations:

Table 10: Expected values for Alfvèn ratio rA, normalized cross-helicity σc, and normalized residual energy σ r for a pure Alfvèn wave outward or inward oriented.
Parameter Definition Expected Value
rA eV∕eB 1
σc (e+ − e− )∕(e+ + e− ) ±1
σr (eV − eB)∕(eV + eB ) 0

B.3.2 Spectral analysis using Elsässer variables

A spectral analysis of interplanetary data can be performed using + z and − z fields. Following Tu and Marsch (1995a) the energy spectrum associated with these two variables can be defined in the following way:

e±j (fk) = 2δT-δz±j,k(δz±j,k)∗, (124 ) n
where δz±j,k are the Fourier coefficients of the j-component among x, y, and z, n is the number of data points, δT is the sampling time, and f = k∕n δT k, with k = 0, 1,2,...,n∕2 is the k-th frequency. The total energy associated with the two Alfvèn modes will be the sum of the energy of the three components, i.e.,
± ∑ ± e (fk) = ej (fk). (125 ) j=x,y,z

Obviously, using Equations (124View Equation and 125View Equation), we can redefine in the frequency domain all the parameters introduced in the previous section.

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