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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

List of Figures

View Image Figure 1:
Turbulence as observed in a river. Here we can see different turbulent wakes due to different obstacles (simple stones) emerging naturally above the water level.
View Image Figure 2:
Three examples of vortices taken from the pictures by Leonardo da Vinci (cf. Frisch, 1995).
View Image Figure 3:
Turbulence as observed in a turbulent water jet (Van Dyke, 1982) reported in the book by Frisch (1995) (photograph by P. Dimotakis, R. Lye, and D. Papantoniu).
View Image Figure 4:
Turbulence in the atmosphere of Jupiter as observed by Voyager.
View Image Figure 5:
High resolution numerical simulations of 2D MHD turbulence at resolution 2048 × 2048 (courtesy by H. Politano). Here, the authors show the current density J (x,y), at a given time, on the plane (x,y).
View Image Figure 6:
Concentration field c(x,y), at a given time, on the plane (x, y). The field has been obtained by a numerical simulation at resolution 2048 × 2048. The concentration is treated as a passive scalar, transported by a turbulent field. Low concentrations are reported in blue while high concentrations are reported in yellow (courtesy by A. Noullez).
View Image Figure 7:
The original pictures by Reynolds which show the transition to a turbulent state of a flow in a pipe, as the Reynolds number increases from top to bottom (see the website Reynolds, 1883).
View Image Figure 8:
Turbulence as measured in the atmospheric boundary layer. Time evolution of the longitudinal velocity and temperature are shown in the upper and lower panels, respectively. The turbulent samples have been collected above a grass-covered forest clearing at 5 m above the ground surface and at a sampling rate of 56 Hz (Katul et al., 1997).
View Image Figure 9:
A sample of fast solar wind at distance 0.9 AU measured by the Helios 2 spacecraft. From top to bottom: speed, number density, temperature, and magnetic field, as a function of time.
View Image Figure 10:
Turbulence as measured at the external wall of a device designed for thermonuclear fusion, namely the RFX in Padua (Italy). The radial component of the magnetic field as a function of time is shown in the figure (courtesy by V. Antoni).
View Image Figure 11:
Magnetic intensity fluctuations as observed by Helios 2 in the inner solar wind at 0.9 AU, for different blow-ups. Some self-similarity is evident here.
View Image Figure 12:
Time evolution of the variables x(t), y (t), and z (t) in the Lorenz’s model (see Equation (21View Equation)). This figure has been obtained by using the parameters Pr = 10, b = 8∕3, and R = 28.
View Image Figure 13:
The Lorenz butterfly attractor, namely the time behavior of the variables z(t) vs. x(t) as obtained from the Lorenz’s model (see Equation (21View Equation)). This figure has been obtained by using the parameters Pr = 10, b = 8∕3, and R = 28.
Watch/download Movie Figure 14: (mpg-Movie; 2363 KB)
Movie: An animation built on SOHO/EIT and SOHO/SUMER observations of the solar-wind source regions and magnetic structure of the chromospheric network. Outflow velocities, at the network cell boundaries and lane junctions below the polar coronal hole, reach up to 10 km s–1 are represented by the blue colored areas (original figures from Hassler et al., 1999).
View Image Figure 15:
Helmet streamer during a solar eclipse. Slow wind leaks into the interplanetary space along the flanks of this coronal structure. Image reproduced from MSFC.
View Image Figure 16:
High velocity streams and slow wind as seen in the ecliptic during solar minimum as function of time [yyddd ]. Streams identified by labels are the same co-rotating stream observed by Helios 2, during its primary mission to the Sun in 1976, at different heliocentric distances. These streams, named “The Bavassano–Villante streams” after Tu and Marsch (1995a), have been of fundamental importance in understanding the radial evolution of MHD turbulence in the solar wind.
View Image Figure 17:
High velocity streams and slow wind as seen in the ecliptic during solar maximum. Data refer to Helios 2 observations in 1979.
View Image Figure 18:
High velocity streams and slow wind as seen in the ecliptic during solar minimum.
View Image Figure 19:
Left panel: a simple sketch showing the configuration of a helmet streamer and the density profile across this structure. Right panel: Helios 2 observations of magnetic field and plasma parameters across the heliospheric current sheet. From top to bottom: wind speed, magnetic field azimuthal angle, proton number density, density fluctuations and normalized density fluctuations, proton temperature, magnetic field magnitude, total pressure, and plasma beta, respectively. Image reproduced by permission from Bavassano et al. (1997), copyright by AGU.
View Image Figure 20:
The magnetic energy spectrum as obtained by Coleman (1968).
View Image Figure 21:
A composite figure of the magnetic spectrum obtained by Russell (1972).
View Image Figure 22:
Magnetic energy spectra, velocity spectra and kinetic energy spectra obtained by Podesta et al. (2007). Image reproduced by permission, copyright by AAS.
View Image Figure 23:
Velocity spectral index vs. heliocentric distance (Roberts, 2007).
View Image Figure 24:
Top panel: Trace of power in the magnetic field as a function of the angle between the local magnetic field and the sampling direction at a spacecraft frequency of 61 mHz. The larger scatter for 𝜃 > 90 B is the result of fewer data points at these angles. Bottom panel: spectral index of the trace, fitted over spacecraft frequencies from 15.98 mHz. Image reproduced by permission from Horbury et al. (2008), copyright by APS.
View Image Figure 25:
Typical interplanetary magnetic field power spectrum at 1 AU. The low frequency range refers to Helios 2 observations (adapted from Bruno et al., 2009) while the high frequency refers to WIND observations (adapted from Leamon et al., 1998). Vertical dashed lines indicate the correlative, Taylor and Kolmogorov length scales.
View Image Figure 26:
Typical two-point correlation function. The Taylor scale λT and the correlation length λC are the radius of curvature of the Correlation function at the origin (see inset graph) and the scale at which turbulent fluctuation are no longer correlated, respectively.
View Image Figure 27:
Estimates of the correlation function from ACE-Wind for separation distances 20 –350 RE and two sets of Cluster data for separations 0.02 –0.04RE and 0.4– 1.2 RE, respectively. Image adapted from Matthaeus et al. (2005).
View Image Figure 28:
Left panel: parabolic fit at small scales in order to estimate λ T. Right panel: exponential fit at intermediate and large scales in order to estimate λC. The square of the ratio of these two length scales gives an estimate of the effective magnetic Reynolds number. Image adapted from Matthaeus et al. (2005).
View Image Figure 29:
Left panel: power density spectra of magnetic field fluctuations observed by Helios 2 between 0.3 and 1 AU within the trailing edge of the same corotating stream shown in Figure 16, during the first mission to the Sun in 1976 and by Ulysses between 1.4 and 4.8 AU during the ecliptic phase. Ulysses observations at 4.8 AU refer to the end of 1991 while observations taken at 1.4 AU refer to the end of August of 2007. While the spectral index of slow wind does not show any radial dependence, the spectral break, clearly present in fast wind and marked by a blue dot, moves to lower and lower frequency as the heliocentric distance increases. Image adapted from Bruno et al. (2009).
View Image Figure 30:
Radial dependence of the frequency break observed in the ecliptic within fast wind as shown in the previous Figure 29. The radial dependence seems to be governed by a power-law of the order of − 1.5 R.
View Image Figure 31:
Power density spectra of the three components of IMF after rotation into the minimum variance reference system. The black curve corresponds to the minimum variance component, the blue curve to the maximum variance, and the red one to the intermediate component. This case refers to fast wind observed at 0.3 AU and the minimum variance direction forms an angle of ∘ ∼ 8 with respect to the ambient magnetic field direction. Thus, most of the power is associated with the two components quasi-transverse to the ambient field.
View Image Figure 32:
Correlation function just for the Z component of interplanetary magnetic field as observed by Helios 2 during its primary mission to the Sun. The blue color refers to data recorded at 0.9 AU while the red color refers to 0.3 AU. Solid lines refer to fast wind, dashed lines refer to slow wind.
View Image Figure 33:
Contour plot of the 2D correlation function of interplanetary magnetic field fluctuations as a function of parallel and perpendicular distance with respect to the mean magnetic field. The separation in r∥ and r⊥ is in units of 1010 cm. Image reproduced by permission from Matthaeus et al. (1990), copyright by AGU.
View Image Figure 34:
σm vs. frequency and wave number relative to an interplanetary data sample recorded by Voyager 1 at approximately 1 AU. Image reproduced by permission from Matthaeus and Goldstein (1982b), copyright by AGU.
Watch/download Movie Figure 35: (avi-Movie; 1752 KB)
Movie: A numerical simulation of the incompressible MHD equations in three dimensions, assuming periodic boundary conditions (see details in Mininni et al., 2003a). The left panel shows the power spectra for kinetic energy (green), magnetic energy (red), and total energy (blue) vs. time. The right panel shows the spatially integrated kinetic, magnetic, and total energies vs. time. The vertical (orange) line indicates the current time. These results correspond to a 1283 simulation with an external force applied at wave number kforce = 10 (movie kindly provided by D. Gómez).
View Image Figure 36:
Alfvénic correlation in fast solar wind. Left panel: large scale Alfvénic fluctuations found by Bruno et al. (1985). Right panel: small scale Alfvénic fluctuations for the first time found by Belcher and Solodyna (1975). Image reproduced by permission, copyright by AGU.
View Image Figure 37:
Alfvénic correlation in fast and slow wind. Notice the different degree of correlation between these two types of wind.
View Image Figure 38:
Histograms of normalized cross-helicity σc showing its evolution between 0.3 (circles), 2 (triangles), and 20 (squares) AU for different time scales: 3 h (top panel), 9 h (middle panel), and 81 h (bottom panel). Image reproduced by permission from Roberts et al. (1987b), copyright by AGU.
View Image Figure 39:
Values of the Alfvén ratio rA as a function of frequency and heliocentric distance, within slow (left column) and fast (right column) wind. Image reproduced by permission from Marsch and Tu (1990a), copyright by AGU.
Watch/download Movie Figure 40: (avi-Movie; 1781 KB)
Movie: A 1283 numerical simulation, as in Figure 35, but with an external force applied at wave number kforce = 3 (movie kindly provided by D. Gómez).
View Image Figure 41:
Scatter plot between the z-component of the Alfvén velocity and the proton velocity fluctuations at about 2 mHz. Data refer to Helios 2 observations at 0.29 AU (left panel) and 0.88 AU (right panel). Image adapted from Bavassano and Bruno (2000).
View Image Figure 42:
Power density spectra ± e computed from ± δz fluctuations for different time intervals indicated by the arrows. Image reproduced by permission from Tu et al. (1990), copyright by AGU.
View Image Figure 43:
Power density spectra − e and + e computed from − δz and + δz fluctuations. Spectra have been computed within fast (H) and slow (L) streams around 0.4 and 0.9 AU as indicated by different line styles. The thick line represents the average power spectrum obtained from all the about 50 e spectra, regardless of distances and wind speed. The shaded area is the 1σ width related to the average. Image reproduced by permission from Tu and Marsch (1990b), copyright by AGU.
View Image Figure 44:
Ratio of e over e+ within fast wind at 0.3 and 0.9 AU in the left and right panels, respectively. Image reproduced by permission from Marsch and Tu (1990a), copyright by AGU.
View Image Figure 45:
Upper panel: solar wind speed and solar wind speed multiplied by σc. In the lower panels the authors reported: σc, rE, e, e+, magnetic compression, and number density compression, respectively. Image reproduced by permission from Bruno and Bavassano (1991), copyright by AGU.
View Image Figure 46:
Ratio of e over e+ within fast wind between 1 and 5 AU as observed by Ulysses in the ecliptic. Image reproduced by permission from Bavassano et al. (2001), copyright by AGU.
View Image Figure 47:
Left column: e+ and e spectra (top) and σc (bottom) during a slow wind interval at 0.9 AU. Right column: kinetic e v and magnetic e B energy spectra (top) computed from the trace of the relative spectral tensor, and spectrum of the Alfvén ratio rA (bottom) Image reproduced by permission from Tu and Marsch (1991).
View Image Figure 48:
Left, from top to bottom: frequency histograms of σr vs. σc (here σC and σR) for fast wind observed by Helios 2 at 0.29, 0.65 and 0.88 AU, respectively. The color code, for each panel, is normalized to the maximum of the distribution. The yellow circle represents the limiting value given by σ2 + σ2 = 1 c r while, the yellow dashed line represents the relation σr = σc − 1, see text for details. Right, from top to bottom: frequency histograms of σr vs. σc (here σC and σR) for slow wind observed by Helios 2 at 0.32, 0.69 and 0.90 AU, respectively. The color code, for each panel, is normalized to the maximum of the distribution. Image reproduced by permission from Bruno et al. (2007), copyright EGU.
View Image Figure 49:
Power density spectra for e+ and e during a high velocity stream observed at 0.3 AU. Best fit lines for different frequency intervals and related spectral indices are also shown. Vertical lines fix the limits of five different frequency intervals analyzed by Bruno et al. (1996). Image reproduced by permission, copyright by AIP.
View Image Figure 50:
Left panel: wind speed profile is shown in the top panel. Power density associated with e+ (thick line) and e (thin line), within the five frequency bands chosen, is shown in the lower panels. Right panel: wind speed profile is shown in the top panel. Values of the angle 𝜃 ± between the minimum variance direction of + δz (thick line) and − δz (thin line) and the direction of the ambient magnetic field are shown in the lower panels, relatively to each frequency band. Image reproduced by permission from Bruno et al. (1996), copyright by AIP.
View Image Figure 51:
Large scale solar wind profile as a function of latitude during minimum (left panel) and maximum (right panel) solar cycle phases. The sunspot number is also shown at the bottom panels. Image reproduced by permission from McComas et al. (2003), copyright by AGU.
View Image Figure 52:
Magnetic field and velocity hourly correlation vs. heliographic latitude. Image reproduced by permission from Smith et al. (1995), copyright by AGU.
View Image Figure 53:
Normalized magnetic field components and magnitude hourly variances plotted vs. heliographic latitude during a complete latitude survey by Ulysses. Image reproduced by permission from Forsyth et al. (1996), copyright by AGU.
View Image Figure 54:
Spectral indexes of magnetic fluctuations within three different time scale intervals as indicated in the plot. The bottom panel shows heliographic latitude and heliocentric distance of Ulysses. Image reproduced by permission from Horbury et al. (1995c), copyright by AGU.
View Image Figure 55:
Spectral exponents for the Bz component estimated from the length function computed from Ulysses magnetic field data, when the s/c was at about 4 AU and ∘ ∼ − 50 latitude. Different symbols refer to different time intervals as reported in the graph. Image reproduced by permission from (from Horbury et al., 1995a).
View Image Figure 56:
Spectral exponents for the Bz component estimated from the length function computed from Helios and Ulysses magnetic field data. Ulysses length function (dotted line) is the same shown in the paper by Horbury et al. (1995a) when the s/c was at about 4 AU and ∼ − 50 ∘ latitude. Image reproduced by permission from Marsch and Tu (1996), copyright by AGU.
View Image Figure 57:
Hourly variances of the components and the magnitude of the magnetic field vs. radial distance from the Sun. The meaning of the different symbols is also indicated in the upper right corner. Image reproduced by permission from Forsyth et al. (1996), copyright by AGU.
View Image Figure 58:
(a) Scale dependence of radial power, (b) latitudinal power, (c) radial spectral index, (d) latitudinal spectral index, and (e) spectral index computed at 2.5 AU. Solid circles refer to the trace of the spectral matrix of the components, open squares refer to field magnitude. Correspondence between wave number scale and time scale is based on a wind velocity of 750 km s–1. Image reproduced by permission from Horbury and Balogh (2001), copyright by AGU.
View Image Figure 59:
(a) Scale dependence of power anisotropy at 2.5 AU plotted as the log10 of the ratio of BR (solid circles), BT (triangles), BN (diamonds), and |B| (squares) to the trace of the spectral matrix; (b) the radial, and (c) latitudinal behavior of the same values, respectively. Image reproduced by permission from Horbury and Balogh (2001), copyright by AGU.
View Image Figure 60:
Power spectra of magnetic field components (solid circles) and magnitude (open squares) from Ulysses (solid line) and Helios 1 (dashed line). Spectra have been extrapolated to 1 AU using radial trends in power scalings estimated from Ulysses between 1.4 and 4.1 AU and Helios between 0.3 and 1 AU. Image reproduced by permission from Horbury and Balogh (2001), copyright by AGU.
View Image Figure 61:
Trace of e+ (solid line) and e (dash-dotted line) power spectra. The central and right panels refer to Ulysses observations at 2 and 4 AU, respectively, when Ulysses was embedded in the fast southern polar wind during 1993 – 1994. The leftmost panel refers to Helios observations during 1978 at 0.3 AU. Image reproduced by permission from Goldstein et al. (1995a), copyright by AGU.
View Image Figure 62:
Normalized cross-helicity and Alfvén ratio at 2 and 4 AU, as observed by Ulysses at ∘ − 80 and ∘ − 40 latitude, respectively. Image reproduced by permission from Goldstein et al. (1995a), copyright by AGU.
View Image Figure 63:
Left panel: values of hourly variance of ± δz (i.e., ± e) vs. heliocentric distance, as observed by Ulysses. Helios observations are shown for comparison and appear to be in good agreement. Right panel: Elsässer ratio (top) and Alfvén ratio (bottom) are plotted vs. radial distance while Ulysses is embedded in the polar wind. Image reproduced by permission from Bavassano et al. (2000a), copyright by AGU.
View Image Figure 64:
2D histograms of normalized cross-helicity σc (here indicated by σC) and normalized residual energy σr (here indicated by σR) for different heliospheric regions (ecliptic wind, mid-latitude wind with strong velocity gradients, polar wind). Image reproduced by permission from Bavassano et al. (1998), copyright by AGU.
View Image Figure 65:
Results from the multiple regression analysis showing radial and latitudinal dependence of the power e+ associated with outward modes (see Appendix B.3.1). The top panel refers to the same dataset used by Horbury and Balogh (2001). The bottom panel refers to a dataset omni-comprehensive of south and north passages free of strong compressive events (Bavassano et al., 2000a). Values of e+ have been normalized to the value ∘ e+ assumed by this parameter at 1.4 AU, closest approach to the Sun. The black line is the total regression, the blue line is the latitudinal contribution and the red line is the radial contribution. Image reproduced by permission from Bavassano et al. (2002a), copyright by AGU.
View Image Figure 66:
e+ (red) and e (blue) radial gradient for different latitudinal regions of the solar wind. The first three columns, labeled EQ, refer to ecliptic observations obtained with different values of the upper limit of TBN defined as the relative fluctuations of density and magnetic intensity. The last two columns, labeled POL, refer to observations of polar turbulence outside and inside 2.6 AU, respectively. Image reproduced by permission from Bavassano et al. (2001), copyright by AGU.
View Image Figure 67:
Time evolution of the power density spectra of z+ and z− showing the turbulent evolution of the spectra due to velocity shear generation (from Roberts et al., 1991).
View Image Figure 68:
Radial evolution of e+ and e spectra obtained from the Marsch and Tu (1993a) model, in which a parametric decay source term was added to the Tu’s model (Tu et al., 1984) that was, in turn, extended by including both spectrum equations for e+ and e and solved them self-consistently. Image reproduced by permission from Marsch and Tu (1993a), copyright by AGU.
View Image Figure 69:
Spectra of e+ (thick line), e (dashed line), and ρ e (thin line) are shown for 6 different times during the development of the instability. For t ≥ 50 a typical Kolmogorov slope appears. These results refer to β = 1. Image reproduced by permission from Malara et al. (2001b), copyright by EGU.
View Image Figure 70:
Top left panel: time evolution of e+ (solid line) and e (dashed line). Middle left panel: density (solid line) and magnetic magnitude (dashed line) variances. Bottom left panel: normalized cross helicity σc. Right panel: Ulysses observations of σc radial evolution within the polar wind (left column is from Malara et al., 2001b, right panel is a courtesy of B. Bavassano).
View Image Figure 71:
The first two rows show magnetic field compression (see text for definition) for fast (left column) and slow (right column) wind at 0.3 AU (upper row) and 0.9 AU (middle row). The bottom panels show the ratio between compression at 0.9 AU and compression at 0.3 AU. This ratio is generally greater than 1 for both fast and slow wind.
View Image Figure 72:
From left to right: normalized spectra of number density, magnetic field intensity fluctuations (adapted from Marsch and Tu, 1990b), and proton temperature (adapted from Tu et al., 1991). Different lines refer to different heliocentric distances for both slow and fast wind.
View Image Figure 73:
From top to bottom: field intensity |B |; proton and alpha particle velocity vp and vα; corrected proton velocity vpc = vp − δvA, where vA is the Alfvén speed; proton and alpha number density np and nα; proton and alpha temperature Tp and T α; kinetic and magnetic pressure Pk and Pm, which the authors call Pgas and Pmag; total pressure Ptot and β = Pgas∕Pmag (from Tu and Marsch, 1995a).
View Image Figure 74:
Correlation coefficient between number density n and total pressure pT plotted vs. the correlation coefficient between kinetic pressure and magnetic pressure for both Helios relatively to fast wind. Image reproduced by permission from Marsch and Tu (1993b).
View Image Figure 75:
Histograms of ρ(N − Pt) and ρ(Pm − Pk ) per solar rotation. The color bar on the left side indicates polar (red), mid-latitude (blue), and low latitude (green) phases. Moreover, universal time UT, heliocentric distance, and heliographic latitude are also indicated on the left side of the plot. Occurrence frequency is indicated by the color bar shown on the right hand side of the figure. Image reproduced by permission from Bavassano et al. (2004), copyright EGU.
View Image Figure 76:
Solar rotation histograms of B-N and B-T in the same format of Figure 75. Image reproduced by permission from Bavassano et al. (2004), copyright EGU.
View Image Figure 77:
Scatter plots of the relative amplitudes of total pressure vs. density fluctuations for polar wind samples P1 to P4. Straight lines indicate the Tu and Marsch (1994) model predictions for different values of α, the relative PBS/W contribution to density fluctuations. Image reproduced by permission from Bavassano et al. (2004), copyright EGU.
View Image Figure 78:
Relative amplitude of density fluctuations vs. turbulent Mach number for polar wind. Solid and dashed lines indicate the M and 2 M scalings, respectively. Image reproduced by permission from Bavassano et al. (2004), copyright EGU.
View Image Figure 79:
Wind speed profile V and |σc|V are shown in the top panel. The lower three panels refer to correlation coefficient, phase angle and coherence for the three components of δV and δB fluctuations, respectively. The successive panel indicates the value of the angle between magnetic field and velocity fluctuations minimum variance directions. The bottom panel refers to the heliocentric distance (from Bruno and Bavassano, 1993).
View Image Figure 80:
Structure functions for the magnetic field intensity Sn (r) for two different orders, n = 3 and n = 5, for both slow wind and fast wind, as a function of the time scale r. Data come from Helios 2 spacecraft at 0.9 AU.
View Image Figure 81:
Structure functions Sn (r) for two different orders, n = 3 and n = 5, for both slow wind and high wind, as a function of the fourth-order structure function S4(r). Data come from Helios 2 spacecraft at 0.9 AU.
View Image Figure 82:
Left panel: normalized PDFs for the magnetic fluctuations observed in the solar wind turbulence by using Helios data. Right panel: distribution function of waiting times Δt between structures at the smallest scale. The parameter β is the scaling exponent of the scaling relation PDF (Δt) ∼ Δt −β for the distribution function of waiting times.
View Image Figure 83:
Left panel: normalized PDFs of velocity fluctuations in atmospheric turbulence. Right panel: distribution function of waiting times Δt between structures at the smallest scale. The parameter β is the scaling exponent of the scaling relation −β PDF (Δt) ∼ Δt for the distribution function of waiting times. The turbulent samples have been collected above a grass-covered forest clearing at 5 m above the ground surface and at a sampling rate of 56 Hz (Katul et al., 1997).
View Image Figure 84:
Left panel: normalized PDFs of the radial magnetic field collected in RFX magnetic turbulence (Carbone et al., 2000). Right panel: distribution function of waiting times Δt between structures at the smallest scale. The parameter β is the scaling exponent of the scaling relation PDF (Δt) ∼ Δt −β for the distribution function of waiting times.
View Image Figure 85:
Differences for the longitudinal velocity δu = u(t + τ ) − u (t) τ at three different scales τ, as shown in the figure.
View Image Figure 86:
Differences for the magnetic intensity Δbτ = B (t + τ) − B(t) at three different scales τ, as shown in the figure.
View Image Figure 87:
We show the kinetic energy spectrum |un (t)|2 as a function of log2 kn for the MHD shell model. The full line refer to the Kolmogorov spectrum k −n2∕3.
View Image Figure 88:
We show the magnetic energy spectrum |bn(t)|2 as a function of log2kn for the MHD shell model. The full line refer to the Kolmogorov spectrum k −n2∕3.
View Image Figure 89:
Time behavior of the real part of velocity variable u (t) n at three different shells n, as indicated in the different panels.
View Image Figure 90:
Time behavior of the real part of magnetic variable b (t) n at three different shells n, as indicated in the different panels.
View Image Figure 91:
In the first three panels we report PDFs of both velocity (left column) and magnetic (right column) shell variables, at three different shells ℓn. The bottom panels refer to probability distribution functions of waiting times between intermittent structures at the shell n = 12 for the corresponding velocity and magnetic variables.
View Image Figure 92:
An example of the linear scaling for the third-order mixed structure functions Y ±, obtained in the polar wind using Ulysses measurements. A linear scaling law represents a range of scales where Yaglom’s law is satisfied. Image reproduced by permission from Sorriso-Valvo et al. (2007), copyright by APS.
View Image Figure 93:
The linear scaling relation is reported for both the usual third-order structure function Y + ℓ and the same quantity build up with the density-mediated variables W + ℓ. A linear relation full line is clearly observed. Data refer to the Ulysses spacecraft. Image reproduced by permission from Carbone et al. (2009a), copyright by APS.
View Image Figure 94:
Left: normalized PDFs of fluctuations of the longitudinal velocity field at four different scales τ. Right: normalized PDFs of fluctuations of the magnetic field magnitude at four different scales τ. Solid lines represent the fit made by using the log-normal model. Image reproduced by permission from Sorriso-Valvo et al. (1999), copyright by AGU.
View Image Figure 95:
Scaling laws of the parameter λ2 (τ) as a function of the scales τ, obtained by the fits of the PDFs of both velocity and magnetic variables (see Figure 94). Solid lines represent fits made by power laws. Image reproduced by permission from Sorriso-Valvo et al. (1999), copyright by AGU.
View Image Figure 96:
From top to bottom: 81 s averages of velocity wind profile in km s–1, magnetic field intensity in nT, the logarithmic value of magnetic (blue line), thermal (red line), and total pressure (black line) in dyne/cm2 and field intensity residuals in nT. The two vertical boxes delimit the two time intervals #1 and #2 which were chosen for comparison. While the first interval shows strong magnetic intermittency, the second one does not. Image reproduced by permission from Bruno et al. (2001), copyright by Elsevier.
View Image Figure 97:
Left column, from top to bottom: we show magnetic field intensity, maximum λ3, intermediate λ2 and minimum λ1 variance components for magnetic field (blue color) and wind velocity relative to the time interval #1 shown in Figure 96. Right below, we show the hodogram on the maximum variance plane λ3 − λ2, as a function of time (blue color line). The red lines are the projection of the blue line. The large arc, from A to B, corresponds to the green segment in the profile of the magnetic field components shown in the upper panel. The same parameters are shown for interval #2 (Figure 96), in the same format, on the right hand side of the figure. The time resolution of the data is 81 s. Image reproduced by permission from Bruno et al. (2001), copyright by Elsevier.
View Image Figure 98:
Trajectory followed by the tip of the magnetic field vector (blue color line) in the minimum variance reference system for interval #1 (left) and #2 (right). Projections on the three planes (red color lines) formed by the three eigenvectors λ1,λ2,λ3, and the average magnetic field vector, with its projections on the same planes, are also shown. The green line extending from label A to label B refers to the arc-like discontinuity shown in Figure 97. The time resolution of the magnetic field averages is 6 s. Image reproduced by permission from Bruno et al. (2001), copyright by Elsevier. (To see animations relative to similar time intervals click on Figures 99 for a timeseries affected by the intermittency phenomenon or at 100 for non-intermittent and intermittent samples.
Watch/download Movie Figure 99: (gif-Movie; 4927 KB)
Movie: Trajectory followed by the tip of the magnetic field vector in the minimum variance reference system during a time interval not characterized by intermittency. The duration of the interval is 2000 × 6 s but the magnetic field vector moves only for 100 × 6 s in order to make a smaller file (movie kindly provided by A. Vecchio).
Watch/download Movie Figure 100: (gif-Movie; 3897 KB)
Movie: Trajectory followed by the tip of the magnetic field vector in the minimum variance reference system during a time interval characterized by intermittent events. The duration of the interval is 2000 × 6 s but the magnetic field vector moves only for 100 × 6 s in order to make a smaller file (movie kindly provided by A. Vecchio).
View Image Figure 101:
Simple visualization of hypothetical flux tubes which tangle up in space. Each flux tube is characterized by a local field direction, and within each flux tube the presence of Alfvénic fluctuations makes the magnetic field vector randomly wander about this direction. Moreover, the large scale is characterized by an average background field direction aligned with the local interplanetary magnetic field. Moving across different flux-tubes, characterized by a different values of |B |, enhances the intermittency level of the magnetic field intensity time series (cf. Bruno et al., 2001).
View Image Figure 102:
Composite figure made adapting original figures from the paper by Chang et al. (2004). The first element on the upper left corner represents field-aligned spatio-temporal coherent structures. A cross-section of two of these structures of the same polarity is shown in the upper right corner. Magnetic flux iso-contours and field polarity are also shown. The darkened area represents intense current sheet during strong magnetic shear. The bottom element of the figure is the result of 2D MHD simulations of interacting coherent structures, and shows intermittent spatial distribution of intense current sheets. In this scenario, new fluctuations are produced which can provide new resonance sites, possibly nucleating new coherent structures.
View Image Figure 103:
Probability distributions of the angular displacements experienced by magnetic vector on a time scale of 6 s at 0.3 and 0.9 AU, for a fast wind, respectively. Solid curves refer to lognormals contributing to form the thick solid curve which best fits the distribution. Image reproduced by permission from Bruno et al. (2004), copyright EGU.)
View Image Figure 104:
Measurements of angular differences of magnetic field direction on time scale of 128 s. Data set is from ACE measurements for the years 1998 – 2004. Exponential fits to two portions of the distribution are shown as dashed curves. Images reproduced by permission from Borovsky (2008), copyright by AGU.
View Image Figure 105:
Distribution function for two time periods. The left panels show the dependence of F (𝜃,ζ) on 𝜃, and the right panels show the dependence of F(𝜃,ζ ) on ζ. The presence of a current sheet makes F (𝜃,ζ) to increases linearly with ζ (dashed lines in the right panels). Image reproduced by permission from Li (2008), copyright by AAS.
View Image Figure 106:
Flatness ℱ vs. time scale τ relative to magnetic field fluctuations. The left column (panels A and C) refers to slow wind and the right column (panels B and D) refers to fast wind. The upper panels refer to compressive fluctuations and the lower panels refer to directional fluctuations. Vertical bars represent errors associated with each value of ℱ. The three different symbols in each panel refer to different heliocentric distances as reported in the legend. Image reproduced by permission from Bruno et al. (2003b), copyright by AGU.
View Image Figure 107:
Flatness ℱ vs. time scale τ relative to wind velocity fluctuations. In the same format of Figure 106 panels A and C refer to slow wind and panels B and D refer to fast wind. The upper panels refer to compressive fluctuations and the lower panels refer to directional fluctuations. Vertical bars represent errors associated with each value of ℱ. Image reproduced by permission from Bruno et al. (2003b), copyright by AGU.
View Image Figure 108:
Values of 2 λ (upper panel) and 2 σ (lower panel) vs. heliocentric distance (see Section 9.2 for description of Castaing distribution and definition of λ and σ). These values have been calculated for the projected low frequency beginning of the inertial range relative to each distance (see text for details). R, T, and N components are indicated by asterisks, crosses and circles, respectively. Image reproduced by permission from Pagel and Balogh (2003), copyright by AGU.
View Image Figure 109:
Radial profile of the pseudoenergy transfer rates obtained from the turbulent cascade rate through the Yaglom relation, for both the compressive and the incompressive case. The solid lines represent the radial profiles of the heating rate required to obtain the observed temperature profile.
View Image Figure 110:
a) Typical interplanetary magnetic field power spectrum obtained from the trace of the spectral matrix. A spectral break at about ∼ 0.4 Hz is clearly visible. b) Corresponding magnetic helicity spectrum. Image reproduced by permission from Leamon et al. (1998), copyright by AGU.
View Image Figure 111:
The fourth-order moment K (f ) of magnetic fluctuations as a function of frequency f is shown. Dashed line refers to data from Helios spacecraft while full line refers to data from Cluster spacecrafts at 1 AU. The inset refers to the number of intermittent structures revealed as da function of frequency. Image reproduced by permission from Alexandrova et al. (2008), copyright by AAS.
View Image Figure 112:
Observed dispersion relations (dots), with estimated error bars, compared to linear solutions of the Maxwell–Vlasov equations for three observed angles between the k vector and the local magnetic field direction (damping rates are represented by the dashed lines). Proton and electron Landau resonances are represented by the black curves Lp,e. Proton cyclotron resonance are shown by the curves Cp. (the electron cyclotron resonance lies out of the plotted frequency range). Image reproduced by permission from Sahraoui et al. (2010a), copyright by APS.
View Image Figure 113:
Top: Angles between the wave vectors and the mean magnetic field as a function of the wave number. Bottom: Frequency-wave number diagram of the identified waves in the plasma rest frame. Magnetosonic (MS), whistler (WHL), and kinetic Alfvén waves (KAW)dispersion relations are represented by dashed, straight, and dotted lines, respectively. Image reproduced by permission from Narita et al. (2011a), copyright by AGU.
View Image Figure 114:
BY component of the IMF recorded within a high velocity stream.
View Image Figure 115:
Magnetic field auto-correlation function at 1 AU. Image reproduced by permission from Matthaeus and Goldstein (1982b), copyright by AGU.
View Image Figure 116:
Some examples of Mexican Hat wavelet, for different values of the parameters τ and ′ t.
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.
View Image 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.
View Image 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.
View Image Figure 120:
Mean field reference system.
View Image Figure 121:
Outline of a top-hat plasma analyzer.
View Image Figure 122:
Unit volume in phase space.
View Image 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.
View Image 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.
View Image 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 124 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.
View Image Figure 126:
Time derivative of the curve B = B1 + B2 shown in Figure 125 assuming the magnetic flux is referred to a unitary surface.