1. Observing MHD Turbulence in the Solar Wind
Advanced School in Space Environment - ASSE 2006:
SOLAR-TERRESTRIAL PHYSICS
10-15 September 2006, L’Aquila, Italy
Observing MHD Turbulence in the Solar Wind by
R. Bruno
INAF-IFSI, Rome, Italy

2. Study by Leonardo da Vinci (1452-1519)
Related to the problem of reducing the rapids in the river Arno in Florence
“Turbulence still remains the last major unsolved problem in classical physics.” Feynman et al. (1977)
“Turbulence is like pornography. It is hard to define, but if you see it, you recognize it immediately.” G.K. Vallis (1999)

3. TURBULENCE Space and time chaotic behavior of a fluid flow
The legacy of
A.N. Kolmogorov
( )

4. The first feature we notice in interplanetary fluctuations is an approximate self-similarity when we look at different scales
Scales of days
Scales of hours and minutes
Scales of seconds

5. self-similarity implies power-laws
The field v(l) is invariant for a scale transformation lrl if there exists a parameter m(r) such that:
The solution of this relation is a power law: v(l)=Clh where h=-logrm(r)

6. As a matter of fact, interplanetary fluctuations do show power laws
Scales of fractions of AUs
f-5/3 fc
A typical IMF power spectrum in interplanetary space at 1 AU
[Low frequency from Bruno et al., 1985, high freq. tail from Leamon et al, 1999]

7. Spectral index of turbulent phenomena is universal
Scales of fractions of AUs
Scales of cms
f-5/3 fc
A typical IMF power spectrum in interplanetary space at 1 AU
[Low frequency from Bruno et al., 1985, high freq. tail from Leamon et al, 1999]
Laboratory experiment on turbulence with low temperature helium gas flow [Maurer et al., 1994]

8. The phenomenology at the basis of these observations follows the energy cascade á la Richardson in the hypothesis of homogeneous and isotropic turbulence 
 = energy transfer rate
integral scales
inertial range
~k-1 
dissipation scales 
homogeneous=statistically invariant under space translation
isotropic=statistically invariant under simultaneous rotation of dv and

9. K41 theory and k-5/3 scaling (A.N. Kolmogorov,1941)
eddy turnover time (generation of new scales) 2 v2 2
if e doesn’t depend on the scale

10. Turbulence is the result of nonlinear dynamics and is described by the NS eq.
Incompressible
Navier-Stokes equation
u  velocity field
P  pressure
n  kinematic viscosity
nonlinear
dissipative
For large Renon-linear regime

11. Turbulence is the result of nonlinear dynamics and is described by the NS eq.
Incompressible
Navier-Stokes equation
u  velocity field
P  pressure
n  kinematic viscosity
nonlinear
nonlinear
dissipative
dissipative
Hydromagnetic flows: same
“structure” of NS equations
Elsässer variables z+ z-
<B>

12. Turbulence is the result of nonlinear dynamics and is described by the NS eq.
Incompressible
Navier-Stokes equation
u  velocity field
P  pressure
n  kinematic viscosity
nonlinear
nonlinear
dissipative
dissipative
Hydromagnetic flows: same
“structure” of NS equations
Nonlinear interactions and the consequent energy cascade need both Z+ and Z-

13. Definition of the Elsässer variables
outward propagating wave B0 k B0
inward propagating wave k

14. Definition of the Elsässer variables
outward propagating wave B0 k B0
inward propagating wave k

15. Different origin for Z+ and Z- modes
Alfvénic radius
VSW = VA
VSW < VA
solar wind
solar wind
15~20 Rs
VSW > VA
VSW > VA
Only Outward waves escape the region inside the Alfvénic radius.
Outside the Alfvénic radius both waves are carried away by the solar wind.
interplanetary space
outward waves: solar origin + local origin
inward waves: local origin only

16. Alfvénic correlations in the solar wind
fast solar wind
slow solar wind

17. Alfvénic correlations in the solar wind
The solar wind is the most accessible medium in which to study collisionless MHD turbulence.
The solar wind turbulence, within fast wind, has a strong Alfvénic character (Alfvénic modes have a longer lifetime than other MHD modes).
Alfvénic turbulence
contributes to solar wind heating and acceleration
accelerates particles to high energies
affects cosmic ray propagation
In the following we will concentrate on Alfvénic turbulence

18. Total energy Magnetic helicity Cross-helicity
When an MHD fluid is turbulent, it is impossible to know the detailed behaviour of v(x,t) and b(x,t), the only description available is the statistical one.
Very useful is the knowledge of the invariants of the ideal equations of motion (dissipative terms ~ 0 )
There are three quadratic invariants of the ideal system (Frisch et al., 1975)
Total energy
Magnetic helicity
Cross-helicity

19. 1) the total energy per unit density E
where b is in Alfvén units
2) the cross helicity Hc
where B is defined via
3) the magnetic helicity Hm
The integrals of these quantities over the entire plasma containing regions are the invariants of the ideal MHD equations

20. However, it is more convenient to use the normalized expressions for cross-helicity and magnetic helicity
σc and σm can vary between +1 and -1
The sign of σc indicates correlation or anticorrelation between δv and δb
The sign of σm indicates left or right hand polarization

21. Definitions and invariants using Elsässer variables
fields
Second order moments
e+ and e- energy
Kinetic energy
Magnetic energy
Cross-helicity

22. Normalized parameters
Elsässer ratio
Alfvén ratio
Normalized parameters
Normalized cross-helicity
Normalized residual energy

23. for an Alfvén wave:
rA=eV/eB=1
C=(e+-e-)/(e++e-) =1
R=(eV-eB)/(eV+eB)=0

24. Helios Two spacecraft, launch 1974 & 1975 ecliptic orbit
perihelion : 0.29AU
aphelion: 1.0 AU
Provide measurements of solar wind at an early stage

25. Interplanetary data
Eclipse , courtesy of G. Fardelli

26. Interplanetary data

27. MHD turbulence: wind speed dependence and radial(time) evolution on the ecliptic

28. Solar wind turbulence: experimental evidence of non-linear cascade
Non-linear interactions still active between 0.3 and 1 AU
first k-5/3 spectrum
Coleman, Ap. J., 153, 371, 1968
Helios 2 data

29. e+ and e– spectra vs. wind speed
Outward fluctuations (e+) are dominant within fast wind.
e+~ e- within slow wind
Notice the value of the spectral slope towards a f –5/3 regime in slow wind.
(Helios data)
Marsch and Tu, JGR, 95, 8211, 1990

30. e+ and e– spectra vs. radial distance for fast wind
0.3 AU
0.9 AU
For increasing distance the e+ and e– spectra approach each other
(e+ decreases faster than e– )
At the same time the spectral slopes evolve, with the development of an extended f –5/3 regime.
(Helios data) e+ e+
f –1
f –5/3 e– e–
Marsch and Tu, JGR, 95, 8211, 1990

31. e+ and e– spectra vs. radial distance for slow wind
0.3 AU
0.9 AU e+ e+
No much radial evolution
spectral slopes always close to f –5/3
(Helios data) e- e-
f –5/3
f –5/3

32. Elsässer ratio vs. radial distance
rE = e–/e+
Fast wind
0.3 AU
0.9 AU
The Elsässer ratio increases going from 0.3 to 0.9 AU.
However, its value depends on the frequency range
(Helios data)
Marsch and Tu, JGR, 95, 8211, 1990

33. Elsässer ratio at large distance
rE = e–/e+
Fast wind
From hourly variances of
z+ andz-
Further ecliptic observations do not indicate any clear radial trend for the Elsässer ratio between 1 and 5 AU.
(Ulysses data ecliptic phase)
Bavassano et al., JGR, 106, 10659, 2001

34. Alfvén ratio vs. radial distance
rA = ev / eb
Fast wind
0.3 AU
0.9 AU
Bruno et al., JGR, 90, 4373, 1985, Marsch and Tu, JGR, 95, 8211, 1990
(Helios data)
Going from 0.3 to 0.9 AU the Alfvén ratio, at hourly frequencies, decreases from ~1 to ~0.5.

35. Alfvén ratio at large distances
rA = ev / eb
Fast wind
radial variation of the Alfvén ratio rA as observed by Helios and Voyagers between 0.3 and 20 AU.
At MHD scales (red line), after the fast decrease inside 1 AU, rA remains nearly unchanged; rA~0.5.
(Helios and Voyagers data)
Roberts et al., JGR, 95, 4203, 1990

36. First conclusion
MHD turbulence in fast wind evolves with distance towards fully developed turbulence
MHD turbulence in slow wind does not suffer radial evolution (already fully developed at 0.3 AU)

37. Which mechanism does generate turbulence
in the ecliptic?

38. The 6 lowest Fourier modes of B and V define the shear profile
Turbulence generation on the ecliptic: velocity shear mechanism (Coleman 1968)
The 6 lowest Fourier modes of B and V define the shear profile
The z+ spectrum evolves slowly
Alfvén modes added
Solar wind turbulence may be locally generated by non-linear MHD processes at velocity-shear layers.
Magnetic field reversals speed up the spectral evolution.
T=3 ~ 1 AU
This process might have a relevant role in driving turbulence evolution in low-latitude solar wind, where a fast-slow stream structure and reversals of magnetic polarity are common features.
A z– spectrum is quickly developed at high k
2D Incompressible simulations by Roberts et al., Phys. Rev. Lett., 67, 3741, 1991

39. MHD turbulence at high latitude

40. A new laboratory: the polar wind
Ulysses, the first spacecraft able to explore the heliosphere at high latitudes (up to 80°), has allowed us to study Alfvénic turbulence under a different solar wind regime, the polar wind.
McComas et al., GRL, 25, 1, 1998

41. Ulysses Launched in 1990 and still operating perihelion : 1.3AU
aphelion: 5.4 AU
Latitudinal excursion: 82°

42. Present location of Ulysses in the heliosphere

43. Polar wind features L O W HIGH
The presence of a polar wind depends on solar activity.
At LOW activity the polar wind fills a large fraction of the heliosphere.
In contrast, polar wind almost disappears at HIGH activity. L O W
HIGH
The polar wind, a relatively homogeneous environment, offers the opportunity of studying how the Alfvénic turbulence evolves under almost undisturbed conditions.
McComas et al., GRL, 29 (9), 2002

44. Polar wind: spectral evolution
Power spectra of z+ and z– at 2 and 4 AU in polar wind clearly indicate a spectral evolution qualitatively similar to that observed in ecliptic wind.
The development of a turbulent cascade with increasing distance moves the breakpoint between the f –1 and f –5/3 regimes to larger scales. z+
f –1
polar wind
2 AU z–
polar wind
4 AU
f –5/3
Goldstein et al., GRL, 22, 3393, 1995

45. Spectral breakpoint: a comparison with ecliptic wind
polar wind
In the polar wind the breakpoint is at smaller scale than at similar distances in the ecliptic wind.
Thus, spectral evolution in the polar wind is slower than in the ecliptic wind.
(data from: Helios, IMP, Pioneer, Voyager, Ulysses)
Horbury et al., Astron. Astrophys., 316, 333, 1996

46. Polar wind: Elsässer and Alfvén ratios vs. radial distance
After an initial increase, rE becomes almost constant (around 0.5).
The predominance of outward modes (z+) is preserved as far as 5 AU.
The imbalance in favour of the magnetic energy does not go beyond a value of ~0.25 for rA.
rE = e–/e+
rA = ev / eb
(data from Ulysses)

47. Radial dependence of e+ and e–
Ulysses polar wind observations show that e+ exhibits the same radial gradient over all the investigated range of distances. In contrast, e– shows a change of slope at ~2.5 AU.
Notice the good agreement of the Ulysses gradients with Helios observations in the trailing edge of ecliptic fast streams.
Ulysses
Helios
(data from Helios and Ulysses)
Bavassano et al., JGR, 105, 15959, 2000

48. Turbulence generation in the polar wind: parametric decay
In the polar wind there aren’t strong velocity gradients  velocity shear mechanism is not relevant.
parametric decay seems to work much better.
A large amplitude Alfvén wave decays into three daughter waves, two Alfvénic ones (forward and backward propagating) and one compressive. Energy mainly goes to the backward Alfvénic mode and to the compressive mode.

49. Turbulence generation in the polar wind: parametric decay
Numerical Simulations of the non-linear growth of parametric decay (e.g., see Malara et al., Phys. Plasmas, 7, 2866, 2000 and Del Zanna, Geophys. Res. Lett., 28, 2585, 2001) have shown that the final state strongly depends on the value of  (thermal to magnetic pressure ratio).
For <1 the instability completely destroys the initial Alfvénic correlation.
For =1 (a value close to solar wind conditions) the instability is not able to go beyond some limit in the disruption of the initial correlation between velocity and magnetic field fluctuations. Final state is C~0.5

50. The parametric instability for =1
Malara et al., Phys. Plasmas, 7, 2866, 2000
The parametric instability for =1
The decay ends in a state in which the initial Alfvénic correlation is partially preserved.
The predicted cross-helicity behaviour qualitatively agrees with that observed by Ulysses.
Ulysses

51. Final Remarks on Alfvénic turbulence
We have a comprehensive view of the Alfvénic turbulence evolution in the 3-D heliosphere inside 5 AU.
In the examined regions the Alfvénic turbulence appears characterized by a predominance of:
outward fluctuations
magnetic fluctuations.
As regards the outward fluctuations, in polar wind their dominant character extends to large distances from the Sun.
The imbalance in favour of magnetic energy does not appear to go beyond an asymptotic value (not understood yet).
polar turbulence evolution is slower than ecliptic turbulence.
ecliptic evolution driven by velocity shear;
polar evolution driven by parametric decay(?)

52. Intermittency in the Solar Wind Fluctuations

53. characteristics of self-similar and intermittent signals
self-similar sample
intermittent sample
statistical properties strongly
depend on where the window
is positioned
Frisch, 1995
Frisch, 1995
INTERMITTENCY: from the latin verb “Intermittere”  to interrupt
statistical properties do not depend on the position of the window
an intermittent signal alternates activity to quiescence

54. Self-similarity implies also rescaling:
if we use the standardized variables (Van Atta & Park, 1975) l
dVl(x)=|V(x+l)-V(x)|
V(x)
V(x+l) x

55. Interplanetary fluctuations are never purely self-similar (scale invariant)
Large scales  PDF Gaussian
Small scales  PDF Gaussian core + stretched tails
PDFs of solar wind speed fluctuations 1 1 2 2 3 3
(Sorriso et al., 1999)
The strongest events do not belong to the Gaussian statistics and represent the intermittent component of the fluctuations

56. 4th order moment or Flatness
Since the PDFs are not Gaussian at all scales, power spectrum is not longer sufficient and we need to investigate higher moments
Solar Wind Speed
4th order moment or Flatness
where
STRUCTURE FUNCTION
For a Gaussian statistics Ft=3
“A random function is intermittent at small scales if the flatness grows without bound at smaller and smaller scales” (Frisch, 1995)
wind speed fluctuations are more intermittent within slow wind

57. Flatness of magnetic field intensity
Also IMF is more intermittent within slow wind
We believe that the reason lies in the different nature of the source regions

58. Flatness of magnetic field componentsBZ
Slow wind
Stretched
In high speed solar wind, perpendicular components are dominated by stochastic Alfvénic fluctuations and the PDFs are nearly Gaussian
(Marsch and Tu, 1994) BZ
Fast wind
Gaussian

59. Intermittency in the mean field reference system
Mean field reference system analysis
Intermittent events mainly affect the component along the Archimedean spiral
Normal components more Alfvénic, less intermittent

60. Intermittency in ordinary fluids
Intermittency in fluids is strongly related to coherent structures (vortexes) whose life time is considerably longer than that of the surrounding fluctuations . (Chainais et al., Phys. Fluids, 11, 3524, 1999)

61. Intermittency in ordinary fluids
Intermittency in fluids is represented by coherent structures (vortexes) whose life time is considerably longer than that of the surrounding fluctuations
Energy is dissipated here at the smallest scales
What is intermittency in the solar wind?

62. Intermittency in the solar wind
(obtained from St2)
Inertial range
Proton
Gyrofrequency
kolmogorov
Intermittency starts within the inertial range at about 2 decades away from the dissipation range.
Then, what causes intermittency in the solar wind magnetic field?
Proton
Gyrofrequency

63. Looking at the nature of intermittent events
To measure Intermittency we adopt the Local Intermittency Measure technique based on wavelet transform (Farge et al1., 1990, Meneveau2, 1991)
a wavelet is a localized function which can be translated and re-scaled
the wavelet function  has zero mean and compact support.
unlike the Fourier transform, wavelet transform allows to unfold a signal both in time and frequency (or space and scale).
In Topological Fluid Mechanics, ed H.L.Moffat, Cambridge Univ. Press, 765, 1990
2) Menevau, C., J. Fluid Mech., 232, 469, 1991

64. The wavelet transform W{f(t)} of a function f(t) is a projection of f(t) on a wavelet basis to obtain wavelet coefficients w(,t).
These coefficients are obtained through a convolution between the analyzed function and a shifted and scaled version of a given wavelet basis
Because of Parceval’s theorem,
|w(,t)|2 represents the energy content of fluctuations f(t+ )-f(t) at the scale  and time t

65. Local Intermittency Measure
In 1991, 2Meneveau showed that the Flatness Factor of wavelet coefficients at a given scale is equivalent to the Flatness Factor of data at the same scale
Thus:
values of FF()>3 allow to localize events which lie outside the Gaussian statistics and cause Intermittency.
2) Menevau, C., J. Fluid Mech., 232, 469, 1991

67. Focusing on a magnetic field intermittent event
Using the Local Intermittency Measure
Minimum Variance Ref. Sys. A B
the discontinuity is 2-D (Ho et al., 1995)
there is no mass-flux across it
it looks like a TD not in pressure balance, possibly the border between adjacent flux tubes
[Bruno et al., 2001]
Fluctuations are Alfvénic on both sides of the discontinuity

68. Helios 2: 6sec magnetic field data at 0.9 AU

69. Spaghetti-like model
(Bruno et al.,2001)
average B
Local B
Structures are convected by the wind and reflect the complicate magnetic topology at the source regions of the wind

70. Radial evolution FAST WIND SLOW WIND 0.3 AU 0.9 AU
Each plot refers to a subinterval of 2000 data points

71. Waiting times between intermittent events localized via LIM (Local Intermittency Measure)
Waiting times distribution
The waiting times are distributed according to a power law
PDF(Δt) ~ Δt-β  long range correlations
Thus, the generating process is not Poissonian ().
Intermittent “coherent” events might be either locally generated by the turbulent energy cascade or generated at the sun and convected by the wind.
(Bruno and Carbone, 2005)

72. Turbulence in the solar wind is a superposition of stochastic fluctuations and coherent structures convected by the wind

73. Using Elsässer variables one identifies two main distinct families: Alfvénic fluctuations and convected structures.
Time scale 1hr
These coherent structures are dominated by magnetic energy and might be a signature of the border between adjacent flux tubes (TDs)
sC2+sR2=1

74. Moreover, these uncompressive structures dominate regions whose local magnetic field is oriented at large angles with respect to the average (large scale) magnetic field
(Each pixel > 3 elements)
At 1 AU about 25% of the 1hr intervals are magnetic dominated structures

75. Final Remarks on intermittency
Solar wind fluctuations are approximately self-similar and their power spectrum resembles a turbulence spectrum (k-5/3)
This approximate self-similarity induces to study higher order moments of the PDF of the fluctuations disclosing the phenomenon of intermittency.
Intermittency affects scales two orders of magnitude away from the dissipation range and it is due to larger scale coherent fluctuations forming the underlying turbulent structure frozen in the wind.
Interplanetary MHD turbulence is a mixture mainly of stochastic propagating (Alfvénic) fluctuations and frozen-in coherent structures convected away by the wind.
Structures are not independent, long range correlations do exist (common generating process at the sun?)

76. “There is growing evidence that there are structures with non-trivial geometry down to very fine scales ……………Leonardo was probably aware of the need to describe turbulent flow as a mixture of coherent and random motion”
[Frisch, ’95]
Studio di Leonardo da Vinci
( )
relativo al problema di eliminare le rapide dall’Arno

77. For those who want to know more about interplanetary turbulence see the following review:
Roberto Bruno and Vincenzo Carbone,
“The Solar Wind as a Turbulence Laboratory”,
Living Rev. Solar Phys., 4, p , 2005