Intermittency of turbulence in the solar wind
V. Carbone, L. Sorriso–Valvo , F. Lepreti , P. Veltri and R. Bruno†
Dipartimento di Fisica, Universitá della Calabria and Istituto Nazionale di Fisica della Materia, sezione di
Cosenza, Italy.
†
Istituto di Fisica dello Spazio Interplanetario/CNR, Rome, Italy.
Abstract.
We review some of the work done to investigate statistical features of turbulence in the inner solar wind, within slow–
speed streams. We present results related to anomalous scaling laws of fluctuations and to the scaling behaviour of Probability
Density Function (pdf). The results are compared with turbulence in other systems, that is neutral fluid flows and laboratory
plasma. Some of the statistical properties are shared with low–resolution cascade models which describe the gross features of
turbulence.
INTRODUCTION
Let us introduce the differences of fields, along the direction of r, between two points separated by a distance
r, that is δ z
r = [z (x + r) z (x)] er which represent
characteristic fluctuations across eddies at the scale r (e r
being the direction of r). These are the main quantities
involved in the study of statistical properties of turbulence. In fact, under suitable hypothesis (homogeneity,
stationarity and isotropy, see Ref. [2]), an exact relation
for the inertial range can be obtained from ideal MHD
equations
Turbulence is a phenomenon in which chaotic dynamics and power law statistics coexist, and is characterised
by randomness in both space and time, unpredictability
and instability to every small perturbation. Flows in turbulent conditions are often characterised by the presence
of structures on all scales, energetically dominant with
respect to the remaining part of the flow. In these conditions low–resolution dynamical models can describe
quite realistically the flow [1]. Here we briefly summarize the work done by studying low–frequency plasma
turbulence using the solar wind as a "wind tunnel" and
spacecrafts as probes. We will compare the results with
other turbulent systems.
<
Low–frequency turbulence in plasmas is the result of
nonlinear dynamics of incompressible Magnetohydrodynamic (MHD) equations, which can be written as
+
z ∇ z = ∇(P=ρ ) + ν ∇2 z + f
3
(2)
being ε the energy transfer rates in the stationary state,
for both Alfvénic fluctuations (brackets represent ensemble averages). This relation is the MHD analogous of
the 4=5–law Kolmogorov’s law [3]. Since the 3–th order
moment is different from zero the energy cascade generates fluctuations with some phase correlations. Moreover ideal incompressible MHD equations (1) are invariant providing lengths and velocities are scaled respectively as r ! λ r and z ! z λ h for each value of h (or
both v ! vλ h and B ! Bλ h ). Then we expect scaling
h
laws where δ z
r δ v δ B r . Since equations are invariants for each value of h, this can be fixed only with
suitable phenomenlogical considerations. The compressible case is a little bit different. Since scaling laws are
expected also for the density fluctuations, velocity and
magnetic fluctuations must have scaling laws with different exponents.
In the fluid–like case, by introducing the pseudo–
2
energies dissipation rate ε (δ z
r ) =Tr , and using the
eddy–turnover times Tr τNL r=δ zr , we obtain the
2 scaling relations (δ z
r ) δ zr ε r, which lead to the
SCALING LAWS FOR
HYDROMAGNETIC TURBULENCE
∂ z
∂t
2 δ z >= 4 ε r
r
(δ zr )
(1)
with the conditionp
∇ z = 0. We use the Elsässer vari
ables z = u B= 4πρ , being u and B respectively the
velocity and magnetic field, P is the total pressure, ρ the
constant mass density, ν represents the kinematic viscosity (assumed to be equal to the magnetic diffusivity) and
f represent some external forcing terms.
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
439
1.0
PDF( b)
1.0
0.1
0.1
= 0.02 h
= 0.02 h
1.0
0.1
0.1
= 0.4 h
= 0.4 h
1.0
0.01
-3
= 5.8 h
-2
-1
0
b
1
0.01
= 5.8 h
2
3 -3
-2
-1
0
v
1
2
PDF( v)
PDF( b)
1.0
PDF( v)
PDF( b)
1.0
PDF( v)
Kolmogorov’s scaling h = 1=3 when ε are constant.
When the charged fluid is magnetically
p dominated, the
Alfvén time τA r=CA (CA = B0 = 4πρ is the Alfvén
velocity related to the average magnetic field) can become smaller than the eddy–turnover times, so that the
nonlinear interactions between opposite travelling eddies, are lowered. This means that the energy cascade is
(τ =τ ) [4], and the scaling
realised in a time Tr τNL
NL A
2
2
relation becomes (δ z r ) (δ z
r ) ε r, which leads to
the Kraichnan scaling law h = 1=4 if the pseudo–energies
dissipation rate are constant.
Homogeneous and isotropic turbulence in MHD has
been described by the pseudo–energies density spectra
which are related to the 2–th order moment of fluctua2
tions kE (k) < (δ z
r ) > (r 1=k). Using the scaling
laws with h = 1=3, a Kolmogorov’s spectrum E (k) k 5=3 is obtained, while in the other case (h = 1=4) the
Iroshnikov–Kraichnan spectrum E k 3=2 is recovered. Satellite observations do not give definite answer
as regarding to the phenomenology preferred by the solar wind [5]. Of course in the usual fluid flows the Kolmogorov’s spectrum is observed in all cases, and this indicates that the predictability lost at dynamical level, can
be reintroduced at a statistical level [3].
To get some insigth of turbulence we have to look at
higher order moments of fluctuations. In fact, since δ z r
are stochastic variables, the Probability Density Functions pd f (δ z
r ) are uniquely determined when the entire
set of moments of fluctuations are known. For a gaussian process the 2–th order moment suffices to fully determine pdf and then to characterize turbulence. It can
be immediately seen what kind of prediction can be
made for the high–order moments. In the fluid–like case
1=3 , so that, by defining the p–th
we have δ z
r (ε r )
p
order moment S (
p r ) =< (δ zr ) >, we obtain S p (r ) (ε ) p=3 r p=3 . In the magnetically dominating case we
1=4 , so that S p (r) (C ε ) p=4 r p=4 .
have δ z
r (ε r )
A
In both cases the scaling exponent ζ p , defined through
ζp
S
p (r ) r , are linear ζ p = ph.
3
FIGURE 1. We report pdfs of fluctuations for velocity (right
hand panels) and magnetic field (left hand panels) at different
scales τ for solar wind data.
mission in the inner heliosphere. The original data were
collected in 81 s. bins and we choose a set of subintervals of 2 days each. The subintervals were selected separately within low speed regions and high speed regions
selected in a standard way according to a threshold velocity [6]. For each subinterval we calculated the velocity
and magnetic increments at a given time scale τ through
δ Vτ = V (t + τ ) V (t ) and δ Bτ = B(t + τ ) B(t ). Of
course in the supersonic solar wind moving at speed V SW ,
the usual Taylor’s hypothesis is verified, and we can get
information on spatial scale r through τ = r=VSW . In the
following we report only results relative to the slow periods. The incompressibility assertion is perhaps correct
in the Alfvén waveband, in fast streams, but it is very approximative in the slow streams, even in the same waveband. Also, at still lower wavelengths, it is certainly false.
Let us consider the pdfs of the normalized quantities
δ uτ = δ Vτ = < δ Vτ2 >1=2 and δ bτ = δ Bτ = < δ B2τ >1=2
(using the ergodic theorem brackets are now time averages). The interest of this normalization is the fact that
pdfs of these fluctuations can be compared as far as the
scaling properties are concerned. In particular if pdfs at
two different scales become identical, the phenomenon is
self–similar [7]. A plot of pdfs calculated for these quantities (see fig. 1) as a function of the scale r shows that:
1) the pdfs are not gaussian, at least at small scales; 2)
the shape of pdfs depends on the scale τ . In particular,
while for large scales the differences are gaussian distributed, as the scale becomes smaller the wings of pdfs
becomes more and more important [8, 9]. This accumulation of high–probability intense fluctuations to small
scales is one of manifestation of intermittency in turbulence. Since the process is not gaussian the energy density does not play any privileged role. Turbulence must
be characterized not only by the second–order moment,
rather by the whole set of moments or by the scaling behaviour of pdfs.
SOLAR WIND LOW–FREQUENCY
FLUCTUATIONS
The satellite observations of both velocity and magnetic field in the interplanetary space, offer us an almost
unique possibility to gain information on the turbulent
MHD state in a very large scale range, say from 1 AU
(Astronomical Units) up to 10 3 km. Here we report some
analyses of plasma measurements of the bulk velocity
V (t ) and magnetic field intensity B(t ). These analysis
are mainly based on plasma measurements as recorded
by the instruments on board Helios 2 during its primary
440
0.1
= 0.35 s
0.01
= 0.5 s
1.0
0.1
=3s
0.01
=8 s
1.0
0.1
= 50 s
0.01
FIGURE 2. Normalized scaling exponents ζp =ζ3 of p–th
order moment for both velocity and magnetic field for solar
wind data.
-2
-1
0
v
1
0.01
= 128 s
2
3 -4 -3 -2 -1
0
b
1
2
3
PDF( b)
PDF( v)
0.01
-3
PDF( b)
PDF( v)
0.01
PDF( b)
PDF( v)
1.0
4
FIGURE 3. Pdfs of fluctuations for velocity field (fluid
flows, left hand panel), and magnetic field (laboratory plasma,
right hand panel) at different scales τ .
Since the pdfs are scale–independent, turbulence appears to be not globally self–similar, so that moments of
fluctuations can have anomalous scaling relations. From
the scaling properties of p–th moments of fluctuations
p
p
< δV
τ > and < δ Bτ > [6] The relative scaling exponents ζ p =ζ3 allow us to compare them with the scaling
exponents obtained in usual fluid flows [10, 6]. In figure 2 the scaling exponents as a function of the order
p are reported. It is evident that the behaviour of ζ p =ζ3
is different from the linear shape of the classical phenomenologies (see also Ref. [11]): the shape of ζ p =ζ3
turns out to be a nonlinear function of p.
The above behaviours of pdfs and moments are quite
universal. We analysed a sample of fluid turbulence collected in the earth’s boundary layer [12] where the longitudinal velocity field v(t ) and the temperature field
T (t ) are recorded. Furthermore some samples of magnetic turbulence collected at the edge of RFX have been
examined. RFX is a device in PAdova (Italy) where the
plasma is confined in a reversed field pinch configuration
[13] designed for thermonuclear fusion. In figures 3 the
pdfs of the normalized velocity δ w τ = δ vτ = < δ v2τ >1=2
(for the fluid flow), and normalized magnetic fluctuations δ bτ = δ Bτ = < δ B2τ >1=2 for RFX, have been reported. The scaling behaviour of pdfs looks to be similar
to what has been observed in the solar wind. The pdfs
are gaussian at large scales, and develop fat tails at small
scales. Furthermore we examined the normalized scaling
exponents for velocity and temperature fields in the fluid
flows. The values of the normalised exponents for the velocity turn out to be the same for the solar wind and for
the fluid sample. It is worthwhile to realize that in this experiment the temperature field acts like a passive scalar,
and is transported by the velocity field. Using the differences ∆ p = p=3 (ζ p =ζ3 ) as a measure of the intensity
of intermittency, it can be seen that the passive scalar is
more intermittent than the velocity field (fig. 4). In the
solar wind the same behaviour is visible for the magnetic
field. Actually magnetic field, which behaves like a "pas-
FIGURE 4. Normalized scaling exponents ζp =ζ3 of p–th
order moment for both velocity and temperature fields, for fluid
flows data.
sive vector", results to be more intermittent than the velocity field, and this seems to be a characteristic of MHD
flows because this is visible also in numerical simulations [14]. As regards laboratory plasma, intermittency
of magnetic fluctuations in RFX [15] depends on the distance from the external wall where measurements have
been performed (fig. 5). We found that intermittency increases going towards the external wall.
FIGURE 5. Normalized scaling exponents ζp =ζ3 for RFX,
calculated from turbulent samples taken at different distances
R from the external wall. Distances are normalized with the
minor radius a = 0:457 m of the torus.
441
vx (km/sec)
10
0
-10
0
1.0*10
0
1.0*10
0
1.0*10
0.0*10
3
2.0*10
3
3.0*10
3
2.0*10
3
2.0*10
3
4.0*10
3
3.0*10
3
3.0*10
3
5.0*10
3
4.0*10
3
4.0*10
3
6.0*10
3
7.0*10
3
5.0*10
3
5.0*10
3
8.0*10
3
6.0*10
3
7.0*10
3
6.0*10
3
7.0*10
3
9.0*10
3
8.0*10
3
8.0*10
3
1.0*10
3
9.0*10
3
9.0*10
4
1.1*10
3
1.0*10
3
1.0*10
4
4
1.1*10
4
1.1*10
vx (km/sec)
5
0
-5
vx (km/sec)
0.0*10
0
0.0*10
vx (km/sec)
4
4
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
n
FIGURE 6. We report the time evolution of velocity fluctuations δ Vτ (t ) at four different scales for solar wind data. Scales
increases from the top to the bottom.
WHAT IS INTERMITTENCY? A
MULTIFRACTAL MODEL
singularities of the gradient of the field.
Because of the idea of self–similarity underlying the
energy cascade process in turbulence [1], a different
point of view can be introduced [16, 9]. That is a model
which tries to characterize the behaviour of the pdfs
through the scaling law of a parameter describing how
the shape of the pdf changes in going towards small
scales. In its simplest form the model can be introduced
by saying that the pdf of the increments δ ψ r (representing here both velocity and magnetic fluctuations) at a
given scale r, is made by a convolution of the typical
Gaussian distribution of widths σ =< δ ψ 2 >1=2 , whose
distribution is given by a function G λ (σ )
Let us look at figure 6, where we reported the time evolution of δ Vτ for four different values of τ for the solar wind data. Fluctuations at large scales appear to be
smooth, while as the scale becomes smaller, intense fluctuations are visible. In fact these intense fluctuations are
not distributed in a continuous way, instead they are relatively rare, and we see that there are periods with relative
quiet activity alternating to small periods where the turbulent activity is very high. This is precisely the meaning
of intermittency in fully developed turbulence. Starting
from this point, it is natural to conjecture that, even if
the fluid cannot be globally self–similar, self–similarity
can be reintroduced as a local property. This is the basis
of the multifractal model of intermittency, cf. e.g. [3], in
which it is conjectured that turbulent flows can be made
of an infinite set of points S h (x), each characterised by a
h
scaling law δ z
τ τ and a local scaling exponent h(x).
The dimension of the set is variable D(h). With this in
mind it can be shown that the high–order moments can
be described by ζ p = minh [ ph + 3 D(h)]. In this way
the departure of ζ p from a linear scaling, and then intermittency, can be characterized through the changing of
generalized dimensions D(h), as h is varied. That is as
p increses, we are probing regions of fluids where even
more rare and intense events exists. These regions are
characterised by a smaller value of h, and by stronger
P (δ ψr ) =
p1
2π
Z
∞
0
Gλ (σ ) exp
δ ψr2 =2σ 2
dσ
σ
(3)
In a purely self–similar situation, where the energy cascade generates only a trivial variation of σ with the scale,
a Gaussian distribution for P(δ ψ r ) is recovered. When
the cascade is not strictly self–similar, the width of the
distribution Gλ is different from zero, and the scaling behaviour of the width of this distribution, namely λ 2 , can
be used to characterize intermittency.
In order to make a quantitative analysis of the energy
cascade leading to the process just described, the distributions have been fitted by using the log–normal ansatz
[16]
442
Gλ (σ ) =
p 1 exp
2πλ
ln2 σ =σ0
2λ 2
!
"file" for the gaussian background, and another for structures. In figure 7 we give an example of that behaviour.
Apart from recognizing the typical structures in the
space [18, 19], some statistics can be made. The interesting statistics is about the time separation of structures.
Let us call ∆t the waiting time between two consecutive
structures, that is between w g (τ ; t ) and wg (τ ; t + ∆t ) at a
scale τ , and let us consider the pdf P(∆t ). In figures 8
we report the pdf for magnetic structures calculated for
solar wind and RFX, and the pdf obtained for velocity
structures in fluid flows. As it can be seen the waiting
times are distributed according to a well defined power
law P(∆t ) ∆t β with some values for β , extended over
at least two decades. A similar investigation in the usual
fluid flows and in the laboratory plasma, shows the same
phenomenon (see fig. 8). This property is very interesting, because this means that the underlying process of
cascade is non–poissonian [20, 21]. In fact waiting times
occurring between isolated poissonian events, must be
distributed according to an exponential function [22].
The power law for P(∆t ) represents the asymptotic behaviour of a Lévy function with characteristic exponent
α = β 1 [23]. This function describes self–affine processes and are obtained from the central limit theorem
by relaxing the hypothesis that the variance of variables
is finite. The power law for waiting times we found is
a clear evidence that long–range correlation (or in some
sense "memory") exists in the underlying cascade process [23].
(4)
The width of the log–normal distribution of σ is given
by λ (r) =< (∆ ln σ )2 >1=2 .
The expression (3) have been fitted on the experimental pdfs for both velocity and magnetic intensity, and the
corresponding values of the parameter λ can be recovered. In figures 1 we plotted, as full lines, the curves relative to the fit, showing that the scaling behaviour of pdfs
in all cases is very well described by the function (3). At
each scale τ , we get a value for the parameter λ 2 (τ ),
which for τ 1 hour, can be fitted with a power law
λ 2 (τ ) = µτ γ . The values of µ and γ obtained in the fitting procedure are µ ' 0:75 0:03 and γ = 0:18 0:03
for the magnetic field, while µ ' 0:38 0:02 and γ =
0:20 0:04 for the velocity field, in the range of scales
τ 0:72 hours.
TURBULENT STRUCTURES AND
NON–POISSONIAN EVENTS
The nonlinear energy cascade towards smaller scales accumulates fluctuations only in relatively small regions of
space, where gradients become singular. These regions
can be viewed as localized zones of fluid where some
phase correlation exists (coherent structures). Structures
continuously appear and disappear apparently in a random fashion, in some random location of fluid, and they
carry the great quantity of energy of flows. The turbulent
flow can be viewed as a superposition of non–gaussian
structures, within the sea of gaussian background.
The presence of structures can be evidenced by using
for example a Wavelets transform. Unlike the Fourier basis, Wavelets allow a decomposition both in time and
frequency (or space and scale) (see for example [17]
and references therein). That is a function f (t ) can be
projected on a wavelet basis with coefficients w(τ ; t ).
Since a Parceval’s theorem exists, the square modulus
jw(τ ; t )j2 represents the energy content of fluctuations
f (t + τ ) f (t ) w(τ ; t ) at the scale τ at time t. It is
useful to introduce a measure of local intermittency [17],
as for example lim = jw(τ ; t )j2 = < jw(τ ; t )j2 > (averages
are made over all times at a given scale τ ). This represent
the energy content of fluctuations at a given scale, with
respect to the standard deviation of fluctuations at that
scale. The whole set of wavelets coefficients can then be
splitted in two set: a "gaussian" set wg (τ ; t ) and a "structure" set ws (τ ; t ). Then w(τ ; t ) = wg (τ ; t ) ws (τ ; t ), according to wheter lim is respectively lesser or greater of
a given threshold (the symbol stands here for union of
disjoint sets). An inverse wavelet transform, performed
separately on wg and ws , gives two separate time series: a
CONCLUSIONS
We reported some of the work done on solar wind turbulence, and we compared the behaviour with other turbulent systems. Intermittency manifests itself through a
breakdown of pure self–similarity of fluctuations, leading to anomalous scaling laws, and non–gaussian tails
for pdfs at small scales. At small scales the statistics of
waiting times between intermittent isolated events, results to be non–poissonian. This indicates that the underlying cascade process which generates these events
conserves memory. In some sense, the cascade continuously transmit to small scales a phase–correlated excitation into varying subsets of the fluid. To what extent
dynamical models can describe all behaviours is an interesting task [21]. A simplified shell model describing
the gross features of MHD turbulence [24] is able to reproduce all statistics observed. Time intermittency, that
is the occurrence of bursts of chaoticity concentrated on
the dissipative shells, generates a break of the global scaling invariance in the shell model, which is responsible for
the observed departure from self–similarity.
443
Complete signal
l.i.m. smaller
than threshold
l.i.m. larger
than threshold
Structures
Gaussian background
FIGURE 7. An example of the procedure used to recognize structures on a given scale. In the top panel of the figure we report
the original time serie (here a sample of velocity field in the solar wind). Then we operate with the lim procedure (see text), and we
obtain the two series at the bottom of the figure.
FIGURE 8. The distribution of waiting times between structures at the smallest scale for the velocity (fluids, left panel) and
magnetic fluctuations (solar wind, central panel and RFX, left panel).
We are grateful to Roland Grappin for the critical
reading of the manuscript.
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