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### 2nd IWF Turbulence Workshop 30.9 – 4.10. 2013, GRAZ

Turbulence in the

heliosphere

Zoltán Vörös

Space Research Institut

Graz, Austria

Project support: P24740-N27

FP7/2007-2013 - 313038/STORM

- Fields and plasma properties in the solar wind show

fluctuations over a wide range of temporal & spatial

scales ranging from solar cycle/rotation scales down to

the electron scales.

Fundamental plasma phenomena, such as turbulence

or/and magnetic reconnection exhibit multi-scale

coupling features. Turbulent interactions are usually supposed to be local in the Fourier/physical/phase space, but can be also nonlocal.

Here we advocate that certain aspects of complex

plasmas can be understood statistically only. However,...

Large number of degrees of freedomPROLOGUE

The solar wind as a plasma laboratory;

Turbulence: textbook phenomenologies;

Are the phenomenologies supported by experiments?

New ingredients

After all, what is turbulence?

OUTLINEProperties of the solar wind near the Earth

n is proton density, Vsw is solar wind speed, B is magnetic field strength, A(He) is He++/H+ ratio, Tp is proton temperature, Te is electron temperature, T is alpha particle temperature, Cs is sound speed, CA is Alfven speed (after Gosling).

The solar wind is a nearly collisionless, supersonic and super-Alfvenic flow of electrons, protons, alpha particles and other less abundant ions, permeated by the interplanetary magnetic field.

There exist large fluctuations around the mean values in all parameters.

Fully developed fluid/plasma turbulence

There is no general theory based on first principles

L

PHENOMENOLOGIES

Isotropic &

homogeneous &

incompress. &

stationary &

no dissipation

The turbulent cascade is a superposition of

‚eddies‘ of sizes

D

D

Fluc-

tuations

[e.g. Frisch, 1995,

Salem et al., ApJ09]

Energy

Eddy interaction

or transfer, or

‚turn-over‘ time

Models depend on how is defined

Energy exchange rate

Fully developed (neutral) fluid turbulence

Kolmogorov phenomenology (1941) –K41

[after Salem et al., ApJ09]

Isotropic &

homogeneous &

incompress. &

stationary &

no dissipation

Since

Bessel-Parseval

theorem

and

Spectral energy density

Fully developed plasma turbulence

There is no general theory based on first principles.

Iroshnikov (1963) & Kraichnan (1965)

phenomenology – IK65

Isotorpic fluctuations; Energy transfer is due to ‚weak‘ interactions between Alfvenic fluctuations moving in opposite directions along B with VA.

v and B fluctuations

are of the same order

of magnitude

‚Weak‘ means that wave packets undergo multiple collisions before they change.

One interaction takes an Alfven time

Therefore, in comparison with K41 energy transfer time is times longer, that is:

and

[e.g. Biskamp, 2003

Salem et al.,ApJ09]

Fully developed turbulence - solar wind

Observations

B

f -1

K41

f -5/3

f -2

[Wicks et al. ,

MNRAS, 2010]

IK65

V

- Problems:
- - competition: other types of fluctuations
- local magnetic field introduces anisotropy
- field and plasma parameters scale differently
- only V shows IK65 scaling
- excess of magnetic over kinetic energy
- non-stationarity & intermittency
- structures in the solar wind

f -3/2

[Salem et al.,

ApJ 2009]

Ev/Eb

excess of magnetic

energy

Other types of fluctuations

For example: pressure anisotropy driven fluctuations

Mirror instability

threshold

Oblique firehose

instability threshold

(Bale et al. 2009)

Anisotropy

B

f -1

Goldreich & Sridhar phenomenology (1995) –GS95

Eddies of size are elongated along mean B, the fluctuations develop in the perpendicular components:

‚Critical balance‘

or strong

interaction

condition:

f -5/3

f -2

[Wicks et al. ,

MNRAS, 2010]

Alfven

time

Eddy turnover time

Spacecraft

const.

[Horbury et al., 2008

Chen et al., 2010]

VSW

Anisotropy

Observations show that a Kolmogorov

cascade develops in perpendicular to B

direction confirming GS95.

However, numerical simulations show (Mason et al. 2008) that large

average magnetic fields

can lead to IK65 perpendicular cascade:

- Actually,recent investigations (Boldyrev PRL 2006; Chen et al. ApJ 2012)
- indicate that Alfvenic fluctuations can be 3-dimensionally anisotropic
- in a scale-dependent manner!
- direction of the mean B
- due to alignment of and near proton gyroscales perp.dir.(1)
- perp.dir.(2) perpendicular to perp.dir.(1).
- Alignment depends on non-universality, compressional effects.

Different scalings V & B

B: K41 or IK65

V: IK65

How to estimate if V and B scale differently??

Possibilities:

- Estimate the scalings and energy transfer for each quantity
- B, V, n, T, E, etc.
- 2. Combine B and V to third order moments with no prescribed spectra.

Third-order moments in hydrodynamics:

Kolmogorov 4/5 law

Theoretical result obtained from Navier-Stokes equation, for incompressible,

isotropic, stationary, homogeneous turbulence with no dissipation:

The third-order moment of speed fluctuations along the bulk flow scales

linearly with separation

Third-order moments in MHD

Elsässer field:

The incompressible MHD equations, containing resemble the

Navier-Stokes eqs. Nonlinear interactions exist between fluctuations

propagating in opposite directions: .

The MHD equations for and are evaluated in two points

and

then vector differences are formed

and combined to second-order correlation tensor

assuming homogeinity, isotropy, stationarity, vanishing dissipation and

projecting to longitudinal direction [Politano and Pouquet]:

Yaglom‘s law

Kolmogorov 4/5 law

Dissipation tensor

kinematic viscosity

Yaglom‘s law

Yaglom‘s law is valid under the

assumption of:

incompressibility, stationarity

homogeinity and isotropy.

Solar wind does not fully comply with...

Sorriso_Valvo

et al. 2007

High-speed polar wind data by Ulysses

at distance 3-4 AU

heliolatitude 55 – 35 degrees

observation length: 20 intervals of 10 days

(Sorriso-Valvo et al., 2007)

V,B correlations lost

Solar wind in ecliptic is more structured

(MacBride et al. 2008)

Attempt to account for compressible

effects (Carbone et al. 2009)

Mean energy transfer rate include density

Carbone et al.

2009

W+

102 J/Kg s

Y+

Stationarity?

to reach convergence estimating 1 year

long data intervals are needed (Podesta et al. 2009)

stationarity has to be studied

Nonlinear cascade is significantly

enhanced by density fluctuations

Excess of magnetic over kinetic energy

Possible explanations:

1.) local dynamo effect

(Grappin et al. 1983)

2.) Compressible fluctuations

can change the energy exchange

rate

3.) Nonlocal coupling in k-space

(Galtier, 2006); nonlocal coupling induces

intermittency (Vörös et al., 2006, 2007)

4.) Radial evolution of Alfvenic turbulence

(Bruno et al., 2007)

Axcess of magnetic energy is associated with magnetic structures

intermittency

e.g. Salem et al. 2009

Inertial range

Radial evolution of Alfvenic fluctuations in the solar wind

Substantial part of the turbulence research in the solar wind

is addressing the question about alfvenicity of fluctuations

Review papers:

Tu, C.-Y., Marsch, E., “MHD structures, waves and turbulence in the solar wind: Observations

and theories”, Space Sci. Rev., 1995.

Bruno R. and V. Carbone, “The Solar Wind as a Turbulence Laboratory”,

Living Rev. Solar Phys., 2005.

Alfven mode propagation

inward outward

Normalized X-helicity

[-1sC+1]

-measures correlations

between V and B

measures predominance

of energy associated with

Z+ or Z-

Normalized residual energy

[-1sR+1]

measures predominance

of kinetic or magnetic energies

Useful quantities

For an Alfvén mode: |sC|=1; sR=0

16

Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr)

0.3 AU

Helios-2

fast wind observations

Alfvénic population

17

Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr)

0.3 AU

Helios-2

fast wind observations

Alfvénic population

0.7 AU

18

Radial evolution of MHD turbulence in terms of sR and sC (scale of 1hr)

0.3 AU

Helios-2

fast wind observations

Alfvénic population

0.7 AU

0.9 AU

A new population, characterized by magnetic energy excess , appears

(Bruno et al., 2007)

19

FAST WIND

SLOW WIND

The radial evolution

of fast wind and slow

wind is different.

Alfvenic fluctuations are

stronger in the fast wind.

As the fast wind expands, the

Alfvenic component decays

due to the nonlinear

cascade. The magnetically

dominated structures become

visible at ~1 AU.

In the slow wind the

Alfvenic fluctuations are

weaker and do not mask

the magnetic structures.

Magnetic structures

introduce/increase intermittency

(Bruno et al., 1999

D‘Amicis et al. 2010)

However:

- only a few events analyzed;

- the radial evolution of other

turbulent related quantities,

e.g. anisotropy, is questionable;

0.3 AU

0.7 AU

0.9 AU

20

(Bruno et al., 2007)

High-order statistics: intermittency & morePDF

Turbulent fluctuations are non-Gaussian.

We have already mentioned:

- Energy transfer rate (4/5 law, Yaglom law): third-order structure function

- Intermittency: scaling fourth-order structure function connection to structures.

Possible strategies:

1. Sturcture functions

if is nonlinear

the process is multifractal andd there is a disitribution of structures which are singular or very irregular

[eg. Burlaga ,1993,

Horbury et al., 1996, Vörös, 1998]

2. Wavelet transform coefficients can be related

to structure functions [e.g.Salem et al 2009, Farge et al. 2006]

which allow to

find singularities or magnetic structures and remove them from the data.

3. Calculation of moments

3rd order:

4. Non-Gaussian PDF models

For example, log-normal, Castaing, kappa,

log-kappa, etc. (Leubner&Vörös, 2005;

Burlaga & Vinas, 2005; Leitner et al. 2009).

4th order:

... and higher moments

What is the shape of PDFs in the solar wind?

after Burlaga & Vinas 2004

close to Gaussian

increasing

peakedness

skewed

PDFs

Intermittency: convected structures or turbulence?

Numerical simulations show that turbulence produces its own highly localized

coherent structures, eg. current sheets, discontinuities (eg. Servidio et al., 2009, 2012).

Other structures, such as discontinuities, shocks, flux tubes originate on

the Sun.

One can localize coherent structures by wavelet thresholding and remove

them from time series, making the PDFs gaussian or the structure functions

to follow K41 or IK65 scalings (Salem et al., 2009).

after wavelet

filtering

However,

if the structures

are produced

by turbulence,

wavelets remove

part of the turbulence

as well.

Intermittency: convected structures or turbulence?

Intermittency associated with structures can be nonlocal, why local wavelet thresholding

is removing localized structures.

An interdependence of Skew and Kurt indicates that large-scale structures influencing small-scale statistics are present resulting in non-locality and non-universal scalings (Warhaft, 2000).

Warhaft [2000, 2002] has shown that small-scale PDFs of a passive scalar field in turbulence are skewed if large-scale scalar gradients exists in turbulent flows. Holzer and Siggia (1994) have shown that large-scale gradients not only skew PDFs, but also increase intermittency.

In MHD flows the magnitude of B fluctuations resembles dynamical properties of passive scalars (Bershaskii & Sreenivasan, 2004).

We have to check how the statistical moments associated with B are related to each other.

In hydrodynamics passive scalars in shear flows obey

Kurt = a . (Skew)2 + b

Moment interdependence near shocks

Skewness

(B)

Kurtosis

(B)

Multiple shock interval in 2000 (Vörös et al. 2006)

Interval with no discontinuities

Comparisons

Kurtosis

Passive scalar statistics

in a fluid flow

(Chatwin, Robinson,

1997)

Skewness

Passive scalar statistics near interplanetary shocks (Vörös et al., 2006)

Passive scalar statistics

in the Earth’s plasma

sheet

Evidence for nonlocal turbulence

interactions in space plasmas

(Vörös et al., 2007)

The importance of structures

2. Turbulence associated with unstable flux tubes

Flux tubes are basic magnetic structures in the solar wind – represent sources of intermittency.

The total pressure profile inside flux tubes depends

on magnetic field twist. Tubes twisted with the angle

of 70o or more are unstable against kink instability

which can lead to reconnection

(Zaqarashvili et al., in prep.)

Twisted flux tubes in an external twisted

magnetic field can be unstable to K-H

instability even in cases of sub-Alfvenic

shears (Zaqarashvili et al., in prep.) and

significantly contribute to turbulence

in the solar wind.

Conclusions

The concept of turbulent cascades and nonlinear interactions

is important to understand energy redistribution over multiple

scales in the solar wind.

More effort is needed, however, to understand, for example,

intermittency, the role of coherent structures, nonlocality, or to identify

the drivers of fluctuations, their evolution, etc.

Intermittent structures can be found e.g. by wavelets, but it is not known

if they are produced by turbulence or not.

Case studies are needed to identify the structures embedded in

turbulent plasma and to establish the interrelationships between

structures and non-Gaussian turbulence.

Epilogue

Turbulence is a phenomenon of instability at high Reynolds numbers…a complete theory of general solutions of the Navier-Stokes equations are called for …it is tied to 3-D (von Neumann, 1949)

OR

Turbulence is a highly excited state of a system with many degrees of freedom (in most cases a continuous medium) to be described statistically. This excited state is far away from thermodynamic equilibrium and is accompanied by intensive energy dissipation. Such states can be found in fluids, plasmas, magnets, dielectrics, etc. the problem of turbulence goes far beyond the limits of hydrodynamics (or MHD) and the Navier-Stokes equation (Zakharov, L‘vov, Falkovich, 1992).

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