The turbulent cascade The turbulent cascade in the solar windin the solar wind

Luca Sorriso-ValvoLICRYL – IPCF/CNR, Rende, Italy

sorriso@fis.unical.it

R. Marino, V. Carbone, R. Bruno, P. Veltri, A. Noullez, B. Bavassano

Reynolds Number

Turbulence

Lv0

• • l

Analysis of longitudinal velocity differences

)(v)(vv xlxl L

e

vR

Since dissipation is efficient only at very small scales, the system dissipates energy by transferring it to small scales

Nonlinear energy cascade (Richardson picture)

Energy injection ()

Non-linear energy transfer ()

Energy dissipation ()

Integral scale L

Inertial range

Dissipative scale ld

eddies

Energy cascade

Fvvvv 2Pt

Re =Non-linear

Dissipative=v L

Ein

Eout

Enl

Energy balance

power-law (observed universal exponent: -5/3)

Navier-Stokes MHD

Elsasser fields

Energy injection ()

Non-linear energy transfer ()

Energy dissipation ()

Integral scale L

Inertial range

Dissipative scale ld

This leads to the Kolmogorov scaling law 3/1rvr

3/5)( kkE

Under the Kolmogorov hypothesis (K41) of constant energy transfer rate, the scaling

parameter is h = 1/3

Phenomenology of fluid turbulence

Kolmogorov spectrum

Introducing the energy dissipation rate

NL

rv

2

rNL v

r

r

vr

3

The characteristic time to realize the cascade (eddy-turnover time) is the lifetime of turbulent

eddies

Phenomenological arguments for magnetically dominated MHD turbulence

r

NL z

r

2

r

r

T

z

A

A c

r

r

zz rr

22

A

NLNLrT

Since the Alfvén time might be shorter than the eddy-turnover time, nonlinear interactions are reduced and the cascade is realized in a time T

When the flow is dominated by a (large-scale) magnetic field, there is one more characteristic time, the Alfvén time A, related to the sweeping of Alfvénic

fluctuations

A different scaling relation for the pseudo-energies transfer rates4/1rzr

2/3)( kkE Kraichnan spectrum

Belc

her

and

Davis

, JG

R,

1971

Energy cascade needs both z+ and z- fluctuations for the non-linear term to exist. Is the turbulent spectrum compatible with the observed Alfvénic

fluctuations ?

Observations indicate that one of the Elsasser fluctuations is approximately zero (Alfvénic turbulence), thus the turbulent non-linear Energy cascade

should be inhibited.

A puzzle for MHD turbulence

z+ z-

Elsasser variables fluctuations

The energy cascade is due to

the nonlinear term of MHD equations

Nonlinear interactions occur between fluctuations

propagating in opposite direction with respect to the

magnetic field.

Evidences of power spectrum in the solar wind attributed to fully developed MHD turbulence

Cole

man,

Ap

J, 1

968

10-5 10-4 10-3 10-2 10-1100

101

102

103

104

105

106

107

0.9AU

0.7AU

0.3AU

trace of magnetic fied spectral matrix

-1.72

-1.70

-1.67

-1.07

-1.06

-0.89 f-1

f-5/3

pow

er d

ensi

ty

frequency

Bavass

ano e

t al.,

JGR

1982

Open question about the existence of a MHD turbulent energy

cascade

DYNAMIC ALIGNMENT: Dobrowolny et al. (PRL, 1980) proposed a possible “solution” of the puzzle: if the two Elsasser fields have the same energy transfer rate (same spectral slope), an initial small unbalance at injection scale (meaning quasi-correlated Alfvénic fluctuations) is maintained along a nonlinear cascade

toward smaller scales. This is enough to explain the simultaneous observation of a turbulent spectrum and the presence of one single Alfvénic “mode”.

An exact law for incompressible MHD turbulent cascade: Politano & Pouquet

Mixed third-order momentLarge-scale

inhomogeneitiesPressure term (anisotropy)

Dissipative term (vanishing in the inertial range)

term including the pseudo-energy dissipation rate tensor

From incompressible MHD equations, an exact relation can be derived for the mixed third-order moment assuming stationarity, homogeneity, isotropy (Kolmogorov 4/5)

Politano & Pouquet, PRE 1998Politano, Pouquet, Carbone,EPL 1998Sorriso-Valvo et al, PRL 2007

IF THE YAGLOM RELATION IS VALID, A NONLINEAR ENERGY CASCADE MUST EXIST.

THE RELATION IS THE ONLY EXACT AND NONTRIVIAL RESULT IN TURBULENCE

Numerical evidences

Sorriso-Valvo et al, Phys. of Plasmas 2002

From 2-dimensional numerical simulation of MHD equations (1024X1024)

A snapshot of the current j from the simulation in the statistically steady state

Ulysses data 1996: results

High latitude ( > 35°)

Running windows of 10 days (2000 data points each) have been used to avoid radial distance and latitudinal variations, as well as non-stationariety effects.

8 minutes averages of velocity, magnetic field and

density are used to build the Elsasser fields Z±

Low solar activity (1994-1996)

The Politano & Pouquet relation is satisfied in several Ulysses samples (polar, fast wind high Alfvénic correlations!)

Although the data may be affected by inhomogeneity, local anisotropy

and compressible effects, the observed P&P scaling law is robust in most periods of Ulysses dataset.

The first REAL evidence that (low

frequency) solar wind can be described in

the framework of MHD turbulence

Sorriso-Valvo et al., PRL (2007)

1. Observation of inertial range scaling

The estimated values of energy transfer rates are about 100J/Kg sec.

For comparison, energy transfer rates per unit mass in usual fluid flows are 1 50 J/Kg s

2. The energy transfer rates

3. The inertial range of SW turbulence

Ev

EB

f-5/3

The velocity spectrum extends to large scales, while magnetic spectral break is often (but not

always) observed around a few hours.

Radial velocity and magnetic field spectra for one sample of wind with Yaglom scaling

1h

In literature: up to 6-12 hours. Our data: up to 1-2 days. What is the actual extent of the inertial range in SW?

Velocity inertial range can locally extend up to

1-2 days! Seen from Yaglom law and from spectral properties.1day

Magnetic contribution to energy transport

Velocity contribution to energy transport

4. MHD or Navier-Stokes? The role of magnetic field

bvbv 22 22222bvbbvvZZY i

It is possible to separate the contributions to the energy cascade: terms

advected by velocity and terms advected by

magnetic field.

In this example, Yaglom scaling is dominated by

the magnetic field at small scales (from 10

mins to 3h), but at large scale only the velocity

advects energy Total energy transport

Separation of scales around the Alfvén time

NON UNIVERSALITY OF SOLAR WIND TURBULENCE!

SMALL SCALES (< hours): Magnetic filed dominates or equilibrium: MHD cascade

LARGE SCALES (> hours): Velocity dominates (Navier-Stokes cascade), or non- negligible magnetic field contribution (MHD cascade)

3.-4. Different behaviour in different samples

5. Compressive turbulence in solar wind: phenomenological scaling law

Energy-transfer rates per unit volume in compressible MHD

wwzw

zz

2

3/1

2

2)()(

wwW

xwxww

Density-weighted Elsasser variables: what scaling law for

third-order mixed moment?

Kowal & Lazarian, ApJ 2007, Kritsuk et al., ApJ 2007, Carbone et al., PRL 2009

Introducing phenomenological variables, dimensionally

including density

Low-amplitude density fluctuations could play a role in the scaling law

5. Compressive turbulence in solar wind: observation from data

Phenomenological compressible P&P enhanced scaling is observed, even in samples where the incompressible law is not verified.

Estimate of the heating rate needed for the solar wind: models for turbulence

Vasquez et al., JGR 2007

Solar wind models Adiabatic expansion, temperature should decrease with helioscentric distance

3/4)( rrT

Spacecraft measurements Temperature decay is slower than expected from adiabatic expansion

rrT )(

]1;7.0[

Is the measured turbulent energy flux enough to explain the observed non-adiabatic expansion of the solar wind?

6. Solar wind heating

Compressive and incompressive dissipation rates, compared with model wind heating (2 temperatures)

estimated energy transfer rate: compressible case

estimated energy transfer rate: incompressible cascade energy transfer

rate required for the observed T

7. The role of cross-helicity

Fast streams (high cross-helicity) have a weak turbulent cascade on only one mode.

Slow streams, in which the two modes coexist and can exchange energy more rapidly, the energy cascade is more efficient (giving larger transfer rates) and is observable on both modes.

This observation reinforces the scenario proposed by Dobrowolny et al. suggesting that MHD cascade is favoured in low cross-helicity samples.

Cross-helicity plays a relevant role in the MHD turbulent cascade in solar wind.

Fast: onlyone mode

fast

slow

Slow: both modes

ConclusionsConclusions

1. The MHD cascade exists in the solar wind.

2. First measurements of energy dissipation rate (statistical analysis in progress).

3. The extent of the inertial range is larger than previously estimated.

4. The role of magnetic field is relevant (but not always) for the cascade, especially at small scales. However, no universal phenomenology is observed (work in progress).

5. Density fluctuations, despite their low-amplitude, enhance the turbulent cascade (more theoretical results are needed).

6. Heating of solar wind could be entirely due to the compressible turbulent cascade.

7. Cross-helicity plays an important role in the scaling: Alfvénic fluctuations inhibit the cascade, in the spirit of dynamic alignment (work in progress).

The turbulent cascade was observed in Ulysses solar wind data through the Politano & Poquet law, providing several results: