Turbulence in the Solar Wind

Turbulence in the Turbulence in the Solar WindSolar WindCharles W. SmithCharles W. Smith

withwithS. Dasso, R.J. Leamon, M.A. S. Dasso, R.J. Leamon, M.A.

Forman, Forman, K. Hamilton, P.A. Isenberg, K. Hamilton, P.A. Isenberg,

B.T. MacBride, B.T. MacBride, W.H. Matthaeus, J.D. W.H. Matthaeus, J.D.

Richardson, J. Tessein, Richardson, J. Tessein, B.J. Vasquez and G.P. ZankB.J. Vasquez and G.P. Zank

The inertial range is a pipeline for transporting magnetic energy from the large scales to the small scales where dissipation can occur.

Interplanetary Turbulence Interplanetary Turbulence SpectrumSpectrum

f -1 “energy containing range”

f -5/3

“inertial range”

f -3

“dissipation range”

0.5 HzFew hours

Ma

gn

etic

Po

wer

Why Turbulence?Why Turbulence? Ultimate dynamics of the solar wind if Ultimate dynamics of the solar wind if left to its own devices.left to its own devices.

Sets the rate of solar wind heating.Sets the rate of solar wind heating. Partial responsibility for the manner of Partial responsibility for the manner of heating.heating.

Controls the distribution of energy in Controls the distribution of energy in spectrum.spectrum.

Builds/destroys correlations responsible Builds/destroys correlations responsible for charged particle scattering.for charged particle scattering. Dictates transverse magnetic fluctuations.Dictates transverse magnetic fluctuations. Directs wave vector away from field-Directs wave vector away from field-alignment.alignment.

Solar Wind HeatingSolar Wind Heating

In the range 0.3 < R < 1.0 AU, Helios observations demonstrate the following:

For VSW < 300 km/s, T ~ R -1.3 0.13

300 < VSW < 400 km/s, T ~ R -1.2 0.09

400 < VSW < 500 km/s, T ~ R -1.0 0.10

500 < VSW < 600 km/s, T ~ R -0.8 0.10

600 < VSW < 700 km/s, T ~ R -0.8 0.09

700 < VSW < 800 km/s, T ~ R -0.8 0.17

We need to back out the heating rate as a point of comparison for inferred heating rates at 1 AU. This involves solving an equation like:

•

•••+−= Qr

T

dr

dT

3

4

Adiabatic expansion yields T ~ R-4/3.

Low speed wind expands without in situ heating!?

High speed wind is heated as it expands.

Low-speed results have been corrected once in situ acceleration was considered.

Observations of TP

Approx. adiabatic prediction


Explaining the Heating Explaining the Heating RateRate


f -5/3


f -3


0.5 HzFew hours

Ma

gn

etic

Po

wer

~ u3/l

Supply-Side Heating Supply-Side Heating TheoryTheory

λα

λββλλλ

α

3

2

32

2

3

1

3

4

'

'

Z

Uk

m

r

T

dr

dT

EZU

ZUr

C

dr

dU

EZ

UZ

r

A

dr

dZ

B

p

PI

PI

+−=

−+−=

+−−=A = 1.1

C = 1.8

α = 1 = β

Constrained by symmetry, Taylor-Karman local phenom., and solar wind conditions.

Z± = v ± b are Elsasser variables.

λ is the similarity scale = correlation length.

T = proton temperature.

Zhou and Matthaeus, JGR, 95, 10,291 (1990); Zank et al., JGR, 101, 17,093 (1996);

Matthaeus et al., PRL, 82, 3444 (1999); Smith et al., JGR, 106, 8253 (2001);

Smith et al., ApJ, 638, 508 (2006)

full bisphere

% bisphere

bisphere

with pickup ionsNon-adiabatic expansion

Voyager 2 observations

Turbulent heating model

T / TS = (V / <V>) 3 – 2

Richardson & Smith, GRL, 30, 1206 (2003)

Inertial Range CascadeInertial Range Cascade

0.5 Hz



f -5/3


f -3


Few hours

Magnetic Power

6.1

re whe

3/53/2

≈

= −

K

Kk

C

kCP

Energy Cascade RateEnergy Cascade Rate

The rate large-scale structures drive the turbulence

The rate of energy cascade through the

inertial range.

The rate of energy dissipation in

the dissipation range.

The rate of turbulent heating of the background plasma.

At 1 AU <> ~3 x 103 Joules/kg-s

Inertial Range Inertial Range CharacteristicsCharacteristics

Strong correlation between Strong correlation between V and V and B.B. Signature of outward propagation.Signature of outward propagation.

Fluctuations perpendicular to the mean BFluctuations perpendicular to the mean B00.. Large variance anisotropy (Large variance anisotropy (BB//BB > 1) > 1) Signature of largely noncompressive Signature of largely noncompressive fluctuationsfluctuations

Wave vectors both parallel and Wave vectors both parallel and perpendicular to Bperpendicular to B00 As shown by Matthaeus et al. and Dasso et al.As shown by Matthaeus et al. and Dasso et al.

5/3 power law index (Kolmogorov)5/3 power law index (Kolmogorov)


Strong correlation between Strong correlation between V V and and B.B. Signature of outward propagation.Signature of outward propagation.

Milano et al., PRL, 93, 2004.


Fluctuations perpendicular to Fluctuations perpendicular to the mean Bthe mean B00.. Large variance anisotropy (Large variance anisotropy (BB

22//BB22

> 1)> 1) Signature of largely Signature of largely noncompressive fluctuationsnoncompressive fluctuations

NIMHD and WCMHD theories seem to imply a β-scaling to the variance anisotropy.

This represents balance between excitation and dissipation of compressive component.

Smith et al., JGR, in press (2006)


Wave vectors parallel and Wave vectors parallel and perpendicular to Bperpendicular to B00

As shown by Matthaeus et al. and As shown by Matthaeus et al. and Dasso et al.Dasso et al.

Matthaeus et al., JGR, 95, 20,673, 1990.

Dasso et al., ApJ, 635, L181-184, 2005.

Slow wind is 2D

Fast wind is 1D


5/3 power law index (Kolmogorov)5/3 power law index (Kolmogorov)0.5 Hz


f -5/3


f -3


Few hours

Mag

netic

Pow

er

6.1

re whe

3/53/2

≈

= −

K

Kk

C

kCP

To apply the Kolmogorov formula [Leamon et al. (1999)]:

1.Pk = CK 2/3 k5/3

2.Fit the measured spectrum to obtain “weight” for the result

• Not all spectra are -5/3! I assume they are!

3.Use fit power at whatever frequency (I use ~10 mHz)

4.Convert P(f) → P(k) using VSW

5.Convert B2 → V2 using VA via (V2 = B2/4)

6.Allow for unmeasured velocity spectrum (RA = ½)

7.Convert 1-D unidirectional spectrum into omnidirectional spectrum

= (2/VSW) [(1+RA) (5/3) PfB (VA/B0)2 /

CK ]3/2 f5/2

Leamon et al., J. Geophys. Res., 104, 22331 (1999)

Beware!Beware! Kolmogorov spectral prediction yields Kolmogorov spectral prediction yields ..

If the fluid is turbulentIf the fluid is turbulent!! A static spectrum could yield a completely A static spectrum could yield a completely irrelevant prediction having nothing to do irrelevant prediction having nothing to do with anything.with anything.

Kolmogorov structure function prediction Kolmogorov structure function prediction measuresmeasures the strength of the nonlinear the strength of the nonlinear terms.terms. Only verification of an active turbulent Only verification of an active turbulent cascade.cascade.

Politano and Pouquet (1998) extended Politano and Pouquet (1998) extended structure function ideas to MHD.structure function ideas to MHD. We have recently applied these ideas to the We have recently applied these ideas to the solar wind at 1 AU.solar wind at 1 AU.

See talk by Forman and poster by MacBride.See talk by Forman and poster by MacBride.

Energy and Dissipation Energy and Dissipation RatesRates

See Forman et al talk, this session.

See MacBride et al. poster, this meeting.

Cascade & dissipation rate is sufficient to dissipate the inertial range in 3-5 days and equilibrate

outward and inward propagating waves.

Power spectrum derivation of ~ 104 Joules/kg-s

The Dissipation RangeThe Dissipation Range

0.5 Hz

If the inertial range is a pipeline, the dissipation range consumes the energy at the end of the process.


f -5/3


f -3


Few hours

Magnetic Power

6.1

re whe

3/53/2

≈

= −

K

Kk

C

kCP

Spectral steepening with dissipation

Inertial range spectrum ~ 5/3

Ion Inertial Scale

Leamon Found:Leamon Found: Dissipation range spectrum highly Dissipation range spectrum highly variable.variable.

Dissipation range has smaller variance Dissipation range has smaller variance anisotropy than inertial range.anisotropy than inertial range. Compressive component more important.Compressive component more important.

Quasi-perpendicular wave vectors are Quasi-perpendicular wave vectors are more aggressively damped than parallel more aggressively damped than parallel vectors.vectors. Cyclotron resonances is responsible for ½ Cyclotron resonances is responsible for ½ 2/3 of energy dissipation.2/3 of energy dissipation.

Hamilton et al., unpublished.

Transition to Transition to DissipationDissipation

Traditional fluid turbulence requires:Traditional fluid turbulence requires: Results from processes contained Results from processes contained withinwithin the the fluid approximation.fluid approximation.

Onset of dissipation scales with Onset of dissipation scales with ~ ~ ((33//))1/41/4..

Dissipation range spectrum is universal Dissipation range spectrum is universal F(F(k).k).

The solar wind is not a traditional The solar wind is not a traditional fluid!fluid! Dissipation results from the breakdown of Dissipation results from the breakdown of the single fluid theory.the single fluid theory.

At scales like (some number of) ion inertial At scales like (some number of) ion inertial scales.scales.

Smith et al., ApJ, 645, L85, 2006.

SummarySummary Large-scale drivers of the turbulent cascade is able Large-scale drivers of the turbulent cascade is able

to account for the rate of heating the solar wind.to account for the rate of heating the solar wind. Issues with the rates determined from the inertial range.Issues with the rates determined from the inertial range.

Dissipation rate suggests that inertial range Dissipation rate suggests that inertial range observations arise observations arise in situin situ.. Variance anisotropy scales with plasma Variance anisotropy scales with plasma ββ.. Compressive component must be explainable via Compressive component must be explainable via

excitation/decay processes buried within turbulence.excitation/decay processes buried within turbulence. Maybe maintaining association with initial conditions…Maybe maintaining association with initial conditions…

Onset of dissipation results from breakdown of fluid Onset of dissipation results from breakdown of fluid theory.theory. Cyclotron damping is only part of the story.Cyclotron damping is only part of the story. Most aggressive dissipation acts on the perpendicular Most aggressive dissipation acts on the perpendicular

wave vectors.wave vectors.

Dissipation range spectrum depends on the rate of Dissipation range spectrum depends on the rate of cascade.cascade. More compressive than inertial range.More compressive than inertial range. More aggressive dissipation of k More aggressive dissipation of k B B00..

Extra SlidesExtra Slides

V & V & B Variation with B Variation with VVSWSW

Basic Solar Wind Basic Solar Wind ScalingsScalings

Leamon Found:Leamon Found: Dissipation range spectrum highly Dissipation range spectrum highly variable.variable.

Dissipation range has smaller variance Dissipation range has smaller variance anisotropy than inertial range.anisotropy than inertial range. Compressive component more important.Compressive component more important.

Quasi-perpendicular wave vectors are Quasi-perpendicular wave vectors are more aggressively damped than parallel more aggressively damped than parallel vectors.vectors. Cyclotron resonances is responsible for ½ Cyclotron resonances is responsible for ½ 2/3 of energy dissipation.2/3 of energy dissipation.

Non-Cyclotron ResonanceNon-Cyclotron Resonance

Leamon et al., ApJ, 507, L181-184, 1998.

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Turbulence in the Solar Wind