1. Turbulence in the Solar Wind
Charles W. Smith
with S. Dasso, R.J. Leamon, M.A. Forman, K. Hamilton, P.A. Isenberg, B.T. MacBride, W.H. Matthaeus, J.D. Richardson, J. Tessein, B.J. Vasquez and G.P. Zank

2. Interplanetary Turbulence Spectrum
f -1 “energy containing range”
f -5/3
“inertial range”
The inertial range is a pipeline for transporting magnetic energy from the large scales to the small scales where dissipation can occur.
Magnetic Power
f -3 “dissipation range”
Few hours
0.5 Hz

3. Why Turbulence?
Ultimate dynamics of the solar wind if left to its own devices.
Sets the rate of solar wind heating.
Partial responsibility for the manner of heating.
Controls the distribution of energy in spectrum.
Builds/destroys correlations responsible for charged particle scattering.
Dictates transverse magnetic fluctuations.
Directs wave vector away from field-alignment.

4. Solar Wind Heating Observations of TP
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:
Observations of TP
Approx. adiabatic prediction
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.

5. Explaining the Heating Rate
  ~ u3/l
f -1 “energy containing range”
f -5/3
“inertial range”
The inertial range is a pipeline for transporting magnetic energy from the large scales to the small scales where dissipation can occur.
Magnetic Power
f -3 “dissipation range”
Few hours
0.5 Hz

6. Supply-Side Heating Theory
full bisphere
% bisphere
bisphere
with pickup ions
Supply-Side Heating Theory
Non-adiabatic expansion
Voyager 2 observations
Turbulent heating model
T / TS = (V / <V>)  3 – 2
Richardson & Smith, GRL, 30, 1206 (2003)
Z± = v ± b are Elsasser variables.
is the similarity scale = correlation length.
T = proton temperature.
A = 1.1
C = 1.8
= 1 = 
Constrained by symmetry, Taylor-Karman local phenom., and solar wind conditions.
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)

7. Inertial Range Cascade
0.5 Hz
The inertial range is a pipeline for transporting magnetic energy from the large scales to the small scales where dissipation can occur.
f -1 “energy containing range”
f -5/3
“inertial range”
f -3 “dissipation range”
Few hours
Magnetic Power

8. Energy 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

9. Inertial Range Characteristics
Strong correlation between V and B.
Signature of outward propagation.
Fluctuations perpendicular to the mean B0.
Large variance anisotropy (B/B > 1)
Signature of largely noncompressive fluctuations
Wave vectors both parallel and perpendicular to B0
As shown by Matthaeus et al. and Dasso et al.
5/3 power law index (Kolmogorov)

10. Inertial Range Characteristics
Strong correlation between V and B.
Signature of outward propagation.
Milano et al., PRL, 93, 2004.

11. Inertial Range Characteristics
NIMHD and WCMHD theories seem to imply a -scaling to the variance anisotropy.
This represents balance between excitation and dissipation of compressive component.
Fluctuations perpendicular to the mean B0.
Large variance anisotropy (B2/B2 > 1)
Signature of largely noncompressive fluctuations
Smith et al., JGR, in press (2006)

12. Inertial Range Characteristics
Dasso et al., ApJ, 635, L , 2005.
Slow wind is 2D
Fast wind is 1D
Matthaeus et al., JGR, 95, 20,673, 1990.
Wave vectors parallel and perpendicular to B0
As shown by Matthaeus et al. and Dasso et al.

13. Inertial Range Characteristics
To apply the Kolmogorov formula [Leamon et al. (1999)]:
Pk = CK 2/3 k5/3
Fit the measured spectrum to obtain “weight” for the result
Not all spectra are -5/3! I assume they are!
Use fit power at whatever frequency (I use ~10 mHz)
Convert P(f) → P(k) using VSW
Convert B2 → V2 using VA via (V2 = B2/4)
Allow for unmeasured velocity spectrum (RA = ½)
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, (1999)
0.5 Hz
f -1 “energy containing range”
f -5/3
“inertial range”
f -3 “dissipation range”
Few hours
Magnetic Power
5/3 power law index (Kolmogorov)

14. Beware! Kolmogorov spectral prediction yields .
If the fluid is turbulent!
A static spectrum could yield a completely irrelevant prediction having nothing to do with anything.
Kolmogorov structure function prediction measures the strength of the nonlinear terms.
Only verification of an active turbulent cascade.
Politano and Pouquet (1998) extended structure function ideas to MHD.
We have recently applied these ideas to the solar wind at 1 AU.
See talk by Forman and poster by MacBride.

15. Energy and Dissipation Rates
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
See Forman et al talk, this session. See MacBride et al. poster, this meeting.

16. The Dissipation Range f -1 “energy containing range” f -5/3
0.5 Hz
If the inertial range is a pipeline, the dissipation range consumes the energy at the end of the process.
f -1 “energy containing range”
f -5/3
“inertial range”
f -3 “dissipation range”
Few hours
Magnetic Power
Spectral steepening with dissipation
Inertial range spectrum ~ 5/3
Ion Inertial Scale

17. Leamon Found: Dissipation range spectrum highly variable.
Hamilton et al., unpublished.
Dissipation range spectrum highly variable.
Dissipation range has smaller variance anisotropy than inertial range.
Compressive component more important.
Quasi-perpendicular wave vectors are more aggressively damped than parallel vectors.
Cyclotron resonances is responsible for ½  2/3 of energy dissipation.

18. Transition to Dissipation
Traditional fluid turbulence requires:
Results from processes contained within the fluid approximation.
Onset of dissipation scales with  ~ (3/)1/4.
Dissipation range spectrum is universal F(k).
The solar wind is not a traditional fluid!
Dissipation results from the breakdown of the single fluid theory.
At scales like (some number of) ion inertial scales.
Smith et al., ApJ, 645, L85, 2006.

19. Summary
Large-scale drivers of the turbulent cascade is able to account for the rate of heating the solar wind.
Issues with the rates determined from the inertial range.
Dissipation rate suggests that inertial range observations arise in situ.
Variance anisotropy scales with plasma .
Compressive component must be explainable via excitation/decay processes buried within turbulence.
Maybe maintaining association with initial conditions…
Onset of dissipation results from breakdown of fluid theory.
Cyclotron damping is only part of the story.
Most aggressive dissipation acts on the perpendicular wave vectors.
Dissipation range spectrum depends on the rate of cascade.
More compressive than inertial range.
More aggressive dissipation of k  B0.

20. Extra Slides

21. V & B Variation with VSW

22. Basic Solar Wind Scalings

23. Leamon Found: Dissipation range spectrum highly variable.
Dissipation range has smaller variance anisotropy than inertial range.
Compressive component more important.
Quasi-perpendicular wave vectors are more aggressively damped than parallel vectors.
Cyclotron resonances is responsible for ½  2/3 of energy dissipation.

24. Non-Cyclotron Resonance
Leamon et al., ApJ, 507, L , 1998.