Examination of the Power Spectrum and Multi-Dimensional Cascade for

Evidence of VB Effects

Charles W. Smith

with

Miriam A. Forman, Benjamin J. MacBride, Joshua E. Stawarz, Jeffrey A. Tessein,

& Bernard J. Vasquez

Outline

• Does VB alter the power spectrum?– Examine 2nd-order struc. fn. technique for applicability.– Apply technique to ACE data.

• Does VB alter the rate of energy cascade?– Examine 3rd-order struc. fn. technique for applicability.– Apply technique to ACE data.

VB (cascade rate) q (power law)

Power Spectrum Index Predictions

• HC Driven:– Grappin et al. (1983)– Matthaeus and Zhou (1989)– Chandran (2008)

• Anisotropy Driven:– Fyfe et al. (1977)– Montgomery & Turner (1981)– Higdon (1984)– Goldreich and Sridhar (1995)– Boldyrev (2005, 2006)

• General Isotropic:– Kolmogorov (1941)– Iroshnikov (1964)– Kraichnan (1965)

How many predictions for the turbulent spectrum do we have?

How few different observational results can we find?

2nd-Order Structure Functions

iii

iii

iiii

ii

RB

tBtBB

tBtBtBtB

tBtBS

22

22

2

2

2

22

22

2nd-order structure function

stationarity

autocorrelation function

…and the Fourier transform of Rii() = diagonal terms of the spectral matrix.

If S2() ~ n,

then E(f) ~ fq

where q=(n+1)

So, n=2/3 leads to q=5/3,

and n=1 leads to q=2.

HC Effects in Total Energy SpectraWe will continue to fit 64 to 640 s lags as this is the smallest scale possible.

1-hr subintervals: 12-hr subintervals:

When we analyze magnetic spectra – they all center on 5/3.

When we analyze velocity spectra – they all center on 3/2.

0 < |C| < 0.2

<q> = 1.680 0.004

0 < |C| < 0.2

<q> = 1.733 0.003

0.8 < |C| < 1.0

<q> = 1.622 0.003

0.8 < |C| < 1.0

<q> = 1.646 0.002

Subset for Fast/Slow WindsFast Winds VSW > 500 km/s Slow Winds VSW < 400 km/s

0.8 < |HC / E| < 1.0

1-hr subsets

<q> = 1.614 0.002

0.8 < |HC / E| < 1.0

1-hr subsets

<q> = 1.732 0.006

However, we can do better!

We can measure the dynamics that lead to spectral form!

…and isn’t this really more important?

Kolmogorov’s 4/5 Law

0V

V)V(VV 2

νPt

ρM

|L|5

4LxVxVL 3

||||3 S

)Lx(V)x(V2

1)L( t

43 LVL …and if isotropic

The N-S / MHD Versions

4LV 3L

4LZ LZ2

L Tr

In MHD it looks like:

Politano and Pouquet, Phys. Rev. E, 57, R21, 1998a.

Politano and Pouquet, GRL, 25, 274, 1998b.

Stawarz et al., The Astrophysical Journal, 697, 1119, 2009.

Also, Stawarz et al. poster, turbulence session, this meeting.

s]-[Joules/kg 106.3 5 SWPheat VT

We use 10 yrs of ACE observations to compare the cascade rate with the inferred rate of proton heating.

There is a surplus of cascade energy (20-40%) available to heat electrons.

HC Effect in Turbulent CascadeAs with our earlier studies, we divide the data into 12-hr samples.

We exclude transients (shocks, CMEs, and upstream regions).

We compute third-moments for Z+ and Z cascade in isotropic formalism.

We (always) shuffle Z+/ into Zout/in according to mean IMF.

We subset the samples according to the bulk energy content.

…and then we subset for cross helicity C.

Above: Distribution function for bulk energy divided into 3 subsets (arrows).

Moments / Lag yield cascade rates.

Positive slope at small C.

Negative slope at large C.

Dominant (outward propagating) Z in blue.

Becomes most negative cascade at high C.

Selective Decay -- What Else?High C intervals are isolated regions of (relatively) high-speed wind.

They evolve in isolation between wind sheer and rarefaction.

Summary• We find no evidence of the dependence of the

energy spectrum on HC.– Not the magnetic or velocity spectrum, either.– Studying only 1 to 10 min fluctuations at 1 AU.

• We find compelling evidence of selective decay at scales greater than 1 min that takes a particular form:– The majority mode shows cascade to large scales

while the minority mode forward cascades.– This occurs in isolated (relatively high-speed) winds.

Extra Slides:

Have we confused the question?Spectral predictions apply to the rugged invariant total energy.

Not to components, individual fields, etc.!

Scale Size of Recent Pubs.

Hamilton et al., J. Geophys. Res., 113, A01106, 2008.

The large-scale fluctuations are given to us by the Sun’s activity and acceleration process.

The small scales evolve at a predictable rate.

There is ample “disagreement” in the published literature of observations, but often involving very different spatial & temporal scales.

Tessein

LeamonHorburyPodesta

MacBrideStawarz

Directional Dependence of Spectral Index

BR-independence:

5/3 spectrum for B

1.4 spectrum for V

Both Horbury et al. (2008) and Podesta (2009) observe steepening to 2 index for B in range 10 to 100 sec.

See poster by Tessein et al., turbulence session, this meeting.

Horbury et al. & Podesta

(left) Horbury et al. (2008)

(right) Podesta (2009)

2 spectral indices

Applying Structure Fn. Formalism

BRzD

BRyD

ISO

LLZTrLZLD

LLZTrLZLD

LLZTrLZLD

cos||4

sin||2

||3

4

2

31

2

32

2

||3

Politano & Pouquet, Phys. Rev. E, 57, R21--R24 (1995).

Politano & Pouquet, Geophys. Res. Lett., 25(3), 273--276 (1998).

In an MHD extension of the Kolmogorov 4/5 law in hydrodynamics:

MacBride et al., The Astrophysical Journal, 679, 1644, 2008.

Selective Decay -- What Else?