Nonuniformity of Spectral Indices in the Inertial Range
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In a recent series of papers using Wind data Podesta et al. (JGR2006, ApJ2007)
showed that the power spectral index for solar wind velocity fluctuations
in the inertial range at 1 AU is approximately -3/2. The inertial range
is the part of the spectrum where energy is transported without
dissipation to smaller scales where it heats the background plasma. That
transport dynamic dictates the rate of in situ heating of the plasma. The
Podesta result stands in contrast to numerous magnetic field results
showing -5/3 spectral indices and suggests significant changes in the
theories that describe energy transport in the inertial range. Although
there exists a broad range of predictions and simulations addressing the
multi-dimensional nature and spectral form of turbulent
magnetohydrodynamic (MHD) power spectra, there are no predictions at
present that embrace this dichotomy. To make matters worse, there are
several theories that predict either of the two different spectral
indices, but those predictions always apply to both magnetic and velocity
spectra uniformly. Spectra may vary as a function of length scale or
orientation relative to the mean interplanetary magnetic field (IMF), but
there is no established theory of turbulence that predicts different
spectra for the two fields.
At the same time numerous theories have been presented and are evolving to
explain the apparent distinction between field-aligned and perpendicular wave
vectors. Reduced MHD (RMHD) theory suggests that parallel and perpendicular
wave vectors are essentially decoupled with some degree of ongoing energy
transport from parallel to perpendicular wave vectors. Goldreich and Sridhar
(ApJ1995) predict that depletion of the parallel wave vectors will become
sufficient as to steepen the measured spectrum to an index of -2 when the IMF
is aligned with the solar wind flow (radial). Both Horbury et al. (PRL2008)
and Podesta (ApJ2009) examine inertial range spectra at the smallest scales of
the inertial range (1 to 10 sec) using wavelet techniques and find evidence of
steepening to -2 spectral indices when the IMF is aligned with the solar wind
flow.
Tessein et al. (ApJ2009) examine 10 years of ACE observations spanning solar
minimum and maximum conditions. They compute second-order structure functions
S2(τ)
~ Aτn
for each hour of data, fit each hourly result to
extract the index n for lags of 1 to 10 minutes corresponding to the
middle scales of the inertial range. They relate the index of S2 to the
index of the underlying power spectrum E(f) ~ f q
according to the
familiar relation q = -(n+1) and gather the computed indices according to
the orientation of the IMF and solar wind speed. They compute means,
errors, and standard deviations for the trace of the IMF fluctuations
BTrace and the trace of the
velocity fluctuations, VTrace, and present the
resulting statistics shown above. Uncertainties represent standard
deviations. On average 5000 independent measurements contribute to each
average, making the error-of-the-mean ~70 times smaller. While they
confirm that the Podesta (JGR2006, ApJ2007) results are independent of IMF
direction, they find no evidence of any statistically significant
dependence of the spectral index on the orientation of the mean IMF as
found at small scales by Horbury et al. (ApJ2008) and Podesta (ApJ2009)
and they find no evidence of the -2 spectral index predicted by Goldreich
and Sridhar.
Although interesting, the analysis raises more questions than it answers: How
can the velocity and magnetic spectra differ? Will the two spectral forms
evolve toward one another and at what rate? How can the parallel and
perpendicular wave vectors possess the same spectral index if they are
essentially decoupled? Will they decouple further as predicted? What is the
underlying and most fundamental dynamic that can explain these spectral forms
when all current evidence indicates that inertial range spectra are generated
in situ and do not originate from a single common solar source? Is it
possible that the smallest scales of the inertial range turbulence are more
evolved than the larger scales while both contribute to the energy-conserving
transport that heats the plasma? Lastly, what does this say for the
predictions of Goldreich and Sridhar who require a non-isotropic energy
transport while velocity and magnetic fields seem to evolve individually?
This item was contributed by
Jeffrey Tessein and Charles Smith (University of New
Hampshire), Miriam Forman (SUNY/Stonybrook), William Matthaeus (University of
Delaware), Benjamin MacBride (UC/Berkeley), and Joseph Borovsky (Los Alamos).
Address questions and comments to
Last modified 31 Jul 2009.