Research Note: Correlation Scale of Energy Containing Structures in the Base of Coronal Holes |
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An essential parameter for models of coronal heating
and fast solar wind acceleration that relay on the dissipation of MHD
turbulence is the characteristic energy-containing length,
$$*λ _{⊥}*, of the squared velocity and magnetic field fluctuations,

We explore two CHs (Figure 1 on the left, the FOV/NST is marked with boxes) for which New Solar Telescope (NST) time series were observed. Both CHs were observed at the times of their crossing the central meridian. The first CH was observed on 2011 August 12, and it is referred to hereafter as CH 2011-08-12. The second CH was observed on 2012 June 4 and is referred to hereafter as CH 2012-06-04.

The solar granulation data were acquired with NST with a broad-band TiO filter centered at 705.7 nm with a band-pass of 1 nm, which allows us to register the solar continuum intensity at this wavelength. The pixel scale of the PCO.2000 camera, 0.0375", is 2.9 times smaller than the telescope diffraction limit of 77 km. KISIP software package for speckle interferometry (Woger and von der Luhe 2007) was utilized to achieve diffraction limited resolution in the reconstructed images. The final CH 2011-08-12 (CH 2012-06-04) data set consists of 82 (659) speckle-reconstructed, aligned and destretched images. The final time cadence was 12 and 13 s, respectively.

Spatial fluctuations of the *transverse* component of the magnetic field
inside a CH are of primary interest in this study. A unique data set for
transverse magnetic fields in a large quiet sun area obtained with the Solar
Optical Telescope/Spectro-Polarimeter (SOT/SP) aboard *Hinode* spacecraft on
2007 March 10 was kindly provided to us by Dr. B. Lites to analyze the squared
transverse magnetic field fluctuations, $$*b _{t}^{2}*.
The maps of the magnetic field
components are shown in their Figure 2 in Lites et al. (2008). We consider the
$$

Our main goal is to find, following Batchelor (1953), "convenient
measures of the linear extent of the region within which velocities
[as well as squared velocities and magnetic field] are appreciably
correlated".
We will use three different approaches for deriving the characteristic length
from the correlation function, $$*B(r)*.
The first approach is to determine the Batchelor integral scale

$$*λ=∫ _{0}^{rmax} B(r) dr*. (1)

We adopted $$*r _{max}* to be 5 Mm to exclude the non-diminishing tail caused by
noise.
The second method is to approximate the correlation function near the origin
(Hinze 1958, Monin and Yaglom 1975, Feder 1989):

$$*B(r)= Const · exp(-r/ς)*. (2)

When $$*B(r)* is fitted with an exponential function, the correlation
function drops by $$*e* times at a scale $$*ς*. This scale is called the
correlation length in the percolation (Feder 1989) and in second-order
phase transitions theories.
Finally, the characteristic length can also be determined via the $$*e*-folding
scale of $$*B(r)* without any approximation of the latter (the scale, where the
*measured* correlation function drops by $$*e* times). We denote this measure
as $$*L*.
The parameters introduced above, $$*λ*, $$*ς*, and $$*L*, are considered
as proxies for the characteristic length, and are calculated for all data sets.

We first focus on the statistical properties of the squared amplitude,
$$*u ^{2}=(u_{x}^{2} + u_{y}^{2})*, of the
$(u$

We then calculated correlation functions of the transverse velocity components,
$$*u _{x}* and $$

We find that the behavior of the parallel and normal correlation functions is
different (see Figure 4). Whereas the normal correlation function,
$$*B _{n}*, is positive for all scales, the parallel function changes sign, being always
situated below the normal correlation function; i.e.,
$$

The correlation functions from the squared magnetic field components are shown
in Figure 5. The most interesting to us is the correlation function of
the squared transverse magnetic field component,
$$*b _{t}^{2}*, plotted with the green
line. On scales below approximately 0.5 Mm, the three correlation functions are
similar. On larger scales, the NST
$$

**Preliminary outcome**

The characteristic lengths of the energy containing structures,
$$*u ^{2}*
and
$$

Thus, the characteristic length of the energy-containing structures in the photosphere lies in the range of 600-2000~km, which is on average an order of magnitude lower than the values used currently in models (Matthaeus et al. 1999, Dmitruk et al. 2001). Taking into account that the nonlinear dissipation terms in the MHD equations (Eq.(1) in Zank et al. (2012), is inversely proportional to the correlation length of energy containing structures at the base of the corona (see Eq.16 in Zank et al. 2012), these results play a critical role in determining the effectiveness of the coronal turbulence transport models in heating the solar corona and hence in driving the solar wind.

It is worthy to note that obtained estimates of the averaged transverse
velocity (about 1.2 km/s) and the characteristic length scale for
$$*u ^{2}*-fluctuations (about 1300 km), combined with the results of
Rudiger, Kitchatinov and Brandenburg (2011) allow us to evaluate the turbulent
magnetic
diffusivity and cross-helicity in the photosphere. Indeed, when the above
estimates are used in the expression for turbulent magnetic
diffusivity
$$