An essential parameter for models of coronal heating and fast solar wind acceleration that rely on the dissipation of MHD turbulence is the characteristic energy-containing length $lambda_{bot}$ of the squared velocity and magnetic field fluctuations
($u^2$ and $b^2$) transverse to the mean magnetic field inside a coronal hole (CH) at the base of the corona. The characteristic length scale defines directly the heating rate. We use a time series analysis of solar granulation and magnetic field measurements inside two CHs obtained with the New Solar Telescope (NST) at Big Bear Solar Observatory. A data set for transverse magnetic fields obtained with the Solar Optical Telescope/Spectro-Polarimeter (SOT/SP) aboard {it Hinode} spacecraft was utilized to analyze the squared transverse magnetic field fluctuations $b_t^2$. Local correlation tracking (LCT) was applied to derive the squared transverse velocity fluctuations $u^2$. We find that for $u^2$-structures, Batchelor integral scale $lambda$ varies in a range of 1800 - 2100 km, whereas the correlation length $varsigma$ and the $e$-folding length $L$ vary between 660 and 1460 km. Structures for $b_t^2$ yield $lambda approx 1600$ km, $varsigma approx 640$ km, and $L approx 620$ km. An averaged (over $lambda, varsigma$, and $L$) value of the characteristic length of $u^2$-fluctuations is 1260$pm$500 km, and that of $b_t^2$ is 950$pm$560 km. The characteristic length scale in the photosphere is approximately 1.5-50 times smaller than that adopted in previous models (3-30$times10^3$ km). Our results provide a critical input parameter for current models of coronal heating and should yield an improved understanding of fast solar wind acceleration.
An abstract mathematical concept of fractal organization of certain complex objects received significant attention in astrophysics during last decades. The concept evolved into a broad field including multi-fractality and intermittency, percolation t
heory, self-organized criticality, theory of catastrophes, etc. Such a strong mathematical and physical approach provide new possibilities for exploring various aspects of astrophysics. In particular, in the solar and stellar magnetism, multi-fractal properties of magnetized plasma turned to be useful for understanding burst-like dynamics of energy release events, conditions for turbulent dynamo action, nature of turbulent magnetic diffusivity, and even the dual nature of solar dynamo. In this review, I will briefly outline how the ideas of multi-fractality are used to explore the above mentioned aspects of solar magnetism.
Results of a statistical analysis of solar granulation are presented. A data set of 36 images of a quiet Sun area on the solar disk center was used. The data were obtained with the 1.6 m clear aperture New Solar Telescope (NST) at Big Bear Solar Obse
rvatory (BBSO) and with a broad-band filter centered at the TiO (705.7 nm) spectral line. The very high spatial resolution of the data (diffraction limit of 77 km and pixel scale of 0.$$0375) augmented by the very high image contrast (15.5$pm$0.6%) allowed us to detect for the first time a distinct subpopulation of mini-granular structures. These structures are dominant on spatial scales below 600 km. Their size is distributed as a power law with an index of -1.8 (which is close to the Kolmogorovs -5/3 law) and no predominant scale. The regular granules display a Gaussian (normal) size distribution with a mean diameter of 1050 km. Mini-granular structures contribute significantly to the total granular area. They are predominantly confined to the wide dark lanes between regular granules and often form chains and clusters, but different from magnetic bright points. A multi-fractality test reveals that the structures smaller than 600 km represent a multi-fractal, whereas on larger scales the granulation pattern shows no multi-fractality and can be considered as a Gaussian random field. The origin, properties and role of the newly discovered population of mini-granular structures in the solar magneto-convection are yet to be explored.
Generation and diffusion of the magnetic field on the Sun is a key mechanism responsible for solar activity on all spatial and temporal scales - from the solar cycle down to the evolution of small-scale magnetic elements in the quiet Sun. The solar d
ynamo operates as a non-linear dynamical process and is thought to be manifest in two types: as a global dynamo responsible for the solar cycle periodicity, and as a small-scale turbulent dynamo responsible for the formation of magnetic carpet in the quiet Sun. Numerous MHD simulations of the solar turbulence did not yet reach a consensus as to the existence of a turbulent dynamo on the Sun. At the same time, high-resolution observations of the quiet Sun from Hinode instruments suggest possibilities for the turbulent dynamo. Analysis of characteristics of turbulence derived from observations would be beneficial in tackling the problem. We analyse magnetic and velocity energy spectra as derived from Hinode/SOT, SOHO/MDI, SDO/HMI and the New Solar Telescope (NST) of Big Bear Solar Observatory (BBSO) to explore the possibilities for the small-scale turbulent dynamo in the quiet Sun.
On the basis of observations of solar granulation obtained with the New Solar Telescope (NST) of Big Bear Solar Observatory, we explored proper motion of bright points (BPs) in a quiet sun area, a coronal hole, and an active region plage. We automati
cally detected and traced bright points (BPs) and derived their mean-squared displacements as a function of time (starting from the appearance of each BP) for all available time intervals. In all three magnetic environments, we found the presence of a super-diffusion regime, which is the most pronounced inside the time interval of 10-300 seconds. Super-diffusion, measured via the spectral index, $gamma$, which is the slope of the mean-squared displacement spectrum, increases from the plage area ($gamma=1.48$) to the quiet sun area ($gamma=1.53$) to the coronal hole ($gamma=1.67$). We also found that the coefficient of turbulent diffusion changes in direct proportion to both temporal and spatial scales. For the minimum spatial scale (22 km) and minimum time scale (10 sec), it is 22 and 19 km$^{2}$ s$^{-1}$ for the coronal hole and the quiet sun area, respectively, whereas for the plage area it is about 12 km$^{2}$ s$^{-1}$ for the minimum time scale of 15 seconds. We applied our BP tracking code to 3D MHD model data of solar convection (Stein et al. 2007) and found the super-diffusion with $gamma=1.45$. An expression for the turbulent diffusion coefficient as a function of scales and $gamma$ is obtained.
We present results of a study of intermittency and multifractality of magnetic structures in solar active regions (ARs). Line-of-sight magnetograms for 214 ARs of different flare productivity observed at the center of the solar disk from January 1997
until December 2006 are utilized. Data from the Michelson Doppler Imager (MDI) instrument on-board the {it Solar and Heliospheric Observatory} (SOHO) operating in the high resolution mode, the Big Bear Solar Observatory digital magnetograph and {it Hinode} SOT/SP instrument were used. Intermittency spectra were derived via high-order structure functions and flatness functions. The flatness function exponent is a measure of the degree of intermittency. We found that the flatness function exponent at scales below approximately 10 Mm is correlated to the flare productivity (the correlation coefficient is - 0.63). {it Hinode} data show that the intermittency regime is extended toward the small scales (below 2 Mm) as compared to the MDI data. The spectra of multifractality, derived from the structure functions and flatness functions, are found to be more broad for ARs of highest flare productivity as compared to that of low flare productivity. The magnetic structure of high-flaring ARs consists of a voluminous set of monofractals, and this set is much richer than that for low-flaring ARs. The results indicate relevance of the multifractal organization of the photospheric magnetic fields to the flaring activity. Strong intermittency observed in complex and high-flaring ARs is a hint that we observe a photospheric imprint of enhanced sub-photospheric dynamics.
Line-of-sight magnetograms for 217 active regions (ARs) of different flare rate observed at the solar disk center from January 1997 until December 2006 are utilized to study the turbulence regime and its relationship to the flare productivity. Data f
rom {it SOHO}/MDI instrument recorded in the high resolution mode and data from the BBSO magnetograph were used. The turbulence regime was probed via magnetic energy spectra and magnetic dissipation spectra. We found steeper energy spectra for ARs of higher flare productivity. We also report that both the power index, $alpha$, of the energy spectrum, $E(k) sim k^{-alpha}$, and the total spectral energy $W=int E(k)dk$ are comparably correlated with the flare index, $A$, of an active region. The correlations are found to be stronger than that found between the flare index and total unsigned flux. The flare index for an AR can be estimated based on measurements of $alpha$ and $W$ as $A=10^b (alpha W)^c$, with $b=-7.92 pm 0.58$ and $c=1.85 pm 0.13$. We found that the regime of the fully-developed turbulence occurs in decaying ARs and in emerging ARs (at the very early stage of emergence). Well-developed ARs display under-developed turbulence with strong magnetic dissipation at all scales.
Low and mid-latitude coronal holes (CHs) observed on the Sun during the current solar activity minimum (from September 21, 2006, Carrington rotation (CR) 2048, until June 26, 2009 (CR 2084)) were analyzed using {it SOHO}/EIT and STEREO-A SECCHI EUVI
data. From both the observations and Potential Field Source Surface (PFSS) modeling, we find that the area occupied by CHs inside a belt of $pm 40^circ$ around the solar equator is larger in the current 2007 solar minimum relative to the similar phase of the previous 1996 solar minimum. The enhanced CH area is related to a recurrent appearance of five persistent CHs, which survived during 7-27 solar rotations. Three of the CHs are of positive magnetic polarity and two are negative. The most long-lived CH was being formed during 2 days and existed for 27 rotations. This CH was associated with fast solar wind at 1 AU of approximately 620$pm 40$ km s$^{-1}$. The 3D MHD modeling for this time period shows an open field structure above this CH. We conclude that the global magnetic field of the Sun possessed a multi-pole structure during this time period. Calculation of the harmonic power spectrum of the solar magnetic field demonstrates a greater prevalence of multi-pole components over the dipole component in the 2007 solar minimum compared to the 1996 solar minimum. The unusual large separation between the dipole and multi-pole components is due to the very low magnitude of the dipole component, which is three times lower than that in the previous 1996 solar minimum.
Parameters of magnetic flux distribution inside low-latitude coronal holes (CHs) were analyzed. A statistical study of 44 CHs based on Solar and Heliospheric Observatory (SOHO)/MDI full disk magnetograms and SOHO/EIT 284AA images showed that the dens
ity of the net magnetic flux, $B_{{rm net}}$, does not correlate with the associated solar wind speeds, $V_x$. Both the area and net flux of CHs correlate with the solar wind speed and the corresponding spatial Pearson correlation coefficients are 0.75 and 0.71, respectively. A possible explanation for the low correlation between $B_{{rm net}}$ and $V_x$ is proposed. The observed non-correlation might be rooted in the structural complexity of the magnetic field. As a measure of complexity of the magnetic field, the filling factor, $ f(r)$, was calculated as a function of spatial scales. In CHs, $f(r)$ was found to be nearly constant at scales above 2 Mm, which indicates a monofractal structural organization and smooth temporal evolution. The magnitude of the filling factor is 0.04 from the Hinode SOT/SP data and 0.07 from the MDI/HR data. The Hinode data show that at scales smaller than 2 Mm, the filling factor decreases rapidly, which means a mutlifractal structure and highly intermittent, burst-like energy release regime. The absence of necessary complexity in CH magnetic fields at scales above 2 Mm seems to be the most plausible reason why the net magnetic flux density does not seem to be related to the solar wind speed: the energy release dynamics, needed for solar wind acceleration, appears to occur at small scales below 1 Mm.
