Turbulent thermal convection is a non-equilibrium process ubiquitous in nature that sets in at a high Rayleigh number (Ra) [1
]. It occurs in several astrophysical environments, such as stellar interiors and planetary atmospheres, and it owns some aspects that are not completely understood. A typical manifestation of this process in the heliosphere is represented by the solar convection [2
In the outermost layers of the Sun’s interior (
), the energy generated by nuclear reactions is transported almost entirely by convection, because radiative transfer becomes inefficient owing to the increase in opacity determined by the recombination of hydrogen and helium [2
]. The convective envelope extends up to the solar surface, where plasma becomes optically thin and photons with energy in the visible range are eventually radiated into space. At a small scale, the region radiating the solar energy, called the photosphere, appears to be permeated by inhomogeneities that give rise to a granular pattern, also known as solar granulation [2
]. The granules, characterized by a spatial dimension of 1–2 Mm and a typical lifetime of 5–10 min, are due to the plasma motions that are driven by the convective overshooting, thus reflecting the turbulent motions underneath the solar surface, where Ra is estimated to be of the order of
]. A photospheric pattern attributable to the granulation was already observed in the early 19th century [3
]. However, it was only in recent decades that our knowledge of solar convection has greatly advanced. Indeed, relatively long series of homogeneous observations are needed to disclose the attributes and various manifestations of solar convection. These series have been achieved either in space or by equipping ground-based telescopes with adaptive optics (AO) systems that compensate the blurring and distortion in the observation due to the Earth’s atmospheric turbulence (seeing). Current 1 m class telescopes operating with such compensation allow acquiring the time series of data of the solar surface that last several tens of minutes by resolving spatial scales down to 100 km. Recently, the first 4 m class solar telescope, the Daniel K. Inouye Solar Telescope (DKIST, [4
]) in Haleakala (Hawaii, USA) has acquired data of the photosphere resolving even smaller spatial scales [5
]. Furthermore, the residual degradation of these data originating from the instrument, due to optical distortions and light scattering, and unaccounted for by the AO systems, can now be removed from the observations with the application of post facto image processing techniques [6
The imaging observations of the photosphere obtained with these methods have provided information about the morphology and evolution of the granulation pattern [7
], its dynamics including vortex flows [9
] and advection by larger scale flows [10
]. Moreover, the analysis of spectral lines measurements, which reveals the properties of the convective plasma logged in the line shape, has shown the height-dependence of the velocity structure of flows in granules and intergranular lanes [11
] and the narrow transition zones where granular flows bend down [13
], as well as the presence of the small-scale turbulent motions therein [14
]. Some studies have also reported convective patterns at scales larger than those of the granulation, in particular of the mesogranulation (with a spatial scale of 5–10 Mm, e.g., [15
]), supergranulation (with a spatial scale of 20–50 Mm, [20
]) and giant cells (with a spatial scale of 100 Mm and larger, [23
]). However, the real presence of some of these larger scale convective patterns is still debated [19
High-resolution observations have also shown that the magnetic field emerging into the solar atmosphere strongly affects the granulation. Generated by a turbulent dynamo process in the Sun’s interior, the magnetic field is maintained by converting the kinetic energy of convective plasma motions into magnetic energy [27
]. Therefore, there is an interplay between the magnetic field and plasma flows, which at small scales also manifest as the so-called quiet-Sun magnetism (see [28
] for a review). The interaction between granules and magnetic fields is not completely clear and there are several scientific open questions on the involved processes, especially related to the emergence of magnetic flux concentrations at the granular spatial scale [30
] and their subsequent diffusion [33
]. In this context, some authors have highlighted the presence of ubiquitous horizontal fields in the internetwork [36
]. Other authors have studied the evolution of coherent magnetic flux structures emerging in the quiet Sun. For example, the emergence of magnetic bipoles in the quiet Sun has been analyzed [40
]; the observation of a small-scale magnetic bipole has been reported from emergence to decay, studying the correlation between its evolution and the underlying convective motions [41
]; flux-sheet emergence events consisting of strongly inclined magnetic fields appearing at the granular scale have recently been analyzed [42
]; and the difference between the granules observed in various regions characterized by different field strengths has been investigated [44
In addition to solar physics studies, high-resolution observations of the Sun’s convection have also been a natural test-bed for the theory of dynamical systems and non-linear dynamics. Nesis et al. [45
] have applied non-linear methods to series of spectral observations to investigate the attractor underlying the granulation phenomenon in the three-dimensional phase space spanned by intensity, Doppler velocity, and line broadening. They have found that the granulation attractor does not fill the entire phase space, as one would expect from the high Reynolds and Ra numbers of the photospheric plasma. In addition, using photospheric observations in the quiet Sun, Viavattene et al. [46
] have recently shown that solar convective motions satisfy the symmetry relations of the Gallavotti–Cohen fluctuation relation for non-equilibrium stationary states.
It is worth mentioning that multi-dimensional time-dependent atmosphere models derived from radiative magneto–hydrodynamic (MHD) simulations are able to reproduce the solar granulation pattern with great fidelity [47
]. This might lead to the conclusion that the atmosphere models derived from simulations are accurate representations of the real Sun [2
]. However, simulations suffer from limitations and uncertainties on the physical parameters employed and can only cover a part of the size range of the solar convection.
The current knowledge derived from both observations and simulations still does not allow us to understand all the processes involved with solar convection. For example, the convection also plays a role in the excitation and upwards propagation of waves into the solar atmosphere [50
]. Indeed, turbulent motions are expected to excite waves that propagate outwards, transporting mechanical energy through the photospheric layers into the chromosphere and corona [51
]. However, while the occurrence of acoustic events has been demonstrated in the quiet Sun [52
], with waves propagating up to the upper atmospheric layers [54
], the excitation of MHD waves by the solar convection, predicted by numerical simulations, has been rarely observed [55
Here, we focused on the nature of the solar convection, aiming to further characterize it by means of the pseudo-Lyapunov exponents.
In the framework of chaotic and dynamical systems, Lyapunov exponents (hereafter also referred to as
exponents) are quantities that characterize the rate of divergence of the trajectories described by the system in the phase space, thus providing information about its chaotic or dissipative state. In other words, if the physical system begins to evolve from two slightly different initial states x
, the divergence between these initial states after n
iterations is given by
If , the trajectories will diverge exponentially and the system is chaotic. Conversely, if , the system is in dissipative regime. Therefore, the exponents indicate the predictability for dynamical systems, considered as an important tool for studying the stability of a dynamical system.
The Lyapunov exponents of the solar granulation were estimated in model atmospheres derived from the numerical simulations as in Kurths and Brandenburg [58
], Steffen and Freytag [59
], by varying the control parameters (e.g., Ra, Prandtl, or Taylor numbers) of computations. In addition, they were derived from the spectral observations of the photosphere by Hanslmeier and Nesis [60
]. The latter study was performed on the photographic spectra of a large sample of granules observed simultaneously with a spatial resolution of ≈0.24″, by using a ground-based telescope, which at the time was lacking AO systems.
In view of the lack of time series of observations, Hanslmeier and Nesis [60
] computed the spatial fluctuations of four parameters derived from the available data and estimated the variation of these fluctuations for increasing samples of data points. In fact, Hanslmeier and Nesis [60
] argued that, by virtue of the ergodic condition, the fluctuations of quantities estimated over a large sample of granules are at some time representative of the fluctuations of the same quantities evaluated over a long time interval at a single point of the studied system. However, Hanslmeier and Nesis [60
] note that the variation in the quantities estimated in their study are not identical to the Lyapunov exponents
that are inherently representative of the temporal evolution of the studied system. Based on that, Hanslmeier and Nesis [60
] referred to the exponents considered in their study as to Lyapunov-like exponents (hereafter referred to as
). The four quantities derived from the observations by Hanslmeier and Nesis [60
] are: the continuum intensity (
), the line-of-sight (LoS) velocity (
), the full width at half maximum (
) and the asymmetry (A
) of the spectral lines in their data.
Hanslmeier and Nesis [60
] reported significant changes of the
exponents computed from various line parameters derived from their observations. These changes occurred in a range of spatial scales spanning from the mesogranular to subgranular scale. In particular, they reported positive values of the
exponents computed from fluctuations of
, when looking at subgranular scales through Fourier filtering. This led the authors to suppose that their measurements were depicting turbulent motions occurring on subgranular spatial scales.
The results obtained by Hanslmeier and Nesis [60
] show a weak convergence and noisy behavior of the computed exponents at the smallest spatial scales investigated by the authors. Moreover, they were derived from a single observation of a sample of granules simultaneously observed at different evolutionary stages, as a proxy of the temporal evolution of an individual convective granular cell, as for the original definition of the Lyapunov exponents
. These facts have inspired us to perform a new study of the
exponents based on current state-of-the-art observations.
In the present work, we analyzed the four spectral line parameters considered by Hanslmeier and Nesis [60
], and derived them from spectro-polarimetric observations taken with state-of-the-art instruments. The studied data were taken in the quiet Sun near disk center, far away from active regions, at two lines originating in the photosphere and with different spatial resolution. To perform our study, we employed the methods proposed by Hanslmeier and Nesis [60
], but we also estimated the exponents from the analysis of the time series of observations as deduced from the definition of the Lyapunov exponents
. However, the Lyapunov exponents computed in our case do not exactly adhere to the same strict definition, so we will refer to these as pseudo-Lyapunov exponents.
In the following sections, after describing the data sets used in this work and the adopted data analysis (Section 2
), we present and discuss the results obtained from our investigation (Section 3
and Section 4
shows the values for the
exponents derived from
fluctuations in the continuum intensity images, displaying up to 10,000 image pixels in one frame from each data set. It can be clearly seen that the three exponents converge to negative values. Moreover, for all the computed exponents, there is a flat region that is representative of stable convergence, which is usually reached when using more than about 2000 image pixels for the calculation. We found such a behavior for all the
exponents computed in this work, so that in the following, we plot the variation of the
exponents computed up to 4000 image pixels only.
The values of the
exponents in Figure 4
differ from each other and depend on the analyzed data set. In order to investigate whether the differences among the computed exponents depend on the diverse spatial resolution of the analyzed observations or other data characteristics, e.g., the variable image contrast C
, we analyzed the relationship between the
exponents computed in each continuum intensity image of the available IBIS and CRISP observations and their contrast C
. Figure 5
(panel a and panel b) shows the variation of the
exponents derived from
fluctuations in the continuum intensity images characterized by the highest and lowest C
values in the IBIS and CRISP series. The dashed areas indicate the uncertainties associated with our
estimates in the IBIS and CRISP data sets, which are ±0.1 and ±0.05, respectively. Figure 5
(panel c and panel d) displays a scatter plot between C
values derived from the
fluctuations in the IBIS and CRISP data series. The arrows point to the values from the scans with the highest and the lowest C
in each series. It can be clearly seen that there is an almost linear relation between the image quality, defined by its contrast C
, and the value of the
exponent. In our study, we analyzed all the observations available from the two data sets, except for the seeing degraded IBIS scans. However, in the following, we only show results derived from the analysis of the observations characterized by the highest C
value in each series. Table 2
summarizes the results obtained.
(panel a) shows the variation of the
exponents as a function of the number of image pixels computed from
in the IBIS, CRISP, and HMI data. It is seen that the convergence of the computed
is to negative values. The values in Figure 6
(panel a) suggest that the computed exponents depend on the spatial resolution of the analyzed observations, obtaining larger
values for the higher spatially resolved CRISP data with respect to the less resolved HMI and IBIS observations. As reported in Table 2
, the average values of the
exponents with their respective uncertainty range between
obtained from the HMI and CRISP data sets, respectively. The dependence of the
value on the data resolution suggested by the above findings is also supported by the relationship between the exponents obtained from the IBIS and HMI data, which are taken by sampling the same Fe I 617.3 nm spectral line. It is worth noting that the small errors in Table 2
derived from HMI data are considerably lower that those for CRISP and IBIS because the HMI observations are unaffected by seeing degradation.
(panel b) displays the
exponents evaluated from
fluctuations. Even in this case, the
exponents stably converge to negative values. These increase with the spatial resolution of the data, ranging between
for the HMI and CRISP observations, respectively.
On the other hand, Figure 6
(panel c) shows the
exponents obtained from
fluctuations. The values of the
exponents are between
for the CRISP and HMI data, respectively. Again, the value of the
exponent from the CRISP data is higher than that from IBIS observations which, in turn, is higher than the one estimated from the HMI observations. Figure 6
(panel d) shows the
exponents evaluated from
fluctuations in the CRISP and IBIS data sets. We found a stable convergence of the computed exponents to negative values from both of the analyzed observations. The values of the
for the CRISP and IBIS data, respectively.
Considering the results from the CRISP and IBIS data, it is seen that the relationship between the
values computed from the two data sets is similar for all the investigated quantities, with a clear dependence on the data sets analyzed, and no signatures of a tendency of the computed exponents to positive values. Indeed, Figure 6
reports negative values for all the
exponents derived from our analysis. Therefore, all together, the values and trends in Figure 6
report on fluctuations of the investigated quantities typical of dissipative systems, which tend to point to a less dissipative regime as spatial resolution increases.
It is worth noting that the results presented in Figure 4
and Figure 6
show that the estimated values increase with the spatial resolution of the analyzed data. However, they also show that the relation among the values obtained from the diverse data depend on other characteristics of the data as well. In particular, we notice that the values estimated from the IBIS observations are closer to the ones derived from HMI data, than those from CRISP observations, in contrast to the diverse spatial resolution between the full-disk HMI and high-resolution IBIS and CRISP data.
We note that the IBIS and CRISP data were taken at the Fe I 617.3 nm and Fe I 630.15 nm lines, respectively, which sampled slightly different heights of the photosphere. Indeed, the formation height of the spectral line sampled by IBIS is slightly deeper in the photosphere than that of the line observed with CRISP. Thus one could expect higher values in the IBIS data relevant to deeper atmospheric layers. The height difference could leave signatures, mostly in the and derived from these data. However, it is not the case. Therefore, the higher value of the obtained from the CRISP observations with respect to the one computed in the IBIS data could depend on other characteristics of the two observations than their spatial resolution and the line formation height. However, in order to investigate if the different spatial resolution is responsible for the gap between the values derived from IBIS and CRISP data, we degraded the CRISP observations to the expected spatial resolution of the IBIS observation and repeated our analysis.
displays the variation of the four
exponents obtained from the IBIS and degraded CRISP data, by considering the highest contrast observation in each series. The variation of the
exponents obtained from the full-resolution CRISP best scan were over-plotted as a reference. We found that the degradation of the CRISP data only slightly affects the values of the
exponents obtained from our analysis. In particular, the
exponents computed in the full-resolution (degraded) CRISP observations resulted as −3.0 (−3.10), −1.02 (−1.08), −5.18 (−5.24), −5.12 (−5.37) from the
, respectively. We noticed that only the variation of the exponents derived from
fluctuations exceeded the uncertainty of our measurements. Therefore, we thus concluded that the difference between the
exponents derived from the IBIS and CRISP data cannot be ascribed to the diverse spatial resolution of the data alone, but should be originated by other characteristics of the data.
In this respect, it is worth noting that the IBIS and CRISP observations differ for the solar features imaged in the FoV. Indeed, the IBIS FoV does not show visible bright points, while the CRISP FoV contains several “chains” of these features along the intergranular lanes. These can be clearly noticed in Figure 1
and in more detail, in Figure 2
, thanks to the higher spatial resolution of the CRISP instrument. These features could leave signatures in the various estimated
exponents. In order to investigate if bright points affect the estimated values, we considered the
exponents derived from sub-FoVs including and lacking such solar features. We found that the values derived from the two data samples differ for quantities that only slightly exceed the uncertainty of our estimates reported in Table 2
, but for the values obtained from
fluctuations, which differ for a factor of 10 in terms of the computed uncertainty. The higher
exponent obtained for the
fluctuations in the FoV containing the bright points could be attributed to the turbulent motions [78
] reported in these features which induce broadening of the spectral profiles. Contrary to what might have been expected by considering the significantly higher intensity of the bright points with respect to the surrounding solar atmosphere, the values derived from
fluctuations estimated in regions including chains of bright points only slightly differed from the ones obtained in solar surface areas clearly lacking such features. This could also be the case for the
fluctuations, since downflow motions are usually observed in these features.
Thus, we suggested that the difference existing between values of the exponents derived from IBIS and degraded CRISP data may be attributed to the characteristics of the measurements and data processing, such as the level of stray-light in the observations and photometric accuracy of the post facto processing, respectively.
It is worth noting that the results presented above were derived from the analysis of the whole FoVs and sub-FoVs displayed in Figure 1
. However, we also computed the
exponents on several sub-FoVs with a size comparable to that shown in Figure 1
, which were extracted at different positions of the IBIS FoV. This was to test the accuracy of our results upon the analysis of different solar surface regions. We found that the values of the
exponents estimated from the diverse sub-FoVs all lay within two times the uncertainty of the exponents derived from the analysis of the whole IBIS FoV reported in Table 2
, and within the uncertainty of our estimates for the sub-FoVs in Figure 1
that are listed in Table 3
presented in the next Section.
3.1. Further Analyses
In order to further investigate the dependence of the computed
exponents on the spatial resolution of the analyzed data, we performed a spatial degradation of the IBIS observations and repeated our analysis on data affected by different degradation levels. To degrade the observation, we used a smoothing function based on Gaussian kernels. We degraded the maps of the continuum intensity
, LoS velocity
derived from the IBIS data to the same spatial scales as used in Hanslmeier and Nesis [60
], specifically at the following:
Subgranular, corresponding to a spatial resolution of 0.36″ ( pixels kernel);
Granular, corresponding to a spatial resolution of 0.90″ ( pixels kernel);
Mesogranular, corresponding to a spatial resolution of 5.04″ ( pixels kernel).
summarizes the results obtained from the analysis of all the IBIS degraded data. Figure 8
(panel a) shows the
exponents computed from
fluctuations in IBIS degraded data. For comparison, we also plotted the values of the
exponents from the full-resolution IBIS data set and from the HMI observations. The
exponents derived from
when degrading the IBIS data with kernels at the subgranular, granular, and mesogranular spatial scales, respectively. There is thus a clear tendency of the
values to increase with the lesser degraded data. Figure 8
(panels b, c, d) display the
exponents derived from the
fluctuations in the IBIS data degraded with the various kernels. We note that the data degradation mostly affects the value of the
exponents computed from
fluctuations, i.e., from the quantities most representative of the convective motions of solar plasma at small scales. For the sake of completeness, it is worth mentioning that we obtained similar results using a Fourier filtering of the data.
We noticed that the value of the exponent derived from the fluctuations in the HMI observations lie between those evaluated from the IBIS data degraded at the subgranular and granular scales. This finding suggests that the actual resolution in the degraded IBIS data could actually be slightly worse than the one expected from the applied kernel. On the other hand, the value from the fluctuations in HMI data is between the values of exponent computed in the IBIS data degraded with the granular and mesogranular kernels, as expected based on a dependence of the computed value on the spatial resolution of the data. As expected, the significantly lower resolution of the HMI data does not allow highlighting any contribution from the sub-granular structures. Indeed, the HMI maps only show granules, without any further spatial detail.
Furthermore, we studied the effect of the small magnetic flux concentrations present in the IBIS data on the computed
exponents. As it is known, weak magnetic fields affect the ascending/descending convective motions, thus modifying the typical convection pattern [28
] and giving rise to abnormally large granules [79
]. For our purpose, we evaluated the
exponents for the two sub-FoVs assumed as representative of magnetic and quiet regions, see Figure 1
(top-left panel) and Figure 3
. Figure 8
(panel e) shows the
exponents computed from
fluctuations in the two sub-FoVs. As a reference, the plot also displays the
values obtained from the analysis of the entire, full-resolution IBIS FoV. The average
exponents for the magnetic and quiet sub-FoVs are
, respectively, to be compared with
obtained from the analysis of the entire IBIS FoV. We note that the values of the
exponent derived from the sub-FoV representative of magnetic regions largely overlap with those derived from the analysis of the quiet and full IBIS FoVs presented above. Figure 8
(panels f, g, h) display the
exponents computed from
fluctuations in the two sub-FoVs. It can be clearly seen that the
exponents computed from all these quantities from the two sub-FoVs largely overlap. This is even more clear when considering the values of the computed exponents with their uncertainties which are summarized in Table 3
These findings show that the presence of weak magnetic fields only slightly affects the value of the exponents derived from all the computed fluctuations. For the parameter, the interaction between convection and small scale magnetic field concentrations manifests itself with lower values than those derived from quiet regions.
The aforementioned results refer to the pseudo-Lyapunov exponents
estimated as proposed by Hanslmeier and Nesis [60
], by using spatial fluctuations of the parameters representative of the studied system as tracers of either the regular or irregular evolution of the parameters describing the studied system. In order to account for the time dependence in the definition of the pseudo-Lyapunov exponents, we also analyzed the fluctuations of the computed parameters over time.
shows the results obtained from the residual maps of
, and A
based on the IBIS observations by using Equation (3
). The different colors show the results obtained from different values of j
, which correspond to different time intervals elapsed between the compared observations.
We note in Figure 9
that the exponents derived from the residuals computed over time assume negative values for all the physical quantities estimated from the IBIS data set. Moreover, we report that the estimated exponents clearly tend towards smaller values at the increase in time elapsed between the compared observations, as for the sampling of a less dissipative regime.
4. Comments and Conclusions
In this work, we analyzed the pseudo-Lyapunov exponents
presented by Hanslmeier and Nesis [60
] using state-of-the-art observations taken with the IBIS, CRISP and HMI instruments. Following Hanslmeier and Nesis [60
], we studied the exponents computed from fluctuations of the continuum intensity
, LoS velocity
, full width at half maximum
of the spectral lines, which are quantities describing the temperature and velocity of the solar plasma sampled by the observations. The analyzed data sets differ in terms of spatial resolution, spectral sampling, and post facto processing. We found a dependence of the
exponents on the spatial resolution and other characteristics of the analyzed data. All the computed exponents are negative, in contrast to previous results reported in the literature.
Our results confirm the dependence of the
exponents on the spatial resolution of the analyzed data reported by Hanslmeier and Nesis [60
]. We showed a clearer signature of this dependence than previously reported in the literature, which could be suggestive of the detection of less dissipative regimes with increasing spatial resolution. In addition, we found higher values of the
exponents obtained from
fluctuations than in previous analyses, as likely due to more accurate photometry of the data analyzed in our study. We further investigated the exponents in magnetic and quiet regions identified in the IBIS observations, by showing that the values obtained from the two regions coincide within the uncertainty of our estimates.
Comparing our results with those reported in Hanslmeier and Nesis [60
], we first noticed that the values computed in our study differ than those previously published. Our results also exhibit a clear convergence and are less noisy than those in Hanslmeier and Nesis [60
], especially for the exponents computed from
fluctuations. Conversely, our values are all negative, in contrast to the results in Hanslmeier and Nesis [60
] that show a tendency towards positive values derived from
fluctuations at some spatial scales through Fourier filtering. We inferred that the significant difference between the results obtained in our study, with respect to those by Hanslmeier and Nesis [60
], may be due to the rather diverse observational techniques employed to obtain the data analyzed in the two compared studies. In fact, the IBIS and CRISP data were acquired with a spectral sampling performed with Fabry–Pérot interferometers, while the HMI observations were obtained with a Lyot filter combined with a Michelson interferometer. In contrast, the data used in Hanslmeier and Nesis [60
] are spectrograms acquired with a grating spectrograph. Furthermore, our data differ from those considered by Hanslmeier and Nesis [60
] in the use of CCD sensors instead of photographic plates and of AO systems and of post facto reduction techniques. The absence of the AO in the acquisition of the observations analyzed by Hanslmeier and Nesis [60
] and the photographic data they had processed could have induced a smearing of the spatial resolution of the studied observations and of the photometry of their measurements: both these effects can impact the estimated exponents. In addition, it is worth noting that Hanslmeier and Nesis [60
] analyzed spectra obtained at a given time, while in our study, we considered measurements obtained from the sampling of a spectral line performed in time. Actually, the different methodology could slightly affect the line shape through which we aimed to infer the properties of the solar convective plasma. In this respect, it is worth noting that the data used in Hanslmeier and Nesis [60
] are similar for the methodology to the ones acquired within the Solar Optical Telescope (SOT, [82
]) aboard the Hinode mission. In addition to the same methodology employed to obtain the data, the SOT/Hinode observations also sample the same spectral line used for the CRISP data analyzed in this study, with images characterized by a slightly lower resolution (approximately 0.3″ for SOT/Hinode with respect to 0.13″ for CRISP). We planned to analyze the SOT/Hinode data in the near future to investigate how the different instrumentation could affect the
exponents determination. Furthermore, we could also analyze the series of HMI observations taken every 45 s, acquired with a different camera than the one used for the data employed in this study, in order to further investigate the exponents derived from time series of observations. Finally, IBIS and CRISP data also allow to derive exponents from line bisector analysis at different line depths that could give more information on the observed processes than those deduced from the simple fitting of the whole spectral line.
The pseudo-Lyapunov exponents
analyzed by Hanslmeier and Nesis [60
] and in this study are not identical to the Lyapunov exponents
that, according to the dynamical systems theory, describe the either chaotic or dissipative regime of a system from the evolution of its trajectories in the phase space. The exponents estimated with the method proposed by Hanslmeier and Nesis [60
] assume that the non-linear nature of the solar convection can be represented at any time by the properties of a large sample of convective cells observed simultaneously at different stages. In order to account for the temporal evolution of system entering the definition of the Lyapunov exponents, we also analyzed the variation of the physical quantities used to represent the solar convection at any point over time. We showed that all the exponents estimated from the residuals of physical quantities analyzed along time assume negative values, which are typical of a dissipative regime.
Current state-of-the-art observations as those analyzed here allow measuring the properties of solar plasma accurately at spatial scales as small as 100 km at the Sun’s surface. However, the resolution and sensitivity of the observations analyzed in our study, as well as both the methods employed to estimate the Lyapunov exponents, have not allowed us to detect any hint of the chaotic regime of solar convection. This regime can only be investigated by resolving even smaller spatial scales than those detected by the observations used in our study, and by using different methods than those employed in this study to measure the divergence of the solar convection in the phase space. The next generation 4 m-class ground-based DKIST and European Solar Telescope (EST, [83
]), as well the Polarimetric and Helioseismic Imager [84
] onboard the Solar Orbiter mission [85
] are expected to provide spectropolarimetric data sets with unprecedented spatial resolution down to 30 km on the solar atmosphere. The observations that will be obtained with these telescopes appear promising for the study of the small-scale physical processes that occur in turbulent regime in greater detail, with methods commonly employed in the dynamical systems theory.