A&A 373, 307-317 (2001)
DOI: 10.1051/0004-6361:20010494
A. Nesis - R. Hammer - M. Roth - H. Schleicher
Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, 79104 Freiburg, Germany
Received 8 December 2000 / Accepted 2 April 2001
Abstract
We investigate the attractor underlying the granular phenomenon by
applying nonlinear methods to series of spectrograms from 1994 and
1999.
In the three-dimensional phase space spanned by intensity, Doppler
velocity, and turbulence (line broadening), the granulation attractor
does not fill the entire phase space, as expected from the high
Reynolds and Rayleigh numbers of the photospheric plasma, but rather
shows a highly structured form. This could be due to the correlations
between intensity, turbulence, and velocity, which represent also the
Reynolds stress.
To obtain insight into the dimensionality of the attractor, we use the
time lag method, a nonlinear method that enables us to get information
about the underlying attractor of a dynamical system (granulation)
from the measurement of one physical quantity only. By applying this
method to the observed Doppler velocities, we show that the
granulation attractor can be described by three independent
variables. The dimension of the granulation attractor seems to be
independent of the appearance of big granules and shear flow.
Furthermore, the power analysis of the Doppler velocity shows power
down to the spatial resolution of the instrument (0.3 arcsec). In
order to decide whether the power at the smallest scales is real or
noise, we use again the time lag method in combination with either a
high pass digital or wavelet filter, which filters out the large wave
numbers. It appears that the power at the smallest scales represents a
real signal.
Key words: Sun: photosphere - Sun: granulation
Observables like Doppler velocity, intensity, and turbulence (line broadening) variations can provide insight into the physics of the granulation, i.e., into the physics of the upper solar convective layers. So far, measurements of these observables have been processed by means of a power and coherence analysis, which is actually connected with the physical concept of modes in linear theories: Nesis et al. (1993) studied the scaling law of small and large granular structures; Komm et al. (1991) and Espagnet et al. (1995) investigated the penetration of the granulation into the photosphere. Furthermore, Nesis et al. (1996) investigated the granular dynamics simultaneously in "plage'' and nonactive regions, and Nesis et al. (1997) repeated the study of the extension of the granular dynamics into the overshoot layers with better observations.
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Figure 1:
Part of our best spectrogram 99S4.Sp66, in the wavelength
region
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Granules are dissipative structures and the end product of the
convective processes in the upper solar layers: a highly nonlinear
dissipative system operating far from thermodynamical equilibrium.
According to Lorenz (1963), Ruelle & Takens (1971),
and Hirsch & Smale (1974), such systems usually approach in
time a strange attractor in its phase space. The attractor of a
dynamical system can easily be obtained if the coupled differential
equations for the relevant variables of the system are known. But, in
many experimental situations neither the relevant variables nor even
their total number are known, so that the attractor of the system is
not a priori accessible. In this case, the attractor can be
reconstructed in an artificial phase space, if a time series of one
single variable is measured (Packard et al. 1980; Takens
1981). The embedding theorem of Takens (1981) ensures
that the attractor is reliably reconstructed in the limit of a
sufficiently large dimension
of the
artificial space. Based on this theorem, different procedures have
been developed in order to determine dynamic as well as static
invariants of attractors from experimental time series of single
variables, e.g., the "correlation integral'' method (Grassberger &
Procaccia 1983).
Nesis et al. (1999) found that during the evolution of granules the two dimensional phase portrait of the turbulence-granulation dynamical system shows notable changes in time, while the power spectra of the turbulence reveal a two-component character. These facts raise the question if the velocity field of the deep photospheric layers results from the superposition of many random events or whether it is governed by some global deterministic mechanism that can be effectively modeled by a small number of dynamical variables. In the latter case the signal may result from the superposition of two or more periodic effects, or it may be chaotic as a consequence of being associated with a low-dimensional attractor.
In the current investigation we address the granulation attractor and its dimension from the observational point of view, using the Doppler velocity, turbulence (line broadening), and intensity variations along the spectrograph slit as input data. Although these observables are functions of space instead of time, the results should nevertheless be reliable in view of the reasonable assumption that an ensemble of spatially neighbouring states is equivalent to a time trajectory of a single state over a sufficiently long period of time. Therefore, any physical state found along the slit at any given instant will be realized in time at a single point. This assumption is related to the ergodic hypothesis, which states that for an ensemble of strictly independent states, a large number of observations made on a single system at N arbitrary instants of time have the same statistical properties as observing simultaneously Narbitrarily chosen systems from the ensemble.
To obtain insight into the dimensionality of the attractor underlying the granulation, we apply the time-lag and "correlation integral'' method (cf. Grassberger & Procaccia 1983) to the Doppler velocity measurements. Here, we are interested to see (i) if the attractor corresponding to the granular Doppler velocity has a low or high dimensionality and (ii) if the dimensionality shows any change within the last 6 years, i.e., from solar minimum to solar maximum. We assert that the granulation attractor is a manifestation of the granulation dynamics and any of its mutual interactions with characteristic physical quantities; especially magnetic field and pressure stratification should be reflected by its dimension.
Furthermore, the investigation of the granular velocity power scaling law at scales comparable to the instrumental resolution confronts us with the problem of their recognition. We use wavelets methods and the correlation integral technique to examine the nature of the signal corresponding to the smallest recorded scales. The behavior of the attractor dimension enables us to distinguish between pure noise and signal.
The current investigation is based on series of high spatial resolution spectrograms taken during periods of exceptional seeing conditions during 1994 May and 1999 August with the German Vacuum Tower Telescope (VTT) in Izaña (Tenerife).
The observational material of May 1994 consists of a series of
long-slit photographic spectrograms taken every 15 s at a fixed
position near the center of the solar disk, covering about 20 min in
total. The exposure time was 4 s and the wavelength range was
-
In this work we
discuss the spectrogram 94.A48. For a detailed description of this
material see Nesis et al. (1996, 1997).
In August 1999 further series of spectrograms were taken, again at a
fixed position near the center of the solar disk every 15 s,
covering about 20 min in total. The associated slit jaw images define
well the position of the spectrograph slit on the solar disk (see
Fig. 1). The spectrograms were recorded by an electronic
Xedar imaging camera, a
pixel CCD system operated
in frame selection mode, with an exposure time of 600 msec. By
binning
pixels we obtained a spatial resolution of
0.13 arcsec per pixel and a spectral resolution
of 1.4 mÅ per pixel. In the current work we discuss
the spectrogram, 99S4.Sp66, which turned out to be of particularly high
quality (Fig. 1). It exhibits an intensity contrast of
7%. The wavelength range was again
-
and included several absorption
lines of different strength.
The quality of the spectra was judged both visually, on the basis of the crispness of line wiggles at large magnification, and numerically, based on the properties of power spectra.
The observables Doppler velocity
,
line broadening
FWHM, and continuum intensity
were measured along
the spectrograph slit at equidistant positions si. The
variations were measured as Doppler shift
fluctuations of the line core about its mean value along the
spectrograph slit, while the line broadening fluctuations FWHM
correspond to the full line width at half maximum (cf. Nesis et al. 1993). In the following the intensity variations
along the spectrograph slit are also given
relative to the mean intensity. By calculating the line profiles we
verified that the lines have a small temperature sensitivity, as
expected for neutral metal lines of ca. 4 eV excitation energy.
Because the absorption lines are to first order also magnetically
insensitive and the line asymmetry is negligible, the FWHM
reflects an unresolved photospheric velocity
field w which unambiguously includes photospheric turbulent
velocity fluctuations; thus in the following we will refer to
as turbulent velocity as opposed to the granular
convective velocity
.
According to Meyer (1993) a wavelet transform presupposes the following necessities: the analyzing wavelet is required to have zero average, wavelets have to be continuous functions, and there should be a single, spatially localized "mother function'', which is translated to cover all positions and dilated to obtain the scale decomposition. Furthermore, a reconstruction formula has to exist for recovering the signal exactly from its wavelet coefficients. All analyzing wavelets are then mutually similar, and scale covariant with one another.
Wavelets can store separately the high and low frequency coefficients by the wavelet transformation (cf. Press et al. 1992). We use this property to remove the spatial high frequency noise. The inverse wavelet transformation, in addition, enables us to control the filtering process in such a way that the main features of the original signal remain intact. The wavelets which we use in this paper are the orthogonal Daubechies functions, which allow to separate a signal into a minimal number of independent coefficients. They are defined on the dyadic grid and are covariant with respect to discrete translations and dilatations only (see Press et al. 1992). As a consequence, the length of the data in pixels has to be a power of two, and the resolved scales are also powers of two. Therefore, in the case of the 94.A48 spectrogram we use only 512 out of the 676 digitized data points along the spectrograph slit.
In order to clarify the distinction between signal and noise in the
power at sub-granular scales 0.40
,
we apply to
the observables a low pass digital filter (cf. Nesis et al. 1992): a gently weighted running mean filter
(cf. Kendall &
Stuart 1968). Since we are interested in signals beyond
about 20 Mm-1 in the wave number domain, corresponding to
0.43 arcsec, we constructed the numerical filter in such a way that
its spatial frequency response shows a transmissivity of
55% of
the original signal at 20 Mm-1. We call the difference between
the measured velocity along the slit
,
and the
output of the low pass filter residual velocity. The residual
velocity, thus, corresponds to the output of a high pass filter.
The dynamical behavior of a complex system of dissipative nature can be studied by means of power spectra. Here, we use the power analysis in association with the digital filter described above and the time lag method in order to find out if the power assigned to the smallest scales is due to the granulation or to noise.
The degree of the correlation between the intensity, Doppler velocity and turbulence is shown in the 3D phase portrait of the granulation (cf. Fig. 4 and Sect. 3).
The existence of a chaotic attractor in a dissipative system can be
demonstrated by analyzing a suitable single-variable time series.
Packard et al. (1980) outline a simple method
(time lag) developed by Ruelle & Takens (1971) for reconstructing a
phase space from one dynamical variable: let
be
measurements of a physical variable at the time
.
From this sequence one can construct a set of
m-dimensional vectors
,
of the
form
A lower limit on the dimension D of the attractor can be calculated
by computing the correlation integral introduced by Grassberger &
Procaccia (1983). It is defined as
For a system with behavior governed by a D-dimensional attractor,
Grassberger & Procaccia (1983) postulated that when ris less than the overall size of the attractor,
,
where
is called the
correlation dimension. They then showed that
.
In fact, for several model systems, Grassberger & Procaccia
(1983) found that
so that
is expected to be a good estimate of the exact dimensionality
:
There are, however, limitations to the Grassberger-Procaccia
algorithm. The range [
]
of values
of r is limited (
and
are the
smallest and largest values of the distance of pairs
,
in particular
is the
diameter of the reconstructed attractor).
Now, to illustrate the choice of the space lag T, let us suppose
that two vectors vi and vj have first components
corresponding to the beginning of a granular structure. According to
Hilborn (1994) the succeeding components of these vectors are
reasonably well correlated as long as
,
where
is an appropriate characteristic space lag of the
autocorrelation decay for the original space series.
Komm et al. (1991) analyzed Doppler velocity space series and
found a zero autocorrelation for
km. Mattig et al. (1969) studied the cross correlation between granular
intensity and velocity series and found the first zero crossing at
725 km, i.e., at a spatial displacement of
725 km. Therefore, in our case
km should be a reasonable characteristic space lag.
Since the maximum embedding dimension is of order m = 7 (see below,
Figs. 7, 10), the condition
results in the choice
of the space lag
,
which is approximately
three times the pixel size of 90 km (cf. Sect. 2.1).
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Figure 2:
Granular Doppler velocity variations
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Figure 3:
Log-log power spectrum of the granular Doppler velocity
variations before ( grey curve,
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Figure 4: 3D phase portrait of the granulation dynamics. The variations along the spectrograph slit of the Doppler velocity, turbulence, and intensity correspond to x, y, z axis, respectively. The planes xy, xz, and yz show qualitatively the correlations of the velocity - turbulence (dash-dot line), velocity - intensity (dash-dot-dot-dot line), and turbulence - intensity (dashed line), respectively. |
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The results that we present and discuss in the following are based on spectrogram 99S4.Sp66 taken in 1999, and spectrogram 94.A48 from 1994. The presentation of the results follows the order of the processing steps presented in Sect. 2.2.
Figure 1 shows clearly the correspondence between granule and intergranular space on the slit jaw picture, and bright and dark stripes on the spectrogram, respectively. Thus, we can trace clearly the Doppler shifts of the lines towards the blue and red parts of spectrum due to convective up- and downflows in granules and in the intergranular space, respectively. The dark line parallel to the abscissa, at 87 arcsec on the ordinate, is due to a fiducial hair across the spectrograph slit.
Figure 2 shows the Doppler velocity variations
(about their mean value) along the spectrograph slit
for the magnetically insensitive (g=0) absorption line Ni I
491.20 nm (cf. abscissa of Fig. 1). The abscissa of the plot
is the length in arcsec along the slit, whereas the full line at the
position 87 arcsec is due to the hair crossing the
slit (cf. Fig. 1). On the ordinate positive velocities denote
the granular upflow, while negative ones mark the downflow within the
intergranular space. In Fig. 2 large amplitude variations
within a distance of
0.7 arcsec or
500 km lead to
shear flows and, thus, to the possibility of shear turbulence (cf. Nesis et al. (1997), for example at the intervals [40, 45]
[52, 56], [74, 80] and [90, 95] on the x-axis. Noteworthy is also the
behavior of the Doppler velocity along the slit in the interval [27, 38]
and around the position 100 arcsec. Here the velocity
shows small amplitude variations and the granulation
on the slit jaw picture looks like a "jelly'' (cf. Fig. 1).
It is important to note that the slope of the power changes at
(
Mm), from
approximately -1.8 in the logarithmic wavenumber range
[0.2, 0.7] to about -6.5 in the range [0.7, 1.3]. This raises
the question of the underlying physical process causing the change of
the slope.
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Figure 5:
Three-dimensional phase portrait of the granulation,
reconstructed by a single observable according to Eq. (1).
The axis denoted by Zi represents this single observable:
filtered
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Figure 6:
A plot of the natural logarithm of the correlation integral
Cm(r) as a function of ![]() ![]() |
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The dotted line parallel to the x-axis marks the convergence value,
which defines the dimension of the attractor. The attractor dimension
is found to be about 3.6. The convergence begins at the embedding
dimension
,
about twice the dimension
of the attractor. We like to notice that white noise would provide a
correlation dimension that does not converge (dotted line with
crosses).
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Figure 7:
The correlation dimension
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Figure 8 shows the two-dimensional phase portrait of the attractor reconstructed for the convective velocity variations gained from spectrogram 94.A48. The x-axis denoted by X(i)shows velocity variations along the slit; the y-axis is the time-lagged version of this observable X(i+T). This plot corresponds to the attractor projection on the X(i), X(i+T)plane of the embedding space. The trajectory is seen to be smooth over long periods, concentrated towards the diagonal. This appears to reflect the fact that the trajectory belongs to a well structured attractor of low dimension for the convective velocity, which will be further discussed below.
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Figure 8:
Two dimensional phase portrait calculated from
the convective velocity variations
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Figure 9: Same as Fig. 6, but for spectrogram 94.A48. |
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Figure 9 shows the natural logarithm of the correlation
integral
C(m)(r) calculated for 11 distances r
and for different integer embedding dimensions m (see
Eq. (2)) for the velocity
,
traced from
spectrogram 94.A48, up to the embedding dimension
m=8. We stop the calculation when the slope of the graph
becomes practically equal to the slope of the preceeding calculation,
as can be seen with the aid of the dotted lines. These indicate also
the range of r where we measure the line inclination. For high
values of the embedding dimension (
)
and small
distances (
), the correlation integral is
almost constant. For large values of
,
on the other
hand, the finite size of the attractor makes
C(m)(r)
saturate at 1.
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Figure 10: Same as Fig. 7, but for spectrogram 94.A48. |
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Figure 10 demonstrates again the convergence of the
correlation dimension
with increasing
value of the embedding dimension m (dark dots). The dashed line
parallel to the x-axis marks the convergence value, which defines also
the dimension of the attractor. For the velocity variations
,
gained from spectrogram 94.A48, the
attractor dimension has again a fractal dimension of about 3.6. The
convergence begins at the embedding dimension
,
which is about twice the
attractor dimension. We like to stress again that a white
noise signal would give a correlation dimension that does not
converge (dotted line with crosses).
Figure 1 shows the dynamical behavior of the highly structured overshoot layers. The granule on the slit jaw pictures and the Doppler shifts of the absorption lines on the associated spectrogram show that the bright (hot) regions represent ascending, whereas the darker (colder) regions signify descending flow. But, on the other hand, theoretical considerations and model calculations of the granulation phenomenon regard these ordered layers as a highly turbulent hydrodynamical system, characterized by many degrees of freedom, thus provoking the continued quest for the physics of the granulation.
Important empirical insight in the physics of the granulation is
provided by the power spectrum of the line of sight velocity
,
measured along the spectrograph slit as shown in
Fig. 2. The one-dimensional power spectrum in
Fig. 3 (grey line) reflects the dynamical behavior of the
granular structures (3.5 Mm down to 300 km). It is characterized by
a change of its slope at
;
and an
enhancement of the power at small granular scales. A literature survey
shows that our finding concerning the change of the slope rests on a
firm empirical basis. The velocity power spectra published by Durrant
& Nesis (1982), Komm et al. (1990), Nesis et al. (1993), Espagnet et al. (1993), and Nesis et al. (1996) exhibit also a change of their slopes at
practically the same granular scale as we find, of the order of
Mm.
This finding means that the dynamical behavior of the granular spatial scales can not be described by a single power law of the form k-q with a constant power exponent q. The slope change points to a division of the granular scales into groups, according to their dynamical behavior, and thus raises the question about the physical reasons underlying this change.
Keil (1980) showed a two-dimensional
power spectrum
of a line of sight velocity measurement in a medium-strength
absorption line. In his Fig. 4, traces parallel to the k-axis
represent the power as a function of the wavenumber at different
frequencies. To compare our one-dimensional power spectrum
(cf. Fig. 3, grey line) with that of Keil (1980,
Fig. 4) we used a trace parallel to the k-axis at a frequency
corresponding to a time period comparable to the mean life time of
granules. This gives again a change of the power slope at a scale of
1.5 Mm: the slope at small scales seems to be steeper than that at
larger scales. According to Keil (1980) the velocity
variations at spatial scales of
2 Mm and smaller, show a great
deal of power at frequencies corresponding to granular lifetimes, as
well as the high-frequency power. However, at spatial scales smaller
than 1.5 Mm the two-dimensional power is practically free from any
oscillatory power.
Thus, the results presented by Keil (1980, Fig. 4) also support our findings concerning the change of the velocity power slope Fig. 3 (grey line) as well as the associated indication of the existence of separate regimes of granular scales with different dynamical behavior. Remarkable, however, is the coincidence of the oscillatory power with one of these two scale regimes.
Muller (1999) pointed out the existence of a critical granular size of the order of 1000 km, which divides the granular structures into two groups with clearly different behavior and different fractal dimensions associated with their geometry. Furthermore, he noticed that according to Mandelbrot (1977) these fractal dimensions prescribe the power law exponent and thus the dynamical state of the corresponding hydrodynamical system.
It is remarkable that our findings concerning the change of the power slope (cf. Fig. 3, grey line) are in a good agreement with those reported by Muller (1999) in spite of the different approach.
In Fig. 3 (grey line) the enhancement of the velocity
power corresponding to small scale velocity fluctuations raises the
question about the fitting of the observed power by a scaling law of
the form k-q with a constant power exponent q.
To address this question we filtered out the small scale velocity
fluctuations: first we applied a low pass digital filter to the
original velocity fluctuations, with a cutoff wavenumber at
.
Then we subtracted the low pass output from
the original signal (the input of the low pass filter). This
difference gives a series of fluctuations along the spectrograph slit
associated with the small scales of the solar surface. The effectivity
of the low pass digital filter to remove high wave numbers is
demonstrated by the similarity between the two power plots in
Fig. 3 (dark line and gray line) up to the
cutoff wavenumber
.
The power
spectrum of the low pass velocity signal (dark line) and the
original velocity fluctuations (gray line) can be expressed by a
power law of the form k-q with the exponent
for small granules [0.331.2] Mm and
for the large granules [1.23.2] Mm.
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Figure 11:
The correlation dimension
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The observational analysis of the dynamics of the granular layers is based mainly on three observables: Doppler velocity fluctuations, turbulence (line broadening), and intensity (see Sect. 2.2). The 3D plot in Fig. 4 shows the phase space spanned by these three observables, while on the three planes we see the correlations between the observables. It reflects in particular the high correlation between intensity and velocity and the moderate one between turbulence and velocity. The poor correlation between intensity and turbulence is likely due to the origin of the turbulence at the borders of granules in the dark intergranular lanes (cf. Nesis et al. 1997).
According to Bradshaw (1976), correlations between the observables describe sensitively the evolution of any convective flow, especially its transition to turbulence. From this point of view the significant correlation between the three observables implies a highly organized flow with few degrees of freedom, i.e., the existence of a low dimensional granulation attractor.
An organized flow with few degrees of freedom could be due to the presence of magnetic fields (e.g., Tennekes & Lumley 1972). Even though our measurements were taken in nonactive regions, weaker intergranular magnetic fields cannot be excluded. In fact, a large body of recent numerical work on magnetoconvection (Cattaneo 1999; Weiss & Proctor 2001) suggests that the solar granulation can act as an efficient dynamo source for small-scale magnetic fields of a strength determined by the balance between the rates of vorticity generation by buoyant and magnetic curvature forces (Galloway et al. 1978). This process would contribute to organizing the flow and reducing thus the number of degrees of freedom.
To learn more about the existence of a granular attractor and its dimension, we applied the time lag method for the reconstruction of the attractor embedding space and the correlation integral to measure its dimension, as described in Sect. 2.2.2.
In Fig. 5 we see the 3D phase portrait of the granular layers, reconstructed according to the time lag method from a single observable, namely the filtered, noise-free, velocity fluctuations. The axis denoted by Zi corresponds to this observable, while the time lag method fills the other dimensions Zi+T, and Zi+2Twith lagged versions of the used observable, respectively. For more details see Sect. 2.2, Eq. (1). The plots on the planes xy, xz, and yz in Fig. 5 corresponds to the 2D phase portrait (the projections) of the attractor and represent the pairwise correlations between the observable Zi and the lagged versions of this observable Zi+T and Zi+2T.
The resemblance between Figs. 4 and 5 is evident and corroborates the existence of a low dimensional attractor for the dynamical system underlying the granulation.
The plots in Fig. 6 display in a log-log presentation the correlation integral C(m) as a function of r for various embedding dimensions m. Actually, by means of the correlation integral (cf. Eq. (2)) we measure the density of the points of the attractor trajectory within a definite distance r in a given embedding space m. We assign, thus, a probability measure to the various distances r. Towards large r the correlation integral approaches a constant value, which means that the calculation has included practically all the points of the trajectory of the attractor within the given embedding space. The saturation of the correlation integral begins at large embedding dimensions, which means that only up to a definite dimension m the calculation finds the same number of points of the attractor trajectory within the small distance r. The correlation integral as a function of r changes its slope with the embedding dimension m. This reflects the influence of the embedding space on the structure of the attractor size.
In Fig. 6 we see clearly the convergence of the slopes of
the correlation integral (cf. Eq. (2)) as a function of
distance r with increasing embedding dimension m. According to
Eq. (4) the convergence limit defines the correlation
dimension
,
which measures the size of the attractor in
the corresponding embedding space.
To find the convergence limit we plotted in Fig. 7 the
correlation dimension
as a function of m.
increases monotonically up to m=6 and then
saturates at the level of
,
which is the
dimension of the attractor in an embedded space of the order of
6, twice the attractor dimension ( e.g., Argyris et al. 1995).
From a physical point of view this finding suggests that the evolution
of the dissipative dynamical system underlying the granular structures
has a low dimension strange attractor. The dimension
indicates furthermore that the corresponding dynamical system
can be described by only 3 to 4 observables. In other words, the
system corresponding to the granulation layers is provided with a low
dimensional (3 to 4) attractor
- a finding that at first glance appears to
contradict the description of parts of the granulation layer as
turbulent.
We applied the time lag method (including the reconstruction of the phase space, the calculation of the correlation integral and of the attractor dimension) first to the velocity fluctuations traced by the spectrogram 99-S4Sp66 taken in 1999 (i.e, near solar maximum). Figures 5-7 exhibit the results of this calculation.
In a second step we applied the same method to the velocity fluctuations traced by the spectrogram 94.A48 taken in 1994 i.e., near solar minimum). The results of this calculation are displayed in Figs. 8-10. Interestingly they turn out to be very similar to those presented in Figs. 5-7. (We notice that the resemblance between Figs. 5 and 8 concerns the 2D plane Zi,Zi+T in Fig. 5 and the 2D plot in Fig. 8). This similarity of the results corresponding to data observed at completely different times and levels of solar activity reveals clearly the existence of a continuity of the physical behavior of the granulation layers over time.
In Fig. 3 (grey line) the enhancement of the velocity
power at small scales raises the question about their convective
nature. To address this question (actually the "noise'' problem of
the velocity power spectrum) we used again the time lag method,
applying it this time to the residual velocity fluctuations (the
output of the high pass filter). Figure 11 shows the
correlation dimension
as a function of the embedding
dimension m for the residual velocity obtained from spectrogram
99-S4Sp66 (dashed line). The correlation dimension for the residual
velocity does not show any saturation with increasing embedding
dimension up to m=8, as it was the case in Figs. 7
and 10. On the other hand, however, the slope of
for the residual velocity (dashed line) is clearly
different from its slope for white noise (dotted line). Thus the
residual velocity variations associated with the velocity power
enhancement at higher wave numbers in Fig. 3 (grey line)
can not be considered as white noise. They reflect rather the
existence of a high dimensional dynamical system (attractor), for
example a turbulent flow within the intergranular space.
Our observational material probes the behavior of the granular dynamics of the deep photospheric layers near minimum and maximum solar activity. It reveals clearly that small and large granules are governed by different dynamics, as evidenced by different power slopes, with a well defined scale separating both types of granules. The application of a high pass digital or wavelet filter on the original data does not affect this result.
It is obvious that the dynamics of the granular structures is
important for the production of photospheric turbulence and for the
interaction with short period oscillations. To obtain insight into
the dynamical system underlying the granulation we applied the time
lag method to a single quantity: the observed velocity variations
.
Based only on this observable we reconstructed
the space within which the attractor of the dynamical system
underlying the granulation is embedded; the attractor dimension
defines the number of degrees of freedom of this system. We found that
the granular structures are produced by
a dynamical system with a low dimension (3-4) attractor.
The significant
correlation between the observed intensity, velocity, and turbulence
variations infers a well organized flow.
The application of the time lag method to two sets of velocity variations observed near solar maximum and minimum, respectively, gave the same result. The reproducibility of the results points to a continuity of the dynamical behavior of the granulation layers and emphasizes the significance of our results.
The request to distinguish velocity variations associated with subgranular scales from variations due to photon statistics, numerical effects, and pixel impurities is evident. Using the time lag method we found that the velocity variations associated with intergranular structures do not behave like white noise. They rather appear to belong to turbulent flow coexisting with the flow of the granular structures or to represent turbulent flows within the intergranular space.
Acknowledgements
We are grateful to the referee, Dr. J. Brooke, for constructive comments.