A&A 381, 265-270 (2002)
V. Carbone1,2 - F. Lepreti1,2 - L. Primavera1,2 -
E. Pietropaolo3 - F. Berrilli4 -
G. Consolini5 - G. Alfonsi6 - B. Bavassano5 - R. Bruno5 - A. Vecchio1 - P. Veltri1,2
1 - Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy
2 - Istituto Nazionale per la Fisica della Materia, Unità di Cosenza, Italy
3 - Dipartimento di Fisica, Univesità di L'Aquila, 67010 L'Aquila, Italy
4 - Dipartimento di Fisica, Università di Roma "Tor Vergata'', 00133 Roma, Italy
5 - Istituto di Fisica dello Spazio Interplanetario-CNR, 00133 Roma, Italy
6 - Dipartimento di Difesa del Suolo, Università della Calabria, 87036 Rende (CS), Italy
Received 3 April 2001 / Accepted 10 April 2001
We propose the application of Proper Orthogonal Decomposition (POD) analysis to the photospheric vertical velocity field obtained through the data acquired by the THEMIS telescope, to recover a proper optimal basis of functions. As first results we found that four modes, which are energetically dominant, are nearly sufficient to reconstruct both the convective field and the field of the "5-min'' oscillations.
Key words: Sun: photosphere - Sun: granulation - Sun: oscillations
Photospheric periodic motions, known as "5-min'' oscillations, have been
observed since 1962 (Leighton et al. 1962). These oscillations
have a period
of about 3-12 min and are usually called p-modes. Models of these
oscillations can be made by assuming that they are driven by pressure
perturbations which provide the restoring force (see for example Stix
1991, and references therein). Usually, the observed line of sight
u(x,y,t) is analysed, in both time and space (the coordinates (x,y)correspond to the position on the field of view), through a Fourier transform
A different phenomenon usually observed on the same time scales on the photosphere is granulation, that is, a cellular pattern which covers the entire solar surface, except in sunspots. This pattern is commonly interpreted as the result of the turbulent convective motions which take place in the subphotospheric layers (see for example Bray et al. 1984). Since the time scales of p-mode oscillations and convective motions are of the same order, to measure convective velocities, the oscillatory contribution should be filtered out. This has become a standard technique by using Fourier transforms in both space and time (Title et al. 1989; Straus et al. 1992). The result is a map of the velocity field obtained by reconstructing the field through a filtered inverse Fourier transform. That is, the contribution of the ridges in the k- plane due to "5-min'' oscillations is removed.
Fourier analysis has some disadvantages. The velocity field is represented as a linear combination of plane waves whose shape is given a priori, each corresponding to a characteristic scale (k-1) and a characteristic period ( ). However, information related to both position and time in physical space is completely hidden. This is an advantage when dealing with waves, but it is a strong disadvantage when dealing with strongly localised spatial structures like the convective structures. On the other hand, even a rough look at a typical granulation pattern of the solar photosphere (or a look at a high resolution map of ridge in the k-plane), yields the impression that most scales are excited (Consolini et al. 1999), and evolve stochastically in time. That is, the granulation is a complex spatio-temporal turbulent effect. We then need information about different scales, information that is often a useful ingredient for modelling and physical insight. This difficulty calls for a representation that decomposes the velocity field into contributions of different scales as well as different locations.
In the present paper we would like to introduce a different technique which allows us to decompose the velocity field, or other fields, thus providing a basis that optimally represents a flow in the energy norm. This is called Proper Orthonormal Decomposition (POD) and it is also known as the Karhunen-Loéve expansion. POD was introduced some time ago in the context of turbulence by Lumley (Lumley 1967; see also Holmes et al. 1998, and references therein), and it is a powerful technique to extract basis functions that represent ensemble averaged structures, such as coherent structures in turbulent flows. Convective structures, which are usually observed on the photosphere, can be seen as a kind of coherent structure within a stochastic field, and this represents the basis for the present paper. In the next section we briefly describe the POD, then we present the result of POD applied to velocity fields observed by THEMIS, and finally we discuss the results we obtained and future perspectives.
POD is designed to yield a complete set of eigenfunctions that are optimal in
energy, when compared to each other (Lumley 1967).
Given an ensemble of fields, in our case the vertical velocity fields
at different times t, namely
we introduce the expansion using the POD as
|Figure 1: We report the time evolution of the maximum cross-correlation max between the velocity field , obtained through a filter by the k- technique, and the field uN(x,y,t) obtained through the POD procedure with a truncation to Nmodes.|
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|Figure 2: Time evolution of the first ten coefficients aj(t) of the POD expansion.|
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|Figure 3: In the left-hand panel we show the spatial pattern , in the plane (x,y), of the POD expansion. In the right-hand panel we show the spatial pattern , in the plane (x,y), related to p-modes.|
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The line of sight velocity fields u(x,y,t) are calculated from monochromatic
images of the solar photosphere acquired on July 1, 1999 with the Italian
Panoramic Monochromator (Cavallini 1998) installed on the
refer to 7 spectral points within the FeI 557.61 nm photospheric absorption
line. This line is formed at a photospheric height of about 370 km
(Komm et al. 1991) and the velocity fields are obtained from
The size of the fields is
the spatial resolution
of the acquired images is about 0.4
while the resolution of the
velocity fields is around 0.7
The analysed time series consist of
32 images which cover a time interval of about 40 min.
First we compared results from POD analysis with the results of the
filter. Images were filtered by the standard technique, and
the granular velocity field
was obtained. Then, as a
measure of the difference between POD analysis and k-,
Fig. 1 we report the time evolution of the maximum
obtained by allowing N to
be variable. Two modes suffice to reproduce the results obtained from
the standard technique, the maximum correlation being about .
true, except for the images which are
at the boundaries of the temporal interval. This is probably due to spurious
boundary effects in the k-
analysis, when the Fourier transform
is applied to non periodic functions like those from observations.
In Fig. 2 we report the time evolution of the first ten
coefficients aj(t) for
j = 0,1,...,9.
It is evident that the j=2,3 modes correspond to the
"5-min'' oscillations. On the contrary, we can expect that the
first two mode are related to the convective overshooting.
This is confirmed by the
spatial shape of the various modes which is represented by the eigenfunctions
In Fig. 3 we report two cases corresponding to
mode (Fig. a), and the
mode (Fig. b). As it can be seen, the first mode is
characterised by the typical spatial scales of convective structures,
while the second map, corresponding
to oscillations, shows on average larger structures. This result agrees,
roughly speaking, with the common finding that oscillations have predominantly
smaller wavevectors than granules.
In Fig. 4 we report the kinetic energy of each mode and the
cumulative energy content of the modes
as a function of the mode number j. As it can be seen, the
first four modes take more than
of the total energy. This
indicates that, even if the j=0,1 modes dominate the energetics of
overshooting motions, and the j=2,3 modes dominate the energetics of
subdominant contributions are important. These subdominant contributions are
due to nonlinear coupling between the most energetic modes. These
contributions cannot be filtered using the standard k-
because a spatial localization of modes is not allowed by the Fourier
analysis. Looking at the time evolution of the various modes, we can recognize
contributions due to both convection and oscillations.
In Fig. 5 we show an example of a velocity field calculated from
Doppler shifts and the corresponding k-
In the same figure the velocity field
|Figure 4: The kinetic energy content of each eigenmode j (upper panel), and the cumulative kinetic energy content of each eigenmode j (lower panel), as a function of the mode number j.|
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|Figure 5: An example line of sight velocity field calculated from Doppler shifts (upper left), the same velocity field filtered with the k- technique (upper right), and the velocity fields reconstructed from the j=0,1 modes (lower left) and from the j=2,3 modes (lower right) respectively.|
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1) Through the POD expansion we found that the first two modes (j=0,1), where the energy is mainly concentrated, describe the pattern of convective structures, and the next two modes (j=2,3) describe "5-min'' oscillations. The pattern recovered from the first two modes is reasonably well correlated to the pattern obtained through the standard k- technique. However only of the total energy is confined in the first four modes, the remaining fraction being spreaded among the next modes. Then, even if the first four modes suffice to reproduce both the convective pattern and the "5-min'' oscillations, this is a strong indication that these most energetic modes interact to produce the modes at higher wave numbers.
2) We are able to reproduce the spatial pattern which contributes to the velocity field at each eigenmode. In other words, we can reconstruct not only the convective overshooting pattern, but also the spatial pattern associated with photospheric oscillations.
What we have found must be considered as early results of a technique which can be successfully applied to photospheric motions to gain insight into the physical behavior of convective and oscillatory motion, as well as their interactions. Indeed, little is known about the interaction between oscillations and convection in the solar atmosphere (Stix 1991). Convection might be a source of damping for oscillations, but it should be also seen as the main contribution to a stochastic excitation of the oscillatory behavior (Lighthill 1952; Goldreich & Keeley 1977). Our results can be used to improve knowledge in this area. Of course, we need longer time series, with a larger spatial extension, in order to better investigate the characteristic frequencies as well as spatial patterns of the p-modes, and in order to investigate the role played by meso-granular and/or super-granular cells. In the future we will build up a low-order model which describe the coupling between the p-modes and the granulation. This can be made using the empirical eigenfunctions to make a Galerkin approximation of equations describing the phenomenon. A future paper will be devoted to this interesting topic.
We thank the THEMIS staff for efficient support in the observations. This work was partially supported by the Italian National Research Council (CNR) grant Agenzia2000 CNRC0084C4.