Mem. S.A.It. Suppl. Vol. 9, 63c© SAIt 2006

Memorie della

Supplementi

POD analysis of photospheric velocity field: solaroscillations and granulation

A. Vecchio1,V. Carbone1,F. Lepreti1,L. Primavera1, P. Veltri1 L. Sorriso-Valvo2,

Th. Straus3

1 Dipartimento di Fisica, Universita della Calabria, and Istituto Nazionale di Fisica dellaMateria, Unita di Cosenza, 87030 Rende (CS), Italy

2 Liquid Crystal Laboratory - INFM/CNR - 87036 Rende (CS), Italy3 INAF - Osservatorio Astronomico di Capodimonte, Napoli, Italy

e-mail: vecchio@fis.unical.it

Abstract. The spatio-temporal dynamics of the solar photosphere is studied by performinga Proper Orthogonal Decomposition (POD) of line of sight velocity fields computed fromhigh resolution data coming from the MDI/SOHO instrument. Using this technique, weare able to identify and characterize the different dynamical regimes acting in the system.Low frequency oscillations, with frequencies in the range 20–130 µHz, dominate the mostenergetic POD modes (excluding solar rotation), and are characterized by spatial patternswith typical scales of about 3 Mm. Patterns with larger typical scales of ' 10 Mm, areassociated to p-modes oscillations at frequencies of about 3000 µHz.

Key words. Sun, solar oscillations, convective turbulence

1. Introduction

The solar photosphere is an interesting ex-ample of a system exhibiting complex spatio-temporal behavior. The early research in pat-tern formation focused on the presence ofsimple periodic structures, while the mainquestions currently addressed concern regimescharacterized by higher complexity, that is,patterns that are more irregular in space andtime. This is often related to the occurrenceof intermediate states between order and turbu-lence (Akhromeyeva et al. 1989). Indeed on thesolar surface the turbulent evolution of gran-ulation coexist with the contribution of solarglobal oscillations with peculiar discrete fre-

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quencies (Leighton et al. 1962; Severnyet al.1976; Brookes et al. 1978; Thomson et al.1995; Cristensen-Dalsgaard 2002). Two mainmodes can be excited, namely acoustic p-modes, the 5 minutes oscillations, (in the range1000–5000 µHz (Ulrich 1970)) and gravita-tional g-modes (in the range 1–200 µHz). TheFourier analysis of spatio-temporal maps ofLOS velocity fields on the solar photosphere,the so-called k-ω spectra, shows ridges cor-responding to the discrete p-modes (Deubner1974, 1975)(see fig.1). On the other hand,discrete frequencies in the low-frequency (g-modes) range are not recognized, but rather thepower observed in this range is spread over acontinuum, commonly attributed to the solarturbulent convection (Harvey 1985).

64 Vecchio: POD of solar photospheric motions

0 2 4 6k [Mm-1]

0

2

4

6

8

ν [m

Hz]

Fig. 1. The k–ω spectrum of the dataset.

2. The POD tecnique

In the present paper, we use the ProperOrthogonal Decomposition (POD) as a power-ful tool to investigate the dynamics of stochas-tic spatio-temporal fields. In astrophysical con-texts, POD has been recently used to ana-lyze the spatio-temporal dynamics of the so-lar cycle (Mininni et al. 2002). Introducedin the context of turbulence (Holmes et al.1996), the POD decomposes a field u(r, t) asu(r, t) =

∑∞j=0 a j(t)Ψ j(r) (1), the eigenfunc-

tions Ψ j being constructed by maximizing theaverage projection of the field onto Ψ j, con-strained to the unitary norm. Averaging leadsto an optimization problem that can be cast as∫

Ω〈u(r, t), u(r′, t)〉Ψ(r′)dr′ = λΨ(r) (2), where

Ω represents the spatial domain and brack-ets represent time averages. The integral equa-tion (2) provides the eigenfunctions Ψ j and acountable, infinite set of ordered eigenvaluesλ j ≥ λ j+1, each representing the kinetic en-ergy of the j-th mode. Thus, POD builds upthe basis functions, which are not given a pri-ori, but rather obtained from observations. Thetime coefficients a j(t) are then computed fromthe projection of the data on the correspondingbasis functions Ψ j(r) so that the sum (1), whentruncated to N terms, contains the largest pos-sible energy with respect to any other linear de-composition of the same truncation order. Thismethod is particularly appropriate when ana-lyzing complex physical systems, where differ-

ent dynamical regimes coexist. POD allows toidentify these regimes and to characterize theirenergetics and their spatial structure.

3. Data analysis

The POD has been applied to the time evolu-tion line of sight velocity field u(x, y, t) (x, y be-ing the coordinates on the surface of the Sun),obtained from MDI mounted on SOHO. Thefield of view is 695 × 695 pixels, with a spa-tial resolution of about 0.6 arcsec. Total timelasts about 15 hours and images are acquiredevery minute, so that we have N = 886 im-ages. In this way we get a set of eigenfunctionsψ j(x, y) and coefficients a j(t), as well as the se-quence of eigenvalues λ j ( j = 0, 1, ,N). Mostof the energy is associated with the first PODmode, accounting for the line of sight compo-nent of solar rotation. As this is not related toturbulence or solar oscillations, this mode willbe ignored throughout this paper. The rest ofthe energy is decreasingly shared by the fol-lowing 885 modes, so that, for example, only4% is associated to the second ( j = 1) PODmode. In laboratory turbulent flows, POD at-tribute almost 90% of total energy to the typ-ical large-scale coherent structures, confinedto j ≤ 2 (Alfonsi and Primavera 2002). Inour case, 90% of energy (excluding the ro-tation) is contained in the first 140 modes.This indicates the absence of dominating large-scale structures and the presence of some tur-bulent dynamics related to nonlinear interac-tions among different modes and structures atall scales. For each value of j two well de-fined frequencies are clearly observed: the firstone in the low frequency range and the sec-ond one in the range where p-modes are usu-ally observed. As an example in fig. (2) we re-port the time evolution of coefficients a j(t) andthe corresponding frequency spectra |a j(ν)|2,obtained through temporal Fourier transformfor three values of j. Concerning the spatialpattern, each value of the mode j is associ-ated to a wide range of wave vectors. This canbe seen in fig. (3) where we show the eigen-functions ψ j(x, y) and the corresponding wavevector spectra |ψ j(k)|2 (where k2 = k2

x + k2y ),

obtained through spatial Fourier transform of

Vecchio: POD of solar photospheric motions 65

ψ j(x, y). Thus, modes can be separated in threedifferent groups, accordingly to grain size, andtemporal behaviour. The 1 ≤ j ≤ 11, most en-ergetic modes display a pattern with very finestructures, recalling the photospheric granula-tion, with a broad spectrum, and clearly showa center-limb modulation. This kind of eigen-functions are associated with time coefficientsdominated by low-frequency oscillations in therange 20–127 µHz. Conversely, eigenfunctionsof the modes j = 12 and j = 13 present acoarser pattern, and their (still broad) spectrashow ridges. In this case high frequency oscil-lations, in the range 3250–3550 µHz, prevail.The further lower energy modes cannot be pre-cisely classified, and seem to present a mixtureof the previous characteristics, both in grainsize and in spectral shape. The amplitudes ofthe two peaks, in time power spectrum, arefound to be of the same order. The spatial pat-tern associated to each mode, although com-plicated, can be quantitatively characterizedby computing the integral scale length L j =∫ ∞

0

∣∣∣ψ j(k)∣∣∣2 k−1dk/

∫ ∞0

∣∣∣ψ j(k)∣∣∣2 dk which repre-

sents the energy containing scale of classicalturbulence (Pope 2000). This allows to esti-mate the typical scale for the first ten finegrained modes L ' 3 Mm, while for thecoarse grained modes L ' 10 Mm. Let usfirst focus on the high-frequency modes. Boththe informations on the involved spatial scalesand the measured frequencies indicate thatsolar p-mode contributions are identified byPOD. Indeed, the frequencies we observe havebeen predicted by helioseismological modelscf. Cristensen-Dalsgaard (2002), and are wellknown from observations using Fourier tech-niques (Deubner 1975; Leibacher et al. 2000).The spatial pattern of about 10 Mm, whichPOD associates with high frequency modes,is in agreement with the horizontal coherencelength attributed to solar p-modes. In order toconfirm this, we compared the k–ω classical re-sults with the POD spectra. Choosing, for ex-ample, the POD mode j = 12, we performeda cut of the k–ω spectrum at the measuredpeak frequency (3254 µHz). We thus obtain awavelength spectrum, reported on figure3, thatcan be compared with the j = 12 eigenfunc-tion spectrum. As can be seen, the spikes ob-

served in the POD spectrum qualitatively cor-respond to the typical ridges structure of thek–ω spectrum. Concerning the low-frequencydominated modes, it is interesting to notethat the measured frequency range, mentionedabove, is compatible with the theoretical re-sults on solar gravitational modes (Cristensen-Dalsgaard 2002). However, because of the res-olution limit, related to the finite length of thetime series, the low frequency oscillations de-tected by POD can not be unambiguosly at-tributed to discrete modes, rather than to a con-tinuum. The evidence of center-limb modula-tion shows that such modes are mainly associ-ated with the contribution of horizontal veloci-ties.

4. Conclusions

We presented the first application of PODon high resolution solar photospheric velocityfields, which represents an example of convec-

Fig. 2. In the left-hand column we report thetime evolution of POD coefficients a j(t) (m/s)for the modes j = 8, 12, 49. In the right-handcolumn we report the corresponding frequencyspectra |a j(ν)|2 (m2/s2).

66 Vecchio: POD of solar photospheric motions

Fig. 3. In the left-hand column we report fromtop to bottom the POD eigenfunctions Ψ j(x, y)for three the modes j = 8, j = 12 and j = 49.In the right hand column we report the wavevector spectra |Ψ j(k)|2 of the corresponding im-ages. Superimposed on the second and thirdspectra, the wave vector spectra obtained fromthe k–ω cut, respectively at ω = 3254 µHz andω = 76 µHz, are represented (dotted lines, inarbitrary units).

tive turbulence in high Rayleigh numbers nat-ural fluids. POD is able to capture the main en-ergetic and spatial features of the solar photo-sphere. Two main oscillatory processes, wellseparated in frequency, are detected in therange 3250–3550 µHz and in the range 20–130 µHz. The high frequency waves are thewell known acoustic p-modes and their prop-erties, as obtained from POD, are in agreementwith previous results based on Fourier tech-niques. On the other hand, low frequency oscil-lations, with frequencies compatible with thoseexpected from the theory of solar g-modes,prevail in the most energetic POD modes (ex-cluding solar rotation), which are characterized

by spatial eigenfunctions with typical scales of' 3 Mm. The clear association between lowfrequency oscillations and small spatial scales,which are close to the solar granulation, is themost interesting result provided by our analy-sis. This suggests the presence of strong cou-pling between low frequency modes and theturbulent convection. As a consequence, theuse of empirical eigenfunctions for the descrip-tion of such system seems appropriate, and afurther improvement of this analysis can be ex-pected from its application to longer time se-ries.

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