4.2 The scales of supergranulation

4.2.1 Length scales

The horizontal scale of the supergranulation velocity field was the very first physical characteristic of the pattern to be measured. Using correlation of the signal, Hart (1956) found a typical length of 26 Mm. Since this pioneering work, this value has oscillated around 30 Mm. To appreciate correctly the values that are given in the literature, one should have in mind that supergranulation is a fluctuating, disordered pattern, hence only its statistical properties make sense. We should also remember that each technique has its own biases and gives values according to these biases.

A first technique to determine the typical length scale of the supergranulation is to measure the position of the maximum of spectral power or the correlation length of the horizontal velocity fields. The auto-correlation of Dopplergrams was first used in the seminal work of Leighton et al. (1962Jump To The Next Citation Point) and Simon and Leighton (1964Jump To The Next Citation Point), who gave the value of 32 Mm for the supergranulation length scale. The following major step was realized with the data from the SOHO/MDI instrument. The major progress made with this instrument has been the tremendous increase of the size (and quality) of the data set leading to very good statistics. Using data collected in May – June 1996, i.e., at solar minimum, Hathaway et al. (2000Jump To The Next Citation Point2002Jump To The Next Citation Point) determined the power spectrum of the line-of-sight velocity, finding a peak at 36 Mm (spherical harmonic ℓ = 120). This peak extends from 20 Mm up to 63 Mm as given by the width at half-maximum.

Using the granule tracking method, Rieutord et al. (2008Jump To The Next Citation Point) also determined the characteristics of the spectral peak of supergranulation. They found a similar length scale of 36 Mm and an extension between 20 Mm and 75 Mm (the epoch is March 2007, also at solar minimum). The data set in this case was much smaller (7.5 h and a field of view of 300 × 200 Mm2), but still a hundred of supergranules were captured, giving good statistics. A similar measurement by Rieutord et al. (2010Jump To The Next Citation Point) using the small field of view of Hinode (76 × 76 Mm2) encompassing only four supergranules, gave a peak at 30 Mm.

Other authors, like DeRosa et al. (2000) and DeRosa and Toomre (2004Jump To The Next Citation Point), used local correlation tracking to determine the horizontal flows from the Doppler signal of SOHO/MDI and identified supergranules with horizontal divergences. From these data, they derived a rather small “diameter” in the 12 – 20 Mm range. Using a similar technique, Meunier et al. (2007cJump To The Next Citation Point) found a mean value for supergranule diameters around 30 Mm. As underlined in these papers, the size of supergranules very much depends on the smoothing procedure used in the data processing.

Another set of independent measurements was performed by Del Moro et al. (2004Jump To The Next Citation Point) using data from local helioseismology. The technique is based on the fact that local helioseismology gives (more easily than the velocity itself) the local horizontal divergence of the flows, as this quantity appears as a difference between wave travel times. Thus, using a similar data set as DeRosa and Toomre (2004Jump To The Next Citation Point), Del Moro et al. (2004) extracted the horizontal divergence from the local propagation of waves and could also determine the statistics of supergranule sizes. They found a mean diameter at 27 Mm with a peak in the distribution at 30 Mm. These latter results have been confirmed by Hirzberger et al. (2008Jump To The Next Citation Point) using an even larger set of data (collecting more than 105 supergranules).

Alternatively, several authors used tesselation algorithms or threshold-based identification techniques to capture individual supergranulation cells and subsequently study their geometrical properties and spatial arrangement. Such techniques have mostly been applied to maps of the chromospheric network (e.g., Hagenaar et al., 1997Jump To The Next Citation PointSchrijver et al., 1997Jump To The Next Citation PointBerrilli et al., 1998), whose relationship to supergranulation is further described in Section 4.6.1. Following this approach, Schrijver et al. (1997Jump To The Next Citation Point) notably found that the patterns of granulation and supergranulation are very similar when properly rescaled. Their results are “nearly compatible with an essentially random distribution of upflow centers”. Comparisons between the spatial arrangement of supergranulation cells and granulation cells were also performed by Berrilli et al. (2004Jump To The Next Citation Point), who found that the supergranules distribution is well represented by a “hard sphere random close packing model” and by Hirzberger et al. (2008Jump To The Next Citation Point), whose result differ markedly from those of Berrilli et al. (2004) and are compatible with a field of “non-overlapping circles with variable diameters”.

To conclude this paragraph, we would like to stress an important difference between the various techniques used to characterize the scale and spatial distribution of supergranules. The first technique consists in determining the scale at which the kinetic energy spectral density or correlation length of horizontal motions is maximal, while the second technique relies on identifying coherent structures using tesselation algorithms and threshold conditions (such as the FWHM of autocorrelation functions) to study the size statistics of the resulting distribution. Unsurprisingly, the two methods provide slightly different values for the supergranulation “length scale”. As noticed by Leighton et al. (1962Jump To The Next Citation Point), threshold-based detection gives an estimate of the size of supergranules, whereas the location of the kinetic energy spectrum is an indication of the average distance between supergranules (assumed as to be the energy-containing structures).

4.2.2 Time scales

After supergranulation was discovered, one of the first questions was that of the lifetime of the structures. Here too, we would have to distinguish the lifetime of the coherent structures and the spectral power in a given time scale. However, this latter quantity, being too difficult to derive, is not available. Thus, the time scales discussed below are based on coherent cellular structures.

Worden and Simon (1976Jump To The Next Citation Point) suggested a lifetime of 36 h for the lifetime of supergranulation and reported a detection of vertical velocity fields only at the edge of supergranulation cells, confirming earlier work by Frazier (1970Jump To The Next Citation Point). Later, Wang and Zirin (1989Jump To The Next Citation Point) showed that supergranulation lifetime estimates depended strongly on the choice of tracer or proxy. They obtained 20 h using Dopplergrams, two days using direct counting techniques of supergranulation cells and 10 h using the tracking of magnetic structures (see also Section 4.6). Here again, SOHO/MDI data have dramatically increased the statistics and thus quality of the determinations. The latest results of Hirzberger et al. (2008Jump To The Next Citation Point) lead to a lifetime around 1.6 ± 0.7 or 1.8 ± 0.9 d, depending on the technique used. These values are somewhat longer than the previous ones, but the length of the time series associated with the size of the sample enable a better representation of long-living supergranules.

4.2.3 Velocity scales

A typical velocity associated with supergranules can be derived from the ratio between the previously discussed typical length and time scales. Taking 30 Mm for the former and 1.7 d for the latter, we find 205 m s–1 as the typical horizontal velocity. This estimate is in reasonable agreement with more direct inferences of the supergranulation velocity field from observations: the original work of Hart (1954) inferred 170 m s–1, Simon and Leighton (1964Jump To The Next Citation Point) mentioned 300 m s–1 and more recently Hathaway et al. (2002Jump To The Next Citation Point) evaluated this amplitude at ∼ 360 m s–1.

The preceding values are obtained from Doppler shifts. They are quite imprecise because they always mix the horizontal and vertical components of the flow. Granule tracking does not suffer from such a problem, however we here face the remaining problem of the scale dependence of the velocity. The obtained values depend on the way data are filtered.

Possibly, the best way to describe the velocity field amplitude of supergranulation is the spectral density of horizontal kinetic energy Eh(k), which describes the relation between the scale and amplitude of the flow. It is defined as

1 ⟨ ⟩ ∫ ∞ -- v2h = Eh (k)dk . (1 ) 2 0
Figure 4View Image provides an example of such a power spectrum. The dimensional value of Eh (k) was derived very recently from Hinode/SOT data by Rieutord et al. (2010Jump To The Next Citation Point). The spectral power density at supergranulation scales is 500 km3 s–2, which is larger than that at granulation scales6. This energy density is related (dimensionally) to the velocity at scale λ = 2π ∕k by the relation ∘ ------- V = kE (k ) λ h. Here, V ≃ 300 m s− 1 λ=36 Mm, which is quite consistent with the direct Doppler measurements of the velocity field at supergranulation scales.

The horizontal velocity needs to be completed by the vertical velocity. This latter quantity is unfortunately much harder to extract, because the signal is noised by the 5 min oscillations and by the presence of magnetic field concentrations at supergranule boundaries, where up and downflows tend to be localised (see Section 4.6 below). November (19891994Jump To The Next Citation Point) advocated that this vertical component was in fact the mesogranulation that he detected some years before on radial velocities at disc centre (November et al., 1981). The rms value of this quantity was then estimated to be 60 m s–1. More recently this quantity was evaluated using the SOHO/MDI data by Hathaway et al. (2002Jump To The Next Citation Point). They derived an estimate of 30 m s–1. This value is in line with the results of Rieutord et al. (2010Jump To The Next Citation Point) obtained from Hinode/SOT data using power spectra of line-of-sight velocities.

View Image

Figure 4: Kinetic energy spectra of solar surface flows. (a) The power spectrum of the line-of-sight velocity using SOHO/MDI Doppler data (Hathaway et al., 2000Jump To The Next Citation Point). The supergranulation peak near ℓ = 120 is clearly visible, while the granulation peak expected around ℓ = 3000 is eroded and effectively shifted to larger scales (ℓ ∼ 1500) due to time-averaging. (b) From the CALAS camera at Pic du Midi. Power spectrum (in relative units) of the horizontal velocity obtained from granule tracking for different time averages (Rieutord et al., 2008Jump To The Next Citation Point). (c) Absolute spectral density in km3 s–2 also derived from granule tracking, but applied to Hinode/SOT data (Rieutord et al., 2010Jump To The Next Citation Point). In (b) and (c) the power spectra are those defined in Equation (1View Equation).

  Go to previous page Go up Go to next page