where the index refers to the horizontal quantities and to the vertical ones. From this equation, we see that a measure of the horizontal divergence and the vertical velocity together with a value of the density scale height (given by a model), allow for an estimation of the vertical velocity scale height.
Combining Dopplergrams and correlation tracking inferences with the above considerations on the continuity equation, November (1994) made the noteworthy prediction that the supergranulation flow should disappear at depths larger than 2.4 Mm below the visible surface. Note that his suggestion that the mesogranulation signal detected in power spectra at a horizontal scale of 7 Mm corresponded to the vertical flow component of convective supergranulation cells was part of the same argument. More recently, Rieutord et al. (2010) did the same exercise with divergences and velocity fields derived from Hinode data and found a vertical velocity scale height of 1 Mm, indicating a very shallow structure.
The advent of local helioseismology in the late 1990s made it possible to probe the supergranulation flow at optically-thick levels. Duvall Jr et al. (1997), using preliminary MDI data, only detected flows at supergranulation scales in the first few Mm below the surface. Duvall Jr (1998) further estimated that the depth of supergranulation was 8 Mm. Zhao and Kosovichev (2003) reported evidence for converging flows at 10 Mm and estimated the supergranulation depth to be 15 Mm. Woodard (2007) reported a detection of the flow pattern down to 5 Mm corresponding to the deepest layers accessible with their data set. Using new Hinode data, Sekii et al. (2007) recently found that a supergranulation pattern, monitored for 12 h in a small field of 80 × 40 Mm2, does not persist at depths larger than 5 Mm. The existence of a return flow at depths larger than 5 Mm has also been suggested but remains unclear (Duvall Jr, 1998; Zhao and Kosovichev, 2003). Note that imaging deep convection using helioseismic techniques is not an easy task. Braun and Lindsey (2003) and Lindsey and Braun (2004) provide a detailed description of the shortcomings and artefacts of helioseismic inversions in this context (see also Gizon and Birch, 2005).
To summarise, the determination of the vertical extent of the supergranulation below the surface is still in a preliminary phase. The few results mentioned above point to a shallow structure but they are affected by large uncertainties associated with both the intrinsic difficulty to perform such measurements and with their weak statistical significance. It is clear that a decisive step forward regarding this problem requires a careful study of the systematics and the processing of a very large amount of data to reduce the impact of the fluctuating nature of the flows.
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