4.4 Multi-scale convection

With access to extended time series of uniform quality data, especially from SOHO/MDI, but also from TRACE and the best Earth-based observatories, it became possible to simultaneously record motions over a range of scales, and to for example study the advection of mesogranulation scale cells by supergranulation scale motions.

Muller et al. (1992) used a 3 h time series from the Pic du Midi observatory to study the evolution and motion of mesogranules. They were able to show that mesogranules are advected by supergranulation scale motions, and also that the evolution of mesogranules is influenced by their location in larger supergranular scale flows. The evolution of supergranular scale flow fields and their influence on mesogranular scale flows were also studied by DeRosa and Toomre (1998), by Shine et al. (2000), and by DeRosa and Toomre (2004Jump To The Next Citation Point). The latter study provides evidence that flows on different scales are to some extent self-similar, and that they have amplitudes that vary smoothly with spatial and temporal scales. Figure 5 of DeRosa and Toomre (2004Jump To The Next Citation Point) illustrates the smooth behavior of the amplitudes characterizing multi-scale solar convection; over a range of time averaging windows extending from 0.1 to 100 hours the velocity amplitude varies smoothly, deviating from a linear fit in the log-log diagram by less than 10%. Figure 14 of DeRosa and Toomre (2004) illustrates the self-similar behavior, in that folding the data with Gaussians of varying size still leaves the size distributions essentially unchanged, after normalization with the filter size.

Other studies also indicate a close similarity and coupling between patterns at different scales. Schrijver et al. (1997a) compared the cellular patterns of the white light granulation and of the chromospheric Ca ii K supergranular network. They matched the patterns to generalized Voronoi foams and concluded that the two patterns are very similar. Roudier et al. (2003a) and Roudier and Muller (2004) demonstrated that a significant fraction of the granules in the photosphere are organized in the form of ‘trees of fragmenting granules’, which consists of families of repeatedly splitting granules, originating from single granules at their beginnings (see also Müller et al., 2001). Trees of fragmenting granules can live much longer than individual granules, with lifetimes typical of mesogranulation; this illustrates that larger scale flows are able to influence and modulate the evolution of smaller scale flows.

A smooth spectrum of motions was anticipated on theoretical grounds from the very beginning, but at the time it was thought that observations showed otherwise; observations were interpreted as though granulation and supergranulation represented distinct scales of motion, with no significant motions at intermediate scales. This lead to theoretical suggestions specifically aimed at explaining such a state of affairs (e.g., Simon and Weiss, 1968).

The numerical experiments by Stein and Nordlund (1989) showed that the topology of convection beneath the solar surface is dominated by effects of stratification, which lead to gentle, expanding and structure-less warm upflows on the one hand, and strong, converging filamentary cool downdrafts, on the other hand. The horizontal flow topology was shown to be cellular, with a hierarchy of cell sizes; granulation being a shallow surface phenomenon associated with the small scale heights near the surface layers, while deeper layers support successively larger cells. The downflows of small cells close to the surface merge into filamentary downdrafts of larger cells at greater depths. It is the radiative cooling at the surface that provides the entropy-deficient material which drives the circulation. The supply of high entropy fluid from the bottom of the convection zone is maintained by diffusive radiative energy transfer from the central, nuclear burning region, and in this sense the heat supply from below maintains the energy flux through the convection zone, but it is the surface that is responsible for generating the entropy contrast patterns that, via the corresponding buoyancy work, maintains the structure of the convective motions.

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Figure 22: Observed and simulated horizontal velocity amplitudes over a wavenumber range extending from global scales to below granulation scales. Observed velocities are from correlation tracking of TRACE and SOHO white light images (Shine, private communication), and from SOHO/MDI Doppler image modeling (Hathaway et al., 2000, and private communication). Simulation results are from Stein and Nordlund (1998) (granulation scales – orange symbols) and Stein et al. (2006aJump To The Next Citation Point,bJump To The Next Citation Point) (supergranulation scales – black symbols).

DeRosa et al. (2002) constructed the first models of solar convection in a spherical geometry that could explicitly resolve both the largest dynamical scales of the system (of order the solar radius) as well as smaller scale convective overturning motions comparable in size to solar supergranulation (20 – 40 Mm). They found that convection within these simulations spans a large range of horizontal scales, especially near the top of the domain, where convection on supergranular scales is apparent. The smaller cells are advected laterally by the larger scales of convection, which take the form of a connected network of narrow downflow lanes that horizontally divide the domain into regions measuring approximately 100 – 200 Mm across. Correspondingly, Stein et al. (2006b,aJump To The Next Citation Point) found a similarly wide range of scales of motion, with a smooth amplitude spectrum consistent with SOHO/MDI observation (Georgobiani et al., 2007Jump To The Next Citation Point), in simulations covering scales from sub-granular to supergranular.

To properly compare velocity amplitudes over a large range of scales it is useful to display a ‘velocity spectrum’, showing the square root of the velocity power per unit logarithmic interval of wavenumber k

∘ ------ V(k ) = kP (k), (42 )
where P(k ) is the traditional ‘power spectrum’ (velocity power per unit linear wave number). The quantity kP (k) is the contribution to the total mean square velocity per unit ln k, and is independent of the unit of wavenumber. It has dimension velocity squared, and its square root is a good measure of the velocity amplitude at various scales.

Figure 22View Image shows a composite of simulated and observed velocities, over a range of scales that extends from global to below granulation scales. Note that the velocities on granular, mesogranular, and supergranular scales are consistent with commonly adopted values (a few  km s–1 on granular scales, a few several hundred  m s–1 on mesogranular scales, and 100 – 200 m s–1 on supergranular scales). Note that the velocity spectrum is approximately linear in k, and that it continues to giant cell scales, with no distinct features on scales larger than granulation.

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Figure 23: Horizontal and vertical components of the convective and oscillatory components (separated by sub-sonic filtering) of the velocity field in a supergranulation scale convection simulation, compared with the convective and oscillatory components of the line-of-sight velocity field, as observed with SOHO/MDI in high resolution mode. The ‘convective’ and ‘oscillatory’ (‘modes’ in the figure) are separated by a line ω = ck, where c = 7 km s–1 (from the same data set as was used in Stein et al., 2006a).
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Figure 24: Subsonic filtered line-of-sight velocities from SOHO, Gaussian filtered to the same effective number of pixel elements, for physical sizes of 400, 200, 100, and 50 Mm. The order of the panels has been scrambled, to make the point that the patterns are very similar, and that it is not obvious how to order the panels in a sequence of increasing size, for example (but see Figure 25View Image).

Because of mass conservation large scale convection motions are necessarily dominated by horizontal motions, while on the other hand large scale solar oscillations are dominated by vertical motions near the solar surface. Doppler measurements from, for example, SOHO/MDI are sensitive to a mix of horizontal and vertical motions into the line-of-sight, in that for example the high-resolution mode of SOHO/MDI extends over a square of size ∼ 210 Mm, which is often also slightly off-set relative to solar disc center. Figure 23View Image illustrates these points.

The velocity spectrum shows that velocities are present over a large range of scales simultaneously, with larger scale motions decreasing in amplitude roughly in inverse proportion to the size. By using a Gaussian filter that leaves approximately the same number of effective resolution elements in picture that vary in size with factors of 2, 4, 8, and 16 one can show that, in addition to being nearly scale-free with respect to the amplitude spectrum, the patterns of motion are also very similar on different scales (cf. Figure 24View Image). The actual sizes of the panels are revealed in Figure 25View Image.

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Figure 25: Subsonic filtered line-of-sight velocities from SOHO, with the same order of panels as in Figure 24View Image, but without applying the Gaussian filter that was used there.

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