3.3 Scale selection

Because of the enormous variation of mass density with depth, conservation of mass plays a central role in determining the convective flow patterns; whenever a fluid parcel moves up or down over a density scale height it must expand or contract with a factor of e. This constraint dictates the scales on which the dominant convective motions occur.

The upper layers of the solar convection zone are indeed highly stratified, with density scale heights that are smaller than the typical horizontal dimensions of granules. The rapid decrease in density with height leads to a marked asymmetry in the convective flows. Fluid moving upward into lower density layers cannot carry all its mass higher. Hence, upward moving fluid must diverge and most of it turns over within a scale height. Downward moving fluid on the other hand is moving into denser regions and gets compressed as it descends. As a result, diverging upflows have their fluctuations smoothed out, while converging downflows have their’s increased and become turbulent. Except within 100 km of the surface, upflows occupy about 2/3 and downflows about 1/3 of the area.

The typical size of granules, and other convective cells deeper inside the convection zone, is set by mass conservation and is a few times the local scale height. To conserve mass, most of the upflow through a convective cell at any given depth must flow out through the sides of the cell over approximately a scale height, H. Hence, approximating the convective cell by a circular cylinder of radius r and height H,

2 πr ρvz ≈ 2 πrH ρvH . (36 )
Thus the horizontal cell size must be of order,
r = 2H (vH ∕vz) . (37 )
A certain minimum vertical velocity is needed to transport sufficient energy to the surface to balance the radiative losses. This can be estimated by balancing the radiative loss from the surface, the solar flux, with the enthalpy flux,
( 5 ) σT 4eff ≈ ρVz -kT + xχ , (38 ) 2
where χ is the hydrogen ionization potential and x is the hydrogen ionization fraction. For an order of magnitude estimate for the required vertical velocity near the surface, assume x ≈ 0.1. This gives a velocity of ∼ 2 km s–1 for the minimum velocity needed to supply the radiative losses from the surface (Nordlund and Stein, 1991Jump To The Next Citation Point). Since the vertical velocity cannot decrease below this value if the granule is to remain bright, the horizontal expansion velocity must increase as the granule size increases to balance the greater volume of fluid being brought to the surface through the larger granule area. However, the horizontal velocities cannot much exceed the sound speed of ≈ 7 km s–1 at the surface. Hence, an upper limit to the size of granules is of order 2r ∼ 4 Mm (H ∼ 300 km near the surface) (Nelson and Musman, 1978Nordlund, 1978). As a result of the increasing scale height with depth, due to the increasing temperature with depth, the scales of motion increase continually with depth. For simulations extending 20 Mm below the surface, the convective cellular structures reach sizes of about 20 – 30 Mm (Figure 6View Image). There is no special feature corresponding to mesogranular cells (although that scale is clearly visible in the slice at 4 Mm depth) or the supergranular scale (visible in slices from a depth of about 8 Mm).

The larger scales flow patterns are in a sense also driven by the surface cooling, albeit in a more indirect manner. Rising plasma expands and spreads outwards, and over each density scale height a large fraction turns over and joins the downflowing plasma. Even from a depth of just a few Mm, only a tiny fraction of the ascending flow reaches the surface and is cooled by radiation cooling; the rest turns over and is only cooled by mixing with downflowing material (Stein and Nordlund, 1989Jump To The Next Citation Point). Conversely, descending flows from the surface are being entrained by adiabatic overturning fluid, and turbulent mixing in the downdrafts thus reduces the entropy contrast gradually with depth (cf. Figure 5View Image).

The hierarchy of horizontal scales associated with the gradual increase of scales with depth implies that smaller scales are being advected horizontally by larger scale patterns, and the descending fluid thus forms a hierarchical structure of merging downdrafts (Figure 7Watch/download Movie). In this hierarchy the granular (surface) scales are driven directly by radiative cooling while the larger scales are driven by the entrained and mixed lower entropy contrast plasma further down in the hierarchy.

On the average, the plasma at any one level has a slightly superadiabatic stratification, which is a mix of almost perfectly isentropic ascending plasma and cooler descending plasma. Formally, this average structure is convectively unstable according to the Schwarzschild criterion (Schwarzschild, 1958), and one may regard the larger scales as being driven by the corresponding convective instability; a slight perturbation at larger scales is self amplifying, since the diverging upflow sweeps smaller scales downdrafts together, thus amplifying the perturbation (Stein and Nordlund, 1989Jump To The Next Citation PointRast, 2003Jump To The Next Citation Point).

View Image

Figure 6: Horizontal slices of vertical velocity (light is downward, dark upward) at depths of 0, 2, 4, 8, 12, and 16 Mm (across then down). Each image is 48 × 48 Mm. Note the gradual and continuously increasing area of the upflows, as outlined by the surrounding downflow lanes, with increasing depth.

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Figure 7: mpg-Movie (37871 KB) Vertical velocity (red upward, blue downward) and streaklines (seen more clearly in the movie) in a vertical cut through the 24 Mm wide simulation domain. Diverging upflows sweep downflows toward each other at the boundaries of the larger, deeper lying upflows (movie by Chris Henze, NASA Advanced Supercomputing Division, Ames Research Center).

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