3.9 Vorticity

The overturning flow at the edges of granules produces horizontal vortex tubes at the interface between the granule and the intergranular lane. The equation for vorticity ω ≡ ∇ × u is obtained by taking the curl of the equation for the velocity (the equation for the momentum, Equation 6View Equation, divided by the density),
∂u-+ (u ⋅ ∇ )u = − 1∇P − ∇ Φ − 1∇ ⋅ τ , (39 ) ∂t ρ ρ visc
The result is (for the case of constant viscosity)
∂ω-+ (u ⋅ ∇ )ω = (ω ⋅ ∇ )u + 1-∇ ρ × ∇P + ν ∇2ω. (40 ) ∂t ρ2
From this equation we see that vorticity is generated where the density and pressure gradients are not parallel, which occurs where radiation transport effects are important, that is near the surface. At the mushroom heads of downdrafts, where there is a change in entropy, so that the density and pressure gradients are not parallel, ring vortices form (Figure 17View Image). These are connected back up to the surface by typically two, but sometimes more, trailing vortices (similar to those from the tips of airplane wings) (Figure 18Watch/download Movie). The equation for the vorticity also shows that existing vorticity is enhanced by stretching and compression, which occurs primarily in turbulent downflows, and it is diminished by expansion in upflows.

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Figure 18: mov-Movie (29987 KB) Magnitude of the vorticity (entrophy) in a single downdraft. Top of the image is the visible solar surface, bottom is 2.5 Mm below the surface. The vertical scale is stretched. Horizontal tickmarks are 237 km apart.
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Figure 19: Maximum horizontal and vertical flow Mach numbers as a function of depth.
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Figure 20: Mach numbers in horizontal planes (contours at Mach number = 1, 1.2, 1.4, 1.6) for vertical (top) and horizontal flow (bottom) superimposed on images of the vertical velocity at the surface (left) and 1 Mm below the surface (right) (velocity scale on right in  km s–1).

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