The Emergence of Solar Supergranulation as a Natural Consequence of Rotationally-Constrained Interior Convection

We investigate how rotationally-constrained, deep convection might give rise to supergranulation, the largest distinct spatial scale of convection observed in the solar photosphere. While supergranulation is only weakly influenced by rotation, larger spatial scales of convection sample the deep convection zone and are presumably rotationally influenced. We present numerical results from a series of nonlinear, 3-D simulations of rotating convection and examine the velocity power distribution realized under a range of Rossby numbers. When rotation is present, the convective power distribution possesses a pronounced peak, at characteristic wavenumber $\ell_\mathrm{peak}$, whose value increases as the Rossby number is decreased. This distribution of power contrasts with that realized in non-rotating convection, where power increases monotonically from high to low wavenumbers. We find that spatial scales smaller than $\ell_\mathrm{peak}$ behave in analogy to non-rotating convection. Spatial scales larger than $\ell_\mathrm{peak}$ are rotationally-constrained and possess substantially reduced power relative to the non-rotating system. We argue that the supergranular scale emerges due to a suppression of power on spatial scales larger than $\ell\approx100$ owing to the presence of deep, rotationally-constrained convection. Supergranulation thus represents the largest non-rotationally-constrained mode of solar convection. We conclude that the characteristic spatial scale of supergranulation bounds that of the deep convective motions from above, making supergranulation an indirect measure of the deep-seated dynamics at work in the solar dynamo. Using the spatial scale of supergranulation in conjunction with our numerical results, we estimate an upper bound of 10 m s$^{-1}$ for the Sun's bulk rms convective velocity.