Motions in the convection zone of our sun which are of supergranular or larger scale will be influenced by the solar rotation and may lead to a redistribution of angular momentum observable as a `differential rotation'. Current models of global-scale convection and actual solar observations from helioseismology show a disparity in the differential rotation profiles that they predict: simulations of convection in spherical shells show angular velocity to be constant on cylinders, whereas present inversions of solar $p$ mode frequency-splitting observations suggest that angular velocity is constant on radii at least at mid-latitudes. We extend previous numerical calculations from the previously studied laminar regime to turbulent regimes in an attempt to reconcile these differences. To achieve sufficient resolution to allow a turbulent state, we restrict the geometry to a local area f-plane model, and study the ensuing turbulent convection in a slab of perfect gas positioned at various latitudes on the sphere. Cases at large supercriticality (Ra < 10^7, Pr > 0.1), and with fast rotation (Ta < 10^7) have been studied with high resolution (up to 256^2 x 130).
As in the non-rotating case, a seemingly laminar surface network of downflows is established resembling solar granulation, although this thermal boundary layer effect still serves to disguise a turbulent interior punctuated by vertically-coherent structures. However, with the addition of significant rotation, the temporal coherence of the upper network is lost, with increased vorticity and inertial action on large scale flows providing strong mechanisms for destruction and creation of new cells. The ubiquitous characterisation of turbulence as vortex tubes reveals a much richer set of interactions in the rotating case: vortex sheets are wrapped up into horizontal tubes which entwine and entangle with strong vertically-coherent tubes. Such vortex interactions between dowflowing, and therefore cyclonic, plumes leads to retarded vertical mixing. In contrast, linear transfer via the Coriolis terms serves to isotropise the small scale motions, equilibrating the vertical and horizontal components of energy and enstrophy. Strong rotational influence (Ro < 1) on turbulent motions leads to alignment of structures within the flow with the axis of rotation in a Taylor-Proudman-like manner. If the rotation vector is not aligned with gravity, then the resultant tilted structures provide correlations between the vertical and horizontal velocities and thus Reynolds stresses which generate mean zonal and meridional shear flows. These flows may be significant, may exhibit observed features such as the spiralling of the mean vector with depth, and in general transport zonal and meridional momentum downwards in the interior, with substantial transport by the turbulent motions. Noteworthy is the lack of strong mean flows in highly turbulent but modest rotation cases and a constant zonal mean in the interior of strongly rotationally-constrained cases, akin to the differential rotation profiles of helioseismology.
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