Turbulent Compressible Convection with Rotation. II. Mean Flows and Differential Rotation

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Abstract
The e †ects of rotation on turbulent, compressible convection within stellar envelopes are studied through three-dimensional numerical simulations conducted within a local f-plane model. This work seeks to understand the types of di †erential rotation that can be established in convective envelopes of stars like the Sun, for which recent helioseismic observations suggest an angular velocity proÐle with depth and latitude at variance with many theoretical predictions. This paper analyzes the mechanisms that are responsible for the mean (horizontally averaged) zonal and meridional Ñows that are produced by convection inÑuenced by Coriolis forces. The compressible convection is considered for a range of Rayleigh, Taylor, and Prandtl (and thus Rossby) numbers encompassing both laminar and turbulent Ñow conditions under weak and strong rotational constraints. When the nonlinearities are moderate, the e †ects of rotation on the resulting laminar cellular convection leads to distinctive tilts of the cell boundaries away from the vertical. These yield correlations between vertical and horizontal motions that generate Reynolds stresses that can drive mean Ñows, interpretable as di †erential rotation and meridional circulations. Under more vigorous forcing, the resulting turbulent convection involves complicated and contorted Ñuid particle trajectories, with few clear correlations between vertical and horizontal motions, punctuated by an evolving and intricate downÑow network that can extend over much of the depth of the layer. Within such networks are some coherent structures of vortical downÑow that tend to align with the rotation axis. These yield a novel turbulent alignment mechanism, distinct from the laminar tilting of cellular boundaries, that can provide the principal correlated motions and thus Reynolds stresses and subsequently mean Ñows. The emergence of such coherent structures that can persist amidst more random motions is a characteristic of turbulence with symmetries broken by rotation and stratiÐcation. Such structure is here found to play a crucial role in deÐning the mean zonal and meridional Ñows that coexist with the convection. Though they are subject to strong inertial oscillations, the strength and type of the mean Ñows are determined by a combination of the laminar tilting and the turbulent alignment mechanisms. Varying the parameters produces a wide range of mean motions. Among these, some turbulent solutions exhibit a mean zonal velocity proÐle that is nearly constant with depth, much as deduced by helioseismology at midlatitudes within the Sun. The solutions exhibit a deÐnite handedness, with the direction of the persistent mean Ñows often prescribing a spiral with depth near the boundaries, also in accord with helioseismic deductions. The mean helicity has a proÐle that is positive in the upper portion of the domain and negative in the lower portion, a property bearing on magnetic dynamo processes that may be realized within such rotating layers of turbulent convection.

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