Reynolds stress and heat flux in spherical shell convection

Abstract

Context. Turbulent fluxes of angular momentum and enthalpy or heat due to rotationally affected convection play a key role in determining differential rotation of stars. Their dependence on latitude and depth has been determined in the past from convection simulations in Cartesian or spherical simulations. Here we perform a systematic comparison between the two geometries as a function of the rotation rate. Aims. Here we want to extend the earlier studies by using spherical wedges to obtain turbulent angular momentum and heat transport as functions of the rotation rate from stratified convection. We compare results from spherical and Cartesian models in the same parameter regime in order to study whether restricted geometry introduces artefacts into the results. In particular, we want to clarify whether the sharp equatorial profile of the horizontal Reynolds stress found in earlier Cartesian models is also reproduced in spherical geometry. Methods. We employ direct numerical simulations of turbulent convection in spherical and Cartesian geometries. In order to alleviate the computational cost in the spherical runs, and to reach as high spatial resolution as possible, we model only parts of the latitude and longitude. The rotational influence, measured by the Coriolis number or inverse Rossby number, is varied from zero to roughly seven, which is the regime that is likely to be realised in the solar convection zone. Cartesian simulations are performed in overlapping parameter regimes. Results. For slow rotation we find that the radial and latitudinal turbulent angular momentum fluxes are directed inward and equatorward, respectively. In the rapid rotation regime the radial flux changes sign in accordance with earlier numerical results, but in contradiction with theory. The latitudinal flux remains mostly equatorward and develops a maximum close to the equator. In Cartesian simulations this peak can be explained by the strong "banana cells". Their effect in the spherical case does not appear to be as large. The latitudinal heat flux is mostly equatorward for slow rotation but changes sign for rapid rotation. Longitudinal heat flux is always in the retrograde direction. The rotation profiles vary from anti-solar (slow equator) for slow and intermediate rotation to solar-like (fast equator) for rapid rotation. The solar-like profiles are dominated by the Taylor-Proudman balance.

Meridional Circulation in Solar and Stellar Convection Zones

“…As the rotation rate of the system decreases, the high-latitude cell extends to lower latitudes, ultimately supplanting the counter cell in the equatorial regions for rotation rates lower than  Ω . The shift from multi-cellular MC profiles in the solar regime to single-celled profiles in the antisolar regime has also been noted by other authors (Guerrero et al 2013a;Gastine et al 2014;Käpylä et al 2014). …”

“…Equatorial and polar rotation rates were calculated by averaging zonal velocity in both longitude and depth, and then over a latitude range of 0°N-20°N latitude and 70°N-90°N, respectively. Anti-solar cases tend to arise when Ro is greater than about 0.163. case B0.5 is just straddling the transition between the solar-like and anti-solar regimes of DR. Gastine et al (2014) and Käpylä et al (2014) have recently shown that the behavior of the DR near the solar/anti-solar transition can depend on the history of the simulation, representing a type of hysteresis. For a given transitional Ro, the simulation may exhibit either solar or anti-solar DR.…”

The existence of the Λ effect in the solar convection zone as indicated by SDO/HMI data

“…We have shown earlier by means of quasilinear turbulence theory that the function V is negative and exists mainly in the supergranulation layer, while H is positive and exists mainly in the bulk of the convection zone. Numerical simulations provide very similar results (Chan 2001;Käpylä et al 2011; for more details see Rüdiger et al 2013). We also know that the positive quantity H in (5) and (6) both contribute to the equatorial acceleration so that it is not trivial to find the sign of (4) as a function of depth.…”

“…Inserting (2) and the second term in (1) into (3), one finds Ω dependent on r but independent of θ. For positive radial shear in the rotation law, on the other hand, a meridional flow toward the equator is induced at the surface, which accelerates the equator by transporting angular momentum (Kippenhahn 1963). Such a meridional flow, however, has not been observed so far.…”

The Solar-Stellar Connection

“…Authors of these studies have been able to lower the diffusivity level low enough to yield more time dependent dynamo solutions possessing both cyclic behavior, regular butterfly diagram and torsional oscillation-like behavior (Käpylä et al 2011(Käpylä et al , 2012Dubé and Charbonneau 2013;Beaudoin et al 2013;Augustson et al 2013). Some have even obtained for the first time buoyant magnetic wreaths (Nelson et al 2011(Nelson et al , 2013.…”