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Register a new userA Large Eddy Simulation of Turbulent Compressible Convection: Differential Rotation in the Solar Convection Zone
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Francis J. Robinson
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We present results of two simulations of the convection zone, obtained by solving the full hydrodynamic equations in a section of a spherical shell. The first simulation has cylindrical rotation contours (parallel to the rotation axis) and a strong meridional circulation, which traverses the entire depth. The second simulation has isorotation contours about mid-way between cylinders and cones, and a weak meridional circulation, concentrated in the uppermost part of the shell. We show that the solar differential rotation is directly related to a latitudinal entropy gradient, which pervades into the deep layers of the convection zone. We also offer an explanation of the angular velocity shear found at low latitudes near the top. A non-zero correlation between radial and zonal velocity fluctuations produces a significant Reynolds stress in that region. This constitutes a net transport of angular momentum inwards, which causes a slight modification of the overall structure of the differential rotation near the top. In essence, the {it thermodynamics controls the dynamics through the Taylor-Proudman momentum balance}. The Reynolds stresses only become significant in the surface layers, where they generate a weak meridional circulation and an angular velocity `bump.
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We study the effect of turbulent drift of a large-scale magnetic field that results from the interaction of helical convective motions and differential rotation in the solar convection zone. The principal direction of the drift corresponds to the direction of the large-scale vorticity vector. Thus, the effect produces a latitudinal transport of the large-scale magnetic field in the convective zone wherever the angular velocity has a strong radial gradient. The direction of the drift depends on the sign of helicity and it is defined by the Parker-Yoshimura rule. The analytic calculations are done within the framework of mean-field magnetohydrodynamics using the minimal tau-approximation. We estimate the magnitude of the drift velocity and find that it can be several m/s near the base of the solar convection zone. The implications of this effect for the solar dynamo are illustrated on the basis of an axisymmetric mean-field dynamo model with a subsurface shear layer. We find that the helicity--vorticity pumping effect can have an influence on the features of the sunspot time--latitude diagram, producing a fast drift of the sunspot activity maximum at the rise phase of the cycle and a slow drift at the decay phase of the cycle.
A mean-field theory of differential rotation in a density stratified turbulent convection has been developed. This theory is based on a combined effect of the turbulent heat flux and anisotropy of turbulent convection on the Reynolds stress. A coupled system of dynamical budget equations consisting in the equations for the Reynolds stress, the entropy fluctuations and the turbulent heat flux has been solved. To close the system of these equations, the spectral tau approach which is valid for large Reynolds and Peclet numbers, has been applied. The adopted model of the background turbulent convection takes into account an increase of the turbulence anisotropy and a decrease of the turbulent correlation time with the rotation rate. This theory yields the radial profile of the differential rotation which is in agreement with that for the solar differential rotation.
Investigation of the turbulent properties of solar convection is extremely important for understanding the multi-scale dynamics observed on the solar surface. In particular, recent high-resolution observations have revealed ubiquitous vortical structures, and numerical simulations have demonstrated links between vortex tube dynamics and magnetic field organization and have shown the importance of vortex tube interactions in the mechanisms of acoustic wave excitation on the Sun. In this paper we investigate the mechanisms of the formation of vortex tubes in highly-turbulent convective flows near the solar surface by using realistic radiative hydrodynamic LES simulations. Analysis of data from the simulations indicates two basic processes of vortex tube formation: 1) development of small-scale convective instability inside convective granules, and 2) a Kelvin-Helmholtz type instability of shearing flows in intergranular lanes. Our analysis shows that vortex stretching during these processes is a primary source of generation of small-scale vorticity on the Sun.
The development of 2D and 3D simulations of solar convection has lead to a picture of convection quite unlike the usually assumed Kolmogorov spectrum turbulent flow. We investigate the impact of this changed structure on the dissipation properties of the convection zone, parametrized by an effective viscosity coefficient. We use an expansion treatment developed by Goodman & Oh 1997, applied to a numerical model of solar convection (Robinson et al. 2003) to calculate an effective viscosity as a function of frequency and compare this to currently existing prescriptions based on the assumption of Kolmogorov turbulence (Zahn 1966, Goldreich & Keeley 1977). The results match quite closely a linear scaling with period, even though this same formalism applied to a Kolmogorov spectrum of eddies gives a scaling with power-law index of 5/3.
The existence of large-scale dynamos in rigidly rotating turbulent convection without shear is studied using three-dimensional numerical simulations of penetrative rotating compressible convection. We demonstrate that rotating convection in a Cartesian domain can drive a large-scale dynamo even in the absence of shear. The large-scale field contains a significant fraction of the total field in the saturated state. The simulation results are compared with one-dimensional mean-field dynamo models where turbulent transport coefficients, as determined using the test field method, are used. The reason for the absence of large-scale dynamo action in earlier studies is shown to be due to the rotation being too slow: whereas the alpha-effect can change sign, its magnitude stays approximately constant as a function of rotation, and the turbulent diffusivity decreases monotonically with increasing rotation. Only when rotation is rapid enough a large-scale dynamo can be excited. The one-dimensional mean-field model with dynamo coefficients from the test field results predicts reasonably well the dynamo excitation in the direct simulations. This result further validates the test field procedure and reinforces the interpretation that the observed dynamo is driven by a turbulent alpha-effect. This result demonstrates the existence of an alpha-effect and an alpha^2-dynamo with natural forcing.
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