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Register a new userDirect Calculation of the Turbulent Dissipation Efficiency in Anelastic Convection
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The current understanding of the turbulent dissipation in stellar convective zones is based on the assumption that the turbulence follows Kolmogorov scaling. This assumption is valid for some cases in which the time frequency of the external shear is high (e.g., solar p modes). However, for many cases of astrophysical interest (e.g., binary orbits, stellar pulsations, etc.), the timescales of interest lie outside the regime of applicability of Kolmogorov scaling. We present direct calculations of the dissipation efficiency of the turbulent convective flow in this regime, using simulations of anelastic convection with external forcing. We show that the effects of the turbulent flow are well represented by an effective viscosity coefficient, we provide the values of the effective viscosity as a function of the perturbation frequency and compare our results to the perturbative method for finding the effective viscosity of Penev et al. that can be applied to actual simulations of the surface convective zones of stars.
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We have adapted the anelastic spectral code of Barranco & Marcus (2006) to simulate a turbulent convective layer with the intention of studying the effectiveness of turbulent eddies in dissipating external shear (e.g. tides). We derive the anelastic equations, show the time integration scheme we use to evolve these equations and present the tests we ran to confirm that our code does what we expect. Further we apply a perturbative approach to find an approximate scaling of the effective eddy viscosity with frequency, and find that it is in general agreement with an estimate obtained by applying the same procedure to a realistic simulation of the upper layers of the solar convective zone.
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.
Earth-like planets have viscoelastic mantles, whereas giant planets may have viscoelastic cores. The tidal dissipation of such solid regions, gravitationally perturbed by a companion body, highly depends on their rheology and on the tidal frequency. Therefore, modelling tidal interactions presents a high interest to provide constraints on planets properties and to understand their history and their evolution, in our Solar System or in exoplanetary systems. We examine the equilibrium tide in the anelastic parts of a planet whatever the rheology, taking into account the presence of a fluid envelope of constant density. We show how to obtain the different Love numbers that describe its tidal deformation. Thus, we discuss how the tidal dissipation in solid parts depends on the planets internal structure and rheology. Finally, we show how the results may be implemented to describe the dynamical evolution of planetary systems. The first manifestation of the tide is to distort the shape of the planet adiabatically along the line of centers. Then, the response potential of the body to the tidal potential defines the complex Love numbers whose real part corresponds to the purely adiabatic elastic deformation, while its imaginary part accounts for dissipation. This dissipation is responsible for the imaginary part of the disturbing function, which is implemented in the dynamical evolution equations, from which we derive the characteristic evolution times. The rate at which the system evolves depends on the physical properties of tidal dissipation, and specifically on how the shear modulus varies with tidal frequency, on the radius and also the rheological properties of the solid core. The quantification of the tidal dissipation in solid cores of giant planets reveals a possible high dissipation which may compete with dissipation in fluid layers.
We extend the analysis of Penev et al. (2007) to calculate effective viscosities for the surface convective zones of three main sequence stars of 0.775Msun, 0.85Msun and the present day Sun. In addition we also pay careful attention to all normalization factors and assumptions in order to derive actual numerical prescriptions for the effective viscosity as a function of the period and direction of the external shear. Our results are applicable for periods that are too long to correspond to eddies that fall within the inertial subrange of Kolmogorov scaling, but no larger than the convective turnover time, when the assumptions of the calculation break down. We find linear scaling of effective viscosity with period and magnitudes at least three times larger than the Zahn (1966, 1989) prescription.
We consider the effect of stratification on systematic, large-scale flows generated in anelastic convection. We present results from three-dimensional numerical simulations of convection in a rotating plane layer in which the angle between the axis of rotation and gravity is allowed to vary. This model is representative of different latitudes of a spherical body. We consider two distinct parameter regimes: (i) weakly rotating and (ii) rapidly rotating. In each case, we examine the effect of stratification on the flow structure and heat transport properties focussing on the difference between Boussinesq and anelastic convection. Furthermore, we show that regimes (i) and (ii) generate very different large-scale flows and we investigate the role stratification has in modifying these flows. The stratified flows possess a net helicity not present in the Boussinesq cases which we suggest, when combined with the self-generated shear flows, could be important for dynamo action.
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