## Publication Abstracts

### Canuto and Minotti 2001

**328**, 829-838, doi:10.1046/j.1365-8711.2001.04911.x.

The goal of this paper is to derive analytic expressions for the turbulent fluxes of momentum (Reynolds stresses), heat and mean molecular weight.

(i) Angular momentum. To solve the angular momentum equation one needs to know the Reynolds stresses R_{ij}, in particular R_{φr}. It is shown that the latter has the form R_{rφ} = -2D_{S}S_{φr} - 2D_{v}V_{φr} - D_{0}Ω_{0} - D_{1}Ω + ..., where 2S_{φr} = sinθrδΩ/δr is the shear and 2rV_{φr} = sinθδ(r^{2}Ω)/δr is the vorticity. The dots indicate buoyancy and meridional currents. The forms of the turbulent diffusivities entering the shear part D_{s}, vorticity part D_{v}, rigid rotation Ω_{0} and differential rotation Ω ≡ Ω(r,θ) are also derived. Previous models have only the shear term. The vorticity term gives rise to a true diffusion-like equation for the angular momentum which now reads δ(r^{2}Ω)/δt = r^{-2}δ(r^{4}D_{s}δΩ/δr)/δr + r^{-2}δ[r^{2}D_{v}δ(r^{2}Ω)]δr + ...

(ii) Mean temperature equation. Differential rotation alters the mean temperature equation. In the stationary case, the new flux conservation law reads (χ is the radiative diffusivity) ∇ + K_{h}χ^{-1}(∇ - ∇_{ad}) + ∇_{Ω} = ∇_{r}, where the new term is given by ∇_{Ω} = (H_{p}/c_{p}χT)R_{rφ}ū_{φ}.

(iii) Tensorial diffusivities. The turbulent flux of a scalar φ (like T and μ) is shown to have the form J_{i}_{φ} = -D_{ij}_{φ} δΦ/δx_{j}, where the D_{ij} are tensorial diffusivities. They are shown to be functions of the external source of energy (e.g. flux of gravity waves), rigid-body rotation, differential rotation, meridional currents, T-μ gradients and Peclet number Pe which characterizes the role of radiative losses.

(iv) Mixing and advection. The tensorial nature of the diffusivities D_{ij} has an immediate consequence: the symmetric part D_{ij}^{s} gives rise to mixing (by diffusion) while the antisymmetric part D_{ij}^{a} gives rise to advection which cannot be represented by a diffusion coefficient. The equation describing a mean scalar field Φ is therefore δΦ/δt + (ū + u^{*})·∇Φ = δ(D_{ij}^{s}δΦ/δx_{j})δx_{i}, u_{i}^{*} = δD_{ij}^{a}/δx_{j}. Thus, even without a mean velocity field u, there is an advective term u^{*} arising from turbulence alone. The advective nature of turbulence was not accounted for in previous studies which have therefore underestimated the full potential of turbulent motion.

(v) Peclet number dependence. Radiative losses are an important part of the physical picture, for they weaken the temperature gradient, and thus reduce the effect of stable stratification and ultimately enhance mixing. The Peclet number dependence is accounted for in the model.

(vi) Shear-induced versus wave-induced mixing. In this formalism, the dichotomy between the two processes no longer exists, since we show that the flux of gravity waves, treated as an external source of energy, is a natural ingredient of the formalism.

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#### BibTeX Citation

@article{ca08900n, author={Canuto, V. M. and Minotti, F.}, title={Mixing and transport in stars. I. Formalism: Momentum, heat and mean molecular weight}, year={2001}, journal={Monthly Notices of the Royal Astronomical Society}, volume={328}, pages={829--838}, doi={10.1046/j.1365-8711.2001.04911.x}, }

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#### RIS Citation

TY - JOUR ID - ca08900n AU - Canuto, V. M. AU - Minotti, F. PY - 2001 TI - Mixing and transport in stars. I. Formalism: Momentum, heat and mean molecular weight JA - Mon. Not. Roy. Astron. Soc. JO - Monthly Notices of the Royal Astronomical Society VL - 328 SP - 829 EP - 838 DO - 10.1046/j.1365-8711.2001.04911.x ER -

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