Mixing and transport in stars - I. Formalism: momentum, heat and mean molecular weight

Details Mixing and transport in stars - I. Formalism: momentum, heat and mean molecular weight Mixing and transport in stars - I. Formalism: momentum, heat and mean molecular weight

Dec 2001

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001mnras.328..829c&link_type=abstract

Monthly Notices of the Royal Astronomical Society, Volume 328, Issue 3, pp. 829-838.

Astronomy and Astrophysics

Astronomy

8

Turbulence, Stars: Rotation

Scientific paper

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 Rij, in particular Rφr. It is shown that the latter has the form Rrφ=-2DsSφr- 2DvVφr-D0Ω0- D1Ω+..., where 2Sφr=sinθr∂Ω/∂r is the shear and 2rVφr=sinθ∂(r2Ω)/∂r is the vorticity. The dots indicate buoyancy and meridional currents. The forms of the turbulent diffusivities entering the shear part Ds, vorticity part Dv, rigid rotation Ω0 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 ∂t(r2Ω)=r-2∂r r4Ds ∂Ω/∂r+r-2∂r r2Dv ∂r (r2Ω)+.... (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) ∇+Khχ-1(∇- ∇ad)+∇Ω=∇r, where the new term is given by ∇Ω=(Hp/cpχT)Rrφūφ. (iii) Tensorial diffusivities. The turbulent flux of a scalar φ (like T and μ) is shown to have the form Jiφ=- Dijφ∂Φ xj, where the Dij 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 Dij has an immediate consequence: the symmetric part Dijs gives rise to mixing (by diffusion) while the antisymmetric part Dija gives rise to advection which cannot be represented by a diffusion coefficient. The equation describing a mean scalar field Φ is therefore ∂Φ∂t +(ū+u*).∇Φ=partxi Dijs∂Φ∂xj ui*=∂xjDija. Thus, even without a mean velocity field ū, 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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