[Paper Review] Anisotropic diffusion and shear instabilities

S. Talon, J. P. Zahn|arXiv (Cornell University)|Sep 2, 1996

Fluid Dynamics and Turbulent Flows3 references20 citations

TL;DR

This paper investigates how anisotropic turbulence—characterized by strong horizontal diffusivity ($D_h$) and weak vertical diffusivity ($D_v$)—reduces the stabilizing effect of molecular weight gradients in stably stratified stellar interiors. By modifying the classical Richardson criterion to include horizontal turbulent diffusion, the authors derive a revised expression for vertical turbulent diffusivity ($D_v$), showing that shear instabilities can be triggered at significantly lower shear rates than predicted by standard models.

ABSTRACT

We examine the role of anisotropic turbulence on the shear instabilities in a stratified flow. Such turbulence is expected to occur in the radiative interiors of stars, due to their differential rotation and their strong stratification, and the turbulent transport associated with it will be much stronger in the horizontal than in the vertical direction. It will thus weaken the restoring force which is caused by the gradient of mean molecular weight ($μ$). We find that the critical shear which is able to overcome the $μ$-gradient is substantially reduced by this anisotropic turbulence, and we derive an estimate for the resulting turbulent diffusivity in the vertical direction.

Motivation & Objective

• To understand how anisotropic turbulence in differentially rotating, stably stratified stellar interiors reduces the stabilizing effect of molecular weight gradients.
• To modify the classical Richardson criterion to account for horizontal turbulent diffusion, which weakens the restoring force from compositional stratification.
• To derive a quantitative estimate for the effective vertical turbulent diffusivity ($D_v$) resulting from shear instabilities in the presence of anisotropic turbulence.
• To assess the implications of this mechanism for the mixing of chemical elements in radiative zones of stars, particularly in rapidly rotating O and B stars.

Proposed method

• Adapts the classical Richardson criterion for shear instability by incorporating horizontal turbulent diffusivity ($D_h$) as a key parameter that reduces the effective stabilizing influence of the molecular weight gradient.
• Uses a mixing-length-type formalism to model turbulent eddies, introducing a dimensionless parameter $\Gamma = v\ell / 6K$ for thermal diffusion and $\Gamma_\mu = v\ell / 6D_h$ for compositional diffusion.
• Derives a modified instability criterion: $\left(\frac{\Gamma}{\Gamma+1}\right)N^{2}_{T} + \left(\frac{\Gamma_{\mu}}{\Gamma_{\mu}+1}\right)N^{2}_{\mu} \leq Ri_c \left(\frac{dU}{dz}\right)^2$, which accounts for turbulent erosion of compositional stratification.
• Applies the critical Reynolds number condition ($Re_c \approx 10$) to constrain the size and lifetime of turbulent eddies, ensuring they are not damped by viscosity.
• Solves the resulting second-order equation for $v\ell$ under the condition of maximum vertical transport, leading to an expression for $D_v \simeq v\ell / 3$.
• Derives the final formula for vertical diffusivity: $D_v \simeq \frac{2Ri_c (dU/dz)^2}{N^{2}_{T}/(K+D_h) + N^{2}_{\mu}/D_h}$, valid in the limit of mild turbulence ($x \ll K$).

Experimental results

Research questions

• RQ1 How does anisotropic turbulence with strong horizontal diffusivity ($D_h$) affect the stability of shear instabilities in stably stratified stellar interiors?
• RQ2 To what extent does horizontal turbulent diffusion reduce the stabilizing influence of molecular weight gradients on shear instabilities?
• RQ3 What is the resulting effective vertical turbulent diffusivity ($D_v$) when horizontal turbulence weakens the compositional stabilizing term?
• RQ4 How does the modified Richardson criterion, including turbulent diffusion, alter the critical shear required to trigger instability?
• RQ5 Can this mechanism explain the observed abundance anomalies in O and B stars, particularly in relation to rotation and chemical mixing?

Key findings

• Anisotropic turbulence with $D_h \gg D_v$ significantly reduces the stabilizing effect of molecular weight gradients, lowering the threshold for shear instability onset.
• The critical Richardson number is effectively reduced due to turbulent erosion of the $\mu$-gradient, allowing instability to occur at lower shear rates than predicted by the standard criterion.
• The derived expression for vertical diffusivity is $D_v \simeq \frac{2Ri_c (dU/dz)^2}{N^{2}_{T}/(K+D_h) + N^{2}_{\mu}/D_h}$, which depends on both thermal and compositional stratification and the horizontal diffusivity.
• Solutions exist only when the shear rate exceeds a threshold given by $\left(\frac{dU}{dz}\right)^2 \geq \frac{\nu Re_c}{6Ri_c}\left[\frac{N^{2}_{T}}{K+D_h} + \frac{N^{2}_{\mu}}{D_h}\right]$, which is substantially weaker than the classical condition.
• The model predicts that in rapidly rotating stars, this mechanism can lead to partial mixing, consistent with observed abundance anomalies in O and B stars.
• The method accounts for turbulent eddy size and lifetime via the Reynolds number constraint ($Re_c \approx 10$), ensuring physical consistency with viscous damping.