Stably stratified turbulence at low Prandtl number in stellar interiors

We extend the scaling relations of strongly (stably) stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stars and gas giants. Such turbulence can be driven by e.g. dynamical shear instabilities in radiative zones of stars. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. PrRb<1, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of the Boussinesq equations in the PrRb≪1 regime. We find that all the standard scaling relations from the unity Pr limit of strongly stratified turbulence simply carry over if the Froude number, Fr, is replaced by a modified Froude number, Fr_M≡Fr/(PrRb)^1/4. The geophysical and astrophysical regimes are thus smoothly connected across the PrRb=1 transition. Applications to vertical transport in stellar radiative zones via turbulent diffusion is discussed.