Critical balance and scaling of strongly stratified turbulence at low Prandtl number

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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 stably stratified regions of stars and gas giants. 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. $Pr\,Rb<1$ , 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 $Pr\,Rb\ll 1$ 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, $Fr_M\equiv Fr/(Pr\,Rb)^{1/4}$ . The geophysical and astrophysical regimes are thus smoothly connected across the $Pr\,Rb=1$ transition. Applications to vertical transport in stellar radiative zones and modification to the instability criterion for the small-scale dynamo are discussed.