On the inference of the state of turbulence and mixing efficiency in stably stratified flows

Scaling arguments are presented to quantify the widely used diapycnal (irreversible) mixing coefficient [STIX]x1D6E4=[STIX]x1D716PE/[STIX]x1D716 in stratified flows as a function of the turbulent Froude number Fr=[STIX]x1D716/Nk . Here, N is the buoyancy frequency, k is the turbulent kinetic energy, [STIX]x1D716 is the rate of dissipation of turbulent kinetic energy and [STIX]x1D716PE is the rate of dissipation of turbulent potential energy. We show that for Fr 1 , [STIX]x1D6E4∝ Fr⁻² , for Fr~ O(1) , [STIX]x1D6E4∝ Fr⁻¹ and for Fr 1 , [STIX]x1D6E4∝ Fr⁰ . These scaling results are tested using high-resolution direct numerical simulation (DNS) data from three different studies and are found to hold reasonably well across a wide range of $Fr$ that encompasses weakly stratified to strongly stratified flow conditions. Given that the $Fr$ cannot be readily computed from direct field measurements, we propose a practical approach that can be used to infer the $Fr$ from readily measurable quantities in the field. Scaling analyses show that Fr∝ (LT/LO)⁻² for LT/LO>O(1) , Fr∝ (LT/LO)⁻¹ for LT/LO~ O(1) , and Fr∝ (LT/LO)-2/3 for LT/LO<O(1) , where LT is the Thorpe length scale and LO is the Ozmidov length scale. These formulations are also tested with DNS data to highlight their validity. These novel findings could prove to be a significant breakthrough not only in providing a unifying (and practically useful) parameterization for the mixing efficiency in stably stratified turbulence but also for inferring the dynamic state of turbulence in geophysical flows.