Kolmogorov or Bolgiano-Obukhov scaling: Universal energy spectra in stably stratified turbulent fluids

Abstract

We set up the scaling theory for stably stratified turbulent fluids. For a system having infinite extent in the horizontal directions, but with a finite width in the vertical direction, this theory predicts that the inertial range can display three possible scaling behavior, which are essentially parametrized by the buoyancy frequency N, or dimensionless horizontal Froude number F_{h}, and the vertical length scale l_{v} that sets the scale of variation of the velocity field in the vertical direction for a fixed Reynolds number. For very low N or very high Re_{b} or F_{h}, and with l_{v}≫l_{h}, the typical horizontal length scale, buoyancy forces are irrelevant and hence, unsurprisingly, the kinetic energy spectra show the well-known K41 scaling in the inertial range. In this regime, the local temperature behaves as a passively advected scalar, without any effect on the flow fields. For intermediate ranges of values of N,F_{h}∼O(1), corresponding to moderate stratification, buoyancy forces are important enough to affect the scaling. This leads to the Bolgiano-Obukhov scaling which is isotropic, when l_{v}∼l_{h}. Finally, for very large N, corresponding to strong stratification, together with a very small l_{v}, the inertial-range flow fields effectively two-dimensionalize. The kinetic energy spectra are predicted to be anisotropic with only the horizontal part of the kinetic energy spectra following the K41 scaling. This suggests an intriguing re-entrant K41 scaling, as a function of stratification, for the horizontal components of the velocity field in this regime. The scaling theory further predicts the scaling of the thermal energy in each of these three scaling regimes. Our theory can be tested in large-scale simulations and appropriate laboratory-based experiments.