Kolmogorov or Bolgiano-Obukhov: Universal scaling 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 behaviour, which are essentially parametrised 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 $l_v$ being of the same order as $l_h$, the typical horizontal length scale, buoyancy forces are irrelevant and hence, unsurprisingly, the kinetic energy spectra shows 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 $N,\,F_h\sim {\cal O}(1)$, corresponding to moderate stratification, buoyancy forces are important enough to affect the scaling. This leads to the Bolgiano-Obukhov scaling which is isotropic, when $l_v\sim l_h$. Finally, for very large $N$ or equivalently for vanishingly small $F_h,\,L_o$, corresponding to strong stratification, together with a very small $l_v$, the system effectively two-dimensionalise; the kinetic energy spectrum is predicted to be anisotropic with only the horizontal part of the kinetic energy spectra follows the K41 scaling, suggesting an intriguing {\em re-entrant} K41 scaling, as a function of stratification, for ${\bf v}_\perp$ 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.