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

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 ${\mathrm{Re}}_b$ or $F_h$, and with $l_v\gg l_h$, 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 $N,F_h\sim O\left(1\right)$, 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 $N$, 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.

Figure 1

Schematic state diagram of stably stratified turbulence in the $L_o-l_v$ plane. The broken straight line indicates $l_v=L_o$. Three distinct scaling regimes are marked: The green patch at the top corresponds to weak stratification with 3D isotropic K41 scaling for the kinetic energy spectra, the red patch in the middle corresponds to moderate stratification with isotropic BO scaling for the kinetic energy spectra, and the blue patch corresponds to strong stratification with 2D K41 scaling for the in-plane part of the kinetic energy spectra. These regions are schematically drawn, have no sharp boundaries, and are surrounded by crossover regions (see text).

Figure 2

Schematic diagram showing scaling regimes of the kinetic energy spectra as functions of wave vector $k$: Regions marked (a), (b), and (c), respectively, are 3D anisotropic K41 scaling, the buoyancy-dominated scaling regime, and 3D anisotropic K41 scaling for the kinetic energy. The buoyancy-dominated region marked (b) can display 3D BO scaling for the kinetic energy spectra or 2D K41 scaling for the in-plane part of the kinetic energy spectra with an effective 2D flow field in the inertial range, depending on the strength of the imposed stratification (see text). Here $L$ is the system size in the horizontal plane.