Nonlinear formation of structures in rotating stratified turbulence

Nonlinear approaches such as direct numerical simulations (DNS) yield a characteristic structuring in homogeneous turbulence as a result of modified dynamics under the effect of rotation and stable stratification. The structures are elongated for dominant rotation or flat when stratification dominates. Nonlinearity is essential for constructing these anisotropic Eulerian features, which we quantify by single and two-point second-order statistics. Linear approaches such as rapid distorsion theory (RDT) and kinematic simulations (KS) do not reproduce these effects at all. However, when looking at Lagrangian statistics, both linear and nonlinear models seem to yield very similar anisotropic trends. In order to investigate this paradox, we consider statistically homogeneous turbulence with vertical stable stratification characterized by the Brunt–Väisälä frequency N and vertical system rotation with frequency Ω in the Boussinesq system of equations. For different values of the ratio 2 Ω/N, we compare Eulerian and Lagrangian statistics. The detailed dynamics of energy is studied by splitting the velocity in toroidal and poloidal modes, which we put in relation with the wave/vortex linear decomposition. From DNS, the results for dominant stratification show a large disequilibrium of anisotropy between the toroidal and poloidal parts. For dominant rotation, angular spectra show an equidistribution of energy between poloidal and toroidal parts, with non-isotropic angular distribution of the energy density down to the smallest scales. Regarding Lagrangian statistics, DNS results are compared with two linear models based on RDT and KS, respectively. The linear models reproduce extremely well the oscillations and confinement of vertical one-point dispersion when stratification is present. Horizontal diffusion laws compare well in a qualitative way. However, quantitative differences can be detected.