Weak, homogeneous turbulence with rotation and stratification: a numerical study of the non-propagating component

Following Scott & Cambon (2024 J. Fluid Mech. vol. 979, A17), henceforth referred to as [I], a spectral approach is used and the flow is expressed as a sum of normal modes, which are of two types: inertial/gravity waves and non-propagating (NP) modes. It was shown in [I] that, for weak (small Rossby or Froude number) turbulence, the NP component of the flow decouples from the waves at leading order and here we focus on the NP part alone. It is demonstrated that the evolution equations of the NP component are equivalent to the three-dimensional, quasi-geostrophic (QG) approximation of geophysical fluid dynamics. For QG turbulence, the seminal paper of Charney (1971 J. Atmos. Sci. vol. 28, pp. 1087–1095), referred to as [II], concluded that, as for two-dimensional turbulence, the energy cascade for QG turbulence should go from smaller to larger scales and that the inertial-range spectrum at wavenumber $k$ should behave as $k^{-3}$ . He also proposed that the energy distribution in spectral space is isotropic if the vertical wavenumber is appropriately scaled and deduced a principle of equipartition in which the average kinetic energy is twice the potential one. We use Charney’s transformation of spectral coordinates to effectively eliminate the parameter $\beta =2{\varOmega} /N$ , where ${\varOmega}$ is the rotation rate and $N$ the Brunt–Vaisala frequency, and give results of numerical calculations concerning the energy distribution. The results mostly agree with [II] at large enough times, although they do not support Charney isotropy. They further suggest self-similarity of the time evolution of the three-dimensional energy distribution in spectral space away from the vertical axis.