Spectral energy scaling in precessing turbulence

We study precessing turbulence, which appears in several geophysical and astrophysical systems, by direct numerical simulations of homogeneous turbulence where precessional instability is triggered due to the imposed background flow. We show that the time development of kinetic energy $\mathcal{K}$ occurs in two main phases associated with different flow topologies: (i) an exponential growth characterizing three-dimensional turbulence dynamics and (ii) nonlinear saturation during which $\mathcal{K}$ remains almost time independent, the flow becoming quasi-two-dimensional. The latter stage, wherein the development of $\mathcal{K}$ remains insensitive to the initial state, shares an important common feature with other quasi-two-dimensional rotating flows such as rotating Rayleigh-Bénard convection, or the large atmospheric scales: in the plane $k_{\parallel }=0$, i.e., the plane associated to an infinite wavelength in the direction parallel to the principal rotation axis, the kinetic energy spectrum scales as $k_{\perp }^{-3}$. We show that this power law is observed for wave numbers ranging between the Zeman “precessional” and “rotational” scales, $k_S^{-1}$ and $k_{\mathrm{\Omega }}^{-1}$, respectively, at which the associated background shear or inertial timescales are equal to the eddy turnover time. In addition, an inverse cascade develops for $(k_{\perp },k)<k_S$, and the spherically averaged kinetic energy spectrum exhibits a $k^{-2}$ inertial scaling for $k_S<k<k_{\mathrm{\Omega }}$.

Figure 1

Time evolution of the kinetic energy growth rate ${\sigma }_K={\left(S\mathcal{K}\right)}^{-1}\dot{\mathcal{K}}={\left(S\mathcal{K}\right)}^{-1}(\mathcal{P}-\mathcal{D})$ for several values of the couple ($\mathrm{\Omega },\varepsilon )$. At the saturation stage, ${\sigma }_K$ slightly fluctuates about zero reflecting a balance between $\mathcal{P}$ and $\mathcal{D}.$ The inset shows the independence of the saturated kinetic energy to the initial Reynolds number ${\text{Re}}_0$.

Figure 2

Left: Schematic of the mean flow. Right: Horizontal cut of vertical vorticity ${\omega }_z$ in the case $(\mathrm{\Omega }=5,\varepsilon =0.17)$ with resolution ${512}^3$. In the exponential growth stage (top snapshot at $\mathit{\text{St}}=10$ and ${\text{Ro}}_{\omega }=0.25)$ one observes large-scale filaments together with intense localized dominant vortices (with highest value of 10), while at the saturation stage (bottom snapshot at $\mathit{\text{St}}=40,{\text{Ro}}_{\omega }=0.45),$ which corresponds to a quasi-2D flow, there are two large columnar vortices (with highest value of 20) emerging from the background of 3D turbulent fluctuations.

Figure 3

Energy spectrum $E^{\left(2\text{D}\right)}(k_{\perp },k_{\parallel }=0)$ 2D modes and $E^{\left(3\text{D}\right)}\left(k_{\perp }\right)$ of 3D modes $k_{\parallel }=1,2$ at the saturation stage for $(\mathrm{\Omega }=20,\varepsilon =0.17)$, with resolution ${512}^3$. (The tiny peak at $k=85$ is a remnant of the precomputation at coarser resolution ${256}^3$; it does not affect our current conclusions.) Solid lines for spectra averaged over $\mathit{\text{St}}\in [20,40]$. One observes that the averaged ${\langle E^{\left(2\text{D}\right)}\left(k_{\perp }\right)\rangle }_t$ and $E^{\left(2\text{D}\right)}(k_{\perp },t)$ at $\mathit{\text{St}}=30$ (dashed line) and$\mathit{\text{St}}=40$ (dots) are almost the same, exhibiting a $k_{\perp }^{-3}$ scaling in the range $k_S\sim 5<k_{\perp }<k_{\mathrm{\Omega }}\sim 30$. At large horizontal scales $(k_{\perp }<k_S)$, the energy is mainly concentrated around the plane $k_{\parallel }=0$, and transferred via an inverse cascade that creates a strong anisotropy, as shown by the top inset displaying the energy ratio $E_h^{\left(2\text{D}\right)}/E_v^{\left(2\text{D}\right)}$ (dashed) and $E_h^{\left(3\text{D}\right)}/E_v^{\left(3\text{D}\right)}$ for $k_{\parallel }=1,2$. The bottom inset shows the flux spectrum indicative of backscatter at large scale.

Figure 4

Average over $\mathit{\text{St}}\in [20,40]$ of the radial spectrum $E(k,t)$ normalized by rotation scaling for $(\mathrm{\Omega },\varepsilon )=(20,0.17),(20,0.10)$, and (5,0.17) with resolutions ${512}^3,{256}^3$, and ${512}^3$. Dashed horizontal lines at $C_{\mathrm{\Omega }}=1.22$ and $C_{\mathrm{\Omega }}=1.87$ for reference [see Eq. (11)]. The left inset shows the nonlinear transfer flux normalized by the dissipation rate for the three precessing turbulence cases. Inverse cascade of energy develops at $k<k_S.$ The right inset displays the time evolution of the ratio of horizontal energy (respectively, integral length scale associated to $k_{\parallel }=0$ mode) to vertical energy (respectively, integral length scale) for $(\mathrm{\Omega }=5,\varepsilon =0.17)$