Spectral energy transfers and kinetic-potential energy exchange in rotating stratified turbulence

Direct numerical simulations of forced homogeneous rotating stratified turbulence are carried out with the same Rossby number and different Froude numbers. We investigated the effects of different stratification on energy transfers across scales and kinetic-potential energy exchange in Fourier space in the inverse energy cascade range. When the stratification is weak, almost all the kinetic energy flux in the inertial range comes from two-dimensional and three-dimensional (2D-3D) coupling interactions and 2D dynamics dominates at very large scales, which are similar to those in the purely rotating case. However, compared with the purely rotating case, the inverse cascade is weakened through the kinetic-potential energy exchange induced by 3D wave-vortical interactions. The total kinetic-potential energy exchange is found to oscillate at later times, which is produced by 2D inertia-gravity waves with fixed wave numbers. When the stratification is strong, the distributions of decomposed kinetic fluxes are different from those in purely rotating case. The magnitude of 2D-3D coupling energy flux becomes small and 3D-mode energy flux is important near the forcing scales. Besides, homo- and heterochiral energy fluxes are different in the inverse cascade. The kinetic-potential energy exchange under strong stratification mainly comes from 3D modes and wave-vortical interactions. Strengths of stratification negligibly affect properties of locality of the kinetic energy cascade, which is infrared local and ultraviolet nonlocal.

Figure 1

(a) Kinetic energy $E_K$ and (b) kinetic energy flux from 3D modes to 2D modes ${\mathrm{\Pi }}_{K,\text{3D}\to \text{2D}}$ as functions of time for all simulations. The solid horizontal line in panel (b) indicates ${\mathrm{\Pi }}_{K,\text{3D}\to \text{2D}}=0$ for reference.

Figure 2

(a) Potential energy $E_P$ and (b) the conversion rate of kinetic energy to potential energy $E_{KP}$ as functions of time for Run2 and Run3.

Figure 3

Time evolutions of different energy fluxes for Run2 (left) and Run3 (right): the kinetic flux from 3D modes to 2D modes, ${\mathrm{\Pi }}_{K,\text{3D}\to \text{2D}}$; the flux from kinetic energy to potential energy in 3D modes and in 2D modes, ${\mathrm{\Pi }}_{KP,\text{3D}\to \text{3D}}$ and ${\mathrm{\Pi }}_{KP,\text{2D}\to \text{2D}}$. Note that ${\mathrm{\Pi }}_{K,\text{3D}\to \text{2D}}$ for Run1 is shown for reference.

Figure 4

Fluxes from kinetic energy to potential energy by interactions among different linear eigenmodes obtained from (a) Run2 and (b) Run3.

Figure 5

(a), (c), (e) The time evolution of the kinetic energy spectrum for (a) Run1, (c) Run2, and (e) Run3 at different times. (b), (d), (f) Three types of kinetic energy spectra from one realization at time $t=480{\tau }_0$ for (b) Run1, (d) Run2, and (f) Run3.

Figure 6

(a), (c) The time evolution of the potential energy spectrum for (a) Run2 and (c) Run3 at different times. (b), (d) Two types of potential energy spectra from one realization at time $t=480{\tau }_0$ for (b) Run2 and (d) Run3.

Figure 7

(a), (c) The time evolution of the spectrum of the conversion rate of kinetic energy to potential energy for (a) Run2 and (c) Run3 at different times. (b), (d) Two types of spectra of the conversion rate from one realization at time $t=480{\tau }_0$ for (b) Run2 and (d) Run3.

Figure 8

The time evolution of (a), (b) the vortical-mode energy spectrum and (c), (d) the wave-mode energy spectrum for (a), (c) Run2 and (b), (d) Run3 at different times.

Figure 9

The time evolution of the kinetic energy flux for (a) Run1, (b) Run2, and (c) Run3 at different times.

Figure 10

The time evolution of the potential energy flux for (a) Run2 and (b) Run3 at different times.

Figure 11

(a), (c), (e) Total kinetic energy flux ${\mathrm{\Pi }}_K\left(k\right)$ and fluxes decomposed on the different 2D-3D interactions ${\mathrm{\Pi }}_{\text{2D}\leftrightarrows \text{2D}}\left(k\right)$, ${\mathrm{\Pi }}_{\text{3D}\leftrightarrows \text{3D}}\left(k\right)$ and ${\mathrm{\Pi }}_{\text{3D}\leftrightarrows \text{2D}}\left(k\right)$ [Eqs. (13, 14, 15)] and (b), (d), (f) ${\mathrm{\Pi }}_{\text{3D}\leftrightarrows \text{2D}}\left(k\right)$ and its three contributions of different classes of triads [Eqs. (16, 17, 18)] for (a), (b) Run1, (c), (d) Run2, and (e), (f) Run3. All the results come from one realization at time $t=480{\tau }_0$.

Figure 12

Total kinetic energy flux ${\mathrm{\Pi }}_K\left(k\right)$, homochiral energy flux ${\mathrm{\Pi }}^{\text{HO}}\left(k\right)$, and heterochiral energy flux ${\mathrm{\Pi }}^{\text{HE}}\left(k\right)$ for (a) Run1, (b) Run2, and (c) Run3 from one realization at time $t=480{\tau }_0$.

Figure 13

Normalized kinetic energy transfer functions $T_K(k_h,k_z)/{\left|T_K\right|}_{\text{max}}$ at different times for (a), (b) Run2 and (c), (d) Run3.

Figure 14

Normalized kinetic to potential energy transfer functions $T_{KP}(k_h,k_z)/{\left|T_{KP}\right|}_{\text{max}}$ at different times for Run2.

Figure 15

Contribution fractions to the kinetic energy flux across scale $l=\pi /15$: (a) from scales $[{\mathrm{\Delta }}^{\left(n\right)},{\mathrm{\Delta }}^{(n+1)})$; (b), (c) from scales $[{\delta }^{(n+1)},{\delta }^{\left(n\right)})$ based on different decomposition methods.