Regime transition in the energy cascade of rotating turbulence

Transition from a split to a forward kinetic energy cascade system is explored in the context of rotating turbulence using direct numerical simulations with a three-dimensional isotropic random force uncorrelated with the velocity field. Our parametric study covers confinement effects in high-aspect-ratio domains and a broad range of rotation rates. The data presented here add substantially to previous works, which, in contrast, focused on smaller and shallower domains. Results indicate that for fixed geometrical dimensions the Rossby number acts as a control parameter, whereas for a fixed Rossby number the product of the domain size along the rotation axis and the forcing wave number governs the amount of energy that cascades inversely. The regime transition criterion hence depends on both control parameters.

Figure 1

Three-dimensional spherically averaged energy spectrum of the initial condition: kf02-a01 (□), kf04-a01 ($—$), kf08-a01 ($+$), kf16-a01 ($—$), kf32-a01 ($\text{––⚬––}$), kf04-a32 ($*$) kf08-a16 ($▵$).

Figure 2

Time evolution of box-averaged kinetic energy (a) and energy dissipation rate (b) for ${\mathrm{Ro}}_{\varepsilon }\approx 0.31$ (weak rotation). Lines corresponding to the same ${\kappa }_f{\mathcal{L}}_{\perp }$ are grouped by color: ${\kappa }_f{\mathcal{L}}_{\perp }=2$ ($\blacksquare$), ${\kappa }_f{\mathcal{L}}_{\perp }=4$ ($\blacksquare$), ${\kappa }_f{\mathcal{L}}_{\perp }=8$ ($\blacksquare$), and ${\kappa }_f{\mathcal{L}}_{\perp }=16$ ($\blacksquare$). Lines corresponding to the same $A_r$ are grouped by type: $A_r=1$ ($\text{––⚬––}$), $A_r=8$ ($—–—$) (cf. Table 1).

Figure 3

Time evolution of box-averaged kinetic energy (a) and energy dissipation rate (b) for ${\mathrm{Ro}}_{\varepsilon }\approx 0.06$ (strong rotation). Lines corresponding to the same ${\kappa }_f{\mathcal{L}}_{\perp }$ are grouped by color: ${\kappa }_f{\mathcal{L}}_{\perp }=2$ ($\blacksquare$), ${\kappa }_f{\mathcal{L}}_{\perp }=4$ ($\blacksquare$), ${\kappa }_f{\mathcal{L}}_{\perp }=8$ ($\blacksquare$), ${\kappa }_f{\mathcal{L}}_{\perp }=16$ ($\blacksquare$), ${\kappa }_f{\mathcal{L}}_{\perp }=32$ ($\blacksquare$). Lines corresponding to the same $A_r$ are grouped by type: $A_r=1$ (—⚬—), $A_r=2$ ($\_\_$), $A_r=4$ ($\text{−}\text{−}\text{−}$), $A_r=8$ ($–\text{−}–$), $A_r=16$ ($\_\_$), $A_r=32$ ($\text{––□––}$) (cf. Table 1).

Figure 4

Time evolution of box-averaged kinetic energy (a) and energy dissipation rate (b) for ${\kappa }_f{\mathcal{L}}_{\perp }=8$ and ${\kappa }_f{\mathcal{L}}_{\parallel }=64$. Different line colors correspond to the range $0.06<{\mathrm{Ro}}_{\varepsilon }<1.25$; see Table 1.

Figure 5

Phase transition diagram for weak and strong rotation and varying geometrical dimensions (a) and for constant geometrical dimension and varying ${\mathrm{Ro}}_{\varepsilon }$ (b). Color scheme in (a) is the same as that in Fig. 3. In (a), the data point for ${\kappa }_f{\mathcal{L}}_{\perp }={\kappa }_f{\mathcal{L}}_{\parallel }=32$ (case kf32-a01) is almost identical to that for case kf04-a08 (${\kappa }_f{\mathcal{L}}_{\perp }=4$; ${\kappa }_f{\mathcal{L}}_{\parallel }=32$) and is therefore not visible.

Figure 6

Phase transition diagram in terms of the combined control parameter ${\mathrm{Ro}}_{\varepsilon }{\kappa }_f{\mathcal{L}}_{\parallel }$ for all data points in Fig. 5. Colored circles represent data from Fig. 5; squares, data from Fig. 5.

Figure 7

Spectral energy flux for ${\mathrm{Ro}}_{\varepsilon }\approx 0.06$ and cases with ${\kappa }_f{\mathcal{L}}_{\parallel }=32$ (a) and ${\kappa }_f{\mathcal{L}}_{\perp }=16$ (b). In (a), ${\kappa }_f{\mathcal{L}}_{\perp }=8$ ($—$), ${\kappa }_f{\mathcal{L}}_{\perp }=16$ ($—$), and ${\kappa }_f{\mathcal{L}}_{\perp }=32$ ($—$). In (b), ${\kappa }_f{\mathcal{L}}_{\parallel }=16$, 32, and 64 ($—$). The arrow denotes the direction of increase.

Figure 8

Three-dimensional spherically averaged energy spectrum for ${\kappa }_f{\mathcal{L}}_{\parallel }=32$ (a) and ${\kappa }_f{\mathcal{L}}_{\perp }=16$ (b) with ${\mathrm{Ro}}_{\varepsilon }\approx 0.06$. Line styles are the same as in Fig. 7, apart from the reference energy spectrum in Fig. 1 with ${\kappa }_f{\mathcal{L}}_{\perp }={\kappa }_f{\mathcal{L}}_{\parallel }=32$ ($–\text{−}–$).

Figure 9

Two-dimensional energy spectrum for ${\mathrm{Ro}}_{\varepsilon }\approx 0.06$ with ${\kappa }_f{\mathcal{L}}_{\perp }={\kappa }_f{\mathcal{L}}_{\parallel }=32$ (case kf32-a01) at $t=30{\tau }_f$. Data are normalized by $\left(2\pi {\kappa }_{\perp }\right)u_f^2/{\kappa }_f^3$ and plotted as ${log}_{10}$. Inset: Highlight of the region around the forcing wave number: ${\kappa }_{\perp }/{\kappa }_f<1.5$ and ${\kappa }_{\parallel }/{\kappa }_f<1.5$.

Figure 10

One-dimensional energy spectra for ${\mathrm{Ro}}_{\varepsilon }\approx 0.06$ and ${\kappa }_f{\mathcal{L}}_{\perp }={\kappa }_f{\mathcal{L}}_{\parallel }=32$ (case kf32-a01) along directions ${\kappa }_{\perp }$ (a) and ${\kappa }_{\parallel }$ (b). Lines represent the time evolution of the energy spectrum: $t=0$ ($—$), and $t=10{\tau }_f$, $20{\tau }_f$ and $30{\tau }_f$ ($—$). A reference line for the scaling laws that best agrees with the presented data is also shown ($\text{−}\text{−}\text{−}$).