Turbulence with combined stratification and rotation: Limitations of Corrsin’s hypothesis

The properties of one-particle and particle-pair diffusion in rotating and stratified turbulence are studied by applying the rapid distortion theory to a kinematic simulation of the Boussinesq equation with a Coriolis term. We discuss the simplified Corrsin hypothesis and restrict the validity of its predictions to pure rotation. We emphasize the existence of two $\tau$ regimes driven by very different physics when rotation is present. Particular attention is given to the locality-in-scale hypothesis for two-particle diffusion in both the horizontal and the vertical directions.

Figure 1

Stratification and rotation direction.

Figure 2

Craya-Hering frame and Fourier space discretization.

Figure 3

Schematic scenario of superposition of rotation and stratification diffusion patterns. Dashed lines show what can be predicted by the simplified Corrsin hypothesis. (a) Pure rotation; (b) pure stratification; (c) rotation and stratification.

Figure 4

Nondimensional one-particle mean square horizontal displacement $⟨{\zeta }_h^2⟩∕L^2$ as a function of $\tau ∕t_d$ in stratification-dominant turbulence with $B=0.2$ for cases B, C, D, E, and F in Table .

Figure 5

Effects of rotation on stratification with $N=500$. Nondimensional one-particle mean square horizontal displacement $⟨{\zeta }_h^2⟩∕L^2$ plotted as a function of $\tau ∕t_d$ for cases A, K, P, and O in Table . From top to bottom $\Omega =0$, 125, 500, and 2500.

Figure 6

One-particle mean square horizontal displacement in rotation-dominant turbulence with $B=2$ for cases P, Q, R, S, and T in Table . (a) $⟨{\zeta }_h^2⟩{\Omega }^2∕u^{\prime 2}$ as a function of $\tau \Omega ∕\pi$. (b) $⟨{\zeta }_h^2⟩∕L^2$ as a function of $\tau ∕t_d$.

Figure 7

Effects of stratification on rotation with $\Omega =500$. Nondimensional one-particle mean square horizontal displacement $⟨{\zeta }_h^2⟩∕L^2$ plotted as a function of $\tau ∕t_d$ for cases J, P, N, M, and L in Table . From top to bottom $N=2000$, 500, 100, 10, and 0.

Figure 8

Nondimensional one-particle mean square vertical displacement $⟨{\zeta }_3^2⟩N^2∕u^{\prime 2}$ as a function of $\tau N∕2\pi$ in stratification-dominant turbulence with $B=0.2$ for (from top to bottom) cases B, C, D, E, and F in Table .

Figure 9

Effects of rotation on stratification with $N=500$. Nondimensional one-particle mean square vertical displacement $⟨{\zeta }_3^2⟩N^2∕u^{\prime 2}$ plotted as a function of $\tau N∕2\pi$ for the same cases as in Fig. 5. From top to bottom $\Omega =0$, 125, 500, and 2500.

Figure 10

Nondimensional one-particle mean square vertical displacement $⟨{\zeta }_3^2⟩{\Omega }^2∕u^{\prime 2}$ as a function of $\tau \Omega ∕\pi$ in rotation-dominant turbulence with $B=2$ for (from top to bottom) cases P, Q, R, S, and T in Table .

Figure 11

Effects of stratification on rotation with $\Omega =500$. Nondimensional one-particle mean square vertical displacement $⟨{\zeta }_3^2⟩{\Omega }^2∕u^{\prime 2}$ plotted as a function of $\tau \Omega ∕\pi$ for the same cases as in Fig. 7. From bottom to top $N=2000$, 500, 100, 10, and 0.

Figure 12

Schematic scenario of one-particle diffusion when rotation and stratification are superposed. (a) For $B>1$ the rotation pattern settles first and the stratification characteristic plateau appears afterward. (b) For $B<1$ the stratification plateau appears before the rotation effects start, so that the rotation does not affect the plateau.

Figure 13

Nondimensional pair mean square relative horizontal separation when $B=0.2$ for cases B, C, D, E, F, H, and I in Table . (a) $⟨{\Delta }_h^r{\left(\tau \right)}^2⟩∕L^2$ as a function of $\tau ∕t_d$. (b) $⟨{\Delta }_v^r{\left(\tau \right)}^2⟩N^2∕\delta v_v^2$ as a function of $\tau N∕2\pi$.

Figure 14

Nondimensional two-particle mean square relative horizontal separation with $B=2$ for cases P to V in Table . $⟨{\Delta }_h^r{\left(\tau \right)}^2⟩∕L^2$ as a function of $\tau ∕t_d$ (middle curves: cases P, R, and T).

Figure 15

Nondimensional two-particle mean square relative vertical separation, $⟨{\left({\Delta }_3^r\right)}^2⟩∕L^2$, as a function of $\tau ∕t_d$ with $B=2$ for cases P to V in Table (middle curves: cases P, R, and T).

Figure 16

Effects of stratification on rotation with $\Omega =500$. Nondimensional two-particle mean square relative separation in horizontal plane, $⟨{\left({\Delta }_h^r\right)}^2⟩∕L^2$, plotted as a function of $\tau ∕t_d$ for cases L, M, N, P, and J in Table . From top to bottom $N=2000$, 500, 100, 10, and 0.

Figure 17

Effects of stratification on rotation with $\Omega =500$. Nondimensional two-particle mean square relative separation in vertical direction, $⟨{\left({\Delta }_v^r\right)}^2⟩{\Omega }^2∕u^{\prime 2}$, plotted as a function of $\tau \Omega ∕\pi$ for the same cases as in Fig. 16. From bottom to top $N=2000$, 500, 100, 10, and 0.

Figure 18

Effects of rotation on stratification with $N=500$. Nondimensional two-particle mean square relative separation in horizontal plane, $⟨{\left({\Delta }_h^r\right)}^2⟩∕L^2$, plotted as a function of $\tau ∕t_d$ for cases A, K, P, and O in Table . From top to bottom $\Omega =0$, 125, 500, and 2500.

Figure 19

Effects of rotation on stratification with $N=500$. Nondimensional two-particle mean square relative separation in vertical direction, $⟨{\left({\Delta }_v^r\right)}^2⟩N^2∕u^{\prime 2}$, plotted as a function of $\tau N∕2\pi$ for the same cases as in Fig. 18. From top to bottom $\Omega =0$, 125, 500, and 2500.

Figure 20

Nondimensional two-particle horizontal diffusivity $\left(d∕dt\right)⟨{\Delta }_h^2⟩t_{\eta }∕{\eta }^2$ as a function of ${\delta }_h∕\eta$. The solid line represents Richardson’s four-thirds diffusivity law. (a) $B=0.2$ for cases B to F in Table . (b) $B=2$ for cases P to T in Table .

Figure 21

Effects of $B$ on two-particle diffusivity in rotating and stratified turbulence. Nondimensional horizontal diffusivity $\left(d∕dt\right)⟨{\Delta }_h^2⟩t_{\eta }∕{\eta }^2$ plotted as a function of $\delta ∕\eta$. From top to bottom $B=0$, 0.5, 2, and 10. Other parameters are $L=1$, $1∕\eta =12.5$, $u^{\prime }=0.35$, and ${\Delta }_0∕\eta =1$. Solid line has a slope of 4/3, representing Richardson’s four-thirds diffusivity law.

Figure 22

${\beta }_h$ as a function of ${\delta }_h$: (a) $B=0.2$ for cases B, G, H, and I in Table ; (b) $B=2$ for cases P, U, and V in Table .