Single-particle Lagrangian statistics from direct numerical simulations of rotating-stratified turbulence

Geophysical fluid flows are predominantly turbulent and often strongly affected by the Earth's rotation, as well as by stable density stratification. Using direct numerical simulations of forced Boussinesq equations, we study the influence of these effects on the motion of fluid particles. We perform a detailed study of Lagrangian statistics of acceleration, velocity, and related quantities, focusing on cases where the frequencies associated with rotation and stratification (RaS), $f$ and $N$, respectively, are held at a fixed ratio $N/f=5$. The simulations are performed in a periodic domain, at Reynolds number $\text{Re}\approx 4000$, and Froude number Fr in the range $0.03\lesssim \text{Fr}\lesssim 0.2$ (with Rossby number $\text{Ro}=5\text{Fr}$). As the intensity of RaS increases, a sharp transition is observed between a regime dominated by eddies to a regime dominated by waves, which corresponds to $\text{Fr}\lesssim 0.07$. For the given runs, this transition to a wave-dominated regime can also be seemingly described by simply comparing the timescales $1/N$ and ${\tau }_{\eta }$, the latter being the Kolmogorov timescale based on the mean kinetic energy dissipation. Due to the known anisotropy induced by RaS, we consider separately the motion in the horizontal and vertical directions. In the regime $N{\tau }_{\eta }<1$, acceleration statistics exhibit well known characteristics of isotropic turbulence in both directions, such as probability density functions with wide tails and acceleration variance approximately scaling as per Kolmogorov's theory. In contrast for $N{\tau }_{\eta }>1$, they behave very differently, experiencing the direct influence of the imposed rotation and stratification. On the other hand, the Lagrangian velocity statistics exhibit visible anisotropy for all runs; nevertheless the degree of anisotropy becomes very strong in the regime $N{\tau }_{\eta }>1$. We observe that in the regime $N{\tau }_{\eta }<1$, rotation enhances the mean-square displacements in horizontal planes in the ballistic regime at short times but suppresses them in the diffusive regime at longer times. This suppression of the horizontal displacements becomes stronger in the regime $N{\tau }_{\eta }>1$, with no clear diffusive behavior. In contrast, the displacements in the vertical direction are always reduced. This inhibition is extremely strong in the $N{\tau }_{\eta }>1$ regime, leading to a scenario where particles almost appear to be trapped in horizontal planes.

Figure 1

The kinetic energy spectrum $E\left(k\right)$ normalized by $U^{2}L$ as a function of $kL$, where $U$ and $L$ are the large-scale velocity and length scales, respectively. The curves correspond to cases listed in Table 1. Solid green lines indicating a spectral slope of $k^{-5/3}$ and $k^{-11/5}$ in panels (a) and (b), respectively, are drawn for reference.

Figure 2

Kolmogorov-scaled acceleration variance, as defined by Eq. (8), plotted as a function of $1/\text{Fr}$. The parallel (vertical) and perpendicular (horizontal) components of accelerations are shown in blue triangles and red circles, respectively. The HIT run corresponds to $1/\text{Fr}=0$. In addition the rotation-only run also corresponds to $1/\text{Fr}=0$, but is shown in solid symbols.

Figure 3

Standardized probability density functions (PDFs) of (a) the perpendicular (horizontal) and (b) the parallel (vertical) components of acceleration, i.e., $a_{\perp }$ and $a_{\parallel }$, respectively. $\sigma$ denotes the corresponding standard deviation. The dotted line shows the standardized Gaussian distribution for comparison. The legend is split over two panels, but applies to each panel individually.

Figure 4

Lagrangian autocorrelation of the perpendicular (horizontal) component of acceleration ${\rho }_{\perp }^a\left(\tau \right)$ against the time lag $\tau$ normalized by the Kolmogorov timescale ${\tau }_{\eta }$ in panel (a) and by the rotation period $2\pi /f$ in panel (b). Inset in panel (a) shows a zoomed view at small-time lag. The legend is split over two panels, but applies to each panel individually.

Figure 5

Lagrangian frequency spectrum of the perpendicular (horizontal) component of acceleration $E_{\perp }^a\left(\omega \right)$ against the frequency $\omega$ normalized by the Kolmogorov timescale ${\tau }_{\eta }$ in panel (a) and by the rotation frequency $f$ in panel (b). The spectra are nondimensionalized using the mean kinetic energy dissipation rate $\langle \epsilon \rangle$. Vertical dotted lines in panel (b) correspond to frequencies $\omega =f$ and $N$ (note $N=5f$). The legend is split over two panels, but applies to each panel individually.

Figure 6

Lagrangian autocorrelation of the parallel (vertical) component of acceleration ${\rho }_{\parallel }^a\left(\tau \right)$ against the time lag $\tau$ normalized by the Kolmogorov timescale ${\tau }_{\eta }$ in panel (a) and by the Brunt-Väisälä period $2\pi /N$ in panel (b). The legend is split over two panels, but applies to each panel individually.

Figure 7

Lagrangian frequency spectrum of the parallel (vertical) component of acceleration $E_{\parallel }^a\left(\omega \right)$ against the frequency $\omega$ normalized by the Kolmogorov timescale ${\tau }_{\eta }$ in panel (a) and by the Brunt-Väisälä frequency $N$ in panel (b). The spectra are nondimensionalized using the mean kinetic energy dissipation rate $\langle \epsilon \rangle$. Vertical dotted lines in panel (b) correspond to frequencies $\omega =f$ and $N$ (with $N=5f$). The legend is split over two panels, but applies to each panel individually.

Figure 8

Lagrangian autocorrelation ${\rho }_{\perp }^u\left(\tau \right)$ of the perpendicular (horizontal) component of the velocity, normalized by the Kolmogorov time scale ${\tau }_{\eta }$ in panel (a) and the rotation period $2\pi /f$ in panel (b). The legend is split over two panels, but applies to each panel individually.

Figure 9

Lagrangian autocorrelation ${\rho }_{\parallel }^u\left(\tau \right)$ of the parallel (vertical) component of the velocity. Similar normalizations as Fig. 6 are used. The legend is split over two panels but applies to each panel individually.

Figure 10

Mean-square displacement as a function of time in the (a) perpendicular (horizontal) direction and (b) parallel (vertical) direction. All quantities normalized by Kolmogorov scales. The dotted lines represent slopes 2 and 1 at short and long times, respectively. The legend is split over two panels but applies to each panel individually.