Inverse centrifugal effect induced by collective motion of vortices in rotating turbulent convection

Coherent vortex structures exist ubiquitously in many flow systems ranging from small-scale turbulence to large-scale geophysical and astrophysical flows [1, 2, 3], and their dynamics plays a crucial role in determining turbulent mixing and transport in those systems. Previous studies of vortex dynamics are mainly focused on isolated vortices [3, 4]. However, densely populated vortices are observed in rapidly rotating turbulence [5, 6], and the resulting vortex interactions may lead to markedly different dynamics compared to that of isolated vortices [7, 8, 9]. Many nonequilibrium statistical systems in nature consisting of densely populated, interacting entities often exhibit collective behavior, i.e., the entities self-aggregate to perform collective motions. Examples include flocking of bird, swarming of bacteria and clustering of active matters, etc [10, 11, 12]. Whether collective behavior of vortices can arise in rotating turbulent flows is thus a question of fundamental interest.

Here we demonstrate both experimentally and numerically the collective motion of vortices in rotating turbulent convection. The resultant long-range correlated vortex dynamics gives rise to the counterintuitive effect of inverse centrifugal motion, i.e., the warm and lighter convective vortices exhibit outward motion from the rotation axis. This intriguing phenomenon occurs in a rapidly rotating regime where the strong centrifugal buoyancy breaks the symmetry in both the population and vorticity magnitude of the vortices. Our study reveals that it is through local hydrodynamic interactions that the densely populated vortices self-aggregate into large-scale vortex clusters, in which the warm cyclonic vortices submit to the collective motion dominated by the strong anticyclones and move outwardly.

The fluid dynamics of rotating turbulent flows is often studied in rotating Rayleigh-Bénard convection (RBC). Despite the considerable progress achieved in studying non-rotating and weakly rotating turbulent RBC [13, 14, 15, 16, 17, 18], some important convection regimes that may exhibit novel vortex dynamics are yet to be explored (see, e.g. [19, 20]). Recent studies report that in rapidly rotating RBC, the convection flows are organized by the Coriolis force into coherent columnar vortices [21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. These columnar vortices are helical structures with their vorticity correlated to the temperature, i.e., cyclones are warm (upwelling) vortices while anticyclones are cold (downwelling) ones if observed from near the bottom boundary [22, 29]. The Taylor-Proudman theorem [31], which predicts that rapid rotation suppresses flow variations along the vertical axis of rotation, provides a comprehensive description of the flow structures of the columnar vortices. From a dynamical viewpoint, however, the horizontal motion of, and the interaction between these vortices remain to be better understood.

Our experimental apparatus was designed for high-precision flow structure measurement in rotating RBC [20, 30]. We used cylindrical cells with an inner diameter $d{=}$240 mm and length $H{=}$63.0 (120.0) mm, yielding the aspect ratio $\Gamma{=}d/H{=}$3.8 (2.0). The experiment was conducted with a constant Prandtl number Pr${=}\nu/{\kappa}{=}4.38$ and in the range $2.0{\times}10^{7}{\leq}$Ra${\leq}2.7{\times}10^{8}$ of the Rayleigh number Ra${=}{\alpha}g{\Delta}TH^{3}/{\kappa}{\nu}$ ($\alpha$ is the isobaric thermal expansion coefficient, $g$ the acceleration of gravity, $\Delta T$ the applied temperature difference, $\kappa$ the thermal diffusivity, and $\nu$ the kinematic viscosity). Rotation rates up to 5.0 rad/s were used. Thus the Ekman number Ek${=}\nu/2{\Omega}H^{2}$ spanned $4.9{\times}10^{-6}{\leq}$Ek${\leq}2.7{\times}10^{-4}$, corresponding to a range of the reduced Rayleigh number $1.06{\leq}\mathrm{Ra/Ra_{c}}{\leq}120$, with $\mathrm{Ra_{c}}{=}8.7$Ek^-4/3 the critical value for the onset of convection [32]. The Froude number Fr${=}{\Omega}^{2}d/2g$ was within $0{<}$Fr${\leq}0.31$. The flow field at a fluid depth of $z{=}H/4$ was measured using the technique of particle image velocimetry (PIV) (See schematic of the experimental set-up in Fig. 1i). In the direct numerical simulation (DNS) we solved the Navier-Stokes equations with the Coriolis and centrifugal forces included, using the multiple-resolution version of the CUPS code [33, 34]. The simulation was performed in a cylindrical domain with $\Gamma{=}4$, $\mathrm{Ra}{=}2.0{\times}10^{7}$ and $1.06{\leq}\mathrm{Ra/Ra_{c}}{\leq}40$.

Figure 1 presents snapshots of the vortex structures over the measured fluid height ($z=H/4$). At a low rotation rate when the centrifugal force is negligible (Fig. 1a), the cyclonic vortices (shown in red color) possess a greater number density and on average larger vorticity in magnitude than that of anticyclones (blue color). Both types of vortices exhibit stochastic horizontal motions as indicated by the vortex trajectories in Fig. 1b, presumably due to the turbulent background flows. The mean-square-displacement (MSD) of the vortices becomes a linear function of time at large times, indicating a Brownian-type, normal diffusive motion [20].

However, at higher rotation rates when the centrifugal force becomes dominant, we observe strong anticyclones with larger population than the cyclones (Fig. 1c). The anticyclones undergo outward radial motions accompanied by stochastic fluctuations along their paths (Fig. 1d), until they move close to the sidewall where their radial motion is terminated by the retrogradely traveling plumes. Compared to the anticyclones, the motion of weak cyclones are much more complex. Figure 1d indicates that in the outer region ($r{\geq}d/4$), the cyclones move toward the cell center while in the inner region ($r{\leq}d/4$) most of them migrate radially outward. In this rapid rotating case, the MSD of both types of vortices indicate superdiffusive behavior (see Supplementary Movies of the vortex motions).

To further quantify the vortex motions, we show the profiles of mean radial velocity ${\langle}u_{r}{\rangle}_{\xi}$ of the vortices measured in the inner region of the cell in Figs. 1e-1h. These velocity profiles reveal four distinct flow regimes depending on the rotation rates: (I) A randomly-diffusive regime exists in the slow rotating limit with Ra one order in magnitude larger than $\mathrm{Ra_{c}}$. In this regime the vortices move in a random manner, yielding ${\langle}u_{r}{\rangle}_{\xi}{\approx}0$ (Fig. 1e). (II) Centrifugation-influenced regime where the magnitude of ${\langle}u_{r}{\rangle}_{\xi}$ increases linearly with $r$ ($3\mathrm{Ra_{c}}{\leq}\mathrm{Ra}{\leq}5\mathrm{Ra_{c}}$). We observe that warm cyclones (cold anticyclones) move radially inward (outward), which is in agreement with the centrifugal effect (Fig. 1f). (III) Inverse-centrifugal regime ($1.5\mathrm{Ra_{c}}{\leq}\mathrm{Ra}{\leq}3\mathrm{Ra_{c}}$) in which there is anomalous outward cyclonic motion (Fig. 1g), and the radial gradients of ${\langle}u_{r}{\rangle}_{\xi}$ for both types of vortices reach a maximum. (IV) Asymptotic regime in the rapid rotation limit ($\mathrm{Ra}{\leq}1.5\mathrm{Ra_{c}}$) where the opposite radial motions of cyclones and anticyclones recover (Fig. 1h).

We formulate a theoretical model that consists a sets of Langevin equations incorporating the centrifugal force, which governs the radial vortex motion in a background of stochastic fluctuations. As shown in Fig. 2, the model provides predictions of the first and second moments of the radial vortex displacements which replicate very well the experimental data in flow regimes (II) and (IV) (see Supplementary Information 2 for detailed discussions of the model). Nonetheless, Figure 2 also suggests that the inverse centrifugal motion of the cyclones in the anomalous regime (III) remains unexplained on the basis of the model. A key question remained to be answered is then what sets the anomalous vortex motion?

To gain insights into the observed phenomenon, we first examine the relative strength of vorticity between the cyclones and anticyclones. Figure 3a shows the vorticity ratio $\gamma_{\omega}$ of the anticyclones to the cyclones. We note that in the randomly-diffusive regime, $\gamma_{\omega}$ is approximately 0.6, meaning that at the measured fluid height $z{=}H/4$ the cyclonic vorticity is overall larger in magnitude than the anticyclonic ones (see Fig. 1a). It is the case because in this regime the downwelling anticyclones generated from the top travel a longer distance to the measured fluid layer than the upwelling cyclones, with their momentum and vorticity largely dissipated by the background turbulence [27]. (The vorticity magnitude of the two types of vortices are equal if measured at $z{=}H/2$. See the visualization from our DNS in the right inset of Fig. 3a.) With increasing $\Omega$ the up- and down-welling vortices evolve into the vertically symmetric, columnar structures [25, 26], thanks to the Taylor-Proudman effect [31]. Hence one would expect that the vorticity strength of the cyclones and anticyclones become comparable, i.e., $\gamma_{\omega}$ approaches unity. Our DNS data indeed show this trend when the centrifugal force is switched off. However, when centrifugal effect is dominant, both the experimental and DNS results reveal that $\gamma_{\omega}$ exceed unity considerably in the inverse-centrifugal regime, indicating an asymmetric vorticity field dominated by the anticyclones (left inset of Fig. 3a). In the asymptotic regime where the severe rotational constraint finally weakens the convective vortices, $\gamma_{\omega}$ eventually returns to unity, and the symmetry of the cyclonic and anticyclonic vorticity restores. Remarkably, we observe that the trend of $\gamma_{\omega}(Ra/Ra_{c})$ is universal and independent of $Ra$ and $\Gamma$.

We demonstrate that the asymmetry of the vorticity field in the anomalous regime results from the centrifugal effect. Figure 3b presents the radial profiles of the mean temperature of the vortices and of the background fluid. These numerical data indicate a noticeable warming of the background fluid in the inner region, owing to the centrifugal effect [35, 36, 37]. As a result, the temperature difference ${\delta}T$ of the cold anticyclones from the background exceeds that of the warm cyclones. Since ${\delta}T$ is proportional to the buoyancy forcing on the vortices, it is predicted to be positively correlated to the vorticity $\omega$ in recent theoretical models [24, 26]. Indeed Fig. 3c shows that ${\delta}T$ and $\omega$ are both larger in magnitudes for anticyclones than for cyclones in the inner region, which explains the asymmetry of the vorticity field ($\gamma_{\omega}$>1).

In the background of the broken symmetry of the vorticity field in the anomalous regime, we show here that it is the long-range correlated vortex motion that gives rise to the inverse centrifugal motion of the cyclones. Figure 4 presents the instantaneous motion of the vortices, with their spatial distribution presented in a Voronoi diagram. One sees that the adjacent vortices often self-organize into vortex clusters, i.e., the vortices move largely in the same direction. We adopt two criteria to identify vortex clusters, i.e., the distance of two neighboring vortices is smaller than 1.5 times the mean vortex diameter and the angle $\phi$ between their velocity vectors is within a threshold ($\phi{\leq}\phi^{\ast}{=}60^{\circ}$). Our analysis over the range $30^{\circ}{\leq}\phi^{\ast}{\leq}75^{\circ}$ confirms that the results of correlated vortex motion are not sensitive to the choice of $\phi^{\ast}$. The direction of the motion of each cyclone $i$ is represented by the angle $\theta$ between its position vector $\vec{r}_{i}$ relative to the rotation axis and its velocity $\vec{u}_{i}$. We find that $\theta$ is strongly dependent on the number ($N$) of vortices in a cluster (inset). For isolated cyclones ($N{=}1$), the most probable direction of motion is radially inward ($\theta_{p}{=}\pi$). However, for clustered cyclones ($N{>}1$) we find $\theta_{p}{=}0$ as they move outward. The standard deviation of $\theta$ decreases monotonically when $N$ increases. Our data reveal that within large clusters the motion of weak cyclones submit to that of strong anticyclones and move outwardly in a collective manner. Their inverse centrifugal motion becomes more unidirectional with the increasing of the cluster size.

Some insights into the physical mechanism responsible for the vortex-cluster formation can be gained through statistical analysis of the cluster size distribution $p(N)$. Figure 5a shows that $p(N)$ for various $Ra/Ra_{c}$ can be well described by $p(N){=}AN^{-b}e^{-N/N_{c}}$. Here $b$ and $N_{c}$ are the fitting parameters with their dependence of $Ra/Ra_{c}$ plotted in the inset. For clusters in small size $N$, $p(N)$ first decays as a power function $N^{-b}$ up to a cutoff size $N_{c}$. Studies of the collective behavior in various natural systems have revealed that local aggregation of interacting entities is the essential ingredient for the power-law decay of the group-size distributions in these systems [38, 39, 40]. In the present vortex system, each vortex is surrounded more likely by counter-rotating vortices in a densely populated state (Fig. 4). Owing to the vortex-pair interaction, adjacent vortices of opposite-sign tend to move in similar directions [41] (see Supplementary Information 3 for details). Moreover, the shield structure forming near the edge of each vortex prevents strong interactions in closer proximity [26, 28, 30], such as vortex merging and annihilation. As a result, isolated vortices are often aggregated into neighboring clusters and move collectively. The power-law exponent, $b{=}1.5{\pm}0.04$, is found to be independent of $Ra/Ra_{c}$ (inset of Fig. 5a), and falls into the range of previous theoretical predictions [38, 42]. For large $N$ we find that $p(N)$ evolves into an exponential tail with the cutoff size $N_{c}$ varying with $Ra/Ra_{c}$ and reaching a maximum at $Ra/Ra_{c}{=}1.62$, where the vorticity ratio $\gamma_{\omega}$ is maximum (see Fig. 3a). The rescaled data $p(N)N_{c}^{1.5}/A$ as a function of $N/N_{c}$ thus collapse into one master curve as shown in Figure 5b.

Dynamical systems consisting of clustered entities often exhibit scale-invariant, collective motions [12]. Here we further analyze the spatial correlation function of vortex velocity fluctuation within a vortex cluster $C(l){=}\sum_{ij}[{\vec{u^{\prime}}_{i}(\vec{r}_{i}{+}\vec{l}){\cdot}\vec{u^{\prime}}_{j}(\vec{r}_{j})\delta(l{-}l_{ij})}]/[C_{0}{\cdot}\sum_{ij}\delta(l{-}l_{ij})]$, where $\vec{u}^{\prime}_{i}{=}\vec{u}_{i}{-}\vec{V}$ is the relative vortex velocity around the mean cluster velocity $\vec{V}{=}\sum_{i}{\vec{u}_{i}}/N$, $l_{ij}$ is the distance between the vortex pair $(i,j)$ and $C_{0}$ is a normalization constant. Figure 5c shows that $C(l)$ decreases as the distance $l$ increases, with the decay length depending on the cluster size $N$. In Fig. 5d we present the correlation function $C(l/L)$, scaled by the cluster length $L$, for clusters with various sizes. This rescaling leads to the converging of the data onto a single curve representing a stretched exponential function, which crosses zero at the correlation length $l_{c}{\approx}0.3L$ for all cluster sizes $N$. Thus the correlated motions of the vortices are long-range and scale-free, i.e. there is no characteristic length scale here except the length $L$ of the cluster. We remark that the scattering of data points at large distances ($l{\approx}L$), owing to insufficient statistics, has negligible influence in the determination of $l_{c}$.