Large-scale Vortices in Rapidly Rotating Rayleigh–Bénard Convection at Small Prandtl number

Figure 1 shows the flow structures for the simulation cases with Ra = 10⁸. The left column displays the horizontal cuts of the axial vorticity at the plane z = 0.25. The right column displays the vertical cuts of the axial vorticity at the plane y = 0.75. Simulation cases with $\widetilde{{Ra}}=1,4,10$ and 90 (cases C1, C3, C5, and C13) are arranged from the top to bottom. Obviously the flow structures are quite different for these cases. For the case $\widetilde{{Ra}}=1$, the flow is near the onset of oscillatory convection. The structure shows up as small vortices of alternately positive and negative axial vorticities. The structure of axial vorticity tends to be antisymmetric about the midplane. The vortices pattern is similar to those observed in simulations and experiments of Chong et al. (2020), but the motions are different. In their simulations and experiments at Pr = 4.38, they observed that small vortices moved as Brownian motions. However, in our simulations small vortices tend to move as following traveling waves (see animation for Figure 14). The reason for the difference is probably that we use a small Prandtl number, such that oscillatory wave can be easily excited in our cases. For the case $\widetilde{{Ra}}=4$, a coexisting large-scale cyclone and anticyclone extend throughout the whole domain. Intense shear flow is driven in the interacting region of these LSVs. Time evolution of the flow structure shows that these LSVs are formed through clustering and merging processes of small vortices (see animation for Figure 15). For the case $\widetilde{{Ra}}=10$, only a large-scale cyclone appears, accompanied by small vortices advected by a large-scale anticyclonic circulation (see animation for Figure 16). For the case of $\widetilde{{Ra}}=90$, the condensation of convective flow disappears, and the flow structure tends to be three-dimensional turbulent (see animation for Figure 17).

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Figure 2 summarizes the simulation cases on the $\widetilde{{Ra}}-{Ro}$ plane. In this figure, three groups of simulation cases with different Rayleigh numbers Ra = 10⁶, 10⁷, and 10⁸ are marked as lines with slopes of 3/4 (Ra increases from the upper left to the bottom right). Four regimes are identified with different symbols on the figure: the square symbol for multiple small vortices (Regime I); the circle plus symbol for the coexisting large-scale cyclone and anticyclone (Regime II); the circle symbol for the large-scale cyclone (Regime III); and the cross symbol for turbulence (Regime IV). The appearance of LSVs highly depends on the system-scale Rossby number Ro and vertical Reynolds number Re_z . Here, Re_z is defined as ${{Re}}_{z}\,=\,u{{\prime\prime} }_{z}{Re}$, where the double prime denotes the rms average of the whole domain. The empirical result shows that two conditions must be satisfied for the emergence of LSVs. First, the system-scale Rossby number must be smaller than a value of order unity. Our simulations suggest Ro ≲ 0.4. Second, the vertical Reynolds number Re_z should be larger than a value of about 400, so that turbulent convection can be developed in the vertical direction. Coriolis force plays an important role in the formation of LSVs. When Ro is large, Coriolis effect is unimportant and the flow is more likely to be turbulent if the Reynolds number is large enough. A large-scale cyclone appears when Ro is small enough. The line Ro = 0.4 in Figure 2 clearly separates Regime IV from Regime III. When Ro further decreases, a large-scale anticyclone emerges and coexists with a large-scale cyclone. In our calculation, the transition from Regime III to Regime II occurs at around Ro = 0.1. Apart from Regime II and Regime III, another regime with multiple small vortices (Regime I) exists when the Rossby number is small. In Regime I, the Rayleigh number is just above the supercritical value. Re_z is smaller than 400 in this regime, which means the flow is more likely to be laminar than turbulent along the vertical direction. The conditions on the appearance of LSVs were also discussed in Favier et al. (2014) and Guervilly et al. (2014), but only for large-scale cyclones in their cases. They have identified that large-scale cyclones appear when the Rossby number is smaller than a critical value and the Reynolds number is greater than a value of about 100. Our simulation contributes to the literature on RBBC by showing that a regime of coexisting large-scale cyclones and anticyclones may appear when the Ro further decreases. Käpylä et al. (2011) and Chan & Mayr (2013) also investigated the condition for the emergence of LSVs in their simulations on compressible convection. Apart from Regime II and III, they reported another regime where anticyclone dominates. However, this regime has not been identified in our current numerical result, probably because of the lack of compressible effect in the Boussinesq flow.

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To investigate how energy is distributed, we first calculate the power spectral density of kinetic energy at different wavenumbers. Since our simulation cases are aperiodic in the vertical direction, we only compute the two-dimensional kinetic energy spectrum P₂(k) on the horizontal space (Chan & Sofia 1996; Cai 2018). For a Boussinesq flow, the kinetic energy density P₂(k) at a specific layer can be evaluated as (Cai 2020)

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \int {P}_{2}(k){dk}=\sum _{m}\sum _{n}{a}_{m,n}^{2}({\left|{{\rm{\Psi }}}_{m,n}\right|}^{2}+{\left|{D}_{z}{{\rm{\Phi }}}_{m,n}\right|}^{2}\\ \quad +{a}_{m,n}^{2}{\left|{{\rm{\Phi }}}_{m,n}\right|}^{2})\,,\end{array}\end{eqnarray} \tag{ 8 }$$

where $k=[{\left({m}^{2}+{n}^{2}\right)}^{1/2}]$ is the horizontal wavenumber (the brackets mean the number is rounded off to an integer), ${a}_{m,n}=(2\pi /L){\left({m}^{2}+{n}^{2}\right)}^{1/2}$, L = 1 is the lateral size of the box, and m, n are the spectral numbers in the x- and y-directions, respectively.

Figure 3 shows the compensated power spectral density kP₂(k) as a function of k for cases C1 ($\widetilde{{Ra}}=1$), C3 ($\widetilde{{Ra}}=4$), C5 ($\widetilde{{Ra}}=10$), and C13 ($\widetilde{{Ra}}=90$), respectively. First, we see that the power spectral densities do not vary significantly at different heights, despite the fact that a faster decay rate is observed for small-scale motions at the midplane in rapidly rotating cases. Second, we notice that the energy spectral densities scale approximately as k⁻³ within the wavenumber range 1 ≤ k ≤ 3 for cases C3 and C5, which is consistent with the scaling in a large-scale condensation.

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In order to study the effects of rotation on the energy transfer of turbulent convection, we decompose the fluid motion into depth-averaged barotropic (2D) and depth-dependent baroclinic (3D) components (Julien et al. 2012; Favier et al. 2014). For example, the 2D barotropic and 3D baroclinic components of velocity are defined as

$$\begin{eqnarray}&&\langle {\boldsymbol{u}}\rangle (t,x,y)={\int }_{0}^{1}{\boldsymbol{u}}(t,x,y,z){dz}\,,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\boldsymbol{u}}^{\prime} (t,x,y,z)={\boldsymbol{u}}(t,x,y,z)-\langle {\boldsymbol{u}}\rangle (t,x,y)\,,\end{eqnarray} \tag{ 10 }$$

respectively. Also, we use the symbol overbar (e.g., $\overline{\langle {u}_{z}\rangle }$) to represent the corresponding temporal averages.

After the decomposition, we can calculate the power spectral densities of the kinetic energies of 2D barotropic and 3D baroclinic components. Figure 4 presents the vertically averaged power spectral density of kP₂(k) as functions of k. The contributions from 2D and 3D components and their summation are shown with blue, brown, and yellow lines, respectively. The Kolmogorov −5/3 scaling law in three-dimensional turbulence and the enstrophy cascade −3 scaling law in two-dimensional turbulence are also shown for references. For the case $\widetilde{{Ra}}=90$ shown in Figure 4(D), the energy contained in the 2D component is small compared to that in 3D component. It is reasonable since in this regime the rotational effect is small and the flow is more likely to be three-dimensional turbulent. The 2D effect starts to play a role, when rotation rate increases and LSVs appear (cases $\widetilde{{Ra}}=4$ and $\widetilde{{Ra}}=10$). Figures 4(B) and (C) clearly shows that more energy is contained in 2D component for large-scale motions (small k). For small-scale motions, more energy is still contained in 3D component. When $\widetilde{{Ra}}$ further decreases to 1 (Figure 4(A)), the energy contained in the 2D component is comparable to that in the 3D component for small-scale motions, and much larger than that in the 3D component for large-scale motions. From the discussion, we see that the 2D effect are more and more important when Ro decreases.

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To further illustrate the transfer of kinetic energy among different scales, we define the kinetic energy transfer coefficients from shell Q to shell K (Alexakis et al. 2005; Mininni et al. 2005; Favier et al. 2014) as

$$\begin{eqnarray}&&{ \mathcal T }(Q,K)=-{\int }_{V}\overline{{{\boldsymbol{u}}}_{K}\cdot ({\boldsymbol{u}}\cdot {\rm{\nabla }}{{\boldsymbol{u}}}_{Q})}{dV}\,,\end{eqnarray} \tag{ 11 }$$

where the shells Q and K are defined on the spectral space with horizontal wavenumbers in the range of k ∈ (Q − 1, Q] and k ∈ (K − 1, K], respectively; u_Q and u_K are the corresponding shell filtered velocities; and V is the volume of computational domain. Positive (negative) ${ \mathcal T }(Q,K)$ means that the kinetic energy is taken from (given to) shell Q and given to (taken from) shell K. The left column of Figure 5 shows ${ \mathcal T }(Q,K)$ four selected cases in the four regimes. For the case $\widetilde{{Ra}}=90$ in Regime IV, the flow is not condensed and the kinetic energy is taken from larger scales to smaller scales. However, for the case $\widetilde{{Ra}}=10$ in Regime III, we see from Figure 5 that energy can be directly transferred from small scales (K ≤ 18) to the largest scale. This nonlocal inverse energy cascade provides energy to sustain the formation of the large-scale cyclone. It is in agreement with the results of the previous numerical simulations on RRBC (Favier et al. 2014; Guervilly et al. 2014). For the case $\widetilde{{Ra}}=4$ in Regime II, despite that energy is taken from small scales (5 ≤ K ≤ 22), we also observe a direct energy transfer from the largest scale to moderate scales (K = 4). The reason might be associated with the appearance of an anticyclone. Compared with a cyclone, the convection is more turbulent within the anticyclone because the effective rotating speed is smaller (Käpylä et al. 2011; Chan & Mayr 2013). With more vigorous turbulent motions, the energy is more likely transferred from large scales to small scales. For the case $\widetilde{{Ra}}=1$ in Regime I, the vertical Reynolds number is small and the Ekman number is high, and the rotation effect dominates the buoyant effect at small scales. Thus, we observe that the kinetic energy transfer is cut off at small scales (K ≤ 17). The right column of Figure 5 shows ${\sum }_{Q}{ \mathcal T }(Q,K)$, which is the total energy transferred from all Q to a single shell K. A negative value of ${\sum }_{Q}{ \mathcal T }(Q,K)$ means that the wavenumber K supplies energy, thus this wavenumber can be thought as a forcing wavenumber. From the right column of Figure 5, we see that the forcing wavenumbers for cases C1, C3, C5, and C13, are 7 ≤ K ≤ 11, 2 ≤ K ≤ 13, 2 ≤ K ≤ 12, and 1 ≤ K ≤ 6, respectively. When rotational effect increases, the forcing wavenumbers tend to shift from large-scale wavenumbers to small-scale wavenumbers. In the turbulence regime (Regime IV), the mode K = 1 is a forcing wavenumber. However, it is no longer a forcing wavenumber in other regimes where the rotational effects are dominant.

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To consider the energy transfer between the 2D and 3D components, we further define the the self- and cross-transfer coefficients as

$$\begin{eqnarray}&&{{ \mathcal T }}_{22}(Q,K)=-{\int }_{V}\overline{\langle {{\boldsymbol{u}}}_{K}\rangle \cdot ({\boldsymbol{u}}{\rm{\nabla }}\cdot \langle {{\boldsymbol{u}}}_{Q}\rangle )}{dV}\,,\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{{ \mathcal T }}_{32}(Q,K)=-{\int }_{V}\overline{\langle {{\boldsymbol{u}}}_{K}\rangle \cdot ({\boldsymbol{u}}{\rm{\nabla }}\cdot {{\boldsymbol{u}}}_{Q}^{\prime} )}{dV}\,,\end{eqnarray} \tag{ 13 }$$

where ${{ \mathcal T }}_{22}(Q,K)$ and ${{ \mathcal T }}_{32}(Q,K)$ measure the interactions of the 2D and 3D components of shell Q with the 2D component of shell K, respectively. Figures 6(A) and (C) show the heatmaps of ${{ \mathcal T }}_{22}(Q,K)$ and ${{ \mathcal T }}_{32}(Q,K)$ for case C5. From Figure 6(A), we note that the local energy transfer from large to small scales is an important process in self-interaction of 2D components. Apart from the local energy transfer, we also observe a significant direct energy transfer process from moderate scales 3 ≤ Q ≤ 14 to the largest scale K = 1. The cross interaction of 3D and 2D components (Figure 6(C)), however, shows that the moderate scales of 3D component (2 ≤ Q ≤ 6) take energy from the largest scale of 2D component (K = 1). The small scales of 3D component (8 ≤ Q ≤ 48), on the other hand, put energy into the largest scale of 2D component (K = 1). To investigate the net effect of self- and cross -transfer of kinetic energy, we have computed the total energy transfer from all shell Q to a single shell K. Figures 6(B) and 6(D) show ${\sum }_{Q}{{ \mathcal T }}_{22}(Q,K)$ and ${\sum }_{Q}{{ \mathcal T }}_{32}(Q,K)$ as a function of K, respectively. Apparently, the 3D component has a net effect of taking energy from shell K = 1 (Figure 6(D)). However, the 2D component shows a net effect of putting energy into the shell K = 1 (Figure 6(B)), indicating that the LSVs are probably maintained by a 2D self-transfer process in this case.

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In this subsection, we discuss the heat transfer in RBBC. To study the 2D and 3D effects on heat transfer, we separate the heat flux into 2D and 3D components. To achieve this, we first split u_z and Θ into two parts, by letting ${u}_{z}=\langle {u}_{z}\rangle +{u}_{z}^{\prime}$ and ${\rm{\Theta }}=\langle {\rm{\Theta }}\rangle +{\rm{\Theta }}^{\prime}$. Here, ${u}_{z}^{\prime}$ and Θ are three-dimensional perturbations from the two-dimensional integrated mean values. Then we can split the convective flux into 2D and 3D components by

$$\begin{eqnarray}&&{F}_{c}={Pe}\overline{\langle {u}_{z}{\rm{\Theta }}\rangle }={F}_{c,2{\rm{D}}}+{F}_{c,3{\rm{D}}},\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}&&{F}_{c,2{\rm{D}}}(x,y)={Pe}\overline{\langle {u}_{z}\rangle \langle {\rm{\Theta }}\rangle }\,,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{F}_{c,3{\rm{D}}}(x,y)={Pe}\overline{\langle u{{\prime} }_{z}{\rm{\Theta }}^{\prime} \rangle }\,.\end{eqnarray} \tag{ 16 }$$

Here, F_c,2D measures the heat flux transported by 2D convection, and F_c,3D measures the heat flux transported by 3D convection. Both F_c,2D and F_c,3D take average on convective flux temporally and vertically, and thus they are functions of x and y. As mentioned earlier, both cyclonic and anticyclonic regions appear in Regimes II and III. The cyclonic and anticyclonic regions can be shown more clearly after taking temporal and vertical averages. As were shown in the animations for Figures 15 and 16, the cyclonic and anticyclonic regions move around with time, therefore the temporal average cannot be taken for too long a period. For both cases, we take the average for a period of about 20 units of time. The nondimensional system rotation period is 4π Ro, thus the averaged time covers about 18 system rotation periods for case C3 and nine system rotation periods for case C5. Within this time period, the cyclones and anticyclones do not drift too much away.

The results of case C3 (Regime II) and C5 (Regime III) are shown in Figures 7 and 8, respectively. The pattern at Figure 7(E) shows clearly the cyclonic (the region marked by red circle) and anticyclonic (the region marked by blue circle) regions on the contour plot of the averaged vertical component of vortical structure $\overline{\langle {\omega }_{z}\rangle }$. Figures 7(A) and (B) show the averaged vertical velocity $\overline{\langle {u}_{z}\rangle }$ and temperature perturbation $\overline{\langle {\rm{\Theta }}\rangle }$. Both $\overline{\langle {u}_{z}\rangle }$ and $\overline{\langle {\rm{\Theta }}\rangle }$ show structures of twisted rolls in the cyclonic and anticyclonic regions. In simulations of rapidly rotating compressible flow, Chan & Mayr (2013) have found that a cyclone has lower temperature, while an anticyclone has higher temperature in the core region. Our result of RBBC shows different temperature structures in these LSVs. One possible reason may be that our RBBC simulations lack of compressible effect. In Boussinesq flow, the effect of density variation is only considered in the buoyancy term, while the variation on the horizontal direction is ignored. However, in the compressible flow, the horizontal variation of density has significant effect on the horizontal variation of temperature, which may help create a lower (higher) temperature core for a cyclone (anticyclone). Figures 7(C) and (D) show the convective fluxes transported by 2D and 3D components. We note that the 2D component F_c,2D almost transports positive convective flux. The 3D component F_c,3D tends to transport positive convective fluxes in the anticyclonic region. However, in the cyclonic region, the convective flux transported by it can either be positive or negative. It indicates the 3D turbulent motions plays a more important role in transporting convective flux in the anticyclonic region rather than the cyclonic regions. Similar behaviors are observed in Figure 8.

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In order to quantitatively compare the efficiencies of heat transportation between the cyclonic and anticyclonic regions, we take averages on F_c,2D and F_c,3D within the disks around the cyclonic and anticyclonic spots (the centers of the red and blue circles in $\overline{\langle {\omega }_{z}\rangle }$), respectively. Figure 7(F) shows the averaged convective fluxes in the cyclonic and anticyclonic regions for case C3 by red and blue curves, respectively. In the anticyclonic region, the contributions of heat transportation by 2D and 3D components are comparable. However, in the cyclonic region, the heat transportation by the 2D component dominates that by the 3D component. For case C5, the result is different. In the cyclonic region, the heat transportation by 2D and 3D components are comparable. On the other hand, in the anticyclonic region, the heat transportation by the 3D component is much higher than that by the 2D component. By comparing cases C3 and C5, we see that the 2D component plays more and more important role in both cyclonic and anticyclonic regions when rotation rate increases. To verify this trend, we have also computed the averaged convective fluxes transported by 2D and 3D components in case C1. It has been found that the convective flux transported by 2D component is about four times that by the 3D component. From the plot of $\overline{\langle {\omega }_{z}\rangle }$ in Figure 9(A), we see that case C1 also contains cyclonic and anticyclonic regions, but we have not found significant difference of convective fluxes transported between these two regions. Although the cyclonic and anticyclonic regions are well separated in an averaged sense (Figure 9(A)), the averaged vertical velocity $\overline{\langle {v}_{z}\rangle }$ has not shown similar distribution (Figure 9(B)). Apart from $\overline{\langle {v}_{z}\rangle }$, both the averaged horizontal velocities $\overline{\langle {v}_{x}\rangle }$ (Figures 9(C) and (E)) and $\overline{\langle {v}_{y}\rangle }$ (Figures 9(D) and (F)) have developed shear structures. The shear velocity in the x-direction is larger than that in the y-direction by an order of magnitude. As a result, the group motions of small vortices are more prominent along the x-direction (see animation for Figure 14). If the shear velocities in the x- and y-directions are comparable, then we would expect that LSVs could be formed. To examine whether the shear flow is preferred in the x-direction somehow in the system or just by chance, we continue the simulation of case C1 by switching the x- and y-directions. From the animation for Figure 18, we see that the shear flow is preferred in one direction by chance.

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From the above discussion, we have come to the the following conclusions. First, as rotation rate increases, the 2D effect increases and the 3D effect decreases in transporting convective flux. Second, in Regimes II and III when LSVs appear, heat transfer by convection is more efficient in the anticyclonic region than in the cyclonic region. It is consistent with the observation that convection in the anticyclone is more turbulent because the effective rotation is smaller (Chan & Mayr 2013). In the simulations of Boussinesq flow, Guervilly et al. (2014) also observed a significant reduction of heat transfer inside the cyclone.

In this subsection, we investigate the effects of rotation on the statistical results of velocities and Nusselt number. To compare the results among different groups, we use the nonrotating case within each group as a reference case and normalize all the statistical values by the corresponding reference values. Figures 10(A)–(C) show the normalized rms velocities u''/u''_∞, $u{{\prime\prime} }_{z}/{u{{\prime\prime} }_{z}}_{\infty }$, and $u{{\prime\prime} }_{h}/{u{{\prime\prime} }_{h}}_{\infty }$ as functions of Ro, respectively. The subscript ∞ denotes the value of the reference nonrotating case. First, we note that the variation of u''/u''_∞ on Ro is not monotonic. It shows an increasing trend with increasing Ro in the Regimes I, II, and IV, while a decreasing trend with increasing Ro in Regime III. The trend reversal in Regime III is due to the appearance of large-scale cyclones, in which the flow is dominant by the horizontal large-scale motions. This can be further verified by looking at Figures 10(B) and (C), where the reversal trend is only observed in the horizontal but not in the vertical velocity profiles. Second, we note that the normalized velocities in different groups almost collapse into a single curve in Regime IV. In this regime, the normalized velocities are mainly affected by the convective Rossby number. In the vicinity of Ro ∼ 1 of Regime IV, the normalized vertical velocity is estimated to approximately obey a scaling of $u{{\prime\prime} }_{z}/{u{{\prime\prime} }_{z}}_{\infty }\propto {{Ro}}^{2/7}$. In other fast rotating regimes, apart from the convective Rossby number, the normalized velocities also depend on Rayleigh numbers. Cases with higher Rayleigh number tend to have higher normalized vertical velocities. Although the curves of $u{{\prime\prime} }_{z}/{u{{\prime\prime} }_{z}}_{\infty }$ for different groups are separated, we still find that $u{{\prime\prime} }_{z}/{u{{\prime\prime} }_{z}}_{\infty }$ approximately obeys a scaling of $u{{\prime\prime} }_{z}/{u{{\prime\prime} }_{z}}_{\infty }\propto {{Ro}}^{5/3}$ in these regimes. Figure 10(D) shows the normalized modified Nusselt number (Nu − 1)/(Nu_∞ − 1), where the Nusselt number is defined as ${Nu}=\int \left[{Pe}\langle \overline{{u}_{z}{\rm{\Theta }}}\rangle -\langle \partial \overline{{\rm{\Theta }}}/\partial z\rangle \right]{dz}+1$. As seen from the figure, (Nu − 1)/(Nu_∞ − 1) always increases with increasing Ro, which indicates that rotation has a negative effect on heat transfer. The slope of (Nu − 1)/(Nu_∞ − 1) has a decreasing trend with increasing Ro. In the rapidly rotating regimes (Ro < 0.4), the curves of (Nu − 1)/(Nu_∞ − 1) in each group approximately follows a scaling (Nu − 1)/(Nu_∞ − 1) ∝ Ro³. It should be mentioned here that the derived scalings of u''_z /u''_z and (Nu − 1)/(Nu_∞ − 1) on Ro are empirical results, which currently lack theoretical explanations. In the low Rossby regime, only a few data points are available for the fits. The robustness of these scalings needs to be checked when more data points are available in the future.

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It has long been found that asymmetry between cyclones and anticyclones emerges in rapidly rotating convection (Chan 2007; Käpylä et al. 2011; Guervilly et al. 2014; Guervilly & Hughes 2017). Guervilly et al. (2014) have discussed several possible mechanisms to explain the asymmetry. The most likely mechanism is that the thermal plumes ejected from the thermal boundary layers tend to induce more cyclonic vorticity. Therefore the clustering of the like-signed cyclonic vorticity favors the formation of the large-scale cyclone. In the low Rossby number limit, both ejection and injection of thermal plumes are allowed in the thermal boundary layer (Vorobieff & Ecke 2002; Sprague et al. 2006). Thus, it is expected that the asymmetry would disappear when Ro is very small (Julien et al. 2012). Guervilly et al. (2014) have studied this mechanism for cases with dominant large-scale cyclone. Here, we extend the discussion of this mechanism across different flow pattern regimes. Following the work of Guervilly et al. (2014), we define the axial vorticity skewness as

$$\begin{eqnarray}&&S(z)=\displaystyle \frac{{\int }_{A}\overline{{\omega }_{z}^{3}}{dA}/{\int }_{A}{dA}}{{\left({\int }_{A}\overline{{\omega }_{z}^{2}}{dA}/{\int }_{A}{dA}\right)}^{3/2}}\,,\end{eqnarray} \tag{ 17 }$$

the z-dependent axial vorticity skewness as

$$\begin{eqnarray}&&S^{\prime} (z)=\displaystyle \frac{{\int }_{A}\overline{{\left({\omega }_{z}-{\int }_{z}{\omega }_{z}{dz}\right)}^{3}}{dA}/{\int }_{A}{dA}}{{\left[{\int }_{A}\overline{{\left({\omega }_{z}-{\int }_{z}{\omega }_{z}{dz}\right)}^{2}}{dA}/{\int }_{A}{dA}\right]}^{3/2}}\,,\end{eqnarray} \tag{ 18 }$$

and the z-invariant axial vorticity skewness as

$$\begin{eqnarray}&&\overline{S}=\displaystyle \frac{{\int }_{A}\overline{{\left({\int }_{z}{\omega }_{z}{dz}\right)}^{3}}{dA}/{\int }_{A}{dA}}{{\left[{\int }_{A}\overline{{\left({\int }_{z}{\omega }_{z}{dz}\right)}^{2}}{dA}/{\int }_{A}{dA}\right]}^{3/2}}\,,\end{eqnarray} \tag{ 19 }$$

where dA = dxdy is the differential area element in the horizontal planes.

Figure 11 shows S(z), $S^{\prime} (z)$, and $\overline{S}$ for cases C1, C3, C5, and C13. For case C13 (Regime IV), S(z) and $S^{\prime} (z)$ have a similar structure (Figure 11(C)), with a distribution of large positive values near the thermal boundary layers and small positive values (almost zeros) in the middle of the box. Figure 1 also shows that small but strong cyclonic vortex structures are created in the thermal boundary layers in this case. These vortex structures can only penetrate a short distance, thus the skewness profiles decrease rapidly away from the boundaries. For case C5 (Regime III), a large-scale cyclone appears so that the z-invariant axial vorticity skewness $\overline{S}$ has a large positive value (Figure 11(C)). S(z) and $S^{\prime} (z)$ have a similar profile near the boundaries. In the middle region, S(z) are larger than $S^{\prime} (z)$ but both of them are positive. From Figure 1, we can see that both cyclonic and anticyclonic vortex structures have been developed in the thermal boundary layers, but the cyclonic vortices are stronger and can penetrate farther away from the boundaries. For case C3 (Regime II), although a pair of cyclone and anticyclone are formed, asymmetry between the cyclone and anticyclone has not completely disappeared ($\overline{S}$ is about one in Figure 11(B)). Similar to case C5, both cyclonic and anticyclonic vortex structures have been developed in the thermal boundary layers in case C3 (Figure 1). However, the anticyclonic vortex structures in case C3 are stronger and can penetrate deeper than those in case C5. As a result, the z-dependent axial vorticity skewness $S^{\prime} (z)$ is close to zero in the middle region (Figure 11(B)). For case C1 (Regime I), anticyclonic structures have similar strengths as cyclonic structures (Figure 1). Since the vortical profile is almost antisymmetric about the middle plane (Figure 1), z-invariant axial vorticity skewness $\overline{S}$ is nearly zero (Figure 11(A)). S(z) is positive in both lower and upper half boxes (Figure 11(A)), which indicates that the vortical profile is not exactly antisymmetric (cyclonic vorticity is stronger than anticyclonic vorticity). $S^{\prime} (z)$ turns to be negative near the boundaries in this case, but its value is too small to draw an affirmative conclusion.

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In the study of Vorobieff & Ecke (2002), they showed that flow patterns are dominated by cyclonic thermal plumes when Ro ∼ 1. When Ro is further reduced, the number of anticyclonic thermal plumes increases but cyclonic thermal plumes are still favored. Our simulation result agrees well with their experimental studies. Our result also supports the mechanism proposed by Guervilly et al. (2014), that is, the preference for cyclonic thermal plumes might be responsible for the asymmetry between cyclones and anticyclones.

From RBBC simulations at Pr = 1, Favier et al. (2014) reported a higher critical convective Rossby number Ro_c1 ≈ 0.6 for the appearances of LSVs. In most of their simulations, the aspect ratios of simulation boxes are higher than one (Γ > 1). Guervilly et al. (2014) has also reported that the aspect ratio has impact on the appearances of LSVs. To investigate the effects of Pr and Γ, we have run several simulations at Pr = 1 with different aspect ratios. The simulation parameters are listed in Table 2. First, we notice that from cases D1–D7 that the criteria on the appearance of LSVs at Pr = 0.1 can also be applied to cases at Pr = 1. That is, for Γ = 1, the appearance of LSVs generally requires Ro < 0.4 and ${{Re}}_{z}\gt 400$. Figure 12(A) shows the flow structure of case D1. In this case, multiple small vortices appear as the vertical Reynolds number is small. Figure 12(B) shows the flow structure of case D3. Since Ro < 0.4 and ${{Re}}_{z}\gt 400$, a large-scale cyclone appears as expected. Figure 12(C) shows the flow structure of case D7. For this case, Ro = 0.55 and there is no evidence of the appearance of LSVs.

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Table 2. Parameters of Simulation Cases at Pr = 1

Case	Γ	ℓc	u''	uz ''	uh ''	Ra	Rez	Ro	E	$\widetilde{{Ra}}$	Nu	Regime
D1	1	0.082	0.0606	0.0208	0.0567	3 × 108	359.95	0.0850	4.905 × 10−6	25	7.463	I
D2	1	0.097	0.1510	0.0430	0.1445	3 × 108	744.42	0.1429	8.249 × 10−6	50	21.259	III
D3	1	0.116	0.2228	0.0612	0.2140	3 × 108	1060.08	0.2403	1.387 × 10−5	100	37.563	III
D4	1	0.128	0.2229	0.0690	0.2116	3 × 108	1195.18	0.3257	1.880 × 10−5	150	46.023	III
D5	1	0.138	0.1497	0.0774	0.1272	3 × 108	1339.81	0.4041	2.333 × 10−5	200	52.596	IV
D6	1	0.146	0.1422	0.0819	0.1148	3 × 108	1419.17	0.4777	2.758 × 10−5	250	56.295	IV
D7	1	0.152	0.1369	0.0850	0.1054	3 × 108	1472.57	0.5477	3.162 × 10−5	300	59.711	IV
D7b	2	0.152	0.1882	0.0825	0.1685	3 × 108	1428.41	0.5477	3.162 × 10−5	300	58.275	III
D7c	3	0.152	0.2067	0.0818	0.1893	3 × 108	1415.96	0.5477	3.162 × 10−5	300	57.983	III
D8	1	0.048	0.0724	0.0144	0.0709	3 × 109	786.16	0.0548	1.000 × 10−6	30	9.246	II

Note. Γ is the lateral-to-height aspect ratio; ℓ_c is the critical wavelength for the onset of convection; u'' is the rms velocity; u_z '' is the rms vertical velocity; u_h '' is the rms horizontal velocity; Ra is the Rayleigh number; Re_z is the vertical Reynolds number; Ro is the convective Rossby number; E is the Ekman number; $\widetilde{{Ra}}$ is the modified Rayleigh number; Nu is the Nusselt number; and the cases are classified into four regimes according to flow patterns. Temporally and spatially (the whole box) averaged values are reported. The grid resolutions are N_x × N_y × N_z = 256 × 256 × 257 for D1–D7, 512 × 512 × 257 for D7b and D8, and 768 × 768 × 257 for D7c.

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In the work of Favier et al. (2014), they performed the same simulation as case D7, except that a different aspect ratio with Γ = 3 was used. However, they have observed LSVs in their simulation. Thus, the aspect ratio probably is important to the formation of LSVs (Favier et al. 2014; Guervilly et al. 2014). To investigate the effect of Γ, we performed two companion simulations D7b and D7c with different aspect ratios Γ = 2 and Γ = 3, respectively. Figures 12(D) and (E) show the flow structures of D7b and D7c. Apparently, large-scale cyclones appear in these two cases. This is consistent with the result of Favier et al. (2014). Therefore, aspect ratio indeed affects the appearance of LSVs. Compared to simulations at Γ = 1, the critical convective Rossby number on the appearance of LSVs is higher for simulations at larger Γ.

Favier et al. (2014) have also reported several cases with Ro ≤ 0.1 at Pr = 1 and Γ = 1. In our simulations at Pr = 0.1, we find that a coexisting large-scale cyclone and anticyclone appear when Ro < 0.1 and Re_z is large enough. One may ask whether this criterion can be applied to simulations at higher Pr. To investigate this problem, we have run a simulation case at Pr = 1 with Ro = 0.0548 and Ra = 3 × 10⁹. This simulation case was also reported in Favier et al. (2014), but here we have used a higher grid resolution. Figure 12(F) shows that a coexisting large-scale cyclone and anticyclone indeed appear in this case. Thus, the criterion obtained in low Pr regimes is probably also valid in high Pr regimes. An affirmative conclusion requires more simulations across different parameter regimes.

Figure 13 presents a summary of simulation results from different studies. As shown in the figure, for the appearance of LSVs, it requires that Ro is smaller than a certain critical value. We find this critical value is about Ro ∼ 0.4 for a unit box, but Favier et al. (2014) reported a higher value of about Ro ∼ 0.6 for a wider box. The variation of this critical Ro on the size of box needs to be further investigated. A conjecture would be that the critical Ro increases slightly with Γ. The appearance of a coexisting cyclone and anticyclone occurs at a lower critical value Ro ∼ 0.1 in a unit box. Interestingly, Stellmach et al. (2014) found a pair of cyclone and anticyclone at Ro = 0.044 and Pr = 1 in a small box with Γ ≈ 0.22. If the critical Ro indeed increases with Γ, then we would expect that the critical Ro for the appearance of a pair of cyclone and anticyclone is slightly lower than 0.1 in Γ = 0.22. The case in Stellmach et al. (2014) has a Ro = 0.044 < 0.1, which still satisfies the criterion for the emergence of a coexisting cyclone and anticyclone. From the simulation results, we speculate that the criteria for the appearances of LSVs could possibly be universal for flows at different Pr.

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