THE EFFECT OF ROTATION ON OSCILLATORY DOUBLE-DIFFUSIVE CONVECTION (SEMICONVECTION)

Figure 3 shows the turbulent compositional flux as a function of time for each of the simulations with Pr = τ = 0.1 and ${R}_{0}^{-1}=1.25$, focusing on the growth and saturation of basic instability of the ODDC. It clearly shows that the growth of the linear instability in simulations of rotating ODDC (with θ = 0) behaves similarly to the nonrotating case. This is not surprising since the overall growth of the linear instability is dominated by the fastest-growing modes, which in this case are completely unaffected by rotation (see Section 3).

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After the initial growth of the linear instability, each simulation reaches a nonlinear saturation (at around t = 300 in each case) and becomes homogeneously turbulent. Figure 3 shows that the compositional flux in the homogeneously turbulent phase is roughly independent of Ta* at low Ta*. For Ta* = 0.01, 0.1, and 1, the mean fluxes during this phase are statistically similar to one another, while the most rapidly rotating simulation (Ta* = 10) reaches a plateau that is slightly higher than the others. Note that the composition flux in the nonrotating simulation (Ta* = 0) behaves differently, because in this case layers begin to form almost immediately after saturation. This causes it to continue to grow after saturation (albeit at a slower rate), never achieving a quasi-steady state as the rotating simulations do. We now look in more detail at the behavior of the low Ta* and high Ta* sets of simulations, respectively.

Figure 4 shows that in low Ta* simulations the homogeneously turbulent phase (where fluxes remain more or less statistically steady) is followed by a series of stepwise increases in the compositional (and thermal) flux, which are indicative of layers that form spontaneously through the γ-instability and then merge progressively over time. In each case, three layers initially form that then merge into two, and ultimately into a single layer with a single interface. This final configuration is statistically stationary. We therefore find the qualitative evolution of layers to be consistent with previous studies of nonrotating layered ODDC (Rosenblum et al. 2011; Wood et al. 2013). However, the progressive increase in rotation rate introduces quantitative differences between rotating and nonrotating cases, even at low Ta*. The rotation rate clearly affects the timescales for layer formation and layer mergers, respectively, with stronger rotation leading to delays in both processes.

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The formation of layers can be understood quantitatively by studying the growth of "layering modes" predicted by γ-instability theory (as in Rosenblum et al. 2011; Stellmach et al. 2011; Mirouh et al. 2012, for example). Each layering mode corresponds to a horizontally invariant, vertically sinusoidal perturbation to the background density profile. To analyze them, we therefore look at the amplitude of the Fourier modes of density perturbations with wavenumbers (0, 0, k_n), where ${k}_{n}=\tfrac{2\pi n}{{L}_{z}}$, where L_z = 100d is the domain height, and n is the number of layers in the process of forming. The evolution of the (0, 0, k₂) and (0, 0, k₃) modes as a function of time for each of the three low Ta* simulations is shown in Figure 5.

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For simulations with Ta* = 0, 0.01, and 0.1, (0, 0, k₃) is the mode that initially grows to have the largest amplitude, which explains why the staircases first form with three layers. We see that, at low Ta*, the (0, 0, k₃) modes all initially grow at roughly the same rate. This is unsurprising since rotation does not have a direct effect on the γ-instability because Coriolis terms only appear in the momentum equation in (8), which is ignored in the mean field theory upon which the γ-instability is based (Mirouh et al. 2012). Rotation could in principle have an indirect influence over layer formation by significantly affecting the turbulent fluxes in the homogeneously turbulent phase and changing the relationship between ${\gamma }_{\mathrm{tot}}^{-1}$ and ${R}_{0}^{-1}$, but as we see in Figure 3 this is not the case in the low Ta* regime.

To understand the delay in the formation of layers, we must instead look at the amplitude that the density perturbations must achieve in order to trigger the onset of layered convection. Indeed, rotation is well known to delay the onset of instability in the case of thermal convection between parallel plates (Chandrasekhar 1961), so by analogy, we expect that the localized positive density gradients caused by the growth of layering modes must be larger to trigger convective overturning and cause the staircase to appear. In the Appendix, we estimate the critical density gradient needed to trigger convection in rotating Rayleigh–Bénard convection. Using this result, we then compute the amplitude the layering modes must achieve as a function of k_n and Ta* to be

$$\begin{eqnarray}&&| {A}_{n}| =\left|\displaystyle \frac{\tfrac{3{\pi }^{4}}{{H}_{n}^{4}}{\left(\tfrac{{H}_{n}^{4}{\mathrm{Ta}}^{* }}{2{\Pr }^{2}{\pi }^{4}}\right)}^{\tfrac{2}{3}}+\tfrac{27{\pi }^{4}}{4{H}_{n}^{4}}+({R}_{0}^{-1}-1)}{2{k}_{n}}\right|.\end{eqnarray} \tag{ 23 }$$

where H_h = L_z/n = 2π/k_n is the nondimensional layer height associated with the layering mode (0, 0, k_n). This amplitude is shown in Figure 5 for each of the (0, 0, k₃) modes in the low Ta* simulations. Consistent with our idea, the layering modes stop growing shortly after achieving their respective critical amplitudes (except for the Ta* = 1 case; see Section 4.3). This indicates that layered convection has commenced, taking the form of turbulent convective plumes bounded by freely moving, stably stratified interfaces.

In each case, the mode (0, 0, k₃) is then overtaken by modes (0, 0, k₂) and ultimately (0, 0, k₁) (not shown here). These multilayer phases are metastable in that they persist over many eddy-turnover times before merging. Snapshots of the three-, two-, and one-layered phases for Ta* = 0 and Ta* = 0.1 are shown in Figures 6(b) and 7(b).

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Rotation also has a strong influence on several aspects of the dynamics of layered convection, including the mean density profile within the layers, the stability of the interfaces, and, as mentioned earlier, the merger timescale. In Figures 6(a) and 7(a), horizontally averaged density profiles show in greater detail the structure of the layers themselves in the three-, two-, and one-layered phases. Stronger rotation is correlated with larger positive density gradients in the layers themselves, which in turn necessarily lead to more stably stratified layer interfaces (at fixed ${R}_{0}^{-1}$).

The increase with rotation rate of the density gradients within the layers is similar to what occurs in rotating Rayleigh–Bénard convection (Julien et al. 1996). It is usually argued that turbulent buoyancy mixing by convection adjusts the mean density gradient (outside of any potential boundary layers) to a state of marginal stability. Combined with the fact that the theoretical critical density gradient for marginal stability increases with Ta* (see Equation (23)), our results are not surprising.

To study this quantitatively, we first calculate the z derivative of a horizontally averaged density profile and then estimate the gradient in each layer by fitting profiles from a range of time steps over which the layer is stable. We repeat this procedure for the three-, two-, and one-layered stages in the low Ta* simulations. Figure 8 shows this data as a function of the product of Pr and the thermal Rayleigh number, which, in our nondimensionalization, is defined as

$$\begin{eqnarray}&&{\mathrm{Ra}}_{T}=\displaystyle \frac{g\alpha | {T}_{0z}-{T}_{0z}^{\mathrm{ad}}| {({H}_{n}d)}^{4}}{{\kappa }_{T}\nu }={H}_{n}^{4}.\end{eqnarray} \tag{ 24 }$$

Also included are theoretical values of the density gradient calculated using Equation (30) in the Appendix, which are given by

$$\begin{eqnarray}&&\displaystyle \frac{d\langle \rho {\rangle }_{H}}{{dz}}=\displaystyle \frac{3{\pi }^{4}}{{H}_{n}^{4}}{\left(\displaystyle \frac{{H}_{n}^{4}{\mathrm{Ta}}^{* }}{2{\Pr }^{2}{\pi }^{4}}\right)}^{\tfrac{2}{3}}+\displaystyle \frac{27{\pi }^{4}}{4{H}_{n}^{4}},\end{eqnarray} \tag{ 25 }$$

where $\langle \cdot {\rangle }_{H}$ represents a horizontal average.

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We indeed find that the density gradients in layers are largest for the simulation with the higher rotation rate (Ta* = 0.1), and that the results from this simulation also fit well with prediction. The case with Ta* = 0.01, on the other hand, shows much greater variability in density gradient, particularly at larger layer heights, making it difficult to determine whether or not the theoretical predictions are valid. From Figure 8 we also see that in both cases the steepening of the density gradient is somewhat greater for smaller layers. However, this dependence of density gradient on layer height is weaker than the theory suggests.

In Figure 4 we see that the simulation with Ta* = 0.1 spends a greater amount of time in the three- and two-layered phases than either of the other two simulations, and it consequently takes about twice as long as the nonrotating run (Ta* = 0) to reach the one-layered phase. The root cause is related to the lower supercriticality of convection within the layers combined with the increased stability of the interfaces. Through inspection of the density profiles of our layered simulations, we see that the positions and shapes of the interfaces in rotating simulations have less variability with time compared with nonrotating simulations. We also see in Figure 4 that the fluxes in the rotating layered systems oscillate less than they do in the nonrotating ones. Wood et al. (2013) already discussed these large-amplitude oscillations in nonrotating simulations and attributed them to the presence of large plumes of fluid punching through interfaces, periodically causing spikes in transport. In our layered rotating simulations (particularly the Ta* = 0.1 case), the absence of large-amplitude oscillations in the layered phase suggests that the motion of these plumes is inhibited, possibly because convection is weaker and the interfacial gradients are stronger. Since the plumes could be key players in the layer merger events, their suppression in the rotating simulations could also explain why the merger timescale is longer.

All of these effects also contribute to the reduction of mean fluxes of temperature and chemical composition through density staircases in rotating ODDC compared with the nonrotating case. To show this quantitatively, Table 1 presents measurements of ${\mathrm{Nu}}_{T}-1$ and ${\mathrm{Nu}}_{\mu }-1$ (with standard deviations) for each value of Ta*. For low Ta* simulations, which clearly form convective layers, the fluxes are measured in the one-layered phase. The simulation with Ta* = 0.01 shows 2% and 7% decreases in thermal and compositional fluxes, respectively, compared to the nonrotating simulation, while the Ta* = 0.1 simulation shows 38% and 44% reductions.

Table 1.  Nondimensional Mean Turbulent Thermal and Compositional Fluxes through the Domain

Ta*	Ta	${\mathrm{Nu}}_{T}-1$	${\mathrm{Nu}}_{\mu }-1$	${\gamma }_{\mathrm{tot}}^{-1}$
0	0	25.3 ± 10.8	149.8 ± 84.4	0.68 ± 0.12
0.01	1	24.5 ± 5.2	139.0 ± 37.5	0.68 ± 0.074
0.1	10	15.5 ± 4.0	82.9 ± 28.4	0.62 ± 0.067
1	100	10.2 ± 2.1	46.5 ± 12.4	0.52 ± 0.061
10	1000	14.4 ± 3.1	61.7 ± 14.7	0.51 ± 0.061
10 (narrow)	1000	44.0 ± 11.2	216.5 ± 58.9	0.62 ± 0.16

Note.  Mean values are calculated by taking time averages of the turbulent fluxes in the ultimate statistically stationary state achieved by the simulation. In each case, Pr = τ = 0.1, ${R}_{0}^{-1}=1.25$, and θ = 0. For the cases with Ta* = 0, 0.01, and 0.1, these fluxes are measured in the one-layered phase. For the cases with Ta* = 1 and 10, the fluxes are measured once the system reaches a statistically steady state (see Figure 4).

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Wood et al. (2013) showed that fluxes in nonrotating layered ODDC follow a power-law scaling that depends on the product of the Prandtl number and the thermal Rayleigh number. Figure 9 shows the mean nondimensional turbulent compositional flux as a function of PrRa_T for each of our low Ta* simulations (Ta* = 0, 0.01, 0.1). To collect this data, average fluxes were computed in the one-, two-, and three-layer phases of each simulation. We clearly see that rotation leads to reduced transport rates in layered convection. This is not entirely surprising because rotation is known to reduce the convective efficiency in Rayleigh–Bénard convection (Rossby 1969). Bearing in mind the very limited amount of data available, we nevertheless attempt to fit it with flux laws of the form ${\mathrm{Nu}}_{T}-1=A{({\mathrm{PrRa}}_{T})}^{a}$ and ${\mathrm{Nu}}_{\mu }-1=B{({\mathrm{PrRa}}_{T})}^{b}$, using a nonlinear least squares fit. The results are presented in Table 2.

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Table 2.  Best Fits for Data Presented in Figure 9

		${\mathrm{Nu}}_{T}-1=A{({\mathrm{PrRa}}_{T})}^{a}$		${\mathrm{Nu}}_{\mu }-1=B{({\mathrm{PrRa}}_{T})}^{b}$
Ta*	Ta	A	a	B	b
0	0	0.076	0.32	0.21	0.36
0.01	1	0.071	0.32	0.22	0.35
0.1	10	0.016	0.37	0.035	0.42

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We find that rotation affects the constants of proportionality (A and B) much more than it affects the exponent of PrRa_T (a and b). For Ta* = 0.01, rotation has a minimal effect on the fluxes in each layered phase, and the relationship between flux and PrRa_T is the same as in the nonrotating case (Wood et al. 2013). For Ta* = 0.1, however, rotation reduces the coefficient A by almost a factor of 5 and increases the exponent a by around 15%. There is evidence, however, that this change in the exponent may be due to the fact that the relative effect of rotation decreases for increasing values of PrRa_T. Indeed, Figure 10 shows an increase in Rossby number as layers merge in low Ta* simulations (see Equation (21) for the definition of Rossby number). This suggests that, for sufficiently large layer heights, rotational effects could become negligible, and the flux law probably tends to the one found by Wood et al. (2013). This will need to be verified in simulations using larger computational domains.

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In contrast to the low Ta* case, the behavior of high Ta* simulations (Ta* = 1 and 10) is radically different from that described in studies of nonrotating ODDC. In Figure 4 we see that neither of the high Ta* simulations shows clear stepwise increases in either the compositional flux or ${\gamma }_{\mathrm{tot}}^{-1}$. Instead we see turbulent fluxes that grow slowly and oscillate rapidly after saturation of the linear instability until they reach a highly variable, statistically stationary state.

The Ta* = 1 and Ta* = 10 simulations are themselves quite different from one another. Figure 11 shows that the growth of step-like density perturbations through the γ-instability occurs for the Ta* = 1 simulation just as they do in the low Ta* cases. However, from Figure 5 we see that their amplitudes never become large enough to trigger the onset of convection. The absence of the standard stepwise increase in the fluxes associated with the transition to layered convection also supports the idea that the latter does not happen in this simulation (see Figure 4). Interestingly, the snapshot of chemical composition shown in Figure 11 reveals that the system is dominated by a large-scale cyclonic vortex. Inspection of the vertical velocity field shows that it is (roughly) constant within the vortex, which is consistent with Taylor-Proudman balance but is inconsistent with a system composed of convective layers separated by interfaces that resist penetrative motion. In some sense, it is perhaps more appropriate to consider Ta* = 1 to be a transitional case rather than a high Ta* case because it displays features of both high and low Ta* regimes.

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At significantly higher Ta* (in this case Ta* = 10), we see from Figure 12 that the growth of perturbations to the density profile is completely suppressed for the duration of the simulation. Instead, after a transitional period, the system becomes dominated by a cyclonic vortex similar to that observed in the Ta* = 1 simulation, albeit with much stronger vorticity. From the snapshots of vertical vorticity, ω_z, in Figure 13 we see that the simulations with Ta* = 1 and 10 have highly concentrated, vertically invariant vortex cores, necessarily surrounded by a more diffuse region of mostly anticyclonic vorticity (since $\int \int {\omega }_{z}(x,y,z){dxdy}=0$ for all z). We find that, based on all available simulations, these large-scale vortices only occur in the high Ta* regime. By comparison, the Ta* = 0.1 simulation shows no large-scale coherent structure in the vorticity field (which is true of the other low Ta* simulations as well). These features are strongly reminiscent of the large-scale vortices found by Guervilly et al. (2014) in rotating Rayleigh–Bénard convection using stress-free boundary conditions. In a parameter study, they found that Reynolds numbers greater than 300 and Rossby numbers less than 0.15 were needed for large-scale vortices to form. Using the Reynolds number from Guervilly et al. (2014) defined as

$$\begin{eqnarray}&&\mathrm{Re}=\displaystyle \frac{{w}_{\mathrm{rms}}{L}_{z}}{\Pr },\end{eqnarray} \tag{ 26 }$$

where w_rms is the rms vertical velocity, we find that values of Re for our simulations are ∼10³. Meanwhile, the Rossby numbers are shown in Figure 10 and are less than 0.1 for high Ta*. This suggests that their vortex formation process may be applicable to our high Ta* simulations despite the significant differences in the systems being studied (ODDC versus Rayleigh–Bénard convection). Also as in Guervilly et al. (2014), we find that whenever large-scale vortices form they always grow to fill the horizontal extent of the domain.¹ Julien et al. (2012) proposed that this may always occur in Cartesian domains using the f-plane approximation regardless of box size. However, they argued that this would be limited in practice by the Rossby radius of deformation in astrophysical objects. Beyond that size, convection or ODDC would likely lead to development of zonal flows in banded structures instead.

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The amount of energy that can be extracted from ODDC to drive large-scale vortices in the high Ta* regime is illustrated in Figure 14, where we see that the vast majority of kinetic energy in the Ta* = 1 and 10 simulations goes into horizontal fluid motions. These results showing ratios of horizontal kinetic energy to total kinetic energy are consistent with those calculated in Guervilly et al. (2014). The total amount of kinetic energy in the vertical fluid motions remains the same, however, in all of the simulations, while it is the total kinetic energy of the system that is much larger for high Ta* than for low Ta* simulations.

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It is worth noting that while the thermal and compositional fluxes (see Figure 4) and vertical velocity stop growing and reach a statistically stationary state (at around t = 3500 in both high Ta* simulations) the total kinetic energy continues to grow (driven by the continued growth of the horizontal kinetic energy) and has not saturated by t = 6000. This can be attributed to the fact that horizontal fluid motions are only limited by viscosity and may only saturate on the global viscous diffusion timescale, which is ∼10⁵ in these simulations.

Unlike in the low Ta* regime where average fluxes are calculated through time integration of the quasi-steady one-layered phase, we choose ranges for time integration of fluxes from t = 3500 to the end of the simulations in the high Ta* regime. From Table 1 we see that the simulation with Ta* = 1 has the weakest transport in either Ta* regime, with 58.1% and 67.4% reductions to thermal and compositional transport, respectively, compared to the nonrotating case. Interestingly, the Ta* = 10 simulation shows a slight increase in flux over the Ta* = 1 case. A possible explanation for this is that the presence of a stably stratified interface separating layers in the Ta* = 1 simulation inhibits vertical motion through the large-scale vortex. This could also suggest that, in the high Ta* regime, increased rotation may actually serve to enhance transport rather than suppress it, through vertical motions whose coherence is strengthened by the vortex. However, by contrast with the layered regime, fluxes in the presence of a large-scale vortex are highly dependent on the aspect ratio of the box. In Table 1 the narrower simulation at high Ta* = 10 has significantly higher fluxes than its wider counterpart. This makes it challenging to scale our results to simulations with larger domains, let alone apply them to more realistic astrophysical situations.

Also noteworthy in Table 1 is that ${\gamma }_{\mathrm{tot}}^{-1}$ in the ultimate statistically steady state is roughly the same across all simulations (namely ${\gamma }_{\mathrm{tot}}^{-1}\approx 0.5\mbox{--}0.65$), with high Ta* simulations having a ${\gamma }_{\mathrm{tot}}^{-1}$ that is at most 15% lower than in the low Ta* regime. This is significantly less than the variability that occurs from the formation of large-scale structures (layers or vortices): Figure 4(b) shows how ${\gamma }_{\mathrm{tot}}^{-1}$ increases from roughly 0.35 in the homogeneously turbulent phase to about 0.6 in the ultimate stages.

In order to study the effect of varying Pr and τ, we show a set of simulations with Pr = τ = 0.3 and ${R}_{0}^{-1}=1.1$. As in Section 4, we have chosen parameters at which layers form in nonrotating ODDC. Figure 15 shows the evolution of the turbulent compositional flux as a function of time for simulations with Ta* = 0, 0.09, 0.9, 9, and 90 (corresponding to Ta = 0, 1, 10, 100, and 1000). Consistent with Section 4, we find that at low Ta* stepwise increases in mixing rates indicate the transition to layered convection, whereas in the high Ta* case we do not. The transition between low and high Ta* is still Ta* ≈ 1 (equivalently Ta = 10 when Pr = 0.3). This shows that Ta* is a more appropriate bifurcation parameter than Ta to determine when ODDC is rotationally dominated. As in Section 4, layer formation only occurs in the low Ta* regime. As with the Ta* = 1 simulation from Section 4, which shows characteristics of both high and low Ta* ODDC, the Ta* = 0.9 simulation here develops a large-scale vortex, as well as layer-like perturbations to the background density profile (without evidence of actual layered convection).

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Large-scale vortices are observed in simulations with Ta* = 0.9 and 9 and look very similar to the corresponding snapshots of the Ta* = 1 and 10 simulations in Figures 11 and 12 from the previous section. Interestingly, however, the large-scale vortex does not form in our most rapidly rotating simulation with Ta* = 90; instead we see multiple small-scale vortices (see Figure 15). Analysis of Re and Ro for this simulation places it in a regime where large-scale vortices should form according to the criteria of Guervilly et al. (2014). This suggests that there may be additional constraints on the formation of large-scale vortices in ODDC that should be determined through a more in-depth survey of parameter space in a future study. Surprisingly, the compositional fluxes for the Ta* = 9 and 90 runs are similar, which is likely a coincidence as we saw that the fluxes in the presence of large-scale vortices depend on domain size.

The simulations we have presented so far were runs with small values of ${R}_{0}^{-1}$, which are conducive to layer formation in nonrotating ODDC. However, there is a range of larger values of ${R}_{0}^{-1}$ where a system is unstable to ODDC, but where layers are not predicted to spontaneously form through the γ-instability. Previous studies have shown that, without exception, simulations in this parameter regime remain nonlayered for as long as they are run. These simulations are dominated by large-scale gravity waves and were studied in depth by Moll et al. (2016) in the context of nonrotating ODDC. In that work they found that the growth of large-scale gravity waves is associated with very moderate (but still nonzero) increases in thermal and compositional transport. However, these increases are very small compared to the increases in turbulent transport due to layers and are likely unimportant for the purposes of stellar and planetary modeling. As a result, turbulent transport by ODDC at ${R}_{0}^{-1}$ greater than the layering threshold ${R}_{L}^{-1}$ can be ignored.

We now address how rotation affects nonlayered ODDC (i.e., ODDC at ${R}_{0}^{-1}\gt {R}_{L}^{-1}$). As in Section 4, we present five simulations with Ta* = 0, 0.01, 0.1, 1, and 10, and with Pr = τ = 0.1 and θ = 0. However, for each of these simulations we now set ${R}_{0}^{-1}=4.25$. For comparison, for the stated values of Pr and τ, ${R}_{L}^{-1}\simeq 1.7$, and the critical inverse density ratio for marginal stability is ${R}_{c}^{-1}=5.5$.

As in Section 4 we find that high ${R}_{0}^{-1}$ simulations can be divided into two general classes of behavior depending on Ta*. As seen in the snapshots in Figure 16, low Ta* simulations are qualitatively similar to nonrotating simulations in that they are dominated by large-scale gravity waves. The strongest gravity wave mode in both the Ta* = 0 and Ta* = 0.01 simulations has three wavelengths in the vertical direction, has one wavelength in the x direction, and is invariant in the y direction. The simulation with Ta* = 0.1 by contrast is dominated by a larger scale mode with a single wavelength in each spatial direction. Despite their qualitative similarity, inspection of the compositional flux in Figure 17 shows large reductions compared to the nonrotating simulation (Ta* = 0), even in the case where Ta* = 0.01. To understand why this is the case, note how Ro is small even in the lowest Ta* simulation. This is because the rms velocities are very small in this regime. Rotation therefore plays a role in the saturation of the gravity waves and acts to reduce their amplitudes, which in turn significantly reduce the mixing rates.

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As seen in the snapshot in Figure 16(e), the Ta* = 10 simulation is dominated by vertically invariant vortices, while the Ta* = 1 simulation is again a transitional case that shows evidence both of gravity waves and of vortices. A significant difference with the results of Section 4, however, is that vortices at low ${R}_{0}^{-1}$ are large-scale, while those at high ${R}_{0}^{-1}$ are small-scale (for the same values of Ta*). This suggests that the formation of large-scale vortices requires a more unstable stratification (which leads to more turbulence) than is present in the high ${R}_{0}^{-1}$ simulations shown here. This is, again, qualitatively consistent with the findings of Guervilly et al. (2014) that large-scale vortices only form for sufficiently high Reynolds number.

The most rapidly rotating simulation (Ta* = 10) shows a slight increase in the compositional flux compared to the nonrotating simulation but remains far less efficient than layered convection. Importantly, as with the nonrotating simulation, layers never form at any point (when ${R}_{0}^{-1}=4.25$). Consequently, fluxes through nonlayered (high ${R}_{0}^{-1}$) systems are effectively diffusive, and the conclusion from Moll et al. (2016), namely that turbulent fluxes are negligible for nonlayered systems, remains valid for all the simulations presented here.

The main result of this study is the discovery of two distinct regimes in rotating ODDC depending on whether the rotation rate Ω is high or low. We find that the most appropriate parameter for determining if a system is in one regime or the other is ${\mathrm{Ta}}^{* }=\tfrac{4{{\rm{\Omega }}}^{2}{d}^{4}}{{\kappa }_{T}^{2}}$, where d is given in Equation (6) and κ_T is the thermal diffusivity. The transition from the regime with slow rotation to the regime that is rotationally dominated occurs consistently at Ta* ≈ 1.

In the low Ta* regime in polar regions (with θ = 0), rotating ODDC behaves in a qualitatively similar way to nonrotating ODDC. The transition to layered convection (or lack thereof) at low Ta* is consistent with the predictions of γ-instability theory made for nonrotating ODDC (Mirouh et al. 2012): at parameters where layers form in nonrotating ODDC, we also observe layer formation in low Ta* simulations. Likewise, in the simulations we ran at nonlayered parameters, we find that low Ta* simulations do not form layers and are dominated by gravity waves like their nonrotating counterparts (Moll et al. 2016). We understand this to be true because the thermal and compositional fluxes immediately after saturation of the primary instability of ODDC are unaffected by rotation in this regime. Since the γ-instability only depends on these fluxes, it is similarly unaffected. Given the limited number of available simulations, we cannot say definitively what effect (if any) rotation has on the layering threshold ${R}_{L}^{-1}$ (the value of the inverse density ratio, ${R}_{0}^{-1}$, below which layers are predicted to form through the γ-instability, and above which they are not), only that it is not significant in the low Ta* simulations presented here, which are far from that threshold. However, we believe that ${R}_{L}^{-1}$ would be relatively unaffected by rotation for Ta* < 1.

Beyond these qualitative similarities with nonrotating simulations, however, rotation in low Ta* simulations has a deleterious effect on thermal and compositional transport in both the layered and nonlayered parameter regimes. For a given layer height, turbulent fluxes through a thermocompositional staircase decrease as rotation increases (e.g., by about 50% in the Ta* = 0.1 simulation presented in Section 4.2). However, our results also suggest that this effect becomes smaller as the layer height increases (through mergers, for example). For reasonably large layer heights, we postulate that rotation has a minimal effect on ODDC, and that the flux laws originally proposed by Wood et al. (2013) actually hold. Turbulent fluxes through nonlayered ODDC in the gravity-wave-dominated phase are reduced by as much as 90% compared with the nonrotating case, but this merely implies that they remain negligible, as discussed by Moll et al. (2016).

Finally, low Ta* simulations at higher colatitude θ are not significantly different from their polar counterparts. Inclination has a negligible effect on the temperature and compositional fluxes, but it may induce differences in the timescales of layer formation and mergers.

In the high Ta* regime, the dynamics are radically different from the nonrotating and low Ta* simulations. Most striking is that layer formation is inhibited at low inverse density ratios (except in the equatorial case). Instead, the dynamics are dominated by vortices aligned with the direction of rotation, which span the domain. Their horizontal scales seem to depend on ${R}_{0}^{-1}$, θ, and Ta*. In polar regions, we observe that some simulations become dominated by a single large-scale cyclonic vortex that grows to fill the domain, similar to those observed by Guervilly et al. (2014) in rotating Rayleigh–Bénard convection. Our preliminary data show that this phenomenon may be limited to low ${R}_{0}^{-1}$, with Ta* between 1 and 10, but the precise conditions necessary for these large-scale vortices to form remain to be determined. We find that large-scale vortices do not occur in the most rapidly rotating simulation (Ta* = 90) at low ${R}_{0}^{-1}$, in any of the high ${R}_{0}^{-1}$ simulations, or in any of the inclined simulations. In these cases, the system dynamics are instead dominated by many smaller-scale vortices of both polarities.

Turbulent fluxes through different types of vortices vary with parameters in a complex manner, and it is therefore difficult to make general statements about them. The fluxes in the presence of large-scale vortices are significant, but they appear to be highly dependent on the dimensions of the domain, which makes it difficult to predict in situ mixing in a star or planet. On the other hand, fluxes in the presence of small-scale vortices are not likely to be dependent on domain size, but their dependence on Ta* and ${R}_{0}^{-1}$ has yet to be extensively studied. The most definitive aspect of the fluxes in simulations that host small-scale vortices is that they are highly dependent on the inclination θ, with higher inclinations causing less efficient transport (except at the equator). This is because velocities are constrained to be along the axis of rotation by Taylor-Proudman effects. It is interesting to note that, in the high Ta* regime, layers form in our equatorial simulation ($\theta =\tfrac{\pi }{2}$). In this run, turbulent fluxes through the layers are comparable to layered fluxes in the low Ta* and nonrotating regimes. All of this suggests that the poles and equator may be regions of strongly enhanced temperature and compositional transport in ODDC, while turbulent mixing at latitudes in between is quenched.

Finally, simulations where Ta* ≈ 1 appear to be edge cases with features of both high and low Ta*. At parameters conducive to layering, simulations with Ta* ≈ 1 show evidence of perturbations to the background density profiles, indicating the growth of the γ-instability. However, we also see the development of large-scale, vertically invariant vortices that prevent actual layered convection from occurring. Also, when Ta* ≈ 1 at nonlayered, gravity-wave-dominated parameters, we see evidence of gravity waves as well as small thin vortices that are nearly vertically invariant.

There are several caveats to these conclusions that should be mentioned. The dimensionality of parameter space that would need to be explored to provide a comprehensive study of rotating ODDC is high and comprises (L_x, L_y, L_z), θ, Pr, τ, Ta*, and finally ${R}_{0}^{-1}$. Computational limitations have forced us to be highly selective on the sets of simulations explored, so this study does not constitute a comprehensive sweep of parameter space. As such, there may be behaviors that occur at unexplored parameters that are not addressed here.

First of all, in the interest of reducing computational expense, we have chosen to run most of our simulations in domains with dimensions (100d)³. With boxes of this size, layers always merge until a single interface remains (as in the nonrotating case). It would be interesting to see in a taller domain if rotation has a role in determining the layer height (i.e., to see if layers stop merging before reaching the one-layered phase). Wider domain sizes may also help to answer questions about the vortices present in high Ta* simulations. Particularly, they may reveal whether large-scale vortices have a characteristic horizontal length or whether they always grow to fill the domain. Wider boxes may also show if there are so far undetected large-scale features emerging in systems dominated by small-scale vortices.

Another area of uncertainty is that the chosen values of ${R}_{0}^{-1}$, ${R}_{0}^{-1}=1.25$ and ${R}_{0}^{-1}=4.25$, are fairly close to the convective and marginal stability thresholds, respectively, making them somewhat extreme cases. While we do not believe that choosing less extreme parameter values would lead to dramatic qualitative changes in the results, we cannot rule this possibility out until further work has been completed.

Finally, for computational reasons, the values of Pr and τ chosen for our simulations (Pr = τ = 0.1 and 0.3) are substantially larger than the values in stellar interiors (where Pr ∼ τ ∼ 10⁻⁶) and the interiors of giant planets like Jupiter and Saturn (where Pr ∼ τ ∼ 10⁻³). Consequently, there may be additional physical effects that occur at low parameter values that are not observed here. However, the values used here may be closer to actual values for ice giants such as Uranus and Neptune, whose equations of state are influenced by the presence of water and methane ices in their atmospheres (Redmer et al. 2011).

As summarized above, our results for the low Ta* regime show that attenuation of the fluxes due to rotational effects is not likely to be significant for astrophysical models or observations. In this regime, we advocate use of the parameterizations presented in Wood et al. (2013) in layered ODDC and Moll et al. (2016) in nonlayered ODDC. However, a potentially observable effect of rotating ODDC in this regime could be related to how rotation affects the structure of layers and interfaces in low Ta* simulations. We have found in our experiments that rotation leads to layer interfaces that are more stably stratified than in nonrotating simulations. It may be possible in the future that such steep density gradients could be observed in a star through asteroseismology. This line of inquiry could also be extended to gas giant planets. In Saturn, for example, it may be possible to detect density gradients using ring seismology (Fuller 2014), and in the future, we may even be able to probe the interior structure of Jupiter through detection of global modes (Gaulme et al. 2011).

In the high Ta* regime, the results of this study have potential observational implications for thermal and compositional transport in stars and planets. The sensitivity of the turbulent heat flux to the inclination in our high Ta* simulations suggests that the transport in a rapidly rotating giant planet could vary substantially with latitude (with higher fluxes at the poles and equator). Indeed, the gas giants in our own solar system are found to have luminosities that are independent of latitude, despite the fact that regions close to the equator receive more solar energy. Since we would expect regions of Jupiter or Saturn's atmosphere that get more radiation from the Sun to have higher luminosities (because they are reradiating more solar energy), the isotropy of the outgoing flux in luminosity suggests that more heat from the interiors of these planets is being radiated at the poles than at other latitudes. Further study is warranted to determine if rapidly rotating ODDC contributes to this effect. The large-scale vortices present in the polar regions of the high Ta* simulations present an intriguing observational potential for regions with strong heat and compositional fluxes and strong collimated vertical flows. However, there is reasonable doubt as to whether these large-scale vortices represent a real physical phenomenon. We only observe them to occur in polar simulations (in a limited range of Ta*), and it is possible that even a slight misalignment between the direction of gravity and the rotating axis could prevent them from forming.

For stars, in the case of semiconvection zones adjacent to convection zones, Moore & Garaud (2015) showed that nonrotating ODDC is always in the layered regime, and that transport through the semiconvective region is so efficient that the latter gets rapidly absorbed into the convection zone. In essence, aside from a fairly short transient period, the star evolves in a similar way taking into account semiconvection, or ignoring it altogether and using the Schwarzchild criterion to determine the convective boundary. Our results suggest that this conclusion remains true for slowly rotating stars. However, if the star is in the high Ta* regime instead, layered convection is suppressed, and transport through the semiconvective region is much weaker and may possibly depend on latitude. This would in turn imply fairly different evolutionary tracks and asteroseismic predictions.