3.1 Time evolution

Figure 1(a) shows the temporal evolution of the energy $E$ for flows with $Ro_{f}=0.1$ and different memory time scale ${\it\tau}_{m}$ of the forcing. As ${\it\tau}_{m}$ increases, we observe a gradual increase of the amplitude of the energy up to an order of magnitude. The time-series for ${\it\tau}_{m}/{\it\tau}_{NL}>4$ are characterised by large signal variations, which require extremely long time-integrations, restricting our runs to moderate Reynolds numbers. Note that even for low ${\it\tau}_{m}$ a steady state is reached after a transient that lasts for about 50 to $100{\it\tau}_{NL}$ turnover times, indicating how expensive computationally it is to reach a steady-state regime in rotating flows forced at large scales.

Figure 1.

Time series of (a) the energy $E$ for the flows with different memory time scales ${\it\tau}_{m}$ of the forcing and of (b) the rates of energy dissipation ${\it\epsilon}$ and injection $\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{f}\rangle$ for the flows with ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ and $\infty$ at Rossby number $Ro_{f}=0.1$ and Reynolds number $Re_{f}=333$ .

Table 1.

Numerical control parameters of the DNS.

The temporal evolution of the energy dissipation rate ${\it\epsilon}={\it\nu}\langle |{\bf\omega}|^{2}\rangle$ and the energy injection rate $\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{f}\rangle$ are presented in figure 1(b). For clear illustration purposes we choose to plot only the two extreme cases of the flows with highly random-in-time forcing ( ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ ) and time-independent forcing ( ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ ) at $Ro_{f}=0.1$ . At relatively early times the two flows reach a steady state (see figure 1 a) and therefore the balance ${\it\epsilon}=\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{f}\rangle$ is satisfied with both flows having the same rates of energy injection and dissipation. However, after a very long time period ( ${\sim}600{\it\tau}_{NL}$ turnover times) the flow with the time-independent forcing deviates eventually to a new statistically steady state. This happens when $\boldsymbol{u}$ becomes correlated with $\boldsymbol{f}$ and then the flow adjusts its dissipation rate such that a new steady state is achieved.

The adjustment of the dissipation rate by the flow owing to the increase of the correlation between the external mechanical force and the velocity field explains why the energy increases as we increase the memory time scale of the forcing. This is a very interesting property of rotating flows from a practical point of view if one wants to minimise or maximise the energy dissipation rate in a potential application such as in turbomachinery.

We should point out here that the energy dissipation rate remains small for the flow, with ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ throughout the duration of the simulation, for two reasons. The first reason is the short memory time scale of the forcing that keeps the $\langle \boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{f}\rangle$ correlation small, and the second reason is the low Rossby number, resulting in a weak turbulent cascade as we show later on.

3.2 The role of helicity

Helicity is common in real flows and it can be created, for example, in planetary atmospheres in the presence of rotation and stratification (Moffatt Reference Moffatt1978; Tobias Reference Tobias2009; Marino et al. Reference Marino, Mininni, Rosenberg and Pouquet2013). In homogenenous non-rotating turbulence it is expected that the helicity spectrum cannot develop if it is initially zero (André & Lesieur Reference André and Lesieur1977) or if an external mechanism does not inject net helicity (Dallas, Fauve & Alexakis Reference Dallas, Fauve and Alexakis2015). In our runs zero net helicity is injected into the flow. Nevertheless, for a given rotation rate (i.e.  $Ro_{f}=0.1$ ) we observe that the relative helicity ${\it\rho}_{H}=H/(|\boldsymbol{u}|\,|{\bf\omega}|)$ increases as the memory time scale of the forcing increases (see figure 2).

Figure 2.

(a) Time series of relative helicity ${\it\rho}_{H}$ and (b) probability density function (PDF) of the absolute value of ${\it\rho}_{H}$ for flows with different forcing memory time scales at Rossby number $Ro_{f}=0.1$ and Reynolds number $Re_{f}=333$ .

It is apparent from figure 2 that the mirror-symmetry breaking depends on the value of ${\it\tau}_{m}$ . Figure 2(a) shows ${\it\rho}_{H}$ to be almost zero at early times for all the flows, and as ${\it\tau}_{m}$ increases the mirror symmetry breaks at later times only for long enough ${\it\tau}_{m}$ . We observe that helicity emerges in the flow as soon as ${\it\tau}_{m}$ becomes of the order of the eddy turnover time, i.e.  ${\it\tau}_{m}/{\it\tau}_{NL}\sim O(1)$ . To analyse further this behaviour of ${\it\rho}_{H}$ we plot the probability density function (PDF) of the time series of the absolute value of relative helicity in figure 2(b). This plot shows an increase in the mean value of $|{\it\rho}_{H}|$ and also a broadening of the tails of the PDFs for longer memory time scales. So, unlike in homogeneous turbulence, helicity can be created in rotating flows by an external force with a sufficiently long memory time scale, even though net helicity is not injected directly into the flow.

To determine whether the breaking of mirror-symmetry, which distinguishes flows with highly random-in-time and time-independent forcings, remains at higher Reynolds numbers we carried out simulations for the two extreme cases of ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ and $\infty$ at $Re_{f}=714$ and $Ro_{f}=0.1$ . Our numerical simulations confirmed the persistence of this behaviour at higher Reynolds numbers. We therefore analyse these higher Reynolds number runs in order to gain further insight into the effects of helicity on the flow.

Figure 3.

Visualisations of the relative helicity ${\it\rho}_{H}$ for the flows at Rossby number $Ro_{f}=0.1$ and Reynolds number $Re_{f}=714$ with (a)  ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ , ${\it\rho}_{H}=-0.018$ and (b)  ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ , ${\it\rho}_{H}=0.45$ .

Visualisations of the relative helicity of the flows with $Re_{f}=714$ are presented in figures 3(a) and 3(b) with the average value of ${\it\rho}_{H}=-0.018$ for ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ and ${\it\rho}_{H}=0.45$ for ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ . The red and blue colours in figure 3 indicate right-hand (positive helicity) and left-hand (negative helicity) circularly polarized helical waves, respectively. An instructive way to explain this further is to decompose the velocity field into circularly polarised helical waves (Constantin & Majda Reference Constantin and Majda1988; Waleffe Reference Waleffe1992)

where $\text{i}\boldsymbol{k}$ , $\boldsymbol{h}_{+}$ and $\boldsymbol{h}_{-}$ are the linearly independent eigenvectors of the curl operator, i.e.  $\text{i}\boldsymbol{k}\times \boldsymbol{h}_{\pm }=\pm |\boldsymbol{k}|\boldsymbol{h}_{\pm }$ . These complex eigenvectors are orthogonal to each other and are fully helical. So, now $\widehat{\boldsymbol{u}}(\boldsymbol{k})$ can be expressed as a linear combination of the eigenvectors $\boldsymbol{h}_{+}$ and $\boldsymbol{h}_{-}$ only as follows

since $\boldsymbol{k}\boldsymbol{\cdot }\widehat{\boldsymbol{u}}(\boldsymbol{k})=0$ . Then, the helicity can be separated into modes of positive and negative helicity, viz.

where $^{\ast }$ denotes the complex conjugate.

The nature of the forcing is clearly imprinted on the flow structure in figure 3. The flow with the highly random-in-time forcing (see figure 3 a) gives a quasi-2D flow with two large columnar vortices, typical at low Rossby numbers due to the Taylor–Proudman theorem. These two vortices are governed by helical waves of opposite polarity. On the other hand, the flow with the time-independent forcing is characterised by helical waves of opposite polarity that travel within the flow, breaking the quasi-2D behaviour at small and intermediate scales that is imposed by rotation (see figure 3 b). Note that the two large-scale vortices are still present but this time they have the same sign of helicity on average. Similar effects of helicity have also been observed on a previous study of decaying rotating turbulence (Morinishi et al. Reference Morinishi, Nakabayashi and Ren2001).

In order to quantify the level of anisotropy of these two runs we consider the 2D energy spectrum, which is defined as

The sum is restricted here at energy in cylinders of radius $k_{\bot }$ and energy in planes $k_{\Vert }$ . Figures 4(a) and 4(b) show the 2D energy spectrum for the flows with ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ and $\infty$ , respectively. The contours of the 2D energy spectrum for an isotropic flow are represented by concentric circles centred at the origin of the axes. Any deviation from the circular pattern indicates the level of anisotropy in the flow. By comparing the two contour plots of $E_{2D}$ , it becomes clear that in figure 4(b) the intermediate and small scales are closer to isotropy, implying that the flow with the time-independent forcing is overall less anisotropic than the flow with the highly random-in-time forcing. This observation is in agreement with the visualisation of figure 3, which prompts us to postulate that the helicity plays a central role in the suppression of anisotropy in the flow.

Figure 4.

The two-dimensional energy spectrum $E_{2D}(k_{\bot },k_{\Vert })$ for the flows at Rossby number $Ro_{f}=0.1$ and Reynolds number $Re_{f}=714$ with (a)  ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ , ${\it\rho}_{H}=-0.018$ and (b)  ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ , ${\it\rho}_{H}=0.45$ .

3.3 Spectral behaviour

In this section we present the spectra of the energy $E(k)$ and the energy flux ${\it\Pi}_{E}(k)$ . The energy spectrum was spherically averaged using the following expression:

and the spectrum of the energy flux was obtained as

The energy flux is a measure that illustrates the direction of the energy cascades. These spectra were time-averaged after the flows had reached a steady-state solution. Note that the flow with the time-independent forcing ( ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ ) was time-averaged only after ${\sim}800{\it\tau}_{NL}$ turnover times, when the new steady state was reached (see figure 1 b).

Note that we did not observe any differences to the scalings of the spectra when averaging over spheres and over cylinders. This has also been shown clearly by Mininni, Alexakis & Pouquet (Reference Mininni, Alexakis and Pouquet2009), who performed simulations at low Rossby numbers and of similar resolutions to ours. So, the scalings of the spectra that we present are also valid for energy spectra as a function of $k_{\bot }$ . However, significant differences in the scalings are observed in the $k_{\Vert }$ direction.

Figure 5.

(a) Energy spectra $E(k)$ compensated by $k^{5/2}$ and (b) the energy flux spectra ${\it\Pi}_{E}(k)$ normalised with the dissipation rate ${\it\epsilon}_{E}$ for flows with different forcing memory time scale at Rossby number $Ro_{f}=0.1$ and Reynolds number $Re_{f}=333$ .

The effects of the memory time scale of the forcing are also apparent on the spectral dynamics of our flows. Figure 5(a) shows the energy spectra of the flows with different memory time scales of the forcing compensated by $k^{5/2}$ . The spectra of these flows obey different power laws which clearly depend on the memory time scale of the forcing. The runs that are forced with the highly random-in-time forcing seem to have a $k^{-5/2}$ scaling. As ${\it\tau}_{m}$ increases the spectra start to deviate gradually from the $k^{-5/2}$ scaling towards a $k^{-2}$ and finally reach a $k^{-5/3}$ scaling for the flow with the time-independent forcing. The $k^{-5/3}$ energy spectrum can be interpreted from the fact that the intermediate and small scales of the flow are closer to isotropy (see figures 4 b and 3 b) and hence we expect the Kolmogorov phenomenology to be valid in this case. All the exponents that we observe here could be related to the various phenomenologies on strong and weak-wave turbulence in the literature, where the interplay between ${\it\tau}_{NL}$ and the time scale of the inertial waves ${\it\tau}_{w}\propto {\it\Omega}^{-1}$ is central to obtain the different energy spectra. These spectral exponents have also been observed in other studies of forced rotating flows (Yeung & Zhou Reference Yeung and Zhou1998; Mininni & Pouquet Reference Mininni and Pouquet2010; Alexakis Reference Alexakis2015). Owing to the moderate resolution of these simulations, the statements about the exact spectral exponents are qualitative. In any case, our results clearly show a dependence of the spectra on the nature of the forcing in rotating turbulence. Simulations integrated for extremely long times with higher Reynolds numbers but also lower Rossby numbers, falling in the same dynamical regime of the two-dimensional parameter space, are necessary to verify whether our results imply a lack of universality in these flows.

The corresponding spectra for the energy flux ${\it\Pi}_{E}(k)$ normalised by the energy dissipation rate ${\it\epsilon}_{E}=2{\it\nu}\sum _{k}k^{2}E(k)$ are shown in figure 5(b). The positive flux in this plot indicates a forward cascade while the negative flux indicates a transfer of energy from the small to the large scales of the flow. In the case of negative flux we do not talk about a cascade because we do not have enough scale separation between the forcing scale and the box size. As the memory time scale of the forcing increases, the forward cascade becomes stronger. On the other hand, the flux of energy towards the large scales increases as the forcing becomes more random-in-time with the flow reaching a quasi-2D state. These observations are in line with the visualisations of figure 3.

We have already seen that as the forcing becomes less time-dependent, helicity increases considerably in our flows, so these changes in the spectra can also be related to the presence of strong net helicity in the flow. This is in agreement with prior studies that have shown the influence of helicity on the energy spectrum by directly injecting helicity into the flow (Mininni & Pouquet Reference Mininni and Pouquet2009, Reference Mininni and Pouquet2010). In contrast, helicity does not seem to have any significant effect on the spectra in non-rotating, homogeneous and isotropic helical turbulence (Dallas et al. Reference Dallas, Fauve and Alexakis2015).

The fact that helicity is not a sign-definite quantity and because we do not inject any net helicity, the sign of helicity in our flows undergoes changes in its inertial range. Therefore, there is either no power-law or it is difficult to define one in our helicity spectra. For this reason, we do not show any helicity spectra here. In the next section, we examine the Rossby number dependence of the dynamics of the flows.

4.1 Global behaviour

In the previous sections we saw that the dynamics depend on the nature of the forcing for a given Rossby number. Here, we investigate the effects of the rotation rate on the flows, focusing on the extreme cases of the forcing being highly random-in-time ( ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ ) and time-independent ( ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ ) for fixed $Re_{f}=333$ . Again here we restrict ourselves to moderate Reynolds numbers because extremely long integration times for the runs with time-independent forcing are inevitable.

For high Rossby number flows the effect of rotation is negligible and the energy is expected to flow to scales smaller than the forcing scale. However, as the Rossby number is decreased and the flow tends to become quasi-2D, there is more and more energy transferred to scales larger than the forcing scale due to an inverse cascade (Pouquet et al. Reference Pouquet, Sen, Rosenberg, Mininni and Baerenzung2013).

We examine the energy and the relative helicity for runs with different Rossby numbers. The triangles and circles denote runs forced with a time-independent forcing and random-in-time forcing, respectively. As the Rossby number is decreased we see that energy increases as expected (see figure 6 a). However, the rate of increase and the values of energy for high enough rotation rates depend on the nature of the mechanical force. Note that the flow with the time-independent forcing has much more energy at small Rossby numbers.

Figure 6.

Rossby number dependence of (a) energy and (b) absolute value of relative helicity for flows with Reynolds number $Re_{f}=333$ . The ○ and ▵ denote runs forced with ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ and ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ , respectively.

The relative helicity also behaves very differently for the two types of flows and this is shown in figure 6(b). The flow with the random-in-time forcing has zero net helicity for all $Ro_{f}$ . However, the flow with the time-independent forcing bifurcates to a state of non-zero helicity for small enough Rossby numbers. The value of $|{\it\rho}_{H}|$ seems to vary discontinuously as $Ro_{f}$ is decreased with the flow bifurcating to a helical state at the critical $Ro_{f}^{crit}\simeq 0.2$ . Thus, the transition from the non-helical to the helical state is a jump bifurcation. In summary, net helicity emerges in the flow only for small enough $Ro_{f}$ and long enough  ${\it\tau}_{m}$ .

Helicity is a pseudoscalar quantity and $H\neq 0$ only if it is directly injected into the flow (i.e.  $\boldsymbol{u}\boldsymbol{\cdot }(\boldsymbol{{\rm\nabla}}\times \boldsymbol{f})\neq 0$ ) by a helical mechanical force or if another pseudoscalar quantity exists related to the pseudovector $\boldsymbol{{\rm\nabla}}\times \boldsymbol{f}$ . In our work, we observe that net helicity emerges in rapidly rotating flows with long enough memory time scale forcings only. So, a pseudovector that relates the rotation vector with the forcing is ${\it\bf\Omega}\times (\boldsymbol{{\rm\nabla}}\times \boldsymbol{f})$ and hence the pseudoscalar quantity that will allow the generation of helicity in a rotating flow is

A similar expression was derived in a different way by Hide (Reference Hide1975) for a rapidly rotating flow in geostrophic balance assuming that the nonlinear term is negligible. Now, from (4.1) we can deduce that no net helicity will be generated for a short memory time scale forcing since $\langle \boldsymbol{{\rm\nabla}}\times \boldsymbol{f}\rangle _{t}=0$ (with $\langle \cdot \rangle _{t}$ denoting an average over time), assuming isotropy and ergodicity. On the other hand, for a forcing with long enough memory time scale $\langle \boldsymbol{{\rm\nabla}}\times \boldsymbol{f}\rangle _{t}=\boldsymbol{g}(\boldsymbol{x})\neq 0$ and therefore $H\neq 0$ for long enough integration time scales, in agreement with our observations.

Moffatt (Reference Moffatt1970) suggested that a random superposition of inertial waves will exhibit a lack of mirror-symmetry if and only if there is a mechanical excitation in a preferred direction in the propagation of the waves with respect to the axis of rotation. Otherwise, the random superposition of inertial waves in equal proportions would give zero net helicity. Based on (4.1), we conjecture that such a mechanism is pertinent to our flows where the angle ${\it\varphi}$ between ${\it\Omega}\boldsymbol{e}_{z}$ and $\boldsymbol{{\rm\nabla}}\times \boldsymbol{f}$ is fixed at time $t=0$ for a time-independent forcing, and hence such a forcing can add a preferred direction of propagation to the inertial waves, inducing the mirror-symmetry breaking in our flows. From the other side, a highly random-in-time forcing can excite inertial waves in all directions in equal proportions since ${\it\varphi}$ is random in time, and this is why the net helicity remains zero for any value of $Ro_{f}$ in this case.

4.2 Spectral behaviour

Here we analyse the energy spectra of the flows at different Rossby numbers. In figure 7(a) we present the energy spectra $E(k)$ compensated by $k^{5/2}$ of flows with highly random-in-time forcing. For $Ro_{f}=0.5$ the energy spectrum is close to the Kolmogorov $k^{-5/3}$ scaling with the effects of the Coriolis force having no significant influence on the dynamics of the flow. However, as the Rossby number decreases ${\it\tau}_{w}\propto {\it\Omega}^{-1}$ becomes the dominant time scale and then the spectrum is changed to the weak-wave turbulence prediction of $E(k)\propto k^{-5/2}$ .

Figure 7.

Energy spectra $E(k)$ compensated by $k^{5/2}$ for the flows with (a)  ${\it\tau}_{m}/{\it\tau}_{NL}=0.5$ and (b)  ${\it\tau}_{m}/{\it\tau}_{NL}=\infty$ at different Rossby numbers and Reynolds number $Re_{f}=333$ .

Similar behaviour is observed for the spectra of the flows with time-independent forcing (see figure 7 b) but with two different characteristics. The first is the significant condensation of energy at large scales for small enough Rossby numbers in comparison to the flows with random-in-time forcing. The second is the transition from the Kolmogorov-like regime with ${\it\tau}_{E}\ll {\it\tau}_{w}$ to the weak-wave turbulence regime with ${\it\tau}_{w}\ll {\it\tau}_{E}$ which occurs at lower Rossby numbers, showing the dependence of this transition on the nature of the mechanical force.

Here, we should point out that weak-wave turbulence theory arguments, which assume uniform and isotropic forcing, predict a $k^{-5/2}$ spectrum but they do not predict condensation of energy at large scales due to an inverse cascade in unbounded domains. This is in agreement with figure 7(a) where there is some energy condensation at large scales but it is not significant in comparison to figure 7(b). However, the energy condensation at large scales in figure 7(b) suggests that weak-wave turbulence theory is not necessarily valid for the small Rossby number flows with time-independent forcing even though $E(k)\propto k^{-5/2}$ .