2.1 Nonlinear interactions in Navier–Stokes

In a rotating frame, the Navier–Stokes equations for an incompressible fluid with velocity $\boldsymbol{u}$ and under the action of a mechanical forcing $\boldsymbol{F}$ read

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}=-(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}-2\unicode[STIX]{x1D6FA}\boldsymbol{z}\times \boldsymbol{u}-\unicode[STIX]{x1D735}{\mathcal{P}}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}+\boldsymbol{F}, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

where ${\mathcal{P}}$ is the total pressure (including the centrifugal force, and normalised by the uniform fluid mass density), $\boldsymbol{z}$ is parallel to the rotation axis, $\unicode[STIX]{x1D6FA}$ is the rotation frequency, and $\unicode[STIX]{x1D708}$ is the kinematic viscosity. The Reynolds number, defined as usual as $Re=UL/\unicode[STIX]{x1D708}$ (where $U$ is the r.m.s. velocity and $L$ is the energy injection scale) quantifies the strength of the nonlinear term against viscous damping.

Using the incompressibility condition given by (2.2), and for $\boldsymbol{F}=0$ , the Fourier transform of (2.1) can be written as

(2.3) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D708}k^{2}+2\unicode[STIX]{x1D6FA}\mathbb{P}_{\boldsymbol{ k}}\boldsymbol{z}\times \right)\boldsymbol{u}_{\boldsymbol{k}}=-\text{i}\mathbb{P}_{\boldsymbol{k}}\mathop{\sum }_{\boldsymbol{p}+\boldsymbol{q}=\boldsymbol{k}}(\boldsymbol{u}_{\boldsymbol{p}}\boldsymbol{\cdot }\boldsymbol{q})\boldsymbol{u}_{\boldsymbol{q}},\end{eqnarray}$$

where $[\mathbb{P}_{\boldsymbol{k}}]_{ij}=\unicode[STIX]{x1D6FF}_{ij}-k_{i}k_{j}/k^{2}$ is the projector operator, which projects in the direction perpendicular to $\boldsymbol{k}$ to enforce incompressibility. All terms on the left-hand side of the equation are linear in $\boldsymbol{u}_{\boldsymbol{k}}$ and do not couple modes with different $\boldsymbol{k}$ . So, after transforming the nonlinear term in (2.1), the resulting convolution on the right-hand side of this equation tells us that only modes $\boldsymbol{p}$ and $\boldsymbol{q}$ in triads satisfying $\boldsymbol{k}=\boldsymbol{q}+\boldsymbol{p}$ can give or receive energy from the mode with wave vector $\boldsymbol{k}$ . Note this is not a unique property of the Navier–Stokes equation but of any nonlinear equation with quadratic nonlinearities.

2.2 Rotating flows and relevant time scales

In the presence of rotation (or of other restitutive forces), waves can be excited with well-defined frequencies for each wave vector, given by the dispersion relation of the waves $\unicode[STIX]{x1D714}(\boldsymbol{k})$ . For a rotating flow, the dispersion relation of inertial waves is

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D714}(\boldsymbol{k})=\pm \frac{2\unicode[STIX]{x1D6FA}k_{z}}{k}.\end{eqnarray}$$

We can then define the Rossby number as

(2.5) $$\begin{eqnarray}Ro=\frac{U}{2L\unicode[STIX]{x1D6FA}},\end{eqnarray}$$

which measures the ratio of the rotation period to the turnover time of the large-scale eddies. As a result, for small Rossby number we can expect waves to be faster than eddies, at least for a range of scales. Indeed, in the absence of forcing and viscous effects, equations (2.1) and (2.2) have waves with dispersion relation (2.4) as exact solutions.

We can thus define several relevant time scales, as in a turbulent flow one can identify different characteristic times for each possible interaction. The sweeping of the small-scale eddies (of size ${\sim}1/k$ ) by the large-scale flow is described by the sweeping time (Chen & Kraichnan Reference Chen and Kraichnan1989)

(2.6) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{S}\sim \frac{1}{Uk}.\end{eqnarray}$$

Note that sweeping does not result in a transfer of energy across scales. Sweeping corresponds to the advection of the small-scale eddies by a large-scale flow, and can act even in the absence of a mean flow (i.e. just a random evolution of the modes at large scales can result in random sweeping). The advection of the small-scale eddies in real space corresponds to a rotation of the Fourier modes by $\text{e}^{\text{i}Ukt}$ . The interaction of similar sized eddies, which result in nonlinear transfer of energy, is described by the nonlinear time scale,

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{NL}\sim \frac{1}{k\sqrt{kE(k)}},\end{eqnarray}$$

where $E(k)$ is the energy spectrum of the flow. Finally, the time scale for the interaction of waves modes is expected to be proportional to the wave period, i.e.

(2.8) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}\sim \frac{1}{|\unicode[STIX]{x1D714}(\boldsymbol{k})|}.\end{eqnarray}$$

While in homogeneous and isotropic turbulence the dominant Eulerian time is the sweeping time (Chen & Kraichnan Reference Chen and Kraichnan1989), when waves are present the dominant time scale can be either the sweeping time, the nonlinear time, or the wave period, depending on which is the fastest at a given scale (Favier et al. Reference Favier, Godeferd and Cambon2010; Servidio et al. Reference Servidio, Carbone, Dmitruk and Matthaeus2011; Clark di Leoni et al. Reference Clark di Leoni, Cobelli, Mininni, Dmitruk and Matthaeus2014, Reference Clark di Leoni, Cobelli and Mininni2015). As a result, these time scales imply that, depending on the shape of the energy spectrum $E(k)$ , the approximation that waves are faster than the eddies for small enough $Ro$ may break down for sufficiently large wavenumbers, or for a subset of the Fourier modes (e.g. for modes with small vertical wavenumber). Below we present a more detailed estimation of which is the dominant time scale for each mode in our numerical simulations.

2.3 Resonant interactions

For modes for which the waves are much faster than the eddies, we can assume wave dynamics dominate the evolution, while the eddies contribute to a slow modulation of the amplitude of the waves. Thus, we can write $\boldsymbol{u}_{\boldsymbol{k}}=\boldsymbol{U}_{\boldsymbol{k}}\text{e}^{\text{i}\unicode[STIX]{x1D714}_{\boldsymbol{k}}t}$ . In practice, this approximation is done after decomposing the modes $\boldsymbol{u}_{\boldsymbol{k}}$ into the helical eigenstates $\boldsymbol{h}_{\pm }(\boldsymbol{k})$ of the linearised (2.1), $\boldsymbol{u}_{\boldsymbol{k}}=a_{+}(\boldsymbol{k})\boldsymbol{h}_{+}(\boldsymbol{k})+a_{-}(\boldsymbol{k})\boldsymbol{h}_{-}(\boldsymbol{k})$ , where the subindices $+$ and $-$ correspond to the two possible polarisations of the waves, see e.g. Waleffe (Reference Waleffe1993). However, for the purpose of the following discussion it is better to work in terms of $\boldsymbol{u}_{\boldsymbol{k}}$ , as those modes are more easily accessed in numerical simulations.

Replacing in (2.3) we obtain

(2.9) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D708}k^{2}+2\unicode[STIX]{x1D6FA}\mathbb{P}_{\boldsymbol{ k}}\boldsymbol{z}\times \right)\boldsymbol{U}_{\boldsymbol{k}}=-\text{i}\mathbb{P}_{\boldsymbol{k}}\mathop{\sum }_{\boldsymbol{p}+\boldsymbol{q}=\boldsymbol{k}}(\boldsymbol{U}_{\boldsymbol{p}}\boldsymbol{\cdot }\boldsymbol{q})\boldsymbol{U}_{\boldsymbol{q}}\text{e}^{-\text{i}[\unicode[STIX]{x1D714}(\boldsymbol{k})-\unicode[STIX]{x1D714}(\boldsymbol{p})-\unicode[STIX]{x1D714}(\boldsymbol{q})]t}.\end{eqnarray}$$

Integrating over several periods of the waves, the nonlinear term can give a non-negligible energy transfer only if triads are resonant, i.e. if $\unicode[STIX]{x1D714}(\boldsymbol{k})=\unicode[STIX]{x1D714}(\boldsymbol{p})+\unicode[STIX]{x1D714}(\boldsymbol{q})$ . In practice, near-resonant triads with

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{r}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})=\frac{\min \{|\unicode[STIX]{x1D714}(\boldsymbol{k})\pm \unicode[STIX]{x1D714}(\boldsymbol{p})\pm \unicode[STIX]{x1D714}(\boldsymbol{q})|\}}{2\unicode[STIX]{x1D6FA}}=O(Ro),\end{eqnarray}$$

are also expected to be relevant (see, e.g. Alexakis Reference Alexakis2015). We will call $\unicode[STIX]{x1D6FE}_{r}$ the resonance factor, as it measures how close to resonance a given triad is in the framework of wave turbulence theory. The minimum and the plus–minus signs added in the latter equation are due to the fact that our modes $\boldsymbol{u}_{\boldsymbol{k}}$ mix both polarisations of the inertial waves.

4.1 Numerical simulations

The code GHOST (Gómez, Mininni & Dmitruk Reference Gómez, Mininni and Dmitruk2005; Mininni et al. Reference Mininni, Rosenberg, Reddy and Pouquet2011) is used to solve (2.1) and (2.2) using a parallel pseudospectral method with a second-order Runge–Kutta scheme for the time evolution. The $2/3$ -rule is used for dealiasing. As will be seen below, computation of the contribution function requires high cadence input/output in time, and a significant amount of storage (note spatial information needs to be saved with twice the frequency of the fastest waves in the system). As a result, only simulations with moderate resolution can be performed. Here we present two simulations using grids of $N^{3}=512^{3}$ points in a three-dimensional periodic box.

Both simulations are identical except for the value of $\unicode[STIX]{x1D6FA}$ . In one of the simulations $\unicode[STIX]{x1D6FA}=4$ , while in the other $\unicode[STIX]{x1D6FA}=8$ . The simulations were started from the fluid at rest, and energy was injected via a mechanical forcing. We chose a Taylor–Green forcing of the form

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{F}=F_{0}[\sin (k_{TG,x}x)\cos (k_{TG,y}y)\cos (k_{TG,x}z)\boldsymbol{x}-\cos (k_{TG,x}x)\sin (k_{TG,y}y)\cos (k_{TG,z}z){\hat{y}}], & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

with $F_{0}=0.277$ , $\boldsymbol{k}_{TG}=(1,1,1)$ (which results in $L=2\unicode[STIX]{x03C0}/k_{TG}=2\unicode[STIX]{x03C0}/\sqrt{3}$ ), and $\unicode[STIX]{x1D708}=6.5\times 10^{-4}$ in dimensionless units (for unit velocity and a box of length $2\unicode[STIX]{x03C0}$ ). While other forcings will presumably produce similar results, Taylor–Green forcing was chosen because it has been reported to result in a larger amplitude of wave modes when compared with random-in-time isotropic forcing (Clark di Leoni et al. Reference Clark di Leoni, Cobelli, Mininni, Dmitruk and Matthaeus2014). The system was let to reach a turbulent steady state with $U\approx 0.9$ , which translates to a Reynolds number of approximately 5000, and a Rossby number of 0.03 for $\unicode[STIX]{x1D6FA}=4$ and of 0.015 for $\unicode[STIX]{x1D6FA}=8$ . Once in the turbulent steady state, the simulations were allowed to run for over 12 large-scale turn over times, the time span over which the following analysis was carried out.

4.2 Energy spectrum and decorrelation times

Before proceeding to the analysis of the contribution function, we first discuss some general properties of the simulations. In figure 1 we show contour levels of the axisymmetric energy spectrum for the simulation with $\unicode[STIX]{x1D6FA}=8$ , as a function of the perpendicular and parallel wavenumbers (with the perpendicular and parallel directions defined with respect to the axis of rotation). Whilst in isotropic turbulence one would expect these contours to be circular, the effect of rotation in these flows impose a clear anisotropy, with a preferred accumulation of energy in modes with $k_{\Vert }\approx 0$ as predicted by Cambon & Jacquin (Reference Cambon and Jacquin1989), Cambon et al. (Reference Cambon, Mansour and Godeferd1997), and Waleffe (Reference Waleffe1992). Moreover, a significant fraction of the energy is in modes with $k_{\Vert }=0$ , for which resonant interactions cannot account.

Three different regions are shaded in figure 1. The white region corresponds to modes with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}<\unicode[STIX]{x1D70F}_{s}<\unicode[STIX]{x1D70F}_{NL}$ . This is the region of ‘fast’ (or ‘wave’) modes, for which the period of the waves is the fastest time scale. The light grey region corresponds to modes with $\unicode[STIX]{x1D70F}_{s}<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}<\unicode[STIX]{x1D70F}_{NL}$ . Although these modes are often considered to be ‘fast’, as shown in Clark di Leoni et al. (Reference Clark di Leoni, Cobelli, Mininni, Dmitruk and Matthaeus2014) these modes are decorrelated in a time scale which is of the order of the sweeping time. In other words, in the Eulerian frame, the dominant time scale for these modes is given by the sweeping, which is the shortest available time. Finally, in the dark grey region the modes have $\unicode[STIX]{x1D70F}_{s}<\unicode[STIX]{x1D70F}_{NL}<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ . This is the region of ‘slow’ modes for which the eddies are faster than the waves. The three shaded regions are shown only as a reference. To compute the value of $\unicode[STIX]{x1D70F}_{NL}$ at each wavenumber using (2.7), an estimation of $E(k)$ is needed. For simplicity, instead of using the measured spectrum, we use the phenomenological expression for non-helical rotating turbulence $E(k)\sim \unicode[STIX]{x1D716}^{1/2}\unicode[STIX]{x1D6FA}^{1/2}k^{-2}$ , where $\unicode[STIX]{x1D716}$ is the energy dissipation rate (Zhou Reference Zhou1995; Müller & Thiele Reference Müller and Thiele2007; Mininni et al. Reference Mininni, Rosenberg and Pouquet2012). In Clark di Leoni et al. (Reference Clark di Leoni, Cobelli, Mininni, Dmitruk and Matthaeus2014) it was shown, from direct computation of the decorrelation times, that this choice results in a good estimation of the dominant time scale for modes laying in the inertial range (i.e. at small and intermediate wavenumbers). At large wavenumbers, where the spectrum drops exponentially as a result of viscous damping, $\unicode[STIX]{x1D70F}_{NL}$ departs from this estimation. However, we will not consider modes in the viscous range, for which also the viscous damping time can be relevant.

The ordering of the time scales described above has implications for the behaviour of the time-correlation function $\unicode[STIX]{x1D6E4}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ . In the white region of figure 1, $\unicode[STIX]{x1D6E4}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ is expected to decay to $1/\text{e}$ of its value for $\unicode[STIX]{x1D70F}=0$ in one wave period of the mode with wave vector $\boldsymbol{k}$ (Favier et al. Reference Favier, Godeferd and Cambon2010; Clark di Leoni et al. Reference Clark di Leoni, Cobelli, Mininni, Dmitruk and Matthaeus2014). This time scale is what we define as the decorrelation time: after this time, the mode $\boldsymbol{k}$ has significantly decorrelated from its previous state. As already mentioned, in the two grey regions the function $\unicode[STIX]{x1D6E4}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ decays to $1/\text{e}$ of its value for $\unicode[STIX]{x1D70F}=0$ in a time equal to $\unicode[STIX]{x1D70F}_{s}$ (Clark di Leoni et al. Reference Clark di Leoni, Cobelli, Mininni, Dmitruk and Matthaeus2014). We will thus consider modes in these three regions to compute the third-order correlators $\unicode[STIX]{x1D6E9}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q},\unicode[STIX]{x1D70F})$ and $\hat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q},\unicode[STIX]{x1D714})$ . In particular, in figure 1 we indicate two fast modes $\boldsymbol{k}=(0,5,5)$ and $(0,0,8)$ , a ‘swept’ mode $(0,0,45)$ , and a slow mode $(0,30,0)$ . These modes will be used in several examples below.

4.3 Analysis of third-order time correlators

Equation (3.2) indicates that the nonlinear interaction with all triads gives rise to the time decorrelation of the mode at a given $\boldsymbol{k}$ . In other words, the apparently random contribution of all nonlinear triads results in the deformation of the structure to the point that the mode decorrelates with itself, and thus transfers its energy to other modes in the allowed triads. As a result, $\unicode[STIX]{x1D6E4}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ decreases for short increments $\unicode[STIX]{x1D70F}$ , and then fluctuates around zero. $-\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}$ should then start from zero for $\unicode[STIX]{x1D70F}=0$ , increase to a maximum, and then fluctuate with the dominant time scale of the mode. Figure 2 shows a partial reconstruction of $-\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{\boldsymbol{k}}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}$ by computing a partial sum over $\boldsymbol{q}$ of the $\unicode[STIX]{x1D6E9}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D70F})$ function for $\boldsymbol{k}=(0,0,8)$ and for $\boldsymbol{k}=(0,30,0)$ . The partial reconstruction is done using (3.2) and (3.5), i.e. we sum $\unicode[STIX]{x1D6E9}$ over the subset of $\boldsymbol{p}$ and $\boldsymbol{q}$ modes available for the analysis and that satisfy the relation $\boldsymbol{p}+\boldsymbol{q}=\boldsymbol{k}$ . Due to the large amount of data required for this computation, we only consider modes in the $(k_{x}=0,k_{y},k_{z})$ plane, and therefore we only sum over the triads with $\boldsymbol{p}\boldsymbol{\cdot }\boldsymbol{x}=\boldsymbol{q}\boldsymbol{\cdot }\boldsymbol{x}=0$ , resulting in the partial reconstruction mentioned above. Nonetheless, this suffices to get the expected behaviour for the time derivative of the decorrelation functions. For the mode $\boldsymbol{k}=(0,0,8)$ , which is a fast mode, the time derivative gets locked to the wave period, while for $\boldsymbol{k}=(0,30,0)$ , which is a slow mode, the dominant time scale is the sweeping time. Here and in the following, except when duly noted, all results shown are for the $\unicode[STIX]{x1D6FA}=8$ simulation.

Figure 2.

Partial reconstruction of $-\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{\boldsymbol{k}}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}$ via the sum over $\boldsymbol{q}$ of the third-order correlator $\unicode[STIX]{x1D6E9}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D70F})$ , from (3.2) (computation of the real part by adding the complex conjugate is implied). The results are for (a) $\boldsymbol{k}=(0,30,0)$ and (b) $\boldsymbol{k}=(0,0,8)$ . The time lag $\unicode[STIX]{x1D70F}$ is normalised by the dominant time scale ( $\unicode[STIX]{x1D70F}_{s}$ in (a) and $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}$ in (b)); the different time scales in these units are also shown as a reference by the horizontal bars in each figure. We recover the expected behaviour for the correlation functions, $\boldsymbol{k}=(0,0,8)$ (which is a fast mode) gets locked to the wave period, while $\boldsymbol{k}=(0,30,0)$ (which is slow) evolves in the sweeping time scale. The integral of these functions gives the correlation function, which decays rapidly on the dominant time scale.

Figure 3.

$|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k}=(0,0,8),\boldsymbol{q},\unicode[STIX]{x1D714})|$ as a function of $\unicode[STIX]{x1D714}$ , for a fixed fast mode $\boldsymbol{k}$ , and for different values of $\boldsymbol{q}$ . Interactions with other wave modes (blue and red curves) show a well tuned spectrum centred around the wave frequency of the mode $\boldsymbol{k}$ , indicating interactions are close to resonance. On the other hand, interactions with eddy modes (green and cyan curves) display a wider spectrum. All subsequent analyses of the contribution function focus on its maximum amplitude and on how well tuned each triad is (i.e. on the width of the peak around the maximum).

4.4 Contribution functions

We now present our analysis of the behaviour of the contribution function $\unicode[STIX]{x1D6E9}$ . Figure 3 shows the value of $|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k}=(0,0,8),\boldsymbol{q},\unicode[STIX]{x1D714})|$ as a function of $\unicode[STIX]{x1D714}$ , for four different values of $\boldsymbol{q}$ (i.e. for four different triads). All of them peak at $\unicode[STIX]{x1D714}_{0}=\unicode[STIX]{x1D714}(\boldsymbol{k})$ , which is the wave frequency of the mode $\boldsymbol{k}$ . This was checked for other values of $\boldsymbol{k}$ as well, and a peak in the corresponding wave frequency was observed in all cases except for the modes with $k_{\Vert }\approx 0$ (i.e. the slow modes), for which no discernible peak is present. Moreover, and in spite of the presence of a peak for modes with $k_{\Vert }>0$ , it is worth noting that the width of the peak depends strongly on the nature of the other modes in the triad. While interactions with other wave modes have most of the power in the peak and are well tuned (i.e. the peak is relatively narrow), interactions with slow modes can have large amplitudes but with a broad spectrum.

This gives a direct way to identify not only the strength of a given triad, but also to measure how resonant the triad is, as more resonant triads are expected to result in a sharper spectrum of $\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})$ by virtue of (3.9). Therefore, to simplify the analysis, we can focus on a few modes $\boldsymbol{k}$ , explore all available values of $\boldsymbol{q}$ on a triad with $\boldsymbol{k}$ , and look only at the maximum value of $\widehat{\unicode[STIX]{x1D6E9}}$ (for all $\unicode[STIX]{x1D714}$ ) and on the relative width of the maximum (i.e. on how well tuned the interaction is around $\unicode[STIX]{x1D714}_{0}$ ).

Figure 4.

Intensity (as a function of $\boldsymbol{q}$ ) of the maximum of the contribution function, $\max _{\unicode[STIX]{x1D714}}\{|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})|\}$ . In each panel, $\boldsymbol{k}$ is fixed, and the maximum of $|\widehat{\unicode[STIX]{x1D6E9}}|$ is plotted for all available triads by varying $\boldsymbol{q}$ . Two $\boldsymbol{k}$ modes are considered, (a) $\boldsymbol{k}=(0,0,8)$ and (b) $\boldsymbol{k}=(0,5,5)$ , both dominated by waves. The black dots in the centre indicate the modes $\boldsymbol{q}=\pm \boldsymbol{k}$ . Two prominent features arise. One is the effect of the anisotropy of the flow, as triads with larger amplitudes are distributed along horizontal bands (i.e. coupling the $\boldsymbol{k}$ modes with smaller vertical wavenumbers). The other is the defect along the line $\boldsymbol{q}=\unicode[STIX]{x1D6FC}\boldsymbol{k}$ , as collinear modes do not contribute to the triads in an incompressible fluid.

In figure 4 we show $\max _{\unicode[STIX]{x1D714}}\{|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})|\}$ for two modes $\boldsymbol{k}=(0,0,8)$ and $(0,5,5)$ , as a function of all possible values of $\boldsymbol{q}$ in the $(0,q_{y},q_{z})$ plane. The anisotropic nature of rotating turbulence makes a stellar apparition here, as the distribution of values is clearly influenced by it. The result indicates that triads which are elongated along the horizontal direction have larger amplitudes, which is compatible with the prediction that energy tends to go towards the slow modes (with $q_{z}\approx 0$ ) as discussed in Waleffe (Reference Waleffe1992). Indeed, the triads with larger amplitudes are located in a horizontal band within $-k_{z}\lesssim q_{z}\lesssim k_{z}$ . There are also strong triads that couple the $\boldsymbol{k}$ mode with modes with larger vertical wavenumber (i.e. triads in the horizontal bands $k_{z}\lesssim |q_{z}|\lesssim 2k_{z}$ ) which are compatible with an anisotropic transfer of a fraction of the energy towards larger wavenumbers (i.e. smaller scales). Finally, collinear modes (i.e. modes with $\boldsymbol{q}=\unicode[STIX]{x1D6FC}\boldsymbol{k}$ ) make no contribution to the triads as a result of the incompressibility of the fluid.

4.5 Normalised contribution functions

Having said this, it can be argued that the strongest triads correspond to modes with $q_{z}\approx 0$ only as a result of the anisotropic energy spectrum shown in figure 1: the modes with small vertical wavenumber have most of the energy, and as a result triads involving those modes will have larger amplitudes. Therefore, we can normalise the triads by the energy of the $\boldsymbol{q}$ mode in the triad, i.e. we can compute $\max _{\unicode[STIX]{x1D714}}\{|\hat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})|\}/E(\boldsymbol{q})$ . If this is done, from (3.2) and (3.8) the normalised contribution function has units of inverse time (i.e. of frequency). This time can thus be interpreted as the time scale of the energy transfer mechanism, as is often done in turbulence theories.

Figure 5.

Intensity of the peak values of the normalised contribution function for each triad, given by $\max _{\unicode[STIX]{x1D714}}\{|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})|\}/E(q_{y},q_{z})$ . In each panel, $\boldsymbol{k}$ is fixed for two modes dominated by waves: (a) $\boldsymbol{k}=(0,0,8)$ , and (b) $\boldsymbol{k}=(0,5,5)$ . The dashed lines represent the modes with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D70F}_{NL}$ . Modes with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}<\unicode[STIX]{x1D70F}_{NL}$ (those above the upper dashed line) have higher frequencies (i.e. shorter time scales), and are thus preferred. Note, however, there is a non-negligible leakage towards slow modes $\boldsymbol{q}$ with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}\gtrsim \unicode[STIX]{x1D70F}_{NL}$ (i.e. modes slightly below the dashed upper curve).

We now turn to the analysis of these normalised contribution functions. In figure 5 we show the peak value of the normalised functions for $\boldsymbol{k}=(0,0,8)$ and $(0,5,5)$ . Two dashed curves indicate the modes for which $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D70F}_{NL}$ , i.e. modes with the eddy turnover time equal to the period of the waves. Modes respectively above and below the upper and lower curves have $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}<\unicode[STIX]{x1D70F}_{NL}$ , and are dominated by the waves. Modes between the two dashed curves have $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}>\unicode[STIX]{x1D70F}_{NL}$ , and are slow modes dominated by the eddies.

Anisotropic effects are still evident in figure 5 after the normalisation, but the role played in the triads by the waves starts to become more clear. Although wave modes have less energy, after normalisation it becomes evident that triadic interactions of the wave modes $\boldsymbol{k}=(0,0,8)$ and $(0,5,5)$ with other wave modes are relatively stronger than interactions with slow modes. If the normalised contribution function is interpreted as an inverse transfer time, it then implies that the transfer between triads involving waves is faster than triads involving slow modes, and thus should be preferred for the interaction. This is true even for modes $\boldsymbol{q}$ with $\unicode[STIX]{x1D70F}_{s}<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}<\unicode[STIX]{x1D70F}_{NL}$ , i.e. for modes with sweeping time faster than the wave period (but with the waves still dominating over the eddies).

Figure 6.

Close up of the geometric distribution of the peak value of the normalised contribution function for each triad, given by $\max _{\unicode[STIX]{x1D714}}\{|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})|\}/E(q_{y},q_{z})$ . As in figure 5, in each panel $\boldsymbol{k}$ is fixed to consider two modes dominated by waves: (a) $\boldsymbol{k}=(0,0,8)$ , and (b) $\boldsymbol{k}=(0,5,5)$ . The dashed curves represent the modes with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D70F}_{NL}$ , and the circles represent the near-resonant modes (according to the theoretical prediction) with $\unicode[STIX]{x1D6FE}_{r}<0.1$ . In good agreement with wave turbulence theories, normalised resonant triads have large amplitudes, but some near-resonant triads are also strong. In panel (b), some of these near-resonant triads have non-negligible coupling with slow modes (see the circles near $q_{y}\approx 10$ and $q_{z}\approx 0$ ) allowing for energy transfer towards these modes.

Large amplitudes (or, equivalently, shorter transfer times) can be seen in figure 5 in the vicinity of $\boldsymbol{q}\approx \boldsymbol{k}$ . Although at first glance it would appear that this is due to local (in Fourier space) interactions being the most prominent, closer inspection reveals that it is also due to the effect of resonances. In figure 6 we show a close up of the normalised contribution functions in figure 5 for small values of $|\boldsymbol{q}|$ . Circles mark the modes that satisfy the theoretical near-resonant condition up to a value of $\unicode[STIX]{x1D6FE}_{r}=0.1$ . It is evident that many of the strongest triads correspond to resonant or near-resonant triads. This is in very good agreement with wave turbulence theories of rotating turbulence, which predict that resonant triads should dominate the coupling between modes (Newell Reference Newell1969). However, this also explains how the system transfers energy towards slow modes, which are inaccessible in weak-wave-turbulence approximations. The data in figure 6 indicate that not only resonant triads are relevant, but that near-resonant triads play an equally important role (at least for the case of a periodic flow). Indeed, large amplitudes can be seen around the circles in figure 6 in a region that is even broader than the fan corresponding to the condition $\unicode[STIX]{x1D6FE}_{r}=0.1$ . Close observation of figure 6(a,b) (as well as the observation of other modes $\boldsymbol{k}$ not shown here) gives rise to the following picture: for $\boldsymbol{k}=(0,0,8)$ , resonant and near-resonant interactions couple this mode with some modes with $q_{z}/q<k_{z}/k$ , thus allowing an energy exchange between these modes. Energy can thus be transferred towards modes with smaller vertical wavenumber, in agreement with the arguments in Cambon & Jacquin (Reference Cambon and Jacquin1989) and Waleffe (Reference Waleffe1993). For $\boldsymbol{k}=(0,5,5)$ , the process is repeated, but now some near-resonant interactions allow for a coupling (and thus a transfer) with slow modes. This is compatible with observations in Smith & Lee (Reference Smith and Lee2005), Alexakis (Reference Alexakis2015), and Gallet (Reference Gallet2015). Moreover, in figure 6(b) a non-negligible coupling with slow modes can be observed even for non-resonant triads (see the region between the two dashed curves with $q_{y}>0$ ), indicating that as energy approaches the slow modes the role of non-resonant interactions may also become more relevant.

4.6 Comparison with small-scale and slow modes

To gain more certainty on the effect of the waves in the triadic interactions, we compare now the previous results with the normalised contribution function for two modes: a small-scale mode with $\boldsymbol{k}=(0,0,45)$ , which is dominated by sweeping (but with the wave period faster than the eddy turnover time), and a small-scale slow mode with $\boldsymbol{k}=(0,30,0)$ , which has zero wave frequency. Here, by small scale, we refer to modes with wavenumbers significantly larger than the forced wavenumber. The resulting normalised contribution functions are shown in figure 7. In both cases, the division given by the curve with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D70F}_{NL}$ is less evident, and a superposition of the modes expected to be resonant or near-resonant (not shown) indicates no clear correlation between the strength of the triad and the theoretical resonant or near-resonant condition. For the mode $\boldsymbol{k}=(0,0,45)$ , the normalised contribution function indicates that coupling is stronger for modes with $q_{z}\approx k_{z}$ , an effect associated with the anisotropy of the flow, while the coupling with slow modes is negligible. The slow mode $\boldsymbol{k}=(0,0,45)$ shows more interesting features. The mode seems to be more strongly coupled with other slow modes $\boldsymbol{q}$ in the vicinity of $\boldsymbol{k}$ , and with modes with large $q_{z}$ (of the order of $|\boldsymbol{k}|$ , compatible with local triadic interactions, although these interactions are non-resonant).

Figure 7.

Peak values of the normalised contribution function for each triad, given by $\max _{\unicode[STIX]{x1D714}}\{|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})|\}/E(q_{y},q_{z})$ . The panels correspond to two fixed values of $\boldsymbol{k}$ : (a) a small-scale ‘fast’ mode $\boldsymbol{k}=(0,0,45)$ , and (b) a small-scale ‘slow’ mode $\boldsymbol{k}=(0,30,0)$ . The dashed curves represent the modes with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D70F}_{NL}$ . The role played by resonant and near-resonant interactions in these cases is less clear.

4.7 Effect of the Rossby number

We can also compare the results in the two simulations with different Rossby number, to quantify the effect of changing the rotation frequency in the intensity of resonant and near-resonant triads. As an illustration, figure 8 shows the geometric distribution of the peak values of the normalised contribution function for the mode $\boldsymbol{k}=(0,0,8)$ in the simulation with larger Rossby number. The same result as in figure 5 is obtained, but the contrast between modes above and below the $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D70F}_{NL}$ curve is less marked. Also, the region of modes dominated by eddies (i.e. of slow modes) increases as expected, and the boundary between the two regions indicated by the change in intensity of the triads moves accordingly. This confirms that the changes in intensity in figure 5 indeed separate triads involving slow and fast modes, and is also consistent with the behaviour expected in a rotating flow as the Rossby number is varied.

Figure 8.

Peak values of the normalised contribution function for each triad, $\max _{\unicode[STIX]{x1D714}}\{|\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k}=(0,0,8),\boldsymbol{q},\unicode[STIX]{x1D714})|\}/E(q_{y},q_{z})$ , for the simulation with weaker rotation (i.e. larger Rossby number). Results are similar to the case with stronger rotation, although the contrast in intensity between triads involving fast and slow waves is less clear.

4.8 Direct measurement of the resonance level of each triad

One of the most important implications of the contribution function is that it allows a direct measurement of how well tuned a triad is, i.e. of how resonant the interaction between three modes is. This was already discussed in the context of figure 3, where we showed that some triads display a narrower peak around the wave frequency than others. But now we can put this observation on firmer grounds.

As it follows from (3.9), for a perfectly resonant triad the contribution function should be a delta distribution centred around $\unicode[STIX]{x1D714}_{0}=\unicode[STIX]{x1D714}(\boldsymbol{k})$ . Near-resonant and non-resonant interactions broaden the peak. This broadening can be measured using the quality factor

(4.2) $$\begin{eqnarray}Q=\frac{\unicode[STIX]{x1D714}_{0}}{\unicode[STIX]{x0394}\unicode[STIX]{x1D714}}.\end{eqnarray}$$

In other words, the $Q$ factor is the inverse relative width of the peak of $\widehat{\unicode[STIX]{x1D6E9}}(\boldsymbol{k},\boldsymbol{q},\unicode[STIX]{x1D714})$ as a function of the frequency. We estimate $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}$ by calculating the width of the peak in the spectrum (see figure 3) at half the amplitude of the maximum value. In the theory of resonators, the $Q$ factor is often interpreted as the ratio of the energy stored to the energy lost in a system. In our case, the larger the $Q$ factor the more resonant the triad is, and the less energy of the mode $\boldsymbol{k}$ is lost (i.e. given) to non-resonant modes.

Figure 9.

Inverse relative bandwidth (quality factor) of the peak in each contribution function, as a function of $\boldsymbol{q}$ , and for fixed $\boldsymbol{k}=(0,0,8)$ , (a) for the simulation with $Ro\approx 0.03$ , and (b) for the simulation with $Ro\approx 0.015$ . Larger quality factors correspond to sharper bandwidths of the triads relative to their central frequency, and thus to more resonant interactions (i.e. the factor quantifies how well tuned a triad is). Circles indicate modes which satisfy the theoretical near-resonance condition with $\unicode[STIX]{x1D6FE}_{r}<0.1$ . A good agreement is observed between the theoretical condition and the quality factor of the contribution function, especially for the flow with smaller Rossby number. Note, however, that relatively large quality factors are observed for branches which are wider than the condition $\unicode[STIX]{x1D6FE}_{r}<0.1$ , indicating again the important role played by near-resonant interactions.

Figure 9 shows this quality factor for each contribution function for fixed $\boldsymbol{k}=(0,0,8)$ and for all possible values of $\boldsymbol{q}=(0,q_{y},q_{z})$ , for the two simulations with different Rossby numbers. Superimposed on the quality factor, circles mark the modes that satisfy the theoretical near-resonant condition up to a value of $\unicode[STIX]{x1D6FE}_{r}=0.1$ . In particular for the simulation with smaller Rossby number, triads with large quality factors (i.e. well tuned triads) more or less coincide with triads with small $\unicode[STIX]{x1D6FE}_{r}$ , especially for the branches in the upper-left and lower-right quadrants in figure 9(b). In other words, the quality factor defined above gives a good measure of how resonant a triad is. Moreover, three features in figure 9 are worth emphasising. First, the $Q$ factor has maximum values of approximately 6, which is of the order of, although a bit smaller than, what is often found in electrical or mechanical resonators. In other words, even resonant triads display broadband peaks in the frequency spectrum. Second, the area covered by triads with the largest values of $Q$ is relatively larger than the area corresponding to the circles with $\unicode[STIX]{x1D6FE}_{r}<0.1$ , confirming the importance of interactions that are even marginally near-resonant. The scaling of this behaviour with Rossby number and domain size can be important for theories of rotating turbulence in infinite domains, in which the behaviour of the resonant and slow manifolds, as well as the relevance of near resonances, are unclear (Cambon et al. Reference Cambon, Rubinstein and Godeferd2004; Chen et al. Reference Chen, Chen, Eyink and Holm2005; Bourouiba Reference Bourouiba2008). Third, there are some modes with relatively large $Q$ values (compared with the mean amplitude of $Q$ for all modes) that do not correspond to resonant or near-resonant modes in wave turbulence theory. Most notably, some of these modes are modes with small $|\boldsymbol{q}|$ and lying in the region of the slow modes.