5.1 The $Po=-0.3745$ case (non-retracing rays)

Figure 4.

For LCa at $\unicode[STIX]{x1D714}_{0}=4500$ and $Po=-0.3745$ , shown are isosurfaces of (a) $w=2$ and (b) ${\mathcal{H}}=\pm 3\times 10^{5}$ , and of the (unstable) BS at the same parameters (computed in the ${\mathcal{I}}$ -symmetric subspace) with (c) $w=2$ and (d) ${\mathcal{H}}=\pm 3\times 10^{5}$ . The online supplementary movie available at https://doi.org/10.1017/jfm.2017.922 shows an animation of ${\mathcal{H}}$ for LCa over one period $\unicode[STIX]{x1D70F}_{\mathit{LCa}}=3.24\times 10^{-4}$ . Isosurfaces of the (e,f) $m=4$ and (g,h) $m=5$ azimuthal Fourier components of $w$ at levels $w=\pm 30$ are shown in two orthogonal views.

In this section, we consider the generic situation, away from triadic resonances of Kelvin modes and away from retracing inertial wave beams. Increasing $\unicode[STIX]{x1D714}_{0}$ with $Po=-0.3745$ fixed, the base state becomes unstable at $\unicode[STIX]{x1D714}_{0}\approx 3992$ in a Hopf bifurcation, resulting in a time periodic limit cycle solution LCa. The structure of LCa is shown in figure 4 for $\unicode[STIX]{x1D714}_{0}=4500$ . The isosurface of $w$ in figure 4(a) resembles that of the basic state, but additional structures are present in the interior emerging from the endwalls. The helicity contours in figure 4(b) show the presence of columnar helical structures in the bulk. These columns are absent in BS (figure 4 d). The unstable BS has been computed by restricting the simulations to the ${\mathcal{I}}$ -invariant subspace, and the $w$ isosurfaces of BS are shown in figure 4(c). The presence of the helical columns in LCa greatly distorts the inertial wave beams. The columnar structures, and their temporal evolution, can be appreciated in the supplementary online movie. The helical columns resemble a rotating wave, but there are significant distortions during the period of the solution.

Figure 5.

Contours of (a) ${\mathcal{H}}(t_{0})$ , (b) ${\mathcal{I}}({\mathcal{H}}(t_{0}))$ and (c) ${\mathcal{H}}(t_{0}+\unicode[STIX]{x1D70F}/2)$ at $z=\unicode[STIX]{x1D6E4}/4$ for LCa at $\unicode[STIX]{x1D714}_{0}=4500$ and $Po=-0.3745$ . Contour levels are ${\mathcal{H}}\in [-7.6\times 10^{5},7.6\times 10^{5}]$ .

Figure 5(a) shows contours of helicity in the plane $z=\unicode[STIX]{x1D6E4}/4$ for LCa at $\unicode[STIX]{x1D714}_{0}=4500$ . The helicity columns rotate, while the overturning flow remains fixed in the table frame, and the interaction between the columns and the overturning flow results in the observed distortions with time. The frequency of the bifurcated solution LCa is $\unicode[STIX]{x1D714}_{\mathit{LCa}}\approx 1.94\times 10^{4}$ , between four and five times $\unicode[STIX]{x1D714}_{0}=4500$ .

The base state is equivariant with respect to the inversion symmetry ${\mathcal{I}}$ . The ‘a’ in LCa signifies that it is not ${\mathcal{I}}$ -equivariant. This can be readily observed in figure 5(a,b) showing the helicity contours of LCa and of the transformed field by ${\mathcal{I}}$ . According to (2.9), the helicity should simply change sign if the solution were symmetric (i.e. ${\mathcal{I}}$ -equivariant). This is not the case, LCa is not ${\mathcal{I}}$ -symmetric. The non-symmetry of LCa can be quantified by the symmetry parameter, which for the case shown in figure 5 is $S=4.12$ . As described at the end of § 2, the limit cycle bifurcating from an ${\mathcal{I}}$ -equivariant steady state (BS) is either ${\mathcal{I}}$ -equivariant (an $F$ -cycle), or the ${\mathcal{I}}$ -symmetry is broken at the bifurcation and the periodic solution (an $S$ -cycle) has a spatio-temporal symmetry ${\mathcal{I}}^{st}$ . This spatio-temporal symmetry is such that applying ${\mathcal{I}}$ is the same as advancing the solution for half a period in time. This is precisely what happens with LCa: the action of ${\mathcal{I}}$ on LCa at a given time (figure 5 b) produces the same result as advancing the solution half a period in time (figure 5 c), i.e. ${\mathcal{I}}(\text{LCa}(t))=\text{LCa}(t+\unicode[STIX]{x1D70F}/2)$ . Therefore, LCa has a spatio-temporal symmetry ${\mathcal{I}}^{st}(\text{LCa}(t))=\text{LCa}(t)$ , and the orbit LCa as a set, is ${\mathcal{I}}$ -symmetric: applying ${\mathcal{I}}$ to a point of LCa results in another point of LCa, ${\mathcal{I}}(\text{LCa})=\text{LCa}$ .

Figure 6.

Relative modal kinetic energies, $e_{m}$ , of LCa at $\unicode[STIX]{x1D714}_{0}=4500$ and $Po=-0.3745$ (yellow circles) and LCs at $\unicode[STIX]{x1D714}_{0}=8000$ and $Po=-0.4$ (green diamonds).

Figure 5 shows that there are four helicity columns in LCa. Figure 6 shows the relative modal kinetic energies of LCa (filled yellow circles), and we indeed observe that, apart from $e_{0}$ and $e_{1}$ which are relative modal energies in the main azimuthal Fourier components of BS, $e_{4}$ and $e_{5}$ are larger than all the others by more than an order of magnitude. The figure also shows that the numerical solution LCa is well resolved. The presence of modal energies $e_{1}$ , $e_{4}$ and $e_{5}$ is suggestive of a triadic resonance. However, it cannot be a resonance between three Kelvin modes because the $(1,1,1)$ spatial structure associated with the overturning base flow BS is not a Kelvin mode, but a forced flow, as discussed earlier. However, it is still possible for two free Kelvin modes to resonate with the forced $(1,1,1)$ flow BS. In order to explore this possibility, we have plotted in figure 4(e–h) isosurfaces of the axial velocity of the azimuthal Fourier components $m=4$ and $m=5$ of LCa. As in the case of pure Kelvin modes (3.2), these Fourier components have a well-defined number of zeros in the radial, axial and azimuthal directions, and we can associate with them the integers $(k,m,n)$ , as is typically done with Kelvin modes. The sign of $m$ remains to be determined, and that depends on the sense of rotation of the Fourier components in the cylinder frame; positive $m$ corresponds to retrograde and negative $m$ corresponds to prograde rotation with respect to $\unicode[STIX]{x1D714}_{0}$ . In the table frame, both components have prograde rotation and their rotation frequencies are $\unicode[STIX]{x1D714}^{\mathit{drift}}=\unicode[STIX]{x1D714}_{\mathit{LCa}}/4=4854.13$ for $m=4$ and $\unicode[STIX]{x1D714}^{\mathit{drift}}=\unicode[STIX]{x1D714}_{\mathit{LCa}}/5=3883.30$ for $m=5$ . According to (3.6), the $m=4$ component rotates prograde ( $\unicode[STIX]{x1D70E}^{\mathit{drift}}=354.13$ ) and the $m=5$ component is retrograde ( $\unicode[STIX]{x1D70E}^{\mathit{drift}}=-616.70$ ) in the cylinder frame. The temporal frequencies in the cylinder frame (3.6) are given by $\unicode[STIX]{x1D70E}_{LCa,4}/\unicode[STIX]{x1D714}_{0}=0.3148$ for $m=4$ , and $\unicode[STIX]{x1D70E}_{LCa,5}/\unicode[STIX]{x1D714}_{0}=0.6852$ for $m=5$ . Therefore, their structure corresponds to modes $(1,-4,1)$ and $(1,5,2)$ . It is worth noting that $\unicode[STIX]{x1D70E}_{LCa,4}/\unicode[STIX]{x1D714}_{0}+\unicode[STIX]{x1D70E}_{LCa,5}/\unicode[STIX]{x1D714}_{0}=1$ , and hence the triadic resonance conditions between these two Fourier modes and the forced $(1,1,1)$ overturning flow BS are fulfilled exactly. This is because for a Hopf bifurcation there is only a single frequency in the table frame, so all the Fourier components have the same temporal frequency.

Now we compare the $m=4$ and $m=5$ azimuthal components of LCa with the Kelvin modes corresponding to the geometry (aspect ratio $\unicode[STIX]{x1D6E4}=1.33$ and tilt angle $\unicode[STIX]{x1D6FC}=3^{\circ }$ ) and Poincaré number of LCa. The frequency of a Kelvin mode $K_{kmn}$ is given by (Marques & Lopez Reference Marques and Lopez2015):

where $a_{k}$ is given by (3.3). The spatial structure of $K_{kmn}$ and the angle $\unicode[STIX]{x1D6FD}$ of the characteristics depend only on the aspect ratio $\unicode[STIX]{x1D6E4}$ ; the frequency $\unicode[STIX]{x1D70E}_{kmn}$ depends also on the tilt angle $\unicode[STIX]{x1D6FC}$ and Poincaré number $Po$ , because the axial angular velocity $\unicode[STIX]{x1D714}_{a}$ of the cylinder depends on them (2.6). We use the detuning parameter $\unicode[STIX]{x1D6FF}_{3}$ (3.12) to quantify how close a pair of free Kelvin modes are to a triadic resonance with the forced $(1,1,1)$ BS. Table 2 shows the pairs of Kelvin modes closest to resonance with BS, with detuning parameter $\unicode[STIX]{x1D6FF}_{3}<0.06$ , and with $k=1,2$ , $n\leqslant 4$ and $|m|\leqslant 10$ . The first section of the table corresponds to close resonances with $k=1$ . The closest resonance is precisely the $(1,-4,1)$ and $(1,5,2)$ case, consistent with the numerical solutions of the Navier–Stokes equations. Of course, the Kelvin modes $K_{1-41}$ and $K_{152}$ are not the same as the $m=4$ and $m=5$ azimuthal Fourier components of LCa. In particular, their structure near the walls, inside the boundary layers, is completely different because the Kelvin modes do not satisfy the physical no-slip boundary conditions. However, the inviscid Kelvin modes and the Fourier components of LCa are strikingly similar in the bulk, and moreover, the values of the frequencies $\unicode[STIX]{x1D70E}$ are also very close:

The last two columns in table 2 show the frequencies of the Kelvin modes in the table frame. Apart from the sign, they are very close to the temporal frequency of LCa, $\unicode[STIX]{x1D714}_{\mathit{LCa}}/\unicode[STIX]{x1D714}_{0}=4.315$ . The sign of $\unicode[STIX]{x1D714}$ in the last two columns, together with $m$ , determines the azimuthal drift of the Kelvin modes in the table frame (3.6), and both are prograde, as are the $m=4$ and $m=5$ Fourier modes of LCa. Furthermore, $K_{1-41}$ and $K_{152}$ are not ${\mathcal{I}}$ symmetric because $m+n$ is odd (3.7). This is consistent with the ${\mathcal{I}}$ symmetry being broken at the Hopf bifurcation. We know that when $m+n$ is odd, the Kelvin modes have a spatio-temporal symmetry (3.9), and this is precisely the symmetry of LCa (see the earlier discussion of figure 5). All of this strongly suggests that at the Hopf bifurcation spawning LCa, a triadic resonance between the forced $(1,1,1)$ flow BS and the free Kelvin modes $K_{1-41}$ and $K_{152}$ is excited, resulting in the instability of BS. $K_{1-41}$ and $K_{152}$ are not exactly tuned to a perfect triadic resonance, but are very close to resonance, with a detuning of only 1.4 %.

Table 2.

Kelvin mode pairs, $K_{k_{1}m_{1}n_{1}}$ and $K_{k_{2}m_{2}n_{2}}$ , closest to resonance with the forced $(1,1,1)$ mode for $\unicode[STIX]{x1D6E4}=1.33$ , $\unicode[STIX]{x1D6FC}=3^{\circ }$ and $Po=-0.3745$ , for $k\leqslant 2$ , $n\leqslant 4$ and $m\leqslant 10$ , and detuning $\unicode[STIX]{x1D6FF}_{3}<0.06$ .

The presence of a triadic resonance is not surprising, because there are many pairs of Kelvin modes with small detuning $\unicode[STIX]{x1D6FF}_{3}$ . By fixing $k=1$ and $\unicode[STIX]{x1D6FF}_{3}<0.06$ there is not only the 1 :  $-4$  : 5 resonance, but also four additional pairs with $|m|\in [8,10]$ . If the $k=1$ condition is relaxed to $k\leqslant 2$ , 15 additional pairs appear, some of them with smaller detuning parameter values than in the 1 :  $-4$  : 5 case, as shown in table 2. Considering larger values of $k$ , $m$ and $n$ identifies more pairs, many of them with arbitrarily small detuning $\unicode[STIX]{x1D6FF}_{3}$ . This is because the Kelvin mode frequencies are dense (Greenspan Reference Greenspan1968). However, these Kelvin modes with large values of $k$ , $m$ and $n$ are quickly damped by viscosity (Greenspan Reference Greenspan1968), and so they are not expected to play any significant dynamical role. It is worth noting that the instability mechanism at the Hopf bifurcation is not given by the closest-to-resonance pair of Kelvin modes (the pair with the minimum value of the detuning parameter). The detuning must be small in order to have resonance, but other factors may play a critical role in the selection, for example the presence or absence of retracing rays, the boundary layer structure, the nonlinearities in the forced $(1,1,1)$ BS, the damping and detuning effects of viscosity, imperfections and noise. The list of possibilities is long, and the only way to decide which resonances will destabilize the base state is to (numerically) solve the Navier–Stokes equations with no-slip boundary conditions. Considering the detuning parameter alone is very limited and does not allow one to discriminate which of the possible resonances will play a role, except perhaps when the geometry and Poincaré number are fine-tuned to a specific low-order (with moderate values of $k$ , $|m|$ and  $n$ ) triadic resonance (e.g. as was done in Lagrange et al. Reference Lagrange, Meunier, Nadal and Eloy2011; Albrecht et al. Reference Albrecht, Blackburn, Lopez, Manasseh and Meunier2015; Marques & Lopez Reference Marques and Lopez2015; Lopez & Marques Reference Lopez and Marques2016b ).

In order to determine if the second group of potential resonances centred around $|m|=9$ play any role in the present problem, we have computed the unstable BS for $\unicode[STIX]{x1D714}_{0}>3992$ (the critical $\unicode[STIX]{x1D714}_{0}$ at which BS become unstable to LCa). We have computed these unstable basic states by restricting the computation to the ${\mathcal{I}}$ -symmetric subspace. This strategy works until BS become unstable in the symmetric subspace at $\unicode[STIX]{x1D714}_{0}\approx 7000$ , where it loses stability at a secondary Hopf bifurcation. The unstable BS is then computed beyond this bifurcation point by using the selective-frequency-damping method (Äkervik et al. Reference Äkervik, Brandt, Henningson, Hœpffner, Marxen and Schlatter2006), which suppresses the oscillations of the bifurcated solution and drives the simulation toward the unstable BS.

Figure 7.

For unstable LCs (computed in the ${\mathcal{I}}$ -subspace), at $\unicode[STIX]{x1D714}_{0}=7000$ and $Po=-0.3745$ , shown are isosurfaces of (a) $w=2$ and (b) ${\mathcal{H}}=\pm 5\times 10^{5}$ , and of the (unstable) BS at the same parameters (computed in the ${\mathcal{I}}$ -symmetric subspace; the bifurcation of LCs from BS in the subspace is slightly subcritical, such that both BS and LCs are stable at $\unicode[STIX]{x1D714}_{0}=7000$ in the subspace), with (c) $w=2$ and (d) ${\mathcal{H}}=\pm 5\times 10^{5}$ . The supplementary movie shows an animation of ${\mathcal{H}}$ for LCs over one period $\unicode[STIX]{x1D70F}_{\mathit{LCs}}=9.50\times 10^{-5}$ . Isosurfaces of the (e,f) $m=8$ and (g,h) $m=9$ azimuthal Fourier components of $w$ at levels $w=\pm 30$ are shown in two orthogonal views.

The structure of the solution that bifurcates at the second Hopf bifurcation of BS is a symmetric limit cycle LCs (an $F$ -cyle), shown in figure 7(a,b) for $\unicode[STIX]{x1D714}_{0}=7000$ (close to the bifurcation point). Also plotted in figure 7(c,d) is the structure of the unstable BS at the same point in parameter space, computed using the selective-frequency-damping method (Äkervik et al. Reference Äkervik, Brandt, Henningson, Hœpffner, Marxen and Schlatter2006). The $w$ isosurface of LCs resembles that of BS, but it is distorted by small structures approximately one third the cylinder height in size. The corresponding helicity isosurfaces for LCs show the presence of helical structures in the bulk, but instead of columns these are also structures whose size is also approximately one third of the cylinder height. These structures are absent in BS. The helical structures, and their temporal evolution, can be appreciated in the supplementary online movie: they rotate prograde and undergo significant distortions during the period of the solution, with frequency $\unicode[STIX]{x1D714}_{s}=66\,139$ . Figure 6 shows the relative modal kinetic energies of LCs (filled green diamonds). Apart from $e_{0}$ and $e_{1}$ which are the modal energies of the main components of BS, $e_{8}$ and $e_{9}$ are larger than all the other modal energies by more than an order of magnitude. The figure also shows that LCs is numerically well resolved. The presence of dominant $e_{1}$ , $e_{8}$ and $e_{9}$ is again suggestive of a triadic resonance. In order to explore this possibility, we have plotted isosurfaces of the axial velocity for the $m=8$ and $m=9$ azimuthal Fourier components of LCs in figure 7(e–h). Their spatial structures in the bulk are the same as for Kelvin eigenmodes $K_{1,\pm 8,2}$ and $K_{1,\pm 9,3}$ ; the sign of the azimuthal number $m$ remains to be determined. The only pair of Kelvin modes in the first part of table 2 with this spatial structure indicates that the system is close to a triadic resonance between the forced $(1,1,1)$ BS, $K_{1-82}$ and $K_{193}$ . The signs of $\unicode[STIX]{x1D714}$ in the last two columns of the table and of $m$ , indicate that these two Kelvin modes are prograde in the table frame, and that they are both ${\mathcal{I}}$ symmetric ( $m+n$ is even), as is the computed LCs. The only discrepancy is in the temporal frequencies in the table frame; for the Kelvin modes the frequency would be approximately 8.4, while the computed frequency of LCs is $\unicode[STIX]{x1D714}_{\mathit{LCs}}/\unicode[STIX]{x1D714}_{0}=9.45$ . This discrepancy of approximately 10 % is of the order of magnitude of the detuning, approximately 4.5 %. Of course, $K_{1-82}$ and $K_{193}$ are not the same as the $m=8$ and $m=9$ azimuthal Fourier components of LCs; boundary layers, viscosity and nonlinearities account for the differences between them. It is worth noting that, as in the 1 :  $-4$  : 5 resonance of LCa, the instability mechanism at the Hopf bifurcation of LCs is not determined by the closest-to-resonance pair of free Kelvin modes.

5.2 The $Po=-0.4$ case (retracing rays)

The change in $Po$ from $Po=-0.3745$ (non-retracing rays) to $Po=-0.4$ (retracing rays), results in a dramatic change in the states and instabilities encountered as $\unicode[STIX]{x1D714}_{0}$ is increased, as illustrated in figure 3. There are three main differences between the two $Po$ scenarios. The first is that in the retracing rays scenario, BS is stable to much larger $\unicode[STIX]{x1D714}_{0}$ values: BS becomes unstable at $\unicode[STIX]{x1D714}_{0}\approx 7240$ , approximately 80 % larger than the critical value at $Po=-0.3745$ , $\unicode[STIX]{x1D714}_{0}\approx 3992$ . The second difference is that the stable bifurcated state is LCs, an ${\mathcal{I}}$ -symmetric limit cycle, instead of being LCa. Moreover, in the non-retracing rays scenario, LCs was never stable, and it had to be computed by restricting the simulations to the ${\mathcal{I}}$ -symmetric subspace. Finally, the third main difference is that the asymmetric branch with LCa, and more complex states, QPa and VLFa (to be described later), still exists but is not connected (via stable states) to BS, and it has shifted to much larger values of $\unicode[STIX]{x1D714}_{0}$ . For $Po=-0.4$ , the LCa branch starts at $\unicode[STIX]{x1D714}_{0}\approx 6800$ , while for $Po=-0.3745$ it starts at $\unicode[STIX]{x1D714}_{0}\approx 3992$ .

Figure 8.

For LCs at $\unicode[STIX]{x1D714}_{0}=8000$ and $Po=-0.4$ , shown are isosurfaces of (a) $w=2$ and (b) ${\mathcal{H}}=\pm 5\times 10^{5}$ , and of (unstable) BS at the same parameters (computed using selective-frequency damping) with (c) $w=2$ and (d) ${\mathcal{H}}=\pm 5\times 10^{5}$ . The supplementary movie shows an animation of ${\mathcal{H}}$ for LCs over one period $\unicode[STIX]{x1D70F}_{\mathit{LCs}}=9.36\times 10^{-5}$ . Isosurfaces of the (e,f) $m=8$ and (g,h) $m=9$ azimuthal Fourier components of $w$ at levels $w=\pm 30$ are shown in two orthogonal views.

Table 3.

Kelvin mode pairs, $K_{k_{1}m_{1}n_{1}}$ and $K_{k_{2}m_{2}n_{2}}$ , closest to resonance with the forced $(1,1,1)$ mode for $\unicode[STIX]{x1D6E4}=1.33$ , $\unicode[STIX]{x1D6FC}=3^{\circ }$ and $Po=-0.4$ , for $k\leqslant 2$ , $n\leqslant 4$ and $m\leqslant 10$ , and detuning $\unicode[STIX]{x1D6FF}<0.06$ .

The structure of LCs at $Po=-0.4$ is shown in figure 8 for $\unicode[STIX]{x1D714}_{0}=8000$ . Figure 8(a–d) shows isosurfaces of axial velocity $w$ and helicity ${\mathcal{H}}$ of LCs, and the corresponding isosurfaces for BS at the same parameter values (computed using selective-frequency damping as BS is unstable even in the ${\mathcal{I}}$ -symmetric subspace). Comparing with LCs at $Po=-0.3745$ and $\unicode[STIX]{x1D714}_{0}=7000$ in figure 7, we observe that the inertial beams emerging from the corner in the basic state BS are much more clearly defined than in the $Po=-0.3745$ case. The structure of the $m=8$ and $m=9$ azimuthal Fourier modes is very similar to the structure of the same Fourier modes in the $Po=-0.3745$ branch. In order to see if the triadic resonance 1:-8:9 is the instability mechanism for the Hopf bifurcation where the stable LCs is spawned, we have computed the pairs of Kelvin modes closest to resonance with the forced BS, with detuning parameter $\unicode[STIX]{x1D6FF}_{3}<0.06$ , and with $k=1,2$ , $n\leqslant 4$ and $|m|\leqslant 10$ for $Po=-0.4$ (the results are presented in table 3). By comparing with table 2, and focusing on the $k=1$ cases in the first section of each table, we find that the pair of Kelvin modes closest to resonance with the forced BS is now $K_{1-82}$ and $K_{193}$ ( $\unicode[STIX]{x1D6FF}_{3}\approx 0.003$ ), while the pair $K_{1-41}$ and $K_{152}$ associated with the triadic resonance leading to LCa in the $Po=-0.3745$ case, has moved now to third position, with a detuning that is 18 times larger ( $\unicode[STIX]{x1D6FF}_{3}\approx 0.054$ ). This example shows how a small variation in the Poincaré number (6 % from $Po=-0.3745$ to $-0.4$ ) results in dramatic changes in the detuning parameter of close-to-resonance pairs of Kelvin modes. In this case, the changes in detuning result in different triadic resonances at the first Hopf bifurcation of BS: 1 :  $-4$  : 5, which breaks the ${\mathcal{I}}$ symmetry, for $Po=-0.3745$ , and 1 :  $-8$  : 9, which preserves the ${\mathcal{I}}$ symmetry, for $Po=-0.4$ . The second section of table 3 shows the additional close-to-resonance pairs with $\unicode[STIX]{x1D6FF}_{3}<0.6$ when we relax the $k$ values to $k\leqslant 2$ . Again, 15 additional Kelvin eigenmode pairs appear with some of them having a smaller detuning parameter value than the 1 :  $-8$  : 9 case.

Figure 9.

For LCa at $\unicode[STIX]{x1D714}_{0}=6800$ and $Po=-0.4$ , shown are isosurfaces of (a) $w=2$ and (b) ${\mathcal{H}}=\pm 8\times 10^{5}$ , and of the (stable) BS at the same parameters with (c) $w=2$ and (d) ${\mathcal{H}}=\pm 4\times 10^{5}$ . The supplementary movie shows an animation of ${\mathcal{H}}$ for LCa over one period $\unicode[STIX]{x1D70F}_{\mathit{LCs}}=2.15\times 10^{-4}$ . Isosurfaces of the (e,f) $m=4$ and (g,h) $m=5$ azimuthal Fourier components of $w$ at levels $w=\pm 90$ are shown in two orthogonal views.

As shown in figure 3, the asymmetric branch with LCa, QPa and VLFa also exists for $Po=-0.4$ . The structure of LCa is shown in figure 9 for $\unicode[STIX]{x1D714}_{0}=6800$ . Comparing with LCa at $Po=-0.3745$ and $\unicode[STIX]{x1D714}_{0}=4500$ in figure 4, we observe that the inertial beams emerging from the corner in BS are much more clearly defined than in the $Po=-0.3745$ case. The spatial structure of the $m=4$ and $m=5$ azimuthal Fourier modes is the same in the two cases $Po=-0.3745$ and $Po=-0.4$ , with quantitative differences due to the large difference in $\unicode[STIX]{x1D714}_{0}$ for the two examples. The first section of table 3 shows that the resonance 1 :  $-4$  : 5 also has a small detuning in the $Po=-0.4$ , and therefore it is very likely to be the triadic resonance that explains the features of LCa on the disconnected asymmetric branch for $Po=-0.4$ .

6.1 Asymmetric branch at $Po=-0.3745$

Figure 10.

Variation of (a) $e_{0}$ and (b) $S$ with $\unicode[STIX]{x1D714}_{0}$ , for the states as indicated (see table 1 for details of these states), for $Po=-0.3745$ .

Figure 10 shows the variety of states, characterized by the relative azimuthally averaged kinetic energy $e_{0}$ and symmetry parameter $S$ , for $Po=-0.3745$ as $\unicode[STIX]{x1D714}_{0}$ is increased up to 20 000. The succession of states and their instabilities share many of the characteristics observed in previous simulations of the transition from the base state (BS) to sustained chaos (SC) (Marques & Lopez Reference Marques and Lopez2015; Lopez & Marques Reference Lopez and Marques2016b ). As detailed in § 5.1, BS first loses stability via a supercritical Hopf bifurcation that breaks the ${\mathcal{I}}$ symmetry, resulting in a (spatio-temporal) ${\mathcal{I}}^{st}$ -symmetric limit cycle LCa, whose spatio-temporal characteristics are consistent with a triadic resonance between BS and two free Kelvin modes, $K_{1-41}$ and $K_{152}$ . Further increasing $\unicode[STIX]{x1D714}_{0}$ , LCa becomes unstable at a Neimark–Sacker bifurcation that is not associated with any new triadic resonances, but corresponds to radial oscillations of the axisymmetric azimuthal mean flow induced by nonlinearities at the higher $\unicode[STIX]{x1D714}_{0}$ (Marques & Lopez Reference Marques and Lopez2015). This $m=0$ streaming flow is not a geostrophic component of the flow; the presence of wave beams generated by the boundary layers introduces an explicit axial variation in the streaming flow, irrespective of whether the beams are retracing or not. The quasiperiodic state spawned at the Neimark–Sacker bifurcation, QPa, loses stability at higher $\unicode[STIX]{x1D714}_{0}$ to a very-low-frequency three-torus state, VLFa. This bifurcation introduces an additional very low frequency. This third frequency is associated with the exchange of energy between the resonant $m=4$ and $m=5$ azimuthal Fourier components and the $m=0$ streaming flow component, and corresponds to recurrent excursions between some of the states obtained in the first bifurcations (BS, LCa and QPa), which are now all unstable. These unstable states only have a small number of unstable directions, and the VLFa flows are trajectories in phase space that are close to heteroclinic orbits between some of these saddle states. In the range $\unicode[STIX]{x1D714}_{0}\in [8400,8800]$ , a sequence of period doubling bifurcations (PDC) of VLFa states is then followed by a short interval in $\unicode[STIX]{x1D714}_{0}$ with intermittent chaotic (IC) states, and then the flow returns to VLFa solutions for $\unicode[STIX]{x1D714}_{0}\sim 10^{4}$ .

Figure 11.

For the sustained chaos (SC) state $\unicode[STIX]{x1D714}_{0}=2\times 10^{4}$ and $Po=-0.3745$ , (a) shows the time series of $S$ over more than half a viscous time; (b,c) show details over $t\in [0.500,0.517]$ of axial velocity at $(r,\unicode[STIX]{x1D703},z)=(0.5,0,-\unicode[STIX]{x1D6E4}/4)$ , $S$ , and the leading relative modal energies $e_{m}$ . The blue vertical lines in (b) indicate the times corresponding to the snapshots of ${\mathcal{H}}$ shown in figure 12.

Figure 12.

Snapshots of the helicity at isolevel ${\mathcal{H}}=\pm 9\times 10^{6}$ , at times as indicated, of SC at $\unicode[STIX]{x1D714}_{0}=2\times 10^{4}$ and $Po=-0.3745$ . See the supplementary movie for an animation of 101 snapshots at times uniformly spaced in the interval $t\in [0.500,0.517]$ .

For $\unicode[STIX]{x1D714}_{0}>10\,400$ , the flow transitions from VLFa to sustained chaos (SC). These states are chaotic trajectories with slow drifts in phase space between the different unstable saddle states, in the same way as in VLFa, but now at the higher $\unicode[STIX]{x1D714}_{0}$ , the transverse intersections between the stable and unstable manifolds of the saddle states are much more prominent, leading to heteroclinic chaos (Guckenheimer & Holmes Reference Guckenheimer and Holmes1997; Guckenheimer et al. Reference Guckenheimer, Krauskopf, Osinga and Sandstede2015). This type of dynamics is consistent with the experimental observation by McEwan (Reference McEwan1970) and Manasseh (Reference Manasseh1992) of resonant collapse. Figure 11(a) shows the time series over more than half a viscous time of the symmetry parameter $S$ , which is chaotic but with well-defined spikes, for a solution at $\unicode[STIX]{x1D714}_{0}=2\times 10^{4}$ . One of the spikes is described in detail in figure 11(b), which shows time series of $S$ (again) together with the axial velocity at a point $w(r=0.5,\unicode[STIX]{x1D703}=0,z=-\unicode[STIX]{x1D6E4}/4,t)$ for $t\in [0.500,0.517]$ , corresponding to the last spike shown in red in figure 11(a). Figure 11(c) shows time series of the dominant modal energies involved around the spike. The now unstable LCa, which was spawned at the 1 :  $-4$  : 5 triadic resonance-induced Hopf bifurcation, is visited for a short time and then the flow collapses to a chaotic state. This process is illustrated in figure 12, showing 15 snapshots of helicity isosurfaces at various times indicated by the blue vertical lines in figure 11(b); the supplementary movie further illustrates this resonant collapse behaviour which consists of 101 snapshots of the helicity ${\mathcal{H}}$ at uniformly spaced times in the interval $t\in [0.500,0.517]$ . Note that there is much going on at very disparate time scales, and the movie is far from being smooth; it would need one or two orders of magnitude more frames and this is prohibitively large to include online. The LCa state with the well-defined helical columns of the 1 :  $-4$  : 5 resonance is clearly visible in figure 12(f). Then, in figure 12(i,j) the flow becomes chaotic, developing small-scale structures throughout, leading to the detuning characterized by the spike in $e_{0}$ (figure 11 c), and then it collapses in figure 12(n) as the large $e_{0}$ drives the system away (detunes) from the support for the 1 :  $-4$  : 5 resonance.

6.2 Symmetric and asymmetric branches at $Po=-0.4$

Figure 13.

Variation of (a) $e_{0}$ and (b) $S$ , for the states as indicated, with $\unicode[STIX]{x1D714}_{0}$ , for $Po=-0.4$ .

Figure 13 shows what happens at $Po=-0.4$ as $\unicode[STIX]{x1D714}_{0}$ is increased. As detailed in § 5.2, LCs is the first state to bifurcate from BS in a supercritical Hopf bifurcation that preserves the ${\mathcal{I}}$ symmetry, and the limit cycle that is spawned, LCs, has spatio-temporal structure consistent with a triadic resonance between BS and two free Kelvin modes $K_{1-82}$ and $K_{193}$ . At $\unicode[STIX]{x1D714}_{0}\approx 8460$ , LCs becomes unstable, and the flow jumps to a remote solution VLFa on the asymmetric branch. The most likely scenario from dynamical systems theory is that LCs undergoes a subcritical Neimark–Sacker bifurcation that breaks ${\mathcal{I}}$ symmetry, and as the bifurcated solutions are unstable, the flow evolves to another stable state that is remote in phase space. The unstable LCs branch, that continues to exist past the Neimark–Sacker bifurcation, can be continued to higher $\unicode[STIX]{x1D714}_{0}$ in the ${\mathcal{I}}$ -invariant subspace, as shown in figure 13. The ${\mathcal{I}}$ -symmetric LCs exhibiting the 1 :  $-8$  : 9 resonance exists and is stable in the range $\unicode[STIX]{x1D714}_{0}\in [7240,8460]$ . For $\unicode[STIX]{x1D714}_{0}>8460$ we have not found any stable ${\mathcal{I}}$ -symmetric solution, and the only stable branch of solutions is the asymmetric branch, associated with the triadic resonance 1 :  $-4$  : 5.

Figure 14.

Temporal evolution of axial velocity $w(0.5,0,-\unicode[STIX]{x1D6E4}/2,t)$ (blue) and $S$ (red), for a simulation at $\unicode[STIX]{x1D714}_{0}=8650$ starting with the LCs state at $\unicode[STIX]{x1D714}_{0}=8450$ as the initial condition.

Figure 14 shows the time series of the axial velocity at a point, $w(0.5,0,-\unicode[STIX]{x1D6E4}/2,t)$ (shown in black) along with the symmetry parameter $S$ (shown in red), for a simulation at $\unicode[STIX]{x1D714}_{0}=8650$ starting with the LCs state at $\unicode[STIX]{x1D714}_{0}=8450$ as the initial condition. The flow appears to have settled early on (within approximately 0.14 viscous times) to an LCs state that seems stable for about the first viscous time. One viscous time corresponds to $\unicode[STIX]{x1D714}_{0}/2\unicode[STIX]{x03C0}$ cylinder rotations; for $\unicode[STIX]{x1D714}_{0}=8650$ this is approximately 1377 cylinder rotations. The 0.14 viscous time corresponds to $0.14\unicode[STIX]{x1D714}_{0}/2\unicode[STIX]{x03C0}\approx 190$ cylinder rotations, and this is approximately the Ekman spin-up time $1/\sqrt{Ek}=\sqrt{4\unicode[STIX]{x1D714}_{0}}$ . However, by monitoring its symmetry, we see that $S$ is growing exponentially during this time. By $t\approx 1$ , $S\approx 10^{-2}$ , and the flow is seen to transition to another state which is quasi-periodic of QPa type with $S\approx 3$ . This state itself is unstable, and at $t\approx 2.3$ , there is another transition to a very-low-frequency VLFa state, which is also unstable, and at $t\approx 2.7$ there is another transition to another VLFa which appears to be stable; we have tracked it for over 2 viscous times (about 3000 cylinder rotations) without any further transitions. This type of evolution in a hysteretic regime is quite common, with the transitory evolution being attracted to unstable saddle states along their stable manifolds only to be diverted away along their unstable manifolds. Very similar behaviour is found in the hysteresis regime of the cubic lid-driven cavity flow (Lopez et al. Reference Lopez, Welfert, Wu and Yalim2017). The stable VLFa solution that was obtained can be continued to lower $\unicode[STIX]{x1D714}_{0}$ , and undergoes a very similar sequence of bifurcations as in the asymmetric branch for $Po=-0.3745$ . At about $\unicode[STIX]{x1D714}_{0}\approx 6800$ , the LCa limit cycle becomes unstable in a cyclic-fold bifurcation (a saddle-node bifurcation of limit cycles), and the flow jumps back to BS for lower $\unicode[STIX]{x1D714}_{0}$ . This is a large hysteretic loop between the symmetric and asymmetric solution branches, covering $\unicode[STIX]{x1D714}_{0}\in [6800,8460]$ . Similar hysteretic behaviour in a precessing cylinder experiment has been reported by Herault et al. (Reference Herault, Gundrum, Giesecke and Stefani2015) who fixed $\unicode[STIX]{x1D714}_{0}\sim 5\times 10^{5}$ and varied $Po\in [0,0.15]$ in a cylinder of aspect ratio $\unicode[STIX]{x1D6E4}=2$ at tilt angle $\unicode[STIX]{x1D6FC}=90^{\circ }$ .

Figure 15.

Power spectral density (PSD) of $w$ at the point $(r,\unicode[STIX]{x1D703},z)=(0.5,0,-\unicode[STIX]{x1D6E4}/4)$ with $Po=-0.4$ for the states (a) LCa at $\unicode[STIX]{x1D714}_{0}=6800$ , (b) QPa at $\unicode[STIX]{x1D714}_{0}=7250$ , (c) VLFa at $\unicode[STIX]{x1D714}_{0}=8650$ , and (d) SC at $\unicode[STIX]{x1D714}_{0}=20\,000$ .

Figure 16.

Phase portraits and Poincaré sections of QPa.

For the various types of solutions obtained with $Po=-0.4$ , figure 15 shows the power spectral density (PSD) of the axial velocity $w$ at the point $(r,\unicode[STIX]{x1D703},z)=(0.5,0,-\unicode[STIX]{x1D6E4}/4)$ . The PSD are plotted in logarithmic scale because the relevant frequencies cover more than three orders of magnitude. Figure 15(a) shows the PSD of the ${\mathcal{I}}^{st}$ -symmetric limit cycle LCa, with a fundamental frequency $f_{1}=\unicode[STIX]{x1D714}_{\mathit{LCa}}/\unicode[STIX]{x1D714}_{0}=4.2977$ plus higher harmonics. The PSD of the quasi-periodic QPa is shown in figure 15(b). There are three large-amplitude peaks at frequencies $f_{1}=4.3144$ , $f_{2}=2.3060$ and $f_{3}=2.0084$ and another one at frequency $f_{4}=0.2976$ , which is approximately an order of magnitude smaller than $f_{1}=\unicode[STIX]{x1D714}_{\mathit{LCa}}/\unicode[STIX]{x1D714}_{0}$ . These frequencies are linearly related, $f_{3}=f_{1}-f_{2}$ and $f_{4}=2f_{2}-f_{1}$ . The frequencies of all the peaks in figure 15(b) are linear combinations of the two frequencies with largest amplitudes, $f_{1}$ and $f_{2}$ , and so QPa is a two-torus. Figure 16(a,b) shows phase portraits of LCa and QPa, where the two-torus structure of QPa is clearly seen. The variables used to illustrate these two views of the phase portraits are the axial velocities at three different points: $w_{1}=w(0.5,0,-\unicode[STIX]{x1D6E4}/4,t)$ , $w_{2}=w(0.5,\unicode[STIX]{x03C0},\unicode[STIX]{x1D6E4}/4,t)$ , and $w_{3}=w(0.5,\unicode[STIX]{x03C0},-\unicode[STIX]{x1D6E4}/4,t)$ . The points in the cylinder corresponding to $w_{1}$ and $w_{2}$ are ${\mathcal{I}}$ -related. Figure 16(a), the phase portrait using the ${\mathcal{I}}$ -related $w_{1}$ and $w_{2}$ shows that LCa and QPa are setwise symmetric; in the variables used, ${\mathcal{I}}$ -symmetry means reflection symmetry with respect to the line $w_{1}+w_{2}=0$ (the grey line in figure 16 a). If LCa and QPa were pointwise ${\mathcal{I}}$ -symmetric, they would be contained onto the line $w_{1}+w_{2}=0$ . In figure 16(a), the phase portrait of LCa has been translated along the symmetry line $w_{1}+w_{2}=0$ for clarity. The phase portrait of the limit cycle LCa is in the interior of the phase portrait of the two-torus QPa, as shown in figure 16(b), using $w_{2}$ and $w_{3}$ . Figure 16(b) also shows a Poincaré section of both LCa and QPa. The Poincaré section of QPa is shown in further detail in figure 16(c), where the successive iterates of the quasiperiodic orbit (corresponding to a time evolution in temporal increments of $2\unicode[STIX]{x03C0}/f_{1}$ ) are numbered sequentially. There are two significant features to be observed. First, that each iterate jumps almost to the opposite point of the Poincaré section. This is a remnant of the half-period-flip ${\mathcal{I}}^{st}$ invariance of LCa which is broken in QPa, and it is this broken half-period flip that is responsible for the two large-amplitude peaks on the PSD of QPa (figure 15 b) that are close to half the frequency $f_{1}$ : $f_{2}/f_{1}=0.5345$ and $f_{2}/f_{1}=0.4655$ . Their closeness to 0.5 is not associated with a period-doubling bifurcation, but rather to a symmetry-breaking bifurcation: the breaking of the spatio-temporal symmetry ${\mathcal{I}}^{st}$ of LCa. The other salient feature of the Poincaré section in figure 16(c) is that after 58 iterations the orbit almost closes in on itself. In fact, $f_{1}/f_{4}=14.497\approx 29/2$ is almost rational, so the quasiperiodic two-torus QPa is almost purely periodic and close to the resonance $29/2$ . The small peak close to zero frequency in figure 15(b) is the beat frequency due to the difference between $f_{1}/f_{4}$ and $29/2$ .

The frequencies $f_{1}$ to $f_{4}$ persist into VLFa, shown in figure 15(c), but they have a broad band about them. This is due to linear combinations with the very low frequency introduced by VLFa, which is an order of magnitude smaller than $f_{4}$ of QPa. Figure 15(d) shows the PSD of the sustained chaos solution SC at $\unicode[STIX]{x1D714}_{0}=2\times 10^{4}$ ; the spectrum is broad band, but among the broad band there is a peak at $f_{1}\approx 4.3$ evident.

The spatio-temporal structure of the solutions in the asymmetric branch is essentially the same for the two $Po$ cases considered, $Po=-0.3745$ and $Po=-0.4$ . Moreover, for $\unicode[STIX]{x1D714}_{0}>8460$ it is the only stable branch of solutions in both cases. We can conclude that the approximate resonance of Kelvin modes $K_{1-41}$ and $K_{152}$ with the forced $(1,1,1)$ BS plays an important role in the dynamics in both cases, being visited from time to time even in the regime of sustained chaos.