3.1 Basic state

The basic state (BS) is steady and axisymmetric. It is stable for sufficiently small $\mathit{Re}$ and $\mathit{Ro}$ , and consists of a bulk flow in solid-body rotation with $v/r\approx \mathit{Re}$ and boundary layers of Ekman type on the top and bottom end walls and a Stewartson-like boundary layer on the side wall. A typical BS at $\mathit{Re}=1.2\times 10^{4}$ and $\mathit{Ro}=0.23$ is presented in figure 1, which shows contours of the azimuthal vorticity, ${\it\eta}$ . The contour levels are cubically spaced so that more levels are concentrated about the zero level. For pure solid-body rotation there is no meridional flow and ${\it\eta}=0$ . Figure 1 shows that this is essentially the case for BS for $r\lesssim 0.7$ and away from the top and bottom Ekman layers. The details of how BS changes with parameters, in particular with $\mathit{Ro}$ , are provided in Gutierrez-Castillo & Lopez (Reference Gutierrez-Castillo and Lopez2015). That study only considered axisymmetric flow, and the only instabilities of BS considered were also axisymmetric. However, as detailed in the following sections, over the wide range of parameters considered, the primary instabilities of BS are not axisymmetric.

Figure 3.

Variations of the modal kinetic energies $E_{m}$ of the various rotating waves $\text{RW}_{m}$ ( $m\in [1,4]$ ) versus $\mathit{Ro}$ for $\mathit{Re}=10^{4}$ . The filled symbols correspond to stable states and open symbols to unstable states (computed in the corresponding subspaces). (b) Is a zoomed-in version of (a).

3.2 Overview of the instabilities

Figure 2 shows a regime diagram in $(\mathit{Ro},\mathit{Re})$ parameter space, summarizing the various types of states obtained following the instabilities of BS. The BS described in the previous section (designated as small filled circles in the figure) is stable for low $\mathit{Ro}$ and $\mathit{Re}$ . As $\mathit{Re}$ and $\mathit{Ro}$ are increased, the BS loses stability in a number of different bifurcations. Generally, the bifurcations are supercritical for $\mathit{Ro}\gtrsim 0.245$ and subcritical for $\mathit{Ro}\lesssim 0.245$ . Flow curvature and shear in the side wall boundary layer, and the flow negotiating the corner in the slower rotating half of the cylinder, where the bottom end wall and the side wall meet, are the primary ingredients for the instabilities. For small $\mathit{Re}$ and large $\mathit{Ro}$ , one class of instabilities leads to low azimuthal wavenumber rotating waves ( $\text{RW}_{L}$ ) concentrated in the bottom corner, and emits inertial wave beams into the interior. These are designated by open squares in figure 2. For larger $\mathit{Re}$ and smaller $\mathit{Ro}$ , high azimuthal wavenumber rotating waves ( $\text{RW}_{H}$ ) are found (designated by open diamonds). These are also concentrated near the bottom corner, but more deeply in the side wall layer. There is an extensive $(\mathit{Ro},\mathit{Re})$ regime in between, where quasiperiodic states (QP) are found that have well-distinguished features of both $\text{RW}_{L}$ and $\text{RW}_{H}$ ; these are designated by large open circles. All the states exist with a range of different azimuthal wavenumbers corresponding to Eckhaus bands. In the following sections, we shall consider a number of one parameter paths in the regime diagram, describing in some detail the various instabilities and flow characteristics.

3.3 Low azimuthal wavenumber rotating waves

With fixed $\mathit{Re}=10^{4}$ , the basic state is stable for $\mathit{Ro}\lesssim 0.265$ . For $\mathit{Ro}$ above that critical value, the BS undergoes a supercritical Hopf bifurcation that breaks axisymmetry, resulting in a rotating wave state with azimuthal wavenumber $m=3$ , $\text{RW}_{3}$ . For larger $\mathit{Ro}$ , a complicated bifurcation process ensues that involves a number of $\text{RW}_{L}$ . A summary of the $\text{RW}_{L}$ solution branches showing their primary modal kinetic energy, $E_{m}$ for $\text{RW}_{m}$ , as functions of $\mathit{Ro}$ are presented in figure 3, where (a) presents the overall picture and (b) is a zoomed-in view near the onset of instability of the BS. Very near the first bifurcation from the BS, the modal kinetic energy $E_{3}$ of the rotating wave $\text{RW}_{3}$ grows linearly with increasing $\mathit{Ro}$ , and then slower than linearly with larger $\mathit{Ro}$ , until $\mathit{Ro}\approx 0.298$ where $\text{RW}_{3}$ loses stability. The unstable $\text{RW}_{3}$ has also been continued to larger $\mathit{Ro}$ by restricting the computations to the $m=3$ symmetry subspace. Starting with the stable $\text{RW}_{3}$ as the initial condition for a slightly larger $\mathit{Ro}$ results in an evolution to a rotating wave with $m=2$ , $\text{RW}_{2}$ . The $\text{RW}_{2}$ solution branch was also continued to higher $\mathit{Ro}$ ; it loses stability for $\mathit{Ro}\gtrsim 0.322$ , and there is a transition to another rotating wave branch with $m=1$ , $\text{RW}_{1}$ . This branch loses stability for $\mathit{Ro}\lesssim 0.275$ , but remains stable for higher $\mathit{Ro}$ , at least up to the highest value considered in this study, $\mathit{Ro}=0.40$ . On the other hand, the $\text{RW}_{2}$ solution branch was also continued to smaller $\mathit{Ro}$ ; it loses stability for $\mathit{Ro}\lesssim 0.275$ and switches to an $m=4$ branch, $\text{RW}_{4}$ . The $\text{RW}_{4}$ loses stability for $\mathit{Ro}\lesssim 0.267$ , and $\mathit{Ro}\gtrsim 0.292$ switching to the $\text{RW}_{3}$ branch.

The bifurcation scenario just presented is typical of an Eckhaus band (Tuckerman & Barkley Reference Tuckerman and Barkley1990). Here, the marginal stability curve is a discrete set of points due to the integer wavenumber $m$ resulting from breaking the azimuthal invariance, $SO(2)$ symmetry, of the BS, which gives the critical $\mathit{Ro}$ for the instability of the BS to a rotating wave with azimuthal wavenumber $m$ . Only the rotating wave $\text{RW}_{m}$ with the lowest critical $\mathit{Ro}$ will be stable at its onset, in this case $\text{RW}_{3}$ , and the rotating waves corresponding to other $m$ will be unstable at their onset. However, these rotating waves become stable at secondary bifurcations and the loci of points in $(m,\mathit{Ro})$ where these occur form the Eckhaus stability boundary. In principle, at a given $\mathit{Ro}$ above critical, all rotating waves with $m$ inside the Eckhaus band are stable, but can become unstable as $\mathit{Ro}$ is increased, and spawn unstable modulated rotating waves. Similar dynamics has been studied in rotating convection problems, where the Eckhaus instability for systems with $SO(2)$ symmetry is further detailed (Lopez et al. Reference Lopez, Marques, Mercader and Batiste2007; Marques & Lopez Reference Marques and Lopez2008).

Figure 4 shows contours of the azimuthal vorticity, ${\it\eta}$ , as well as its non-axisymmetric component, ${\it\eta}-{\it\eta}_{0}$ , where ${\it\eta}_{0}$ is the $m=0$ Fourier component of ${\it\eta}$ in the meridional plane ${\it\theta}=0$ and the plane $z=-0.4$ which is close to the bottom end wall. The four rotating wave states shown in the figure are for $\mathit{Re}=10^{4}$ and $\mathit{Ro}=0.29$ , and are inside the Eckhaus band where all four are stable. The boundary layer structure of the rotating wave states is very similar to that of the basic state (see figure 1), the main difference being a slight bulge in the lower corner region. This bulge is more readily appreciated when the axisymmetric component is removed; it is seen to be localized in the corner region and has azimuthal wavenumber $m$ , as is evident from the plots in the $z=-0.4$ plane. This bulge structure rotates in the azimuthal direction at a constant rate without change of shape. In the corner region at any fixed point in a reference frame rotating with the cylinder mean rotation rate, the passage of the bulge provides a localized disturbance that emits a wave beam into the interior along a cone. This cone forms an angle ${\it\beta}$ with a plane orthogonal to the cylinder axis given by the dispersion relation $\cos {\it\beta}={\it\omega}_{R}/2$ . The dispersion relation is obtained in the inviscid flow limit of infinitesimal perturbations to solid-body rotation (Greenspan Reference Greenspan1968), where ${\it\omega}_{R}$ is the frequency of the perturbation in the frame of reference rotating with the cylinder mean rotation rate ${\it\Omega}$ , and non-dimensionalized with ${\it\Omega}$ . When the localized disturbance is axisymmetric, the resulting wave beams are axisymmetric cones, but when the localized disturbance is not axisymmetric, the wave beams are spirals on the cone. The contours shown in the $z=-0.4$ plane in the third column of figure 4 illustrate the spiral nature of the wave beams, and the plots in the meridional plane in the second column of the figure show the cone structure.

Figure 4.

Contours of (a) ${\it\eta}$ , (b) ${\it\eta}-{\it\eta}_{0}$ in the meridional plane ${\it\theta}=0$ and (c) ${\it\eta}-{\it\eta}_{0}$ in the plane $z=-0.4$ , for the rotating waves $\text{RW}_{m}$ , $m\in [1,4]$ , at $\mathit{Re}=10^{4}$ , $\mathit{Ro}=0.29$ . There are ten cubically spaced contour levels in the ranges ${\it\eta}\in [-5,5]$ and ${\it\eta}-{\it\eta}_{0}\in [-1,1]$ , with the positive being red (dark grey) and the negative being yellow (light grey).

Figure 5.

Isosurfaces of $w-w_{0}$ , at levels $\pm 0.001$ , of (a)  $\text{RW}_{1}$ , (b)  $\text{RW}_{2}$ , (c)  $\text{RW}_{3}$ and (d)  $\text{RW}_{4}$ at $\mathit{Re}=10^{4}$ and $\mathit{Ro}=0.29$ .

Table 1.

Frequencies in the stationary and rotating frames of reference and angles of the inertial wave beam cones for different stable RW states and the unstable LC, all at $\mathit{Re}=10^{4}$ and $\mathit{Ro}=0.29$ .

Figure 5 shows three-dimensional isosurfaces of the non-axisymmetric component of the axial velocity, $w-w_{0}$ , for the four rotating wave states, illustrating the bulges localized near the corner (almost horizontal rollers that are more inclined for larger $m$ ). The resulting wave beams are emitted from the corner along the cone, with reflections off either the top end wall or the axis, depending on the cone angle for each case. Since our simulations are conducted in the stationary frame of reference, to predict the cone angle we need to convert the precession frequency of the rotating wave in the stationary frame ${\it\omega}_{S}$ (which is obtained directly from our simulations) to ${\it\omega}_{R}$ , the frequency a local observer sees in the rotating frame of reference. Recall that a structure with azimuthal wavenumber $m$ takes $2{\rm\pi}m/{\it\omega}_{S}$ time units to rotate $2{\rm\pi}$ in the stationary frame of reference. To calculate the frequency in the rotating frame of reference, the background rotation speed (non-dimensionalized to 1) has to be subtracted from the angular speed of the rotating wave ( ${\it\omega}_{S}/m$ ) to obtain the angular speed of the rotating wave in the rotating frame of reference ( ${\it\omega}_{R}/m$ ), leading to ${\it\omega}_{R}={\it\omega}_{S}-m$ . Table 1 lists ${\it\omega}_{S}$ , ${\it\omega}_{R}$ and the corresponding cone angle ${\it\beta}$ for the four rotating waves shown in figures 4 and 5. The linear inviscid dispersion relation predicts the cone angle ${\it\beta}$ well, even though the simulations are viscous, with $\mathit{Re}=10^{4}$ , and highly nonlinear with $\mathit{Ro}=0.29$ . In the stationary frame of reference, the precession rates of the rotating waves vary considerably, with $\text{RW}_{1}$ precessing retrograde with respect to the sense of mean rotation of the cylinder (hence the negative frequency), and the other three are prograde. However, when viewed in the rotating frame, all are retrograde with a much smaller variation in frequencies. For the lowest $m=1$ , the cone angle is such that the wave beam almost perfectly retraces itself as it is emitted from the corner region $(r,z)=(1,-0.5)$ and is reflected back from the axis at the top end wall $(r,z)=(0,0.5)$ . For increasing $m$ , the wave beams do not penetrate all the way to the axis; for $\text{RW}_{4}$ , there is a clear quiescent zone near the axis. This is consistent with the analysis of Wood (Reference Wood1981), who showed that the radial extent of the quiescent axial zone increases with $m$ .

It is of interest to compare the rotating wave states with the axisymmetric limit cycle that bifurcates from the BS, which was first described in Gutierrez-Castillo & Lopez (Reference Gutierrez-Castillo and Lopez2015). We again fix $\mathit{Re}=10^{4}$ and increase $\mathit{Ro}$ , but compute in the axisymmetric subspace, since the BS is first unstable to non-axisymmetric disturbances. In doing so, the BS becomes unstable via a Hopf bifurcation at $\mathit{Ro}\approx 0.28$ , spawning an axisymmetric limit cycle (LC), which like the BS in this parameter regime, is unstable to non-axisymmetric perturbations. Figure 6(a) shows a snapshot of the azimuthal vorticity ${\it\eta}$ of LC at $\mathit{Ro}=0.29$ . Its time average, $\langle {\it\eta}\rangle$ , shown in figure 6(b) is very similar in structure to the BS, as is to be expected near the Hopf bifurcation. What is particularly of interest is the structure of the Hopf mode, which can be approximated by ${\it\eta}-\langle {\it\eta}\rangle$ and compared directly with the Hopf modes associated with the rotating waves, which are approximated by ${\it\eta}-{\it\eta}_{0}$ . Figure 6(c) shows ${\it\eta}-\langle {\it\eta}\rangle$ of the LC and it is clear that it has very similar structure to ${\it\eta}-{\it\eta}_{0}$ of the rotating waves, shown in the second column of figure 4. The corner region where the side wall and lower end wall meet is the centre of localized unsteadiness. For the LC, the pulsing in the corner is axisymmetric and an axisymmetric wave beam is emitted into the bulk. As the LC is axisymmetric, the frequency of oscillation is the same in the stationary frame and in any rotating frame (reported in the last row of table 1).

Figure 6.

Contours of (a) ${\it\eta}$ , (b) $\langle {\it\eta}\rangle$ and (c) ${\it\eta}-\langle {\it\eta}\rangle$ in a vertical plane for the unstable LC solution at $\mathit{Re}=10^{4}$ , $\mathit{Ro}=0.29$ . There are ten contour levels in the range ${\it\eta}\in [-5,5]$ , $\langle {\it\eta}\rangle \in [-5,5]$ and ${\it\eta}-\langle {\it\eta}\rangle \in [-1,1]$ , cubically spaced, with the positive being red (dark grey) and the negative being yellow (light grey).

To make the connection between the LC and rotating waves more succinct, figure 7 shows snapshots at six equally spaced phases in one period of ${\it\eta}-\langle {\it\eta}\rangle$ for the LC and ${\it\eta}-{\it\eta}_{0}$ for $\text{RW}_{1}$ , $\text{RW}_{2}$ , $\text{RW}_{3}$ and $\text{RW}_{4}$ ; the period for each case is different, as indicated by their frequencies, reported in table 1. Looking at a given meridional plane over time, it is difficult to distinguish between the LC and $\text{RW}_{m}$ cases (except perhaps for the $\text{RW}_{m}$ with larger $m$ due to the small quiescent axial zone). This type of duality between limit cycles and rotating waves in axisymmetric systems has been examined previously, such as in Marques, Lopez & Shen (Reference Marques, Lopez and Shen2002), Lopez & Marques (Reference Lopez and Marques2003), Lopez (Reference Lopez2006).

Figure 7.

Contours of ${\it\eta}-\langle {\it\eta}\rangle \in [-1,1]$ for the unstable LC and ${\it\eta}-{\it\eta}_{0}$ for the stable $\text{RW}_{1}$ , $\text{RW}_{2}$ , $\text{RW}_{3}$ and $\text{RW}_{4}$ in a meridional plane ${\it\theta}=0$ at $\mathit{Re}=10^{4}$ and $\mathit{Ro}=0.29$ ; snapshots at six equally spaced phases over their respective periods are shown.

Now, we fix $\mathit{Ro}=0.28$ ( $\text{RW}_{1}$ , $\text{RW}_{2}$ , $\text{RW}_{3}$ and $\text{RW}_{4}$ are all stable for $\mathit{Re}=10^{4}$ at this $\mathit{Ro}$ ), and consider a one-parameter sweep increasing $\mathit{Re}$ from a smaller value where the BS is stable. At $\mathit{Re}\approx 9.2\times 10^{3}$ , the BS loses stability in a supercritical Hopf bifurcation and $\text{RW}_{2}$ emerges. Figure 8(a) shows how the modal energies $E_{0}$ and $E_{2}$ vary with $\mathit{Re}$ . $\text{RW}_{2}$ remains stable until $\mathit{Re}\approx 1.1\times 10^{4}$ , at which point it undergoes a secondary Hopf bifurcation, spawning a quasiperiodic (QP) state, the details of which are discussed in § 3.5. We already know that $\text{RW}_{1}$ , $\text{RW}_{3}$ and $\text{RW}_{4}$ are also stable in the neighbourhood of $\mathit{Re}\sim 10^{4}$ for this $\mathit{Ro}$ , but the fate of these rotating waves has not been pursued any further. The point we wish to make here is that $\text{RW}_{2}$ is the state that is selected in this one-parameter sweep and that there is no obvious reason for its selection – for example, figure 3(b) shows that at $(\mathit{Re},\mathit{Ro})=(10^{4},0.28)$ , $\text{RW}_{2}$ is not the most energetic of the four rotating waves. Repeating the one-parameter sweep, but with $\mathit{Ro}=0.27$ , we find similar behaviour, but with $\text{RW}_{1}$ being spawned from the BS in a supercritical Hopf bifurcation at $\mathit{Re}\approx 1.01\times 10^{4}$ , and then undergoing a secondary Hopf bifurcation to a QP state at $\mathit{Re}\approx 1.08\times 10^{4}$ (see figure 8 b). It should be noted that figure 3(b) shows that at $(\mathit{Re},\mathit{Ro})=(10^{4},0.27)$ , $\text{RW}_{3}$ and $\text{RW}_{4}$ are the only rotating waves that have bifurcated from the BS and are stable, and yet the one-parameter sweep performed selected $\text{RW}_{1}$ at $\mathit{Re}$ slightly above $10^{4}$ . Similar behaviour is found for the one-parameter sweep with $\mathit{Ro}=0.26$ .

Figure 8.

Variation of principal modal kinetic energies for various states with increasing $\mathit{Re}$ for (a) $\mathit{Ro}=0.28$ and (b) $\mathit{Ro}=0.27$ ; $E_{H}$ corresponds to the modal kinetic energy for the high azimuthal wavenumber component of the various QP states, and HNL are QP states that are highly nonlinear (described in § 3.5).

In order to gain insight into the nature of the QP states that result from secondary Hopf bifurcations from RW states as $\mathit{Re}$ is increased, it is convenient to first examine the rotating wave states at lower $\mathit{Ro}$ , which have much higher azimuthal wavenumbers than the rotating waves $\text{RW}_{L}$ examined so far. As a group, the high- $m$ rotating waves are designed as $\text{RW}_{H}$ (individually, $H$ is replaced with the corresponding $m$ ). The QP states will be shown to be mixed modes of $\text{RW}_{L}$ and $\text{RW}_{H}$ .

3.4 High azimuthal wavenumber rotating waves

Figure 2 indicates that $\text{RW}_{H}$ are found at the low end of the $\mathit{Ro}$ range considered. Fixing $\mathit{Ro}=0.20$ and conducting a one-parameter sweep with increasing $\mathit{Re}$ , we find that the BS is stable up to $\mathit{Re}\approx 1.6\times 10^{4}$ . A further small increase in $\mathit{Re}$ results in a jump to a QP state. Continuation from the QP state to lower $\mathit{Re}$ reveals a series of rotating waves with high azimuthal wavenumbers. This is a clear indication that the BS at $\mathit{Ro}=0.20$ loses stability in a subcritical bifurcation. Figure 9 shows the modal kinetic energies of the states encountered in this one-parameter sweep. The $\text{RW}_{H}$ found have $m=36$ , $41$ and $42$ with increasing $\mathit{Re}$ ; one was obtained from the other as initial condition going from higher to lower $\mathit{Re}$ . The transients involved are very slow (of the order of a viscous time or longer), as is typically found when undertaking one-parameter paths within an Eckhaus band of states (Lopez et al. Reference Lopez, Marques, Mercader and Batiste2007). This, together with the cost of following multiple rotating wave branches with a large range of high azimuthal wavenumbers, makes a detailed study prohibitive.

Figure 9.

Variation of principal modal kinetic energies for various states with increasing $\mathit{Re}$ for $\mathit{Ro}=0.20$ ; $E_{H}$ corresponds to the modal kinetic energy for the high azimuthal wavenumber component of the various states.

Figure 10.

Contours of (a) ${\it\eta}$ and (b) ${\it\eta}-{\it\eta}_{0}$ at ${\it\theta}=0$ , (c) ${\it\eta}-{\it\eta}_{0}$ at $z=-0.4$ and (d) isosurfaces of $w-w_{0}$ , at levels $\pm 0.001$ , for $\text{RW}_{41}$ at $\mathit{Re}=1.55\times 10^{4}$ and $\mathit{Ro}=0.20$ . There are ten cubically spaced contour levels in the range ${\it\eta}\in [-5,5]$ and ${\it\eta}-{\it\eta}_{0}\in [-1,1]$ .

A typical example of a high azimuthal wavenumber rotating wave is $\text{RW}_{41}$ at $\mathit{Ro}=0.20$ and $\mathit{Re}=1.55\times 10^{4}$ ; figure 10 shows contours of ${\it\eta}$ in a meridional plane and of ${\it\eta}-{\it\eta}_{0}$ in a meridional plane and in the plane $z=-0.4$ . Since the modal kinetic energy $E_{41}$ is five orders of magnitude smaller than $E_{0}$ , the ${\it\eta}$ contours of $\text{RW}_{41}$ (figure 10 a) are virtually indistinguishable from those of the BS (not shown) at the same point in parameter space, where both are stable. The contours of ${\it\eta}-{\it\eta}_{0}$ however, show that the perturbation of $\text{RW}_{41}$ away from the BS is localized in the bottom half of the side wall boundary layer and concentrated in the lower corner region, much as is the case for the $\text{RW}_{L}$ . The three-dimensional isocontours of $w-w_{0}$ for $\text{RW}_{41}$ (figure 10 d) show a clear distinction from the $w-w_{0}$ isocontours of $\text{RW}_{L}$ (figure 5). Apart from the large difference in azimuthal wavenumber $m$ , for $\text{RW}_{L}$ , the low- $m$ spirals very near the corner have a small negative helix angle, whereas the perturbation structures of $\text{RW}_{41}$ have a large positive helix angle. Another distinction between the two classes of rotating waves is that all the $\text{RW}_{L}$ emitted inertial wave beams into the interior, whereas no wave beams are evident for $\text{RW}_{41}$ . In the stationary frame of reference, the $m=41$ rotating wave structure has a very fast frequency ${\it\omega}_{S}=39.47$ ; this is two orders of magnitude faster than that of the $\text{RW}_{L}$ . However, in the frame of reference rotating at the mean rotation rate of the cylinder, $\text{RW}_{41}$ has ${\it\omega}_{R}=-1.53$ , which is in the middle of the range of ${\it\omega}_{R}$ for the $\text{RW}_{L}$ , suggesting that inertial wave beams at an angle of approximately $40^{\circ }$ should be emitted from the corner region, but these are not evident. This is likely a result of the enlargement of the quiescent axial zone for the much larger azimuthal wavenumber $m=41$ (Wood Reference Wood1981), and perhaps the larger $m$ structures are also subjected to more viscous dissipation as their spatial gradients are larger (Cortet, Lamriben & Moisy Reference Cortet, Lamriben and Moisy2010; Machicoane et al. Reference Machicoane, Cortet, Voisin and Moisy2015). Also, it should be noted that the relative strength of $\text{RW}_{41}$ , $E_{41}/E_{0}$ , is approximately two orders of magnitude smaller than that of the $\text{RW}_{L}$ , and this also contributes to the muting of any associated inertial wave beams. The other $\text{RW}_{H}$ found in the low- $\mathit{Ro}$ regime are similar to $\text{RW}_{41}$ .

3.5 Quasiperiodic mixed modes

The parameter regime where rotating waves $\text{RW}_{L}$ and $\text{RW}_{H}$ are found is quite small, and most of the regime diagram (figure 2) where the BS is unstable is dominated by QP states. Near where rotating waves exist, the QP states are seen to emerge following secondary bifurcations from the rotating waves. The QP states have oscillations associated with two distinct azimuthal wavenumbers that correspond to $\text{RW}_{H}$ and $\text{RW}_{L}$ (and in some cases to $m=0$ from LC). It is tempting to think of this as resulting from a double-Hopf bifurcation where at a codimension-2 point (i.e. a single point in $(\mathit{Re},\mathit{Ro})$ parameter space), a $\text{RW}_{L}$ and a $\text{RW}_{H}$ bifurcate simultaneously from the BS, and in some neighbourhood of the codimension-2 point there is a mixed mode $\text{QP}_{L,H}$ . This type of scenario has been observed in other differentially rotating cylinder systems (Lopez et al. Reference Lopez, Hart, Marques, Kittelman and Shen2002; Marques et al. Reference Marques, Lopez and Shen2002; Marques, Gelfgat & Lopez Reference Marques, Gelfgat and Lopez2003), but in the present problem the situation is much more complicated, primarily due to the $\text{RW}_{L}$ and $\text{RW}_{H}$ coming in Eckhaus bands, as well as due to the subcritical nature of the instability of the BS at the lower end of the $\mathit{Ro}$ range.

Figure 11 presents an example of a mixed mode $\text{QP}_{0,36}$ , showing contours of ${\it\eta}$ and ${\it\eta}-{\it\eta}_{0}$ in different planes and isosurfaces of $w-w_{0}$ . $QP_{0,36}$ is a mixed mode with contributions from LC that drives inertial wave beams and $\text{RW}_{36}$ which does not. Contours of ${\it\eta}$ in a meridional plane (figure 11 a) clearly show the inertial waves, and are very similar to the ${\it\eta}$ contours of the pure LC shown in figure 6. Removing the axisymmetric component, the contours of ${\it\eta}-{\it\eta}_{0}$ shown in figure 11(b,c) and the isosurfaces of $w-w_{0}$ shown in figure 11(d) are very similar to those of a pure $\text{RW}_{H}$ , such as those of $\text{RW}_{41}$ shown in figure 10.

Figure 11.

Contours of (a) ${\it\eta}$ and (b) ${\it\eta}-{\it\eta}_{0}$ at ${\it\theta}=0$ , (c) ${\it\eta}-{\it\eta}_{0}$ at $z=-0.4$ and (d) isosurfaces of $w-w_{0}$ , at levels $\pm 0.001$ , for $\text{QP}_{0,36}$ at $\mathit{Re}=1.35\times 10^{4}$ , $\mathit{Ro}=0.23$ . There are ten cubically spaced contour levels in the range ${\it\eta}\in [-5,5]$ and ${\it\eta}-{\it\eta}_{0}\in [-1,1]$ .

Figure 12 is an example of a mixed mode $\text{QP}_{2,39}$ . In this case, there is a mix of $\text{RW}_{2}$ and $\text{RW}_{39}$ . The contours of ${\it\eta}$ shown in figure 12(a) are very similar to those of $\text{QP}_{0,36}$ in figure 11(a); however, subtracting off the axisymmetric component ( ${\it\eta}-{\it\eta}_{0}$ shown in figure 12 b,c) does not remove the inertial wave beams as it did for $\text{QP}_{0,36}$ , as they are driven by the $m=2$ component of $\text{QP}_{2,39}$ . This is further illustrated by the three-dimensional isosurfaces of the axial velocity. Figure 12(d) shows isosurfaces of the full solution minus its axisymmetric component, $w-w_{0}$ , and is rather complicated. However, plotting only the $m=2$ component of $w$ , $w_{2}$ in figure 12(e) and only the $m=39$ component, $w_{39}$ in figure 12(f) clearly shows that $\text{QP}_{2,39}$ is indeed a mixed mode of $\text{RW}_{2}$ and $\text{RW}_{39}$ ; compare these isosurfaces with those of $\text{RW}_{2}$ in figure 5(b) and $\text{RW}_{41}$ in figure 10(d). The quantitative differences between the low and high azimuthal wavenumber components of the QP and the corresponding $\text{RW}_{L}$ , or LC, and $\text{RW}_{H}$ are due primarily to the different points in $(\mathit{Re},\mathit{Ro})$ parameter space where the comparisons are made.

Figure 12.

Contours of (a) ${\it\eta}$ , $(b,c)$ ${\it\eta}-{\it\eta}_{0}$ , and isosurfaces of (d) $w-w_{0}$ , (e) $w_{2}$ and (f) $w_{39}$ for $\text{QP}_{2,39}$ at $\mathit{Re}=1.50\times 10^{4}$ and $\mathit{Ro}=0.28$ . There are ten cubically spaced contour levels in the range ${\it\eta}\in [-5,5]$ and ${\it\eta}-{\it\eta}_{0}\in [-1,1]$ , and the isosurfaces are at $\pm 0.001$ .

To obtain the frequencies associated with the QP states, we follow the procedure used for $\text{RW}_{L}$ and $\text{RW}_{H}$ . The fast Fourier transform of the time series of $w$ at a point close to the bottom corner of the cylinder provides a temporal spectrum of the flow in the stationary frame of reference, ${\it\omega}_{S}$ , which for a QP flow consists of two frequencies and all their linear combinations. These two frequencies are associated with either the $\text{RW}_{L}$ or LC and the $\text{RW}_{H}$ components of which QP is comprised. To calculate the corresponding frequencies in the rotating frame of reference, ${\it\omega}_{R}$ , two intermediate steps are needed. First, we produce a movie of the low and high $m$ components of the QP independently, with a very high time resolution over a short time interval, to determine if they are retrograde or prograde with the mean cylinder rotation in the stationary frame of reference; if it is prograde (retrograde) the associated frequency is positive (negative). Such movies are also useful for verifying which frequency is associate with the low and high $m$ components. Then, to determine the frequency of the low and high $m$ components of the QP in the rotating frame of reference, $m$ is subtracted from the corresponding frequency in the stationary frame of reference.

Figure 13(a) shows the mean modal kinetic energy for each azimuthal wavenumber $m$ in a $\text{QP}_{1,31}$ state for which two peaks at $m=1$ and $m=31$ are evident (the peak at $m=0$ corresponding to the mean flow is always dominant). Figure 13(c) shows the corresponding temporal spectrum of that state, determined from the time series of $w$ at $(r,{\it\theta},z)=(0.98,0,-0.4)$ . It also consists of two peaks and all their linear combinations; these are at ${\it\omega}_{S}=2.53$ associated with the $m=1$ component of $\text{QP}_{1,31}$ (verified to be prograde) and ${\it\omega}_{S}=29.20$ associated with the $m=31$ component of $\text{QP}_{1,31}$ (also prograde). The two frequencies associated with $\text{QP}_{1,31}$ in the rotating frame of reference are ${\it\omega}_{R}=1.53$ and ${\it\omega}_{R}=-1.80$ respectively.

Figure 13.

Azimuthal ( $E_{m}$ mean modal kinetic energies) and temporal (power spectral density, PSD of $w$ at $(r,{\it\theta},z)=(0.98,0,-0.4)$ ) spectra for $(a,c)$ $\text{QP}_{1,31}$ at $\mathit{Re}=1.40\times 10^{4}$ , $(b,d)$ HNL at $\mathit{Re}=1.55\times 10^{4}$ , both cases at $\mathit{Ro}=0.27$ .

Some of the QP states have spatial and temporal spectra that are broad band, and they are not straightforward to decompose into simple mixed modes as done previously. Such cases are termed highly nonlinear (HNL). The spatial and temporal broad-band spectra of a HNL at $\mathit{Re}=1.55\times 10^{4}$ and $\mathit{Ro}=0.27$ are shown in figure 13(b,d), respectively. This HNL is found between two different QP, $\text{QP}_{1,31}$ and $\text{QP}_{2,32}$ , as $\mathit{Re}$ is increased with $\mathit{Ro}=0.27$ fixed (see figure 8 b). Figure 14 shows ${\it\eta}$ and ${\it\eta}-{\it\eta}_{0}$ contours and isocontours of $w-w_{0}$ of this HNL. The meridional plots are similar to those of the various QP states previously described. The contours of ${\it\eta}-{\it\eta}_{0}$ at $z=-0.4$ show a strong $m=1$ modulation of the high- $m$ wall structure; this can be interpreted as coming from both the $m=1$ component of $\text{QP}_{1,31}$ and the nonlinear interaction between the $m=31$ component of $\text{QP}_{1,31}$ and the $m=32$ component of $\text{QP}_{2,32}$ . The wave beam in this HNL seems to be driven by the $m=1$ component of HNL. The wave beam cone is clearly identified in the $w-w_{0}$ isosurfaces (figure 14 d), but the $m=1$ wave beam is not as clean as it is for $\text{RW}_{1}$ (figure 5 a). This is mainly due to the broad-band nature of its temporal spectrum. The HNL were found to persist for long times in parameter regimes separating some QP states with differing primary azimuthal wavenumbers.

Figure 14.

Contours of (a) ${\it\eta}$ and (b) ${\it\eta}-{\it\eta}_{0}$ at ${\it\theta}=0$ , (c) ${\it\eta}-{\it\eta}_{0}$ at $z=-0.4$ and (d) isosurfaces of $w-w_{0}$ at levels $\pm 0.001$ , for HNL at $\mathit{Re}=1.55\times 10^{4}$ , $\mathit{Ro}=0.27$ . There are ten cubically spaced contour levels in the range ${\it\eta}\in [-5,5]$ and ${\it\eta}-{\it\eta}_{0}\in [-1,1]$ .