3.2.1. Increasing the Rossby number

We now examine the case of a flow characterized by a Rossby number larger than that of the reference case, but having similar Ekman number. This case has $\mathit{Ro}=0.17$ , $\mathit{Ek}=2.5\times 10^{-4}$ , and a corresponding internal Reynolds number $\mathit{Re}_{i}=121$ , which is well beyond the critical value of $\mathit{Re}_{i,c}\approx 22.4$ . Linear stability analysis predicts a wavenumber 5 configuration to be the most unstable structure at this flow condition (figure 4), which has an eigenmode structure similar to that shown in figure 5(b).

This three-dimensional flow case has been computed using 40 azimuthal Fourier modes, as greater spatial resolution is required to capture time-dependent disturbances in the shear layer. This case was also computed using 24 Fourier modes, which described a very similar saturated solution to the case computed using 40 Fourier modes. The energy time series of the non-zero wavenumbers is shown in figure 9(a). Again, the flow is initialized with the axisymmetric base flow solution perturbed by white noise. A wavenumber 5 structure arises during the initial stages of the flow. However, the dominance of the $k=5$ structure is short-lived as the energy in wavenumber 2 becomes larger in comparison at approximately $t=80$ . Subsequently, the energy in $k=2$ and its harmonics plateaus over time while the energies of the odd wavenumbers decay. At $t\simeq 200$ the energies of the odd wavenumbers switch to an exponential growth behaviour, extracting energy from the fundamental mode, and eventually saturate at $t\approx 800$ with comparable amplitudes to the even wavenumbers. Eventually, the flow saturates to a state which is dominated by $k=2$ distorted by a wavenumber 1 structure (marked as ‘ $2^{+1}$ ’ in figure 4). The comparable energies in wavenumbers 1 and 2 causes both wavenumbers to impose their symmetries on weaker higher-wavenumber structures and therefore the similarity in energy oscillations typically exhibited by the group of $k+k_{peak}n$ wavenumbers is no longer valid.

The flow development represented through axial vorticity contours at mid-depth is illustrated in figure 9(b). A flow indistinguishable from the axisymmetric base flow solution is observed in the first frame at $t=5$ . A non-axisymmetric structure begins to evolve around the disk periphery at $r=1$ as the energies in the dominant wavenumbers increase in amplitude. At $t=35$ , four vortices are seen to form on one half of the initially axisymmetric shear layer. As time evolves, the vortices grow larger in size and the remaining shear layer also begins to roll up. Eventually, six vortices are observed at $t=45$ that are arranged irregularly, with the two vortices at the bottom of the frame being clustered close together. This pair of vortices coalesces at a future time and the structure adopts a pentagonal configuration. This initial process is reflected in the energy time series shown in figure 9(a). That is, the $k=5$ structure is dominant during the exponential growth regime. It is noted that the wavenumber 5 structure at $t=50$ is an asymmetric pentagon, and that a regular pentagon was not observable at other time steps. This is due to the time dependence of the flow as the energies in many of the wavenumbers are still varying at comparable amplitude.

At future times, the nonlinear effects cause the vortices to coalesce, resulting in a dipolar structure at $t=145$ . The two vortices exhibit low vorticity and encircle an oval-shaped interior of higher vorticity. This eventually becomes distorted as one of the vortices breaks and deforms into a strand as seen at $t=1175$ . The transition from a symmetric dipole state to a distorted state is reflected in the energy time series whereby increases in the wavenumber 1 structure are observed until it becomes comparable to the energy contained in the wavenumber 2 structure. Again, the constant energies displayed by the dominant modes indicate that the structure is unchanging and drifts steadily about the axis of rotation. A similar asymmetric dipole structure was obtained numerically by Früh & Nielsen (Reference Früh and Nielsen2003) in the same system using a quasi-two-dimensional model. A similar structure was also captured experimentally by van de Konijnenberg et al. (Reference van de Konijnenberg, Nielsen, Juul Rasmussen and Stenum1999). However, they observed a smooth vortex on one side and an unstable structure on the opposite side where small vortices continuously formed and dissipated. The differences in flow conditions and apparatus in that study may be reasons why the time-dependence is not seen here.

Distinct alternating weak and strong bands of axial vorticity are observable in the vortices and the interior region at $t=145$ and $t=1175$ . Examination of the discretized domain underlying the axial vorticity contours demonstrated that these waves are significantly larger than the radial and azimuthal grid spacings. Thus, these structures are a physical feature of the flow which we suspect to be inertial waves. Inertial waves arise in rotating flows due to the Coriolis force acting as a restoring force and exhibit frequencies in the range $0<{\it\omega}_{d}/\overline{{\it\Omega}}<2$ , where ${\it\omega}_{d}$ and $\overline{{\it\Omega}}$ are angular velocities of the drifting polygonal structure and the background flow, respectively (e.g. see Lopez & Marques Reference Lopez and Marques2011, Reference Lopez and Marques2014, and references therein). In this system, the waves are stationary with respect to the rotating polygonal structure and satisfy ${\it\omega}_{d}/\overline{{\it\Omega}}<2$ . These waves appear to originate from the disk–tank interface where the Stewartson and Ekman layers interact. This interaction may be related to the generation of these waves – a feature common with other configurations involving Ekman and Stewartson layers (e.g. Hart & Kittelman Reference Hart and Kittelman1996; Sauret et al. Reference Sauret, Cébron, Le Bars and Le Dizès2012; Lopez & Marques Reference Lopez and Marques2014). Instabilities of the Ekman and Stewartson layers have been reported as the driving mechanism in rotating spherical shells (Noir, Jault & Cardin Reference Noir, Jault and Cardin2001; Calkins et al. Reference Calkins, Noir, Eldredge and Aurnou2010) and differentially rotating configurations (Lopez & Marques Reference Lopez and Marques2010), respectively. It should be noted that visually similar wave patterns have been also observed in prior studies which have investigated spontaneous emergence of inertia–gravity waves from the interaction of two counter rotating vortices (e.g. Viúdez & Dritschel Reference Viúdez and Dritschel2006; Snyder et al. Reference Snyder, Muraki, Plougonven and Zhang2007; Wang, Zhang & Snyder Reference Wang, Zhang and Snyder2009). A thorough review of inertia–gravity waves from atmospheric jets and fronts can be found in Plougonven & Zhang (Reference Plougonven and Zhang2014).

The same flow conditions simulated at a lower azimuthal resolution of 24 Fourier modes yielded a similar energy time series and similar vorticity and velocity contours. Thus, these small-scale wave patterns are not a numerical artefact. It is expected that these features would be very difficult to capture experimentally using laboratory techniques such as laser Doppler velocimetry and particle image velocimetry, since the azimuthal vorticity values are so much smaller than the values of axial vorticity in the flow. However, an altimetric imaging velocimetry technique has been able to successfully capture such structures (e.g. Afanasyev, Rhines & Lindahl Reference Afanasyev, Rhines and Lindahl2008). Despite the experimental limitation, the dipole captured here by direct numerical simulation is in agreement with the wavenumber range at these parameters reported from the experiments of Früh & Read (Reference Früh and Read1999): they reported wavenumbers 2–4 in the vicinity of $\mathit{Ro}=0.17$ and $\mathit{Ek}=2.5\times 10^{-4}$ .

Despite the complex patterns observed in the $r{-}{\it\theta}$ plane, the saturated flow still maintains reflective symmetry about the horizontal mid-plane. This symmetry is consistent with the axisymmetric solution at the same parameters ( $\mathit{Re}_{\mathit{Ek}}=21.5$ , figure 4). Figure 10(a–c) illustrates contours of the axial velocity, axial vorticity and azimuthal vorticity on the vertical semi-plane of the tank ( $r{-}z$ plane). All three contours are illustrated at the same arbitrary azimuthal angle and show reflective symmetry about the mid-depth. The axial velocity exhibits Ekman pumping at the disk–tank interface in addition to recirculation zones on either side. The recirculation zones are bounded by oblique waves which vary throughout the domain. This is in contrast to the reference flow case where the structures are not significantly inclined (see figure 8 c). These inclined structures can also be seen in the axial vorticity, and especially in the azimuthal vorticity, where low vorticity contours have been emphasized. Visualizations of the flow over time reveal that these inclined structures drift at the same rate as the primary polygonal structure (i.e. they are stationary relative to the rotating structure). In addition, the wave patterns can be seen in an iso-surface of axial vorticity as illustrated in figure 10(d). Closer inspection of the reference case flow also reveals chevron-type patterns in the iso-surface of axial vorticity (figure 8 c), though they are not as prominent. The weaker wave patterns observed in the reference case flow may be due to the lower Rossby number. Increases to the Rossby number drives a stronger axial jet flow in the Stewartson layers while decreasing the Ekman number produces thinner Stewartson layers, both of which increases the susceptibility of instability (Vo et al. Reference Vo, Montabone and Sheard2014, Reference Vo, Montabone and Sheard2015) and hence the emergence of inertial waves (Lopez & Marques Reference Lopez and Marques2010). This result is evident throughout the parameter space explored in this paper.

Figure 10.

Flow condition of $\mathit{Ro}=0.17$ and $\mathit{Ek}=2.5\times 10^{-4}$ at $t=1629$ . (a–c) Contours of (a) axial velocity, (b) axial vorticity and (c) azimuthal vorticity are displayed on the entire $r{-}z$ plane. Axial vorticity and azimuthal vorticity contour levels are as in figure 8 while axial velocity contour levels are plotted in the range $u_{z}=\pm 0.1|\mathit{Ro}|({\it\Omega}+{\it\omega})$ , where dark and light contour shading represent low and high values, respectively. (d) An isometric view of an axial vorticity iso-surface of ${\it\omega}_{z}=1.88$ is shown. The arrow indicates the chevron-type patterns identified in the flow.

The next section also investigates a flow condition that is well beyond the onset of instability, though achieved by decreasing the Ekman number of the reference flow case rather than by increasing the Rossby number.

3.2.2. Decreasing the Ekman number

The effect of decreasing the Ekman number to increase $\mathit{Re}_{i}$ is examined at flow conditions of $\mathit{Ro}=0.05$ and $\mathit{Ek}=8\times 10^{-5}$ . This reduction in Ekman number causes the investigated point to be at an internal Reynolds number of $\mathit{Re}_{i}=83.6$ , significantly beyond the instability threshold $\mathit{Re}_{i,c}\approx 22.4$ . Therefore, it is expected that, as in the previous section, the unstable wavenumber from the three-dimensional simulations will be different to the wavenumber predicted by linear stability analysis due to nonlinear effects. More specifically, the nonlinear state will adopt a lower azimuthal wavenumber than its linear prediction.

As indicated in the regime diagram (figure 4), the linear preferred wavenumber increases with decreasing Ekman number at small Rossby numbers. The predicted wavenumber here is 8, as compared to 5 for the reference case. Since the growth rates of the most unstable wavenumber is associated with the mode I instability, the eigenmode exhibits a central octagon exhibiting high values of vorticity with a thin ring of lower vorticity surrounding it.

Here the three-dimensional flow has been resolved using 128 azimuthal Fourier modes. The energy time series of each non-zero wavenumber is shown in figure 11(a). The three-dimensional solution has been initialized with the axisymmetric base flow solution perturbed with white noise. It can be seen that the energies contained in the wavenumber 6 and 8 structures are initially the highest throughout the linear growth regime. The appearance of the wavenumber 6 structure in the linear regime rather than the dominant $k=8$ mode is explained by the small difference (less than 5 %) in their respective growth rates, and the possibility that the initial white-noise seeding may unevenly seed the unstable azimuthal wavenumbers. The competition between $k=6$ and $k=8$ structures weakens beyond $t\approx 100$ as nonlinear effects become significant. At this time, the energy in $k=8$ and subsequently $k=6$ begin to decrease, and the flow becomes dominated by a stable $k=3$ configuration by $t\approx 160$ . Unlike the $k=2^{+1}$ mode obtained earlier, the second dominant mode for this flow is a harmonic of the dominant wavenumber. Thus, an undisrupted triangular structure is expected to emerge in the three-dimensional flow.

Figure 11.

Flow conditions of $\mathit{Ro}=0.05$ and $\mathit{Ek}=8\times 10^{-5}$ . Time series of (a) the energies contained in the first 127 azimuthal wavenumbers and (b) the averaged wavenumber weighted by energy. The dominant wavenumber transitions from $k=8$ to 6 to 3 over time represented by bold lines. The dashed lines enclosing $2.5\leqslant k^{\ast }<3.5$ and $7.5\leqslant k^{\ast }<8.5$ represent integer wavenumbers 3 and 8, respectively.

An alternative interpretation of the energy time series is achieved by considering the average wavenumber weighted by energy over time, as illustrated in figure 11(b) and calculated as

(3.1) $$\begin{eqnarray}k^{\ast }={\displaystyle \frac{\displaystyle \mathop{\sum }_{k=1}^{N}E_{k}k}{\displaystyle \mathop{\sum }_{k=1}^{N}E_{k}}},\end{eqnarray}$$

where $k$ is an integer azimuthal wavenumber, $N$ is the number of Fourier modes used in the simulation and $E_{k}$ is the energy contained in the $k$ th wavenumber. The energy associated with the base flow ( $k=0$ ) is omitted in calculating $k^{\ast }$ . The purpose of this quantity is to identify the dominant wavenumber that would be observed in the flow at a specific time, provided the amplitudes are not so small as to be swamped by the base flow. This parameter also helps to illustrate the wavenumber vacillation process when there are multiple wavenumbers competing with very large energies. The plot demonstrates a brief saturation of a wavenumber 8 structure ( $7.5\leqslant k^{\ast }<8.5$ ) in the time interval $35\lesssim t\lesssim 100$ . As the nonlinear effects become significant, the apparent wavenumber decreases towards $k^{\ast }=3~(2.5\leqslant k^{\ast }<3.5)$ and is sustained for $t\gtrsim 220$ . These changes in dominant wavenumber are consistent with the energies demonstrated in figure 11(a).

The three-dimensional wavenumber 3 structure achieved at saturation is represented in the contours of axial vorticity at two different depths, as shown in figure 12(a,b). The structure is composed of three concentrated regions of low vorticity migrating around the $r=1$ circumference. As described earlier, the contours near the horizontal boundary ( $z/H=0.1$ ) illustrate a circular ring of low vorticity attributed to the discontinuity at the disk–tank interface. The small-scale wavy structures described earlier (see figure 9 b) are evident in the interior and satellite vortices. As with the previous cases, this flow maintains its reflective symmetry about the horizontal mid-plane, as observed in the azimuthal vorticity contours in figure 12(c). An iso-surface of axial vorticity at $t=469$ is illustrated in figure 12(d). The iso-surface illustrates three vortices arranged in a triangle extending the entire depth of the tank. Additionally, distinct chevron-type features can be observed on the inner wall of the axial vorticity iso-surface.

Figure 12.

Flow condition of $\mathit{Ro}=0.05$ and $\mathit{Ek}=8\times 10^{-5}$ at $t=469$ . (a,b) Contours of axial vorticity at two different heights of (a) $z/H=0.1$ and (b)  $z/H=0.5$ . The flow is reflectively symmetric about the $z/H=0.5$ . Axial vorticity contour levels are as in figure 5. (c) Contours of azimuthal vorticity displayed on the entire $r{-}z$ plane with levels plotted as in figure 8. (d) An isometric view of an axial vorticity iso-surface of ${\it\omega}_{z}=1.96$ is shown. The arrow indicates the chevron-type patterns identified in the flow.

Increasing the flow conditions further beyond the instability onset by decreasing the Ekman number has shown similar characteristics to those exhibited by increasing the Rossby number. That is, the intensification of nonlinear effects encourages coalescence of vortices which yield wavenumbers that are smaller than the linear predictions. The flows presented so far in this paper have all demonstrated reflective symmetry about the horizontal mid-plane and steady saturated states. The next section details transitions between chaotic, steady and time-periodic flows observed at an even higher  $\mathit{Re}_{i}$ .

3.2.3. High- $\mathit{Re}_{i}$ regime

Unsteady saturated and time-dependent flows have been reported in experimental and numerical studies at high- $\mathit{Re}_{i}$ conditions. Früh & Read (Reference Früh and Read1999) experimentally investigated a similar system and observed weak fluctuations, modulated oscillations, highly irregular flow, and period-doubled solutions in their positive- $\mathit{Ro}$ parameter space. Flows with weak fluctuations were discovered near the onset of instability at low $\mathit{Ek}$ while all other time-dependent flows were realized at high $\mathit{Re}_{i}$ values (both $\mathit{Ro}$ and $\mathit{Ek}$ being small or large). An extended experimental and numerical study conducted by Früh & Nielsen (Reference Früh and Nielsen2003) found time-dependent behaviour only for flows with three or fewer vortices for $\mathit{Re}\gtrsim 150$ (at constant $\mathit{Ek}=3\times 10^{-4}$ ). A similar qualitative result was reported by van de Konijnenberg et al. (Reference van de Konijnenberg, Nielsen, Juul Rasmussen and Stenum1999), who found symmetric and steady flows for $k\geqslant 4$ and time-dependent flow for $k\leqslant 3$ in a parabolic bowl. Periodic deformations via slight oscillations of the triangular interior boundary for $k=3$ was seen while a periodic formation and decay of small vortices were obtained for $k=2$ . A decrease in wavenumber corresponded to an increase in Reynolds number. In their numerical simulations of the axisymmetric flows in this differential-disk system, Vo et al. (Reference Vo, Montabone and Sheard2014, Reference Vo, Montabone and Sheard2015) found no evidence of unsteady solutions throughout their computed Rossby and Ekman number ranges ( $0.01\lesssim \mathit{Ro}\lesssim 0.6$ and $5\times 10^{-5}\lesssim \mathit{Ro}\lesssim 3\times 10^{-3}$ , respectively).

In unpublished experimental flow visualizations from particle tracking carried out in 1998, a typical travelling wavenumber 2 flow was revealed at $\mathit{Ro}=0.77$ and $\mathit{Ek}=1\times 10^{-3}$ . The apparatus used is described in Früh & Read (Reference Früh and Read1999). Increasing the Rossby number to $\mathit{Ro}=0.924$ while holding $\mathit{Ek}$ constant produced a flow which appears axisymmetric in the horizontal plane with an indication of a pair of toroidal Taylor-like vortices in the vertical plane. Images of the experimental flow visualization are shown in figure 13(a,b). To elucidate the flow structure and dynamics in this regime, this section investigates high- $\mathit{Re}_{i}$ flows based on the flow conditions investigated experimentally. Thereafter, the high- $\mathit{Re}_{i}$ condition is held constant while $\mathit{Ro}$ and $\mathit{Ek}$ are varied to examine whether or not the characteristics present at the high- $\mathit{Re}_{i}$ regime are dependent only on $\mathit{Re}_{i}$ , or if they are further a function of both $\mathit{Ro}$ and $\mathit{Ek}$ . Additionally, any transitions in the temporal properties of the flow will be described.

Figure 13.

Flow conditions of (a,c)  $\mathit{Ro}=0.77$ , $\mathit{Ek}=1\times 10^{-3}$ and (b,d)  $\mathit{Ro}=0.924$ , $\mathit{Ek}=1\times 10^{-3}$ . Visual comparison between (a,b) experimental particle tracking and (c,d) the axial vorticity contours from a three-dimensional simulation. Axial vorticity contour levels as for figure 5.

The flow condition of $\mathit{Ro}=0.77$ and $\mathit{Ek}=1\times 10^{-3}~(\mathit{Re}_{i}=193)$ is considered first. An axisymmetric simulation of this flow condition demonstrates saturation towards a steady-state flow. The three-dimensional direct numerical simulation generates a wavenumber 2 structure first arising in the linear regime and remaining dominant during its nonlinear development. The energy in the wavenumber 1 structure increases gradually over time and eventually plateaus as the second most energetic non-zero wavenumber. Axial vorticity contours in the horizontal mid-plane illustrated in figure 13(c) exhibit excellent agreement with the experimental observations. Both the experimental visualization and three-dimensional simulation illustrate an elliptical vortex located at the centre of the tank with two vortices situated on opposite sides of the ellipse.

In contrast, significantly different dynamics are observed when the Rossby number is increased to $\mathit{Ro}=0.924~(\mathit{Re}_{i}=323)$ . For comparison against experimental visualization, a top-down view of the axial vorticity contours at mid-depth is provided in figure 13(d). The flow exposes a structure which is almost axisymmetric, especially in the central region, which is visually similar to that observed experimentally. A numerical simulation for the axisymmetric solution at this same flow condition was computed. At this higher Rossby number, the axisymmetric solution illustrates an unsteadiness in the axial jets supplied by Ekman pumping from the rapid disk rotation emanating from the disk–tank interfaces. The presence of stronger axial jets driven by faster spinning disks (larger $\mathit{Ro}$ ) causes an instability to develop as a breaking of the hyperbolic zone at the point where the jets meet. This unsteadiness in turn breaks the reflective symmetry about the horizontal mid-plane and the flow saturates to a perfect periodicity in time. This periodic flapping of the axial jets is not seen in the saturated three-dimensional flow, rather the axial jets become three-dimensional and fluctuate irregularly. This irregular fluctuation is shown in figure 14, which displays snapshots of an axial vorticity iso-surface over time. The iso-surface is chosen to conveniently highlight the breaking of the axial jets around the periphery of the disks.

Figure 14.

Flow conditions of $\mathit{Ek}=1\times 10^{-3}$ and $\mathit{Ro}=0.924$ . (a) An oblique view of an iso-surface of axial vorticity in the three-dimensional flow. (b) Snapshots of axial vorticity over time shown through a close-up of the flow corresponding to the marked box in (a). The flow is shown in a stationary frame of reference relative to the rotating flow.

Figure 15.

Flow conditions of a constant $\mathit{Re}_{i}=323$ at (a–c) $\mathit{Ek}=1\times 10^{-3},\mathit{Ro}=0.924$ , (d–f) $\mathit{Ek}=7\times 10^{-4},\mathit{Ro}=0.707$ , (g–i) $\mathit{Ek}=5\times 10^{-4},\mathit{Ro}=0.549$ , (j–l) $\mathit{Ek}=3\times 10^{-4},\mathit{Ro}=0.375$ , (m–o) $\mathit{Ek}=7\times 10^{-5},\mathit{Ro}=0.126$ . (a,d,g,j,m) Energy time series, (b,e,h,k,n) the axial vorticity contours illustrated in the $r{-}{\it\theta}$ mid-plane, (c,f,i,l,o) iso-surfaces of axial vorticity demonstrating the axial dependence of the flow structures. The axial vorticity plots are represented at times corresponding to the ending time of their respective energy time series.

Since the axisymmetric solution is periodic rather than steady state, the three-dimensional direct numerical simulation is not initialized with a prescribed base flow but rather begins with the axisymmetric boundary conditions (tank and disk rotation) and white noise. This is applicable to the remaining flows simulated in this section. The three-dimensional direct numerical simulation exhibits chaotic behaviour in the energies of all wavenumbers, as shown in figure 15(a). The saturated flow is described by a dominant wavenumber 2 structure with its first harmonic ( $k=4$ ) being the second dominant mode. The erratic oscillations in energy are caused by the irregular fluctuations in the axial vorticity strands stemming from the disk–tank interface when viewed in the $r{-}z$ plane. In this chaotic state, the reflective symmetry about the horizontal mid-plane is lost. Additional visualizations of the flow are also shown in figure 15(a) via contours and an iso-surface of axial vorticity to illustrate the depth dependence of the flow.

Other flows at $\mathit{Re}_{i}=323$ are investigated by decreasing both $\mathit{Ek}$ and $\mathit{Ro}$ . The energies of the non-zero wavenumbers, axial vorticity contours and iso-surfaces are illustrated in figure 15. As $\mathit{Ek}$ and $\mathit{Ro}$ decrease, the flow transitions from being chaotic at $\mathit{Ek}=1\times 10^{-3}$ to steady state at $5\times 10^{-4}\leqslant \mathit{Ek}\leqslant 7\times 10^{-4}$ . It should be noted that as the Ekman number is decreased, the energy in the wavenumber 1 structure increasingly becomes comparable to the wavenumber 2 mode while remaining the second dominant mode. This causes the initial dipole to rotate off-axis and broadens one of its vortices. Additionally, the central structure becomes thinner, loses symmetry and breaks apart with decreasing Ekman number. Despite the comparable energies in the leading modes, the energies maintain a plateau which demonstrates a steady-state solution. That is, the rigid instability structure drifts about the axis of rotation.

Further decreasing below $\mathit{Ek}=3\times 10^{-4}$ causes the wavenumber 1 structure to become dominant and compete with the wavenumber 2 structure. The vortex which was broadened due to the increase in energy in the wavenumber 1 structure eventually deforms into a strand of vorticity that no longer encloses a region of fluid (figure 15 d,e). The strand of vorticity folds back into itself partially and connects with the opposing vortex adopting an S-shaped appearance for the structure as a whole. In the case of $\mathit{Ek}=3\times 10^{-4}$ , the energies in the non-zero wavenumbers saturate to a clear periodic undulation over time. This undulation also exists at the lower $\mathit{Ek}=7\times 10^{-5}$ although it is weaker and not as obvious on a logarithmic scale and not perfectly periodic. That is, the temporal behaviour of the flow transitions from a periodic to an irregular state from $\mathit{Ek}=3\times 10^{-4}$ to $\mathit{Ek}=7\times 10^{-5}$ . The decrease in Ekman number also demonstrates more prominent waves in the flow.

At these smaller Ekman numbers, the unsteadiness is caused by a horizontal flapping of the axial vorticity strand. This periodic motion is illustrated in figure 16(a) over a time span coinciding with the period demonstrated in the energy oscillations for $\mathit{Ek}=3\times 10^{-4}$ . The strand of axial vorticity thickens as its tail curls towards the edge of the disk and becomes thinner as the tail kicks away from the disk. The opposing vortex does not deform significantly throughout this motion and the flow maintains a reflective symmetry about the horizontal mid-plane. A similar periodic motion is observed for $\mathit{Ek}=7\times 10^{-5}$ . However, the opposing vortex is much thinner and deformations are more prominent at this Ekman number as compared to $\mathit{Ek}=3\times 10^{-4}$ . In addition, there is a continuous formation and decay of vorticity at its periphery, which may be the cause of the imperfect undulations demonstrated in the energy time series. A similar feature was also observed by van de Konijnenberg et al. (Reference van de Konijnenberg, Nielsen, Juul Rasmussen and Stenum1999), although they reported the feature as being periodic and the vortices appeared coherent.

Figure 16.

(a) Flow conditions of $\mathit{Ek}=3\times 10^{-4}$ and $\mathit{Ro}=0.375$ . Snapshots of axial vorticity over a time span corresponding to the period associated with the $k=1$ energy fluctuations. (b) Flow conditions of $\mathit{Ek}=1\times 10^{-3}$ and $\mathit{Ro}=0.924$ . Erratic changes in the flow are illustrated through snapshots of axial vorticity over two undulations in the dominant wavenumber 2 structure. The arrows in the insets indicate regions of interest as described in the text.

In stark contrast to the periodic and irregular flow observed at low $\mathit{Ek}$ , the fluctuations in energy for $\mathit{Ek}=1\times 10^{-3}$ are not caused by irregular wavering of the axial vorticity strand but rather erratic oscillations of the axial jets and continuous deformation of the axial vorticity patches situated at the periphery of the disk. The flow deformations over approximately two undulations of the energy in $k=2$ are illustrated in figure 16(b).

To our knowledge, this study is the first to report the wavering of the axial vorticity strand and recognize its association with periodic change in the energy of the dominant wavenumber. Bergeron et al. (Reference Bergeron, Coutsias, Lynov and Nielsen2000) were able to obtain flow states at large Reynolds number conditions which illustrated time-periodic behaviour when visualized with vorticity contours and through measurements of the global enstrophy. However, they did not provide a physical explanation as to why these flows exhibit this periodic behaviour. At least a single strand of axial vorticity can be seen in their figures 17 and 18, rather than a completely closed vortex, much like that shown in figure 16(a). Their time evolution snapshots appear to demonstrate a modulation of this axial vorticity strand, similar to the instability described here. The change they observe may be related to the wavering in the tail of the axial vorticity strand in addition to changing its size. This could be a possible explanation for their time-dependent behaviour. It must be noted that their time-periodic states were obtained with free-slip boundary conditions whereas no-slip boundary conditions are used in this study. They note that at very large Reynolds numbers, the no-slip boundary condition causes a continuous generation of vorticity erupting from the inner rod used to drive the rotation of the disks which leads to chaotic flow. Similar chaos was observed by Früh & Nielsen (Reference Früh and Nielsen2003), who reported that time-dependent flow may originate from a co-existence of global modes or from the generation of small vortices off the inner rod.

The pure and S-shaped dipole structures captured in these simulations resemble structures in Venus’s polar vortices. A dipole structure at the north pole of Venus was first mapped by the Pioneer Venus Orbiter in the 1970s in the middle atmosphere (Taylor et al. Reference Taylor, Diner, Elson, McCleese, Martonchik, Delderfield, Bradley, Schofield, Gille and Coffey1979, Reference Taylor, Beer, Chahine, Diner, Elson, Haskins, McCleese, Martonchik, Reichley, Bradley, Delderfield, Schofield, Farmer, Froidevaux, Leung, Coffey and Gille1980). The more recent European Space Agency’s Venus Express mission captured an S-shaped dipole structure at the south pole via the Visible and Infrared Thermal Imaging Spectrometer (Piccioni et al. Reference Piccioni, Drossart, Sanchez-Lavega, Hueso, Taylor, Wilson, Grassi, Zasova, Moriconi, Adriani, Lebonnois, Coradini, Bezard, Angrilli, Arnold, Baines, Bellucci, Benkhoff, Bibring, Blanco, Blecka, Carlson, Di Lellis, Encrenaz, Erard, Fonti, Formisano, Fouchet, Garcia, Haus, Helbert, Ignatiev, Irwin, Langevin, Lopez-Valverde, Luz, Marinangeli, Orofino, Rodin, Roos-Serote, Saggin, Stam, Titov, Visconti and Zambelli2007), though this is a thermal feature rather than a vorticity structure. The dynamics of Venus’s south polar vortex are very complex, with its internal structure constantly varying on a daily time scale (Luz et al. Reference Luz, Berry, Piccioni, Drossart, Politi, Wilson, Erard and Nuccilli2011; Garate-Lopez et al. Reference Garate-Lopez, Hueso, Sanchez-Lavega, Peralta, Piccioni and Drossart2013).

It should be noted that three-dimensional simulations have been performed in the present work at several flow conditions additional to those described in §§ 3.1–3.3 (these additional cases are included in figure 4). The simulations establish trends which describe saturated modes being equivalent to the wavenumber predicted by linear stability analysis for flows in the vicinity of the instability onset, and as the internal Reynolds number is increased either by increasing the Rossby number or decreasing the Ekman number, the structures of the stable flow state exhibit azimuthal structures that are different (lower wavenumber) from those predicted by linear stability analysis. The comparable energy in each azimuthal wavenumber structure exacerbates nonlinear effects, which causes unstable structures of varying wavenumbers to compete and interact.

In addition, it was also found (though not shown here), that wavenumbers associated with the mode II instability had little influence on the stability of the flow in the nonlinear regime. The energy in the mode II wavenumbers is observed to briefly dominate in the linear regime as for the linear stability analysis, but in the nonlinear regime their energy is seen to quickly decay. Thus, the mode II instability is not expected to dictate the structure of real flows. The next section investigates the sensitivity of a flow and the hysteretic nature of the primary instabilities present in these flows.

3.3.1. Perturbing the axisymmetric solution

The saturated three-dimensional flows observed in the previous sections were initialized from the evolved steady-state axisymmetric solution perturbed with white noise. Since many of the prior energy time series demonstrated a single stable azimuthal wavenumber of very large energy compared to the other wavenumbers, it is expected that different initial conditions of the same flow condition will achieve the same stable solution. In this section, the steady-state axisymmetric base flows are initially seeded with various unstable linear mode solutions, with and without white noise.

A flow characterized by $\mathit{Ro}=0.5$ and $\mathit{Ek}=3\times 10^{-3}$ is considered for investigation here. The large growth rates inherent in the high- $\mathit{Ro}$ and high- $\mathit{Ek}$ flow allows for shorter computation times in reaching a saturated solution. According to the linear stability analysis, the most unstable wavenumber is predicted to be $k=3$ . The growth rates of this flow and its linear instability structure are shown in figure 17(a) and its inset, respectively.

Figure 17.

Flow condition of $\mathit{Ro}=0.5$ and $\mathit{Ek}=3\times 10^{-3}$ . (a) The growth rates for a range of wavenumbers obtained through linear stability analysis. The dashed line represents neutral stability, where points above and below symbolize unstable and stable wavenumbers, respectively. The inset shows representative axial vorticity contours of linear instabilities of finite amplitude superimposed onto its respective steady-state axisymmetric base flow shown at $z/H=0.5$ . Nonlinear effects have been neglected, and contour levels are as in figure 5. (b) Energies of the first 23 non-zero wavenumbers over time. The inset shows typical contours of axial vorticity at $z/H=0.5$ of a saturated flow regardless of the initial conditions provided they are axisymmetric. Equi-spaced contour levels are plotted in the range $2\overline{{\it\Omega}}\pm 5{\it\omega}$ .

A three-dimensional direct numerical simulation has been initialized with its steady-state axisymmetric base flow and white noise. The energies contained in each azimuthal wavenumber over time for the first 23 non-zero wavenumbers are shown in figure 17(b). The three-dimensional solution agrees with the linear prediction as the energy in the wavenumber 3 structure is dominant during the initial stages of the flow development. The structures of the linear and nonlinear wavenumber 3 structures are very similar in appearance. As nonlinear effects become apparent, the even wavenumbers slowly increase in energy over time and eventually surpass the wavenumber 3 structure, resulting in a dominant wavenumber 2 flow.

To see the effects of different initial conditions, the steady-state axisymmetric base flow was seeded with an instability mode of a single wavenumber. Eigenvectors for $k=2,3,4$ and $k=6$ were separately seeded into the initial steady-state axisymmetric base flow and the energies of the leading wavenumbers were monitored until the flow evolved into a saturated state. As the non-seeded flow indicated a wavenumber 2 structure at saturation (figure 17 b), it is expected that all seeded flows will ultimately produce the same wavenumber provided the axisymmetric flow is insensitive to initial conditions. For all eigenvector-seeded cases, the wavenumber 2 structure and its harmonics prevail and demonstrate saturated energy profiles similar to that obtained by the non-seeded case. For example, a wavenumber 3 seeded eigenvector illustrates a triangular structure at the initial state which evolves into a dipole structure at the final state due to nonlinear interactions as the energies in the odd wavenumbers decay. The axial vorticity contours of a typical nonlinear solution for the saturated wavenumber 2 at this flow condition is depicted in the inset of figure 17(b).

The sensitivity of axisymmetric flow solutions at other flow conditions has also been analysed. A lower limit of $\mathit{Ro}=0.1$ and $\mathit{Ek}=2\times 10^{-4}$ was considered for investigation as lesser forcing conditions require a longer time integration and greater computational resources to simulate the smaller growth rates and flow structures. The tested flow solutions have all shown the same insensitivity characteristics as $\mathit{Ro}=0.5$ and $\mathit{Ek}=3\times 10^{-3}$ (figure 17). This type of insensitivity has also been observed by Bergmann et al. (Reference Bergmann, Tophøj, Homan, Hersen, Andersen and Bohr2011), who studied a different type of rotating flow. In that study, the sensitivity of the flow was examined through the testing of various initial conditions involving a flow initialized from rest, initialized from manually disrupting a developed triangular structure and initialized with a high rotation rate such that the flow was axisymmetric. The resultant flow for all three cases exhibited a triangle, which suggests that the flow was insensitive to initial conditions.

3.3.2. Perturbing the non-axisymmetric solution

To further study the flow sensitivity, the effect of hysteresis was investigated by changing the forcing conditions of non-axisymmetric saturated flows. This change in forcing is to mimic, to an extent, experimental conditions which have experienced strong hysteresis and have shown dependence on the direction of the change of experimental conditions (e.g. Früh & Read Reference Früh and Read1999; van de Konijnenberg et al. Reference van de Konijnenberg, Nielsen, Juul Rasmussen and Stenum1999; Bergeron et al. Reference Bergeron, Coutsias, Lynov and Nielsen2000; Früh & Nielsen Reference Früh and Nielsen2003).

Two saturated flow conditions of $\mathit{Ro}=0.0325$ and $\mathit{Ro}=0.02375$ at $\mathit{Ek}=8\times 10^{-5}$ have been interchanged. The Rossby number is changed instantly from $\mathit{Ro}=0.0325$ to $0.02375$ and vice versa at an arbitrary time after a stable saturated flow has been reached. These flow conditions were chosen such that they belong in the lower parameter regime, in contrast to the previous section, to illustrate the lower growth rates and that the hysteresis effects can occur over a minor change in $\mathit{Ro}$ . Additionally, large $\mathit{Ek}$ conditions do not display a large variety of unstable wavenumbers and therefore are not suitable for this type of sensitivity study. The Rossby number was varied instead of the Ekman number because it demonstrated a rapid change in wavenumber over a small range. The energy time series of these saturated flow solutions are depicted in figure 18. Stable wavenumbers of $k=4$ and $k=5$ are exhibited in the saturated state for $\mathit{Ro}=0.0325$ and $\mathit{Ro}=0.02375$ , respectively, which differ from the linearly predicted state of $k=7$ for both flows (refer to figure 4).

Figure 18.

The energy time series for $\mathit{Ek}=8\times 10^{-5}$ , for (a)  $\mathit{Ro}=0.0325$ with the first 39 non-zero wavenumbers with an inset illustrating a close-up region of the competing modes just beyond the linear regime, and (b)  $\mathit{Ro}=0.02375$ with the first 31 non-zero wavenumbers. Saturated stable states of $k=4$ and $k=5$ are obtained for each flow, respectively.

The non-axisymmetric solution of $\mathit{Ro}=0.0325$ and $\mathit{Ek}=8\times 10^{-5}$ at $t=2350$ is used as the initial condition. This time is marked by a vertical dashed line as shown in figure 19(a), and corresponds to a time where the dominant mode has saturated. The axial vorticity structure at this nominated time ( $t=2350$ ) and the final time ( $t=3450$ ) display no significant differences. At $t=2350$ , the flow is dominated by a wavenumber 4 structure with the energy in its harmonics saturating over time while all other wavenumbers are decaying. As the Rossby number is decreased to $\mathit{Ro}=0.02375$ , the energies in each azimuthal wavenumber drop gradually over a short time period. The energies in each wavenumber then demonstrate the same trends established before $t=2350$ . That is, a stable square configuration is observed and the harmonics of $k=4$ approach a plateau. The wavenumber 5 structure, which was determined to be most unstable mode for $\mathit{Ro}=0.02375$ (see figure 18 b), continues to decay. Thus, bi-stability is evident. The contours of axial vorticity at $z/H=0.5$ are illustrated in the inset of figure 19(a).

Figure 19.

The Rossby number is changed from (a) $\mathit{Ro}=0.0325\rightarrow \mathit{Ro}=0.02375$ and (b) $\mathit{Ro}=0.02375\rightarrow \mathit{Ro}=0.0325$ with $\mathit{Ek}=8\times 10^{-5}$ . Energies of various non-zero wavenumbers over time with the change in $\mathit{Ro}$ occurring instantaneously at $t=2350$ and $t=3500$ , respectively, which have been marked by a vertical dashed line. The insets show the axial vorticity contours extracted at $z/H=0.5$ at $t=3750$ and $t=9150$ , respectively. Contour levels are as in figure 5.

The energy time series for the reverse case of increasing the Rossby number from $\mathit{Ro}=0.02375$ to $\mathit{Ro}=0.0325$ at $\mathit{Ek}=8\times 10^{-5}$ is shown in figure 19(b). The change in flow conditions is marked by a vertical dashed line at $t=3500$ . The increase in $\mathit{Ro}$ causes a sharp increase in energy for all of the azimuthal wavenumbers. The flow continues to sustain the wavenumber 5 structure over a long period of time with energies from $k=4$ gradually decreasing. At approximately $t=8000$ , the energies in $k=4$ increase and eventually become dominant at $t=8450$ . Thus, the stable configuration observed is the same as that obtained from initializing the flow from the steady-state axisymmetric base flow with a flow condition of $\mathit{Ro}=0.0325$ and $\mathit{Ek}=8\times 10^{-5}$ .

This pair of flow conditions has demonstrated that multiple stable saturated states may exist at a single flow condition, depending on how it is approached. This hysteretic behaviour is in agreement with the strong hysteresis observed in previous studies of similar shear layer systems (Früh & Read Reference Früh and Read1999; Bergeron et al. Reference Bergeron, Coutsias, Lynov and Nielsen2000; Hollerbach et al. Reference Hollerbach, Futterer, More and Egbers2004; Aguiar et al. Reference Aguiar, Read, Wordsworth, Salter and Hiro Yamazaki2010).

3.3.3. Bifurcation analysis

The hysteretic nature of the three different linear instability transitions, namely modes I, II and III (see Vo et al. Reference Vo, Montabone and Sheard2014), has been investigated. The Stuart–Landau model (§ 2.3) has been applied to the most unstable linear wavenumber and its harmonics. The nonlinear behaviour is determined through the sign of the $l$ parameter. The restriction of simulating a particular wavelength and its harmonics is achieved numerically through a truncation of the azimuthal range of the domain to exactly fit the wavelength of interest. As a consequence of the axisymmetry of the domain, only the considered wavenumber and its harmonics are able to fit perfectly in the truncated domain. Each numerical study has been initiated with the axisymmetric base flow solution seeded with the most unstable eigenmode scaled to very small amplitudes.

The transitional behaviour of the mode I instability of $\mathit{Ro}=0.05$ and $\mathit{Ek}=3\times 10^{-4}$ is investigated. A plot of the growth rate against wavenumber is shown in figure 5(a), which illustrates the existence of only the mode I instability and is most unstable to a wavenumber 5 configuration. The mode transition is determined to be supercritical, which can be deduced from figure 20. Initially, the amplitude of the wavenumber 5 instability is small and grows exponentially in time until it plateaus at approximately $t=440$ . This exponential trend is representative of the linear regime, which is required to obtain an accurate gradient near the vertical axis in figure 20(b). The gradient at the vertical axis corresponds to the $l$ parameter in the Stuart–Landau equation. The negative slope (having $l=4.55\times 10^{-6}$ ) indicates that the mode evolution is supercritical. The $\text{d}\log |{\it\Lambda}|/\text{d}t$ curve approaches the horizontal axis as the mode saturates. The intercept of the vertical axis, which represents the growth rate of the mode in the linear regime, has a value of 0.0187, identical to that predicted by linear stability analysis (see figure 5 a).

Figure 20.

Flow conditions described by $\mathit{Ro}=0.05$ and $\mathit{Ek}=3\times 10^{-4}$ . (a) Amplitude of the most unstable isolated mode corresponding to wavenumber 5 against time. (b) The rate of change in amplitude over time against the square of the amplitude. A negative linear gradient at the vertical axis indicates that the transition is supercritical.

A range of other flow conditions that are dominated by the mode I instability have also been investigated. The same Stuart–Landau model consistently finds the transition of the mode I instability to be supercritical. This confirms and extends the experimentally determined supercritical behaviour detected for low-wavenumber instability by Früh & Read (Reference Früh and Read1999), van de Konijnenberg et al. (Reference van de Konijnenberg, Nielsen, Juul Rasmussen and Stenum1999) and Bergeron et al. (Reference Bergeron, Coutsias, Lynov and Nielsen2000). In addition, the Stuart–Landau model has been applied to a variety of flow conditions that encourage the growth of the mode II and III instability modes. The $l$ parameter has been determined to be positive in all flow cases, which suggests that the weakly nonlinear mode evolution behaviour in this type of rotating flow is supercritical.