Taylor rolls on tour: slow drift of turbulent large-scale structures in flows with continuous symmetries

Abstract

In Rayleigh–Bénard convection and Taylor–Couette flow cellular patterns emerge at the onset of instability and persist as large-scale coherent structures in the turbulent regime. Their long-term dynamics has been thoroughly characterised and modelled for the case of turbulent convection, whereas turbulent Taylor rolls have received much less attention. Here we present direct numerical simulations of axisymmetric Taylor–Couette flow in the corotating regime and reveal a transition to spatio–temporal chaos as the system size increases. Beyond this transition, Taylor rolls suddenly undergo erratic drifts evolving on a very slow time scale. We estimate an effective diffusion coefficient for the drift and compare the dynamics with analogous motions in Rayleigh–Bénard convection and Poiseuille flow, suggesting that this spontaneous diffusive displacement of large coherent structures is common among different types of wall-bounded turbulent flows.

1. Introduction

The smallest eddies in turbulent flows are dictated by the fluid’s kinematic viscosity ( $\nu$ ) and dissipation, whereas the largest ones are shaped by the flow geometry, boundary conditions and source of driving. Very large coherent motions in the flow field (superstructures) carry a substantial part of the kinetic energy, which increases as the Reynolds number (Re) increases (Smits et al. 2011). Understanding their role in transport and mixing is an active field of research, with many open questions relevant for predicting and modelling environmental fluid flows (Dauxois et al. 2021). In systems with linear instabilities, such as Rayleigh–Bénard convection (RBC) and Taylor–Couette flow (TCF), the origin of turbulent superstructures can be traced down to the onset of hydrodynamic instability. For TCF, Taylor rolls emerge from the primary instability of circular Couette flow (Taylor 1923), and then undergo a sequence of bifurcations (Coles 1965; Fenstermacher et al. 1979; Prigent et al. 2006), which increases the spatio–temporal complexity of the flow as Re increases (Feldmann et al. 2023). Seemingly, they persist in the form of turbulent Taylor rolls up to the highest Re investigated to date (Lathrop et al. 1992; Ravelet et al. 2010; Huisman et al. 2014; Ostilla-Mónico et al. 2016a ; Sacco et al. 2019). We refer to Grossmann et al. (2016) for a recent review of turbulent Taylor–Couette flow.

Turbulent Taylor rolls have been uncovered in experiments and direct numerical simulations (DNS) by taking temporal averages of the velocity field (Dong 2007; Ravelet et al. 2010; Ostilla-Mónico et al. 2013; Huisman et al. 2014; Ostilla-Mónico et al. 2016a , b ), based upon the assumptions that the rolls remain stable and do not travel in the axial direction ( $z$ ). In most laboratory experiments, the cylinders are bounded by solid end walls, whereas in DNS axially periodic boundary conditions (BC) are usually employed. This renders $z$ homogeneous and enables the usage of short computational domains, which typically accommodate one or two pairs of Taylor rolls (Dong 2007; Brauckmann & Eckhardt 2013; Ostilla-Mónico et al. 2016a , b ; Sacco et al. 2019). In these computations, typical observation times do not exceed a few hundred convective time units. This raises the question of whether Taylor rolls remain stable and stationary up to arbitrarily long times. In cylindrical RBC cells, for example, the characteristic large-scale circulations (LSC) are known to undergo spontaneous diffusive meandering in the naturally homogeneous (i.e. azimuthal) direction (Sun et al. 2005; Brown & Ahlers 2006; Xi et al. 2006; Brown & Ahlers 2008). Slow dynamics of the LSC – clearly separated from the time scale of the turbulent fluctuations – was also shown more recently in doubly periodic DNS of rectangular RBC at Rayleigh numbers up to ${R\hspace {-0.1em}a} = 10^7$ (Pandey et al. 2018). Similarly, Kreilos et al. (2014) found slow spanwise displacements of velocity streaks in turbulent boundary layer and Poiseuille flows.

In this paper, we reveal a transition giving rise to slow, large-scale dynamics in axisymmetric TCF. Beyond a critical domain size, spatio–temporal chaos emerges and the Taylor rolls undergo erratic drifts in $z$ . Compared with the cylinder rotation, the drift speeds are small, but large roll displacements can occur on a slow time scale. We show that the drift statistics are consistent with a Wiener process and characterise the motion with an effective diffusion coefficient of the order of $\nu$ .

2. Computer experiments

We perform axisymmetric DNS of corotating Taylor–Couette flow (table 1) with periodic BC in $z$ for moderate Reynolds numbers, allowing for both large computational domains and long integration times at affordable computing costs. We integrate the incompressible Navier–Stokes equations (subject to no-slip BC in $r$ ) forward in time ( $t$ ) using our pseudo-spectral DNS code nsCouette (López et al. 2020). The equations are formulated in cylindrical coordinates ( $r,\theta ,z$ ) and rendered dimensionless using $d$ , ${\nu }/{d}$ and ${d^2}/{\nu }$ (i.e. the characteristic viscous time scale of the problem), as unit length, unit speed and unit time, respectively. In axisymmetric DNS, the $\theta$ -dependence is dropped, but all three velocity components are computed.

Table 1. Taylor-roll drift dynamics for $\Gamma = 24$ and $\eta = 0.99$ . Listed are control parameters ( ${R\hspace {-0.1em}a}, {R\hspace {-0.1em}e_{i}}, {R\hspace {-0.1em}e_{o}}, {R\hspace {-0.1em}e_{\hspace {-0.1em}s}}, {R_{\hspace {-0.1em}\Omega }}$ ), response parameters ( $N\hspace {-0.20em}u_{\hspace {-0.1em}s}$ , $N\hspace {-0.20em}u$ , $R\hspace {-0.1em}e_{\tau }$ ), standard deviations $V(\alpha )^{{1}/{2}}$ of the drift speed, $\alpha =v_R$ , the net axial flux, $\alpha =\overline {u}_z$ , and the effective diffusion coefficient, $D_R$ , estimated as in figure 2. NoFX stands for no axial flux ( ${\overline u}_z=0$ ) enforced in the simulation.

Motivated by an exact analogy (Eckhardt et al. 2020) between two-dimensional RBC and axisymmetric TCF in the narrow-gap limit ( $\eta ={r_i}/{r_o}\rightarrow {1}$ ), we set $\eta = 0.99$ and vary ${R\hspace {-0.1em}e_{\hspace {-0.1em}s}} = {Ud}/{\nu }$ , ${R_{\hspace {-0.1em}\Omega }} = {2d\Omega }/{U}$ and $\Gamma = {L_z}/{d}$ . Here, $d=r_o-r_i$ , is the gap width between the inner ( $i$ ) and the outer ( $o$ ) cylinder with radius $r_{i/o}$ , $L_z$ is their axial length, $U = u_{\theta ,i}-\eta \,u_{\theta ,o}$ and $\Omega = {u_{\theta ,o}}/{r_o}$ , where $u_{\theta ,{i}/{o}}$ denotes the azimuthal speed of the cylinders. An important response parameter is the Nusselt number ( $N\hspace {-0.20em}u_{\hspace {-0.1em}s}$ ), which quantifies the transport of angular momentum across the fluid layer (Eckhardt et al. 2007; Brauckmann et al. 2016; Eckhardt et al. 2020). According to the exact Navier–Stokes mapping of Eckhardt et al. (2020), flows are mathematically identical as long as ${R\hspace {-0.1em}a}={R\hspace {-0.1em}e_{\hspace {-0.1em}s}}^2\,{R_{\hspace {-0.1em}\Omega }} (1-{R_{\hspace {-0.1em}\Omega }} )= \text {const.}$ , with ${N\hspace {-0.20em}u} = 1+{({N\hspace {-0.20em}u_{\hspace {-0.1em}s}}-1)}/{(1-{R_{\hspace {-0.1em}\Omega }})} $ and ${N\hspace {-0.20em}u}$ being the usual Nusselt number in RBC.

In a first set of DNS (compiled later in figure 3 a), we fix all parameters but $\Gamma$ to investigate the onset of spatio–temporal chaos with respect to the lateral domain size. In a second set (compiled in table 1), we fix $\Gamma$ and explore the effect of shear ( $\textit{Re}_{s\hspace {-0.1em}}$ ) and rotation ( $R_\Omega$ ). The initial conditions are chosen to trigger the desired number of Taylor rolls ( $N_R$ ) necessary to maintain their aspect ratio constant throughout all DNS runs ( ${N_R}/{\Gamma } = 1$ ). This is important because the dynamics is known to depend on ${N_R}/{\Gamma }$ (Ostilla-Mónico et al. 2015, 2016a ; Wang et al. 2020; Zwirner et al. 2020). The highest friction Reynolds number ( ${R\hspace {-0.1em}e_{\tau }} = {u_\tau d}/{\nu }$ , where $u_\tau$ is the friction velocity at the cylinder walls) measured in all DNS is 408 (table 1). The spatial resolution in terms of wall units (i.e. based on $Re_{\tau }$ and denoted by ${}^+$ ) is at least $0.07 \leqslant \Delta r^+\leqslant 4.03$ and $\Delta z^+= 4.89$ , which is state of the art in DNS of wall-bounded turbulence (Ostilla-Mónico et al. 2016a , b ; Feldmann et al. 2021).

3. Drift dynamics

Figure 1. Spatio–temporal dynamics of Taylor rolls for different domain sizes ( $\Gamma$ ). Shown are contours of the wall-normal velocity component ( $u_r$ ) at midgap position extracted from DNS ( ${R\hspace {-0.1em}e_{\hspace {-0.1em}s}}= 9475$ and ${R_{\hspace {-0.1em}\Omega }}=0.14$ in all cases). Arrows represent 2000 convective time units. (a) Stationary Taylor rolls in a small (subcritical) domain. (b–d) Axially drifting Taylor rolls in larger (supercritical) domains exhibiting large excursions on a slow time scale. The black line ( $z_R$ ) plotted on top of the $u_r$ space–time data in (d) represents the temporal evolution of the phase angle of the dominant Fourier mode (here $k_z = 12$ , corresponding to 12 pairs of rolls). It serves as a proxy for the collective axial displacement of the entire stack of rolls. (e) Temporal evolution of the modal kinetic energy, $\langle E_{k_{z}}\rangle _r$ , contained in mode number $k_z$ for the same case as in (d); angled brackets denote averaging in the radial direction.

Figure 2. Time series of the axial displacement of Taylor rolls ( $z_R$ ) for different $\Gamma$ at ${R\hspace {-0.1em}e_{\hspace {-0.1em}s}}= 9475$ , ${R_{\hspace {-0.1em}\Omega }}= 0.14$ . (a) Chaotic small-scale oscillations about a fixed axial location in short (subcritical) domains ( $\Gamma \in [4,8]$ ). The simulations span more than 200 viscous time units ( ${d^2}/{\nu }$ ), without reflecting any change of this behaviour; except for the initial transients in the first two viscous time units of the simulations, which we discard for all further analyses in all cases. (b) Huge erratic axial drifts in a larger (supercritical) domain close to the critical point ( $\Gamma = 10$ ). (c) Large erratic axial drifts with qualitatively similar dynamics in larger (supercritical) domains ( $\Gamma \in [16,24,48]$ ). (d) Temporal evolution of the displacement variance, $V(z_R)$ , for the data sets shown in (c). The slope of the corresponding linear fit (broken lines) serves as an estimate for the effective diffusion coefficient for the drift dynamics.

Small domains restrict the dynamics of the system, resulting in nearly stationary rolls. This is apparent from the space–time diagram of the wall-normal velocity ( $u_r$ ) for $\Gamma = 8$ (figure 1 a). If we now enlarge the domain ( $\Gamma \geqslant 10$ , ${N_R}/{\Gamma }=1$ ), the Taylor rolls undergo large, erratic, collective drifts in $z$ that evolve on a slow time scale (figure 1 b–d). In a domain with $N_R = 24$ rolls, for example, the most energetic axial mode is always $k_z = 12$ (figure 1 e), confirming that the space–time representation of $u_r$ is indeed a robust way to identify Taylor rolls and to track their dynamics. Every few viscous time units (e.g. at $t\approx 17.5$ ), the competition with neighbouring modes (here $k_z\in \{11,13\}$ ) represents rare attempts to switch to another state with 11 or 13 pairs of rolls. These attempts, however, remain unsuccessful in all our simulations.

To analyse the drift dynamics quantitatively, we compute axial Fourier spectra of $u_r$ (in particular the $u_r$ midgap space–time data as exemplarily shown in figure 1 d) and use the phase angle information of the dominant axial mode (here $k_z= 12$ ) to approximate the collective displacement of the Taylor rolls ( $z_R$ ), as done earlier by Sacco et al. (2019). The temporal evolution of $z_R$ (figure 1 d) aligns well with $u_r$ , thereby confirming the suitability of $z_R$ to quantify the collective axial drift of the Taylor rolls. For $\Gamma \leqslant 8$ , the rolls first undergo slow transient drifts in the beginning of the simulation and then ultimately oscillate with tiny amplitudes and high frequencies about a statistically steady state (figure 2 a). This fast dynamics of the rolls was reported earlier for three-dimensional turbulent TCF in a domain accommodating one pair of rolls (Sacco et al. 2019). By contrast, for $\Gamma = 10$ , the rolls wander more than $ 100d$ before turning back for the first time, and continue moving erratically thereafter (figure 2 b). With further increasing $\Gamma$ , these excursions persist but become less extreme (figure 2 c).

We quantify the Taylor-roll motion statistically by computing the variance of the axial displacement, $V(z_R, t) = \langle z_R^{2}\rangle _{t} - \langle z_R\rangle ^{2}_t$ , where angled brackets denote temporal averaging up to time $t$ . For $\Gamma \leqslant 8$ , the fast dynamics of the rolls is centred around a fixed location and $V(z_R)$ quickly saturates to a constant, in agreement with Sacco et al. (2019), who reported Gaussian fluctuations of $z_R$ with constant variance. By contrast, for $\Gamma \geqslant 10$ , $V(z_R)$ grows approximately linearly with time, as in a Wiener process (figure 2 d). The drift of the rolls can thus be characterised with an effective diffusion coefficient, $D_R$ , as the slope of a linear fit to the variance. For all our analyses, we use time series of at least 200 viscous time units and we generally discard the first $2 {d^2}/{\nu }$ to exclude initial transients.

4. Transition to large erratic drift dynamics

Figure 3. Transition to large, erratic drift dynamics of Taylor rolls. (a) Beyond a critical domain size (grey line, $\Gamma _c= 9.99$ from a power-law fit to the data), the Taylor-roll drift can be characterised by an effective diffusion coefficient, $D_R$ , as shown in figure 2(d); here for $Ra = 10^7$, $Re_s = 9475$ and $R_\Omega = 0.14$ . Note, that $\Gamma$ must take even integer values in order to maintain a unit aspect ratio of the Taylor rolls. (b) For a fixed domain size ( $\Gamma = 24$ ), the Taylor-roll drift starts at a critical Rayleigh number, ${R\hspace {-0.1em}a}_c\approx 6 \times 10^5$ , and becomes more pronounced as Ra increases. The grey region denotes the uncertainty in determining ${R\hspace {-0.1em}a}_c$ ; it spans the interval between the last run with $D_R=0$ and the first run with $D_R\gt 0$ .

Figure 4. Transition from temporal to spatio–temporal chaos for increasing $\Gamma$ . Shown are premultiplied energy spectra, $\varepsilon (\lambda )$ , at ${R\hspace {-0.1em}e_{\hspace {-0.1em}s}}= 9475$ and ${R_{\hspace {-0.1em}\Omega }}= 0.14$ from a subcritical ( $\Gamma = 8$ ) and a supercritical ( $\Gamma = 24$ ) domain. Spectra for other $\Gamma$ look very similar and are thus not shown here. (a) Premultiplied spectra of the modal kinetic energy (as, for example, in figure 1 e) versus axial wavelengths, $\lambda _z={2\pi }/{\kappa _z}$ , where $\kappa _z$ is the axial wavenumber and angled brackets denote temporal averaging. (b) Premultiplied temporal Fourier spectra of the modal kinetic energy (as, for example, in figure 1 e) for the dominant mode (here, for example, for $k_z= 4$ in the case of $\Gamma = 8$ ) versus temporal wavelengths, $\lambda _t={2\pi }/{\omega }$ , where $\omega$ is the angular frequency. FFT means fast Fourier transform.

The dependence of the diffusion coefficient $D_R$ on the domain size $\Gamma$ is shown in figure 3 for a fixed Rayleigh number ( ${R\hspace {-0.1em}a}=10^7$ ). Our data suggest a divergence of $D_R$ near a critical point ( $\Gamma _c= 9.99$ ) followed by a monotonic decrease as $\Gamma$ increases. To examine the nature of this transition, we compare spatial and temporal Fourier spectra from sub- and supercritical domains (figure 4). For $\Gamma = 8$ , the axial spectrum of $u_r$ presents discrete peaks at wavelength $\lambda _z = 2d$ and its harmonics only (figure 4 a). This implies that the flow state consists of four perfectly synchronised copies of one pair of Taylor rolls. In fact, when comparing this state to those obtained for $\Gamma \in \{2,4\}$ , the same Nusselt numbers, $\langle {N\hspace {-0.20em}u_{\hspace {-0.1em}s}}\rangle = 12.82474\pm 0.00002$ , and spectra (not shown here) are recovered. By contrast, the states obtained for $\Gamma \geqslant 10$ exhibit continuous spatial spectra (e.g. for $\Gamma = 24$ in figure 4 a), indicating spatial defects in the roll structure, i.e. there are no identical rolls in the entire stack. A similar transition to spatio–temporal chaos was reported before for axially oscillated (Avila et al. 2007) and hydromagnetic (Guseva et al. 2015) Taylor–Couette flow. For both, however, no slow large-scale drift of the roll patterns was reported, possibly due to much shorter simulation times.

The transition to spatio–temporal chaos also alters the temporal spectra (figure 4 b). For $\Gamma \lt \Gamma _c$ , the temporal spectrum is continuous, indicating temporal chaos, and exhibits a peak at $2\times 10^{-3}{d^2}/{\nu }$ (i.e. approximately 20 convective time units) before falling sharply. This peak is associated with the fast, small-displacement dynamics with $D_R=0$ and Gaussian statistics reported by Sacco et al. (2019). For $\Gamma \gt \Gamma _c$ , the temporal spectrum features an additional broad peak at approximately $ 0.1 {d^2}/{\nu }$ , corresponding to the slow drift dynamics characterised by a Wiener process (i.e. $D_R \gt 0$ and linearly increasing variance). The transition to spatio–temporal chaos is also reflected in the mean Nusselt number, but only in the third digit; $\langle {N\hspace {-0.20em}u_{\hspace {-0.1em}s}}\rangle = 12.76\pm 0.04$ for all $\Gamma \geqslant 10$ . We note that while much effort has been dedicated to remove drifts in the analysis of turbulent dynamics of wall-bounded flows (Willis et al. 2013; Budanur et al. 2015), here the onset of spatio–temporal chaos appears intrinsically linked to the slow, erratic drift dynamics.

5. Dependence of the drift dynamics on the flow configuration

We exploit the analogy between TCF and RBC (Bradshaw 1969; Veronis 1970; Prigent et al. 2006; Eckhardt et al. 2007, 2020) to demonstrate that the drift dynamics is found throughout the centrifugally unstable corotating regime. According to the exact Navier–Stokes mapping of Eckhardt et al. (2020), axisymmetric TCF systems in the narrow-gap limit ( $\eta \rightarrow 1$ ) are exactly identical if ${R\hspace {-0.1em}a} = {R\hspace {-0.1em}e_{\hspace {-0.1em}s}}^{2}\,{R_{\hspace {-0.1em}\Omega }} (1-{R_{\hspace {-0.1em}\Omega }} ) = \text {const.}$ , i.e. the large-scale drift dynamics is identical as well. Indeed, for moderate outer cylinder rotation ( ${R_{\hspace {-0.1em}\Omega }}\in \{0.14, 0.30\}$ ), the drift statistics are similar (table 1, compare rows 3 and 4) and the same is true for the corrected Nusselt number, ${N\hspace {-0.20em}u} = 1 + {({N\hspace {-0.20em}u_{\hspace {-0.1em}s}}-1)}/{(1-{R_{\hspace {-0.1em}\Omega }})}$ . We attribute the small deviations to small, yet finite, curvature effects ( $\eta = 0.99 \lt 1$ ), which are not included in the analogy. For very slow outer cylinder rotation ( ${R_{\hspace {-0.1em}\Omega }} = 0.05$ ), the drift statistics deviate noticeably (table 1, compare rows 3, 4 and 6). This is as expected because the exact analogy breaks down in the limit of a stationary outer cylinder ( ${R_{\hspace {-0.1em}\Omega }}= 0$ ).

Next, we fix ${R_{\hspace {-0.1em}\Omega }}= 0.14$ and $\Gamma = 24$ , and vary $R\hspace {-0.1em}e_{\hspace {-0.1em}s}$ , thereby varying the Rayleigh number (figure 3 b). At low Ra, the rolls are stationary, as observed for low $\Gamma$ in § 3. Similarly, at a critical point ( ${R\hspace {-0.1em}a}_c\approx 6\times 10^5$ ), the rolls begin to drift in the axial direction. As Ra is increased beyond this critical point, the diffusion coefficient of the drift increases. This is contrary to the effect of increasing $\Gamma$ , but consistent with RBC experiments (Xi et al. 2006), where the rate of erratic rotations of the LSC increases tenfold as Ra increases from $10^9$ to $10^{10}$ . A two-dimensional parametric study of the combined effect of $\Gamma$ and $R\hspace {-0.1em}a$ would be interesting, but expensive and thus out of the scope of this work.

The axial drift of the Taylor rolls is associated with a net mass flux in $z$ with mean speed $\overline {u}_z$ . This flux is strongly correlated to the drift speed ( $v_R = \dot{z}_R = {\partial z_R}/{\partial t}$ ) of the rolls (figure 5 a), and raises the question of whether the roll displacement causes the net axial flux or vice versa. The fact that $v_R$ is approximately $500{d}/{U}$ ahead of $\overline {u}_z$ suggests the former (figure 5 b). We probe this hypothesis by enforcing $\overline {u}_z= 0$ , as in laboratory experiments of TCF with end walls. In our DNS with axially periodic BC, we enforce $\overline {u}_z =0$ by imposing an appropriate adverse pressure gradient at each time step. This technique was previously applied to successfully compare axially periodic simulations to lab experiments for Taylor–Couette flow in the counter-rotating regime (Edwards et al. 1991), with radial heating (Ali & Weidman 1990) and with axially oscillating inner cylinder (Marques & Lopez 1997; Avila et al. 2007).

Figure 5. Taylor-roll dynamics with and without axial flux constraint for otherwise identical parameters ( ${R\hspace {-0.1em}e_{\hspace {-0.1em}s}} = 9475$ , ${R_{\hspace {-0.1em}\Omega }} = 0.14$ , $\Gamma = 24$ ). (a) Time series of the drift speed of the Taylor rolls ( $v_R = \dot{z}_R = {\partial z_R}/{\partial t}$ ) and the net axial flux ( $\overline {u}_z$ ) for the case in figure 1(d). (b) Close-up to the data in (a). As a visual reference, the black dash represents 500 convective time units. (c) Time series from a simulation with no axial flux (NoFX, i. e. $\overline {u}_z =0$ ) enforced.

As a result of suppressing the axial mass flux, $v_R$ , $V(v_R)$ and $D_R$ are substantially reduced (figure 5 c, table 1, compare rows 4 and 5), but when rescaled, the drift dynamics remains qualitatively unaltered (figure 6 a). Specifically, $V(z_R)$ still increases linearly with time (figure 6 e), although at a slower pace.

6. Discussion

Figure 6. Comparison of large-scale drifts ( $z_R$ ) among different set-ups and fluid systems. (a) Taylor-roll drift in Taylor–Couette flow with no axial flux (NoFX, $\bar {u}_z = 0$ ); same run as in figure 5(c). Azimuthal meandering of a single convection roll in cylindrical Rayleigh–Bénard convection for laboratory experiments lasting 11 days ( ${R\hspace {-0.1em}a} = 10^{10}$ , Brown & Ahlers (2006)) and 33 days ( ${R\hspace {-0.1em}a}=10^9$ , Xi et al. (2006)), respectively. Spanwise streak displacement in Poiseuille flow DNS (Kreilos et al. 2014). (b–e): Corresponding displacement variance, $V(z_R)$ , including linear fits (broken lines) to demonstrate an approximate linear growth with time and to estimate an effective diffusion coefficient ( $D_R$ ).

We have shown that axisymmetric Taylor–Couette flows exhibit a transition to spatio–temporally chaotic Taylor rolls that drift erratically in the axial direction. At long time scales, the drift motion is diffusive and can lead to very large displacements. Future works should clarify whether this dynamics persists in three-dimensional TCF simulations and in experiments with end walls. We note that even with walls, flow patterns can drift in $z$ with phase being created/annihilated near the walls (Heise et al. 2008).

In figure 6 the roll displacements extracted from our DNS are compared to the azimuthal meandering of the large-scale circulation in circular Rayleigh–Bénard convection (Brown & Ahlers 2006; Xi et al. 2006) and spanwise streak displacements in Poiseuille flow (Kreilos et al. 2014). For the sake of comparison, we converted the rotation angle of the LSC to a length scale as $z_R(t)=R\,\theta(t)$ using the radius ( $R$ ) of the RBC cell. Additionally, we rescaled all drift signals to the viscous time unit, which is also the relevant one of the exact analogy (Eckhardt et al. 2020). The qualitative agreement is remarkable and suggests that this slow dynamics might be inherent to large-scale motions in many fluid systems. However, longer RBC and Poiseuille flow runs would be needed to confirm the Wiener statistics found here for Taylor–Couette flow, and to estimate the corresponding diffusion coefficients (figure 6 b–e). Additional statistical analyses and modelling strategies previously applied to three-dimensional RBC (Brown & Ahlers 2007, 2008) could help elucidate further aspects of the drift dynamics reported here and deepen this comparison. A particularly intriguing question is whether the rare cessation events reported for Rayleigh–Bénard convection have a counterpart in Taylor–Couette flow. This would correspond to a sudden decay and resurgence of Taylor rolls at a different location (possibly with a change of the number of rolls). While such events were not observed here, our data suggests that they may occur; see, for example, the event at $t\approx 17.5$ in figure 1 (e).

We stress that the shortest time series considered here (figure 2) correspond to $200{R\hspace {-0.1em}e_{\hspace {-0.1em}s}} = {\mathcal {O}(10^6 )}$ convective time units and to $ 200\sqrt {{R\hspace {-0.1em}a}} = {\mathcal {O}(10^5 )}$ free-fall time units in RBC ( ${R\hspace {-0.1em}a} = 10^7$ ). These observation times are comparable to those used to characterise large-scale states in RBC (Pandey et al. 2018; Wang et al. 2020), but they are several orders of magnitudes longer compared with typical observation times in high-Re TCF studies (Brauckmann & Eckhardt 2013; Huisman et al. 2014; Ostilla-Mónico et al. 2016a , b ; Sacco et al. 2019). In the Taylor–Couette apparatus of Huisman et al. (2014), for example, this would correspond to a measurement time of two weeks.