3.1. ECSs and state space geometry

It is easier to understand the state space geometry for this problem and the structure of the chaotic set associated with turbulent flow using coordinates (observables) that are invariant under rotation around the axis. We found the following coordinates to provide a convenient projection: the normalized energy

(3.1)\begin{equation} \mathcal{E} = \frac{1}{Re^2V}\int_{V} {\boldsymbol u}^2 \, \textrm{d}V, \end{equation}

rate of energy dissipation

(3.2)\begin{equation} \mathcal{D} = \frac{1}{Re^2V}\int_{V} \boldsymbol\omega^2 \, \textrm{d}V \end{equation}

and helicity

(3.3)\begin{equation} \mathcal{H} = \frac{1}{Re^2V} \int_V {\boldsymbol u}\boldsymbol{\cdot}\boldsymbol{\omega} \, \textrm{d}V. \end{equation}

Here $Re=|Re_i-Re_o|/2$ is the characteristic scale for the velocity, $V$ is the volume of the flow domain and $\boldsymbol {\omega }=\boldsymbol {\nabla }\times \boldsymbol {u}$ is the vorticity. Note that $\mathcal {E}$ and $\mathcal {D}$ are invariant under both $R_\phi$ and $K_z$, while $\mathcal {H}$ is invariant under $R_\phi$ and changes sign under $K_z$. The running average of helicity for the turbulent flow initialized as described in the previous section is shown in figure 2 as a blue curve. It is clear that this turbulent trajectory (we will refer to it as ${\boldsymbol u}^a(t)$) explores distinct but connected regions of state space. For $1\lesssim t\lesssim 40$ and $53\lesssim t\lesssim 57$, ${\boldsymbol u}^a(t)$ is confined to the part of the chaotic set (which we will refer to as lobe 1) that is centred at $\mathcal {H}=0$. For $41\lesssim t\lesssim 53$, ${\boldsymbol u}^a(t)$ is confined to the part of the chaotic set (which we will refer to as lobe 2) that is centred at $\mathcal {H}=\mathcal {H}_2\approx -0.003$. The chaotic set has several other lobes, as will be discussed shortly.

Figure 2.

Helicity for ${\boldsymbol u}^a(t)$ (blue), ${\boldsymbol u}^b(t)$ (red), ${\boldsymbol u}^c(t)$ (green). Shown is the running average taken over a window of one time unit. Note that $10$ time units are removed from the signal of ${\boldsymbol u}^c(t)$ at $t=40$, during which the flow remains in lobe 3. The mean helicity of each lobe is illustrated with black dashed lines.

The relation between these lobes and ECSs can be seen in a low-dimensional projection of the state space onto $\mathcal {E}$, $\mathcal {D}$ and $\mathcal {H}$ shown in figure 3. The portions of turbulent trajectory ${\boldsymbol u}^a(t)$ confined to lobe 1 (lobe 2) are shown as a blue (red) cloud of points, representing different snapshots of the flow. For each ECS ${\boldsymbol u}_n$ that is not reflection symmetric, both ${\boldsymbol u}_n$ and $K_z{\boldsymbol u}_n$ are shown in figure 3. Since all three coordinates are invariant with respect to rotations around the axis, a family of temporally periodic solutions corresponding to each $\mathrm {TW}_{}$, i.e. $R_\phi {\boldsymbol u}_n$ with $\phi \in [0,2{\rm \pi} )$, is mapped to a single point. Similarly, a family of temporally quasi-periodic solutions corresponding to each $\mathrm {RPO}_{}$ is mapped to a single closed curve. A large number of computed ECSs ($\mathrm {TW}_{01}$, $\mathrm {RPO}_{01}$–$\mathrm {RPO}_{12}$ and $\mathrm {RPO}_{19}$) are collocated with lobe 1, as illustrated in figure 4(a), but we could not find any ECSs collocated with lobe 2. Instead, all initial conditions in lobe 2 converged to ECSs that lie outside of both lobes 1 and 2. Indeed, there is no guarantee that Newton iterations converge to a solution that is close to the near recurrence used as the corresponding initial condition. At first glance, this appears to suggest that some of the ECSs we found are dynamically relevant and some are not.

Figure 3.

Three-dimensional projection of the infinite-dimensional state space (a) and the corresponding two-dimensional projections (b–d). The coordinates used are the energy $\mathcal {E}$, the rate of energy dissipation $\mathcal {D}$ and the helicity $\mathcal {H}$. The individual lobes of the chaotic set are shown as clouds of points with different colour: lobe 1 shown in blue; lobe 2 in red; lobe 3 in green. The computed $\mathrm {RPO}_{}$s ($\mathrm {TW}_{}$s) are shown as solid lines (filled circles) and their symmetry-related copies as dotted lines (open circles) of the same colour.

Figure 4.

Two-dimensional projection of (a) lobe 1 and (b) lobe 3. In each lobe, the chaotic set (grey point cloud) is in the background. The $\mathrm {RPO}_{}$s and $\mathrm {TW}_{}$s collocated with each lobe in this projection are overlaid as solid curves and filled circles, respectively.

Interestingly, most of the ECSs lying outside of lobes 1 and 2 were found to be grouped in the same region of the state space. This grouping suggests that these solutions may be a part of the skeleton underlying a different lobe of the same chaotic set or, perhaps, a different chaotic set. In fact, by using an $\mathrm {RPO}_{}$ lying in that region of the state space to initialize the flow, we computed a second turbulent trajectory, ${\boldsymbol u}^b(t)$, which remained confined to a region of the state space centred at $\mathcal {H}=\mathcal {H}_3\approx -0.013$ for 60 time units. The corresponding running average of helicity is shown in red in figure 2 and snapshots of ${\boldsymbol u}^{b}(t)$ are shown as a green point cloud in figure 3. Here $\mathrm {RPO}_{13}$–$\mathrm {RPO}_{18}$ and $\mathrm {RPO}_{20}$ as well as $\mathrm {TW}_{02}$ are collocated with this set, referred to as lobe 3 below, as figure 4(b) illustrates.

To determine whether lobe 3 is dynamically connected to lobes 1 and 2, we continued the turbulent trajectory ${\boldsymbol u}^a(t)$ further in time. The running average of helicity for a portion of this trajectory (we will refer to it as ${\boldsymbol u}^c(t)$) is shown in green in figure 2. It is clear that this trajectory visits lobes 1 and 2 as well as the symmetric (under reflection $K_z$) copies of lobes 2 and 3. This suggests that lobe 1 as well as lobes 2 and 3 and their symmetric copies are all dynamically connected. The turbulent trajectory ${\boldsymbol u}^c(t)$ will not be analysed in the remainder of this paper. We will simply note that, after several hundred time units, turbulent flow eventually escapes the chaotic set composed of the five lobes mentioned previously and settles on what appears to be a stable quasiperiodic state. Hence, in our system, turbulence is a long-lived transient, similar to what was found in the same geometry at a larger aspect ratio (Hochstrate et al. Reference Hochstrate, Abshagen, Avila, Will and Pfister2010).

In the subsequent discussion, we will refer to the portion of ${\boldsymbol u}^a(t)$ restricted to lobes 1 and 2 of the chaotic set as ${\boldsymbol u}^1(t)$ and ${\boldsymbol u}^2(t)$, respectively. We will also identify ${\boldsymbol u}^3(t)={\boldsymbol u}^b(t)$, since ${\boldsymbol u}^b(t)$ remains in lobe 3 for the entire interval over which it was computed. The characteristic flow fields in each lobe are compared in figure 5, where the velocity field associated with ${\boldsymbol u}^1(t)$, ${\boldsymbol u}^2(t)$ or ${\boldsymbol u}^3(t)$ has been averaged in both time and $\theta$. Inside each lobe, we find two cellular vortical structures. In lobe 1, the two vortices are symmetric. In lobes 2 and 3, one vortex is notably stronger than the other. This asymmetric flow structure illustrates the reflection symmetry breaking and explains the relative arrangement of the three lobes along the $\mathcal {H}$ axis in figure 3.

Figure 5.

Mean flow field in (a) lobe 1, (b) lobe 2 and (c) lobe 3. In this figure and all following two-dimensional slices, the arrows show the in-plane ($r$ and $z$) components of the flow and the colour indicates the out-of-plane ($\theta$) component.

At first glance, the spatial structure of the mean flow in different lobes appears to be similar to the structure of either single-cell (A1) or double-cell (N2) stationary cellular flows found for this (or similar) geometry in previous studies, which considered the case $Re_o=0$ (Cliffe Reference Cliffe1983; Pfister et al. Reference Pfister, Schmidt, Cliffe and Mullin1988; Furukawa et al. Reference Furukawa, Watanabe, Toya and Nakamura2002). Furthermore, N2 was found to be stable for $Re_i\lesssim 133$ and is replaced by A1 (or its symmetric copy) at a higher $Re_i$. The onset of time-dependence at $Re_i\approx 892$ does not appear to alter the structure of the flow dramatically (Marques & Lopéz Reference Marques and Lopéz2006). Weak counter-rotation stabilizes symmetric flows (such as N2) at the expense of the asymmetric ones (such as A1), but does not notably modify the structure of the flows either (Schulz et al. Reference Schulz, Pfister and Tavener2003).

The set-up considered here leads to a (nearly) symmetric flow in lobe 1 that has one qualitative difference with N2: the sign of the azimuthal vorticity $\omega _\theta$ is opposite to that found in previous studies. In our case, the radial velocity is the largest and is positive near the end-caps, as figure 5(a) illustrates. For N2, the radial velocity is the largest and is positive at the centre plane $z=0$; it represents a strong central jet of angular momentum (Marques & Lopéz Reference Marques and Lopéz2006; Altmeyer et al. Reference Altmeyer, Do, Marques and Lopez2012) which is not found in lobe 1. These differences are likely due to our choice of the outer cylinder rotating independently of the end-caps; in previous studies of TCF, the end-caps were assumed to rotate with the same angular velocity as the outer cylinder.

3.2. A measure of closeness

Figure 3 suggests that most of the ECSs we found are embedded into either lobe 1 or lobe 3 of the chaotic set and, therefore, are dynamically relevant. However, low-dimensional projections can be misleading. To determine that an ECS is truly dynamically relevant, we need to ensure

1. (i) that turbulent flow comes close to an ECS, at least occasionally, in the full state space;

2. (ii) that turbulent flow remains in the neighbourhood of the ECS over a characteristic time scale of the flow; and

3. (iii) that turbulent flow shadows (i.e. evolves in a manner similar to) the ECS over time, while inside the neighbourhood of the ECS.

We will investigate whether these three conditions are satisfied in the remainder of the paper but, first, we need to define what ‘close’ means in either the physical space or the corresponding high-dimensional state space. The term ECS reflects the visual similarity between (spatial) structures commonly observed in a turbulent flow and snapshots of (numerically) exact solutions of Navier–Stokes in the three-dimensional physical space. Visual similarity is, however, too qualitative to be of much use for the purposes of predicting dynamics, even on relatively short time scales; we need a more quantitative definition of similarity.

Similarity is a relative term; before it is defined, we need to determine a yard stick for what is considered dissimilar. For parameters considered here, turbulent TCF ${\boldsymbol u}(t)$ is characterized by a fluctuation $\tilde {\boldsymbol u}(t)={\boldsymbol u}(t)-\bar {\boldsymbol u}$ about an axially symmetric mean flow $\bar {\boldsymbol u}=\langle {\boldsymbol u}(t)\rangle _{t}$ that is small compared with this mean flow, i.e. $\|\tilde {\boldsymbol u}(t)\|\ll \|\bar {\boldsymbol u}\|$. Hence, the characteristic magnitude $\sigma =\langle \|\tilde {\boldsymbol u}(t)\|\rangle _{t}$ of the fluctuations about the mean flow – effectively the ‘radius’ of the chaotic set – is a more appropriate scale for the dissimilarity in the physical space, or distance in the state space, between different flow states than the magnitude $\|\bar {\boldsymbol u}\|$ of the mean flow.

Since dynamics partition the chaotic set into three different lobes, there are many different ways to define $\sigma$. To avoid ambiguity, we define it using a turbulent trajectory that lies entirely inside lobe 1. Fortunately, the diameters of all three lobes are quite similar. Next we define the (normalized) distance from an arbitrary point ${\boldsymbol u}$ in the state space to the family of solutions $R_\phi {\boldsymbol u}_n(\tau )$,

(3.4)\begin{equation} D_n({\boldsymbol u}) = \min_{\phi, \tau} \sigma^{{-}1}\| {\boldsymbol u} - R_\phi{\boldsymbol u}_n(\tau) \|, \end{equation}

where $\tau$ defines the temporal phase along the $\mathrm {RPO}_{}$ (this dependence is omitted for $\mathrm {TW}_{}$s for which time evolution is equivalent to rotation). The distance

(3.5)\begin{equation} D_n^j(t) = D_n({\boldsymbol u}^j(t)) \end{equation}

measures how similar (close in the state space) the turbulent flow field snapshot ${\boldsymbol u}^j(t)$ is to the family of solutions $R_\phi {\boldsymbol u}_n(\tau )$. We will use $j=a,b$ when referring to the entire turbulent trajectories ${\boldsymbol u}^a(t)$ and ${\boldsymbol u}^b(t)$ and $j=1,2,3$ when referring to the portion of those trajectories confined to lobes 1, 2 or 3. We will drop the indices when this does not cause confusion. The distances from both turbulent trajectories, ${\boldsymbol u}^a(t)$ and ${\boldsymbol u}^b(t)$, to various ECSs are shown in figure 6, with the minimal values listed in table 1. The lengths of both turbulent trajectories correspond to $O(10^4)$ periods of the shortest $\mathrm {RPO}_{}$ we found, so they are likely long enough to sample a significant fraction of all three lobes.

Figure 6.

(a) The distance $D^a_n(t)$ between the turbulent trajectory ${\boldsymbol u}^a(t)$ confined to both lobes 1 and 2 and every ECS ${\boldsymbol u}_n$. All ECSs except for $\mathrm {RPO}_{05}$ and $\mathrm {RPO}_{15}$ are shown in grey. (b) The distance $D^b_n(t)$ between the turbulent trajectory ${\boldsymbol u}^b(t)$ inside lobe 3 and ECSs collocated with this lobe. All ECSs except for $\mathrm {RPO}_{15}$ are shown in grey. The dashed line indicates our chosen threshold $\bar {D}$ below which two states are considered ‘close’.

Let us first consider the turbulent trajectory ${\boldsymbol u}^a(t)$ confined to lobes 1 and 2, which was used to generate the initial conditions for the Newton search. The distances to all the ECSs (as well as their symmetry-related copies) are shown in grey in figure 6(a), except for two solutions: $\mathrm {RPO}_{05}$ (in blue) and $\mathrm {RPO}_{15}$ (in red). All of the distances jump sharply at $t\approx 1$, 42, 54 and 57, as can be clearly seen by focusing on the distance to $\mathrm {RPO}_{05}$ and $\mathrm {RPO}_{15}$, characterized, respectively, by a low (high) average value of $|\mathcal {H}|$. These jumps correspond to the turbulent trajectory moving between lobes 1 and 2. When the distance to $\mathrm {RPO}_{05}$ is low (high), the turbulent flow is inside lobe 1 (lobe 2). Recall that $\mathrm {RPO}_{15}$ lies inside lobe 3, which is closer to lobe 2, so the distance to this $\mathrm {RPO}_{}$ shows the opposite trend.

Such behaviour is an example of intermittency, which is a characteristic feature of all turbulent flows, but is particularly apparent in transitional flows (Chaté & Manneville Reference Chaté and Manneville1987) where it commonly manifests itself as spatiotemporal alternation between laminar and turbulent behaviour (Eckhardt et al. Reference Eckhardt, Schneider, Hof and Westerweel2007). In the present case, intermittency represents a competition between different flow patterns shown in figure 5(a,b) – a phenomenon which has been observed in TCF at large $\varGamma$ (Tsameret & Steinberg Reference Tsameret and Steinberg1994) as well as in many other non-equilibrium systems.

Figure 6(a) shows that the turbulent trajectory ${\boldsymbol u}^a(t)$ spends most of the time in lobe 1 (e.g. for $1\lesssim t\lesssim 42$). This is the likely reason most of the ECSs we found using a Newton search also lie in that lobe. Indeed, each of the fourteen ECSs ($\mathrm {TW}_{01}$, $\mathrm {RPO}_{01}$–$\mathrm {RPO}_{12}$ and $\mathrm {RPO}_{19}$) that is collocated with lobe 1 in the projection shown in figure 3 was also found to be collocated with this lobe in the full state space, with $D_n^1(t)\approx 0.6$ on average (and a minimum of around $0.2$) during the intervals when the turbulent flow is inside lobe 1.

The situation is similar for the turbulent trajectory ${\boldsymbol u}^b(t)$ confined to lobe 3. For the eight ECSs collocated with this lobe in the low-dimensional projection ($\mathrm {TW}_{02}$, $\mathrm {RPO}_{13}$–$\mathrm {RPO}_{18}$ and $\mathrm {RPO}_{20}$), the distance is $D_n^3(t)\approx 0.7$ on average and gets as low as $0.3$ or so, according to figure 6(b). The spike around $t=10$ appears to represent an ‘extreme’ event, where the turbulent trajectory explores the periphery of lobe 3. The discussion in the remainder of the paper will focus mainly on lobe 1, which contains most of the ECSs found.

To get a sense of the relationship between the magnitude of the distance in the state space and the similarity of the flow fields in the physical space, we compare snapshots of the turbulent flow field ${\boldsymbol u}^a(t)$ with a nearby ECS for various values of $D(t)$ in figure 7. The flow fields appear visually similar for $D(t)\lesssim \bar {D}$ with $\bar {D}=0.4$, while for $D(t)\gtrsim \bar {D}$ the differences become apparent. Based on this comparison, we define two flow states to be similar (dissimilar), or close (far) in the state space, for $D(t)<\bar {D}$ ($D(t)>\bar {D}$). The corresponding threshold $\bar {D}$ is shown as the dashed horizontal lines in figure 6. While using such an ‘ocular norm’ to define closeness of different flow states may appear somewhat arbitrary, it is quite common in analysing experimental data (De Lozar et al. Reference De Lozar, Mellibovsky, Avila and Hof2012).

Figure 7.

Comparison of (a) the turbulent flow field ${\boldsymbol u}^a(t)$ and (b) $\mathrm {RPO}_{04}$ at an instant where $D^a_{7}(t) = 0.34$ is below threshold. Comparison of (c) the turbulent flow field ${\boldsymbol u}^a(t)$ and (d) $\mathrm {RPO}_{12}$ at an instant where $D^a_{15}(t) = 0.78$ is above threshold. To represent the flow structure in the entire flow domain, here and below, we show two level sets of $u_\theta$, one with a positive value (in red) and one with a negative value (in blue). In all plots, the mean flow has been subtracted off.

Instead of relying on a somewhat subjective metric based on qualitative similarity of different flow fields, the distance threshold can be chosen based on a partition of the chaotic set induced by the set of computed ECSs. Let us partition lobe 1 into tubular neighbourhoods of different ECSs with sizes determined by $\bar {D}$. To get a sense of how large a fraction of lobe 1 is covered by the union of these neighbourhoods, we computed the probability $P$ that turbulent flow ${\boldsymbol u}^a(t)$, restricted to lobe 1, is in the neighbourhood of any of the fourteen ECSs collocated with this lobe. As figure 8(b) illustrates, these neighbourhoods cover nearly 100 % of lobe 1 for $\bar {D}\gtrsim 1$, approximately 40 % for $\bar {D}=0.4$, and less than 1 % for $\bar {D}\lesssim 0.2$.

Figure 8.

(a) The minimal distance, $D^{min}_{nk}$, between $\mathrm {RPO}_{}$s collocated with lobe 1. Braces on the right of the panel group together $\mathrm {RPO}_{}$s that are connected by continuation in $Re_i$. (b) The probability that a snapshot of the turbulent trajectory will lie in a neighbourhood of the ECS shown in panel (a), as a function of tubular neighbourhood radius, $\bar {D}$. The dashed line indicates our chosen threshold below which two states are considered ‘close’.

Alternatively, $\bar {D}$ can be chosen based on the separation between different ECSs. At small $\bar {D}$, their neighbourhoods will not overlap. As $\bar {D}$ is increased, different neighbourhoods start to overlap more and more. As figure 8(a) shows, the minimal distance

(3.6)\begin{equation} D^{min}_{nk} = \min_{\phi,t,\tau} \sigma^{{-}1}\| {\boldsymbol u}_k(t) - R_\phi{\boldsymbol u}_n(\tau) \|, \end{equation}

between pairs of ECSs is typically around 0.4 or larger, so our choice of $\bar {D}$ should be sufficient to distinguish different ECSs. The minimal distance between some pairs of ECSs (e.g. $\mathrm {RPO}_{09}$–$\mathrm {RPO}_{11}$), however, is as low as 0.2, which means that individual snapshots of these solutions may not be visually distinct.

The results presented in this section show that a snapshot of turbulent flow ${\boldsymbol u}^j(t)$ is close to a family of ECSs $R_\phi {\boldsymbol u}_n$ provided $D_n^j(t)<\bar {D}$, where $\bar {D}=0.4$. As figure 6(a) illustrates, the turbulent trajectory ${\boldsymbol u}^a(t)$ comes close to multiple $\mathrm {RPO}_{}$s and one $\mathrm {TW}_{}$ collocated with lobe 1; this happens quite frequently. Similarly, figure 6(b) illustrates that turbulent trajectory ${\boldsymbol u}^b(t)$ comes close to multiple $\mathrm {RPO}_{}$s and one $\mathrm {TW}_{}$ collocated with lobe 3. In particular we find that condition (i) for dynamical relevance is satisfied for all ECSs collocated with lobe 1, with the possible exception of $\mathrm {RPO}_{19}$ (cf. table 1). Figure 8(a) shows that $\mathrm {RPO}_{19}$ lies far from all other solutions collocated with lobe 1. This suggests that $\mathrm {TW}_{01}$ and $\mathrm {RPO}_{01}$–$\mathrm {RPO}_{12}$ are actually embedded in the primary lobe and may play a dynamically important role, while $\mathrm {RPO}_{19}$ does not.

3.3. Shadowing of solutions

Next we turn to checking conditions (ii) and (iii) of dynamical relevance using time intervals where $D^j_n(t)<\bar {D}$ for a given $n$. While condition (ii) is easy to verify by direct inspection, verifying shadowing condition (iii) requires more care. Formally, shadowing is defined for chaotic trajectories that come infinitesimally close to an unstable solution (Gaspard Reference Gaspard2005). In our case, none of the computed ECS families are approached particularly closely by turbulent flow on time scales accessible to numerical simulations or experiments. One, therefore, has to define shadowing in practically meaningful terms. Yalnız et al. (Reference Yalnız, Hof and Budanur2020) proposed a topological approach to define shadowing of UPOs in a turbulent flow confined to a subspace with only discrete symmetries. The approach is based on computing the number of connected components and holes for temporally discretized trajectories over one temporal period of the solution and comparing the corresponding persistence diagrams using the bottleneck distance. We define shadowing events differently for several reasons. First of all, the period of an ECS is not a proper time scale: in practice, ECSs with longer periods will never be shadowed fully. Instead, an ECS ${\boldsymbol u}_n$ will typically be shadowed for an interval of time comparable to the inverse of the escape rate

(3.7)\begin{equation} \gamma_n = \sum_k\lambda_{n,k}, \end{equation}

where the sum goes over the unstable directions associated with the ECS and $\lambda _{n,k}$ are the corresponding Floquet exponents. We only require an ECS to be shadowed for an interval equal to its expected escape time $\gamma _n^{-1}$. The mean escape times are $0.024$ for lobe 1 and $0.006$ for lobe 3.

Furthermore, the approach used here is both less complicated and allows one to define shadowing for both $\mathrm {RPO}_{}$s and $\mathrm {TW}_{}$s. It is essentially a generalization of the approach proposed by Suri et al. (Reference Suri, Kageorge, Grigoriev and Schatz2020) for UPOs and relies on the skew product decomposition of the dynamics (Fiedler et al. Reference Fiedler, Sandstede, Scheel and Wulff1996; Sandstede, Scheel & Wulff Reference Sandstede, Scheel and Wulff1999) in the vicinity of relative solutions induced by continuous symmetries. For each $\mathrm {RPO}_{}$, the computed ECS ${\boldsymbol u}_n(\tau )$ represents a family of solutions that lie on a two-torus in the state space; all points of this two-torus can be parameterised as $R_\phi {\boldsymbol u}_n(\tau )$, where $0<\phi <2{\rm \pi} /N$ and $0<\tau < T$. Let $R_\phi {\boldsymbol u}_n(\tau )$ be a point in the family of solutions that is closest to a point ${\boldsymbol u}^a(t)$ on the turbulent trajectory, such that

(3.8)\begin{equation} \{\tau(t), \phi(t)\} = \arg\min_{\tau,\phi} \sigma^{{-}1}\|{\boldsymbol u}^a(t) - R_\phi{\boldsymbol u}_n(\tau)\|, \end{equation}

where $\phi$ is constrained to the interval $[0,2{\rm \pi} /N)$ for solutions with an $N$-fold discrete rotational symmetry (e.g. $N=2$ for $\mathrm {RPO}_{01}$ and $\mathrm {RPO}_{15}$). In this decomposition, coordinates $\tau$ and $\phi$ describe the evolution along the group manifold (time translations and rotations), while $D_n^a$ describes the evolution transverse to the group manifold. Below, we will drop the subscript and superscript of $D$ since the context makes it clear which turbulent solution and ECS is discussed.

For $D=0$, a trajectory ${\boldsymbol u}(t)$ will have ${\rm d}\tau /{\rm dt}=1$ and ${\rm d}\phi /{\rm dt}=0$ for $0<\tau < T$. Once per period, both $\tau (t)$ and $\phi (t)$ will experience a jump by, respectively, $T$ and $\varPhi$ (see table 1). The discontinuities in $\tau (t)$ and $\phi (t)$ can be removed using coordinates $\tilde \tau$ and $\tilde \phi$ such that

(3.9)\begin{equation} \left. \begin{gathered} \tau = \tilde{\tau}\ \mathrm{mod}\ T,\\ \tilde\phi ={\phi}\ \mathrm{mod}\ \varPhi. \end{gathered} \right\} \end{equation}

Then, for $D=0$, $\textrm {d}\tilde \tau /\textrm {d}t=1$ and $\textrm {d}\tilde \phi /\textrm {d}t=0$ for all $t$. For any trajectory that has a small but non-zero $D$, the dynamics of $\tau$ and $\phi$ should be qualitatively similar: $\textrm {d}\tilde \tau /\textrm {d}t\approx 1$ and $\textrm {d}\tilde \phi /\textrm {d}t\approx 0$. The evolution of $D$, $\tau$ and $\phi$ can therefore be used to define a set of natural criteria for shadowing of an $\mathrm {RPO}_{}$.

Specifically, we will consider a turbulent trajectory to shadow an $\mathrm {RPO}_{}$ if, for a temporal interval $I(t)=[t-\gamma _n^{-1}/2,t+\gamma _n^{-1}/2]$, all three coordinates evolve similarly to the way they would for the $\mathrm {RPO}_{}$. In practice, we define similarity in terms of normalized deviations

(3.10)\begin{equation} \left. \begin{gathered} E_\tau = \min_{t_0} \frac{1}{T\gamma_n^{{-}1}} \int_{I(t)} |t'+t_0-\tilde \tau(t')|^2 \,\textrm{d}t'\\ E_\phi = \min_{\phi_0} \frac{N}{{\rm \pi}\gamma_n^{{-}1}} \int_{I(t)} |\phi_0-\tilde \phi(t') |^2\, \textrm{d}t' \end{gathered} \right\} \end{equation}

of $\tilde {\tau }$ and $\tilde {\phi }$ from straight lines with slope one and zero, respectively. The turbulent trajectory is considered to be shadowing an $\mathrm {RPO}_{}$, if the following three conditions are satisfied:

1. (a) $D(t')<\bar {D}$ for $t'\in I(t)$;

2. (b) $E_\tau <\bar {E}_\tau$; and

3. (c) $E_\phi <\bar {E}_\phi$

for an appropriate choice of the thresholds $\bar {E}_\tau$ and $\bar {E}_\phi$. For $\mathrm {TW}_{}$s, the two coordinates describing the evolution along the group manifold are not independent, $\textrm {d}\tau /\textrm {d}\phi =T/\varPhi$. In this case, shadowing is determined by two of the three conditions, (a) and (b).

We chose $\bar {E}_\tau =0.0001$ and $\bar {E}_\phi =0.0003$, so that conditions (a–c) are satisfied with roughly equal probability. Figure 9 summarizes the results for both the turbulent trajectory ${\boldsymbol u}^a(t)$ confined to lobes 1 and 2 and the turbulent trajectory ${\boldsymbol u}^b(t)$ confined to lobe 3. In the former case, we find that turbulence shadows both $\mathrm {RPO}_{}$s and a TW. Moreover, if an ECS is shadowed, then so is its reflected copy. This suggests that turbulence does not break reflection symmetry in a statistical sense, i.e. the probability of shadowing any ECS is comparable to the probability of shadowing its reflected copy. As figure 3 illustrates, this is not entirely unexpected, since lobe 1 is fairly symmetric with respect to the reflection.

Figure 9.

A summary of the shadowing events (marked with black bars) for each of the two turbulent trajectories, (a) ${\boldsymbol u}^a(t)$ and (b) ${\boldsymbol u}^b(t)$. Each ECS is represented by two rows, one for the numerically converged solution ${\boldsymbol u_{n}}$ (grey) and one for its symmetric copy $K_{z} {\boldsymbol u_{n}}$ (white). Red bars on the right represent the escape times $\gamma _n^{-1}$ to scale. Curly braces on the far right group together $\mathrm {RPO}_{}$s that are related via continuation in $Re_i$.

For instance, consider a shadowing event for $\mathrm {RPO}_{05}$ shown in figure 10 and its symmetry-related copy in figure 11 (the corresponding movies are included as supplementary material). The same format is used here and below to illustrate shadowing events. Panel (a) shows the evolution of the three coordinates $\tau$, $\tilde {\phi }$ and $D$ with black (white) circles indicating instances when the corresponding shadowing criterion is satisfied (not satisfied). Red lines denote the temporal intervals when the criterion based on the corresponding coordinate is satisfied; for an interval to be considered an instance of shadowing, all three criteria must be met simultaneously. The grey bar shows the length of the interval $\gamma _n^{-1}$ for comparison with the temporal period of the ECS. Panels (b) and (c) compare, respectively, the spatial structure (contours of constant $u_\theta$ at the values denoted in yellow on the colourbar in panels (d,e)) of the turbulent flow and the corresponding ECS in the entire domain at the instant denoted by the vertical black line in panel (a). Finally, panels (d) and (e) compare the velocity fields in the yellow cross-section shown in panels (b) and (c), respectively. In all panels, the turbulent mean flow is subtracted off.

Figure 10.

Turbulent flow ${\boldsymbol u}^a(t)$ shadowing $\mathrm {RPO}_{05}$. (a) Evolution of coordinates $\tau$, $\tilde {\phi }$ and $D$. Black (white) circles indicate instances when the corresponding shadowing criterion is satisfied (not satisfied). Red lines denote the temporal intervals during which a given criterion is satisfied. The grey bar shows the escape time, $\gamma _n^{-1}$. Snapshots of (b) the turbulent flow and (c) the ECS at the instant marked with the black vertical line in panel (a). Flow fields in the yellow cross-section for (d) the turbulent flow and (e) the ECS. A movie of this shadowing event is included as supplementary material available at https://doi.org/10.1017/jfm.2021.522.

Figure 11.

A shadowing event for the reflected copy of $\mathrm {RPO}_{05}$ in lobe 1. A corresponding movie is available as supplementary material.

Note that, for both $\mathrm {RPO}_{05}$ and its symmetry-related copy, the respective $\mathrm {RPO}_{}$ is shadowed continuously for several periods (interval of around $6\gamma _n^{-1}$). During the shadowing intervals, all three criteria are satisfied to a high accuracy: the slope of $\tau (t)$ is near unity, the slope of $\tilde {\phi }(t)$ is near zero and $D(t)$ stays below the threshold $\bar {D}$. As discussed previously, for $D(t)<\bar {D}$ the turbulent flow is visually almost indistinguishable from the corresponding ECS in the entire domain. Figure 10(b,c) illustrates this particularly convincingly for $\mathrm {RPO}_{05}$. The similarity is not just qualitative, but quantitative, as figure 10(d,e) shows. For small $D(t)$, it is expected that the turbulent flow evolves in the same manner as the nearby $\mathrm {RPO}_{}$. However, when $D(t)$ briefly increases above threshold, the slopes of $\tau (t)$ and $\tilde {\phi }(t)$ remain virtually unchanged, suggesting that turbulence continues shadowing the same member of the $\mathrm {RPO}_{}$ family even when the corresponding flows are not very similar. Analogous statements apply to the reflected version of $\mathrm {RPO}_{05}$, as figure 11 illustrates.

Figure 12 presents a sequence of several shadowing events for $\mathrm {RPO}_{01}$. These shadowing events are of duration comparable to those for $\mathrm {RPO}_{05}$ and its reflection, measured in units of $\gamma _n^{-1}$, but correspond to quite a few periods of $\mathrm {RPO}_{01}$. Not only that, turbulence also visits the neighbourhood of $\mathrm {RPO}_{01}$ much more frequently, as figure 9(a) demonstrates. In fact, this solution is by far the most frequently visited $\mathrm {RPO}_{}$ in lobe 1. Note that $\mathrm {RPO}_{01}$ is symmetric with respect to $K_zR_{{\rm \pi} /2}$, so reflection is equivalent to a rotation by ${\rm \pi} /2$. Hence, both $\mathrm {RPO}_{01}$ and its reflection belong to the same solution family, and both rows in figure 9(a) contain identical sequences of shadowing events. Also, just like in the case of $\mathrm {RPO}_{05}$, turbulent flow is essentially indistinguishable from $\mathrm {RPO}_{01}$ during the shadowing episodes.

Figure 12.

A shadowing event for $\mathrm {RPO}_{01}$ in lobe 1. A corresponding movie is available as supplementary material.

Figure 13 illustrates shadowing of $\mathrm {TW}_{01}$. Rather unexpectedly, $\mathrm {TW}_{01}$ is found to be shadowed even more frequently than $\mathrm {RPO}_{01}$. In fact, when turbulent trajectory ${\boldsymbol u}^a(t)$ is inside lobe 1, it spends more time near $\mathrm {TW}_{01}$ than any other ECS according to figure 9(a). Moreover, when $\mathrm {TW}_{01}$ is shadowed, then typically so is $\mathrm {RPO}_{01}$. This is not particularly surprising, as $\mathrm {RPO}_{01}$ is rather compact, and the two ECSs lie very close to each other, with the distance between them being roughly a half of $\bar {D}$ according to figure 8(a). This is a consequence of $\mathrm {RPO}_{01}$ being born in a Hopf bifurcation of $\mathrm {TW}_{01}$, which happens to lie close in the parameter space. It should also be mentioned that the length of the shadowing intervals for $\mathrm {TW}_{01}$ tends to be quite large (e.g. around $8\gamma _n^{-1}$ for the event shown in figure 13) compared with typical shadowing intervals for the $\mathrm {RPO}_{}$s embedded in lobe 1.

Figure 13.

A shadowing event for $\mathrm {TW}_{01}$ in lobe 1. A corresponding movie is available as supplementary material.

We have not found any shadowing events for $\mathrm {RPO}_{19}$ or its reflection, even though it appears to be embedded inside lobe 1. Indeed, the smallest value of $D(t)$ for this ECS is $0.42$, which is above our threshold $\bar {D}$. Every other $\mathrm {RPO}_{}$ (as well as $\mathrm {TW}_{01}$) embedded in lobe 1 is being shadowed by turbulence. Recall that ${\boldsymbol u}^a(t)$ visits both lobes 1 and 2. As expected, shadowing events are confined to the temporal intervals when turbulent flow is inside lobe 1. The shadowing criteria (a–c) are not satisfied when the turbulent trajectory is inside lobe 2 as we have not identified ECSs in that region of the state space.

For completeness, we have also performed a similar analysis for the turbulent trajectory ${\boldsymbol u}^b(t)$ which lies in lobe 3. It was found to shadow $\mathrm {RPO}_{13}$–$\mathrm {RPO}_{16}$ and $\mathrm {RPO}_{18}$. Since this lobe breaks the reflection symmetry, none of the reflected copies of these $\mathrm {RPO}_{}$s were shadowed. An example of shadowing for $\mathrm {RPO}_{15}$ is shown in figure 14. It is worth emphasizing that ECSs in lobe 3 are more unstable than those in lobe 1, as illustrated by both the larger number of unstable directions and the shorter escape time. As a result, there are fewer instances of shadowing and each shadowing event is nominally shorter. For instance, even though $\mathrm {RPO}_{15}$ is shadowed for almost three (very short) periods in figure 14, this interval corresponds to only 0.022 in non-dimensional units, compared with around $0.1$ for $\mathrm {RPO}_{01}$ and $\mathrm {RPO}_{05}$.

Figure 14.

A shadowing event for $\mathrm {RPO}_{15}$ in lobe 3. A corresponding movie is available as supplementary material.