# Exact coherent structures and shadowing in turbulent Taylor-Couette flow

Michael C. Krygier1<EMAIL_ADDRESS>Joshua L. Pughe-Sanford1  Roman O.
Grigoriev1  1School of Physics, Georgia Institute of Technology, Atlanta, GA
30332, USA

###### Abstract

We investigate a theoretical framework for modeling fluid turbulence based on
the formalism of exact coherent structures (ECSs). Although highly promising,
existing evidence for the role of ECSs in turbulent flows is largely
circumstantial and comes primarily from idealized numerical simulations. In
particular, it remains unclear whether three-dimensional turbulent flows in
experiment shadow any ECSs. In order to conclusively answer this question, a
hierarchy of ECSs should be computed on a domain and with boundary conditions
exactly matching experiment. The present study makes the first step in this
direction by investigating a small-aspect-ratio Taylor-Couette flow with
naturally periodic boundary condition in the azimuthal direction. We describe
the structure of the chaotic set underlying turbulent flow driven by counter-
rotating cylinders and present direct numerical evidence for shadowing of a
collection of unstable relative periodic orbits and a traveling wave, setting
the stage for further experimental tests of the framework.

###### keywords:

turbulence theory, chaos, nonlinear dynamical systems

##  1 Introduction

Fluid turbulence has a unique place in science and engineering, due to both
its ubiquity and tremendous practical importance as well as its resistance to
progress despite a long history of systematic investigation. A statistical
description, which dominated early theoretical studies, brought some advances,
such as the Kolmogorov’s scaling law (Kolmogorov, 1941) and the law of the
wall (Von Kármán, 1930; Nikuradse, 1932). These advances, however, are based
on general concepts such as dimensional analysis, spatial uniformity, and/or
isotropy, that are not directly related to the equations governing fluid flow
and shed little light onto the nature of the turbulent cascades or momentum
transport in wall-bounded flows. Moreover, statistical approaches provide
minimal insight into the prediction and control of fluid turbulence.

Existing statistical models of fluid turbulence fail to predict even such
simple quantities as the mean energy dissipation (e.g., in isotropic
turbulence) or momentum flux (e.g., in a wall-bounded flow). The calculation
of the friction coefficient for pipe flow turbulence is a good example, where
one has to rely on empirically derived Moody charts (Moody, 1944). Statistical
description also fails to account for the presence of coherent structures,
which are well-known to play an important role in turbulence (Hussain, 1983).
Indeed, coherent structures break both spatial uniformity and isotropy and
introduce new spatial and temporal scales, invalidating the entire foundation
of statistical description.

The most promising alternative approach is to build a dynamical description of
turbulence firmly based on the equations governing fluid flow. Unlike other
classical field theories, such as electromagnetism, whose governing equations
are linear, fluid turbulence is governed by the Navier-Stokes equation which
is strongly nonlinear, making analytical solutions intractable. Recent
advances in numerical methods brought a realization that coherent structures
represent unstable solutions of Navier-Stokes with simple temporal dependence,
with the earliest example provided by Nagata (1990). This started a revolution
in our understanding of fluid turbulence (Kawahara et al., 2012). Termed exact
coherent structures (ECSs), such solutions have been shown to play a key role
in the transition from laminar flow to turbulence (Kerswell, 2005; Eckhardt et
al., 2008) and self-sustaining processes in boundary layers (Waleffe & Wang,
2005). In some cases, a single ECS was found to reproduce certain statistical
properties of weakly turbulent flow. A time-periodic solution obtained by
Kawahara & Kida (2001) for plane Couette flow was found to reproduce, with
fairly high accuracy, both the mean flow profile and the fluctuations in all
three components of the velocity.

It is therefore natural to ask whether these results are coincidental or some
collection of ECSs can in fact provide a dynamical and statistical description
of fluid turbulence. The idea that turbulence can be thought of as a
deterministic walk through a repertoire of patterns (which we now associate
with different ECSs) goes back to Eberhard Hopf (Hopf, 1942, 1948). In Hopf’s
view, the snapshot of turbulent flow in the physical space corresponds to a
point in the associated infinite-dimensional state (or phase) space. This
point traces out a one-dimensional trajectory as the flow evolves in time.
This trajectory is confined, due to dissipation, to a finite-dimensional set
embedded within this state space. This set can be either an attractor (for
sustained turbulence) or a repeller (for transient turbulence).

It took two more decades to flesh out the details of Hopf’s picture, when
mathematical foundations of deterministic chaos, geometry of chaotic sets, and
ergodic theory were developed (Lorenz, 1963; Mandelbrot, 1967; Arnold & Avez,
1968). In particular, for uniformly hyperbolic chaotic systems without
continuous symmetries, unstable periodic orbits (UPOs) are dense in the
chaotic set (Gaspard, 2005), which has two important implications. The first
one is dynamical: chaotic trajectories shadow nearby UPOs. The second one is
statistical: temporal averages over a chaotic trajectory can be computed as a
sum over an infinite hierarchy of UPOs (Arnold & Avez, 1968), with the weight
of each term predicted by periodic orbit theory (Auerbach et al., 1987;
Cvitanović, 1988; Lan, 2010).

Turbulent fluid flows are chaotic, at least in so far as the sensitive
dependence on initial conditions is concerned, so it is natural to ask whether
the properties of shadowing and ergodicity also apply to turbulence.
Originally conjectured by Hopf, the shadowing property is widely assumed
(Cvitanović, 2013) in the studies exploring the ECS-based framework, although
there is very little direct evidence in its favor.

Most of the evidence for turbulent flows visiting neighborhoods of ECSs has
been obtained using numerical simulations of minimal flow units and,
therefore, has to be taken with a grain of salt. For instance, unstable
equilibria and UPOs were found to be visited in forced two-dimensional flow
(Kazantsev, 1998; Chandler & Kerswell, 2013; Lucas & Kerswell, 2015),
isotropic turbulence (van Veen et al., 2006), and plane Couette flow
(Cvitanović & Gibson, 2010; Kreilos & Eckhardt, 2012). Traveling waves (TWs)
were found to be visited in plane Poiseuille flow (Park & Graham, 2015) and
pipe flow (Schneider et al., 2007; Kerswell & Tutty, 2007; Dennis & Sogaro,
2014) and relative periodic orbits (RPOs) in pipe flow (Willis et al., 2013;
Budanur et al., 2017). Experiments provide more direct and solid evidence. For
instance, TWs were found to be visited in pipe flows (Hof et al., 2004; De
Lozar et al., 2012) and unstable equilibria in forced quasi-two-dimensional
flows (Suri et al., 2017, 2018).

Visits by turbulence to a neighborhood of an ECS, however, do not imply
shadowing, since, in practice, no ECSs are visited particularly closely.
Shadowing requires that turbulent flow remain in the neighborhood of an ECS
for an extended interval of time comparable to the characteristic escape time.
It also requires that turbulent flow evolve in the same manner as the ECS.
Such evidence is currently limited to UPOs in flows without continuous
symmetries, e.g., Kolmogorov flow in two dimensions (Suri et al., 2020) and
three dimensions (Yalnız et al., 2020). However, it is not completely
understood how the presence of symmetries affects the shadowing property.

To sum up, while a dynamical description of fluid turbulence based on exact
coherent structures is promising, it is yet to be properly validated in both
numerical and experimental setting, especially for flows with continuous
symmetries. A key challenge is the disconnect between experiments that are
conducted on large spatial domains and typically for open flows such as pipe,
channel, or plane Couette flow and numerics: ECSs are typically computed on
minimal flow units with unphysical (e.g., spatially periodic) boundary
conditions and/or under unrealistic constraints (e.g., in highly symmetric
subspaces). To ensure an apples-to-apples comparison, the dynamical
description should be validated in a geometry and under conditions where
numerical simulations match experiment as closely as possible. Hence, this
study focuses on a weakly turbulent Taylor-Couette flow (TCF) between
concentric cylinders, which is easy to realize in practice. This is a closed
flow whose boundary conditions in the azimuthal direction are naturally
periodic, reflecting continuous rotational symmetry. Furthermore, we do not
restrict ECSs to lie in any symmetry subspace.

Figure 1:  Geometry of the Taylor-Couette flow driven by counter-rotating
cylinders. The fluid domain is shaded purple.

The TCF considered in this study is characterized by four nondimensional
parameters. Two of these are geometrical: $\Gamma=h/d$ and $\eta=r_{i}/r_{o}$,
where $h$ is the height of the fluid layer and $r_{i}$ and $r_{o}$ are the
radii of the inner and outer cylinder, respectively, as shown in Figure 1. The
other two are the Reynolds numbers $Re_{i}=\Omega_{i}r_{i}d/\nu$ and
$Re_{o}=\Omega_{o}r_{o}d/\nu$, where $d=r_{o}-r_{i}$ is the gap between the
cylinders, $\nu$ is the kinematic viscosity of the fluid, and $\Omega_{i}$ and
$\Omega_{o}$ are the angular velocities of the two cylinders. We focus on
flows driven by counter-rotating cylinders ($Re_{i}=1200$, $Re_{o}=-1200$) in
the wide-gap ($\eta=0.5$), small-aspect-ratio ($\Gamma=1$) geometry with
stationary end-caps.

Previous studies of wide-gap, small-aspect-ratio TCF mainly focused on the
flows driven by inner cylinder rotation with the outer cylinder and end-caps
being stationary. The bifurcations between steady flows associated with the
changes in $Re_{i}$, $\Gamma$, and $\eta$ have been studied by Benjamin &
Mullin (1981); Mullin (1982); Cliffe (1983); Aitta et al. (1985); Pfister et
al. (1988) and Mullin et al. (2002). Time-dependent flows have been
investigated by Lorenzen et al. (1983); Pfister et al. (1991); Furukawa et al.
(2002). The onset of turbulent flows has been considered by Streett & Hussaini
(1991); Pfister et al. (1992) for $\Gamma>1$, Marques & Lopéz (2006) for
$\Gamma=1$, and Buzug et al. (1992, 1993) for $\Gamma<1$. The effect of the
outer cylinder rotation ($Re_{o}\neq 0$) on the bifurcation sequence was
considered by Schulz et al. (2003) and Altmeyer et al. (2012), but neither
study explored Reynolds numbers sufficiently high to observe turbulent flows.

Turbulence in counter-rotating TCF has been investigated mainly in small-gap
($0.9\lesssim\eta<1$), large-aspect-ratio ($\Gamma\gg 1$) geometry (Coles,
1965; Andereck et al., 1986). A number of TWs, both stable and unstable, have
been identified in this regime for $Re_{o}\lesssim-1200$. These TWs take the
shape of rotating spiral waves (Meseguer et al., 2009), ribbons (Deguchi &
Altmeyer, 2013), or spatially localized vortex pairs (Deguchi et al., 2014).
However, no unstable RPOs of TCF have been found so far.

The main objective of the present numerical study is to validate the key
assumptions which underlie the dynamical description of fluid turbulence in a
setting which is straightforward to replicate in experiment (Hochstrate et
al., 2010; Hoffmann et al., 2013; Heise et al., 2013). Specifically, we are
looking to compute a library of ECSs (both TWs and RPOs) that are dynamically
prominent and analyze turbulent flows for close passes to each ECS to identify
shadowing events, if there are any. The paper is structured as follows.
Numerical methods for solving the Navier-Stokes equation and computing ECSs
are described in Section 2. Our results are presented in Section 3 and
analyzed in 4. Finally, Section 5 presents our conclusions.

##  2 Mathematical Formulation

###  2.1 Direct numerical simulation

The flow is governed by the Navier-Stokes equation and the incompressibility
condition which can be written in nondimensional form using the gap $d$ and
the diffusive time scale $d^{2}/\nu$ as the length and time scale,
respectively:

$\displaystyle\partial_{t}{\bf u}+({\bf u}\cdot\nabla){\bf u}$
$\displaystyle=-\nabla p+\nabla^{2}{\bf u},$ $\displaystyle\nabla\cdot{\bf u}$
$\displaystyle=0.$ (1)

Here ${\bf u}=(u_{r},u_{\theta},u_{z})$ and $p$ are the nondimensional
velocity and pressure. No-slip boundary conditions are imposed on all the
walls:

$\displaystyle{\bf u}(r_{i},\theta,z,t)$ $\displaystyle=(0,Re_{i},0),$
$\displaystyle{\bf u}(r_{o},\theta,z,t)$ $\displaystyle=(0,Re_{o},0),$
$\displaystyle{\bf u}(r,\theta,\pm h/2,t)$ $\displaystyle=(0,0,0).$ (2)

The last relation describes the boundary conditions at the top and bottom of
the fluid layer.

Direct numerical simulations (DNS) of TCF were performed using a
pseudospectral code (Avila et al., 2008; Mercader et al., 2010; Avila, 2012)
which solves the governing equations in cylindrical coordinates
$(r,\theta,z)$. The velocity field ${\bf u}$ at location $(r,\theta,z)$ and
time $t$ is given by

${\bf
u}(r,\theta,z,t)=\mathrm{Re}\sum_{k=0}^{N_{r}}\sum_{l=0}^{N_{z}}\sum_{m=0}^{N_{\theta}/2}{\bf
U}^{klm}(t)T_{k}(\rho)T_{l}(\zeta)e^{im\theta},$ (3)

where $\rho=(2r-r_{i}-r_{o})/d$ and $\zeta=2z/h$. $N_{r}$, $N_{z}$, and
$N_{\theta}$ are the number of spectral modes in the three coordinate
directions, $T_{k}(\cdot)$ is the Chebyshev polynomial of order $k$, and
$\mathrm{Re}$ denotes the real part. The solution is advanced in time using a
second order stiffly stable time-splitting scheme (Hugues & Randriamampianina,
1998). Advection terms are evaluated on the spatial grid
$(r_{k},z_{l},\theta_{m})$ in physical space, where

$\displaystyle r_{k}$
$\displaystyle=\frac{(r_{o}-r_{i})\cos(k\pi/N_{r})+r_{i}+r_{o}}{2},\;k=0,\ldots,N_{r},$
$\displaystyle z_{l}$
$\displaystyle=\frac{\Gamma\cos(l\pi/N_{z})}{2},\;l=0,\ldots,N_{z}$ (4)

are Chebyshev collocation points and $\theta_{m}=2\pi m/N_{\theta}$ with
$m=0,\cdots,N_{\theta}-1$. The Helmholtz and Poisson equations are solved
efficiently using a complete diagonalization of the operators in both the
radial and axial direction for each Fourier mode (Orszag & Patera, 1983).

We set $N_{r}=32$, $N_{\theta}=128$, $N_{z}=48$ (which corresponds to
$3(N_{r}+1)(N_{z}+1)N_{\theta}=620928$ degrees of freedom) and used a time
step $dt=O(10^{-6})$ to accurately resolve the spatial structure and temporal
dependence of turbulent flow and all computed ECSs. The spatial resolution was
chosen such that the magnitude of the spectral coefficients ${\bf U}^{klm}$
decreases by at least four orders of magnitude for ECSs (and at least three
orders of magnitude for turbulent flows) as $k$, $l$, or $m$ increases from
the smallest to the largest value.

###  2.2 Computation of Exact Coherent Structures

Under the boundary conditions (2.1), TCF is invariant under arbitrary
rotations $R_{\phi}$ about the $z$-axis, and a reflection $K_{z}$ about the
mid-plane $z=0$, where

$\displaystyle R_{\phi}{\bf u}(r,\theta,z,t)$ $\displaystyle={\bf
u}(r,\theta+\phi,z,t),$ (5) $\displaystyle K_{z}{\bf u}(r,\theta,z,t)$
$\displaystyle=(u_{r},u_{\theta},-u_{z})(r,\theta,-z,t).$ (6)

These transformations form a symmetry group
$\mathcal{G}=\mathrm{SO}(2)\times\mathrm{Z}_{2}$. The presence of continuous
rotational symmetry and the lack of reflection symmetry in $\theta$ imply that
the dynamically relevant ECSs in Taylor-Couette flow are relative, e.g.,
relative periodic orbits (RPOs) and relative equilibria (TWs), which
correspond to time-periodic and stationary states in a co-rotating reference
frame. In particular, RPOs satisfy

$\displaystyle{\bf u}(T)-R_{\Phi}{\bf u}(0)={\bf 0},$ (7)

where $\Phi$ and $T$ are the solution’s rotational shift and period,
respectively. The angular velocity of this co-rotating reference frame is
$\Omega=\Phi/T$. TWs also satisfy (7) for $T=\Phi/\Omega$ at arbitrary $\Phi$.
For both RPOs and TWs, some of which have an $N$-fold discrete rotational
symmetry, we arbitrarily restrict $0\leq\Phi<2\pi/N$ to make the definition of
rotational shift unique. To find an ECS, the nonlinear equation (7) is solved
for ${\bf u}(0)$, $T$ and/or $\Phi$ using a custom Newton-GMRES solver that
takes advantage of a hookstep algorithm (Viswanath, 2007, 2009). A relative
residual

$\displaystyle\varepsilon=\frac{\|{\bf u}(T)-R_{\Phi}{\bf u}(0)\|}{\|{\bf
u}(0)\|},$ (8)

was used as the stopping condition for the solver: solutions are considered
converged for $\varepsilon<10^{-11}$. A large set of initial conditions for
the solver was generated using the natural measure of the flow, as described
below.

Turbulent flow was established using the following protocol. With fluid
initially stationary in the entire flow domain, the outer cylinder angular
velocity was set to a value $\Omega_{o}$ corresponding to the target
$Re_{o}=-1200$ (with the inner cylinder stationary). The flow was then allowed
to evolve for 8.8 time units until a (steady and azimuthally uniform)
asymptotic state was established. All the Fourier modes except $m=0$ were then
weakly perturbed in order to break the azimuthal symmetry of the flow, after
which the inner cylinder angular velocity was set to a value $\Omega_{i}$
corresponding to the target $Re_{i}=1200$. The flow was then evolved for 1.6
time units until a statistically stationary state was reached.

Different turbulent field snapshots were then used as initial conditions to
the Newton solver to generate a set of ECSs. To increase the likelihood that
the computed ECSs are dynamically relevant, we initialized the Newton solver
using deep minima of the recurrence function (Kawahara & Kida, 2001;
Viswanath, 2007; Cvitanović & Gibson, 2010)

$G(t,\tau)=\min_{\phi}\left\lVert{\bf u}(t+\tau)-R_{\phi}{\bf
u}(t)\right\rVert,$ (9)

where $\left\lVert\cdot\right\rVert$ is the $L_{2}$-norm. Some of these minima
correspond to close passes to RPOs or TWs, and the corresponding flow fields
${\bf u}(t)$, time delays $\tau$, and rotation angles $\phi$ represent good
initial conditions for the solver. An alternative approach to identifying
dynamically relevant initial conditions relies on dynamic mode decomposition
(Page & Kerswell, 2020). Converged solutions were also numerically continued
in $Re_{i}$ using pseudo-arclength continuation (Allgower & Georg, 2003); some
branches turned around, yielding several additional solutions. For this
reason, all of the solutions used in this study were computed for
$Re_{i}=1200\pm 3$. Their properties are summarized in subsection 2.2.

$n$ | ${\bf u}_{n}$ | $T$ | $\Phi$ | $N_{\lambda}^{u}$ | $\gamma_{n}^{-1}$ | $\min_{t}D_{n}^{1}(t)$ | $\min_{t}D_{n}^{2}(t)$ | $\min_{t}D_{n}^{3}(t)$ |  Discrete
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | Symmetry
1 | TW01 | $0.130\,462\,104\,447\,339\,7$ | $\pi$ | 10 | $0.0121$ | $0.190\,526\,715\,431\,270\,7$ | $1.169\,502\,672\,335\,630\,5$ | – | $K_{z}R_{\pi/2}$
2 | TW02 | $0.012\,703\,950\,422\,983\,7$ | $\pi$ | 10 | $0.0026$ | $1.434\,840\,118\,873\,064\,0$ | $0.679\,578\,692\,841\,247\,7$ | $0.271\,578\,012\,965\,765\,6$ | $R_{\pi}$
3 | TW03 | $0.045\,878\,363\,908\,670\,1$ | $2\pi$ | 43 | $0.0004$ | $1.370\,408\,063\,701\,976\,4$ | $1.575\,895\,721\,688\,945\,3$ | – | $K_{z}R_{\pi}$
4 | RPO01 | $0.015\,862\,418\,270\,623\,2$ | $0.354\,129\,719\,612\,447$ | 4 | $0.0228$ | $0.184\,355\,120\,870\,628\,4$ | $1.164\,873\,295\,783\,716\,7$ | – | $K_{z}R_{\pi/2}$
5 | RPO02 | $0.050\,147\,398\,308\,782\,9$ | $0.827\,541\,005\,771\,14$ | 9 | $0.0063$ | $0.220\,488\,092\,117\,342\,2$ | $1.152\,064\,316\,967\,270\,0$ | – | –
6 | RPO03 | $0.050\,615\,223\,616\,385\,2$ | $0.881\,718\,176\,320\,125$ | 4 | $0.0095$ | $0.179\,154\,276\,255\,741\,9$ | $1.163\,967\,778\,511\,791\,7$ | – | –
7 | RPO04 | $0.051\,205\,516\,284\,300\,1$ | $0.967\,587\,754\,859\,682$ | 5 | $0.0205$ | $0.224\,736\,363\,993\,117\,7$ | $1.134\,687\,841\,336\,545\,3$ | – | –
8 | RPO05 | $0.026\,156\,588\,394\,435\,7$ | $3.411\,832\,631\,727\,47$ | 5 | $0.0227$ | $0.137\,068\,648\,600\,970\,7$ | $1.119\,025\,035\,569\,536\,1$ | – | –
9 | RPO06 | $0.006\,284\,243\,701\,610\,5$ | $3.268\,769\,835\,409\,118\,141$ | 6 | $0.0247$ | $0.219\,736\,311\,259\,922\,7$ | $1.172\,928\,391\,234\,360\,4$ | – | –
10 | RPO07 | $0.021\,344\,863\,939\,384\,4$ | $0.158\,087\,406\,999\,197$ | 4 | $0.0210$ | $0.299\,507\,550\,021\,691\,1$ | $1.150\,828\,239\,808\,853\,4$ | – | $R_{\pi}$
11 | RPO08 | $0.049\,269\,494\,275\,489\,2$ | $0.909\,119\,027\,478\,775$ | 8 | $0.0124$ | $0.202\,775\,500\,129\,422\,3$ | $1.164\,153\,232\,043\,731\,1$ | – | –
12 | RPO09 | $0.019\,141\,306\,724\,221\,1$ | $3.494\,867\,890\,402\,29$ | 3 | $0.0214$ | $0.229\,732\,411\,033\,357\,9$ | $1.156\,032\,609\,010\,358\,3$ | – | –
13 | RPO10 | $0.019\,440\,140\,828\,386\,1$ | $3.461\,503\,077\,609\,76$ | 4 | $0.0152$ | $0.240\,445\,280\,182\,263\,4$ | $1.148\,569\,453\,460\,023\,3$ | – | –
14 | RPO11 | $0.019\,643\,535\,960\,662\,7$ | $3.427\,092\,869\,685\,03$ | 5 | $0.0159$ | $0.231\,541\,776\,838\,380\,5$ | $1.143\,798\,338\,197\,079\,3$ | – | –
15 | RPO12 | $0.021\,701\,947\,815\,931\,3$ | $0.144\,164\,982\,451\,532$ | 5 | $0.0193$ | $0.295\,238\,976\,289\,250\,2$ | $1.159\,623\,214\,979\,420\,3$ | – | $R_{\pi}$
16 | RPO13 | $0.008\,103\,889\,592\,645\,75$ | $1.161\,672\,755\,862\,89$ | 9 | $0.0051$ | $1.354\,905\,310\,903\,809\,8$ | $0.766\,082\,325\,560\,366\,8$ | $0.269\,044\,047\,553\,645\,6$ | $R_{\pi}$
17 | RPO14 | $0.044\,638\,061\,905\,787$ | $5.001\,635\,468\,406\,78$ | 9 | $0.0077$ | $1.449\,823\,004\,924\,008\,2$ | $0.881\,404\,768\,411\,266\,8$ | $0.275\,803\,769\,268\,608\,5$ | –
18 | RPO15 | $0.007\,326\,796\,978\,907\,245\,2$ | $1.329\,826\,099\,82$ | 10 | $0.0045$ | $1.408\,744\,472\,553\,043\,6$ | $0.702\,609\,854\,778\,348\,6$ | $0.183\,798\,006\,433\,209\,6$ | $R_{\pi}$
19 | RPO16 | $0.006\,309\,599\,525\,084\,22$ | $1.642\,944\,704\,706\,32$ | 10 | $0.0039$ | $1.518\,062\,719\,357\,508\,3$ | $1.027\,099\,151\,045\,074\,0$ | $0.390\,298\,098\,211\,943\,0$ | –
20 | RPO17 | $0.012\,141\,228\,497\,869\,5$ | $0.318\,603\,061\,149\,668$ | 15 | $0.0016$ | $1.528\,851\,647\,413\,972\,4$ | $1.006\,578\,232\,729\,234\,1$ | $0.581\,079\,877\,344\,509\,6$ | $R_{\pi}$
21 | RPO18 | $0.055\,794\,884\,392\,551\,425$ | $2.291\,242\,678\,434\,820\,8$ | 7 | $0.0084$ | $1.440\,237\,700\,366\,485\,9$ | $0.815\,497\,623\,748\,926\,9$ | $0.224\,759\,359\,697\,336\,8$ | –
22 | RPO19 | $0.045\,096\,829\,918\,751\,004$ | $0.748\,862\,783\,800\,688\,71$ | 12 | $0.0021$ | $0.424\,853\,237\,360\,272\,3$ | $1.314\,821\,946\,394\,937\,1$ | – | –
23 | RPO20 | $0.036\,123\,079\,700\,446\,183$ | $3.939\,561\,286\,622\,311\,087$ | 9 | $0.0084$ | $1.422\,450\,728\,421\,804\,4$ | $0.719\,778\,559\,348\,493\,8$ | $0.243\,801\,340\,193\,396\,7$ | –

Table 1:  Properties of ECSs found in TCF for $\Gamma=1$, $\eta=0.5$,
$Re_{i}\approx 1200$ and ${Re_{o}={-1200}}$: the temporal period $T$ and shift
$\Phi$, the number of unstable directions $N_{\lambda}^{u}$, the inverse of
the escape rate $\gamma_{n}^{-1}$, the minimal distance to the turbulent
trajectory in lobes 1, 2, and 3, and the discrete symmetries, if any.

We used the shortest ECS (RPO06) to verify that our numerical solutions are
fully resolved. Specifically, after converging this solution using the
standard spatial and temporal discretization, we recomputed it using a finer
discretization and used the corresponding relative residual (8) to quantify
the accuracy of the computed solution. Doubling the spatial resolution in all
directions, while keeping the temporal resolution the same, yielded an
acceptably small value $\varepsilon=6.1\times 10^{-4}$. Doubling the temporal
resolution, while keeping the spatial resolution the same, yielded an even
smaller value $\varepsilon=1.2\times 10^{-6}$.

##  3 Results

###  3.1 Exact coherent structures 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

$\mathcal{E}=\frac{1}{Re^{2}V}\int_{V}{\bf u}^{2}\,dV,$ (10)

rate of energy dissipation

$\displaystyle\mathcal{D}=\frac{1}{Re^{2}V}\int_{V}\boldsymbol{\omega}^{2}\,dV,$
(11)

and helicity

$\mathcal{H}=\frac{1}{Re^{2}V}\int_{V}{\bf u}\cdot\boldsymbol{\omega}\,dV.$
(12)

Here $Re=|Re_{i}-Re_{o}|/2$ is the characteristic scale for the velocity, $V$
is the volume of the flow domain, and $\boldsymbol{\omega}=\nabla\times{\bf
u}$ 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 ${\bf u}^{a}(t)$) explores distinct but connected regions
of state space. For $1\lesssim t\lesssim 40$ and $53\lesssim t\lesssim 57$,
${\bf u}^{a}(t)$ is confined to the part of the chaotic set (which we will
refer to as lobe 1) that is centered at $\mathcal{H}=0$. For $41\lesssim
t\lesssim 53$, ${\bf u}^{a}(t)$ is confined to the part of the chaotic set
(which we will refer to as lobe 2) that is centered 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 ${\bf u}^{a}(t)$ (blue), ${\bf u}^{b}(t)$ (red), ${\bf
u}^{c}(t)$ (green). Shown is the running average taken over a window of $1$
time unit. Note that $10$ time units are removed from the signal of ${\bf
u}^{c}(t)$ at $t=40$, during which the flow remains in lobe 3. The mean
helicity of each lobe are 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 subsection 2.2. The portions of turbulent trajectory
${\bf u}^{a}(t)$ confined to lobe 1 (lobe 2) are shown as a blue (red) cloud
of points, representing different snapshots of the flow. For each ECS ${\bf
u}_{n}$ that is not reflection-symmetric, both ${\bf u}_{n}$ and $K_{z}{\bf
u}_{n}$ are shown in subsection 2.2. Since all three coordinates are invariant
with respect to rotations around the axis, a family of temporally periodic
solutions corresponding to each TW, i.e., $R_{\phi}{\bf u}_{n}$ with
$\phi\in[0,2\pi)$, is mapped to a single point. Similarly, a family of
temporally quasi-periodic solutions corresponding to each RPO is mapped to a
single closed curve. A large number of computed ECSs (TW01, RPO01-RPO12, and
RPO19) 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 (top) and the corresponding two-dimensional projections (bottom). 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 color; lobe 1 shown
in blue, lobe 2 in red, and lobe 3 in green. The computed RPOs (TWs) are shown
as solid lines (filled circles) and their symmetry-related copies as dotted
lines (open circles) of the same color.

(a)  (b)

Figure 4:  Two-dimensional projection of (a) lobe 1 and (b) lobe 3. In each
lobe, the chaotic set (gray point cloud) is in the background. RPOs and TWs
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 RPO lying in that region of the state space to initialize the flow, we
computed a second turbulent trajectory, ${\bf u}^{b}(t)$, which remained
confined to a region of the state space centered 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 ${\bf u}^{b}(t)$ are shown as a green point cloud in subsection
2.2. RPO13-RPO18 and RPO20 as well as TW02 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 ${\bf u}^{a}(t)$ further in time. The
running average of helicity for a portion of this trajectory (we will refer to
it as ${\bf u}^{c}(t)$) 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 ${\bf u}^{c}(t)$ will not be analyzed 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.,
2010).

In the subsequent discussion, we will refer to the portion of ${\bf u}^{a}(t)$
restricted to lobes 1 and 2 of the chaotic set as ${\bf u}^{1}(t)$ and ${\bf
u}^{2}(t)$, respectively. We will also identify ${\bf u}^{3}(t)={\bf
u}^{b}(t)$, since ${\bf u}^{b}(t)$ 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 ${\bf
u}^{1}(t)$, ${\bf u}^{2}(t)$, or ${\bf u}^{3}(t)$ 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 subsection 2.2.

(a)  (b)

(c)

Figure 5:  Mean flow field in (a) lobe 1, (b) lobe 2, and (c) lobe 3. In this
figure and all following 2D slices, the arrows show the in-plane ($r$ and $z$)
components of the flow and the color 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, 1983; Pfister
et al., 1988; Furukawa et al., 2002). 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, 2006). 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., 2003).

The setup 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 positive near the end-caps, as Figure
5(a) illustrates. For N2, the radial velocity is the largest and positive at
the center plane $z=0$; it represents a strong central jet of angular momentum
(Marques & Lopéz, 2006; Altmeyer et al., 2012) 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

subsection 2.2 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 neighborhood 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 neighborhood 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 “exact coherent structure” 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 Taylor-Couette flow ${\bf u}(t)$ is characterized by a fluctuation
$\tilde{\bf u}(t)={\bf u}(t)-\bar{\bf u}$ about an axially symmetric mean flow
$\bar{\bf u}=\langle{\bf u}(t)\rangle_{t}$ that is small compared with this
mean flow, i.e., $\|\tilde{\bf u}(t)\|\ll\|\bar{\bf u}\|$. Hence, the
characteristic magnitude $\sigma=\langle\|\tilde{\bf u}(t)\|\rangle_{t}$ 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 $\|\bar{\bf u}\|$ 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 ${\bf u}$ in the state space to
the family of solutions $R_{\phi}{\bf u}_{n}(\tau)$:

$\displaystyle D_{n}({\bf u})=\min\limits_{\phi,\tau}\sigma^{-1}\|{\bf
u}-R_{\phi}{\bf u}_{n}(\tau)\|,$ (13)

where $\tau$ defines the temporal phase along the RPO (this dependence is
omitted for TWs for which time evolution is equivalent to rotation). The
distance

$\displaystyle D_{n}^{j}(t)=D_{n}({\bf u}^{j}(t))$ (14)

measures how similar (close in the state space) the turbulent flow field
snapshot ${\bf u}^{j}(t)$ is to the family of solutions $R_{\phi}{\bf
u}_{n}(\tau)$. We will use $j=a,b$ when referring to the entire turbulent
trajectories ${\bf u}^{a}(t)$ and ${\bf u}^{b}(t)$ 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, ${\bf u}^{a}(t)$ and ${\bf u}^{b}(t)$, to
various ECSs are shown in subsection 2.2, with the minimal values listed in
subsection 2.2. The lengths of both turbulent trajectories correspond to
$O(10^{4})$ periods of the shortest RPO we found, so they are likely long
enough to sample a significant fraction of all three lobes.

(a)  (b)

Figure 6:  (a) The distance $D^{a}_{n}(t)$ between the turbulent trajectory
${\bf u}^{a}(t)$ confined to both lobes 1 and 2 and every ECS ${\bf u}_{n}$.
All ECSs except for RPO05 and RPO15 are shown in gray. (b) The distance
$D^{b}_{n}(t)$ between the turbulent trajectory ${\bf u}^{b}(t)$ inside lobe 3
and ECSs collocated with this lobe. All ECSs except for RPO15 are shown in
gray. The dashed line indicates our chosen threshold $\bar{D}$ below which two
states are considered “close.”

Let us first consider the turbulent trajectory ${\bf u}^{a}(t)$ 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 gray in subsection 2.2(a), except for two
solutions: RPO05 (in blue) and RPO15 (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 RPO05 and RPO15, 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 RPO05 is low
(high), the turbulent flow is inside lobe 1 (lobe 2). Recall that RPO15 lies
inside lobe 3, which is closer to lobe 2, so the distance to this RPO shows
the opposite trend.

Such behavior is an example of intermittency, which is a characteristic
feature of all turbulent flows, but is particularly apparent in transitional
flows (Chaté & Manneville, 1987) where it commonly manifests itself as
spatiotemporal alternation between laminar and turbulent behavior (Eckhardt et
al., 2007). In the present case, intermittency represents a competition
between different flow patterns shown in Figure 5(a) and (b) – a phenomenon
which has been observed in Taylor-Couette flow at large $\Gamma$ (Tsameret &
Steinberg, 1994) as well as in many other non-equilibrium systems.

subsection 2.2(a) shows that the turbulent trajectory ${\bf u}^{a}(t)$ 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 Newton search also lie in that
lobe. Indeed, each of the fourteen ECSs (TW01, RPO01-RPO12, and RPO19) that is
collocated with lobe 1 in the projection shown in subsection 2.2 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 ${\bf u}^{b}(t)$
confined to lobe 3. For the eight ECSs collocated with this lobe in the low-
dimensional projection (TW02, RPO13-RPO18, and RPO20), the distance is
$D_{n}^{3}(t)\approx 0.7$ on average and gets as low as $0.3$ or so, according
to subsection 2.2(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.

(a)  (b)

(c)  (d)

Figure 7:  Comparison of (a) the turbulent flow field ${\bf u}^{a}(t)$ and (b)
RPO04 at an instant where $D^{a}_{7}(t)=0.34$ is below threshold. Comparison
of (c) the turbulent flow field ${\bf u}^{a}(t)$ and (d) RPO12 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.

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 ${\bf u}^{a}(t)$ with a
nearby ECS for various values of $D(t)$ in subsection 2.2. 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
subsection 2.2. While using such an “ocular norm” to define closeness of
different flow states may appear somewhat arbitrary, it is quite common in
analyzing experimental data (De Lozar et al., 2012).

(a)  (b)

Figure 8:  (a) The minimal distance, $D^{\text{min}}_{nk}$, between RPOs
collocated with lobe 1. Braces on the right of the panel group together RPOs
that are connected by continuation in $Re_{i}$. (b) The probability that a
snapshot of the turbulent trajectory will lie in a neighborhood of the ECS
shown in (a), as a function of tubular neighborhood radius, $\bar{D}$. The
dashed line indicates our chosen threshold below which two states are
considered “close.”

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 neighborhoods 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 neighborhoods, we computed the probability
$P$ that turbulent flow ${\bf u}^{a}(t)$, restricted to lobe 1, is in the
neighborhood of any of the fourteen ECSs collocated with this lobe. As Figure
8(b) illustrates, these neighborhoods cover nearly 100% of lobe 1 for
$\bar{D}\gtrsim 1$, about 40% for $\bar{D}=0.4$, and less than 1% for
$\bar{D}\lesssim 0.2$.

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

$D^{\text{min}}_{nk}=\min\limits_{\phi,t,\tau}\sigma^{-1}\|{\bf
u}_{k}(t)-R_{\phi}{\bf u}_{n}(\tau)\|,$ (15)

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., RPO09-RPO11) 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
${\bf u}^{j}(t)$ is close to a family of ECSs $R_{\phi}{\bf u}_{n}$ provided
$D_{n}^{j}(t)<\bar{D}$, where $\bar{D}=0.4$. As subsection 2.2(a) illustrates,
the turbulent trajectory ${\bf u}^{a}(t)$ comes close to multiple RPOs and one
TW collocated with lobe 1; this happens quite frequently. Similarly,
subsection 2.2(b) illustrates that turbulent trajectory ${\bf u}^{b}(t)$ comes
close to multiple RPOs and one 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 RPO19 (cf. subsection
2.2). Figure 8(a) shows that RPO19 lies far from all other solutions
collocated with lobe 1. This suggests that TW01 and RPO01-RPO12 are actually
embedded in the primary lobe and may play a dynamically important role, while
RPO19 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,
2005). 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. (2020) proposed a topological approach to define shadowing of
unstable periodic orbits 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 ${\bf u}_{n}$ will typically be shadowed for
an interval of time comparable to the inverse of the escape rate

$\displaystyle\gamma_{n}=\sum_{k}\lambda_{n,k},$ (16)

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 RPOs and TWs. It is essentially a generalization of
the approach proposed by Suri et al. (2020) for unstable periodic orbits and
relies on the skew product decomposition of the dynamics (Fiedler et al.,
1996; Sandstede et al., 1999) in the vicinity of relative solutions induced by
continuous symmetries. For each RPO, the computed ECS ${\bf u}_{n}(\tau)$
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 $R_{\phi}{\bf
u}_{n}(\tau)$, where $0<\phi<2\pi/N$ and $0<\tau<T$. Let $R_{\phi}{\bf
u}_{n}(\tau)$ be a point in the family of solutions that is closest to a point
${\bf u}^{a}(t)$ on the turbulent trajectory, such that

$\\{\tau(t),\phi(t)\\}=\arg\min\limits_{\tau,\phi}\sigma^{-1}\|{\bf
u}^{a}(t)-R_{\phi}{\bf u}_{n}(\tau)\|,$ (17)

where $\phi$ is constrained to the interval $[0,2\pi/N)$ for solutions with an
$N$-fold discrete rotational symmetry (e.g., $N=2$ for RPO01 and RPO15). 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 ${\bf u}(t)$ will have $d\tau/dt=1$ and $d\phi/dt=0$
for $0<\tau<T$. Once per period, both $\tau(t)$ and $\phi(t)$ will experience
a jump by, respectively, $T$ and $\Phi$ (see subsection 2.2). The
discontinuities in $\tau(t)$ and $\phi(t)$ can be removed using coordinates
$\tilde{\tau}$ and $\tilde{\phi}$ such that

$\displaystyle\tau$ $\displaystyle=\tilde{\tau}\ \mathrm{mod}\ T,$
$\displaystyle\tilde{\phi}$ $\displaystyle={\phi}\ \mathrm{mod}\ \Phi.$ (18)

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

Specifically, we will consider a turbulent trajectory to shadow an 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 RPO. In
practice, we define similarity in terms of normalized deviations

$\displaystyle E_{\tau}$
$\displaystyle=\min_{t_{0}}\frac{1}{T\gamma_{n}^{-1}}\int_{I(t)}|t^{\prime}+t_{0}-\tilde{\tau}(t^{\prime})|^{2}dt^{\prime}$
$\displaystyle E_{\phi}$
$\displaystyle=\min_{\phi_{0}}\frac{N}{\pi\gamma_{n}^{-1}}\int_{I(t)}|\phi_{0}-\tilde{\phi}(t^{\prime})|^{2}dt^{\prime}$
(19)

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

  1. (a)

$D(t^{\prime})<\bar{D}$ for $t^{\prime}\in I(t)$,

  2. (b)

$E_{\tau}<\bar{E}_{\tau}$,

  3. (c)

$E_{\phi}<\bar{E}_{\phi}$

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

(a)  (b)

Figure 9:  A summary of the shadowing events (marked with black bars) for each
of the two turbulent trajectories, (a) ${\bf u}^{a}(t)$ and (b) ${\bf
u}^{b}(t)$. Each ECS is represented by two rows, one for the numerically
converged solution ${\bf u_{n}}$ (gray) and one for its symmetric copy
$K_{z}{\bf u_{n}}$ (white). Red bars on the right represent the escape times
$\gamma_{n}^{-1}$ to scale. Curly braces on the far right group together RPOs
that are related via continuation in $Re_{i}$.

We chose $\bar{E}_{\tau}=0.0001$ and $\bar{E}_{\phi}=0.0003$, so that
conditions (a)-(c) are satisfied with roughly equal probability. subsection
2.2 summarizes the results for both the turbulent trajectory ${\bf u}^{a}(t)$
confined to lobes 1 and 2 and the turbulent trajectory ${\bf u}^{b}(t)$
confined to lobe 3. In the former case, we find that turbulence shadows both
RPOs 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 subsection 2.2
illustrates, this is not entirely unexpected, since lobe 1 is fairly symmetric
with respect to the reflection.

(a)

(b)  (c)

(d)  (e)

Figure 10:  Turbulent flow ${\bf u}^{a}(t)$ shadowing RPO05. (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 gray 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 (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.

For instance, consider a shadowing event for RPO05 shown in subsection 2.2 and
its symmetry-related copy in subsection 2.2 (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 gray 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 color bar in panels (d)
and (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.

(a)

(b)  (c)

(d)  (e)

Figure 11:  A shadowing event for the reflected copy of RPO05 in lobe 1. A
corresponding movie is available as supplementary material.

Note that, for both RPO05 and its symmetry-related copy, the respective 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. subsection 2.2(b-c) illustrates this particularly convincingly for
RPO05. The similarity is not just qualitative, but quantitative, as subsection
2.2(d-e) shows. For small $D(t)$, it is expected that the turbulent flow
evolves in the same manner as the nearby 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 RPO family even when the corresponding flows are not very
similar. Analogous statements apply to the reflected version of RPO05, as
subsection 2.2 illustrates.

(a)

(b)  (c)

(d)  (e)

Figure 12:  A shadowing event for RPO01 in lobe 1. A corresponding movie is
available as supplementary material.

subsection 2.2 presents a sequence of several shadowing events for RPO01.
These shadowing events are of duration comparable to those for RPO05 and its
reflection, measured in units of $\gamma_{n}^{-1}$, but correspond to quite a
few periods of RPO01. Not only that, turbulence also visits the neighborhood
of RPO01 much more frequently, as subsection 2.2(a) demonstrates. In fact,
this solution is by far the most frequently visited RPO in lobe 1. Note that
RPO01 is symmetric with respect to $K_{z}R_{\pi/2}$, so reflection is
equivalent to a rotation by $\pi/2$. Hence, both RPO01 and its reflection
belong to the same solution family, and both rows in subsection 2.2(a) contain
identical sequences of shadowing events. Also, just like in the case of RPO05,
turbulent flow is essentially indistinguishable from RPO01 during the
shadowing episodes.

(a)

(b)  (c)

(d)  (e)

Figure 13:  A shadowing event for TW01 in lobe 1. A corresponding movie is
available as supplementary material.

subsection 2.2 illustrates shadowing of TW01. Rather unexpectedly, TW01 is
found to be shadowed even more frequently than RPO01. In fact, when turbulent
trajectory ${\bf u}^{a}(t)$ is inside lobe 1, it spends more time near TW01
than any other ECS according to subsection 2.2(a). Moreover, when TW01 is
shadowed, then typically so is RPO01. This is not particularly surprising, as
RPO01 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 RPO01 being born in a Hopf bifurcation
of TW01, which happens to lie close in the parameter space. It should also be
mentioned that the length of the shadowing intervals for TW01 tends to be
quite large (e.g., around $8\gamma_{n}^{-1}$ for the event shown in subsection
2.2) compared with typical shadowing intervals for the RPOs embedded in lobe
1.

We have not found any shadowing events for RPO19 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 RPO (as well as TW01) embedded in lobe 1 is being shadowed by
turbulence. Recall that ${\bf u}^{a}(t)$ 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.

(a)

(b)  (c)

(d)  (e)

Figure 14:  A shadowing event for RPO15 in lobe 3. A corresponding movie is
available as supplementary material.

For completeness, we have also performed a similar analysis for the turbulent
trajectory ${\bf u}^{b}(t)$ which lies in lobe 3. It was found to shadow
RPO13-RPO16, and RPO18. Since this lobe breaks the reflection symmetry, none
of the reflected copies of these RPOs were shadowed. An example of shadowing
for RPO15 is shown in subsection 2.2. 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 RPO15 is shadowed for almost three (very
short) periods in subsection 2.2, this interval corresponds to only 0.022 in
nondimensional units, compared with around $0.1$ for RPO01 and RPO05.

##  4 Discussion

We have considered a new regime of small-aspect-ratio Taylor-Couette flow
where the two cylinders rotate in opposite directions, so both centrifugal and
shear instabilities may induce and sustain turbulent flows. For the Reynolds
numbers considered, we find that turbulent flows explore several distinct
regions of the state space. These regions correspond to five lobes of the
associated chaotic set that are dynamically connected to each other.

Discovery of lobe 3 was prompted by the observation that a large number of
ECSs were found in the same region of the state space outside of lobes 1 and
2. While it is expected to find clusters of ECSs in regions of the state space
inhabited by turbulence, to the best of our knowledge, this is the first
example where a cluster of ECSs was found to predict the presence of a chaotic
set supporting turbulent flow. ECSs inside lobe 3 were found using both
parameter continuation of ECSs inside lobe 1 and using a Newton-Krylov solver
with initial conditions confined to lobe 1. This suggests that computation of
large sets of ECSs is a fairly robust way of identifying initial conditions
that lead to turbulence, at least of transient variety.

The majority of the computed ECSs were found to be collocated with either lobe
1 or lobe 3. Somewhat surprisingly, none of the ECSs were found to be
collocated with lobe 2, although this may be due to the initial conditions for
the Newton-Krylov solver lying mostly in lobe 1. Furthermore, some of the ECSs
(e.g., TW03 computed via continuation) were found to lie outside of all three
lobes of the chaotic set, which is not entirely surprising. Of the ECSs
collocated with one of the lobes, the majority are RPOs, although TW01 and
TW02 are also collocated with lobes 1 and 3, respectively. More importantly,
most of the ECSs collocated with one or the other lobe are found to be
shadowed by turbulent trajectories. A similar result appears to also hold for
turbulent pipe flow, although the respective study (Budanur et al., 2017) has
only verified condition (a) for shadowing of a single RPO.

Closeness and shadowing are often assumed to be synonymous in the literature,
which has fostered confusion in the field. In fact, it is possible for the
turbulent trajectory to be close to an ECS and not shadow it. In particular,
closeness implies that $D$ is small regardless of whether the two solutions
co-evolve. On the other hand, shadowing implies co-evolution, not just
closeness, so that the ECS describes the dynamics of turbulent flow over some
period of time. Hence, shadowing of an ECS implies that the ECS is dynamically
relevant, while closeness to an ECS does not. Of course, for infinitesimally
small $D$, closeness and shadowing do become equivalent. However, in practice
turbulence never approaches any ECS infinitesimally closely. Our results
suggest that Euclidean distance $D$ between turbulent flow and an ECS family
becoming small compared with its mean value is neither a necessary nor a
sufficient condition for co-evolution. As Figures 2.2-2.2 illustrate, one
routinely finds co-evolution in terms of variables $\tau$ and $\phi$
parameterising the group manifold, (i.e., conditions (b) and (c) being
satisfied) for values of $D$ that are not particularly small. Similarly, small
values of $D$ do not guarantee that $\tau$ and $\phi$ evolve as they should
when turbulence shadows an RPO. Nonetheless, there is a strong correlation
between the three conditions (a), (b), and (c): when one is satisfied, more
often than not so are the other two.

More importantly, given that $D$ never becomes particularly small, we find
many convincing examples of shadowing for both RPOs and TWs on accessible time
scales, despite our library of ECSs being relatively small. This is critical
for the practical utility of an ECS-based framework for a dynamical
description of turbulent flow. If the computed ECSs were rarely shadowed by
turbulence, that would imply a serious problem with this framework, indicating
that dynamically important solutions have not been found and possibly do not
even correspond to either RPOs or TWs. Admittedly, it is quite possible that
(relative) quasi-periodic solutions may be more dynamically relevant than,
say, the periodic ones. Of course, such a possibility cannot be excluded based
on our results. Indeed, even for lobe 1, the neighborhoods of a dozen or so
embedded RPOs and TWs (plus their reflections), cover less than a half of this
lobe, despite their relatively generous size, as illustrated in Figure 8(b).

On the other hand, we found that multiple solutions may be shadowed
simultaneously. As subsection 2.2 illustrates, TW01 and RPO01 are frequently
shadowed at the same time. The same is true of RPO02 and RPO08 or RPO07 and
RPO12. This is not coincidental as the respective solutions are themselves
close. For instance, RPO07 and RPO12 shadow each other for their entire
duration, as illustrated by a movie provided in the Supplementary Material.
Close solutions are all related via parameter continuation; groups of related
solutions are indicated with brackets in Figure 8(a) and subsection 2.2(a).
For uniformly hyperbolic systems where periodic orbits are dense (Gaspard,
2005), simultaneous shadowing of multiple unstable solutions is expected. The
chaotic set underlying turbulence is not uniformly hyperbolic, as explained
below. Despite this breakdown of uniform hyperbolicity, we find some RPOs to
lie close enough to be shadowed simultaneously.

Unexpectedly, we found that the ECS that is shadowed the most by turbulent
flow is a TW, not an RPO. This result is potentially quite significant, as it
suggests that an extension of periodic orbit theory to systems with continuous
symmetries may have to include contributions from solutions other than RPOs.
Since turbulence spends a large fraction of time in the neighborhood of TW01,
excluding the contribution from this ECS to the average of any observable
would greatly impact the corresponding temporal mean. Of course, it is
possible that it is RPO01, rather than TW01, that plays the important
dynamical and statistic role, with TW01 being shadowed because RPO01 is.
Nonetheless, we chose a set of parameters that happens to be close to the
bifurcation where RPO01 is born. As a result, RPO01 and TW01 are not very
distinct. A similar analysis to ours would have to be repeated further away
from the bifurcation (e.g., at a higher $Re_{i}$) to determine which solution
plays a more important role.

Finally, we find that the shadowing property is robust to small changes in
system parameters. As many of the solutions listed in subsection 2.2 were
found through continuation in $Re_{i}$, there is a small deviation in this
parameter from that describing turbulent flow. This variation emulates the
discrepancies one might see when comparing numerically computed solutions to
turbulent flows in experiment.

##  5 Conclusions

We set out to determine whether ECSs of one type or another are present and
being shadowed by turbulent flow in a geometry with continuous symmetries and
boundary conditions exactly matching a realistic experimental setup. Both
questions were answered affirmatively for a small-aspect-ratio TCF with
counter-rotating cylinders. We found two dozen RPOs and TWs (not counting
their symmetry-related copies) and determined that many of them are shadowed
on accessible time scales, some rather frequently. The majority of ECSs we
found are RPOs, so it is not surprising that it is this type of ECSs that is
shadowed the most frequently. What was unexpected is that the single most
shadowed solution is a TW.

These results are quite significant, as they provide clear and unambiguous
evidence supporting Hopf’s picture of turbulence as a deterministic walk
through neighborhoods of various unstable solutions of the Navier-Stokes
equation. The shadowing property implies that we can predict the evolution of
turbulent flow over some interval of time, justifying the use of the ECS-based
framework for a deterministic, dynamical description of turbulence. The length
of this interval is not universal and is determined, as could be expected, by
the degree of instability of the solution that is being shadowed.

Another key property of chaotic dynamics that has not been addressed fully is
ergodicity. The number of ECSs we computed is insufficient for their
neighborhoods to cover any of the lobes of the chaotic set, so we cannot make
any conclusive statements regarding this property, except for one. For ECSs
embedded into both lobes 1 and 3, the number of unstable degrees is not
constant. As shown by Kostelich et al. (1997), this implies that the dynamics
are not uniformly hyperbolic in either lobe, so we should not expect the
ergodic property to hold. This result is not exclusive to the flow we
considered; the same observation applies to both pipe flow (Willis et al.,
2013; Budanur et al., 2017), 2D Kolmogorov flow (Chandler & Kerswell, 2013;
Lucas & Kerswell, 2015) and its quasi-2D experimental realization Suri et al.
(2018, 2020). The implications of this for a dynamical theory of turbulence
remain unclear.

This study focused almost exclusively on the dynamical significance of various
ECSs. In fact, an ECS-based framework can also be used to connect a
deterministic, dynamical description of fluid turbulence with a more
traditional, statistical description. In particular, partition of the chaotic
set into neighborhoods of various ECSs can be used to compute temporal means
of any observable as a weighted sum over different ECSs. At present it remains
unclear what types of ECSs should be included in the sum and how the weights
should be computed. Initial studies (Kazantsev, 1998; Chandler & Kerswell,
2013; Lucas & Kerswell, 2015) aiming to address such issues had various
limitations and were not conclusive, so further work in this area is needed.
We will consider this issue in a subsequent publication.

[Acknowledgements]The authors would like to thank Marc Avila for sharing his
Taylor-Couette flow code and to gratefully acknowledge financial support by
ARO under grant W911NF-15-1-0471 and by NSF under grant CMMI-1725587.

[Declaration of interests]The authors report no conflict of interest.

[Supplementary data]Movies illustrating various shadowing events are available
at
http://www.cns.gatech.edu/~roman/tcf/.

[Data availability statement]Exact coherent structures and their key
properties are available on GitHub at
https://github.com/cdggt/tcf/tree/main/eta0.50.

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