**1. Introduction**

In shear flows, turbulence tends to first appear in spatially localized patches that are interspersed by quiescent, laminar regions, a phenomenon commonly referred to as spatio-temporal intermittency. The resulting flow pattern chaotically changes in time and unless the entire flow relaminarises, it never settles to a steady state. One of the earliest reports of laminar turbulent intermittency dates back to Osborne Reynolds and his study of pipe flow [1]. The corresponding turbulent "flashes" or "puffs" are quasi-one-dimensional, meaning that they tend to fill out the radial-azimuthal pipe cross-section, whilst being localized in the streamwise direction [2]. Puffs have a well defined mean length; however, their spacing and hence the size of the laminar gaps is irregular and continuously changes. The resulting overall flow pattern can be accurately modeled as one dimensional [3]. In flows that are extended in two spatial dimensions, but strongly confined in the third (such as channel and Couette flows), turbulence forms elongated stripes [4–7]. Here turbulence fills the wall normal gap and is localized in the extended streamwise and spanwise directions. The resulting laminar-turbulent intermittent stripe pattern can be regarded as quasi-two-dimensional.

In quasi-one- and two-dimensional cases alike, individual patches of turbulence have finite lifetimes and eventually decay. Early propositions that individual turbulent patches (or turbulence in small domains) become sustained at a critical point [8–11] turned out to be incorrect. Despite their often long lifetimes individual patches remain transient and eventually decay following a memoryless process [12–16]. In line with other contact processes such as directed percolation [17] and coupled map lattices [18,19] and as pointed out in the context of shear flows [20,21] spatial proliferation of active sites can give rise to a phase transition to sustained turbulence. Specifically it has been demonstrated for pipe flow [22,23] that turbulence becomes sustained via a contact process where individual localized patches remain transient but can seed new patches before they decay. Also puff splitting has been found to be a memoryless process, a circumstance that allowed to determine the critical point for pipe flow as the balance point between lifetimes and splitting rates [23].

A key remaining question regarding the onset of turbulence, for both one-dimensional and two-dimensional cases alike, is whether the transition is of first or second order (in the context of contact processes and phase transitions in statistical physics, see [24]). In a second-order phase transition, the turbulent fraction decreases continuously to zero as the Reynolds number is decreased toward the critical point, whereas in a first-order phase transition the turbulent fraction jumps from a finite value to zero at the critical point. Hence first-order transitions are referred to as discontinuous and second-order transitions as continuous. In both cases, however, the laminar flow is linearly stable and because of the hysteresis the flow must be initialized with turbulence to measure the transition. While for pipe flow the transition is presumed continuous [3], this so far could not be shown explicitly due to the excessive time scales that prohibit to reach a statistical steady state sufficiently close to the critical point [25]. In a circular Couette experiment of large azimuthal and small axial aspect ratio, where flow patterns like in pipe flow can only evolve in one spatial dimension, the transition has been shown to be continuous [26] and to fall into the directed percolation (DP) universality class.

In an earlier study Bottin and Chatté [8] characterized the transition to turbulence in an experimental study of planar Couette flow in a moderately large aspect-ratio (190*d* × 35*d* in the streamwise and spanwise direction, where *d* is the gap). In this two dimensional setting, the turbulent fraction was about 30% close to the onset of sustained turbulence and dropped dramatically to zero (laminar flow) as the Reynolds number was reduced. The authors suggested that the onset of turbulence in plane Couette flow corresponds to first-order phase transition. Duguet et al. [27] did direct numerical simulations of a larger system (400*d* × 178*d*), but with substantially shorter observation times (2 <sup>×</sup> 104 advective time units), and reported similar results. More recently, Chantry et al. [28] examined numerically the onset of turbulence in Waleffe flow. In contrast to Couette flow, in this case stress-free boundary conditions are applied at the walls and the flow is driven by a sinusoidal body force. The choice of boundary conditions greatly reduces computational cost and allowed direct numerical simulations of a very large aspect-ratio system (1280*d* × 1280*d*) for very long observation times (exceeding 2 <sup>×</sup> 10<sup>6</sup> advective time units). Their simulations compellingly show that transition in this simple model system falls in the universality class of two-dimensional directed percolation. While suggestive, it nevertheless remains unclear if for quasi-two-dimensional Couette type flows the transition is either of first or second order. For a recent review of the flow patterns and dynamics of wall-bounded flows extended in two directions, see Tuckerman et al. [7].

In Taylor–Couette flow between two counter-rotating cylinders, the flow dynamics is qualitatively similar to plane Couette flows [4,5,16,29,30] provided that the laminar velocity profile is linearly stable. Indeed, in the narrow-gap limit η = *ri*/*ro* → 1, where *ri* and *ro* are the radii of the inner and outer cylinders, Taylor–Couette flow turns into rotating plane Couette flow [31]. For fully turbulent flows, the dynamics of Taylor–Couette flow converges to that of rotating plane Couette flow already for moderately small gaps η ≥ 0.9 [32]. By contrast, the dynamics of transition for exactly counter-rotating cylinders is alike that of plane Couette flow only for very narrow gaps η ≥ 0.993 [33]; for larger gaps the linear centrifugal (Rayleigh) instability occurs at lower Reynolds number than the subcritical transition. We note that a new linear instability of counter-rotating Taylor–Couette flow was recently discovered [34], however this instability occurs for extremely high Reynolds numbers (for η > 0.9, <sup>|</sup>*R eo*|>108, where *Reo* is the Reynolds number of the outer cylinder) and disappears in the narrow gap limit. This instability is far away in parameter space of the experiments performed here, with *Reo* = O - 103 . In Figure 1 we show a regime diagram of counter-rotating Taylor–Couette flow of radius ratio of η = 0.98. In the infinite-cylinders case, the onset of Taylor vortices is at *Rei* = 292 when the

outer cylinder does not rotate (*Reo* = 0). For increasing counter-rotation of the outer cylinder, the linear stability threshold rises to higher *Rei* and the stability boundary previously measured with our experimental setup [35] is in excellent quantitative agreement with the linear stability analysis of the infinite-cylinder case (solid line in Figure 1), and to a lesser extent also with the experimental measurements of Prigent and Dauchot [36]. For moderately strong counter-rotation (*Reo* < −800), turbulence can be triggered via finite amplitude perturbations well below the linear instability. Such perturbations occurred naturally in the experimental setup of Prigent and Dauchot [36], whereas in our setup a progressively growing band of hysteresis between the onset of linear instability and the decay of sub-critical turbulence can be observed.

**Figure 1.** Stability diagram of counter-rotating Taylor–Couette flow with radius ratio η = 0.98 and stationary lids, *Re*lids = 0. The solid line shows the linear stability boundary in the infinite–cylinder case. As the the Reynolds number of the outer cylinder (*Reo*) is decreased, the linear instability of the laminar, circular Couette flow is shifted to higher Reynolds number of the inner cylinder (*Rei*). The empty symbols denote our experimental measurements of the onset of instability, obtained by increasing *Rei* at fixed *Reo*, which we reported previously in [35]. Subcritical turbulence in the form of turbulent stripes and spots is found in the shaded region starting at *Reo* ≈ −800; the full symbols mark the relaminarization of subcritical turbulence and were obtained by decreasing *Rei* at fixed *Reo*, in order to detect hysteresis. In this paper, the subcritical transition at *Reo* = −1000 is analyzed in more detail (statistically) to shed light on the nature of this phase transition to turbulence. For comparison the data of Prigent et al. and coworkers (diamonds) [5,36,37] for a similar radius ratio η are shown, indicating the sensitivity of the flow to finite amplitude perturbations.

In Avila and Hof [35], the critical Reynolds number for self-sustained turbulence was measured by quasi-statically decreasing *Rei* in steps of 1 min. This measurement procedure is suited to obtain a rough estimate of the transition border, but does not take into account the stochastic nature of turbulence decay. Measurements of the lifetimes statistics are required here, as previously performed in a small aspect ratio Taylor–Couette flow (55*d* × 34*d*) [16]. Compared to all previous quasi-two dimensional Couette or Taylor–Couette experiments, our system's streamwise-spanwise area is at least 12 times larger (311*d* × 263*d*), see Table 1. This allows us to study the nature of the turbulence transition with a reduced influence of finite-size effects. We show that lifetimes are exponentially distributed below the critical point and that the increase of the turbulent fraction beyond the critical point is continuous and therefore of second order.

**Table 1.** Summary of experiments (first four rows) and direct numerical simulations (last four rows) of plane Couette and Taylor–Couette flows in the sub-critical regime. Only published works in which lifetimes were determined statistically and/or the turbulent fraction close to onset was measured are listed. The systems investigated by Lemoult et al. [26] and Shi et al. [38] are quasi-one-dimensional (strongly confined in the spanwise direction). In their experiments and DNS the minimum measurable turbulent fraction is constrained by the streamwise length (instead of the area).

