**2. Experimental Methods**

The Taylor–Couette experiment used in this study consists of two concentric cylinders with radii *ri* = (110.25 ± 0.025) mm and *ro* = (112.53 ± 0.05) mm leading to a radius ratio η = 0.98 and an azimuthal length of 311 gap width *d* = *ro* − *ri* = 2.28 mm. The Reynolds number of the inner (outer) cylinder with angular velocity ω*<sup>i</sup>* (ω*o*) is defined as *Rei* = ω*irid*/ν (*Reo* = ω*orod*/ν), where ν is the kinematic viscosity of the working fluid. The azimuthal direction is in our system the streamwise direction and is naturally periodic (in contrast to Couette flow experiments); this eliminates end effects in the streamwise direction. The axial (spanwise) direction is bounded by the axial lids and has a length of 263*d*. The lids can be rotated independently of the cylinders. Their Reynolds number is based on the radius of the outer cylinder (*Re*lid = ωlid*rod*/ν). In many Taylor–Couette experiments, the lids are attached to the outer cylinder to reduce the Ekman pumping, see, e.g., [16,39–41]. For example, spiral patterns are less influenced by the axial lids, when the lids co-rotate with the outer cylinder, than when they are stationary [42]. The effect of axial boundary conditions was investigated systematically in experiments [43] and in simulations [44], that showed that rotating the lids at angular speeds between the inner and outer cylinder leads to laminar flows closest to circular Couette flow. For our setup and selected parameter regime, *Re*lids = −800 minimized end effects, but the spatio-temporal dynamics was identical for lids attached to the outer cylinder *Re*lids = −1000, and for stationary lids *Re*lids = 0, because of the large height-to-gap aspect ratio. Rotating the lids merely led to a slight stabilization of the laminar flow and hence to a small shift of the onset of turbulence to slightly higher *Rei*.

The viscosity of the working fluid silicone oil was determined by measuring the onset of Taylor vortices for stationary outer cylinder as the inner cylinder rotation was increased. Specifically, the value of the viscosity was selected to match the critical inner Reynolds number obtained with a linear stability analysis of laminar, circular Couette flow between infinite cylinders (*Rei*,*<sup>c</sup>* = 292 at *Reo* = 0 ). The accuracy of this method and of our experiment is verified in the excellent agreement obtained with the linear stability results throughout the counter-rotating regime. In particular, the discrepancy is less than 1% in *Rei* when comparing the experimentally measured and the theoretical stability curves. For the visualization of the flow the working fluid silicone oil was seeded with aluminium platelets.

The turbulent fraction was determined by analyzing the images from a high speed camera used to monitor the flow. The flow was seeded with highly reflective aluminum platelets (Eckart, Effect Pigments, STAPA WM Chromal V/80 Aluminum) in a concentration below 1% in weight (and volume). In turbulent flows these tracers are randomly oriented and reflect light efficiently. Turbulent flow patches appear therefore brighter than laminar regions. In our image processing code we use this difference in the light intensity to distinguish laminar from turbulent regions by thresholding. The

turbulent fraction is calculated at each instant of time in the spatio-temporal diagrams (see, e.g., Figure 2) as axial length covered by turbulent flow in comparison to the axial length of the field of view. Further details of the image analysis are provided in [35]. Videos were typically recorded with 80 Hz and the resolution in the axial direction was 1920 pixels and in the azimuthal direction between 5 and 1080 pixels, from which only 3 were used for the generation of the spatio-temporal diagrams and hence the quantitative analysis. The measurements shown in this paper consist of three independent sets of experiments with slightly different viscosities and different field of views of the camera, each of them optimized for the specific analysis. For the measurements shown in Figures 1 and 3, the working fluid silicon oil has a viscosity of ν = (4.65 ± 0.02)*cSt*. The field of view of the camera in Figure 3 was (50*d* × 80*d*), corresponding to about 10% of the total area and was located 46*d* above the lower lid. For the measurements in Figures 4 and 5 the viscosity was ν = (4.55 ± 0.02)*cSt* and the field of view consisted of a line of 3 pixel width and an axial length of 245*d*, which started 5*d* above the lower lid. For the measurements in Figures 6 and 7 the viscosity was ν = (4.41 ± 0.02)*cSt* and the field of view was (5*d*× 170*d*) and started 25*d* above the lower lid. More details of the setup and the image analysis and processing that are omitted here can be found in [35].

**Figure 2.** (**a**,**c**) Spati-temporal dynamics of two selected lifetime measurements at *Rei* = 530, *Re*lids = −800. (**b**,**d**) Corresponding instantaneous (black solid line) and averaged (red thick line) turbulent fraction. The average turbulent fraction is calculated in windows of about 9 s (moving-average technique) to illustrate the long-time dynamics and is used to detect the relaminarization of the flow. The left green line marks the time of the reduction in *Rei* and the right green line the decay of turbulence. The determined lifetime corresponds to the time interval between the two green lines.

**Figure 3.** Snapshots of typical flow patterns in counter-rotating Taylor–Couette flow. (**a**) The linear instability arises in the counter-rotating regime in the form of laminar spirals (snapshot taken at *Rei* = 560, *Reo* = −700). (**b**) Laminar spirals can coexist with turbulent spots frequently decaying and arising, or they can aligne into stripes, as shown here (*Rei* = 700, *Reo* = −700). (**c**) Laminar-turbulent intermittency in the form of subcritical turbulent stripes (*Rei* = 600,*Reo* = −1000). (**d**) For decreasing *Rei* the regions of laminar flow around the turbulent stripes increase in area (*Rei* = 540,*Reo* = −1000). The field of view corresponds here to about 10% of the total system size area. The axial lids are stationary in all snapshots (*Re*lids = 0). All snapshots were taken in the statistically steady regime.

**Figure 4.** Spatio-temporal dynamics of subcritical turbulence *Re*lids = −800 (**a**) and *Reo* = −1000 (**b**) following a reduction in *Rei*. Turbulent stripes dominate the dynamics at *Rei* = 630, prior to an abrupt reduction to *Rei* = 530 (green line). The turbulent fraction decreases immediately after the reduction in *Rei*, but it takes about 20 s for the flow to adjust into a (metastable) statistically steady state. The long-time dynamics of these two cases are is displayed in Figure 2a,b, respectively. The axial direction is in dimensionless units (i.e., normalized with the gap width *d*).

**Figure 5.** Lifetime statistics at *Reo* = −1000 and *Re*lids = −800. Shown is the survival probability of turbulence (in a logarithmic scale) as a function of time for several *Rei*, as indicated in the legend. The symbols denote individual measurements, which are sorted in increasing survival time to construct the survival probability function. In all cases, the initial condition was a turbulent flow at *Rei* = 630 and the rotation of the cylinder was suddenly changed to the desired *Rei*.

**Figure 6.** Excerpt of the spatio-temporal dynamics of turbulent stripes (*Reo* = −1000 and *Re*lids = 0) above the critical point for the onset of sustained turbulence. (**a**) *Rei* = 525, (**b**) *Rei* = 532.

**Figure 7.** Second-order phase transition in counter-rotating Taylor–Couette flow (*Reo* = −1000 and *Re*lids = 0). The turbulent fraction increases smoothly from a minimum of 7% at *Rei* = 525 up to 60% at *Rei* = 600. The error bars indicate a 1% deviation in *Rei*, which estimated from the discrepancy between linear stability analysis and experiment in Figure 1. The black line is a fit of the form *Tf* = *a* - *Rei* − *Rei*,*<sup>c</sup>* β*DP* , with <sup>β</sup>*DP* <sup>=</sup> 0.583, to the data points in the vicinity of the critical point (524 < *Rei* < 540). The fit parameters are *a* = 0.0667 and *Rei*,*<sup>c</sup>* = 524.1. Below the critical point (grey region), turbulence is transient.

#### **3. Results**

The experiments reported in this work were performed at *Reo* = −1000 as indicated by the red line in Figure 1. The dynamics obtained at this selected *Reo* is representative for the subcritical regime and hence also for other *Reo*.
