*3.1. Global Stability and Coherent Structures Close to Onset*

In the present subsection, we recall some key results of Ref. [36] together with some updated predictions. The investigation of the onset of turbulence starts with the determination of the global Reynolds number *Reg*, defined as the highest Reynolds number below which no turbulence can survive (at least in the thermodynamic limit, i.e., over infinite observation times in unbounded domains). Since the flow is subcritical, using a given type of initial condition for this task can lead to overestimates of *Reg*. The commonly adopted strategy, both in experiments and numerics, is that of an adiabatic descent [39] initiated from a turbulent state at sufficiently high Reynolds number. In the limit where the waiting time between successive diminutions of *Re* is sufficient long, the value at which turbulence gets extinct is a good approximation of *Reg*. Figure 3 displays information about *Reg* depending on the radius ratio *η*. For *L<sup>θ</sup>* = 2*π* (*n* = 1), *Reg* increases monotonically with decreasing *η*. For larger *Lθ*, *Reg* is always smaller than for the case with *L<sup>θ</sup>* = 2*π* and the same value of *η*, with a now decreasing trend for *Reg*(*η*) which is even more marked once *η* ≤ 0.3. The values of *L<sup>θ</sup>* needed to obtain this curve robustly are all listed in Table 1. As for the case of artificially extended aCf at *η* = 0.1, the result for *L<sup>θ</sup>* = 128*π* is plotted in the figure. The parameter range strictly below *η* = 0.1 has not been investigated.

**Figure 3.** Radius ratio *η* dependency of the global critical Reynolds number Re*g*. The plot includes the pCf limit *η* → 1 from Ref. [21] (labeled "\*1"), as well as DNS data from Ref. [36] for *η* = 0.5 and 0.8 (labeled "\*2"). Triangles: original aCf with *L<sup>θ</sup>* = 2*π* is plotted using triangles; circles: artificially extended aCf (*L<sup>θ</sup>* > 2*π*).

The fact that artificially extended systems display a lower threshold in *Re* indicates that some specific spatiotemporal regimes, specific to large *L<sup>θ</sup>* and not allowed for in narrow domains, are able to maintain themselves against relaminarization. As in Ref. [36], we can compare typical snapshots of the velocity fields in the corresponding regime in order to highlight the qualitative differences. Figures 4 and 5 display instantaneous snapshots of the radial velocity at mid-gap (i.e., *r* = (*rin* + *rout*)/2) at respectively *η* = 0.3 and 0.1, one very close to *Reg* (left column) and the other slightly above it (right column). Each row corresponds to a different value of the integer *n* (*n* = 1, 16, 48, and 64), i.e., another value of *Lθ*. When *n* = 1, the one-dimensional intermittency is reminiscent of the dynamics in cylindrical pipe flow [40]. The differences between different values of *η* emerge only for higher *n*. For *η* = 0.3, the stripe patterns exhibit an obliqueness typical of most laminar-turbulent patterns [25,26,37,41]. However, it is visually clear that the situation is different for *η* = 0.1, with shorter structures and less pronounced obliqueness. It is not immediately clear whether the effective dimensionality of the proliferation process is rather one or two. These issues can be addressed using the determination of critical exponents, as will be done in the next subsection.
