**3. Results**

The entire adiabatic descent is shown using a space-time diagram of the crossflow energy shown in Figure 2a

$$E\_{cf} = \frac{1}{2} \int (\mu\_y^2 + \mu\_z^2) dy\tag{5}$$

evaluated at an arbitrary value of *z* (here *z* " *Lz*{2). The space variable is expressed in a frame moving in the streamwise direction with the mean bulk velocity *Ub*p*G*q for that particular value of *ReG <sup>τ</sup>*. Since *ReG <sup>τ</sup>* is lowered over the course of time, this allows one to capture the different flow regimes preceding full relaminarization. The intensity of turbulence, measured here by the value of *Ec f* , is seen to gradually increase as *ReG <sup>τ</sup>* is lowered. At high *ReG <sup>τ</sup>*, the so-called featureless turbulence occupies the full domain, as shown in Figure 2b at *ReG <sup>τ</sup>* " 100 using isocontours of *τ*<sup>1</sup> p*x*, *z*q " *τ*p*x*, *z*q ´ *τlam*. As *ReG τ* is lowered, turbulence self-organizes into the recognizable pattern regime [17] shown in Figure 2c for *ReG <sup>τ</sup>* " 80. As *ReG <sup>τ</sup>* is further reduced the turbulent zones become sparser (see Figure 2d for *ReG <sup>τ</sup>* " 60). The spatially localized turbulent regions emerge as narrow stripes throughout the process of decreasing *ReG <sup>τ</sup>* while the gaps between them constantly increase in size. The emerging patterns never feature an array of strictly parallel stripes like in former computational approaches [19,31,41], instead they feature competing orientations as in pCf [4], see Figure 2b–d. In this regime the pattern travels with a streamwise convection velocity slightly slower than *Ub*p*G*q. Within the quasi-laminar gaps, *Ec f* reaches very low values, at least an order of magnitude less than in the core of the turbulent stripes. The lower *ReG <sup>τ</sup>*, the lower these values. Below *ReG <sup>τ</sup>* " 50 the stripe pattern eventually breaks up to form independent turbulent bands of finite length, all parallel to each other [34], as shown in Figure 2e for *ReG <sup>τ</sup>* " 40. The new resulting pattern as a whole shows negligible spanwise advection, while it propagates in *x* with a velocity close to Ě*ub* [42]. The independent turbulent bands show enhanced motility in both directions *x* and *z*. This motion relative to the frame of reference causes the tilt of the stripes seen in Figure 2a for *ReG <sup>τ</sup>* ą 50 as well as the apparent increase of thickness.

In pipe flow it was noted recently [43] that the emergence of spatial localization does not imply the proximity to the transitional point (below which turbulence is not sustained) as long as the statistics about the size of the laminar gaps fail at displaying power-laws tails. The laminar gaps are estimated as the streamwise distance *lx* between local maxima of *τ* (values lower than *τ*`*σ*p*τ*q, with *σ* the standard deviation, have been discarded). The cumulative distribution (CDF) of the laminar gap size is shown in Figure 3 in lin-log coordinates. For all values of *Re<sup>τ</sup>* shown, it shows an exponential tails and no algebraic part. Exponential distributions are a hallmark of spatio-temporal intermittency, unlike critical phenomena which are characterized by algebraic/power law related to the scale invariance property. The entire regime of channel flow for 39 ď *ReG <sup>τ</sup>* ď 100 can be described as being spatiotemporally intermittent, and is hence far above any critical point. Please note that the critical point of pPf is estimated to approximately *Recl* " 660 [18] i.e., *ReG <sup>τ</sup>* « 36 and falls outside the range of parameters investigated here.

**Figure 2.** (**a**) Space-time diagram of *Ec f* p*x* ´ *Ub*p*G*q *t*, *t*q for *z* " *Lz*{2 during the adiabatic descent protocol, in a frame travelling in the *x*-direction at the mean bulk velocity *Ub*p*G*q. Vertical axis: time with corresponding values of *ReG <sup>τ</sup>* values indicated. (**b–e**) isocontours of *τ*<sup>1</sup> p*x*, *z*q for *ReG <sup>τ</sup>* " 100, 80, 60, 40.

**Figure 3.** CDF of laminar gap size for *ReG <sup>τ</sup>* " 80, 60, 50, 40.
