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tokens: as local instances of a decreasing mean field. By way of illustration of the key features of the moderate- and ultra-high-beta regimes, we refer the reader to Fig. 13 . This shows two numerical experiments, one with β Ω/ S , and the other with β > Ω/ S , in which the sign of the shear is alternated several times, so the system goes through multiple successive periods of decaying (firehose-unstable) and growing (mirror-unstable) mean field (this is analogous to successive instances of the experiments described in Sections 3.1 and 3.2 ). The key point is that at ultra-high β, periods of firehose growth and decay alternate, the system spends most of its evolution close to the firehose threshold, the mirror instability is almost completely suppressed, while the mean magnitude of the magnetic field barely changes at all; in contrast, at moderate β, the periods of decaying/growing mean field feature (relatively) lively firehose/mirror fluctuations which, however, do not survive long past the point when they cease to be driven by the external shear. In the context of the emergence of the threshold value of β, we have also carried out a β scan of the driven firehose turbulence, to complement a previous study by Kunz et al. ( 2014a ), where only one value of β was used. It has turned out (Section 2.1 ) that the saturation level of firehose fluctuations depends on β, viz. , , so when β ≳ Ω/ S , we have . The latter saturation regime takes the system on the order of one shear time to achieve and follows after a long phase of secular growth, during which the pressure anisotropy is pinned at marginal stability purely by the increase in the fluctuation energy, with no appreciable scattering of particles. This is broadly in line with the earlier theoretical argument of Schekochihin et al. ( 2008 ) and Rosin et al. ( 2011 ), although unlike in their study, the fluctuations are oblique, rather than parallel. We do not as yet have a complete theoretical understanding of this regime, or of the β and S dependence of the saturated firehose amplitude. Figure 13. Time evolution of the magnitude of the mean field (solid black), mean magnitude of the field 〈 B 〉 (bold red), firehose fluctuation level (solid blue) and mirror fluctuation level (dashed blue) in two runs where the sign of shear was alternated several times: (a) the case of β Ω/| S |, (b) the case of β > Ω/| S |. Note the much-delayed onset (amounting to near-complete suppression) of mirror instability and near constancy of 〈 B 〉 in the ultra-high-β case. These plots should be compared to Figs 11 and 12 (a,c). 4.2 Implications for cosmic magnetogenesis As large-scale dynamical motions that drive pressure anisotropies are invariably associated with changing the macroscopic magnetic field and since the resulting instabilities give rise to microscopic magnetic fluctuations, the implications of all this for the long-standing problem of the origin (and maintenance) of cosmic magnetism are the obviously