tokens: for a cold flow such that (equation B1 in Appendix B , with ω = 1). This expression provides the two solid blue curves shown in Fig. 4 , computed in two extreme cases for the location of the Alfvén surface: near the SM surface with x SM = 2, namely (lower curve) and much further out with for the upper one (as in typical near-equipartition cold flows). For a given magnetic lever arm λ (mostly determined by ξ), κ must be large enough. This rules out all super-SM solutions located at the left-hand side of these curves. It can be seen right away that the parameter space of super-A solutions for near-equipartition fields ( n = 0) will be very small, with mass loads κ smaller than 0.1, disc ejection efficiencies ξ smaller than 0.08, and magnetic lever arms larger than ∼7, in agreement with Ferreira ( 1997 ). On the other hand, MRI-like driven flows at small μ seem to allow mass loads larger than unity with small λ. These isothermal flows could be of great interest for reproducing dense outflows (ξ > 0.1) with low asymptotic speeds ( u p ∞ ∼ Ω K o r o ). In other words, a weakly magnetized disc is better suited for a massive ejection than a near-equipartition disc, not only by providing the necessary bending at the disc surface through the MRI-like mechanism, but also by bringing the Alfvén surface closer to the disc (see the discussion on the role of Ψ A in Appendix B ). Nevertheless, not all super-SM solutions located at the right-hand side of the solid curves in Fig. 4 can become super-A. For a given ξ, λ is roughly determined and so is the cylindrical radius r A = r o (λ/ω) 1/2 of the Alfvén point. But equation ( 21 ) shows that κ also depends on p and μ. For a given toroidal current density p , the SM constraint provides μ so that κ is fixed. On the other hand, p also determines the radial magnetic field component at the disc surface and thereby the initial jet bending (see Fig. B3 ). As a consequence, playing with p not only affects the disc vertical equilibrium (SM point) but also this initial jet angle. Not all couples (κ, λ) fulfil the Grad–Shafranov equation ( 30 ). If a solution is not possible, this means that there is no altitude A of the Alfvén point that can be found starting from the conditions provided at the base of the jet (SM point). Changing the value of p leads to a slight modification of κ, λ as well as the jet angle at the SM point, allowing thereby to (possibly) meet the A condition. This translates into an adaptation of the altitude A of the Alfvén point (thus the angle Ψ A ) according to the disc conditions. The necessary condition highlights this aspect. When λ is large, r A / r o is large and gravity plays a negligible role so that is satisfied whenever λ > 3 (dashed line in Fig. 4 ). But gravity cannot be neglected anymore at lower values of λ, as can be seen in equation ( 31 ). In that case, providing requires to increase sin Ψ A , namely to bring the Alfvén surface closer to the disc surface. The closer it is to the
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