We thank the referee for providing thoughtful comments on the manuscript. The
authors acknowledge financial support from the CNES French space agency and
PNHE program of French CNRS. All the computations presented in this paper were
performed using the GRICAD infrastructure (https://gricad.univ-grenoble-
alpes.fr), which is supported by Grenoble research communities.

## References

  * Anderson et al. (2003) Anderson, J. M., Li, Z.-Y., Krasnopolsky, R., & Blandford, R. D. 2003, The Astrophysical Journal, 590, L107, aDS Bibcode: 2003ApJ…590L.107A
  * Anderson et al. (2005) Anderson, J. M., Li, Z.-Y., Krasnopolsky, R., & Blandford, R. D. 2005, The Astrophysical Journal, 630, 945
  * Anderson et al. (2006) Anderson, J. M., Li, Z.-Y., Krasnopolsky, R., & Blandford, R. D. 2006, The Astrophysical Journal Letters, 653, L33
  * Asada & Nakamura (2012) Asada, K. & Nakamura, M. 2012, The Astrophysical Journal, 745, L28
  * Bally et al. (2007) Bally, J., Reipurth, B., & Davis, C. J. 2007, Observations of Jets and Outflows from Young Stars, conference Name: Protostars and Planets V Pages: 215 ADS Bibcode: 2007prpl.conf..215B
  * Barnier et al. (2022) Barnier, S., Petrucci, P.-O., Ferreira, J., et al. 2022, Astronomy & Astrophysics, Volume 657, id.A11, <NUMPAGES>19</NUMPAGES> pp., 657, A11
  * Barniol Duran et al. (2017) Barniol Duran, R., Tchekhovskoy, A., & Giannios, D. 2017, Monthly Notices of the Royal Astronomical Society, 469, 4957
  * Blandford & Payne (1982) Blandford, R. D. & Payne, D. G. 1982, Monthly Notices of the Royal Astronomical Society, 199, 883
  * Blandford & Znajek (1977) Blandford, R. D. & Znajek, R. L. 1977, Monthly Notices of the Royal Astronomical Society, 179, 433
  * Boccardi et al. (2017) Boccardi, B., Krichbaum, T. P., Ros, E., & Zensus, J. A. 2017, Astronomy and Astrophysics Review, 25, 4
  * Bogovalov (1997) Bogovalov, S. V. 1997, Astronomy and Astrophysics, v.323, p.634-643, 323, 634
  * Bollen et al. (2017) Bollen, D., Van Winckel, H., & Kamath, D. 2017, Astronomy & Astrophysics, Volume 607, id.A60, <NUMPAGES>17</NUMPAGES> pp., 607, A60
  * Bonito et al. (2011) Bonito, R., Orlando, S., Miceli, M., et al. 2011, The Astrophysical Journal, 737, 54, aDS Bibcode: 2011ApJ…737…54B
  * Cabrit (2007) Cabrit, S. 2007, 243, 203, conference Name: Star-Disk Interaction in Young Stars
  * Casse & Ferreira (2000a) Casse, F. & Ferreira, J. 2000a, Astronomy and Astrophysics, v.353, p.1115-1128 (2000), 353, 1115
  * Casse & Ferreira (2000b) Casse, F. & Ferreira, J. 2000b, Astronomy and Astrophysics, v.361, p.1178-1190 (2000), 361, 1178
  * Casse & Keppens (2002) Casse, F. & Keppens, R. 2002, The Astrophysical Journal, 581, 988, aDS Bibcode: 2002ApJ…581..988C
  * Casse & Keppens (2004) Casse, F. & Keppens, R. 2004, The Astrophysical Journal, 601, 90, aDS Bibcode: 2004ApJ…601…90C
  * Ceccobello et al. (2018) Ceccobello, C., Cavecchi, Y., Heemskerk, M. H. M., et al. 2018, Monthly Notices of the Royal Astronomical Society, 473, 4417, aDS Bibcode: 2018MNRAS.473.4417C
  * Cheung et al. (2007) Cheung, C. C., Harris, D. E., & Stawarz, L. 2007, The Astrophysical Journal, 663, L65
  * Contopoulos & Lovelace (1994) Contopoulos, J. & Lovelace, R. V. E. 1994, The Astrophysical Journal, 429, 139
  * Corbel et al. (2003) Corbel, S., Nowak, M. A., Fender, R. P., Tzioumis, A. K., & Markoff, S. 2003, Astronomy and Astrophysics, 400, 1007
  * Coriat et al. (2011) Coriat, M., Corbel, S., Prat, L., et al. 2011, Monthly Notices of the Royal Astronomical Society, 414, 677, aDS Bibcode: 2011MNRAS.414..677C
  * Doi et al. (2018) Doi, A., Hada, K., Kino, M., Wajima, K., & Nakahara, S. 2018, The Astrophysical Journal, 857, L6, aDS Bibcode: 2018ApJ…857L…6D
  * Fanaroff & Riley (1974) Fanaroff, B. L. & Riley, J. M. 1974, Monthly Notices of the Royal Astronomical Society, 167, 31P, aDS Bibcode: 1974MNRAS.167P..31F
  * Fender & Gallo (2014) Fender, R. & Gallo, E. 2014, Space Science Reviews, 183, 323, aDS Bibcode: 2014SSRv..183..323F
  * Fendt (2006) Fendt, C. 2006, The Astrophysical Journal, 651, 272
  * Ferreira (1997) Ferreira, J. 1997, Astronomy and Astrophysics, 319, 340
  * Ferreira & Casse (2004) Ferreira, J. & Casse, F. 2004, The Astrophysical Journal Letters, 601, L139
  * Ferreira et al. (2006) Ferreira, J., Dougados, C., & Cabrit, S. 2006, Astronomy and Astrophysics, Volume 453, Issue 3, July III 2006, pp.785-796, 453, 785
  * Ferreira & Pelletier (1995) Ferreira, J. & Pelletier, G. 1995, Astronomy and Astrophysics, v.295, p.807, 295, 807
  * Gallo et al. (2004) Gallo, E., Corbel, S., Fender, R. P., Maccarone, T. J., & Tzioumis, A. K. 2004, Monthly Notices of the Royal Astronomical Society, 347, L52, aDS Bibcode: 2004MNRAS.347L..52G
  * Gracia et al. (2006) Gracia, J., Vlahakis, N., & Tsinganos, K. 2006, Monthly Notices of the Royal Astronomical Society, 367, 201, aDS Bibcode: 2006MNRAS.367..201G
  * Heyvaerts & Norman (1989) Heyvaerts, J. & Norman, C. 1989, The Astrophysical Journal, 347, 1055
  * Heyvaerts & Norman (2003a) Heyvaerts, J. & Norman, C. 2003a, The Astrophysical Journal, 596, 1270
  * Heyvaerts & Norman (2003b) Heyvaerts, J. & Norman, C. 2003b, The Astrophysical Journal, 596, 1256, aDS Bibcode: 2003ApJ…596.1256H
  * Jacquemin-Ide et al. (2019) Jacquemin-Ide, J., Ferreira, J., & Lesur, G. 2019, Monthly Notices of the Royal Astronomical Society, 490, 3112
  * Komissarov & Falle (1998) Komissarov, S. S. & Falle, S. A. E. G. 1998, Monthly Notices of the Royal Astronomical Society, 297, 1087
  * Krasnopolsky et al. (1999) Krasnopolsky, R., Li, Z.-Y., & Blandford, R. 1999, The Astrophysical Journal, 526, 631
  * Krasnopolsky et al. (2003) Krasnopolsky, R., Li, Z.-Y., & Blandford, R. D. 2003, The Astrophysical Journal, 595, 631
  * Laing & Bridle (2013) Laing, R. A. & Bridle, A. H. 2013, Monthly Notices of the Royal Astronomical Society, 432, 1114, aDS Bibcode: 2013MNRAS.432.1114L
  * Laing & Bridle (2014) Laing, R. A. & Bridle, A. H. 2014, Monthly Notices of the Royal Astronomical Society, 437, 3405, aDS Bibcode: 2014MNRAS.437.3405L
  * Laing et al. (1994) Laing, R. A., Jenkins, C. R., Wall, J. V., & Unger, S. W. 1994, 54, 201, conference Name: The Physics of Active Galaxies ADS Bibcode: 1994ASPC…54..201L
  * Lee et al. (2017) Lee, C.-F., Ho, P. T. P., Li, Z.-Y., et al. 2017, Nature Astronomy, 1, 0152, aDS Bibcode: 2017NatAs…1E.152L
  * Li et al. (1992) Li, Z.-Y., Chiueh, T., & Begelman, M. C. 1992, The Astrophysical Journal, 394, 459, aDS Bibcode: 1992ApJ…394..459L
  * Liska et al. (2018) Liska, M., Hesp, C., Tchekhovskoy, A., et al. 2018, Monthly Notices of the Royal Astronomical Society, 474, L81, aDS Bibcode: 2018MNRAS.474L..81L
  * Lister et al. (2013) Lister, M. L., Aller, M. F., Aller, H. D., et al. 2013, The Astronomical Journal, 146, 120, aDS Bibcode: 2013AJ….146..120L
  * Lister et al. (2009) Lister, M. L., Cohen, M. H., Homan, D. C., et al. 2009, The Astronomical Journal, 138, 1874, aDS Bibcode: 2009AJ….138.1874L
  * Louvet et al. (2018) Louvet, F., Dougados, C., Cabrit, S., et al. 2018, Astronomy & Astrophysics, Volume 618, id.A120, <NUMPAGES>28</NUMPAGES> pp., 618, A120
  * Matsakos et al. (2009) Matsakos, T., Massaglia, S., Trussoni, E., et al. 2009, Astronomy and Astrophysics, Volume 502, Issue 1, 2009, pp.217-229, 502, 217
  * Matsakos et al. (2008) Matsakos, T., Tsinganos, K., Vlahakis, N., et al. 2008, Astronomy and Astrophysics, 477, 521
  * Merloni & Fabian (2003) Merloni, A. & Fabian, A. C. 2003, Monthly Notices of the Royal Astronomical Society, 342, 951, aDS Bibcode: 2003MNRAS.342..951M
  * Mignone et al. (2007) Mignone, A., Bodo, G., Massaglia, S., et al. 2007, The Astrophysical Journal Supplement Series, 170, 228, aDS Bibcode: 2007ApJS..170..228M
  * Mingo et al. (2019) Mingo, B., Croston, J. H., Hardcastle, M. J., et al. 2019, Monthly Notices of the Royal Astronomical Society, 488, 2701, aDS Bibcode: 2019MNRAS.488.2701M
  * Miyoshi & Kusano (2005) Miyoshi, T. & Kusano, K. 2005, Journal of Computational Physics, 208, 315
  * Murphy et al. (2010) Murphy, G. C., Ferreira, J., & Zanni, C. 2010, Astronomy and Astrophysics, Volume 512, id.A82, <NUMPAGES>14</NUMPAGES> pp., 512, A82
  * Murphy et al. (2009) Murphy, G. C., Zanni, C., & Ferreira, J. 2009, 13, 117, conference Name: Protostellar Jets in Context ADS Bibcode: 2009ASSP…13..117M
  * Okamoto (2001) Okamoto, I. 2001, Monthly Notices of the Royal Astronomical Society, 327, 55, aDS Bibcode: 2001MNRAS.327…55O
  * Okamoto (2003) Okamoto, I. 2003, The Astrophysical Journal, 589, 671, aDS Bibcode: 2003ApJ…589..671O
  * Ostriker (1997) Ostriker, E. C. 1997, The Astrophysical Journal, 486, 291, aDS Bibcode: 1997ApJ…486..291O
  * Ouyed et al. (2003) Ouyed, R., Clarke, D. A., & Pudritz, R. E. 2003, The Astrophysical Journal, 582, 292
  * Ouyed & Pudritz (1993) Ouyed, R. & Pudritz, R. E. 1993, The Astrophysical Journal, 419, 255, aDS Bibcode: 1993ApJ…419..255O
  * Ouyed & Pudritz (1997a) Ouyed, R. & Pudritz, R. E. 1997a, The Astrophysical Journal, 482, 712
  * Ouyed & Pudritz (1997b) Ouyed, R. & Pudritz, R. E. 1997b, The Astrophysical Journal, 484, 794
  * Ouyed & Pudritz (1999) Ouyed, R. & Pudritz, R. E. 1999, Monthly Notices of the Royal Astronomical Society, 309, 233
  * Park et al. (2019) Park, J., Hada, K., Kino, M., et al. 2019, The Astrophysical Journal, 887, 147, aDS Bibcode: 2019ApJ…887..147P
  * Pelletier & Pudritz (1992) Pelletier, G. & Pudritz, R. E. 1992, The Astrophysical Journal, 394, 117
  * Perucho (2020) Perucho, M. 2020, Monthly Notices of the Royal Astronomical Society, 494, L22
  * Perucho & Martí (2007) Perucho, M. & Martí, J. M. 2007, Monthly Notices of the Royal Astronomical Society, 382, 526
  * Polko et al. (2010) Polko, P., Meier, D. L., & Markoff, S. 2010, The Astrophysical Journal, 723, 1343
  * Polko et al. (2014) Polko, P., Meier, D. L., & Markoff, S. 2014, Monthly Notices of the Royal Astronomical Society, 438, 959
  * Porth & Fendt (2010) Porth, O. & Fendt, C. 2010, The Astrophysical Journal, 709, 1100, aDS Bibcode: 2010ApJ…709.1100P
  * Porth & Komissarov (2015) Porth, O. & Komissarov, S. S. 2015, Monthly Notices of the Royal Astronomical Society, 452, 1089
  * Pudritz et al. (2006) Pudritz, R. E., Rogers, C. S., & Ouyed, R. 2006, Monthly Notices of the Royal Astronomical Society, 365, 1131
  * Ramsey & Clarke (2011) Ramsey, J. P. & Clarke, D. A. 2011, The Astrophysical Journal Letters, 728, L11
  * Ramsey & Clarke (2019) Ramsey, J. P. & Clarke, D. A. 2019, Monthly Notices of the Royal Astronomical Society, 484, 2364
  * Ray et al. (2007) Ray, T., Dougados, C., Bacciotti, F., Eislöffel, J., & Chrysostomou, A. 2007, Toward Resolving the Outflow Engine: An Observational Perspective (eprint: arXiv:astro-ph/0605597), conference Name: Protostars and Planets V Pages: 231 ADS Bibcode: 2007prpl.conf..231R
  * Ray & Ferreira (2021) Ray, T. P. & Ferreira, J. 2021, New Astronomy Reviews, 93, 101615, aDS Bibcode: 2021NewAR..9301615R
  * Reipurth & Bally (2001) Reipurth, B. & Bally, J. 2001, Annual Review of Astronomy and Astrophysics, Vol. 39, p. 403-455 (2001)., 39, 403
  * Sheikhnezami et al. (2012) Sheikhnezami, S., Fendt, C., Porth, O., Vaidya, B., & Ghanbari, J. 2012, The Astrophysical Journal, 757, 65, aDS Bibcode: 2012ApJ…757…65S
  * Staff et al. (2015) Staff, J. E., Koning, N., Ouyed, R., Thompson, A., & Pudritz, R. E. 2015, Monthly Notices of the Royal Astronomical Society, 446, 3975
  * Staff et al. (2010) Staff, J. E., Niebergal, B. P., Ouyed, R., Pudritz, R. E., & Cai, K. 2010, The Astrophysical Journal, 722, 1325
  * Stepanovs & Fendt (2016) Stepanovs, D. & Fendt, C. 2016, The Astrophysical Journal, 825, 14, aDS Bibcode: 2016ApJ…825…14S
  * Stute et al. (2014) Stute, M., Gracia, J., Vlahakis, N., et al. 2014, Monthly Notices of the Royal Astronomical Society, 439, 3641, aDS Bibcode: 2014MNRAS.439.3641S
  * Stute et al. (2008) Stute, M., Tsinganos, K., Vlahakis, N., Matsakos, T., & Gracia, J. 2008, Astronomy and Astrophysics, 491, 339
  * Tabone et al. (2020) Tabone, B., Cabrit, S., Pineau des Forêts, G., et al. 2020, Astronomy and Astrophysics, 640, A82
  * Tabone et al. (2018) Tabone, B., Raga, A., Cabrit, S., & Pineau des Forêts, G. 2018, Astronomy & Astrophysics, Volume 614, id.A119, <NUMPAGES>13</NUMPAGES> pp., 614, A119
  * Tchekhovskoy & Bromberg (2016) Tchekhovskoy, A. & Bromberg, O. 2016, Monthly Notices of the Royal Astronomical Society, 461, L46, aDS Bibcode: 2016MNRAS.461L..46T
  * Tchekhovskoy et al. (2010) Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2010, New Astronomy, 15, 749, aDS Bibcode: 2010NewA…15..749T
  * Teşileanu et al. (2014) Teşileanu, O., Matsakos, T., Massaglia, S., et al. 2014, Astronomy & Astrophysics, Volume 562, id.A117, <NUMPAGES>9</NUMPAGES> pp., 562, A117
  * Tudor et al. (2017) Tudor, V., Miller-Jones, J. C. A., Patruno, A., et al. 2017, Monthly Notices of the Royal Astronomical Society, 470, 324, aDS Bibcode: 2017MNRAS.470..324T
  * Tzeferacos et al. (2009) Tzeferacos, P., Ferrari, A., Mignone, A., et al. 2009, Monthly Notices of the Royal Astronomical Society, 400, 820, aDS Bibcode: 2009MNRAS.400..820T
  * Tzeferacos et al. (2013) Tzeferacos, P., Ferrari, A., Mignone, A., et al. 2013, Monthly Notices of the Royal Astronomical Society, 428, 3151, aDS Bibcode: 2013MNRAS.428.3151T
  * Ustyugova et al. (1995) Ustyugova, G. V., Koldoba, A. V., Romanova, M. M., Chechetkin, V. M., & Lovelace, R. V. E. 1995, The Astrophysical Journal, 439, L39, aDS Bibcode: 1995ApJ…439L..39U
  * Ustyugova et al. (1999) Ustyugova, G. V., Koldoba, A. V., Romanova, M. M., Chechetkin, V. M., & Lovelace, R. V. E. 1999, The Astrophysical Journal, 516, 221, aDS Bibcode: 1999ApJ…516..221U
  * Vlahakis & Königl (2003) Vlahakis, N. & Königl, A. 2003, The Astrophysical Journal, 596, 1080, aDS Bibcode: 2003ApJ…596.1080V
  * Vlahakis & Tsinganos (1997) Vlahakis, N. & Tsinganos, K. 1997, Monthly Notices of the Royal Astronomical Society, 292, 591
  * Vlahakis et al. (2000) Vlahakis, N., Tsinganos, K., Sauty, C., & Trussoni, E. 2000, Monthly Notices of the Royal Astronomical Society, 318, 417, aDS Bibcode: 2000MNRAS.318..417V
  * Walker et al. (2018) Walker, R. C., Hardee, P. E., Davies, F. B., Ly, C., & Junor, W. 2018, The Astrophysical Journal, 855, 128
  * Weber & Davis (1967) Weber, E. J. & Davis, Jr., L. 1967, The Astrophysical Journal, 148, 217, aDS Bibcode: 1967ApJ…148..217W
  * Zanni et al. (2007) Zanni, C., Ferrari, A., Rosner, R., Bodo, G., & Massaglia, S. 2007, Astronomy and Astrophysics, Volume 469, Issue 3, July III 2007, pp.811-828, 469, 811
  * Zanni & Ferreira (2013) Zanni, C. & Ferreira, J. 2013, Astronomy & Astrophysics, Volume 550, id.A99, <NUMPAGES>20</NUMPAGES> pp., 550, A99
  * Zensus (1997) Zensus, J. A. 1997, Annual Review of Astronomy and Astrophysics, Volume 35, 1997, pp. 607-636., 35, 607

##  Appendix A Long time scales evolution

In numerical simulations, the time increment is fixed by the
Courant–Friedrichs–Lewy condition $\Delta t<\Delta x/C_{max}$, where $C_{max}$
is the maximal wave speed in the cell (in our case $u+v_{FM}$) and $\Delta x$
is the cell size. In a standard simulation, the time increment of all cells is
chosen by taking the absolute minimum of all the time increments in the full
computational domain. In our simulations this time step is set by the smallest
cells around the inner spherical boundary at $R_{d}$, which also happen to
have the strongest field and the highest Alfvén speed. However, these cells
near the source are also the ones that converge the fastest to a stationary
solution. Thus the cells that we consider are converging to a steady state
(meaning the relative variation of the density in one integration step is
smaller than an arbitrarily small parameter) are not used to determine the
time increment of the cells that are still evolving in time. The time
increment used to evolve the evolving cells is computed by taking a minimum
over only the cells that have not converged yet. The cells that have converged
to a steady state are still integrated in time using their own local time
increment so as to ensure the stability of the integration and to be able to
capture any perturbation that could possibly alter their steady condition.
Since the cells that converge the fastest are those characterized by the
shortest time increment, the time increments used to evolve the cells that are
still evolving in time and have not converged yet to a steady state becomes
larger and larger. It is important to point out that the stationary solutions
obtained with this time boost are also a solution of the standard non-
accelerated algorithm.

In Figure 20 is represented the gain in computing time obtained thanks to the
time boost. The acceleration factor, defined as the ratio between the physical
time reached using the time boost and the physical time that would have been
achieved using the standard CFL condition, is plotted versus the progressive
numbering of the outputted files. Without the acceleration due to our handling
of the CFL condition, the time interval between two outputs would have been
constant, and the physical time of the solution would have been proportional
to the output number. Any increase of the acceleration factor means that
another batch of cells has converged. That means the time increment of the
cells that are still evolving in time becomes larger, thus increasing the
reached time scales. This increase is clearly visible after the 300th output.
Using the output number as a proxy for the computational cost of a simulation,
this figure clearly shows that at the end of the integration the time boost
enables us to reach time scales at least two or three orders of magnitude
larger than using a standard CFL condition, without increasing the
computational cost of the simulation. Analogously, without employing the time
boost we would have required two or three orders of magnitude more CPU hours
to reach the same time scales. Our approach enabled us to produce simulations
that would have consumed way more computing time otherwise. The reference
simulation K2 consumed 725 CPU hours, but without the time boost it would have
required almost two million CPU hours. It enabled us to work on simply 64
processors kindly provided by GRICAD (Grenoble Alpes Recherche -
Infrastructure de Calcul Intensif et de Données).

For the evolution of the acceleration factor with the mass load, we can see in
Table 1 that the simulations with $\kappa$ the closest to that of Blandford &
Payne (1982) converge the fastest, reaching larger timescales at the end of
the computation. Nevertheless, all seem to reach comparable convergence
speeds: in Figure 20 we see that the simulation K5 reaches an acceleration
factor similar to that of K2 at the final output. For the evolution with the
magnetic field, we can clearly see that the higher the $\alpha$ and thus the
flatter the profile of the vertical magnetic field, the slower the simulation
converges. For higher values of $\alpha$, the jet is initially more collimated
as $B_{r}/B_{z}$ is higher on the disk. Thus the field lines further on the
disk have a higher impact, retarding the global convergence. That is why the
simulation A5 with $\alpha=15/16$ has not converged yet. As an instability
develops, some cells that were previously stable become unstable, hence a
decrease in acceleration.

Figure 20: Evolution of the acceleration for the simulations K2, K5 and A5.

##  Appendix B Boundary conditions

The plots represented in Figure 21 illustrate the injection boundary
conditions we have chosen, for the reference simulation K2. The Bernouillli
invariant $E$, the poloidal magnetic field $B_{\Phi}$, the vertical magnetic
field $B_{z}$, the mass to magnetic flux ratio $\eta=\mu_{0}\rho v_{p}/B_{p}$,
the rotation speed of the magnetic surfaces $\Omega_{\star}=\Omega-\eta
B_{\Phi}/(\mu_{0}\rho r)$, the speed of sound $C_{s}=\sqrt{\gamma P/\rho}$ and
vertical Alfvénic speed $V_{A_{z}}=B_{z}/\sqrt{\mu_{0}\rho}$ are plotted on
the first cell over the injection boundary. The toroidal magnetic field goes
to zero on the axis for symmetry reasons, and $\textbar
B_{\Phi}/B_{z}\textbar\gtrsim 1$ on the disk : The JED magnetic field is
weakly toroidal. The launching conditions are very cold as
$V_{A_{z}}/C_{s}\sim 10^{2}$ on both the source and the disk.

The reader may observe the power law dependency with the magnetic flux on the
whole disk ($\Psi>10$, after the black vertical line) for all parameters but
the electric current. This is directly induced by the self-similar ansatz.
However, the torroidal magnetic field $B_{\Phi}$ breaks the power law
dependency and shows a swift decrease for $\Psi>2.10^{6}$. Of all eight
variables, only $B_{R}$ and $B_{\Phi}$ are free at the injection boundary (see
Equation (8)), and need to cross a characteristic surface to be fixed. For the
toroidal current it is the Alfvénic surface. As all magnetic surfaces over
$\Psi\gtrsim 10^{9}$ never cross the Alfvénic surface, the current can never
be fixed. Thus, the simulation cannot ever be stationary in this region.

As the disk is a self-similar Jet-Emitting Disk, all dimensionless parameters
are assumed to be independent from the radius (Blandford & Payne (1982)). All
these parameters are regrouped in section 2.4.

Figure 21: Conditions on the lowermost cells of the reference simulation K2.
Here all parameters are traced along the first cell above the lower boundary :
$R=1\text{ and }\theta\in[0;\pi/2]$ on the source and then $\theta=\pi/2\text{
and }R\in[0;R_{ext}]$ on the disk. Are represented the Bernouilli invariant
$E$, the toroidal and vertical magnetic field $B_{\Phi}$ and $B_{z}$, the mass
to magnetic flux ratio $\eta$, the rotation speed of magnetic surfaces
$\Omega_{*}$, the speed of sound $C_{s}$ and the vertical Alfvénic speed
$V_{A_{z}}$, over the magnetic flux $\Psi$. The black vertical line
corresponds to the flux anchored at ($R=1$,$\theta=\pi/2$), at the source/disk
interface.

As explained in section 2.3.2.2, we defined a spline function $f(\theta)$
equal to zero on axis ($\theta=0$) and one at the inner disk radius
$(\theta=\pi/2)$ to smoothly connect the axis values with the inner disk ones
: $f(\theta)\equiv(3\sin^{2}\theta-2\sin^{3}\theta)^{3/2}$.

The injection speed ($V_{R}$ on the source, $-V_{\theta}$ on the disk) is
fixed at $\kappa\mu^{2}=0.1$ even when varying $\kappa$. However, even though
the injection Mach number $M_{s}=u_{p}/C_{s}$ is assumed to be the same all
along the boundary, in order to account for the varying inclination of the
magnetic surfaces, its value is modified with the parameter $\alpha$ from
simulation to simulation. Its variation with $\alpha$ is shown in Table 1.

##  Appendix C Rankine-Hugoniot jump conditions

In this section, we write the Rankine-Hugoniot jump conditions valid for
standing, adiabatic, recollimation shocks. Contrary to Ouyed & Pudritz (1993),
we take into account the toroidal magnetic field since the shocks arise when
that component is dominant. The local jump $[A]=A_{2}-A_{1}$ between a pre-
shock quantity $A_{1}$ and its post-shock value $A_{2}$ are expressed in the
rest frame as

$\displaystyle[\rho u_{\perp}]=0$ $\displaystyle=0$ $\displaystyle=0$
$\displaystyle=0$ (21) $\displaystyle=0$ $\displaystyle=0$

where $H=C_{s}^{2}/(\Gamma-1)$ is the enthalpy and $u_{\perp},B_{\perp}$
(resp. ${\bf u}_{\parallel},{\bf B}_{\parallel}$ are the normal (resp.
tangential) components to the shock surface. The shock is axisymmetric so the
tangential component of the magnetic field ${\bf
B}_{\parallel}=B_{t}\@vec{e}_{t}+B_{\phi}\@vec{e}_{\phi}$, whereas the
poloidal component is ${\bf
B}_{p}=B_{t}\@vec{e}_{t}+B_{\perp}\@vec{e}_{\perp}$, with the unit vectors
$(\@vec{e}_{\perp},\@vec{e}_{t},\@vec{e}_{\phi})$ defining a local orthonormal
coordinate system. Since these jump conditions express the conservation of
mass, angular momentum and energy in ideal MHD, the five MHD invariants along
a given magnetic surface ($\eta,\Omega_{*},L,E,S$) are thus obviously also
conserved (see Fig.7).

In the case of a shock, the mass flux through the surface is non zero, which
requires $B_{\perp 2}=B_{\perp 1}\neq 0$ and leads to

$\@vec{B}_{\parallel 2}=\frac{m^{2}-1}{m^{2}/\chi-1}\@vec{B}_{\parallel 1}$
(22)

where $m=u_{\perp}/V_{Ap,\perp}=u_{p}/V_{Ap}$ is the Alfvénic Mach number of
the incoming (pre-shock) flow and $\chi=\rho_{2}/\rho_{1}=u_{\perp 1}/u_{\perp
2}$ is the shock compression rate. This equation shows that there are three
non-trivial discontinuities with $\chi\geq 1$: (1) an oblique shock with
$m^{2}>\chi>1$, (2) a normal shock with $m^{2}=\chi>1$ (requiring
$\@vec{B}_{\parallel 1}=0$) and (3) an Alfvén shear discontinuity with
$m^{2}=\chi=1$ (allowing an arbitrary jump between the two tangential field
components). The oblique shock is the only case studied here.

After some algebra all post-shock quantities can be expressed as function of
the pre-shock ones, in particular

$\displaystyle\frac{B_{\phi 2}}{B_{\phi
1}}=\frac{B_{t2}}{B_{t1}}=\chi\frac{m^{2}-1}{m^{2}-\chi}$
$\displaystyle\frac{u_{\phi 2}}{u_{\phi
1}}=\frac{m^{2}-1}{m^{2}-\chi}\frac{m^{2}r_{A}^{2}-\chi
r^{2}}{m^{2}r_{A}^{2}-r^{2}}$ (23) $\displaystyle\frac{P_{2}}{P_{1}}=1+\Gamma
m^{2}_{s}(\chi-1)\left(\frac{1}{\chi}+\frac{b^{2}}{2}\frac{2\chi-m^{2}(1+\chi)}{(\chi-m^{2})^{2}}\right)$
$\displaystyle\frac{T_{2}}{T_{1}}=\frac{1}{\chi}\frac{P_{2}}{P_{1}}$

where the sonic Mach number $m_{s}=u_{\perp}/C_{s}$ and magnetic shear
$b^{2}=(B_{\parallel}/B_{\perp})^{2}$ are computed in the pre-shock region. Of
particular interest are the relative variations of the toroidal magnetic field
component $\delta B_{\phi}=B_{\phi 2}/B_{\phi 1}-1$ and the plasma angular
velocity $\delta\Omega=\Omega_{2}/\Omega_{1}-1$, as well as the total
deflection angle of the poloidal magnetic surface $\delta i=i_{2}-i_{1}$ where
$\tan i=B_{t}/B_{\perp}$, which read

$\displaystyle\delta B_{\phi}$ $\displaystyle=$
$\displaystyle(\chi-1)\frac{m^{2}}{m^{2}-\chi}$ $\displaystyle-\delta\Omega$
$\displaystyle=$
$\displaystyle\frac{\chi-1}{m^{2}-\chi}\frac{m^{2}(r^{2}-r_{A}^{2})}{m^{2}r_{A}^{2}-r^{2}}\leq\frac{\chi-1}{m^{2}-\chi}$
(24) $\displaystyle\tan\delta i$ $\displaystyle=$
$\displaystyle\frac{m^{2}(\chi-1)}{m^{2}-\chi}\frac{\tan
i_{1}}{1+\chi\tan^{2}i_{1}\frac{m^{2}-1}{m^{2}-\chi}}$

These quantities are plotted in Fig. 9. The compression rate $\chi$ is the
solution of the cubic polynomial equation

$-A\chi^{3}+B\chi^{2}-C\chi+D=0$ (25)

with

$\displaystyle A$ $\displaystyle=$ $\displaystyle
1+b^{2}+\frac{1+\chi_{o}}{\Gamma m^{2}_{s}}$ $\displaystyle B$
$\displaystyle=$
$\displaystyle\chi_{o}(1+b^{2})+2m^{2}\left(1+\frac{1+\chi_{o}}{\Gamma
m^{2}_{s}}-b^{2}\frac{\chi_{o}-3}{4}\right)$ $\displaystyle C$
$\displaystyle=$ $\displaystyle
m^{2}\left(2\chi_{o}+b^{2}\frac{1+\chi_{o}}{2}+m^{2}\left(1+\frac{1+\chi_{o}}{\Gamma
m^{2}_{s}}\right)\right)$ $\displaystyle D$ $\displaystyle=$
$\displaystyle\chi_{o}m^{4}$

where $\chi_{o}=(\Gamma+1)/(\Gamma-1)$ is the maximal compression ratio for a
hydrodynamic shock. Equation 25 has one positive root only for an incoming
super-FM flow, namely for $n_{\perp}=u_{\perp}/V_{fm,\perp}$ larger than
unity.

We are dealing here with supersonic ($m_{s}>>1$) and super-A ($m>>1$) cold
jets, where the dominant magnetic field is the toroidal one
($b^{2}\simeq(B_{\phi}/B_{\perp})^{2}>>1$). The FM Mach number in the normal
direction writes then $n_{\perp}\simeq mV_{Ap,\perp}/V_{A\phi}=m/b$, which
leads to the simplified equation for $\chi$

$\frac{\chi_{o}-3}{2}\chi^{2}+\left(\frac{1+\chi_{o}}{2}+n_{\perp}^{2}\right)\chi-\chi_{o}n_{\perp}^{2}=0\
.$ (26)

It shows that whenever jets reach a very large FM Mach number $n_{\perp}$, a
large compression rate $\chi\simeq\chi_{o}$ is possible. But this is never
achieved in our case. indeed, the poloidal FM Mach number $n=u_{p}/V_{FM,p}$
($>n_{\perp}$) writes

$n^{2}=\omega_{A}\frac{B_{pA}r_{A}^{2}}{B_{p}r^{2}}\frac{1-1/m^{2}}{1-r_{A}^{2}/r^{2}}\left(\frac{u_{p}}{\Omega_{*}r_{A}}\right)^{3}\sim\omega_{A}\left(\frac{u_{p}}{\Omega_{*}r_{A}}\right)^{3}$
(27)

where $\omega_{A}=\Omega_{*}r_{A}/V_{Ap,A}$ is the fastness parameter
introduced in F97 (ratio at the Alfvén point of the speed of the magnetic
rotator to the poloidal Alfvén speed). For magneto-centrifugal jets like ours,
with $m^{2}>>1,r>>r_{A}$ and achieving their maximal velocity
$u_{p}\sim\sqrt{2}\Omega_{*}r_{A}$, the FM Mach number is
$n^{2}\sim\omega_{A}$, which is larger than but of the order of unity (see
also Krasnopolsky et al. 2003). As a consequence, we expect rather weak shocks
as illustrated by the small values of $\chi$ achieved along the various shocks
(see Fig. 9).

Figure 22: Distribution at $t_{end}$ along the main recollimation shock of the
compression ratio for simulation K2. The yellow curve is the theoretical
solution $\chi_{th}$ of Eq. 25, computed using the pre-shock quantities, while
the blue curve is the ratio $\chi=\rho_{2}/\rho_{1}$.

Figure 22 plots the theoretical solution $\chi_{th}$ of Eq. 25 (in yellow)
computed along the main recollimation shock of our simulation K2 and compares
it with the ratio $\chi=\rho_{2}/\rho_{1}$ (in blue) directly measured (see
Fig. 4). The correspondence is very good, with discrepancies remaining below a
few percent. The two regions where larger differences are obtained correspond
to the positions where the two smaller shocks (triggered at the spine-jet
interface) merge with the main shock: the orange one near $r\sim 500$ and the
cyan one near $r\sim 900$.

The shocks were detected by following all magnetic field lines anchored on the
disk and looking for discontinuities. This is not obvious in a discrete grid.
To do so, we computed the derivative of the toroidal magnetic field ($\delta
B_{\phi}$) over the curvilinear abscissa along the field line, as shocks are
best seen with the electric poloidal current and explored its local extrema.
We checked that a different approach, based on the calculation of the
refraction angle $\delta i$ of the poloidal magnetic surface, produces very
similar results. This gave us the shock locations used to produce the plots in
Figures 2 and 17. Since PLUTO has a shock capturing scheme, each shock is
resolved and has a finite width. To determine the shock width, we checked that
the density was growing within the shock as expected. Then, still following
the field line, we looked for the closest local minimum and maximum in
density. The positions of these extrema allowed to compute the values of the
pre-shock and the post-shock quantities, respectively. Those were finally used
to compute the parameters leading to the Figures 9 and 22.