2.1. Units and normalisation

The code units and normalisation used previously in Rezgui et al. (Reference Rezgui, Marzougui, Woodring, Svoboda and Lili2019) and (Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022) is also applied in this work. To accurately describe the physical problem, we have selected fiducial values that yield effective scales. In light of this, lengths are expressed in units of inner disc radius $r_{in}$ . Typically, it is assumed that $r_{in}$ is a few radii from the central object. Velocities are measured in units of Keplerian speed at $r_{in}$ : $V_{k,in} = \sqrt{GM\big /r_{in}}$ . Therefore, the unit of time is defined by $t_{in}= r_{in}\big /V_{k,in}$ . The index in refers to a number value at the inner disc radius at $z = 0$ and time $t = 0$ . Densities are expressed in units of ${\unicode{x03C1}}_{d,in}$ with respect to the midplane of the disc at its inner radius. From the state equation, the pressures are determined in units of ${\unicode{x03B5}}^2{\unicode{x03C1}}_{d,in}V_{k,in}^2$ , giving thereby $P_{in} = {\unicode{x03B5}}^2$ . While from the equation of momentum, the magnetic field is expressed in units of $\sqrt{{\unicode{x03BC}}_0{\unicode{x03C1}}_{d,in}V_{k,in}^2}$ giving then $B_{in}={\unicode{x03B5}}\sqrt{2{\unicode{x03BC}}_0}$ , where ${\unicode{x03BC}}_0$ is the initial disc magnetisation. We define the disc aspect ratio $c_s\big /V_{k,in}$ by the ratio of the isothermal sound speed to the Keplerian speed, both evaluated at disc midplane.

The typical number values for a YSO of mass $ M=1$ M $_\odot$ are adopted in this study. For the purposes of comparison with earlier works documented in the literature (Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2009; Murphy, Ferreira, & Zanni Reference Murphy, Ferreira and Zanni2010; Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013; Stepanovs & Fendt Reference Stepanovs and Fendt2014) investigating stellar sources, we may assume $r_{in}=0.1$ AU. The corresponding units in YSO are listed in the Appendix A for reference guide.

2.2. Initial conditions

The initial configuration used in our calculations involves a thin disc, rotating at a slightly sub-Keplerian speed, traversed by a weak purely vertical magnetic field. We adopt the alpha prescription (Shakura & Sunyaev Reference Shakura and Sunyaev1973) where the viscosity is assumed to be proportional to the height scale of the disc h and the sound speed $c_s$ , that is, $\alpha_v = 3{\unicode{x03BD}}_v\big / 2c_s h$ .

It is worth noting that the disc and its hydrostatic corona are in pressure balance. In addition, the coronal density is considered obviously several orders of magnitude below the disc density. Equation (16) derives the toroidal velocity from the radial equilibrium, whereas equation (17) derives the radial speed from the angular momentum conservation equation. The launch of a robust jet from the inner part of the disc takes place in a short timescale if we initially impose an accretion movement according to the relation described in equation (18) (Murphy, Ferreira, & Zanni Reference Murphy, Ferreira and Zanni2010; Sheikhnezami et al. Reference Sheikhnezami, Fendt, Porth, Vaidya and Ghanbari2012; Rezgui et al. Reference Rezgui, Marzougui, Woodring, Svoboda and Lili2019, Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022).

2.2.1. Initial disc structure

We take into account a thin accretion disc with epsilon aspect ratio ${\unicode{x03B5}} = c_s\big /V_k$ . The hydrostatic vertical equilibrium is solved to determine the disc density and pressure. Disc density is determined by:

(14) \begin{equation} {\unicode{x03C1}}_d = {\unicode{x03C1}}_{d,in}\Big(\frac{2}{5{\unicode{x03B5}}^2}\Big[\frac{r_{in}}{R}-(1-\frac{5{\unicode{x03B5}}^2}{2})\frac{r_{in}}{r}\Big]\Big)^\frac{3}{2} \end{equation}

where R is the spherical radius defined by $R=\sqrt{r^2 + z^2}$ .

While, as initial disc pressure distribution, we prescribe:

(15) \begin{equation} P_d = P_{d,in}\Big(\frac{{\unicode{x03C1}}_{d,in}}{{\unicode{x03C1}}_d}\Big)^\frac{5}{3} \end{equation}

where $P_{d,in}={\unicode{x03B5}}^2{\unicode{x03C1}}_{d,in}V_{k,in}^2$

Following Murphy, Ferreira, & Zanni (Reference Murphy, Ferreira and Zanni2010), the toroidal velocity is given by:

(16) \begin{equation} U_{{\unicode{x03D5}}, d}= \sqrt{\frac{GM}{r}}\Big[\sqrt{1-\frac{5{\unicode{x03B5}}^2}{2}} + \frac{2}{15}{\unicode{x03B5}}^2\alpha_v^2{\unicode{x03B8}} \Big(1-\frac{6z^2}{5{\unicode{x03B5}}^2r^2}\Big)\Big] \end{equation}

We write the radial and vertical components of the velocity as follows:

(17) \begin{equation} U_{r, d}= -\alpha _v{\unicode{x03B5}}^2\sqrt{\frac{GM}{r}}\Big[10-\frac{32}{3}\alpha_v^2{\unicode{x03B8}}-{\unicode{x03B8}} \Big(5-\frac{z^2}{{\unicode{x03B5}}^2r^2}\Big)\Big] \end{equation}

with $ {\unicode{x03B8}} = \frac{11}{15}\Big(1+ \frac{64}{25}\alpha _v^2\Big)^{-1}$

Note that

(18) \begin{equation} U_{z,d}= \frac{z}{r}U_{r,d} \end{equation}

2.2.2 Coronal area structure

On top of the disc, a hydrostatic spherically symmetric atmosphere is prescribed, whereby we apply a polytropic pressure-density relationship (Zanni et al. Reference Zanni, Ferrari, Rosner, Bodo and Massaglia2007; Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013; Stepanovs, Fendt, & Sheikhnezami Reference Stepanovs, Fendt and Sheikhnezami2014):

(19) \begin{equation} {\unicode{x03C1}}_c ={\unicode{x03C1}}_{a,in}\Big(\frac{r_{in}}{R}\Big)^\frac{1}{{\unicode{x03B4}}-1} \end{equation}

(20) \begin{equation} P_c = {\unicode{x03C1}}_{a,in}\frac{{\unicode{x03B4}}-1}{{\unicode{x03B4}}} \frac{GM}{r_{in}}\Big(\frac{r_{in}}{R}\Big)^\frac{{\unicode{x03B4}}}{{\unicode{x03B4}}-1} \end{equation}

We refer the index c to the disc corona. We point out that ${\unicode{x03B4}}$ is a constant which quantifies the initial density contrast between disc and corona ( ${\unicode{x03C1}}_{a,in}= {\unicode{x03B4}} {\unicode{x03C1}}_{d,in}$ ), and it has been assumed in all our calculations equal to $10^{-4}$ as previous studies cited above.

2.2.3 Magnetic field distribution

Following previous simulations documented in the literature (Murphy, Ferreira, & Zanni Reference Murphy, Ferreira and Zanni2010; Rezgui et al. Reference Rezgui, Marzougui, Woodring, Svoboda and Lili2019, Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022), we prescribe the initial magnetic field by a magnetic flux function ${\unicode{x03C8}}$ expressed by:

(21) \begin{equation} {\unicode{x03C8}}(r,z) = 4B_{z,0}r_{in}^2\Big(\frac{r}{r_{in}}\Big)^\frac{1}{4}\frac{m^\frac{7}{4}}{(m^2+z^2/r^2)^\frac{7}{8}}\end{equation}

Here $B_{z,0}$ and the parameter m measure respectively the vertical field strength at $r=r_{in}$ and $z=0$ , and the height scale on which the initial magnetic field bends. The magnetic field components are calculated based on the following relations: $rB_r=-\frac{\partial {\unicode{x03C8}}}{\partial z}$ and $ rB_z = \frac{\partial {\unicode{x03C8}}}{\partial r}$ . Following Murphy, Ferreira, & Zanni (Reference Murphy, Ferreira and Zanni2010) we use in all our cases of simulations the initial value of the bending parameter m equal to 0.935.

2.2.4 Resistivity and viscosity distributions

We employ the same expression of the resistivity and viscosity as in Zanni & Ferreira (Reference Zanni and Ferreira2009) and Murphy, Ferreira, & Zanni (Reference Murphy, Ferreira and Zanni2010):

(22) \begin{equation} {\unicode{x03BD}}_v = \frac{2}{3}\alpha_v \Big[c_s^2(r)_{z=0}+\frac{2}{5}\Big(\frac{GM}{R}-\frac{GM}{r}\Big)\Big]\sqrt{\frac{r^3}{GM}} \end{equation}

Where $c_s(r)_{z=0}$ is the isothermal sound speed calculated on the midplane of the disc.

2.3. Numerical grid

The computing domain spans a rectangular grid region (Fig. 1), with purely uniform spacing applied in both the radial and vertical directions. The grid cells are $(512 \times 1\,536)$ , resulting in a high resolution of $\Delta_r=\Delta_z= 0.078$ on a physical domain spanning from 0 to 40 $r_{in}$ radially and from 0 to 120 $r_{in}$ vertically.

Figure 1. Schematic representation of the computational domain covering purely uniform spacing in radial and vertical directions. A zoom in on the bottom left corner of the grid is also shown to illustrate the sink, which is treated as an internal boundary. See the text for details.

We apply boundary conditions identical to those previously used in (Rezgui et al. Reference Rezgui, Marzougui, Woodring, Svoboda and Lili2019, Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022). This maintains axial symmetry on the rotation axis and equatorial symmetry for the disc midplane. To the ghost cells at the upper r and z boundaries, we employ an outflow boundary condition (zero gradient). The sink zone, which is defined by the area where the central object is located (Fig. 1, yellow box), requires a specific treatment. Since the interaction between disc and central object is outside the purview of this investigation, we exclude this area (bottom-left corner) from the computational domain by cutting out several cells in r and z directions. Inside the sink, the equations are not evolved. We follow the same strategy as previous studies in literature, where the sink is treated as an internal boundary condition (Murphy, Ferreira, & Zanni Reference Murphy, Ferreira and Zanni2010; Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2009, Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013). This latter has been discussed in our previous article Rezgui et al. (Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022) (see Section 2.3). The most important point to consider is that we impose the non-positivity of the poloidal velocity in this area to ensure that no artificial outflows are permitted to leave the sink.

2.4. Simulation parameters

Our numerical calculations are governed by six nondimensional parameters for the physical quantities listed below:

• The saturated thermal conduction parameter ${\unicode{x03D5}}_s$

• The disc magnetisation ${\unicode{x03BC}} = B^2 \big /2P$

• The sound-to-Keplerian speed ratio ${\unicode{x03B5}}= c_s \big /V_k$

• The initial density contrast between disc and corona ${\unicode{x03B4}}$

• The magnetic Prandtl number $P_m$

• The initial geometry of the magnetic field defined by the bending parameter m

To investigate the thermal conduction contribution in the angular momentum transport and energy budget of the disc–jet system, we carried out five cases of numerical simulations based on the variation of the saturation parameter ${\unicode{x03D5}}_s$ . We adopt the same range values (Table 1, section on the left) of this parameter used in Rezgui et al. (Reference Rezgui, Marzougui, Woodring, Svoboda and Lili2019) and (Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022). We point out that this range of values is close to those used in the studies of the advection-dominated accretion flows (ADAF systems) (Shadmehri Reference Shadmehri2008; Abbassi, Ghanbari, & Najjar Reference Abbassi, Ghanbari and Najjar2008; Ghanbari, Abbassi, & Ghasemnezhad Reference Ghanbari, Abbassi and Ghasemnezhad2009; Faghei Reference Faghei2012). In addition, we have shown in Rezgui et al. (Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022) that it can be applied to YSOs such as the HH jets (Reipurth et al. Reference Reipurth, Davis, Bally, Raga, Bowler, Geballe, Aspin and Chiang2019; Ahmane et al. Reference Ahmane, Mignone, Zanni, Massaglia and Bouldjderi2020). We also show in Table 1 (section on the right) the values of the other five parameters that we kept constant in all our calculations.

Table 1. Range values of ${\unicode{x03D5}}_s$ (section on the left) and Input Parameters (section on the right).

We ran the code for advanced time steps. In fact, the simulations of cases 1 and 2 are performed up to $t = 700$ which indicates that the disc has completed 112 periods of rotation at its inner radius. While the final time step of the simulations of cases 3–5 is marked by $t = 200$ roughly translated into 32 periods of rotation at the same disc radius. We have shown in our previous articles (Rezgui et al. Reference Rezgui, Marzougui, Woodring, Svoboda and Lili2019, Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022) that the time $t = 200$ is sufficient to study the evolution of the disc–jet system, emphasising that the outflow was efficiently accelerated and reached the fast magnetosonic surface.

3.1. Transport of angular momentum

The prevailing notion is that for successful jet formation, the magnetic torque must change its sign at the disc surface. This indicates that the predominant involvement in transferring angular momentum lies with the induced toroidal magnetic field component. As a result, the accretion flow is enabled inside the disc. Subsequently, the centrifugal force accelerates the material outward into the surrounding medium, where the jet extracts a portion of the angular momentum and energy from the disc.

We explore the angular momentum transport by means of its fluxes through the same control volume used to quantify the accretion and ejection rates. Following previous works (Zanni et al. Reference Zanni, Ferrari, Rosner, Bodo and Massaglia2007; Murphy, Ferreira, & Zanni Reference Murphy, Ferreira and Zanni2010, Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2009, Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013), the integration domain (Fig. 2) is considered as a disc sector delimited by two surfaces $S_{in}$ and $S_{out}$ normal to the equatorial plane and identified at $r_{in}$ and $r_{out}$ respectively. On top of the disc, we define an inclined surface, denoted $S_c$ through which the material escapes outwardly. Following Murphy, Ferreira, & Zanni (Reference Murphy, Ferreira and Zanni2010), we used $r_{in} = 1.4$ and $r_{out} = 5$ for calculation.

Figure 2. Schematic representation of the control volume used to compute the fluxes of the angular momentum and energy transport.

The accretion angular momentum flux is defined as $\dot{J}_{acc} = \dot{J}_{acc,kin} + \dot{J}_{acc,mag}$ where

(23) \begin{equation} \dot{J}_{acc,kin} = \int_{S_{in}} r_{in} {\unicode{x03C1}} U_{\unicode{x03D5}} \boldsymbol{U} \cdot dS - \int_{S_{out}} r_{out} {\unicode{x03C1}} U_{\unicode{x03D5}} \boldsymbol{U} \cdot dS\end{equation}

and

(24) \begin{equation} \dot{J}_{acc,mag} = \int_{S_{in}} r_{in} B_{\unicode{x03D5}} \boldsymbol{B} \cdot dS - \int_{S_{out}} r_{out} B_{\unicode{x03D5}} \boldsymbol{B} \cdot dS\end{equation}

The kinetic part $\dot{J}_{acc,kin}$ determines the flux of angular momentum inside the integration domain resulting from the inflow motion, while the magnetic part $\dot{J}_{acc,mag}$ defines the magnetic torque responsible for the radial transport of angular momentum inside the accretion disc.

The angular momentum balance can be written as follows (Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013):

(25) \begin{equation} \dot{J}_{acc} = 2 \dot{J}_{jet} + \dot{J}_{visc}, \end{equation}

which shows that the angular momentum flux due to inflow is driven by the jet torque $\dot{J}_{jet}$ , defined as the torque exerted on the disc by the jet, or the viscous torque $\dot{J}_{visc}$ which is due to viscous stresses. The ejection torque is also the sum of two parts: $\dot{J}_{jet} = \dot{J}_{jet,kin} + \dot{J}_{jet,mag}$ where

(26) \begin{equation} \dot{J}_{jet,kin} = \int_{S_c} r {\unicode{x03C1}} U_{\unicode{x03D5}} \boldsymbol{U} \cdot dS\end{equation}

and

(27) \begin{equation} \dot{J}_{jet,mag} = \int_{S_c} r B_{\unicode{x03D5}} \boldsymbol{B} \cdot dS\end{equation}

The viscous torque is defined as:

(28) \begin{equation} \dot{J}_{visc} = \int_{S_{in}} r \Pi \cdot dS - \int_{S_{out}} r \Pi \cdot dS\end{equation}

where $ \Pi$ is the viscous stress tensor introduced in equation (5).

We start by investigating the angular momentum transport inside the disc. We aim to shed light on the magnetic and kinetic contributions to the total accretion angular momentum. Fig. 3 shows the time evolution of the ratios $\dot{J}_{acc,kin} \big / \dot{J}_{acc}$ and $\dot{J}_{acc,mag} \big / \dot{J}_{acc}$ for all cases of simulations until time step $t=200$ . The conspicuous finding to emerge from this figure is that the kinetic torque represents the major contribution to $\dot{J}_{acc}$ while the magnetic part represents a very small contribution to the total accretion angular momentum. We recorded, for example, that the kinetic and magnetic contributions to the total accretion angular momentum of case 1 at $t=200$ are approximately 99% and 1%, respectively. These findings are consistent with previous works in the literature (Zanni et al. Reference Zanni, Ferrari, Rosner, Bodo and Massaglia2007; Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2009).

Figure 3. Time evolutions of the ratios $\dot{J}_{acc,kin}\big /\dot{J}_{acc}$ (top panel) and $\dot{J}_{acc,mag}\big /\dot{J}_{acc}$ (bottom panel) for the cases 1–5; case 1 ( ${\unicode{x03D5}}_s= 0.002$ ), case 2 ( ${\unicode{x03D5}}_s= 0.004$ ), case 3 ( ${\unicode{x03D5}}_s= 0.006$ ), case 4 ( ${\unicode{x03D5}}_s= 0.008$ ), case 5 ( ${\unicode{x03D5}}_s= 0.01$ ).

Moreover, we find lower values of $\dot{J}_{acc,mag}\big /\dot{J}_{acc}$ than those recorded in these cited papers, which confirms that the magnetic torque acts to reduce the angular momentum. We also find that the ratio $\dot{J}_{acc,mag}\big /\dot{J}_{acc,kin}$ does not exceed $2 \% $ for all our simulations, which is lower than that previously revealed in Zanni et al. (Reference Zanni, Ferrari, Rosner, Bodo and Massaglia2007) (about $10 \%$ ). Inside the disc, the kinetic torque plays therefore the important role in the angular momentum transport.

Interestingly, increasing the saturation parameter ${\unicode{x03D5}}_s$ leads to an increase in the amplitude of the ratio $\dot{J}_{acc,kin}\big /\dot{J}_{acc}$ and to a decrease in the ratio $\dot{J}_{acc,mag}\big /\dot{J}_{acc}$ until the time step $t=200$ . Thermal conduction acts then to further promote accretion, which makes the inner area of the disc denser, proving the presence of a sufficient material that can be ejected outward. This effect is reversed at advanced times of the simulations where the jet moves far enough from the disc. We observe in Fig. 4 particularly from $t= 500$ a decrease in the amplitude of $\dot{J}_{acc,kin}\big /\dot{J}_{acc}$ in presence of thermal conduction. This observation aligns with our prior work (Rezgui et al. Reference Rezgui, Marzougui, Woodring, Svoboda and Lili2019) where we showed that thermal conduction acts, at $t=500$ , to decelerate the disc’s angular velocity $U_{\unicode{x03D5}}$ (note that the kinetic torque profile depends on $U_{\unicode{x03D5}}$ ). This has been attributed to the dominance of gravity over centrifugal force, allowing the inflow towards the central object. It is also evident to notice the plateau reached by the quantity $\dot{J}_{acc,kin}\big /\dot{J}_{acc}$ identified for the case 2 from $t = 400$ until the final time step of our calculations. This finding pleads in favour of the quasi-stationarity of the solutions in presence of thermal conduction.

Figure 4. Time evolution of the ratio $\dot{J}_{acc,kin}\big /\dot{J}_{acc}$ of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) and case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

It is commonly accepted that the differential motion between the disc layers leads to the transfer of angular momentum from the inner parts of the disc to the outer parts, since the viscosity causes the adjacent layers of fluid in the disc to move at different speeds. Irrespective of the specific mechanism accountable for viscosity in accretion discs, previous studies (Murphy, Ferreira, & Zanni Reference Murphy, Ferreira and Zanni2010; Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013) have indicated that this process does not significantly affect the evolution of the disc-jet system. However, we will briefly examine the impact of thermal conduction on the temporal evolution of the viscous torque $\dot{J}_{visc}$ .

Fig. 5 depicts its profile for all our cases of simulations. After a few oscillations at the initial time steps, we observe that the viscous torque gradually decreases over time, and it is obvious to notice that it converges towards very low values. This behaviour indicates that its role in the angular momentum transport becomes less important as the accretion–ejection structure becomes fully established. Increasing the saturation parameter ${\unicode{x03D5}}_s$ leads to an increase in the amplitude of the viscous torque. This can be explained by the fact that the gas in the inner regions experiences a torque that causes it to lose more angular momentum and moves inward, while the gas in the outer regions experiences a torque that causes it to gain more angular momentum and moves outwards. Hence, within the disc’s inner region, gravity prevails over centrifugal force due to increased braking of its rings. This mechanism enables the accretion of additional material onto the central object. Nevertheless, the thermal conduction effect on the time evolution of $\dot{J}_{visc}$ is not substantial at advanced time steps of our calculations, as previously mentioned.

Figure 5. Temporal evolution of the viscous torque $\dot{J}_{visc}$ for cases 1–5 up to $t=200$ (top panel) and for cases 1–2 up to $t=700$ (bottom panel); case 1 ( ${\unicode{x03D5}}_s= 0.002$ ), case 2 ( ${\unicode{x03D5}}_s= 0.004$ ), case 3 ( ${\unicode{x03D5}}_s= 0.006$ ), case 4 ( ${\unicode{x03D5}}_s= 0.008$ ), case 5 ( ${\unicode{x03D5}}_s= 0.01$ ).

Within the same context, we also place a similar emphasis on examining the impact of thermal conduction on the ejection torque, which pertains to the torque exerted by the outflow on the disc. The relative contributions of the magnetic and kinetic parts to the total ejection angular momentum $\dot{J}_{jet}$ differ from those within the disc. In fact, we show in Fig. 6 the evolution in time of the ratios $\dot{J}_{jet,mag} \big /\dot{J}_{jet}$ and $\dot{J}_{jet,kin} \big /\dot{J}_{jet}$ respectively, for cases 1–2 until the final time step of our simulations.

Figure 6. Temporal evolution of the ratios $\dot{J}_{jet,mag} \big /\dot{J}_{jet}$ (top panel) and $\dot{J}_{jet,kin} \big /\dot{J}_{jet}$ (bottom panel) for cases 1–2 up to $t=700$ ; case 1 ( ${\unicode{x03D5}}_s= 0.002$ ), case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

One of the essential findings from the first sight of these curves is that the main contribution to the total ejection angular momentum is attributed to the magnetic part, in contrast to what is revealed inside the disc. This feature becomes prominent starting at time step $t=200$ where $\dot{J}_{jet,mag}$ represents approximately 60–80% of the total ejection angular momentum. This result is in concordance with the previous simulations conducted by Tzeferacos et al. (Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2009). Even more crucial is that at advanced time steps, particularly from $t=350$ , thermal conduction plays an essential role by enhancing the influence of the magnetic component $\dot{J}_{jet,mag} \big /\dot{J}_{jet}$ while decreasing the impact of the kinetic component $\dot{J}_{jet,kin} \big /\dot{J}_{jet}$ .

In order to give more insight about this behaviour, it is therefore essential to hearken back to the thermal conduction role in the magnetocentrifugal mechanism (Blandford & Payne Reference Blandford and Payne1982) discussed in our previous paper Rezgui et al. (Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022) where the investigation of the vertical dependence of the ratio $\vert B_{\unicode{x03D5}}\vert \big /B_p$ revealed that the toroidal magnetic acceleration becomes stronger in presence of thermal conduction which suggests that the magnetic field has a complete control over the ejected plasma and compels it to corotate with the field lines. We emphasised that the magnetic energy stored in the toroidal component of the Lorentz force at the disc surface propels the plasma along the magnetic field lines. This finding serves as a compelling argument supporting the notion that thermal conduction gives more weight to the contribution of the magnetic component $\dot{J}_{jet,mag}$ to the total ejection angular momentum.

In the same vein, it is valuable to assess the effectiveness of the magnetocentrifugal mechanism by examining the ratio between the magnetic $\dot{J}_{jet,mag}$ and the kinetic $\dot{J}_{jet,kin}$ torque. Fig. 7 provides a visualisation of the temporal evolution of this ratio for cases 1 and 2. As the jet propagates along the axis of rotation of the disc, we observe a noticeable increase in this ratio. Beyond a certain time step, specifically $t = 200$ , the magnetic component dominates the kinetic component, with the ratio exceeding unity. In case 1 of our simulations, we identify a peak at $t=300$ where the magnetic torque $\dot{J}_{jet,mag}$ is found to be $3.8$ times greater than the kinetic torque $\dot{J}_{jet,kin}$ . Concretely, we record an improvement of approximately $23.7\% $ in the integral of this ratio from $t = 400$ up to the final time step of our runs in case 2 relative to case 1. A higher ratio signifies a larger amount of specific angular momentum accessible at the disc’s surface, resulting in an increase in centrifugal acceleration and, consequently, higher poloidal terminal speeds for the plasma.

Figure 7. Time evolution of the ratio $\dot{J}_{jet,mag}\big /\dot{J}_{jet,kin}$ of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) and case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

It presents compelling evidence and detailed support for the notion that the accretion–ejection structure is primarily influenced by the magnetic force, with the magnetic contribution becoming increasingly dominant over the kinetic contribution in the presence of thermal conduction at advanced time steps.

It is interesting to present a detailed illustration of the angular momentum distribution in the disc-jet system over time. To this end, we compute the angular momentum flux $\dot J= r {\unicode{x03C1}} U_{\unicode{x03D5}} \boldsymbol{U} - r B_{\unicode{x03D5}} \boldsymbol{B}$ for cylindrical coordinates within each cell of the computation grid. In Fig. 8, we show the log-magnitude of $\dot J$ per cell sample of our reference case 1 across varying time steps. Notably, from the time step $t=50$ , our findings vividly demonstrate the establishment and evolution of the accretion-ejection structure within the system. The discernible variation in the magnitude of $\dot J$ unequivocally indicates the extraction of angular momentum from the disc surface by the jet. Simultaneously, our observations highlight the concurrent accretion of matter towards the central object.

Figure 8. Log-magnitude distribution of the angular momentum flux $ \dot J= r {\unicode{x03C1}} U_{\unicode{x03D5}} \boldsymbol{U} - r B_{\unicode{x03D5}} \boldsymbol{B}$ in the disc–jet system of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) for dynamical time steps (from top to bottom): $t = 0$ , $t = 50$ , $t = 300$ and $t = 700$ .

In Fig. 9, we extend our visualisation methodology to depict the temporal evolution of the log-magnitudes concerning the magnetic and kinetic constituents of the angular momentum flux within the intricate dynamics of the disc-jet system. Employing distinct monochromatic colour gradients (green for the kinetic component and blue for the magnetic component) in the first and second rows respectively, the third row of this illustration presents a compelling visual decomposition into these components, offering insights into their spatial coexistence. This comprehensive analysis corroborates our findings discussed above, emphasising the individual contributions of each component to the overall angular momentum.

Figure 9. Time evolution of log-magnitude of the kinetic component log $\big \vert \dot{J}_{kin} \big \vert $ (top panel), log-magnitude of the magnetic component log $\big \vert \dot{J}_{mag} \big \vert $ (central panel), and visual overlay of the sum of two components: log $ \big \vert \dot{J}_{kin} \big \vert$ + log $\big \vert \dot{J}_{mag} \big \vert$ (bottom panel) in the disc–jet system of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ).

Notably, the magnetic component (illustrated in the second row) primarily exerts influence within the jet region, while the kinetic component (shown in the first row) exhibits relevance across the entire structure. The third row’s composite visualisation delineates the dominance of the kinetic component within the accretion disc and unveils the concurrent presence of both components within the jet, shedding light on their coexistence and respective roles within distinct regions of the system. This detailed depiction affords a deeper understanding of the nuanced interplay between magnetic and kinetic elements in shaping the angular momentum dynamics within the disc-jet paradigm.

3.2. Energy budget of the disc–jet system

Within this section, our examination focuses on the role of thermal conduction in the energetics of the disc–jet system. Similarly to the analysis of angular momentum transfer, we explore the energy fluxes through the control volume introduced in Fig. 2.

The theoretical global energy budget can be written as:

(29) \begin{equation} \dot{E}_{lib} = \dot{E}_{jet} + \dot{E}_{rad}\end{equation}

where $ \dot{E}_{lib} = \dot{E}_{acc} + \dot{E}_{visc} + \dot{E}_{cond}$ is the actual power that is liberated within the disc.

$\dot{E}_{acc}$ is the total accretion power (accretion energy flux) which is defined as the sum of the mechanical (gravitational and kinetic), magnetic and enthalpy energy fluxes which are expressed as follows:

(30) \begin{equation} \dot{E}_{acc,grav} = \int_{S_{in}} \Phi_g {\unicode{x03C1}} \boldsymbol{U} \cdot dS - \int_{S_{out}} \Phi_g {\unicode{x03C1}} \boldsymbol{U} \cdot dS\end{equation}

(31) \begin{equation} \dot{E}_{acc,kin} = \int_{S_{in}} \frac{1}{2} U^2 {\unicode{x03C1}} \boldsymbol{U} \cdot dS - \int_{S_{out}} \frac{1}{2} U^2 {\unicode{x03C1}} \boldsymbol{U} \cdot dS\end{equation}

(32) \begin{equation} \dot{E}_{acc,mag} = \int_{S_{in}} \boldsymbol{E} \times \boldsymbol{B} \ \cdot dS - \int_{S_{out}} \boldsymbol{E} \times \boldsymbol{B} \cdot dS\end{equation}

(33) \begin{equation} \dot{E}_{acc,enth} = \int_{S_{in}} \frac{{\unicode{x03B3}}}{{\unicode{x03B3}}-1} P \boldsymbol{U} \cdot dS - \int_{S_{out}} \frac{{\unicode{x03B3}}}{{\unicode{x03B3}}-1} P \boldsymbol{U} \cdot dS\end{equation}

$\dot{E}_{visc}$ is the flux of energy due to viscosity and defined as:

(34) \begin{equation} \dot{E}_{visc} = \int_{S_{in}} {\boldsymbol{u}} \Pi \cdot dS - \int_{S_{out}} {\boldsymbol{u}} \Pi \cdot dS\end{equation}

$\dot{E}_{cond}$ is the flux of energy due to thermal conduction and can be written as:

(35) \begin{equation} \dot{E}_{cond} = \int_{S_{in}} \nabla \cdot \boldsymbol{F}_c \cdot dS - \int_{S_{out}} \nabla \cdot \boldsymbol{F}_c \cdot dS\end{equation}

$\dot{E}_{jet}$ is the jet power characterising the energy extracted by the outflow. It is calculated as the sum of all energy fluxes (mechanical, Poynting, and enthalpy) leaving the disc surface:

(36) \begin{equation} \dot{E}_{jet} = \dot{E}_{jet,kin} + \dot{E}_{jet,grav} + \dot{E}_{jet,mag} + \dot{E}_{jet,enth}\end{equation}

where

(37) \begin{equation} \dot{E}_{jet,kin} = \int_{S_c} \frac{1}{2} U^2 {\unicode{x03C1}} \boldsymbol{U} \cdot dS\end{equation}

(38) \begin{equation} \dot{E}_{jet,grav} = \int_{S_c} \Phi_g {\unicode{x03C1}} \boldsymbol{U} \cdot dS\end{equation}

(39) \begin{equation} \dot{E}_{jet,mag} = \int_{S_c} \boldsymbol{E} \times \boldsymbol{B} \ \cdot dS\end{equation}

(40) \begin{equation} \dot{E}_{jet,enth} = \int_{S_c} \frac{{\unicode{x03B3}}}{{\unicode{x03B3}}-1} P \boldsymbol{U} \cdot dS\end{equation}

$\dot{E}_{rad}$ represents the power released into heat by both Joule and viscous terms, which is eventually emitted as radiation from the surface of the disc. Following Murphy, Ferreira, & Zanni (Reference Murphy, Ferreira and Zanni2010) our calculation does not account for radiation. However, we ensure that the Joule and viscous heating effects are balanced by the cooling term $\zeta_{cool}$ (equation (6)).

In the preceding section of this paper, we showed that the contribution of the viscous torque to the angular momentum transport within the disc is not significant. Therefore, we will not delve, as prior studies (Murphy, Ferreira, & Zanni Reference Murphy, Ferreira and Zanni2010; Tzeferacos et al. Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013), into an in-depth analysis of the effects of the viscous power on the energy balance of the disc-jet system. Instead, our primary focus will be on comparing the power generated by the jet with the power derived from accretion, as well as the energy flux associated with thermal conduction. By concentrating our efforts on these specific aspects, we aim to shed light on the intricate interplay between these sources of energy in the system under study.

We commence by elucidating how the existence of thermal conduction influences the total accretion power and then unravel the diverse contributions to $\dot{E}_{acc}$ . We show in Fig. 10 the time evolution of the total accretion power for the cases considered in this work. At initial time steps, a remarkable observation is the sudden surge in the amplitude of the accretion power, particularly from $t= 100$ , indicating the successful operation of the inflow mechanism, effectively delivering material to the disc. Subsequently, after a few fluctuations, particularly from $t = 300$ , an intriguing correlation emerges between the saturation parameter ${\unicode{x03D5}}_s$ and the accretion power. As the saturation parameter increases, so does the amplitude of the accretion power. Specifically, we record an improvement of approximately $2\% $ in the integral accretion power from $t=400$ up to the final time step of our calculations in case 2 relative to case 1. This relationship demonstrates the influence of thermal conduction within the system, as it enhances the energy content within the control volume and consequently, boosts the overall energy output. Notice that at initial time steps, the influence of thermal conduction on the temporal evolution of $\dot{E}_{acc}$ remains imperceptible, as during this period, the system resides in a transitional regime.

Figure 10. Temporal evolution of the total accretion power $\dot{E}_{acc}$ for cases 1–5 up to $t=200$ (top panel) and for cases 1–2 up to $t=700$ (bottom panel); case 1 ( ${\unicode{x03D5}}_s= 0.002$ ), case 2 ( ${\unicode{x03D5}}_s= 0.004$ ), case 3 ( ${\unicode{x03D5}}_s= 0.006$ ), case 4 ( ${\unicode{x03D5}}_s= 0.008$ ), case 5 ( ${\unicode{x03D5}}_s= 0.01$ ).

We examine the significance of the energy arising from the enthalpy flux and mechanical power in relation to the overall accretion power. As illustrated in Fig. 11, our findings indicate that the dominant contribution originates from the mechanical flux (top panel). However, it is noteworthy that the enthalpy energy contribution (bottom panel) follows closely and gradually becomes more prominent as time progresses. Remarkably, the cumulative sum of these two contributions in each time step converges to a value approaching unity, providing compelling evidence for the negligible impact of magnetic contributions.

Figure 11. Temporal evolution of the ratios $\dot{E}_{acc,mec}\big / \dot{E}_{acc}$ (top panel) and $\dot{E}_{acc,enth}\big / \dot{E}_{acc}$ (bottom panel) for cases 1–2 up to $t=700$ ; case 1 ( ${\unicode{x03D5}}_s= 0.002$ ), case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

For example, in our reference case 1, we observed a notable trend in the contributions of mechanical and enthalpy flux to the total accretion power. Specifically, the mechanical contribution slowly decreased from an initial value of 100 to $44 \%$ while the enthalpy contribution correspondingly increased from 0 to $56.5 \%$ . This observation can be attributed to the increasing significance of thermal energy transport towards the central object as time progresses. The data suggest that as the system evolves, the role of thermal energy becomes increasingly prominent, leading to a gradual shift in the relative importance of the different energy sources involved in accretion.

Furthermore, Fig. 11 reveals that the increase of the saturation parameter leads to changes in the amplitudes of two key ratios. On one hand, the amplitude of ratio $\dot{E}_{acc,mec}\big / \dot{E}_{acc}$ shows a clear increase, while on the other hand, the amplitude of ratio $\dot{E}_{acc,enth}\big / \dot{E}_{acc}$ experiences a decrease. This behaviour further proves the thermal conduction impact in enhancing the pumping of material towards the immediate vicinity of the central object and in concurrently diminishing the amount of energy dissipated locally. Consequently, it prevents enthalpy from accumulating within the disc.

It is important to thoroughly investigate the contribution of energy resulting from thermal conduction in order to comprehend the overall liberated energy within the disc. The temporal evolution of the ratio $\dot{E}_{cond}\big / \dot{E}_{lib}$ , as illustrated in Fig. 12, demonstrates the significance of this contribution. Even though we observed a maximum percentage of approximately $20 \%$ for case 1 at the final time of our simulation, it is crucial not to underestimate its impact. Furthermore, increasing the saturation parameter clearly leads to an improvement of this contribution. Moreover, it is interesting to note that the increase in energy due to thermal conduction over time leads to a corresponding rise in enthalpy energy, as we have previously highlighted (Fig. 11, bottom panel).

Figure 12. Time evolution of the ratio $\dot{E}_{cond}\big / \dot{E}_{lib}$ of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) and case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

We show in Fig. 13 the temporal evolution of the ratio between the bipolar jet and liberated power $2\dot{E}_{jet}\big / \dot{E}_{lib}$ of cases 1–2 of our simulations. The data presented in Fig. 13 clearly demonstrate that the outflow is not able to extract the complete energy, as indicated by the maximum recorded percentage of approximately $55 \%$ . This stands in contrast to cold jets, where the enthalpy flux is consistently negligible and the majority of the accretion power is liberated through the outflow channel. Two primary factors contribute to this behaviour. Firstly, the disc is traversed by a very weak magnetic field that lacks the capability to harness a substantial fraction of the available power. Secondly, as the amount of energy dissipated locally within the disc increases, the enthalpy flux depicted in Fig. 11, transported by accretion towards the central object, becomes increasingly significant at the expense of the jet power (Fig. 13). We reveal that the integral ratio $2\dot{E}_{jet}\big / \dot{E}_{lib}$ exhibits a decrease of approximately $13.8\%$ in case 2 relative to case 1. This finding suggests a distinctive trend wherein the ratio between the bipolar jet and liberated power experiences a discernible reduction over the recorded time period ([100, 700]) when increasing the saturation parameter. Therefore, it can be inferred that there is a direct impact on the amount of energy that is extracted by the bipolar outflow from the disc.

Figure 13. Time evolution of the ratio between the bipolar jet and liberated power $2\dot{E}_{jet}\big / \dot{E}_{lib}$ of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) and case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

It is worth noting that these findings align well with the earlier research of Tzeferacos et al. (Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013). While their study did not address thermal conduction, it did emphasise the significance of the enthalpy accretion flux and reported a range of 58–87% for the ratio $2\dot{E}_{jet}\big / \dot{E}_{lib}$ . Consequently, the values observed in our investigation are plausible.

In order to shed light on the energy dynamics within the jet, it is crucial to illuminate the diverse factors that contribute to the total jet power, as this allows for a more comprehensive analysis and a closer approximation of the major contributors. The evaluation of individual components that make up the total jet power provides valuable insights into the underlying mechanisms governing jet formation and propagation. We show in Fig. 14 the temporal evolution of the ratios $\dot{E}_{jet,mag}\big / \dot{E}_{jet}$ (top panel) and $\dot{E}_{jet,enth}\big / \dot{E}_{jet}$ (bottom panel) for cases 1 and 2.

Figure 14. Temporal evolution of the ratios $\dot{E}_{jet,mag}\big / \dot{E}_{jet}$ (top panel) and $\dot{E}_{jet,enth}\big / \dot{E}_{jet}$ (bottom panel) for cases 1–2 up to $t=700$ ; case 1 ( ${\unicode{x03D5}}_s= 0.002$ ), case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

The data presented in Fig. 14 (top panel) provide compelling evidence that the primary contributor to the total jet power is the Poynting flux. Notably, the ratio $\dot{E}_{jet,mag}\big / \dot{E}_{jet}$ exhibits pronounced fluctuations during the initial time steps of our simulations, surpassing unity and indicating the non-stationary solutions. It is worth mentioning that this behaviour aligns with previous studies on cold jets. For instance, Zanni et al. (Reference Zanni, Ferrari, Rosner, Bodo and Massaglia2007) reported a similar pattern with their runs, wherein the amplitude of this ratio exceeded unity even at the final time step.

However, we observe that these fluctuations disappear as the jet progresses into the surrounding medium. Remarkably, our findings demonstrate that our solutions attain stability from $t = 300$ onwards, since the sum of the energy components approaches 1 (see Table 2). Additionally, our analysis reveals that thermal conduction does not exert a significant influence on the temporal evolution of the ratio $\dot{E}_{jet,mag}\big / \dot{E}_{jet}$ , except for the solutions in case 2, which achieve stability at a later time point of $t = 350$ .

Building upon our previous findings (Fig. 11), which demonstrated the significant role of the accretion enthalpy flux in contributing to the total accretion power, it becomes evident that the jet enthalpy contribution to the overall jet power also holds considerable importance. This observation is substantiated by the profile of the ratio $\dot{E}_{jet,enth}\big / \dot{E}_{jet}$ (bottom panel of Fig. 14), where we observe a progressive increase in the jet enthalpy contribution over time. Remarkably, this ratio reaches for case 1 a maximum percentage of approximately $60 \% $ at the final time of our simulations. This value under scrutiny in this study is not only substantial, but also plausible when compared to the outcomes presented in the research conducted by Tzeferacos et al. (Reference Tzeferacos, Ferrari, Mignone, Zanni, Bodo and Massaglia2013). Notably, the latter work involved simulations that did not account for thermal conduction. In particular, the authors documented a significant jet enthalpy contribution up to $33 \% $ . Thus, our findings highlight the critical significance of including jet enthalpy as a key element in understanding and assessing the total jet power.

Furthermore, we note that the increase in the saturation parameter correlates with an enhancement in the jet enthalpy contribution. This observation further confirms the link between the saturation parameter and the jet temperature, as previously discussed in our paper Rezgui et al. (Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022). The amplified jet enthalpy contribution associated with higher saturation parameter suggests an intensified thermal energy deposition within the jet. Consequently, controlling the saturation parameter becomes crucial for accurately characterising the jet’s energetics and assessing its overall power. These findings underscore the intricate relationship between the saturation parameter, jet temperature, and the resulting enthalpy contribution, shedding light on the underlying mechanisms governing the jet’s thermodynamic behaviour.

In our analysis, we also have carefully considered the contribution of gravity flux to the overall jet power. Fig. 15 illustrates the temporal evolution of the ratio $\dot{E}_{jet,grav}\big / \dot{E}_{jet}$ , revealing distinct characteristics in its profile. Initially, oscillations are observed, denoting a transition phase, but stability is achieved after $t = 200$ , as evidenced by the plateau in the ratio’s profile. Notably, the gravitational contribution is negative, as the jet remains within the potential well of the central object at the disc surface. This behaviour also aligns with previous simulations conducted by Zanni et al. (Reference Zanni, Ferrari, Rosner, Bodo and Massaglia2007). Moreover, we find no discernible influence of the variation of the saturation parameter on the gravitational contribution.

In Table 2, we present a comprehensive overview of the various contributions to the total jet power at the final time step of our simulations for cases 1 and 2. This table offers valuable insights into the individual components that contribute to the overall jet power.

We present in Fig. 16 the temporal evolution of the ratio between the jet Poynting and kinetic flux $\dot{E}_{jet,mag}\big / \dot{E}_{jet,kin}$ . A higher value of this ratio signifies a greater availability of magnetic energy per particle at the disc surface, facilitating the acceleration of the outflow. The Poynting flux originating from the disc surface is effectively converted into kinetic energy along the outflow, as evidenced by the super-fast magnetosonic nature of the jet, which is demonstrated in our previous paper Rezgui et al. (Reference Rezgui, Marzougui, Lili, Preiner and Ceccobello2022) where we showed that our solutions can effectively cross the MHD critical surfaces and reach the upper end of the computational domain. The data depicted in Fig. 16 highlight that the magnetic energy per particle at the disc surface manifests greater availability during the initial temporal stages compared to that recorded at the later time steps (from $t=300$ up to the end of the integration).

Remarkably, we discern two distinct phases characterising the thermal conduction impact on the temporal evolution of the ratio $\dot{E}_{jet,mag}\big / \dot{E}_{jet,kin}$ . At initial time steps, we pinpoint that increasing the saturation parameter leads to a decrease in the amplitude of this ratio. However, a noteworthy shift occurs from time step $t=400$ , where an inverse relationship emerges; an increase in ${\unicode{x03D5}}_s$ thereafter corresponds to an elevation in this ratio. In particular, we note an enhancement of approximately $11.2\%$ in the integral of $\dot{E}_{jet,mag}\big / \dot{E}_{jet,kin}$ from time step $t=400$ up to the end of our calculations in case 2 when compared to case 1. This feature indicates an improved efficacy of the magnetocentrifugal mechanism due to thermal conduction at advanced time steps.

To further elucidate the energy distribution dynamics within the disc-jet system, we adopt a methodology akin to our approach for depicting angular momentum distribution (as observed in Figs. 8 and 9). In fact, by computing the energy flux $ \dot E= \frac{1}{2} U^2 {\unicode{x03C1}} \boldsymbol{U} + \Phi_g {\unicode{x03C1}} \boldsymbol{U} + \boldsymbol{E} \times \boldsymbol{B} + \frac{{\unicode{x03B3}}}{{\unicode{x03B3}}-1} P \boldsymbol{U}$ , encompassing mechanical, magnetic and enthalpy terms within individual cells across our computational grid, we point out crucial insights. We visualise in Fig. 17 the logarithmic magnitude of $\dot E$ per cell sample of our reference case 1. Intriguingly, we discern that the inner region of the disc exhibits a progressive gain in energy over time, indicative of the efficacy of the ongoing accretion process. Concurrently, we underscore the extraction of energy from the disc’s surface by the bipolar jet, with this energy extraction displaying an incremental trend over time. This illustration offers a clear depiction of the energy redistribution dynamics, emphasising the exchange mechanisms between the accretion disc and the jet.

Table 2. Values of the different contributions to the total jet power $\dot{E}_{jet}$ calculated at $t = 700$ .

Figure 15. Time evolution of the ratio $\dot{E}_{jet,grav}\big / \dot{E}_{jet}$ of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) and case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

Figure 16. Temporal evolution of the ratio between the jet Poynting and kinetic flux $\dot{E}_{jet,mag}\big / \dot{E}_{jet,kin}$ of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) and case 2 ( ${\unicode{x03D5}}_s= 0.004$ ).

Figure 17. Log-magnitude distribution of the energy power $ \dot E= \frac{1}{2} U^2 {\unicode{x03C1}} \boldsymbol{U} + \Phi_g {\unicode{x03C1}} \boldsymbol{U} + \boldsymbol{E} \times \boldsymbol{B} + \frac{{\unicode{x03B3}}}{{\unicode{x03B3}}-1} P \boldsymbol{U}$ in the disc–jet system of case 1 ( ${\unicode{x03D5}}_s= 0.002$ ) for dynamical time steps (from top to bottom): $t = 0$ , $t = 50$ , $t = 300$ and $t = 700$ .

In Fig. 18 of our study, we expand the scope of our visualisation to encompass the temporal evolution of log-magnitudes pertaining to the mechanical $ \dot{E}_{mec} = \frac{1}{2} U^2 {\unicode{x03C1}} \boldsymbol{U} + \Phi_g {\unicode{x03C1}} \boldsymbol{U}$ , magnetic $\dot{E}_{mag} = \boldsymbol{E} \times \boldsymbol{B} $ , and enthalpy $\dot{E}_{enth} = \frac{{\unicode{x03B3}}}{{\unicode{x03B3}}-1} P \boldsymbol{U} $ components constituting the energy power $\dot E$ within the disc-jet system. Utilising distinct monochromatic colour gradients – green representing the mechanical component, blue depicting the magnetic component, and red signifying the enthalpy component – the first, second, and third rows of this representation respectively illustrate a comprehensive decomposition into these individual energy components. Examination of log-magnitude of each component underscores the importance of the enthalpy and mechanical elements throughout the entirety of the disc-jet system. Notably, the magnetic component exerts its significant influence primarily from the disc’s surface, where the jet begins to develop, to the end of our grid. This visualisation reaffirms our findings, emphasising the distinctive contributions made by each component to the overall energy $\dot E$ . The composite in the fourth row distinctly reveals the simultaneous presence of all three components within the jet, illuminating their coexistence and interplay.

Upon scrutiny of Fig. 9, we unveiled a direct correlation between the temporal escalation in energy attributed to thermal conduction and the concurrent amplification in enthalpy energy. We noted that this observation emphasises that as local energy dissipation intensifies within the disc, the significance of enthalpy accretion flux grows at the expense of jet power. In order to elucidate this phenomenon further, we undertake a visualisation of the log-magnitude of $\dot E$ for both case 1 and case 2 at time steps $t=300$ and $t=700$ (Fig. 19). Our examination reveals a discernible trend wherein the material power within the ejection zone of case 2 consistently registers lower values compared to those in case 1 over time. This evidence suggests that the augmentation of the saturation parameter ${\unicode{x03D5}}_s$ leads to a gradual reduction in the jet’s power, elucidating an important correlation between this parameter and the jet power within the system.