3 Results

In this section we discuss a range of models of increasing degree of complexity. We first consider in Sect. 3.1 the simplest model that contains neither magnetic field nor mass sink at the centre to show how the pressure gradient resulting from the (fixed) entropy distribution drives a disc outflow. It is further shown that the outflow is significantly reduced if accretion onto the central object is allowed, but restored again if the disc can generate a large-scale magnetic field (Sect. 3.2). Having thus demonstrated the importance of the large-scale magnetic field in our model, we discuss in Sect. 3.3 its structure, largely controlled by the dynamo action but affected by the outflow. Model parameters used in these sections are not necessarily realistic as we aim to illustrate the general physical nature of our solutions. We present a physically motivated model in Sect. 4 where the set of model parameters is close to that of a standard accretion disc around a protostellar object. The physical nature of our solutions is discussed in Sects. 3.4-3.6.

   
3.1 Nonmagnetic outflows

We illustrate in Fig. 2 results obtained for a model without any magnetic fields and without mass sink. A strong outflow develops even in this case, which is driven mostly by the vertical pressure gradient in the transition layer between the disc and the corona, in particular by the term $T\vec{\nabla} s$ in Eq. (21). The gain in velocity is controlled by the total specific entropy difference between the disc and the corona, but not by the thickness d of the transition layer in the disc profile (4).

The flow is fastest along the rotation axis and within a cone of polar angle of about $30^\circ$, where the terminal velocity $u_z\approx3$ is reached. The conical shape of the outflow is partly due to obstruction from the dense disc, making it easier for matter to leave the disc near the axis. Both temperature and density in nonmagnetic runs without mass sink are reduced very close to the axis where the flow speed is highest.

The general flow pattern is sensitive to whether or not matter can accrete onto the central object. We show in Fig. 3 results for the same model as in Fig. 2, but with a mass sink given by Eq. (6) with $\tau _{\rm star}=0.01$. This can be compared with earlier work on thermally driven winds by Fukue (1989) who also considered polytropic outflows, but the disc was treated as a boundary condition. In Fukue (1989), outflows are driven when the injected energy is above a critical value. The origin of this energy injection may be a hot corona. The critical surface in his model (see the lower panel of his Fig. 2) is quite similar to that found in our simulation (our Fig. 3), although our opening angle was found to be larger than in Fukue's (1989) model. (Below we show, albeit with magnetic fields, that smaller values of $\beta$ do result in smaller opening angles, see Fig. 9, which would then be compatible with the result of Fukue (1989).)

As could be expected, the mass sink hampers the outflow in the cone (but not at $\varpi \ga 0.5$). The flow remains very similar to that of Fig. 3 when $\tau_{\rm star}$ is reduced to 0.005. Thus, the nonmagnetic outflows are very sensitive to the presence of the central sink. As we show now, magnetized outflows are different in this respect.

   
3.2 Magnetized outflows

In this section we discuss results obtained with magnetic fields, first without mass sink at the centre and then including a sink. We show that the effects of the sink are significantly weaker than in the nonmagnetic case.

Figure 4: A magnetic model without mass sink at the centre: velocity vectors and poloidal magnetic field lines (white) superimposed on a grey scale representation of temperature (in terms of h) for a run with $\alpha _0=-0.3$ (resulting in roughly dipolar magnetic symmetry) at time t=269. In contrast to the nonmagnetic model of Fig. 2, a conical shell has developed that is cooler and less dense than its exterior. The conical shell intersects $z=\pm 1$ at around $\varpi \approx 1.2$. The dashed line shows the surface where the poloidal velocity equals $({c_{\rm s}}^2+v_{\rm A,pol}^2)^{1/2}$, with $v_{\rm A,pol}$ the Alfvén speed from the poloidal magnetic field (fast magnetosonic surface with respect to the poloidal field). The disc boundary is shown with a thin black line; $\beta =0.1$.

Figure 5: Profiles of vertical velocity and sound speed along the axis, $\varpi =0$, for the nonmagnetic model of Fig. 2 with smoothing radius r0=0.05(dotted) and two runs with magnetic field, with r0=0.05 as in Fig. 4 (solid) and r0=0.02 (dashed, t=162). The shaded area marks the location of the disc. Terminal wind speeds are reached after approximately three disc heights. The presence of a magnetic field and a deeper potential well (smaller r0) both result in faster outflows. The models shown here have no mass sink at the centre and $\beta =0.1$.

Figure 6: Results for a larger domain, $[0,8]\times [-2,30]$, with the same parameters as in Fig. 4, averaged over times $t=200 \dots 230$, $\beta =0.1$. Left panel: velocity vectors, poloidal magnetic field lines and grey scale representation of hin the inner part of the domain. Right panel: velocity vectors, poloidal magnetic field lines and normalized specific enthalpy $h/\vert\Phi \vert$ in the full domain.

A magnetized outflow without the central mass sink, shown in Fig. 4, is similar to that in Fig. 2, but is denser and hotter near the axis, and the high speed cone has a somewhat larger opening angle. In addition, the outflow becomes more structured, with a well pronounced conical shell where temperature and density are smaller than elsewhere (the conical shell reaches $z=\pm 1$ at $\varpi \approx 1.2$ in Fig. 4). Here and in some of the following figures we also show the fast magnetosonic surface with respect to the poloidal field. In Sect. 3.5 we show that this surface is close to the fast magnetosonic surface. As shown in Fig. 5, the outflow becomes faster inside the cone ( $u_z \approx 5$ on the axis). As expected, we find that deeper potential wells, i.e. smaller values of r₀ in Eq. (5) and (23), result in even faster flows and in larger opening angles.

Our results are insensitive to the size and symmetry of the computational domain: we illustrate this in Fig. 6 with a larger domain of the size $[0,8]\times [-2,30]$. The disc midplane in this run is located asymmetrically in zin order to verify that the (approximate) symmetry of the solutions is not imposed by the symmetry of the computational domain. Figure 6 confirms that our results are not affected by what happens near the computational boundaries.

Figure 7: As in Fig. 4, but with a mass sink at the centre, $\tau _{\rm star}=0.01$. The outflow speed near the axis and the opening angle of the conical shell (now reaching $z=\pm 1$ at $\varpi \approx 0.5$) are reduced in comparison to that in the model without mass sink shown in Fig. 4, but the general structure of the outflow is little affected. Averaged over times $t=130 \dots 140$, $\beta =0.1$. (Reference model.)

Unlike the nonmagnetized system, the magnetized outflow changes only comparatively little when the mass sink is introduced at the centre. We show in Fig. 7 the results with a sink ( $\tau _{\rm star}=0.01$) and otherwise the same parameters as in Fig. 4. As could be expected, the sink leads to a reduction in the outflow speed near the axis; the flow in the high speed cone becomes slower. But apart from that, the most important effects of the sink are the enhancement of the conical structure of the outflow and the smaller opening angle of the conical shell. A decrease in $\tau_{\rm star}$ by a factor of 2 to 0.005 has very little effect, as illustrated in Figs. 8 and 10.

Increasing the entropy contrast (while keeping the specific entropy unchanged in the corona) reduces the opening angle of the conical shell. Pressure driving is obviously more important in this case, as compared to magneto-centrifugal driving (see Sect. 3.5). A model with $\beta =0.02$ (corresponding to a density and inverse temperature contrast of about 50:1 between the disc and the corona) is shown in Fig. 9. At $\vartheta = 60^\circ$, the radial velocity $u_{\rm r}$ is slightly enhanced relative to the case $\beta =0.1$ (contrast 10:1); see Fig. 10. At $\vartheta = 30^\circ$, on the other hand, the flow with the larger entropy contrast reaches the Alfvén point close to the disc (at $r\approx0.27$ as opposed to $r \ga 1$in the other case), which leads to a smaller terminal velocity.

Figure 8: As in Fig. 7, but with a stronger mass sink at the centre, $\tau _{\rm star}=0.005$. The flow pattern is very similar to that of Fig. 7 where the time scale of the sink is twice as large. Averaged over times $t=122 \dots 132$, $\beta =0.1$.

Figure 9: As in Fig. 7, but with a larger entropy contrast, $\beta =0.02$. The opening angle of the conical shell is reduced in comparison to that of Figs. 7 and 8 where the entropy contrast is smaller (the shell crosses $z=\pm 1$ at $\varpi \approx 0.15$). Averaged over times t=140...240, $\tau _{\rm star}=0.01$.

We conclude that the general structure of the magnetized flow and its typical parameters remain largely unaffected by the sink, provided its efficiency $\tau_{\rm star}^{-1}$ does not exceed a certain threshold. It is plausibly the build-up of magnetic pressure at the centre that shields the central object to make the central accretion inefficient. This shielding would be even stronger if we included a magnetosphere of the central object. We discuss in Sect. 4.1 the dependence of our solution on the geometrical size of the sink and show that the general structure of the outflow persists as long as the size of the sink does not exceed the disk thickness.

Figure 10: Spherical radial velocity component, $u_{\rm r}$ (dashed), and poloidal Alfvén speed, $v_{\rm A,pol}$ (solid), as functions of spherical radius at polar angles $\vartheta = 30^\circ$ (thick lines) and $\vartheta = 60^\circ$ (thin lines) for the models of Figs. 7 (top panel), 8 (middle panel) and 9 (bottom panel).

   
3.3 Magnetic field structure

The dynamo in most of our models has $\alpha_0<0$, consistent with results from simulations of disc turbulence driven by the magneto-rotational instability (Brandenburg et al. 1995; Ziegler & Rüdiger 2000). The resulting field symmetry is roughly dipolar, which seems to be typical of $\alpha\Omega$ disc dynamos with $\alpha_0<0$ in a conducting corona (e.g., Brandenburg et al. 1990). We note that the dominant parity of the magnetic field is sensitive to the magnetic diffusivity in the corona: a quadrupolar oscillatory magnetic field dominates for $\alpha_0<0$if the disc is surrounded by vacuum (Stepinski & Levy 1988).

For $\alpha_0<0$, the critical value of $\alpha_0$ for dynamo action is about 0.2, which is a factor of about 50 larger than without outflows. Our dynamo is then only less than twice supercritical. A survey of the dynamo regimes for similar models is given by Bardou et al. (2001).

Figure 11: Radial profiles of magnetic pressure from the toroidal (solid) and poloidal (dotted) magnetic fields and thermal pressure (dashed) for the model of Fig. 6. Shown are the averages over the disc volume (lower panel) and over a region of the same size around z=8 in the corona (upper panel). $\beta =0.1$.

The initial magnetic field (poloidal, mixed parity) is weak ( $p_{\rm mag}\equiv{\vec{B}}^2/(2\mu_0)\approx10^{-5}$), cf. Fig. 11 for comparison with the gas pressure], but the dynamo soon amplifies the field in the disc to $p_{\rm mag,tor}\approx10$, and then supplies it to the corona. As a result, the corona is filled with a predominantly azimuthal field with $p_{\rm mag,tor}/p\approx100$ at larger radii; see Fig. 11. We note, however, that the flow in the corona varies significantly in both space and time. Magnetic pressure due to the toroidal field $B_\varphi$, $p_{\rm mag,tor}$, exceeds gas pressure in the corona outside the inner cone and confines the outflow in the conical shell. The main mechanisms producing $B_\varphi$ in the corona are advection by the wind and magnetic buoyancy (cf. Moss et al. 1999). Magnetic diffusion and stretching of the poloidal field by vertical shear play a relatively unimportant rôle.

The field in the inner parts of the disc is dominated by the toroidal component; $\vert B_\varphi/B_z\vert\approx3$ at $\varpi\la0.5$; this ratio is larger in the corona at all $\varpi$. However, as shown in Fig. 11, this ratio is closer to unity at larger radii in the disc.

As expected, $\alpha _0>0$ results in mostly quadrupolar fields (e.g., Ruzmaikin et al. 1988). As shown in Fig. 12, the magnetic field in the corona is now mainly restricted to a narrow conical shell that crosses $z=\pm 1$ at $\varpi \approx 0.6$. Comparing this figure with the results obtained with dipolar magnetic fields (Fig. 4), one sees that the quadrupolar field has a weaker effect on the outflow than the dipolar field; the conical shell is less pronounced. However, the structures within the inner cone are qualitatively similar to each other.

The magnitude and distribution of $\alpha$ in Eq. (14) only weakly affect magnetic field properties as far as the dynamo is saturated. For a saturated dynamo, the field distribution in the dynamo region ( $0.2<\varpi<1.5$, |z|<0.15) roughly follows from the equipartition field, $B\simeq(\varrho\mu_0 v_0^2)^{1/2}$with $v_0={c_{\rm s}}$. In other words, nonlinear states of disc dynamos are almost insensitive to the detailed properties of $\alpha$ (e.g., Beck et al. 1996; Ruzmaikin et al. 1988).

A discussion of disc dynamos with outflows, motivated by the present model, can be found in Bardou et al. (2001). It is shown that the value of magnetic diffusivity in the corona does not affect the dynamo solutions strongly. Moreover, the outflow is fast enough to have the magnetic Reynolds number in the corona larger than unity, which implies that ideal integrals of motion are very nearly constant along field lines; see Sect. 3.6. The most important property is the sign of $\alpha$ as it controls the global symmetry of the magnetic field.

Figure 12: As in Fig. 4, but with $\alpha _0=+0.3$, at time t=132. The magnetic field geometry is now mostly quadrupolar because $\alpha _0>0$.

   
3.4 Mass and energy loss

The mass injection and loss rates due to the source, sink and wind are defined as

$$\dot{M}_{\rm source} = \int q_\varrho^{\rm disc} ~ {\rm d} {}... ... \dot{M}_{\rm sink} = \int q_\varrho^{\rm star} ~ {\rm d} {}V,$$ (36)

and

$$\dot{M}_{\rm wind}=\oint\varrho {\vec{u}}\cdot{\rm d} {}{\vec{S}},$$ (37)

respectively, where the integrals are taken over the full computational domain or its boundary. About 1/3 of the mass released goes into the wind and the rest is accreted by the sink, in the model with $\tau _{\rm star}=0.01$ and $\beta =0.1$ of Fig. 7. Reducing $\tau_{\rm star}$ by a factor of 2 (as in the model of Fig. 8), only changes the global accretion parameters by a negligible amount ($\la$$10 \%$).

Figure 13: Azimuthally integrated mass flux density, represented as a vector $2\pi \varpi \varrho (u_\varpi , u_z)$, in the simulation of Fig. 7 with a dipolar magnetic field and a mass sink at the centre. Shades of grey show the distribution of the mass source in the disc, $q_\varrho ^{\rm disc}$. The disc boundary is shown with a white line.

The mass loss rate in the wind fluctuates on a time scale of 5 time units, but remains constant on average at about $\dot M_{\rm wind}\approx3$, corresponding to $6\times10^{-7}~M_\odot~{\rm yr}^{-1}$, in the models of Figs. 7 and 8. The mass in the disc, $M_{\rm disc}$, remains roughly constant.

The rate at which mass needs to be replenished in the disc, $\dot M_{\rm source}/M_{\rm disc}$, is about 0.4. This rate is not controlled by the imposed response rate of the mass source, $\tau_{\rm disc}^{-1}$, which is 25 times larger. So, the mass source adjusts itself to the disc evolution and does not directly control the outflow. We show in Fig. 13 trajectories that start in and around the mass injection region. The spatial distribution of the mass replenishment rate  $q_\varrho ^{\rm disc}$shown in Fig. 13 indicates that the mass is mainly injected close to the mass sink, and $q_\varrho ^{\rm disc}$ remains moderate in the outer parts of the disc. (Note that the reduced effect of the mass sink in the magnetized flow is due to magnetic shielding rather than to mass replenishment near the sink - see Sect. 3.2.)

The angular structure of the outflow can be characterized by the following quantities calculated for a particular spherical radius, r=8, for the model of Fig. 6: the azimuthally integrated normalized radial mass flux density, $\dot{M}(\vartheta)/M_{\rm disc}$, where

$$\dot{M}(\vartheta)=2\pi r^2\varrho u_{\rm r}\sin\vartheta,\quad M_{\rm disc}=\int_{\rm disc}\rho~{\rm d} {}V,$$ (38)

the azimuthally integrated normalized radial angular momentum flux density, $\dot{J}(\vartheta)/J_{\rm disc}$, where

$$\dot{J}(\vartheta)=2\pi r^2\varrho \varpi u_\varphi u_{\rm r}... ...J_{\rm disc}=\int_{\rm disc}\rho \varpi u_\varphi~{\rm d} {}V,$$ (39)

the azimuthally integrated normalized radial magnetic energy flux (Poynting flux) density, $\dot{E}_{\rm M}(\vartheta)/E_{\rm M}$, where

$$\dot{E}_{\rm M}(\vartheta)=2\pi r^2{({\vec{E}}\times{\vec{B}}... ...E_{\rm M}=\int_{\rm disc}{{\vec{B}}^2\over2\mu_0}~{\rm d} {}V,$$ (40)

and the azimuthally integrated normalized radial kinetic energy flux density, $\dot{E}_{\rm K}(\vartheta)/E_{\rm K}$, where

$$\dot{E}_{\rm K}(\vartheta)=2\pi r^2({\textstyle{1\over2}}\var... ...nt_{\rm disc}{\textstyle{1\over2}}\rho{\vec{u}}^2~{\rm d} {}V.$$ (41)

Here, $M_{\rm disc},\ J_{\rm disc},\ E_{\rm M}$ and $E_{\rm K}$ are the mass, angular momentum, and magnetic and kinetic energies in the disc. A polar diagram showing these distributions is presented in Fig. 14 for the model of Fig. 6. Note that this is a model without central mass sink where the flow in the hot, dense cone around the axis is fast. The fast flow in the hot, dense cone carries most of the mass and kinetic energy. A significant part of angular momentum is carried away in the disc plane whilst the magnetic field is ejected at intermediate angles, where the conical shell is located.

Figure 14: Dependence, on polar angle $\vartheta$, of azimuthally integrated radial mass flux density $\dot{M}(\vartheta)$ through a sphere r=8 (solid, normalized by the disc mass $M_{\rm disc}\approx 12$), azimuthally integrated radial angular momentum flux density $\dot{J}(\vartheta)$ (dashed-dotted, normalized by the disc angular momentum $J_{\rm disc}\approx 4.3$), azimuthally integrated radial Poynting flux density $\dot{E}_{\rm M}(\vartheta)$ (divided by 5, dashed, normalized by the magnetic energy in the disc $E_{\rm M}\approx 0.6$), and azimuthally integrated radial kinetic energy flux density $\dot{E}_{\rm K}(\vartheta)$ (divided by 10, dash-3dots, normalized by the kinetic energy in the disc $E_{\rm K}\approx 9.7$), for the model of Fig. 6. The unit of all the quantities is $[t]^{-1}~{\rm rad}^{-1}$.

Figure 15: Angular momentum (normalized by the maximum angular momentum in the disc) for the model with $\tau _{\rm star}=0.01$ and $\beta =0.1$, shown in Fig. 7. The maximum value is $l/l_{\rm disc,max} \approx 2.5$. The dashed line shows the fast magnetosonic surface with respect to the poloidal field (cf. Fig. 4), the solid line the Alfvén surface where the poloidal velocity equals the poloidal Alfvén speed, and the dotted line the sonic surface. Averaged over times t=130...140.

Figure 16: The ratio of the poloidal magneto-centrifugal and pressure forces, $\vert\FF_{\rm pol}^{\rm(mc)}\vert/\vert\FF_{\rm pol}^{\rm(p)}\vert$ as defined in Eq. (42), is shown with shades of grey, with larger values corresponding to lighter shades. The maximum value is $\vert\FF_{\rm pol}^{\rm(mc)}\vert/\vert\FF_{\rm pol}^{\rm(p)}\vert\approx 411$. Superimposed are the poloidal magnetic field lines. The dashed line shows the fast magnetosonic surface with respect to the poloidal field (cf. Fig. 4). The parameters are as in the model of Fig. 7. Averaged over times t=130...140.

Figure 17: The ratio ${\vec{J}}\cdot {\vec{B}}/ \vert{\vec{J}}\vert~ \vert{\vec{B}}\vert$ in the corona for the model with $\tau _{\rm star}=0.01$ and $\beta =0.1$, shown in Fig. 7. For a force-free magnetic field, ${\vec{J}}= C {\vec{B}}$, this ratio is $\pm 1$. Averaged over times t=130...140.

Figure 18: As in Fig. 16, but for the model of Fig. 9, i.e. with larger entropy contrast, $\beta =0.02$. Averaged over times t=140...240.

The radial kinetic and magnetic energy flux densities, integrated over the whole sphere, are $L_{\rm K}\equiv\int_0^\pi\dot{E}_{\rm K}(\vartheta)~{\rm d} {}\vartheta\approx0.54~\dot{M}_{\rm wind}c_*^2$ and $L_{\rm M}\equiv\int_0^\pi\dot{E}_{\rm M}(\vartheta)~{\rm d} {}\vartheta\approx0.03~\dot{M}_{\rm wind}c_*^2$, respectively, where $c_*\approx2.8$ is the fast magnetosonic speed (with respect to the poloidal field) at the critical surface (where u_z=c_*) on the axis. Thus, $\dot{M}_{\rm wind}c_*^2$ can be taken as a good indicator of the kinetic energy loss, and the magnetic energy loss into the exterior is about 3% of this value. These surface-integrated flux densities (or luminosities) are, as expected, roughly independent of the distance from the central object.

   
3.5 Mechanisms of wind acceleration

The magnetized outflows in our models with central mass sink have a well-pronounced structure, with a fast, cool and low-density flow in a conical shell, and a slower, hotter and denser flow near the axis and in the outer parts of the domain. Without central mass sink, there is a high speed, hot and dense cone around the axis.

The magnetic field geometry (e.g., Fig. 7) is such that for $\varpi > 0.1$, the angle between the disc surface and the field lines is less than $60^\circ$, reaching $\approx 20^\circ$ at $\varpi \approx 1$-1.5, which is favourable for magneto-centrifugal acceleration (Blandford & Payne 1982; Campbell 1999, 2000, 2001). However, the Alfvén surface is so close to the disc surface in the outer parts of the disc that acceleration there is mainly due to pressure gradient. The situation is, however, different in the conical shell containing the fast wind. As can be seen from Fig. 15, the Alfvén surface is far away from the disc in that region and, on a given field line, the Alfvén radius is at least a few times larger than the radius of the footpoint in the disc. This is also seen in simulations of the magneto-centrifugally driven jets of Krasnopolsky et al. (1999); see their Fig. 4. The lever arm of about 3 is sufficient for magneto-centrifugal driving to dominate. As can be seen also from Fig. 10, the flow at the polar angle $\vartheta = 60^\circ$ is mainly accelerated by pressure gradient near the disc surface (where the Alfvén surface is close to the disc surface). However, acceleration remains efficient out to at least r=1within the conical shell at $\vartheta\approx30^\circ$. This can be seen in the upper and middle panels of Fig. 10 (note that the conical shell is thinner and at a smaller $\vartheta$in the model with larger entropy contrast, and so it cannot be seen in this figure, cf. Fig. 9). These facts strongly indicate that magneto-centrifugal acceleration dominates within the conical shell.

Another indicator of magneto-centrifugal acceleration in the conical shell is the distribution of angular momentum (see Fig. 15), which is significantly larger in the outer parts of the conical shell than in the disc, which suggests that the magnetic field plays an important rôle in the flow dynamics. We show in Fig. 16 the ratio of the "magneto-centrifugal force'' to pressure gradient, $\vert\FF_{\rm pol}^{\rm(mc)}\vert/\vert\FF_{\rm pol}^{\rm(p)}\vert$, where the subscript "pol'' denotes the poloidal components. Here, the "magneto-centrifugal force'' includes all terms in the poloidal equation of motion, except for the pressure gradient (but we ignore the viscous term and the mass production term, the latter being restricted to the disc only),

$${\vec{F}}^{\rm (mc)}=\varrho\left(\Omega^2\vec{\varpi}-\vec{\nabla}\Phi\right)+{\vec{J}}\times{\vec{B}},$$ (42)

and ${\vec{F}}^{(p)}=-\vec{\nabla}p$. The large values of the ratio in the conical shell confirm that magneto-centrifugal driving is dominant there. On the other hand, the pressure gradient is strong enough in the outer parts of the disc to shift the Alfvén surface close to the disc surface, leading to pressure driving. This is also discussed in Casse & Ferreira (2000b) who point out that, although the criterion of Blandford & Payne (1982) is fulfilled, thermal effects can be strong enough to lead to pressure driving. According to Ferreira (1997), a decrease of the total poloidal current $I_{\rm pol} = 2 \pi \varpi B_\varphi / \mu_0$ along a field line is another indicator of magneto-centrifugal acceleration. We have compared the poloidal current $I_{\rm pol}^{\rm (top)}$ at the Alfvén point, or at the top of our box if the Alfvén point is outside the box, with the poloidal current $I_{\rm pol}^{\rm (surf)}$ at the disc surface, and find that outside the conical shell this ratio is typically $\ge$0.8, while along the field line that leaves the box at $(\varpi,z)=(0.6,1)$, we get $I_{\rm pol}^{\rm (top)} / I_{\rm pol}^{\rm (surf)} \approx 0.18$, i.e. a strong reduction, which confirms that magneto-centrifugal acceleration occurs inside the conical shell. We note, however, that the changing sign of $B_\varphi$ and therefore of $I_{\rm pol}$ makes this analysis inapplicable in places, and the distribution of angular momentum (Fig. 15) gives a much clearer picture.

As further evidence of a significant contribution from magneto-centrifugal driving in the conical shell, we show in Fig. 17 that the magnetic field is close to a force-free configuration in regions where angular momentum is enhanced, i.e. in the conical shell and in the corona surrounding the outer parts of the disc. These are the regions where the Lorentz force contributes significantly to the flow dynamics, so that the magnetic field performs work and therefore relaxes to a force-free configuration. The radial variation in the sign of the current helicity  ${\vec{J}}\cdot{\vec{B}}$ is due to a variation in the sign of the azimuthal magnetic field and of the current density. Such changes originate from the disc where they imprint a corresponding variation in the sign of the angular momentum constant, see Eq. (46). These variations are then carried along magnetic lines into the corona. The locations where the azimuthal field reverses are still relatively close to the axis, and there the azimuthal field relative to the poloidal field is weak compared to regions further away from the axis.

Figure 19: Flow parameters as functions of height z, along the field line with its footpoint at $(\varpi _{\rm fp},z_{\rm fp})=(0.17,0.15)$, for the model of Fig. 7. Upper left: poloidal Alfvén Mach number, $M_{\rm A}\equiv u_{\rm pol}/v_{\rm A,pol}$(dashed), and a similar quantity that includes the poloidal Alfvén speed as well as the sound speed (fast magnetosonic Mach number with respect to the poloidal field), $M_{\rm FM}\equiv u_{\rm pol}/(c_{\rm s}^2+v_{\rm A,pol}^2)^{1/2}$(dotted) - the two curves are practically identical; upper right: ratio of toroidal ( $B_{\rm tor}$) and poloidal ( $B_{\rm pol}$) magnetic fields; lower left: toroidal velocity $u_{\rm tor}$ (solid) and poloidal velocity $u_{\rm pol}$ (dash-3dots), in units of the toroidal velocity at the footpoint, $u_{\rm tor,fp}$; lower right: density $\varrho$ in units of the density at the footpoint, $\varrho _{\rm fp}$. The position of the Alfvén (and fast magnetosonic) point on the field line is indicated by the vertical line. This figure is useful to compare with Fig. 3 of Ouyed et al. (1997).

Pressure driving is more important if the entropy contrast between the disc and the corona is larger (i.e. $\beta$ is smaller): the white conical shell in Fig. 16 indicative of stronger magneto-centrifugal driving shifts to larger heights for $\beta =0.02$, as shown in Fig. 18. We note, however, that there are periods when magneto-centrifugal driving is dominant even in this model with higher entropy contrast, but pressure driving dominates in the time averaged picture (at least within our computational domain).

We show in Fig. 19 the variation of several quantities along a magnetic field line that has its footpoint at the disc surface at $(\varpi,z)=(0.17,0.15)$ and lies around the conical shell. Since this is where magneto-centrifugal driving is still dominant, it is useful to compare Fig. 19 with Fig. 3 of Ouyed et al. (1997), where a well-collimated magneto-centrifugal jet is studied. Since our outflow is collimated only weakly within our computational domain, the quantities are plotted against height z, rather than $z/\varpi$ as in Ouyed et al. (1997) ($z/\varpi$ is nearly constant along a field line for weakly collimated flows, whereas approximately $z/\varpi \propto z$ along a magnetic line for well-collimated flows). The results are qualitatively similar, with the main difference that the fast magnetosonic surface in our model almost coincides with the Alfvén surface in the region around the conical shell where the outflow is highly supersonic. Since we include finite diffusivity, the curves in Fig. 19 are smoother than in Ouyed et al. (1997), who consider ideal MHD. A peculiar feature of the conical shell is that the flow at $z \la 1$ is sub-Alfvénic but strongly supersonic. The fast magnetosonic surface is where the poloidal velocity $v_{\rm pol}$ equals the fast magnetosonic speed for the direction parallel to the field lines,

$$v_{\rm pol}^2 ={1\over2}\left( {c_{\rm s}^2{+}v_{\rm A}^2 +... ...s}^2{+}v_{\rm A}^2)^2 - 4c_{\rm s}^2v_{\rm A,pol}^2}}\right),$$ (43)

with $v_{\rm A,pol}$ the Alfvén speed from the poloidal magnetic field. This surface has the same overall shape as the fast magnetosonic surface with respect to the poloidal field, albeit in some cases it has a somewhat larger opening angle around the conical shell and is located further away from the disc in regions where the toroidal Alfvén speed is enhanced.

   
3.6 Lagrangian invariants

Axisymmetric ideal magnetized outflows are governed by five Lagrangian invariants, the flux ratio,

$$k = \varrho u_z/B_z = \varrho u_\varpi/B_\varpi,$$ (44)

the angular velocity of magnetic field lines,

$$\widetilde\Omega = \varpi^{-1} (u_\varphi - k B_\varphi/\varrho),$$ (45)

the angular momentum constant,

$$\ell=\varpi u_\varphi-\varpi B_\varphi/(\mu_0 k),$$ (46)

the Bernoulli constant,

$$e={\textstyle{1\over2}}{\vec{u}}^2+h+\Phi-\varpi\widetilde\Omega B_\varphi/(\mu_0 k),$$ (47)

and specific entropy s (which is a prescribed function of position in our model). In the steady state, these five quantities are conserved along field lines, but vary from one magnetic field line to another (e.g., Pelletier & Pudritz 1992; Mestel 1999), i.e. depend on the magnetic flux within a magnetic flux surface, $2\pi a$, where $a=\varpi A_\varphi$ is the flux function whose contours represent poloidal field lines.

Figure 20: The four Lagrangian invariants k(a), $\widetilde\Omega(a)$, $\ell (a)$, and e(a)as a function of the flux function afor the model of Fig. 6. All points in the domain $0<\varpi <8$, $0.2\leq z\leq 30$are shown (provided $0<a\le 0.25$). The dots deviating from well-defined curves mostly originate in $8\protect\la z\leq30$. Note that specific entropy as fifth Lagrangian invariant is trivially conserved in our model, because entropy is spatially constant throughout the corona and temporally fixed.

We show in Fig. 20 scatter plots of k(a), $\widetilde\Omega(a)$, $\ell (a)$, and e(a)for the model of Fig. 6. Points from the region $0.2\le z\la 8$ collapse into a single line, confirming that the flow in the corona is nearly ideal. This is not surprising since the magnetic Reynolds number is much greater than unity in the corona for the parameters adopted here. For $8\la z\le 30$, there are departures from perfect MHD; in particular, the angular velocity of magnetic field lines, $\widetilde\Omega$, is somewhat decreased in the upper parts of the domain (indicated by the vertical scatter in the data points). This is plausibly due to the finite magnetic diffusivity which still allows matter to slightly lag behind the magnetic field. As this lag accumulates along a stream line, the departures increase with height z. Since this is a "secular'' effect only, and accumulates with height, we locally still have little variation of k and $\widetilde\Omega$, which explains why magneto-centrifugal acceleration can operate quite efficiently.

The corona in our model has (turbulent) magnetic diffusivity comparable to that in the disc. This is consistent with, e.g., Ouyed & Pudritz (1999) who argue that turbulence should be significant in coronae of accretion discs. Nevertheless, it turns out that ideal MHD is a reasonable approximation for the corona (see Fig. 20), but not for the disc. Therefore, magnetic diffusivity is physically significant in the disc and insignificant in the corona (due to different velocity and length scales involved), as in most models of disc outflows (see Ferreira & Pelletier 1995 for a discussion). Thus, our model confirms this widely adopted idealization.