COUNTERROTATION IN MAGNETOCENTRIFUGALLY DRIVEN JETS AND OTHER WINDS

The basic equations governing ideal MHD outflows are the momentum, mass, and magnetic flux conservation, the frozen-in law for infinite conductivity, and the first law of thermodynamics. In steady and axisymmetric conditions (Tsinganos 1982; Heyvaerts & Norman 1989) there are five integrals, quantities that are conserved along a given (poloidal) stream/field line. These are the magnetic flux A, the mass to magnetic flux ratio, the energy flux E(A), the angular momentum flux L(A), and the angular frequency or corotation frequency Ω(A). In the following, we denote cylindrical coordinates with (z, ϖ, φ) and spherical coordinates with (r, θ, φ).

The total angular momentum flux and the corotation frequency can be expressed in terms of the azimuthal and poloidal components of the magnetic and velocity fields. The toroidal component of the induction equation combined with the frozen flux condition gives the corotation frequency Ω,

$$\begin{equation} \Omega (A)= \frac{1}{\varpi }\left(V_\varphi - \frac{V_{\rm p}}{B_{\rm p}}B_\varphi \right). \end{equation} \tag{ 1 }$$

The angular momentum flux is obtained by integrating the momentum equation in the azimuthal direction,

$$\begin{equation} L(A)= \varpi \left(V_\varphi - \frac{B_{\rm p}}{4\pi \rho V_{\rm p}}B_\varphi \right). \end{equation} \tag{ 2 }$$

The poloidal Alfvén speed and the cylindrical Alfvén radius are defined as

$$\begin{equation} V_{\rm A}^2 = \frac{B^2_ {\rm p}}{4 \pi \rho },\quad \varpi _{*}^2 = \frac{L}{\Omega }. \end{equation} \tag{ 3 }$$

For a smooth crossing of the Alfvén surface, where V_p = V_A, the cylindrical radius along a given field line must adjust to this value, ϖ = ϖ_*.

From Equations (1)–(3), we derive the azimuthal components as functions of the free integrals and the poloidal components of the fields, using the Alfvén speed and radius:

$$\begin{equation} V_{\varphi } =\frac{L}{\varpi } \frac{\frac{ V_{\rm p}^2}{ V_{\rm A}^2} - \frac{ \varpi ^2}{ \varpi _*^2}}{\frac{ V_{\rm p}^2}{ V_{\rm A}^2}- 1} \,,\qquad \frac{B_{\varphi }}{\sqrt{4\pi \rho }} =-\frac{L}{\varpi }\frac{V_{\rm p}}{V_{\rm A}} \frac{\frac{ \varpi ^2}{ \varpi _*^2}-1}{\frac{ V_{\rm p}^2}{ V_{\rm A}^2}-1} \,. \end{equation} \tag{ 4 }$$

With these expressions, we show that a reversal of the rotation velocity is possible. Note that the toroidal components depend only on the poloidal ones with the free MHD integrals being independent of the thermodynamics of the flow.

Consider the flow along a given stream/field line as seen in Figure 1. Assuming that the flow remains everywhere super-Alfvénic after crossing the corresponding critical surface, the denominator of Equation (4) is always positive, V²_p/V_A² − 1 > 0. The sign of the toroidal velocity is then given by the numerator,

$$\begin{equation} {\rm sgn}(V_{\varphi }) = {\rm sgn} \left({\frac{ V_{\rm p}^2}{ V_{\rm A}^2} - \frac{ \varpi ^2}{ \varpi _*^2}}\right). \end{equation} \tag{ 5 }$$

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The initially positive rotation velocity will change sign if the velocity of the flow along the flux tube is less than the threshold value,

$$\begin{equation} V_{\rm p}< V_{\rm A}\frac{ \varpi }{ \varpi _\star } = V_{\rm threshold} \,. \end{equation} \tag{ 6 }$$

Note that this value is always above the local Alfvén speed because we expect ϖ > ϖ_⋆.

Along a given flux tube, we can write the magnetic and mass flux conservation as

$$\begin{equation} B_{\rm p}\delta S= B_{\star }\delta S_{\star } = {\rm constant} \end{equation} \tag{ 7 }$$

and

$$\begin{equation} \rho V_{\rm p}\delta S= \rho _{\star } V_{\star }\delta S_{\star }= {\rm constant} \,. \end{equation} \tag{ 8 }$$

Note that δS is the surface element perpendicular to the poloidal velocity (see Figure 1), such that one can always replace $\bm {V}_{\rm p}$ by any of its components using the following equation,

$$\begin{equation} \bm {V}_{\rm p}\cdot \delta \bm {S} = V_{\rm p}\delta S= V_{z}\delta S_{z} = V_{r}\delta S_{r}= V_{i}\delta S_{i} \,, \end{equation} \tag{ 9 }$$

where V_i is the ith component of the velocity and δS_i is the cross section perpendicular to the ith direction. The calculation is simpler if one works with the poloidal velocity. Nevertheless, it is more straightforward to compare the z component of the velocity and the cross section δS_z, which is in the ϖ direction, since these are the observed quantities.

Using these two equalities in the numerator of the azimuthal velocity we successively get

$$\begin{eqnarray} &&\frac{ V_{\rm p}^2}{ V_{\rm A}^2} - \frac{ \varpi ^2}{ \varpi _{\star }^2} = \frac{ 4\pi \rho V_{\rm p} V_{\rm p}}{ B_{\rm p}^2} - \frac{ \varpi ^2}{ \varpi _{\star }^2} = \frac{ 4\pi \rho _\star V_{\star } \frac{ \delta S_{\star }}{ \delta S} V_{\rm p}}{ B_{\star }^2 \frac{ \delta S_{\star }^2}{ \delta S^2 }} - \frac{ \varpi ^2}{ \varpi _{\star }^2 } \,. \nonumber\\ \end{eqnarray} \tag{ 10 }$$

Since at the Alfvén surface we have 4πρ_⋆V²_⋆ = B²_⋆, the above expression simplifies further,

$$\begin{equation} \frac{ V_{\rm p}^2}{ V_{\rm A}^2} - \frac{ \varpi ^2}{ \varpi _{\star }^2} = \frac{\frac{ \delta S_{\star }}{ \delta S}V_{\rm p}}{\frac{ \delta S_{\star }^2}{ \delta S^2 }V_{\star }} - \frac{ \varpi ^2}{ \varpi _{\star }^2} = \frac{ \delta S}{ \delta S_{\star }} \frac{ V_{\rm p}}{ V_{\star }} - \frac{ \varpi ^2}{ \varpi _{\star }^2} \,. \end{equation} \tag{ 11 }$$

Thus, the sign of this last expression determines the sign of the azimuthal velocity in the super-Alfvénic region:

$$\begin{equation} {\rm sgn}(V_{\varphi }) = {\rm sgn}\left(\frac{ V_{\rm p}}{ V_{\star }} - \frac{ \varpi ^2}{ \varpi _{\star }^2} \frac{ \delta S_{\star }}{ \delta S}\right). \end{equation} \tag{ 12 }$$

This gives a new expression for the threshold value defined in Equation (6),

$$\begin{equation} V_{\rm threshold} = V_{\star }\frac{ \varpi ^2}{ \varpi _\star ^2} \frac{ \delta S_{\star }}{ \delta S} \,. \end{equation} \tag{ 13 }$$

If the flow remains super-Alfvénic, reversal of the rotation takes place when the velocity drops below some threshold value. The flow can decelerate either because it expands and this provokes adiabatic cooling, or because its kinetic energy drops due to some other mechanism such as radiative losses. The reversal might also occur if the threshold value increases. This may be the case when the cross section of the tube decreases significantly while the flux tube itself widens. This is illustrated in Figure 2, namely, the toroidal velocity becomes negative at the location where the poloidal speed curve crosses the line of the threshold velocity.

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Counterrotation is also obviously obtained if the flow becomes sub-Alfvénic. In this case, the denominator of Equation (4) becomes negative and the reversal of the toroidal velocity is accompanied by a reversal of the toroidal magnetic field too. This may happen after a shock, which is probably similar to the suggestion of Fendt (2011) that the reversal of the toroidal field is due to the presence of shocks.

Last, we also note that in the sub-Alfvénic part of the flow close to the source, rotation may also change sign if there is a strong local acceleration and if the velocity increases above the threshold value (instead of below in the super-Alfvénic part).

Another way of exploring the counterrotation possibilities is to express the numerator of the rotation speed from Equation (4) using the density ratio,

$$\begin{equation} \frac{ V_{\rm p}^2}{ V_{\rm A}^2} - \frac{ \varpi ^2}{ \varpi _{\star }^2} = \frac{ \rho _{\star }}{ \rho } - \frac{ \varpi ^2}{ \varpi _{\star }^2} \,. \end{equation} \tag{ 14 }$$

The sign of this quantity determines the sign of V_φ, i.e., reversal will occur if $\rho >\rho _{\star }\frac{\varpi ^2_{\star }}{\varpi ^2}$.

For meridionally self-similar models (Sauty et al. 2011), it is more convenient to use the radial component of the velocity, i.e., Equation (9). The criterion for a change of sign in the flow rotational velocity is now given by

$$\begin{equation} V_{r}<V_{r,\star }\frac{\cos \theta }{\cos \theta _\star } \,. \end{equation} \tag{ 15 }$$

This is particularly interesting for analytical stellar jet or wind models.

Similarly, for radially self-similar models, reversal of the rotation occurs where

$$\begin{equation} \vert V_{\theta } \vert < \vert V_{\theta, \star } \vert \frac{\sin \theta }{\sin \theta _\star } \,. \end{equation} \tag{ 16 }$$

This formula applies instead to analytical disk wind models.

If the physical quantities are uniform across the jet, then δS is proportional to ϖ². The condition then reduces to the much simpler expression, V < V_⋆. The threshold value is exactly the Alfvén speed at the Alfvén surface.

To compare this criterion with observations and numerical simulations, it is useful to work with the vertical component of the velocity using Equation (9) because this is the jet velocity component measured. The surface element is then replaced by its projection along z, δS_z, which is the cross section of the flux tube in the horizontal ϖ direction that corresponds to the jet radius as it is measured,

$$\begin{equation} V_{z} < V_{\star }\frac{ \varpi ^2}{ \varpi _\star ^2} \frac{ \delta S_{z, \star }}{ \delta S_z} \,. \end{equation} \tag{ 17 }$$

If we average the quantities at a given altitude z, we may have a criterion for the averaged velocity using the previous equation, namely,

$$\begin{equation} {\bar{V}_{z}} = \frac{ \int {V_{z} \delta S_z}}{ \varpi ^2} < \frac{ \int {V_{\star } \delta S_\star }}{ \varpi _\star ^2} = {\bar{V}_{\star }} \,. \end{equation} \tag{ 18 }$$

The criterion (i.e., $\bar{V}_{\rm }<\bar{V}_{\star }$) is equivalent to the criterion for transversally uniform flow.

This may be more adequate to analyze observational data and jet models such as the one proposed by Lovelace et al. (1991). In fact, Equation (16) of Lovelace et al. (1991) shows that an increase of the average jet velocity induces a decrease of the rotational one, which could become negative.

We have performed a numerical simulation in which several reversals of the toroidal velocity are observed. We have adopted the initial conditions of Matsakos et al. (2012), model NIPH, in which two analytical solutions for the stellar and disk wind components are combined and a constant external pressure is applied beyond a specific radius. We have carried out the temporal integration of the MHD equations using PLUTO (Mignone et al. 2007) in a resolution of 512 × 2048 up to a final time that corresponds to ∼100 crossing times of the simulation box.

During the initial transient phase, a strong shock propagates upward due to the fact that the initial state is not in equilibrium. Together with the shock, a significant reversal of the toroidal velocity is also observed, although this phenomenon is not directly related with the above analysis. Once the system reaches a quasi-steady state, several layers demonstrate an opposite V_φ. These counterrotating regions remain roughly at the same location for large time intervals before they vanish and reappear. They obviously remain transient phenomena.

The left panel of Figure 3 displays contours of the toroidal velocity on the left and the poloidal component on the right. We have selected a streamline (ϖ(z), gray and white solid lines of the left panel) that crosses the region where the rotation reverses, alongside which the cylindrical radius, the angular momentum fluxes; the toroidal, poloidal, and Alfvén speeds; as well as the poloidal Alfvénic Mach number are shown on the right panel of Figure 3.

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The plots suggest that the V_φ component becomes negative at the location where the poloidal velocity decreases approaching the Alfvén speed. Note that the flow remains always super-Alfvénic. The dot–dash vertical lines indicate the change of sign of the rotation. In this region, the poloidal velocity reaches the local threshold value which varies along the flow. Although the jet is not an exact steady flow, the simulation results can be explained based on the analysis that describes the super-Alfvénic regime.

In Figure 3, we plot the total angular momentum flux L and its two components (see Equation (2)), namely, the hydrodynamical part L_HYD = ϖV_φ and the magnetic one L_MAG = −ϖ(B_p/4πρV_p)B_φ. The total angular momentum flux is clearly not conserved because the configuration is not in an exact steady state. However, the reversal of the toroidal velocity corresponds to an exchange between the two components where the magnetic part becomes dominant. L drops rapidly up to s = 190. In this first part, the reversal of the velocity is strongly time dependent and related to the shock as suggested by Fendt (2011). Above s = 190, L is almost constant and in this region our steady-state criterion for the reversal is valid. It is a nice illustration of the above theory.