2 The wind equations

The conservation laws of the general-relativistic magnetohydrodynamics have been derived by Bekenstein & Oron (1978). Camenzind (1986b) used these results to obtain the equations governing a stationary axisymmetric wind. In this section, we first recall these equations and we apply them to the particular case of a flow occurring in the equatorial plane. Then we derive a new formulation for this problem where the wind solution is a level contour of a Bernoulli-like function. A very similar formulation was earlier studied by Sakurai (1985) for the non-relativistic case. The similarity allows us to compare our results with those of the classical case.

2.1 Assumptions and basic equations

Any stationary and axisymmetric space-time can be represented by the following metric:

$${\rm d}s^2 = g_{tt} {\rm d}t^2 + 2 g_{t\phi} {\rm d}t {\rm ... ... + g_{\phi\phi} {\rm d}\phi^2 + g_{ab} {\rm d}x^a {\rm d}x^b$$ (1)

where the coordinates t and $\phi$ correspond to the two symmetries of the space-time defined by the two Killing fields $k=\partial_{\rm t}$ (stationarity) and $m=\partial_\phi$(axisymmetry). The metric coefficients g_ab depend only on the two remaining coordinates x^a (a=1,2). Note that in the whole paper we will make use of the (-+++) signature for the metric. The electromagnetic field is described by the field tensor  $F^{\mu\nu}$and the dual field tensor $F^{*\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}$( $\epsilon^{\mu\nu\rho\sigma}$ being the Levi-Civita alternating tensor) which satisfy the Maxwell equations

$$\nabla_\mu F^{*\mu\nu} = 0.$$ (2)

The motion of the plasma is governed by the energy- and momentum conservation equation

$$\nabla_\mu T^{\mu\nu} = 0,$$ (3)

where the energy-momentum tensor $T^{\mu\nu}$ is made up of the fluid part

$$T^{\mu\nu}_{\rm matter} = \rho \frac{{h}}{{c^2}} u^\mu u^\nu + P g^{\mu\nu}$$ (4)

and of the electromagnetic part

$$% T^{\mu\nu}_{\rm em} = \frac{1}{4\pi}\left( \frac{{b^2}}{... ...mu u^\nu + \frac{b^2}{2} g^{\mu\nu} - b^\nu b^\mu \right).$$ (5)

$b^\mu$ is the magnetic field according to a comoving observer, written as

$$b^\mu = F^{*\mu\nu} u_\nu$$ (6)

with $b^2=b_\mu b^\mu$. In the comoving frame $b^\mu$ reduces to the common magnetic field

$$b^\mu_{\rm co} = F^{*\mu0} = (0,\vec B).$$ (7)

All dissipative effects (heat conduction, viscosity, cooling by radiation, etc.) have been neglected so that the flow is adiabatic. Ideal MHD is also assumed, which means that the proper electric field as seen in the plasma frame vanishes

$$F^{\mu\nu} u_\mu = 0.$$ (8)

The following quantities appear in these equations: $u^\mu$ is the 4-velocity, $\rho$ is the comoving mass density, P is the pressure and h is the specific enthalpy, which is given by (assuming a constant adiabatic index $\gamma$)

$$h = {c^2} + \frac{\gamma}{\gamma-1}\frac{P}{\rho}\cdot$$ (9)

The last assumption is that the particle number (or mass) is conserved

$$\nabla_\mu \left(\rho u^\mu\right) = 0.$$ (10)

Before writing the wind equations, it is useful to define the specific angular momentum of the flow

$$l = -\frac{u_\phi}{u_t} {c^2}.$$ (11)

As a consequence of the symmetries, the non vanishing elements of the electromagnetic tensor are given by

$$- F_{1t} = {c} \Omega(P)\frac{\rho}{\eta(P)}\sqrt{-g} u^2,$$ F_t1 = (12)

$$- F_{2t} = -{c} \Omega(P)\frac{\rho}{\eta(P)}\sqrt{-g} u^1,$$ F_t2 = (13)

$$F_{\phi1} -F_{1\phi} = -c \frac{\rho}{\eta(P)}\sqrt{-g} u^2,$$ = (14)

$$F_{\phi2} -F_{1\phi} = -c \frac{\rho}{\eta(P)}\sqrt{-g} u^{1},$$ = (15)

$$F_{21} = c \frac{\rho}{\eta(P)}\sqrt{-g} \left(\Omega(P) {u^t} - {u}^\phi \right),$$ F₁₂ = (16)

where the angular frequency of the streamline $\Omega(P)$ at its footpoint and the mass flux per unit flux tube $\eta(P)$ are constant along each flow line P. The corresponding magnetic field is

$$\frac{\rho}{\eta(P)} \left[ 1+\frac{{u_t u^t}}{{c^2}} \left( 1-\frac{{\Omega(P)l}}{{c^2}} \right) \right] ,$$ b^t = (17)

$$b^\phi \frac{\rho}{\eta(P)} \left[ \Omega(P) + \frac{{u_t u}^\phi}{{c^2}} \left( 1-\frac{{\Omega(P) l}}{{c^2}} \right) \right],$$ = (18)

$$\frac{\rho}{\eta(P)} \frac{{u_t}}{{c^2}} \left(1-\frac{{\Omega(P)l}}{{c^2}}\right) u^1 ,$$ b¹ = (19)

$$\frac{\rho}{\eta(P)} \frac{{u_t}}{{c^2}} \left(1-\frac{{\Omega(P)l}}{{c^2}}\right) u^2.$$ b² = (20)

It is also useful to give the expression of the classical magnetic field in the frame of the central object for the comparison with the classical case. In this frame, it is related to the dual field tensor by $B_\mu=F_{*\mu0}$. Then

$$-\frac{{1}}{{c^2}}\frac{\rho}{\eta(P)} \left(\Omega(P)u^t-u^\phi\right) g_{t\phi},$$ B^t = (21)

$$B^\phi \frac{{1}}{{c^2}}\frac{\rho}{\eta(P)} \left(\Omega(P)u^t-u^\phi\right) g_{tt} ,$$ = (22)

$$-\frac{{1}}{{c^2}}\frac{\rho}{\eta(P)} \left( g_{tt} + \Omega(P) g_{t\phi} \right) u^1,$$ B¹ = (23)

$$-\frac{{1}}{{c^2}}\frac{\rho}{\eta(P)} \left( g_{tt} + \Omega(P) g_{t\phi} \right) u^2.$$ B² = (24)

Because the flow is stationary and axisymmetric, the total angular momentum L(P) and the total energy $E_{\rm tot}(P)$ are also conserved along each flow line P which provides us with two new equations

 

$$L(P) = -\frac{{u_t}}{{c^2}}\cdot \biggl\lbrace \frac{{h}}{{c^2}} ... ...^2}} l +\left( g_{t\phi} + \Omega(P) g_{\phi\phi} \right) \right] \biggr\rbrace$$ (25)

and

$$E_{\rm tot}(P) = -\frac{{u_t}}{{c^2}}\cdot\biggl\lbrace h + \Omeg... ...2}} l + \left( g_{t\phi} + \Omega(P) g_{\phi\phi}\right) \right]\biggr\rbrace .$$ (26)

It is convenient to write the total energy as $E_{\rm tot}(P) = {c^2}+E(P)+\Omega(P)L(P)$ so that the energy conservation can have the simpler form:

$$E(P) + {c^2} = -h \frac{{u_t}}{{c^2}} \left[1-\frac{{\Omega(P)l}}{{c^2}}\right].$$ (27)

Each flow line P is completely determined by the four constants $\Omega(P)$, $\eta(P)$, L(P) and E(P). The light surface is defined by

$$g_{tt} + 2 g_{t\phi}\Omega(P) + g_{\phi\phi}\Omega^2(P) = 0$$ (28)

and the Alfvén point is fixed by two conditions

$$\frac{{1}}{{c^2}} \left[ g_{tt} + 2 g_{t\phi}\Omega(P) + g_{\phi\phi}\Omega^2(P) \right]_{\rm A} = - M_{\rm A}^2,$$ (29)

$$\frac{{1}}{{c^2}} \frac{\left( g_{t\phi} + \Omega(P) g_{\phi\phi}\right)_{\rm A}} {M_{\rm A}^2} = \frac{L(P)}{E(P)+{c^2}},$$ (30)

where the "Mach'' number M is given by

$$M^2 = \frac{4\pi\eta^2(P)\,h}{\rho {c^2}}\cdot$$ (31)

One sees immediately from (28) and (29) that the Alfvén point stays always inside the light surface (because of  $M_{\rm A}>0$).

2.2 The wind equations in the equatorial plane

We use now the spherical coordinates (x¹=r and $x^2=\theta$) and limit our study to the equatorial plane $\theta=\frac{\pi}{2}$. The specification of the flow line P is dropped from here on and $\Omega$, $\eta$, E, and L are used instead of $\Omega(P)$, $\eta(P)$, E(P), and L(P). Because of the symmetry, $u^\theta$and $B^\theta$ vanish in this plane but this is not necessarily the case for their derivatives. Then the conservation of mass (10) can be written

$$\sqrt{-g} s(r) \rho u^r = \dot{m},$$ (32)

where the function s(r) depends on the geometry of the flux tubes. In the simple case where $\partial_\theta u^\theta = 0$ and $\partial_\theta B^\theta=0$, we have $s(r)={\rm const.}$ (constant opening angle). Otherwise we have

$$\left. \partial_\theta u^\theta \right\vert _{\theta=\frac{\pi}{2}} = \frac{s'(r)}{s(r)}u^r .$$ (33)

The conservation of angular momentum (25) and the conservation of energy (27) read

 

$$L = \left[ g_{\phi\phi}\frac{h}{c^2} - \frac{{g}_{{t}\phi}^{{2}} ... ...}} - {g_{tt} g}_{\phi\phi}}{{c^2}} \frac{\Phi^2\rho}{4\pi\dot{m}^2} \right] u^t$$ (34)

and

 

$$-h \left[\frac{{g_{tt}} +{\Omega g}_{{t}\phi}}{{c^2}} u^t + \frac{{g}_{{t}\phi} + {\Omega g}_{\phi\phi}}{{c^2}} u^\phi \right],$$ (35)

where instead of using $\eta$ we have introduced the magnetic flux $\Phi=\dot{m}/\eta$. The Eqs. (32), (34) and (35) are completed by the normalization of the four velocity

$$g_{tt} \left( u^t \right)^2 + 2 g_{t\phi} u^t u^\phi + g_{\... ...left(u^\phi\right)^2 + g_{\rm rr} \left(u^r\right)^2 = -c^2$$ (36)

and the equation of state. We assume here, like in Sakurai (1985), a polytropic relation $P=\kappa \rho^\gamma$ so that the specific enthalpy is given by

$$h = c^2 + \frac{\gamma}{\gamma-1}\kappa\rho^{\gamma-1}.$$ (37)

The system of Eqs. (32), (34)-(37) describes entirely the flow determined by the six constants $\Omega,E,L,\Phi,\dot{m},\kappa$ and the free function s(r). In addition the two supplementary conditions

 

$$-\frac{{1}}{{c^2}} \left( g_{tt} + 2\Omega g_{t\phi} + \Omega^2 g_{\phi\phi} \right)_{\rm A} M_{\rm A}^2$$ =

$$\frac{4\pi\dot{m}^2}{\Phi^2} \frac{h_{\rm A}}{\rho_{\rm A}c^2}$$ = (38)

and

$$- \frac{{1}}{{c^2}} \left( \frac{ g_{t\phi} + \Omega g_{\p... ...+ \Omega^2 g_{\phi\phi}} \right)_{\rm A} = \frac{L}{E+c^2}$$ (39)

(all quantities with index A are computed at the Alfvén point) must be fulfilled, so that the flow remains regular at the Alfvén point.

The classical limit (for a weak gravitational field and for velocities small compared to the speed of light) of this system of equations in the case where $s(r)={\rm const.}=1$ gives exactly the Eqs. (1) to (6) in Sakurai (1985). Notice that E, $\Phi$ and $\Omega$ have the same meaning in both papers whereas we use here different notations for the mass flux $\dot{m}$, the total angular momentum Land the polytropic constant $\kappa$ which are respectively f, $\Omega {r}_{\rm A}^{ 2}$ and K in Sakurai (1985). Notice also that the relation (39) between L and $r_{\rm A}$ tends towards $L=\Omega {r}_{\rm A}^{{2}}$ in the classical limit, so that all notations are fully consistent.

2.3 A Bernoulli-like formulation

Following Sakurai (1985) we use the dimensionless variables $x=r/r_{\rm A}$ and $y=\rho/\rho_{\rm A}$. The metric coefficients are also normalized to become dimensionless $\tilde g_{tt}(x)= {g_{tt}/c^2}$, $\tilde g_{t\phi}(x)= {g}_{{t}\phi}/({c r})$, $\tilde g_{\phi\phi}(x)= g_{\phi\phi}/r^2$, $\tilde{g}_{rr}(x)=g_{rr}$and $\sqrt{-\tilde{g}}(x)=\sqrt{-g}/r^2$. These coefficients may be not only functions of x but also of some parameters defining the metric (they are given for the Minkowski, Schwarzschild and Kerr metric in Appendix A). We define $\tilde{\varpi}^2 =\left({\tilde g}{^2}_{{t}\phi}-{\tilde g_{tt}\tilde g}_{\phi\phi}\right)/{c^2}$. We normalize s by $\tilde{s}(x)=s/s_{\rm A}$ and define a dimensionless specific enthalpy by $\tilde{h}=h/c^2$. Concerning four-vectors like $u^\mu$or $B^\mu$ we will use the definitions $\tilde{A}^t=A^t$, $\tilde{A}^\phi=r\sin{\theta} A^\phi$, $\tilde{A}^r=A^r$ and $\tilde{A}^\theta=r A^\theta$ so that the spatial part of these vectors is now given in the usual basis $\partial_r, \frac{1}{r}\partial_\theta, \frac{1}{r\sin{\theta}}\partial_\phi$, where all components have the same dimension. We introduce four normalized parameters

$$\beta' \frac{1}{c^2}\left( \frac{\dot{m}}{s_{\rm A}r_{\rm A}^2\rho_{\rm A}} \right)^2,$$ = (40)

$$\Theta' \frac{\gamma\kappa\rho_{\rm A}^{\gamma-1}}{c^2},$$ = (41)

$$\omega' \frac{\left(\Omega r_{\rm A}\right)^2}{c^2},$$ = (42)

$$\frac{E}{c^2},$$ E' = (43)

and we are now able to rewrite the system of Eqs. (32) and (34) to (37):

$$\sqrt{\beta'} = \sqrt{-\tilde{g}}\ \tilde{s}\ x^2 y \frac{\tilde{u}^r}{c}$$ (44)

 

$$\frac{K_{\rm A}}{M_{\rm A}^2}\left(E'+1\right) = x\left\lbrace \l... ...tilde{h}(1)}{M_{\rm A}^2} \tilde{\varpi}^2 x y\right] \tilde{u}^t \right\rbrace$$ (45)

 

$$E'+1 = - \tilde{h}\left[\left( \tilde g_{tt} + \sqrt{\omega'}\til... ...i} + \sqrt{\omega'}\tilde g_{\phi\phi} x \right)\frac{\tilde{u}^\phi}{c}\right]$$ (46)

 

$$-1 = \tilde g_{tt} \left(\tilde{u}^t\right)^2 + 2\tilde g_{t\phi}... ...tilde{u}^\phi}{c}\right)^2 + \tilde{g}_{rr}\left(\frac{\tilde{u}^r}{c}\right)^2$$ (47)

$$\tilde{h}(y) = 1+\frac{\Theta}{\gamma-1}y^{\gamma-1}.$$ (48)

In (46) and (45) the constants $L/r_{\rm A}c$ and $\Phi^2\rho_{\rm A}/4\pi\dot{m}^2$ have been eliminated using the conditions (38) and (39) at the Alfvén point, which now read

$$-\left( \tilde g_{tt}(1) + 2\sqrt{\omega'}\tilde g_{t\phi}(1) + \omega'\tilde g_{\phi\phi}(1) \right) M_{\rm A}^2$$ =

$$\frac{4\pi\dot{m}^2}{\Phi^2} \frac{\tilde{h}(1)}{\rho_{\rm A}}$$ = (49)

and

$$- \frac{\tilde g_{t\phi}(1) + \sqrt{\omega'}\tilde g_{\phi\phi}(1)} {M_{\rm A}^2} = \frac{L}{r_{\rm A} c}\frac{1}{E'+1}$$ (50)

so that

$$\frac{\Phi^2\rho_{\rm A}}{4\pi\dot{m}^2} = \frac{\tilde{h}(... ...{\rm A} c} = \frac{K_{\rm A}}{M_{\rm A}^2}\left(E'+1\right)$$ (51)

with $K_{\rm A} = \tilde g_{t\phi}(1) + \sqrt{\omega'}\tilde g_{\phi\phi}(1)$.

The component $\tilde{u}^{r}$ can be expressed from (44) and substituted into (47) to provide a first relation between $\tilde{u}^t$ and $\tilde{u}^\phi$. A second relation between these two components is given by (45) when subtracting $K_{\rm A}/M_{\rm A}^2\times$ (46). It allows us to express all components of the four velocity as functions of only xand y. Then the remaining Eq. (46) becomes a Bernoulli-like equation $\tilde{H}(x,y)={\rm const.}$ as (10) of Sakurai (1985). We do not further elaborate on the different steps which lead to this final expressions:

$$\tilde{h}(y) = 1+\frac{\Theta'}{\gamma-1}y^{\gamma-1},$$ (52)

$$\tilde{u}^t = \sqrt{K(x,y)}\frac{N(x,y)}{\sqrt{\mathcal{D}(x,y)}},$$ (53)

$$\frac{\tilde{u}^\phi}{c} = \sqrt{K(x,y)}\frac{D(x,y)}{\sqrt{\mathcal{D}(x,y)}},$$ (54)

$$\frac{\tilde{u}^r}{c} = \frac{\sqrt{\beta'}}{\sqrt{-\tilde{g}}\tilde{s}x^2y},$$ (55)

 

$$\tilde{H}(x,y)+1 \tilde{\varpi}^2 x \tilde{h}(y)\,\sqrt{K(x,y)} \frac{\left\vert\mathcal{N}(x,y)\right\vert}{\sqrt{\mathcal{D}(x,y)}}$$ =

= E'+1, (56)

where we have introduced the following auxiliary functions:

$$1 + \frac{\tilde{g}_{rr}}{-\tilde{g}} \frac{\beta'}{\tilde{s}^2 x^4 y^2},$$ K(x,y) = (57)

$$\left[ \left(M_{\rm A}^2+\sqrt{\omega'}K_{\rm A}\right) \tilde g_... ...\rm A} \tilde g_{t\phi} \right] \tilde{h}(y) -\tilde{h}(1)\tilde{\varpi}^2 x y,$$ N(x,y) = (58)

$$-\left[ \left(M_{\rm A}^2+\sqrt{\omega'}K_{\rm A}\right) \tilde g... ... g_{tt} \right] \tilde{h}(y) -\sqrt{\omega'}\tilde{h}(1)\tilde{\varpi}^2 x^2 y,$$ D(x,y) =

$$\mathcal{N}(x,y) \left( \tilde g_{tt} + 2\sqrt{\omega'}x\ \tilde g_{t\phi} + \omega' x^2\ \tilde g_{\phi\phi} \right) \tilde{h}(1) y + M_{\rm A}^2\tilde{h}(y),$$ = (59)

$$\mathcal{D}(x,y) -\left[ \tilde g_{\phi\phi} D^2(x,y) + 2\tilde g_{t\phi} D(x,y) N(x,y) +\tilde g_{tt} N^2(x,y)\right].$$ = (60)

Equation (56) is the Bernoulli equation we will now consider. The solution y(x) of the wind equations appears as the level contour E' of the surface H(x,y). Notice that the classical limits of $\beta'$, $\Theta '$ and $\omega '$ are not exactly the corresponding $\beta$, $\Theta$ and $\omega$ parameters used by Sakurai (1985), who made the choice of normalizing these quantities with the value of the gravitational potential at the Alfvén point $GM/r_{\rm A}$ (see Eqs. (11a), (11b) and (11c) of Sakurai 1985) whereas we used c² to make the definition of these parameters more general. However a simple relation applies between Sakurai's and our parameters: $\beta/\beta'=\Theta/\Theta'=\omega/\omega'=r_{\rm A}/r_{\rm g}$ where $r_{\rm g}=GM/c^2$ is the gravitational radius. This is also valid for the classical limit of the Bernoulli function $\tilde{H}(x,y)$ and the definition used by Sakurai (1985).

Before studying $\tilde{H}(x,y)$ in the following section, we have to note that this function is not defined everywhere in the region x>0, y>0 as it is the case in the classical limit. The function $\mathcal{D}(x,y)$ must be strictly positive so that $\tilde{H}(x,y)$is well defined and the velocities are not imaginary. The domain where this condition applies is determined in Appendix C. In the sub-Alfvénic region (y>1) this domain lies always inside the light surface, its location is given more precisely in Appendix B.