A&A 394, 329-338 (2002)
DOI: 10.1051/0004-6361:20021135
P. Varnière - M. Tagger
DSM/DAPNIA/Service d'Astrophysique (CNRS URA 2052), CEA Saclay, 91191 Gif-sur-Yvette, France
Received 9 January 2002 / Accepted 30 July 2002
Abstract
We present a detailed calculation of the mechanism by which
the Accretion-Ejection Instability can extract accretion energy and
angular momentum from a magnetized disk, and redirect them to its
corona. In a disk threaded by a poloidal magnetic field of the order of
equipartition with the gas pressure, the instability is composed of a
spiral wave (analogous to galactic ones) and a Rossby vortex. The
mechanism detailed here describes how the vortex, twisting the footpoints of
field lines threading the disk, generates Alfvén waves propagating to
the corona. We find that this is a very efficient mechanism, providing
to the corona (where it could feed a jet or a wind) a substantial
fraction of the accretion energy.
Key words: accretion, accretion disks - instabilities - MHD - waves - galaxies: jets
MHD models have shown that jets can be very efficient to carry away the accretion energy and angular momentum extracted by turbulence from accretion disks (Blandford & Payne 1982; Lovelace et al. 1987; Pelletier & Pudritz 1992), if the disk is threaded by a poloidal magnetic field. This fits with the observation that accretion and ejection are intimately connected in objects ranging from protostellar disks to X-ray binaries and AGNs. However these models, based on self-similar analytical computations or on numerical simulations, most often start at the upper surface of the disk. Although more recent works (Ferreira & Pelletier 1993a, 1993b, 1995; Casse & Ferreira 2000) find solutions connecting continuously the disk and the jet, these solutions are heavily constrained by conflicting requirements. These can be traced, in part, to the fact that disk models, whether they rely on specific instability mechanisms or on the assumption of a turbulent viscosity, imply that the accretion energy and angular momentum are transported radially outward. They must thus somehow be redirected upward to feed the jet.
The Accretion-Ejection Instability (AEI) of magnetized accretion disks, presented by Tagger and Pellat (1999, hereafter TP99), could provide a solution to this difficulty. It occurs in the inner region of the disk, in the configuration assumed by the MHD models of jets (i.e. a disk threaded by a poloidal field of the order of equipartition with the gas thermal pressure), and it has the unique property that energy and momentum extracted from the disk can be emitted vertically as Alfvén waves propagating along magnetic field lines to the corona of the disk. Thus they could provide a source for a jet or a wind formed in the corona.
This ability to emit the energy and angular momentum as Alfven wave was recognized in TP99, and indeed justified the name given to the instability. However this possibility was shown only in a WKB approximation, valid away from the region (the corotation radius, where the wave rotates at the same velocity as the gas) where Alfvén wave emission is most efficient. The WKB result was found divergent at corotation, providing a good indication that this mechanism of vertical emission could be quite efficient.
The goal of this paper is to present a more general derivation, valid in the corotation region. We use a description of the waves in three dimensions (whereas TP99 was basically a model averaged over the disk thickness). We are thus able to give an explicit computation of the Alfvén wave emission mechanism and of its efficiency. The main unknown to be solved for at this stage is the fraction of the accretion energy and angular momentum, extracted from the disk by the instability, which will end up emitted to the corona.
The result we present is quite limited: the constraints of giving an
analytical derivation force us to use a very simplified magnetic field
geometry, namely an initially constant and vertical field,
and a more realistic one would certainly affect the result.
Appendix A is dedicated to the case of a radially varying
B_{z} field. We show that the present computation may be apply in the case
of a slowly variable B field.
On the other hand, the result we obtain is interesting in itself: we
will show that, in linear theory, the flux of the Alfvén waves is
again divergent at corotation. Although we discuss how it can be
regularized, our interpretation will be that the efficiency of the
mechanism is indeed quite high, but that we are reaching the limits of linear theory and that the true result will most certainly be determined
by self-consistent non-linear effects. The present work should thus be
viewed as an exploration of the complex physics involved and of its
potential efficiency, which will then have to be treated by non-linear
simulations.
Figure 1: The structure of the instability is shown here schematically as a function of radius. It is formed of a standing spiral density wave in the inner part of the disk, coupled to a Rossby vortex it excites at its corotation radius. The spiral grows by extracting energy and angular momentum from the disk, and depositing them in the Rossby vortex; the latter in turn generates Alfvén waves propagating toward the corona of the disk. | |
Open with DEXTER |
We will give here a short review of the main characteristics of the AEI, and refer the interested reader to TP99, or to Varnière et al. (2002) and Rodriguez et al. (2002) where a detailed comparison with the low-frequency QPO of black-hole binaries is given. Non-linear MHD simulations were performed by Caunt & Tagger (2001). The AEI is essentially a spiral instability, similar to galactic ones but driven by magnetic stresses rather than self-gravity. It affects the inner region of an accretion disk threaded by a poloidal magnetic field of the order of equipartition with the thermal pressure of the gas ( of the order of unity), i.e. the configuration and magnitude used in most MHD jet models.
The instability is composed of a spiral density wave and a Rossby wave it generates at its corotation radius (the radius where the wave rotates at the same velocity as the gas). The spiral forms a standing pattern between the inner edge of the disk and the corotation radius. Because of differential rotation, it couples to the Rossby wave, i.e. the spiral and Rossby waves exchange energy and angular momentum in the corotation region where, because of differential rotation they loose their separate identities.
In this process the Rossby wave also forms a standing vortex, and the spiral grows by storing in it the energy and angular momentum it extracts from the inner region of the disk (thus causing accretion).
In a thin disk in vacuum, the process stops there. This is the case in galaxies and recently Fridman et al. (2001) observe and analyse such a vortex in NGC 157. However one must remember that the Rossby vortex represents a torsion of the footpoints of the field lines threading the disk. Thus if the disk is covered by a low-density corona, this torsion will propagate vertically along the field lines as Alfvén waves: these will thus in turn transport a fraction of the accretion energy and angular momentum to the corona, where they could provide a source for a jet or an outflow. The whole process is shown schematically in Fig. 1.
As in previous works, we consider the very simple setup of an infinitely thin disk threaded by a vertical magnetic field adding this time the hypothesis that it is radially constant. Here however the disk is embedded in a low density corona. This simplified geometry will be enough to fully characterize the emission of Alfvén waves in the corona. The equilibrium field, , is assumed to be of the order of equipartition with the gas pressure (plasma ) in the disk. The case of a radially varying B_{0} will be studied in Appendix A.
We choose to present here results with a constant B^{0} because assuming that B^{0} depends on r creates at equilibrium a radial magnetic pressure force which can easily become dominant in the corona, since the other forces (gravity, Coriolis, pressure) act on a very low density. As a result a realistic equilibrium will in general include a flow along the field lines, and lead to a complex configuration limiting our ability to extract analytical results unless artificial assumptions are made.
On the other hand instability requires, as found in TP99:
We work in cylindrical coordinates
.
We consider linear perturbations, described by Lagrangian displacements
with:
From this we write the induction equation,
We consider perturbations varying as exp
(so that m will be
the number of arms of the spiral). We define a de-dimensionalized
perpendicular gradient,
The spiral can thus be amplified if positive energy is emitted beyond corotation (as a spiral wave emitted outward: this is the Swing mechanism, responsible for the amplification of galactic spirals), or stored in a Rossby vortex at the corotation radius: in TP99 this was shown to result, in a magnetized disk, in the Accretion-Ejection Instability. This corotation resonance introduces, in the variational form of TP99, a pole (a denominator proportional to , singular at corotation). This pole contributes, following the Landau prescription familiar in plasma physics, an imaginary term which represents the energy exchanged with the Rossby vortex.
In TP99 it was also found that, when one takes into account a low-density corona above the disk, Alfvén waves are emitted toward the corona; their contribution was computed in a WKB approximation, valid only away from corotation. This had of course a limited interest, since we expect the Alfvén waves to be strongest at corotation where the torsional motion associated with the vortex is strongest. Here we will present a more general formulation, showing how the energy flux of the Alfvén waves appears as an additional imaginary term in the variational form, generated at the vertical boundaries of the disk. This will allow us to compute explicitly the Alfvén wave flux at corotation.
In terms of the Lagrangian displacement, the linearized Euler equation
for a perturbation varying as
can be
written as:
Linearizing the contribution of magnetic stresses,
,
we find:
In order to obtain the new variational form, we first write a quadratic form built from the divergence and the curl of times the Euler equation, leaving unspecified for the moment.
We first apply the operator
to
the Euler equations, i.e. we compute:
In order to obtain the variational form we will first integrate these
equations vertically. Assuming that the disk is covered by a low-density corona, we integrate between
and
,
chosen well into the corona so that at that height only the
Alfvén
waves propagate and their vertical propagation can be described in the WKB
approximation^{}.
We write the integral:
F | |||
= | 0 | (16) |
After some algebra, integrating by parts and grouping
terms we get:
The right-hand side is formed of boundary terms, obtained in the integrations by parts. All these terms are easily shown to be imaginary (i.e. correspond to growth or damping), if the boundaries are far enough that the radial and vertical derivatives can be estimated in a WKB approximation, , and if waves do propagate at the boundaries, i.e. if the wavevectors are real so that waves can effectively transport energy away.
The first bracket corresponds to the flux of the wave at the radial boundaries; as explained in TP99, a wide range of boundary conditions at the inner disk edge allow the waves to be reflected without loss of energy, i.e. this term vanishes or remains real at . At it gives the flux of an outgoing wave, responsible for the usual Swing amplification of spiral waves (driven by self-gravity in galaxies, by pressure in the Papaloizou-Pringle instability (Papaloizou & Pringle 1985), or by magnetic stresses in Tagger et al. 1990).
The last term is new and corresponds to a flux at the vertical boundaries, i.e. the flux of the Alfvén waves emitted vertically. This is confirmed by the fact that this term is associated with the torsional () component of the perturbations.
In our variational form, Eq. (17), the corotation
resonance does not appear explicitly as it did in the equivalent form of
TP99: we do not get denominators containing
,
vanishing at
corotation. But the corotation resonance must of course be present,
since the physics described here is more general than in TP99: it is
hidden here in the singularity of :
Eq. (15) has a
singular point at corotation (where
vanishes) because the
highest-order radial derivative of
is proportional to
(assuming, as will be checked a posteriori, that the vertical derivative
vanishes at corotation), while the terms in
are proportional to
.
We thus turn, in the vicinity of corotation, to a Frobénius
expansion of the form:
= | (18) | ||
= | (19) |
(22) |
(23) |
The manner in which
and
project on the solutions which are
regular and singular at corotation depends on the global solution, which
must be obtained numerically as in TP99, of the problem with its
boundary conditions. Here we will only use the fact that
has a
singular contribution (whose Frobénius expansion starts with
)
which makes the quadratic form, Eq. (17), non variational at corotation. Combining the singular
terms, we find their contribution to the integral:
(25) |
Our goal now is to compute how this singular vorticity generates Alfvén waves, transmitting to the corona a part of the accretion energy and angular momentum extracted from the disk. This flux is readily identified, in Eq. (17), as the last term which represents a contribution from the lower and upper boundaries of our integration domain: this is thus the flux emitted vertically, and we expect that the large amplitude and the singularity of will give a strong contribution at corotation.
The flux of Alfvén waves to the corona appears in our variational
form, Eq. (17), as a surface term taken at the lower and
upper boundaries of our integration domain:
Let us first consider the case of a disk in vacuum: the Alfvén
velocity in the corona goes to infinity, so that Eqs. (20), (21) reduce to
and
.
In a radial WKB approximation, this gives the
result familiar in spiral wave theory, that
varies above the disk
as
,
where k is the horizontal wavenumber, so that vanishes exponentially. On the other hand one finds that
stays
constant with z. The full vertical solution, valid across the disk, was given
by Tagger et al. (1992). If the coronal density is now small but
non vanishing, Eq. (20) shows that far above the disk
must be of the order of
,
where
k^{-1} is the radial scalelength of ,
and
is the
Alfvén velocity in the corona, large but not infinite. Thus now
is negligible in
Eq. (21), while the vertical derivative of
must be
retained. In a WKB approximation in z we now expand this derivative as:
(28) |
(29) |
The second surprise is that k^{2}_{z} is proportional to
,
so
that we find k_{z} real only on one side of corotation, depending on
the sign of the derivative of
.
Thus the singular
perturbation will propagate only on one side of corotation! This can be
understood by returning to Eq. (21) and writing it now
in terms of
,
still neglecting the contribution of .
We get:
(32) |
The meaning of Eq. (31) becomes simple: it describes how Rossby waves, forming the singular part of our full solution for , will now propagate upward along the field lines as what we might call Rossby-Alfvén waves. If the radial derivative of is positive (this is the instability criterion of TP99), the wave propagates only beyond corotation and thus carries a positive energy flux, so that its formation in the disk and propagation along the field lines destabilize the negative energy spiral wave inside corotation, which is the main component of our instability.
This makes the physics of the coupling between the instability and Alfvén waves much more complex than the resonance, found by Curry & Pudritz (1996), between normal modes of the Magneto-Rotational Instability (MRI) and Alfvén waves. The main reason is that they work in a vertical WKB approximation, assuming that the modes have a fixed vertical wavenumber k_{z}; it is worth remembering here the result of a more complete vertical solution (Tagger et al. 1992): the AEI corresponds to solutions with n_{z}=0 nodes across the disk height, whereas the MRI corresponds to solutions with . For , most unstable when , a vertical WKB approximation can be used. Keeping k_{z} fixed results in finding a resonance away from corotation, at , and thus ignoring the coupling with the Rossby vortex. It also ignores the difficulties, encountered here, associated with the vertical variation of the Alfvén velocity.
We can now compute the flux of Alfvén waves, appearing in the
right-hand side of Eq. (17):
(35) |
(36) |
(37) |
In the body of the paper we have studied the case of a vertical and constant unperturbed field. In Appendix A we discuss the effect of a field which is still straight but depends on the radius. The analytical computation cannot be fully completed, mainly because no simple equilibrium exists without a flow along the field lines; but we show that within reasonable bounds (that the magnetic stress term does not exceed the centrifugal and gravitation forces) our conclusions should not change.
On the other hand a realistic model should also include the curved geometry of the magnetic field, as obtained in jet models. In this case the situation becomes much more complex because in general the three basic MHD waves become coupled by the geometry, in addition to the coupling by differential rotation studied here. In a straight field the Alfvén and slow magnetosonic wave are decoupled: this has allowed us to defer the consideration of slow magnetosonic waves, together with vertical motion, to separate work. In a curved field the Alfvén wave will include vertical motion, and we should in principle include all three components of the displacement and all three MHD waves.
The mixing can still be weak if the scales are very different. For instance, the characteristic wavelength of the slow wave is of the order of , i.e. of the disk scale height if we are not too close to corotation. Thus if the magnetic field is curved on a large scale (of the order of r), the mixing of the waves is weak and our conclusion should not change much. In particular the Alfvén wave is excited much more efficiently than the slow wave, for which we expect no singular source analogous to the Rossby vortex for the Alfvén wave.
On the other hand more elaborate jet models (e.g. Casse & Ferreira 2000) show that a slow magnetosonic point forms above the disk. The field lines are sharply bent in its vicinity. But there the Alfven velocity is already much higher than the sound velocity. This disparity should again maintain a separation between slow and Alfvén waves, and allow the latter to propagate the vorticity from the Rossby wave. A realistic computation goes far beyond our present abilities. But we note that the coupling of waves by geometric effects might introduce interesting new channels to deposit energy and momentum from the wave in the corona.
We have presented a computation of the flux of Alfvén waves emitted to the corona of a magnetized disk by the Accretion-Ejection Instability. This means that we have justified here the name chosen by TP99: the instability is a spiral density wave, which grows by extracting energy and angular momentum from the disk (thus causing accretion) and transferring them radially outward to the Rossby vortex at corotation; a significant fraction, given by Eq. (38), of this flux is then transmitted vertically to the corona as Alfvén waves. We expect that, if the Alfvén waves can deposit their energy and momentum in the corona, this would be an ideal mechanism to feed a wind or jet directly from the accretion process in the disk.
The amplification of the wave (and thus the flux deposited by the spiral in the vortex) and the flux transmitted to Alfvén waves are both linked to the singularity of the vortex. This allows us to give in a very simple form a result of paramount importance in the physics of accretion disks and jets: an estimate of the fraction of the accretion energy, extracted from the inner region of the disk, which will end up in the corona where it might feed a jet. This fraction is of the order of unity if the coronal density is not too low (typically 10^{-4} of the density in the disk would be sufficient, in an X-ray binary).
In order to obtain analytically a physically consistent result, we have had to use a very artificial configuration of the equilibrium magnetic field, vertical and independent of r. On the other hand this has allowed us, proceeding rigorously by perturbation of a variational form, to obtain an exact result clarifying the role and the physical nature of the singularity of the Rossby vortex at corotation. We can thus expect that these results would survive less stringent assumptions on the equilibrium configuration.
However this must be taken carefully: our final result is in fact divergent at the corotation radius, and regularized by the effect of the finite thickness of the disk, or by the growth rate of the instability. In both cases, it depends on the density in the corona of the disk. Thus we believe that the end result should be obtained from a self-consistent, non-linear description where the growth of the instability itself affects the evolution of the magnetic geometry and the mass loading of the corona.
In this respect it is worth mentioning one of the results of stationary MHD jets models: in these models, as the gas is accelerated along the field lines it passes a slow magnetosonic point where the field lines bend outwards. Magnetocentrifugal acceleration can then proceed and leads to the formation, higher up and further out, of an alfvénic point. The slow magnetosonic point is thus associated with the mass loading of the field lines, and the alfvénic point to the acceleration. By analogy we can thus expect that, while the Alfvén waves described in the present work allow to accelerate the gas, the instability can also generate slow magnetosonic waves which will lift the gas above the disk. The coupling of the instability to the slow wave will be the object of a forthcoming paper.
In the body of the present article we made the simple assumption that the equilibrium field was vertical and constant, allowing us to get an analytical derivation of the Alfven waves flux. In this Appendix we give the general derivation in the case where the field is still straight along z but depends on s. We will follow the same computation as in the main text, referring to the corresponding equations.
The first modification appears in the contribution of magnetic stresses
.
We now have to take into account
the equilibrium current, j^{0}:
After some algebra, integrating by parts and grouping
terms we get the equivalent of Eq. (17)
Taking into account the gradient of B^{0} makes the analytical derivation of the dispersion relation and Alfvén flux much more complex and we will not attempt it here.
If we make the additional assumption that B_{0} depends on r only weakly, i.e. that the current j^{0} is
weak, we can get rid of the influence of the terms containing
g. This assumption is equivalent to requiring that the
magnetic stress term is at most of the order of the centrifugal
force, which should be the case in a realistic jet model
including an equilibrium flow along the field lines. Using
as the Alfvén velocity in the corona we obtain
the condition: