11institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
22institutetext: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge, CB3 OHA, United Kingdom 33institutetext: AIM, CEA, CNRS,
Université Paris-Saclay, Université de Paris, F-91191 Gif-sur-Yvette, France
44institutetext: IRAP, Université de Toulouse, CNRS, UPS, CNES, Toulouse,
France

# Are Low-Frequency Quasi-Periodic Oscillations seen in accretion flows the
disk response to a jet instability?

J. Ferreira 11 G. Marcel 22 P.-O. Petrucci 11 J. Rodriguez 33 J. Malzac 44 R.
Belmont 33 M. Clavel 11 G. Henri 11 S. Corbel 33 M. Coriat 44

(Received month dd, yyyy; accepted month dd, yyyy)

Low Frequency Quasi-Periodic Oscillations or LF QPOs are ubiquitous in black
hole X-ray binaries and provide strong constraints on the accretion-ejection
processes. Although several models have been proposed so far, none has been
proven to reproduce all observational constraints and no consensus has emerged
yet.

We make the conjecture that disks in binaries are threaded by a large scale
vertical magnetic field that splits it into two radial zones. In the inner Jet
Emitting Disk (JED), a near equipartition field allows to drive powerful self-
collimated jets, while beyond a transition radius, the disk magnetization is
too low and a Standard Accretion Disk (SAD) is settled. In a series of papers,
this hybrid JED-SAD disk configuration has been shown to successfully
reproduce most multi-wavelength (radio and X-rays) observations, as well as
the concurrence with the LFQPOs for the archetypal source GX 339-4.

We first analyze the main QPO scenarios provided in the literature: 1) a
specific process occurring at the transition radius, 2) the accretion-ejection
instability and 3) the solid-body Lense-Thirring disk precession. We recall
their main assumptions and shed light on some severe theoretical issues that
question the capability to reproduce LF QPOs. We then argue that none of these
models could be operating under the JED-SAD physical conditions.

We finally propose an alternative scenario where LF QPOs would be the disk
response to an instability triggered in the jets, near a magnetic
recollimation zone. Such a situation could account for most Type-C QPO
phenomenology and is consistent with the global behavior of black hole
binaries. The calculation of this non-destructive jet instability remains
however to be done. If the existence of this instability is numerically
confirmed, then it could also naturally account for the jet wobbling
phenomenology seen in various accreting sources, around compact objets and
young forming stars.

###### Key Words.:

Accretion, accretion discs – Magnetohydrodynamics (MHD) – X-rays: binaries –
Galaxies: active – ISM: jets and outflows

##  1 Introduction

Black Hole X-ray Binaries (hereafter BH XrB) are binary systems where a
stellar-mass black hole accretes matter from a low-mass companion star. The
incoming mass forms an accretion disk around the black hole which is detected
mainly in X-rays. All systems are X-ray transients, spending sometimes years
in a barely detectable quiescence and suddenly increasing their luminosity by
factors up to $10^{6}$ during months, until returning back to quiescence (see
e.g. Remillard & McClintock 2006; Done et al. 2007; Dunn et al. 2010). These
episodes of intense activity are accompanied by drastic spectral changes, with
two main canonical spectral states. In the hard state, the spectrum is
dominated by a (cut-off) power-law like emission usually attributed to inverse
Comptonization that peaks at a few 100 keV and shows a quite intense (10 to
20% rms) rapid (subsecond) variability. In the soft state, the spectrum is
characterized by a thermal black body emission of a few keV only and exhibits
much less variability. A complete outbursting cycle can then be followed in a
Hardness-Intensity Diagram (HID), where most BH sources make a q-shaped
evolutionary track (see e.g. Dunn et al. 2010). Starting at low emission and
in the hard state, an outburst consists first of a tremendous increase of
luminosity in the hard state, a sharp transition to a softer (low hardness)
state while maintaining a roughly constant luminosity, followed by a
luminosity decrease while remaining in this soft state, another transition
back to the hard state at roughly the same luminosity and a final decay back
the initial state (finishing the q-shape).

But the HID only probes accretion onto the black hole and does not convey all
the phenomenology of these sources. It is now well known that jets, possibly
in the form of bipolar self-confined outflows, are also emitted by the central
regions. These jets are mainly detected in radio bands and show a clear
correlation with the underlying X-ray emission, albeit only when the source is
in the hard state (see e.g. Corbel et al. 2000, 2003, 2013; Gallo et al. 2003;
Merloni et al. 2003; Falcke et al. 2004; Fender et al. 2004 and the review by
Fender & Gallo 2014). While explaining only the behavior seen in the HID was
already quite challenging per se, including the jet production mechanism (and
its associated power leakage in the hard states) remains an open issue. There
have been several propositions of course (e.g. Esin et al. 1997; Meyer-
Hofmeister et al. 2005; Yuan et al. 2005; Ferreira et al. 2006b; Begelman &
Armitage 2014; Kylafis & Belloni 2015) but, to date, there is no clear
consensus on which scenario is the most likely to explain the complex multi-
wavelength (radio to X-rays) time dependent behavior of BH XrBs.

The analyzis of the fast timing variations seen in hard X-rays offers however
another independent means to grasp the BH XRBs phenomenology (see Motta 2016;
Ingram & Motta 2019 for recent reviews). The intrinsic variability of the
sources is highly dependent of their spectral state, reaching 20% in the hard
states, going down to 1-10% level in the hard-to-soft and soft-to-hard
intermediate states and often consistent with zero in the soft state. Besides,
the power density spectra have complex and state dependent shapes. On top of a
broad band continuum, they can exhibit narrow features, called quasi-periodic
oscillations or QPOs, of centroid frequency $\nu$ and extent $\Delta\nu$.
There are three main types of low frequency ($\nu<50$ Hz) QPOs that seem
intimately linked to the spectral state (Remillard et al. 1999; Casella et al.
2004; Belloni et al. 2005): Type-C, the most common, are detected in hard
states, Type-B appear during (and define) the intermediate states while Type-A
are rare, only seen in the high soft state. Type-A QPOs are very weak and
broad with $Q=\nu/\Delta\nu<3$ and a frequency $\nu_{A}\sim 6-8$ Hz. Type-B
QPOs are more prominent and narrow, with $Q\geq 6$ and $\nu_{B}\sim 1-6$ Hz.
Finally, Type-C QPOs are strong and narrow, with $Q\geq 10$ and a frequency
$\nu_{C}$ ranging from mHz to $\sim 10$ Hz.

Although there is a time sequence from one QPO type to another, it is not
clear whether all types should be explained by the same model. For instance
Type-B QPOs are stronger in more face-on objects and reveal a time proximity
with transient radio flares, which has lead to propose a link between their
appearance and the launching of discrete relativistic ejections (Fender et al.
2009; Motta et al. 2015; Stevens & Uttley 2016; Russell et al. 2019; Homan et
al. 2020). But these two properties are not shared by all QPOs. Indeed, Type-A
QPOs are detected in a jetless phase while Type-C QPOs are characterized by
three extra properties (Remillard & McClintock 2006; Motta 2016). First, the
frequency $\nu_{C}$ is independent of the energy band and source inclination,
although always associated to the inner (hot) accretion flow (e.g van den
Eijnden et al. 2017). Second, the time evolution of the QPO frequency is
tightly correlated with the variation of the soft ($<10$ keV) source count
rate and hard X-ray power-law index, as well as the innermost radius of the
cold (black body) accretion flow. However, the QPO frequencies are always much
lower than the Keplerian frequency of the inner cold disk radius (e.g. Muno et
al. 1999; Markwardt et al. 1999; Sobczak et al. 2000; Rodriguez et al. 2002;
Vignarca et al. 2003; Rodriguez et al. 2004b; Remillard & McClintock 2006;
Marcel et al. 2020). Finally, phase lags are also commonly observed between
different energy bands (e.g. 2-5 versus 13-30 keV, Casella et al. 2004). The
lag of a Type-C QPO is strongly correlated with its frequency and depends also
on the object inclination: near zero but with an increasing hard lag (soft
photons arriving first) with increasing QPO frequency for low inclination
sources whereas high-inclination sources turn to soft lags at larger QPO
frequencies (e.g. Motta et al. 2015; van den Eijnden et al. 2017; de Ruiter et
al. 2019; Zhang et al. 2020; Ma et al. 2021).

Explaining Type-C LFQPOs is a challenging task and there have been already
several models proposed in the literature. Yet, there is still no consensus as
all QPO models are still facing major theoretical issues. Moreover, and more
importantly, no model embraces the global picture of these accreting systems.
Indeed, as intriguing as QPOs may be, they are only an epiphenomenon of the
main accretion-ejection process. They must therefore be understood within, and
be part of, a global framework that addresses also the other observational
constraints namely, spectral evolution, jet formation and quenching.

Hereafter, we focus on such a framework, the hybrid disk configuration
proposed by Ferreira et al. (2006b), which has been recently successfully
confronted to observations in a series of papers (Marcel et al. 2018b, a,
2019, 2020, 2022; Ursini et al. 2020; Marino et al. 2020; Barnier et al.
2022). In this framework, a large scale vertical magnetic field $B_{z}$ is
assumed to thread the inner disk regions, say below $10^{4}r_{g}$ where
$r_{g}=GM/c^{2}$ is the gravitational radius. At any given radius, its
dynamical importance is measured at the disk midplane by the magnetization
$\mu(r)=B_{z}^{2}/(\mu_{o}P_{tot})$, where $P_{tot}$ is the sum of the gas and
radiation pressure. It has been argued that this magnetization is a decreasing
function of the radius and that the disk is separated into two distinct
regions (Ferreira et al. 2006b; Petrucci et al. 2008). Beyond a transition
radius $r_{J}$ the magnetization is low and the disk is assumed to be in a
Standard Accretion Disk (hereafter SAD) mode, where most of the disk angular
momentum is carried radially away by a MRI-driven turbulence (Shakura &
Sunyaev 1973; Balbus & Hawley 1991; Balbus 2003). Although winds are possible
and even expected when a large scale $B_{z}$ field is present, their influence
on the disk energetics is rather small (Zhu & Stone 2018; Jacquemin-Ide et al.
2021). As a consequence, the SAD region accretes at a highly subsonic speed
and is optically thick. On the contrary, the region below $r_{J}$ and
extending down to the innermost stable circular orbit $r_{ISCO}$ is in a Jet
Emitting Disk (hereafter JED) mode, where a sizable fraction of the disk
angular momentum is carried away vertically by two self-confined magnetically-
driven jets. The JED model is a generalization and an extension of the
Blandford & Payne (1982) jet model, addressing both the mass loading issue and
causal connection with the underlying disk. This inner JED region is
characterized by a magnetic field near equipartition (a constant $\mu$ around
$0.1-0.8$), leading to a supersonic accretion speed (Ferreira & Pelletier
1995; Ferreira 1997). As a consequence, the JED region is much less dense than
the outer SAD and becomes optically thin and geometrically slim.

As estimated in Marcel et al. (2018a), the radial JED-SAD transition is
achieved when the inner disk magnetization in the SAD reaches a value
$\mu_{c}\sim 10^{-3}$. This critical value triggers the inward radial
transition to a JED and leads to a sudden drop in density and increase in
accretion speed as the jet torque becomes dominant. Postulated in 2006, this
radial JED-SAD disk configuration seems to be qualitatively consistent with
current 3D global MHD simulations, both in non-relativistic (Jacquemin-Ide et
al. 2021) and relativistic regimes (Igumenshchev et al. 2003; Tchekhovskoy et
al. 2011; McKinney et al. 2012; Avara et al. 2016; Liska et al. 2020 to cite
only a few)111In GRMHD simulations, this inner zone has usually been termed a
Magnetically Arrested Disk or MAD, a model initially defined as
$B_{z}B_{r}^{+}/\mu_{o}\sim\rho\Omega_{K}^{2}rh$, namely where the poloidal
laminar magnetic field would be so strong that it would provide a support
against gravity. Assuming $B_{r}^{+}\sim B_{z}$, this requires a disk
magnetization $\mu\sim r/h$ (see Narayan et al. 2003 and references therein).
On the other hand, a JED is a sub-Keplerian disk with $\mu$ of order unity,
enough to launch centrifugally-driven jets. More numerical efforts must be
done to assess the differences between a JED and a MAD..

This framework allows to reproduce most of the available data for the
archetypical source GX 339-4. Indeed, by allowing the disk accretion rate
$\dot{m}$ and the transition radius $r_{J}$ to vary independently with time, 4
cycles of activity have been successfully reproduced (see Marcel et al. 2020
and references therein). It is remarkable that time evolutions of the pair
$(r_{J},\dot{m})$ are able to reproduce each individual spectra and the HID of
GX 339-4, but also the radio emission (although in a more qualitative way).
Moreover, it has been also firmly established that the frequency of the
detected Type-C LFQPOs follows the scaling law

$\nu_{QPO}=\nu_{K}(r_{J})/\chi$ (1)

where $\nu_{K}(r_{J})$ is the keplerian orbital frequency at $r_{J}$ and
$\chi$ is a constant factor $\sim 100$ (actually varying from 70 to 130,
depending on the outbursting cycle). This result, verified for more than two
decades in frequencies, clearly advocates for a strong connexion between the
physical source of the LFQPO and the transition radius $r_{J}$ (Marcel et al.
2020).

To our knowledge, and despite several simplifications, this framework is
currently the only one providing a clear picture addressing most of the
observational accretion-ejection constraints. However, it is not clear yet if
the observed timing properties, and in particular the LFQPOs, can fit inside
this paradigm. This is the purpose of this paper.

We first critically analyze in section 2 the main QPO scenarios provided in
the literature: a specific process triggered at the transition radius, the
accretion-ejection instability (Tagger & Pellat 1999) and solid-body Lense-
Thirring disk precession model (Ingram et al. 2009). We recall their main
results but highlight also (sometimes severe) theoretical issues (see Ingram &
Motta (2019); Marcel & Neilsen (2021) for observational issues). Putting these
issues aside, it will be moreover argued that none of these scenarii can
provide an explanation for Type-C QPOs fulfilling Eq.(1) within the JED-SAD
framework. In section 3, we finally propose a novel scenario where the
observed LFQPOs would be the inner JED response to a jet instability occurring
away from the disk. It will be shown in Section 4 that such a situation not
only provides a qualitative explanation for the LFQPO phenomenology, but it
may also provide insights into the behavior of astrophysical jets. We conclude
in section 5.

##  2 Comments on some models for LF QPOs

Before proposing a new scenario for the LFQPO (Sect. 3) we first discuss in
this section several existing models. We only address those which, in our
view, are the most representative in the literature and refer instead the
interested reader to the reviews of Done et al. (2007) and Ingram & Motta
(2019) for an exhaustive list. Note that all models require the existence of a
transition radius usually associated to the innermost black body disk. When
applicable, we discuss also some of these models in the context of the JED-SAD
framework.

###  2.1 A specific process occurring at the transition radius

Since the QPO frequency in hard X-ray emission is tightly correlated with the
transition radius (labelled $r_{J}$ in the JED-SAD framework), it is natural
to first look at a physical process that would be triggered at that specific
location. The problem of course is the factor $\chi\sim 100$ which requires to
look for a very slow, secular process.

It is known for instance that any misalignment between the black hole spin and
the disk leads to a Lense-Thirring (hereafter LT) precession due to the
dragging of inertial frames (see Bardeen & Petterson 1975 and references
therein). As a consequence, a test particle located at a radius $r$ from a
black hole of dimensionless angular momentum $j$ would undergo a relativistic
precession at the LT-frequency $\nu_{LT}(r)\simeq\frac{jc}{\pi
r_{g}}(r/r_{g})^{-3}$ which is steeply decreasing with the distance. Proposed
initially for kHz QPOs in XrBs (Stella & Vietri 1998; Stella et al. 1999), one
could also imagine that the outskirts of a hot inner flow, specifically a hot
ring near the transition radius with the SAD, could provide an observable
LFQPO. However, the ratio
$\nu_{LT}/\nu_{K}(r_{J})=2j(r_{J}/r_{g})^{-3/2}\propto r_{J}^{-3/2}$ keeps
decreasing with the distance. This is in contradiction with the observational
constraint Eq. (1), where this ratio remains roughly constant regardless of
$r_{J}$.

Motivated by the similarity between the observed power-density spectra of BH
XrBs and the response function to external perturbations of driven mechanical
and electrical systems, Psaltis & Norman (2000) computed the response of a
narrow (of $\delta r/r\sim 0.01$), geometrically thin accretion disk annulus
to a broad spectrum of isothermal perturbations imposed outside of it. That
ring corresponds to an abrupt change of disk properties, probably again the
transition radius. They found that some resonances appear superimposed on the
incoming spectrum, leading to observable QPOs. Since they neglected the radial
pressure forces, the predicted mode frequencies depend mostly on the
gravitational field around the compact object and only weakly on the
hydrodynamic properties of the flow itself. As a result, the selected
frequencies are related to the epicyclic and LT frequencies at the transition
radius, which are both unable to explain LFQPOs due to the huge factor $\chi$.

Kato & Manmoto (2000) proposed a model of trapped oscillations that would be
triggered at the transition radius between the outer SAD and an inner
Advection Dominated Accretion Flow or ADAF (Ichimaru 1977; Narayan & Yi 1994,
see also the review by Yuan & Narayan 2014). Such a situation implies a very
narrow transition zone where the rotation profile needs to become slightly
super Keplerian (Abramowicz et al. 1998). As a consequence the epicyclic
frequency $\kappa^{2}$ becomes negative, leading to a local instability
(Rayleigh criterion, when the stabilizing effects due to the strong
inhomogeneities are negligible). The authors then argue that the slowly
growing amplitude of the perturbations would remain trapped around the
transition radius, leading to local QPOs.

Assuming a particular radial profile for the disk temperature within the
transition region, they solved the local dispersion relation of these
inertial-acoustic modes. They showed that it is possible to fine-tune the
radial profile so that the eigenvalue could match the QPO frequency at a given
transition radius. However, and this is quite uncomfortable, this fine-tuning
must be done for every radius and for a given set of disk parameters. It seems
therefore difficult to reconcile this result with the generic behavior deduced
from observations222This fine-tuning has nothing to do with a ”p-disk model”,
which is a multi-blackbody disk model where the temperature exponent is left
free (Mineshige et al. 1994; Kubota et al. 2005).. Not only there is no reason
why such a fine-tuning should occur, but the model relies also on strong
approximations. Of particular importance is the neglect of the accretion
motion, which would unavoidably lower the local growth of perturbations by
advecting them.

This remark is even more critical within the JED-SAD framework, which assumes
the presence of a large scale $B_{z}$ field everywhere. Whether or not such an
instability would be present near $r_{J}$ in this case requires a novel
investigation. However, the disk material undergoes a transsonic transition at
$r_{J}$ (Ferreira et al. 2006b; Marcel et al. 2018a; Scepi et al. 2019), since
the SAD accretes at subsonic speeds whereas the JED is supersonic. Moreover,
in order for a JED to maintain a constant disk magnetization $\mu$, any extra
magnetic flux that would be carried in by the accreting flow must be expelled
out. This is exactly what happens in some GRMHD simulations, where a magnetic
Rayleigh-Taylor instability is seen to grow up and expel the magnetic flux
excess as magnetic bubbles (McKinney et al. 2012; Marshall et al. 2018). Quite
interestingly, this is done quasi-periodically but on a time scale comparable
to a few Keplerian orbits at $r_{J}$, inconsistent with LFQPOs. The existence
of such a messy, transsonic JED-SAD transition casts therefore serious doubts
on the possibility to allow the development of any secular instability at that
location.

###  2.2 The accretion-ejection instability

The accretion-ejection instability (AEI) has been first studied by Tagger &
Pellat (1999) in the context of accretion disks threaded by a large scale
$B_{z}$ field, and then proposed to explain QPOs by Varnière et al. (2002) and
Rodriguez et al. (2002). A non-axisymmetric ideal MHD instability is found to
be triggered whenever the vertical field reaches equipartition along with some
radial profile. These spiral waves travel back and forth between the disk
inner radius $r_{in}$ (initially assumed to be the innermost SAD radius) and
some radius $r_{co}$, defined as the co-rotation between the wave phase speed
and the disk material rotation. The waves transport angular momentum
outwardly, allowing the ”cavity” between $r_{in}$ and $r_{co}$ to accrete
rapidly. Although spiral waves are also emitted beyond $r_{co}$, accretion in
this outer region is nevertheless assumed to carry on thanks to the usual MRI-
driven turbulence. At the co-rotation radius, waves are evanescent but a
resonance with a vertical Alfvén wave allows the transfer of some energy and
angular momentum in the vertical direction. A magnetized azimuthally localized
vortex is thus expected to grow at $r_{co}$, opening the possibility for some
ejection there (but not demonstrated yet). Such a vortex is invoked to be the
locus of enhanced dissipation (a hot spot), leading to a QPO with a frequency
$\nu_{K}(r_{co})$, as long as some geometrical occultation is also present of
course (so QPOs can be seen only in high inclination sources).

At first sight, the AEI would perfectly fit at the interface between the JED
and SAD zones, due to the build-up of a large scale $B_{z}$ field near
equipartition in the JED. But in order for the AEI to reproduce the LFQPO
behavior encapsulated in Eq. (1), the ratio of the co-rotation radius to the
innermost disk radius (equal to $r_{J}$ within the JED-SAD framework) must
then be $\chi^{2/3}>>1$ (independent of $r_{J}$). Such a distance between the
inner boundary and the co-rotation radius cannot be achieved within the AEI
framework (see for instance Fig. 4 in Varnière et al. 2002). Indeed, as the
magnetization increases, the wavelength becomes larger, widening the ”cavity”
and pushing out to larger radii the co-rotation radius. But this leads also to
a widening of the forbidden zone around the corotation radius and the
amplification mechanism becomes less efficient. As a result, the growth rate
has a maximum for $\mu$ only close to 1 (Tagger & Pellat 1999). In other
words, the factor $\chi\sim 100$ for Type-C QPOs cannot be achieved within the
AEI framework for reasonable values of the magnetic field.

It must also be realized that the complex situation invoked for the AEI has
been computed with a highly simplified setup so far. For instance, the
toroidal component of the magnetic field has been neglected, which forbids of
course the launching of jets and winds that would seriously affect the disk
angular momentum transport. The unavoidable MRI-driven turbulence has been
also discarded, while it provides both another torque and wave dissipation,
two ingredients that would again lower the efficiency of the mechanism. On the
other hand, despite the assumption of a perfect wave reflection at the inner
disk boundary, the maximum AEI growth rate is already quite low, namely of
order $V_{A}/r\sim\mu^{1/2}\Omega_{K}h/r$ only (Tagger & Pellat 1999). This
means that the physical ingredients listed above (jets, turbulence) would
probably seriously lower it or even quench the instability. Finally, the
probable presence of a magnetic Rayleigh-Taylor instability at the edge of the
JED, as seen in MAD simulations (McKinney et al. 2012; Marshall et al. 2018)
questions the very existence of the AEI at that location. Because of all these
extra ingredients, assuming a perfectly reflecting boundary at $r_{J}$ appears
rather dubious indeed.

Given all these difficulties and in particular the unreachable factor $\chi$,
we think that AEI is very unlikely the source of Type-C QPOs within the JED-
SAD framework. However, for Type-A and/or Type-B LFQPOs, that are observed
only when $r_{J}$ is near the ISCO and relativistic effects important (see
Varnière et al. 2012 and references therein), the AEI may remain a possible
candidate.

###  2.3 Solid-body Lense-Thirring disk precession model

The solid-body precession of the inner disk due to the Lense-Thirring effect
is certainly the most popular model invoked to explain Type-C LFQPOs (Ingram
et al. 2009; Ingram & Motta 2019). This model relies on a geometrical general
relativity effect, occurring whenever the black hole spin is misaligned with
the disk axis. The resulting frame drag causes important structural changes in
the surrounding accretion disk (precession) as it struggles to adapt its
equatorial plane to the periodic changes in the local gravitational field. The
model of Ingram et al. (2009) for QPOs has been designed within the framework
of Esin et al. (1997): an inner hot, optically thin and geometrically thick
flow (ADAF) is settled until a transition radius $r_{t}$ beyond which the disk
becomes the usual cold, optically thick and geometrically thin SAD. Within
this framework, a LFQPO correlated with $r_{t}$ (hence also with the disk
black body) could arise if the inner geometrically thick disk (ADAF) modulates
the X-ray flux. But this can only happen if a significant portion of the
ADAF’s volume is precessing as a solid body, namely with a unique precession
frequency $\nu_{prec}$.

As shown by Ingram et al. 2009, if that volume extends down to the innermost
stable orbit, not only the precession frequency $\nu_{prec}$ would exhibit a
too strong dependence on the black hole spin, but it would also provide
frequencies far too high. As a consequence, it is usually assumed that the
innermost disk regions, up to a few $r_{g}$, are actually aligned with the
black hole spin, probably due to a Bardeen-Petterson effect (Bardeen &
Petterson 1975; Papaloizou & Lin 1995; Lubow et al. 2002). Alignment of the
inner zone is actually always seen in all numerical simulations of tilted
black holes (see e.g. Fragile et al. 2007; Fragile 2009), although the
presence of large scale magnetic fields seems to seriously modify the estimate
of this inner radius and even question the physical mechanism (McKinney et al.
2013; Krolik & Hawley 2015; Liska et al. 2018, 2019; Chatterjee et al. 2020;
Liska et al. 2021). Nevertheless, assuming an inner radius of a few $r_{g}$
and a solid-body LT-precession up to an arbitrary transition radius $r_{t}$,
Ingram et al. (2009) have shown that the precession frequency
$\nu_{prec}(r_{t})$ could indeed cover the observed frequency range of Type-C
QPOs from $\sim 0.01$ to $\sim 10$ Hz. Note that the actual value of
$\nu_{prec}$ is a direct function of the external radius $r_{t}$, which
remained a free parameter in this work. Several other observational features
related to QPOs have been obtained within this framework (see the review of
Ingram & Motta 2019).

However, as appealing as this scenario is, the proof that a solid-body LT-
precession is settled over a significant radial range remains controversial in
numerical GR simulations. Indeed, while solid body precession has been clearly
shown in hydrodynamical flows (see eg. Fragile & Anninos 2005; Dyda & Reynolds
2020), the trend is much less clear when magnetized flows are considered.
There has been some works claiming that there is indeed a volume undergoing a
solid-body LT-precession (e.g. Fragile et al. 2007; Liska et al. 2018, 2019)
and others where no evidence has been found (eg. McKinney et al. 2013;
Sorathia et al. 2013; Krolik & Hawley 2015; Chatterjee et al. 2020). The main
reason of this discrepancy is probably the difference in the large scale
magnetic field accumulated in the central disk regions (namely the value of
the achieved disk magnetization $\mu$). Indeed, not only a vertical field
triggers an MRI-driven turbulence but it also launches jets from the disk. And
it is well known that both effects, turbulence and jets, produce extra torques
that may prevent the enforcement of solid-body precession by the Lense-
Thirring torque (Sorathia et al. 2013; McKinney et al. 2013). It is also
possible that high tilts ($65^{o}$) would be able to tear off the inner disk
up at a radius $r_{t}$ and enforce a solid-body precession (Liska et al.
2021). However, it would be a rather strong implication of the model if, in
order to reproduce LFQPOs, high tilts would always be required.

In any case, we would like to stress that providing the proof that a
significant portion of the disk, say from $\sim 5r_{g}$ to $r_{out}\sim 20$ or
even up to $100r_{g}$, is actually precessing as a solid-body is a formidable
numerical task. It would require for instance to show that the local
precession angle is the same for the whole range of radii, and that it evolves
linearly in time such that $P(t)-P(0)=\dot{P}t$, with $\dot{P}=2\pi\nu_{QPO}$
allowing to measure the QPO frequency (see e.g. Fig 2 in Dyda & Reynolds
2020). But this demands to maintain disk conditions at $r_{out}$ (i.e.
accretion rate) rather constant for a duration $\Delta T$ that must be at the
very least 3 or 4 times the QPO period, namely $\Delta T\sim
1-3\,(r_{out}/r_{g})^{3/2}10^{3}r_{g}/c$. For instance, a 0.1 Hz QPO would
require a converged simulation that would last for a few times $10^{5}r_{g}/c$
for a 10 solar masses BH. While such long times have been recently achieved in
a few simulations (e.g. Chatterjee et al. 2020; Liska et al. 2021), constant
disk conditions in the outer regions are still not met (to our knowledge).
This situation is due to the initial conditions, which do not include a steady
outer accretion disk. It is therefore very difficult to assess whether a
solid-body LT-precession is actually settled in and clearly deserves further
numerical simulations, long enough and tailored to maintain an outer cold thin
disk.

Notwithstanding theses current difficulties, one could ask if this scenario
could fit within our JED-SAD framework. It requires of course the black hole
spin to be misaligned with the disk angular momentum vector, and there is no
reason why it should not be (although large tilts are not expected to always
be the rule). As a result of the LT-torque, the innermost disk region (up to a
few $r_{g}$ according to GRMHD simulations) should always be aligned with the
BH spin, whether the disk is in a JED or in a SAD accretion mode (note however
that this alignment may be modified from the usual Bardeen-Petterson mechanism
by the presence of large scale magnetic fields Liska et al. 2019, 2021). But
what happens then beyond this innermost aligned region, when there is an inner
JED settled up to a large $r_{J}$?

According to McKinney et al. (2013) and references therein, one can answer
this question by comparing the LT-torque with the other local torques acting
on the disk. The local LT torque acting on a disk annulus can be estimated as
$\Gamma_{LT}=\Omega_{LT}L\sin\beta$, where $\Omega_{LT}$ is the LT-precession
pulsation, $L=\Sigma\Omega_{K}r^{2}$ is the disk angular momentum per unit
area and $\beta$ the black hole tilt angle (see e.g. McKinney et al. 2013 and
references therein). In a JED, the dominant torque is the magnetic braking
provided by the two jets and writes
$\Gamma_{jets}=-2r\frac{B_{\phi}^{+}B_{z}}{\mu_{o}}$, which leads to a ratio

$\frac{\Gamma_{LT}}{\Gamma_{jets}}=\sin\beta\frac{2j}{q\mu}\frac{r}{h}\left(\frac{r}{r_{g}}\right)^{-3/2}$
(2)

In this expression, $q=-B_{\phi}^{+}/B_{z}$ is the magnetic shear measured at
the disk surface, $\mu$ is the disk magnetization measured at the disk
midplane (note that it does not include turbulent fields) and $h/r$ the local
disk aspect ratio. Since a hot JED verifies $q\mu\sim 1$ and $h/r\sim 0.1$
(Marcel et al. 2018b), this expression shows that the LT torque should never
be dominant in a JED beyond a few $r_{g}$. Please note that this conclusion
stems only from our assumption of the existence of a near equipartition
laminar magnetic field.

This result can also be understood in other terms. In order for a solid-body
LT-precession to take place, the disk must behave as a whole entity. This
situation requires that a strong causal connection is maintained between its
two radial boundaries. Namely, that bending waves can propagate back and forth
and enforce the same precession rate over the entire volume. This ”wave-like”
regime has been estimated in hydrodynamical situations to require
$\alpha_{v}<h/r$, namely a turbulent Shakura-Sunyaev $\alpha_{v}$ parameter
smaller than the disk aspect ratio (Papaloizou & Lin 1995; Lubow et al. 2002).
Since MRI provides a scaling $\alpha_{v}\sim 10\mu^{1/2}$ (Salvesen et al.
2016), it shows that weakly magnetized ($\mu<10^{-4}$), hot ($h/r>0.2$) flows
could be in this regime and undergo indeed a solid body LT-precession. This
was the situation envisioned initially (with an ADAF as the hot inner flow)
and possibly achieved in some weakly magnetized GRMHD numerical simulations.
But a JED has a near equipartition field ($\mu\sim 1$) and accretes at
supersonic speeds due to its dominant jet torque. This challenges any upstream
wave propagation and certainly forbids thereby the establishment of a solid
body LT-precession over its entire volume 333Note that the JED thermal balance
calculations done by Marcel et al. (2018a) lead to a disk aspect ratio $h/r$
varying slowing with the radius (consistent with both the ion and electronic
temperature profiles). On the other hand and for simplicity, a constant
accretion Mach number (larger than unity) has been assumed in the JED,
consistent with self-similar calculations. In a more realistic situation one
would expect also a varying accretion Mach number. However, observations do
require a supersonic accretion flow throughout the hot accretion flow (Marcel
& Neilsen 2021; Kawamura et al. 2022), which means that the accretion speed
(hence the jet torque) must also adapt. Thus, although there is a caveat of
our calculations here, we believe it does not affect the main arguments
presented in this section..

How much of the JED volume, located beyond the innermost region aligned with
the BH, could be actually undergoing a solid-body precession remains an open
issue. We stress however that in order to recover the observational
correlation encapsulated in Eq.(1), that volume should reach a radius
$r_{t}\equiv r_{J}$, which is doubtful according to both the previous causal
argument and torque estimates. In the case of high tilts leading to disk
tearing at a radius $r_{t}$ (Liska et al. 2021), this radius should then also
coincide with the transition to an outer SAD accretion mode. Within our JED-
SAD framework, it is unclear to us why $r_{J}$, which marks the transition
from $\mu$ near unity in the JED to $\mu\sim 10^{-3}$ in the inner SAD, should
always be coincident with the tearing radius $r_{t}$.

Note finally that Marcel & Neilsen (2021) reached a similar conclusion based
only on observational constraints. Indeed, in order to reproduce the observed
X-ray spectra during the most luminous hard states, the hot flow must accrete
at sonic to supersonic speeds (see e.g. Marcel & Neilsen 2021; Kawamura et al.
2022), unreachable with typical viscous torques. Since Type-C QPOs are
prevalent in these luminous states, they concluded that solid-body LT-
precession is unlikely the driving mechanism.

The existence of a BH-aligned precessing inner JED region translates of course
into an inner jet precession and could therefore contribute to produce Type-A
and/or Type-B QPOs (Stevens & Uttley 2016; Liska et al. 2019, 2021; Kylafis et
al. 2020; Ma et al. 2021). And it could also provide the geometrical effect
that is invoked to explain the influence of the source inclination on both the
QPO amplitude and lags (e.g. Motta et al. 2015; Heil et al. 2015; van den
Eijnden et al. 2017). But according to the above discussion, we doubt that
solid body LT-precession could be the generic physical mechanism responsible
for Type-C QPOs in all XrBs and believe that another mechanism needs therefore
to be found (see also Nathan et al. 2022).

##  3 LFQPOs as the disk signature of a jet instability

Type-C LFQPOs are detected in the hard energy band which is associated to a
corona or hot inner flow (see however discussion in Rodriguez et al. 2004a,
2008), but their frequencies show a tight correlation with the inner standard
accretion disk radius, which is the JED-SAD transition radius $r_{J}$ in our
view (Eq.1). The difficulty is to reconcile this correlation with a factor
$\chi\sim 100$, which requires a secular process.

In the JED-SAD framework, two bipolar self-confined jets are magnetically
launched from the inner JED. Thus, instead of looking for a secular
instability within the disk itself, we propose that these QPOs are the disk
response to some instability triggered in the jets themselves, away from the
disk. This idea could naturally reconcile the Type-C QPOs’ low frequency
(long-term or large scales behavior) with their apparent link with the
transition radius $r_{J}$. There are several aspects that must be considered:
(i) How can a jet instability still impact the underlying disk ? (ii) Why
would it have an influence on the JED spectrum ? (iii) What kind of jet
instability would then be necessary ?

###  3.1 Causal connection with the underlying disk

Let us first assume that jets launched below $r_{J}$ are indeed prone to some
global instability. Since jets are clearly observed up to large scales, this
instability must not lead to jet disruption. We thus only require that its
non-linear stage leads to some local plasma and electric current re-
organisation, ending up mostly into jet wobbling. It is that wobbling that
defines a frequency, which is then expected to be conveyed backwards to the
disk through waves. If this instability is triggered in the causally connected
jet region, namely before the Fast-Magnetosonic (FM) surface, then FM waves
can indeed propagate upstream and reach the disk (on a time of the order of
the Keplerian orbital time scale at $r_{J}$, e.g. Ferreira & Casse 2004) so
that we expect them to lead only to some broad band noise. But much longer
time and/or spatial scales must be at play in order to trigger Type-C QPOs.

We are thus lead to assume that this jet wobbling occurs beyond the FM
surface. In that case, waves can no longer propagate upstream within the jet.
However, the JED-SAD framework requires the existence of a large scale
vertical field threading the whole accretion disk. Although the super-FM jet
is defined with the magnetic flux threading the JED, there is still magnetic
field around it, threading the SAD. Such a field defines a magnetic sheath
inside which the inner jet is propagating (see sketch in Fig.1). This outer
sheath being sub-FM (as for instance in the simulations of Murphy et al.
2010), waves can still propagate downwards and reach thereby the disk. The
path followed by these waves is not straightforward as the medium is
inhomogeneous and waves are known to undergo some refraction. However, we
expect modes to propagate preferentially along the sheath down to the
transition radius $r_{J}$ (much alike a surface mode). Moreover, once these
waves reach the sub-FM zone of the inner jet, they will start to act as a
lateral source localized at the limiting surface anchored at $r_{J}$. This
will lead, in turn, to perturbations able this time to propagate within the
jet and reaching the whole JED extension.

Our main assumption is therefore the existence of a jet instability triggered
at a distance $z_{I}$ leading to a subsequent jet wobbling. This lateral jet
displacement bounces back on the magnetic sheath, which triggers the
propagation of perturbations. How long it takes for these perturbations to
reach the disk is a difficult question, as it requires to follow the path of
these waves. Note however that once the instability is triggered, the
perturbations in the jet itself are advected downstream, leaving behind the
same physical conditions that have led to the triggering of the instability.
As a consequence, it will grow again near $z_{I}$ and lead to another unstable
regime with a global jet displacement. As long as the conditions for the jet
instability are met, one should thus see this going on, defining thereby a jet
wobbling frequency $\nu_{I}$ which is mostly related to the time scale
required for the lateral jet displacement. It is that frequency that will be
observed as a LF QPO in the disk.

Figure 1: Sketch of the JED-SAD hybrid disk configuration and its associated
outflows in the hard state. The Jet Emitting Disk is settled from the ISCO
$r_{isco}$ up to the transition radius $r_{J}$, beyond which a Standard
Accretion Disk is established. A vertical large scale magnetic field is
threading the whole region, up to the axis. The red zone is the relativistic
Blandford-Znajek spine, the blue zone is the sub to mildly relativistic
Blandford-Payne jet launched from the JED and the grey shaded area is the
outer magnetic sheath, with only unsteady sub-Alfvénic outflow (wind). The
optimal location for the jet instability is the altitude $z_{I}$ where the
outer magnetic surface anchored at $r_{J}$ is starting to recollimate toward
the axis, defining the jet radius $r_{jet}$.

###  3.2 Impact on the accretion flow: LF QPOs

The jet instability acts like a hammer hitting the sheath with a
characteristic frequency $\nu_{I}$. The information (wobbling frequency and
released energy) is transported backwards by FM waves travelling along the
magnetic sheath surrounding the jet near $r_{J}$. A fraction of this energy is
then expected to be spread all over the JED, through a lateral ”shower” of FM
modes triggered at the magnetic sheath near the disk. How is that supposed to
affect the emitted hard X-ray spectrum and lead to an observable LFQPO?

The fluctuations, typically 1-20% seen in hard X-rays, can be understood
through two independent processes, both related to these incoming waves. Since
they are FM waves, some vertical compression of the hot JED material is
naturally expected, leading to some enhanced dissipation (thus emission) at
the disk surface. The second aspect is related to the possible modifications
of the jet torque acting upon the disk. These waves are indeed expected to
introduce also fluctuations on the local toroidal magnetic field, hence on the
torque due to the jet. As a consequence, one should see fluctuations of the
JED accretion rate and on the related release of accretion energy, responding
with the same quasi-periodic frequency $\nu_{I}$.

However, to get a proper QPO with a rather large quality factor, one needs to
have an almost resonant cavity. As shown by Cabanac et al. (2010), the JED
itself may play this role, acting also as a low bandpass medium: its response
to a white noise excitation is a flat-top noise power spectral density at low
frequencies and a red noise at high frequency. In other words, if the jet acts
like a hammer, the bell would be the JED444This is somehow similar to the
Psaltis & Norman (2000) approach: the hot inner JED acts as a filtering cavity
of all incoming perturbations.. A QPO is therefore expected to arise at a
frequency $\nu_{I}$, the same at each energy bin and related to the dynamical
(epicyclic or keplerian) frequency at $r_{J}$ (see below). It is however
unclear how the existence of the outer magnetic sheath and intrinsic
fluctuations of the dissipation within the JED will affect the calculations
done by Cabanac et al. (2010). This should deserve further investigations.

###  3.3 Which instability could lead to jet wobbling ?

An MHD jet is a helical magnetic structure that can be seen as an ensemble of
poloidal magnetic surfaces, nested around each other and in pressure
equilibrium with the surrounding medium. In our JED-SAD framework, the jet is
mostly the super-FM outflow emitted by the underlying JED established between
the ISCO $r_{isco}$ and $r_{J}$ (blue region in Fig.1). Below $r_{isco}$,
magnetic field lines are brought in and concentrated over the plunging region
and black hole horizon (red region in Fig.1), leading to the production of a
relativistic tenuous spine through the Blandford-Znajek process (Blandford &
Znajek 1977). It is however unclear whether the existence of this spine has
any dynamical impact on the outer Blandford-Payne jet. Indeed, radio
observations can be accounted for using only the power carried by this outer
flow component (Marcel et al. 2018a, 2019, 2020). But, in any case, if some
jet instability is to give rise to a LFQPO related to $r_{J}$, it must involve
the distant thus non-relativistic jet region. We thus neglect below the effect
of the spine (see however Barnier et al. 2022).

Jets are prone to many instabilities such as pressure driven (PD), current
driven (CD) and Kelvin-Helmholtz (KH) instabilities (plus any combination of
them, Appl & Camenzind 1992; Baty & Keppens 2002, for a review see e.g. Hardee
2013). All these instabilities can either trigger radially localized, internal
and surface modes or long-wavelength body modes. The final non-linear outcome
of these instabilities can go from a simple internal redistribution of jet
quantities (leading to a non-linear jet stabilization) to the production of
internal shocks and/or MHD turbulence. In this latter case, a significant
energy dissipation may eventually end up to a complete disruption of the jet
on a finite distance. There is no consensus on which instability would be
dominant in a jet as their existence and relative strength (growth rate)
depend on the jet radial stratification, which is still poorly understood.

In this paper, we envision a non-destructive instability that gives birth to a
long-wavelength instability at an altitude $z_{I}$, namely a global body-mode
that involves lateral displacements of the whole outflow of radius $r_{jet}$.
The jet radius $r_{jet}$ is the radius achieved at $z_{I}$ of the magnetic
surface anchored at $r_{J}$. As discussed above, this displacement will bounce
at the jet/magnetic sheath interface and trigger upstreaming (toward the disk)
propagations of MHD disturbances. The jet displacement itself is advected
downstream the flow and become potentially detectable farther out as jet
wobbling. This allows another sequence (instability, jet displacement,
bouncing) to set in at $z_{I}$, so that the whole process will be quasi-
periodic with a characteristic frequency $\nu_{I}$. The altitude $z_{I}$ is
therefore the lowest distance from the source where such an instability can
take place. The frequency $\nu_{I}$ is related to the time scale involved for
allowing FM waves to travel across the jet radius at $z_{I}$, so that it can
behave as a whole (body mode). The frequency can thus be estimated as

$\nu_{I}\sim\left.\frac{V_{FM}}{2r}\right|_{r_{jet}}$ (3)

where $V_{FM}$ is the FM phase speed at the jet radius $r_{jet}$. Assuming
that the jet is cold and dominated by the toroidal magnetic field leads to

$V_{FM}\simeq
V_{A\phi}=\frac{B_{\phi}}{\sqrt{\mu_{o}\rho}}=\frac{m}{m^{2}-1}\left(1-\frac{r_{A}^{2}}{r^{2}}\right)\Omega_{*}r\simeq\frac{\Omega_{*}r}{m}$
(4)

evaluated at $r=r_{jet}\gg r_{J}$ (Pelletier & Pudritz 1992). In this
expression, $\Omega_{*}\simeq\Omega_{K}(r_{J})$ is the angular velocity of the
magnetic surface (MHD invariant), $r_{A}$ is the Alfvén radius where the flow
becomes super-Alfvénic and $m=u_{p}/V_{Ap}$ is the Alfvénic Mach number,
assumed to be much larger than unity.

In order for the characteristic frequency $\nu_{I}$ to be identical to the
LFQPO frequency $\nu_{C}$, the condition

$m\sim\pi\chi$ (5)

on the Alfvénic Mach number $m$ must be fulfilled, where we made use of
Eq.(1). This is possible as long as magnetically driven jets from accretion
disks are able to reach a value $m$ between $\sim 220$ and 400, for $\chi$
ranging from $\sim 70$ to 130 (Marcel et al. 2020). This is indeed the case,
as shown for instance in Fig. 8 in Ferreira (1997). It corresponds to tenuous
MHD solutions that become super-FM (namely $n=u_{p}/V_{FM}$ of a few) and that
expand the most, with $r_{jet}/r_{J}$ reaching values of a few hundreds up to
a thousand (see his Fig.6). Note that Eq.(5) is not a fine tuning but instead
a selection (and a property) of the underlying MHD solution established within
the JED. The fact that such MHD solutions have been already computed is quite
promising for our scenario.

Jets are highly inhomogeneous media, with both tremendous magnetic and
velocity gradients whose effects on the development of an instability are not
fully understood yet (see e.g. Bondeson et al. 1987; Birkinshaw 1991; Kersalé
et al. 2000; Frank et al. 2000; Appl 1996; Mizuno et al. 2009, 2011, 2014; Kim
et al. 2015, 2016, 2018). We note however that no instability is expected
below the Alfvén surface. Indeed, the stabilizing magnetic tension due to the
poloidal magnetic field is be too important. Besides, just after the Alfvén
point, MHD jets undergo a huge lateral expansion leading to
$|B_{\phi}|>>B_{p}$ and $m>>1$, which reduces drastically the growth rate of
any CD and KH modes, contrary to cylindrical jet configurations (see e.g.
Rosen & Hardee 2000; Moll et al. 2008; McKinney & Blandford 2009; Porth &
Komissarov 2015; Kim et al. 2016). We thus expect the triggering of CD or KH
instabilities only at an altitude $z_{I}$ where the jet radius has achieved
some limiting value (i.e. a quasi-cylindrical jet), determined by the
transverse equilibrium with its surrounding medium.

Our analyzis points to the capital role played by the magnetic surface
anchored at $r_{J}$. That surface corresponds to the last self-confined
magnetic surface and thus to a maximum of the electric current flowing inside
the jet. But since a sub-FM flow (probably even sub-Alfvénic) is expected to
be established in the outer magnetic sheath (the grey portion of the magnetic
flux anchored beyond $r_{J}$ in Fig.1), a velocity shear is also necessarily
present. Moreover, since the lateral equilibrium must be fulfilled, the outer
(gas plus magnetic) pressure must balance both the inner jet ram pressure and
magnetic deconfining force acting on the magnetic sheath. A density increase
is thus to be expected in this region. These two elements (inner high speed
jet surrounded by a denser and slower outflow) are very similar to those found
at the spine-jet interface and clearly observed in GRMHD simulations. In
accordance with our discussion, a CD kink instability leading to jet wobbling
and wiggles is indeed seen to develop (see e.g. McKinney & Blandford 2009;
Tchekhovskoy et al. 2011). Besides, the instability does not lead to jet
destruction, probably because of the existence of this velocity shear (Mizuno
et al. 2014; Kim et al. 2016, 2018).

We argued above that the optimal location for triggering a jet instability
leading to jet wobbling would be the altitude $z_{I}$ where the jet radius
would reach its maximal size. This is a situation expected in self-similar
Blandford-Payne jets and their generalization, as these type of jets always
undergo a magnetic recollimation toward the axis (Contopoulos & Lovelace 1994;
Ferreira 1997; Polko et al. 2010). Now, the existence of this intrinsic
recollimation may cause a pressure mismatch at the jet-sheath interface (the
white zone depicted in Fig.1), leading to a possible ”recollimation”
instability (Matsumoto & Masada 2013; Matsumoto et al. 2017). This new 3D
instability, first associated to a Rayleigh-Taylor instability, is actually a
variant of the centrifugal instability emerging along curved streamlines
(Gourgouliatos & Komissarov 2018a, b). Whether it could lead to jet
destruction and not only jet wobbling as assumed here remains to be fully
assessed. However, it has been recently shown that it could be quenched with
the presence of some external azimuthal magnetic field (Matsumoto et al.
2021).

As a final note, we remark that if we assume the recollimation point to be the
locus of the jet instability, then the non-relativistic jet solutions that
fulfill all constraints (super-FM, large radius and Eq.(5)) require a quite
tiny mass loading parameter along that surface, namely a disk ejection
efficiency $\xi$, defined as $\dot{M}_{acc}\propto r^{\xi}$ in the JED,
satisfying $5\,10^{-3}<\xi<10^{-2}$ (see Figs. 6 and 8 in Ferreira 1997).
These values are consistent with jet speed estimates done for Cygnus X-1
(Petrucci et al. 2010). Note that this range stems from exact mathematical
solutions but that more realistic ones should deviate from a strict self-
similarity. Indeed, although the JED outermost magnetic surface should fulfill
that constraint, the mass loading (or ejection index $\xi$) can increase
progressively toward the center.

##  4 Discussion

###  4.1 Numerical simulations

Our wobbling jet scenario is quite generic and should therefore be observed in
numerical jet simulations. However, as already pointed out, not only the
physical conditions envisioned here are quite demanding but they also require
full 3D simulations. Note that this jet wobbling should not be confused with
jet precession due to the black hole tilt and leading to a Bardeen-Petterson
like (probably modified by large scale magnetic fields) alignment of the
innermost disk regions (McKinney et al. 2013; Liska et al. 2019, 2021).
Although such precession may provide the correct physical ground for high
frequency (kHz) QPOs, it cannot explain LF QPOs discussed here (see also
Kylafis et al. 2020). Instead, we rely on a body mode instability in the
super-FM regime, involving the whole jet radius and leading to quasi-periodic
variations in the jet direction.

Although the stability of two-component magnetized jets are currently under
study (see e.g. Millas et al. 2017 and references therein), they usually focus
on the interplay between the inner relativistic spine and the outer slower and
denser flow (usually referred to as the sheath in these works). We also note
that with a much simpler jet profile, kink instabilities can indeed be
obtained, affecting the large scale jet morphology and giving rise to
interesting quasi-periodic radiative signatures (e.g. Tchekhovskoy & Bromberg
2016; Barniol Duran et al. 2017; Dong et al. 2020 see also the large scale 2D
simulations of Chatterjee et al. 2019 showing pinch instabilities). However,
we stress that a self-confined magnetized outflow must carry its own electric
poloidal current and must therefore be seen as made of three components
embedded in an external medium, not just two: the axial spine, the jet and the
outer sheath.

In our view, the role of the central spine in the whole outflow collimation
remains to be assessed (see for instance Barnier et al. 2022). The jet
corresponds here to the super-FM Blandford-Payne outflow launched from the
JED. The outer sheath is built up from the unavoidable magnetic flux threading
the SAD region, which is settled beyond the JED. Note that such a magnetized
sheath must be understood as being part of the whole outflow, since it must
carry the return electric current (at the jet-sheath interface) that insulates
the jet from the external medium (or cocoon). Understanding the propagation
and stability properties of such stratified outflows is of great importance
for accreting objects. But addressing both the large spatial and time scales
involved, while also accounting for the complex jet stratification imposed by
the underlying hybrid disk configuration, remains a numerical challenge.

###  4.2 Jet disruption and JED disappearance

Super-FM jets from JEDs are expected to start wobbling at some altitude
$z_{I}$, but would they be disrupted? This is a difficult question to answer
theoretically, as the non-linear saturation could be a simple readjustment of
the gradients and the smearing out of the instability. There are for instance
circumstances where a CD instability leads to the stabilisation of KH modes
(Baty & Keppens 2002). Based on the observation that Type-C QPOs are
maintained as long as there are steady compact jets (see however discussion in
Fender et al. 2009; Fender & Gallo 2014), we assumed in our simplified picture
that this instability is not leading to jet disruption. This requires that the
jet is able to propagate up to a distance $L=xr_{jet}$ with, say $x$ of a few
hundreds, on a time $T=L/u_{p,max}$ shorter than the characteristic growth
time of the instability $\tau=\nu_{I}^{-1}$ (e.g. Kim et al. 2016). In this
expression, $u_{p,max}$ is the maximum poloidal speed of the jet and is thus
associated to the ISCO radius $r_{isco}$ (for simplicity, we neglect here the
effect of the fast spine). As a consequence, the instability will lead to jet
destruction if $\tau<T$, which provides the condition
$u_{p,max}/u_{p}(r_{j})<x/2n$, where $n=u_{p}/V_{VM}$ is the FM Mach number
measured at the outer jet region. This crude estimate shows right away that as
$r_{J}\rightarrow r_{isco}$ it becomes harder to maintain stability: the
thinner the jet and the more fragile, prone to destruction, it gets. This is
qualitatively consistent for instance, with the fact that as the jet radius
shrinks, the turbulent boundary layer due to a KH surface mode becomes of the
order of the jet radius, leading thereby to its disruption (Baty & Keppens
2006; Kim et al. 2016). We thus expect a non-destructive jet wobbling whenever
$r_{J}\gg r_{isco}$.

On the contrary, as the JED shrinks (hard-to-soft transition), the resulting
jet would become too narrow to survive the instability. That would be
coincident with a dramatic jet restructuration until its complete disruption
and disappearance (assumed to occur at the ”jet line”). However, before the
complete jet disappearance, reconnecting events of the magnetic structure are
expected to lead to the dramatic release of jet energy, possibly associated
with discrete ejecta and subsequent radio flares (Homan et al. 2020; Wood et
al. 2021).

These evidences of a link between the appearance of Type-B QPOs and the launch
of discrete ejecta fits quite well within our jet instability scenario for
QPOs. As $r_{J}\rightarrow r_{isco}$, we expect the jet to become also
narrower due to the (relatively stronger) external magnetic pressure (Spruit
et al. 1997). This translates into a QPO frequency
$\nu_{I}\sim\Omega_{K}(r_{J})/(2m)$ (Eq. 3) that could change while the
transition radius $r_{J}$ remains almost constant. Indeed, a smaller opening
of the magnetic field lines leads to a smaller jet acceleration, hence to a
smaller Alfvénic Mach number $m$. As a consequence, the QPO frequency may
(slightly) increase while there is no detectable change in the inner cold disk
radius (Kara et al. 2019; Ma et al. 2021). This is an aspect of jet
collimation that has not been fully investigated yet.

###  4.3 Detectable signatures: IR QPOs, jet wobbling

Within the framework of wobbling jets, getting QPOs at the same frequency and
at energy bands usually associated with jet emission (ie OIR) appears rather
natural. Indeed, the jet instability is triggered at an altitude $z_{I}$ (the
recollimation zone) and gives rise to perturbations that propagate both up-
and downstream. It will take a time $\Delta t_{X}$ for the upstream waves to
travel down to the disk and give rise to an observable QPO in the X-ray band.
But since jet wobbling is quasi-periodically produced, it may well give rise
to a OIR QPO at a distance $z_{IR}>z_{I}$ after a time $\Delta
t_{IR}\sim(z_{IR}-z_{I})/u_{p}$, where $u_{p}$ is the flow speed (and possibly
to some variability signature in radio bands Tetarenko et al. 2019). There is
therefore no obvious reason why $\Delta t_{X}$ should always be shorter than
$\Delta t_{IR}$, so we do not expect any clear trend for the time lag between
X and OIR QPOs (see also Veledina et al. 2013, 2015).

This is consistent with OIR QPOs lagging behind X-ray QPOs with $\sim 0.1$ sec
in several XrBs (Gandhi et al. 2010; Kalamkar et al. 2016; Gandhi et al.
2017), but also with the non detection of any lag in MAXI J1535-571
(Vincentelli et al. 2021) or even X-rays possibly lagging behind the optical
QPO in MAXI J1820+070 (Paice et al. 2021). Moreover, there are evidences of
different properties between the two QPOs, like the rms-flux relation
(Vincentelli et al. 2018) or the time evolution of the power spectral density
(Vincentelli et al. 2019). Although a common origin for X and OIR QPOs is
quite natural within our framework, it seems plausible that some filtering
effect is occurring as perturbations are propagating downstream the jet (see
also Hynes et al. 2003).

Probing jet wobbling in BH XrB jets may be quite hard to achieve because of
the lack of resolution (see however Miller-Jones et al. 2019). In the context
of AGN, this requires instead a long term monitoring of jets. It is well known
that AGN jets display non-radial motions, indicative of accelerated, non-
ballistic motions (see e.g. Lister et al. 2016; Boccardi et al. 2017). But jet
wobbling has been inferred in various BL Lac sources implying time scales from
2 to 20 yr (e.g. Agudo et al. 2012; Arshakian et al. 2020 and references
therein). Walker et al. (2018) have also found evidences of such a pattern in
the radio galaxy M87, with a period of $\sim 9$ yr. When scaled down to a 10
solar mass BH, this value provides a frequency around 2 Hz, which is
consistent with the range of LF QPOs in BH XrBs.

In the realm of Young Stellar Objects, the situation is quite similar.
Protostellar jets seen in the optical are thought to arise from the innermost
disk regions, where orbital periods are ranging from few days to a year or so
(Ferreira et al. 2006a; Ray et al. 2007 and references therein). On the other
hand, these jets harbor knots that resemble bow shocks whose separations are
indicative of time scales ranging from years to tens of years (e.g.López-
Martín et al. 2003; Agra-Amboage et al. 2011; Ellerbroek et al. 2014; Lee et
al. 2017; Tabone et al. 2017). Such time scales are too short to be explained
by perturbations due to a companion star, and too long to be related to any
dynamical time scale at the launching radius (see for instance jet wobbling in
HH30, Louvet et al. 2018 and references therein). Instead of relating these
knots to time variable ejection events from the source, we propose to relate
them to the same jet instability. Such instability could be very similar to
the one invoked here to explain LF QPOs around compact objects, although
leading possibly to jet disruption with a subsequent ballistic motion in some
sources.

##  5 Conclusion

Building upon the success of the JED-SAD framework to reproduce the radio
emission and X-rays spectral energy distributions during various cycles of the
archetypal source GX 339-4, we addressed in this paper the question of the
origin of Type-C LFQPOs.

We first critically analyzed the most common models invoked for such QPOs: 1)
models looking for a specific process triggered at the transition radius
$r_{J}$, 2) the accretion-ejection instability and 3) the solid-body Lense-
Thirring disk precession model. We showed that the first two types of models
are facing major theoretical issues, unsolved yet, and that the published
versions of these models do not account for the observed tight correlation
with the disk transition radius (Eq. 1). We also argued that these models
could not be operating within the JED-SAD framework.

We then discussed the case of the solid-body LT-precession model, which is
invoked whenever the black hole spin is misaligned with the disk angular
momentum vector. We argued that evidences in GRMHD simulations for such a
solid-body behavior were not fully convincing, especially in the case of
highly magnetized accretion flows. This casts thereby doubts on the capability
of this model to operate within the JED-SAD framework (at least for moderate
black hole tilts). As suggested however by current numerical simulations, the
innermost JED region, up to a few $r_{g}$, would probably be aligned with the
black hole spin, leading to an inner jet precession (namely a precession of
both the Blandford-Znajek axial spine and the innermost zones of the
surrounding Blandford-Payne jet). But whether some fraction of the JED,
located beyond the aligned region, would undergo a solid-body LT-precession
remains an open issue that deserves further numerical simulations, long enough
and tailored to maintain an outer cold thin disk. We argued however that such
an effect would hardly be efficient up to the transition radius $r_{J}$ with
the outer SAD, as required by observations.

We propose instead that the JED-SAD framework offers the conditions allowing
for a jet wobbling, namely a non-destructive long wavelength 3D body mode
probably triggered by a kink or recollimation instability in the super-FM jet
regime. Such a wobbling, triggered away from the disk, will nevertheless
affect the disk through FM waves travelling upstream along the surrounding
magnetic sheath, as well as, of course, downstream the jet. This outer
magnetic sheath is the key ingredient allowing to connect the fast inner jet,
which is hammering quasi-periodically against the sheath, to the underlying
resonant disk (JED). This scenario offers thereby a unique explanation to the
existence of QPOs seen both in disk (X-rays) and jet (UV and OIR) emission
signatures.

The theoretical foundations of this scenario are threefold: i) existence of
exact calculations of MHD accretion-ejection flows with the correct required
properties; ii) self-consistent thermal disk balance calculations, with
thorough confrontations to observations; iii) current knowledge on MHD
instabilities in stratified super-FM flows. Notwithstanding these facts and
hints, the proposed dynamical mechanism as well as its radiative consequences
remain to be firmly established. To do so, global 3D jet simulations must be
designed so that such an instability could be indeed observed, with its up-
streaming perturbations heading toward the disk. On the radiative side, the
work of Cabanac et al. (2010) should be extended in order to incorporate the
existence of these incoming perturbations, and possibly address also the
question of the time lags and their energy dependence.

We finally note that polarization measurements by IXPE, just launched in Dec.
2021 or, if selected, and in a more distant future, by eXTP, may bring great
enlightenment on LFQPOs. Indeed, if swings in polarization angle are detected,
this will provide stringent conditions that all the models and scenarios
discussed above will have to explain. This is provided of course that such
constraints are usable, as required integration times are much bigger than the
periods to probe (but see for instance Ingram & Maccarone 2017).

###### Acknowledgements.

JF would like to thank Adam Ingram and Matthew Liska, as well as the referee,
Chris Done, for interesting discussions that allowed to improve the quality of
the paper. We acknowledge financial support from the CNES French space agency
and PNHE program of French CNRS.

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