Magnetic Flux Plays an Important Role during a Black Hole X-Ray Binary Outburst in Radiative Two-temperature General Relativistic Magnetohydrodynamic Simulations

Black hole (Bh) X-ray binaries cycle through different spectral states of accretion over the course of months to years. Although persistent changes in the Bh mass accretion rate are generally recognized as the most important component of state transitions, it is becoming increasingly evident that magnetic fields play a similarly important role. In this article, we present the first radiative two-temperature general relativistic magnetohydrodynamics simulations in which an accretion disk transitions from a quiescent state at an accretion rate of Ṁ∼10−10ṀEdd to a hard-intermediate state at an accretion rate of Ṁ∼10−2ṀEdd . This huge parameter space in mass accretion rate is bridged by artificially rescaling the gas density scale of the simulations. We present two jetted BH models with varying degrees of magnetic flux saturation. We demonstrate that in “standard and normal evolution” models, which are unsaturated with magnetic flux, the hot torus collapses into a thin and cold accretion disk when Ṁ≳5×10−3ṀEdd . On the other hand, in “magnetically arrested disk” models, which are fully saturated with vertical magnetic flux, the plasma remains mostly hot with substructures that condense into cold clumps of gas when Ṁ≳1×10−2ṀEdd . This suggests that the spectral signatures observed during state transitions are closely tied to the level of magnetic flux saturation.


Introduction
Black hole X-ray binaries (BhXRBs) are observed in various spectral states of accretion (e.g., Esin et al. 1997;Fender et al. 2004;Remillard & McClintock 2006;Belloni 2010;Belloni & Motta 2016).Most of the time, BhXRBs are quiescent, with low luminosities and a hard X-ray spectrum.However, according to our current understanding, thermal-viscous instabilities (Lightman & Eardley 1974;Scepi et al. 1976) in the outer accretion disk drive periodic outbursts during which the gas supply falling into the black hole increases by several orders of magnitude (e.g., Lasota 2001).When BhXRBs go into outburst, their luminosity increases by several orders of magnitude before, and their spectrum changes shape.Similar cycles might exist in active galactic nuclei (AGN; e.g., Noda & Done 2018) albeit they occur on much longer timescales and thus cannot be observed.It is also well known that the (quasiperiodic) variability and jet production are tightly linked to the spectral state of a black hole (e.g., Ingram & Motta 2019).However, due to the complexity of the physics involved, such as radiation-matter interaction and thermal decoupling between ions and electrons, our ability to model outbursts is extremely limited.
Spectral states are classified by two parameters (e.g., Kalemci et al. 2022).The first parameter is the luminosity with respect to the Eddington limit (L Edd = 4πGM Bh c/κ es ∼ 1.3 × 10 38 M Bh / M e erg s −1 , where κ es is the opacity due to electron scattering), which represents the point at which radiation pressure overcomes gravity.The second parameter is the hardness of the spectrum, which is typically defined as the luminosity ratio between the hard and soft X-ray bands and quantifies how much the spectrum deviates from a thermal blackbody.During an outburst, the BhXRB first transitions from the quiescent state to the hardintermediate state, during which the luminosity rises rapidly and the spectrum remains dominated by hard Comptonized emission.Then, the source transitions to the high-soft state characterized by a soft spectrum and the absence of a jet.After spending (typically) weeks to months in the high-soft state, the source transitions back through several (jetted) intermediate states to the quiescent state.This transition from the high-soft state to the quiescent state occurs at a much lower luminosity than the other way around for reasons that are not well understood (e.g., Begelman & Armitage 2014;Scepi et al. 2021).
Most general relativistic magnetohydrodynamic (GRMHD) simulations to date address accretion in the quiescent state.While BhXRBs indeed spend most of their time in the quiescent state, they accrete most of their gas (and hence grow most rapidly) in the hard-intermediate and high-soft states (e.g., Fabian 2012).However, simulating accretion disks in these luminous states is numerically challenging due to the presence of dynamically important radiation fields and thermal decoupling between ions and electrons.Presently, only a handful of GRMHD codes are able to model radiation (e.g., McKinney et al. 2013;Saḑowski et al. 2013;Fragile et al. 2014;Ryan et al. 2017;White et al. 2023).In addition, since radiative cooling makes such accretion disks thinner, one needs a much higher resolution to resolve them.For example, to resolve a disk that is 2 times thinner on a spherical grid without static or adaptive mesh refinement requires a factor 32 more computational time.These factors make such simulations extremely expensive and complex.Due to recent algorithmic and computational advances, radiative GRMHD simulations of accretion disks accreting above a few percent of the Eddington limit (i.e., very thin disks) came within the realm of possibility (e.g., Ohsuga & Mineshige 2011;Mishra et al. 2016Mishra et al. , 2020Mishra et al. , 2022;;Fragile et al. 2018;Morales Teixeira et al. 2018;Lančová et al. 2019;Liska et al. 2022bLiska et al. , 2023)).These recent advances supplement earlier work that attempted to tackle the physics driving accretion in the luminous states using an ad hoc cooling function in place of first-principles radiation (e.g., Noble et al. 2009;Avara et al. 2016;Hogg & Reynolds 2017, 2018;Bollimpalli et al. 2023;Nemmen et al. 2023;Scepi et al. 2024).
Recently, radiative two-temperature (2T) GRMHD simulations of accretion disks accreting at L ∼ 0.35 L Edd demonstrated that in systems where no vertical magnetic flux is present, a thin and cold accretion disk forms, possibly explaining the high-soft state (Liska et al. 2022b).However, in the presence of dynamically important large-scale vertical magnetic flux that saturates the disk, the accretion disk enters the magnetically arrested disk state (e.g., "MAD," Narayan et al. 2003;Tchekhovskoy et al. 2011;McKinney et al. 2012).In MADs, magnetic flux impedes smooth inflow of gas, and instead, gas accretion proceeds much more chaotically (e.g., Begelman et al. 2022).Moreover, the typically cold and slender accretion disk undergoes truncation, transforming into a two-phase coronal plasma characterized by the coexistence of cold gas clumps enveloped by a hotter and sparser gas environment within a (truncation) radius of approximately r  20 r g (Liska et al. 2022b).These magnetically truncated disks provide a promising model for the luminous-hard state.First, the decoupling of the plasma into a two-phase medium might explain why hard X-Ray emission dominates over the thermal blackbody emission from the disk.Second, the presence of cold plasma down to the innermost stable circular orbit (ISCO) provides an interesting explanation for the observed relativisticbroadened iron-reflection lines in the luminous-hard state (e.g., Reis et al. 2010), which before this work was thought to only feature hot gas unable to produce such lines.In between the zero-magnetic flux and MAD regime, where vertical magnetic flux is present but does not saturate the disk, Lančová et al. (2019) demonstrated that a hot plasma with both inflowing and outflowing components sandwiches a cold and thin accretion disk.Such puffy disk models can potentially describe BhXRBs in some intermediate spectral states (e.g., between the luminous-hard and high-soft states), which launch relativistic jets but show no clear evidence of significant disk truncation (e.g., Kara et al. 2019).
However, none of the previously discussed work addresses at which accretion rate a hot quiescent-state torus transitions to a thin (truncated) hard-intermediate state disk and what role magnetic fields play in that process.In this article we present the first radiative 2T GRMHD simulations spanning 8 orders of magnitude in mass accretion rate.These simulations demonstrate a transition from a hot torus in the quiescent state to either a magnetically truncated (e.g., Liska et al. 2022b) or puffy accretion disk (e.g., Lančová et al. 2019) in the (hard-) intermediate state, depending on the amount of magnetic flux saturation.In Section 3 we describe our radiative 2T GRMHD code and numerical setup before presenting our results in Section 4 and concluding in Section 5.

Numerical Setup
To model the rise from quiescence to the hard-intermediate state, we use the GPU-accelerated GRMHD code H-AMR (Liska et al. 2018(Liska et al. , 2022a)).H-AMR evolves the radiative 2T GRMHD equations (e.g., Saḑowski et al. 2013;Saḑowski et al. 2017) on a spherical grid in Kerr-Schild coordinates.Similar to Liska et al. (2022b), we model the radiation field as a second fluid using the M1-closure approximation and, in addition, also evolve the photon number density to get a better estimate for the radiation temperature (Saḑowski & Narayan 2015).Radiative processes such as bremsstrahlung, synchrotron, bound-free, iron line emission, and scattering (including Comptonization) are included, assuming an M Bh = 10 M e black hole surrounded by plasma with solar abundances (X = 0.70, Y = 0.28, Z = 0.02).The associated energy-averaged gray opacities are provided in McKinney et al. (2017, Equations (C16), (D7), and (E1)).
At each time step, the dissipation rate is calculated by subtracting the internal energy provided by the entropy equation from the internal energy provided by the energy equation (e.g., Ressler et al. 2015).Subsequently, the total energy dissipation is divided as a source term between the electron and ions based on a reconnection heating model (Rowan et al. 2017).This deposits a fraction δ e  0.5 of the dissipation into the electrons, which varies between δ e ∼ 0.2 in less magnetized regions to δ e ∼ 0.5 in highly magnetized regions.Coulomb collisions (Stepney 1983) are taken into account through an implicit source term (e.g., Saḑowski et al. 2017).To avoid the jet funnel becoming devoid of gas and to keep the GRMHD scheme stable, we floor the density in the drift frame of the jet (Ressler et al. 2015) such that the ratio of the density and magnetic pressure r  12.5 p c b .We use a spherical grid in the Kerr-Schild foliation with coordinates = ( ) x r log 1 , x 2 = θ, and x 3 = j with a resolution of N r × N θ × N j = 420 × 192 × 192 for our SANE model and N r × N θ × N j = 560 × 192 × 192 for our MAD model.This adequately resolves the fastest growing magnetorotational instability (MRI) wavelength by 16 cells in all three dimensions.We place the outer boundary at R out = 10 3 r g for our SANE model and at R out = 10 4 r g for our MAD model.We also maintain at least five cells within the event horizon such that the inner boundary is causally disconnected from the rest of the computational domain.We speed up the simulations approximately threefold by introducing four levels of local adaptive time stepping (Liska et al. 2022a).To prevent cell squeezing around the polar axis from slowing down our simulations (e.g., Courant et al. 1928), we use four levels of static mesh derefinement (Liska et al. 2018(Liska et al. , 2022a) ) to reduce the j resolution to N j = [96,48,24,12] within θ  [30°, 15°, 7°.5, 3°.75] from each pole.This maintains a cell aspect ratio of roughly |Δr|: |Δθ|: |Δj| ∼ 1: 1: 2 throughout the grid, which is sufficient to capture the three-dimensional nature of the turbulence.

Physical Setup
To understand the effects of magnetic flux saturation on the transition from the quiescent to the hard-intermediate state, we include two models in the "standard and normal evolution" (SANE; Narayan & Yi 1994) and MAD (Narayan et al. 2003) regimes.We assume a rapidly spinning black hole with spin parameter a = 0.9375.Our SANE model (XRB SANE) features a standard Fishbone and Moncrief torus (Fishbone & Moncrief 1976) with inner radius r in = 6 r g , radius of maximum pressure at = r r 12 g max , and outer radius r out ∼ 50r g .Our MAD model (XRB MAD), on the other hand, features a much larger torus with r in = 20 r g and = r r 41 g max whose outer edge lies at r out ∼ 800 r g .These tori are pretty standard choices in the GRMHD community (e.g., Porth et al. 2019;Chatterjee et al. 2023).We thread the SANE model with magnetic vector potential, and the MAD model with magnetic vector potential, where ρ is the gas density.In both cases this produces a poloidal magnetic flux loop that is approximately the size of the torus.Since the MAD torus is much larger than the SANE torus, only the MAD torus contains sufficient magnetic flux to enter the MAD state.There is no toroidal magnetic flux present in the initial conditions.In both cases, we normalize the resulting magnetic field such that b where p gas max and p b max are the maximum gas and magnetic pressure in the torus.For the purpose of calculating the initial torus solution we set the adiabatic index γ = 5/3 for our SANE model and γ = 13/9 for our MAD model.We subsequently distribute, according to our heating prescription involving a magnetic reconnection model (Rowan et al. 2017), the total pressure between the ions and electrons before we selfconsistently evolve their entropy and adiabatic indices (e.g., Saḑowski et al. 2017).
To make GRMHD simulations of BhXRB outbursts feasible, we artificially shorten the relevant timescales by introducing a rescaling method that, as a function of time, sets the accretion rate to a predetermined value.However, before we apply this method, we first run the simulation for t = 10 4 r g /c in 2T nonradiative GRMHD to get the accretion disk into a quasisteady state.We subsequently restart the simulation in radiative 2T GRMHD and renormalize the density (ρ) every time step (Δt ∼ 10 −2 r g /c) with a factor ζ such that the running average of the mass accretion rate over Δt = 10 3 r g /c at r = 5r g , is scaled to a time-dependent "target" mass accretion rate, Target 10 Edd t 10 4 10 4 via the rescaling factor, 2 is the Eddington accretion rate and η NT = 0.178 is the analytically derived radiative efficiency for a radiatively efficient thin accretion disk (Novikov & Thorne 1973).In addition to the density (ρ), we also rescale the internal energy density (u g ), radiation energy density (E rad ), and magnetic energy density (b 2 ) with the same prefactor ζ.This leads to a doubling of the black hole mass accretion rate every t = 10 4 r g /c.Note that this approach automatically increases the total amount magnetic flux at the event horizon r Bh , Bh Bh by a factor z , such that the normalized magnetic flux, remains constant.When f ∼ 40-50 we expect that the disk turns MAD and flux bundles get ejected from the black hole (e.g., Tchekhovskoy et al. 2011;McKinney et al. 2012).We achieve inflow-outflow equilibrium over the mass accretionrate doubling time (Δt = 10 4 r g /c) up to approximately r ∼ 15r g in our SANE model and r ∼ 30 r g in our MAD model.This is the radius within which our models are expected to converge to their steady accretion-rate analogs with a constant ζ.
Note that constant-ζ GRMHD simulations typically exhibit a factor ∼5 variability in the dimensionless accretion rate during the first t ∼ 10,000 r g /c when seeded from a torus that has not yet fully developed MRI turbulence (e.g., Porth et al. 2019).Thus, to span the given range in  M, we argue that rescaling  M in a single ultralong duration GRMHD model is superior to running a suite of ∼25-30 GRMHD models with a constant ζ that each start from an equilibrium torus and only last for a duration of Δt = 10,000 r g /c.Namely, the variability of  M (Figure 1) during each Δt = 10,000 r g /c time interval (after t = 10,000 r g /c) is at most a factor 3, which is less than the variability during the first Δt = 10,000 r g /c.In other words, we argue that using a well-converged turbulent accretion state at, for example, an accretion rate Edd as the initial condition for a GRMHD model with an accretion rate Edd is preferred over restarting the entire simulation from scratch.

Results
We evolve both models for t ∼ 260,000-280,000 r g /c, during which the targeted mass accretion rate increases by 8 orders of magnitude.As illustrated in Figure 1(a), the black hole mass accretion rate closely follows the targeted mass accretion rate for both models.However, as illustrated in Figure 1(b), while the normalized magnetic flux threading the Bh event horizon stays constant in our MAD model, it increases by a factor of ∼3 in our SANE model.This is because a significant fraction of the initial gas reservoir accretes or gets ejected in the form of winds, leading to a relative larger increase in the dimensionless magnetic flux compared to the mass accretion rate.The rapid variability of the magnetic flux observed in our MAD model is a well-known characteristic of MADs (e.g., Tchekhovskoy et al. 2011;McKinney et al. 2012) caused by flux bundles being ejected from the black hole event horizon through magnetic reconnection (e.g., Ripperda et al. 2022).
The contour plots of density and electron temperature in Figure 2 illustrate three different stages of the artificially induced state transition.An accompanying animation also illustrating the ion temperature is included in the supplementary materials and on our YouTube playlist. 6In the first stage Edd ), the radiative efficiency is low, and radiative cooling plays a negligible role.The ions in the plasma are significantly hotter than the electrons because our heating prescription typically injects only a fraction δ e ∼ 0.2-0.4 of the dissipative heating into the electrons (and a fraction δ i ∼ 0.6-0.8into the ions).In the second stage Edd ), radiative cooling of the electrons becomes efficient, leading to a drop in electron temperature but no other structural change (see also Chatterjee et al. 2023).In the third stage Edd ), Coulomb collisions become efficient.This allows the ions to cool by transferring their energy to the radiation-emitting electrons and eventually, leads to a rapid collapse of the hot torus.
In our SANE model, this collapse results in a geometrically thin accretion disk surrounded by hot magnetic-pressure supported gas outside of r  3 r g .Thus, the disk is only truncated very close to the black hole.The production of hot nonthermal electrons within the ISCO was predicted by Hankla et al. (2022).Interestingly, this hot coronal gas rather than the thin accretion disk seems to be responsible for the majority of  M (Figure 3).This is similar to the puffy accretion disks first presented in other radiative GRMHD simulations (Lančová et al. 2019) and in pure MHD simulations of weakly to moderately magnetized disks (Jacquemin-Ide et al. 2021).On the other hand, in our MAD model, the hot torus transitions into a two-phase medium with cold optically thick patches of gas surrounded by hot, optically thin plasma.These cold patches of gas are visible for 20 r g  r  100 r g and do not reach the event horizon.Since this work was performed at a rather low resolution, we were forced to terminate both our SANE and MAD models around Edd because the scale height of the disk dropped below h/r  0.1 and the cold plasma became underresolved.Nevertheless, this work addresses at which accretion rate the hot torus transitions into a (truncated) cold accretion disk.Future work featuring at least an-order-of-magnitude higher effective resolution will be necessary to understand how the disk further evolves as we keep increasing the mass accretion rate.
These findings diverge from the magnetically truncated accretion disk models detailed in Liska et al. (2022b), where a slender disk was truncated at r ∼ 20 r g and cold patches of gas reached the event horizon.The absence of this thin disk in our simulations may be attributed to the considerably higher saturation of magnetic flux within our torus, distinguishing it from the disk in Liska et al. (2022b).This discrepancy could feasibly result in a significantly larger truncation radius.Consequently, if the truncation radius in our MAD model lies much farther out, it is plausible that our simulation's duration is insufficient to capture the formation of a thin accretion disk.A larger truncation radius (assuming the cold clumps of gas were sufficiently resolved, which might not be true) might consequently also explain why no cold plasma reaches the event horizon.Namely, as proposed in Liska et al. (2022b), magnetic reconnection can potentially evaporate the cold clumps of gas before they reach the event horizon.This is less likely to happen if the magnetic truncation radius moves closer in, and hence, the cold clumps have less time to evaporate.
In Figure 4  R being the radiation stress-energy tensor.This collapse manifests itself as a rapid decrease in the density scale height of the disk, with 〈〈x〉〉 ρ denoting a density-weighted average.On the other hand, in our MAD model, the radiative efficiency asymptotes to η rad ∼ 1.2η NT .This has been observed in other radiative MADs (e.g., Curd & Narayan 2023) and could, pending future analysis, potentially be explained by the presence of a dynamically important magnetic field that injects energy into the accreting gas, which is not accounted for in Novikov & Thorne (1973).In addition, there is only a marginal factor ∼2 decrease in the disk scale height after the formation of cold plasma because magnetic pressure is able to stabilize the accretion disk against runaway thermal collapse (e.g., Saḑowski 2016;Jiang et al. 2019).To better understand the outflows, we subdivide the plasma into jet and wind components at the b 2 /(ρc 2 ) = 1 boundary.Using this definition, we can define the wind-and jet-driven outflow efficiencies in terms of total energy accretion rate  E and energy accretion rate in the jet  E jet : While both the wind and jet efficiency remain constant within a factor unity in our MAD model, the increase of the normalized magnetic flux in our SANE model causes the jet to become significantly more efficient over time (e.g., η jet ∝ f 2 ).
To better understand when, during an outburst, certain physical processes become important, we plot in Figures 5(a Edd .Meanwhile the plasma transitions in Figures 5(e), (f) from a quasirelativistic adiabatic index γ ∼ 1.5 to a nonrelativistic γ ∼ 5/3.Even at accretion rates that are typically associated with radiatively inefficient accretion Edd ), future work will need to test if electron cooling (see also Dibi et al. 2012;Yoon et al. 2020) and/or a self-consistent adiabatic index can change the spectral signatures compared to equivalent nonradiative single-temperature GRMHD models (e.g., Mościbrodzka et al. 2014).

Discussion and Conclusion
In this article we addressed for the first time the transition from the quiescent to the hard-intermediate state using radiative 2T GRMHD simulations.By rescaling the black hole mass accretion rate across 8 orders of magnitude, these simulations demonstrated that radiative cooling and Coulomb coupling become increasingly important and eventually lead to a transition to a two-phase medium.While the hot torus in SANE models transitions to a thin accretion disk surrounded by a sandwich-like corona above ´- Edd , reminiscent of a puffy (Lančová et al. 2019) or magnetically elevated (Begelman & Silk 2017) disk, the MAD torus transitions to a two-phase medium with cold clumps of gas embedded in a hot corona (see also Bambic et al. 2024 Edd .The formation of a two-phase plasma consisting of cold clumps surrounded by hot gas was already demonstrated in our previous work (Liska et al. 2019(Liska et al. , 2022b)).However, while in Liska et al. (2022b) a cold accretion disk was present beyond r  20r g , no cold accretion disk forms in our MAD models.The absence of a truncated disk could potentially be explained by the amount and location of the excess magnetic flux that does not thread the event horizon.Namely, while in Liska et al. (2022b) the magnetic flux peaked around r t ∼ 20-25r g , which is coincident with the magnetic truncation radius, the magnetic flux in this work does not peak until r t ∼ 40r g .This suggests that the torus is MAD much farther out, and thus the magnetic truncation radius is larger than in Liska et al. (2022b).Since the timescale for the disks to come into inflow equilibrium at r t ∼ 40r g exceeds t  20,000r g /c, it is conceivable that a truncated disk did not have sufficient time to form in our MAD models.Future work will need to address how the magnetic truncation radius evolves as the disk gets more saturated with vertical magnetic flux.
We expect that the MAD and SANE models will exhibit markedly distinct spectral and temporal variability characteristics.For instance, MAD models are projected to give rise to truncated accretion disks, which may correspond to highluminosity spectral states displaying evidence of disk truncation.Even though cold clumps of gas can break off the inner disk and reach the event horizon as demonstrated in Liska et al. (2022b), the thermal emission will be suppressed due to the much smaller coverage fraction in magnetically truncated disks.Consequently, we anticipate the emission to be predominantly governed by the hot coronal plasma.Conversely, in SANE models, we anticipate a considerably more pronounced thermal spectrum emanating from the accretion disk as it extends down to the ISCO.The magnitude of hard X-ray emission will hinge upon optical depth of the coronal plasma.As suggested by Liska et al. (2022b), it is improbable for accretion disks devoid of vertical magnetic flux to generate adequate hot plasma for producing hard X-ray emission.Such models with a net zero magnetic flux are likely pertinent to the high-soft state.Nevertheless, in SANE models featuring some vertical magnetic flux, puffy accretion disks can form (Lančová et al. 2019) where hot plasma sandwiches the accretion disk and (potentially) hardens the (otherwise) soft disk spectrum.
The goal of this article was to study the transition from the quiescent state to the hard-intermediate state, both of which feature radio jets.Thus, we have not considered models that do not produce any jets, such as models with a purely toroidal field (e.g., Liska et al. 2022b), and instead only considered a jetted However, the jet power is set by the total amount of magnetic flux threading the black hole (P jet ∝ Φ 2 ), and thus an SANE jet at a much higher accretion rate can easily outperform an MAD jet at a lower accretion rate.Thus, an interesting possibility to be explored in future work would include a model where the magnetic flux does not increase proportional to F µ  M but is truncated at a maximum value z F µ F F ( ) min , 0 max .This would cause the disk to transition from an MAD disk in the quiescent state to an SANE disk in the hard-intermediate state where, at least initially, the magnetic pressure is still dynamically important (e.g., Begelman & Silk 2017;Dexter & Begelman 2019;Lančová et al. 2019).In upcoming work, we will employ ray-tracing calculations to compare both our existing models and future models featuring a truncated magnetic flux against multiwavelength observations, which offer constraints on the truncation radius and the size/geometry of coronal structures in actual astrophysical systems (e.g., Ingram & Done 2011;Fabian et al. 2014;Plant et al. 2014;García et al. 2015;Kara et al. 2019).
There are several theoretical and observational arguments that support this "truncated flux" scenario.First, for systems to remain MAD during a 2-4 order of magnitude increase in  M, they would need to advect 1-2 orders of magnitude of additional magnetic flux onto the Bh (e.g., F µ  M in an MAD).Especially when the outer disk becomes geometrically thin, it is unclear if this is physically possible since theoretical arguments suggest thin disks might not be able to advect magnetic flux loops (e.g., Lubow et al. 1994).Second, observations suggest that the disk truncation radius in the hard-intermediate state appears (e.g., Reis et al. 2010;Kara et al. 2019) to be rather small (r t  5r g ).This is inconsistent with recent GRMHD simulations, which demonstrated that even when the disk only contained a factor ∼1.5 of excess magnetic flux (above the MAD limit); this led to a truncation radius r t ∼ 20r g (e.g., Liska et al. 2019Liska et al. , 2022b)).Third, lowfrequency quasiperiodic oscillations, which are ubiquitous in Interestingly, the MAD model maintains a significantly higher radiative efficiency, presumably due to more efficient synchrotron emission.
the hard-intermediate state (e.g., Ingram & Motta 2019), are most likely seeded by a precessing disk, which tears off from a larger nonprecessing disk (e.g., Stella & Vietri 1998;Ingram et al. 2009Ingram et al. , 2016;;Musoke et al. 2023).This has been observed in radiative and nonradiative GRMHD simulations where a tilted thin accretion disk is threaded by a purely toroidal magnetic field (e.g., Liska et al. 2022aLiska et al. , 2023;;Musoke et al. 2023) and in similar GRMHD simulations where the accretion disk is threaded by a below-saturation-level vertical magnetic field (e.g., Liska et al. 2021).However, there are no numerical simulations that have shown any disk tearing or precession where the disk is saturated by vertical magnetic flux (e.g., Fragile et al. 2023).The main problem is that for a disk to tear (and precess), the warping of spacetime needs to substantially exceed the viscous torques holding the disk together (e.g., Nixon & King 2012;Nealon et al. 2015;Doǧan et al. 2018;Doğan & Nixon 2020;Raj et al. 2021).However, the viscous torques stemming from equipartition-strength magnetic fields within the truncation radius might be too strong for a disk to tear.
While our simulations incorporate the effects of radiation and thermal decoupling between ions and electrons, they still rely on a rather simplistic heating prescription for electrons extracted from particle-in-cell models (Rowan et al. 2017).
Since, absent any Coulomb collisions, the cooling rate in a given magnetic field will be determined by the temperature and density of the radiation-emitting electrons, the radiative efficiency at lower accretion rates can become sensitive to the used heating prescription (e.g., Chael et al. 2018).For example, in our models roughly a fraction δ e ∼ 20%-40% of the dissipation ends up in the electrons.If this electron-heating fraction is smaller/bigger, we expect for the radiative efficiency to drop/rise and for the collapse to a two-phase medium to occur later/earlier.Similarly, other microphysical effects, typically not captured by the ideal MHD approximation, such as thermal conduction between the corona and disk (e.g., Meyer & Meyer-Hofmeister 1994;Liu et al. 1999;Meyer-Hofmeister & Meyer 2011;Cho & Narayan 2022;Bambic et al. 2024) and a nonunity magnetic Prandtl number (e.g., Balbus & Henri 2008), could alter the transition rate to a twophase medium.
In addition, it was recently demonstrated that the physics driving accretion in luminous black holes (e.g., with L  0.01 L Edd ), which are misaligned with the black hole spin axis, is fundamentally different.Namely, dissipation of orbital energy is driven by nozzle shocks induced by strong warping (Kaaz et al. 2023;Liska et al. 2023) instead of MRI-driven turbulence (e.g., Balbus & Hawley 1991, 1998).These nozzle shocks form perpendicular to the line of nodes, where the disk's midplane intersects the equatorial plane of the black hole and increase the radial speed of the gas by 2-3 orders of magnitude in luminous systems that are substantially misaligned.This could, at a given accretion rate, lead to a decrease in the disk's density, potentially delaying the formation of a thin accretion disk.We expect to address outbursts of warped accretion disks using a simulation campaign performed at much higher resolutions.Numerically, this article has also introduced a method to study outbursts by artificially rescaling the density as a function of time.This solves the issue that the physical processes in the outer disk that drive such drastic fluctuations in the mass accretion rate occur over timescales that are too long to simulate (real outbursts typically take weeks to months, while our simulations last for t ∼ 10-15 s).Future applications of this method might include (i) ultraluminous accretion disks, which decay from super-Eddington to sub-Eddington accretion rates; (ii) the transition from the hard-intermediate state to the highsoft state, where the magnetic flux threading the black hole drops while the accretion rate remains constant; and (iii) the transition from the high-soft state to the quiescent state, characterized by a gradual drop in the mass accretion rate.
We also plan to address the structure and dynamics of the thin and truncated disks as we keep increasing the density scale with a dedicated simulation campaign performed at a much higher resolution.
we plot the time evolution of the bolometric luminosity (panels (a) and (b)), density scale height (panels (c) and (d)), and outflow efficiencies (panels (c) and (d)) as functions of the mass accretion rate.While the luminosity increases from L = 10 −15 L Edd to L = 10 −2 L Edd , the radiative efficiency increases by 3-5 orders of magnitude.Similar to results presented in the radiative GRMHD simulations of Ryan et al. (2017) and Dexter et al. (2021), the MAD model is significantly more radiatively efficient, especially at low accretion rates.This is caused by more efficient synchrotron cooling in the highly magnetized gas of an MAD.Around = the SANE model collapses into a thin accretion disk, and we observe a rapid order-of-magnitude rise in the radiative efficiency to the Novikov & Thorne (NT; Novikov & Thorne 1973) limit of η rad ∼ η NT ∼ 0.18.Here,

Figure 1 .
Figure 1.Panel (a): the event horizon mass accretion rate  M closely follows the target mass accretion rate  M target (red) for both the SANE (black) and MAD (blue) models.Panel (b): the normalized magnetic flux f maintains saturation value in the MAD model and stays a factor 2.0 below saturation in the SANE model.

Figure 2 .Λ
Figure2.The SANE (upper panels) and MAD (lower panels) models at three different accretion rates.The left hemisphere illustrates the electron temperature (T e ), while the right hemisphere illustrates the density (ρ).The disk-jet boundary (b 2 /(ρc 2 ) = 1) is demarcated by a white line, and the last scattering surface (τ es = 1) is demarcated by a magenta line.The inset in the left hemisphere gives the mass accretion rate and luminosity in Eddington units.See our high-resolution YouTube playlist (https://www.youtube.com/playlist?list=PLDO1oeU33Gwm1Thyw0iHC14BbvBWaG5cE) and the lower-resolution associated animation of the density, electron temperature, and ion temperature for both models.At very low accretion rates (left panels), the electron temperature is determined by the heating rate and adiabatic evolution of the electrons.At intermediate accretion rates (middle panels), the electrons cool efficiently, but there is no noticeable change in the disk structure.At accretion rates of-   M M 10 2Edd the torus collapses into a thin accretion disk (SANE) sandwiched by a hot plasma or forms a magnetically truncated accretion disk (MAD).The in-line animations show the density (left panels), plasma (middle panels), and electron temperature (right panels) for the SANE (upper panels) and MAD (lower panels) models.These animations last 196 s and span t = 0 r g /c until t = 295,000 r g /c and include the accretion rate, luminosity, and radiative efficiency in Eddington units.(An animation of this figure is available.)

Figure 3 .
Figure 3.A contour plot of density with velocity streamlines in black and the last scattering surface in magenta for model XRB SANE.Similar to the puffy accretion disk models presented in Lančová et al. (2019) the majority of gas accretion seems to be driven by inflows outside of the disk's midplane.

Figure 4 .
Figure 4. Panels (a), (b): as the mass accretion rate rises, the luminosity increases from L ∼ 10 −15 -10 −13 L EDD to L ∼ 10 −2 L EDD .This is driven by both an increase in the mass accretion rate and radiative efficiency.Panels (c), (d): the density scale height of the disk stays relatively constant until the accretion rate exceeds ~-  M M 10 3 EDD , at which point the disk collapses.At this point, Coulomb collisions are efficient, and both ions can cool through the radiation-emitting electrons.Panels (e), (f): the jet (blue), wind (black), radiative (purple), and NT73 (dashed green) efficiency as functions of  M .Interestingly, the MAD model maintains a significantly higher radiative efficiency, presumably due to more efficient synchrotron emission.

Figure 5 .
Figure 5. Panels (a), (b): in both models at r = 10r g the electron temperature T e drops for

Figure 7 .Figure 8 .
Figure 7. Same as Figure 2 but for an M = 6.5 × 10 9 M e AGN.The in-line animation shows the density (left panels), plasma-(middle panels), and electron temperature (right panels) for the SANE (upper panels) and MAD (lower panels) models.These animations last 195 s and span t = 0 r g /c until t = 292,500 r g /c and include the accretion rate, luminosity, and radiative efficiency in Eddington units.High-resolution versions are also available on our YouTube playlist (https://www.youtube.com/playlist?list=PLDO1oeU33Gwm1Thyw0iHC14BbvBWaG5cE).(An animation of this figure is available.)