Variability from Nonaxisymmetric Fluctuations Interacting withStanding Shocks in Tilted Black Hole Accretion Disks

Ken B. HeniseyNatural Science Division, Pepperdine University, Malibu, CA 90263, USA

Omer M. BlaesDepartment of Physics, University of California, Santa Barbara, CA 93106, USA

and

P. Chris FragileDepartment of Physics and Astronomy, College of Charleston, Charleston, SC 29424, USA

ABSTRACT

We study the spatial and temporal behavior of fluid in fully three-dimensional, general rel-ativistic, magnetohydrodynamical simulations of both tilted and untilted black hole accretionflows. We uncover characteristically greater variability in tilted simulations at frequencies simi-lar to those predicted by the formalism of trapped modes, but ultimately conclude that its spatialstructure is inconsistent with a modal interpretation. We find instead that previously identified,transient, over-dense clumps orbiting on roughly Keplerian trajectories appear generically inour global simulations, independent of tilt. Associated with these fluctuations are acoustic spiralwaves interior to the orbits of the clumps. We show that the two nonaxisymmetric standing shockstructures that exist in the inner regions of these tilted flows effectively amplify the variabilitycaused by these spiral waves to markedly higher levels than in untilted flows, which lack standingshocks. Our identification of clumps, spirals, and spiral-shock interactions in these fully generalrelativistic, magnetohydrodynamical simulations suggests that these features may be importantdynamical elements in models which incorporate tilt as a way to explain the observed variabilityin black hole accretion flows.

Subject headings: accretion, accretion disks — black hole physics — MHD — turbulence — waves —X-rays: binaries

1. Introduction

Much current theoretical modeling of black holeaccretion flows is done in the context of the mag-netorotational instability (MRI; Balbus & Haw-ley 1991) which drives MHD turbulence and facil-itates the transport of angular momentum. How-ever, there are few, if any, observationally testablemanifestations of MRI turbulence. Because tur-bulence is inherently time-dependent, one wouldhope that variability would be a promising avenueto explore, especially given that it is ubiquitous

in observations across the electromagnetic spec-trum of accreting black hole sources. Particularlyin the X-rays, observed variability power spectraconsist of continua that can be fit by broken powerlaws, as well as discrete quasi-periodic oscillations(QPOs). In black hole X-ray binaries, both theamplitude and slopes of the continua, as well asthe presence and character of the QPOs, dependon the observed X-ray spectral state (e.g. Remil-lard & McClintock 2006). Low-frequency QPOsare observed in both the low/hard and the steep

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power law states, while high-frequency QPOs areonly observed in the steep power law state. Thehigh/soft or thermal state exhibits much less X-rayvariability overall, and what variability it does ex-hibit appears to be associated with the hard X-ray,nonthermal component (Churazov et al. 2001).

Global simulations of the accretion flow withfully developed MRI turbulence have been usedto model the expected continuum variability un-der a variety of assumptions that relate varia-tions in local fluid quantities to variations of whatmight be observed (e.g. Noble & Krolik 2009and references therein). QPOs have also beenlooked for in such simulations, with mixed re-sults. For example, Reynolds & Miller (2009)failed to find the trapped inertial waves predictedby hydrodynamic disk models (e.g. Wagoner1999; Kato 2001) in their global MHD simula-tions of accretion disks. However, similar simu-lations by O’Neill et al. (2011) identified radiallyextended, low-frequency quasi-periodic behaviorin the azimuthal magnetic field associated withdynamo cycles within the MRI turbulence itself,although how that would manifest itself observa-tionally is unclear. Hydrodynamically unstableinner tori that re-establish themselves after mag-netically mediated accretion episodes have beenobserved in MHD simulations (Machida & Mat-sumoto 2008), and may be a mechanism for low-frequency QPOs. High-frequency QPOs in themass fluxes have also been claimed in at least oneMHD simulation (Kato 2004b). Schnittman et al.(2006) applied various post-processing emissionmodels and ray tracing calculations to a globalMHD simulation to make predictions of observedvariability power spectra, and found that transientQPOs could be seen, but they were likely insignifi-cant as they would only appear in certain observerdirections at any given time. On the other hand,Dolence et al. (2012) recently discovered transientQPOs in their radiative post-processing of a globalMHD simulation appropriate for the Galactic cen-ter source Sgr A∗. These QPOs appear to orig-inate from m = 1 non-axisymmetric structuresformed within the inner accretion flow.

All the simulations used in these studies rep-resent flows in which the angular momentum ofthe accretion flow is aligned with the spin of theblack hole, or the black hole is not spinning atall. Tilted accretion flows, in which the angu-

lar momenta of the accretion flow and the spin-ning black hole are not aligned, have additionalcomplexity in their dynamics (Fragile et al. 2007;Fragile & Blaes 2008), and therefore also likely intheir variability. Orbits of fluid elements in tiltedflows are generally eccentric, not circular, and theradial gradient of eccentricity can give rise to apair of non-axisymmetric standing shocks whichcompletely alter the character of the innermostparts of the flow. Provided the flow is not too geo-metrically thin, Lense-Thirring torques can causeglobal, rigid-body precession of the otherwise dif-ferentially rotating and turbulent flow. Tilted ac-cretion flows are probably common around super-massive black holes in galactic nuclei, where thefuel source is unlikely to know about the spin ofthe black hole, and in X-ray binaries if the spinof the black hole is misaligned with respect to thebinary orbital angular momentum.

Motivated in part by predictions that hydro-dynamic inertial modes can be excited by eccen-tric orbits and warps in a tilted disk (Kato 2004a,2008; Ferreira & Ogilvie 2008), we examined vari-ability power spectra of the local fluid density, ra-dial velocity, and vertical velocity in a simulationof a flow with an initial tilt angle of 15 arounda Kerr black hole with a dimensionless spin pa-rameter a/M = 0.9, as well as an untilted flowaround the same black hole (Henisey et al. 2009).In the tilted simulation, we identified a partic-ularly strong, coherent feature in all three fluidvariables that had the same frequency over an ex-tended range of radii. The frequency of this fea-ture was consistent with it being a trapped inertialmode, but we were unable to convincingly demon-strate that this was its true nature. It appearedto be associated with an overdense clump orbitingwith the background flow, as well as inertial andinertial-acoustic waves. We therefore suggestedthat this was a preliminary confirmation that diskwarps and eccentricity might excite inertial modeseven in the presence of MRI turbulence. Dexter &Fragile (2011) later did a radiative post-processinganalysis of this tilted simulation and failed to findany manifestation of this discrete frequency fea-ture in the simulated light curves that might beobserved at infinity.

Despite our failure in finding a radiative sig-nal of this QPO, there is still much to be gleanedfrom these simulations. The variability dynam-

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ics of turbulent accretion flows, and in particu-lar tilted accretion flows, remains very poorly un-derstood. To that end, this paper extends ourprevious analysis to a series of simulations, withlonger durations and with 0, 10 and 15 tiltangles. The tilted accretion flows generally ex-hibit discrete frequency features that have broadradial extents. However, these features are tran-sient, and the frequencies that are present comeand go as the simulations progress. The featuresdo not appear to be manifestations of trapped in-ertial waves, but are instead due to transient co-orbiting clumps that have associated radially ex-tended waves. Such clumps are also present inthe untilted simulations, but they are not accom-panied by the same high-power radially extendedstructures. The radial power arises in the tilted ge-ometries owing to the standing shocks in the innerparts of the flow which amplify the variability ofeach clump’s associated wave. This amplificationoccurs because the positive density fluctuations inthe waves are significantly enhanced at the pointswhere they cross the shocks.

We present the basis for these conclusions as fol-lows. In Section 2, we provide an overview of thesimulations used in our analysis. In Section 3, wedescribe the time-averaged structure of the innerflow regions, focusing in particular on the standingshocks in the tilted flows. In Section 4, we extendour previous study (Henisey et al. 2009) and com-ment on the radial distribution of variability in ourdatasets. We present our analysis of the physicsbehind the variability in Section 5, which consistsof two parts. In Section 5.1, we present visual-izations of the three-dimensional structure of thevariability at a given frequency in the context ofthe nonaxisymmetric background. Motivated bythe dominant m = 1 modes originally identified inour previous work, we break down the variabilityinto prograde and retrograde m = 1 structures inSection 5.2 and discuss a possible mechanism bywhich these structures form. We summarize ourconclusions in Section 6.

2. Simulations

This work aims to characterize variability infour distinct general relativistic, magnetohydro-dynamic simulation datasets of black hole accre-tion flows generated by the Cosmos++ code (An-

ninos et al. 2005). We continue our previous study(Henisey et al. 2009) of two such simulations orig-inally introduced in Fragile et al. (2007): 90h, anuntilted configuration with an initial fluid angu-lar momentum aligned with the spin axis of ana/M = 0.9 Kerr black hole, and 915h, a tilted flowwith initial fluid angular momentum misalignedfrom the spin axis of the hole by 15. We alsostudy two new high-resolution simulations, 910hand 915h-64, both exploring tilted geometries withdisk-spin misalignments of 10 and 15, respec-tively. These simulations all run on essentially thesame grids in which the simulation domains reachequivalent peak resolutions of 1283 zones (Fragileet al. 2007).

Each black hole spacetime is modeled us-ing a Kerr-Schild coordinate system with spin-parameter a/M = 0.9 and which is tilted toachieve the desired angle with respect to thegrid midplane. We embed each spacetime withan analytic solution for a torus whose inneredge and pressure maximum lie in the grid mid-plane at 15RG and 25RG, respectively, whereRG ≡ GM/c2, and has an initial specific angu-lar momentum distribution given by a power-lawwith exponent q = 1.68 (Chakrabarti 1985). Eachtorus was then seeded with different initial ran-dom perturbations. For this reason, simulations915h and 915h-64 are fully independent, despitehaving identical misalignments and unperturbedinitial tori. Weak poloidal magnetic field loopshaving βmag ≡ P/Pmag ≥ 10 lie along the isobarswithin the torus and ultimately seed the MRI.Integration from these initial conditions utilizesCosmos++’s internal-energy, artificial viscosityformulation while the magnetic fields evolve byan advection-split form, using a hyperbolic diver-gence cleanser to maintain a roughly divergence-free field. For more detail, see Fragile & Anninos(2005) and Fragile et al. (2007).

Throughout this paper, an “orbit” refers tothe orbital period of a circular geodesic in theequatorial plane of the black hole at the radius25RG of the pressure maximum of the initial torus:1Porb ' 791GM/c3. We begin our timing analy-sis four orbits after the start of each simulation,so that MRI turbulence is fully developed in theinner regions of the flow and the average mass ac-cretion rate as a function of radius reaches a moreor less uniform profile inside of 15RG. Datasets

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from simulations 90h, 910h, and 915h then in-clude information from the next six orbits, that is,from time t = 4 to 10Porb of the evolution. The915h-64 dataset, on the other hand, includes 12orbits of data split into two epochs labeled 915h-64a and 915h-64b, encompassing t = 4 to 10Porb

and t = 10 to 16Porb, respectively. A summary ofthe differences between these five datasets appearsin Table 1.

3. Background Flow Structure

A tilted accretion flow around a spinning blackhole has no axis of symmetry, and the spatialstructure of the flow can be quite complex. Fragileet al. (2007) identified two high density arms thatthey called ”plunging streams” in tilted 915h sim-ulation originating at around 4RG and dominatingthe accretion in the innermost regions of the flow.Fragile & Blaes (2008) showed that two standingshocks accompany these overdense arms and ex-tend outward to larger radii. They also demon-strated a sharp decrease in angular momentuminside 6RG and indicated enhanced energy dissipa-tion inside roughly 10RG in comparison with theuntilted flow, phenomena that they suggested aredirectly related to the shock structures.

As we shall see, these shock structures play animportant role in the variability of tilted accre-tion flows. We therefore spend some time herediscussing their structure in some detail for eachof the tilted simulation datasets, using the rate ofchange of entropy as a diagnostic. For an idealgas up to an arbitrary constant of integration, wemay define the entropy such that s ≡ log (p/ργ),

Table 1: Dataset Summary

Dataset Simulation Tilt Time Domaina

90h 90h 0 4 ≤ t < 10910h 910h 10 4 ≤ t < 10915h 915h 15 4 ≤ t < 10

915h-64ab 915h-64 15 4 ≤ t < 10915h-64bb 915h-64 15 10 ≤ t < 16

a In units of the orbital period of a test particleat 25RG.

b Though computationally continuous, we dividethe analysis of simulation 915h-64 into two, 6orbit epochs.

where γ is the fluid’s adiabatic index, and all quan-tities are measured in the fluid rest frame. In oursimulations, γ = 5/3. Since the spatial locationof a fluid element with rapidly increasing entropystrongly suggests the location of a shock front,we locate spatial maxima in the average rate ofchange of specific entropy calculated at each gridzone,

∆s

∆t≡⟨uµ∇µsut

⟩t

, (1)

where uµ is the fluid 4-velocity, ut is the time com-ponent of that 4-velocity, and ∇µ is a covariantderivative, which in the case of the scalar operands, reduces to partial derivatives in the four space-time coordinates. Here and throughout this pa-per, the angle brackets indicate averages over thesubscripting spacetime coordinates and imply thatthe resulting expression remains a function of theunaveraged coordinates. Equation (1) is equiva-lent to ∆s/∆t = 〈∂s/∂t+ ~v · ∇s〉t, where ~v is thefluid’s coordinate 3-velocity (that is, vi = ui/ut)and ∇ is a 3-gradient in the spatial coordinates.In this form, it is more clear that this representsthe average Lagrangian rate of change of the en-tropy with respect to coordinate time, that is, thetotal change in entropy of fluid elements as theypass through a given grid zone over the total sim-ulation time [

∫(ds/dt)dt]/

∫dt or simply ∆s/∆t.

Since the accretion flow experiences roughly 24 ofLense-Thirring precession during the time evolu-tion in each of our datasets, features identified bytime-averaged quantities like those defined abovemay be smeared by at most that angle.

In Figure 1, we plot two contours: first, a semi-transparent contour inside of which 〈ρ〉t/〈ρ〉t,θ,φ >2.5, that is, where the time-averaged density at agiven point in space is 2.5 times larger than theshell-averaged density at the same radius; second,a solid contour inside of which the time-averagedspecific entropy generation rate ∆s/∆t > 0.070in the same arbitrary units defined above. Forcomparison, Table 2 details the average generationrates per coordinate volume, again in the same ar-bitrary units, both inside the contours and outside(but still within 15RG). Our contour choice clearlycontains dramatically larger specific entropy gen-eration rates than the rest of the simulation vol-umes and therefore effectively locates these quiteplanar shock surfaces. Note that the sense of fluid

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orbital motion is right-handed about the verticalaxes shown in Figure 1, and that at a given radius,the density is maximal just behind the standingshocks.

We characterize the extent and strengths of theshocks by studying the shell and time-averaged La-grangian specific entropy generation rates, shownin Figure 2. For completeness, this figure also in-cludes a panel for the untilted simulation whichexhibits generation rates at least 50 times smallerthan those in the tilted simulations; this is consis-tent with the absence of standing shocks in thatdataset. Each tilted simulation can be charac-terized by a transition from an almost constantplateau at smaller radii to the almost constantbaseline at larger radii. This transition occursmore sharply in the less tilted 910h dataset. Ad-ditionally, the center of that transition region oc-curs at a smaller radius, around 7RG, than in 915hand both epochs of 915h-64 where it takes placeat around 10RG. As a result, the total time andvolume integrated specific entropy generation rateinside the transition region is smallest in 910h andlargest in 915h-64b. Since orbital time scales awayfrom the hole are not necessarily small comparedto the simulations’ durations, one can see subtledifferences at large radii in the entropy genera-tion rate between 915h-64a and 915h-54b (panelsd and e, respectively) which indicate continuingdisk evolution.

In the 15 tilted simulations, we measure com-pression ratios across these shock surfaces as highas 4.7, depending on location. This is compara-ble to what we would expect: for a constant 5/3adiabatic index, as our simulations assume, hydro-dynamic shocks should have a compression ratio ofat most 4, neglecting magnetic fields and relativ-

Table 2: Specific entropy generation rate ∆s/∆tper coordinate volume

Dataset Outsidea Insidea

910h 0.0011 0.123915h 0.0027 0.132915h-64a 0.0024 0.138915h-64b 0.0017 0.154

a The average generation rate per coordinatevolume inside and outside of the contours∆s/∆t = 0.07 in Figure 1 and with r < 15RG.

ity. We find compression ratios as high as 4.0 inthe 10 simulation (Generozov et al. 2012).

Fragile & Blaes (2008) also presented compar-ative plots of characteristic velocities in 90h and915h. Figure 3 reproduces these plots, only nowthe time-averages cover the interval t = 4 to10Porb instead of the smaller interval t = 7 to10Porb, and we have also added curves for thenon-radial component of the velocity and the testparticle orbital velocity. Fragile & Blaes (2008)stated that the sharp upturn in vr indicated thestart of the plunging region, and pointed out thatthis occurred at larger radii in the tilted simu-lation. Note, however, that fluid elements out-side of roughly 4RG in the 915h tilted simulationmake many complete orbits before truly plungingtoward the hole. The high density arms shownin Figure 1 are therefore mostly due to post-shock compression, and are not literally plungingstreams, at least outside roughly 4RG. As we dis-cuss below, we posit that this shock compression isat the heart of the enhanced variability from spiralwaves associated with transient orbiting clumps.

4. Power Density Spectra

Our first step in probing the three-dimensionalstructure of variability in our simulated accretionflows is to understand the radial distribution ofspectral power. Interesting structures that rotate,oscillate, or otherwise vary at characteristic fre-quencies may be identified in frequency space asradially extended regions with enhanced power.To this end, we follow the method outlined in ourprevious work (Henisey et al. 2009) and computethe shell-averaged spectral power in fluid densityfluctuations,

Pρ(f, r) ≡ 〈|ρ|2〉θ,φ/〈ρ2〉t,θ,φ, (2)

where ρ is the temporal Fourier transform of the(prewhitened, windowed, and padded) fluid den-sity and is a function of space and frequency.We plot the resulting spectra for each of the fivedatasets in Figure 4.

Two features in these spectra warrant discus-sion. First, in Henisey et al. (2009) we showedthat transient clumps traveling on roughly Kep-lerian trajectories exist amid the MRI turbulencein simulations 90h and 915h. They appear as en-hanced power along the curve of orbital frequency

5

a. b.

c. d.

Fig. 1.— Three-dimensional contours in time-averaged density normalized to the time and shell-averageddensity 〈ρ〉t/〈ρ〉t,θ,φ (semi-transparent gray) and Lagrangian specific entropy generation rate ∆s/∆t in arbi-trary units [solid gray; see Equation 1] drawn at 2.5 and 0.07, respectively, from 910h (a), 915h (b), 915h-64a(c), and 915h-64b (d). Axes extend to 15RG. The regions enclosed in the entropy generation rate contouridentify the location of standing shock surfaces within the flow.

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a.

b. c.

d. e.

Fig. 2.— Time and shell-averaged specific entropy generation rate as a function of radius in 90h (a), 910h(b), 915h (c), 915h-64a (d), and 915h-64b (e).

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a. b.

Fig. 3.— The time and shell-averaged radial (solid red) and non-radial (dashed red; r√

(vθ)2 + (vφ sin θ)2)velocities, the time and shell-averaged sound speed (blue), and the test particle Keplerian orbital velocity(black) for simulations 90h (a) and 915h (b). Note that the non-radial component of the velocity very closelytracks the orbital velocity (dashed red and black, respectively) as expected. Time-averages span the intervalt = 4 to 10Porb and shell averages include a weighting by the time-averaged density. In both simulations,we avoid characterizing the region between 4 and 10RG as “plunging” since fluid elements therein orbit thehole many times before making their final plunge toward the horizon.

versus radius. In Figure 4, we now clearly see thatthese orbiting clumps arise generically in all fiveof our datasets.

Second, a set of features that we previouslyidentified in 915h in Henisey et al. (2009) likewiseappear here strongly in each of the tilted simula-tions. Specifically, the tilted datasets exhibit sig-nificant power in vertical streaks extending inwardin radius from the orbital frequency curve. For theremainder of this work, we will refer to such poweras “intraorbital”, denoting that it is power at aspecific frequency and at all radii inward of thefrequency’s corotation radius. Importantly, theseintraorbital, coherent structures coincide with thepower enhancements along the orbital frequencycurve associated with orbiting clumps. In the nextsection, we explore the relationship between theseintraorbital features and their respective orbitalclumps.

In Henisey et al. (2009), we concentrated muchof our study on the 118Hz feature in the 915hsimulation. Here we note that strong verticaltracks also appear in 910h at 150Hz, in 915h-64aat 140Hz, and in 915h-64b at 60Hz. Noting that915h-64a and 915h-64b represent different epochsof the same simulation and that 915h and 915h-64a differ only in there initial random perturba-tions, variety in the position of and power in these

simulations’ vertical streaks indicates that whilethese structures are long lived with respect to thelocal orbital period, they are nonetheless transientfeatures: they come and go on time scales smallerthan but comparable to the simulation durations.While these intraorbital streaks appear exclusivelyin the tilted simulations, features that are similarin structure though markedly weaker (by perhapsas large a factor as 30) in spectral power existwithin the untilted flow. One such example ap-pears at 110Hz. Note that in all cases (both tiltedand untilted), a local minimum seems to occuralong the vertical track at roughly the midpointbetween the corotation radius and innermost sta-ble circular orbit (ISCO).

These intraorbital power features contribute togenerally enhanced variability in each of the tiltedsimulations at all frequencies greater than approx-imately 50Hz (equivalently, the orbital frequencyat roughly 15RG) and radially inward of the or-bital frequency curve. There appears to be a qual-itative trend of increasing intraorbital power withincreasing tilt and with more advanced flow evo-lution, though we have not attempted to quanti-tatively assess its significance.

To summarize, Figure 4 confirms that tran-sient over-dense clumps orbiting with the back-ground flow are generic in our simulations. The

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a.

b. c.

d. e.

Fig. 4.— Shell-averaged power spectra of density fPρ as a function of frequency f and coordinate radius r(cf. Equation 2) for datasets 90h (a), 910h (b), 915h (c), 915h-64a (d), and 915h-64b (e). Over-plots includethe orbital frequency (solid) and its harmonics (dotted), the geodesic radial epicyclic frequency and the ISCOradius (dashed), and the geodesic vertical epicyclic frequency (triple-dot dashed).

9

mechanism responsible for the formation of theseclumps remains unexplained. In terms of ourglobal variability study however, it is sufficientthat these clumps exist and that, above some fre-quency threshold, they are associated with radi-ally extended power at the same frequency as theclump’s orbital frequency. This intraorbital spec-tral power is significantly enhanced in the tilteddatasets compared to that observed in the untiltedsimulation. Beyond identifying these variabilityfeatures, however, there is little more to gleanfrom this shell-averaged analysis. The next sec-tion, therefore, studies the full three-dimensionalbehavior of these structures at specific frequen-cies and aims to understand the variability in thecontext of the background structure discussed inSection 3.

5. Structural Analysis

We now study the variability discussed in theprevious section by analyzing its spatial struc-ture and temporal behavior in more detail. InSection 5.1, we visualize the orbiting clumps andtheir associated intraorbital variability within thecontext of the flow’s nonaxisymmetric backgroundand analyze the temporal phase relationshipsacross the spatial extent of the high-power vari-ability. In Section 5.2, we then put forward aunified physical model that explains those spec-tral features in each dataset.

5.1. Spatial structure of variability

The coherent structures identified in Section 4represent density perturbations throughout thedisk varying at specific frequencies. We can there-fore visualize these structures by filtering away anyfluctuations that occur on greater or smaller timescales. Mathematically, we construct a frequency-filtered density,

ρω(t, ~r) = Rω cosωt+ Iω sinωt, (3)

where Rω ≡ 2Re[ρ(ω,~r)] and Iω ≡ 2Im[ρ(ω,~r)] aretwice the real and imaginary parts of the Fouriertransform of the density time-series. The factor of2 accounts for the fact that Fourier transforms ofreal-valued functions like our density are even infrequency space, splitting total power evenly be-tween positive and negative frequencies. With theabove definitions then, we can restrict our study

to positive frequencies. This density representsonly those perturbations of the flow that vary withangular frequency ω, and therefore, although thetime t directly relates to the simulation coordi-nate time, we only need to study the function’sevolution through a single period 0 ≤ t < 2π/ω.Figure 5 shows snapshots in time of contours offrequency-filtered density normalized in a man-ner consistent with the spectra in Figure 4, thatis, ρω

√ω/2π〈ρ2〉t,θ,φ. For each dataset, we have

evaluated the filtered density at frequencies thatexhibit significant intraorbital power in Figure 4.We have selected moments within the oscillationperiod that most clearly illustrate that this in-traorbital power is spatially concentrated near thebackground’s over-dense arms. The contours lie at0.125 for datasets 910h, 915h, and 915h-64a and0.175 for dataset 915h-64b.

Variability at the given frequencies in eachtilted dataset spatially manifests itself in twoforms: a clump orbiting near the corotation radiusand peaking in amplitude after passing the shockfronts, and a pulsation within each over-densearm just past the shocks and inside the corotationradius. The latter feature is responsible for thepreviously identified intraorbital power.

Four explanations for this variability structureseem plausible. First, after interacting with theshock, the orbiting clump may, by some mech-anism, liberate some small amount of materialwhich would fall along the arms toward the hole.Second, after interacting with the shock, the or-biting clump may send a pressure wave down theover-dense arm. Third, standing inertial wavesmay be trapped within the over-dense arms and beexcited and maintained by the clump-shock inter-action. Fourth, a coherent acoustic spiral patternproduced and maintained by an orbiting clumpmay interact with the radially extended shock, in-creasing the amplitude of both the clump and spi-ral independently.

In light of our conclusions in Section 3, we ruleout the first explanation since sloughed off mate-rial cannot simply fall past the otherwise roughlyKeplerian flow circulating around the hole. Sim-ilarly, a close examination of the temporal phaserelationships within the pulses forces us to discardthe second and third explanations. Specifically,Figure 6 shows six successive snapshots in timeof the 118Hz variability, that is, of the frequency-

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a. b.

c. d.

Fig. 5.— Three-dimensional visualizations of the spatial structure of the variability in density for oscillationsat 150Hz in 910h (a), at 118Hz in 915h (b), at 140Hz in 915h-64a (c), and at 60Hz in 915h-64b (d).Specifically, we depict contours of the frequency-filtered density (see Equation 3) normalized in a mannerconsistent with the spectra in Figure 4, that is, ρω

√ω/2π〈ρ2〉t,θ,φ. These surfaces lie at positive (red) and

negative (blue) 0.125 in 910h, 915h, and 915h-64a and 0.175 in 915h-64b. For reference, semitransparentsurfaces indicate density (light gray) and specific entropy generation rate (dark gray) contours identical tothose in Figure 1. Axes extend to 15RG.

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filtered density given by Equation 3. The timeinterval between each frame is 1/40th of the oscil-lation’s period. By construction, these structuresvary at the filter frequency, in this case, 118Hz, re-gardless of any local characteristic frequencies likethe orbital frequency. Notice that the red variabil-ity structure inside the over-dense arm closest tothe viewer (arrow 3) appears to start at small ra-dius and moves outward toward larger radii. Like-wise, two additional structures (labeled 2 and 4)appear to recede away from the center. Since thesecond and third hypotheses predict an inwardlypropagating wave and a standing wave within thearms, respectively, this behavior rules out bothmodels.

In the fourth hypothesis however, a radially ex-tended spiral pattern rotates bodily about the holeat the same frequency (here 118Hz) as its associ-ated corotating clump. Depending on the shape ofthe spiral relative to a shock surface, the point ofinteraction between the two may appear to moveinward or outward in time along the shock’s face.For example, if some segment of the spiral is trail-ing, the interaction point between it and the sta-tionary shock surface will appear to move out-ward in time, akin to structure 3 in Figure 6. Inthis way, a spiral-shock interaction model permitsall the complex temporal relationships depicted inFigure 6. Of our four hypotheses, this remains theonly plausible explanation.

At the spiral-shock interaction point, the acous-tic wave passing through the shock will experiencean enhancement in its density amplitude (as de-fined by half the difference between the maximumand minimum density in the wave) equal to theshock compression ratio discussed in Section 3. Inthe 15 tilted simulations, the ratios were as highas 4.7 while in the 10 tilted simulation they wereas large as 4.0 (Generozov et al. 2012). We there-fore expect density power in the acoustic waves(which goes as the square of the amplitude) to beenhanced by a factor of roughly 22 in the 15 tiltedsimulation, and 16 in the 10 tilted simulations, ascompared to the untilted simulation. This quanti-tatively matches the larger intraorbital power inthe vertical streaks in the tilted simulations ascompared to the untilted simulation in the den-sity power spectra shown in Figure 4. This spiral-shock interaction model can explain both the ap-parent motion of the variability along the back-

ground over-dense arms and the magnitude of theincreased variability at the point of interaction.

5.2. Prograde and retrograde decomposi-tion

Of those models described above that aim toexplain the apparent movement of perturbationsalong the over-dense arms, only that in which arotating acoustic spiral accompanies the orbitingclumps identified in Section 4 and visualized insection 5.1 is consistent with the observed tempo-ral relationships. In an axisymmetric backgroundakin to that in our untilted simulation, for ex-ample, such a spiral pattern should be both sta-tionary in a frame corotating with the clump andlargely m = 1 in its azimuthal symmetry. By as-suming these characteristics, this section furtherdecomposes the variability in our simulations to lo-cate these spiral structures and better understandtheir physical nature.

Since the structures visualized in the previoussection are largely m = 1, we first isolate thiscomponent of the variability from the frequency-filtered time series in Equation 3. In other words,we project the spatial functions Rω and Iω ontothe two m = 1 spatial basis functions so that

ρω,m=1 = (Rω,c cosφ+Rω,s sinφ) cosωt

+ (Iω,c cosφ+ Iω,s sinφ) sinωt, (4)

where

Rω,c =1

π

∫ 2π

0

Rω cosφdφ (5a)

Rω,s =1

π

∫ 2π

0

Rω sinφdφ (5b)

Iω,c =1

π

∫ 2π

0

Iω cosφdφ (5c)

Iω,s =1

π

∫ 2π

0

Iω sinφdφ. (5d)

Using basic trigonometric identities, one can com-bine these terms in a straight-forward manner tofind

ρω,m=1 = Aω cos(φ− ωt− αω)

+Bω cos(−φ− ωt− βω), (6)

12

a.

1 -

2 -

3 -4

5

b.

1 -

2 -

3 - 4

5

c.

1 -

2 -

3 - 4

5

d.

1 -2 -

3- 4

5

e.

1 - 2 -

3- 4

5

f.

1 -2 -

3- 4

5

Fig. 6.— Successive three-dimensional snapshots in time of the 118Hz pattern in the 915h simulation givenby normalized frequency-filtered density (see Figure 5). By construction, this variability is independent ofany local characteristic frequencies like the orbital frequency. Each frame is separated by 1/40th of a fulloscillation period. Semi-transparent surfaces (light and dark gray) and positive (red) and negative (blue)contours lie at the same levels as in Figure 5. Numbered arrows track the location of the trailing (1) andleading (5) edges of the orbiting clump as well as three edges of the flashes in the over-dense arms: one whichadvances outwardly from the hole (3) and two that recede away from it (2 and 4). Axes extend to 15RG.

13

where

A2ω =

1

4

[(Rω,c + Iω,s)

2

+ (Rω,s − Iω,c)2]

(7a)

B2ω =

1

4

[(Rω,c − Iω,s)2

+ (Rω,s + Iω,c)2]

(7b)

tanαω =Rω,s − Iω,cRω,c + Iω,s

(7c)

tanβω =−Rω,s − Iω,cRω,c − Iω,s

. (7d)

Given the form of the arguments within each co-sine, we can now interpret the coefficients Aω andBω as the amplitudes of the two rotating compo-nents, a prograde pattern and a retrograde pat-tern, respectively, at each (r, θ) pair.

Four basic motions can result from this inter-pretation on a given ring of constant r and θ.Clearly, if Aω or Bω is zero, one would observea purely retrograde or prograde motion, respec-tively. Similarly, Aω = Bω indicates a standingwave structure. Less intuitively, the more generalcase in which 0 < Bω < Aω turns out to be aprograde pattern that no longer exhibits a con-stant amplitude or rotation speed. Like two sinu-soidal waves of differing amplitudes crossing on astring, the two components constructively inter-fere at some moments producing a large densityperturbation (of magnitude |Aω| + |Bω|) that ro-tates in a prograde manner at an instantaneousangular speed smaller than the angular frequencyof the oscillation. In the same way, at the mo-ments when the two components destructively in-terfere, they produce a lower amplitude perturba-tion (having magnitude |Aω| − |Bω|) that againrotates in a prograde fashion but now with an in-stantaneous angular speed greater than the angu-lar frequency of the oscillation.

Keeping these possible motions in mind, we re-turn to our untilted and tilted disks. In the ax-isymmetric background of the untilted simulation,the decomposition of a clump with a corotatingspiral ought to return a purely prograde pattern.On the other hand, we suggested in Section 5.1that the standing shocks associated with the non-axisymmetric background amplify the density ofthe clump and its corotating spiral in the tilted

simulations. We therefore expect to find a pre-dominantly prograde spiral throughout the regioninterior to the clump accompanied by a weaker butnonzero retrograde component. This latter piecemerely accounts for the fact that the complete per-turbation has a higher magnitude and slower ro-tation speed just past the standing shocks and alower amplitude and faster rotation speed awayfrom the shocks. This result aligns well with thestructure and evolution depicted in Figures 5 and6, respectively.

We plot in Figure 7 the power in each rotatingcomponent, 〈|Aω|2〉θ and 〈|Bω|2〉θ at the indicatedfrequencies and as a function of radius, normaliz-ing to match Figure 4. As anticipated, the m = 1variability in the untilted simulation is dominatedby a prograde pattern at all radii and shows verylittle power in the retrograde term. We take thisas evidence that the weak intraorbital variabilityfound in the power spectra in Figure 4 for the un-tilted simulation arises from prograde spirals coro-tating with Keplerian clumps.

For the tilted simulations, Figure 7 shows thatthe clump is likewise dominated by progrademotion. We interpret the prograde pattern as,in some sense, the average power profile of theclump’s associated spiral. Additionally though,we also see an enhanced retrograde componentand interpret the relative retrograde to progradepower as the degree to which the shock amplifyingbackground deforms the spiral as it rotates aroundthe disk. We emphasize that one must only inter-pret the retrograde power as a mathematical tool;it cannot be any kind of physical structure. If itwere a real, propagating perturbation, it would betraveling upstream, against the background fluidflow at speeds which far exceed the local soundspeed. Nonetheless, the largely prograde natureof the variability corroborates the clump-spiral hy-pothesis in both untilted and tilted configurationsand further indicates that the standing shocks as-sociated with tilted geometries enhance variabilityby effectively amplifying passing density fluctua-tions.

Shock amplification and tilt aside, we charac-terize the nature of these apparently ubiquitous,prograde spiral patterns by comparing the struc-ture of their prograde components to diskoseismicpredictions. Isothermal diskoseismic models of un-tilted, flat disks indicate that short wavelength

14

a. b.

c. d.

e. f.

Fig. 7.— Total power at specified frequencies (black; 〈R2ω + I2ω〉θ,φ) in 90h (a), 910h (b), 915h (c and d),

915h-64a (e), and 915h-64b (f) compared with the power in the m = 1 projection at the indicated frequencies(blue; 〈A2

ω + B2ω〉θ). The projected power is further decomposed into its prograde (light red; 〈A2

ω〉θ) andretrograde (dark red; 〈B2

ω〉θ) contributions. Note that all curves have been multiplied by ω/2π〈ρ2〉t,θ,φ tomatch the normalization of the spectra in Figure 4. Triangles mark the test particle corotation radius (black)and m = 1 Lindblad resonances (light red). Light red, horizontal lines mark propagation regions for n = 0,m = 1 modes.

15

WKB perturbations with local radial wave num-ber k obey the dispersion relation (Okazaki et al.1987),

k2 =(ω2 − κ2r)(ω2 − nκ2θ)

ω2c2s, (8)

where ω ≡ ω−mΩ is the Doppler-shifted wave fre-quency, κr is the radial epicyclic frequency, and nis a non-negative integer that indicates the numberof vertical nodes in the oscillation. Adopting thehypothesis that our prograde spiral patterns rep-resent n = 0, m = 1 acoustic perturbations hav-ing angular frequency ω, then the region aroundthe corotation radius bounded by the inner andouter Lindblad resonances is an evanescent regionwith local radial decay constant adisp such thatk = iadisp. If the WKB approximation holds, wecan directly calculate the local radial decay rateof the amplitude of the prograde spiral using aderivative of the power in Figure 7, specifically,

a =1

2

d

dr

log

[ω

2π

〈A2ω〉θ

〈ρ2〉t,θ,φ

]. (9)

If our spirals are acoustic in nature, then up toits sign, this calculated radial decay rate shouldroughly match the theoretical decay constantadisp, that is, the imaginary component of thewave number from Equation 8. In Figure 8, weplot both a and adisp. While there is noise, it isnoteworthy that the measured decay rate in eachsimulation is positive inside corotation, negativeoutside corotation, and approximately equal inmagnitude to that predicted by the dispersion re-lation in Equation 8. We believe that this furthersupports the hypothesis that intraorbital poweris the manifestation of spirals excited by orbitingclumps that are acoustic in nature.

To aid in the visualization of these spiral pat-terns, Figure 9 depicts a snapshot in time ofthe location (in an average sense) of the max-ima of the prograde pattern at each radius, thatis, αω ≡ arctan [〈Rω,s − Iω,c〉θ/〈Rω,c + Iω,s〉θ] in-spired by the definition of αω in Equation 7. Weinterpret the shape of these curves as the av-erage shape of the clump-spiral structure. Onecan imagine that these plots evolve in time bynoting that the prograde spiral will rotate coun-terclockwise through the stationary backgroundflow. Breaking down their structure, we see thatthe prograde spirals from each simulation share

the same, qualitative, three-segment geometry: atrailing segment at small radii, a leading segmentaround an intermediate radius near the inner Lind-blad resonance, and another trailing segment ter-minating at the corotating clump.

The first and last segments of these spirals seemconsistent with the outward moving density fluc-tuations located in the over-dense arm and iden-tified in Figure 6 by labels 2, 3, and 4. Since2 and 3 appear over roughly the same range ofradii in their respective arms, our m = 1 spi-ral model would predict that the two perturba-tions should be spatially symmetric though halfa period out of phase in time. Looking closelyhowever, we see that while 2 is retreating awayfrom the hole and 3 is growing toward larger radii,they are both nonetheless simultaneously positive.This indicates that feature 2 leads 3 in phase byslightly less than half a period. This behaviormust be due to m > 1 contributions to the overallclump/spiral structure, though we have not ana-lyzed these higher order effects in detail.

Given the above, one might anticipate that themiddle, leading segment ought to have an associ-ated inward moving density enhancement in eachover-dense arm. The absence of such features inFigure 6 has a two-fold explanation. First, Fig-ure 9 only roughly depicts the location of the spi-ral’s maximum at each radius and not the max-ima’s relative amplitudes. As seen in Figure 7,the density perturbations ubiquitously reach lo-cal minima near the Lindblad resonances and,correspondingly, the leading segments of the spi-rals must be relatively weak. These perturbationstherefore have little effect in Figure 6. Second,the retrograde component (not shown in Figure 9for purposes of clarity) is oriented such that thewhole leading segment reaches its maxima all atonce and without any apparent motion. Althoughwe strongly suspect that there is some connectionbetween the common shape of these spirals andthe location of the inner Lindblad resonances, it isunclear what mechanism might be responsible forthe apparent correlations.

To summarize, orbiting clumps and their ac-companying spiral patterns are generic in thesesimulations. Although these features do little toincrease the overall variability in untilted diskgeometries, the standing shocks characteristic oftilted simulations interact strongly with the or-

16

a. b.

c. d.

e. f.

Fig. 8.— Local radial decay rate a (red; see Equation 9) in the prograde, m = 1 component of variabilitycompared with the local radial decay constant adisp (blue; positive inside corotation, negative outside coro-tation) predicted from the dispersion relation in Equation 8 at specified frequencies in 90h (a), 910h (b),915h (c and d), 915h-64a (e), and 915h-64b (f).

17

a. b.

c. d.

e. f.

Fig. 9.— Snapshots of the spatial position of the maxima of the prograde pattern of the m = 1 mode powerat each radius (red; solid inward and dotted outward of corotation) for patterns at the indicated frequenciesin 90h (a), 910h (b), 915h (c and d), 915h-64a (e), and 915h-64b (f). For reference, we plot the approximatelocation of the background density maxima (light gray) and shock surfaces (dark gray) as well as circlesmarking the test-particle corotation radius (solid blue) and inner and outer Lindblad resonances (dashedblue). The prograde pattern rotates counterclockwise in time with respect to the density maxima and shocksurfaces.

18

biting spirals, producing regions of higher densitywhere the two come into contact. Since the trail-ing segments of spiral pattern are more tightlywound than the standing shocks, the high den-sity region appears to move outwardly along theshock surfaces. This mechanism explains the com-plex behavior shown in Figure 6 and provides anexplanation for the characteristically higher vari-ability at suborbital frequencies seen exclusivelyin tilted geometries.

6. Conclusions

In our previous paper (Henisey et al. 2009), weidentified variability at frequencies comparable tothose expected for inertial modes (i.e. below thelocal radial epicyclic frequency) in the inner partsof a GRMHD simulation of a tilted accretion flow(915h). This appeared to provide preliminary con-firmation that warps and eccentricity might ex-cite inertial modes even in the presence of MRIturbulence in accretion disks (Kato 2004a, 2009;Ferreira & Ogilvie 2008), whereas in shearing boxand global untilted flow simulations, such inertialmodes are destroyed by MRI turbulence (Arraset al. 2006; Reynolds & Miller 2009). However, ourmore complete analysis here of simulation 915h,along with other tilted simulations 910h and 915h-64, shows that the three-dimensional structure ofthe enhanced variability at suborbital frequenciesis not consistent with the excitation of trapped in-ertial modes.

The variability that we observe is in fact presentin both untilted and tilted simulations, and orig-inates from transient over-dense clumps of mate-rial co-orbiting with the background flow and ran-domly located at nearly all radii. Exhibiting apredominantly m = 1 azimuthal structure, suchfluctuations are likely only to manifest themselvesin simulations which incorporate a complete 2πtreatment of the azimuthal angle. We note thatDolence et al. (2012) also identified m = 1 struc-tures in their global GRMHD simulations that ap-peared to be associated with the QPO they identi-fied in their radiative post-processing of these sim-ulations. It may be that these structures are sim-ilar to the clumps that we have identified in ourwork. In any case, the formation mechanism forthese clumps remains unclear. Given that they or-bit with the background flow, it may be that they

are generated by some instability associated witha corotation resonance (e.g. Fu & Lai 2011), butthis requires further study.

We find that these orbiting clump structuresare accompanied by extended acoustic spiral waveswith pattern speeds equal to the orbital angularvelocity of the clump. Exhibiting power which ex-ponentially decreases away in radius from the or-bit of the clump, these spirals are consistent withforced acoustic perturbations inside the otherwiseevanescent region surrounding the corotation res-onance (Okazaki et al. 1987). In the untilted sim-ulation 90h, these spiral perturbations are quiteweak. As shown by Fragile & Blaes (2008), how-ever, converging eccentric orbits in tilted accretionflows produce two standing shock surfaces whichprecede two over-dense arms. The compressionof the orbiting flow at these shocks amplifies thedensity amplitude of the otherwise weak acousticspirals, thereby greatly enhancing their variabilitypower signature. We therefore hypothesize that,given a greater tilt, any moderately thick accretionflow will exhibit greater variability at frequenciescomparable to the orbital frequencies at all radiiinward of perhaps 10 to 15RG, that is, the furthestradial extent of the shocks.

These findings are quite relevant to a recentclass of models which produce a low-frequencyQPO in a hot, somewhat tilted, inner flow that isqualitatively consistent with observations of blackhole X-ray binaries (Ingram et al. 2009; Ingram &Done 2011). Such tilted flows naturally precess ata Lense-Thirring precession frequency determinedby the hot flow’s outermost edge (Fragile et al.2007). Therefore, in these low-frequency QPOmodels, we predict increased continuum variabilityat frequencies greater than the orbital frequencymeasured at either the hot flow’s outer cutoff ra-dius or the termination radius of the standingshocks, whichever is closer to the hole.

Over short intervals of time, the variability dueto the clumps and spirals occurs at discrete fre-quencies depending on which clumps happen to bepresent. While these frequencies are comparableto high-frequency QPOs in black hole X-ray bi-naries, the transient nature of the clumps impliesthat they cannot produce robust high-frequencyQPOs, and are likely instead to just contributeto enhanced continuum variability as we notedabove. We have not identified any other poten-

19

tial high-frequency QPO candidates in our simu-lations. We appear to be in the good companyof many other accretion simulations to date, andwe suspect that either the QPO is too weak todetect in current numerical simulations or, morelikely, that the QPO requires physics which is notyet captured by our numerical simulations. AsReynolds & Miller (2009) point out, current globalsimulations lack the necessary treatment of radi-ation physics to accurately simulate the near Ed-dington limited accretion which characterizes thesteep power-law spectral state uniquely associatedwith the high-frequency QPO. Therefore, in thecoming years when new numerical methods meetmore advanced computational technologies, vari-ability studies akin to those we have presentedhere will become increasingly relevant in under-standing the complex dynamics associated withblack hole accretion flows.

We thank Lars Bildsten for useful conversa-tions, and Aleksey Generozov and Julia Wilson forhelp in analyzing the density compression ratios ofthe shocks. This work was supported in part byNSF grant AST-0707624. PCF acknowledges sup-port from the National Science Foundation undergrants AST-0807385 and PHY11-25915.

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