arXiv subjects

Gordon I. Ogilvie

Publications and source records attributed to Gordon I. Ogilvie.

At least 19 recordsLinked to original sources

arXiv2026

Resonant three-mode coupling in asteroseismology I: Bifurcations, chaos, and DAV white-dwarf outbursts

Resonant three-mode coupling is a fundamental nonlinear process in stellar oscillations and other wave systems. We investigate the global dynamics of a linearly overstable parent mode coupled to two damped daughter modes. The principal control parameters are frequency detuning $Δ$ and daughter damping rate $Γ$, normalized by the parent growth rate. We combine high-resolution maps of the largest Lyapunov exponent in the $(Γ,Δ)$ plane with linear stability analysis, Floquet calculations, and numerical continuation. The resulting parameter space includes stable equilibria, simple and period-doubled limit cycles, chaotic attractors, formally unbounded solutions, and higher-period `shrimp' islands within chaotic regions. Successive period doublings lead to chaos, while boundaries of isolated periodic windows exhibit type-I intermittency. At large $Γ$, the main bifurcation boundaries approach straight lines, indicating that the dynamics is governed primarily by $r=Δ/(2Γ)$. We connect finite-$Γ$ bifurcations to an earlier one-dimensional return map. As $ε\rightarrow0$, this map recovers the principal period-doubling sequence, the onset of chaos near $r\simeq0.38$, and the asymptotic positions of prominent periodic windows. We also examine the effects of unequal daughter damping rates. Applying these results to sporadic outbursts in cool DAV white dwarfs, we find that estimated $g$-mode growth rates, damping rates, and detunings place resonant triplets in both periodic and chaotic regimes. Thus, instability of the three-mode equilibrium need not produce a regular limit cycle: deterministic chaos and intermittency may contribute to irregular outburst amplitudes and recurrence times.

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arXiv2026

Disc breaking and parametric instability in warped accretion discs driven by differential precession

We present the first local simulations of accretion disc breaking (or disc tearing) with differential precession. Precession can occur due to a non-spherical potential (e.g. tilted circumbinary discs), or relativistic effects (e.g. tilted discs around spinning black holes). This tends to induce a twist in the disc, potentially causing it to break into rings. We extend earlier work on free (unforced) warps in Held+2026 [arXiv:2511.18655v1] to include differential precession in the shearing box, and carry out quasi-2D, isothermal, high-resolution ($N = 32$ cells per scale-height $H$) numerical simulations in which an imposed precession drives a warp in a tilted disc in either the wavelike ($α< H/r$) or diffusive ($α> H/r$) regime. Our key finding is that, at sufficiently large tilts and precession amplitudes, the disc breaks into multiple (4-5) mutually tilted rings exhibiting a rich morphology: from wide ($\sim 10H$), flat rings to compact tori of width $\sim H$. Interactions between rings occur as a result of oblique shocks that form in the gaps between them, leading to instances of disc rejoining. Smaller tilts and precession amplitudes lead instead to parametric instability of inertial waves in the disc. We develop a linear viscous theory of bending waves under the influence of an imposed precession. Finally, we present a systematic study of the effects of grid-based dissipation on tilted discs, and find that numerical dissipation damps the tilt and induces precession. This effect decreases with resolution as $\propto (1/N)^β$, $2.3 \lesssim β\lesssim 3.9$.

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arXiv2026

Tilted, warped, and eccentric disks

We review some of the interesting consequences that tilts, warps, and eccentricities can introduce into the dynamics, thermodynamics, and observational appearance of accreting systems, with an emphasis on disks around black holes and compact stars. We begin with a review of the two types of precession that are associated with eccentric and tilted orbits in general relativity and Newtonian gravity. We then discuss the types of accretion systems that may manifest tilted or eccentric disks. In separate sections we discuss first tilted and then eccentric disks, each section covering relevant and interesting observational, theoretical, and numerical results. Next, we explore potential connections between the phenomenology of quasi-periodic oscillations and either tilted or eccentric disks. Finally, we present some concluding thoughts and discuss future directions this research might take.

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arXiv2025

Non-linear spiral waves in accretion discs

We derive a simple, accurate, non-linear, global equation governing spiral density waves in thin, non-self-gravitating, inviscid accretion discs. These discs may have any slowly varying surface density or temperature profile. For specific 'self-similar' disc profiles, solutions to our equation match (novel) smooth non-linear exact spiral solutions derived via a separate method, which highlight that non-linear spiral waves need not shock. Indeed, at low amplitudes, we find that dispersion can overcome wave steepening, and may prevent the inner spiral wakes excited by low mass planets (below roughly 1% of a thermal mass) embedded in protoplanetary discs from shocking. At high amplitudes, we find a simple universal description of non-linear spiral waves with shocks, as well as caps on the possible amplitude and wave action flux of non-linear spirals both with and without shocks, depending on how many arms they have. We further find that highly non-linear spirals are far more loosely wound than their linear counterparts. These developments shed light on why two-armed spirals are prevalent across a range of astrophysical systems which don't necessarily possess an intrinsic twofold symmetry, and why they appear surprisingly loosely wound in observations. These results are supported by very high-resolution numerical simulations.

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arXiv2025

Disc breaking and parametric instability in warped accretion discs

We present the first local simulations of disc breaking/tearing in a warped accretion disc. Warps can arise due to a misalignment between the disc and the rotation axis of the central object, or a misalignment with the orbital plane of a binary (or planetary) companion. Warped discs can break into rings, as found in observations of circumbinary protoplanetary discs and global simulations of tilted discs around spinning black holes. In this work we isolate the mechanism of disc breaking in high-resolution, quasi-2D, local (shearing box), hydrodynamic simulations of a Keplerian disc. We consider the evolution of a free (unforced) warp in the wavelike ($α< H/r$) regime. At large warp amplitudes ($ψ_{\text{max}} \gtrsim 1$) the disc breaks into four rings on timescales of around 20 orbits which are separated by gaps of around $\sim 10H_0$. The warp exhibits a rich tapestry of small-scale dynamics, including horizontal sloshing motions, vertical oscillations or bouncing, warp steepening, and shocks. The shocks act as a source of enhanced dissipation which facilitates gap opening and thus disc breaking. At smaller warp amplitudes $ψ_{\text{max}} \lesssim 1$, for which we also develop a quasi-linear theory, the disc does not break, but instead exhibits hydrodynamic parametric instability. We also investigate the effect of viscosity: at small warp amplitudes the parametric instability is damped and the warp propagates as a pure bending wave, while at large warp amplitudes the emerging gaps are partially filled by viscous diffusion.

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arXiv2025

Instability and vertical eccentricity variation in global hydrodynamic disk simulations

Many dynamical interactions can induce eccentricities in astrophysical accretion disks. Disk eccentricities in turn seed a variety of instabilities, even in ideal hydrodynamics. We use 3D nonlinear simulations and 2+1D linear calculations to characterize local and global instabilities in strongly distorted disks. On local scales, our simulations show the growth of parametrically excited inertial waves, which drive wave turbulence. The inertial waves' growth rates and localizations agree with the predictions of local theory. On global scales, we observe the growth of a separate family of low-frequency, vertically structured modes that compare favorably with eigenmodes computed from the linear theory of an eccentric background state. These low-frequency modes interact nonlinearly with the inertial wave turbulence driven by parametric instability, and they induce variation in eccentricity profiles that are initially uniform in the vertical direction. Extrapolating from our vertically local framework, we postulate that these secondary distortions may correspond to the corrugation of an initially planar eccentric disk. Our simulations demonstrate that strong disk eccentricities drive numerous dynamical phenomena even in a purely hydrodynamic, Newtonian framework.

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arXiv2025

High-resolution models of the vertical shear instability

(abridged) Context: The vertical shear instability (VSI) is a promising mechanism to generate turbulence and transport angular momentum in protoplanetary discs. While most recent work has focused on adding more complex physics, the saturation properties of the instability in radially extended discs and its convergence as a function of resolution are still largely unknown. We tackle the question of VSI saturation and associated turbulence using radially extended fully 3D global disc models at very high resolution so as to capture both the largest VSI scales and the small-scale turbulent cascade. We use the GPU-accelerated code Idefix to achieve resolutions of up to 200 points per scale height in the 3 spatial directions, with a full 2pi azimuthal extent and disc aspect ratio H/R=0.1. Results: We demonstrate that large-scale transport properties are converged with 100 points per scale height, leading to a Shakura-Sunyaev alpha=1.3e-3. Inner boundary condition artifacts propagate deep inside the computational domain, leading to reduced alpha in these regions. The large-scale corrugation wave zones identified in 2D models survive in 3D, albeit with less coherence. Our models show no sign of long-lived zonal flows, pressure bumps or vortices, in contrast to lower-resolution simulations. Finally, we show that the turbulent cascade resulting from VSI saturation can be interpreted in the framework of critically balanced rotating turbulence. Conclusion: The VSI leads to vigorous turbulence in protoplanetary discs, associated with outward angular momentum transport but without any significant long-lived features that could enhance planet formation. The innermost regions of VSI simulations are always polluted by boundary-condition artifacts affecting the first VSI wave train, so radially extended domains should be used in a more systematic manner.

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arXiv2025

The Laplace Surface of a Circumplanetary Disc

The classical Laplace surface defines the location of circular particle orbits that do not undergo nodal precession around a planet with some obliquity. Close to the planet the surface coincides with the equator of the planet, while far from the planet it coincides with the orbital plane of the planet. We determine the shape of the Laplace surface of a circumplanetary disc that results from accretion of circumstellar gas and experiences the effects of gas pressure, self-gravity, and viscosity, as well as the gravitational effects due to the planetary spin and the star. We apply the linear theory of warped discs in the wavelike regime for a small-obliquity planet such as Jupiter. As a result of dissipation, a disc that begins slightly away from its Laplace surface will evolve to it. Because of pressure effects in typically warm circumplanetary discs, the disc is highly flattened compared with the classical Laplace surface, meaning that it is much less warped but still significantly tilted. For the case of Jupiter, the disc does not align anywhere with the equator of the planet. For such discs, the effects of self-gravity and viscosity on warping are typically small. The disc tilt is intermediate between the planet's equatorial and orbital planes. Circumplanetary discs that are much cooler than expected can undergo warping and alignment with the planet's equator at small radii. The results have implications for the orbital evolution of satellites in the solar system that are observed to be somewhat aligned with their classical Laplace surface.

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arXiv2025

Gravitational instability and affine dynamics of gaseous astrophysical discs

We develop several aspects of the theory of gaseous astrophysical discs in which the gravity of the disc makes a significant contribution to its structure and dynamics. We show how the internal gravitational potential can be expanded in powers of the aspect ratio of the disc (or of a structure within it) and separated into near and far contributions. We analyse the hydrostatic vertical structure of a wide family of disc models, both analytically and numerically, and show that the near contribution to the internal gravitational potential energy can be written in an almost universal form in terms of the surface density and scaleheight. We thereby develop an affine model of the dynamics of (generally non-hydrostatic) self-gravitating discs in which this contribution to the energy acts as a gravitational pressure in the plane of the disc. This combines with and significantly reinforces the gas pressure, allowing us to define an enhanced effective sound speed and Toomre stability parameter Q for self-gravitating discs. We confirm that this theory fairly accurately reproduces the onset of axisymmetric gravitational instability in discs with resolved vertical structure. Among other things, this analysis shows that the critical wavelength is on the order of twenty times the scaleheight, helping to justify the validity of the affine model. The weakly nonlinear theory also typically exhibits subcritical behaviour, with equilibrium solutions of finite amplitude being found in the linearly stable regime Q > 1 for adiabatic exponents less than 1.50.

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arXiv2025

The vertical shear instability in protoplanetary discs as an outwardly travelling wave. I. Linear theory

We revisit the global linear theory of the vertical shear instability (VSI) in protoplanetary discs with an imposed radial temperature gradient. We focus on the regime in which the VSI has the form of a travelling inertial wave that grows in amplitude as it propagates outwards. Building on previous work describing travelling waves in thin astrophysical discs, we develop a quantitative theory of the wave motion, its spatial structure and the physical mechanism by which the wave is amplified. We find that this viewpoint provides a useful description of the large-scale development of the VSI in global numerical simulations, which involves corrugation and breathing motions of the disc. We contrast this behaviour with that of perturbations of smaller scale, in which the VSI grows into a nonlinear regime in place without significant radial propagation.

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arXiv2025

Nonlinear analysis of gravitational instability in a 3D gaseous disc

Astrophysical discs which are sufficiently massive and cool are linearly unstable to the formation of axisymmetric structures. In practice, linearly stable discs of surface density slightly below the threshold needed for this instability often form spiral structures, and can subsequently fragment or exhibit a state of self-sustained turbulence, depending on how rapidly the disc cools. This has raised the question of how such turbulence is possible in the linearly stable regime. We suggest a nonlinear mechanism for this phenomenon. We find analytically weakly nonlinear axisymmetric subcritical solitary equilibria which exist in linearly stable 3D discs that are close to the instability threshold. The energy of these 'soliton' solutions is only slightly higher than that of a uniform disc, and the structures themselves are expected to be unstable to non-axisymmetric perturbations. In this way, these subcritical solitary equilibria highlight a nonlinear instability and provide a possible pathway to a turbulent state in linearly stable discs.

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arXiv2024

Instability and warping in vertically oscillating accretion discs

Many accretion discs have been found to be distorted: either warped due a misalignment in the system, or non-circular as a result of orbital eccentricity or tidal deformation by a binary companion. Warped, eccentric, and tidally distorted discs are not in vertical hydrostatic equilibrium, and thus exhibit vertical oscillations in the direction perpendicular to the disc, a phenomenon that is absent in circular and flat discs. In extreme cases, this vertical motion is manifested as a vertical `bouncing' of the gas, potentially leading to shocks and heating, as observed in recent global numerical simulations. In this paper we isolate the mechanics of vertical disc oscillations by means of quasi-2D and fully 3D hydrodynamic local (shearing-box) models. To determine the numerical and physical dissipation mechanisms at work during an oscillation we start by investigating unforced oscillations, examining the effect of initial oscillation amplitude, as well as resolution, boundary conditions, and vertical box size on the dissipation and energetics of the oscillations. We then drive the oscillations by introducing a time-dependent gravitational potential. A key result is that even a purely vertically oscillating disc is (parametrically) unstable to developing inertial waves, as we confirm through a linear stability analysis. The most important of these has the character of a bending wave, whose radial wavelength depends on the frequency of the vertical oscillation. The nonlinear phase of the instability exhibits shocks, which dampen the oscillations, although energy can also flow from the bending wave back to the vertical oscillation.

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arXiv2024

Horseshoes and spiral waves: capturing the 3D flow induced by a low-mass planet analytically

The key difficulty faced by 2D models for planet-disc interaction is in appropriately accounting for the impact of the disc's vertical structure on the dynamics. 3D effects are often mimicked via softening of the planet's potential; however, the planet-induced flow and torques often depend strongly on the choice of softening length. We show that for a linear adiabatic flow perturbing a vertically isothermal disc, there is a particular vertical average of the 3D equations of motion which exactly reproduces 2D fluid equations for arbitrary adiabatic index. There is a strong connection here with the Lubow-Pringle 2D mode of the disc. Correspondingly, we find a simple, general prescription for the consistent treatment of planetary potentials embedded within '2D' discs. The flow induced by a low-mass planet involves large-scale excited spiral density waves which transport angular momentum radially away from the planet, and 'horseshoe streamlines' within the co-orbital region. We derive simple linear equations governing the flow which locally capture both effects faithfully simultaneously. We present an accurate co-orbital flow solution allowing for inexpensive future study of corotation torques, and predict the vertical structure of the co-orbital flow and horseshoe region width for different values of adiabatic index, as well as the vertical dependence of the initial shock location. We find strong agreement with the flow computed in 3D numerical simulations, and with 3D one-sided Lindblad torque estimates, which are a factor of 2 to 3 times lower than values from previous 2D simulations.

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arXiv2023

Tidally excited gravity waves in the cores of solar-type stars: resonances and critical-layer formation

We simulate the propagation and dissipation of tidally induced nonlinear gravity waves in the cores of solar-type stars. We perform hydrodynamical simulations of a previously developed Boussinesq model using a spectral-element code to study the stellar core as a wave cavity that is periodically forced at the outer boundary with a given azimuthal wavenumber and an adjustable frequency. For low-amplitude forcing, the system exhibits resonances with standing g-modes at particular frequencies, corresponding to a situation in which the tidal torque is highly frequency-dependent. For high-amplitude forcing, the excited waves break promptly near the centre and spin up the core so that subsequent waves are absorbed in an expanding critical layer, as found in previous work, leading to a tidal torque with a smooth frequency-dependence. For intermediate-amplitude forcing, we find that linear damping of the waves gradually spins up the core such that the resonance condition can be altered drastically. The system can evolve towards or away from g-mode resonances, depending on the difference between the forcing frequency and the closest eigenfrequency. Eventually, a critical layer forms and absorbs the incoming waves, leading to a situation similar to the high-amplitude case in which the waves break promptly. We study the dependence of this process on the forcing amplitude and frequency, as well as on the diffusion coefficients. We emphasize that the small Prandtl number in the centre of solar-like stars facilitates the development of a differentially rotating core owing to the nonlinear feedback of waves. Our simulations and analysis reveal that this important mechanism may drastically change the phase of gravity waves and thus the classical picture of resonance locking in solar-type stars needs to be revised.

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arXiv2023

Parametric instability in warped astrophysical discs: growth, saturation and feedback

Attempts to understand the dynamics of warped astrophysical discs have garnered significant attention, largely motivated by the growing catalogue of observed distorted systems. Previous studies have shown that the evolution of the warp is crucially regulated by the internal flow fields established by the undulating geometry. These are typically modelled as laminar horizontal, shearing flows which oscillate back and forth at approximately the orbital frequency. However this shearing motion is known to be susceptible to a hydrodynamic, parametric instability of inertial waves which might modify the warped dynamics. Whilst the linear growth phase is well understood, the subsequent nonlinear saturation combined with the self-consistent feedback onto the warp has not been studied. In this work, we implement a novel numerical setup using the recent ring model framework of Fairbairn and Ogilvie, within the Lagrangian code GIZMO. We formally identify several locally growing modes in the simulation, as predicted by a three-mode coupling analysis of the instability, and find decent agreement with the theoretical growth rates. We understand the saturation mechanism as a wave breaking process which suppresses the growth of shorter wavelength parametric couplings first, whilst allowing the longest mode to dominate the final quasi-steady, wavelike turbulence. The Reynolds stresses, transporting energy from the warp to the small scales, can be effectively modelled using a time-dependent, anisotropic viscous alpha model which closely captures the amplitude and phase evolution of the warp. Consequently, this model might help inform future global studies which are commonplace but typically don't resolve the parametric instability.

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arXiv2022

A New Window to Tidal Asteroseismology: Non-linearly Excited Stellar Eigenmodes and the Period Spacing Pattern in KOI-54

We revisit the Tidally Excited Oscillations (TEOs) in the A-type, main-sequence, eccentric binary KOI-54, the prototype of heartbeat stars. Although the linear tidal response of the star is a series of orbital-harmonic frequencies which are not stellar eigenfrequencies, we show that the non-linearly excited non-orbital-harmonic TEOs are eigenmodes. By carefully choosing the modes which satisfy the mode-coupling selection rules, a period spacing ($ΔP$) pattern of quadrupole gravity modes ($ΔP \approx 2520-2535$ sec) can be discerned in the Fourier spectrum, with a detection significance level of $99.9\%$. The inferred period spacing value agrees remarkably well with the theoretical $l=2,m=0$ g modes from a stellar model with the measured mass, radius and effective temperature. We also find that the two largest-amplitude TEOs at $N=90, 91$ harmonics are very close to resonance with $l=2,m=0$ eigenmodes, and likely come from different stars. Previous works on tidal oscillations primarily focus on the modeling of TEO amplitudes and phases, the high sensitivity of TEO amplitude to the frequency detuning (tidal forcing frequency minus the closest stellar eigenfrequency) requires extremely dense grids of stellar models and prevents us from constraining the stellar physical parameters easily. This work, however, opens the window of real tidal asteroseismology by using the eigenfrequencies of the star inferred from the non-linear TEOs and possibly very-close-to-resonance linear TEOs. Our seismic modeling of these identified eigen g-modes shows that the best-matching stellar models have ($M \approx 2.20, 2.35 M_{\odot}$) and super-solar metallicity, in good agreement with previous measurements.

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arXiv2022

Standing solitary waves as transitions to spiral structures in gravitationally unstable accretion disks

Astrophysical disks that are sufficiently cold and dense are linearly unstable to the formation of axisymmetric rings as a result of the disk's gravity. In practice, spiral structures are formed, which may in turn produce bound fragments. We study a nonlinear dynamical path that can explain the development of spirals in a local model of a gaseous disk on the subcritical side of the gravitational instability bifurcation. Axisymmetric equilibria can be radially periodic or localized, in the form of standing solitary waves. The solitary solutions have an energy slightly larger than a smooth disk. They are further unstable to non-axisymmetric perturbations with a wide range of azimuthal wavenumbers. The solitary waves may act as a pathway to spirals and fragmentation.

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arXiv2022

Hydrodynamics of warps in the local model of astrophysical discs

We show how the local approximation of astrophysical discs, which is the basis for the well known model of the shearing box, can be used to study many aspects of the dynamics of warped discs. In the local model, inclination of the orbit of a test particle with respect to the reference orbit corresponds to a vertical oscillation of the particle at the orbital frequency. Warping of a disc corresponds to a locally axisymmetric corrugation of the midplane of the disc that oscillates vertically at the orbital frequency, while evolution of the warp corresponds to a modulation of the complex amplitude of the vertical oscillation. We derive a conservation law for this amplitude that is the local equivalent of the conservation of angular momentum and therefore governs the evolution of the warp. For lengthscales that are long compared to the vertical scaleheight of the disc, the known non-resonant and resonant regimes of warp dynamics, including the diffusive and wavelike regimes of Keplerian discs, occur in the local model in the same way as in a global view of warped discs.

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