The scale of the problem is difficult to overstate. AGB stars shine with luminosities of 100 to 1000 times that of the Sun and have effective temperatures below 3000 kelvin, giving them radii of several hundred solar radii. Red supergiants are even more extreme: hotter, between 3450 and 4100 kelvin, with radii that can exceed 1000 solar radii. Despite their vast size, their atmospheres are astonishingly thin and dynamic. Convection, the process that transports heat from the interior to the visible surface by bulk motion of hot gas, is not a local phenomenon in these stars. In the Sun, convection manifests as millions of small granules covering the photosphere. In giant envelopes, the pressure scale height is so large that individual convective plumes grow to sizes comparable to the stellar radius itself. Observations with optical interferometers have now directly imaged these giant cells on stars such as Betelgeuse, Antares and the AGB star π1 Gruis, revealing a handful of enormous bright and dark patches that evolve on weekly to yearly timescales.

Understanding such convection demands a fully three-dimensional, time-dependent treatment, because the process is non-local, non-linear, and spans wildly disparate length scales. Traditional one-dimensional stellar evolution models rely on mixing length theory, a prescription invented by Böhm-Vitense in 1958 that compresses all of convection into a single tunable parameter. For the Sun and solar-like stars, calibrated mixing lengths work reasonably well. But in the coolest and most luminous stars, where the convective plumes span a large fraction of the star, turbulence becomes supersonic, and radiative opacity varies by orders of magnitude, the one-dimensional approximation breaks down dramatically. The review emphasizes that only global simulations, often called star-in-a-box models, which encompass the majority of the convective envelope plus the near circumstellar environment, can capture the physics at play.

Two numerical codes anchor this effort. The first, CO5BOLD, developed over decades and first applied to global giant-star simulations by Freytag and collaborators in 2002, solves the coupled equations of compressible hydrodynamics and non-local radiative energy transport on a three-dimensional Cartesian grid embedded in a fixed, spherically symmetric gravitational field. It uses a finite-volume approach with directional splitting and an approximate Riemann solver of Roe type, along with a short-characteristics method for radiative transfer built on Rosseland mean opacity tables that merge high-temperature OPAL data with low-temperature PHOENIX data. The code can be extended to magnetohydrodynamics and can handle rotation through a corotating frame. The second code, Athena++, developed by Stone and colleagues, takes a different route: it employs adaptive mesh refinement, task-based dynamic execution, and a Godunov-type shock-capturing scheme, solving the time-dependent, frequency-integrated radiative transfer equation directly over discrete ordinates, with radiation fully coupled to the gas energy and momentum equations. For red supergiants, Athena++ simulations use spherical polar coordinates covering 70.6 percent of the face-on hemisphere, with an accurate gravitational potential that includes the mass of the simulated envelope itself, a detail that matters because these loosely bound atmospheres are anything but negligible in mass.

What do these simulations actually show? The picture that emerges is spectacularly turbulent. The surfaces of simulated AGB and red supergiant stars are dominated by a few giant convective cells, with wide upflows surrounded by narrow, turbulent downdrafts, evolving on timescales from weeks to years. Non-stationary convection in the deep interior generates sound waves. These waves travel outward, slow down and compress as the temperature and sound speed drop, and their amplitudes grow with the steep decline in density until they steepen into shocks. Just above the visible surface, the resulting shock network is extraordinary: turbulent pressure can exceed twenty times the gas pressure, shock waves propagate with average Mach numbers of 3 to 4, and local peaks reach Mach numbers of 8 to 10. Individual shocks above the photosphere can span 200 to 250 solar radii, a width comparable to the star’s own radius, at temperatures around 2500 kelvin. These shocks intermittently levitate gas to distances where dust can condense, providing the crucial first step in the wind-acceleration mechanism that drives the massive mass-loss rates of 10 to the minus 6 to 10 to the minus 4 solar masses per year observed in AGB stars.

Remarkably, the simulations show that radial pulsations are not imposed on these models but emerge spontaneously from the convective motions themselves, a result first demonstrated for AGB stars by Freytag and colleagues in 2017 and confirmed in later work by Ahmad and coauthors in 2023. The extracted pulsation periods agree with those observed in long-period variable stars. This interplay between convection and pulsation leaves an observable fingerprint in stellar spectra. Spectral lines form over a range of atmospheric depths, and each line carries the imprint of the local velocity field through its shift, width and asymmetry. The tomographic method, originally conceived to study shock waves in pulsating Cepheids and Mira variables, exploits this by sorting spectral lines according to their formation depth and cross-correlating them with observed spectra. In Mira stars, the technique beautifully reproduces the classic Schwarzschild scenario, in which a shock front marching upward through the atmosphere produces a sequence from red-shifted absorption to line doubling and finally blue-shifted emission.

Red supergiants, it turns out, play by different rules. Applied to years of high-resolution spectra of the prototypical red supergiants μ Cephei and Betelgeuse, the tomographic method revealed no such clean shock scenario. Instead, the atmospheric layers exhibit a phase shift between velocity and temperature variations, tracing a hysteresis loop in the temperature-velocity plane. Crucially, synthetic spectra computed from three-dimensional CO5BOLD simulations show the same hysteresis behavior with timescales matching the observed photometric variability. The interpretation is elegant: the loop is the signature of convective turnover, in which hot rising material reaches the upper atmosphere, cools, and falls back, modulated by acoustic waves generated by the turbulent convective flow below. The method even provided insight into Betelgeuse’s famous Great Dimming of early 2020, when the star faded to magnitude 1.6. Tomography identified two successive shocks, in February 2018 and January 2019, that expanded part of the atmosphere and, through increased molecular opacity in the cooler gas, obscured the star, an explanation consistent with, and complementary to, the dust-veil picture from direct imaging.

The Athena++ simulations have yielded equally consequential results on the theoretical side. Goldberg and colleagues in 2022 performed the first three-dimensional calibration of mixing length theory in the red supergiant regime, recovering surprisingly high convective efficiencies with a mixing length parameter of roughly 3 to 4, and showing that radiation pressure provides about one-third of the support against gravity in the envelope. The simulations also revealed an inverse correlation between density and radial velocity in the outer layers, uncharacteristic of ordinary convection, and an atmosphere far more extended than one-dimensional models predict. In these luminous layers, where the opacity locally exceeds the Eddington opacity, the usual correlation between dense sinking material and convection flips entirely: cold, opaque, dense regions move outward. This has direct implications for how red supergiants die. When such a star explodes as a Type II-P supernova, its shock must traverse this wildly inhomogeneous envelope, whose surface undulates over radial ranges of 10 to 20 percent of the stellar radius. The shock therefore breaks out at different radii and different times around the star, smearing the ultraviolet flash of shock breakout over roughly 3 to 6 hours instead of the sub-hour duration predicted by spherical models, and reducing peak luminosities by factors of 2 to 10. This means astronomers cannot use breakout rise times to measure progenitor radii via light-travel-time effects, but they can instead use breakout durations to probe the scale of convective inhomogeneities themselves.

The convergence of three-dimensional radiation-hydrodynamics modeling, interferometric imaging, ALMA observations of molecular envelopes, spectropolarimetric monitoring of surface magnetism, and tomographic spectroscopy is transforming our understanding of the late lives of stars. Convection, once a fudge factor in one-dimensional codes, is now a quantitatively constrained, observable, three-dimensional process that shapes winds, dust formation, pulsation, magnetism and the very first light of supernovae. As future ultraviolet missions such as ULTRASAT catch hundreds of shock breakouts at high cadence, and as ever more powerful simulations push toward even greater realism, the giant convective cells churning on the surfaces of dying stars will continue to yield their secrets, one shock wave, one spectral line, and one hysteresis loop at a time.

Subject of Research: Three-dimensional radiation-hydrodynamics simulations of convection in the atmospheres of asymptotic giant branch and red supergiant stars

Article Title: Signatures of convection in the atmospheres of cool evolved stars

Article References: Signatures of convection in the atmospheres of cool evolved stars. (n.d.). https://doi.org/10.1007/s41115-024-00020-w

Image Credits: AI Generated

DOI: 10.1007/s41115-024-00020-w

Keywords: stellar convection, red supergiants, AGB stars, radiation hydrodynamics, CO5BOLD, Athena++, tomography, Betelgeuse, shock breakout, supernovae, mixing length theory, stellar mass loss

The review, led by Hubert Klahr, Hans Baehr, Julio David Melon Fuksman, and Thomas Pfeil of the Max Planck Institute for Solar System Research, arrives at a moment when observational facilities such as the Atacama Large Millimeter/submillimeter Array are delivering breathtaking images of planet-forming disks. Rings, gaps, spiral arms, and hints of long-lived vortices appear routinely in the data, but interpreting these structures requires models that faithfully capture the underlying physics. The authors argue that the first step is deceptively simple: before adding magnetic fields, self-gravity, or dust feedback, a simulation must correctly reproduce the pure hydrodynamic behavior of an irradiated disk, and that behavior is far from trivial.

Disks around young stars are born as byproducts of star formation, a buffer for the excess angular momentum of a collapsing molecular cloud core. During the first tens of thousands of years, the star acquires most of its mass through the disk, a phase in which the disk’s own self-gravity supplies the main torque driving accretion. But after this main accretion phase, enough material remains to build planetary systems over the following ten million years, until winds and photoevaporation strip the gas away. Crucially, for planet formation, an overly vigorous accretion flow toward the star can wash away nascent planets, so understanding when and how disks settle into a calmer state matters enormously.

One of the review’s central themes is baroclinicity, the misalignment between surfaces of constant pressure and constant density in a disk. Because stellar irradiation heats the disk surface while viscous dissipation may warm the midplane, disks develop both radial and vertical temperature gradients. In such a configuration, pressure and density contours are inclined with respect to one another, a situation familiar from Earth’s atmosphere and oceans that also drives instabilities in disks. These effects were long missed in simulations because they are too weak to emerge at low resolution or with overly dissipative numerical schemes. Only recently, with increased computing power, have researchers been able to confirm the analytic predictions of thermal baroclinic instability theory with full numerical experiments.

The review catalogues the family of instabilities that radiation hydrodynamic simulations must capture. The vertical shear instability, or VSI, arises from the vertical gradient of rotation in a baroclinic disk and operates best when thermal relaxation is extremely fast, since rapid cooling prevents the stable vertical stratification from suppressing the unstable shear. The convective overstability, in contrast, is strongest when the thermal relaxation time is comparable to the orbital period; it grows from epicyclic oscillations of gas parcels displaced radially in a weakly convective environment. The Goldreich-Schubert-Fricke instability, a close cousin of the VSI inherited from the theory of rotating stars, operates alongside the convective overstability, and recent work has shown that any disk unstable to one is unstable to the other, because both depend equally on the disk’s baroclinicity. A third mechanism, the subcritical baroclinic instability, generates long-lived anticyclonic vortices from radial entropy gradients combined with thermal relaxation, though it lacks a linear growth-rate prediction and is therefore harder to use as a code benchmark.

To help researchers validate their codes, the authors lay out a systematic testing procedure. The method begins by constructing an equilibrium disk model, typically through a so-called 1+1-dimensional calculation that solves for vertical hydrostatic balance at each radius while enforcing energy conservation with flux-limited diffusion and physically motivated dust opacities. This equilibrium serves both as an initial condition for multidimensional simulations and as the basis for perturbation theory: analytic linear analysis yields predicted growth rates for unstable modes. A code that reproduces those growth rates in the linear regime can then be trusted to explore the nonlinear turbulence that follows. If a code fails to match the predicted growth, the review notes, this signals insufficient resolution or a numerically dissipative scheme rather than a failure of the underlying theory.

The practical details of such tests are demanding. In local axisymmetric simulations carved from a global disk model, the authors demonstrate that resolving roughly 256 cells per pressure scale height is required to reproduce growth rates down to ten thousandths of the orbital frequency. Numerical schemes matter as well: high-order reconstruction methods combined with accurate Riemann solvers recover the predicted linear growth, whereas more diffusive approximate solvers can suppress instability entirely. Even the handling of cooling, implemented through thermal relaxation of the pressure toward an equilibrium value, must be treated carefully, with operator splitting modified so that relaxation times shorter than the dynamical step remain stable. These benchmarks, the authors argue, should become standard practice for any group embarking on radiation hydrodynamic disk simulations.

Beyond linear tests, the review surveys how different radiative transfer approximations shape the outcomes of full three-dimensional simulations. Flux-limited diffusion, the workhorse of earlier decades, is computationally efficient and accurate in optically thick regions but smears out shadows and introduces unphysical diffusion where radiation streams freely. The M1 two-moment method, implemented in codes such as PLUTO, preserves the direction of radiative fluxes and captures shadowing, but it artificially merges crossing beams of light, which can overestimate midplane temperatures by more than 40 percent in single-group calculations. A newer half-moment scheme reduces that error to a few percent, while discrete ordinates and Monte Carlo methods offer the greatest accuracy at the highest cost. The review’s message is that no single method suits every problem, and the choice must weigh computational expense against the physics one needs to capture, whether self-shadowing, scattering, or frequency-dependent heating.

The consequences of getting radiation transport right extend deep into planet formation theory. Global simulations of the vertical shear instability show that the resulting turbulence generates stresses whose strength depends on the square of the radial temperature gradient and on the local cooling time, challenging the classical assumption that turbulent viscosity scales simply with gas pressure. VSI turbulence can spawn long-lived anticyclonic vortices, provided simulations span the full azimuthal extent of the disk, and these storm systems are expected to act as efficient traps for pebbles and dust, potentially accelerating planetesimal formation. In irradiated disks, radiation hydrodynamical studies reveal that the instability can be localized to the surface layers when dust depletion lengthens midplane cooling times, producing a quiescent midplane beneath a vigorously turbulent atmosphere, a stratification with direct consequences for where dust can settle and planets can grow.

The review closes with a forward-looking agenda. The authors recommend abandoning fixed-temperature disk models in favor of self-consistent thermal evolution, since every realistic disk structure is subject to thermal baroclinic instabilities that fixed-temperature setups either suppress artificially or misrepresent. Future work must couple evolving dust populations to the opacity and cooling calculations, treat the separate temperatures of gas, dust, and radiation, and develop well-balanced or low-Mach-number schemes that resolve subsonic fluctuations in a supersonically rotating medium. The payoff is substantial: radiation hydrodynamic simulations calibrated against linear theory will underpin the interpretation of molecular line kinematics and scattered-light images from current and next-generation telescopes, transforming stunning pictures of planet-forming disks into quantitative tests of how worlds are born.

Subject of Research: Numerical radiation hydrodynamics methods and stability tests for simulating protoplanetary disks around young stars

Article Title: Numerical radiation hydrodynamics for circumstellar disks

Article References: Klahr, H., Baehr, H., Melon Fuksman, J. D., & Pfeil, T. (2026). Numerical radiation hydrodynamics for circumstellar disks. Living Reviews in Computational Astrophysics, 12(1), Article 3. https://doi.org/10.1007/s41115-026-00026-6

Image Credits: AI Generated

DOI: 10.1007/s41115-026-00026-6

Keywords: protoplanetary disks, radiation hydrodynamics, planet formation, vertical shear instability, convective overstability, baroclinic instability, radiative transfer, turbulence, numerical simulations, stellar irradiation, computational astrophysics, disk stability