Stellar Convection Research Articles

ENERGY CONSERVATION AND GRAVITY WAVES IN SOUND-PROOF TREATMENTS OF STELLAR INTERIORS. II. LAGRANGIAN CONSTRAINED ANALYSIS

The speed of sound greatly exceeds typical flow velocities in many stellar and planetary interiors. To follow the slow evolution of subsonic motions, various sound-proof models attempt to remove fast acoustic waves whilst retaining stratified convection and buoyancy dynamics. In astrophysics, anelastic models typically receive the most attention in the class of sound-filtered stratified models. Generally, anelastic models remain valid in nearly adiabatically stratified regions like stellar convection zones, but may break down in strongly sub-adiabatic, stably stratified layers common in stellar radiative zones. However, studying stellar rotation, circulation, and dynamos requires understanding the complex coupling between convection and radiative zones, and this requires robust equations valid in both regimes. Here we extend the analysis of equation sets begun in Brown Vasil & Zweibel 2012, which studied anelastic models, to two types of pseudo-incompressible models. This class of models has received attention in atmospheric applications, and more recently in studies of white-dwarf supernovae progenitors. We demonstrate that one model conserves energy but the other does not. We use Lagrangian variational methods to extend the energy conserving model to a general equation of state, and dub the resulting equation set the Generalized Pseudo-Incompressible (GPI) model. We show that the GPI equations suitably capture low frequency phenomena in both convection and radiative zones in stars and other stratified systems, and we provide recommendations for converting low-Mach number codes to this equation set.

Differential rotation and meridional flows in stellar convection zones

AbstractDifferential rotation and meridional flow are key ingredients in flux transport dynamo models of the solar activity cycle. As the subsurface flow pattern is not sufficiently constrained by observations, it is a major source of uncertainty in solar and stellar dynamo models. We discuss the current mean field theory of stellar differential rotation and meridional flows and its predicitons for the Sun and stars on the lower main sequence.

Are There Pulsationally Unstable Low-degree p1 Modes in the Sun?

By using a non-local time-dependent theory of stellar convection, the solar non-adiabatic pulsations of the low- and intermediate-degree (l < 25) modes are calculated. The results show that the non-radial p1 modes of l = 1–5 are pulsationally unstable. However, the adjacent g, f, p2-p5 modes and the p1 modes of l > 5 are stable. From the analysis of the diagram of integrated work it is discovered that the excitation of oscillations comes from the radiation zone beneath the convective region. Whether the sun possesses unstable low-degree p1 modes is of significant importance for clarifying the excitation mechanism of solar five-minute oscillations.

Zonal flow regimes in rotating anelastic spherical shells: An application to giant planets

The surface zonal winds observed in the giant planets form a complex jet pattern with alternating prograde and retrograde direction. While the main equatorial band is prograde on the gas giants, both ice giants have a pronounced retrograde equatorial jet.We use three-dimensional numerical models of compressible convection in rotating spherical shells to explore the properties of zonal flows in different regimes where either rotation or buoyancy dominates the force balance. We conduct a systematic parameter study to quantify the dependence of zonal flows on the background density stratification and the driving of convection.In our numerical models, we find that the direction of the equatorial zonal wind is controlled by the ratio of the global-scale buoyancy force and the Coriolis force. The prograde equatorial band maintained by Reynolds stresses is found in the rotation-dominated regime. In cases where buoyancy dominates Coriolis force, the angular momentum per unit mass is homogenized and the equatorial band is retrograde, reminiscent to those observed in the ice giants. In this regime, the amplitude of the zonal jets depends on the background density contrast with strongly stratified models producing stronger jets than comparable weakly stratified cases. Furthermore, our results can help to explain the transition between solar-like (i.e. prograde at the equator) and the “anti-solar” differential rotations (i.e. retrograde at the equator) found in anelastic models of stellar convection zones.In the strongly stratified cases, we find that the leading order force balance can significantly vary with depth. While the flow in the deep interior is dominated by rotation, buoyancy can indeed become larger than Coriolis force in a thin region close to the surface. This so-called “transitional regime” has a visible signature in the main equatorial jet which shows a pronounced dimple where flow amplitudes notably decay towards the equator. A similar dimple is observed on Jupiter, which suggests that convection in the planet interior could possibly operate in this regime.

Convective line shifts for theGaiaRVS from the CIFIST 3D model atmosphere grid

To derive space velocities of stars along the line of sight from wavelength shifts in stellar spectra requires accounting for a number of second-order effects. For most stars, gravitational redshifts, convective blueshifts, and transverse stellar motion are the dominant contributors. We provide theoretical corrections for the net velocity shifts due to convection expected for the measurements from the Gaia Radial Velocity Spectrometer (RVS). We used a set of three-dimensional time-dependent simulations of stellar surface convection computed with CO5BOLD to calculate spectra of late-type stars in the Gaia RVS range and to infer the net velocity offset that convective motions will induce in radial velocities derived by cross-correlation. The net velocity shifts derived by cross-correlation depend both on the wavelength range and spectral resolution of the observations. Convective shifts for Gaia RVS observations are less than 0.1 km/s for late-K-type stars, and they increase with stellar mass, reaching about 0.3 km/s or more for early F-type dwarfs. This tendency is the result of an increase with effective temperature in both temperature and velocity fluctuations in the line-forming region. Our simulations also indicate that the net RVS convective shifts can be positive (i.e. redshifts) in some cases. Overall, the blueshifts weaken slightly with increasing surface gravity, and are enhanced at low metallicity. Gravitational redshifts amount up to 0.7 km/s and dominate convective blueshifts for dwarfs, but become much weaker for giants.

3D hydrodynamical simulations to interpret observations of stellar surfaces of red supergiant stars

As red supergiants are the largest and brightest stars, they are ideal targets for the new generation of sensitive, high resolution instrumentation that provides spectrophotometric, interferometric, astrometric, and imaging observables. The interpretation of the complex stellar surface images requires numerical simulations of stellar convection that take into account multi-dimensional time-dependent radiation hydrodynamics with realistic input physics. We show the results obtained with the synergy between the radiative hydrodynamics code CO5BOLD and the post-processing radiative transfer code Optim3D. Such simulations support a proper and quantitative analysis of these observations, and the observations provide the theoretical work with strong constraints.

NEW SCALING FOR THE ALPHA EFFECT IN SLOWLY ROTATING TURBULENCE

Using simulations of slowly rotating stratified turbulence, we show that the alpha effect responsible for the generation of astrophysical magnetic fields is proportional to the logarithmic gradient of kinetic energy density rather than that of momentum, as was previously thought. This result is in agreement with a new analytic theory developed in this paper for large Reynolds numbers. Thus, the contribution of density stratification is less important than that of turbulent velocity. The alpha effect and other turbulent transport coefficients are determined by means of the test-field method. In addition to forced turbulence, we also investigate supernova-driven turbulence and stellar convection. In some cases (intermediate rotation rate for forced turbulence, convection with intermediate temperature stratification, and supernova-driven turbulence) we find that the contribution of density stratification might be even less important than suggested by the analytic theory.

Non-LTE line formation of Fe in late-type stars – I. Standard stars with 1D and 〈3D〉 model atmospheres

We investigate departures from LTE in the line formation of Fe for a number of well-studied late-type stars in different evolutionary stages. A new model of Fe atom was constructed from the most up-to-date theoretical and experimental atomic data available so far. Non-local thermodynamic equilibrium (NLTE) line formation calculations for Fe were performed using 1D hydrostatic MARCS and MAFAGS-OS model atmospheres, as well as the spatial and temporal average stratifications from full 3D hydrodynamical simulations of stellar convection computed using the Stagger code. It is shown that the Fe I/Fe II ionization balance can be well established with the 1D and mean 3D models under NLTE including calibrated inelastic collisions with H I calculated from the Drawin's (1969) formulae. Strong low-excitation Fe I lines are very sensitive to the atmospheric structure; classical 1D models fail to provide consistent excitation balance, particularly so for cool metal-poor stars. A better agreement between Fe I lines spanning a range of excitation potentials is obtained with the mean 3D models. Mean NLTE metallicities determined for the standard stars using the 1D and mean 3D models are fully consistent. Also, the NLTE spectroscopic effective temperatures and gravities from ionization balance agree with that determined by other methods, e.g., infrared flux method and parallaxes, if one of the stellar parameters is constrained independently.

COMPARING THE EFFECT OF RADIATIVE TRANSFER SCHEMES ON CONVECTION SIMULATIONS

We examine the effect of different radiative transfer schemes on the properties of 3D simulations of near-surface stellar convection in the superadiabatic layer, where energy transport transitions from fully convective to fully radiative. We employ two radiative transfer schemes that fundamentally differ in the way they cover the 3D domain. The first solver approximates domain coverage with moments, while the second solver samples the 3D domain with ray integrations. By comparing simulations that differ only in their respective radiative transfer methods, we are able to isolate the effect that radiative efficiency has on the structure of the superadiabatic layer. We find the simulations to be in good general agreement, but they show distinct differences in the thermal structure in the superadiabatic layer and atmosphere.

ENERGY CONSERVATION AND GRAVITY WAVES IN SOUND-PROOF TREATMENTS OF STELLAR INTERIORS. PART I. ANELASTIC APPROXIMATIONS

Typical flows in stellar interiors are much slower than the speed of sound. To follow the slow evolution of subsonic motions, various sound-proof equations are in wide use, particularly in stellar astrophysical fluid dynamics. These low-Mach number equations include the anelastic equations. Generally, these equations are valid in nearly adiabatically stratified regions like stellar convection zones, but may not be valid in the sub-adiabatic, stably stratified stellar radiative interiors. Understanding the coupling between the convection zone and the radiative interior is a problem of crucial interest and may have strong implications for solar and stellar dynamo theories as the interface between the two, called the tachocline in the Sun, plays a crucial role in many solar dynamo theories. Here we study the properties of gravity waves in stably-stratified atmospheres. In particular, we explore how gravity waves are handled in various sound-proof equations. We find that some anelastic treatments fail to conserve energy in stably-stratified atmospheres, instead conserving pseudo-energies that depend on the stratification, and we demonstrate this numerically. One anelastic equation set does conserve energy in all atmospheres and we provide recommendations for converting low-Mach number anelastic codes to this set of equations.

Ak-ω MODEL FOR TURBULENTLY THERMAL CONVECTION IN STARS

Both observations and numerical simulations show that stellar convective motions are composed of semi-regular flows of convective rolling cells and the fully developed turbulence. Although the convective rolling cells are crucial for the properties of the stellar convection that transports heat and mixes materials in the stellar interior, their contributions have not been included in turbulent convection models proposed up to now. We simplify the structure of the convective rolling cells as a cellular pattern moving circle by circle with different angular velocities around the center, estimating their typical size by solving approximately for the temperature difference over the stationary temperature background and their average shear of velocity by evaluating approximately their kinetic energy transformed by themselves working as thermal engines from the heat involved in the convective rolling cells. We obtain a steady state solution in the fully local equilibrium which is similar to what is obtained in the standard mixing-length theory, by applying such model assumptions to the standard $k$-$\varepsilon$ model and properly choosing the model parameter $c_{\varepsilon 3}$. Accordingly, we propose a $k$-$\omega$ model to include the transport effect of turbulence in stars. Preliminary results of their applications to the sun and other stars with different masses and in different evolutionary stages show good agreements with results of the standard mixing-length theory and results of numerical simulations for the stellar convection.

Driving magnetic activity: differential rotation, flow structures, and surface patterns

AbstractThe interplay between stellar rotation and turbulent flows is a major ingredient for vertical angular momentum transport in stellar convection zone. Combined with the centrifugal force and the buoyancy force due to pole-equator temperature gradients one can expect a large-scale flow structure that is usually referred to as differential rotation and meridional flows. I review such observations for stars other than the Sun, mostly for stars significantly more active, and ask the question whether such observations can constrain the dynamo process.

Wave-driven mass loss in the last year of stellar evolution: setting the stage for the most luminous core-collapse supernovae

ABSTRACT During the late stages of stellar evolution in massive stars (C fusion and later), the fusion luminosity in the core of the star exceeds the star's Eddington luminosity. This can drive vigorous convective motions which in turn excite internal gravity waves. The local wave energy flux excited by convection is itself well above Eddington during the last few years in the life of the star. We suggest that an interesting fraction of the energy in gravity waves can, in some cases, convert into sound waves as the gravity waves propagate (tunnel) towards the stellar surface. The subsequent dissipation of the sound waves can unbind up to several M⊙ of the stellar envelope. This wave-driven mass loss can explain the existence of extremely large stellar mass-loss rates just prior to core collapse, which are inferred via circumstellar interaction in some core-collapse supernovae (e.g. SNe 2006gy and PTF 09uj, and even Type IIn supernovae more generally). An outstanding question is understanding what stellar parameters (mass, rotation, metallicity and age) are the most susceptible to wave-driven mass loss. This depends on the precise internal structure of massive stars and the power spectrum of internal gravity waves excited by stellar convection.

Simulations of the solar near-surface layers with the CO5BOLD, MURaM, and Stagger codes

Radiative hydrodynamic simulations of solar and stellar surface convection have become an important tool for exploring the structure and gas dynamics in the envelopes and atmospheres of late-type stars and for improving our understanding of the formation of stellar spectra. We quantitatively compare results from three-dimensional, radiative hydrodynamic simulations of convection near the solar surface generated with three numerical codes CO5BOLD, MURaM, and STAGGER and different simulation setups in order to investigate the level of similarity and to cross-validate the simulations. For all three simulations, we considered the average stratifications of various quantities (temperature, pressure, flow velocity, etc.) on surfaces of constant geometrical or optical depth, as well as their temporal and spatial fluctuations. We also compared observables, such as the spatially resolved patterns of the emerging intensity and of the vertical velocity at the solar optical surface as well as the center-to-limb variation of the continuum intensity at various wavelengths. The depth profiles of the thermodynamical quantities and of the convective velocities as well as their spatial fluctuations agree quite well. Slight deviations can be understood in terms of differences in box size, spatial resolution and in the treatment of non-gray radiative transfer between the simulations. The results give confidence in the reliability of the results from comprehensive radiative hydrodynamic simulations.

Numerical calculation of convection with reduced speed of sound technique

Context. The anelastic approximation is often adopted in numerical calculation with low Mach number, such as stellar internal convection. This approximation requires frequent global communication, because of an elliptic partial differential equation. Frequent global communication is negative factor for the parallel computing with a large number of CPUs. Aims. The main purpose of this paper is to test the validity of a method that artificially reduces the speed of sound for the compressible fluid equations in the context of stellar internal convection. The reduction of speed of sound allows for larger time steps in spite of low Mach number, while the numerical scheme remains fully explicit and the mathematical system is hyperbolic and thus does not require frequent global communication. Methods. Two and three dimensional compressible hydrodynamic equations are solved numerically. Some statistical quantities of solutions computed with different effective Mach numbers (due to reduction of speed of sound) are compared to test the validity of our approach. Results. Numerical simulations with artificially reduced speed of sound are a valid approach as long as the effective Mach number (based on the reduced speed of sound) remains less than 0.7.

EFFECT OF UNCERTAINTIES IN STELLAR MODEL PARAMETERS ON ESTIMATED MASSES AND RADII OF SINGLE STARS

Accurate and precise values of radii and masses of stars are needed to correctly estimate properties of extrasolar planets. We examine the effect of uncertainties in stellar model parameters on estimates of the masses, radii and average densities of solar-type stars. We find that in the absence of seismic data on solar-like oscillations, stellar masses can be determined to a greater accuracy than either stellar radii or densities; but to get reasonably accurate results the effective temperature, log g and metallicity must be measured to high precision. When seismic data are available, stellar density is the most well determined property, followed by radius, with mass the least well determined property. Uncertainties in stellar convection, quantified in terms of uncertainties in the value of the mixing length parameter, cause the most significant errors in the estimates of stellar properties.

Total-variation-diminishing implicit–explicit Runge–Kutta methods for the simulation of double-diffusive convection in astrophysics

We put forward the use of total-variation-diminishing (or more generally, strong stability preserving) implicit–explicit Runge–Kutta methods for the time integration of the equations of motion associated with the semiconvection problem in the simulation of stellar convection. The fully compressible Navier–Stokes equation, augmented by continuity and total energy equations, and an equation of state describing the relation between the thermodynamic quantities, is semi-discretized in space by essentially non-oscillatory schemes and dissipative finite difference methods. It is subsequently integrated in time by Runge–Kutta methods which are constructed such as to preserve the total variation diminishing (or strong stability) property satisfied by the spatial discretization coupled with the forward Euler method. We analyse the stability, accuracy and dissipativity of the time integrators and demonstrate that the most successful methods yield a substantial gain in computational efficiency as compared to classical explicit Runge–Kutta methods.

Spectral diagnostics of late-type stars: Non-LTE and &lt;3D&gt; approach

We determine effective temperature, metallicity, and microturbulence for a number of well-studied late-type stars. We use the new NLTE atomic model of Fe, and discuss the results for the MARCS models, as well as for the spatial and temporal averages of full 3D hydrodynamical simulations of stellar convection. It is shown that, contrary to the mean 3D models, certain limitations shall be imposed on the line formation and spectrum synthesis calculations with classical hydrostatic 1D models to obtain physically-realistic results.

Simulations of stellar convection with CO5BOLD

High-resolution images of the solar surface show a granulation pattern of hot rising and cooler downward-sinking material – the top of the deep-reaching solar convection zone. Convection plays a role for the thermal structure of the solar interior and the dynamo acting there, for the stratification of the photosphere, where most of the visible light is emitted, as well as for the energy budget of the spectacular processes in the chromosphere and corona. Convective stellar atmospheres can be modeled by numerically solving the coupled equations of (magneto)hydrodynamics and non-local radiation transport in the presence of a gravity field. The CO5BOLD code described in this article is designed for so-called “realistic” simulations that take into account the detailed microphysics under the conditions in solar or stellar surface layers (equation-of-state and optical properties of the matter). These simulations indeed deserve the label “realistic” because they reproduce the various observables very well – with only minor differences between different implementations. The agreement with observations has improved over time and the simulations are now well-established and have been performed for a number of stars. Still, severe challenges are encountered when it comes to extending these simulations to include ideally the entire star or substellar object: the strong stratification leads to completely different conditions in the interior, the photosphere, and the corona. Simulations have to cover spatial scales from the sub-granular level to the stellar diameter and time scales from photospheric wave travel times to stellar rotation or dynamo cycle periods. Various non-equilibrium processes have to be taken into account. Last but not least, realistic simulations are based on detailed microphysics and depend on the quality of the input data, which can be the actual accuracy limiter. This article provides an overview of the physical problem and the numerical solution and the capabilities of CO5BOLD, illustrated with a number of applications.

Numerical simulation of core convection by a multi-layer semi-implicit spherical spectral method

A semi-implicit multi-layer spherical spectral method for simulating stellar core convection is described. The fully compressible three-dimensional hydrodynamic equations with rotation and energy generation are solved. Prognostic variables are expressed as finite sums of spherical harmonics in the horizontal directions and handled by the finite difference method in the radial direction. The stratified approximation is used to simplify the nonlinearity to quadratic. A multi-layer scheme is employed to overcome the time step problem arising from shrinking grid sizes in the physical space near the center of the star. Despite of the different spectral truncations in different layers, round-off conservation of the total mass and total angular momentum of the whole domain can be maintained, and were confirmed numerically. The code is parallelized; with 12 processors the speedup factor is about 9. The solutions of model core convection with and without rotation are discussed.