Numerical solution of a non-linear conservation law applicable to the interior dynamics of partially molten planets

The energy balance of a partially molten rocky planet can be expressed as a non-linear diffusion equation using mixing length theory to quantify heat transport by both convection and mixing of the melt and solid phases. In this formulation the effective or eddy diffusivity depends on the entropy gradient, $\partial S/\partial r$, as well as entropy. First we present a simplified model with semi-analytical solutions, highlighting the large dynamic range of $\partial S/\partial r$, around 12 orders of magnitude, for physically-relevant parameters. It also elucidates the thermal structure of a magma ocean during the earliest stage of crystal formation. This motivates the development of a simple, stable numerical scheme able to capture the large dynamic range of $\partial S/\partial r$ and provide a flexible and robust method for time-integrating the energy equation. We then consider a full model including energy fluxes associated with convection, mixing, gravitational separation, and conduction that all depend on the thermophysical properties of the melt and solid phases. This model is discretised and evolved by applying the finite volume method (FVM), allowing for extended precision calculations and using $\partial S/\partial r$ as the solution variable. The FVM is well-suited to this problem since it is naturally energy conserving, flexible, and intuitive to incorporate arbitrary non-linear fluxes that rely on lookup data. Special attention is given to the numerically challenging scenario in which crystals first form in the centre of a magma ocean. Our computational framework is immediately applicable to modelling high melt fraction phenomena in Earth and planetary science research. Furthermore, it provides a template for solving similar non-linear diffusion equations arising in other disciplines, particularly for non-linear functional forms of the diffusion coefficient

Reversal and amplification of zonal flows by boundary enforced thermal wind

Zonal flows in rapidly-rotating celestial objects such as the Sun, gas or ice giants form in a variety of surface patterns and amplitudes. Whereas the differential rotation on the Sun, Jupiter and Saturn features a super-rotating equatorial region, the ice giants, Neptune and Uranus harbour an equatorial jet slower than the planetary rotation. Global numerical models covering the optically thick, deep-reaching and rapidly rotating convective envelopes of gas giants reproduce successfully the prograde jet at the equator. In such models, convective columns shaped by the dominant Coriolis force typically exhibit a consistent prograde tilt. Hence angular momentum is pumped away from the rotation axis via Reynolds stresses. Those models are found to be strongly geostrophic, hence a modulation of the zonal flow structure along the axis of rotation, e.g. introduced by persistent latitudinal temperature gradients, seems of minor importance. Within our study we stimulate these thermal gradients and the resulting ageostrophic flows by applying an axisymmetric and equatorially symmetric outer boundary heat flux anomaly ($Y_{20}$) with variable amplitude and sign. Such a forcing pattern mimics the thermal effect of intense solar or stellar irradiation. Our results suggest that the ageostrophic flows are linearly amplified with the forcing amplitude $q^\star$ leading to a more pronounced dimple of the equatorial jet (alike Jupiter). The geostrophic flow contributions, however, are suppressed for weak $q^\star$, but inverted and re-amplified once $q^\star$ exceeds a critical value. The inverse geostrophic differential rotation is consistently maintained by now also inversely tilted columns and reminiscent of zonal flow profiles observed for the ice giants. Analysis of the main force balance and parameter studies further foster these results

Tracing cosmic evolution with clusters of galaxies

The most successful cosmological models to date envision structure formation as a hierarchical process in which gravity is constantly drawing lumps of matter together to form increasingly larger structures. Clusters of galaxies currently sit atop this hierarchy as the largest objects that have had time to collapse under the influence of their own gravity. Thus, their appearance on the cosmic scene is also relatively recent. Two features of clusters make them uniquely useful tracers of cosmic evolution. First, clusters are the biggest things whose masses we can reliably measure because they are the largest objects to have undergone gravitational relaxation and entered into virial equilibrium. Mass measurements of nearby clusters can therefore be used to determine the amount of structure in the universe on scales of 10^14 to 10^15 solar masses, and comparisons of the present-day cluster mass distribution with the mass distribution at earlier times can be used to measure the rate of structure formation, placing important constraints on cosmological models. Second, clusters are essentially ``closed boxes'' that retain all their gaseous matter, despite the enormous energy input associated with supernovae and active galactic nuclei, because the gravitational potential wells of clusters are so deep. The baryonic component of clusters therefore contains a wealth of information about the processes associated with galaxy formation, including the efficiency with which baryons are converted into stars and the effects of the resulting feedback processes on galaxy formation. This article reviews our theoretical understanding of both the dark-matter component and the baryonic component of clusters. (Abridged)Comment: 54 pages, 15 figures, Rev. Mod. Phys. (in press

Curvature pressure in a cosmology with a tired-light redshift

A hypothesis of curvature pressure is used to derive a static and stable cosmology with a tired-light redshift. The idea is that the high energy particles in the inter-galactic medium do not travel along geodesics because of the strong electrostatic forces. The result is a reaction back on the medium that is seen as an additional pressure. Combined with the explanation of the Hubble redshift as a gravitational interaction results in a static and stable cosmology. The predicted Hubble constant is 60.2 km/s/Mpc, the predicted background microwave temperature is 3 degrees and quasar luminosity functions and angular size distributions are shown to be consistent with the model. Since most observations that imply dark matter rely on redshift data it is argued that there is no dark matter. Observations of quasar absorption lines, supernovae light curves and the Butcher-Oemler effect are discussed. The curvature pressure is important for stellar structure and may explain the solar neutrino deficiency.Comment: 27 pages, no figures. This is a rewritten version of astro-ph/9803009 Accepted by Australian J. Phys. Changes to title, typo's and updated reference

Mass transport by buoyant bubbles in galaxy clusters

We investigate the effect of three important processes by which AGN-blown bubbles transport material: drift, wake transport and entrainment. The first of these, drift, occurs because a buoyant bubble pushes aside the adjacent material, giving rise to a net upward displacement of the fluid behind the bubble. For a spherical bubble, the mass of upwardly displaced material is roughly equal to half the mass displaced by the bubble, and should be ~ 10^{7-9} solar masses depending on the local ICM and bubble parameters. We show that in classical cool core clusters, the upward displacement by drift may be a key process in explaining the presence of filaments behind bubbles. A bubble also carries a parcel of material in a region at its rear, known as the wake. The mass of the wake is comparable to the drift mass and increases the average density of the bubble, trapping it closer to the cluster centre and reducing the amount of heating it can do during its ascent. Moreover, material dropping out of the wake will also contribute to the trailing filaments. Mass transport by the bubble wake can effectively prevent the build-up of cool material in the central galaxy, even if AGN heating does not balance ICM cooling. Finally, we consider entrainment, the process by which ambient material is incorporated into the bubble. AbridgedComment: Accepted for publication in MNRAS. 17 pages, 4 figures, 2 tables. Formatted for letter paper and adjusted author affiliations

Stellar Models with Magnetism and Rotation: Mixing Length Theories and Convection Simulations

Some low-mass stars appear to have larger radii than predicted by standard 1D structure models; prior work has suggested that inefficient convective heat transport, due to rotation and/or magnetism, may ultimately be responsible. In this thesis, we explore this possibility using a combination of 1D stellar models, 2D and 3D simulations, and analytical theory. First, we examine this issue using 1D stellar models constructed using the Modules for Experiments in Stellar Astrophysics (MESA) code. We begin by considering standard models that do not explicitly include rotational/magnetic effects, with convective inhibition modelled by decreasing a depth-independent mixing length theory (MLT) parameter αMLT. We provide formulae linking changes in αMLT to changes in the interior specific entropy, and hence to the stellar radius. Next, we modify the MLT formulation in MESA to mimic explicitly the influence of rotation and magnetism, using formulations suggested by Stevenson (1979) and MacDonald and Mullan (2014) respectively. We find rapid rotation in these models has a negligible impact on stellar structure, primarily because a star’s adiabat, and hence its radius, is predominantly affected by layers near the surface; convection is rapid and largely uninfluenced by rotation there. Magnetic fields, if they influenced convective transport in the manner described by MacDonald and Mullan (2014), could lead to more noticeable radius inflation. Finally, we show that these non-standard effects on stellar structure can be fabricated using a depth-dependent αMLT: a non-magnetic, non-rotating model can be produced that is virtually indistinguishable from one that explicitly parameterises rotation and/or magnetism using the two formulations above. We provide formulae linking the radially-variable αMLT to these putative MLT reformulations. We make further comparisons between MLT and simulations of convection, to establish how heat transport and stellar structure are influenced by rotation and magnetism, by looking at the entropy content of 2D local and 3D global convective calculations. Using 2D “box in a star” simulations, created using the convection code Dedalus, we investigate changes in bulk properties of the specific entropy for increasingly stratified domains. We observe regions stable against convection near the bottom boundary, resulting in the specific entropy in the bulk of the domain exceeding the bottom boundary value: this could be a result of physical effects, such as increased amounts of viscous dissipation for more supercritical, highly stratified cases, but may also be influenced by the artificial boundary conditions imposed by these local simulations. We then turn to 3D global simulations, created using the convection code Rayleigh, and investigate these same properties as a function of rotation rate. We find the average of the shell-averaged specific entropy gradient in the middle third of the domain to scale with rotation rate in a similar fashion to the scaling law derived via MLT arguments in Barker et al. (2014), i.e., |⟨ds/dr⟩| ∝ Ω^4/5.This research has been supported by the European Research Council, from the European Union’s Horizon 2020 research and innovation programme, under grant agreement No. 337705 (CHASM), and by a Consolidated Grant from the UK STFC (ST/J001627/1)