Three-dimensional solutions for the geostrophic flow in the Earth's core

In his seminal work, Taylor (1963) argued that the geophysically relevant limit for dynamo action within the outer core is one of negligibly small inertia and viscosity in the magnetohydrodynamic equations. Within this limit, he showed the existence of a necessary condition, now well known as Taylor's constraint, which requires that the cylindrically-averaged Lorentz torque must everywhere vanish; magnetic fields that satisfy this condition are termed Taylor states. Taylor further showed that the requirement of this constraint being continuously satisfied through time prescribes the evolution of the geostrophic flow, the cylindrically-averaged azimuthal flow. We show that Taylor's original prescription for the geostrophic flow, as satisfying a given second order ordinary differential equation, is only valid for a small subset of Taylor states. An incomplete treatment of the boundary conditions renders his equation generally incorrect. Here, by taking proper account of the boundaries, we describe a generalisation of Taylor's method that enables correct evaluation of the instantaneous geostrophic flow for any 3D Taylor state. We present the first full-sphere examples of geostrophic flows driven by non-axisymmetric Taylor states. Although in axisymmetry the geostrophic flow admits a mild logarithmic singularity on the rotation axis, in the fully 3D case we show that this is absent and indeed the geostrophic flow appears to be everywhere regular.Comment: 29 Pages, 8 figure

Impacts of fragmented accretion streams onto Classical T Tauri Stars: UV and X-ray emission lines

Context. The accretion process in Classical T Tauri Stars (CTTSs) can be studied through the analysis of some UV and X-ray emission lines which trace hot gas flows and act as diagnostics of the post-shock downfalling plasma. In the UV band, where higher spectral resolution is available, these lines are characterized by rather complex profiles whose origin is still not clear. Aims. We investigate the origin of UV and X-ray emission at impact regions of density structured (fragmented) accretion streams.We study if and how the stream fragmentation and the resulting structure of the post-shock region determine the observed profiles of UV and X-ray emission lines. Methods. We model the impact of an accretion stream consisting of a series of dense blobs onto the chromosphere of a CTTS through 2D MHD simulations. We explore different levels of stream fragmentation and accretion rates. From the model results, we synthesize C IV (1550 {\AA}) and OVIII (18.97 {\AA}) line profiles. Results. The impacts of accreting blobs onto the stellar chromosphere produce reverse shocks propagating through the blobs and shocked upflows. These upflows, in turn, hit and shock the subsequent downfalling fragments. As a result, several plasma components differing for the downfalling velocity, density, and temperature are present altoghether. The profiles of C IV doublet are characterized by two main components: one narrow and redshifted to speed $\approx$ 50 km s$^{-1}$ and the other broader and consisting of subcomponents with redshift to speed in the range 200 $\approx$ 400 km s$^{-1}$. The profiles of OVIII lines appear more symmetric than C IV and are redshifted to speed $\approx$ 150 km s$^{-1}$. Conclusions. Our model predicts profiles of C IV line remarkably similar to those observed and explains their origin in a natural way as due to stream fragmentation.Comment: 11 pages, 10 figure

Three-dimensional hydrodynamical CO5BOLD model atmospheres of red giant stars VI. First chromosphere model of a late-type giant

Although observational data unequivocally point out to the presence of chromospheres in red giant stars, no attempts have been made so far to model them using 3D hydrodynamical model atmospheres. We therefore compute an exploratory 3D hydrodynamical model atmosphere for a cool red giant in order to study the dynamical and thermodynamic properties of its chromosphere, as well as the influence of the chromosphere on its observable properties. 3D radiation hydrodynamics simulations are carried out with the CO5BOLD model atmosphere code for a star with the atmospheric parameters (Teff=4010 K, log g=1.5, [M/H]=0.0), which are similar to those of the K-type giant star Aldebaran (alpha Tau). ... we compute the emergent continuum intensity maps at different wavelengths, spectral line profiles of Ca II K, the Ca II infrared triplet line at 854.2nm, and H alpha, as well as the spectral energy distribution (SED) of the emergent radiative flux. The initial model quickly develops a dynamical chromosphere characterised by propagating and interacting shock waves. The peak temperatures in the chromospheric shock fronts reach values on the order of up to 5000 K although the shock fronts remain quite narrow. Like for the Sun, the gas temperature distribution in the upper layers is composed of a cool component due to adiabatic cooling in the expanding post-shock regions and a hot component due to shock waves. For this red giant model, the hot component is a rather flat high-temperature tail, which nevertheless affects the resulting average temperatures significantly. The simulations show that the atmospheres of red giant stars are dynamic and intermittent. Consequently, many observable properties cannot be reproduced with one-dimensional static models but demand for advanced 3D HD modelling. Furthermore, including a chromosphere in the models might produce significant contributions to the emergent UV flux.Comment: 14 pages, 8 figures, A&A (2017, accepted

Full sphere hydrodynamic and dynamo benchmarks

Convection in planetary cores can generate fluid flow and magnetic fields, and a number of sophisticated codes exist to simulate the dynamic behaviour of such systems. We report on the first community activity to compare numerical results of computer codes designed to calculate fluid flow within a whole sphere. The flows are incompressible and rapidly rotating and the forcing of the flow is either due to thermal convection or due to moving boundaries. All problems defined have solutions that allow easy comparison, since they are either steady, slowly drifting or perfectly periodic. The first two benchmarks are defined based on uniform internal heating within the sphere under the Boussinesq approximation with boundary conditions that are uniform in temperature and stress-free for the flow. Benchmark 1 is purely hydrodynamic, and has a drifting solution. Benchmark 2 is a magnetohydrodynamic benchmark that can generate oscillatory, purely periodic, flows and magnetic fields. In contrast, Benchmark 3 is a hydrodynamic rotating bubble benchmark using no slip boundary conditions that has a stationary solution. Results from a variety of types of code are reported, including codes that are fully spectral (based on spherical harmonic expansions in angular coordinates and polynomial expansions in radius), mixed spectral and finite difference, finite volume, finite element and also a mixed Fourier–finite element code. There is good agreement between codes. It is found that in Benchmarks 1 and 2, the approximation of a whole sphere problem by a domain that is a spherical shell (a sphere possessing an inner core) does not represent an adequate approximation to the system, since the results differ from whole sphere results

Cascades and transitions in turbulent flows

Turbulence is characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal 3D/2D isotropic cases characterized by the presence of a strictly direct/inverse energy cascade, respectively. We review recent works from a unified point of view and we present a classification of all known transfer mechanisms. Beside the classical cases of direct and inverse cascades, the different scenarios include: split cascades to small and large scales simultaneously, multiple/dual cascades of different quantities, bi-directional cascades where direct and inverse transfers of the same invariant coexist in the same scale-range and finally equilibrium states where no cascades are present, including the case when a condensate is formed. We classify all transitions as the control parameters are changed and we analyse when and why different configurations are observed. Our discussion is based on a set of paradigmatic applications: helical turbulence, rotating and/or stratified flows, MHD and passive/active scalars where the transfer properties are altered as one changes the embedding dimensions, the thickness of the domain or other relevant control parameters, as the Reynolds, Rossby, Froude, Peclet, or Alfven numbers. We discuss the presence of anomalous scaling laws in connection with the intermittent nature of the energy dissipation in configuration space. An overview is also provided concerning cascades in other applications such as bounded flows, quantum, relativistic and compressible turbulence, and active matter, together with implications for turbulent modelling. Finally, we present a series of open problems and challenges that future work needs to address.Comment: accepted for publication on Physics Reports 201