Inertial waves in a differentially rotating spherical shell

We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in E^{1/3} with the Ekman number E, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with E^{1/2} as Ekman boundary layers. The behavior of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.Comment: 31 pages, 16 figures, accepted for publication in Journal of Fluid Mechanic

Tidal inertial waves in the differentially rotating convective envelopes of low-mass stars - I. Free oscillation modes

Star-planet tidal interactions may result in the excitation of inertial waves in the convective region of stars. In low-mass stars, their dissipation plays a prominent role in the long-term orbital evolution of short-period planets. Turbulent convection can sustain differential rotation in their envelope, with an equatorial acceleration (as in the Sun) or deceleration, which can modify the waves' propagation properties. We explore in this first paper the general propagation properties of free linear inertial waves in a differentially rotating homogeneous fluid inside a spherical shell. We assume that the angular velocity background flow depends on the latitudinal coordinate only, close to what is expected in the external convective envelope of low-mass stars. We use i) an analytical approach in the inviscid case to get the dispersion relation, from which we compute the characteristic trajectories along which energy propagates. This allows us to study the existence of attractor cycles and infer the different families of inertial modes; ii) high-resolution numerical calculations based on a spectral method for the viscous problem. We find that modes that propagate in the whole shell (D modes) behave the same way as with solid-body rotation. However, another family of inertial modes exists (DT modes), which can propagate only in a restricted part of the convective zone. Our study shows that they are less common than D modes and that the characteristic rays and shear layers often focus towards a wedge - or point-like attractor. More importantly, we find that for non-axisymmetric oscillation modes, shear layers may cross a corotation resonance with a local accumulation of kinetic energy. Their damping rate scales very differently from what we obtain for standard D modes and we show an example where it is independent of viscosity (Ekman number) in the astrophysical regime in which it is small.Comment: 17 pages, 15 figures, accepted for publication in A&

Gravito-inertial waves in a differentially rotating spherical shell

The gravito-inertial waves propagating over a shellular baroclinic flow inside a rotating spherical shell are analysed using the Boussinesq approximation. The wave properties are examined by computing paths of characteristics in the non-dissipative limit, and by solving the full dissipative eigenvalue problem using a high-resolution spectral method. Gravito-inertial waves are found to obey a mixed-type second-order operator and to be often focused around short-period attractors of characteristics or trapped in a wedge formed by turning surfaces and boundaries. We also find eigenmodes that show a weak dependence with respect to viscosity and heat diffusion just like truly regular modes. Some axisymmetric modes are found unstable and likely destabilized by baroclinic instabilities. Similarly, some non-axisymmetric modes that meet a critical layer (or corotation resonance) can turn unstable at sufficiently low diffusivities. In all cases, the instability is driven by the differential rotation. For many modes of the spectrum, neat power laws are found for the dependence of the damping rates with diffusion coefficients, but the theoretical explanation for the exponent values remains elusive in general. The eigenvalue spectrum turns out to be very rich and complex, which lets us suppose an even richer and more complex spectrum for rotating stars or planets that own a differential rotation driven by baroclinicity.Comment: 33 pages, 14 figures, accepted for publication in Journal of Fluid Mechanic

Self-gravity at the scale of the polar cell

We present the exact calculus of the gravitational potential and acceleration along the symmetry axis of a plane, homogeneous, polar cell as a function of mean radius a, radial extension e, and opening angle f. Accurate approximations are derived in the limit of high numerical resolution at the geometrical mean of the inner and outer radii (a key-position in current FFT-based Poisson solvers). Our results are the full extension of the approximate formula given in the textbook of Binney & Tremaine to all resolutions. We also clarify definitely the question about the existence (or not) of self-forces in polar cells. We find that there is always a self-force at radius except if the shape factor a.f/e reaches ~ 3.531, asymptotically. Such cells are therefore well suited to build a polar mesh for high resolution simulations of self-gravitating media in two dimensions. A by-product of this study is a newly discovered indefinite integral involving complete elliptic integral of the first kind over modulus.Comment: 4 pages, 4 figures, A&A accepte

A substitute for the singular Green kernel in the Newtonian potential of celestial bodies

The "point mass singularity" inherent in Newton's law for gravitation represents a major difficulty in accurately determining the potential and forces inside continuous bodies. Here we report a simple and efficient analytical method to bypass the singular Green kernel 1/|r-r'| inside the source without altering the nature of the interaction. We build an equivalent kernel made up of a "cool kernel", which is fully regular (and contains the long-range -GM/r asymptotic behavior), and the gradient of a "hyperkernel", which is also regular. Compared to the initial kernel, these two components are easily integrated over the source volume using standard numerical techniques. The demonstration is presented for three-dimensional distributions in cylindrical coordinates, which are well-suited to describing rotating bodies (stars, discs, asteroids, etc.) as commonly found in the Universe. An example of implementation is given. The case of axial symmetry is treated in detail, and the accuracy is checked by considering an exact potential/surface density pair corresponding to a flat circular disc. This framework provides new tools to keep or even improve the physical realism of models and simulations of self-gravitating systems, and represents, for some of them, a conclusive alternative to softened gravity.Comment: Accepted for publication in A&A; 7 pages, color figure

Forming Circumbinary Planets: N-body Simulations of Kepler-34

Observations of circumbinary planets orbiting very close to the central stars have shown that planet formation may occur in a very hostile environment, where the gravitational pull from the binary should be very strong on the primordial protoplanetary disk. Elevated impact velocities and orbit crossings from eccentricity oscillations are the primary contributors towards high energy, potentially destructive collisions that inhibit the growth of aspiring planets. In this work, we conduct high resolution, inter-particle gravity enabled N-body simulations to investigate the feasibility of planetesimal growth in the Kepler-34 system. We improve upon previous work by including planetesimal disk self-gravity and an extensive collision model to accurately handle inter-planetesimal interactions. We find that super-catastrophic erosion events are the dominant mechanism up to and including the orbital radius of Kepler-34(AB)b, making in-situ growth unlikely. It is more plausible that Kepler-34(AB)b migrated from a region beyond 1.5 AU. Based on the conclusions that we have made for Kepler-34 it seems likely that all of the currently known circumbinary planets have also migrated significantly from their formation location with the possible exception of Kepler-47(AB)c.Comment: 6 pages, 5 figures, accepted for publication in ApJ

Pyruvate carboxylase deficiency type C; variable presentation and beneficial effect of triheptanoin

Pyruvate carboxylase is a mitochondrial enzyme essential for the tricarboxylic acid cycle (TCA), gluconeogenesis and fatty-acid synthesis. Pyruvate carboxylase deficiency (PCD) mostly presents with life-limiting encephalopathy (types A/B). A milder type C presentation is rare, with a comparatively favourable prognosis. Therapies remain essentially supportive. Triheptanoin is an odd-chain triglyceride, with the potential to replenish TCA intermediates (anaplerosis), and its metabolites cross the blood–brain-barrier. Outcomes of triheptanoin treatment in PCD types A/B have been disappointing, but have not been reported in type C. Here, we present two new patients with PCD type C, and report the response to treatment with triheptanoin in one. Patient 1 (P1) presented with neonatal-onset lactic acidosis and recurrent symptomatic lactic acidosis following exercise and during illnesses, with frequent hospitalisations. Speech development was delayed. MRI-brain showed delayed cerebral myelination. Patient 2 (P2) presented with episodic ketoacidosis, hyperlactataemia and hypoglycaemia at 2 years of age, with gross motor delay and mild global volume loss on MRI brain. Treatment with triheptanoin was commenced in P1 at 3 years of age with up-titration to 35 mL/day (25% of daily energy intake) over 6 months, due to transient diarrhoea. Dietary long-chain triglycerides were restricted, with fat-soluble vitamin supplementation. Subsequently, hospitalisations during intercurrent illnesses decreased, post-exertional hyperlactataemia resolved and exercise tolerance improved. Continued developmental progress was observed, and repeat MRI 18 months after initiation showed improved myelination. Triheptanoin was well-tolerated and appeared efficacious during 2 years' follow-up, and has potential to restore energy homeostasis and myelin synthesis in PCD type C

Novel Characteristics of Valveless Pumping

This study investigates the occurrence of valveless pumping in a fluidfilled system consisting of two open tanks connected by an elastic tube. We show that directional flow can be achieved by introducing a periodic pinching applied at an asymmetrical location along the tube, and that the flow direction depends on the pumping frequency. We propose a relation between wave propagation velocity, tube length, and resonance frequencies associated with shifts in the pumping direction using numerical simulations. The eigenfrequencies of the system are estimated from the linearized system, and we show that these eigenfrequencies constitute the resonance frequencies and the horizontal slope frequencies of the system; 'horizontal slope frequency' being a new concept. A simple model is suggested, explaining the effect of the gravity driven part of the oscillation observed in response to the tank and tube diameter changes. Results are partly compared with experimental findings.Art. no. 22450