291 research outputs found
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
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