404 research outputs found
On the dynamics of radiative zones in rotating stars
In this lecture I try to explain the basic dynamical processes at work in a
radiative zone of a rotating star. In particular, the notion of baroclinicity
is thoroughly discussed. Attention is specially directed to the case of
circulations and the key role of angular momentum conservation is stressed. The
specific part played by viscosity is also explained. The old approach of
Eddington and Sweet is reviewed and criticized in the light of the seminal
papers of Busse 1981 and Zahn 1992. Other examples taken in the recent
literature are also presented; finally, I summarize the important points.Comment: 21 pages 5 figure
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
More concerning the anelastic and subseismic approximations for low-frequency modes in stars
Two approximations, namely the subseismic approximation and the anelastic
approximation, are presently used to filter out the acoustic modes when
computing low frequency modes of a star (gravity modes or inertial modes). In a
precedent paper (Dintrans & Rieutord 2001), we observed that the anelastic
approximation gave eigenfrequencies much closer to the exact ones than the
subseismic approximation. Here, we try to clarify this behaviour and show that
it is due to the different physical approach taken by each approximation: On
the one hand, the subseismic approximation considers the low frequency part of
the spectrum of (say) gravity modes and turns out to be valid only in the
central region of a star; on the other hand, the anelastic approximation
considers the Brunt-Vaisala frequency as asymptotically small and makes no
assumption on the order of the modes. Both approximations fail to describe the
modes in the surface layers but eigenmodes issued from the anelastic
approximation are closer to those including acoustic effects than their
subseismic equivalent.
We conclude that, as far as stellar eigenvalue problems are concerned, the
anelastic approximation is better suited for simplifying the eigenvalue problem
when low-frequency modes of a star are considered, while the subseismic
approximation is a useful concept when analytic solutions of high order
low-frequency modes are needed in the central region of a star.Comment: 5 pages 3 fig, to appear in MNRA
Self-consistent 2D models of fast rotating early-type star
This work aims at presenting the first two-dimensional models of an isolated
rapidly rotating star that include the derivation of the differential rotation
and meridional circulation in a self-consistent way.We use spectral methods in
multidomains, together with a Newton algorithm to determine the steady state
solutions including differential rotation and meridional circulation for an
isolated non-magnetic, rapidly rotating early-type star. In particular we
devise an asymptotic method for small Ekman numbers (small viscosities) that
removes the Ekman boundary layer and lifts the degeneracy of the inviscid
baroclinic solutions.For the first time, realistic two-dimensional models of
fast-rotating stars are computed with the actual baroclinic flows that predict
the differential rotation and the meridional circulation for intermediate-mass
and massive stars. These models nicely compare with available data of some
nearby fast-rotating early-type stars like Ras Alhague ( Oph), Regulus
( Leo), and Vega ( Lyr). It is shown that baroclinicity drives
a differential rotation with a slow pole, a fast equator, a fast core, and a
slow envelope. The differential rotation is found to increase with mass, with
evolution (here measured by the hydrogen mass fraction in the core), and with
metallicity. The core-envelope interface is found to be a place of strong shear
where mixing will be efficient.Two-dimensional models offer a new view of
fast-rotating stars, especially of their differential rotation, which turns out
to be strong at the core-envelope interface. They also offer more accurate
models for interpreting the interferometric and spectroscopic data of
early-type stars.Comment: 16 pages, 17 figures, to appear in Astronomy and Astrophysic
Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum
We investigate the asymptotic properties of inertial modes confined in a
spherical shell when viscosity tends to zero. We first consider the mapping
made by the characteristics of the hyperbolic equation (Poincar\'e's equation)
satisfied by inviscid solutions. Characteristics are straight lines in a
meridional section of the shell, and the mapping shows that, generically, these
lines converge towards a periodic orbit which acts like an attractor.
We then examine the relation between this characteristic path and
eigensolutions of the inviscid problem and show that in a purely
two-dimensional problem, convergence towards an attractor means that the
associated velocity field is not square-integrable. We give arguments which
generalize this result to three dimensions. We then consider the viscous
problem and show how viscosity transforms singularities into internal shear
layers which in general betray an attractor expected at the eigenfrequency of
the mode. We find that there are nested layers, the thinnest and most internal
layer scaling with -scale, being the Ekman number. Using an
inertial wave packet traveling around an attractor, we give a lower bound on
the thickness of shear layers and show how eigenfrequencies can be computed in
principle. Finally, we show that as viscosity decreases, eigenfrequencies tend
towards a set of values which is not dense in , contrary to the
case of the full sphere ( is the angular velocity of the system).
Hence, our geometrical approach opens the possibility of describing the
eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers
(), which are out of reach numerically, and this for a wide
class of containers.Comment: 42 pages, 20 figures, abstract shortene
MHD simulations of the solar photosphere
We briefly review the observations of the solar photosphere and pinpoint some
open questions related to the magnetohydrodynamics of this layer of the Sun. We
then discuss the current modelling efforts, addressing among other problems,
that of the origin of supergranulation.Comment: 10 pages, 6 figures; 4th French-Chinese Meeting on Solar Physics
Understanding Solar Activity: Advances and Challenges, 4th French-Chinese,
Nice, Franc
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