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 $E^{1/3}$-scale, $E$ 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 $[0,2\Omega]$, contrary to the case of the full sphere ($\Omega$ 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 ($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide class of containers.Comment: 42 pages, 20 figures, abstract shortene

Water maser variability over 20 years in a large sample of star-forming regions: the complete database

Context. Water vapor emission at 22 GHz from masers associated with star-forming regions is highly variable. Aims. We present a database of up to 20 years of monitoring of a sample of 43 masers within star-forming regions. The sample covers a large range of luminosities of the associated IRAS source and is representative of the entire population of H2O masers of this type. The database forms a good starting point for any further study of H2O maser variability. Methods. The observations were obtained with the Medicina 32-m radiotelescope, at a rate of 4-5 observations per year. Results. To provide a database that can be easily accessed through the web, we give for each source: plots of the calibrated spectra, the velocity-time-flux density plot, the light curve of the integrated flux, the lower and upper envelopes of the maser emission, the mean spectrum, and the rate of the maser occurrence as a function of velocity. Figures for just one source are given in the text for representative purposes. Figures for all the sources are given in electronic form in the on-line appendix. A discussion of the main properties of the H2O variability in our sample will be presented in a forthcoming paper.Comment: 11 pages, 9 figures, to be published in Astronomy and Astrophysics; all plots in appendix (not included) can be downloaded from http://www.arcetri.astro.it/~starform/water_maser_v2.html or http://www.ira.inaf.it/papers/masers/water_maser_v2.htm

Water masers in the massive protostar IRAS 20126+4104: ejection and deceleration

We report on the first multi-epoch, phase referenced VLBI observations of the water maser emission in a high-mass protostar associated with a disk-jet system. The source under study, IRAS 20126+4104, has been extensively investigated in a large variety of tracers, including water maser VLBA data acquired by us three years before the present observations. The new findings fully confirm the interpretation proposed in our previous study, namely that the maser spots are expanding from a common origin coincident with the protostar. We also demonstrate that the observed 3-D velocities of the maser spots can be fitted with a model assuming that the spots are moving along the surface of a conical jet, with speed increasing for increasing distance from the cone vertex. We also present the results of single-dish monitoring of the water maser spectra in IRAS 20126+4104. These reveal that the peak velocity of some maser lines decreases linearly with time. We speculate that such a deceleration could be due to braking of the shocks from which the maser emission originates, due to mass loading at the shock front or dissipation of the shock energy.Comment: 11 pages, 8 figures. Accepted for publication in A&

Waves attractors in rotating fluids: a paradigm for ill-posed Cauchy problems

In the limit of low viscosity, we show that the amplitude of the modes of oscillation of a rotating fluid, namely inertial modes, concentrate along an attractor formed by a periodic orbit of characteristics of the underlying hyperbolic Poincar\'e equation. The dynamics of characteristics is used to elaborate a scenario for the asymptotic behaviour of the eigenmodes and eigenspectrum in the physically relevant r\'egime of very low viscosities which are out of reach numerically. This problem offers a canonical ill-posed Cauchy problem which has applications in other fields.Comment: 4 pages, 5 fi

Numerical simulations of the kappa-mechanism with convection

A strong coupling between convection and pulsations is known to play a major role in the disappearance of unstable modes close to the red edge of the classical Cepheid instability strip. As mean-field models of time-dependent convection rely on weakly-constrained parameters, we tackle this problem by the means of 2-D Direct Numerical Simulations (DNS) of kappa-mechanism with convection. Using a linear stability analysis, we first determine the physical conditions favourable to the kappa-mechanism to occur inside a purely-radiative layer. Both the instability strips and the nonlinear saturation of unstable modes are then confirmed by the corresponding DNS. We next present the new simulations with convection, where a convective zone and the driving region overlap. The coupling between the convective motions and acoustic modes is then addressed by using projections onto an acoustic subspace.Comment: 5 pages, 6 figures, accepted for publication in Astrophysics and Space Science, HELAS workshop (Rome june 2009

Viscous dissipation by tidally forced inertial modes in a rotating spherical shell

We investigate the properties of forced inertial modes of a rotating fluid inside a spherical shell. Our forcing is tidal like, but its main property is that it is on the large scales. Our solutions first confirm some analytical results obtained on a two-dimensional model by Ogilvie (2005). We also note that as the frequency of the forcing varies, the dissipation varies drastically if the Ekman number E is low (as is usually the case). We then investigate the three-dimensional case and compare the results to the foregoing model. These solutions show, like their 2D counterpart, a spiky dissipation curve when the frequency of the forcing is varied; they also display small frequency intervals where the viscous dissipation is independent of viscosity. However, we show that the response of the fluid in these frequency intervals is crucially dominated by the shear layer that is emitted at the critical latitude on the inner sphere. The asymptotic regime is reached when an attractor has been excited by this shear layer. This property is not shared by the two-dimensional model. Finally, resonances of the three-dimensional model correspond to some selected least-damped eigenmodes. Unlike their two-dimensional counter parts these modes are not associated with simple attractors; instead, they show up in frequency intervals with a weakly contracting web of characteristics. Besides, we show that the inner core is negligible when its relative radius is less than the critical value 0.4E^{1/5}. For these spherical shells, the full sphere solutions give a good approximation of the flows (abridged abstract).Comment: 32 pages, 19 figs, accepted in J. Fluid Mec