Gravito-inertial modes in a differentially rotating spherical shell
Giovanni M. Mirouh, Clément Baruteau, Michel Rieutord, Jérôme Ballot
GGravito-inertial modes in a differentially rotating spherical shell
Giovanni M. Mirouh , , a , Cl´ement Baruteau , , Michel Rieutord , , and J´erˆome Ballot , Universit´e de Toulouse, UPS-OMP, IRAP, Toulouse, France CNRS, IRAP, 14 avenue Edouard Belin, 31400 Toulouse, France
Abstract.
While many intermediate- and high-mass main sequence stars are rapidly and di ff erentially rotating,the e ff ects of rotation on oscillation modes are poorly known. In this communication we present a first study ofaxisymmetric gravito-inertial modes in the radiative zone of a di ff erentially rotating star. We consider a simplifiedmodel where the radiative zone of the star is a linearly stratified rotating fluid within a spherical shell, withdi ff erential rotation due to baroclinic e ff ects. We solve the eigenvalue problem with high-resolution spectralcomputations and determine the propagation domain of the waves through the theory of characteristics. Weexplore the propagation properties of two kinds of modes: those that can propagate in the entire shell and thosethat are restricted to a subdomain. Some of the modes that we find concentrate kinetic energy around short-periodshear layers known as attractors. We describe various geometries for the propagation domains, conditioning thesurface visibility of the corresponding modes. Gravito-inertial modes are low-frequency modes restoredby buoyancy and Coriolis forces. In rotating stars they maybe excited by internal mechanisms, such as the κ -mechanism,or for planet-harbouring stars by tidal e ff ects. Determin-ing how tidally-excited waves deposit their energy and an-gular momentum helps predicting the orbital evolution ofclose-in planets. Inertial modes in a di ff erentially rotatingconvective layer have been studied [1] as well as gravito-inertial modes in a radiative shell with solid-body rotation[2], but the influence of di ff erential rotation on gravito-inertial modes in a radiative zone is still unknown.We wish to have more insights into the mode propertiesof high- and intermediate-mass main sequence stars, suchas δ Scuti, Slowly Pulsating B, and β Cephei stars. Highrotation rates have been detected in most of these stars [3],while the radial di ff erential rotation increases throughouttheir evolution [4]. The oscillation properties of rapidlyand di ff erentially rotating stars are less constrained thanthose of slow rotators. For instance, regular patterns arehardly found in δ Scuti spectra [5]. The fundamental pa-rameters are also modified by rotation, and may explainstars detected outside of their expected instability domain[6].Gravito-inertial modes probe the internal layers of thestellar radiative zone, around the convective core. A bettercharacterisation of the oscillations will help constrain thephysical model, in particular the core size.
We consider a viscous fluid enclosed in a spherical shell.We impose a linear background temperature gradient throughthe radiative shell, resulting in a linear stable stratification. a e-mail: [email protected] The Brunt-V¨ais¨al¨a frequency then reads n ( r ) = N × r with N a constant [see 7, for more details]. According to [8],with no-slip boundary conditions on both sides, this leadsto the following shellular di ff erential rotation profile Ω ( r ): Ω ( r ) Ω ( R ) = + R (cid:90) r n ( r (cid:48) ) r (cid:48) dr (cid:48) = + N (cid:32) − r R (cid:33) . (1)We also make use of the Boussinesq approximation. Weconsider a radiative zone going from r = η R = . R to r = R , R being the stellar radius.We determine the properties of the linear oscillations inthis model using two methods [1, 2, and references therein].1. We solve the eigenvalue problem of the stellar oscil-lations, namely the linearised equations of motion, en-ergy, and mass conservation, including all dissipativeterms.2. We compute the paths of characteristics of the associ-ated adiabatic case [e.g. 9].The dissipative properties of the fluid are characterisedby the Prandtl and the Ekman numbers, which are respec-tively defined as:Pr = νκ , E = νΩ ( R ) R , (2)where ν is the kinematic viscosity and κ the thermal di ff u-sivity of the fluid.In stars, Pr ∼ − and E ∼ − − − . However,given practical resolution limitations [10], we set Pr = − and E = − for a first numerical exploration. Vary-ing E and Pr parameters changes the minimum length-scaleat which the energy of a mode may be focused along theattractor of characteristics. a r X i v : . [ a s t r o - ph . S R ] N ov PJ Web of Conferences -6 -5 -4 -3 -2 -1 00.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
Nr=300 L=600 η =0.350 N =1| ω | =1.1273 τ =-4.61x10 -4 -6 -5 -4 -3 -2 -1 00.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 Nr=200 L=400 η =0.350 N =2| ω | =0.7363 τ =-5.45x10 -3 -6 -5 -4 -3 -2 -1 00.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 Nr=300 L=600 η =0.350 N =3| ω | =2.8995 τ =-4.41x10 -3 -6 -5 -4 -3 -2 -1 00.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 Nr=300 L=600 η =0.350 N =4.6| ω | =0.4873 τ =-6.73x10 -3 Fig. 1.
Meridional slices of kinetic energy obtained by solvingthe dissipative linearised hydrodynamics equations, with attrac-tors (green) and turning surfaces (red) overplotted. The energyis plotted on a logarithmic scale and normalised to its maximumvalue. The various geometries are discussed in the main text.
Depending on the parameters and the position in the star,the solutions of the adiabatic problem may be evanescentor oscillatory. Therefore, for a given set of Brunt-V¨ais¨al¨afrequency N and wave frequency ω , eigenmodes may oc-cupy only a fraction of the spherical shell. This permitsa classification of the modes, depending on whether themode can propagate in the whole shell or in only part of it.Figure 1 shows the kinetic energy distribution of someaxisymmetric ( m =
0) modes for various values of N and ω , in a meridional plane. The energy is focused alongshear layers, which correspond to the trajectory of charac-teristics computed in the non-di ff usive case. The top leftmode spans the whole shell, and the induced surface tem-perature perturbation might be detected by observations.The top right mode shows a turning surface: the propa-gation domain is limited to a subdomain of the shell. Forthis mode, the temperature perturbation only reaches thesurface around the equator, making the oscillation visi-ble only for large inclinations of the rotation axis on theline of sight. The bottom left mode also has a turning sur-face, but the oscillations are confined into an inner sub-domain that does not reach the surface. This mode shouldnot be visible in lightcurves. Notice how the kinetic en-ergy is spread through the subdomain with no convergenceon a short-period attractor. Finally, the bottom right panelshows an oscillation trapped in a wedge. The wedge is anacute angle created by turning surfaces and shell bound-aries. It results in a focusing of the kinetic energy in thewedge, and impacts the visibility and dissipation proper-ties of the mode. The occurence and properties of wedgetrapped modes must be studied more thoroughly. We have classified oscillation modes into di ff erent cat-egories in a ( N , ω ) plane according to their propagationproperties. Further details will be presented in a forthcom-ing paper.We also compute the Lyapunov exponents, quantifying theconvergence of the characteristics towards a short-periodattractor. The faster the convergence, the more damped themode and the less likely it is expected to be observed [7]. For the first time, we compute the oscillations of a di ff er-entially rotating radiative region of star, where di ff erentialrotation is part of the baroclinic flow triggered by the com-bined e ff ects of rotation and stable stratification.We have given a first view of axisymmetric eigenmodeswhich may propagate in such a background flow. The nextsteps include investigating non-axisymmetric modes andthe use of more realistic Brunt-V¨ais¨al¨a frequency profiles,before tackling more realistic configurations with two-dim-ensional compressible stellar models, as the ESTER mod-els [4]. References
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