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Steady axisymmetric flow in the Taylor-Proudman regime. In the left panel, the coloured contours represent the differential rotation δΩ(r, θ) = Ω(r, θ)−Ω 0 normalised to the rotation rate of the outer sphere Ω 0 while in the right panel, we show the norm of the total meridional velocity field #» U tot = U r + V f (r) #» e r + U θ #» e θ . The parameters of the simulation are E = 10 −4 , P r (N 0 /Ω 0 ) 2 = 10 −4 and Re c = 10 −2 (simulation 1.3 of Table 2). The black lines show the streamlines of the total meridional circulation i.e. taking the effect of the contraction into account. Outside the tangent cylinder, the dashed lines correspond to a counterclockwise circulation and the full lines to a clockwise one while inside the tangent cylinder, the dashed lines represent a downward circulation and the full ones, an upward circulation.
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Context. Stars experience rapid contraction or expansion at different phases of their evolution. Modelling the transport of angular momentum and the transport of chemical elements occurring during these phases remains an unsolved problem.
Aims. We study a stellar radiative zone undergoing radial contraction and investigate the induced differential...
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... wind balance Eq. (24), such a regime is possible if the buoyancy force is weak enough. In the following of the paper, the regimes of cylindrical rotation will be called Taylor-Proudman regimes although strictly speaking the Taylor-Proudman theorem tells that all the velocity components must be cylindrical. The meridional cut in the left panel of Fig. 1 shows the structure of the flow for a simulation in this regime (E = 10 −4 , P r (N 0 /Ω 0 ) 2 = 10 −4 , Re c = 10 −2 ). The differential rotation exhibits a cylindrical profile, the rotation rate being maximum near the tangent cylinder. For this particular simulation, the maximal amplitude of the normalised differential rotation (Ω(r, ...
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... 1.75 to 60.2. Two additional simulations have been also performed at E = 10 −4 and E = 10 −6 , the other parameters being fixed. The different runs and their associated parameters are summarised in Table 6. Figure 9 shows the steady flow obtained for a density contrast ρ i /ρ 0 = 20.9. It can be compared to the Boussinesq simulation displayed in Fig. 1, a simulation performed with the same non-dimensional numbers but without density stratification. As we can see, the differential rotation is still cylindrical and its amplitude is similar to the Boussinesq case. This property is con- firmed at other density ratios (not shown). The main difference in the AM distribution concerns the ...
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... that the equatorial boundary layer is practically suppressed at high density stratification which in turn explains the lack of a strong vertical jet near the equator. The behaviour close to the inner sphere of three different fields, the radial gradient of angular velocity ∂Ω(r, θ)/∂r, the cylindrical and vertical velocity fields, is displayed in Fig. 10. In the Boussinesq case, we observe that an equatorial boundary layer is present and, just like in the classical spherical Couette flow, it is associated with a strong vertical jet along the tangent cylinder. However, when the density contrast increases, the jumps of the three fields near the inner sphere are practically suppressed ...
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... we study in the viscous regime the modification to the differential rotation and the meridional flow induced by the introduction of a varying density across the shell. Simulations are performed for the various density contrasts and contraction Reynolds numbers Re c listed in Table 7. Figure 11 shows the coloured contours of the rotation rate Ω and the streamlines of the meridional circulation in the steady state for two anelastic simulations with ρ i /ρ 0 = 20.9. The other parameters are the same as the ones used in the Boussinesq cases represented in Fig. 3. Similar to the Boussinesq case, the differential rotation profile is still radial and the amplitude of differential rotation still increases with Re c (first and third panels of Fig. 11). ...
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... in the steady state for two anelastic simulations with ρ i /ρ 0 = 20.9. The other parameters are the same as the ones used in the Boussinesq cases represented in Fig. 3. Similar to the Boussinesq case, the differential rotation profile is still radial and the amplitude of differential rotation still increases with Re c (first and third panels of Fig. 11). However, the amplitude of the differential rotation is decreased compared to the Boussinesq case. Indeed, for ρ i /ρ 0 = 20.9, the maximum value of the rotation contrast between the inner and the outer spheres is nearly four times smaller than in the Boussinesq case when Re c = 10 −2 , and almost twice as small as when Re c = 10. The ...
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... and the outer spheres is nearly four times smaller than in the Boussinesq case when Re c = 10 −2 , and almost twice as small as when Re c = 10. The total meridional circulation, visible in the second and fourth panel of Fig. 3, is dominated by the radial contraction field. As in the Boussinesq case, a circulation Article number, page 13 of 24 Fig. 10. Radial partial derivative of the ratio between the azimuthal velocity field U φ and the radius r (left panel), cylindrical radial velocity field U s = U r sin θ + U θ cos θ (middle panel) and vertical velocity field U z = U r cos θ − U θ sin θ (right panel), as a function of radius. In each panel, the top row corresponds to the ...
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... and outer spheres. Thus, in the left panel we have ρ i /ρ 0 = 10.8 in dashed lines, 20.9 in dash-dotted lines and 60.2 in dotted lines (runs 2.2 to 2.4 of Table 6) while in the middle and right panels, the two black curves are respectively obtained for ρ i /ρ 0 = 1.75 (plain curve) and ρ i /ρ 0 = 20.9 (dashed lines) (runs 2.1 and 2.3 of Table 6). Fig. 11. Stationary differential rotation δΩ(r, θ)/Ω 0 (first and third panels) and norm of the total meridional velocity field #» U tot = U r + V f (r) #» e r + U θ #» e θ (second and fourth panels) with its associated streamlines in black, in the viscous anelastic regime. The parameters of the simulation shown in the two left panels are E = ...
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... and 60.2. Middle Panel: numerical solutions are now represented for Re c = 10 −2 , 10 −1 , 1 and 10 (green, blue, purple and red respectively) for ρ i /ρ 0 = 20.9 (runs 1.3, 2.3, 3.3 and 4.3 of Table 7). Right Panel: same but when Re c ≥ 1, the solution in dashed lines is numerically estimated by taking the full contraction term in Eq. (55) In Fig. 12, we show the agreement between the analytical expression (56) and the numerical simulations. In all panels of this figure, the profiles of the differential rotation found in the numerical calculations for various values of the parameters are plotted at a particular latitude (here, θ = π/8) as a function of radius. We also overplot in ...
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... inner and outer spheres is fixed to ρ i /ρ 0 = 20.9 but Re c is increased. It is quite clear from the left panel that the agreement between the numerical and analytical solutions is perfect for the low Reynolds number cases. When the Reynolds number is increased, a departure from the analytical solution appears, as observed in the middle panel of Fig. 12, where the profile of Eq. (56), scaled with Re c , is shown in dashed lines. Indeed in this case, the level of differential rotation becomes comparable to Ω 0 and the first part of the contraction term V 0 r 2 0 ρ 0 /ρr 3 ∂(rU φ )/∂r in Eq. (54) is no longer negligible, leading to a correction to Eq. (56). When this correction is ...
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... to Ω 0 and the first part of the contraction term V 0 r 2 0 ρ 0 /ρr 3 ∂(rU φ )/∂r in Eq. (54) is no longer negligible, leading to a correction to Eq. (56). When this correction is applied, the agreement between the numerical solutions at higher Re c and the new analytical expression (not given here) is recovered, as shown in the last panel of Fig. ...
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... meridional cut displayed in the left panel of Fig. 13 shows that, as in the Boussinesq case, the differential rotation profile is neither radial nor cylindrical. But comparing Fig. 13 to its Boussinesq counterpart Fig. 5 reveals clear differences between the two simulations that only differ by their density stratification. Firstly, the amplitude of the rotation contrast taken between the ...
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... meridional cut displayed in the left panel of Fig. 13 shows that, as in the Boussinesq case, the differential rotation profile is neither radial nor cylindrical. But comparing Fig. 13 to its Boussinesq counterpart Fig. 5 reveals clear differences between the two simulations that only differ by their density stratification. Firstly, the amplitude of the rotation contrast taken between the inner and outer spheres is smaller in the anelastic case. Secondly, the distribution of the differential rotation within the ...
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... and in particular the region of high rotation rates localised at the inner sphere close to the equator that was clearly visible in the Boussinesq case is now absent. Quantitatively, along the inner sphere, the rotation rates of the equator and the pole only differ by 11% while this contrast was 67% in the Boussinesq case. From the right panel of Fig. 13, it is obvious that the meridional flow in the anelastic case is also smoother and not focused towards the equator at the inner sphere. That comes as no surprise since the meridional flow is similar in the Eddington-Sweet and TaylorProudman regimes and we have already found in the anelastic Taylor-Proudman case that the circulation is ...
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... reasonable approximation for the differential rotation between the inner and outer spheres obtained in the anelastic Eddington-Sweet regime. However, as for the Boussinesq case, we found that the agreement is better when the contribution of the outer Ekman layer is taken away by subtracting the numerical Taylor-Proudman solution. This is done in Fig. 14 which shows that the global differential rotation does scale approximately with the inverse of the integral of the density between the two spheres. It also shows that the rotation contrast is practically independent of the latitude except for the lowest density ratio ρ i /ρ 0 = 1.75. The latitudinal variation of the inner sphere ...
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... local maximum seems to be related to the strong vertical jet which advects AM towards the equator, this jet being in turn associated with the presence of an equatorial boundary layer. Figure 15 shows the radial profiles of the gradient of the angular velocity together with the cylindrical radial and vertical velocity fields (respectively left, middle and right panels) near the inner sphere at θ = π/4 and at the equator (respectively top and bottom rows), both for the Boussinesq (blue curve) and anelastic cases (black curves). As in the Taylor-Proudman regime, we observe that the strong equatorial boundary layer and the vertical jet that are present in the Boussinesq case nearly disappear in the anelastic simulations. ...
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... in the Boussinesq case nearly disappear in the anelastic simulations. Thus, contrary to the Boussinesq case, the meridional flow is no longer focused towards the equator. The different AM advections between the Boussinesq and anelastic cases thus appears to explain the smoother, more radial differential rotation observed in the anelastic case. Fig. 14. Differential rotation between the inner and outer spheres ∆Ω = Ω(r i ) − Ω(r 0 ) normalised to the top value Ω 0 obtained after subtracting the numerical Taylor-Proudman contribution, plotted as a function of the inverse of the integral of the background density profile ρ(r) normalised to the outer sphere value ρ 0 for different ...
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Citations
Context. The transport of angular momentum and chemical elements within evolving stars remains poorly understood. Asteroseismic and spectroscopic observations of low-mass main sequence stars and red giants reveal that their radiative cores rotate orders of magnitude slower than classical predictions from stellar evolution models and that the abundances of their surface light elements are too small. Magnetohydrodynamic (MHD) turbulence is considered a primary mechanism to enhance the transport in radiative stellar interiors but its efficiency is still largely uncertain.
Aims. We explore the transport of angular momentum and chemical elements due to azimuthal magnetorotational instability, one of the dominant instabilities expected in differentially rotating radiative stellar interiors.
Methods. We employed 3D MHD direct numerical simulations in a spherical shell of unstratified and stably stratified flows under the Boussinesq approximation. The background differential rotation was maintained by a volumetric body force. We examined the transport of chemical elements using a passive scalar.
Results. We provide evidence of magnetorotational instability for purely azimuthal magnetic fields in the parameter regime expected from local and global linear stability analyses. Without stratification and when the Reynolds number Re and the background azimuthal field strength are large enough, we observed dynamo action driven by the instability at values of the magnetic Prandtl number Pm in the range 0.6 − 1, which is the smallest ever reported in a global setup. When considering stable stratification at Pm = 1, the turbulence is transitional and becomes less homogeneous and isotropic upon increasing buoyancy effects. The transport of angular momentum occurs radially outward and is dominated by the Maxwell stresses when stratification is large enough. We find that the turbulent viscosity decreases when buoyancy effects strengthen and scales with the square root of the ratio of the reference rotation rate Ω a to the Brunt–Väisälä frequency N . The chemical turbulent diffusion coefficient scales with stratification similarly to the turbulent viscosity, but is lower in amplitude so that the transport of chemicals is slower than the one of angular momentum, in agreement with recent stellar evolution models of low-mass stars.
Conclusions. We show that the transport induced by azimuthal magnetorotational instability scales somewhat slowly with stratification and may enforce rigid rotations of red giant cores on a timescale of a few thousand years. In agreement with recent stellar evolution models of low-mass stars, the instability transports chemical elements less efficiently than angular momentum.
Context. Magnetars are highly magnetized neutron stars that can produce a wide diversity of X-ray and soft gamma-ray emissions that are powered by magnetic dissipation. Their magnetic dipole is constrained in the range of 10 ¹⁴ –10 ¹⁵ G by the measurement of their spin-down. In addition to fast rotation, these strong fields are also invoked to explain extreme stellar explosions, such as hypernovae, which are associated with long gamma-ray bursts and superluminous supernovae. A promising mechanism for explaining magnetar formation is the amplification of the magnetic field by the magnetorotational instability (MRI) in fast-rotating protoneutron stars (PNS). This scenario is supported by recent global incompressible models, which showed that a dipole field with magnetar-like intensity can be generated from small-scale turbulence. However, the impact of important physical ingredients, such as buoyancy and density stratification, on the efficiency of the MRI in generating a dipole field is still unknown.
Aims. We assess the impact of the density and entropy profiles on the MRI dynamo in a global model of a fast-rotating PNS. The model focuses on the outer stratified region of the PNS that is stable to convection.
Methods. Using the pseudo-spectral code MagIC, we performed 3D Boussinesq and anelastic magnetohydrodynamics simulations in spherical geometry with explicit diffusivities and with differential rotation forced at the outer boundary. The thermodynamic background of the anelastic models was retrieved from the data of 1D core-collapse supernova simulations from the Garching group. We performed a parameter study in which we investigated the influence of different approximations and the effect of the thermal diffusion through the Prandtl number.
Results. We obtain a self-sustained turbulent MRI-driven dynamo. This confirms most of our previous incompressible results when they are rescaled for density. The MRI generates a strong turbulent magnetic field and a nondominant equatorial dipole, which represents about 4.3% of the averaged magnetic field strength. Interestingly, an axisymmetric magnetic field at large scales is observed to oscillate with time, which can be described as a mean-field α Ω dynamo. By comparing these results with models without buoyancy or density stratification, we find that the key ingredient explaining the appearance of this mean-field behavior is the density gradient. Buoyancy due to the entropy gradient damps turbulence in the equatorial plane, but it has a relatively weak influence in the low Prandtl number regime overall, as expected from neutrino diffusion. However, the buoyancy starts to strongly impact the MRI dynamo for Prandtl numbers close to unity.
Conclusions. Our results support the hypothesis that the MRI is able to generate magnetar-like large-scale magnetic fields. The results furthermore predict the presence of a α Ω dynamo in the protoneutron star, which could be important to model in-situ magnetic field amplification in global models of core-collapse supernovae or binary neutron star mergers.
Context. Some contracting or expanding stars are thought to host a large-scale magnetic field in their radiative interior. By interacting with the contraction-induced flows, such fields may significantly alter the rotational history of the star. They thus constitute a promising way to address the problem of angular momentum transport during the rapid phases of stellar evolution.
Aims. In this work, we aim to study the interplay between flows and magnetic fields in a contracting radiative zone.
Methods. We performed axisymmetric Boussinesq and anelastic numerical simulations in which a portion of the radiative zone was modelled by a rotating spherical layer, stably stratified and embedded in a large-scale (either dipolar or quadrupolar) magnetic field. This layer is subject to a mass-conserving radial velocity field mimicking contraction. The quasi-steady flows were studied in strongly or weakly stably stratified regimes relevant for pre-main sequence stars and for the cores of subgiant and red giant stars. The parametric study consists in varying the amplitude of the contraction velocity and of the initial magnetic field. The other parameters were fixed with the guidance of a previous study.
Results. After an unsteady phase during which the toroidal field grew linearly and then back-reacted on the flow, a quasi-steady configuration was reached, characterised by the presence of two magnetically decoupled regions. In one of them, magnetic tension imposes solid-body rotation. In the other, called the dead zone, the main force balance in the angular momentum equation does not involve the Lorentz force and a differential rotation exists. In the strongly stably stratified regime, when the initial magnetic field is quadrupolar, a magnetorotational instability is found to develop in the dead zones. The large-scale structure is eventually destroyed and the differential rotation is able to build up in the whole radiative zone. In the weakly stably stratified regime, the instability is not observed in our simulations, but we argue that it may be present in stars.
Conclusions. We propose a scenario that may account for the post-main sequence evolution of solar-like stars, in which quasi-solid rotation can be maintained by a large-scale magnetic field during a contraction timescale. Then, an axisymmetric instability would destroy this large-scale structure and this enables the differential rotation to set in. Such a contraction-driven instability could also be at the origin of the observed dichotomy between strongly and weakly magnetic intermediate-mass stars.
Context. Most exoplanets detected so far are close-in planets, which are likely to be affected by tidal dissipation in their host star. To obtain a complete picture of the evolution of star–planet systems, we need to consider the effect of tides within stellar radiative and convective zones.
Aims. We aim to provide a general formalism allowing us to assess tidal dissipation in stellar radiative zones for late- and early-type stars, including stellar structure with a convective core and an envelope like in F-type stars. This allows us to study the dynamics of a given system throughout the stellar evolution. On this basis, we investigate the effect of stellar structure and evolution on tidal dissipation in the radiative core of low-mass stars.
Methods. We developed a general theoretical formalism to evaluate tidal dissipation in stellar radiative zones that is applicable to early- and late-type stars. From the study of adiabatic oscillations throughout the star, we computed the energy flux transported by progressive internal gravity waves and the induced tidal torque. By relying on grids of stellar models, we studied the effect of stellar structure and evolution on the tidal dissipation of F-, G-, and K-type stars from the pre-main sequence (PMS) to the red giant branch (RGB).
Results. For a given star–planet system, tidal dissipation reaches a maximum value on the PMS for all stellar masses. On the main sequence (MS), it decreases to become almost constant. The dissipation is then several orders of magnitude smaller for F-type than for G- and K-type stars. During the subgiant phase and the RGB, tidal dissipation increases by several orders of magnitude, along with the expansion of the stellar envelope. We show that the dissipation of the dynamical tide in the convective zone dominates the evolution of the system during most of the PMS and the beginning of the MS, as the star rotates rapidly. Tidal dissipation in the radiative zone then becomes the strongest contribution during the subgiant phase and the RGB as the density at the convective-radiative interface increases. For similar reasons, we also find that the dissipation of a metal-poor star is stronger than the dissipation of a metal-rich star during the PMS, the subgiant phase, and the RGB. The opposite trend is observed during the MS. Finally, we show that the contribution of a convective core for the most massive solar-type stars is negligible compared to that of the envelope because the mass distribution of the core does not favor the dissipation of tidal gravity waves.
