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# Normalised mean zonal velocity φ // 2 at various dimensionless heights z = √ 1 − s 2 in homoeoidal shell geometries (with aspect ratios η pol = ηeq = 0.5) for libration forcings. The DNS profiles are computed at z = z0 for s ηeq, and at z0 = 0 for s > ηeq, giving a single profile for each DNS. Gray area shows the tangent cylinder s ηeq. Forcings with ωin = ω and in = on the no-slip inner boundary, which is subject to the same forcing as the outer boundary (except for the stress-free case, labelled SF). Black dashed curves indicate the theoretical profiles in the shell geometry. (a) Spherical shell with a no-slip (cyan and orange solid curves) or a stress-free (red and pink solid curves) inner boundary. Blued dotted dashed curves illustrate the full-sphere analytical profiles. DNS performed at E = 10 −6 , ω = 3, and = 10 −6 for the two forcings. (b) Homoeoidal (i.e. η pol = ηeq) spheroidal shells subject to longitudinal librations (solid coloured curves). DNS performed at E = 2.5 × 10 −6 , ω = π, and = 5 × 10 −4 .

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The generation of mean flows is a long-standing issue in rotating fluids. Motivated by planetary objects, we consider here a rapidly rotating fluid-filled spheroid, which is subject to weak perturbations of either the boundary (e.g. tides) or the rotation vector (e.g. in direction by precession, or in magnitude by longitudinal librations). Using bo...

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## Citations

... The Ekman number scalings we have observed are very different from the zonal flows generated by tidal forcing in a full sphere in the experiments (with a deformable no-slip boundary) of, for example, Morize et al. (2010), where the zonal velocity scales as 2 E −3/10 , or for those produced by libration-driven inertial waves, as studied in Cébron et al. (2021) and Lin & Noir (2021), where it instead scales as 2 E 0 or 2 E −1/10 , respectively (though these also result from self-interaction, so the zonal velocity also scales as 2 ). We underline that the best fitting laws here seem quite different from the ones emerging in Paper I in E −3/2 and E −1/2 , presumably because of (3, 3). ...

In close exoplanetary systems, tidal interactions drive orbital and spin evolution of planets and stars over long timescales. Tidally-forced inertial waves (restored by the Coriolis acceleration) in the convective envelopes of low-mass stars and giant gaseous planets contribute greatly to the tidal dissipation when they are excited and subsequently damped (e.g. through viscous friction), especially early in the life of a system. These waves are known to be subject to nonlinear effects, including triggering differential rotation in the form of zonal flows. In this study, we use a realistic tidal body forcing to excite inertial waves through the residual action of the equilibrium tide in the momentum equation for the waves. By performing 3D nonlinear hydrodynamical simulations in adiabatic and incompressible convective shells, we investigate how the addition of nonlinear terms affects the tidal flow properties, and the energy and angular momentum redistribution. In particular, we identify and justify the removal of terms responsible for unphysical angular momentum evolution observed in a previous numerical study. Within our new set-up, we observe the establishment of strong cylindrically-sheared zonal flows, which modify the tidal dissipation rates from prior linear theoretical predictions. We demonstrate that the effects of this differential rotation on the waves neatly explains the discrepancies between linear and nonlinear dissipation rates in many of our simulations. We also highlight the major role of both corotation resonances and parametric instabilities of inertial waves, which are observed for sufficiently high tidal forcing amplitudes or low viscosities, in affecting the tidal flow response.

... The Ekman number scalings we have observed are very different from the zonal flows generated by tidal forcing in a full sphere in the experiments (with a deformable no-slip boundary) of, for example, Morize et al. (2010), where the zonal velocity scales as 2 E −3/10 , or for those produced by libration-driven inertial waves, as studied in Cébron et al. (2021) and Lin & Noir (2021), where it instead scales as 2 E 0 or 2 E −1/10 , respectively (though these also result from self-interaction, so the zonal velocity also scales as 2 ). We underline that the best fitting laws here seem quite different from the ones emerging in Paper I in E −3/2 and E −1/2 , presumably because of . ...

In close exoplanetary systems, tidal interactions drive orbital and spin evolution of planets and stars over long timescales. Tidally-forced inertial waves (restored by the Coriolis acceleration) in the convective envelopes of low-mass stars and giant gaseous planets contribute greatly to the tidal dissipation when they are excited and subsequently damped (e.g. through viscous friction), especially early in the life of a system. These waves are known to be subject to nonlinear effects, including triggering differential rotation in the form of zonal flows. In this study, we use a realistic tidal body forcing to excite inertial waves through the residual action of the equilibrium tide in the momentum equation for the waves. By performing 3D nonlinear hydrodynamical simulations in adiabatic and incompressible convective shells, we investigate how the addition of nonlinear terms affects the tidal flow properties, and the energy and angular momentum redistribution. In particular, we identify and justify the removal of terms responsible for unphysical angular momentum evolution observed in a previous numerical study. Within our new set-up, we observe the establishment of strong cylindrically-sheared zonal flows, which modify the tidal dissipation rates from prior linear theoretical predictions. We demonstrate that the effects of this differential rotation on the waves neatly explains the discrepancies between linear and nonlinear dissipation rates in many of our simulations. We also highlight the major role of both corotation resonances and parametric instabilities of inertial waves, which are observed for sufficiently high tidal forcing amplitudes or low viscosities, in affecting the tidal flow response.

... The Ekman number scalings we have observed are very different from the zonal flows generated by tidal forcing in a full sphere in the experiments (with a deformable no-slip boundary) of, for example, Morize et al. (2010), where the zonal velocity scales as 2 E −3/10 , or for those produced by libration-driven inertial waves, as studied in Cébron et al. (2021) and Lin & Noir (2021), where it instead scales as 2 E 0 or 2 E −1/10 , respectively (though these also result from self-interaction, so the zonal velocity also scales as 2 ). We underline that the best fitting laws here seem quite different from the ones emerging in Paper I in E −3/2 and E −1/2 , presumably because of . ...

In close exoplanetary systems, tidal interactions drive orbital and spin evolution of planets and stars over long timescales. Tidally-forced inertial waves (restored by the Coriolis acceleration) in the convective envelopes of low-mass stars and giant gaseous planets contribute greatly to the tidal dissipation when they are excited and subsequently damped (e.g. through viscous friction), especially early in the life of a system. These waves are known to be subject to nonlinear effects, including triggering differential rotation in the form of zonal flows. In this study, we use a realistic tidal body forcing to excite inertial waves through the residual action of the equilibrium tide in the momentum equation for the waves. By performing 3D nonlinear hydrodynamical simulations in adiabatic and incompressible convective shells, we investigate how the addition of nonlinear terms affects the tidal flow properties, and the energy and angular momentum redistribution. In particular, we identify and justify the removal of terms responsible for unphysical angular momentum evolution observed in a previous numerical study. Within our new set-up, we observe the establishment of strong cylindrically-sheared zonal flows, which modify the tidal dissipation rates from prior linear theoretical predictions. We demonstrate that the effects of this differential rotation on the waves neatly explains the discrepancies between linear and nonlinear dissipation rates in many of our simulations. We also highlight the major role of both corotation resonances and parametric instabilities of inertial waves, which are observed for sufficiently high tidal forcing amplitudes or low viscosities, in affecting the tidal flow response.

... The profiles displayed in figure 16(b) show that the mean azimuthal "wind" in the system considerably increases with the forcing amplitude. This effect is quantitatively investigated in figure 16(c), showing that the maximum of |V θ | increases proportionally to the square of the forcing amplitude a, a result consistent with other studies (see, e.g., the recent work of Cebron et al. (2021)). No relevant scaling could be found, however, for the radial location of these maxima. ...

We present an experimental and numerical study of the nonlinear dynamics of an inertial wave attractor in an axisymmetric geometrical setting. The rotating ring-shaped fluid domain is delimited by two vertical coaxial cylinders, a conical bottom and a horizontal wave generator at the top: the vertical cross-section is a trapezium, while the horizontal cross-section is a ring. Forcing is introduced via axisymmetric low-amplitude volume-conserving oscillatory motion of the upper lid. The experiment shows an important result: at sufficiently strong forcing and long time scale, a saturated fully nonlinear regime develops as a consequence of an energy transfer draining energy towards a slow two-dimensional manifold represented by a regular polygonal system of axially oriented cyclonic vortices undergoing a slow prograde motion around the inner cylinder. We explore the long-term nonlinear behaviour of the system by performing a series of numerical simulations for a set of fixed forcing amplitudes. This study shows a rich variety of dynamical regimes, including a linear behaviour, a triadic resonance instability, a progressive frequency enrichment reminiscent of weak inertial wave turbulence and the generation of a slow manifold in the form of a polygonal vortex cluster confirming the experimental observation. This vortex cluster is discussed in detail, and we show that it stems from the summation and merging of wave-like components of the vorticity field. The nature of these wave components, the possibility of their detection under general conditions and the ultimate fate of the vortex clusters at even longer time scale remain to be explored.