Abstract
Tidal dissipation is responsible for circularizing the orbits and synchronizing the spins of solar-type close binary stars, but the mechanisms responsible are not fully understood. Previous ...work has indicated that significant enhancements to the theoretically predicted tidal dissipation rates are required to explain the observed circularization periods (
P
circ
) in various stellar populations and their evolution with age. This was based partly on the common belief that the dominant mechanism of tidal dissipation in solar-type stars is turbulent viscosity acting on equilibrium tides in convective envelopes. In this paper, we study tidal dissipation in both convection and radiation zones of rotating solar-type stars following their evolution. We study equilibrium tide dissipation, incorporating a frequency-dependent effective viscosity motivated by the latest hydrodynamical simulations, and inertial wave (dynamical tide) dissipation, adopting a frequency-averaged formalism that accounts for the realistic structure of the star. We demonstrate that the observed binary circularization periods can be explained by inertial wave (dynamical tide) dissipation in convective envelopes. This mechanism is particularly efficient during pre-main-sequence phases, but it also operates on the main sequence if the spin is close to synchronism. The predicted
P
circ
due to this mechanism increases with the main-sequence age in accordance with observations. We also demonstrate that both equilibrium tide and internal gravity-wave dissipation are unlikely to explain the observed
P
circ
, even during the pre-main sequence, based on our best current understanding of these mechanisms. Finally, we advocate more realistic dynamical studies of stellar populations that employ tidal dissipation due to inertial waves.
I present results from the first global hydrodynamical simulations of the elliptical instability in a tidally deformed gaseous planet (or star) with a free surface. The elliptical instability is ...potentially important for tidal evolution of the shortest-period hot Jupiters. I model the planet as a spin–orbit aligned or anti-aligned, and non-synchronously rotating, tidally deformed, homogeneous fluid body. A companion paper presented an analysis of the global modes and instabilities of such a planet. Here I focus on the non-linear evolution of the elliptical instability. This is observed to produce bursts of turbulence that drive the planet towards synchronism with its orbit in an erratic manner. If the planetary spin is initially anti-aligned, the elliptical instability also drives spin–orbit alignment on a similar time-scale as the spin synchronization. The instability generates differential rotation inside the planet in the form of zonal flows, which play an important role in the saturation of the instability, and in producing the observed burstiness. These results are broadly consistent with the picture obtained using a local Cartesian model (where columnar vortices played the role of zonal flows). I also simulate the instability in a container that is rigid (but stress-free) rather than free, finding broad quantitative agreement. The dissipation resulting from the elliptical instability could explain why the shortest-period hot Jupiters tend to have circular orbits inside about 2–3 d, and predicts spin synchronization (and spin–orbit alignment) out to about 10–15 d. However, other mechanisms must be invoked to explain tidal circularization for longer orbital periods.
ABSTRACT
Turbulent convection is thought to act as an effective viscosity in damping equilibrium tidal flows, driving spin and orbital evolution in close convective binary systems. Compared to ...mixing-length predictions, this viscosity ought to be reduced when the tidal frequency |ωt| exceeds the turnover frequency ωcv of the dominant convective eddies, but the efficiency of this reduction has been disputed. We re-examine this long-standing controversy using direct numerical simulations of an idealized global model. We simulate thermal convection in a full sphere, and externally forced by the equilibrium tidal flow, to measure the effective viscosity νE acting on the tidal flow when |ωt|/ωcv ≳ 1. We demonstrate that the frequency reduction of νE is correlated with the frequency spectrum of the (unperturbed) convection. For intermediate frequencies below those in the turbulent cascade (|ωt|/ωcv ∼ 1−5), the frequency spectrum displays an anomalous 1/ωα power law that is responsible for the frequency reduction νE∝1/|ωt|α, where α < 1 depends on the model parameters. We then get |νE| ∝ 1/|ωt|δ with δ > 1 for higher frequencies, and δ = 2 is obtained for a Kolmogorov turbulent cascade. A generic |νE| ∝ 1/|ωt|2 suppression is next found for higher frequencies within the dissipation range of the convection (but with negative values). Our results indicate that a better knowledge of the frequency spectrum of convection is necessary to accurately predict the efficiency of tidal dissipation in stars and planets resulting from this mechanism.
Convection is thought to act as a turbulent viscosity in damping tidal flows and in driving spin and orbital evolution in close convective binary systems. This turbulent viscosity should be reduced, ...compared to mixing-length predictions, when the forcing (tidal) frequency exceeds the turnover frequency cv of the dominant convective eddies. However, two contradictory scaling laws have been proposed and this issue remains highly disputed. To revisit this controversy, we conduct the first direct numerical simulations of convection interacting with the equilibrium tidal flow in an idealized global model of a low-mass star. We present direct computations of the turbulent effective viscosity, E, acting on the equilibrium tidal flow. We unexpectedly report the coexistence of the two disputed scaling laws, which reconciles previous theoretical (and numerical) findings. We recover the universal quadratic scaling in the high-frequency regime . Our results also support the linear scaling in an intermediate regime with . Both regimes may be relevant to explain the observed properties of close binaries, including spin synchronization of solar-type stars and the circularization of low-mass stars. The robustness of these two regimes of tidal dissipation, and the transition between them, should be explored further in more realistic models. A better understanding of the interaction between convection and tidal flows is indeed essential to correctly interpret observations of close binary stars and short-period planetary orbits.
Abstract
We simulate the nonlinear hydrodynamical evolution of tidally excited inertial waves in convective envelopes of rotating stars and giant planets modeled as spherical shells containing ...incompressible, viscous, and adiabatically stratified fluid. This model is relevant for studying tidal interactions between close-in planets and their stars, as well as close low-mass star binaries. We explore in detail the frequency-dependent tidal dissipation rates obtained from an extensive suite of numerical simulations, which we compare with linear theory, including with the widely employed frequency-averaged formalism to represent inertial wave dissipation. We demonstrate that the frequency-averaged predictions appear to be quite robust and are approximately reproduced in our nonlinear simulations spanning the frequency range of inertial waves as we vary the convective envelope thickness, tidal amplitude, and Ekman number. Yet, we find nonlinear simulations can produce significant differences with linear theory for a given tidal frequency (potentially by orders of magnitude), largely due to tidal generation of differential rotation and its effects on the waves. Since the dissipation in a given system can be very different both in linear and nonlinear simulations, the frequency-averaged formalism should be used with caution. Despite its robustness, it is also unclear how accurately it represents tidal evolution in real (frequency-dependent) systems.
The spin axis of a rotationally deformed planet is forced to precess about its orbital angular momentum vector, due to the tidal gravity of its host star, if these directions are misaligned. This ...induces internal fluid motions inside the planet that are subject to a hydrodynamic instability. We study the turbulent damping of precessional fluid motions, as a result of this instability, in the simplest local computational model of a giant planet (or star), with and without a weak internal magnetic field. Our aim is to determine the outcome of this instability, and its importance in driving tidal evolution of the spin–orbit angle in precessing planets (and stars). We find that this instability produces turbulent dissipation that is sufficiently strong that it could drive significant tidal evolution of the spin–orbit angle for hot Jupiters with orbital periods shorter than about 10–18 d. If this mechanism acts in isolation, this evolution would be towards alignment or anti-alignment, depending on the initial angle, but the ultimate evolution (if other tidal mechanisms also contribute) is expected to be towards alignment. The turbulent dissipation is proportional to the cube of the precession frequency, so it leads to much slower damping of stellar spin–orbit angles, implying that this instability is unlikely to drive evolution of the spin–orbit angle in stars (either in planetary or close binary systems). We also find that the instability-driven flow can act as a system-scale dynamo, which may play a role in producing magnetic fields in short-period planets.
ABSTRACT
Turbulent convection is thought to act as an effective viscosity (νE) in damping tidal flows in stars and giant planets. However, the efficiency of this mechanism has long been debated, ...particularly in the regime of fast tides, when the tidal frequency (ω) exceeds the turnover frequency of the dominant convective eddies (ωc). We present the results of hydrodynamical simulations to study the interaction between tidal flows and convection in a small patch of a convection zone. These simulations build upon our prior work by simulating more turbulent convection in larger horizontal boxes, and here we explore a wider range of parameters. We obtain several new results: (1) νE is frequency dependent, scaling as ω−0.5 when ω/ωc ≲ 1, and appears to attain its maximum constant value only for very small frequencies (ω/ωc ≲ 10−2). This frequency reduction for low-frequency tidal forcing has never been observed previously. (2) The frequency dependence of νE appears to follow the same scaling as the frequency spectrum of the energy (or Reynolds stress) for low and intermediate frequencies. (3) For high frequencies (ω/ωc ≳ 1 − 5), νE ∝ ω−2. 4) The energetically dominant convective modes always appear to contribute the most to νE, rather than the resonant eddies in a Kolmogorov cascade. These results have important implications for tidal dissipation in convection zones of stars and planets, and indicate that the classical tidal theory of the equilibrium tide in stars and giant planets should be revisited. We briefly touch upon the implications for planetary orbital decay around evolving stars.
Tidal friction is thought to be important in determining the long-term spin-orbit evolution of short-period extrasolar planetary systems. Using a simple model of the orbit-averaged effects of tidal ...friction, we study the evolution of close-in planets on inclined orbits, due to tides. We analyse the effects of the inclusion of stellar magnetic braking by performing a phase-plane analysis of a simplified system of equations, including the braking torque. The inclusion of magnetic braking is found to be important, and its neglect can result in a very different system history. We then present the results of numerical integrations of the tidal evolution equations, where we find that it is essential to consider coupled evolution of the orbital and rotational elements, including dissipation in both the star and planet, to accurately model the evolution. The main result of our integrations is that for typical Hot Jupiters, tidal friction aligns the stellar spin with the orbit on a similar time as it causes the orbit to decay. This tells us that if a planet is observed to be aligned, then it probably formed coplanar. This reinforces the importance of Rossiter–McLaughlin effect observations in determining the degree of spin-orbit alignment in transiting systems. We apply these results to the only observed system with a spin-orbit misalignment, XO-3, and constrain the efficiency of tidal dissipation (i.e. the modified tidal quality factors Q′) in both the star and the planet in this system. Using a model in which inertial waves are excited by tidal forcing in the outer convective envelope and dissipated by turbulent viscosity, we calculate Q′ for a range of F-star models, and find it to vary considerably within this class of stars. This means that using a single Q′, and assuming that it applies to all stars, is probably incorrect. In addition, we propose an explanation for the survival of two of the planets on the tightest orbits, WASP-12 b and OGLE-TR-56 b, in terms of weak dissipation in the star, as a result of their internal structures and slow rotation periods.
Abstract
We study how stably stratified or semi-convective layers alter the tidal dissipation rates associated with the generation of internal waves in planetary interiors. We consider if these ...layers could contribute to the high rates of tidal dissipation observed for Jupiter and Saturn in our solar system. We use an idealized global spherical Boussinesq model to study the influence of stable stratification and semi-convective layers on tidal dissipation rates. We carry out analytical and numerical calculations considering realistic tidal forcing and measure how the viscous and thermal dissipation rates depend on the parameters relating to the internal stratification profile. We find that the strongly frequency-dependent tidal dissipation rate is highly dependent on the parameters relating to the stable stratification, with strong resonant peaks that align with the internal modes of the system. The locations and sizes of these resonances depend on the form and parameters of the stratification, which we explore both analytically and numerically. Our results suggest that stable stratification can significantly enhance the tidal dissipation in particular frequency ranges. Analytical calculations in the low-frequency regime give us scaling laws for the key parameters, including the tidal quality factor
Q
′
due to internal gravity waves. Stably stratified layers can significantly contribute to tidal dissipation in solar and extrasolar giant planets, and we estimate substantial tidal evolution for hot Neptunes. Further investigation is needed to robustly quantify the significance of the contribution in realistic interior models, and to consider the contribution of inertial waves.
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