scholarly journals Convective turbulent viscosity acting on equilibrium tidal flows: new frequency scaling of the effective viscosity

2020 ◽  
Vol 497 (3) ◽  
pp. 3400-3417 ◽  
Author(s):  
Craig D Duguid ◽  
Adrian J Barker ◽  
C A Jones
Keyword(s):  
Effective Viscosity ◽  
Turbulent Viscosity ◽  
Low Frequency ◽  
Giant Planets ◽  
Turbulent Convection ◽  
Tidal Dissipation ◽  
Orbital Decay ◽  
Tidal Theory ◽  
Tidal Flows ◽  
Hydrodynamical Simulations

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.

scholarly journals Tidal flows with convection: frequency-dependence of the effective viscosity and evidence for anti-dissipation

2019 ◽  
Author(s):  
Craig D Duguid ◽  
Adrian J Barker ◽  
C A Jones
Keyword(s):  
High Frequency ◽  
Effective Viscosity ◽  
Large Scale ◽  
Binary Systems ◽  
Orbital Evolution ◽  
Turbulent Convection ◽  
Hot Jupiters ◽  
Tidal Flows ◽  
Tidal Frequency ◽  
Hydrodynamical Simulations

Abstract Tidal interactions are important in driving spin and orbital evolution in planetary and stellar binary systems, but the fluid dynamical mechanisms responsible remain incompletely understood. One key mechanism is the interaction between tidal flows and convection. Turbulent convection is thought to act as an effective viscosity in damping large-scale tidal flows, but there is a long-standing controversy over the efficiency of this mechanism when the tidal frequency exceeds the turnover frequency of the dominant convective eddies. This high frequency regime is relevant for many applications, such as for tides in stars hosting hot Jupiters. We explore the interaction between tidal flows and convection using hydrodynamical simulations within a local Cartesian model of a small patch of a convection zone of a star or planet. We adopt the Boussinesq approximation and simulate Rayleigh-Bénard convection, modelling the tidal flow as a background oscillatory shear flow. We demonstrate that the effective viscosity of both laminar and turbulent convection is approximately frequency-independent for low frequencies. When the forcing frequency exceeds the dominant convective frequency, the effective viscosity scales inversely with the square of the tidal frequency. We also show that negative effective viscosities are possible, particularly for high frequency tidal forcing, suggesting the surprising possibility of tidal anti-dissipation. These results are supported by a complementary high-frequency asymptotic analysis that extends prior work by Ogilvie & Lesur. We discuss the implications of these results for interpreting the orbital decay of hot Jupiters, and for several other astrophysical problems.

scholarly journals On the frequency-dependent effective viscosity of laminar and turbulent flows

2019 ◽  
Vol 82 ◽  
pp. 35-42
Author(s):  
G.I. Ogilvie
Keyword(s):  
Fluid Flow ◽  
Turbulent Flows ◽  
Effective Viscosity ◽  
Three Dimensional ◽  
Giant Planets ◽  
Frequency Dependent ◽  
Tidal Dissipation ◽  
Jean Paul ◽  
Phd Thesis ◽  
Laminar And Turbulent Flows

The efficiency of tidal dissipation in convective zones of stars and giant planets depends, in part, on the response of a three-dimensional fluid flow to the periodic deformation due to the equilibrium tide — a problem considered by Jean-Paul Zahn in his PhD thesis. We review recent results on this problem and present novel calculations based on some idealized models.

scholarly journals Tidal dissipation in evolving low-mass and solar-type stars with predictions for planetary orbital decay

2020 ◽  
Vol 498 (2) ◽  
pp. 2270-2294
Author(s):  
A J Barker
Keyword(s):  
Gravity Waves ◽  
Wave Breaking ◽  
Effective Viscosity ◽  
Main Sequence ◽  
Internal Gravity Waves ◽  
Stellar Structure ◽  
Tidal Dissipation ◽  
Dominant Mechanism ◽  
Hot Jupiters ◽  
Orbital Decay

ABSTRACT We study tidal dissipation in stars with masses in the range 0.1–1.6 M⊙ throughout their evolution, including turbulent effective viscosity acting on equilibrium tides and inertial waves (IWs) in convection zones, and internal gravity waves in radiation zones. We consider a range of stellar evolutionary models and incorporate the frequency-dependent effective viscosity acting on equilibrium tides based on the latest simulations. We compare the tidal flow and dissipation obtained with the conventional equilibrium tide, which is strictly invalid in convection zones, finding that the latter typically overpredicts the dissipation by a factor of 2–3. Dissipation of IWs is computed using a frequency-averaged formalism accounting for realistic stellar structure for the first time, and is the dominant mechanism for binary circularization and synchronization on the main sequence. Dissipation of gravity waves in the radiation zone assumes these waves to be fully damped (e.g. by wave breaking), and is the dominant mechanism for planetary orbital decay. We calculate the critical planetary mass required for wave breaking as a function of stellar mass and age, and show that this mechanism predicts destruction of many hot Jupiters but probably not Earth-mass planets on the main sequence. We apply our results to compute tidal quality factors following stellar evolution, and tidal evolutionary time-scales, for the orbital decay of hot Jupiters, and the spin synchronization and circularization of binary stars. We also provide predictions for shifts in transit arrival times due to tidally driven orbital decay of hot Jupiters that may be detected with NGTS, TESS, or PLATO.

scholarly journals Efficiency of tidal dissipation in slowly rotating fully convective stars or planets

2020 ◽  
Vol 497 (4) ◽  
pp. 4472-4485 ◽  
Author(s):  
Jérémie Vidal ◽  
Adrian J Barker
Keyword(s):  
Frequency Spectrum ◽  
Effective Viscosity ◽  
Binary Systems ◽  
Tidal Flow ◽  
Orbital Evolution ◽  
Model Parameters ◽  
Turnover Frequency ◽  
Tidal Dissipation ◽  
Tidal Flows ◽  
Turbulent Cascade

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.

scholarly journals Tidal dissipation in stars and giant planets: Jean-Paul Zahn's pioneering work and legacy

2019 ◽  
Vol 82 ◽  
pp. 5-33 ◽  
Author(s):  
S. Mathis
Keyword(s):  
Binary Stars ◽  
Planetary Systems ◽  
Giant Planets ◽  
Tidal Dissipation ◽  
Turbulent Friction ◽  
Jean Paul ◽  
Tidal Flows ◽  
Order Of Magnitude ◽  
Celestial Bodies ◽  
Work Done

In this lecture opening the session focused on tides in stellar and planetary systems, I will review the Jean-Paul Zahn's key contributions to the theory of tidal dissipation in stars and fluid planetary layers. I will first recall the general principles of tidal friction in celestial bodies. Then, I will focus on the theories of the stellar equilibrium and dynamical tides founded by Jean-Paul and their predictions for the evolution of binary stars. I will underline their essential legacy for ongoing studies of tidal dissipation in stars hosting planets and in fluid planetary regions. I will also discuss his pioneering work on the turbulent friction applied on tidal flows by stellar convection and the corresponding still unsolved challenging problems. Next, I will present the results we obtained on tidal dissipation in the potential dense rocky/icy core of gaseous giant planets such as Jupiter and Saturn within the Encelade international team. This mechanism provides important keys to interpret the high-precision astrometric measurements of the rates of tidal orbital migration of the moons of these planets, which are found to be larger than expected. This corresponds to a Jovian and Saturnian tidal frictions which are higher by one order of magnitude than the usually used values calibrated on formation scenarios. Finally, I will review the work done by Jean-Paul and Michel Rieutord on potential Ekman boundary layers associated to tidal flows. As a consequence, a coherent physical modeling of tides is now mandatory to understand the properties and the evolution of stellar and planetary systems. To progress on this forefront research subject, we are walking on the path first drawn by Jean-Paul.

scholarly journals Pebble-isolation mass: Scaling law and implications for the formation of super-Earths and gas giants

2018 ◽  
Vol 612 ◽  
pp. A30 ◽  
Author(s):  
Bertram Bitsch ◽  
Alessandro Morbidelli ◽  
Anders Johansen ◽  
Elena Lega ◽  
Michiel Lambrechts ◽  
...  
Keyword(s):  
Turbulent Viscosity ◽  
Scaling Law ◽  
Stokes Number ◽  
Giant Planets ◽  
Mass Scaling ◽  
The Core ◽  
Pressure Bump ◽  
Hydrodynamical Simulations ◽  
Gas Accretion ◽  
Migration Pathways

The growth of a planetary core by pebble accretion stops at the so-called pebble isolation mass, when the core generates a pressure bump that traps drifting pebbles outside its orbit. The value of the pebble isolation mass is crucial in determining the final planet mass. If the isolation mass is very low, gas accretion is protracted and the planet remains at a few Earth masses with a mainly solid composition. For higher values of the pebble isolation mass, the planet might be able to accrete gas from the protoplanetary disc and grow into a gas giant. Previous works have determined a scaling of the pebble isolation mass with cube of the disc aspect ratio. Here, we expand on previous measurements and explore the dependency of the pebble isolation mass on all relevant parameters of the protoplanetary disc. We use 3D hydrodynamical simulations to measure the pebble isolation mass and derive a simple scaling law that captures the dependence on the local disc structure and the turbulent viscosity parameter α. We find that small pebbles, coupled to the gas, with Stokes number τf < 0.005 can drift through the partial gap at pebble isolation mass. However, as the planetary mass increases, particles must be decreasingly smaller to penetrate the pressure bump. Turbulent diffusion of particles, however, can lead to an increase of the pebble isolation mass by a factor of two, depending on the strength of the background viscosity and on the pebble size. We finally explore the implications of the new scaling law of the pebble isolation mass on the formation of planetary systems by numerically integrating the growth and migration pathways of planets in evolving protoplanetary discs. Compared to models neglecting the dependence of the pebble isolation mass on the α-viscosity, our models including this effect result in higher core masses for giant planets. These higher core masses are more similar to the core masses of the giant planets in the solar system.

scholarly journals Survival of planets around shrinking stellar binaries

2015 ◽  
Vol 112 (30) ◽  
pp. 9264-9269 ◽  
Author(s):  
Diego J. Muñoz ◽  
Dong Lai
Keyword(s):  
Binary Stars ◽  
Orbital Evolution ◽  
Eclipsing Binaries ◽  
Target List ◽  
Tidal Dissipation ◽  
Compact Binaries ◽  
Kepler Mission ◽  
Orbital Decay ◽  
Erratic Behavior ◽  
Circumbinary Planets

The discovery of transiting circumbinary planets by the Kepler mission suggests that planets can form efficiently around binary stars. None of the stellar binaries currently known to host planets has a period shorter than 7 d, despite the large number of eclipsing binaries found in the Kepler target list with periods shorter than a few days. These compact binaries are believed to have evolved from wider orbits into their current configurations via the so-called Lidov–Kozai migration mechanism, in which gravitational perturbations from a distant tertiary companion induce large-amplitude eccentricity oscillations in the binary, followed by orbital decay and circularization due to tidal dissipation in the stars. Here we explore the orbital evolution of planets around binaries undergoing orbital decay by this mechanism. We show that planets may survive and become misaligned from their host binary, or may develop erratic behavior in eccentricity, resulting in their consumption by the stars or ejection from the system as the binary decays. Our results suggest that circumbinary planets around compact binaries could still exist, and we offer predictions as to what their orbital configurations should be like.

Tidal dissipation modelling in gaseous giant planets at the time of space missions

2021 ◽  
Author(s):  
Hachem Dhouib ◽  
Stéphane Mathis ◽  
Florian Debras ◽  
Aurélie Astoul ◽  
Clément Baruteau
Keyword(s):  
Internal Structure ◽  
Differential Rotation ◽  
Giant Planets ◽  
Solid Core ◽  
Large Radius ◽  
Mixing Zone ◽  
Convective Envelope ◽  
Tidal Dissipation ◽  
Hot Jupiters ◽  
Tidal Waves

&lt;p&gt;Gaseous giant planets (Jupiter and Saturn in our solar system and hot Jupiters around other stars) are turbulent rotating magnetic objects that have strong and complex interactions with their environment (their moons in the case of Jupiter and Saturn and their host stars in the case of hot Jupiters/Saturns). In such systems, the dissipation of tidal waves excited by tidal forces shape the orbital architecture and the rotational dynamics of the planets.&lt;/p&gt; &lt;p&gt;During the last decade, a revolution has occurred for our understanding of tides in these systems. First, Lainey et al. (2009, 2012, 2017) have measured tidal dissipation stronger by one order of magnitude than expected in Jupiter and Saturn. Second, unexplained broad diversity of orbital architectures and large radius of some hot Jupiters are observed in exoplanetary systems. Finally, new constraints obtained thanks to &lt;em&gt;Kepler&lt;/em&gt;/K2 and TESS indicate that tidal dissipation in gaseous giant exoplanets is weaker than in Jupiter and in Saturn (Ogilvie 2014, Van Eylen et al. 2018, Huber et al. 2019).&lt;/p&gt; &lt;p&gt;Furthermore, the space mission JUNO and the grand finale of the CASSINI mission have revolutionized our knowledge of the interiors of giant planets. We now know, for example, that Jupiter is a very complex planet: it is a stratified planet with, from the surface to the core, a differentially rotating convective envelope, a first mixing zone (with stratified convection), a uniformly rotating magnetised convective zone, a second magnetized mixing zone (the diluted core, potentially in stratified convection) and a solid core (Debras &amp; Chabrier 2019). So far, tides in these planets have been studied by assuming a simplified internal structure with a stable rocky and icy core (Remus et al. 2012, 2015) and a deep convective envelope surrounded by a thin stable atmosphere (Ogilvie &amp; Lin 2004) where mixing processes, differential rotation and magnetic field were completely neglected.&lt;/p&gt; &lt;p&gt;Our objective is thus to predict tidal dissipation using internal structure models, which agree with these last observational constrains. In this work, we build a new ab-initio model of tidal dissipation in giant planets that coherently takes into account the interactions of tidal waves with their complex stratification induced by the mixing of heavy elements, their zonal winds, and (dynamo) magnetic fields. This model is a semi-global model in the planetary equatorial plane. We study the linear excitation of tidal magneto-gravito-inertial progressive waves and standing modes. We take into account the buoyancy, the compressibility, the Coriolis acceleration (including differential rotation), and the Lorentz force. The tidal waves are submitted to the different potential dissipative processes: Ohmic, thermal, molecular diffusivities, and viscosity. We here present the general formalism and the potential regimes of parameters that should be explored. The quantities of interest such as tidal torque, dissipation, and heating are derived. This will pave the way for full 3D numerical simulations that will take into account complex internal structure and dynamics of gaseous giant (exo-)planets in spherical/spheroidal geometry.&lt;/p&gt; &lt;p&gt;&amp;#160;&lt;/p&gt;

scholarly journals Detectability of embedded protoplanets from hydrodynamical simulations

2020 ◽  
Vol 492 (3) ◽  
pp. 3440-3458 ◽  
Author(s):  
E Sanchis ◽  
G Picogna ◽  
B Ercolano ◽  
L Testi ◽  
G Rosotti
Keyword(s):  
Surface Density ◽  
Three Dimensional ◽  
Giant Planets ◽  
Evolutionary Models ◽  
Upper Limits ◽  
Disc Material ◽  
Silicate Feature ◽  
Hydrodynamical Simulations ◽  
Cq Tau ◽  
Absolute Magnitudes

ABSTRACT We predict magnitudes for young planets embedded in transition discs, still affected by extinction due to material in the disc. We focus on Jupiter-sized planets at a late stage of their formation, when the planet has carved a deep gap in the gas and dust distributions and the disc starts to being transparent to the planet flux in the infrared (IR). Column densities are estimated by means of three-dimensional hydrodynamical models, performed for several planet masses. Expected magnitudes are obtained by using typical extinction properties of the disc material and evolutionary models of giant planets. For the simulated cases located at 5.2 au in a disc with a local unperturbed surface density of 127 $\mathrm{g} \, \mathrm{cm}^{-2}$, a 1MJ planet is highly extinct in the J, H, and Kbands, with predicted absolute magnitudes ≥ 50 mag. In the L and Mbands, extinction decreases, with planet magnitudes between 25 and 35 mag. In the Nband, due to the silicate feature on the dust opacities, the expected magnitude increases to ∼40 mag. For a 2MJ planet, the magnitudes in the J, H, and Kbands are above 22 mag, while for the L, M, and Nbands, the planet magnitudes are between 15 and 20 mag. For the 5MJ planet, extinction does not play a role in any IR band, due to its ability to open deep gaps. Contrast curves are derived for the transition discs in CQ Tau, PDS 70, HL Tau, TW Hya, and HD 163296. Planet mass upper limits are estimated for the known gaps in the last two systems.

scholarly journals How much does turbulence change the pebble isolation mass for planet formation?

2018 ◽  
Vol 615 ◽  
pp. A110 ◽  
Author(s):  
S. Ataiee ◽  
C. Baruteau ◽  
Y. Alibert ◽  
W. Benz
Keyword(s):  
Aspect Ratio ◽  
Planet Formation ◽  
Turbulent Viscosity ◽  
Analytical Calculation ◽  
Radial Pressure ◽  
Stokes Number ◽  
High Viscosity ◽  
Necessary Condition ◽  
Pressure Maximum ◽  
Hydrodynamical Simulations

Context. When a planet becomes massive enough, it gradually carves a partial gap around its orbit in the protoplanetary disk. A pressure maximum can be formed outside the gap where solids that are loosely coupled to the gas, typically in the pebble size range, can be trapped. The minimum planet mass for building such a trap, which is called the pebble isolation mass (PIM), is important for two reasons: it marks the end of planetary growth by pebble accretion, and the trapped dust forms a ring that may be observed with millimetre observations. Aims. We study the effect of disk turbulence on the PIM and find its dependence on the gas turbulent viscosity, aspect ratio, and particles Stokes number. Methods. By means of 2D gas hydrodynamical simulations, we found the minimum planet mass to form a radial pressure maximum beyond the orbit of the planet, which is the necessary condition to trap pebbles. We then carried out 2D gas plus dust hydrodynamical simulations to examine how dust turbulent diffusion impacts particles trapping at the pressure maximum. We finally provide a semi-analytical calculation of the PIM based on comparing the radial drift velocity of solids and the root mean square turbulent velocity fluctuations around the pressure maximum. Results. From our results of gas simulations, we provide an expression for the PIM vs. disk aspect ratio and turbulent viscosity. Our gas plus dust simulations show that the effective PIM can be nearly an order of magnitude larger in high-viscosity disks because turbulence diffuse particles out of the pressure maximum. This is quantified by our semi-analytical calculation, which gives an explicit dependence of the PIM with Stokes number of particles. Conclusions. Disk turbulence can significantly alter the PIM, depending on the level of turbulence in regions of planet formation.

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