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

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.

On the interaction between fast tides and convection

“…The interaction between equilibrium tides and convection is therefore likely to be dominated by terms considered in previous Boussinesq models (e.g. Ogilvie & Lesur 2012;Duguid et al 2020b;Vidal & Barker 2020a). These usually indicate weak dissipation of equilibrium tides, such that tidal dissipation in pre-and main-sequence stars and giant planets is probably instead due to inertial and internal gravity waves (e.g.…”

“…Recent local (Ogilvie & Lesur 2012;Braviner 2015;Duguid et al 2020a,b) and global (Vidal & Barker 2020b,a) numerical simulations have provided strong evidence that νE ∝ (ω/ωc) −2 for fast tides (though not for the reasons originally proposed), due to Reynolds stresses involving correlations between convective flow components. As a result, tidal evolution due to equilibrium tide dissipation is predicted to be weak in most applications involving pre-and main-sequence stars and giant planets (Duguid et al 2020b;Barker 2020), though it is probably still the dominant mechanism in giant stars (e.g. Verbunt & Phinney 1995;Mustill & Villaver 2012;Beck et al 2018;Sun et al 2018;Price-Whelan & Goodman 2018).…”

On a new formulation for energy transfer between convection and fast tides with application to giant planets and solar type stars

“…This term, however, is present in the energy conservation equation for the fluctuations even when a linear analysis of the tides is carried out, as it comes from the u j ∂V i /∂ x j term in Navier-Stokes equation. In Goodman & Oh (1997), it is eliminated on the assumption that it does not contribute to dissipation and, in Ogilvie & Lesur (2012) and Duguid, Barker, & Jones (2020), it cancels out for the particular form of the flow chosen to model the tides.…”

“…Numerical simulations have attempted to measure the turbulent viscosity and its period dependence in local models (Penev et al 2009;Ogilvie & Lesur 2012;Duguid, Barker, & Jones 2020), and the first global simulations have been published very recently (Vidal & Barker 2020a,b). Interestingly, the simulations (in the four more recent publications) show that the turbulent viscosity actually becomes negative at large forcing frequencies.…”

“…Interestingly, the simulations (in the four more recent publications) show that the turbulent viscosity actually becomes negative at large forcing frequencies. This suggests that the standard picture of convective turbulence dissipating the tides is dubious when the period of the tides is smaller than the turnover timescale, even though negative viscosities are only obtained for unrealistically low tidal periods (Duguid, Barker, & Jones 2020;Vidal & Barker 2020b).…”