We study the tidal forcing, propagation and dissipation of linear inertial waves in a rotating fluid body. The intentionally simplified model involves a perfectly rigid core surrounded by a deep ocean consisting of a homogeneous incompressible fluid. Centrifugal effects are neglected, but the Coriolis force is considered in full, and dissipation occurs through viscous or frictional forces. The dissipation rate exhibits a complicated dependence on the tidal frequency and generally increases with the size of the core. In certain intervals of frequency, efficient dissipation is found to occur even for very small values of the coefficient of viscosity or friction. We discuss the results with reference to wave attractors, critical latitudes and other features of the propagation of inertial waves within the fluid, and comment on their relevance for tidal dissipation in planets and stars.
Deep Dive into Tidal dissipation in rotating fluid bodies: a simplified model.
We study the tidal forcing, propagation and dissipation of linear inertial waves in a rotating fluid body. The intentionally simplified model involves a perfectly rigid core surrounded by a deep ocean consisting of a homogeneous incompressible fluid. Centrifugal effects are neglected, but the Coriolis force is considered in full, and dissipation occurs through viscous or frictional forces. The dissipation rate exhibits a complicated dependence on the tidal frequency and generally increases with the size of the core. In certain intervals of frequency, efficient dissipation is found to occur even for very small values of the coefficient of viscosity or friction. We discuss the results with reference to wave attractors, critical latitudes and other features of the propagation of inertial waves within the fluid, and comment on their relevance for tidal dissipation in planets and stars.
Tidal interactions determine the fate of short-period extrasolar planets. They affect their spin and orbital parameters, and in extreme cases may lead to the destruction of planets as a result of orbital decay or intense tidal heating. Tidal evolution is also important in close binary stars and in the satellite systems of the planets of the solar system.
The efficiency of these processes is often parametrized by the tidal quality factor Q, which is a dimensionless inverse measure of the dissipative properties of the body considered as a forced oscillator (e.g. Goldreich & Soter 1966). The tidal Q of fluid bodies such as stars and giant planets is difficult to calculate from first principles, even in linear theory. The tidal disturbance generally consists of two parts. One (the ’equilibrium tide’) is a quasi-hydrostatic bulge that is carried around the body by a smooth velocity field, while the other (the ‘dynamical tide’) consists of internal waves that are excited by the low-frequency tidal forcing and may have a short wavelength. Dissipation of the equilibrium tide can occur through its interaction with turbulent convection, but the damping rate is uncertain, especially when the tidal period is short compared to the convective timescale (Zahn 1966b(Zahn , 1977;;Goldreich & Keeley 1977;Goldreich & Nicholson 1977;Goodman & Oh 1997;Penev et al. 2007). This approach has perhaps been most successful in application to the circularization of binaries containing a giant star (Verbunt & Phinney 1995), in which case the tidal period exceeds the convective timescale. To study dynamical tides one should consider the excitation, propagation and dissipation of low-frequency internal waves in rotating, stratified fluids. Calculations have been made for a variety of objects, using different simplifications and approximations, by Zahn (1970Zahn ( , 1975Zahn ( , 1977)), Savonije & Papaloizou (1983, 1984), Savonije, Papaloizou & Alberts (1995), Savonije & Papaloizou (1997), Papaloizou & Savonije (1997), Lubow, Tout & Livio (1997), Terquem et al. (1998), Goodman & Dickson (1998), Witte & Savonije (1999), Savonije & Witte (2002), Ogilvie & Lin (2004, 2007), Wu (2005a,b), Papaloizou & Ivanov (2005) and Ivanov & Papaloizou (2007). This approach has perhaps been most successful for early-type stars, in which gravity (or inertia-gravity) waves are excited near the base of the radiative envelope and propagate outwards until they are dissipated by radiative damping; in this case estimates can be made rather simply for the dissipation rate (or tidal torque) that are independent of the details of the wave damping mechanism (Goldreich & Nicholson 1989).
In recent work on dynamical tides in rotating giant planets and stars (Ogilvie & Lin 2004, 2007) we have emphasized the role of inertial waves in convective regions, which are nearly adiabatically stratified and do not support gravity waves (see also Wu 2005a,b;Papaloizou & Ivanov 2005;Ivanov & Papaloizou 2007). Inertial waves (Greenspan 1968) propagate in a uniformly rotating fluid at frequencies ω smaller in magnitude than twice the spin frequency Ω, as seen in a frame rotating with the fluid. The group velocity of an inertial wave is proportional to its wavelength and is inclined at an angle λ = arcsin |ω/2Ω| to the rotation axis. The behaviour of rays propagating within a spherical annulus (a thick spherical shell) is very complicated and sensitive to the value of λ (Rieutord, Georgeot & Valdettaro 2001). It generally involves the focusing of rays towards limit cycles known as wave attractors. Another important feature is the existence of a critical latitude (equal to λ) at which the rays are tangent to the core and a singularity is introduced into solutions of the inviscid wave equations. Numerical investigations indicate that the tidally forced disturbances in this frequency range are concentrated in narrow beams whose width diminishes as the viscosity ν is reduced (Ogilvie & Lin 2004). In certain intervals of ω/Ω, a dissipation rate that appears to be asymptotically independent of ν may be achieved. However, the dissipation rate varies in a complicated way with the tidal frequency; presumably this occurs because of the sensitivity of the ray propagation to the value of λ, and depends on the waves being reflected from the inner and outer boundaries.
In an attempt to understand aspects of this behaviour, we have investigated a variety of reduced models. Using a prototypical partial differential equation for internal waves (Ogilvie 2005), we studied the response to periodic forcing in a two-dimensional domain in which the rays are focused towards a single wave attractor, as occurs in the experiments of Maas et al. (1997) and Manders & Maas (2003). We constructed an asymptotic analytical solution and showed that a non-zero dissipation rate is achieved in the astrophysically relevant limit of small viscosity (i.e. small Ekman number). This limiting dissipation rate is independent
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