Origin of tidal dissipation in Jupiter.: II.: The value of Q

The process of tidal dissipation inside Jupiter is not yet understood. Its tidal quality factor ( Q) is inferred to lie between 10(5) and 10(6). Having studied the structure and properties of inertial modes in a neutrally buoyant, coreless, uniformly rotating sphere in a companion paper, we examine here their effects on tidal dissipation. The rate of dissipation caused by resonantly excited inertial modes depends on the following three parameters: how well they are coupled to the tidal potential, how strongly they are dissipated ( by the turbulent viscosity), and how densely distributed they are in frequency. We find that as a function of tidal frequency, the Q-value exhibits large fluctuations, with its maximum value set by the group of inertial modes that satisfy delta omega similar to gamma, where delta omega is the group's typical offset from an exact resonance and gamma are the modes' turbulent damping rates. These are intermediate-order inertial modes with wavenumber lambda similar to 60, and they are excited to a small surface displacement amplitude of order 10(3) cm. The Q-value drops much below the maximum value whenever a lower order mode happens to be in resonance. In our model, inertial modes shed their tidally acquired energy very close to the surface within a narrow latitudinal zone ( the singularity belt''), and the tidal luminosity escapes freely out of the planet. The strength of coupling between the tidal potential and inertial modes is sensitive to the presence of density discontinuities inside Jupiter. In the case of a discreet density jump, as may be caused by the transition between metallic and molecular hydrogen, we find a time-averaged Q similar to 10(7), with a small but nonnegligible chance (similar to 10%) that the current Q-value falls within the empirically determined range. But when such a jump does not exist, Q similar to 10(9). Even though it remains unclear whether tidal dissipation due to resonant inertial modes is the correct answer to the problem, it is impressive that even our simple treatment taking planetary rotation into account already leads to a 3-5 orders of magnitude stronger damping than when rotation is ignored. Moreover, our conclusions are not affected by the presence of a small solid core, a different prescription for the turbulent viscosity, or nonlinear mode coupling, but they depend critically on the static stability in the upper atmosphere of Jupiter. This is currently uncertain. Lastly, we compare our results with those from a competing work by Ogilvie & Lin and discuss the prospect of extending this theory to exo-Jupiters, which appear to possess Q-values similar to that of Jupiter.