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 $10^6$. We examine effects of inertial-modes on tidal dissipation in a neutrally bouyant, core-less, uniformly rotating planet. 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 have a typical offset from an exact resonance of order their turbulent damping rates. 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. 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 $Q \sim 10^7$. Even though it remains unclear whether tidal dissipation due to resonant inertial-modes is the correct answer to the problem, it is impressive that our simple treatment here already leads to three to five orders of magnitude stronger damping than that from the equilibrium tide. 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.