Nonlinear r-modes in a spherical shell: issues of principle

We use a simple physical model to study the nonlinear behaviour of the r-mode instability. We assume that r-modes (Rossby waves) are excited in a thin spherical shell of rotating incompressible fluid. For this case, exact Rossby wave solutions of arbitrary amplitude are known. We find that: (a) These nonlinear Rossby waves carry ZERO physical angular momentum and positive physical energy, which is contrary to the folklore belief that the r-mode angular momentum and energy are negative. We think that the origin of the confusion lies in the difference between physical and canonical quantities. (b) Within our model, we confirm the differential drift reported by Rezzolla, Lamb and Shapiro (1999). Radiation reaction is introduced into the model by assuming that the fluid is electrically charged; r-modes are coupled to electromagnetic radiation through current (magnetic) multipole moments. We study the coupled equations of charged fluid and Maxwell field dynamics and find that: (c) To linear order in the mode amplitude, r-modes are subject to the CFS instability, as expected. (d) Radiation reaction decreases the angular velocity of the shell and causes differential rotation (which is distinct from but similar in magnitude to the differential drift reported by Rezzolla et al.) prior to saturation of the r-mode growth. This is contrary to the phenomenological treatments to date, which assumed that, prior to the saturation of the r-mode amplitude, the loss of stellar angular momentum is accounted for by the r-mode growth. This establishes, for the first time, that radiation reaction leads not only to overall loss of angular momentum, but also to differential rotation. (e) We show that for l = 2 r-mode electromagnetic radiation reaction is equivalent to gravitational radiation reaction in the lowest post-Newtonian order. Based on our electromagnetic calculations, we conclude that inertial frame dragging, both from the background rotation and from the r-mode itself, will modify the r-mode frequency by a factor ∼ RSchwarzschild/Rstar, in qualitative agreement with Kojima (1998).