Differential rotation of nonlinear r-modes

The differential rotation of r-modes is investigated within the nonlinear theory up to second order in the mode amplitude in the case of a slowly rotating, Newtonian, barotropic, perfect-fluid star. We find a nonlinear extension of the linear r-mode, which represents differential rotation that produces large scale drifts of fluid elements along stellar latitudes. This solution includes a piece induced by first-order quantities and another one that is a pure second-order effect. Since the latter is stratified on cylinders, it cannot cancel differential rotation induced by first-order quantities, which is not stratified on cylinders. It is shown that, unlike the situation in the linearized theory, r-modes do not preserve the vorticity of fluid elements at second order. It is also shown that the physical angular momentum and energy of the perturbation are, in general, different from the corresponding canonical quantities.