  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.
  (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 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 assume that the loss of stellar angular momentum is accounted for by the
r-mode growth. We demonstrate, for the first time, that r-mode radiation
reaction leads 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.
