All rotating stars possess a class of circulation modes (r-modes) that are driven toward instability by gravitational radiation reaction. In the hot, rapidly rotating neutron stars that come fresh out of a supernova explosion, this destabilizing force is strong enough to dominate the internal viscous dissipation of the star. So it is expected that the amplitude of the r-mode will grow enormously within ten minutes of the birth of the star. The growth of the r-mode converts rotational energy into mode energy, while gravitational radiation removes angular momentum from the star. Eventually the r-mode should saturate, although no one really knows by what process.
The strength of the radiation emitted and the amount of the angular momentum removed depend critically on the maximum mode amplitude. In collaboration with Lee Lindblom and Joel E. Tohline, I investigated the growth of the r-modes by solving numerically the nonlinear hydrodynamic equations that describe their evolution. In our simulations, the circulation velocities of the mode grow to be comparable to the rotation velocity of the star; then strong shocks develop which damp the mode quickly. Before they do so, the star loses about 40 percent of its initial angular momentum. The nonlinear evolution creates a strong differential rotation, which is concentrated near the surface and especially near the poles of the star.
The numerical code used in our simulation solves the Newtonian equations of hydrodynamics in a rotating reference frame, using the Fortran 90 code developed at LSU by Joel Tohline to study a variety of astrophysical problems.
This movie shows the final stages of the nonlinear evolution of the stellar model. Even if the r-mode is primarily a circulation mode, at high amplitudes a correlated perturbation appears also in the density. The density perturbation grows and finally creates cresting waves on the surfaces. The strong shocks associated with the cresting waves dissipate enough energy to damp the r-modes, and destroy the density perturbation pattern.
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