Linear vs. Quasi-Linear vs. Nonlinear Calculations

Linear Regime:

The above show the density contours for the linear eigefunction of an (n,n') = (3/2,0) polytrope with T/|W| = 0.327. The greyscale on the right was found using our nonlinear 3-dimensional hydrodynamics code and the greyscale on the left found using our linearized hydrodynamics code. The visual comparison is good. Other results are:

• eigenvalues for the bar mode for this model found by the two simulations agree to 2 digits;
• there is also good agreement between the linear and nonlinear simulations for the higher order modes (large m-modes);

Quasi-Linear vs Nonlinear Regime:

The bar mode deforms the oblate spheroid into a bar. The disturbance increases in amplitude until the mode saturates. The mode saturates when the time scale for the torque to drive the unstable object to the marginally stable axisymmetric configuration becomes comparable to the growth time scale for the bar mode. The above quasi-linear structure above is based on this assumption. The velocity field in the frame of the bar and a greyscale representation of the density are shown.

The above is for the equilibrium model used in the linear/quasi-linear analysis. The general appearance is in agreement with the quasi-linear theory. Other properties such as the bar mass are within 1 % of the values predicted by the quasi-linear theory.