Linear vs. Quasi-Linear vs. Nonlinear Calculations

Linear Regime:

The above greyscale plots 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 by a fully nonlinear 3-dimensional hydrodynamics code and the greyscale on the left was found by using a linearized hydrodynamics code. The visual comparison is very good. Other resuilts are:

• eigenvalues found in the two simulations agreed to 2 digits.
• for the cases where nonlinear simulations have yielded results, we find that there is also good agreement the linear and nonlinear simulations for the higher order modes (large m-modes).
• exploration of parameter space using the linear code showed that results based on the classical incompressible fluids yielded good insight into the properties of compressible fluids.
• one-armed spirals were stable for the range of initial condtiions studied.

The understanding of nonaxisymmetric dynamic instabilities in the linear regime appears to be well-in-hand.

Quasi-Linear Regime:

The bar mode deforms the oblate spheroid into a bar. The disturbance increases in amplitude until the mode saturates. We argue that 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. Given this assumption, we calculate the expected quasi-linear structure (see above). On the left is the velocity field in the frame of the bar and on the right is a greyscale (linear) representation of the density.

Notice that off-center density maxima form and that vortices form around the off-center maxima. The fluid does not circulate through the maxima in the frame of the bar. The bar might be a nascent binary system. Further work on the evolution of such bars is in progress (a la Tohline & Cazes 1998).

Nonlinear Regime

The above plots are for the (n,n') = (3/2,0) and T/|W| = 0.327 simulation. The general appearance is in reasonable 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.

The combination of linear and quasi-linear theories allows us to model the nonlinear evolution of the bar mode instability (reasonably well).

This is of interest because of the difficulty in performing nonlinear simulation; it is not possible to explore thoroughly the available parameter space using nonlinear simulations.