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We consider the
secular evolution of cooling protostar sequences.
Because, we are concerned primarily with hydrodynamic instabilities,
we use a simplified treatment
of the thermal physics. We consider barotropic
equations-of-state (EOS);
in particular, we use a pseudo-polytropic
EOS, i.e.,
where K, the polytropic constant, is decreased exponentially with time and n, the polytropic index, is held constant. The polytropic constant for an incompressible gas is n = 0, a monatomic gas is n = 3/2 and a diatomic gas is n = 5/2.
For a pseudo-polytropic gas, only the mass and momentum conservation equations,
need be solved. We use our combination of linear, quasi-linear, and nonlinear numerical codes to study the fission problem.
Linear Simulations:
The linear evolution equations for perturbations of the form,
are
The linearized problem is solved on a cylindrical grid. The
spatial derivatives are discretized
while the time evolution is performed using a
Fourth-order Runge-Kutta algorithm.
An equilibrium is found by setting the time derivatives
to 0, specifying an angular momentum distribution, and polytropic
index n, and then using the Self Consistent Field technique.
The evolution equations are then solved by following an initially random
perturbation (of small size ~ 10-5)
away from the equilibrium state.
Nonlinear Simulations
The fully nonlinear simulations use the 3-dimensional forms of the hydrodynamic equations. The nonlinear problem is solved as an initial value problem by following the evolution of the same axisymmetric equilibrium states used in the linear analysis. The computational code was initially written by J. Tohline (LSU); it was modified by Durisen et al. (IU) so that in its current form is second-order in space and time.
The unstable modes could grow from numerical noise due to the discreteness of the grid. However, to speed the process along, random perturbations of size 10-5 are applied. The equations are solved on a cylindrical grid with a transport scheme 2nd order in space and time.
Quasi-Linear Evolution
The linear and nonlinear simulations are bridged using a quasi-linear theory. The quasi-linear coupling comes from the gravitational self-interaction torque due to the linear eigenfunction (the spiral pattern leads to the self-interaction torque) and has the form
