To investigate the stability properties of polar disks we performed two-dimensional hydrodynamical simulations for flat polytropic gaseous self-gravitating disks which were perturbed by a central S0-like component .
Our disk was constructed to resemble that of the proto-typical galaxy NGC 4650A .
This central perturbation induces initially a stationary two-armed tightly-wound leading spiral in the polar disk .
For a hot disk ( Toomre parameter Q > 1.7 ) , the structure does not change over the simulation time of 4.5 Gyr .
In case of colder disks , the self-gravity of the spiral becomes dominant , it decouples from the central perturbation and grows , until reaching a saturation stage in which an open trailing spiral is formed , rather similar to that observed in NGC 4650A .
The timescale for developing non-linear structures is 1-2 Gyr ; saturation is reached within 2-3 Gyr .
The main parameter controlling the structure formation is the Toomre parameter .
The results are surprisingly insensitive to the properties of the central component .
If the polar disk is much less massive than that in NGC 4650A , it forms a weaker tightly-wound spiral , similar to that seen in dust absorption in the dust disk of NGC 2787 .
Our results are derived for a polytropic equation of state , but appear to be generic as the adiabatic exponent is varied between \gamma = 1 ( isothermal ) and \gamma = 2 ( very stiff ) .