  For solving two-dimensional incompressible flow in the vorticity form by the
fourth-order compact finite difference scheme and explicit strong stability
preserving (SSP) temporal discretizations, we show that the simple
bound-preserving limiter in [Li H., Xie S., Zhang X., SIAM J. Numer. Anal., 56
(2018)]. can enforce the strict bounds of the vorticity, if the velocity field
satisfies a discrete divergence free constraint. For reducing oscillations, a
modified TVB limiter adapted from [Cockburn B., Shu CW., SIAM J. Numer. Anal.,
31 (1994)] is constructed without affecting the bound-preserving property. This
bound-preserving finite difference method can be used for any passive
convection equation with a divergence free velocity field.
