  We study a class of initial value problems (IVPs) involving perturbations on
a density stratified, quiescent, viscous liquid layer with a free-surface. The
geometry is a two-dimensional, rectangular configuration taking into account
surface-tension and gravity. Linearised predictions are obtained by solving the
IVP analytically for free-surface and vortical initial perturbations. The
viscous spectrum comprises a discrete spectrum of propagating surface and
internal modes and damped vorticity modes. Continuous spectra may also exist
depending on whether the domain is bounded or unbounded. It is shown that
independent of stratification, the vorticity modes in the spectrum play an
important role. For an unstratified pool of a highly viscous liquid, the
vorticity modes are found to make a large contribution to the temporal
evolution of the transient vorticity layer, at the free-surface. It is
demonstrated that the temporal evolution of localised vortical perturbations
sufficiently deep inside the pool, cannot be captured by surface modes
(capillarity-gravity modes) and require contributions from the vorticity modes.
The free-surface signature of a deep, localised, vortical perturbation is
however found to be negligible. On a stratified pool of viscous liquid, such
vortical initial perturbations are shown to have significant projections on
vorticity as well as internal gravity modes. In the infinite depth limit of an
unstratified pool, the contribution from the continuous spectrum vorticity
modes is analysed vis-a-vis that from their finite depth, discrete spectrum
counterparts. Several analytical predictions are compared against direct
numerical simulations (DNS) obtaining excellent agreement. Our results extend
classical ones due to Lamb (1932), Yih (1960), Prosperetti (1976), Prosperetti
and Cortelezzi (1982) and receive support from nonlinear simulations.
