  The transient dynamics of a wake vortex, modelled by a strong swirling
$q$-vortex, are examined with an emphasis on exploring optimal transient growth
constructed by continuous eigenmodes associated with continuous spectra. The
pivotal contribution of the viscous critical-layer eigenmodes (Lee & Marcus, J.
Fluid Mech., vol. 967) amongst the entire eigenmode families to optimal
perturbations is numerically confirmed, based on a spectral collocation method
for a radially unbounded domain that ensures correct analyticity and far-field
behaviour. The consistence of the numerical method against numerical
sensitivity provides reliability of results as well as flexibility for tuning.
Both axisymmetric and helical perturbation cases with axial wavenumbers of
order unity or less are considered in the study through both linearised theory
and non-linear simulations, yielding results that align with literature on both
energy growth curves and optimal perturbation structures. Additionally, the
initiation process of transient growth is discussed to reveal its
practicability. Inspired by ice crystals in contrails, the role of backward
influences of inertial particles on the carrier vortex flow, especially via
particle drag, is underscored. In the pursuit of optimal transient growth, the
particles are initially distributed at the periphery of the vortex core to
disturb the vortex. Two-way coupled vortex-particle simulations conclusively
demonstrate clear indications of optimal transient growth during continual
vortex-particle interactions, reinforcing the robustness and significance of
the transient growth process in the original non-linear vortex system over
finite time periods.
