Plankton dynamics are controlled by an often subtle interplay between biological and physical processes. Among the latter, fluid transport is known to play a prominent role. Field studies have, e.g., provided evidence of the effects of turbulent-convection upwelling and downwelling motions on phytoplankton survival. Recent numerical investigations have emphasized, in addition, that relatively large-scale coherent flow features on the vertical can considerably hinder survival and thus negatively impact plankton blooms. In nutrient-rich polar marine environments phytoplankton growth is critically limited by light availability, especially in waters that are partially covered by ice. In these conditions, the heterogeneity of the light intensity distribution, in association with a large-scale coherent fluid flow, can give rise to complex biological dynamics. In the Arctic ocean, several studies reported under-ice phytoplankton blooms that were initiated by the onset of ice melt. Nevertheless, it is still only partially known how such blooms are controlled by the interaction between different factors, such as the increase of light transmittance, leads (openings in the ice), convective mixing, and biological processes. Under-ice blooms are expected to become more common in the future, due to increasingly thinner and dynamic ice coverage, and thus more frequent lead formation. This could significantly alter primary production, and have important consequences on local marine food webs. In this work we consider an advection-reaction-diffusion model of phytoplankton light-limited vertical dynamics in the presence of convective transport, intended as an idealized representation of nonuniformly ice-covered polar waters. Specifically, we assume that the incident light intensity at the surface is horizontally modulated by the presence of opaque obstacles, giving rise to regions of the water column that are characterized by different production regimes. We focus on the impact of advection, and more generally of the different transport processes occurring in the fluid, on the average biomass. By means of numerical simulations we show that convective motions may be harmful to under-ice blooms, in agreement with previous findings. In the present setup, such effect depends on the positions of the surface obstacles with respect to the upwelling and downwelling flow regions. We further find, however, that the sinking speed, due to the density difference between phytoplankton organisms and water, also plays an important role, which depends on how it adds to the flow. While small, the sinking speed has a measurable impact on the growth dynamics of the population and can even be critical for its survival, which may have ecological relevance, as different phytoplankton species have different densities and, hence, different settling velocities.