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This paper considers the initial stage of radiatively driven convection, when the perturbations from a quiescent but time-dependent background state are small. Radiation intensity is assumed to decay exponentially away from the surface, and we consider parameter regimes in which the depth of the water is greater than the decay scale of$$e$$of the radiation intensity. Both time-independent and time-periodic radiation are considered. In both cases, the background temperature profile of the water column is time-dependent. A linear analysis of the system is performed based on these time-dependent profiles. We find that the perturbations grow in time according to$$\exp [(\sigma (t) t)]$$, where$$\sigma (t)$$is a time-dependent growth rate. An appropriately defined Reynolds number is the primary dimensionless number characterising the system, determining the wavelength, vertical structure and growth rate of the perturbations. Simulations using a Boussinesq model (the Stratified Ocean Model with Adaptive Refinement) confirm the linear analysis.