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The onset of convection in a rotating plane layer due to a vertical temperature gradient is studied in this paper. The background stratification is modulated by lateral temperature variations and stable stratification aimed at understating the Earth's outer core convection subject to thermal core--mantle interaction. At the top boundary, sinusoidal and Gaussian temperature variations are imposed apart from the reference case of isothermal condition used in the classical Rayleigh--Benard convection. The additional modulating conditions break the top-bottom flow symmetry leading to flow localisation and asymmetry that exhibit modified temporal dynamics unlike that of the classical Rayleigh Benard cells. The threshold for convection is lowered with flows occurring in surplus heat flux regions caused by the imposed conditions. Despite flow suppression in the stable layer, rapid rotation favours the penetration of convection rolls with smaller wavelengths. The lateral variations in temperature imposed at the top boundary enhance such axial penetration with a laterally varying penetrative extent resulting in a modified clustered flow structure unlike the reference case. With both modulating conditions imposed, the onset of overstable modes is favoured for low Prandtl numbers, a regime which is relevant to the Earth's core conditions. With rapid rotation, a novel mode of travelling wave instability occurs at onset of convection, the propagation direction of which is controlled by the lateral temperature gradients at the top boundary. The onset of oscillatory modes is suppressed by the imposition of the modulating conditions indicated by the significant lowering of the transition Prandtl number.

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