On the stability of the near shore waters of a lake when subject to solar heating

During the day, water in a lake or reservoir absorbs solar radiation according to Beer's law. As the depth decreases towards the shore, more of the radiation penetrates to the bottom, leading to a region of warmer water attached to the lower boundary which is a potentially unstable temperature profile. This situation is modelled by a fluid contained in a triangular domain with a horizontal upper surface. The fluid is subject to internal heating associated with Beer's law and a bottom boundary heat flux associated with the radiation that is not absorbed by the water column. Previous work suggests that the preferred mode for the instability consists of longitudinal rolls with their axes aligned with the slope. As the bottom slope becomes small, the stability problem becomes independent of the base flow and the originally threedimensional (3-D) problem is reduced to a two-dimensional (2-D) problem between horizontal and parallel plates. A quasi-static, linear stability analysis suggests that, for geophysical parameters, the model is locally unstable in a region centred away from the shore