  We study from the numerical point of view, instabilities developed in a fluid layer with a free surface, in a cylindrical container which is non-homogeneously heated from below. In particular we consider the case in which the applied heat is localized around the origin. An axysimmetric basic state appears as soon a non-zero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions depending on vertical and lateral temperature gradients and on the shape of the heating function. We find different kinds of instabilities: extended patterns growing on the whole domain which include those known as targets and spirals waves. Spirals are present even for infinite Prandtl number. Localized structures both at the origin and at the outer part of the cylinder may appear either as Hopf or stationary bifurcations. An overview of the developed instabilities as a function of the dimesionless parameters is reported in this article. 
Spiral Instabilities in Rayleigh-B\'enard Convection under Localized   Heating