Stability of a time-periodic flow incorporating a simple model for noise

A horizontal layer of Boussinesq fluid is known to be susceptible to vortex instability when the lower surface of the fluid is subjected to a time-periodic heating of sufficient strength. In this paper a quasi-steady technique is used to trace the stability of the layer right from a standard linear theory to the problem when the vortices are so strong so as to change the thermal profile at leading order. Furthermore, this system is used to assess the possible influence of the noise that inevitably arises in many physical and laboratory situations. Even though in many problems a fluid flow is assumed to be time-periodic with one dominant driving frequency, in practice many cases are actually contaminated by small components at higher frequencies, which are often multiples of the primary. Here we make a crude model of noise by taking the temperature of the lower surface of the fluid layer to be the sum of two time-dependent periodic functions of differing frequencies. It is shown that the presence of the second frequency can have a dramatic effect on the evolution of a vortex and only a small element of noise appears to be sufficient to generate vortices that penetrate as much as three times further into the layer compared with the situation when the noise is absent.