The impact of source fluctuations in a filling box with interior diffusive mixing

Abstract

We explore the interior stratification that develops in a reservoir of fluid in which there is a localised source of destabilising (positive) buoyancy and a distributed source of stabilising (negative) buoyancy at the surface of the fluid, combined with an interior turbulent diffusivity. The source of destabilising buoyancy forms a descending turbulent plume that entrains ambient fluid, and this, coupled with the interior turbulent diffusivity, leads to mixing of the interior fluid. When the ratio $Pe$ of the plume source volume flux to the diffusive flux across the whole reservoir is large, the reservoir is well mixed except for a narrow surface boundary layer, while for smaller $Pe$ , the reservoir is more stratified, consistent with the results of Hughes et al. (J. Fluid Mech., vol. 581, 2007, pp. 251–276) and Hughes & Griffiths (Annu. Rev. Fluid Mech., vol. 40, issue 1, 2008, pp. 185–208). We explore the evolution of the system when the source buoyancy flux varies in time, exploring both sudden changes and oscillatory changes in the forcing. Provided that the time scale of the change in the forcing is comparable to or smaller than the response time of the reservoir, as the source buoyancy flux decreases the plume may become trapped in the upper part of the reservoir, leading to a much shallower mixed zone. However, in the case of a source whose strength oscillates in time, as the source buoyancy flux increases, again the deeper water becomes involved in the mixing and circulation. We discuss briefly how these results may provide insight into the impact of changes in the forcing on mixing in deep ocean waters.