Two-layer model for shallow horizontal convective circulation

Abstract

This paper discusses experiments and a theoretical model for the convective circulation driven by a surface buoyancy flux in a horizontal layer of fluid. The layer is closed at one end and, at the other end, the buoyancy has a fixed value over a given depth. Such circulation occurs in side arms of cooling lakes used for waste-heat disposal from power generation. Some geophysical circulations, such as in the Red Sea, are also of the above type.The experiments were done in a 35 ft long flume using heat transfer between the heated water and the atmosphere to generate the surface buoyancy flux. The observed circulation was characterized by two distinct layers flowing in opposite directions and separated by a density interface. For upper-layer depths less than about half the total depth at the open end, the downflow was observed to be concentrated near the closed end. Circulation flowrates and vertical temperature profiles were measured.The theoretical model uses a ‘two-layer’ approach. The mass, momentum, and buoyancy conservation equations are integrated vertically on each side of the interface. Mass and buoyancy transfer across the interface are neglected. The interfacial shear stress is proportional to the square of the difference of the average layer velocities. For small layer densimetric Froude numbers, the free surface is shown to be approximately horizontal and the problem reduces to one ordinary differential equation for the thickness of the upper layer. General solutions of this interface equation are presented for horizontal and sloping bottoms. Different configurations are possible depending on the nature of the singular points which occur in the phase plan.For the convective circulation, the interface is shown to go through a singular point. This condition leads to a simple analytical solution for the circulation flowrate in the case of constant surface buoyancy flux and horizontal bottom. This solution compares well with the experimental data and with measurements on the Red Sea circulation.

Circulation and Exchange in Choked Marginal Seas

“…To predict the exchange state that occurs for a particular forcing and to forecast the forcing required to drive the system to maximal exchange, it is necessary to use a further constraint along with some assumptions about the nature of circulation in the marginal sea. Although this has been attempted in nonrotating systems (e.g., Brocard and Harleman 1980;Finnigan andIvey 1999, 2000;Whitehead et al 2003;Grimm and Maxworthy 1999), the only previous attempts to do so in a rotating system are apparently those of Siddall et al (2002Siddall et al ( , 2004, and references therein) and Price and Yang (1998). The latter consider a marginal sea with a deep, quiescent lower layer that spills out and over a shallow sill.…”

A model for the estimation of convective exchange in the littoral region of a shallow lake during cooling

“…Other examples include convective circulation in side-arms of cooling lakes (Sturm & Kennedy, 1980;Brocard & Harleman, 1980) the inverse estuary problem (Harashima & Watanabe, 1987) and convection induced by differential deepening (Imberger, 1985). The hydrodynamics become even more intricate when these flows are analyzed as unsteady, developing flows (Patterson & Imberger, 1980).…”

“…This schematization was retained in the formulation of equation 3a and 3b, even though the formation of the undercurrent implies that a two layer system develops. A partial justification for this erroneous practice is that the two-layer system is a weak one because, as shown by Brocard & Harleman (1980) as well as a visualization experiment by Horsch & Stefan (1987) significant downflow from the upper to the lower layers occurs.…”

Impact of exchange flows on wetland flushing

“…To date, freshwater exchange flows have been mostly studied in the lab or through numerical modeling. The earlier studies focus on the steady baroclinic circulation in sidearms of cooling lakes and its role in improving heat dissipation [e.g., Brocard and Harleman, 1980]. More recent work considers the exchange flows in natural sidearms and characterizes their temporal variability and mixing properties [Farrow and Patterson, 1993;Sturman et al, 1996;Sturman and Ivey, 1998].…”