The formation and evolution of a diffusive interface in a
stable salt-stratified layer cooled from above is studied in a
two-dimensional geometry by direct numerical simulation. For a typical
example with realistic parameters, the evolution of the flow is computed
up to the moment where three layers can be distinguished. Focus is on
the development of the first mixed layer. The convective velocity
scaling as proposed by Hunt (1984) and previously proposed expressions
for the interfacial heat flux (Huppert 1971; Fernando 1989a)
are shown to correspond well with the results of the simulation. The
evolution of the first layer can be well described by an entrainment
relation based on a local balance between kinetic and potential energy
with mixing efficiency γ. The new entrainment relation is shown
to fit the numerical results well and an interpretation of γ in
terms of the overall energy balances of the flow is given.Previously, two rival mechanisms have been proposed that determine
the final thickness of the first layer (Turner 1968; Fernando 1987).
One of the distinguishing features of both mechanisms is whether a
transition in entrainment regime – as the first layer develops
– is a necessary condition for the mixed layer to stop growing.
Another is the presence of a buoyancy jump over the interface before
substantial convection in the second layer occurs. From the numerical
results, we find a significant buoyancy jump even before the thermal
boundary layer ahead of the first layer becomes unstable. Moreover, the
convective activity in the second layer is too small to be able to stop
the growth of the first layer. We therefore favour the view proposed by
Fernando (1987) that a transition in entrainment regime determines the
thickness of the first layer. Following this, a new one-dimensional
model of layer formation is proposed. Important expressions within this
model are verified using the results of the numerical simulation. The
model contains two constants which are determined from the numerical
results. The results of the new model fit experimental results quite
well and the parameter dependence of the thickness of the first layer
is not sensitive to the values of the two constants.
The Taylor-expansion method of moments for the particle system with bimodal distribution