5.2 Theory
5.3.3 Time evolution of patch thickness
A summary of experimental results of the time evolution of the convecting patch thickness h is shown in figure 5.11 for a range of Rρ and initial patch thickness. The growth of h is plotted on a logarithmic scale against the time, t, from the onset of convection. In this figure all the straight lines have slope = 1/2. The results show very good agreement with the predicted h t
1 2
power law (5.18) at small times. At large times the growth drops below the t1 2 power law. This is expected to occur as we assumed a constant density gradient in deriving (5.18). At large times the density gradient and Rρoutside the patch will have increased because the S and T gradients have weakened. The prediction of Stern & Turner (1969) was that a convecting patch would grow as t32
; this is clearly not consistent with the data. For thin ‘two sided’ interfaces the ∆S43
law is a reasonable assumption, but the present results show that it does not describe these ‘one sided’ interfaces, where there are deep linear gradients next to the sharp interfaces at the edges of the convecting patch. This difference should not be surprising as the
5.3 Laboratory experiments 101
∆S4 3flux law was originally derived for sharp interfaces bounded above and below by well- mixed regions where salt fingers can penetrate all the way across the interface and be driven by the∆S between top and bottom boundaries. In deep linear gradients this will not be the case, as discussed in 1.2.3.2
Two additional experiments were performed to test other assumptions of (5.18). In the first experiment, ‘crossed’ linear T and S gradients were made with Rρ 1 5. The bottom half of the tank was then mixed, as sketched in figure 5.10 b). The depth of the mixed re- gion easily exceeded the criterion of (5.11) and started to convect. However, as there was no buoyant convection from the bottom boundary, there was no entrainment across the interface and the mixed-layer depth remained constant to within 1 cm over a duration of 12 hours. The buoyancy flux of the salt fingers would only act to increase the density in this mixed layer and hence increase the stability, rather than drive strong convection that could entrain fluid from the interface. The second experiment used the same density of the T and S solutions as in other experiments but now three well-mixed layers were produced, as sketched in 5.10 (c). In this case convection occurred in all three layers. The interfaces were not observed to move over a 12 hour period. This illustrates the difference between our one sided interfaces above and below a mixed region, and the thin two sided interfaces that occur between the well-mixed regions. These two experiments show that for growth of a mixed layer to occur, there must be smooth gradients above and below the convecting region.
The data from the growth of the layer thickness shown in figure 5.11 can be used to estimate the buoyancy flux of B as a function of Rρ by rearranging (5.17) to give
B N
2h
4η dh
dt (5.20)
where 0 η 0 2. Only values of h and dh
dt at small times are used to calculate the average value of B. Asηis not a well known quantity we made an independent measurement of B by conducting a second set of experiments that measured the rate of change of a linear density gradient due to the ‘up gradient’ salt finger fluxes. In these experiments we used the same Rρ
and Szas in earlier experiments but there was no mixed region. The top and bottom boundaries of the tank result in flux divergences that change the local density gradients. Syringe samples of fluid were taken at equally spaced depths through the tank and their densities measured on
an Anton Paar densitometer (model DMA 602). Thus we were able to measure the rate of the change of density gradient with time. The buoyancy flux per unit area due to irreversible processes (such as mixing or salt fingers) is defined by Winters et al. (1995) as
B g ρoD ∂ ∂t ρz dz (5.21) where D is the depth of the tank, g is the acceleration due to gravity andρo the mean density. The buoyancy flux defined by (5.21) is an averaged value within the tank. In figure 5.12 we plot experimental values of a Nusselt number, defined as
NuB
Bρo gκTαTz
(5.22)
whereκT is the molecular diffusivity of the T component (salt). Typically we would calculate B from the average for 5 - 7 density profiles. The large error bars represent the scatter in the de- termination of the rate of change of density profiles. The error was mainly due to the difficulty in accurately measuring the small changes in density caused by the low fluxes. The buoyancy flux could be measured more accurately if many repeated, continuous density profiles could be taken rather than with interpolating between 10 density samples to determine the profile. At present it is very difficult to record continuous density profiles using the sugar/salt system.
The Nusselt numbers inferred from (5.21) are in close agreement with those inferred from (5.20), if η 0 2. This close agreement shows that the buoyancy flux into the well-mixed region is the same as the buoyancy flux in the linear gradients, rather than being controlled by the density step. An empirical formula for NuB that fits the data shown in (5.12) is
NuB 11 3 Rρ 1
1 8 (5.23)
for the range 1 13 Rρ 1 7.