2.1. Release of dye into experiments

In some experiments presented in this paper, dye is introduced into the tank to visualise the flow. Two methods that were used to introduce dye into the tank are described herein.

For the primary method, which is used to produce horizontal lines of dyed fluid, fluid is slowly injected or extracted from nozzles on the left-hand side of the tank which point towards the interior of the tank (figure 1). The nozzles are simple fittings attached to the end of tubing, and are attached to the side of the tank at heights of 25 and 45 cm above the base of the tank. To produce the horizontal lines, the experiment is paused and the pumps and mechanical agitation are turned off. Fluid is extracted slowly through the nozzles using a syringe, which is then mixed with coloured food dye, and it is then reinjected slowly back in at the same depth. Before the dye is reinjected, its density is checked and if necessary adjusted so that it matches the original density. The fluid spreads slowly and laterally upon reinjection, and is left to do so for an hour, upon which the spreading has mostly stopped. At this point the experiment is resumed and the pumps and mechanical agitation are turned back on.

The other method, used only once in § 3.2, involves the release of a cloud of dye into the turbulent region. Dye is mixed with fluid of salinity $S_B$ and squirted downwards into the turbulent boundary region near the base of the tank using a syringe attached to a long, thin needle. This injection of fluid produces a small turbulent jet that dissipates quickly in the turbulence of the mechanical agitation.

3.1. Steady state

The final stratification generally does not depend on the starting stratification. The system is deemed to have reached a steady state once the salinity profiles are steady and the flux of salinity supplied to the base of the tank matches the flux of salinity removed from the top of the tank. We then check that these salinities do not vary beyond measurement error over an interval of approximately $2$ hours. An example illustrating the convergence to steady state for the data in experiment 1 (BM) is shown in figure 2. Here, we plot the salinity profiles as a function of time. After approximately five hours it can be seen that the salinity profiles converge to a nearly linear vertical profile. The convergence to steady state is then quantified by calculating the root-mean-square (r.m.s.) error between each profile $S_n(z)$ taken at time $t_n$ and the mean of a selection of converged profiles $\bar {S}(z) = \varSigma _{n=N}^M ({S_n(z)}/({M-N})$:

(3.1)\begin{equation} {RMSE}_n = \frac{ \sqrt{\displaystyle \frac{1}{H}\int_0^H[ \bar{S}(z)-S_n(z) ]^2\,{\rm d}{z}}}{\displaystyle \frac{1}{H} \int_0^H\bar{S}(z)\,{\rm d}{z}}.\end{equation}

After six hours, the error decreases to a value of approximately $5\,\%$ and remains smaller than this value for the duration of the experiment. This pattern is generally true for all steady-state experiments.

Figure 2.

Experimental data (experiment 1) taken showing convergence of system to steady state. (a) Vertical salinity profiles as the stratification develops. The profiles are chronologically coloured blue–red–purple, and within each coloured subset also differentiated in order by: crosses and open circles; solid and dashed lines. A steady profile taken as an average of the profiles once the experiment has converged to steady state is plotted as a thicker black line. The salinities of the top and bottom fluid inputs are marked with red and blue arrows, respectively, for reference. (b) Salinity of the extracted fluid (solid) and the injected fluid (dashed) for the top (red) and bottom (blue) sets of nozzles as a function of time. (c) The r.m.s. error (as described in the text) between the profiles and the mean of steady profiles as a function of time.

We also plot (figure 2a) the net supply of salinity to the base of the tank (blue dashed and blue solid lines) and the net removal of salinity from the top of the tank (red dashed and red solid lines) as a function of time. These data also appear to converge to a steady state after six hours. The difference between the top and bottom buoyancy fluxes is smaller than $4\,\%$ of the mean value.

3.2. Boundary mixing flow description

Experiment 2 (BM) converges to steady state after approximately 9.5 hours. At this time, two lines of red dye are injected into the tank interior at heights of 25 and 45 cm. The reinjected cloud of dye spreads slowly and laterally at this depth and then, after approximately one hour, the experiment is resumed, which we label as time $t = 0$ mins in figure 3, or time $t=9.5$ hours in figure 2. The dye study runs for a period of approximately three hours from the time of injection, which is contained fully within the data shown in figure 2.

Figure 3.

Illustration of flow patterns in the BM, steady-state experiment (experiment 2). (a) Snapshots of the front view, see figure 1 of the evolution in time of two lines of dye along with two later injections of green dye. Two lines of red dye are initially injected into the tank interior away from the mixing grid and laterally spread with no vertical mixing, with images detailing their state at times $t =$ (ai) 0 mins, (aii) 40 mins, (aiii) 80 mins after the injection of red dye. After 120 minutes, green dye is released at the bottom of the mixing zone at the right-hand side of the tank and turbulently mixes (av–aviii). (aiv) Schematic summary of the flow. Fluid in the interior of the tank remains quiescent, and does not mix or move. Fluid in the turbulent mixing region mixes in eddy structures, which in turn drives oscillatory movement in the fluid that is near the edge of the turbulent region, which can be seen to oscillate towards and away from the turbulent mixing region. (b) False-colour time series of the red dye intensity of a vertical line in the interior, approximately 40 cm from the left wall of the tank, containing the red dye lines (after subtracting background and cropping the intruding green dye). (c) True-colour time series of a horizontal line (chosen at middepth), starting from when the stronger injection of green dye, into the BM region, reaches midway up the tank ($t = 130$ mins). Note that the movement of the rake can be seen in this time series as an oscillating black structure.

In figure 3 we present a series of photographs showing the evolution of the dye. For two hours the lines of dye in the interior fluid spread laterally towards the mixing region, and tiny amounts of dye are mixed in the turbulent region but this cannot be seen in the photographs. The bulk of the red lines of dye show no net vertical movement and very little mixing (figures 3ai–3aiii); this can be seen in a time series of the dye concentration of a vertical line in figure 3(b). Afterwards. green dye is squirted into the bottom of the turbulent mixing region using a long, thin needle attached to a syringe in two phases: at first, a weak pulse of green dye is released (for image analysis in figure 6) and four minutes later a stronger pulse of green dye is released (for clarity in the true-colour photographs that can be seen in figure 3(av–avii). We observe that the green dye mixes rapidly and is confined to the width of the turbulent mixing region and, within sixteen minutes, also mixes vertically through the depth of the tank. Throughout this process, there is negligible lateral transport. This can be inferred from the stationary lines of red dye, which both do not move vertically or mix. Over a longer time scale, there is an additional, slow exchange of fluid between the boundary and the interior, and this can be seen over the next fifty minutes, where a time series of a horizontal line of pixels (figure 3c), taken at mid depth, shows the width of the green zone growing very slowly over time. The edge of the green dyed fluid has a series of intrusion-like structures that oscillate towards and away from the BM region.

The flow appears to be partitioned into three regions. Near the oscillating mixing rake, on the right-hand side of the tank, there is a turbulent region with width of approximately 25–40 cm, marked by the horizontal extent of the red lines. On the left-hand side of the tank, far from the mechanical mixing, is a largely quiescent region where there is no mixing or net flow, but where there appears to be small oscillatory vertical movement from surface gravity waves. Between the quiescent interior and the turbulent mixing zone is an intermediate region in which there is a series of intrusions which move into and out of the BM zone but there remains a distinct interface between the green and clear fluid. In figure 3(aiv) we summarise this picture schematically. The vertical transport of salinity appears to be dominated by the turbulent diffusive mixing in the turbulent boundary region, with negligible transport in the interior region.

3.3. Whole-tank mixing flow description

We now illustrate the flow patterns in the case of WT mixing. In this case the mixing rake traverses the length of the tank with a speed approximately similar to that in the case of BM.

A single line of dye is introduced in a similar fashion to that described in § 2.1. Five photographs (figure 4ai–av) illustrate the evolution of this line of dye, which mixes vertically and laterally as the rake continuously stirs the fluid.

Figure 4.

Images detailing the evolution of a dye line in steady state in the case of WT mixing (experiment 3). (ai–av) Photographs showing the spreading of a dye line with WT mixing for thirteen minutes after introduction. (b) False-colour time series of the dye concentration of a vertical line in the case of WT mixing. (avi) Schematic summary of the flow. Circular arrows represent the eddies that are left in the wake of the rake's traversal. The dissipation of these eddies leads to mixing of the fluid, which can be visualised by the spreading of the red dye, represented with jagged arrows.

The rake traverses the entirety of the tank approximately every half-minute. The rake traverses at a constant speed through the cloud of dye, which is in turn mixed by the dissipation of eddies that are left in the trailing wake. The eddies appear to be uniform in intensity across the horizontal extent of the tank, and we observe that the dye mixes at approximately the same rate across the tank subject to these eddies. A time series of a vertical line of pixels shows the vertical spread of the dye as a function of time (figure 4b). Dye reaches the bottom of the tank after approximately two minutes and the top of the tank after six minutes. A schematic summarising the flow patterns is shown in figure 4(avi).

4.1. Calculating the bulk diffusivity for BM

In both cases of boundary and WT mixing, mixing by the mechanical turbulence transports salt through the depth of the tank. If we describe this with a tank-averaged effective diffusion process then the effective diffusivity can be calculated based on the steady-state linear salinity stratification and the buoyancy fluxes supplied at the upper and lower boundaries.

The salt fluxes on the top and bottom boundaries are given by $Q_TS_t, \ Q_B ( S_{B} - S_b )$, respectively, where $S_t, \ S_b$ are the salinities of output fluid from the top and bottom boundaries of the tank, respectively. As noted previously, the difference between these salt fluxes is less than 4 % once the experiment reaches steady state. The effective diffusive mixing, $\bar {\kappa }$, drives a salinity flux over the cross-sectional area of the tank, $BL$, with flux $-\bar {\kappa } BL ({{\rm \Delta} S}/{{\rm \Delta} z})$ where ${{\rm \Delta} S}/{{\rm \Delta} z}$ is the salinity gradient.

The effective area-averaged diffusivity $\bar {\kappa }$ may be calculated by averaging the boundary salinity flux at the top and bottom of the domain as given by

(4.1)\begin{equation} \frac{1}{2}[ Q_T ( S_t ) + Q_B ( S_{B} - S_b ) ] ={-}\bar{\kappa} BL \frac{{\rm \Delta} S}{{\rm \Delta} z}.\end{equation}

For each value of $\bar {\kappa }$, the salinity gradient ${{\rm \Delta} S}/{{\rm \Delta} z}$ is calculated. We then fit the linear profile $S(z)$ through the steady-state salinity profiles, minimising the error between the fitted profile and the experimental data as given by the r.m.s. error between the data and the matched salinity profile $\mathcal {S}(z)$:

(4.2)\begin{equation} {RMSE} = \frac{1}{N} \sum_n^N \frac{ \sqrt{\displaystyle \frac{1}{H}\int_0^H[ \mathcal{S}(z)-S_n(z) ]^2 \,{\rm d}{z}}}{\displaystyle \frac{1}{H}\int_0^H\mathcal{S}(z)\,{\rm d}{z}}.\end{equation}

Over the range of $\bar {\kappa }$, a range of possible salinity profiles $S(z)$ can match the data within the experimental error. An example of the resulting fit is shown in figure 5(a) (experiment 1).

Figure 5.

(a) Example of fitting linear solution to the experimental data in experiment 1 (steady state), resulting in $\bar {\kappa } = 0.12 \text { cm}^2\ \text {s}^{-1}$. (b) The r.m.s. error as a function of the calculated value for $\bar {\kappa }$ using (4.1). The salinity of the basal input fluid takes the values $S_B = 0.04 \text { g cm}^{-3}$ (experiment 4), $0.082 \text { g cm}^{-3}$ (experiments 1, 2), $0.12 \text { g cm}^{-3}$ (experiment 5).

For experiments 1, 2, 4 and 5 given in table 1, this r.m.s. error is less than 10 % for values of $\bar {\kappa }_{BM}$ in the range

(4.3)\begin{equation} \bar{\kappa}_{BM} = (0.13 \pm 0.03)\text{ cm}^2\ \text{s}^{{-}1}.\end{equation}

We note that there is a small difference in the salinity at the top and base of the tank relative to the linear gradient model associated with vertical turbulent transport with effective eddy diffusivity $\kappa$. Near the base and top of the tank, there is some additional turbulent mixing owing to the injection and extraction of fluid that drives the net buoyancy flux. However, for simplicity, in developing the model, we assume the diffusivity is constant across the tank.

In the case of WT mixing (experiment 3), a similar process is carried out to estimate the diffusivity. Again, we measure the vertical distribution of salinity and balance the interior diffusive flux with the salinity fluxes supplied at the boundaries. This gives that the bulk diffusivity for the WT mixing set-up is

(4.4)\begin{equation} \bar{\kappa}_{WT} = ( 0.29 \pm 0.06 )\text{ cm}^2\ \text{s}^{{-}1}.\end{equation}

4.2. Dye mixing in the WT mixing experiments

We now investigate the mixing of dye released into the interior of the tank, comparing its mixing with a simple diffusion model using the values estimated for the diffusivity from the buoyancy transport (§ 4.1).

We assume that the light intensity anomaly is linearly correlated with dye concentration within the values used (Cenedese & Dalziel Reference Cenedese and Dalziel1998). We choose a vertical strip within the tank and find the vertical distribution of dye as a function of time. We assume that the dye concentration, $C$, evolves according to the diffusion equation with diffusivity $\bar {\kappa }$:

(4.5)\begin{equation} {\frac{\partial{C}}{\partial{t}}} = \bar{\kappa}\left( {\frac{\partial^2{2}}{\partial{C}^2}}{x} + {\frac{\partial^2{2}}{\partial{C}^2}}{z} \right) . \end{equation}

In the case of a non-uniform horizontal line of dye, this leads to the prediction that the concentration has the form

(4.6)\begin{equation} C(x, z, t) = F(x,t)\exp \left(-\frac{(z-z_0)^2}{4\bar{\kappa}(t-t_0) }\right),\end{equation}

provided the dye has not spread to the top and bottom boundaries of the tank. Here, $F(x,t)$ is an amplitude term that is a function of the initial distribution of dye, which is approximately uniform over the measured section. Additionally, the dye appears to mix rapidly horizontally and as an approximation we focus on the vertical distribution of the dye.

The vertical distribution of the dye within the tank at each time can be fitted with a Gaussian profile to measure the thickness of the line of dye for comparison with the model. According to the model, the variance, $\sigma ^2$, should increase linearly in time,

(4.7)\begin{equation} \sigma^2 = 2 \bar{\kappa} (t - t_0) .\end{equation}

Examples of the fits of the Gaussian profiles to the experimental data are shown in Appendix A in figure 11.

We look at mixing of the red line of dye released into the interior of the tank in the WT mixing (experiment 3). The measured variances (solid lines) are shown in figure 6(a) (blue) as a function of time along with the theoretical variance (dashed lines) (4.7), where $\bar {\kappa } = \bar {\kappa }_{WT}(1 \pm 0.1)$ is the tank-averaged diffusivity calculated from salinity transport (4.4) with error margins shown by the dotted lines. The estimate of the diffusion coefficient associated with the mixing of the dye is consistent with the diffusivity estimated for the salinity transport for the first 500 seconds of the experiment.

Figure 6.

(a) Variance of a Gaussian fit to the spreading of dye lines (solid) in the case of BM (red, experiment 2) and WT mixing (blue, experiment 3), compared with the theoretical spreading (dashed) with corresponding effective diffusivity $\bar {\kappa }$ and $\pm 0.03 \ \text {cm}^2\ \text {s}^{-1}$ error margins (dotted). (b) Variance of a Gaussian fit to the spreading of green dye in the BM region (solid, experiment 2), compared with the theoretical spreading (dashed) with the intensified effective diffusivity, or boundary diffusivity, $\kappa _B$ and $\pm 0.08\ \text {cm}^2\ \text {s}^{-1}$ error margins (dotted).

4.3. Dye mixing in the BM experiments

First we look at the vertical mixing of red dye released into the interior of the tank (experiment 2). The variance of the vertical distribution of dye is shown in figure 6 by a solid line. This appears to be constant in time. This is compared with the theoretical variance (4.7) (dashed line), where $\bar {\kappa } = \bar {\kappa }_{BM}$ is the tank-averaged diffusivity calculated from salinity transport (4.3). By comparison, we see that the dye shows no signs of mixing on this time scale.

Secondly, we look at the vertical mixing of the green dye in the turbulent boundary region in the case of BM (experiment 2). The intensified diffusivity in the boundary region, given by (1.1), $\kappa _B = \bar {\kappa }({L}/{L_B})$, where $L = 120\text { cm}$, depends on the width of the turbulent BM zone which we estimate to be $L_B= (30 \pm 5) \text { cm}$ from figure 3(c). The variance of the vertical distribution of the green dye injected in the BM region (as shown in figure 3) is plotted (solid line) in figure 6(b) as a function of time, along with the theoretical variance (4.7) (dashed and dotted lines), using the value $\kappa _B = (0.56 \pm 0.08)\ \textrm{cm}^2\ \textrm{s}^{-1}$ as measured from the salinity field. The agreement is good suggesting that the local dye and salt mixing processes are equivalent.

6.1. Analytical model

Previous studies (Hopfinger & Toly Reference Hopfinger and Toly1976; Ivey & Corcos Reference Ivey and Corcos1982; Thorpe Reference Thorpe1982; Phillips et al. Reference Phillips, Shyu and Salmun1986) have proposed models for the mixing length scale $L_B$ and turbulent diffusivity $\kappa _B$ in terms of the experimental parameters. The dynamics within the turbulent mixing region can be described by the time-averaged equations for the local mass conservation and by the vorticity equation (see Woods Reference Woods1991).

In the transient experiments, we envisage that the removal of the buoyancy flux supplied to the base of the tank results in an eventual decrease in the buoyancy of fluid parcels in the lower part of the BM region subject to the continued mixing from above. This leads to an upwards, buoyancy-driven flow in the boundary region, and an equal and opposite flow in the interior as seen by the downward migration of the lines of dye in the interior of the fluid. Intrusions form as a result of any divergence of the flow in the boundary region, which then change the spacing between interior isopycnals in a manner consistent with the changing system stratification, even though there is no evidence of mixing of the interior fluid parcels themselves. The flow in each region is exactly such that both regions appear to mix with an effective diffusivity, which is the same as the mean diffusivity, ${\kappa _B L_B}/({L_I + L_B})$.

We continue by assuming that the salinity is horizontally uniform with the exception being in the boundary layers which form at the top and bottom boundaries. We partition the tank into two regions; the BM zone with a uniform turbulent diffusivity and the quiescent interior as shown in figure 8. In the experiments, the horizontal lines of dye are injected and continue to remain horizontal as the stratification evolves, and so we assume that the vertical velocity is a function of depth only. Furthermore, in the steady-state experiments, the vertical buoyancy gradient is uniform and so we make the simplification that the turbulent diffusivity is independent of height. The convergence of flow in the boundary drives an intrusion into the interior, and by conservation of flux requires that the net vertical volume flux in the boundary region and interior is zero:

(6.1)\begin{equation} {\frac{\partial}{\partial{z}}} ( W_BL_B ) ={-}U ={-} {\frac{\partial}{\partial{z}}} ( W_IL_I ),\end{equation}

where $U$ is the lateral flow into the interior from the boundary boundary, $W_I$, $W_B$ are the vertical velocities in the interior and boundary regions and $L_I$, $L_B$ are the widths of these two flow domains and subscript $I$ refers to the interior and subscript $B$ refers to the boundary. Similar to the approach taken by Munk & Wunsch (Reference Munk and Wunsch1998), integrating over a fluid control volume in the boundary region with diffusivity $\kappa _B$ gives that

(6.2)\begin{equation} {\frac{\partial}{\partial{t}}}(L_B S ) + {\frac{\partial}{\partial{z}}}( W_B L_B S ) - {\frac{\partial}{\partial{z}}} \left( L_B \kappa_B {\frac{\partial{S}}{\partial{z}}} \right) ={-}US, \quad L_I \leq x \leq L_I + L_B. \end{equation}

Combining this with (6.1) leads to the result

(6.3)\begin{equation} {\frac{\partial{S}}{\partial{t}}} + W_B(z,t){\frac{\partial{S}}{\partial{z}}} = \kappa_B {\frac{\partial^2{2}}{\partial{S}^2}}{z}, \quad L_I \leq x \leq L_I + L_B,\end{equation}

and similarly in the interior with molecular diffusivity, $\kappa _I \ll \kappa _B$, we have

(6.4)\begin{equation} {\frac{\partial{S}}{\partial{t}}} + W_I(z,t){\frac{\partial{S}}{\partial{z}}} = \kappa_I {\frac{\partial^2{2}}{\partial{S}^2}}{z}, \quad x \leq L_I.\end{equation}

The fluid parcels in the boundary region undergo diffusive mixing whilst in the ambient there is negligible molecular diffusion. Note that there may be transport of fluid between the interior and boundary regions according to (6.1), and that the vertical velocities may vary with height and may be divergent as a result.

Figure 8.

Overview of the idealised one-dimensional model describing the BM experiments where the domain is partitioned into a diffusive region with a stagnant ambient. Equations are derived by assuming that salinity is always horizontally uniform and balancing diffusive and advective fluxes across a fluid slice.

By considering a fluid slice across the whole domain, there is no net advective flux (by (6.1)) and so by adding (6.2) multiplied by the length, $L_B$, and the complement of (6.2) multiplied by $L_I$ for the interior, we find that the divergence of the diffusive salinity flux suggests that the global stratification evolves according to the relation

(6.5)\begin{equation} {\frac{\partial{S}}{\partial{t}}} = \frac{\kappa_B L_B + \kappa_I L_I}{L_I + L_B} {\frac{\partial^2{2}}{\partial{S}^2}}{z} \approx \frac{\kappa_B L_B}{L_I + L_B} {\frac{\partial^2{2}}{\partial{S}^2}}{z}.\end{equation}

This equation suggests that the salinity evolves as if the diffusivities are averaged over the length of the tank. In the situation that the BM dominates, $\kappa _B L_B \gg \kappa _I L_I$, as in the present paper, where $\kappa _I$ corresponds to the molecular mixing value, and so we can neglect the mixing in the interior. The assumption that the isopycnals remain horizontal relies on the time scale of the stratification, ${1}/{N}$ to be smaller than the time scale for turbulent mixing across the depth of basin in the boundary-mixed zone, ${H^2}/{\kappa _B}$, and this applies in the present experiments until the system becomes virtually uniformly mixed through the depth of the experimental tank.

In general, (6.3) may be rearranged so that the boundary velocity $W_B$ may be expressed as

(6.6)\begin{equation} W_B = \frac{\displaystyle \kappa_B {\frac{\partial^2{2}}{\partial{S}^2}}{z} - {\frac{\partial{S}}{\partial{t}}} }{\displaystyle {\frac{\partial{S}}{\partial{z}}} } . \end{equation}

In the case that the system is in steady state, ${\partial S}/{\partial t} = 0$ and, by (6.5), we also have that ${\partial ^2 S}/{\partial z^2} = 0$ and thus we have no vertical velocity in the boundary and also none in the interior, as was previously shown in the steady-state experiments. This model suggests that it is only possible to have vertical velocities if the buoyancy stratification is transitioning over time.

6.2. Boundary conditions

To describe the transient experiments (experiments 6 and 7), we impose the boundary condition that there is no buoyancy flux at the bottom boundary:

(6.7)\begin{equation} {\frac{\partial{S}}{\partial{z}}}(0, t) = 0.\end{equation}

At the top boundary the interior diffusive flux balances the flux for a matched inflow at a volume flux $Q$ of salinity $S_{T} = 0$ and outflow of salinity $S(H, t)$:

(6.8)\begin{equation} -\kappa_B B L_B{\frac{\partial{S}}{\partial{z}}}(H,t) = QS(H, t), \end{equation}

where $B$ is the width of the tank.

We look for a separable solution to find

(6.9)\begin{equation} S(z,t) = \sum_n A_n\exp(-\gamma_n t)\cos(\lambda_n z), \quad \lambda_n > 0,\end{equation}

where

(6.10)\begin{equation} \lambda_n(\kappa_B L_B)^2 = \frac{\gamma_n ( L_I + L_B )}{\kappa_B L_B} ,\end{equation}

and, in this case, $\lambda _n(\kappa _B L_B) H$ satisfies

(6.11)\begin{equation} \tan \lambda_n H = \frac{Q H}{\kappa_B B L_B} \frac{1}{\lambda_n H}.\end{equation}

With an initial stratification $S(z,0) = S_0(z)$, the coefficients $A_n$ may be calculated using Sturm–Liouville theory to get that

(6.12)\begin{equation} A_n = \frac{\displaystyle \int_0^HS_0(z)\cos(\lambda_n z)\,{\rm d}{z}}{\displaystyle \int_0^H\cos^2(\lambda_n z) \,{\rm d}{z}}. \end{equation}

At long times the leading-order solution may be approximated by only the slowest decaying mode $n=0$. Using (6.4) we can then express the interior velocity as

(6.13)\begin{equation} W_I = \frac{\gamma_0}{\lambda_0} \cot(\lambda_0 z),\end{equation}

excluding the bottom boundary layer. From (6.13) it follows that the streamfunction $\psi$ satisfying $\boldsymbol {u} = \boldsymbol {\nabla } \times \psi \hat {\boldsymbol {y}}$ is

(6.14)\begin{equation} \psi = \frac{\gamma_0}{\lambda_0} \cot(\lambda_0 z)x,\end{equation}

and the height of a fluid parcel $z(t)$ is given by

(6.15)\begin{equation} t - t_0 = \frac{1}{\gamma_0} \log \left(\frac{\cos(\lambda_0 z)}{\cos(\lambda_0 z_0)}\right),\end{equation}

where the height $z=z_0$ at time $t = t_0$.

6.3. Comparison of the leading-order mode and the observations

The next mode has $\lambda _1 \approx \lambda _0 + {{\rm \pi} }/{H}$ and so $\gamma _1 - \gamma _0 = ({\kappa _B L_B}/({L_B + L_I})) ({{\rm \pi} ^2}/{H^2} + 2({{\rm \pi} }/{H}) \lambda _0 ) \approx 5 \times 10^{-4}\ \text {s}^{-1}$. Our samples are generally taken at times $t_m \gg 10^{3}\ \text {s}$ and so the leading-order mode becomes a good approximation.

We can check the accuracy of the model presented in § 6 by calculating the r.m.s. difference between the leading-order model solution $S^0(z,0) = A_0\exp (-\gamma _0 t)\cos (\lambda _0 t)$ and the $M-1$ experimental samples, $S_m(z)$ taken at time $t_m$, by

(6.16)\begin{equation} {Error} = \frac{1}{M-1}\sum_{m=2}^M \frac{\sqrt{\displaystyle \frac{1}{H}\int_0^H[ S^0(z,0)-S_m(z)\exp(\gamma_0 t_m) ]^2\, {\rm d}{z}}}{\displaystyle \frac{1}{H}\int_0^HS^0(z, 0)\,{\rm d}{z}}. \end{equation}

The first ($m=1$) data sample is used to find the initial condition for $S^0(z,t)$. We find that, at subsequent times, the difference between the model solution and the experimental data is less than 9 % for $\kappa _B$ within the range $\kappa _B L_B = 11.2\pm 0.8 \ \text {cm}^3\ \text {s}^{-1}$. This is consistent, although more constrained, than the value calculated in § 4. In figure 9, we scale the salinity profiles onto time $t=0$, $S_m(z)\exp (\gamma _0 t_m)$, and compare them with the leading-order model solution, $S^0$.

Figure 9.

Salinity profiles from (a) experiment 7 and (b) experiment 6, scaled onto time $t = 0$, $S_m(z)\exp (\gamma _0 t_m)$, using $\kappa _B L_B = 11.2 \text { cm}^3\ \text {s}^{-1}$. Overlain is the leading-order model solution from § 6, where the leading coefficient $A_0$ is calculated from the first ($m = 1$) profile at time $t = 0$.

The best fit value for $\kappa _B L_B$ is consistent with the value found in § 4. The salinity profiles from this model are consistent with the experimental profiles and are displayed in Appendix B in figure 12. This is further discussed in Appendix B.

6.4. Velocity profiles

We now compare the horizontal and vertical evolution of dye streaks in the experiment with the predictions of the model. Using the interior streamfunction given by (6.14) we predict the vertical and horizontal locations of interior dye parcels in experiment 7 with $\kappa _B L_B = 11.2 \text { cm}^3\ \text {s}^{-1}$.

As shown in figure 7, green dye downwells in the interior, horizontally advects towards the mixing zone and intrudes back into the interior after vertically mixing. In figure 7(aiv), a series of points are marked which track the dye front at $t = 80 \text { mins}$. These can be integrated in time using the streamfunction (6.14) and the subsequent points at times $t = 140 \text { mins}$ and $t = 200 \text { mins}$ are given in figures 7(av) and 7(avi).

We can plot a time series of a vertical line in the tank to track the vertical positions of the green and red dye lines in experiment 7. In figure 10(a) this is plotted and overlain with the vertical positions of the top and bottom edges of the dye line given by (6.15). Furthermore, in figure 10(b), we plot the horizontal position of the tip of the red dye line and this is compared with the model solution. There is reasonably good correspondence of the model prediction with the experimental observations providing support for the theoretical picture proposed herein.

Figure 10.

(a) Time series of a vertical line in experiment 7 showing the time evolution of the vertical positions of the red and green lines. Overlain is the vertical tracking of the top and bottom edges of the dye lines using (6.15). (b) Time evolution of the horizontal distance of the tip of the red dye line plotted against model prediction (which follows the streamline described by $\psi = \exp (-2.8) \text { cm}^2\ \text {s}^{-1}$).