3.1 No turbulent mixing

When the ice is growing in quiescent water, heat is transferred by conduction. We ignore free convection in the liquid. In this case $Q_{\ell }$ is the sum of the conductive heat transfer through and the rate of change of enthalpy of the liquid, so that

where the subscript $\ell$ refers to the liquid properties, ${\it\delta}$ is the thickness of the thermal boundary layer above the ice and ${\rm\Delta}T_{\ell }=T_{\infty }-T_{f}$ , $T_{\infty }$ being the temperature of the liquid far field. Introducing

from the heat balance in a control volume in the liquid region above the ice–water interface, with ${\it\alpha}_{\ell }$ given by

the conductive term in (3.5) can be rewritten as

Substitution of (3.2)–(3.5), and (3.8) into (3.1) gives, for the heat balance in the control volume,

which can be approximated, based on the largest terms, as

This equation is non-dimensionalized by taking the length scale to be the depth $D$ of the liquid, the temperature difference scale to be the temperature difference across the solid at the onset of turbulent mixing and the time scale to be $D^{2}/{\it\alpha}_{\ell }$ , which corresponds to the characteristic time for thermal diffusion over the distance $D$ . This yields

where variables with a superscript $\ast$ are in non-dimensional form. $\mathscr{C}={\it\rho}_{\ell }c_{\ell }/{\it\rho}_{s}c_{s}$ is the ratio of the volumetric heat capacity of the liquid to that of the solid and $A={\it\alpha}_{\ell }/{\it\alpha}_{s}$ is the ratio of the thermal diffusivity of the solid to that of the liquid. The typical values of $\mathscr{C}$ and $A$ for the laboratory experiment and for the geophysical application are listed in table 1.

3.2 Turbulent mixing

For turbulent flow over a flat plate at constant temperature, Reynolds analogy relates the convective heat flux $q_{T}$ to the properties of the momentum boundary layer. In Reynolds analogy, the heat flux and momentum flux at the plate in a turbulent boundary layer are considered equivalent since they are both influenced by the turbulent motion above the plate. The expression for $q_{T}$ (see White Reference White1974, p. 564) is

where $C_{h}$ is a heat transfer coefficient (Stanton number) given empirically by

$U_{\infty }$ is the velocity of the liquid in the far field, $Pr$ is the Prandtl number and $c_{f}$ is the coefficient of friction defined as

where $u_{\ast }$ is the friction velocity. We introduce the coefficient $G$ in the expression for $C_{h}$ to substitute for the constant term $12.8(Pr^{0.68}-1)$ . In the context of our experiment, this term is the term we are trying to constrain. For the turbulent mixing phase in our experiments, $Q_{\ell }$ is augmented by $q_{T}$ , and hence the energy balance for the control volume becomes

By using the same length, temperature difference and time scales as in (3.11) and by additionally using $U_{\infty }$ as the velocity scale, this expression is non-dimensionalized to obtain

where $RePr=Pe$ . Table 1 lists typical values of $Pe$ for the sub-ice shelf cavity and the laboratory set-up.

4.1 Ice thickness

Measured ice thickness $h_{e}$ versus time is shown in figures 5 and 6, from experiments 1–10 and experiment 0 respectively. In figure 5 the time $t=0$ corresponds to the onset of the turbulent shear flow. In figure 6, the shaded region along the line plot has a total width of 0.5 mm and represents the error in the measurements. The error was estimated by taking the standard deviation of 10 repeated measurements of the ice thickness at $r=R$ during Phase 1 of a typical experiment. The ice thickness measurements in experiment 0 and in Phase 1 of experiments 1–10 are assigned the same error estimate.

The error in Phase 2 measured by the same method, using Phase 2 measurements from a typical experiment, is 0.9 mm. The error in ice thickness measurements in Phase 2 is larger than in Phase 1 because the ice–water interface becomes wavy when ice melts in the presence of the turbulent flow. The wavy pattern consists of spiral crests and troughs with wavelength 12–16 mm and amplitude 1–3 mm. It is difficult to visually identify the ice thickness along the diameter from the side-view pictures of the tank due to the waviness of the ice surface.

Ice grows at an almost constant rate when the water is undisturbed, as in experiment 0 and in Phase 1 of experiments 1–10. In Phase 2, mixing by the turbulent shear flow transports warm water from the far field to the ice–water interface, which promotes heat transfer to the ice. The ice then responds in one of three ways, each of which we have observed as a transient at our measurement location $r=R$ : (1) attenuated ice growth, (2) partial melting and (3) complete melting. Attenuated growth refers to ice growing at a rate slower than during Phase 1. Partial melting refers to only a fraction of the ice thickness from Phase 1 changing phase into liquid, such that a residual thinner ice layer still remains in the tank. Complete melting refers to the whole ice layer from Phase 1 changing phase into liquid. Following this transient response, regrowth of ice at the same rate as in Phase 1 is observed.

Figure 7.

Changes in the liquid layer following the onset of turbulent mixing. At the lower end of experimental $Re_{R}$ , the steps shown in this diagram occur over several minutes, whereas at the upper end of experimental $Re_{R}$ , they occur in a few seconds. (a) A stratified layer initially separates the turbulent layer from the ice surface. The interface between the stratified layer and turbulent layer is dome shaped. (b) The turbulent layer progresses downwards, eroding the stratified layer. (c) The turbulent layer has reached the ice–water interface and causes the ice to melt. The thickness of ice melted increases with radius. A spiral wavy profile develops on the ice surface during melting.

Figure 7 shows the sequence of structures that are observed in the ice–water system following the onset of turbulent mixing. A thermally stratified water layer initially separates the growing ice from the turbulent flow, as the turbulence develops beneath the rotating lid. The interface between the stratified layer and the turbulent flow is dome shaped because the turbulent shear stress ${\it\tau}$ increases proportionally to $r^{2}$ and is therefore weaker near the centre of the tank. The dome-shaped interface was imaged in the experiments by inserting dye at the top of the water column into the turbulent layer and monitoring the evolution of the turbulent layer. In our lowest $Re_{R}$ experiment, the stratified layer persists in the presence of the shear flow, thereby preventing turbulence from reaching the ice–water interface. Ice growth is attenuated in this case, but not stopped. At the other extreme, in our highest $Re_{R}$ experiments, the turbulent mixing is strong enough to erode the stratified layer entirely almost immediately after the onset of the turbulent shear flow. When the turbulence comes in direct contact with the ice–water interface, it produces complete melting at high $Re_{R}$ and partial melting at intermediate $Re_{R}$ . In the partial melting cases, the thickness of ice melted increases with radial distance from the tank centre. In table 3, the transient behaviour the ice adopts at $r=R$ in response to turbulent mixing in experiments 1–10 is given along with the corresponding $Re_{R}$ .

The development of the spiral wavy pattern on the ice–water interface when ice melts in our experiments has been explained by Gilpin et al. (Reference Gilpin, Hirata and Cheng1980), as follows. Although the ice thickness is approximately uniform at the end of Phase 1, there are nevertheless small-amplitude deviations from uniform thickness due to random perturbations and minor design flaws in the cooling apparatus. Gilpin et al. (Reference Gilpin, Hirata and Cheng1980) found that such an interface will be unstable to growth in the presence of turbulence when the heat flux from the liquid to the solid is large, which is the case in our experiments during transient melting. The mechanism for the instability involves flow separation downstream of an irregularity in the ice, which causes the heat transfer at a crest to be smaller than the heat transfer at a valley. The amplitude of the irregularity thus grows, which further amplifies the irregularity in the shear flow, producing a growing set of undulations on the ice–water interface as it melts. The wavelength of the undulations increases with $u_{\ast }$ , which is proportional to $r$ in our experiments. The dependence of the undulation wavelength on distance from the tank centre gives rise to the spiral profile of the ice–water interface undulations that we observe.

Figure 8.

Temperature recorded by thermistors A, B, C, D, E, F and G over the course of a typical experiment. The horizontal axis denotes time relative to the onset of turbulent mixing. The vertical dashed line indicates the time at which ice forms a thin layer on the bottom plate.

4.2 Temperature

Thermistors A–G are used to measure the temperature at the heights indicated in figure 3. The resistance $\mathscr{R}$ of a thermistor is related to its temperature $T$ according to the Steinhart–Hart equation,

where $a_{1}$ , $a_{2}$ and $a_{3}$ are the Steinhart–Hart coefficients and are unique to each thermistor. We obtained these coefficients by calibration prior to our series of experiments.

Temperature is recorded starting from the instant nitrogen starts to circulate inside the basal cooling plate. The evolution of temperature at the thermistor locations A–G in a typical experiment (in this case, experiment 6) is shown in figure 8. Occasional outliers in the thermistor readings have been deleted and the data points have been replaced by interpolating neighbouring values. These outliers are due to the thermistor connections occasionally freezing, causing sudden resistance increases which are seen as sudden temperature drops. Following the onset of turbulent mixing, there is an increase in temperature at A because warm water transported to the bottom of the tank causes melting of ice. At that time, thermistors B, C, D, E, F and G record the same temperature, signalling that mixing results in a homogeneous distribution of temperature in the turbulent shear flow. Note that after about 40 minutes of cooling, the temperature at B departs from the temperatures at C, D, E, F and G as thermistor B is engulfed by the growing ice.

Vertical profiles of temperature in the same experiment at the indicated times during Phase 1 and Phase 2 are shown in figure 9(a,b) respectively. Because the ice–water interface is at $T_{f}=0\,^{\circ }\text{C}$ , the measured ice thickness at any time can be checked by interpolating $T_{f}$ in the temperature time series recorded by thermistors A–G. The temperature within the ice can safely be assumed to increase linearly from the temperature of the plate to $T_{f}$ , because heat transfer in the ice is by conduction and its growth rate is slow enough that the ice is in thermal equilibrium with its boundaries. The linear distributions of ice temperature are shown by the straight lines in the left half of figure 9(a,b). Temperature profiles above the ice–water interface in Phase 1 (figure 9 a) are exponential fits to the temperature measurements. For clarity, the temperature data points corresponding to only one profile have been included in each figure. The fitted liquid layer temperature profiles in Phase 1 are characteristic of heat transfer in the liquid by conduction only.

Figure 9.

Vertical profiles of temperature at different times relative to the onset of turbulent mixing during (a) Phase 1 and (b) Phase 2 of a typical experiment. The temperature data points are shown for the $t=-30$  min profile in (a) and for the $t=15$  min profile in (b). In (b), a dashed line is drawn between the location of the ice–water interface and the location of the first thermistor above the ice–water interface to indicate a possible temperature profile in that layer. Liquid temperature profiles in Phase 1 are in blue and liquid temperature profiles in Phase 2 are in red.

We saw no evidence of natural convection in the liquid layer during Phase 1. For a liquid water layer over an ice–water interface at 0 °C, natural convection onsets at Rayleigh numbers above 1700, which has been confirmed experimentally by Boger & Westwater (Reference Boger and Westwater1967). The Rayleigh number for this system can be expressed as

where ${\it\beta}$ is the thermal expansion coefficient of water, $d$ is the convecting layer depth and ${\rm\Delta}T$ is the temperature difference across $d$ . Boger & Westwater (Reference Boger and Westwater1967) take $d$ to be the thickness of the liquid layer between the ice–water interface and the height at which the water is at 4 °C, where it has maximum density. Interpolating the value of $d$ from the temperature measurements in Phase 1 of our experiments and using values from table 2 for the physical properties of water, we found that the Rayleigh number in Phase 1 varies from 250 to approximately 850. It therefore remains below the critical Rayleigh number at which natural convection would occur.

As the liquid cools by conduction in Phase 1 of an experiment, the temperature $T$ in its thermal boundary layer can be modelled according to the relation

The thickness ${\it\delta}$ of the thermal boundary layer can be obtained by taking the reciprocal of the fit coefficient of an exponential fit to this relationship. Figure 10 shows the evolution of ${\it\delta}$ calculated in this way in Phase 1 of the same experiment. The initial value of ${\it\delta}$ is non-zero because, prior to the formation of ice, a thermal boundary layer was already present in the liquid due to cooling from the bottom by the basal cooling plate. The thickness of the thermal boundary layer increases from its initial value as Phase 1 of the experiment proceeds. This indicates that, for a control volume in the liquid above the ice–water interface, the heat loss by conduction to the ice is larger than the enthalpy decrease of the control volume due to the movement of the ice–water interface into it. At the end of Phase 1, ${\it\delta}$ asymptotes to a uniform value. At this stage, the energy balance in the control volume above the ice–water interface is at steady state, that is, conductive heat loss to the ice is balanced by enthalpy decrease due to the upward movement of the ice–water interface. A simple one-dimensional model for the energy balance in the control volume gives ${\it\delta}={\it\alpha}_{\ell }/(\text{d}h_{e}/\text{d}t)$ . The predicted value of ${\it\delta}$ from this model for our case is 20 mm, which is approximately twice as large as the steady-state value of ${\it\delta}$ from figure 10. The rate of growth of ice, $\text{d}h_{e}/\text{d}t$ , is very small in our experiment (about $5.5\times 10^{-3}~\text{mm}\,\text{s}^{-1}$ ). There is very little liquid convective motion in the control volume above the ice–water interface in response to the very slowly moving interface, and hence the thermal boundary layer is thinner than the theoretical prediction.

During Phase 2 of these experiments, all of the thermistors in the liquid typically record nearly the same temperature at a given time after the initial thermal stratification in the liquid has been destroyed by turbulent mixing. A vertical line through the mean of the temperatures measured by the thermistors in the liquid is drawn in figure 9(b) to represent a homogeneous vertical temperature profile in the liquid layer.

Figure 10.

Boundary layer thickness ${\it\delta}$ calculated from exponential fit through non-dimensionalized vertical temperature time series.

4.3 Velocity

Application of the heat balance shown in (3.15) to the control volume around the ice–water interface in figure 4 requires knowledge of the fluid velocity in the far field. Since the temperature distribution in the liquid is nearly homogeneous when there is turbulent mixing, buoyancy forces in the liquid are weak during this phase of the experiments. The circulation in the far field is thus due to the shear induced by the rotating lid only.

The velocity of the shear-driven turbulent flow above the flat bottom surface of the tank is measured for the purpose of relating the fluid velocity in the far field to the lid velocity. We denote by $\overline{U_{{\it\theta}}}$ the mean of the azimuthal velocity component and by $\overline{U_{r}}$ the mean of the radial velocity component of the flow. The vertical profiles of $\overline{U_{{\it\theta}}}$ corresponding to different lid angular velocities are shown in figure 11. They were obtained by horizontally averaging the horizontal component of the velocity vectors from PIV measurements in a vertical strip at $r=R$ . Figure 12 shows $\overline{U_{r}}$ at different radial distances, including $r=R$ , at a height of 0.5 and 7 cm above the basal cooling plate. The radial component of the velocity vectors from PIV measurements in a horizontal sector at these heights were averaged to obtain these profiles of $\overline{U_{r}}$ .

Figure 11.

Mean azimuthal velocity in the fluid column at $r=R$ for different angular velocities of the lid.

Figure 12.

Mean radial velocity at heights of (a) 0.5 cm and (b) 7 cm above the bottom plate. Positive values correspond to inward direction. The vertical dashed line in (a) and (b) is at $r=R$ where the ice thickness measurements are taken.

The $\overline{U_{{\it\theta}}}$ plots show the presence of a thin boundary layer near the bottom plate. For higher angular frequencies of rotation ${\it\Omega}$ of the lid, $\overline{U_{{\it\theta}}}$ has a maximum within the boundary layer. We interpret this maximum to be due to the transfer of angular momentum from the flow near the wall along the bottom plate to the flow in the interior of the tank along the bottom plate. Above the boundary layer, there is a core region with uniform $\overline{U_{{\it\theta}}}$ that extends almost to the top of the fluid column. This uniform core can therefore be considered to be in solid body rotation. The velocity in the thin boundary layer near the rough underside surface of the lid has been omitted from the profile as it was difficult to obtain accurate measurements of velocity in that thin layer by PIV due to light reflections from the rough grid degrading the quality of the images. The far field $\overline{U_{{\it\theta}}}$ is 34 % of the lid velocity at the lowest lid $Re$ and 53 % of the lid velocity at the highest lid $Re$ . $\overline{U_{r}}$ is 3–4 times larger inside the bottom boundary layer than in the interior of the fluid column. The turbulent flow between a rotating disk and a stationary disk has been studied experimentally by Itoh et al. (Reference Itoh, Yamada, Imao and Gonda1992) and Cheah et al. (Reference Cheah, Iacovides, Jackson, Ji and Launder1994) and numerically using large-eddy simulation (LES) by Andersson & Lygren (Reference Andersson and Lygren2006). Itoh et al. (Reference Itoh, Yamada, Imao and Gonda1992) also report the presence of an inner core in which $\overline{U_{{\it\theta}}}$ is homogeneous. Denoting $K=(\overline{U_{{\it\theta}}}/{\it\Omega}r)$ , they found $K$ in the range 31–42 % for local $Re$ ( $={\it\Omega}r^{2}/{\it\nu}$ ) from $1.6\times 10^{5}$ to $8.8\times 10^{5}$ , which corresponds to $1.3\times 10^{4}$ to $7.1\times 10^{4}$ with the definition of $Re$ in (4.1). $\overline{U_{r}}$ in their experiment was directed inwards in the boundary layer near the stationary plate and was zero in the inner core. In our experiments, the larger values of $K$ at $Re$ one order of magnitude smaller and the non-zero $\overline{U_{r}}$ in the inner core can be attributed to the roughness of the top boundary, which affects the circulation in the tank by causing enhanced mixing.

Figure 13.

(a) Torque on the lid for a water depth of 10 cm: ▫, measurements; ——, line of best fit. (b) Friction velocity $u_{\ast }$ calculated from torque measurements.

4.4 Friction velocity

The heat balance in (3.15) also requires knowledge of the friction velocity $u_{\ast }$ of the shear-driven flow. Here $u_{\ast }$ is defined as

where ${\it\tau}$ is the shear stress on the lid, which is given by

with $C_{D}$ being the drag coefficient associated with the lid. Taking $\text{d}F$ to be the incremental change in force along an incremental change in radial distance $\text{d}r$ from the centre and ${\it\Gamma}$ to be the torque on the lid, ${\it\Gamma}$ and $\text{d}F$ are related to $C_{D}$ by

and

so that

Hence $u_{\ast }$ can be determined from ${\it\Gamma}$ according to the relation

The torque on the lid for different angular velocities of rotation is shown in figure 13(a). The line of best fit is a weighted-by-value two-parameter polynomial. The friction velocity derived from the torque measurements using (4.10) is shown in figure 13(b).

5.1 Ice thickness

Our heat balance for the ice–water interface can be integrated in time to model the evolution of ice thickness in the experiments conducted with (3.10) used for Phase 1 and (3.15) used for Phase 2. In what follows, the modelled ice thickness is denoted by $h_{m}$ . The values listed in table 2 for the properties of liquid water and solid ice are used in the integration. ${\rm\Delta}T_{s}$ and ${\rm\Delta}T_{\ell }$ in the heat balance are calculated in the following way:

$T_{A}$ refers to temperature measurements at thermistor A, which is located in a small hole in the basal cooling plate, and $T_{G}$ refers to temperature measurements at thermistor G, which is located 5.25 cm above the basal cooling plate. The fluid velocity in the far field, $U_{\infty }$ , is determined using the measurements of $\overline{U_{{\it\theta}}}$ and $\overline{U_{r}}$ interpolated at the angular velocity of the lid at a height $z=7$  cm:

The friction velocity $u_{\ast }$ , which is used in calculating the coefficient of friction $c_{f}$ defined in (3.14), is determined from the calibration shown in figure 13(b). The equations are integrated by a second-order Runge–Kutta method, with the initial condition for $h_{m}$ being zero. Because the temperature measurements were taken at intervals of 5 s, the time step for integration is also 5 s.

Figure 14.

Misfit to (3.15) versus $G$ .

Figure 15.

Comparison of $h_{e}$ (thinner solid line with shaded error region) with $h_{m}$ (thicker solid line) from (a) experiment 8, (b) experiment 5, (c) experiment 3 and (d) experiment 1. For $t<0$ , the error in $h_{e}$ is 0.5 mm and for $t>0$ , it is 0.9 mm. The hatched region in (c) is discussed in the text. For the $h_{m}$ plots, the blue portion corresponds to ice thickness evolution in Phase 1 while the red portion corresponds ice thickness evolution in Phase 2.

In Phase 2, the heat flux $q_{T}$ from the turbulent flow at the ice–water interface depends on the coefficient $G$ . For the $Pr$ of water at 0 °C, which is listed in table 1, $G$ becomes 62.7. We denote this value by $G_{0}$ . The expression for $G$ is an empirical expression derived for a turbulent boundary layer in air over a perfectly flat plate (White Reference White1974). Using a range of values of $G$ , including $G_{0}$ , we evaluate $h_{m}$ during Phase 2 of experiments 2–10. We also calculate the root mean square (r.m.s) difference ${\rm\Delta}h_{RMS}$ between $h_{e}$ and $h_{m}$ at the corresponding times. The omission of experiment 1 from this comparison will be explained when interpreting figure 15(d). The mean of ${\rm\Delta}h_{RMS}$ for the range of values of $G$ considered is plotted in figure 14. Its minimum occurs when $G=36.0$ . The heat flux from the turbulent layer at the ice–water interface is therefore more closely approximated using this value of $G$ , which will be denoted by $G^{\prime }$ . The fact that $G^{\prime }$ is smaller than $G_{0}$ indicates that heat transfer from a turbulent flow at an ice–water interface is more efficient than at a flat plate. This enhanced heat transfer can be attributed to the ice surface not being uniformly smooth, especially during melting when it develops a wavy profile, since a rough surface has a greater surface area than a flat surface and hence allows for greater heat transfer. With the new value $G^{\prime }$ , the heat transfer coefficient $C_{h}$ from the turbulent flow at the ice–water interface given in (3.12) is related to ( $u_{\ast }/U_{\infty }$ ) by the power law $C_{h}=0.028(u_{\ast }/U_{\infty })^{1.09}$ .

Figure 16.

Contour plot of $h_{m}$ for $Re_{R}$ corresponding to experiments 2–10. The white line represents the onset of turbulent mixing, the dashed vertical black lines indicate the $Re_{R}$ of the experiments, and the contour plot is filled by linearly interpolating modelled ice thickness values at the distinct $Re_{R}$ of the experiments.

$G^{\prime }$ is substituted in (3.15) to calculate $h_{m}$ for Phase 2 of experiments 1–10. Figure 15 shows the comparison between $h_{e}$ and $h_{m}$ in experiments 8, 5, 3 and 1. There is good agreement between $h_{e}$ and $h_{m}$ in experiments 8, 5 and 3 but not in Phase 2 of experiment 1. After the onset of turbulent mixing in experiment 1, a stratified layer remained between the turbulent layer and the ice–water interface up to $t=33$  min. This was evidenced by the behaviour of dye inserted into the turbulent layer, which showed that a clear, stratified layer over the ice–water interface prevented the dyed turbulent layer from reaching the ice surface. Accordingly, the ice growth in that time interval is modelled using (3.10). For this case, the measured rate of ice growth is larger than predicted, a difference which occurs because the stratified layer over the ice–water interface inhibits heat transfer from the liquid far field. Ice grows below the turbulent layer from $t=0$ to $t=33$  min, at which time it reaches the turbulence. For $t>33$  min, $h_{m}$ is modelled using (3.15). The model predicts melting whereas the experimental measurements indicate attenuated growth. The disagreement between $h_{e}$ and $h_{m}$ in experiment 1 reveals a limitation of our model. At the lower end of the range of $Re_{R}$ we investigate, the applied shear stress from the lid is low, and consequently the turbulence is too weak to erode the stratified layer quickly. Our model, however, assumes that the stratified layer becomes eroded quickly at low values of $Re_{R}$ .

Figure 17.

Contour plot of absolute difference between $h_{e}$ and $h_{m}$ for experiments 2–10. The distinct $Re_{R}$ of the experiments are marked by the dashed vertical black lines. Data corresponding to the $Re_{R}$ of the experiments are linearly interpolated to make the contour plot.

A stratified layer is present in experiments 2–4 for a short time interval following the onset of turbulent mixing. Although the theoretical model given by (3.10) is incomplete for this configuration, we use it to approximate $h_{m}$ until the time when the turbulent layer comes into contact with the ice. $h_{m}$ is consistently lower than $h_{e}$ in that time interval, as shown in the hatched region of figure 15(c) for experiment 3. In the determination of $G^{\prime }$ previously discussed, $h_{e}$ and $h_{m}$ from time intervals when a stratified layer was present were not used.

The contour plot of $h_{m}$ calculated using the theoretical model with $G^{\prime }$ for experiments 2–10 is shown in figure 16. Experiment 1 is omitted because it is a case for which our model is not valid. The region corresponding to experiment 1 in figure 16 is hatched. The absolute difference between $h_{e}$ from figure 5 and $h_{m}$ at corresponding times is shown in figure 17. The absolute difference is generally close to the error margin of 0.5 mm for Phase 1 and 0.9 mm for Phase 2, which indicates good agreement between the model and the experiment. The hatched area in figure 17 corresponds to experiment 1 and time intervals when a stratified layer was present during Phase 2 of the experiments. The absolute difference between $h_{e}$ and $h_{m}$ in these intervals during Phase 2 was omitted from the contour plot because the model assumes that the stratified layer is eroded more rapidly and hence deviates from the experimental measurements.

We did not observe evidence of the radial component of the flow near the bottom of the tank during turbulent mixing affecting the evolution of ice in our set of experiments. The radial component of the flow is stronger at higher $Re_{R}$ . In all experiments in which there is transient melting, the rate at which the ice melts always increases with distance from the centre of the tank. During the subsequent refreezing, the rate of ice growth is always uniform at all radial distances. These observations suggest that the far-field flow has a stronger influence on the evolution of ice than the radial flow near the bottom of the tank.