2.1 Thermal convection

In understanding mass transfer due to turbulent compositional convection, very useful insight can be gained by examining the experimental measurements on heat transfer by natural convection tabulated by Holman (Reference Holman2010).

For laminar natural convection on an isothermal boundary of temperature $T_{w}$ facing downward at an angle $\unicode[STIX]{x1D703}$ to the vertical, the Nusselt number $Nu$ is given by

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle Nu\approx 0.56\,(Ra\cos \unicode[STIX]{x1D703})^{1/4} & \displaystyle\end{eqnarray}$$

(Holman Reference Holman2010, p. 344), where $Nu=qL(k(T_{w}-T_{f}))^{-1}$ , $q$ is the heat flux from the boundary, $L$ is the length of the boundary, $k$ is the thermal conductivity of the fluid, $T_{f}$ is the far-field temperature of the fluid, $Ra=g\unicode[STIX]{x1D6FC}(T_{w}-T_{f})L^{3}(\unicode[STIX]{x1D705}\unicode[STIX]{x1D708})^{-1}$ is the Rayleigh number, $g$ is the gravitational acceleration, $\unicode[STIX]{x1D6FC}$ is the thermal expansion coefficient of the fluid, $\unicode[STIX]{x1D705}$ is the thermal diffusivity of the fluid, and $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid. Equation (2.1) was measured in water by Fujii & Imura (Reference Fujii and Imura1972) for values of $\unicode[STIX]{x1D703}$ from $0^{\circ }$ to $86.8^{\circ }$ and values of $Ra\cos \unicode[STIX]{x1D703}$ from $10^{5}$ to $10^{11}$ . In (2.1), the heat transfer is seen to depend on the component of gravitational acceleration up the sloping boundary, $g\cos \unicode[STIX]{x1D703}$ .

For turbulent natural convection on a constant heat flux boundary facing downward at an angle $\unicode[STIX]{x1D703}$ , the Nusselt number at high flux Rayleigh numbers $Ra_{f}$ is empirically correlated by

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle Nu\approx 0.17\,(Ra_{f}\cos ^{2}\unicode[STIX]{x1D703})^{1/4} & \displaystyle\end{eqnarray}$$

(Holman Reference Holman2010, p. 345), where $Ra_{f}=g\unicode[STIX]{x1D6FC}qL^{4}(k\unicode[STIX]{x1D705}\unicode[STIX]{x1D708})^{-1}$ . Equation (2.2) was measured in air by Vliet & Ross (Reference Vliet and Ross1975) for values of $\unicode[STIX]{x1D703}$ from $0^{\circ }$ to $80^{\circ }$ and values of $Ra_{f}\cos ^{2}\unicode[STIX]{x1D703}$ that ranged from $10^{10}$ to $10^{15}$ . Vliet & Ross (Reference Vliet and Ross1975) attributed the dependence of the heat transfer on the angle $\unicode[STIX]{x1D703}$ to a combination of the component of gravitational acceleration up the sloping boundary, and ‘the damping of the turbulence due to the thermally stable condition of the boundary layer flow’.

Since $Ra_{f}=Ra\,Nu$ , (2.2) is equivalent to

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle Nu\approx (0.17)^{4/3}Ra^{1/3}\cos ^{2/3}\unicode[STIX]{x1D703}=0.094\,Ra^{1/3}\cos ^{2/3}\unicode[STIX]{x1D703}. & \displaystyle\end{eqnarray}$$

It is important to note that the one-third power law dependence in (2.3) implies that the turbulent heat flux is independent of the length of the boundary and independent of position on the boundary (Holman Reference Holman2010, p. 337).

By assuming that the heat transport is continuous across the laminar–turbulent transition we can combine (2.1) and (2.3) to give a critical Rayleigh number where the transition will occur:

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle Ra_{c}\sim {\displaystyle \frac{2.0\times 10^{9}}{\cos ^{5}\unicode[STIX]{x1D703}}}. & \displaystyle\end{eqnarray}$$

This critical Rayleigh number corresponds to a critical transition distance up the sloping boundary of

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle L_{c}\sim {\displaystyle \frac{1260}{\cos ^{5/3}\unicode[STIX]{x1D703}}}\left({\displaystyle \frac{\unicode[STIX]{x1D705}\unicode[STIX]{x1D708}}{g\unicode[STIX]{x1D6FC}(T_{w}-T_{f})}}\right)^{1/3}. & \displaystyle\end{eqnarray}$$

It is noted that this distance is measured in the wall-parallel direction and is not the vertical height.

2.2 Dissolution of a sloping solid surface by turbulent compositional convection

Kerr & McConnochie (Reference Kerr and McConnochie2015) presented a theoretical analysis that predicted the ablation velocity and interface conditions of a dissolving vertical solid surface based on empirical heat transfer relationships equivalent to (2.3). The model was itself an extension of previous work investigating a horizontal surface dissolving into fluid above it (Kerr Reference Kerr1994b ).

The model envisages thermal and compositional boundary layers that grow diffusively until a time $\unicode[STIX]{x1D70F}$ when they are periodically removed by the eddies associated with turbulent compositional convection (cf. Lick Reference Lick1965; Howard Reference Howard and Görtler1966). The model can be generalised for a surface that is facing downward at an angle $\unicode[STIX]{x1D703}$ to the vertical by including the angle dependence that is given in (2.3).

We consider a solid surface with composition $C_{s}$ and far-field temperature $T_{s}$ in contact with a semi-infinite fluid with a far-field composition $C_{f}$ and far-field temperature $T_{f}$ . There are three unknown quantities at the interface between the solid and the fluid: the interface temperature $T_{i}$ , interface composition $C_{i}$ and the dissolution velocity $V$ . The analysis of Kerr & McConnochie (Reference Kerr and McConnochie2015), together with the heat transfer relationship given in (2.3), result in $T_{f}-T_{i}$ and the velocity scale ${\mathcal{V}}$ being given by

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle T_{f}-T_{i}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{s}L_{s}+\unicode[STIX]{x1D70C}_{s}c_{s}(T_{i}-T_{s})}{\unicode[STIX]{x1D70C}_{f}c_{f}}}\,\left({\displaystyle \frac{D}{\unicode[STIX]{x1D705}_{f}}}\right)^{1/2}\left({\displaystyle \frac{C_{f}-C_{i}}{C_{f}-C_{s}}}\right) & \displaystyle\end{eqnarray}$$

and

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{V}}={\displaystyle \frac{V}{\unicode[STIX]{x1D6FE}}}=\cos ^{2/3}\unicode[STIX]{x1D703}\,\left({\displaystyle \frac{g(\unicode[STIX]{x1D70C}_{f}-\unicode[STIX]{x1D70C}_{i})D^{2}}{\unicode[STIX]{x1D707}}}\right)^{1/3}\left({\displaystyle \frac{C_{f}-C_{i}}{C_{f}-C_{s}}}\right), & \displaystyle\end{eqnarray}$$

where subscript $f$ corresponds to fluid properties, subscript $i$ corresponds to properties at the interface, subscript $s$ corresponds to solid properties, $\unicode[STIX]{x1D70C}$ is the density, $L$ is the latent heat of fusion, $c$ is the specific heat, $D$ is the compositional diffusivity and $\unicode[STIX]{x1D707}$ is the fluid dynamic viscosity. $\unicode[STIX]{x1D6FE}$ is a constant of proportionality equal to approximately 0.094 (from (2.3)) or $0.090\pm 0.004$ (from Kerr & McConnochie (Reference Kerr and McConnochie2015)).

Equations (2.6) and (2.7) can be solved for the three unknowns using the thermodynamic constraint that the interface temperature and salinity must lie on the liquidus curve. However, from inspection it can be seen that the angle $\unicode[STIX]{x1D703}$ is predicted to have no effect on the interface conditions, while the dissolution velocity is predicted to be proportional to $\cos ^{2/3}\unicode[STIX]{x1D703}$ .

A correction for the moving solid–liquid boundary has been applied which results in (2.6) and (2.7) being valid when $(C_{f}-C_{i})\approx (C_{i}-C_{s})$ . This correction increases the compositional boundary layer thickness to account for the fact that the interface will retreat as the boundary layer grows. As a result, the compositional boundary layer thickness $h_{c}$ is more accurately given by $\sqrt{D\unicode[STIX]{x1D70F}}+V\unicode[STIX]{x1D70F}$ than by $\sqrt{D\unicode[STIX]{x1D70F}}$ . When $(C_{f}-C_{i})\ll (C_{i}-C_{s})$ the velocity of the boundary is negligible and the final term in (2.6) and (2.7) should be replaced with $(C_{f}-C_{i})/(C_{i}-C_{s})$ . Furthermore, when $(C_{f}-C_{i})\gg (C_{i}-C_{s})$ the model breaks down as the turbulent dissolution undergoes a transition to turbulent melting (see appendix of Kerr (Reference Kerr1994b )).

The model also assumes that the compositional buoyancy released at the interface dominates the thermal buoyancy. As in Kerr & McConnochie (Reference Kerr and McConnochie2015), this condition can be described based on the ratio of compositional to thermal buoyancy ${\mathcal{R}}$ as:

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{R}}={\displaystyle \frac{\unicode[STIX]{x1D6FD}(C_{f}-C_{s})\unicode[STIX]{x1D70C}_{f}c_{f}}{\unicode[STIX]{x1D6FC}[\unicode[STIX]{x1D70C}_{s}L_{s}+\unicode[STIX]{x1D70C}_{s}c_{s}(T_{i}-T_{s})]}}\gg 1, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}$ is the compositional expansion coefficient. This condition is generally met in the polar oceans as $\unicode[STIX]{x1D6FC}$ approaches zero at low temperatures.

When considering compositional convection, (2.5) needs to be adjusted to reflect properties of the solute rather than then thermal field. As such, (2.5) should be rewritten as:

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle L_{c}\sim {\displaystyle \frac{1260}{\cos ^{5/3}\unicode[STIX]{x1D703}}}\left({\displaystyle \frac{D\unicode[STIX]{x1D708}}{g\unicode[STIX]{x1D6FD}(C_{i}-C_{f})}}\right)^{1/3}. & \displaystyle\end{eqnarray}$$

For a vertical ice wall dissolving into the polar oceans at temperatures from $-1$ to $5\,^{\circ }\text{C}$ , using values of the physical constants and interface conditions from Kerr & McConnochie (Reference Kerr and McConnochie2015), equation (2.9) suggests that the compositional boundary layer undergoes a transition from laminar to turbulent flow at a distance from the base $L_{c}$ of 2–4 cm. In comparison, laboratory experiments using vertical ice walls have found that the transition to turbulence actually occurs at heights of 10–30 cm (Josberger & Martin Reference Josberger and Martin1981; Carey & Gebhart Reference Carey and Gebhart1982; Johnson & Mollendorf Reference Johnson and Mollendorf1984; Kerr & McConnochie Reference Kerr and McConnochie2015).

A possible explanation of this discrepancy lies in the observation of bidirectional flow in the laminar region next to a dissolving ice face (Josberger & Martin Reference Josberger and Martin1981). Due to the higher molecular diffusivity of heat than salt, the thermal boundary layer is significantly thicker than the haline boundary layer. In the laminar region this leads to a rising haline boundary layer close to the wall and a sinking thermal boundary layer further away from the wall. Once the flow is turbulent the two boundary layers are relatively well mixed and the boundary layer is one-directional. The interaction between the two boundary layers in the laminar region could have an important impact on the laminar–turbulent transition that is not included in the previous thermal experiments that were used to derive (2.5).

3.1 Ablation velocity

Equation (2.7) predicts that the ablation velocity will scale with $\cos ^{2/3}\unicode[STIX]{x1D703}$ . Table 1 gives the experimental conditions and results used to test this prediction and to evaluate the constant of proportionality $\unicode[STIX]{x1D6FE}$ .

Figure 2.

The measured ablation velocity as a function of the velocity scale given in (2.7). The straight line gives the predicted ablation velocity with $\unicode[STIX]{x1D6FE}=0.086$ . The error bars show the estimated experimental uncertainty of $\pm 5\,\%$ based on repeated experiments.

Table 1.

Measured ablation velocities $V$ with the experimental slope angle, far-field temperature $T_{f}$ , far-field salinity $C_{f}$ , the velocity scale from (2.7)  ${\mathcal{V}}$ and calculated values of $\unicode[STIX]{x1D6FE}$ . The average value of $\unicode[STIX]{x1D6FE}$ across all experiments is 0.086.

The constant of proportionality $\unicode[STIX]{x1D6FE}$ is evaluated to be approximately $0.086\pm 0.005$ based on the experimental data given in table 1. This is slightly lower but comparable to the equivalent constant for a vertical ice wall reported by Kerr & McConnochie (Reference Kerr and McConnochie2015) of $0.090\pm 0.004$ . There is no systematic variation of $\unicode[STIX]{x1D6FE}$ with $\unicode[STIX]{x1D703}$ suggesting that the difference is due to experimental uncertainty rather than any further dependence on $\unicode[STIX]{x1D703}$ .

Figure 2 shows the measured ablation velocity $V$ plotted against the velocity scale ${\mathcal{V}}$ . The average value of $\unicode[STIX]{x1D6FE}$ from the experiments given in table 1 is used to plot the predicted ablation velocity (black line). The experimental measurements agree with the predicted ablation velocity to within the experimental uncertainty which is primarily due to the resolution of the photographs used to make the measurements.

To remove any potential effects of the far-field conditions the ablation velocity has been normalised by the expected ablation velocity under the same ambient fluid conditions but a vertical ice wall. This normalised ablation velocity is plotted in figure 3.

Figure 3.

The normalised ablation velocity as a function of $\cos ^{2/3}\unicode[STIX]{x1D703}$ . The ablation velocity has been normalised by the predicted ablation velocity given the same far-field conditions but a vertical wall. The error bars show the estimated experimental uncertainty of $\pm 5\,\%$ based on repeated experiments.

3.2 Interface temperature

Table 2 gives the measured ( $T_{b}$ ) and predicted ( $T_{i}$ ) interface temperatures with the far-field conditions and slope angle. Figure 4 shows the difference between the predicted and measured interface temperatures as a function of angle $\unicode[STIX]{x1D703}$ . The values for $\unicode[STIX]{x1D703}=0^{\circ }$ are taken from experiment E of Kerr & McConnochie (Reference Kerr and McConnochie2015).

Table 2.

Measured ( $T_{b}$ ) and predicted ( $T_{i}$ ) interface temperatures with the experimental slope angle $\unicode[STIX]{x1D703}$ , far-field temperature $T_{f}$ and far-field salinity $C_{f}$ . The values for $\unicode[STIX]{x1D703}=0^{\circ }$ are taken from experiment E of Kerr & McConnochie (Reference Kerr and McConnochie2015).

Figure 4.

Difference between the experimental and predicted interface temperature as a function of slope angle $\unicode[STIX]{x1D703}$ . The two dashed lines show the expected $0.1\,^{\circ }\text{C}$ experimental uncertainty based on previous experiments (Kerr & McConnochie Reference Kerr and McConnochie2015). The value for $\unicode[STIX]{x1D703}=0^{\circ }$ is the average difference between the measured and predicted interface temperature from the experiments of Kerr & McConnochie (Reference Kerr and McConnochie2015).

Based on previous experiments (Kerr & McConnochie Reference Kerr and McConnochie2015), we estimate the uncertainty in measurements of the interface temperature to be approximately $0.1\,^{\circ }\text{C}$ . This experimental uncertainty is shown by the dashed lines in figure 4. The difference between our experimental interface temperatures and that predicted by (2.6) is generally less than the experimental uncertainty, suggesting that (2.6) is able to accurately predict the interface temperature over a range of slopes.

The experimental results and (2.6) show that the interface temperature is independent of the slope angle $\unicode[STIX]{x1D703}$ . The interface temperature depends on the relative transport of heat and salt to the interface which is primarily controlled by the ratio of the thermal and haline diffusivity. Given that these physical quantities are unaffected by the geometry of the ice face, it is unsurprising that the interface temperature is also unaffected by the geometry of the ice.

3.3 Transition length

Finally, the experimental results are used to test (2.9). As noted in § 2.1, the theoretical transition length is smaller than that which is observed in the case of a vertical ice wall dissolving into salt water. This discrepancy could be due to the instability being suppressed by the bidirectional flow caused by the opposing buoyancy of the temperature and salinity fields (Josberger & Martin Reference Josberger and Martin1981). Such an effect cannot occur in the thermal experiments used to derive (2.9) as the boundary layer flow is purely one-directional. When comparing our experimental measurements of the transition length to (2.9) we have used a non-dimensionalised length scale defined as

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle L_{c}^{\ast }=L_{c}\,\left({\displaystyle \frac{g(\unicode[STIX]{x1D70C}_{f}-\unicode[STIX]{x1D70C}_{i})}{D\unicode[STIX]{x1D707}}}\right)^{1/3}. & \displaystyle\end{eqnarray}$$

The term $(g(\unicode[STIX]{x1D70C}_{f}-\unicode[STIX]{x1D70C}_{i})/(D\unicode[STIX]{x1D707}))^{1/3}$ is approximately equal to $56\,000~\text{m}^{-1}$ for our experiments but varies slightly due to different far-field conditions. From (2.9) we expect $L_{c}^{\ast }=A\,\cos ^{-5/3}\unicode[STIX]{x1D703}$ where the coefficient $A$ will be determined from the experimental results.

Figure 5.

Experimental values of the non-dimensional transition length plotted against $\cos ^{-5/3}\unicode[STIX]{x1D703}$ . The solid line shows $L_{c}^{\ast }=86\,840\,\cos ^{-5/3}\unicode[STIX]{x1D703}\,-81\,670$ and the dashed line shows $L_{c}^{\ast }=8506\,\cos ^{-7.37}\unicode[STIX]{x1D703}$ .

Table 3.

Measured transition length $L_{c}$ with the experimental slope angle $\unicode[STIX]{x1D703}$ , far-field temperature $T_{f}$ , far-field salinity $C_{f}$ and the non-dimensionalised transition length $L_{c}^{\ast }$ , from (3.1).

Table 3 gives the dimensional and non-dimensionalised transition lengths with the experimental conditions for each experiment. Figure 5 shows the non-dimensional transition length plotted against $\cos ^{-5/3}\unicode[STIX]{x1D703}$ . The error bars shown in figure 5 indicate the range of locations where turbulent fluctuations first appeared. When the slope is steep the onset of turbulent fluctuations was visually obvious and consistent in time. However, for shallow slope angles, pulses of turbulence originated over a wide range of heights and grew less rapidly.

The solid line in figure 5 shows $L_{c}^{\ast }=86\,840\,\cos ^{-5/3}\unicode[STIX]{x1D703}\,-81\,671$ . In order for $L_{c}\sim \cos ^{-5/3}\unicode[STIX]{x1D703}$ , as predicted by (2.9), a significant offset needs to be added to the equation. The dashed line shows $L_{c}^{\ast }=8506\,\cos ^{-7.37}\unicode[STIX]{x1D703}$ . This was obtained from fitting a power law to the experimental data.

Figure 5 suggests that, within our experimental parameters, the non-dimensional transition length can be given by

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle L_{c}^{\ast }=86\,840\,\cos ^{-5/3}\unicode[STIX]{x1D703}\,-81\,671 & \displaystyle\end{eqnarray}$$

or

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle L_{c}^{\ast }=8506\,\cos ^{-7.37}\unicode[STIX]{x1D703}. & \displaystyle\end{eqnarray}$$

The difference between the two power laws given in (3.2) and (3.3) is significant. For example, at similar far-field conditions to our experiments and a slope angle of $\unicode[STIX]{x1D703}=60^{\circ }$ , equation (3.2) would predict a transition length of 3.4 m while (3.3) would predict a transition length of 25 m. Without measurements of the transition length at shallower slope angles it is difficult to assess which power law is more accurate. If (3.3) is valid, then the presence of both salinity and temperature boundary layers greatly enhances the sensitivity of slope angle on the transition length compared to what was predicted by (2.9).