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The article delves into the modelling of ice disc response to radial water flow, particularly in the context of sea ice thickening. It builds upon previous work by Pantling et al. (2024), extending the analysis into three dimensions. The authors develop and validate three models—inviscid well-mixed film, inviscid flow with a developing thermal boundary layer, and viscous flow with a thermal boundary layer—to predict ice thickness changes. Experimental setups in a cold room were used to test these models under varying water temperatures. The viscous flow model with a thermal boundary layer was found to best match the experimental data, providing valuable insights into the mechanisms of ice thickening. The study highlights the potential of this technique to mitigate Arctic sea ice loss and suggests further research directions, including the consideration of longer time scales and the salinity of water and ice.
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Abstract
Arctic sea ice is melting rapidly, and the Arctic is likely to experience its first ice-free summer in the next few decades unless action is taken locally. One proposed method of reducing or perhaps reversing the melting of Arctic sea ice is pumping seawater onto the surface of the sea ice where it should freeze faster and thicken the ice. This may in turn enable it to last longer or even survive the summer melting period, reflecting more sunlight and becoming stronger multi-year ice with increased resistance to future melting. Despite appearing to be a relatively simple physical problem, the technique has not been researched in depth. Here, the response of ice to water being pumped over its surface is investigated theoretically and experimentally for radial axisymmetric water flow. The dominant heat transfer mechanisms during the period shortly after placement of water onto ice are conduction through the ice away from the water–ice interface and heat transfer from the water to the interface. During this initial period of evolution, advection and radiation to the atmosphere are much smaller in magnitude and hence not included. The heat transfer from the water flow to the interface is modelled for three flows: a well-mixed uniform film flow; a uniform flow with a developing thermal boundary layer; and a laminar, viscous flow with a developing thermal boundary layer. Predictions from these models are compared with data from laboratory experiments using various initial water temperatures. The predictions of the model with a fully developed, laminar viscous flow and a developing thermal boundary layer for the evolution of the ice profile were found to be closest to the data obtained from laboratory experiments with water supplied at 0.5, 1.0 and 1.5 \(^\circ\)C.
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1 Introduction
The rapid melting of Arctic sea ice could have severe impacts on the planet, damaging both meteorological systems (e.g. Overland et al. (2015)) and ecological ones (e.g. Post et al. (2013)). Jahn et al. (2024) show that the first ice-free Arctic could happen in the 2020–2030s under all the Shared Socioeconomic Pathways considered by the IPCC and “are likely to occur by 2050.” Therefore, there is interest as to whether there are actions over and above those considered by the IPCC which might stem the rate of loss of Arctic sea ice (Bodansky and Hunt 2020). As sea ice thickens, it becomes progressively more insulating which slows the rate of new ice growth at the base of the ice. Desch et al. (2017) describe wind-powered pumps across the Arctic pumping seawater through and on top of existing sea ice to bypass the insulating effect and thicken it from the surface. Using a heat balance across the whole Arctic over a winter, Desch et al. (2017) predict that adding \(1\,\textrm{m}\) of water on top of sea ice would increase its thickness by \(0.7\,\textrm{m}\) (due to reduced natural freezing at the ice’s base) and that thickening ice to \(1.5\,\textrm{m}\) should allow it to survive the Arctic summer. However, the effect of pumping the water onto the ice was not investigated and is a central question to the technique that needs to be answered.
Pantling et al. (2024) considered melting of ice and freezing of water resulting from water flowing over ice down an inclined channel. The modelling considered conduction through the ice away from the ice–water interface, the latent heat of solidification absorbed/released when melting ice/freezing water and three different assumptions for the heat transfer from the water flow to the ice–water interface. In experiments, using water introduced at less than 1 \(^\circ\)C, the model that agreed most closely with laboratory experiments was based on viscous and thermal boundary layers determining the heat transfer from the water. Uniform flows with and without a developing thermal boundary layer were also considered, but those predictions for the resulting ice profiles matched less closely to experimental data. For water introduced at 2.0 \(^\circ\)C, the models did not match the experimental data well; this was due in part to the larger degree of melting near the point of injection which in turn increased the error arising from the assumptions in the model of flow of a thin film and heat transfer at a stationary ice–water interface.
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This paper extends the previous work of Pantling et al. (2024) into three dimensions by considering radial axisymmetric flow of freshwater over fresh ice. In reality, seawater is saline and sea ice is a permeable mushy layer of fresh ice containing interstitial brine; these additional considerations are discussed at the end of the paper. First, the three models of Pantling et al. (2024), described above, are updated to consider radial flow over a flat disc of ice. A set-up to investigate the behaviour experimentally in a cold room was built, and experiments were conducted at three different initial water temperatures. The models were then analysed with respect to these experimental results to determine which is most appropriate. The paper’s aim is to improve understanding of the potential for thickening Arctic sea ice and concludes with further work that could be done to progress the research.
2 Modelling
2.1 Constants and nomenclature
Thermal diffusivity of ice, \(\alpha _\textrm{i}\)
\(1.18 \times 10^{-6}\)
\(\mathrm {m^2s^{-1}}\)
Thermal diffusivity of water, \(\alpha _\textrm{w}\)
\(0.132 \times 10^{-6}\)
\(\mathrm {m^2s^{-1}}\)
Thermal conductivity of ice, \(\lambda _\textrm{i}\)
2.22
\(\mathrm {Wm^{-1}K^{-1}}\)
Thermal conductivity of water, \(\lambda _\textrm{w}\)
0.56
\(\mathrm {Wm^{-1}K^{-1}}\)
Density of ice, \(\rho _\textrm{i}\)
916.2
\(\mathrm {{kg}m^{-3}}\)
Density of water, \(\rho _\textrm{w}\)
1000
\(\mathrm {{kg}m^{-3}}\)
Specific heat capacity of ice, \(c_{p,\textrm{i}}\)
2100
\(\mathrm {J{kg}^{-1}K^{-1}}\)
Specific heat capacity of water, \(c_{p, \textrm{w}}\)
4200
\(\mathrm {J{kg}^{-1}K^{-1}}\)
Latent heat of fusion of water, L
334, 000
\(\mathrm {J{kg}^{-1}}\)
Kinematic viscosity of water, \(\nu _\textrm{w}\)
\(1.792 \times 10^{-6}\)
\(\mathrm {m^2 s^{-1}}\)
Freezing temperature of water, \(T_{\textrm{f}}\)
0
\(^\circ\)C
Acceleration due to gravity, g
9.81
\(\mathrm {m s^{-2}}\)
Heat transfer coefficient between water and ice, H
\(5494.8\, v\)
\(\mathrm {Wm^{-2}K^{-1}}\)
Physical properties taken from The Engineering ToolBox (2023) and the empirically determined heat transfer coefficient from Li et al. (2016).
Radial distance from the centre
r
Perpendicular distance from the ice surface
z
Time since initial water injection
t
Change in ice thickness
\(\eta (r, t)\)
Temperature of the ice
\(T_\textrm{i}(r, z, t)\)
for \(z\le \eta\)
Temperature of the water
\(T_\textrm{w}(r, z, t)\)
for \(z\ge \eta\)
Velocity of the flow parallel to the ice surface
v
Depth of the flow
h
Radius of the ice disc
R
Initial depth of the ice disc
d
Conductive heat flux through the ice
\(q_\textrm{i}\)
Heat flux from the water
\(q_\textrm{w}\)
Note that where a parameter is dependent on various positions in space and time the convention of f(r, z, t) will be followed. For example, \(T_\textrm{i}(0, \eta , 0^+)\) represents the temperature of the ice at the disc centre (\(r=0\)), at the interface (\(z=\eta\)), instantaneously after the water is injected (\(t=0^+\))
2.2 Modelling set-up
The modelling set-up for the development of an ice profile with axisymmetric flow over its surface is shown in Fig. 1. Much of the analysis follows Pantling et al. (2024) with changes simply reflecting the change in geometry. The water flows radially outwards from the origin at temperature \(T_\textrm{w0}\), velocity v and with height h. The change in ice thickness \(\eta\) may be either positive or negative depending on the balance of heat transfers to and from the ice front. The ice is treated as semi-infinite such that the temperature signal does not fully penetrate the ice, which is valid provided \(d \gtrsim \sqrt{\alpha _\textrm{i} t}\), where d is the initial depth of the ice, \(\alpha _\textrm{i}\) is the thermal diffusivity of ice and t is time. Also, time and length scales are considered to be sufficiently small that heat transfer from the water to the interface and conduction from the interface through the ice are much greater than heat transfer to the atmosphere by advection or radiation. Similarly, the radial water velocity, v, is taken to be sufficiently large that advection of heat radially by the water greatly exceeds the rate of conduction in the radial direction through both the water and the ice (i.e. \(v \gg \sqrt{\alpha _\textrm{w}/t}\) and \(v \gg \sqrt{\alpha _\textrm{i}/t}\), where \(\alpha _\textrm{w}\) is the thermal diffusivity of water). The time taken for the flow to reach steady state is also assumed to be negligible relative to the timescale for diffusion through the water or ice. Combining these assumptions gives the condition that \(\alpha _\textrm{w,i}/v^2 \ll t < d^2/\alpha _\textrm{i}\) for the model to be valid, constraining the timescale and initial ice depth. The water is assumed to flow axisymmetrically; hence, the modelling cannot be applied once rivulets form. The rate of addition or loss of mass flux arising from melting or freezing is assumed to be much smaller than the supplied flow rate of water which is therefore assumed to be constant.
Fig. 1
Schematic diagram illustrating the model for water flowing radially over ice of semi-infinite thickness
The rate of solidification of water on the surface of ice is determined by the difference between heat transfer to and away from the water–ice interface, so that
where \(\rho _\textrm{i}\) is the density of ice and L is the latent heat of fusion. The change in ice thickness \(\eta\) may vary in both space r and time t as can the conductive heat transfer through the ice and the heat transfer from the water, \(q_\textrm{i}\) and \(q_\textrm{w}\), respectively.
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As the heat diffusion radially in the ice has been assumed to be negligible, the temperature profile in the ice only varies in the z direction to the same degree of approximation. The interface between the ice and the water remains close to the freezing temperature for all space and time, \(T_\textrm{i}(r, \eta , t) = T_\textrm{f}\), and, assuming the ice to be semi-infinite in z, the far-field temperature is the initial ice temperature \(T_\textrm{i0}\). The temperature profile in the ice is therefore given by
where \(\lambda _\textrm{i}\) is the thermal conductivity ice. Given the time taken for the flow to reach the edge of the ice disc is taken to be negligible, the origin of time is independent of radial position and can be taken to be the time that the water is first introduced.
The heat flux to the interface from the water can be described by
where H is a heat transfer coefficient and \(T_\textrm{w}\) is the water temperature.
Within the model, at the instant the water makes contact with the ice there must be a temperature gradient in the ice that tends to infinity as shown by equation (3). On the other hand, the heat transfer from the flow to the water–ice interface is finite, and therefore, the first response must be freezing at a rate that tends to infinity.
The temperature gradient in the ice quickly weakens, reducing the rate of conductive heat transfer and hence the rate of freezing (Pantling et al. 2024). In Sects. 2.3–2.4, the heat flux from the water, \(q_\textrm{w}\), is investigated for three different flow assumptions.
2.3 Radial flow of an inviscid film
If the Reynolds number, \(Re = vh/\nu _w\), is much greater than unity, then the flow can be treated as inviscid (Schlichting and Gersten 2017). Consider the case of water being injected from above by a pipe of radius \(r_0\) from height \(h_0\) onto a disc of ice with pipe exit speed \(v_0\). The water will accelerate under gravity, thereby thinning before impacting the surface. It will then mix before flowing radially as a thin film over the ice surface as shown in Fig. 2.
Fig. 2
Set-up for determining the behaviour of the inviscid film
During free fall, the flow will accelerate from \(v_0\) to \(v = \sqrt{v_0^2 + 2\,g h_0}\). It is assumed that the impact region is of similar radius to that of the injection pipe (which is much smaller than that of the ice disc being considered) and that outside of this region the water flows radially as a thin film of height h(r). Considering the case of water being released from a height above the surface which is much greater than the film height (i.e. \(h_0 \gg h\)) or at sufficient exit velocity \(v_0\) that \(v^2 / 2\,g\Delta (h+\eta ) \gg 1\) the effect of gravity on the height of the film can be neglected for \(r \gg r_0\). Applying Bernoulli’s equation to the film, \(p_0 = p_a + \frac{1}{2} \rho _\textrm{w} v^2\), where \(p_0\) is the total pressure and \(p_a\) is atmospheric pressure, the velocity of the film v is constant. For the case that the rate of addition/loss of mass by melting/freezing is much smaller than the injected flow rate, the mass flow rate \(\dot{m}=2\pi \rho _\textrm{w}hvr\) is also constant. Therefore, applying constant density (as water is assumed to be incompressible) and constant velocity (from Bernoulli), \(hr=\textrm{constant}\); the fluid height is inversely proportional to the radial position (i.e. \(h \propto 1/r\)). Two different assumptions for the heat transfer from the water film to the ice–water interface are now considered.
2.3.1 Inviscid flow of a vertically well-mixed film
If the Reynolds number is sufficiently high that there is significant vertical turbulent mixing throughout the full depth of the film, then there will be a bulk temperature of the film that decreases as the flow spreads radially outwards. A constant velocity gives a constant heat transfer coefficient H which can be determined empirically (Li et al. 2016). The radial temperature profile in the water can then be determined by considering an energy balance on a control volume encompassing an annulus of the flow between r and \(r+\updelta r\) as shown in Fig. 3, where \(T_\textrm{w}\) is the temperature of the water at position r and time t.
Fig. 3
Schematic diagram showing the control volume for deriving the radial water temperature profile
The water temperature as a function of radius is shown in Fig. 4 for \(\dot{m}=5.4 \times 10^{-3} \mathrm {kg s^{-1}}\) and \(v=1.2\,\mathrm {m s^{-1}}\), giving \(H=6700 \; \mathrm {Wm^{-2}K^{-1}}\). The characteristic length scale for heat transfer from the water to the interface in (7), \(l \sim \sqrt{\dot{m} c_{p, \textrm{w}}/\pi H} = 0.032 \; \textrm{m}\), is much shorter than in the case of channel flow (Pantling et al. 2024), and hence, the temperature decreases with radius to close to the freezing temperature more steeply. Due to the flow rate per unit area of ice decreasing with radius, the temperature decreases exponentially with the square of radius, \((T_\textrm{w}-T_\textrm{f}) \propto (T_\textrm{w0}-T_\textrm{f}) \exp {(-r^2)}\), in contrast to the exponential decrease seen in the case of the channel flow, \((T_\textrm{w}-T_\textrm{f}) \propto (T_\textrm{w0}-T_\textrm{f}) \exp {(-x)}\) (Pantling et al. 2024). The difference in the rate at which the temperature approaches the freezing temperature is stark; the temperature reduces to \(1\%\) of its initial value by \(0.071\,\textrm{m}\) for the radial flow, whereas in the case of channel flow this takes over \(1.5 \; \textrm{m}\) for the same flow rate. The steeper temperature reduction is due to the water flowing over a larger area with increasing radius.
Fig. 4
Change in water temperature as a function of radius due to heat transfer from the water to the ice–water interface
The radial temperature profile of the water (7) can be substituted into equation (4) for heat transfer from the water, which along with equation (3) for conduction through the ice can be substituted into the equation for the rate of ice thickness change (1) giving
The equation for the radial ice profile with vertically well-mixed film flow has a very similar form to the equivalent channel model, but with a dependence on \(\exp {(-r^2)}\) rather than \(\exp {(-x)}\) in the heat transfer from the water term (Pantling et al. 2024). The steeper reduction heat transfer from the water (which drives any melting of ice) means that there will be a smaller region of melting at the centre of the disc compared to the channel flow. The conduction term (which drives freezing) is unchanged by the flow regime, and so, the behaviour at very large radius, when the water has cooled to practically its freezing temperature, is identical to large distance for the channel flow.
The change in ice thickness for 1 \(^\circ\)C water injected onto a -10 \(^\circ\)C ice disc is shown in Fig. 5 (again for \(\dot{m}=5.4 \times 10^{-3} \mathrm {kg s^{-1}}\) and \(H=6700 \; \mathrm {Wm^{-2}K^{-1}}\)). At short times, there is strong conduction through the ice and this dominates over heat transfer from the water, and so, there is freezing. The temperature of the water is greatest at the centre of the ice, and hence, this is where the new ice first begins to melt. The freezing to melting transition, where \(\partial \eta / \partial t = 0\), is indicated by the dashed black line. As time progresses, the new ice all melts, giving the white region of no net ice thickness change, and then, the original ice begins to melt too. Once melting has started at any given radius, there is no mechanism in the model for freezing to resume and the ice melts continuously due to the constant supply of warm water. Further away from the centre, the water is cooler, and hence, there is less melting. For radii greater than \(\sim 1.5 \, l\), where l is the characteristic length introduced above, the water is so close to its freezing temperature that melting does not occur until very long times.
Fig. 5
Contour map of the change in ice thickness with radius and time using the inviscid, well-mixed film model
The ice profiles after five minutes with three different water temperatures (for \(\dot{m}=5.4 \times 10^{-3} \mathrm {kg s^{-1}}\) and \(H=6700\,\mathrm {Wm^{-2}K^{-1}}\)) are shown in Fig. 6. New and original ice has melted in the centre where heat transfer from the water dominates over conduction through the ice. The depth of the dip in the middle depends on the initial water temperature with warmer water melting more ice. However, by approximately \(2 \, l\) the flow at all initial temperatures has cooled such that the rate of conduction through the ice is significantly greater than the rate of heat transfer from the water to the interface so that the ice thickness is almost constant with radius.
Fig. 6
Radial ice thickness profile from the inviscid, well-mixed film model for different water temperatures
2.3.2 Inviscid flow with a developing thermal boundary layer
Cases where the flow is still turbulent but where a thermal boundary layer develops on the ice–water interface in the film can be described by the thermal boundary layer equation for radial flow from a point source
As with flow over an isothermal plate, for example, the temperature is constant outside of the thermal boundary layer (Incropera et al. 2007). Taking the order of the temperature changes in the system to be \(\Delta T_0\) and the thermal boundary layer thickness to be \(\delta _{if,TBL}\)
Interestingly, this scaling analysis results in the same dependence on distance from the source as the equivalent channel flow in Pantling et al. (2024). As the flow spreads outwards, more enthalpy is drawn in from above the thermal boundary layer, but this is balanced by the increase in cooling at the interface. The heat flux through the thermal boundary layer is given by
The heat transfer through the water (13) and conductive heat flux through the ice (3) can be substituted into equation (1) for the rate of change of ice thickness which can then be integrated to give an equation describing the development of the ice profile
Figure 7 shows progression of the ice profile (for \(v=1.2\,\mathrm {ms^{-1}}\) and \(T_\textrm{i0}=-10\)\(^\circ\)C). Freezing is only predicted for very short times when conduction is strong, but it is quickly overcome by heat transfer from the water at all radii. The dashed black line indicating peak the freezing to melting transition is very steep showing that heat transfer from the water dominates even as the thermal boundary layer thickens. The white region of no ice thickness change is similarly steep, and both progress linearly with time for this model. As time progresses the original ice rapidly melts even towards the edge of the ice disc where the thermal boundary is at its thickest.
Fig. 7
Contour map of the change in ice thickness with radius and time using the inviscid flow, thermal boundary layer model
The ice profiles for three temperatures, again for \(v=1.2 \; \mathrm {ms^{-1}}\) and \(T_\textrm{i0}=-10\)\(^\circ\)C, are shown in Fig. 8 (note the different vertical scale). The model predicts melting across the whole ice disc for all three initial water temperatures after five minutes. Close to the centre of the disc, the melting tends very steeply to infinity for any initial water temperature and even 0.5 \(^\circ\)C water is predicted to melt \(\sim 1.5\,\textrm{mm}\) of original ice at the edge of the disc. At very large radii, the thermal boundary layer would thicken sufficiently that the heat transfer from the water would be exceeded by conduction leading to freezing.
Fig. 8
Radial ice thickness profile from the inviscid flow, thermal boundary layer model with different water temperatures
2.4 Viscous radial flow with a developing thermal boundary layer
We now consider the case of a laminar, viscous flow. The Navier–Stokes equations govern the velocity profile of the water in both the radial and vertical directions, using the thin film approximation (Oron et al. 1997)
where p is the static pressure, g is the acceleration due to gravity and \(\mu _\textrm{w}\) is the dynamic viscosity of water. Noting that hydrostatic pressure drives the flow, solving (16) and applying atmospheric pressure (\(p_\textrm{a}\)) at the surface (\(z=h\)), one may write
$$\begin{aligned} p = p_\textrm{a} + \rho _\textrm{w} g (h - z) \;. \end{aligned}$$
(17)
The pressure profile in the radial direction can then be determined by substituting (17) into (15) to give the change of fluid height with radius, \(\textrm{d} h/\textrm{d} r\), which is independent of z. Hence, integrating twice with respect to z with boundary conditions of no slip at the water–ice interface (i.e. \(u=0\) when \(z=0\)) and no shear at the free water surface (i.e. \(\partial u/\partial z=0\) when \(z=h\)) gives the water velocity profile
where \(\nu _\textrm{w}\) is the kinematic viscosity of water.
With the assumptions that the flow has reached steady state and that the rate of mass transfer to/from the flow by melting/freezing is much smaller than the mass flux of water injected, the volume flux, Q, is constant for all times and radial positions and is defined by
$$\begin{aligned} Q= & \int \limits _0^h 2\pi r u\,\textrm{d}z \end{aligned}$$
From (20), the height profile as a function of radius can be calculated by rearranging and then integrating between the height h at radius r and the finite radius of the disc R where the water flows off the edge of the surface, and hence, the height is zero to give
Substituting Eq. (21) and its derivative with respect to r back into the equation for the velocity profile of the water (18) gives an explicit velocity profile equation
For scaling analysis of the thermal boundary layer equation, the scale of the characteristic velocity must be found. Evaluating Eq. (22) over the scale of the thermal boundary layer thickness \(z \sim \delta _{vf,TBL} \ll h\) gives
Interestingly, the heat transfer from the water has very little dependence on the flow rate (\(q_\textrm{w} \propto 1/Q^{1/6}\)); hence, the ice profile will also only have a weak dependence on the flow rate. The ice profile for the viscous flow with a thermal boundary layer is calculated by substituting (3) and (27) into (1) and then integrating leading to
The ice profile development for the viscous, thermal boundary layer model (Fig. 9), for \(\kappa = 1.88 \; \mathrm {ms^{-1}}\) and \(T_\textrm{i0} = -10\)\(^\circ\)C, shows a much larger region of freezing than the inviscid, thermal boundary model (Fig. 7). The freezing to melting transition, indicated by the dashed black line, follows \(r \propto t^{3/4}\); hence, the freezing persists for longer. Melting is predicted towards the centre, but this is limited to a smaller area, and even after \(300\,\textrm{s}\) there is no net melting at a radius of \(2.5\,\textrm{cm}\). Due to the rapid changeover between melting and freezing, the crossover region from freezing to melting is much smaller than with the inviscid thermal boundary layer (Fig. 7). The behaviour of the viscous, thermal boundary layer model is important as it best predicted the experimental data in the case of channel flow (Pantling et al. 2024) and so is expected to be the most accurate model for the case of radial flow too.
Fig. 9
Contour map of the change in ice thickness with radius and time using the viscous, thermal boundary layer model
The ice profiles resulting from the viscous, thermal boundary layer model (for \(\kappa = 1.88\,\mathrm {ms^{-1}}\) and \(T_\textrm{i0} = -10\)\(^\circ\)C) are shown in Fig. 10. The thermal boundary layer is thicker than in the laminar flow case (Sect. 2.3.2) leading to a smaller rate of heat transfer from the water and less melting. The steep gradient of \(\eta\) from melting very close to the origin and freezing further out shows that the thicker viscous, thermal boundary layer results in much weaker heat transfer from the water to the ice–water interface. The thicker thermal boundary layer results in the predictions for 0.5, 1.0 and 1.5 \(^\circ\)C water being closer together than for the inviscid flow, thermal boundary layer model. As the thermal boundary layer is so much thicker, the heat transfer from the water is significantly smaller resulting in freezing being predicted for all three temperatures at radii \(\gtrsim 6\,\textrm{cm}\).
Fig. 10
Radial ice thickness profile from the viscous, thermal boundary layer model with different water temperatures with \(\kappa = 1.88\,\textrm{ms}^{-1}\)
2.5 Comparison of models for the development of the radial ice profile
The predictions of the resulting change in ice thickness as a result of water flowing axisymmetrically from a point source are shown for 1 \(^\circ\)C water (and other physical parameters defined previously) in Fig. 11. The inviscid, well-mixed film model has a bulk water temperature that decreases exponentially with radius and a heat transfer from the water that reduces similarly. The model predicts melting up to a radius of around \(0.04\,\textrm{m}\) with a finite dip in the centre and almost constant growth towards the edge of the ice disc. The inviscid flow, thermal boundary layer model predicts at least \(4\,\textrm{mm}\) of melting across the entire ice disc with the melting at radii less than \(0.025\,\textrm{m}\) exceeding \(9.5\,\textrm{mm}\). The viscous flow, thermal boundary layer model has melting that tends to infinity at the centre of the disc, but the very steep gradient leads to a relatively small degree of melting beyond \(0.01\,\textrm{m}\). Here the gradient becomes more shallow as the thermal boundary layer thickens and there is a much less abrupt transition from negative to positive ice thickness change, which occurs at \(\sim 0.03\,\textrm{m}\).
Fig. 11
Radial ice thickness profiles using the three models after five minutes
The inviscid film models break down for \(r < r_0\) as the heat transfer is not modelled for the impact region. Similarly the viscous, thermal boundary layer model breaks down as \(r\rightarrow 0\) because \(h \rightarrow \infty\) as a result of assuming the flow injection area is infinitesimally small, which is non-physical. Thermal buoyancy driven convection is neglected throughout which may break down near the centre where the film is at its thickest. It should also be noted that the thermal boundary layer models are only valid when the thermal boundary layer thickness is less than the fluid height (i.e. \(\delta _\textrm{if,TBL}<h\) and \(\delta _\textrm{vf,TBL}<h\)) which will break down at large radius.
3 Experiments
3.1 Experimental set-up
Experiments were conducted in the Temperature Controlled Laboratory (referred to as the cold room hereafter) at the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the University of Cambridge, see Fig. 12. The cold room provides ambient temperatures down to \(-40 \pm 1\)\(^\circ\)C making it ideal for experiments to validate the models derived in Sect. 2. An experimental rig, shown in Fig. 13, was designed to hold a disc of fresh ice, inject fresh water onto its centre and measure its profile using a digital micrometer. Ice was frozen in a metal circular tin with a thin acrylic base inserted, surrounded on the sides and top by polystyrene to act as an insulator. Insulating the tin on the sides and top enabled heat transfer to occur primarily at the base of the container. Therefore, the ice grew from this surface upwards, preventing two undesirable effects observed when the container was not insulated. Firstly, air bubbles could escape upwards from ice freezing on the bottom leading to more solid ice, free from large voids. Secondly, freezing on the sides was reduced; hence, a hump in the middle of the disc did not form from water freezing from the sides, expanding and then forcing the remaining water upwards. It was necessary to melt and refreeze the ice (by applying warm water) in order to obtain a reasonably flat surface. The ice was smoothed a final time before the experiment by applying a room temperature flat disc of aluminium which melted the top surface until flat. These processes gave an ice surface that was visually deemed to be sufficiently smooth to be suitable for the experiments (deformations \(\lesssim 0.5\,\textrm{mm}\)). During the experiments, the rig was placed into a large bucket to collect the run off water.
Fig. 12
Photograph of the DAMTP cold room used for the experiments
Once the ice had been suitably smoothed, water was pumped from a container onto the centre of the ice disc using a Watson Marlow 505 S Peristaltic Pump. The temperature of the water in the container was measured in order to avoid placing a thermocouple at the pipe exit and disturbing the flow. Separate experiments were done to measure the water temperature difference between the pipe inlet and exit in the cold room. The relationship established determined the container water temperature which needed to be used for a given desired water temperature injected onto the ice. The flow rate of the pump was \(5.4\,\mathrm {{cm}^3 s^{-1}}\), with a pipe of diameter \(4.5 \; \textrm{mm}\) from a height \(7.5 \; (\pm 1) \; \textrm{cm}\) giving \(v=1.2 \; \mathrm {ms^{-1}}\). The \(11 \; \textrm{cm}\) radius of the disc was chosen such that the water flowed as a thin film over the whole surface without forming rivulets. An experiment time of \(5 \; \textrm{minutes}\) was chosen with the ice frozen to an initial depth of at least \(20 \; \textrm{mm}\). After the experiment was finished, all excess water was poured off of the ice and it was allowed to cool. The ice profile was then measured using a micrometer mounted to slide horizontally (as shown in Fig. 13), the uncertainty in this measurement is estimated to be \(\pm 0.5 \; \textrm{mm}\) and the uncertainty of the radial position is estimated to be \(\pm 1 \; \textrm{mm}\).
4 Results
Three experiments were conducted injecting water at 0.5, 1.0 and 1.5 \(^\circ\)C (average temperature ± 0.1 \(^\circ\)C between three repeats) onto ice discs at -10 \(^\circ\)C (cold room temperature, maintained to ± 1 \(^\circ\)C). The water was pumped over the surface for five minutes and flowed radially outwards, and although the flow was not perfectly axisymmetric, it was observed to cover the whole disc in \(\sim 20-30 \; \textrm{s}\). Any experiment where the surface was not completely covered was discarded. Assumptions in the models that advection of the water greatly exceeds conduction in the water/ice, that the ice is semi-infinite, and that the flow is in steady state are appropriate for the experiments since \(\alpha _\textrm{w}/v^2 = 9.2 \times 10^{-7} \; \textrm{s}\), \(\alpha _\textrm{i}/v^2 = 8.2 \times 10^{-6} \; \textrm{s}\) and \(d^2/\alpha _\textrm{i} = 340 \; \textrm{s}\) giving \(\alpha _\textrm{w}/v^2< \alpha _\textrm{i}/v^2 \ll t < d^2/\alpha _\textrm{i}\). It takes over \(3.5 \; \textrm{hours}\) for conduction through the ice to reduce such that it is equal to convective heat transfer to the air which can therefore be reasonably neglected (based on a heat transfer coefficient of \(10 \; \mathrm {W m^{-2} K^{-1}}\) from Engineers Edge (2015) and a temperature difference of 10 \(^\circ\)C). The vast majority of the pumped water flowed off the ice disc in the experiments, changes in ice thickness were \(\mathcal {O}(1 \; \textrm{mm})\) giving a total volume change, \(\sim \pi R^2 \eta\), which is \(\sim 30\) times smaller than the pumped water volume, Qt, at the end of the experiment. Hence, the approximation of constant mass flow rate in the models is also valid. The pipe exit velocity and free fall height \(h_0\) were sufficient to neglect gravity in the film as \(v^2/2\,g\Delta (h+\eta ) \approx 30 \gg 1\). In the experiments, the Reynolds number, \(Re = vh/\nu _\textrm{w}\), is \(\sim 670\), much smaller than the critical Reynolds number, \(Re_\textrm{c} \approx 2000\) (Schlichting and Gersten 2017); hence, the flow is laminar and should be best represented by the viscous model. For the viscous thin film approximation to be valid, the reduced Reynolds number \((v h/\nu _\textrm{w})(h/r)\) must be small. Given the experimental parameters, this drops to roughly 1/2 by \(3 \; \textrm{cm}\); at smaller radii, the approximation will be less valid. The viscous model’s thermal boundary layer thickness exceeds the predicted fluid depth at a radius of \(0.082 \; \textrm{m}\), at this point the model breaks down and the ice thickness change will be predicted less accurately.
The experimental results are analysed with respect to the predictions from the three models developed in Sect. 2. At the centre of the ice disc, some modelling assumptions break down; hence, none of the models captures the observed bowl shape of the ice at the centre of the disc. However, outside of this region the data closely match with the predictions from the viscous, thermal boundary layer model.
4.1 Results for 0.5 \(^\circ\)C water
The experimental results for 0.5 \(^\circ\)C water are shown in Fig. 14. As expected, there is some small net melting at the centre of the disc with freezing observed further away from the centre. The results show the slight asymmetry of the freezing, particularly in Experiment 1 where the left-hand side has increased ice thickness (compared to the right). This suggests slower flow on that side and hence a smaller heat flux from the water so that it cooled and froze faster. The inviscid, well-mixed film model is similar in magnitude to the data, but fails to capture its shape. The inviscid flow, thermal boundary layer model greatly over-predicts the melting for all radii. The viscous, thermal boundary layer model captures both the magnitude and shape of the experimental data well for almost all radii, except in the very centre where the model is based on a point source, whereas in reality there is a finite radius of supply.
Fig. 14
Comparison of the experimental results and model predictions for the change in ice thickness when injecting 0.5 \(^\circ\)C water onto the ice disc
Figure 15 shows the results from experiments with 1.0 \(^\circ\)C water. The experimental data are more repeatable than with 0.5 \(^\circ\)C because the uncertainty in measurement of supply temperature (\(\pm 0.1\)\(^\circ\)C) is a smaller proportion of the bulk temperature difference between supply and freezing temperature. In Experiment 3, there is a small region with larger ice growth, again suggesting that there was slower flow in this region leading to reduced heat flux from the water and more freezing. Again the inviscid, well-mixed film model fails to predict the experimental data predicting the onset of melting much further from the centre than is observed. The inviscid flow, thermal boundary layer model again predicts significant melting across the whole disc which was not observed. The majority of the experimental data match closely to the viscous flow, thermal boundary layer model, slightly under-predicting the amount of freezing at the disc’s edge.
Fig. 15
Comparison of the experimental results and model predictions for the change in ice thickness when injecting 1.0 \(^\circ\)C water onto the ice disc
The results for 1.5 \(^\circ\)C water being injected onto the ice are shown in Fig. 16. The data are again very repeatable between all three experiments, and there is very little to indicate that the flow was not axisymmetric as the measurements along the two radii in each experiment align closely. Net melting is observed for a larger region at the centre of the disc as expected, but there is a similar amount of freezing towards its edge. The two inviscid models do not match to the data. The viscous flow, thermal boundary model again slightly under-predicts the freezing towards the edge of the disc, but matches well to the data. In these experiments, the melting at the centre of the disc was so deep it was beyond the measuring range of the micrometer (\(\sim 15\,\textrm{mm}\)).
Fig. 16
Comparison of the experimental results and model predictions for the change in ice thickness when injecting 1.5 \(^\circ\)C water onto the ice disc
The ice thickness change, \(\eta\), can be non-dimensionalised by the characteristic length scale for diffusion in the ice, \(\sqrt{\alpha _\textrm{i} t}\), giving
The equation is now of the form \(\overline{\eta }_\textrm{vf,TBL} = \frac{2}{\sqrt{\pi }} St- \phi\) where St is the Stefan number which is constant (for constant initial ice temperature) and represents the dimensionless freezing due to conduction through the ice. The dimensionless term \(\phi\) encapsulates the melting due to heat transfer from the water which is dependent on time, the initial water temperature and the radial position. The values of St and \(\phi\) are given by the following expressions
Non-dimensionalising allows the data from all temperatures to be analysed simultaneously, and a linear relationship on the axis would suggest a viscous, thermal boundary layer model is appropriate. A close match of the y-intersect would suggest that the conductive heat transfer through the ice has been modelled reasonably accurately and parallelism to the gradient of the model would indicate that the thermal boundary layer thickness in the model is appropriate. The experimental data, along with a line of best fit, are plotted against the viscous, thermal boundary layer model in Fig. 17. For each temperature, an average has been taken of each radial direction across the three experiments (giving six radial data points) in order to simplify the graph. The uncertainty of the radial position (\(\pm 1 \; \textrm{mm}\)) and initial water temperature (\(\pm 0.1\)\(^\circ\) C) have been combined for the uncertainty in the non-dimensional radius, \(\phi\), by adding percentage uncertainties (\(\% \varepsilon _{\phi } = \% \varepsilon _{\Delta T_\textrm{w}} + 2/3 \, \% \varepsilon _{r}\)).
Fig. 17
Comparison of the non-dimensionalised experimental results and model predictions for the change in ice thickness as a function of non-dimensional radius
The experimental data show a reasonably linear relationship on the non-dimensional axis. The experimental data and predictions from the viscous, thermal boundary layer model are in reasonably close agreement. The line of best fit through the data intersects the vertical axis higher than the prediction from theory suggesting that the rate of conduction through the ice (the Stefan number) has been underestimated. The ice cannot be colder than the cold room temperature, but it is possible the temperature control was consistently inaccurate and there is a systematic error in the ice temperature. Meanwhile, the shallower gradient of the line of best fit suggests the thermal boundary layer thickness has been underestimated; the scaling analysis in equation (25) could have included an \(\mathcal {O}(1)\) scaling factor which propagated through the analysis could account for the difference in gradient. However, the overall trend of the data does match to the model indicating that the underlying physics has been captured. The close match between the experimental data and viscous model was also found in Pantling et al. (2024) as is observed in these experiments, indicating that this model is most suitable for the laminar flow experiments conducted.
6 Conclusions
The purpose of this paper was to develop and investigate the validity of a model predicting the change in ice thickness when water flows axisymmetrically as a thin film over an ice disc. The work built on that of Pantling et al. (2024) for flow over ice down an inclined channel. Equivalent models for the three flows (a vertically well-mixed flow; a uniform flow with a developing thermal boundary layer; and a viscous shear flow with a developing thermal boundary layer) were developed for the axisymmetric radial flow over ice. Experiments were conducted at various water temperatures and analysed with respect to predictions from the modelling. All the models break down and fail to capture the behaviour at the very centre of the ice disc. However, outside of this small region the experimental data at all three initial water temperatures (0.5, 1.0 and 1.5 \(^\circ\)C) are reasonably accurately predicted by the viscous flow, thermal boundary layer model. Non-dimensional axes were chosen to present the data from all the experiments simultaneously; the linear relationship shown confirms that a viscous flow with a thermal boundary layer and the model developed is appropriate.
For the application of thickening Arctic sea ice, there are various extensions to this work that could be carried out as described in Pantling et al. (2024). The modelling in this paper expands the two-dimensional flow into three dimensions as suggested. Further work should consider longer time and length scales where rivulets may form before the flow freezes and heat transfer to the atmosphere by advection/radiation are important. The low Reynolds number may not be representative of the conditions at the field scale where water speeds and distances can be expected to be multiple orders of magnitude larger; hence, the Reynolds number may be super-critical. In this case, the inviscid models may be more appropriate. The modelling here is in fact more appropriate to naled ice (aufeis) growth, where water seeps over an ice surface (Schohl and Etiema 1986). However, in that case heat transfers to the atmosphere rather than in the ice or water dominate. For the application of thickening sea ice, the salinity of the water and ice must be considered; this may cause dissolution of the original ice (if it is less salty) and the saline water will freeze into a mushy layer (see Worster (2000)). When sea ice is treated as a mushy layer, all the additional complexity can be accounted for including porosity/permeability, salinity and temperature. The rejection of brine during the freezing process and after the water has frozen may be crucial to the strength of the formed ice and its durability. Separately, sea ice is often covered in snow, and hence, the flow of salt water through snow must be researched for the ice thickening technique to be understood.
Acknowledgements
The authors would like to thank Stefan Savage at the Department of Engineering and Mark Hallworth at the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, for support in constructing laboratory equipment and providing access and safety monitoring to the cold room.
Declarations
Conflict of interest
The authors have no financial or proprietary interests in any material discussed in this article.
Ethical approval
Not applicable.
Consent for publication
Not applicable.
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