# Ice melting in saltwater: formation of a stratified system in laboratory experiments

Brivaël Collin, Louis-Alexandre Couston, Sylvain Joubaud, Romain Volk

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Brivaël Collin, Louis-Alexandre Couston, Sylvain Joubaud, Romain Volk. Ice melting in saltwater: formation of a stratified system in laboratory experiments. 2025. hal-05254012

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Ice melting in saltwater: formation of a stratified
system in laboratory experiments

Brivaël Collin¹*, Louis-Alexandre Couston², Sylvain Joubaud¹,
Romain Volk¹

¹ENS de Lyon, CNRS, LPENSL, UMR5672, 69342, Lyon cedex 07,
France.

²Université Claude Bernard Lyon 1, ENS de Lyon, CNRS, LPENSL, UMR5672, 69342, Lyon cedex 07, France.

*Corresponding author(s). E-mail(s): brivael.collin@ens-lyon.fr;
Contributing authors: louis.couston@ens-lyon.fr;
sylvain.joubaud@ens-lyon.fr; romain.volk@ens-lyon.fr;

Abstract

We report results on the melting of a tubular ice cuboid in saltwater heated from below. We develop a meter-scale experimental setup capable of detecting the ice-water interface and measuring the temperature, salinity, and velocity fields simultaneously, offering a route to quantitative understanding of multi-component systems with phase changes. We show that during melting, a two-layer system can form spontaneously due to the competition between the destabilizing temperature gradient and the stabilizing salinity gradient. The upper layer, close to the ice, is quiescent while the lower layer is convective. Such a layered system displays a large vertical gradient of kinetic energy, which is characterized by means of a novel multi-step PIV approach that allows us to increase the signal-to-noise ratio in the velocity fields obtained by particle image velocimetry. Using a staircase profiling method, we enable reliable and non-intrusive temperature and salinity measurements. Using the measured salinity and temperature profiles of the two-layer system, we derive a model for the melting of ice which compares quantitatively well with experimental data.
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Keywords: Ice melting, Stratified flow, Buoyancy-driven flow

Contents

<table><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>2</td><td>Description of the experiment</td><td>4</td></tr><tr><td>2.1</td><td>Underlying principle</td><td>4</td></tr><tr><td>2.2</td><td>Clear Ice Making</td><td>5</td></tr><tr><td>2.3</td><td>Heat flux measurements using shadowgraphy</td><td>6</td></tr><tr><td>3</td><td>Multi time-step PIV for a two-layer system</td><td>9</td></tr><tr><td>4</td><td>Measuring Temperature & Salinity Profiles</td><td>12</td></tr><tr><td>4.1</td><td>Probe Calibration</td><td>12</td></tr><tr><td>4.2</td><td>Staircase profiling</td><td>14</td></tr><tr><td>5</td><td>Discussion and conclusions</td><td>16</td></tr></table>
# 1 Introduction

The physics of multiphase binary materials, featuring phase changes, is challenging to predict, yet is of key interest to many fields, including alloy formation [1], energy production [2] and polar oceanography [3]. The hydrodynamics, heat fluxes, composition and geometry of the liquid domain all evolve simultaneously, influencing each other through buoyancy, advection, diffusion and phase change. Thus, deciphering the physics of multiphase binary materials with phase changes typically requires many measurements, which have proved very challenging in the past for laboratory experiments [4–6].

The case of ice melting in saltwater, which is highly significant to polar oceanography, is particularly challenging because the underlying fluid dynamics is in the high Reynolds number regime [3]. The flow has a broad separation of temporal and spatial scales, which is difficult to capture in laboratory experiments and has prevented previous works from comprehensively characterizing the physics of ice melting [4]. For example, investigating the interaction of ice blocks with a saltwater flow, [7] only used sensors at fixed locations to measure temperature and salinity, [8, 9] limited their analysis to shadowgraph images, and [10] relied largely on particle image velocimetry (PIV) images. Numerical investigations have access to every piece of information about the flow, such that simulations of binary fluids with phase changes have experienced continued success in recent years thanks to the development of multiphase flow solvers [11–13]. However, numerical simulations are constrained in terms of physical parameters such as the diffusivity of scalars or momentum, and remain limited to moderate Reynolds or Rayleigh numbers. Thus, laboratory experiments are still critical to validate computational results and explore realistic regions of the parameter space.

Past laboratory experiments on ice melting in saltwater significantly improved our understanding of the melting of icebergs, ice shelves and sea ice, despite limited measurements. Notably, Martin & Kaufman [7, 14] made one of the first quantitative description of saltwater convection beneath a floating ice block, showing the interplay between the velocity $\vec{v}$, temperature $T$ and salinity $S$ fields. Using fixed thermocouples and conductimeters, they measured vertical profiles for temperature and salinity, which they suggested indicated a three-layer organization with diffusion, double-diffusive convection, and thermal convection dominating alternatively. Double diffusion is a weak form of convection, which arises from the fact that the diffusivity of temperature $\kappa_T$ is much larger than the diffusivity of salt $\kappa_S$, i.e. with $\kappa_T/\kappa_S \simeq 175.6$ [15]. More recent experiments have explored different configurations such as a floating horizontal ice block heated from the side [16], a vertical ice wall [17] or a fixed ice cylinder [18]. They have shown that depending on the geometry, boundary conditions, and control parameter values, the system can exhibit diffusive, convective, or double-diffusive behaviors and experience melting rates varying by multiple orders of magnitude. Laboratory experiments on ice melting in freshwater have also shown that there can be a large variety of melt-induced flow regimes depending on the water temperature. In fact, water density $\rho$ has an equation of state $\rho(S, T) \simeq \rho_0(1+\beta(S, T)S - \alpha(S, T)T)$ [19] with $\rho_0$ the reference water density, $\beta$ the haline contraction coefficient and $\alpha$ the thermal expansion coefficient, which is nonlinear. While $\beta$ is always positive, $\alpha$ can
be negative in the low salinity limit, such that cooling due to ice melting can lead to both buoyant and dense flows [9, 18].

Building on those works, we describe an experimental setup which allows us to characterize the main features of a saltwater flow interacting with a melting ice block in the case of strong stratification. Information on the density field and ice melting rate is obtained from shadowgraphy [20] and a custom PIV method using tuples instead of image pairs to reduce noise in velocity estimation [21, 22]. Shadowgraphy and PIV measurements are combined with temperature and salinity profiles using a moving thermistor and conductivity probe. This integrated approach enables a nearly complete system characterization and the development of a simple model predicting melt rates.

The paper is organized as follows. Section 2 presents the experimental setup, the techniques used to make clear, bubble-free ice, and the qualitative behavior of the system through shadowgraphy imaging. Section 3 discusses the PIV setup and the post-processing routines implemented to image a multiscale stratified flow. Section 4 presents our methodology to obtain temperature and salinity profiles. Finally, section 5 discusses how the observations collectively help understand and predict the formation and evolution of the system during melting into a stably-stratified layer overlying a convective zone.

# 2 Description of the experiment

## 2.1 Underlying principle

The experimental setup aims to investigate the melting of a freshwater ice cube, floating on top of hot, salty water, which we will refer to as “bulk water”. The experiment is performed in a closed Plexiglas tank measuring 40 × 40 × 20 cm, filled with saltwater of initial salinity $S_0$, and heated from below as shown in figure 1.a. To minimize unwanted thermal transfers, the entire experiment is housed in a cold room maintained at the same temperature as the forcing temperature of the heat source, denoted by $T_\infty$. Additionally, a polystyrene insulating lid is placed over the tank to reduce thermal transfers from the surrounding air to the top of the ice. The heat source is a metallic plate maintained at room temperature using a water circulation loop fed from a large 200 L water reservoir in thermal equilibrium within the cold room. Altogether, the setup ensures that all components of the system are at the same temperature and at rest prior to the introduction of the ice cuboid. The value $T_\infty$ is determined from two Pt-100 thermistors placed upstream and downstream of the heating plate.

The tank is filled up to $H_b = 37.5$ cm of salt water with adjustable salinity and temperature. For the experiment presented here, the initial salinity is set to $S_0 = 9.4$ g.L$^{-1}$ and the entire system (tank, room, bulk water) is initially in thermal equilibrium at $T_\infty = 10.7$ °C, such that the bulk water density is $\rho_b = 1.067$ g.cm$^{-3}$ with viscosity $\nu = 1.3 \times 10^{-6}$ m$^2$.s$^{-1}$ and thermal diffusivity $\kappa_T = 1.43 \times 10^{-7}$ m$^2$.s$^{-1}$. The experiment starts with the introduction of a floating ice cuboid initially thermalized at 0°C, which covers almost all the water surface area and extends about 10 cm in depth. The fast diffusion of the cold temperature anomaly from the ice-water interface into the bulk water generates an unstable thermal gradient with Rayleigh number
<figure><img src="image_3.png" /><figcaption>**Fig. 1** (a) Schematic of the experiment. As the floating ice melts, a meltwater layer forms on top of the salty bulk water, which is heated from below. (b) Typical shadowgraph of an experiment, displaying the layered structure. $T_∞$ = 10.7°C and $S_0$ = 9.4 g.L⁻<fnref n="1" />.</figcaption></figure>

$Ra_b = \rho_0 g \alpha (T_\infty - T_i) H_b^3 / (\nu \kappa_T) \approx 10^{12}$, where $T_i \approx 0°C$ is the temperature of the ice-water interface. The thermal gradient then leads to a convective flow with a typical velocity $v_b \approx 2.3 \text{ mm.s}^{-1}$ (see section 3).

The melting of ice is enhanced by the flow and a substantial amount of meltwater, initially solid, is released. Since the meltwater is fresh, its density $\rho_m(\approx \rho_0)^1$ is smaller than the bulk water density $\rho_b$. The two-layer meltwater-bulk water system is gravitationally stable, and prevails unless convection entrains the top meltwater layer into the bulk. The stratification and convection strengths can be compared by the convective Richardson number $Ri_c = (\rho_b - \rho_0)gH_m/(\rho_b v_b^2)$. In the present situation, we have $Ri_c \approx 1200 \gg 1$, such that the system is expected to maintain a fresh water layer on top of the bulk water with a strong temperature and salinity stratification. The predicted layering is observed in figure 1.b that displays a shadowgraphy image of the system during melting, with the formation of a two-layer system below the ice block clearly visible.

## 2.2 Clear Ice Making

We study the melting of an ice block with dimensions close to 40 × 20 × 10 cm, which almost fits the cross section of the tank. To prevent solid impurities, we use clear demineralized water. In order to avoid the presence of air bubbles in the ice during freezing, which may lead to the formation of an insulating air cushion below the ice during experiments, we employ the apparatus shown in figure 2.a and grow the ice from the bottom up. The mold is placed inside a conventional freezer at −18° C on a thermally conductive aluminum plate to enhance the heat transfer from the

<footnote n="1">¹The temperature effects on density being an order of magnitude lower than the salinity effects in the present situation, we reasonably approximate $\rho_m$ to the reference density $\rho_0$</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 2** (a) Ice-making apparatus used for our experiments. (b) Ice cuboids made using a regular freezing method from all sides (top) and a directional and agitated freezing method (bottom).</figcaption></figure>

bottom, while the other sides are insulated from the freezer using a polystyrene cap. In addition we use a small water pump placed at the top, near the interface, which creates a circulation that delays the freezing of the top interface and forces bubbles forming on the ice-water freezing front to detach and escape through the free surface. The result can be seen in figure 2.b where we compare a cuboid made using freezing from all sides (no pump, no cap, top image) and a cuboid made using our directional and agitated freezing method (bottom image).

Before each experiment, after freezing, the ice temperature is brought close to 0°C by allowing it to rest at room temperature (~20°C). The ice is shaped into a cuboid using a hand-held ski waxer and drilled to create a conduit for the temperature and salinity probe, which will be described in section 4.

## 2.3 Heat flux measurements using shadowgraphy

We image the ice cuboid using nearly parallel light and a large field of view to track ice volume changes (through melting) and gain information on the meltwater layer and its mixing with the bulk water through spatio-temporal variations of the refractive index [23], which depends on local temperature and salinity.

Our optical setup builds on the work by Machicoane [24] and is illustrated in figure 3. We position a small light-emitting diode (LED) near the focal point of a low-cost plastic Fresnel lens (diameter 36 cm and focal length 45 cm) to produce a nearly collimated beam with a divergence angle estimated as 1.9°. The shadowgraph images are then visualized on a semi-transparent screen 3 cm downstream of the tank and recorded using a 12-bit Basler acA2440-75um camera with a resolution of 2464 × 2056 pixels, corresponding to a pixel/mm ratio of 5.99 and a spatial resolution of 0.167 mm. As shadowgraphy is combined with PIV measurements using a green laser, we use a
<figure><img src="image_5.png" /><figcaption>**Fig. 3** Lateral view of the experimental setup showing the blue LED source, the Fresnel lens, the semi-transparent screen, and the shadowgraphy camera on the right. The PIV camera (on the left) is placed below the shadowgraphy light beam, in front of the tank, to capture the PIV images produced with the green laser. A salinity and temperature sensor is placed on a vertical translation plate to record profiles during the experiment.</figcaption></figure>

blue 425 nm light source in combination with a low-pass interference filter (with a cutoff wavelength of 488 nm) placed in front of the camera lens. Images are recorded at a frame rate of 0.1 Hz, which is a compromise between temporal resolution and data storage as a melting experiment lasts several hours, with an exposure time ($\tau = 3$ ms) adjusted to maximize contrast without any saturation.

We apply a standard contour detection algorithm<fnref n="2" /> to the shadowgraphs (such as the one displayed in figure 1.b) to find the solid/liquid interface. Even though the interface is initially horizontal, a slight tilt can emerge as a result of heterogeneous heat transfers (see figure 1.b and discussions in section 5). We account for the tilt of the ice-water interface by defining two cropping rectangles on both sides of the image and we define the mean ice-water interface position and ice-water interface drop as the average and difference between the positions measured through the two subdomains, i.e., $y_i(t) = (y_1(t) + y_2(t))/2$ and $d_i(t) = (y_1(t) - y_2(t))/2$, respectively.

A series of such cropped images (right rectangle) taken every $10^4$ seconds during melting is displayed in figure 4. The transparent ice cuboid can be seen at the top, and the ice-water interface, which is clearly visible, moves upward as melting proceeds. We show the measured relative position of the ice-water interface $y_i(t) - y_i(0)$ as a function of time in figure 5. The gray region surrounding the solid black line corresponds to the interval $[\min(y_1, y_2) - e, \max(y_1, y_2) + e]$, where $e = 2$ mm is the estimated error regarding the detection of the interface. Since the ice is floating on top of its liquid meltwater phase<fnref n="3" />, the interface movement $y_i(t) - y_i(0)$ is proportional to the total height of ice that has melted, $h(t)$. Assuming uniform melting, one has the relation $y_i(t) - y_i(0) = (\rho_i/\rho_0)h(t)$, where $\rho_i = 920$ kg.m$^{-3}$ is the ice density, from which the melting rate $\dot{h} = dh/dt$ can be extracted.

<footnote n="2">²Note that the shadow of the PIV camera is visible together with a bright line corresponding to the loose transition between meltwater and bulk water.</footnote>

<footnote n="3">³The temperature and salinity effects on the buoyancy of the ice are small at the beginning of the experiment [$\rho(S_0, T_\infty) - \rho_0$]/$\rho_0 \approx 0.1$ and become more and more negligible as the ice melts and floats in its own meltwater.</footnote>
<figure><img src="image_6.png" /><figcaption>Fig. 4 Chronophotography of the shadowgraphs obtained for an experiment with $T_{\infty} = 10.7^{\circ}C$ and $S_0 = 9.4 \text{ g.L}^{-1}$. As the ice melts, a two-layer structure appears in the liquid, featuring two different kinds of dynamics.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5 Relative mean position of the ice interface over time $y_i(t) - y_i(0)$. The experimental data is shown by the solid black line. Two predictions using a fully convective model or a fully diffusive model for the heat transfer are shown by the dotted light blue line and the green dashed line, respectively. The control parameters are $T_{\infty} = 10.7^{\circ}C$ and $S_0 = 9.4 \text{ g.L}^{-1}$.</figcaption></figure>

The melting rate is a key geometric variable that is directly linked to the heat flux from the water to the ice, denoted by $Q_w$. Energy balance at the solid/liquid interface indeed requires

$$SL\rho_i\dot{h} = Q_w - Q_i \quad (1)$$

where $L = 3.3 \times 10^5 \text{ J.kg}^{-1}$ is the latent heat of fusion of freshwater, and $Q_i$ is the conductive heat flux inside the ice [25]. As the ice cuboid was thermalized at $0^{\circ}C$ before the experiment, diffusion is negligible in the ice. Thus, equation (1) reduces to $SL\rho_i\dot{h} = Q_w$ [24] and the interface position is linked to the time history of $Q_w$, which can be parameterized assuming diffusion or convection are dominant.

Assuming that the heat flux $Q_w$ is purely diffusive, one can use the implicit analytical solution derived by Woods et al. [26], which applies for a binary melt (here
salt water) at rest and in contact with a pure solid (here freshwater ice). Using our notations, the solution reads $h_{diff}(t) = 2\lambda\sqrt{\kappa_T t}$, where $\lambda = 7.62 \times 10^{-2}$ is the dimensionless phase-change rate, which depends on the problem parameters through an implicit equation. The interface displacement predicted assuming diffusion is plotted on figure 5 as the green dash-dotted line. It underestimates the melting rate by at least one order of magnitude, suggesting that the heat transfer may rather be convective. This result is consistent with the shadowgraphs displayed in figure 4, which show small-scale turbulent structures in the bulk water.

An alternative parameterization for the heat flux consists in leveraging classical thermal convection laws, i.e. discarding salinity-induced stratification effects. Given the typical Rayleigh number of our experiment $Ra_b$ (defined in section 2.1), it is appropriate to use the empirical prediction from King et al. [27] for the Nusselt number, i.e., $Nu = 0.16Ra^{2/7}$. The heat flux is obtained from the definition $Nu = Q_w/Q_w^{diff}$, which compares the actual heat flux to the diffusive heat flux $Q_w^{diff} = Sk(T_\infty - T_i)/y_i$, where $k = 0.6 \text{ W.m}^{-1}.\text{K}^{-1}$ is the water heat conductivity and $T_i = 0^{\circ}\text{C}$ for the interfacial temperature$^4$. The melt rate is obtained using again the energy balance equation at the interface with the new expression for $Q_w$:

$$ \dot{h}_{conv} = 0.16k \frac{(T_{\infty} - T_i)^{9/7}}{\mathcal{L}\rho_i} \left[ \frac{\alpha g}{\sqrt{y_i \kappa_T \nu}} \right]^{2/7} \quad (2) $$

Assuming $y_i^{-1/7}(t) \simeq y_i^{-1/7}(0)^5$, the melt rate $\dot{h}_{conv}$ is constant with time, meaning we have $h_{conv}(t) = \dot{h}_{conv}t$. The prediction for the ice-water interface position based on the fully-convective scaling law for the heat flux is plotted in figure 5 as the dotted blue line. The new model overestimates melting by at least one order of magnitude, such that the interface moves much faster than in the experiment.

The diffusive and convective laws are unable to predict the melt rate by at least one order of magnitude. A thorough inspection of the fluid dynamics, which is highly heterogeneous (as suggested by the shadowgraphy results), is required.

## 3 Multi time-step PIV for a two-layer system

The Particle Image Velocimetry (PIV) setup, shown in figure 3, consists of a green laser sheet created with a CW Laser (Opus 532 from Laser quantum, 5W@532 nm) and a Powell lens, which illuminates a vertical plane inside the tank. The laser sheet is positioned parallel to both the rear wall of the tank and the shadowgraphy screen to optimize visibility and minimize distortions. Before the experiment, the bulk water is seeded with 8 µm poly-amide particles (density $\rho_p = 1.1 \times 10^3 \text{ kg.m}^{-3}$), ensuring that their sedimentation velocity ($\sim 10^{-5} \text{ m.s}^{-1}$) remains negligible compared to the characteristic flow velocity ($\sim 10^{-3} \text{ m.s}^{-1}$). Empirical observations suggest that mixing between the bulk water and the meltwater layer is sufficient to seed the top

<footnote>$^4T_i$ is taken equal to the ice-water equilibrium interface in freshwater for the purpose of our scaling, since we expect that $T_i - T_\infty$ experiences negligible salinity-induced changes as long as $T_\infty$ is on the order of one Celsius degrees or more.</footnote>

<footnote>$^5[y_i^{-1/7}(0) - y_{i,max}^{-1/7}]/y_i^{-1/7}(0) \simeq 0.018$</footnote>
layer during experiments (cf. figure 6.a). Sequences of images are recorded with a second Basler acA4096-30 um camera with a resolution of 4112 × 2176 pixels and a large field of view 388 × 224 mm in the horizontal and vertical directions, corresponding to a pixel-to-mm ratio of 5.99 and a spatial resolution of 0.167 mm. As the laser plane and shadowgraphy plane are parallel, a joint calibration is performed with vertical positions measured from the bottom of the tank, and horizontal positions from the left side of the tank. We are then able to locate the ice-water interface position on the PIV images at any time, using the shadowgraph data, allowing us to create an evolving mask on said images to remove the ice domain from the velocity field and only consider the displacement in the liquid part of the experiment.

The experiment exhibits several time scales from the eddy turnover time of the flow structures up to the melting time of the ice cuboid, so that we set the exposure time to $\tau_P = 15$ ms to ensure maximum contrast without any motion blur and record tuples of $n = 10$ images, separated by a time interval $\delta t = 100$ ms, every 10 seconds. Recording series of images in bursts ensures that we can set an optimal value for the time interval for the PIV both in the bulk water layer where thermal convection occurs and in the meltwater layer where fluid motions are expected to be weak. PIV computations are conducted in Python using the `pyciv` and `fluidimage` libraries [28, 29]. The computation is done using an initial interrogation window of 64 × 64 pixels and then using a second pass with a window of 32 × 32 pixels. For each point, we use a correlation threshold of 0.3 below which the results are discarded.

The 2D displacement field $d\vec{X}_j$, and the correlation field $c_j$, are computed for image pairs formed between the first and the $j + 1$-th image in the tuple (figure 6.b). We expect that each computed velocity field $\vec{v}_j(x, y, t) = d\vec{X}_j/jdt$ ($j \in [1, n-1]$) corresponds to the same instantaneous physical velocity field, since the dynamics of the system is slow enough and $dt$ is short. Our database thus embeds a high level of redundancy, which can be used to improve the signal to noise ratio in all regions of the field of view. We test three different techniques and assess whether or not they improve the measurement compared to the reference case, which is based on the velocity field $\vec{v}_1$ computed from the (1, 2) image pair. We apply each method to each $(x, y, t)$ coordinate independently.

* The first method consists in selecting the image pair with the highest correlation coefficient for the considered location, i.e.

$$ \vec{v}_{\text{cmax}}(x, y, t) = \frac{d\vec{X}_j}{jdt}, \quad \text{where } c_j = \max\{c_k : k \in [1, n-1]\}. \qquad (3) $$

* The second method consists in taking the weighted average of all measured displacements, yielding

$$ \vec{v}_{\text{avg}}(x, y, t) = \frac{1}{\sum_{j} c_j} \sum_{j=1}^{n-1} c_j \frac{d\vec{X}_j}{jdt}, \qquad (4) $$
<figure><img src="image_8.png" /><figcaption>Fig. 6 (a) Example image used for the PIV. The inset shows that the top meltwater layer is seeded with particles originating from the lower bulk water layer. (b) Adaptive timestep PIV principle. We use a tuple of n = 10 images, rather than a single pair, to compute the displacement field and correlation (dX̃j, c_j, jdt).</figcaption></figure>

with displacements associated with a correlation smaller than 0.3 replaced by zero.

• The third method is directly inspired from the noise reduction technique described in Machicoane et al. [21, 22] and developed for Particle Tracking Velocimetry. Assuming that displacement fields are computed with some level of noise b̃(x, y, t), we have in the limit of short time intervals

$$d\vec{X}_j(x, y, t) = \vec{v}(x, y, t) \times jdt + \vec{b}(x, y, t), \qquad (5)$$

where $\vec{v}(x, y, t)$ is the flow velocity field. An unbiased velocity field, $\vec{v}_{\text{unb}}$, can be obtained by performing a linear regression of $d\vec{X}_j$ as a function of $jdt$ and taking the resulting slope. While promising, the last method is computationally intensive when used at all positions of the grid and all time steps (~$10^7-10^8$ operations for the whole dataset). Therefore, the method is more suitable for computing unbiased estimates of the moments of the velocity field (mean flow, rms velocity field, ...) [21, 22]. Here, we are interested in computing the kinetic energy per unit mass averaged over the horizontal direction and over some time interval, such that we compute the square of the displacement field, i.e.

$$d\vec{X}_j^2 = \vec{v}^2 \times (jdt)^2 + 2jdt \times \vec{v} \cdot \vec{b} + \vec{b}^2. \qquad (6)$$

Assuming that $\vec{v}$ and $\vec{b}$ are uncorrelated, the equation above can be averaged as $\langle d\vec{X}_j^2 \rangle = \langle \vec{v}^2 \rangle \times (jdt)^2 + \langle \vec{b}^2 \rangle$, where $\langle \cdot \rangle$ denotes averaging over the x axis and the moving time window $[t, t+\delta t]$. In practice, we use $\delta t \simeq 2 \times 10^3$ s, which corresponds to the time needed to perform a temperature and salinity profile with the moving probe (see part 4).

We compare all three methods in Figure 7, using time frame $t = 5.3 \times 10^4$ s, i.e. such that the system is in a quasi-static regime, for illustrative purposes. Figure 7.a displays the evolution of $\langle d\vec{x}_j^2 \rangle(y)$ with $j^2$ (third method), computed at three different altitudes y in both the high-velocity region (bottom of the tank) and low-velocity region (top of
the tank). As can be observed, all three curves are well approximated by linear fits of the form $a+b \cdot j^2$, which validates the assumption regarding the decorrelation of $\vec{b}$ and $\vec{v}$, extending the method developed by Machicoane et al. to PIV measurements. We derive the depth-dependent profiles of $\vec{v}_{\text{unb}}$ from the fits performed at each altitude. We show $\langle\vec{v}_1^2\rangle$, $\langle\vec{v}_{\text{cmax}}^2\rangle$, $\langle\vec{v}_{\text{avg}}^2\rangle$, and $\langle\vec{v}_{\text{unb}}^2\rangle$ as functions of $y$ in figure 7.b, using both linear (left figure) and logarithmic (right) scales for the horizontal axis. All estimates based on the three advanced methods are parallel to the raw estimate $\langle\vec{v}_1^2\rangle$ obtained from images 1-2. However, their performance differs in terms of noise reduction. In the upper layer, where we expect almost no motion, the base case $\langle\vec{v}_1^2\rangle$ yields a noise level of $0.16 \text{ mm}^2 \cdot \text{s}^{-2}$, while $\langle\vec{v}_{\text{cmax}}^2\rangle$ and $\langle\vec{v}_{\text{avg}}^2\rangle$ are almost equal with a noise level of $0.08 \text{ mm}^2 \cdot \text{s}^{-2}$ and $0.06 \text{ mm}^2 \cdot \text{s}^{-2}$, respectively. Finally, the linear regression method yields the best result, with a noise level $\sim 10^{-2} \text{ mm}^2 \cdot \text{s}^{-2}$ in the upper layer, i.e. an order of magnitude smaller than with the other methods, and a curve that appears much smoother than any of the other three estimates. We estimate the typical noise level of the velocity field for the raw method and first two advanced methods relative to the third (best) advanced method as $\sqrt{|\langle\vec{v}_{\text{method}}^2\rangle - \langle\vec{v}_{\text{unb}}^2\rangle|}$, where $|\cdot|$ is the absolute value operator and $\bullet$ the spatial average. We find $0.30 \text{ mm} \cdot \text{s}^{-1}$ for the raw estimate, $0.22 \text{ mm} \cdot \text{s}^{-1}$ for the best correlation method, and $0.18 \text{ mm} \cdot \text{s}^{-1}$ for the weighted correlation method.

If the linear regression method most effectively reduces noise, it is more computationally intensive so we use the linear regression method only when considering 1D profiles, and otherwise apply the best correlation method and remove the average noise level so that we consider $\langle\vec{v}_{\text{cmax}}^2\rangle' = \langle\vec{v}_{\text{cmax}}^2\rangle - \sqrt{|\langle\vec{v}_{\text{cmax}}^2\rangle - \langle\vec{v}_{\text{unb}}^2\rangle|}$. Figure 8.a shows the kinetic energy $\langle\vec{v}_{\text{cmax}}^2\rangle'$ averaged over time window $[t, t + \delta t]$ with $t = 5.3 \times 10^4$ s. The figure clearly shows the two-layer structure of the water column, with a single large convection cell moving clockwise in the bottom layer and little kinetic energy in the top layer. Figure 8.b shows the horizontally-averaged kinetic energy profile with depth $\langle\vec{v}_{\text{cmax}}^2\rangle'$ as a function of time. The rising ice-water interface mask (white region) moves upward based on the ice-water interface position inferred from the shadowgraphy measurements. Here, the top diffusive layer (with little kinetic energy) corresponds precisely to the part of the domain where solid ice turned into liquid water. The convective bottom layer, which occupies the same region at all times, appears to slowly weaken as the experiment progresses.

# 4 Measuring Temperature & Salinity Profiles

## 4.1 Probe Calibration

We developed a conductivity and temperature probe to further characterize the two-layer organization of our system and extract fluxes in diffusive layers. Here, we measure the temperature and salinity with a LFS1K0.1305.6W.B.010-6 conductivity-temperature sensor from Innovative Sensor Technology. The temperature sensor is a Pt-1000: its resistance varies with temperature as $R(T) = R_0(1 + a_1T)$ with $R_0 = 1000 \text{ } \Omega$ and $a_1 = 3.9083 \times 10^{-3} \text{ } \circ\text{C}^{-1}$. The resistance is integrated into a home-made bridge that delivers a 0 – 5 V voltage proportional to the temperature in the
<figure><img src="image_9.png" /><figcaption>**Fig. 7** (a) Linear regression $\langle d\vec{X}_j^2 \rangle = \langle \vec{v}_{unb}^2 \rangle \times (jdt)^2 + \langle \vec{v}_{amb}^2 \rangle$ protocol (third method) to get unbiased value for the velocity field's second order moment, averaged over time window $[t, t + \delta t]$ with $t = 5.3 \times 10^4$ s and $\delta t = 2 \times 10^3$ s. The results are shown for 3 vertical positions. (b) Comparison of the mean kinetic energy profile with depth $\langle \vec{v}_{method}^2 \rangle(y)$ over the same time window, obtained with the raw method and three advanced methods. The results are shown using both linear (left) and logarithmic (right) scales for the horizontal axis.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 8** (a) Kinetic energy $\langle \vec{v}_{cmax}^2 \rangle'$ averaged over time window $[t, t + \delta t]$ with $t = 5.3 \times 10^4$ s and $\delta t = 2 \times 10^3$ s for $T_\infty = 10.7^\circ C$, $S_0 = 9.4 \text{ g.L}^{-1}$. (b) Horizontally-averaged kinetic energy $\langle \vec{v}_{cmax}^2 \rangle'$ for the same experiment as a function of time. Gray shading at the early times highlights results obtained before the ice cuboid is placed atop the bulk water.</figcaption></figure>
range $T \in [-10, 50]^{\circ}$C. The calibration of the temperature probe is shown in figure 9.a, the results being satisfactorily reproduced by a linear fit.

The salinity $S$ is obtained from the measurement of the resistivity of a 4-wire conductivity sensor. The measurement is done at 1kHz with a home-made circuit with AD8436 RMS to DC converters whose outputs deliver voltages proportional to the rms values of the voltage applied to the sensor and its corresponding current. The conductivity of the saltwater $C(S, T)$ is computed after acquisition and then converted into salinity using an appropriate salinity-temperature calibration. The function $C(S, T)$ is modeled following Bradshaw et al. [30] who demonstrated that the ratio $C(S, T)/C(S, T_0)$, where $T_0$ is a reference temperature, is well approximated by a second order polynomial of temperature only, i.e.

$$ \frac{C(S, T)}{C(S, T_0)} = B_1 + B_2T + B_3T^2, \qquad (7) $$

with salinity and conductivity related through the polynomial relation (using temperature $T = T_0$ for conductivity)

$$ S = A_1 + A_2C(S, T_0) + A_3(C(S, T_0))^2. \qquad (8) $$

The coefficients of equation (8) are obtained from a first temperature calibration performed at fixed salinity $S_0 = 10$ g.L$^{-1}$, and a second calibration performed at fixed temperature $T_0 = 8.9^{\circ}$C with solutions of known salinity. The results of both calibrations are shown in figures 9.b and 9.c and yield the coefficients $B_1 = 7.89 \times 10^{-1}$, $B_2 = 2.24 \times 10^{-2} \,^{\circ}\text{C}^{-1}$, $B_3 = 1 \times 10^{-4} \,^{\circ}\text{C}^{-2}$, and $A_1 = -1.386 \times 10^{-1}$ g L$^{-1}$, $A_2 = 7.227 \times 10^{-1}$ g L$^{-1}/(\text{mS cm}^{-1})$, $A_3 = 5.2 \times 10^{-2}$ g$^2$ L$^{-2}/(\text{mS}^2 \text{cm}^{-2})$.

The voltages for the temperature and conductivity measurements are acquired with a NI PXI-4472 data acquisition system (DAQ) controlled using MATLAB (8 channels @ 100kHz, [-5,5] V, 23 bits). Data acquisition is performed continuously during the experiment at a frequency of 1 kHz, which is the minimum sampling frequency supported by the DAQ.

## 4.2 Staircase profiling

We perform temperature and salinity measurements along the main vertical axis of the experiment in the middle of the tank ($x = 20$ cm, $z = 10$ cm). The probe is mounted on a rigid arm connected to a Zaber linear stage controlled in Matlab with a travel length of up to 50 cm. The probe is placed at the top of the system and moves up and down within the water column, as well as through the ice within a cylindrical conduit.

We perform multiple vertical profiles throughout the experiment. Since the flow is relatively weak, the entrainment of ambient fluid in the wake of the probe can affect the local dynamics. We minimize the entrainment effect by performing a staircase-like profile. The key parameters include the vertical extent of the profile $L_s = 12$ cm, the number of steps or recording locations $N_{steps}$, the recording time at each step $\tau_s = 100$ s, the distance between two steps $\delta z = 8$ mm and the velocity of the probe between steps $u_p = 0.5$ mm/s. We choose $\tau_s$ so that it is much larger than the response time of
<figure><img src="image_11.png" /><figcaption>**Fig. 9** (a) Calibration of the Pt – 1000 resistance measurement with respect to temperature with fixed salinity $S_0$ = 10 g.L$^{-1}$. (b) Calibration of the electrode cell conductivity with respect to temperature, following equation (7) and with fixed salinity $S_0$. (c) Calibration of the electrode cell conductivity with respect to salinity, following equation (8) and with fixed temperature $T_0$ = 8.9°C.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10** (a) Position of the temperature and salinity probe as a function of time. An example staircase profile consists in the ensemble of steps taken between the two vertical dashed lines. (b) Temperature as a function of time for all steps of a staircase profile. A quasi steady state is reached after roughly 40 seconds.</figcaption></figure>

the probe, and $N_{steps}$ such that each staircase profile is much shorter (~30 minutes) than the typical melting timescale (~15 hours). The resulting position of the probe as a function of time is shown in figure 10.a. Figure 10.b shows that $\tau_s$ is long enough for the probe measurement to reach a quasi constant average value after reaching each new location at $t_i$. The short transient lasts about 20 s and has a relative amplitude of up to 10% for both salinity and temperature measurements. We consider measurements with $t - t_i \in [40, 100]$ s only for the computation of local temperature and salinity averages.

Typical temperature and salinity profiles are shown in figures 11.a and 11.b, where each marker shows the mean value obtained when averaging over $t \in [t_i + 40, t_i + 100]$ s. The solid black line and gray shading show the mean position and vertical spread
<figure><img src="image_13.png" /><figcaption>**Fig. 11** (a) Temperature and (b) salinity profiles extracted from a single staircase profile centered about time $t = 5.3 \times 10^4$ s. (c) Corresponding density profile $\rho(S, T)$ obtained with the Gibbs-SeaWater (GSW) Oceanographic Toolbox. $T_\infty = 10.7^\circ C$, $S_0 = 9.4$ g.L$^{-1}$.</figcaption></figure>

of the ice-water interface at time $5.3 \times 10^4$ s, which lies mid-way within the duration of the staircase profile. At $5.3 \times 10^4$ s, about 5 cm of ice has melted, such that a layered structure is visible and the profiles exhibit three distinct regions. Toward the bottom (up to 350 mm), we observe a well-mixed region where convection occurs, with homogeneous temperature and salinity. In the middle region (between 350 mm and 405 mm), the temperature and salinity fields are stratified, decreasing toward zero close to the ice-water interface. The temperature gradient is closer to being constant (straight line for the temperature profile) than the salinity gradient in the middle stratified region, which corresponds to the diffusive region without kinetic energy, because temperature diffuses faster than salt. Finally, above the ice-water interface, the measured temperature and salinity are homogeneous and equal to 0 because the sensor is moving through a conduit that quickly fills with fresh meltwater. We compute the density $\rho(S, T)$ from the temperature and salinity measurements using the Gibbs-SeaWater (GSW) oceanographic toolbox [31]. The density profile, shown in figure 11.c, shows that the water column is stably stratified in the middle region and closely follows the salinity curve. This density profile shows that the salinity gradient stabilizing effect prevails on the temperature gradient destabilizing effect.

# 5 Discussion and conclusions

The melting of an ice cuboid in saltwater generates salinity and temperature gradients, which lead to the development of an ice-water interface and a bulk water-meltwater
<figure><img src="image_14.png" /><figcaption>**Fig. 12** Temperature (red circles; bottom axis), salinity (blue diamonds; top axis), and kinetic energy (black triangles with red edges; bottom axis after ad-hoc constant rescaling) profiles at early (left plot), intermediate (middle), and late (right) times. The control parameters are $T_\infty = 10.7^\circ\text{C}$, $S_0 = 9.4 \text{ g.L}^{-1}$ and measurements lead to the estimates $v_b^2 = 5.4 \text{ mm}^2\text{s}^{-2}$ and $T_b = 9.5^\circ\text{C}$. The vertical black dashed line highlights the absence of kinetic energy in the top layer.</figcaption></figure>

interface. Both interfaces can be tracked over time on shadowgraphy images and are plotted together with vertical profiles of salinity, temperature, and kinetic energy $\langle \vec{v}_{\text{umb}}^2 \rangle$ in figure 12. The kinetic energy is rescaled on the temperature axis by multiplying it by $T_b/v_b^2$ where $v_b^2 = 5.4 \text{ mm}^2\text{s}^{-2}$ is the typical bulk kinetic energy and $T_b = 9.5^\circ\text{C}$ the typical bulk temperature. The profiles correspond to the experiment with $T_\infty = 10.7^\circ\text{C}$, $S_0 = 9.4 \text{ g.L}^{-1}$ and are shown at three times: $t = 0.3 \times 10^4 \text{ s}$ (short time after starting the experiment), $t = 2.3 \times 10^4 \text{ s}$ (intermediate time), and $t = 5.3 \times 10^4 \text{ s}$ (long time, when about 50% of the initial ice cuboid has melted). As can be observed, the kinetic energy profile is not sharp around the meltwater-bulk water interface, which is a sign of penetrative convection with enhanced mixing of salt and temperature between the layers. When the ice cuboid is placed floating on top of the bulk water, heat is transferred from the bulk water to the ice, creating a large temperature gradient (figure 12 left) so that the water column becomes unstable, triggering Rayleigh-Bénard convection. As heat transfer continues, the ice melts. The melting progressively creates a stable layer of cold freshwater (figure 12 middle and right). The kinetic energy and heat and salinity fluxes are then strongly depth dependent, such that a fully diffusive or fully convective model is inappropriate.

The presence of a top stable, and likely diffusive, layer yet suggests that an alternative predictive model may be derived. In Figure 13.a, we illustrate our proposed model
<figure><img src="image_15.png" /><figcaption>Fig. 13 (a) Schematic of the diffusive meltwater layer model, which can be used to predict the melting rate of an ice cuboid in saltwater when convection in the bottom layer cannot entrain the top meltwater layer. (b) Measured displacement of the ice-water interface $y_i(t) - y_i(0)$ (solid black line) and meltwater-bulk water interface $y_{KE}(t) - y_i(0)$ based on a kinetic energy threshold (solid cyan) over time. The displacement of the ice-water interface $\tilde{y}_i(t) - y_i(0)$ predicted with the diffusive meltwater layer model is shown by the dashed brown line.</figcaption></figure>

for estimating the melt rate. In agreement with the results of Figure 12, we consider that the meltwater-bulk water interface position does not change with time, such that it is always located at $y_i(0)$. The temperature profile is assumed linear in the stable layer $y \in [y_i(0), y_i(t)]$ (with $T(y_i(t)) = T_f^0$) and $T(y_i(0)) = T_b$) and $T = T_b$ in the convective region $y \le y_i(0)$ (cf. solid red line in figure 13.a). We note that the assumption of a quasi-static diffusive equilibrium for temperature in the upper layer is reasonable as the melting time $5 \times 10^4$ s (corresponding to $\ell = 50$ mm of melted ice) is several times longer than the diffusive time for temperature $\tau_\ell = \ell^2/\kappa_T = 1.7 \times 10^4$ s. The heat transfer in the diffusive layer $Q_w$ is then $Q_w = Sk(T_b - T_f^0)/[y_i(t) - y_i(0)]$ and can be combined with $SL\rho_i\dot{h} = Q_w$ to get the first-order differential equation for $y_i(t)$

$$ \frac{d(y_i(t) - y_i(0))}{dt} = \frac{k(T_b - T_f^0)}{\rho_0 L} \frac{1}{y_i(t) - y_i(0)}, \quad (9) $$

which can be solved as $\tilde{y}_i(t) - y_i(0) = \sqrt{2k(T_b - T_f^0)/\rho_0 L}\sqrt{t}$. We show in figure 13.b that the diffusive meltwater layer model is in good agreement with the experimental data up to $t = 1.5 \times 10^4$ s, which is when 30% of ice has melted.

In conclusion, we have successfully developed an experimental setup of a rectangular ice cuboid floating and melting in an initially quiescent and homogeneous saltwater tank, under controlled conditions with comprehensive data acquisition. By placing the experiment in a cold room whose temperature matched the forcing temperature of the
experiment, we minimized parasitic heat transfer. During melting, the ice-water inter-
face and velocity field are measured using shadowgraphy and the PIV system, and a
moving temperature and conductivity probe gives access to temperature and salinity
profiles in the vertical direction. To minimize disruption to the flow and ensure reliable
data, the profiling is conducted in a staircase pattern. Altogether, these measurements
 unlock a quantitative characterization of all salient features of the flow and derivation
 of a predictive model in good agreement with experimental data up to intermediate
times.

The melting of ice in saltwater should be further explored. Notably, the impact of
the forcing temperature and initial salinity on the system dynamics should be inves-
tigated. We expect that the combination of experimental techniques and metrology,
which proved effective for the present complex configuration, could be applied to a
wide range of systems involving phase change and mixing in binary liquids.

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