The effects of salinity and inclination on the morphology of melting ice

This study experimentally investigates how water salinity and ice inclination affect the melting of ice blocks in quiescent saline water, revealing five distinct surface morphologies and demonstrating that increased salinity produces smaller, more uniform scallops while inducing a non-monotonic relationship with melt rate, whereas inclination has minimal impact on the overall melting speed.

The Gist

Imagine you have a giant, rectangular block of ice sitting in a bathtub. Now, imagine two things changing: the water gets saltier (like turning the tap to the ocean), and you tilt the ice block at different angles. What happens to the ice? Does it melt evenly? Does it get smooth, or does it get bumpy?

This paper is essentially a detective story about how ice melts in salty water, and why the shape of the melting ice changes depending on how salty the water is and how steep the ice is tilted. The researchers wanted to understand this because it helps us predict how real glaciers and icebergs melt in the ocean, which is crucial for understanding rising sea levels.

Here is the story of their findings, broken down into simple concepts:

1. The Two Forces at Play: Heat vs. Salt

Think of the melting ice as a tug-of-war between two invisible teams:

• Team Heat: The warm water wants to melt the ice. As it touches the ice, the water near the surface gets cold. Cold fresh water is heavy, so it wants to sink.
• Team Salt: The ice is fresh, but the ocean is salty. When the ice melts, it releases fresh water. Fresh water is lighter than salty water, so it wants to float up.

Depending on how salty the water is, one team wins, or they get stuck in a stalemate. This creates three different "mood rings" for the ice:

• Low Salt (Team Heat Wins): The cold water sinks. The ice melts faster at the top.
• High Salt (Team Salt Wins): The fresh meltwater floats up. The ice melts faster at the bottom.
• Medium Salt (The Stalemate): Both forces are fighting equally. This is where the most interesting shapes appear.

2. The Five "Ice Outfits"

The researchers found that the ice doesn't just melt into a flat block. It dresses up in five distinct "outfits" (morphologies) depending on the conditions:

• The "Top-Melting" Outfit: Happens when the water isn't very salty. The ice melts faster at the top, like a candle burning down from the wick.
• The "Bottom-Melting" Outfit: Happens in very salty water. The fresh water floats up, pulling heat with it, so the bottom of the ice gets eaten away faster.
• The "Incurved" Outfit: If you tilt the ice too much, the edges melt slower than the middle, making the ice look like a shallow bowl or a smile.
• The "Channelized" Outfit (The Runnels): In low-salt water on a tilted block, the ice carves deep, vertical grooves, looking like a washboard or a ribbed muscle.
  • The Secret Ingredient: The researchers discovered that tiny air bubbles trapped inside the ice act like elevator cars. They get stuck in the grooves and ride up, dragging heat with them and digging the channels deeper. It's like a self-reinforcing loop where the bubble helps dig the hole, and the hole helps the bubble go higher.
• The "Scalloped" Outfit (The Bumpy Surface): This happens in medium-salty water. The ice surface becomes covered in small, round dimples, looking like the scales of a fish or the surface of a golf ball.

3. The "Goldilocks" Zone for Bumps

The most fascinating discovery was about the scallops (the bumpy dimples).

• If the water is too fresh or too salty, the ice stays relatively smooth.
• But in the middle ground (medium salinity), the battle between sinking cold water and rising fresh water creates a perfect storm for these bumps to form.
• The Trend: As the water gets saltier within this "Goldilocks" zone, the bumps get smaller, shallower, and more uniform. It's like the salt is "ironing out" the roughness, making the pattern more organized but less dramatic.

4. The Speed of Melting (The Surprising Twist)

You might think that adding more salt always makes ice melt faster or slower in a straight line. But the researchers found something weird: The melting speed is non-monotonic.

Imagine a graph where the line goes down, hits a low point, and then goes back up.

• At low salt, it melts fast.
• As you add salt, the melting actually slows down to a minimum point (around a specific density ratio).
• Then, as you add even more salt, it starts melting fast again.

It's like a traffic jam in the middle of the ocean: the two forces (heat and salt) cancel each other out so effectively that the melting process gets stuck in neutral for a while before speeding up again.

5. How They Took the "Selfies"

To see these tiny changes, the scientists couldn't just look with their eyes. They used a high-tech version of 3D scanning.

• They projected a pattern of light stripes onto the ice.
• As the ice melted and changed shape, the stripes got wavy and distorted.
• By taking thousands of photos and using a clever math trick (combining "spatiotemporal phase shifting" and "orthogonal sampling moiré"—which sounds like a mouthful but is basically a way to filter out noise and see the tiny details), they reconstructed a perfect 3D map of the ice surface as it melted.

Why Does This Matter?

This isn't just about ice cubes in a lab. Real glaciers and icebergs in the ocean are constantly melting.

• If we don't understand how the slope of the ice and the saltiness of the ocean affect the melting, our computer models for predicting sea-level rise will be wrong.
• The study shows that small-scale details (like those bumpy scallops or the air bubbles) matter a lot. They change how fast the ice disappears.

In a nutshell: Ice melting in the ocean is a complex dance between heat and salt. Depending on the steps (salinity and angle), the ice can melt smoothly, carve deep grooves, or grow bumpy scales. And sometimes, the dance slows down completely before speeding up again. Understanding these steps helps us predict the future of our melting planet.

Technical Summary

1. Problem Statement and Motivation

The melting of ice sheets and icebergs in oceanic environments is a critical factor in global sea-level rise and ocean circulation. However, current predictive models often underestimate observed melt rates by up to two orders of magnitude. Recent observations (e.g., at Thwaites Glacier) indicate that melt rates are highly sensitive to small-scale surface structures and interfacial slopes, which are not fully captured by existing bulk-parameterization models.

The specific scientific gap addressed in this study is the lack of controlled experimental data linking salinity, inclination angle, and surface morphology (pattern formation) during the melting of ice. While previous studies have examined thermal or solutal driving forces separately, the complex interplay between thermal and saline buoyancy forces (the "competing regime") and how they dictate surface roughness (scallops vs. channels) under varying inclinations remains poorly understood.

2. Methodology

Experimental Setup:

• Apparatus: Rectangular ice blocks were submerged in a quiescent water tank (50 cm × 38 cm × 78 cm).
• Variables:
  • Salinity ($S_\infty$): Varied from 0 g/kg (freshwater) to 35 g/kg (oceanic conditions).
  • Inclination ($\theta$): Varied from -18° to 50° relative to the vertical.
  • Temperature: Ambient water temperature ($T_\infty$) was maintained between 18°C and 21°C to avoid the density anomaly of water near 4°C.
• Ice Blocks: Three sizes were used, with the primary set being 32 cm × 23 cm × 12 cm. The ice was made opaque by mixing white marker ink into the water before freezing to facilitate optical measurement.
• Flow Regimes: The experiments covered Rayleigh numbers ($Ra$) of $O(10^7)$ and explored three distinct buoyancy regimes defined by the density ratio $R_\rho = \Delta\rho_S / \Delta\rho_T$:
  • Temperature-driven ($R_\rho \ll 1$): Meltwater is denser than ambient water.
  • Salinity-driven ($R_\rho \gg 1$): Meltwater is less dense due to low salinity.
  • Competing ($R_\rho \approx 1$): Thermal and solutal forces oppose each other.

Measurement Technique:

• Fringe Projection Profilometry (FPP): The authors utilized a high-resolution imaging system (Photron camera + Optoma projector) to reconstruct the 3D surface morphology of the melting ice.
• Algorithm: A novel combination of Spatiotemporal Phase-Shifting (ST-PSM) and Orthogonal Sampling Moiré (OSM) methods, termed OST-PSM, was developed. This hybrid approach was crucial for mitigating stripe artifacts and handling low-contrast images caused by the melting ice surface.
• Data Analysis: Surface profiles were analyzed using a watershed algorithm to segment and quantify features like "scallops" (dimples) and "channels." Metrics included wavelength, amplitude, area, and migration velocity.

3. Key Contributions

1. Morphological Classification: The study identifies and classifies five distinct surface morphologies based on the $S_\infty$-$\theta$ parameter space:
   • Scalloped: Rough, dimpled patterns.
   • Channelized: Vertical grooves carved into the ice.
   • Top-melting: Enhanced melting at the upper edge.
   • Bottom-melting: Enhanced melting at the lower edge.
   • Incurved: Concave surfaces where edges melt slower than the center.
2. Novel Measurement Application: This is the first application of FPP with OST-PSM to study pattern formation on dissolving/melting bodies, allowing for high-fidelity, time-resolved 3D topography measurements.
3. Mechanism of Channel Formation: The authors propose that vertical channels arise from a Rayleigh–Bénard-type instability (dense thermal layer overlying lighter ambient water) and are significantly enhanced by air bubbles released from the melting ice, which preferentially rise through the channels, accelerating local melt rates.
4. Non-Monotonic Melt Rate: The study quantifies the Nusselt number ($Nu$) and reveals a non-monotonic relationship with salinity, identifying a minimum melt rate at $R_\rho \approx 3$.

4. Key Results

A. Morphology Regimes:

• Channelized Regime: Observed primarily in low-salinity, low-inclination conditions ($R_\rho < 0.93$). Channels form due to thermal instability and are deepened by rising bubbles. They are absent in high-salinity conditions where the meltwater is less dense than the ambient fluid, suppressing the instability.
• Scalloped Regime: Occurs in the competing regime ($2 \lesssim R_\rho \lesssim 6$).
  • Salinity Effect: Increasing salinity (within this regime) leads to scallops that are smaller, shallower, and more uniform in size.
  • Inclination Effect: Higher inclination angles shift the critical $R_\rho$ range where scallops form.
  • Dynamics: Scallops exhibit downward migration. The migration velocity decreases as salinity increases.
• Top/Bottom Melting:
  • Top-melting: Dominates in the temperature-driven regime ($R_\rho \lesssim 2$) for low inclinations.
  • Bottom-melting: Dominates in the salinity-driven regime ($R_\rho \gtrsim 6$), where buoyant plumes rise from the bottom.
• Incurved Morphology: Observed for all inclinations $\theta > 10^\circ$. The authors attribute this to edge effects and sidewall interactions rather than a primary flow regime.

B. Heat Transfer and Melt Rate:

• Non-Monotonic Behavior: The Nusselt number ($Nu$) shows a distinct minimum around $R_\rho \approx 3$. This is attributed to the cancellation of opposing thermal and solutal buoyancy forces, which reduces the overall convective heat transfer.
• Inclination Independence: Unlike previous oceanic studies that suggested a $\cos(\theta)^{2/3}$ scaling for melt rates, this study found little to no systematic dependence of the overall melt rate on the inclination angle. The authors attribute this discrepancy to the different ambient conditions (higher temperatures allowing for downward flows) compared to polar ocean conditions.

5. Significance and Implications

• Climate Modeling: The findings challenge existing parametrizations of ice-ocean interaction. The discovery that melt rates do not strictly follow inclination-based scaling laws in the presence of varying salinity suggests that current models may need to incorporate complex, non-linear interactions between thermal and solutal boundary layers.
• Pattern Formation Physics: The study provides a mechanistic explanation for the formation of "scallops" and "channels" on icebergs and glacier faces, linking them directly to the density ratio ($R_\rho$) and the role of entrapped air bubbles.
• Methodological Advancement: The successful implementation of OST-PSM for melting interfaces offers a robust tool for future fluid dynamics research involving phase-change boundaries, where surface roughness and low contrast typically hinder optical measurements.

In conclusion, this work demonstrates that the morphology and melt rate of ice in saline water are governed by a complex interplay of buoyancy forces and inclination, with salinity playing a more dominant role in determining surface roughness and melt rate minima than previously assumed.