A combined experimental and direct numerical simulation (DNS) investigation is undertaken to study the laminar boundary-layer (BL) flow adjacent to a melting vertical ice face at two far-field water salinities ($S_\infty =0$ and 34 ‰) and a range of far-field temperatures ($T_\infty$). Wall-normal distributions of vertical velocity and temperature within the BL are measured by a modified molecular tagging velocimetry and thermometry technique. Experimental data match with DNS only when a nonlinear equation of state (EoS) for density is used rather than a linear EoS. For all $S_\infty =0$, i.e. freshwater cases, the flow remains uni-directional, although the flow reverses direction at $T_\infty =4^{\,\circ} \text{C}$. A bi-directional flow, however, exists for $S_\infty =$ 34 g kg−1, where an inner salinity-driven upward flow of fresher water is accompanied by a downward-flowing temperature-driven outer flow. Although the contribution of temperature to density relative to salinity is small $({\approx}1/40)$, the thermal BL region is larger owing to higher diffusivity. This results in increased total buoyancy force when the buoyancy is integrated across the BL, which combined with effects of wall shear stress on salinity BL and a freer thermal BL growth reveals that buoyancy from temperature contributes almost equally to the overall flow. Melt rates ($V$) also show differing features in uni- and bi-directional flows. The uni-directional flows exhibit the standard scaling of increasing velocity magnitude and BL thickness, and decreasing $V$ with distance along the flow direction. Such scalings are not followed in the bi-directional flows. These show a more uniform $V$ with height, which is attributed to the counteracting effects of an upward-growing salinity BL and a downward-growing temperature BL, combined with the necessity of maintaining salinity and temperature flux balance at the ice–water interface.

Roll patterns on floating ice shelves have been suggested to arise from viscous buckling under compressive stresses. A model of this process is explored, allowing for a power-law fluid rheology for ice. Linear stability theory of uniformly compressing base flows confirms that buckling modes can be unstable over a range of intermediate wavelengths when gravity does not play a dominant role. The rate of compression of the base flow, however, ensures that linear perturbations have wavelengths that continually shorten with time. As a consequence, linear instability only ever arises over a certain window of time $t$, and its strength can be characterised by finding the net amplification factor a buckling mode acquires for $t\to \infty$, beginning from a given initial wavenumber. Bi-axial compression, in which sideways straining flow is introduced to prevent the thickening of the base flow, is found to be more unstable than purely two-dimensional (or uni-axial) compression. Shear-thinning enhances the degree of instability in both uni-axial and bi-axial flow. The implications of the theoretical results for the glaciological problem are discussed.

Accurate knowledge of basal topography is required for numerical modelling efforts to predict how Earth’s ice sheets will respond to continued warming. The widely used BedMachine v3 dataset has limitations with respect to its use in modelling studies, particularly in estimating uncertainties. Machine learning approaches offer promise in addressing this gap, with quantile regression forests (QRFs) especially suited to geospatial data. Here, we apply a novel QRF approach to map the basal topography of Greenland’s ice sheet using airborne radio echo sounding (RES) data. Compared to BedMachine, our model reduces the root-mean-squared-error of ice depth predictions by 18%, from 232 to 190 m. It also significantly improves uncertainty calibration: 89.8% of new observations fall within our 90% prediction interval, versus 68% for BedMachine. The QRF model achieves a lower continuous ranked probability score (92 m vs. 130 m), indicating improved balance between accuracy and uncertainty. Our volume estimate for the Greenland ice sheet is 0.7% higher than BedMachine’s, though we emphasise differences in the predicted shape of subglacial features like outlet glacier troughs. This approach offers a computationally efficient, accessible method for deriving subglacial topography from RES data, while providing better-calibrated uncertainty estimates than existing models.

This study investigates the interactions between flexural-gravity waves and interfacial waves in a two-layer fluid, focusing on wave blocking. Both liquid layers are of finite depth bounded on top by a viscoelastic thin plate. Both liquids are incompressible and inviscid, and their flows are two-dimensional and potential. Linear wave theory and a linear equation of a thin floating viscoelastic plate of constant thickness are used. We analyse the phenomenon of wave blocking and Kelvin–Helmholtz (KH) instability in a two-layer fluid with a discontinuous background mean flow. A quartic dispersion relation for frequency as a function of wavenumber and other parameters of the problem is derived. Two cases of uniform current and layers moving with different velocities are studied. Wave blocking occurs when roots of the dispersion relation coalesce without accounting for plate viscosity, leading to zero group velocity. Our findings indicate that wave blocking can occur for both flexural-gravity and interfacial waves under various frequency and current speed conditions, provided that plate viscosity is absent. The role of different parameters and the flow velocities of the upper and lower layers are investigated in the occurrence of wave blocking and KH instability. The loci of the roots of the dispersion relation involving plate viscosity depict that no root coalescence occurs irrespective of the values of wavenumber and frequency in the presence of plate viscosity. The amplitude ratio of the interfacial wave elevation to that of floating viscoelastic plate deflection exhibits the dead-water phenomenon as a density ratio approaches unity.

Understanding firn densification is essential for interpreting ice core records, predicting ice sheet mass balance, elevation changes and future sea-level rise. Current models of firn densification on the Antarctic ice sheet (AIS), such as the Herron and Langway (1980) model are either simple semi-empirical models that rely on sparse climatic data and surface density observations or complex physics-based models that rely on poorly understood physics. In this work, we introduce a deep learning technique to study firn densification on the AIS. Our model, FirnLearn, evaluated on 225 cores, shows an average root-mean-square error of 31 kg m−3 and explained variance of 91%. We use the model to generate surface density and the depths to the $550\,\mathrm{kg\,m}^{-3}$ and $830\,\mathrm{kg\,m}^{-3}$ density horizons across the AIS to assess spatial variability. Comparisons with the Herron and Langway (1980) model at ten locations with different climate conditions demonstrate that FirnLearn more accurately predicts density profiles in the second stage of densification and complete density profiles without direct surface density observations. This work establishes deep learning as a promising tool for understanding firn processes and advancing towards a universally applicable firn model.

The proposed Thermal Sidewall Ice Corer (TSIC) is designed to accurately sample horizontal ice layers of scientific interest, such as tephra layers, basal ice and shear zones, and retrieve ice cores back to the surface. The system features a bending core barrel with a thermal coring head, which bends as it extends from the drill body, enabling it to penetrate horizontal interlayers while maintaining a horizontal position until the ice core is extracted. The bending core barrel is driven by screw pairs, powered by a motor, to apply drilling load and pulling force. As the barrel bends, the ice cores are broken inside and transported to the surface along with the drill via a winch. A camera system has been incorporated into the TSIC to precisely locate the target layer. The corer is suitable for ice boreholes with diameters ranging from 135 to 170 mm, capable of retrieving ice cores with a diameter of 20–30 mm, and achieving a maximum penetration rate of 2 m h−1. The maximum length of ice samples that can be retrieved in a single drilling run is 500 mm. The coring performance for horizontal sampling has been validated through the development and testing of a prototype in the laboratory.

Melting icebergs provide nearly half of the total freshwater flux from ice shelves to the ocean, but the availability of accurate, data-constrained melting rate parametrisations limits the correct representation of this process in ocean models. Here, we investigate the melting of a vertical ice face in a warm, salt-stratified environment in a laboratory setting. Observations of the depth-dependent melting rates ${m}$ and boundary layer flow speed $U$ are reported for a range of initially uniform far-field ambient temperatures $T_a$ above ${10}\,^{\circ }{\rm C}$. Ice scallops are characteristic features observed in all experiments, with the width of the scallops consistent with the theory of double-diffusive layers. The morphology of the scallops changes from symmetric about the scallop centre in the colder experiments to asymmetric in the warmer experiments. Observed melting rates are consistent with a melting rate scaling of the form ${m}\propto U\,\Delta T_a$ proposed by previous work in less extreme parameter regimes, where $\Delta T_a$ is the magnitude of thermal driving between the ambient and ice–fluid interface. Our results indicate that ice scalloping is closely linked to the naturally convecting flow of the ambient fluid. Depth-averaged melting rates depend on the buoyancy frequency in the ambient fluid, and double-diffusive convection promotes a turbulent-flux regime distinct from that explained previously in an unstratified regime. These findings have implications for parametrising melting rates of icebergs and glaciers in numerical models or potential freshwater harvesting operations, and provide insights into the interplay between stratification and ice melting.

A variational principle is proposed to derive the governing equations for the problem of ocean wave interactions with a floating ice shelf, where the ice shelf is modelled by the full linear equations of elasticity and has an Archimedean draught. The variational principle is used to form a thin-plate approximation for the ice shelf, which includes water–ice coupling at the shelf front and extensional waves in the shelf, in contrast to the benchmark thin-plate approximation for ocean wave interactions with an ice shelf. The thin-plate approximation is combined with a single-mode approximation in the water, where the vertical motion is constrained to the eigenfunction that supports propagating waves. The new terms in the approximation are shown to have a major impact on predictions of ice shelf strains for wave periods in the swell regime.

Flux conditions are semi-analytical expressions that can be used to determine the flux at the grounding line of marine ice sheets. In the glaciology literature, such flux conditions have initially been established for the Weertman and Coulomb friction laws. However, the extension to more general and complex friction laws, such as the Budd friction law, for which basal friction depends on both the basal velocity and the effective pressure, is a topic of recent research. Several studies have also shown that hybrid friction laws, which consider a transition between a power-law friction far from the grounding line and a plastic behaviour close to it, were good candidates for improved modelling of marine ice sheets. In this article, we show that the flux conditions previously derived for the Weertman and Coulomb friction laws can be generalised to flux conditions for the Budd friction law with two different effective-pressure models. In doing so, we build a bridge between the results obtained for these two friction laws. We provide a justification for the existence and uniqueness of a solution to the boundary-layer problem based on asymptotic developments. We also generalise our results to hybrid friction laws, based on a parametrisation of the flux condition. Finally, we discuss the validity of the assumptions made during the derivation, and we provide additional explicit expressions for the flux that stay valid when the bedrock slopes are important or when the friction coefficients are relatively small.

The propagation of a finite blister of fluid beneath an elastic sheet is controlled by the local dynamics at the peeling periphery of the current. Previous works have described constant volume elastic blisters by considering the peeling due to curvature at these edges. Here, we show that an along-slope component of gravity fundamentally changes the dynamics by removing the role of curvature at the trailing, upslope edge. The local dynamics of this trailing edge is instead controlled by shear stress in the sheet, as in the elastic Landau–Levich problem, and thereby allows for a receding edge, in contrast to propagation by peeling for which only an advancing contact line is possible. Using an asymptotic analysis, we show that this receding edge condition allows for a new, nearly translating regime in which the body of the blister moves at an approximately constant speed, leaving behind a thin layer of fluid. This prediction is verified by detailed numerical modelling of the two-dimensional downslope spreading. We conclude by discussing the applicability of these results in the rapid spreading of subglacial meltwater.

Ice-infiltrated sediment, or frozen fringe, is responsible for phenomena such as frost heave, ice lenses and metres of debris-rich ice under glaciers. Understanding frozen fringes is important as frost heave is responsible for damaging infrastructure at high latitudes and sediment freeze-on at the base of glaciers can modulate subglacial friction, influencing the rate of global sea level rise. Here we describe the thermomechanics of liquid water flow and freezing in ice-saturated sediments, focusing on the conditions relevant for subglacial environments. The force balance that governs the frozen fringe thickness depends on the weight of the overlying material, the thermomolecular force between ice and sediments across liquid premelted films and the water pressure required by Darcy flow. We combine this mechanical model with an enthalpy method that conserves energy across phase change interfaces on a fixed computational grid. The force balance and enthalpy model together determine the evolution of the frozen fringe thickness and our simulations predict frost heave rates and ice lens spacing. Our model accounts for premelting at ice–sediment contacts, partial ice saturation of the pore space, water flow through the fringe, the thermodynamics of the ice–water–sediment interface and vertical force balance. We explicitly account for the formation of ice lenses, regions of pure ice that cleave the fringe at the depth where the interparticle force vanishes. Our model results allow us to predict the thickness of a frozen fringe and the spacing of ice lenses in subaerial and subglacial sediments.

Supraglacial lakes play a central role in storing melt water, enhancing surface melt and ultimately in driving ice flow and ice shelf melt through injecting water into the subglacial environment and through facilitating fracturing. Here, we develop a model for the drainage of supraglacial lakes through the dissipation-driven incision of a surface channel. The model consists of the St Venant equations for flow in the channel, fed by an upstream lake reservoir, coupled with an equation for the evolution of channel elevation due to advection, uplift and downward melting. After reduction to a ‘stream power’-type hyperbolic model, we show that lake drainage occurs above a critical rate of water supply to the lake due to the backward migration of a shock that incises the lake seal. The critical water supply rate depends on advection velocity and uplift (or more precisely, drawdown downstream of the lake) as well as model parameters such as channel wall roughness and the parameters defining the relationship between channel cross-section and wetted perimeter. Once lake drainage does occur, it can either continue until the lake is empty, or terminate early, leading to oscillatory cycles of lake filling and draining, with the latter favoured by large lake volumes and relatively small water supply rates.

The problem of hydroelastic wave diffraction by a surface-piercing vertical circular cylinder mounted on the bottom of an ice-covered channel is considered. The ice sheet is modelled as an elastic thin plate with homogeneous properties, while the linearized velocity potential theory is adopted to describe the motion of the fluid. The solution starts from the Green function satisfying all other boundary conditions apart from that on the body surface. This is obtained through applying a Fourier transform in the longitudinal direction of the channel and adopting an eigenfunction expansion in the vertical direction. The boundary conditions on the side walls and ice edges are imposed through an orthogonal product. Through the Green function, the velocity potential due to a surface-piercing structure with arbitrary shape can be expressed through a source distribution formula derived in this work, in which only integrals over the body surface and its interaction line with the ice sheet need to be retained. For a vertical circular cylinder, the unknown source distribution can be expanded further into a Fourier series in the circumferential direction, and then the analytical solution of the velocity potential can be obtained further. Extensive results and discussions are provided for the hydrodynamic forces and vertical shear forces on the cylinder, as well as the deflection and strain of the ice sheet. In particular, the behaviour of the solution near one of the natural frequencies of the channel is investigated in detail.

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach and reattach. These problems are of particular importance in glaciology, where they arise in the study of grounding lines and subglacial cavities. In this work, we propose a novel numerical method for solving viscous contact problems based on a mixed formulation with Lagrange multipliers of a variational inequality involving the Stokes equations. The advection equation for evolving the geometry of the domain occupied by the fluid is then solved via a specially-built upwinding scheme, leading to a robust and accurate algorithm for viscous contact problems. We first verify the method by comparing the numerical results to analytical results obtained by a linearised method. Then we use this numerical scheme to reconstruct friction laws for glacial sliding with cavitation. Finally, we compute the evolution of cavities from a steady state under oscillating water pressures. The results depend strongly on the location of the initial steady state along the friction law. In particular, we find that if the steady state is located on the downsloping or rate-weakening part of the friction law, then the cavity evolves towards the upsloping section, indicating that the downsloping part is unstable.

Ice breaking has become one of the main problems faced by ships and other equipment operating in an ice-covered water region. New methods are always being pursued and studied to improve ice-breaking capabilities and efficiencies. Based on the strong damage capability, a high-speed water jet impact is proposed to be used to break an ice plate in contact with water. A series of experiments of water jet impacting ice were performed in a transparent water tank, where the water jets at tens of metres per second were generated by a home-made device and circular ice plates of various thicknesses and scales were produced in a cold room. The entire evolution of the water jet and ice was recorded by two high-speed cameras from the top and front views simultaneously. The focus was the responses of the ice plate, such as crack development and breakup, under the high-speed water jet loads, which involved compressible pressure ${P_1}$ and incompressible pressure ${P_2}$. According to the main cause and crack development sequence, it was found that the damage of the ice could be roughly divided into five patterns. On this basis, the effects of water jet strength, ice thickness, ice plate size and boundary conditions were also investigated. Experiments validated the ice-breaking capability of the high-speed water jet, which could be a new auxiliary ice-breaking method in the future.

The problem of interaction of a uniform current with a submerged horizontal circular cylinder in an ice-covered channel is considered. The fluid flow is described by linearized velocity potential theory and the ice sheet is treated as a thin elastic plate. The potential due to a source or the Green function satisfying all boundary conditions apart from that on the body surface is first derived. This can be used to derive the boundary integral equation for a body of arbitrary shape. It can also be used to obtain the solution due to multipoles by differentiating the Green function with its position directly. For a transverse circular cylinder, through distributing multipoles along its centre line, the velocity potential can be written in an infinite series with unknown coefficients, which can be determined from the impermeable condition on a body surface. A major feature here is that different from the free surface problem, or a channel without the ice sheet cover, this problem is fully three-dimensional because of the constraints along the intersection of the ice sheet with the channel wall. It has been also confirmed that there is an infinite number of critical speeds. Whenever the current speed passes a critical value, the force on the body and wave pattern change rapidly, and two more wave components are generated at the far-field. Extensive results are provided for hydroelastic waves and hydrodynamic forces when the ice sheet is under different edge conditions, and the insight of their physical features is discussed.

The flexural-gravity symmetric waves propagating in an ice channel with a lead of open water in the ice cover are investigated within the linear theory of hydroelasticity. The ice sheets are modelled as either elastic or as rigid plates. Deflections of the elastic ice sheets are described by the linear elastic plate equation. The flexural-gravity waves propagating along the channel and their dispersion relations are obtained by the normal mode method. It is shown that the dispersion relations for elastic ice approach the dispersion relations for rigid ice as the rigidity of ice increases or the wavelength decreases. The region in the plane ice thickness/wavenumber, where the ice cover can be treated as rigid with a prescribed accuracy, is determined. The effects of the ice thickness and width of the lead on the phase speeds are studied to determine critical speeds of a vehicle moving along the channel. It is shown that wave modes have several critical speeds, which may merge for some parameters of the channel. The presence of the lead significantly changes the characteristics of waves in an ice channel. Short waves, which propagate in a channel fully covered with ice, weakly depend on gravity and the presence of water in the channel. However, short waves propagating in the same channel but with a lead of open water are gravity-dominated waves localised in the lead. We conclude that flexural-gravity waves in an ice channel with a lead are less pronounced than in the channel completely covered by ice.

Grounding lines exist where land-based glacial ice flows on to a body of water. Accurately modelling grounding-line migration at the ice–ocean interface is essential for estimating future ice-sheet mass change. On the interior of ice sheets, the shores of subglacial lakes are also grounding lines. Grounding-line positions are sensitive to water volume changes such as sea-level rise or subglacial-lake drainage. Here, we introduce numerical methods for simulating grounding-line dynamics in the marine ice sheet and subglacial-lake settings. Variational inequalities arise from contact conditions that relate normal stress, water pressure and velocity at the base. Existence and uniqueness of solutions to these problems are established using a minimisation argument. A penalty method is used to replace the variational inequalities with variational equations that are solved using a finite-element method. We illustrate the grounding-line response to tidal cycles in the marine ice-sheet problem and filling–draining cycles in the subglacial-lake problem. We introduce two computational benchmarks where the known lake volume change is used to measure the accuracy of the numerical method.