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.

James Croll's Physical Theory of Secular Changes of Climate emerged during an age of revolution in geology that included the rise of the glacial theory and the search for its underlying causes. According to Croll, periods of high eccentricity are associated with the persistence of long glacial epochs, within which glaciations occur in alternate hemispheres when winter is at aphelion every ~11,000 years; however, astronomical forcing is only able to produce glaciation by means of physical agencies (climate feedbacks) that amplify the small effects of varying seasonal irradiation. Croll understood the importance of interglacial deposits because they provided evidence for the occurrence of multiple glaciations within his long glacial epochs. He was aware of the limitations of the terrestrial record and suggested that deep-sea sediments would contain a continuous succession of glacial-interglacial cycles. Contrary to a widespread view, however, Croll was not envisaging the advent of palaeoceanographic exploration avant la lettre, but instead was drawing attention to the inadequacy of the land record as a testbed of his astronomical theory. Yet, the marine record did eventually deliver a test of astronomical theories almost exactly 100 years after the publication of his 1875 book Climate and Time in their Geological Relations. Here, we provide an historical account of the technological and scientific developments that led to this and a summary of insights on astronomically paced climate changes from marine, terrestrial and ice core records. We finally assess Croll's ideas in the context of our current understanding of the theory of ice ages.

Wave radiation and diffraction problems of a body floating in an open water channel confined by two semi-infinite ice sheets are considered. The linearized velocity potential theory is used for fluid flow and a thin elastic plate model is adopted for the ice sheet. The Green function, which satisfies all the boundary conditions apart from that on the body surface, is first derived. This is obtained through applying Fourier transform in the longitudinal direction of the channel, and matched eigenfunction expansions in the transverse plane. With the help of the derived Green function, the boundary integral equation of the potential is derived and it is shown that the integrations over all other boundaries, including the bottom of the fluid, free surface, ice sheet, ice edge as well as far field will be zero, and only the body surface has to be retained. This allows the problem to be solved through discretization of the body surface only. Detailed results for hydrodynamic forces are provided, which are generally highly oscillatory owing to complex wave–body–channel interaction and body–body interaction. In depth investigations are made for the waves confined in a channel, which does not decay at infinity. Through this, a detailed analysis is presented on how the wave generated by a body will affect the other bodies even when they are far apart.

The scattering of flexural-gravity waves in a thin floating plate is investigated in the presence of compression. In this case, wave blocking occurs, which is associated with both a zero in the group velocity and coalition of two or more roots of the related dispersion relation. There exists a region in the frequency space in which there are three real roots of the dispersion equation and hence three propagating modes. This multiplicity leads to mode conversion when scattering occurs. In one of these modes, the energy propagation direction is opposite to the wavenumber, making enforcement of the Sommerfeld radiation condition challenging. The focus here is on a canonical problem in flexural-gravity wave scattering, the scattering of waves by a crack. Formulae are developed that apply uniformly at all frequencies, including through the blocking frequencies. This solution is developed by tracking the movement of the dispersion relation roots carefully in the complex plane. The mode conversion is verified by the scattering matrix of the process and through an energy identity. This energy identity for the case of more than one progressive modes is established using Green's theorem and later applied in the scattering matrix to identify the incident and transmitted waves in the scattering process and derive the radiation condition. Appropriate scaling of the reflection and transmission coefficients are provided with the energy identity. The solution method is illustrated with numerical examples.

In the linear approximation, we study wave motions of a compressed elastic ice sheet caused by the motion of a two-dimensional dipole in the water beneath the sheet. The fluid flow is described by the potential theory, while the ice sheet is modelled through a thin elastic plate floating on the water surface. The solution for the vertical displacement of the ice sheet is derived for a transient dipole undergoing arbitrary two-dimensional motion. Three cases are considered in detail when the dipole moves horizontally with a uniform speed at some depth or horizontally oscillates, or moves and oscillates. The formulae for the plate displacement are derived for the fluid of finite depth, but then analysed in detail for the infinitely deep case. We show that the character of the solutions is different in the different domains of the parameter plane and classify the possible cases. Then we calculate the wave patterns on the plate for the different regimes of dipole motion and typical values of plate parameters. The studied problem can be considered as the simplified model of motion of a circular cylinder in a water under an ice cover. In the last section we compare the characteristics of wave motions onsetting in the far-field zone of the flow around a circular cylinder and its dipole approximation and show that the difference in the wave characteristics and force loads for these two cases is small and quickly vanishes when the ice plate thickness increases. In conclusion, we present estimates of amplitudes and wavelengths of wave perturbations for the real oceanic conditions.

Flexural-gravity wave interactions with multiple cracks in an ice sheet of infinite extent are considered, based on the linearized velocity potential theory for fluid flow and thin elastic plate model for an ice sheet. Both the shape and location of the cracks can be arbitrary, while an individual crack can be either open or closed. Free edge conditions are imposed at the crack. For open cracks, zero corner force conditions are further applied at the crack tips. The solution procedure starts from series expansion in the vertical direction based on separation of variables, which decomposes the three-dimensional problem into an infinite number of coupled two-dimensional problems in the horizontal plane. For each two-dimensional problem, an integral equation is derived along the cracks, with the jumps of displacement and slope of the ice sheet as unknowns in the integrand. By extending the crack in the vertical direction into the fluid domain, an artificial vertical surface is formed, on which an orthogonal inner product is adopted for the vertical modes. Through this, the edge conditions at the cracks are satisfied, together with continuous conditions of pressure and velocity on the vertical surface. The integral differential equations are solved numerically through the boundary element method together with the finite difference scheme for the derivatives along the crack. Extensive results are provided and analysed for cracks with various shapes and locations, including the jumps of displacement and slope, diffraction wave coefficient, and the scattered cross-section.

The hydroelastic waves in a channel covered by an ice sheet, without or with crack and subject to various edge constraints at channel banks, are investigated based on the linearized velocity potential theory for the fluid domain and the thin-plate elastic theory for the ice sheet. An effective analytical solution procedure is developed through expanding the velocity potential and the fourth derivative of the ice deflection to a series of cosine functions with unknown coefficients. The latter are integrated to obtain the expression for the deflection, which involves four constants. The procedure is then extended to the case with a longitudinal crack in the ice sheet by using the Dirac delta function and its derivatives at the crack in the dynamic equation, with unknown jumps of deflection and slope at the crack. Conditions at the edges and crack are then imposed, from which a system of linear equations for the unknowns is established. From this, the dispersion relation between the wave frequency and wavenumber is found, as well as the natural frequency of the channel. Extensive results are then provided for wave celerity, wave profiles and strain in the ice sheet. In-depth discussions are made on the effects of the edge condition, and the crack.