Wrinkling composite sheets.

We examine the buckling shape and critical compression of confined inhomogeneous composite sheets lying on a liquid foundation. The buckling modes are controlled by the bending stiffness of the sheet, the density of the substrate, and the size and the spatially dependent elastic coefficients of the sheet. We solve the beam equation describing the mechanical equilibrium of a sheet when its bending stiffness varies parallel to the direction of confinement. The case of a homogeneous bending stiffness exhibits a degeneracy of wrinkled states for certain lengths of the confined sheet; we explain this degeneracy using an asymptotic analysis valid for long sheets, and show that it corresponds to the switching of the sheet between symmetric and antisymmetric buckling modes. This degeneracy disappears for spatially dependent elastic coefficients. Medium length sheets buckle similarly to their homogeneous counterparts, whereas the wrinkled states in large length sheets concentrate the bending energy towards the soft regions of the sheet.

Stochastic paleoclimatology: Modeling the EPICA ice core climate records.

We analyze and model the stochastic behavior of paleoclimate time series and assess the implications for the coupling of climate variables during the Pleistocene glacial cycles. We examine 800 kiloyears of carbon dioxide, methane, nitrous oxide, and temperature proxy data from the European Project for Ice Coring in Antarctica (EPICA) Dome-C ice core, which are characterized by 100 ky glacial cycles overlain by fluctuations across a wide range of timescales. We quantify this behavior through multifractal time-weighted detrended fluctuation analysis, which distinguishes near-red-noise and white-noise behavior below and above the 100 ky glacial cycle, respectively, in all records. This allows us to model each time series as a one-dimensional periodic nonautonomous stochastic dynamical system, and assess the stability of physical processes and the fidelity of model-simulated time series. We extend this approach to a four-variable model with intervariable coupling terms, which we interpret in terms of possible interrelationships among the four time series. Within the framework of our coupling coefficients, we find that carbon dioxide and temperature act to stabilize each other and methane and nitrous oxide, whereas the latter two destabilize each other and carbon dioxide and temperature. We also compute the response function for each pair of variables to assess the model performance by comparison to the data and confirm the model predictions regarding stability amongst variables. Taken together, our results are consistent with glacial pacing dominated by carbon dioxide and temperature that is modulated by terrestrial biosphere feedbacks associated with methane and nitrous oxide emissions.

Impurity effects in thermal regelation.

When a particle is placed in a material with a lower bulk melting temperature, intermolecular forces can lead to the existence of a "premelted" liquid film of the lower melting temperature material. Despite the system being below the melting temperatures of both solids, the liquid film is a consequence of thermodynamic equilibrium, controlled by intermolecular, ionic and other interactions. An imposed temperature gradient drives the translation of the particle by a process of melting and refreezing known as "thermal regelation". We calculate the rate of regelation of spherical particles surrounded by premelted films that contain ionic impurities. The impurities enhance the rate of motion thereby influencing the dynamics of single particles and distributions of particles, which we describe in addition to the consequences in natural and technological settings.

Predicting critical transitions in multiscale dynamical systems using reservoir computing.

We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results.

Surface phase transitions in ice: from fundamental interactions to applications.

Interfaces divide all phases of matter and yet in most practical settings it is tempting to ignore their energies and the associated implications. There are many reasons for this, not the least of which is the introduction of a new pair of canonically conjugate variables-interfacial energy and its counterpart the surface area. A key set of questions surrounding the treatment of multiphase flows concerns how and when we must account for such effects. I begin this discussion with an abbreviated review of the basic theory of lower-dimensional phase transitions and describe a range of situations in which the bulk behaviour of a two-phase (and in some cases two-component) system is dominated by surface effects. Then I discuss a number of settings in which the bulk and surface behaviour can interact on equal footing. These can include the dynamic and thermodynamic behaviour of floating sea ice, the freezing and drying of colloidal suspensions (such as soil) and the mechanisms of protoplanetesimal formation by inter-particle collisions in accretion discs. This article is part of the theme issue 'The physics and chemistry of ice: scaffolding across scales, from the viability of life to the formation of planets'.

Intrinsic Pink-Noise Multidecadal Global Climate Dynamics Mode.

Understanding multidecadal variability is an essential goal of climate dynamics. For example, the recent phenomenon referred to as the "global warming hiatus" may reflect a coupling to an intrinsic, preindustrial, multidecadal variability process. Here, using a multifractal time-series method, we demonstrate that 42 data sets of 79 proxies with global coverage exhibit pink-noise characteristics on multidecadal timescales. To quantify the persistence of this behavior, we examine high-resolution ice core and speleothem data to find pink noise in both pre- and postindustrial periods. We examine the spatial structure with an empirical orthogonal function analysis of the monthly averaged surface temperature from 1901 to 2012. The first mode clearly shows the distribution of ocean heat flux sinks located in the eastern Pacific and the Southern Ocean and has pink-noise characteristics on a multidecadal timescale. We hypothesize that this pink-noise multidecadal spatial mode may resonate with externally driven greenhouse gas forcing, driving large-scale climate processes.

Theory of the Sea Ice Thickness Distribution.

We use concepts from statistical physics to transform the original evolution equation for the sea ice thickness distribution g(h) from Thorndike et al. into a Fokker-Planck-like conservation law. The steady solution is g(h)=N(q)h(q)e(-h/H), where q and H are expressible in terms of moments over the transition probabilities between thickness categories. The solution exhibits the functional form used in observational fits and shows that for h≪1, g(h) is controlled by both thermodynamics and mechanics, whereas for h≫1 only mechanics controls g(h). Finally, we derive the underlying Langevin equation governing the dynamics of the ice thickness h, from which we predict the observed g(h). The genericity of our approach provides a framework for studying the geophysical-scale structure of the ice pack using methods of broad relevance in statistical mechanics.

Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses.

Droplets deform soft substrates near their contact lines. Using confocal microscopy, we measure the deformation of silicone gel substrates due to glycerol and fluorinated-oil droplets for a range of droplet radii and substrate thicknesses. For all droplets, the substrate deformation takes a universal shape close to the contact line that depends on liquid composition, but is independent of droplet size and substrate thickness. This shape is determined by a balance of interfacial tensions at the contact line and provides a novel method for direct determination of the surface stresses of soft substrates. Moreover, we measure the change in contact angle with droplet radius and show that Young's law fails for small droplets when their radii approach an elastocapillary length scale. For larger droplets the macroscopic contact angle is constant, consistent with Young's law.

Grain boundary melting in ice.

We describe an optical scattering study of grain boundary premelting in water ice. Ubiquitous long ranged attractive polarization forces act to suppress grain boundary melting whereas repulsive forces originating in screened Coulomb interactions and classical colligative effects enhance it. The liquid enhancing effects can be manipulated by adding dopant ions to the system. For all measured grain boundaries this leads to increasing premelted film thickness with increasing electrolyte concentration. Although we understand that the interfacial surface charge densities q(s) and solute concentrations can potentially dominate the film thickness, we cannot directly measure them within a given grain boundary. Therefore, as a framework for interpreting the data we consider two appropriate q(s) dependent limits; one is dominated by the colligative effect and other is dominated by electrostatic interactions.

Density functional theory for Baxter's sticky hard spheres in confinement.

It has recently been shown that a free energy for Baxter's sticky hard-sphere fluid is uniquely defined within the framework of fundamental measure theory (FMT) for the inhomogeneous hard-sphere fluid, provided that it obeys scaled-particle theory and the Percus-Yevick (PY) result for the direct correlation function [H. Hansen-Goos and J. S. Wettlaufer, J. Chem. Phys. 134, 014506 (2011)]. Here, combining FMT weighted densities with a new vectorial weighted density, we regularize the divergences of the associated strongly confined limit. The free energy that emerges is exact in the zero-dimensional limit, leaves the underlying equation of state unaffected, and yields a direct correlation function distinct from that of PY. Comparison with simulation data for both the bulk pair-correlation function and the density profiles in confinement shows that the new theory is significantly more accurate than the PY-based results.

Nonlinear threshold behavior during the loss of Arctic sea ice.

In light of the rapid recent retreat of Arctic sea ice, a number of studies have discussed the possibility of a critical threshold (or "tipping point") beyond which the ice-albedo feedback causes the ice cover to melt away in an irreversible process. The focus has typically been centered on the annual minimum (September) ice cover, which is often seen as particularly susceptible to destabilization by the ice-albedo feedback. Here, we examine the central physical processes associated with the transition from ice-covered to ice-free Arctic Ocean conditions. We show that although the ice-albedo feedback promotes the existence of multiple ice-cover states, the stabilizing thermodynamic effects of sea ice mitigate this when the Arctic Ocean is ice covered during a sufficiently large fraction of the year. These results suggest that critical threshold behavior is unlikely during the approach from current perennial sea-ice conditions to seasonally ice-free conditions. In a further warmed climate, however, we find that a critical threshold associated with the sudden loss of the remaining wintertime-only sea ice cover may be likely.

Premelting controlled active matter in ice.

Self-propelled particles can undergo complex dynamics due to a range of bulk and surface interactions. When a particle is embedded in a host solid near its bulk melting temperature, the latter may melt at the surface of the former in a process known as interfacial premelting. The thickness of the melt film depends on the temperature, impurities, material properties and geometry. A temperature gradient is accompanied by a thermomolecular pressure gradient that drives the interfacial liquid from high to low temperatures and hence the particle from low to high temperatures, in a process called thermal regelation. When the host material is ice and the embedded particle is a biological entity, one has a particularly different form of active matter, which addresses interplay between a wide range of problems, from extremophiles of both terrestrial and exobiological relevance to ecological dynamics in Earth's cryosphere. Of basic importance in all such settings is the combined influence of biological activity and thermal regelation in controlling the redistribution of bioparticles. Therefore, we recast this class of regelation phenomena in the stochastic framework of active Ornstein-Uhlenbeck dynamics and make predictions relevant to this and related problems of interest in biological and geophysical problems. We examine how thermal regelation compromises paleoclimate studies in the context of ice core dating and we find that the activity influences particle dynamics during thermal regelation by enhancing the effective diffusion coefficient. Therefore, accurate dating relies on a quantitative treatment of both effects.

A fundamental measure theory for the sticky hard sphere fluid.

We construct a density functional theory (DFT) for the sticky hard sphere (SHS) fluid which, like Rosenfeld's fundamental measure theory (FMT) for the hard sphere fluid [Y. Rosenfeld, Phys. Rev. Lett. 63, 980 (1989)], is based on a set of weighted densities and an exact result from scaled particle theory (SPT). It is demonstrated that the excess free energy density of the inhomogeneous SHS fluid Φ(SHS) is uniquely defined when (a) it is solely a function of the weighted densities from Kierlik and Rosinberg's version of FMT [E. Kierlik and M. L. Rosinberg, Phys. Rev. A 42, 3382 (1990)], (b) it satisfies the SPT differential equation, and (c) it yields any given direct correlation function (DCF) from the class of generalized Percus-Yevick closures introduced by Gazzillo and Giacometti [J. Chem. Phys. 120, 4742 (2004)]. The resulting DFT is shown to be in very good agreement with simulation data. In particular, this FMT yields the correct contact value of the density profiles with no adjustable parameters. Rather than requiring higher order DCFs, such as perturbative DFTs, our SHS FMT produces them. Interestingly, although equivalent to Kierlik and Rosinberg's FMT in the case of hard spheres, the set of weighted densities used for Rosenfeld's original FMT is insufficient for constructing a DFT which yields the SHS DCF.

Dynamics of colloidal particles in ice.

We use x-ray photon correlation spectroscopy (XPCS) to probe the dynamics of colloidal particles in polycrystalline ice. During freezing, the dendritic ice morphology and rejection of particles from the ice created regions of high particle density, where some of the colloids were forced into contact and formed disordered aggregates. The particles in these high density regions underwent ballistic motion, with a characteristic velocity that increased with temperature. This ballistic motion is coupled with both stretched and compressed exponential decays of the intensity autocorrelation function. We suggest that this behavior could result from ice grain boundary migration.

Analytical solution of stochastic resonance in the nonadiabatic regime.

We generalize stochastic resonance to the nonadiabatic limit by treating the double-well potential using two quadratic potentials. We use a singular perturbation method to determine an approximate analytical solution for the probability density function that asymptotically connects local solutions in boundary layers near the two minima with those in the region of the maximum that separates them. The validity of the analytical solution is confirmed numerically. Free from the constraints of the adiabatic limit, the approach allows us to predict the escape rate from one stable basin to another for systems experiencing a more complex periodic forcing.

Maximal potential energy transport: a variational principle for solidification problems.

We analyze numerically the mechanisms controlling the spacing of chimneys--channels devoid of solid--in two-dimensional mushy layers formed by solidifying a binary alloy. Chimneys are the principal conduits through which buoyancy effects material transport out of the mushy layer and into the liquid from which it formed. Experiments show a coarsening of chimney spacing; we pursue the hypothesis that the spacing adjusts to optimize material transport and hence maximize the rate of removal of potential energy stored in the mushy layer. The optimal solute flux increases approximately linearly with the mush Rayleigh number. However, for spacings below a critical value, the chimneys collapse and solute fluxes cease, revealing a hysteresis between chimney convection and no flow. The results are consistent with a variational principle controlling the dynamics of this dissipative system.

Streaks to rings to vortex grids: generic patterns in transient convective spin up of an evaporating fluid.

We observe the transient formation of a ringed pattern state during spin up of an evaporating fluid on a time scale of order a few Ekman spin up times. The ringed state is probed using infrared thermometry and particle image velocimetry and it is demonstrated to be a consequence of the transient balance between Coriolis and viscous forces which dominate inertia, each of which are extracted from the measured velocity field. The breakdown of the ringed state is quantified in terms of the antiphasing of these force components which drives a Kelvin-Helmholtz instability and we show that the resulting vortex grid spacing scales with the ring wavelength. This is the fundamental route to quasi-two-dimensional turbulent vortex flow and thus may have implications in astrophysics and geophysics wherein rotating convection is ubiquitous.

Light scattering from an isotropic layer between uniaxial crystals.

We develop a model for the reflection and transmission of plane waves by an isotropic layer sandwiched between two uniaxial crystals of arbitrary orientation. In the laboratory frame, reflection and transmission coefficients corresponding to the principal polarization directions in each crystal are given explicitly in terms of the [Formula: see text] axis and propagation directions. The solution is found by first deriving explicit expressions for reflection and transmission amplitude coefficients for waves propagating from an arbitrarily oriented uniaxial anisotropic material into an isotropic material. By combining these results with Lekner's (1991 J. Phys.: Condens. Matter3 6121-33) earlier treatment of waves propagating from isotropic media to anisotropic media and employing a matrix method we determine a solution to the general form of the multiple reflection case. The example system of a wetted interface between two ice crystals is used to contextualize the results.