# Publications by Andrew Wells

## Thermal Convection over Fractal Surfaces

ArXiV [physics.flu-dyn] (2019)

## Thermal convection over fractal surfaces

Journal of Fluid Mechanics Cambridge University Press **907** (2020) A12

We use well resolved numerical simulations with the Lattice Boltzmann Method to study Rayleigh-B´enard convection in cells with a fractal boundary in two dimensions for P r = 1 and Ra ∈ [10^7 , 10^10]. The fractal boundaries are functions characterized by power spectral densities S(k) that decay with wavenumber, k, as S(k) ∼ k^p (p < 0). The degree of roughness is quantified by the exponent p with p < −3 for smooth (differentiable) surfaces and −3 ≤ p < −1 for rough surfaces with Hausdorff dimension D_f =1/2 (p + 5). By computing the exponent β in power law fits Nu ∼ Ra^β, where Nu and Ra are the Nusselt and the Rayleigh numbers for Ra ∈ [10^8, 10^10], we observe that heat transport scaling increases with roughness over the top two decades of Ra ∈ [10^8, 10^10]. For p = −3.0, −2.0 and −1.5 we find β = 0.288 ± 0.005, 0.329 ± 0.006 and 0.352 ± 0.011, respectively. We also observe that the Reynolds number, Re, scales as Re ∼ Ra^ξ , where ξ ≈ 0.57 over Ra ∈ [10^7, 10^10], for all p used in the study. For a given value of p, the averaged Nu and Re are insensitive to the specific realization of the roughness.

## Modeling sea ice

Notices of the American Mathematical Society American Mathematical Society **67** (2020) 1535-1555

## A synthesis of thermodynamic ablation at ice-ocean interfaces from theory, observations and models

Ocean Modelling Elsevier **154** (2020) 101692

Thermodynamic ablation of ice in contact with the ocean is an essential element of ice sheet and ocean interactions but is challenging to model and quantify. Building on earlier observations of sea ice ablation, a variety of recent theoretical, experimental and observational studies have considered ice ablation in contrasting geometries, from vertical to near-horizontal ice faces, and reveal different scaling behaviour for predicted ablation rates in different dynamical regimes. However, uncertainties remain about when the contrasting results should be applied, as existing model parameterisations do not capture all relevant regimes of ice–ocean ablation. To progress towards improved models of ice–ocean interaction, we synthesise current understanding into a classification of ablation types. We examine the effect of the classification on the parameterisation of turbulent fluxes from the ocean towards the ice, and identify the dominant processes next to ice interfaces of different orientation. Four ablation types are defined: melting and dissolving based on ocean temperatures, and shear-controlled and buoyancy-controlled regimes based on the dynamics of the near-ice molecular sublayer. We describe existing observational and modelling studies of sea ice, ice shelves, and glacier termini, as well as laboratory studies, to show how they fit into this classification. Two sets of observations from the Ross and Ronne Ice Shelf cavities suggest that both the buoyancy-controlled and shear-controlled regimes may be relevant under different oceanographic conditions. Overall, buoyancy-controlled dynamics are more likely when the molecular sublayer has lower Reynolds number, and shear for higher Reynolds number, although the observations suggest some variability about this trend.

## The dynamics of a subglacial salt wedge

Journal of Fluid Mechanics Cambridge University Press **895** (2020) A20

Marine-terminating glaciers, such as those along the coastline of Greenland, often release meltwater into the ocean in the form of subglacial discharge plumes. Though these plumes can dramatically alter the mass loss along the front of a glacier, the conditions surrounding their genesis remain poorly constrained. In particular, little is known about the geometry of subglacial outlets and the extent to which seawater may intrude into them. Here, the latter is addressed by exploring the dynamics of an arrested salt wedge – a steady-state, two-layer flow system where salty water partially intrudes a channel carrying fresh water. Building on existing theory, we formulate a model that predicts the length of a non-entraining salt wedge as a function of the Froude number, the slope of the channel and coefficients for interfacial and wall drag. In conjunction, a series of laboratory experiments were conducted to observe a salt wedge within a rectangular channel. For experiments conducted with laminar flow (Reynolds number <i>Re < 800</i>), good agreement with theoretical predictions are obtained when the drag coefficients are modelled as being inversely proportional to <i>Re</i>. However, for fully turbulent flows on geophysical scales, these drag coefficients are expected to asymptote toward finite values. Adopting reasonable drag coefficient estimates for this flow regime, our theoretical model suggests that typical subglacial channels may permit seawater intrusions of the order of several kilometres. While crude, these results indicate that the ocean has a strong tendency to penetrate subglacial channels and potentially undercut the face of marine-terminating glaciers.

## Modelling binary alloy solidification with adaptive mesh refinement

Journal of Computational Physics: X **5** (2020)

© 2019 The solidification of a binary alloy results in the formation of a porous mushy layer, within which spontaneous localisation of fluid flow can lead to the emergence of features over a range of spatial scales. We describe a finite volume method for simulating binary alloy solidification in two dimensions with local mesh refinement in space and time. The coupled heat, solute, and mass transport is described using an enthalpy method with flow described by a Darcy-Brinkman equation for flow across porous and liquid regions. The resulting equations are solved on a hierarchy of block-structured adaptive grids. A projection method is used to compute the fluid velocity, whilst the viscous and nonlinear diffusive terms are calculated using a semi-implicit scheme. A series of synchronization steps ensure that the scheme is flux-conservative and correct for errors that arise at the boundaries between different levels of refinement. We also develop a corresponding method using Darcy's law for flow in a porous medium/narrow Hele-Shaw cell. We demonstrate the accuracy and efficiency of our method using established benchmarks for solidification without flow and convection in a fixed porous medium, along with convergence tests for the fully coupled code. Finally, we demonstrate the ability of our method to simulate transient mushy layer growth with narrow liquid channels which evolve over time.

## Solidification of binary aqueous solutions under periodic cooling. Part 1. Dynamics of mushy-layer growth

Journal of Fluid Mechanics Cambridge University Press **870** (2019) 121-146

We present studies of the solidification of binary aqueous solutions that undergo time-periodic cooling from below. We develop an experiment for solidification of aqueous NH4Cl solutions, where the temperature of the cooling boundary is modulated as a simple periodic function of time with independent variations of the modulation amplitude and frequency. The thickness of the mushy layer exhibits oscillations about the background growth obtained for constant cooling. We consider the deviation given by the difference between states with modulated and fixed cooling, which increases when the modulation amplitude increases but decreases with increasing modulation frequency. At early times, the deviation amplitude is consistent with a scaling argument for growth with quasi-steady modulation. In situ measurements of the mush temperature reveal thermal waves propagating through the mushy layer, with amplitude decaying with height within the mushy layer, whilst the phase lag behind the cooling boundary increases with height. This also leads to phase lags in the variation of the mushy-layer thickness compared to the boundary cooling. There is an asymmetry of the deviation of mushy-layer thickness: during a positive modulation (where the boundary temperature increases at the start of a cycle) the peak thickness deviation has a greater magnitude than the troughs in a negative modulation mode (where the boundary temperature decreases at the start of the cycle). A numerical model is formulated to describe mushy-layer growth with constant bulk concentration and turbulent heat transport at the mush–liquid interface driven by compositional convection associated with a finite interfacial solid fraction. The model recovers key features of the experimental results at early times, including the propagation of thermal waves and oscillations in mushy-layer thickness, although tends to overpredict the mean thickness.

## Solidification of binary aqueous solutions under periodic cooling. Part 2. Distribution of solid fraction

Journal of Fluid Mechanics Cambridge University Press **870** (2019) 147-174

We report an experimental study of the distributions of temperature and solid fraction of growing NH4Cl–H2O mushy layers that are subjected to periodical cooling from below, focusing on late-time dynamics where the mushy layer oscillates about an approximate steady state. Temporal evolution of the local temperature T(z, t) at various heights in the mush demonstrates that the temperature oscillations of the bottom cooling boundary propagate through the mushy layer with phase delays and substantial decay in the amplitude. As the initial concentration C0 increases, we show that the decay rate of the thermal oscillation with height also decreases, and the propagation speed of the oscillation phase increases. We interpret this as a result of the solid fraction increasing with C0, which enhances the thermal conductivity but reduces the specific heat of the mushy layer. We present a new methodology to determine the distribution of solid fraction φ(z) in mushy layers for various C0, using only measurements of the temperature T(z, t). The method is based on the phase behaviour during thermal modulation, and opens up a new approach for inferring mushy-layer properties in geophysical and engineering settings, where direct measurements are challenging. In our experiments, profiles of the solid fraction φ(z) exhibit a cliff–ramp–cliff structure with large vertical gradients of φ near the mush–liquid interface and also near the bottom boundary, but much more gradual variation in the interior of the mushy layer. Such a profile structure is more pronounced for higher initial concentration C0. For very low concentration, the solid fraction appears to be linearly dependent on the height within the mush. The volume-average of the solid fraction, and the local fluctuations in φ(z) both increase as C0 increases. We suggest that the fast increase of φ(z) near the bottom boundary is possibly due to diffusive transport of solute away from the bottom boundary and the depletion of solute content near the basal region

## Mushy layer growth and convection, with application to sea ice

Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Royal Society **377** (2019)

Sea ice is a reactive porous medium of ice crystals and liquid brine, which is an example of a mushy layer. The phase behaviour of sea ice controls the evolving material properties and fluid transport through the porous ice, with consequences for ice growth, brine drainage from the ice to provide buoyancy fluxes for the polar oceans, and sea-ice biogeochemistry. We review work on the growth of mushy layers and convective flows driven by density gradients in the interstitial fluid. After introducing the fundamentals of mushy-layer theory, we discuss the effective thermal properties including the impact of salt transport on mushy-layer growth. We present a simplified model for diffusively controlled growth of mushy layers with modest cooling versus the solutal freezing-point depression. For growth from a cold isothermal boundary, salt diffusion modifies mushy layer growth by around 5-20% depending on the far-field temperature and salinity. We also review work on the onset, spatial localisation and nonlinear development of convective flows in mushy layers, highlighting recent work on transient solidification and models of nonlinear convection with dissolved solid-free brine channels. Finally, future research opportunities are identified, motivated by geophysical observations of ice growth.

## Salinity control of thermal evolution of late summer melt ponds on Arctic sea ice

Geophysical Research Letters American Geophysical Union **45** (2018) 8304-8313

The thermal evolution of melt ponds on Arctic sea ice was investigated through a combination of autonomous observations and two‐dimensional high‐resolution fluid dynamics simulations. We observed one relatively fresh pond and one saline pond on the same ice floe, with similar depth. The comparison of observations and simulations indicates that thermal convection dominates in relatively fresh ponds, but conductive heat transfer dominates in salt‐stratified ponds. Using a parameterized surface energy balance, we estimate that the heat flux to the ice is larger under the saline pond than the freshwater pond when averaged over the observational period. The deviation is sensitive to assumed wind, varying between 3 and 14 W/m2 for winds from 0 to 5 m/s. If this effect persists as conditions evolve through the melt season, our results suggest that this imbalance potentially has a climatologically significant impact on sea‐ice evolution.

## Frazil-ice growth rate and dynamics in mixed layers and sub-ice-shelf plumes

Cryosphere European Geosciences Union **12** (2018) 25-38

The growth of frazil or granular ice is an important mode of ice formation in the cryosphere. Recent advances have improved our understanding of the microphysical processes that control the rate of ice-crystal growth when water is cooled beneath its freezing temperature. These advances suggest that crystals grow much faster than previously thought. In this paper, we consider models of a population of ice crystals with different sizes to provide insight into the treatment of frazil ice in large-scale models. We consider the role of crystal growth alongside the other physical processes that determine the dynamics of frazil ice. We apply our model to a simple mixed layer (such as at the surface of the ocean) and to a buoyant plume under a floating ice shelf. We provide numerical calculations and scaling arguments to predict the occurrence of frazilice explosions, which we show are controlled by crystal growth, nucleation and, gravitational removal. Faster crystal growth, higher secondary nucleation and slower gravitational removal make frazil-ice explosions more likely. We identify steady-state crystal size distributions, which are largely insensitive to crystal growth rate but are affected by the relative importance of secondary nucleation to gravitational removal. Finally, we show that the fate of plumes underneath ice shelves is dramatically affected by frazil-ice dynamics. Differences in the parameterization of crystal growth and nucleation give rise to radically different predictions of basal accretion and plume dynamics; and can even impact whether a plume reaches the end of the ice shelf or intrudes at depth.

## Frazil-ice growth rate and dynamics in mixed layers and sub-ice-shelf plumes

Cryosphere Discussions European Geosciences Union **12** (2017) 25-38

The growth of frazil or granular ice is an important mode of ice formation in the cryosphere. Recent advances have improved our understanding of the microphysical processes that control the rate of ice-crystal growth when water is cooled beneath its freezing temperature. These advances suggest that crystals grow much faster than previously thought. In this paper, we consider models of a population of ice crystals with different sizes to provide insight into the treatment of frazil ice in large-scale models. We consider the role of crystal growth alongside the other physical processes that determine the dynamics of frazil ice. We apply our model to a simple mixed layer (such as at the surface of the ocean) and to a buoyant plume under a floating ice shelf. We provide numerical calculations and scaling arguments to predict the occurrence of frazil-ice explosions, which we show are controlled by crystal growth, nucleation and, gravitational removal. Faster crystal growth, higher secondary nucleation and slower gravitational removal make frazil-ice explosions more likely. We identify steady-state crystal size distributions, which are largely insensitive to crystal growth rate but are affected by the relative importance of secondary nucleation to gravitational removal. Finally, we show that the fate of plumes underneath ice shelves is dramatically affected by frazil-ice dynamics. Differences in the parameterization of crystal growth and nucleation give rise to radically different predictions of basal accretion and plume dynamics; and can even impact whether a plume reaches the end of the ice shelf or intrudes at depth.

## Turbulent plumes from a glacier terminus melting in a stratified ocean

Journal of Geophysical Research: Oceans American Geophysical Union **121** (2016) 4670-4696

The melting of submerged faces of marine-terminating glaciers is a key contributor to the glacial mass budget via direct thermodynamic ablation and the impact of ablation on calving. This study considers the behavior of turbulent plumes of buoyant meltwater in a stratified ocean, generated by melting of either near-vertical calving faces or sloping ice shelves. We build insight by applying a turbulent plume model to describe melting of a locally planar region of ice face in a linearly stratified ocean, in a regime where subglacial discharge is insignificant. The plumes rise until becoming neutrally buoyant, before intruding into the ocean background. For strong stratifications, we obtain leading-order scaling laws for the flow including the height reached by the plume before intrusion, and the melt rate, expressed in terms of the background ocean temperature and salinity stratifications. These scaling laws provide a new perspective for parameterizing glacial melting in response to a piecewise-linear discretization of the ocean stratification.

## The impact of imperfect heat transfer on the convective instability of a thermal boundary layer in a porous media

Journal of Fluid Mechanics Cambridge University Press (2016)

We consider convective instability in a deep porous medium cooled from above with a linearised thermal exchange at the upper surface, thus determining the impact of using a Robin boundary condition, in contrast to previous previous studies using a Dirichlet boundary condition. With the linearised surface exchange, the thermal flux out of the porous layer depends linearly on the temperature difference between the effective temperature of a heat sink at the upper boundary and the temperature at the surface of the porous layer. The rate of this exchange is characterised by a dimensionless Biot number, Bi, determined by the effective thermal conductivity of exchange with the heat sink relative to the physical thermal conductivity of the porous layer. For a given temperature difference between the heat sink at the upper boundary and deep in the porous medium, we find that imperfectly cooled layers with finite Biot numbers are more stable to convective instabilities than perfectly cooled layers which have large, effectively infinite Biot numbers. Two regimes of behaviour were determined with contrasting stability behaviour and characteristic scales. When the Biot number is large the near-perfect heat transfer produces small corrections of order 1/Bi to the perfectly conducting behaviour found when the Biot number is infinite. In the insulating limit as the Biot number approaches zero, a different behaviour was found with significantly larger scales for the critical wavelength and depth of convection both scaling proportional to 1/ √ Bi

## Channelization of plumes beneath ice shelves

Journal of Fluid Mechanics Cambridge University Press **758** (2015) 109-134

We study a simpli ed model of ice-ocean interaction beneath a oating ice shelf, and investigate the possibility for channels to form in the ice shelf base due to spatial variations in conditions at the grounding line. The model combines an extensional thin- film description of viscous ice flow in the shelf, with melting at its base driven by a turbulent ocean plume. Small transverse perturbations to the one-dimensional steady state are considered, driven either by ice thickness or subglacial discharge variations across the grounding line. Either forcing leads to the growth of channels downstream, with melting driven by locally enhanced ocean velocities, and thus heat transfer. Narrow channels are smoothed out due to turbulent mixing in the ocean plume, leading to a preferred wavelength for channel growth. In the absence of perturbations at the grounding line, linear stability analysis suggests that the one dimensional state is stable to initial perturbations, chiefly due to the background ice advection.

## Solidification of a disk-shaped crystal from a weakly supercooled binary melt

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics American Physical Society **92** (2015)

The physics of ice crystal growth from the liquid phase, especially in the presence of salt, has received much less attention than the growth of snow crystals from the vapor phase. The growth of so-called frazil ice by solidification of a supercooled aqueous salt solution is consistent with crystal growth in the basal plane being limited by the diffusive removal of the latent heat of solidification from the solid-liquid interface, while being limited by attachment kinetics in the perpendicular direction. This leads to the formation of approximately disk-shaped crystals with a low aspect ratio of thickness compared to radius, because radial growth is much faster than axial growth. We calculate numerically how fast disk-shaped crystals grow in both pure and binary melts, accounting for the comparatively slow axial growth, the effect of dissolved solute in the fluid phase, and the difference in thermal properties between solid and fluid phases. We identify the main physical mechanisms that control crystal growth and show that the diffusive removal of both the latent heat released and the salt rejected at the growing interface are significant. Our calculations demonstrate that certain previous parametrizations, based on scaling arguments, substantially underestimate crystal growth rates by a factor of order 10–100 for low aspect ratio disks, and we provide a parametrization for use in models of ice crystal growth in environmental settings.

## Steady turbulent density currents on a slope in a rotating fluid

JOURNAL OF FLUID MECHANICS **746** (2014) 405-436

## Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

JOURNAL OF FLUID MECHANICS **716** (2013) 203-227

## Mushy-layer dynamics in micro and hyper gravity

Physics of Fluids **24** (2012)

We describe the results of experiments on mushy layers grown from aqueous ammonium chloride solution in normal, micro, and hyper gravity environments. In the fully developed chimney state, the chimney plume dynamics differ strikingly when conditions change from micro to hyper gravity. In microgravity, we find fully arrested plume motion and suppressed convection. As gravity exceeds Earth conditions, we observe a host of phenomena, ranging from arched plumes that undergo forced Rayleigh-Taylor instabilities to in-phase multiple plume oscillatory behavior. For the same initial solute concentrations and fixed boundary cooling temperatures, we find that, in runs of over two hours, the averaged effects of microgravity and hypergravity result in suppressed growth of the mushy layers, a phenomenon caused by a net enhancement of convective heat and solute transport from the liquid to the mushy layers. These behaviors are placed in the context of the theory of convecting mushy layers as studied under normal laboratory conditions. © 2012 American Institute of Physics.

## Finite-sample-size effects on convection in mushy layers

JOURNAL OF FLUID MECHANICS **704** (2012) 89-108