Salt fingering staircases and the three-component Phillips effect

Understanding the dynamics of staircases in salt fingering convection presents a long-standing theoretical challenge to fluid dynamicists. Although there has been significant progress, particularly through numerical simulations, there are a number of conflicting theoretical explanations as to the driving mechanism underlying staircase formation. The Phillips effect proposes that layering in stirred stratified flow is due to an antidiffusive process, and it has been suggested that this mechanism may also be responsible for salt fingering staircases. However, the details of this process, as well as mathematical models to predict the evolution and merger dynamics of staircases, have yet to be developed. We generalise the theory of the Phillips effect to a three-component system (e.g. temperature, salinity, energy) and demonstrate the first regularised nonlinear model of layering based on mixing-length parameterisations. The model predicts both the inception of layering and its long-term evolution through mergers , whilst generalising, and remaining consistent with, previous results for double-diffusive layering based on flux ratios. Our model of salt fingering is formulated using spatial averaging processes and closed by a mixing length parameterised in terms of the kinetic energy and the salt and temperature gradients. The model predicts a layering instability for a bounded range of parameter values in the salt fingering regime. Nonlinear solutions show that an initially unstable linear buoyancy gradient develops into layers, which merge through a process of stronger interfaces growing at the expense of weaker interfaces. Mergers increase the buoyancy gradient across interfaces, and increase the buoyancy flux through the staircase.Comment: 24 pages, 8 figure

Topographic controls on gravity currents in porous media

We present a theoretical and experimental study of the propagation of gravity currents in porous media with variations in the topography over which they flow, motivated in part by the sequestration of carbon dioxide in saline aquifers. We consider cases where the height of the topography slopes upwards in the direction of the flow and is proportional to the nth power of the horizontal distance from a line or point source of a constant volumetric flux. In two-dimensional cases with n>1/2, the current evolves from a self-similar form at early times, when the effects of variations in topography are negligible, towards a late-time regime that has an approximately horizontal upper surface and whose evolution is dictated entirely by the geometry of the topography. For n<1/2, the transition between these flow regimes is reversed. We compare our theoretical results in the case n=1 with data from a series of laboratory experiments in which viscous glycerine is injected into an inclined Hele-Shaw cell, obtaining good agreement between the theoretical results and the experimental data. In the case of axisymmetric topography, all topographic exponents n>0 result in a transition from an early-time similarity solution towards a topographically controlled regime that has an approximately horizontal free surface. We also analyse the evolution over topography that can vary with different curvatures and topographic exponents between the two horizontal dimensions, finding that the flow transitions towards a horizontally topped regime at a rate which depends strongly on the ratio of the curvatures along the principle axes. Finally, we apply our mathematical solutions to the geophysical setting at the Sleipner field, concluding that topographic influence is unlikely to explain the observed non-axisymmetric flow

Marine ice sheet dynamics: the impacts of ice-shelf buttressing

Marine ice sheets are continent-scale glacial masses that lie partially submerged in the ocean, as applies to significant regions of Antarctica and Greenland. Such ice sheets have the potential to destabilise under a buoyancy-driven instability mechanism, with considerable implications for future sea level. This paper and its companion present a theoretical analysis of marine ice sheet dynamics under the effect of a potentially dominant control of the buttressing force generated by lateral stresses on the downstream floating component of the ice sheet (the ice shelf). The analysis reveals critical conditions under which ice-shelf buttressing suppresses the buoyancy-driven collapse of an ice sheet and elucidates the implications of lateral stresses on grounding-line control and overall ice-sheet structure. Integrations of a suitably simplified quasi-two-dimensional model are conducted, yielding analytical results that provide a quick assessment of steady-state balances for a given ice-sheet configuration. An analytical balance equation describing the spectrum of marine ice sheet flow regimes spanning zero to strong ice-shelf buttressing is developed. It is determined that the dynamics across this spectrum exhibits markedly different flow regimes and structural characteristics. For sufficient buttressing, the grounding line occurs near to where a lateral-drag controlled section of the ice shelf meets the bedrock, implying an independent control of the grounding line by the ice shelf. The role of basal stresses is relegated to controlling only the thickness of the ice sheet upstream of the grounding line, with no significant control of the grounding line itself. It is further demonstrated that lateral stresses are responsible for inducing additional secondary contacts between the ice shelf and the bedrock downstream of the grounding line, resulting in a rich variety of additional steady states. These inducements generate a further stabilising mechanism that can fully suppress grounding-line retreat and eliminate otherwise irreparable hysteresis effects. The results provide a conceptual framework for numerical and observational interpretation of marine ice sheet dynamics, and clarifies the manner in which ice shelves can control grounding-line positions independently. It is thus indicated that a full resolution of the fine details of the flow of ice shelves and the processes controlling their erosion and disintegration is necessary for the confident forecasting of possible ice-sheet collapse over the course of the next few centuries

Fluid invasion of an unsaturated leaky porous layer

We study the flow and leakage of gravity currents injected into an unsaturated (dry), vertically confined porous layer containing a localized outlet or leakage point in its lower boundary. The leakage is driven by the combination of the gravitational hydrostatic pressure head of the current above the outlet and the pressure build-up from driving fluid downstream of the leakage point. Model solutions illustrate transitions towards one of three long-term regimes of flow, depending on the value of a dimensionless parameter D, which, when positive, represents the ratio of the hydrostatic head above the outlet for which gravity-driven leakage balances the input flux, to the depth of the medium. If D⩽0, the input flux is insufficient to accumulate any fluid above the outlet and fluid migrates directly through the leakage pathway. If 0<D⩽1, some fluid propagates downstream of the outlet but retains a free surface above it. The leakage rate subsequently approaches the input flux asymptotically but much more gradually than if D⩽0. If D>1, the current fills the entire depth of the medium above the outlet. Confinement then fixes gravity-driven leakage at a constant rate but introduces a new force driving leakage in the form of the pressure build-up associated with mobilizing fluid downstream of the outlet. This causes the leakage rate to approach the injection rate faster than would occur in the absence of the confining boundary. This conclusion is in complete contrast to fluid-saturated media, where confinement can potentially reduce long-term leakage by orders of magnitude. Data from a new series of laboratory experiments confirm these predictions

Stratified gravity currents in porous media

We consider theoretically and experimentally the propagation in porous media of variable-density gravity currents containing a stably stratified density field, with most previous studies of gravity currents having focused on cases of uniform density. New thin-layer equations are developed to describe stably stratified fluid flows in which the density field is materially advected with the flow. Similarity solutions describing both the fixed-volume release of a distributed density stratification and the continuous input of fluid containing a distribution of densities are obtained. The results indicate that the density distribution of the stratification significantly influences the vertical structure of the gravity current. When more mass is distributed into lighter densities, it is found that the shape of the current changes from the convex shape familiar from studies of the uniform-density case to a concave shape in which lighter fluid accumulates primarily vertically above the origin of the current. For a constant-volume release, the density contours stratify horizontally, a simplification which is used to develop analytical solutions. For currents introduced continuously, the horizontal velocity varies with vertical position, a feature which does not apply to uniform-density gravity currents in porous media. Despite significant effects on vertical structure, the density distribution has almost no effect on overall horizontal propagation, for a given total mass. Good agreement with data from a laboratory study confirms the predictions of the model

Ice fabrics in two-dimensional flows: beyond pure and simple shear

Ice fabrics – the distribution of crystal orientations in a polycrystal – are key for understanding and predicting ice flow dynamics. Despite their importance, the characteristics and evolution of fabrics produced outside of the deformation regimes of pure and simple shear flow has largely been neglected, yet they are a common occurrence within ice sheets. Here, we use a recently developed numerical model (SpecCAF) to classify all fabrics produced over a continuous spectrum of incompressible two-dimensional deformation regimes and temperatures. The model has been shown to accurately predict ice fabrics produced in experiments, where the ice has been deformed in either uniaxial compression or simple shear. Here we use the model to reveal fabrics produced in regimes intermediate to pure and simple shear, as well as those that are more rotational than simple shear. We find that intermediate deformation regimes between pure and simple shear result in a smooth transition between a fabric characterised by a girdle and a secondary cluster pattern. Highly rotational deformation regimes are revealed to produce a weak girdle fabric. Furthermore, we provide regime diagrams to help constrain deformation conditions of measured ice fabrics. We also obtain predictions for the strain scales over which fabric evolution takes place at any given temperature. The use of our model in large-scale ice flow models and for interpreting fabrics observed in ice cores and seismic anisotropy provides new tools supporting the community in predicting and interpreting ice flow in a changing climate

Fluid invasion of an unsaturated leaky porous layer

We study the flow and leakage of gravity currents injected into an unsaturated (dry), vertically confined porous layer containing a localized outlet or leakage point in its lower boundary. The leakage is driven by the combination of the gravitational hydrostatic pressure head of the current above the outlet and the pressure build-up from driving fluid downstream of the leakage point. Model solutions illustrate transitions towards one of three long-term regimes of flow, depending on the value of a dimensionless parameter D, which, when positive, represents the ratio of the hydrostatic head above the outlet for which gravity-driven leakage balances the input flux, to the depth of the medium. If D⩽0, the input flux is insufficient to accumulate any fluid above the outlet and fluid migrates directly through the leakage pathway. If 01, the current fills the entire depth of the medium above the outlet. Confinement then fixes gravity-driven leakage at a constant rate but introduces a new force driving leakage in the form of the pressure build-up associated with mobilizing fluid downstream of the outlet. This causes the leakage rate to approach the injection rate faster than would occur in the absence of the confining boundary. This conclusion is in complete contrast to fluid-saturated media, where confinement can potentially reduce long-term leakage by orders of magnitude. Data from a new series of laboratory experiments confirm these predictions

Roles of Aquaporins in Setaria viridis Stem Development and Sugar Storage

Setaria viridis is a C4 grass used as a model for bioenergy feedstocks. The elongating internodes in developing S. viridis stems grow from an intercalary meristem at the base, and progress acropetally toward fully expanded cells that store sugar. During stem development and maturation, water flow is a driver of cell expansion and sugar delivery. As aquaporin proteins are implicated in regulating water flow, we analyzed elongating and mature internode transcriptomes to identify putative aquaporin encoding genes that had particularly high transcript levels during the distinct stages of internode cell expansion and maturation. We observed that SvPIP2;1 was highly expressed in internode regions undergoing cell expansion, and SvNIP2;2 was highly expressed in mature sugar accumulating regions. Gene co-expression analysis revealed SvNIP2;2 expression was highly correlated with the expression of five putative sugar transporters expressed in the S. viridis internode. To explore the function of the proteins encoded by SvPIP2;1 and SvNIP2;2, we expressed them in Xenopus laevis oocytes and tested their permeability to water. SvPIP2;1 and SvNIP2;2 functioned as water channels in X. laevis oocytes and their permeability was gated by pH. Our results indicate that SvPIP2;1 may function as a water channel in developing stems undergoing cell expansion and SvNIP2;2 is a candidate for retrieving water and possibly a yet to be determined solute from mature internodes. Future research will investigate whether changing the function of these proteins influences stem growth and sugar yield in S. viridis