Spatiotemporal Dynamics of Ice Streams Due to a Triple-Valued Sliding Law

We show that a triple-valued sliding law can be heuristically motivated by the transverse spatial structure of an ice-stream velocity field using a simple one-dimensional model. We then demonstrate that such a sliding law can lead to some interesting stream-like patterns and time-oscillatory solutions. We find a generation of rapid stream-like solutions within a slow ice-sheet flow, separated by narrow internal boundary layers (shear margins), and analyse numerical simulations in two horizontal dimensions over a homogeneous bed and including longitudinal shear stresses. Different qualitative behaviours are obtained by changing a single physical parameter, a mass source magnitude, leading to changes from a slow creeping flow to a relaxation oscillation of the stream pattern, and to steady ice-stream-like solution. We show that the adjustment of the ice-flow shear margins to changes in the driving stress in the one-dimensional approximation is governed by a form of the Ginzburg–Landau equation and use stability analysis to understand this adjustment. In the model analysed here, the width scale of the stream is not set spontaneously by the ice flow dynamics, but rather, it is related to the mass source intensity and spatial distribution.Earth and Planetary Science

Interaction and Variability of Ice Streams under a Triple-Valued Sliding Law and Non-Newtonian Rheology

Ice streams are regions of fast flowing glacier ice that transport a significant portion of the total ice flux from present ice sheets. The flow pattern of ice streams can vary both temporally and spatially. In particular, ice streams can become stagnant and change their path. We study the dynamics of ice streams using an idealized model of an isothermal and power law viscous ice flow that includes horizontal (lateral) shear stresses. The basal sliding law is assumed to be triple-valued. We investigate the spatiotemporal patterns formed because of the flow over a flat bed, fed from an upstream mass source. The ice flows from the mass source region through one or two gaps in a prescribed upstream topographic ridge which restricts the flow, leading to the formation of one or two ice streams. We find a relation between the parameters of the ice rheology and the width of the ice stream shear margins and show how these parameters can affect the minimum width of an ice stream. We also find that complex asymmetric spatiotemporal patterns can result from the interaction of two ice streams sharing a common mass source. The rich spatiotemporal variability is found to mostly be a result of the triple-valued sliding law, but non-Newtonian effects are found to play a significant role in setting a more realistic shear margin width and allowing for relevant time scales of the variability.Earth and Planetary Science

Instability of radially spreading extensional flows. Part 1. Experimental analysis

We present laboratory experiments that show that fingering patterns can emerge when circular interfaces of strain-rate-softening fluids displace less viscous fluids in extensionally dominated flows. The fingers were separated by regions in which the fluid appeared to be torn apart. Initially, the interface had a large dominant wavenumber, but some of the fingers progressively merged so that the number of fingers gradually declined in time. We find that the transition rate to a lower wavenumber during this cascade is faster the larger is the discharge flux of the displacing fluid. At late times, depending on the discharge flux, the pattern either converged into an integer wavenumber or varied stochastically within a finite range of wavenumbers, implying convergence to a fractional wavenumber. In that stage of the evolution we find that the average wavenumber declines with the discharge flux of the displacing fluid.Israel Science Foundation (grant no. 1368/16

Lubricated axisymmetric gravity currents of power-law fluids

The motion of glaciers over their bedrock or drops of fluid along a solid surface can become unstable when these substrates are lubricated. Previous studies modeled such systems as coupled gravity currents (GCs) consisting of one fluid that lubricates the flow of another fluid, and having two propagating fronts. When both fluid are Newtonian and discharged at constant flux, global similarity solutions were found. However, when the top fluid is strain-rate softening experiments have shown that each fluid front evolved with a different exponent. Here we explore theoretically and numerically such lubricated GCs in a model that describes the axisymmetric spreading of a power-law fluid on top of a Newtonian fluid, where the discharge of both fluids is power law in time. We find that the model admits similarity solutions only in specific cases, including the purely Newtonian case, for a certain discharge exponent, at asymptotic limits of the fluids viscosity ratio, and at the vicinity of the fluid fronts. Generally, each fluid front has a power-law time evolution with a similar exponent as a non-lubricated GC of the corresponding fluid, and intercepts that depend on both fluid properties. Consequently, we identify two mechanisms by which the inner lubricating fluid front outstrips the outer fluid front. Many aspects of our theory are found consistent with recent laboratory experiments. Discrepancies suggest that hydrofracturing or wall slip may be important near the fronts. This theory may help to understand the dynamics of various systems, including surges and ice streams

Dynamics and spatiotemporal variability of ice streams

Ice sheets evolve over a wide range of time scales. They grow by snowfall, spread gravitationally, and diminish through melting or iceberg calving at the ice-sheet margin. The evolution of ice-sheets can be substantially affected by the rate of ice transport from their interior to their margins, and ice streams are the dominant transport mechanism in present ice sheets. Ice streams are bands of fast flowing glacier ice whose flow pattern varies both temporally and spatially. In particular ice-streams can become stagnant, reactivate, and flow in varying paths. In this thesis I investigate the dynamics that leads to ice-stream formation and their spatiotemporal variability. The two major dynamical factors I study are the frictional stress at the base of the ice and the non-Newtonian ice rheology. Both of these components are poorly constrained from observations, and may affect the stability of ice flow: the shear-thinning rheology of ice through shear instability, and the frictional bottom stress through the generation of melt water in the basal porous sediments that can lubricate the motion of the overlying ice. While we do not find a flow instability or ice-stream formation caused by the shear-thinning rheology, we do find that a triple-valued bottom sliding law can lead to ice-stream formation in our model and can account for various observed spatiotemporal characteristics of ice-streams. In particular the flow under such a sliding law can generate both steady and oscillatory ice stream solutions, independently of the shear thinning ice rheology. We then analyze the motion of the ice-stream shear-margins by linking the leading order dynamics of ice-streams to the Landau-Ginzburg reaction-diffusion equation. Next, we study the consequences of the non-Newtonian ice rheology on ice flow under a triple-valued sliding law, and investigate the dependence of the ice-stream shear-margin width on the rheology. Finally, we study the spatiotemporal variability due to the interaction of two ice streams. We demonstrate that a spatially symmetric two-stream pattern can be unstable under an asymmetric perturbation, which results in a flow with asymmetric patterns that are maintained through the competition of the two ice-streams over a shared mass source. The rich spatiotemporal variability is found to mostly be a result of the triple valued sliding law, but non Newtonian effects are found to play a significant role in setting a more realistic shear margin width and allowing for relevant time scales of the variability

Spontaneous Generation of Pure Ice Streams Via Flow Instability: Role of Longitudinal Shear Stresses and Subglacial Till

A significant portion of the ice discharge in ice sheets is drained through ice streams, with subglacial sediment (till) acting as a lubricant. The known importance of horizontal friction in shear margins to ice stream dynamics suggests a critical role of longitudinal stresses. The effects of subglacial till and longitudinal stresses on the stability of an ice sheet flow are studied by linear stability analysis of an idealized ice-till model in two horizontal dimensions. A power law-viscous constitutive relation is used, explicitly including longitudinal shear stresses. The till, which has compressible viscous rheology, affects the ice flow through bottom friction. We examine the possibility that pure ice streams develop via a spontaneous instability of ice flow. We demonstrate that this model can be made intrinsically unstable for a seemingly relevant range of parameters and that the wavelengths and growth rates that correspond to the most unstable modes are in rough agreement with observed pure ice streams. Instabilities occur owing to basal friction and meltwater production at the ice-till interface. The most unstable wavelength arise because of selective dissipation of both short and long perturbation scales. Longitudinal stress gradients stabilize short transverse wavelengths, while Nye diffusion stabilizes long transverse wavelengths. The selection of an intermediate unstable wavelength occurs, however, only for certain parameter and perturbation structure choices. These results do not change qualitatively for a Newtonian ice flow law, indicating no significant role to shear thinning, although this may very well be due to the restrictive assumptions of the model and analysis.Earth and Planetary Science