Data assimilation problems in glaciology Daniel Shapero A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2017 Reading Committee: Randall J. Leveque, Chair Ian R. Joughin Benjamin E. Smith Anne Greenbaum Program Authorized to Offer Degree: Applied Mathematics © Copyright 2017 Daniel Shapero University of Washington Abstract Data assimilation problems in glaciology Daniel Shapero Chair of the Supervisory Committee: Dr. Randall J. Leveque Department of Applied Mathematics Rising sea levels due to mass loss from Greenland and Antarctica threaten to inun- date coastal areas the world over. For the purposes of urban planning and hazard mitigation, policy makers would like to know how much sea-level rise can be antici- pated in the next century. To make these predictions, glaciologists use mathematical models of ice sheet flow, together with remotely-sensed observations of the current state of the ice sheets. The quantities that are observable over large spatial scales are the ice surface elevation and speed, and the elevation of the underlying bedrock. There are other quantities, such as the viscosity within the ice and the friction co- efficient for sliding over the bed, that are just as important in dictating how fast the glacier flows, but that are not observable at large scales using current meth- ods. These quantities can be inferred from observations by using data assimilation methods, applied to a model of glacier flow. In this dissertation, I will describe my work on data assimilation problems in glaciology. My main contributions so far have been: computing the bed stress underneath the three biggest Greenland outlet glaciers; developing additional tools for glacier modelling and data assimi- lation in the form of the open-source library icepack; and improving the statistical methodology through the user of total variation priors. Contents 1 Introduction3 1.1 Outline of the dissertation...........................5 1.2 Quantities of interest..............................6 2 Glacier physics 11 2.1 Stokes equations................................. 13 2.2 Mass balance................................... 18 2.3 Approximations................................. 20 2.4 Linearization................................... 27 3 Data assimilation 33 3.1 History...................................... 33 3.2 The adjoint method............................... 35 3.3 Regularization.................................. 45 3.4 Data assimilation since MacAyeal...................... 52 3.5 Basal shear stress in Greenland........................ 54 4 icepack 73 4.1 Key deal.II classes................................ 74 4.2 Design of icepack................................ 77 4.3 Testing PDE solvers............................... 81 5 Priors and regularization 85 5.1 Ice shelf rheology................................ 86 5.2 Temperature and damage evolution..................... 93 5.3 Total variation.................................. 95 5.4 Bayesian inference............................... 104 5.5 Prior selection.................................. 112 5.6 Hyperparameter selection........................... 116 6 Ice shelves 119 6.1 The Ross Ice Shelf................................ 119 6.2 Inferred rheology................................ 120 7 Conclusion 127 1 2 CONTENTS Chapter 1 Introduction In this dissertation, I will describe my work on data assimilation problems in glaciol- ogy. The chief motivation for studying this subject is for making predictions of sea- level rise in the coming century (Church et al., 2013). Changes in Earth’s climate due to rising levels of atmospheric carbon dioxide may in turn affect the mechanical and thermodynamic state of Earth’s glaciers and ice sheets, which could then melt into the oceans. The largest potential contributors to sea-level rise are Earth’s two large ice sheets, Greenland and Antarctica. Greenland contains enough ice to raise global sea levels by 7 m, and Antarctica by 58 m (Vaughan et al., 2013). Fully melt- ing either ice sheet in even the next several thousand years is highly unlikely, so the question becomes how much melting will occur. This depends on various human factors, for example, how successful efforts are to curb the use of CO2-producing fossil fuels. The results of such efforts, depending as they do on human behavior, are hard to predict. For the purposes of informing policy makers, the best one can do is to consider a range of scenarios – complete cessation of fossil fuel use on one end, and business-as-usual increase in fossil fuel use on the other. While Greenland is the smaller of the two ice sheets, it is at a lower latitude and thus experiences more surface melting. Antarctica, on the other hand, experiences very little surface melting at all, having a yearly average surface temperature of 60◦ C. It was once thought that, under a warming climate, Antarctica would grow because− the hydrological cycle would accelerate, bringing more snow accumulation to the ice sheet. Losses from Greenland could still offset this growth in Antarctica, resulting in a net sea-level rise. This type of projection only takes into account the thermodynamics of ice sheets, and not their mechanics, i.e. how fast the ice is flowing into the ocean. In the 1990s and early 2000s, substantial acceleration in the flow of glaciers along the Amundsen Sea Embayment and in the Antarctic Peninsula bore witness to the necessity of studying glacier flow in order to understand the total volume loss from Earth’s ice sheets. As a result, the Intergovernmental Panel on Climate Change (IPCC) has highlighted the dynamics of ice sheets and glaciers as one of the chief sources of uncertainty in predictions of sea-level rise (Church et al., 2013). One of the more alarming ramifications of glacier dynamics is the possibility that many areas of Earth’s ice sheets are unstable. Due to a mechanism 3 4 CHAPTER 1. INTRODUCTION known as marine ice sheet instability, an initial retreat of glaciers resting on reverse bed slopes may trigger even more rapid retreat. In order to predict sea levels in the coming century and beyond, glaciologists often use numerical simulations of ice-sheet flow. Given measurements of the cur- rent state of the ice sheet and estimates of climate forcing, its future state can be simulated assuming that ice flow can be described using continuum mechanics and thermal physics. The simulated final state can then be taken as representative of a possible future state of the ice sheet (Seddik et al., 2012). By running several simulations with different climate forcing, one can establish a range of potential scenarios for how much mass the ice sheets may lose or gain in the next century or beyond. These simulations have been run with varying degrees of success and form the basis for the aggregated projections of the IPCC, one of which is shown in figure 1.1. Figure 1.1: Projections of sea level in the 21st century relative to pre-industrial level. The blue curve represents a scenario where atmospheric CO2 is curbed, the red curve where it is not. Reproduced from Church et al.(2013). In order to run such a simulation in the first place, one needs to know the initial ice sheet state. As it turns out, we need to know more to initialize an ice sheet forecast than we can observe easily using common remote sensing techniques. One way around this problem is to do “spin-up” experiments. A model spin- up consists of picking some reasonable initial conditions, propagating the model forward in time for several thousand years until the ice sheet is roughly in steady state, and taking this state as the start of the simulation (Martin et al., 2011). There 1.1. OUTLINE OF THE DISSERTATION 5 are several problems with this approach. First, there is no guarantee that the ice sheet state obtained at the end of a model spin-up is, in any way, representative of the current state. Second, the climate forcing used to spin up the model might not accurately capture the true climate forcing experienced by the real ice sheets. The introduction of a modern climate to an ice sheet spin up using some approximation of past climate may result in unphysical transients at the beginning of the true simulation. Finally, the computational cost of the spin-up may exceed the cost of the simulation we wish to do in the first place; several millenia of spin-up may be necessary for only a centurial-scale projection. Instead, we can borrow a page from the meteorologists’ book and try to leverage existing measurements to the greatest extent possible. While some fields may not be directly observable at large scales, presumably these quantities have some effect on the fields which are measurable. For example, we cannot directly measure the friction coefficient that dictates how much resistance a glacier encounters as it slips over the bed underneath it. Nonetheless, a glacier flowing over a very resistive bed will likely flow slower than a glacier flowing over a slippery bed, all other factors being equal. We can then ask which spatial distribution of bed friction is most consistent with the velocities we did observe at the ice surface. This idea is the essence of inverse problems or data assimilation. 1.1 Outline of the dissertation In order to use data assimilation methods to infer the complete present state of the ice sheet, we must use the physics of how the various fields we would like to esti- mate relate to each other. For example, the velocities of ice flowing over crystalline bedrock, which has a very high friction coefficient, would tend to be lower than that of ice flowing over water-saturated sediment. Similarly, the mechanical hardness of warm ice is lower than that of cold ice, all factors being held equal, so the strain rates experienced by warm ice would tend to be higher. In chapter §2, I will give an overview of the physics of glacier flow. The velocity of a glacier can be modelled as the solution of an elliptic system of partial differential equations, the Stokes equa- tions, which describe slow viscous fluid flow.