G. W. Evatt

Subglacial floods beneath ice sheets

Subglacial floods (jökulhlaups) are well documented as occurring beneath present day glaciers and ice caps. In addition, it is known that massive floods have occurred from ice-dammed lakes proximal to the Laurentide ice sheet during the last ice age, and it has been suggested that at least one such flood below the waning ice sheet was responsible for a dramatic cooling event some 8000 years ago. We propose that drainage of lakes from beneath ice sheets will generally occur in a time-periodic fashion, and that such floods can be of severe magnitude. Such hydraulic eruptions are likely to have caused severe climatic disturbances in the past, and may well do so in the future.

The expected lifetime of an extraction project

When a mining company begins extraction from a finite resource, it does so in the presence of numerous uncertainties. One key uncertainty is the future price of the commodity being extracted, since a large enough drop in price can make a resource no longer cost-effective to extract, resulting in the mine being closed down. By specifying a stochastic price process, and implementing a financial-type model which leads to the use of partial differential equations, this paper creates the framework for efficiently capturing the probability of a mine remaining open throughout its planned extraction period, and derives the associated expected lifetime of extraction. An approximation to the abandonment price is described, which enables a closed-form solution to be derived for the probability of operational success and expected lifetime. This approximation compares well with the full solution obtained using a semi-Lagrangian numerical technique.

Cauldron subsidence and subglacial floods

Abstract Ice cauldrons are depressions which form at the surface of ice sheets when an underlying subglacial lake empties, in particular when subglacial volcanic eruptions occur. Notable examples of such cauldrons occur on the surface of the Vatnajökull ice cap in Iceland. More generally, cauldrons will form when a subglacial lake empties during a jökulhlaup. The rate of subsidence of the ice surface is related to the rate at which the subglacial water empties from the lake. We use a viscous version of classical beam theory applied to the ice sheet to determine the relation between the subsidence rate and flood discharge. We use the results to make inferences concerning ring fracture spacings in cauldrons, the consequent effect on flood discharge dynamics and the likely nature of subsidence events in the Antarctic Ice Sheet.

The Determination of a Dynamic Cut-Off Grade for the Mining Industry

Prior to extraction from a mine, a pit is usually divided up into 3-D ‘blocks’ which contain varying levels of estimated ore-grades. From these, the order (or ‘pathway’) of extraction is decided, and this order of extraction can remain unchanged for several years. However, because commodity prices are uncertain, once each block is extracted from the mine, the company must decide in real-time whether the ore grade is high enough to warrant processing the block further in readiness for sale, or simply to waste the block. This paper first shows how the optimal cut-off ore grade—the level below which a block should be wasted—is not simply a function of the current commodity price and the ore grade, but also a function of the ore-grades of subsequent blocks, the costs of processing, and the bounds on the rates of processing and extraction. Secondly, the paper applies a stochastic price uncertainty, and shows how to derive an efficient mathematical algorithm to calculate and operate a dynamic optimal cut-off grade criterion throughout the extraction process, allowing the mine operator to respond to future market movements. The model is applied to a real mine composed of some 60,000 blocks, and shows that an extra 10% of value can be created by implementing such an optimal regime.

Rothlisberger channels with finite ice depth and open channel flow

Abstract The theoretical basis of subglacial channel dynamics can be traced back to the work of Röthlisberger (1972) and Nye (1953). Röthlisberger (1972) considered the channels’ behaviour to be governed by a mix between water friction melting back the channel walls and the viscous closure of the surrounding ice; Nye (1953) derived a viscous closure rate for the ice. While their modelling is evidently well constructed, two aspects of their work have gone undeveloped. The first is the consideration of a finite glacier depth within the viscous closure law, instead of the assumption of an infinite glacier depth. The second is the allowance of a region of open channel flow, so that a channel’s water may transition from a region of closed channel flow to one where the water is exposed to the atmosphere. This paper helps close these two gaps, showing how Nye’s equation for the rate of ice closure can be modified, and how the point of transition between closed and open channel flow may be determined.

A potential hidden layer of meteorites below the ice surface of Antarctica.

Antarctica contains some of the most productive regions on Earth for collecting meteorites. These small areas of glacial ice are known as meteorite stranding zones, where upward-flowing ice combines with high ablation rates to concentrate large numbers of englacially transported meteorites onto their surface. However, meteorite collection data shows that iron and stony-iron meteorites are significantly under-represented from these regions as compared with all other sites on Earth. Here we explain how this discrepancy may be due to englacial solar warming, whereby meteorites a few tens of centimetres below the ice surface can be warmed up enough to cause melting of their surrounding ice and sink downwards. We show that meteorites with a high-enough thermal conductivity (for example, iron meteorites) can sink at a rate sufficient to offset the total annual upward ice transport, which may therefore permanently trap them below the ice surface and explain their absence from collection data.

Perpetual Options on Multiple Underlyings

Abstract We study three classes of perpetual option with multiple uncertainties and American-style exercise boundaries, using a partial differential equation-based approach. A combination of accurate numerical techniques and asymptotic analyses is implemented, with each approach informing and confirming the other. The first two examples we study are a put basket option and a call basket option, both involving two stochastic underlying assets, whilst the third is a (novel) class of real option linked to stochastic demand and costs (the details of the modelling for this are described in the paper). The Appendix addresses the issue of pricing American-style perpetual options involving (just) one stochastic underlying, but in which the volatility is also modelled stochastically, using the Heston (1993) framework.

Optimal costless extraction rate changes from a non-renewable resource

This paper considers the role of costless decisions relating to the extraction of a non-renewable resource in the presence of uncertainty. We begin by deriving a size scale of the extractable resource, above which the solution to the valuation and optimal control strategy can be described by analytic solutions; we produce solutions for a general form of operating cost function. Below this critical resource size level the valuation and optimal control strategy must be solved by numerical means; we present a robust numerical algorithm that can solve such a class of problem. We also allow for the embedding of an irreversible investment decision (abandonment) into the optimisation. Finally, we conduct experimentation for each of these two approaches (analytical and numerical), and show how they are consistent with one another when used appropriately. The extensions of this papers techniques to renewable resources are explored.

Joint economic and physical constraints on nuclear power: how much uranium would be needed to decarbonize electricity supply?

A new uranium supply and usage analysis takes into account growth in physical demand for electricity, both before and after decarbonization. It also models the economic willingness of investors to build the required reactors as a profit-maximizing decision. The analysis computes the optimum path of investment and disinvestment in one or more uranium-based fuel cycles as dynamic profit maximization under physical constraints, and the resulting economic decisions are compared with physical decarbonization goals. Investor decisions prove to be insensitive to the electricity price (and hence to higher carbon tax) above a low threshold. Above that price threshold, their decisions on reactor investment are dominated by the total supply of uranium and the speed at which reactors can be built. In order to pay for the very large generating capacity needed, profit-maximizing investors seem to need of the order of 36 M tons of uranium reserves to use (some seven times the worlds presently identified reserves). Even given this supply, they will limit their maximum investment in capacity if the build rate of reactors is too slow. If the world does choose to decarbonize, it will be necessary to manage uranium prospecting, and reactor building, purchase and operation as an interdependent system, and there will be a steep task of uranium exploration over the next decade or two.

Capital Ideas: Optimal Capital Reserve Strategies for a Bank and Its Regulator

We formulate a continuous-time model of a deposit taking bank, operating subject to capital adequacy regulation, where the banks loans are exposed to default risk. The banks objective is to maximise their market value of equity by appropriately controlling loan and equity issuance, dividend payments, and endogenous closure. We show how the bank responds to the span of capital adequacy requirements and default variances. Of interest to regulators, we also show how to find the capital requirement for which the probability of early closure (through either insolvency or endogenous closure) is minimised, and a level at which lending is maximised. An analytical formula is provided to help approximate the location of these points. Furthermore, we find that, whilst the banks optimal capital structure varies in a non-monotonic fashion with default variance, the level of capital requirement that minimises the probability of early closure and maximises lending is fairly robust to changes in the banks default variance/risk.We formulate a general model of a commercial bank and its regulator, where the bank’s loans are exposed to default risk. The bank’s objective is to maximise equity value by appropriately controlling the rate at which new loans are issued, early closure, and dividend payments. The regulator’s objective is to reduce the probability of the bank’s early closure, which they achieve by appropriately controlling the bank’s minimum capital requirements. We show that the regulator can in fact minimise this probability of closure, which is achieved via suitably balancing the risk of insolvency (associated with lower capital requirements) and the risk of endogenous closure (associated with higher capital requirements). We model this situation in continuous time, where the default risk is modelled as a stochastic jump process. Both analytic and numerical results are presented, thus allowing for the full non-linearity of the model to be understood. A discussion upon the model’s practical implementation and extension is also provided.