Journal of Glaciology, Vol. 00, No. 000, 0000 1 An enthalpy formulation for glaciers and ice sheets Andy ASCHWANDEN,1;2∗ Ed BUELER,3;4, Constantine KHROULEV,4 Heinz BLATTER2 1Arctic Region Supercomputing Center, University of Alaska Fairbanks, USA E-mail: [email protected] 2Institute for Atmospheric and Climate Science, ETH Zurich, Switzerland 3Department of Mathematics and Statistics, University of Alaska Fairbanks, USA 4Geophysical Institute, University of Alaska Fairbanks, USA ABSTRACT. Polythermal conditions are found on all scales, from small valley glaciers to ice sheets. Conventional temperature-based \cold-ice" methods do not, however, account for the latent heat stored as liquid water within temperate ice. Such schemes are not energy- conserving when temperate ice is present. Temperature and liquid water fraction are, how- ever, functions of a single enthalpy variable: A small enthalpy change in cold ice is a change in temperature, while a small enthalpy change in temperate ice is a change in liquid water frac- tion. The unified enthalpy formulation described here models the mass and energy balance for the three-dimensional ice fluid, for a surface runoff layer, and for a subglacial hydrology layer all coupled together in a single energy-conserving framework. It is implemented in the Parallel Ice Sheet Model. Results for the Greenland ice sheet are compared with those from a cold-ice scheme. This paper is intended to be an accessible foundation for enthalpy formulations in glaciology. 1. INTRODUCTION a) Polythermal glaciers contain both cold ice (temperature be- low the pressure melting point) and temperate ice (temper- ature at the pressure melting point). This poses a thermal problem similar to that in metals near the melting point and to geophysical phase transition processes such as in mantle b) convection or permafrost thawing. In such problems part of the domain is below the melting point (solid) while other parts are at the melting point and are solid/liquid mixtures. Generally the liquid fraction may flow through the solid phase. For ice specifically, viscosity depends both on temperature and liquid water fraction, leading to a thermomechanically- coupled and polythermal flow problem. Distinct thermal structures in polythermal glaciers have been observed (Blatter and Hutter, 1991). Glaciers with ther- temperate cold mal layering as in Figure 1a are found in cold regions such as the Canadian Arctic (Blatter, 1987; Blatter and Kappen- Figure 1. Schematic view of the two most commonly found poly- berger, 1988), so it is referred to as Canadian-type in the thermal structures: a) Canadian-type and b) Scandinavian-type. following. The bulk of ice is cold except for a temperate layer The dashed line is the cold-temperate transition surface, a level set of the enthalpy field. near the bed which exists mainly due to dissipation (strain) heating. The Greenlandic and the Antarctic ice sheets exhibit such a thermal structure (L¨uthiand others, 2002; Siegert and others, 2005; Motoyama, 2007; Parrenin and others, 2007). a body is simultaneously occupied by all constituents and Figure 1b illustrates a thermal structure commonly found on that each constituent satisfies balance equations for mass, Svalbard (e.g. Bj¨ornssonand others, 1996; Moore and oth- momentum, and energy (Hutter, 1993). Exchange terms be- ers, 1999) and in the Scandinavian mountains (e.g. Schytt, tween components couple these equations. Here we derive an 1968; Hooke and others, 1983; Holmlund and Eriksson, 1989), enthalpy equation from a mixture theory which uses sepa- where surface processes in the accumulation zone form tem- rate mass and energy balance equations, but which specifies perate ice. This type will be called Scandinavian. momentum balance for the mixture only. This is suitable for A theory of polythermal glaciers and ice sheets based on most polythermal ice masses. For wholly-temperate glaciers, mixture concepts is now relatively well understood (Fowler however, a mixture approach with separate momentum bal- and Larson, 1978; Hutter, 1982; Fowler, 1984; Hutter, 1993; ances might be more appropriate (Hutter, 1993). Greve, 1997a). Mixture theories assume that each point in Two types of thermodynamical models of ice flow can be distinguished. So-called \cold-ice" models approximate the ∗Present address: Arctic Region Supercomputing Center, Univer- energy balance by a differential equation for the temperature sity of Alaska Fairbanks, Fairbanks, USA. variable. The thermomechanically-coupled models compared 2 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets by Payne and others (2000) and verified by Bueler and oth- the same enthalpy-based framework, especially taking into ers (2007) were cold-ice models, for example. Such models account pressure variations in subglacial hydrology. Thus the do not account for the full energy content of temperate ice, enthalpy field unifies the treatment of conservation of energy which has varying solid and liquid fractions but is entirely for intra-glacial, supra-glacial, and sub-glacial liquid water. at the pressure-melting temperature. A cold-ice method is An apparently-new basal water layer energy balance equa- not energy-conserving when temperate ice is present because tion, a generalization of parameterizations appearing in the changes in the latent heat content in temperate ice are not literature, arises from our analysis (subsection 3.6). reflected in the temperature state variable. Enthalpy methods are frequently used in computational Cold-ice methods have disadvantages specifically relevant fluid dynamics (e.g. Meyer, 1973; Shamsundar and Sparrow, to ice dynamics. Available experimental evidence suggests 1975; Furzeland, 1980; Voller and Cross, 1981; White, 1982; that an increase in liquid water fraction from zero to one Voller and others, 1987; Elliott, 1987; Nedjar, 2002) but are percent in temperate ice softens the ice by a factor of approx- relatively new to ice sheet modeling. In geophysical prob- imately three (Duval, 1977; Lliboutry and Duval, 1985). Such lems, enthalpy methods have been applied to magma dynam- softening has ice-dynamical consequences including enhanced ics (Katz, 2008), permafrost (Marchenko and others, 2008), strain-heating and associated increased flow. These feedback shoreline movement in a sedimentary basin (Voller and oth- mechanisms are already seen in cold-ice models (Payne and ers, 2006), and sea ice (Bitz and Lipscomb, 1999; Huwald and others, 2000) but they increase in strength when models track others, 2005; Notz and Worster, 2006). While enthalpy is a liquid water fraction. nontrivial function of temperature, water content, and salin- Because liquid water can be generated within temperate ity in a sea ice model (Notz and Worster, 2006), for example, ice by dissipation heating, a polythermal model can compute here it will be a function of temperature, water content, and a more physical basal melt rate. Indeed, a cold-ice method pressure. which is presented at some time step with energy being added Calvo and others (1999) derived a simplified variational to temperate ice must either instantaneously transport the formulation of the enthalpy problem based on the shallow ice energy to the base, as a melt rate, or it must immediately approximation on a flat bed and implemented it in a flow- lose the energy. Transport of the temperate ice, carrying la- line finite element ice sheet model. Aschwanden and Blat- tent energy stored in the liquid water fraction, presumably ter (2009) derived a mathematical model for polythermal increases downstream basal melt rates relative to their values glaciers based on an enthalpy method. Their model used a upstream. Thus the more-complete energy conservation by a brine pocket parametrization scheme to obtain a relation- polythermal model improves the modeling of basal melt rates ship between enthalpy, temperature and liquid water frac- both in space and time. Significantly, fast ice flow is controlled tion, but our theory here suggests no such parameterization by the presence of pressurized water at the ice base, and es- is needed. They demonstrated the applicability of the model pecially its time-variability (Schoof, 2010b). Basal water can to Scandinavian-type thermal structures. In the above stud- also be transported laterally and refreeze at significant rates, ies, however, the flow is not thermomechanically-coupled, and especially over highly-variable bed topography (Bell and oth- instead a velocity field is prescribed. Recently Phillips and ers, 2011). Models of these processes must correctly locate others (2010) proposed an enthalpy-based, highly-simplified the basal melt rate sources. two-column \cryo-hydrologic" model to calculate the poten- One type of polythermal model decomposes the ice domain tial warming effect of liquid water stored at the end of the into disjoint cold and temperate regions and solves separate melt season within englacial fracture. temperature and liquid water fraction equations in these re- The current paper is organized as follows: Enthalpy is de- gions (Greve, 1997a). Stefan-type matching conditions are fined and its relationships to temperature and liquid wa- applied at the cold-temperate transition surface (CTS). The ter fraction are determined by the use of mixture theory. CTS is a free boundary in such models and may be treated Enthalpy-based continuum mechanical balance equations for with front-tracking methods (Greve, 1997a; Nedjar, 2002). mass, energy, and momentum are stated, along with constitu-