An enthalpy formulation for glaciers and ice sheets

Transcription

1 Journal of Glaciology, Vol. 00, No. 000, An enthalpy formulation for glaciers and ice sheets Andy ASCHWANDEN, 1,2 Ed BUELER, 3,4, Constantine KHROULEV, 4 BLATTER 2 1 Arctic Region Supercomputing Center, University of Alaska Fairbanks, USA andy.aschwanden@arsc.edu 2 Institute for Atmospheric and Climate Science, ETH Zurich, Switzerland 3 Department of Mathematics and Statistics, University of Alaska Fairbanks, USA 4 Geophysical Institute, University of Alaska Fairbanks, USA Heinz 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 energyconserving when temperate ice is present. Temperature and liquid water fraction are, however, 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 fraction. 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 Polythermal glaciers contain both cold ice temperature below the pressure melting point) and temperate ice temperature 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 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 thermomechanicallycoupled and polythermal flow problem. Distinct thermal structures in polythermal glaciers have been observed Blatter and Hutter, 1991). Glaciers with thermal layering as in Figure 1a are found in cold regions such as the Canadian Arctic Blatter, 1987; Blatter and Kappenberger, 1988), so it is referred to as Canadian-type in the following. The bulk of ice is cold except for a temperate layer near the bed which exists mainly due to dissipation strain) heating. The Greenlandic and the Antarctic ice sheets exhibit such a thermal structure Lüthi and others, 2002; Siegert and others, 2005; Motoyama, 2007; Parrenin and others, 2007). Figure 1b illustrates a thermal structure commonly found on Svalbard e.g. Björnsson and others, 1996; Moore and others, 1999) and in the Scandinavian mountains e.g. Schytt, 1968; Hooke and others, 1983; Holmlund and Eriksson, 1989), where surface processes in the accumulation zone form temperate ice. This type will be called Scandinavian. A theory of polythermal glaciers and ice sheets based on mixture concepts is now relatively well understood Fowler and Larson, 1978; Hutter, 1982; Fowler, 1984; Hutter, 1993; Greve, 1997a). Mixture theories assume that each point in Present address: Arctic Region Supercomputing Center, University of Alaska Fairbanks, Fairbanks, USA. a) b) temperate cold Figure 1. Schematic view of the two most commonly found polythermal structures: a) Canadian-type and b) Scandinavian-type. The dashed line is the cold-temperate transition surface, a level set of the enthalpy field. a body is simultaneously occupied by all constituents and that each constituent satisfies balance equations for mass, momentum, and energy Hutter, 1993). Exchange terms between components couple these equations. Here we derive an enthalpy equation from a mixture theory which uses separate mass and energy balance equations, but which specifies momentum balance for the mixture only. This is suitable for most polythermal ice masses. For wholly-temperate glaciers, however, a mixture approach with separate momentum balances might be more appropriate Hutter, 1993). Two types of thermodynamical models of ice flow can be distinguished. So-called cold-ice models approximate the energy balance by a differential equation for the temperature variable. The thermomechanically-coupled models compared

2 2 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets by Payne and others 2000) and verified by Bueler and others 2007) were cold-ice models, for example. Such models do not account for the full energy content of temperate ice, which has varying solid and liquid fractions but is entirely at the pressure-melting temperature. A cold-ice method is not energy-conserving when temperate ice is present because changes in the latent heat content in temperate ice are not reflected in the temperature state variable. Cold-ice methods have disadvantages specifically relevant to ice dynamics. Available experimental evidence suggests that an increase in liquid water fraction from zero to one percent in temperate ice softens the ice by a factor of approximately three Duval, 1977; Lliboutry and Duval, 1985). Such softening has ice-dynamical consequences including enhanced strain-heating and associated increased flow. These feedback mechanisms are already seen in cold-ice models Payne and others, 2000) but they increase in strength when models track liquid water fraction. Because liquid water can be generated within temperate ice by dissipation heating, a polythermal model can compute a more physical basal melt rate. Indeed, a cold-ice method which is presented at some time step with energy being added to temperate ice must either instantaneously transport the energy to the base, as a melt rate, or it must immediately lose the energy. Transport of the temperate ice, carrying latent energy stored in the liquid water fraction, presumably increases downstream basal melt rates relative to their values upstream. Thus the more-complete energy conservation by a polythermal model improves the modeling of basal melt rates both in space and time. Significantly, fast ice flow is controlled by the presence of pressurized water at the ice base, and especially its time-variability Schoof, 2010b). Basal water can also be transported laterally and refreeze at significant rates, especially over highly-variable bed topography Bell and others, 2011). Models of these processes must correctly locate the basal melt rate sources. One type of polythermal model decomposes the ice domain into disjoint cold and temperate regions and solves separate temperature and liquid water fraction equations in these regions Greve, 1997a). Stefan-type matching conditions are applied at the cold-temperate transition surface CTS). The CTS is a free boundary in such models and may be treated with front-tracking methods Greve, 1997a; Nedjar, 2002). Because they require an explicit representation of the CTS as a surface, however, such methods are somewhat cumbersome to implement. Their surface representation scheme may impose restrictions on the shape topology and geometry) of the CTS, as when Greve 1997a) describes the CTS by a single vertical coordinate in each column of ice, for example. Enthalpy methods have an advantage here in the sense that the CTS is a level set of the enthalpy variable. No explicit surface-representation scheme is required and no a priori restrictions apply to CTS shape. Transitions between polythermal structures caused by changing climatic conditions can be modeled even if nontrivial CTS topology arises at intermediate stages. For example, increasing surface energy input in the accumulation zone could cause a transition from Canadian to Scandinavian type through intermediate states involving temperate layers sandwiching a cold layer. A practical advantage of enthalpy schemes is that the state space of the evolving ice sheet is simpler because the energy state of the ice fluid and its boundary layers is described by a single scalar field. Indeed we show that energy fluxes in supraglacial runoff and subglacial aquifers can be treated in the same enthalpy-based framework, especially taking into account pressure variations in subglacial hydrology. Thus the enthalpy field unifies the treatment of conservation of energy for intra-glacial, supra-glacial, and sub-glacial liquid water. An apparently-new basal water layer energy balance equation, a generalization of parameterizations appearing in the literature, arises from our analysis subsection 3.6). Enthalpy methods are frequently used in computational fluid dynamics e.g. Meyer, 1973; Shamsundar and Sparrow, 1975; Furzeland, 1980; Voller and Cross, 1981; White, 1982; Voller and others, 1987; Elliott, 1987; Nedjar, 2002) but are relatively new to ice sheet modeling. In geophysical problems, enthalpy methods have been applied to magma dynamics Katz, 2008), permafrost Marchenko and others, 2008), shoreline movement in a sedimentary basin Voller and others, 2006), and sea ice Bitz and Lipscomb, 1999; Huwald and others, 2005; Notz and Worster, 2006). While enthalpy is a nontrivial function of temperature, water content, and salinity in a sea ice model Notz and Worster, 2006), for example, here it will be a function of temperature, water content, and pressure. Calvo and others 1999) derived a simplified variational formulation of the enthalpy problem based on the shallow ice approximation on a flat bed and implemented it in a flowline finite element ice sheet model. Aschwanden and Blatter 2009) derived a mathematical model for polythermal glaciers based on an enthalpy method. Their model used a brine pocket parametrization scheme to obtain a relationship between enthalpy, temperature and liquid water fraction, but our theory here suggests no such parameterization is needed. They demonstrated the applicability of the model to Scandinavian-type thermal structures. In the above studies, however, the flow is not thermomechanically-coupled, and instead a velocity field is prescribed. Recently Phillips and others 2010) proposed an enthalpy-based, highly-simplified two-column cryo-hydrologic model to calculate the potential warming effect of liquid water stored at the end of the melt season within englacial fracture. The current paper is organized as follows: Enthalpy is defined and its relationships to temperature and liquid water fraction are determined by the use of mixture theory. Enthalpy-based continuum mechanical balance equations for mass, energy, and momentum are stated, along with constitutive equations. Boundary conditions are carefully addressed using the new technique in Appendix A. At this level of generality the enthalpy formulation could be used in any ice flow model, but we describe its implementation, using specific simplifications appropriate to whole ice sheet models, in the Parallel Ice Sheet Model Khroulev and the PISM Authors, 2011). PISM is applied to the Greenland ice sheet, and the results are discussed. 2. ENTHALPY METHOD Consider a mixture of ice and water with corresponding partial densities ρ i and ρ w mass of the component per unit volume of the mixture). The mixture density is the sum ρ = ρ i + ρ w. 1) The liquid water fraction, often called water moisture) content, is the ratio ω = ρw ρ. 2)

3 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 3 We treat the mixture of ice and water as incompressible, but of course the bulk densities of ice ˆρ i = 910 kg m 3) and water ˆρ w = 1000 kg m 3) are distinct. However, for liquid water fractions smaller than 5 % the change in mixture density due to changes in liquid water fraction are smaller than 0.5 %. It is therefore reasonable to set ρ ˆρ i. Define the barycentric mixture) velocity v by ρv = ρ i v i + ρ wv w, 3) T T m ω 1 where v i and v w are the velocity of ice and water, respectively Greve and Blatter, 2009). In Cartesian components we denote v = u, v, w). The specific enthalpy, H, is commonly defined as e.g. Moran and Shapiro, 2006): H = u + p/ρ, 4) where u is the specific internal energy and p is the pressure. Specific enthalpy has SI units J kg 1. If a material is heated under constant pressure, and without volume changes, then enthalpy represents inner energy Alexiades and Solomon, 1993), so that in this paper enthalpy is a synonym for internal energy. For cold ice the temperature T is below the pressure melting point T mp). The specific enthalpy H i of cold ice is defined as T H i = C i T ) d T, 5) T 0 where C i T ) is the measured heat capacity of ice at atmospheric pressure. If the reference temperature T 0 is below all modeled ice temperatures then enthalpy values will be positive, though positivity is not important for correctness. The specific enthalpy of liquid water H w is defined as H w = T mp) T 0 C i T ) d T + L + T T mp) C w T ) d T, 6) where C wt ) is the heat capacity of water and L is the latent heat of fusion. Functions H i T ) and H wt, p) are defined by Equations 5) and 6), respectively, so that the enthalpy of liquid water exceeds that of cold ice by at least L. Indeed, if T T mp) then H wt, p) H i T mp)) + L. Supercooled liquid water with T < T mp) is, however, allowed by Equation 6). Experiments suggest that the heat capacity C i T ) of ice is approximately a linear function of temperature in the range of temperatures commonly found in glaciers and ice sheets Petrenko and Whitworth, 1999, and references therein). In fact for most glacier-modeling purposes it suffices to approximate C i T ) by a constant value independent of temperature. The heat capacity of liquid water, C wt ), is also nearly constant within the relevant temperature range. Thus one may define simpler functions H i T ) and H wt, p), but until section 4 we continue with the general forms above. The enthalpy density ρh volumetric enthalpy) of the mixture is given by ρh = ρ i H i + ρ wh w. 7) From Equations 1), 2), 5), 6), and 7), H = HT, ω, p) = 1 ω)h i T ) + ωh wt, p). 8) Equation 8) describes the specific enthalpy of all mixtures, including cold ice, temperate ice, and liquid water. H s Figure 2. At fixed pressure p the temperature of the ice/liquid water mixture is a function of enthalpy, T = T H, p) solid line), as is the liquid water fraction, ω = ωh, p) dotted line). Points H sp) and Hl p) are the enthalpy of pure ice and pure liquid water, respectively, at temperature T mp). In cold ice we have T < T mp) and ω = 0. Temperate ice is a mixture which includes a positive amount of solid ice. Neglecting the possibility of supercooled liquid in the mixture, we have T = T mp) and 0 < ω < 1 in temperate ice. Denote the enthalpy of ω = 0 ice at the pressure-melting temperature by H sp) = T mp) H l H T 0 C i T ) d T. 9) For mixtures with a positive solid fraction, Equation 8) now reduces to two cases { Hi T ), T < T mp), H = 10) H sp) + ωl, T = T mp) and 0 ω < 1. We now observe that an inverse function to H i T ) exists under the reasonable assumption that C i T ) is positive. This inverse is denoted T i H) and it is defined for H H sp). Note H i / T = C i T ) and thus T/ H i = C i T ) 1. At this point we could define ice as cold if a small change in enthalpy leads to a change in temperature alone, and temperate if a small change in enthalpy leads to a change in liquid water fraction alone Aschwanden and Blatter, 2005, 2009). The following functions invert relation 8) for the range of enthalpy values relevant to glacier and ice sheet modeling: { Ti H), H < H sp), T H, p) = 11) T mp), H sp) H, { 0, H < Hsp), ωh, p) = L 1 12) H H sp)), H sp) H. Schematic plots of temperature and water content as functions of enthalpy are shown in Figure 2, with points H s and H l p) = H wt mp), p) = H sp)+l noted. Equations 11) and 12) will only be applied to cold or temperate ice mixtures), and therefore H < H l p) in all cases in the sequel. By Equations 11) and 12), if the pressure and enthalpy are given then temperature and liquid water fraction are determined. By Equation 8), if the pressure, temperature, and liquid water fraction are given then the enthalpy is determined. It follows that pressure and enthalpy are a preferred pair of state variables in a thermomechanically-coupled glacier or ice sheet flow model. From now on, temperature and liquid water fraction are only diagnostically computed, when needed, from this pair of state variables.

4 4 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 3. CONTINUUM MODEL 3.1. Balance equations General balance equation The balance of a quantity ψ describing particles fluid) which move with velocity v is given by ψ = ψ v + φ) + π, 13) where ψ v and φ are the advective and the non-advective flux density, respectively, is the tensor product, and π is a production term Equation 2.13) in Liu, 2002). When ψ is a scalar quantity, ψ v = ψv. Mass balance For the solid and liquid components within the ice mixture, we only allow for mass to be exchanged between components by melting and freezing processes. Chemical creation of water is not considered.) Define M w and M i as the exchange rates between the components, which are production terms in Equation 13). Local conservation of mixture mass implies M i + M w = 0. Setting ψ = ρ i, v = v i, φ = 0, and π = M i = M w in Equation 13) for the ice component, and similarly for the water component, ρ i + ρ iv i ) = M w, 14) ρ w + ρ wv w) = M w. 15) Adding Equations 14) and 15) yields the balance for the mixture ρ + ρv) = 0, 16) where v is the barycentric velocity Equation 3)). As noted already, however, the mixture density ρ ˆρ i is approximately constant. Exact constancy of the density would imply that the mixture is actually incompressible, in the sense v = 0. 17) We now adopt Equation 17), incompressibility, as the mass conservation equation for the mixture. Enthalpy balance Denote the total enthalpy flux of the ice and liquid water components, including any advective and conductive mechanisms, by Φ i and Φ w respectively. We define the residual enthalpy fluxes q i, q w to be the fluxes relative to advection by the barycentric velocity v: q i = Φ i ρ i H i v, 18) q w = Φ w ρ wh wv. 19) A residual flux arises if the velocity of a component is different from the mixture velocity, for instance in the case of Darcy flow of the liquid component in temperate ice. A residual flux also arises if liquid water diffuses in temperate ice. Empirical constitutive relations for these residual fluxes q i, q w are addressed below. Within the mixture, Σ w and Σ i are the enthalpy exchange rates between the components. Local conservation of energy implies Σ i + Σ w = 0. There are also dissipation heating rates Q w and Q i for the components, but we will only give an equation for the total heating rate Q = Q w + Q i for the mixture, in Equation 42) below. For the ice component, setting ψ = ρ i H i, φ = q i, and π = Q i Σ w in Equation 13) gives ρ i H i ) = ρ i H i v + q i ) + Q i Σ w. 20) Similarly for the water component, ρ wh w) = ρ wh wv + q w) + Q w + Σ w. 21) The smoothness of the modeled fields ρ i H i and ρ wh w is closely-related to the empirical form chosen for the enthalpy fluxes q i and q w. Differentiability of these fields follows from the the presence of diffusive terms in enthalpy flux parameterizations. While we assume that the enthalpy solution is spatially-differentiable, its derivatives may be large in magnitude, corresponding to enthalpy jumps in practice. Summing Equations 20) and 21) and using Equation 7) gives a balance for the total enthalpy flux: ρh) = ρhv + q i + q w) + Q. 22) Setting q = q i + q w, using notation d/dt = / + v for the material derivative, and recalling Equation 17), we arrive at ρ dh dt = q + Q 23) for the mixture. Equation 23) is our primary statement of conservation of energy for ice. It has the same form as the temperature-only equations already implemented in many coldice glacier and ice sheet models. Momentum balance Let T be the Cauchy stress tensor. Recall that T D = T + pi defines the deviatoric stress tensor, where p = 1/3) tr T is the pressure. Glacier flow is commonly-assumed to be a gravity-driven incompressible Stokes flow acceleration is neglected) where pressure gradients are balanced only by deviatoric stress gradients. Setting ψ = ρv, φ = T, and π = ρg, the balance of linear momentum follows from Equation 13), namely 0 = T + ρg or T D + p + ρg = 0. 24) Balance of angular momentum implies symmetry T = T T Liu, 2002) Constitutive equations Enthalpy flux The residual heat flux in cold ice, namely that flux which is not advection at the mixture velocity, is given by Fourier s law e.g. Paterson, 1994), q i = k i T ) T, 25) where k i T ) is the thermal conductivity of cold ice. Combining Equations 11) and 25) allows us to write the heat flux in terms of the enthalpy gradient in cold ice, q i = k i T ) T i H)) = k i T )C i T ) 1 H. 26) The last expression in Equation 26) can be written entirely in terms of enthalpy, however, and this is a key step in an enthalpy gradient method Pham, 1995; Aschwanden and Blatter, 2009). In fact, let thus C i H) = C i T i H)) and k i H) = k i T i H)), 27) K i H) = k i H)C i H) 1. 28) Fourier s law for cold ice now has the enthalpy form q i = K i H) H. 29)

5 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 5 The residual heat flux in temperate ice is the sum of sensible and latent heat fluxes Greve, 1997a), q = q s + q l. 30) The sensible heat flux q s in temperate ice is conductive, arising from variations in the pressure-melting temperature. Now, the mixture conductivity, written in terms of enthalpy and pressure, is kh, p) = 1 ωh, p)) k i H) + ωh, p) k w 31) where k w is the thermal conductivity of liquid water; compare Equation 3) in Notz and Worster 2006). By Fourier s law the sensible heat flux has enthalpy form q s = kh, p) T mp). 32) The latent heat flux q l in temperate ice is proportional to the mass flux of liquid water j relative to the barycenter: q l = Lj = Lρ w v w v). 33) Flux j is sometimes called the diffusive water flux Greve and Blatter, 2009), but we prefer to identify it as residual until a diffusive constitutive relation is specified. A constitutive equation for j is required for implementability. Hutter 1982) suggested a general formulation, involving liquid water fraction, its spatial gradient, deformation, and gravity. Two specific realizations have been proposed, a Fick-type diffusion Hutter, 1982) and a Darcy-type flow Fowler, 1984). Laboratory experiments and field observations to identify the constitutive relation are scarce, however. A drop in liquid water fraction was observed close to the bed at Falljökull in Iceland Murray and others, 2000) and on the upper plateau of the Vallée Blanche in the Massif du Mont- Blanc Vallon and others, 1976), and these observations suggests a gravity-driven flow regime may exist at higher water fractions. They support the flexible Hutter 1982) form. But we admit that even the functional form of q l is poorlyconstrained at the present state of experimental knowledge. Greve 1997a) writes the simplest possible form by setting the latent heat flux q l to zero in temperate ice. Our assumption that the enthalpy field has finite spatial derivatives within the ice explains our regularizing choice, namely q l = k 0 ω = K 0 H, 34) where k 0 and K 0 = L 1 k 0 are small positive constants. This form may be justified by a Fick-type diffusion Hutter, 1982), or even by observing that numerical schemes approximating the hyperbolic equation arising from the choice q l = 0 will nonetheless generate some numerical diffusion Greve, 1997a). Mathematically, this regularization implies that the enthalpy field Equation 41) is coercive, which explains why the same regularization is used by Calvo and others 1999). From Equations 29), 30), 32), and 34), we have now expressed the heat flux in terms of enthalpy and pressure, { K q = i H) H, cold, 35) kh, p) T mp) K 0 H, temperate. Ice flow The deviatoric stress tensor ) T D and the strain rate tensor D = 1/2) v + v T are related via the effective viscosity η by T D = 2η D. 36) Laboratory experiments suggest that glacier ice behaves like a power-law fluid e.g. Steinemann, 1954; Glen, 1955), so its effective viscosity η is a function of an effective stress τ e, a rate factor softness) A, and a power n, 2η = A 1 τ 2 e + ɛ 2) 1 n)/2. 37) Here 2τe 2 = T D ij TD ij summation convention). The small constant ɛ units of stress) regularizes the flow law at low effective stress, avoiding the problem of infinite viscosity at zero deviatoric stress Hutter, 1983). In terrestrial glacier applications, ice behaves similarly to metals and alloys with a homologous temperature near one. In cold ice it is common to express the rate factor with the Arrhenius law, A ct, p) = A 0 e Qa+pV )/RT, 38) where A 0, Q a, V, R are constant or have additional temperature dependence; Paterson, 1994). For temperate ice, however, little is known about the effect of liquid water on viscosity. In experimental studies, Duval and others 1977, 1985) derived the following function for the rate factor in temperate ice, A t ω, p), valid for water fractions less than 0.01 or 1 %), A t ω, p) = A ct mp), p) ω), 39) as given by Greve and Blatter 2009). Though Equations 38) and 39) use particular forms based on existing experiments, from any such forms we may write the rate factor as a function of the enthalpy variable and pressure: { AcT H, p), p), H < H sp), AH, p) = 40) A t ωh, p), p), H sp) H < H l p) Field equation for enthalpy Inserting Equation 35) into 23) yields the enthalpy field equation of the ice mixture, the heat conduction term of which depends on whether the mixture is cold H < H sp)) or temperate H H sp)): ρ dh dt = { K i H) H kh, p) T mp) + K 0 H }) + Q. 41) The rate Q at which dissipation of strain releases heat into the mixture is Q = tr D T D). 42) The most complete set of field equations considered in this paper consists of Equations 41) and 42) along with incompressibility Equation 17) and the Stokes Equations 24). One also includes the temperature-dependent power-law constitutive relation described by Equation 40). Section 4 introduces specific simplifications to this complete model which make it more suitable to whole ice sheet scale implementations On jump equations at boundaries The mixture mass density ρ) and mixture enthalpy density ρh) are well-defined in the ice as well as in the air and bedrock, where we assume they have value zero. That is, the air and bedrock are assumed to be water-free in our simplified view, as depicted in Figure 3. The density fields satisfy the stated field equations within the ice, but we now consider their jumps at the surface and base because they are not continuous there. These jump equations then provide boundary conditions for a well-posed conservation of energy problem for the ice. At a general level, suppose that a smooth, non-material, singular surface separates a region V into two sub-regions

6 6 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets n ice ρ = ρ w + ρ i, ρh = ρ wh w + ρ i H i ) bedrock ρ = 0, ρh = 0) air ρ = 0, ρh = 0) Figure 3. Jump condition 45) is applied to fields ρ and ρh, the mass and enthalpy densities of the ice/liquid water mixture. These fields are defined in air, ice, and bedrock. They undergo jumps at the ice upper surface z = h) and the ice base z = b). The normal vector n points into the ice domain. V ±. Let n be the unit normal field on, pointing from V to V +. The jump [[ψ]] of a tensor or scalar field ψ on is defined as [[ψ]] = ψ + ψ, 43) where ψ ± are the one-sided limits of ψ in V ±. In this paper we must allow advection of ψ at some velocity v within and along tangent to). For simplicity of presentation we assume ψ is a scalar in this subsection. We suppose the flow along has area density λ, and also that there is production of ψ in the surface at rate π. As shown in Appendix A by a pillbox argument, general balance Equation 13) implies a jump condition with additional terms: [[ψv n w )]] + [[φ n]] + λ n + λ v ) = π, 44) where w is the normal speed of the surface. Equation 45) generalizes the better-known Rankine-Hugoniot jump condition Liu, 2002). We identify the positive sides of surfaces as the ice side throughout this paper. Because ψ = ρ and ψ = ρh are zero outside of the ice, and suppressing the + superscript for ice-side quantities, Equation 44) states ψ v n w )+φ φ ) n+ λ + λv) = π. 45) Equation 44) arises as a large-scale approximation of the balance of processes occurring in a thin active layer, e.g. of a few meters thickness, confined around the ideal boundary surface and bounded on either side by three-dimensional material. Thus Equation 44) is both a jump condition describing the three-dimensional field ψ and a balance equation for the field λ within the boundary surface. In the latter view the first two jump terms in 44) are production terms, with the source being the adjacent three-dimensional field. The density of ψ may also vanish in the layer λ = 0). The velocity v of the thin layer material e.g. liquid water in runoff or subglacial aquifer, if present) is generally much larger than the bulk ice flow velocity v e.g. of the ice mixture). A similar magnitude contrast may apply to the thin layer production rate π versus the three-dimensional production rate π in Equation 13). The thin layer considered in Equation 44) and/or 45) might have relatively uniform thickness, but our view also allows a distributed hydrologic network at the ice base or surface whose thickness is highly-variable but small compared to the average ice thickness. This distributed network must have parameterized) large-scale behavior similar to a thin layer, but further consideration of this issue is beyond our scope. Stating jump Equations like 44) do not close the continuum model in any case. Specific constitutive relations for the thin active layer are still needed, namely more-orless empirical supraglacial runoff and sub-glacial hydrology models. See subsection 5.1. In the next two subsections we describe both a supraglacial runoff layer and subglacial hydrology layer within the same continuum modeling framework, in which each layer is governed by a pair of jump equations for the fields ρ and ρh. The analogy between these two layers is deliberate. A momentum jump condition is included for completeness Upper surface Jump equation for mass The ice upper free surface = {z = ht, x, y)} is where the mixture density jumps from ρ = 0 in the air to ρ = ˆρ i in the ice Figure 3). Assume that is differentiable. Define N 2 h = 1 + ) 2 h + x ) 2 h = 1 + h 2. 46) y A unit normal vector for pointing from air V ) into ice V + ) is n = N 1 h h x, h ) y, 1 = N 1 h h, 1), 47) and the normal surface speed is w = 0, 0, h ) n = N 1 h h. 48) Consider the solid mixture component at the upper surface. It accumulates perpendicularly at a mass flux rate a i kg m 2 s 1 ). Ice also forms from freezing of the liquid which is already present in the thin layer at a rate µ i. Let ψ = ρ i, φ = 0, v = mixture velocity), λ = 0, and π = µ i + a i in Equation 45): ρ i v n w ) = µ i + a i. 49) Similarly for the water component, but additionally defining λ = ˆρ wη r, where η h is the effective runoff layer thickness, and v = v h = runoff velocity), we have ρ w v n w ) + ˆρwη h) + ˆρ wη h v h ) = µ w + a w, 50) where a w is essentially the rainfall mass rate and µ w is the thin layer mass exchange refreeze) rate. The sum a = a i + a w is the mixture snowfall-rainfallsublimation function, the mass rate for exchange with the atmosphere. Conservation of water mass in the supraglacial layer implies that µ i + µ w = 0 because no water is created or destroyed e.g. by chemistry). Define the runoff-included) surface mass balance M h = a ˆρwη h) ˆρ wη h v h ) 51) SI units kg m 2 s 1 ). The accumulation-ablation function M h, along with ice motion, determine the rate of movement of the ice surface. Indeed, adding Equations 49) and 50) and using definition 51) gives the surface kinematical equation ρv n w ) = M h. 52)

7 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 7 Multiplying through by ρ 1 N h and using Equations 47) and 48) gives the equivalent, possibly more familiar form h + u h x + v h y w = ρ 1 N h M h. 53) The right side is an ice-equivalent surface mass balance with SI units m s 1. Equation 52) describes the jump of ρ from air, where it is zero to incompressible ice, thus generically) across a firn layer. Models which describe the firn layer explicitly could replace Equation 52) by separate kinematical equations for atmosphere-firn surface and the firn-ice surface below it. Jump equation for enthalpy Snow and rain deliver enthalpy to the ice surface, and in general it is transported along the surface as well. Enthalpy is also delivered through non-latent atmospheric fluxes sensible, turbulent, and radiative) whose sum is q atm. These all affect the jump in the enthalpy density ρh. Note H is not defined in the air but we define ρh to have value zero there because ρ = 0 in the air. For the ice component we assume snowfall occurs at a welldefined temperature T snow which varies in time and space, with corresponding enthalpy Hi snow = H i T snow). Denote the enthalpy exchange rate for ice formed by freezing of liquid water as Σ i units J m 2 s 1 ). We make the simplifying assumption that λ = 0 for the solid component; we assume only liquid runoff is mobile. Equation 45) with ψ = ρ i H i, φ = 1 ω)q, φ = 1 ω)q atm, and π = Σ i +a i H it snow), and with other symbols defined as for Equation 49), gives ρ i H i v n w ) + 1 ω) q q atm) n = Σ i + a i H snow i. 54) For the liquid component we make the simplifying assumptions that both runoff and rainfall contain liquid water exactly at the melting temperature. We account for any liquid fraction in snowfall as part of the liquid component here, that is, it is grouped with rainfall in a w. Denote the enthalpy exchange rate for liquid water formed by melting as Σ w. Let Hl atm = H l p atm ) be the enthalpy of any liquid water present at the surface, a constant because we ignore spatial variations in atmospheric pressure and we neglect the supercooled case. With φ = ωq, φ = ωq atm, and λ = ˆρ wη h Hl atm in Equation 45), we have ρ wh w v n w ) + ω + q q atm l ) n + ˆρwη hh atm l ) ) ˆρ wη h Hl atm v h = Σ w + a whl atm. 55) Local conservation of energy for any portion of the surface implies that Σ i +Σ w = 0. Adding the last two equations gives an enthalpy boundary condition for the mixture: ρh v n w ) + q q atm) n = a i Hi snow + a wh atm l ˆρwη hhl atm ) ) ˆρ wη h Hl atm v h. 56) Using Equations 51) and 52) to simplify Equation 56), with Hl atm factored out of derivatives, the result is a balance for energy fluxes at the ice mixture) upper surface: ) M h H Hl atm ) = a i Hi snow Hl atm ) + q atm q n. 57) Equation 57) is the surface energy balance equation written for use as the boundary enthalpy condition in an ice sheet model. It is true at each instant, but its yearly averages are important too, and they may be more easily-modeled in practice. Restricted cases of Equation 57), which the reader may find more familiar, are addressed next. In all these restricted cases we neglect, for simplicity, the typically-small residual enthalpy flux q n into the ice. First, in the case where there is no runoff η h = 0, M h = a ) and positive snowfall a i, Equation 57) is equivalent to a Dirichlet condition for enthalpy, ) ) a i H = a Hi snow + 1 a i a H atm l + qatm n a. 58) This Equation says the surface enthalpy is the mass-rateaveraged enthalpy delivered by mixture precipitation, plus the non-latent atmospheric energy fluxes. A second case, that of a bare ice zone, illuminates the fact that Equation 57) is true at each instant but its timeaverages may be more valuable for modeling. At times of melting and runoff M h 0 and a i = 0), Equation 57) reduces to M h H Hl atm ) = q atm n. 59) That is, the atmospheric fluxes melt the surface ice, with enthalpy H, to liquid water with enthalpy Hl atm. This equation could be used to determine the runoff M h if q atm is given. Despite Equation 59), however, for most modeling purposes we may use the yearly-average ice surface enthalpy H as a boundary condition for the enthalpy field equation. This is because, in the bare ice zone, the enthalpy of near-surface ice will take on a value much closer to the enthalpy value corresponding to the yearly-average atmospheric temperature. Indeed we can compute the yearly-average from Equation 57) if we note that during the majority of the year there is no runoff, no melt, and no rain, and, in this bare-ice zone case, only a thin layer of seasonal snow. This implies that for most of the year we have M h a a i so H H snow i + qatm n a. 60) This Equation also follows from Equation 58). Of course, 60) is also the equation that applies if no runoff, melt, or rain occurs at any time in the year, as on the majority of the Antarctic surface, for example. As a final case, consider the equilibrium line where M h 0, averaged over the course of the year. It follows from Equation 57) that, if all quantities interpreted as their yearly-averages, a i H atm l H snow i ) q atm n. 61) Thus the non-latent) atmospheric energy fluxes are consumed completely in melting the snow. At the equilibrium line the yearly-average ice surface enthalpy H is therefore not determined by Equation 57). Note, however, that one side of the equilibrium line is the bare ice zone where Equation 60) may be applied. Our analysis does not include the energy released by compaction of firn. A firn process model could include production terms in either or both of the component balances 54) and 55), with resulting modifications to Equation 57). Alternately, a model could lump the firn-compaction heat into the term q atm in the above balances. In any case, firn is not part of the incompressible ice mixture so treating firn as part of the ice sheet requires caution.

8 8 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets Jump equation for momentum At the surface tangential stresses wind stress) and atmospheric pressure are small. It is standard to assume a stressfree surface Greve and Blatter, 2009, Equation 5.23)) T n = 0. 62) 3.6. Ice base Jump equation for mass The analysis of the basal layer is similar to the runoff layer at the upper surface, but pressure, frictional heating, and geothermal flux play new roles. On the other hand, at the ice base we assume there is no mass accumulation; all mass comes from solid or liquid water which is either present in the ice or in the layer. The subglacial layer may actually contain a mobile mixture of liquid water, ice fragments, and mineral sediments, but for simplicity we suppose that only pressuremelting-temperature liquid water moves in the layer. We do not model sediment transport. In Equation 64) below we parameterize the subglacial aquifer by an effective layer thickness η b and an effective subglacial liquid velocity v b. When the ice basal temperature is below the pressure-melting point, η b = 0 and v b = 0. We assume this aquifer has a liquid pressure p b, called the pore-water pressure if the aquifer penetrates a permeable till. This pressure does not need to be defined where η b = 0. All fields η b, v b, p b may vary in time and space. The general enthalpy formulation in this paper does not include a model closure, a process model, for the hydrology. Such a closure would relate the flux η b v b to p b, as in Darcy flow, for example, and it would model the pore water pressure p b c.f. Clarke, 2005). See subsection 5.1 for an simplified example treatment. Assuming the base = {z = bt, x, y)} is a differentiable surface, define Nb 2 = 1 + b 2 and n = N 1 b b, +1). The surface unit) normal vector n points from bedrock V ) into ice V + ). The normal surface speed of the ice base is w = N 1 b b/). For the mass balance of the solid mixture component, let ψ = ρ i, φ = 0, λ = 0, and π = µ i in Equation 45), where µ i is the mass exchange rate for freezing of liquid into solid: ρ i v n w ) = µ i. 63) Similarly for the liquid component, let ψ = ρ w, φ = 0, λ = ˆρ wη b, v = v b, and π = µ w in Equation 45), where η b is the basal water layer thickness, v b is the basal water velocity tangent to, and µ w is the mass exchange rate for melting: ρ w v n w ) + ˆρwη b) Define the ice base mass balance rate M b = ˆρwη b) + ˆρ wη b v b ) = µ w. 64) ˆρ wη b v b ), 65) having units of kg m 2 s 1. This quantity is analogous to the runoff-modified surface mass balance M h defined in Equation 51), but, as already noted, at the ice base there is no accumulation from external supply. The negative of M b is called the basal melt rate. Our notation directly acknowledges that this melt rate must be balanced by transport or storage in the subglacial hydrological layer. Equation 65) occurs in the literature; it is Equation 4) in Johnson and Fastook 2002), for example. Local conservation of water mass in each portion of the ice basal layer implies µ i + µ w = 0. Adding Equations 63) and 64), using the expressions for n and w, and multiplying through by ρ 1 N b gives a basal kinematical equation ρv n w ) = M b or b + u b x + v b y w = ρ 1 N b M b. 66) Jump equation for enthalpy The ice mixture enthalpy density ρh jumps from zero in the bedrock to its value in the ice. For simplicity we do not track the energy content of water deep within the bedrock aquifers or permafrost below a thin, active subglacial layer). We assume that the energy content of the bedrock is well-described by its temperature field. Heat is supplied to the subglacial layer by the geothermal flux q lith. Also, the stress applied by the lithosphere to the base of the ice is f = T n, with shear component τ b = f f n)n = shear traction, Liu, 2002), so the frictional heating arising from sliding is F b = τ b v, a positive surface flux, if v is the ice basal velocity. Let Σ i, Σ w be the enthalpy density exchange rates in the subglacial layer, for freezing and melting, respectively. For the energy balance of the ice component, substitution of ψ = ρ i H i, φ + = 1 ω)q, φ = 1 ω)q lith, λ = 0, and π = Σ i + 1 ω)f b in Equation 45) yields ρ i H i v n w ) + 1 ω) q q lith ) n = Σ i + 1 ω)f b. 67) For the water component we use analogous choices to those in Equation 64): substitute ψ = ρ wh w, φ + = ωq, φ = ωq lith, λ = ˆρ wh l p b )η b, v = v b, and π = Σ w + ωf b in Equation 45). This gives ρ wh w v n w ) + ω q q lith ) n + ˆρwH lp b )η b ) + ˆρ wh l p b )η b v b ) = Σ w + ωf b. 68) Though component Equations 67) and 68) have terms for heating in each component, representing a distribution between the phases, Equation 71) for the mixture is used in modeling, and not 67) or 68) directly. The distribution of heat to the components is included to clarify the derivation of 71) below. Similar to Equation 65), we define the ice base hydrological heating rate, a surface heat flux with units W m 2, by Q b = ˆρwH lp b )η b ) ˆρ wh l p b )η b v b ). 69) This is the rate at which hydrological storage and transport mechanisms deliver latent heat at a subglacial location. Regarding the signs, if the basal water layer grows in thickness, / > 0 in Equation 69), then heat is removed from the ice mixture above the subglacial layer.) Adding component equations 67) and 68) and simplifying using local conservation of energy Σ i +Σ w = 0), and applying definition 69), gives ρh v n w ) + q q lith ) n = F b + Q b. 70) Recalling Equation 66) we rewrite the above as M b H = F b + Q b q q lith ) n. 71) Note that H on the left side of Equation 71) is the value of the enthalpy of the ice mixture at its base, which is generally different from the enthalpy of the subglacial liquid water = H l p b ), as in 69)).

9 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 9 Liquid water which is transported in the subglacial aquifer can undergo phase change merely because of changes in the pressure-melting temperature in this water see subsection of Clarke 2005) and this situation is addressed by Equation 71). Conversely, the subglacial pressure changes because of energy fluxes. Indeed, from Equations 65) and 69), with dp b /dt = p b / + v b p b and γ = H l p b )/ p b, we have ˆρ wη b γ dp b dt = H lp b )M b Q b. 72) Using Equation 72) to eliminate Q b from Equation 71) we get ˆρ wη b γ dp b dt = F b M b H H l p b )) q q lith ) n. 73) We emphasize that Equation 73) does not determine pressure changes dp b /dt without a closure, a connection between η b and p b. We define a point in the basal surface = {z = b} to be cold if H < H sp) and η b = 0, and temperate otherwise. If all points of an open region R of are cold then Equation 65) implies M b = 0 and Equation 73) implies Q b = 0, so Equation 71) then says q n = q lith n + F b. 74) Thus the heat entering the base of the ice mixture combines geothermal flux and frictional heating. In this cold and dry case the frictional heating F b parameterizes only classic hard bed sliding Clarke, 2005). Equation 74) is the standard heat flux boundary condition for the base of cold ice. The subglacial aquifer is subject to the obvious positivethickness inequality η b 0. The condition η b = 0 is unstable, however, if the ice base is temperate. In fact Equation 71) implies that any heat imbalance, however slight, can immediately generate a nonzero melt rate if there is temperate ice at the base. The temperate base case is where either H H sp) or η b > 0 by the previous definition. This case simplifies to H H sp) and η b 0 if the subglacial liquid is assumed to be at the pressure-melting temperature. That is, if η b > 0 then heat will move so that the base of the ice mixture has H H sp). Nonetheless, supercooled subglacial water is allowed in Equation 71) and in 73). The temperate base case permits interpreting Equation 71) and/or 73) as a basal melt rate calculation. The basal melt rate itself M b ) can be either positive basal ice melts) or negative subglacial liquid freezes onto the base of the ice). From Equation 73) we can now compute the basal melt rate M b = F b q q lith ) n ˆρ wη b γdp b /dt). 75) H l p b ) H A commonly-adopted simplified view both neglects pressure variation in determining the subglacial liquid enthalpy and assumes that the basal pressure p of the ice mixture is at the same value. In fact, for this paragraph let p 0 be the common pressure both of the subglacial liquid and of the basal ice. The ice base enthalpy is H = H sp 0 ) + ωl and the subglacial liquid enthalpy is H l p 0 ) = H sp 0 ) + L. By constancy of the pressure, Equations 65) and 69) imply we also have Q b = H l p 0 )M b. With these simplifications, Equation 71) says that M b = F b q q lith ) n. 76) 1 ω)l Equation 76) appears in the literature in many places. For example, it matches Equation 5.40) in Greve and Blatter 2009) in the case ω = 0 noting a substitution M b = ρa b and a change the sign of the normal vector n). Similar notational changes give Equation 5) in Clarke 2005). We believe that Equations 71), Equation 73), and 75) are each more fundamental than 76), however, as they apply even in cases with highly-variable subglacial aquifer pressure p b. An additional possibility, which is not addressed in this paper is that liquid water could enter the thin subglacial layer from aquifers deeper in the bedrock; compare Equation 9.123) in Greve and Blatter 2009). The possibility that the subglacial aquifer could be connected directly to the supraglacial runoff layer by moulin drainage is also not modeled here. Jump equation for momentum The normal stress is continuous across the ice-bedrock transition, T lith n = T n. 77) Because this does not provide a boundary condition, sliding needs to be implemented in some other way, for example either assuming a power law type sliding parameterization Weertman, 1971) or a Coulomb type sliding friction Schoof, 2010a). See Clarke 2005) for further discussion. 4. SIMPLIFIED THEORY The mathematical model derived in the previous sections is general enough to apply to a wide range of numerical glacier and ice sheet models. It is neither limited to a particular stress balance nor a particular numerical scheme. In this section, however, we propose simplified parameterizations and shallow approximations which make an enthalpy formulation better suited to current modeling practice Hydrostatic simplifications The pressure p of the ice mixture appears in all relationships between enthalpy, temperature, and liquid water fraction section 2). Simplified forms apply to these relationships in those flow models which make the hydrostatic pressure approximation Greve and Blatter, 2009) p = ρgh z). 78) Recall also the Clausius-Clapyron relation T mp) p = β 79) where β is constant Lliboutry, 1971; Harrison, 1972). In this subsection we derive consequences of 78) and 79). Within temperate ice the heat flux term in enthalpy field Equation 41) is transformed: k T mp) + K 0 H) = β k p) + K 0 H) [ = βρg k h) k ] + K 0 H). 80) z Recall that k = kh, p) varies within temperate ice while K 0 is constant subsection 3.2). For small values of the liquid water fraction c.f. subsection 4.4), a further simplification identifies the mixture thermal conductivity with the constant thermal conductivity of ice, kh, p) = 1 ω)k i + ωk w k i 81)

10 10 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets see Equation 31)). In temperate ice the enthalpy flux divergence term therefore combines an essentially-geometric additive factor, relating to the curvature of the ice surface, with a small true) diffusivity for K 0 > 0: k T mp) + K 0 H) = βρgk i 2 h+ K 0 H). 82) Defining Γ = βρgk i 2 h, enthalpy field Equation 41) simplifies to this one, again with cases for cold and temperate ice: ρ dh dt = { Ki H) K 0 } H ) + { } 0 + Q. 83) Γ The geometric factor Γ can be understood physically as a differential heating or cooling rate of a column of temperate ice by its neighboring columns with differing surface heights. It represents a small contribution to the overall heating rate, however, especially relative to dissipation heating Q Aschwanden and Blatter, 2005). In terms of typical values, with a strain rate 10 3 a 1 and a deviatoric stress 10 5 Pa Paterson, 1994) we have Q J m 3 s 1. On the other hand, if the ice upper surface changes slope by in horizontal distance 1 km then 2 h 10 6 m 1. Using the constants from Table 1 below, we have Γ J m 3 s 1, three orders of magnitude smaller than Q. Of course, the curvature 2 h does not really have a typical value because, even in the simplest models Bueler and others, 2005), it has unbounded magnitude at ice sheet divides and margins. Examination of a typical digital elevation model for Greenland suggests 2 h < 10 6 m 1 for more than 95% of the area, however. For these reasons we implement the simplest model in subsection 5.1 below: Γ = 0 in Equation 83). Regardless of the magnitude of Γ, Equation 83) is parabolic in the mathematical sense. That is, its highest-order spatial derivative term is uniformly elliptic Ockendon and others, 2003). We expect that min{k i H), K 0 } = K 0. In any case we can safely assume a positive minimum value of this coefficient c.f. coercivity in Calvo and others, 1999). It follows that the initial/boundary value problem formed by Equation 83), along with boundary conditions which are of mixed Dirichlet and Neumann type subsection 4.5), is well-posed Ockendon and others, 2003). Equation 83) is strongly advectiondominated in many common modeling situations, however, and numerical schemes should be constructed accordingly Shallow enthalpy equation In addition to hydrostatic approximation, some ice sheet models use the small aspect ratio of ice sheets to simplify their model equations. Heat conduction can be shown to be important in the vertical direction in such shallow models, while it is negligible in horizontal directions Greve and Blatter, 2009). With this simplification, and also taking Γ = 0 for the reasons specified above, Equation 83) becomes ρ dh dt = z { Ki H) K 0 } H z ) + Q. 84) This equation is used in the PISM implementation Simplified relation among H, p, T, ω As noted in section 2, the heat capacity C i T ) of ice is approximately constant in the range of temperatures relevant to glacier and ice sheet modeling. Denote this constant heat capacity of ice by c i so C i T ) = c i + O T ) where T = T T 0. Here O T ) K T for a constant K which is, for instance, a bound on C i / T in the range of relevant temperatures. Equations 5) and 6) become H i = c i T T 0 ) + O T 2 ), 85) H w = c i T mp) T 0 ) + L + O T 2 ). 86) We now drop the terms O T 2 ) above and recall that the mixture enthalpy is H = 1 ω)h i +ωh w and that ω = 0 if T < T mp). Thus and HT, ω, p) = H sp) = c i T mp) T 0 ). 87) { ci T T 0 ), T < T m, H sp) + ωl, T = T m and 0 ω < 1. 88) In our simplified implementation, formula 88) replaces Equation 10). Instead of general forms 11) and 12), we have these inverses: { c 1 T H, p) = i H + T 0, H < H sp), 89) T mp), H sp) H, { 0, H < Hsp), ωh, p) = L 1 90) H H sp)), H sp) H. Though this formulation is quite simplified, it is similar to that employed in other models, including Lliboutry 1976), Katz 2008), and Calvo and others 1999). Equations 88), 89), and 90) are used in the PISM implementation subsection 5.1) A minimal drainage model Observations do not support a particular upper bound on liquid water fraction ω, but reported values above 3% are rare Pettersson and others, 2004, and references therein). At sufficiently-high ω values drainage occurs simply because liquid water is denser than ice. Available parameterizations, especially the absence of flow laws for ice with ω > 1% Lliboutry and Duval, 1985), also suggest keeping ω in the experimentally-tested range. Finally, a parameterized drainage mechanism is not required to implement an enthalpy formulation, but large strain heating rates will raise liquid water fraction to unreasonable levels without it. For such reasons Greve 1997b) introduced a drainage function. We adapt it to the enthalpy context. Let Dω) t be the dimensionless mass fraction removed in time t by drainage of liquid water to the ice base. Note that Dω) is a rate, and the time step and drainage rate must satisfy Dω) t 1, so that no more than the whole mass is drained in one timestep. The mass removed in the timestep is ˆρ wdω) t. Though Greve 1997b) proposes Dω) = + for ω 0.01, choosing Dω) to be finite increases the regularity of the enthalpy solution. Figure 4 shows an example drainage function which is zero for ω 0.01 and which is finite. At ω = 0.02 the rate of drainage is a 1 while above ω = 0.03 the drainage rate of 0.05 a 1 is sufficient to move the whole mass, if it should melt, to the base in 20 years. Though application of this function allows ice with ω > 0.01, we observe in modeling use that such values are rare; see section 5. We cap the ice softness computed from flow law 39) at the ω = 0.01 level to stay in the experimentally-tested range.

11 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 11 D ω) a ω Figure 4. A finite drainage function. Drained water is transported to the base by an un-modeled mechanism. If included at every stage in the analysis in section 3, the consequences of drainage would be pervasive, as the ice mixture can no longer be treated as incompressible, for example. Such limitations are, however, consequences of incomplete experimental knowledge of drainage processes, and not intrinsic in an enthalpy formulation. The basal melt rate M b is increased by the drainage, h M b = M b + b ˆρ wdω) dz. 91) We modify incompressibility equation 17) to globally conserve mass in the sense that the vertical velocity is computed using the new basal melt rate, w = M b + b u, v) z z=b b u x + v ) dz. 92) y We also modify equation 75), or its simplified version 76), by using this new basal melt rate. Because incompressibility along with the surface kinematical equation are usually combined in shallow ice sheet models to give a verticallyintegrated mass continuity equation, the latter equation also picks up the basal melt rate from the drainage model Bueler and Brown, 2009). We observe that the direct drainage modifications to the vertical velocity and surface elevation rate are of less significance to ice dynamics than are the effect of drainage-generated changes in basal melt rate on modeled basal strength e.g. till yield stress; c.f. Bueler and Brown, 2009). Finally, a back-of-the-envelope calculation suggests the importance of drainage to an ice dynamics model. Consider the Jakobshavn drainage basin on the Greenland ice sheet, with yearly-average surface mass balance of approximately 0.4 m a 1 ice-equivalent Ettema and others, 2009), area approximately m 2, and average surface elevation approximately 2000 m. If the surface of this region were to drop uniformly by 1 m in a year, then the gravitational potential energy released is sufficient to fully melt 2 km 3 of ice. This energy will be dissipated by strain heating at locations where strain rates and stresses, and basal frictional heating, are highest. It follows that a model which has three-dimensional grid cells of typical size 1 km 3 = 10 km 10 km 10 m, for example) should sometimes generate liquid water fractions far above 0.01 if drainage is not included. Under abrupt, major changes in basal resistance or calving-front backpressure, with corresponding changes to strain rates and thus strain heating, it is possible that some grid cells of 1 km 3 size are completely melted if time steps of order one year are taken. Model time steps should be shortened so that generated liquid water is drained over many times steps Simplified boundary conditions Mass In all ice sheet and glacier models the surface process components should report the runoff-modified net accumulation M h and a separated snowfall rate a i. In a Stokes model the surface and basal kinematical Equations 52) and 66) are boundary conditions used in determining the mixture velocity field and the surface elevation changes, while in shallow models one may rewrite 92) to give a map-plane mass continuity equation Bueler and Brown, 2009, for example). The basal boundary condition for mass balance is the basal melt rate addressed next). Enthalpy Generally Equation 57) for the upper surface is a Robin boundary condition Ockendon and others, 2003) involving both the surface value of the enthalpy and its normal derivative. In addition to the accumulation and snowfall rates already needed for the mass conservation boundary condition, the surface process components should at least report a snow temperature T snow. There must be a model to compute q atm, the non-latent heat flux from the atmosphere, generally including sensible, turbulent, and radiative fluxes. Modeling these quantities is non-trivial, and simplified models are frequently appropriate, but this is beyond our scope. As suggested earlier, Equation 57) is often simplified by neglecting the residual heat flux downward into the ice, q n = 0. Subsection 3.5 describes the cases of accumulation zone, ablation zone, and equilibrium line. In general, regarding yearlyaverage values, we may regard Equation 57) as yielding a Dirichlet boundary condition for the enthalpy field Equation 41). As a consequence, models for the atmosphere and the melt/runoff layer should report the quantities necessary to compute the right-hand sides of Equations 58) and 60). The runoff layer thickness η h itself, if modeled in a surface processes model, is needed primarily to determine the surface mass balance M h. Surface type is also important, that is, whether there is a positive thickness layer of firn throughout the year or whether the surface is bare ice in the melt season. Figure 5 describes the application of the basal boundary condition and the associated computation of the basal melt rate. Note that in the temperate base case, where η b may be zero or positive and where H H sp), either Equation 75) or 76) allows us to compute a nonzero basal melt rate M b. In doing so, however, we are left needing independent information on the boundary condition to the enthalpy field equation 41) itself. Because that equation is parabolic, a Neumann boundary condition may be used but the no boundary condition applicable to pure advection problems is not mathematically acceptable. Because the basal melt rate calculation will balance the energy at the base, a reasonable model is to apply an insulating Neumann boundary condition K 0 H n = 0 in the case of a positive-thickness layer of temperate ice, and apply a Dirichlet boundary condition H = H sp) if the basal temperate ice has zero thickness. In the discretized implementation, whether there is a positive-thickness layer is diagnosed using a criterion like H+ z) > H sp+ z)). Momentum Equation 62) applies at the ice surface. As noted earlier, a sliding relation or no-slip condition is required at the base, an issue beyond the scope of this paper.

12 12 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets yes cold base) Eqn 74) is Neumann b.c. for Eqn 41); M b = 0 H = H sp) is Dirichlet b.c. for Eqn 41) q = K i H) H at ice base η b > 0 & H < H sp)? no H < H sp)? no no yes H := H sp) positive thickness of temperate ice at base? compute M b from Eqn 75) or 76) yes H n = 0 is Neumann b.c. for Eqn 41) q = kh, p) T mp) at ice base Figure 5. Decision chart for each basal location. Determines basal boundary condition to enthalpy field equation 41), identifies the expression for upward heat flux at the ice base, and computes basal melt rate M b. The basal value of the ice mixture enthalpy is H On bedrock thermal layers Ice sheet energy conservation models may include a thin layer of the earth s lithosphere bedrock), of at most a few kilometers. This is justified because the heat flux q lith = k lith T, which enters into the subglacial layer energy balance Equation 71), varies in response to changes in the insulating thickness of overlain ice. Because the top few kilometers of the lithosphere generally have a homologous temperature far below one, we may treat the bedrock as a heat-conducting solid whose internal energy is fully described by its temperature. Because this layer is shallow, horizontal conduction is neglected. An observed geothermal flux field q geo is part of the model data. If a bedrock thermal layer is present then q geo forms the Neumann lower boundary condition for that layer. Though the numerical scheme for a bedrock thermal model could be implicitly-coupled with the ice above, a simpler view with adequate numerical accuracy comes from splitting the conservation of energy timestep into a bedrock timestep followed by an ice timestep. Thus, for the duration of the timestep the temperature of the top of the lithosphere is held fixed as a Dirichlet condition for the bedrock thermal layer, the flux q lith is computed as an output of the bedrock thermal layer model, and the ice thermal model sees this flux as a part of its basal boundary condition subsection 4.5). 5. RESULTS 5.1. Implementation in PISM The simplified enthalpy formulation of the last section was implemented in the open-source Parallel Ice Sheet Model PISM). Because other aspects of PISM are well-documented in the literature Bueler and others, 2007; Bueler and Brown, 2009; Winkelmann and others, 2010) and online at we only outline enthalpy-related aspects here. The results below used PISM trunk revision Section 4 already identifies the significant continuum equations implemented by PISM, but we note some features that update previous work. Conservation of energy Equation 84) replaces the temperature-based equation e.g. Equation 7) in Bueler and Brown, 2009)). Equations 76) and 91) are used to compute basal melt. This drainage and basal melt model replaces the one described by Bueler and Brown 2009). The finite difference scheme used here for conservation of energy is essentially the same as the one described by Bueler and others 2007), but with the temperature variable replaced by enthalpy. The combined advection-conduction problem in the vertical is solved by an implicit time step, avoiding timestep restrictions based on vertical ice velocity c.f. Bueler and others, 2007). For the reasons named in subsection 4.6, the bedrock thermal layer model is only explicitly-coupled to ice enthalpy problem. The previous lateral diffusion scheme for stored basal water Bueler and Brown, 2009) is not used, so the basal hydrology model is very simple: Generated basal melt is stored in place, it is limited to a maximum of 2 m of stored water, and refreeze is allowed. This stored water drains at a fixed rate of 1 mm year 1 in the absence of active melting or refreeze Application to Greenland Our goals in applying the enthalpy formulation to Greenland are deliberately limited. We demonstrate differences in thermodynamical variables between the enthalpy formulation and a temperature-based method. We examine the basal temperature, the thickness of the basal temperate layer, and the basal melt rate for a modeled steady state. A more comprehensive sensitivity analysis addressing non-steady conditions, sliding, spinup choices, and so on, is beyond the scope of this paper. The majority of the Greenland ice sheet has a Canadian thermal structure, if there is any temperate ice in the ice column at all. For comparison, Aschwanden and Blatter 2009) have modeled a Scandinavian thermal structure for Storgläciaren with a related enthalpy method. To make our results easier to interpret and compare to the literature, we use the EISMINT experimental setup for Greenland Huybrechts, 1998). It provides bedrock and surface elevation, parameterizations for surface temperatures and precipitation, and degree-day factors. Our restriction to the non-sliding shallow ice approximation is in keeping with EIS- MINT model expectations Hutter, 1983). 1 Because there is no sliding, F b = 0 in Equation 76). 1 The addition of a sliding model requires a membrane stress balance Bueler and Brown, 2009). At minimum, this introduces a necessarily-parameterized subglacial process model relating the availability of subglacial water to the basal shear stress. Interpreting the consequences of such subglacial model choices, while

13 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 13 Table 1. Parameters used in subsection 5.2. Variable Name Value Unit E enhancement factor 3 - β Clausius-Clapeyron K Pa 1 c i heat capacity of ice 2009 J kg K) 1 c w... of liquid water 4170 J kg K) 1 c r... of bedrock 1000 J kg K) 1 k i conductivity of ice 2.1 J m K s) 1 k r... of bedrock 3.0 J m K s) 1 ˆρ i bulk density of ice 910 kg m 3 ˆρ w... of liquid water 1000 kg m 3 ˆρ r... of bedrock 3300 kg m 3 L latent heat of fusion J kg 1 g gravity 9.81 m s 2 Two experiments were carried out: 1. a run using an enthalpy formulation CTRL) 2. a run using a temperature-based method COLD) For CTRL the rate factor in temperate ice is calculated according to Equation 39). The temperate ice diffusivity in Equation 84) was chosen to be K 0 = kg m s) 1, which is 1/10 times the value K i = k i /c i for cold ice. This choice may be regarded as a regularization of the enthalpy equation to make it uniformly-parabolic, but the results of Aschwanden and Blatter 2009) suggest this value is already in the range where the CTS is relatively stable under further reduction of K 0. Additional parameters common to both runs are listed in Table 1. Both runs started from the observed geometry and the same heuristic estimate of temperature at depth. Then simulations on a 20 km horizontal grid ran for 230 ka. At this time the grid was refined to 10 km, then they ran for an additional 20 ka. We used an unequally-spaced vertical grid, with spacing from 5.08 m at the ice base to 34.9 m at the top of the computational domain, and a 2000 m thick bedrock thermal layer with 40 m spacing. Figure 6 shows that simulated basal temperatures are broadly similar. Note that, by our definition, ice in CTRL is temperate if H H sp) while ice in COLD is temperate if its pressureadjusted temperature is within K of the melting point. Both methods predict a majority of the basal area to be cold, with COLD giving greater temperate basal area and temperate volume fraction; see Table 2. The thickness of the basal temperate layer Figure 7) is also greater for COLD than for CTRL. Where there is a significant amount of basal temperate ice, typical modeled thicknesses vary from 25 to 50 m in CTRL, with only small near-outlet areas approaching 100 m. critical for understanding fast ice dynamics, is beyond the scope of this paper about enthalpy formulations. Table 2. Measurements at the end of each run. CTRL COLD total ice volume [10 6 km 3 ] temperate ice volume fraction temperate ice area fraction total ice enthalpy [10 23 J kg 1 ] The greater amount of temperate ice from COLD, both in volume and extent, may surprise some readers, but there are reasonable explanations. One is that, in parts of the ice in run CTRL where the liquid water content is significant e.g. approaching 1%), the greater softness from Equation 39) implies that shear deformation and associated dissipation heating are enhanced in the bottom of the ice column and do not continue to increase in the upper ice column as the ice is carried toward the outlet. Note that our result is also consistent with those of Greve 1997b), who observes much larger temperate volume and thickness from a cold-ice simulation e.g. Table 2 in Greve, 1997b). Figure 8 shows that basal melt rates are higher for CTRL than for COLD. This is easily explained by recalling the deficiency of cold-ice methods which was identified in the introduction. Namely, at points within the three-dimensional ice fluid where ice temperature has reached the pressure-melting temperature, and where dissipation heating continues, generated water must either be drained immediately or it must be lost as modeled energy. Neither approach is physical. The method used in COLD is the one described by Bueler and Brown 2009), in which only a fraction of the generated water, especially that portion generated near the base, is drained immediately as basal melt Bueler and Brown, 2009, section 2.4). This cold-ice method fails to generate as much basal melt as it should because it cannot generate it in an energyconserving way. Though an energy-conserving polythermal method is obviously preferable, such methods nonetheless require drainage models, though these are presently constrained by few observations. Regarding the spatial distribution of basal melt rate seen in both runs, note that straightforward conservation of energy arguments c.f. the back-of-the-envelope calculation in subsection 4.4) suggest that in outlet glacier areas the amount of energy deposited in near-base ice by dissipation heating greatly exceeds that from geothermal flux. In any case, these runs used the uniform constant value of 50 mw m 2 from EISMINT-Greenland Huybrechts, 1998), so spatial variations in our thermodynamical results are not explained by variations in the geothermal flux. 6. CONCLUSIONS Introduction of an enthalpy formulation into an ice sheet model, replacing temperature as the thermodynamical state variable, is not a completely trivial process. The enthalpy field Equation 41) is similar enough to the temperature version to make it seem straightforward, but the possibility of liquid water in the ice mixture raises modeling issues, including choice of temperate ice rheology and how to model intraglacial transport drainage). One must interpret the enthalpy equation boundary conditions in the presence of mass and energy sources and sinks at the boundaries, both supraglacial runoff and subglacial hydrology. We have revisited conservation of mass and energy in boundary process models, of necessity, though particular choices of closure relations is generally beyond the scope of our work. At boundaries we use a new technique which views jumps and thin-layer transport models as complementary views of the same equation. Supraglacial and subglacial boundary conditions follow from this technique. A more-complete basal melt rate model appears, Equation 71). This view is wellsuited for coupling to surface mass and energy balance processes, firn processes, runoff flow, subglacial hydrology, and

14 14 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets Figure 6. Pressure-adjusted temperature at the base for the control run left) and the cold-mode run right). Hatched area indicates where the ice is temperate. Values are in degrees Celsius and contour interval is 2 C. The dashed line is the cold-temperate transition surface. so on. We have identified which fields are needed as boundary conditions for an enthalpy formulation, possibly provided through such coupling. The enthalpy formulation can be used to simulate both fully-cold and fully-temperate glaciers. Neither prior knowledge of the thermal structure nor a parameterization of the cold-temperate transition surface CTS) is required. Our application to Greenland shows that an enthalpy formulation can be used for continent-scale ice flow problems. The magnitude and distribution of basal melt rate, a thermodynamical quantity critical to modeling fast ice dynamics in a changing climate, can be expected to be more realistic in an energy-conserving enthalpy formulation than in most existing ice sheet models. ACKNOWLEDGMENTS The authors thank J. Brown and M. Truffer for stimulating discussions. This work was supported by Swiss National Science Foundation grant no , by NASA grant no. NNX09AJ38G, and by a grant of HPC resources from the Arctic Region Supercomputing Center as part of the Department of Defense High Performance Computing Modernization Program. REFERENCES Alexiades, V. and A. D. Solomon, Mathemtical Modeling of Melting and Freezing Processes, Hemisphere Publishing Corporation, Washington. Aschwanden, A. and H. Blatter, Meltwater production due to strain heating in Storglaciären, Sweden, J. Geophys. Res., 110. Aschwanden, A. and H. Blatter, Mathematical modeling and numerical simulation of polythermal glaciers, J. Geophys. Res., 114F1), Bell, R., F. Ferraccioli, T. Creyts, D. Braaten, H. Corr, I. Das, D. Damaske, N. Frearson, T. Jordan, K. Rose, M. Studinger and M. Wolovick, Widespread persistent thickening of the East Antarctic Ice Sheet by freezing from the base, Science, ), Bitz, C. M. and W. H. Lipscomb, An energy-conserving thermodynamic model of sea ice, J. Geophys. Res., 104, Björnsson, H., Y. Gjessing, S. E. Hamran, J. O. Hagen, O. Liestøl, O. F. Pálsson and B. Erlingsson, The thermal regime of sub-polar glaciers mapped by multi-frequency radio echo sounding, J. Glaciol., 42140), Blatter, H., On the thermal regime of an arctic valley glacier, a study of the White Glacier, Axel Heiberg Island, N.W.T., Canada, J. Glaciol., 33114), Blatter, H. and K. Hutter, Polythermal conditions in Artic glaciers, J. Glaciol., 37126), Blatter, H. and G. Kappenberger, Mass balance and thermal regime of Laika ice cap, Coburg Island, N.W.T., Canada, J. Glaciol., 34116), Bueler, E. and J. Brown, Shallow shelf approximation as a sliding law in a thermomechanically coupled ice sheet model, J. Geophys. Res., 114, F Bueler, E., J. Brown and C. Lingle, Exact solutions to the thermomechanically coupled shallow-ice approximation: effective tools for verification, J. Glaciol., 53182), Bueler, E., C. S. Lingle, J. A. Kallen-Brown, D. N. Covey and L. N. Bowman, Exact solutions and numerical verification for isothermal ice sheets, J. Glaciol., 51173), Calvo, N., J. Durany and C. Vázquez, Numerical approach of temperature distribution in a free boundary model for polythermal ice sheets, Numer. Math., 834),

15 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 15 Figure 7. Thickness of the basal temperate ice layer for the control run left) and the cold-mode run right). Values are in meters and contour interval is 25 m. The dashed line is the cold-temperate transition surface. Dotted areas indicate where the bed is temperate but the ice immediately above is cold. Clarke, G. K. C., Subglacial processes, Annu. Rev. Earth Planet. Sci., 33, Duval, P., The role of water content on the creep of polycrystalline ice, IAHS-AISH, 118, Elliott, C. M., Error Analysis of the Enthalpy Method for the Stefan problem, IMA J. Numer. Anal., 7, Ettema, J., M. R. van den Broeke, E. van Meijgaard, W. J. van de Berg, J. L. Bamber, J. E. Box and R. C. Bales, Higher sur- Figure 8. Basal melt rate for the control run left) and the cold-mode run right). Values are in millimeters per year. The dashed line is the cold-temperate transition surface.