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 HeinzBLATTER2

1Arctic Region Supercomputing Center, University of Alaska Fairbanks, USA

E-mail: andy.aschwanden@arsc.edu2Institute for Atmospheric and Climate Science, ETH Zurich, Switzerland

3Department of Mathematics and Statistics, University of Alaska Fairbanks, USA4Geophysical 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

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

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-

mal layering as in Figure 1a are found in cold regions such

as the Canadian Arctic (Blatter, 1987; Blatter and Kappen-

berger, 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 (Luthi 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. Bjornsson and others, 1996; Moore and oth-

ers, 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 tem-

perate 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, Univer-sity of Alaska Fairbanks, Fairbanks, USA.

a)

b)

temperate cold

Figure 1. Schematic view of the two most commonly found poly-

thermal 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 be-

tween components couple these equations. Here we derive an

enthalpy equation from a mixture theory which uses sepa-

rate 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 bal-

ances 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 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets

by Payne and others (2000) and verified by Bueler and oth-

ers (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 approx-

imately 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 la-

tent 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 es-

pecially 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 oth-

ers, 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 re-

gions (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 cumber-

some 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 re-

strictions apply to CTS shape. Transitions between polyther-

mal structures caused by changing climatic conditions can be

modeled even if nontrivial CTS topology arises at intermedi-

ate 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 equa-

tion, 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 prob-

lems, enthalpy methods have been applied to magma dynam-

ics (Katz, 2008), permafrost (Marchenko and others, 2008),

shoreline movement in a sedimentary basin (Voller and oth-

ers, 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 salin-

ity 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 flow-

line finite element ice sheet model. Aschwanden and Blat-

ter (2009) derived a mathematical model for polythermal

glaciers based on an enthalpy method. Their model used a

brine pocket parametrization scheme to obtain a relation-

ship between enthalpy, temperature and liquid water frac-

tion, 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 stud-

ies, 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 poten-

tial 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 de-

fined and its relationships to temperature and liquid wa-

ter fraction are determined by the use of mixture theory.

Enthalpy-based continuum mechanical balance equations for

mass, energy, and momentum are stated, along with constitu-

tive equations. Boundary conditions are carefully addressed

using the new technique in Appendix A. At this level of gen-

erality 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 par-

tial densities ρi and ρw (mass of the component per unit

volume of the mixture). The mixture density is the sum

ρ = ρi + ρw. (1)

The liquid water fraction, often called water (moisture) con-

tent, is the ratio

ω =ρw

ρ. (2)

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(ρi = 910 kg m−3) and

water(ρw = 1000 kg m−3) are distinct. However, for liquid

water fractions smaller than 5 % the change in mixture den-

sity due to changes in liquid water fraction are smaller than

0.5 %. It is therefore reasonable to set ρ ≈ ρi.

Define the barycentric (mixture) velocity v by

ρv = ρivi + ρwvw, (3)

where vi and vw are the velocity of ice and water, respec-

tively (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 “in-

ternal energy”.

For cold ice the temperature T is below the pressure melt-

ing point Tm(p). The specific enthalpy Hi of cold ice is defined

as

Hi =

T∫T0

Ci(T ) dT , (5)

where Ci(T ) is the measured heat capacity of ice at atmo-

spheric pressure. If the reference temperature T0 is below all

modeled ice temperatures then enthalpy values will be posi-

tive, though positivity is not important for correctness. The

specific enthalpy of liquid water Hw is defined as

Hw =

Tm(p)∫T0

Ci(T ) dT + L+

T∫Tm(p)

Cw(T ) dT , (6)

where Cw(T ) is the heat capacity of water and L is the latent

heat of fusion.

Functions Hi(T ) and Hw(T, 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 ≥ Tm(p)

then Hw(T, p) ≥ Hi(Tm(p)) + L. Supercooled liquid water

with T < Tm(p) is, however, allowed by Equation (6).

Experiments suggest that the heat capacity Ci(T ) of ice is

approximately a linear function of temperature in the range of

temperatures commonly found in glaciers and ice sheets (Pe-

trenko and Whitworth, 1999, and references therein). In fact

for most glacier-modeling purposes it suffices to approximate

Ci(T ) by a constant value independent of temperature. The

heat capacity of liquid water, Cw(T ), is also nearly constant

within the relevant temperature range. Thus one may define

simpler functions Hi(T ) and Hw(T, p), but until section 4 we

continue with the general forms above.

The enthalpy density ρH (volumetric enthalpy) of the mix-

ture is given by

ρH = ρiHi + ρwHw. (7)

From Equations (1), (2), (5), (6), and (7),

H = H(T, ω, p) = (1− ω)Hi(T ) + ωHw(T, p). (8)

Equation (8) describes the specific enthalpy of all mixtures,

including cold ice, temperate ice, and liquid water.

H

ωT

Hs Hl

Tm

1

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 = ω(H, p) (dotted line). Points

Hs(p) and Hl(p) are the enthalpy of pure ice and pure liquid water,

respectively, at temperature Tm(p).

In cold ice we have T < Tm(p) 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 = Tm(p) and 0 < ω < 1 in temperate

ice. Denote the enthalpy of ω = 0 ice at the pressure-melting

temperature by

Hs(p) =

Tm(p)∫T0

Ci(T ) dT . (9)

For mixtures with a positive solid fraction, Equation (8) now

reduces to two cases

H =

{Hi(T ), T < Tm(p),

Hs(p) + ωL, T = Tm(p) and 0 ≤ ω < 1.(10)

We now observe that an inverse function to Hi(T ) exists

under the reasonable assumption that Ci(T ) is positive. This

inverse is denoted Ti(H) and it is defined for H ≤ Hs(p).

Note ∂Hi/∂T = Ci(T ) and thus ∂T/∂Hi = Ci(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:

T (H, p) =

{Ti(H), H < Hs(p),

Tm(p), Hs(p) ≤ H,(11)

ω(H, p) =

{0, H < Hs(p),

L−1(H −Hs(p)), Hs(p) ≤ H.(12)

Schematic plots of temperature and water content as func-

tions of enthalpy are shown in Figure 2, with points Hs and

Hl(p) = Hw(Tm(p), p) = Hs(p)+L noted. Equations (11) and

(12) will only be applied to cold or temperate ice (mixtures),

and therefore H < Hl(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 de-

termined. It follows that pressure and enthalpy are a pre-

ferred 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 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets

3. CONTINUUM MODEL

3.1. Balance equations

General balance equationThe balance of a quantity ψ describing particles (fluid) which

move with velocity v is given by

∂ψ

∂t= −∇ · (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 balanceFor 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 wa-

ter is not considered.) Define Mw and Mi as the exchange

rates between the components, which are production terms

in Equation (13). Local conservation of mixture mass im-

plies Mi + Mw = 0. Setting ψ = ρi, v = vi, φ = 0, and

π = Mi = −Mw in Equation (13) for the ice component, and

similarly for the water component,

∂ρi

∂t+∇· (ρivi) = −Mw, (14)

∂ρw

∂t+∇· (ρwvw) = Mw. (15)

Adding Equations (14) and (15) yields the balance for the

mixture∂ρ

∂t+∇· (ρv) = 0, (16)

where v is the barycentric velocity (Equation (3)). As noted

already, however, the mixture density ρ ≈ ρi 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 balanceDenote the total enthalpy flux of the ice and liquid water

components, including any advective and conductive mech-

anisms, by Φi and Φw respectively. We define the residual

enthalpy fluxes qi,qw to be the fluxes relative to advection

by the barycentric velocity v:

qi = Φi − ρiHiv, (18)

qw = Φw − ρwHwv. (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. Empiri-

cal constitutive relations for these residual fluxes qi,qw are

addressed below.

Within the mixture, Σw and Σi are the enthalpy exchange

rates between the components. Local conservation of energy

implies Σi + Σw = 0. There are also dissipation heating rates

Qw and Qi for the components, but we will only give an

equation for the total heating rate Q = Qw +Qi for the mix-

ture, in Equation (42) below. For the ice component, setting

ψ = ρiHi, φ = qi, and π = Qi − Σw in Equation (13) gives

∂(ρiHi)

∂t= −∇ · (ρiHiv + qi) +Qi − Σw. (20)

Similarly for the water component,

∂(ρwHw)

∂t= −∇ · (ρwHwv + qw) +Qw + Σw. (21)

The smoothness of the modeled fields ρiHi and ρwHw is

closely-related to the empirical form chosen for the enthalpy

fluxes qi and qw. Differentiability of these fields follows from

the the presence of diffusive terms in enthalpy flux param-

eterizations. While we assume that the enthalpy solution is

spatially-differentiable, its derivatives may be large in mag-

nitude, corresponding to enthalpy “jumps” in practice.

Summing Equations (20) and (21) and using Equation (7)

gives a balance for the total enthalpy flux:

∂(ρH)

∂t= −∇ · (ρHv + qi + qw) +Q. (22)

Setting q = qi +qw, using notation d/dt = ∂/∂t+v · for the

material derivative, and recalling Equation (17), we arrive at

ρdH

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 cold-

ice glacier and ice sheet models.

Momentum balanceLet T be the Cauchy stress tensor. Recall that TD = 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 ne-

glected) where pressure gradients are balanced only by devi-

atoric stress gradients. Setting ψ = ρv, φ = −T, and π = ρg,

the balance of linear momentum follows from Equation (13),

namely 0 = −∇ ·T + ρg or

−∇ ·TD +∇p+ ρg = 0. (24)

Balance of angular momentum implies symmetry T = TT

(Liu, 2002).

3.2. Constitutive equations

Enthalpy fluxThe 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),

qi = −ki(T )∇T, (25)

where ki(T ) is the thermal conductivity of cold ice. Combin-

ing Equations (11) and (25) allows us to write the heat flux

in terms of the enthalpy gradient in cold ice,

qi = −ki(T )∇ (Ti(H)) = −ki(T )Ci(T )−1∇H. (26)

The last expression in Equation (26) can be written en-

tirely 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

Ci(H) = Ci(Ti(H)) and ki(H) = ki(Ti(H)), (27)

thus

Ki(H) = ki(H)Ci(H)−1. (28)

Fourier’s law for cold ice now has the enthalpy form

qi = −Ki(H)∇H. (29)

Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 5

The residual heat flux in temperate ice is the sum of sen-

sible and latent heat fluxes (Greve, 1997a),

q = qs + ql. (30)

The sensible heat flux qs in temperate ice is conductive, aris-

ing from variations in the pressure-melting temperature. Now,

the mixture conductivity, written in terms of enthalpy and

pressure, is

k(H, p) = (1− ω(H, p)) ki(H) + ω(H, p) kw (31)

where kw 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

qs = −k(H, p)∇Tm(p). (32)

The latent heat flux ql in temperate ice is proportional to

the mass flux of liquid water j relative to the barycenter:

ql = Lj = Lρw (vw − 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 implementabil-

ity. Hutter (1982) suggested a general formulation, involv-

ing 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 observa-

tions to identify the constitutive relation are scarce, however.

A drop in liquid water fraction was observed close to the bed

at Falljokull in Iceland (Murray and others, 2000) and on the

upper plateau of the Vallee Blanche in the Massif du Mont-

Blanc (Vallon and others, 1976), and these observations sug-

gests 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 ql is poorly-

constrained at the present state of experimental knowledge.

Greve (1997a) writes the simplest possible form by setting the

latent heat flux ql to zero in temperate ice. Our assumption

that the enthalpy field has finite spatial derivatives within the

ice explains our regularizing choice, namely

ql = −k0∇ω = −K0∇H, (34)

where k0 and K0 = L−1k0 are small positive constants. This

form may be justified by a Fick-type diffusion (Hutter, 1982),

or even by observing that numerical schemes approximat-

ing the hyperbolic equation arising from the choice ql = 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,

q =

{−Ki(H)∇H, cold,

−k(H, p)∇Tm(p)−K0∇H, temperate.(35)

Ice flowThe deviatoric stress tensor TD and the strain rate tensor

D = (1/2)(∇v +∇vT

)are related via the effective viscosity

η by

TD = 2ηD. (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 τe, a

rate factor (softness) A, and a power n,

2η = A−1(τ2e + ε2

)(1−n)/2. (37)

Here 2τ2e = TD

ijTDij (summation convention). The small con-

stant ε (units of stress) regularizes the flow law at low effec-

tive 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,

Ac(T, p) = A0e(−Qa+pV )/RT , (38)

where A0, Qa, V, R are constant (or have additional temper-

ature dependence; Paterson, 1994). For temperate ice, how-

ever, little is known about the effect of liquid water on viscos-

ity. In experimental studies, Duval and others (1977, 1985)

derived the following function for the rate factor in temperate

ice, At(ω, p), valid for water fractions less than 0.01 (or 1 %),

At(ω, p) = Ac(Tm(p), p) (1 + 181.25ω) , (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:

A(H, p) =

{Ac(T (H, p), p), H < Hs(p),

At(ω(H, p), p), Hs(p) ≤ H < Hl(p).(40)

3.3. 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 < Hs(p))

or temperate (H ≥ Hs(p)):

ρdH

dt= ∇·

({Ki(H)∇H

k(H, p)∇Tm(p) +K0∇H

})+Q. (41)

The rate Q at which dissipation of strain releases heat into

the mixture is

Q = tr(D ·TD

). (42)

The most complete set of field equations considered in this

paper consists of Equations (41) and (42) along with incom-

pressibility Equation (17) and the Stokes Equations (24). One

also includes the temperature-dependent power-law constitu-

tive 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.

3.4. 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 simpli-

fied 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 con-

tinuous 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 σ separates a region V into two sub-regions

6 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets

air(ρ = 0, ρH = 0)

ice(ρ = ρw + ρi, ρH = ρwHw + ρiHi)

bedrock(ρ = 0, ρH = 0)

n

n

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 σ, pointing from V −

to V +. The jump [[ψ]] of a tensor or scalar field ψ on σ 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

vσ within and along (tangent to) σ. For simplicity of presen-

tation we assume ψ is a scalar in this subsection. We suppose

the flow along σ has area density λσ, and also that there

is production of ψ in the surface σ at rate πσ. As shown in

Appendix A by a pillbox argument, general balance Equation

(13) implies a jump condition with additional terms:

[[ψ(v ·n− wσ)]] + [[φ ·n]] +∂λσ∂t

+∇· (λσvσ) = πσ, (44)

where wσ is the normal speed of the surface. Equation (45)

generalizes the better-known Rankine-Hugoniot jump condi-

tion (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− wσ)+(φ−φ−) ·n+∂λσ∂t

+∇· (λσvσ) = πσ. (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 σ and bounded on either side by three-dimensional

material. Thus Equation (44) is both a jump condition de-

scribing the three-dimensional field ψ and a balance equation

for the field λσ 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 (λσ = 0). The ve-

locity vσ 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 πσ 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-or-

less 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 gov-

erned 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.

3.5. Upper surface

Jump equation for massThe ice upper free surface σ = {z = h(t, x, y)} is where the

mixture density jumps from ρ = 0 in the air to ρ = ρi in the

ice (Figure 3). Assume that σ is differentiable. Define

N2h = 1 +

(∂h

∂x

)2

+

(∂h

∂y

)2

= 1 + |∇h|2. (46)

A unit normal vector for σ pointing from air (V −) into ice

(V +) is

n = N−1h

(∂h

∂x,∂h

∂y,−1

)= N−1

h (∇h,−1) , (47)

and the normal surface speed is

wσ =

(0, 0,

∂h

∂t

)·n = −N−1

h∂h

∂t. (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 σ at a rate µi. Let ψ = ρi,

φ = 0, v = (mixture velocity), λσ = 0, and πσ = µi + a⊥i in

Equation (45):

ρi (v ·n− wσ) = µi + a⊥i . (49)

Similarly for the water component, but additionally defining

λσ = ρwηr, where ηh is the effective runoff layer thickness,

and vσ = vh = (runoff velocity), we have

ρw (v ·n− wσ) +∂(ρwηh)

∂t+∇· (ρwηhvh) = µw + a⊥w , (50)

where a⊥w is essentially the rainfall mass rate and µw is the

thin layer mass exchange (refreeze) rate.

The sum a⊥ = a⊥i + a⊥w is the mixture snowfall-rainfall-

sublimation function, the mass rate for exchange with the

atmosphere. Conservation of water mass in the supraglacial

layer implies that µi + µw = 0 because no water is created

or destroyed (e.g. by chemistry). Define the (runoff-included)

surface mass balance

Mh = a⊥ − ∂(ρwηh)

∂t−∇ · (ρwηhvh) (51)

(SI units kg m−2 s−1). The accumulation-ablation function

Mh, 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

ρ(v ·n− wσ) = Mh. (52)

Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 7

Multiplying through by ρ−1Nh and using Equations (47) and

(48) gives the equivalent, possibly more familiar form

∂h

∂t+ u

∂h

∂x+ v

∂h

∂y− w = ρ−1NhMh. (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 enthalpySnow 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 (sen-

sible, turbulent, and radiative) whose sum is qatm. 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 well-

defined temperature Tsnow which varies in time and space,

with corresponding enthalpy Hsnowi = Hi(Tsnow). Denote the

enthalpy exchange rate for ice formed by freezing of liquid

water as Σi (units J m−2 s−1). We make the simplifying as-

sumption that λσ = 0 for the solid component; we assume

only liquid runoff is mobile. Equation (45) with ψ = ρiHi,

φ = (1−ω)q, φ− = (1−ω)qatm, and πσ = Σi+a⊥i Hi(Tsnow),

and with other symbols defined as for Equation (49), gives

ρiHi (v ·n− wσ) + (1− ω)(q− qatm

)·n

= Σi + a⊥i Hsnowi . (54)

For the liquid component we make the simplifying assump-

tions that both runoff and rainfall contain liquid water ex-

actly 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 Σw. Let

Hatml = Hl(patm) 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− = ωqatm, and λσ = ρwηhHatml in Equa-

tion (45), we have

ρwHw (v ·n− wσ) + ω(q− qatm

l

)·n +

∂(ρwηhHatml )

∂t

+∇·(ρwηhH

atml vh

)= Σw + a⊥wH

atml . (55)

Local conservation of energy for any portion of the surface

implies that Σi +Σw = 0. Adding the last two equations gives

an enthalpy boundary condition for the mixture:

ρH (v ·n− wσ) +(q− qatm

)·n = a⊥i H

snowi

+ a⊥wHatml − ∂(ρwηhH

atml )

∂t−∇ ·

(ρwηhH

atml vh

). (56)

Using Equations (51) and (52) to simplify Equation (56), with

Hatml factored out of derivatives, the result is a balance for

energy fluxes at the ice (mixture) upper surface:

Mh(H −Hatml ) = a⊥i

(Hsnow

i −Hatml

)+(qatm − q

)·n. (57)

Equation (57) is the surface energy balance equation writ-

ten 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 prac-

tice. 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 (ηh = 0, Mh =

a⊥) and positive snowfall a⊥i , Equation (57) is equivalent to

a Dirichlet condition for enthalpy,

H =

(a⊥ia⊥

)Hsnow

i +

(1− a⊥i

a⊥

)Hatm

l +qatm ·na⊥

. (58)

This Equation says the surface enthalpy is the mass-rate-

averaged 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 time-

averages may be more valuable for modeling. At times of

melting and runoff (Mh � 0 and a⊥i = 0), Equation (57)

reduces to

Mh(H −Hatml ) = qatm ·n. (59)

That is, the atmospheric fluxes melt the surface ice, with en-

thalpy H, to liquid water with enthalpy Hatml . This equation

could be used to determine the runoff Mh if qatm 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 cor-

responding 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 Mh ≈ a⊥ ≈ a⊥i so

H ≈ Hsnowi +

qatm ·na⊥

. (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 Mh ≈ 0,

averaged over the course of the year. It follows from Equation

(57) that, if all quantities interpreted as their yearly-averages,

a⊥i(Hatm

l −Hsnowi

)≈ qatm ·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 deter-

mined 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 com-

paction 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). Alter-

nately, a model could lump the firn-compaction heat into the

term qatm 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 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets

Jump equation for momentumAt the surface tangential stresses (wind stress) and atmo-

spheric pressure are small. It is standard to assume a stress-

free surface (Greve and Blatter, 2009, Equation (5.23))

T ·n = 0. (62)

3.6. Ice base

Jump equation for massThe 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 pressure-

melting-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 ηb and an effective subglacial

liquid velocity vb. When the ice basal temperature is below

the pressure-melting point, ηb = 0 and vb = 0. We assume

this aquifer has a liquid pressure pb, called the “pore-water

pressure” if the aquifer penetrates a permeable till. This pres-

sure does not need to be defined where ηb = 0. All fields

ηb,vb, pb 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 ηbvb to pb, as in Darcy

flow, for example, and it would model the pore water pressure

pb (c.f. Clarke, 2005). See subsection 5.1 for an simplified

example treatment.

Assuming the base σ = {z = b(t, x, y)} is a differentiable

surface, define N2b = 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 wσ =

N−1b (∂b/∂t).

For the mass balance of the solid mixture component, let

ψ = ρi, φ = 0, λσ = 0, and πσ = µi in Equation (45), where

µi is the mass exchange rate for freezing of liquid into solid:

ρi (v ·n− wσ) = µi. (63)

Similarly for the liquid component, let ψ = ρw, φ = 0, λσ =

ρwηb, vσ = vb, and πσ = µw in Equation (45), where ηb is

the basal water layer thickness, vb is the basal water velocity

tangent to σ, and µw is the mass exchange rate for melting:

ρw (v ·n− wσ) +∂(ρwηb)

∂t+∇· (ρwηbvb) = µw. (64)

Define the ice base mass balance rate

Mb = −∂(ρwηb)

∂t−∇ · (ρwηbvb) , (65)

having units of kg m−2 s−1. This quantity is analogous to the

runoff-modified surface mass balance Mh defined in Equation

(51), but, as already noted, at the ice base there is no accu-

mulation from external supply. The negative of Mb 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 µi + µw = 0. Adding Equations (63)

and (64), using the expressions for n and wσ, and multiplying

through by ρ−1Nb gives a basal kinematical equation ρ(v ·n−wσ) = Mb or

∂b

∂t+ u

∂b

∂x+ v

∂b

∂y− w = −ρ−1NbMb. (66)

Jump equation for enthalpyThe 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 as-

sume 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 qlith. 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 Fb = −τ b ·v, a positive surface

flux, if v is the ice basal velocity.

Let Σi, Σw 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 =

ρiHi, φ+ = (1 − ω)q, φ− = (1 − ω)qlith, λσ = 0, and πσ =

Σi + (1− ω)Fb in Equation (45) yields

ρiHi (v ·n− wσ) + (1− ω) (q− qlith) ·n= Σi + (1− ω)Fb. (67)

For the water component we use analogous choices to those

in Equation (64): substitute ψ = ρwHw, φ+ = ωq, φ− =

ωqlith, λσ = ρwHl(pb)ηb, vσ = vb, and πσ = Σw + ωFb in

Equation (45). This gives

ρwHw (v ·n− wσ) + ω (q− qlith) ·n +∂(ρwHl(pb)ηb)

∂t

+∇· (ρwHl(pb)ηbvb) = Σw + ωFb. (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

Qb = −∂(ρwHl(pb)ηb)

∂t−∇ · (ρwHl(pb)ηbvb) . (69)

This is the rate at which hydrological storage and transport

mechanisms deliver latent heat at a subglacial location. (Re-

garding the signs, if the basal water layer grows in thickness,

∂/∂t > 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 (Σi+Σw = 0), and applying

definition (69), gives

ρH (v ·n− wσ) + (q− qlith) ·n = Fb +Qb. (70)

Recalling Equation (66) we rewrite the above as

MbH = Fb +Qb − (q− qlith) ·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 (=

Hl(pb), as in (69)).

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

4.1.1 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 dpb/dt = ∂pb/∂t+ vb ·∇pb and γ = ∂Hl(pb)/∂pb,

we have

ρwηbγdpbdt

= Hl(pb)Mb −Qb. (72)

Using Equation (72) to eliminate Qb from Equation (71) we

get

ρwηbγdpbdt

= Fb −Mb (H −Hl(pb))− (q− qlith) ·n. (73)

We emphasize that Equation (73) does not determine pres-

sure changes dpb/dt without a closure, a connection between

ηb and pb.

We define a point in the basal surface σ = {z = b} to be

cold if H < Hs(p) and ηb = 0, and temperate otherwise. If

all points of an open region R of σ are cold then Equation

(65) implies Mb = 0 and Equation (73) implies Qb = 0, so

Equation (71) then says

q ·n = qlith ·n + Fb. (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 Fb 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 positive-

thickness inequality ηb ≥ 0. The condition “ηb = 0” is unsta-

ble, however, if the ice base is temperate. In fact Equation

(71) implies that any heat imbalance, however slight, can im-

mediately generate a nonzero melt rate if there is temperate

ice at the base.

The temperate base case is where either H ≥ Hs(p) or

ηb > 0 by the previous definition. This case simplifies to “H ≥Hs(p) and ηb ≥ 0” if the subglacial liquid is assumed to be

at the pressure-melting temperature. That is, if ηb > 0 then

heat will move so that the base of the ice mixture has H ≥Hs(p). 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 (−Mb) 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

−Mb =Fb − (q− qlith) ·n− ρwηbγ(dpb/dt)

Hl(pb)−H. (75)

A commonly-adopted simplified view both neglects pres-

sure 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 p0 be the com-

mon pressure both of the subglacial liquid and of the basal

ice. The ice base enthalpy is H = Hs(p0) + ωL and the sub-

glacial liquid enthalpy is Hl(p0) = Hs(p0) +L. By constancy

of the pressure, Equations (65) and (69) imply we also have

Qb = Hl(p0)Mb. With these simplifications, Equation (71)

says that

−Mb =Fb − (q− qlith) ·n

(1− ω)L. (76)

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 “−Mb =

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 pb.

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 Equa-

tion (9.123) in Greve and Blatter (2009). The possibility that

the subglacial aquifer could be connected directly to the supra-

glacial runoff layer by moulin drainage is also not modeled

here.

Jump equation for momentumThe normal stress is continuous across the ice-bedrock tran-

sition,

Tlith ·n = T ·n. (77)

Because this does not provide a boundary condition, slid-

ing 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.

4.1. Hydrostatic simplifications

The pressure p of the ice mixture appears in all relation-

ships between enthalpy, temperature, and liquid water frac-

tion (section 2). Simplified forms apply to these relationships

in those flow models which make the hydrostatic pressure

approximation (Greve and Blatter, 2009)

p = ρg(h− z). (78)

Recall also the Clausius-Clapyron relation

∂Tm(p)

∂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∇Tm(p) +K0∇H) = −β∇· (k∇p) +∇· (K0∇H)

= −βρg[∇· (k∇h)− ∂k

∂z

]+∇· (K0∇H). (80)

Recall that k = k(H, p) varies within temperate ice while K0

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,

k(H, p) = (1− ω)ki + ωkw ≈ ki (81)

10 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets

(see Equation (31)). In temperate ice the enthalpy flux diver-

gence term therefore combines an essentially-geometric addi-

tive factor, relating to the curvature of the ice surface, with

a small (true) diffusivity for K0 > 0:

∇· (k∇Tm(p) +K0∇H) = −βρgki∇2h+∇· (K0∇H). (82)

Defining Γ = −βρgki∇2h, enthalpy field Equation (41) sim-

plifies to this one, again with cases for cold and temperate

ice:

ρdH

dt= ∇·

({Ki(H)

K0

}∇H

)+

{0

Γ

}+Q. (83)

The geometric factor Γ can be understood physically as

a differential heating or cooling rate of a column of tem-

perate ice by its neighboring columns with differing surface

heights. It represents a small contribution to the overall heat-

ing 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 105 Pa (Pa-

terson, 1994) we have Q ≈ 3× 10−6 J m−3 s−1. On the other

hand, if the ice upper surface changes slope by 0.001 in hori-

zontal distance 1 km then |∇2h| ≈ 10−6 m−1. Using the con-

stants from Table 1 below, we have |Γ| ≈ 2×10−9 J m−3 s−1,

three orders of magnitude smaller than Q. Of course, the cur-

vature ∇2h 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. Ex-

amination of a typical digital elevation model for Greenland

suggests |∇2h| < 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{Ki(H),K0} = K0. 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 Dirich-

let and Neumann type (subsection 4.5), is well-posed (Ock-

endon and others, 2003). Equation (83) is strongly advection-

dominated in many common modeling situations, however,

and numerical schemes should be constructed accordingly.

4.2. Shallow enthalpy equation

In addition to hydrostatic approximation, some ice sheet mod-

els use the small aspect ratio of ice sheets to simplify their

model equations. Heat conduction can be shown to be impor-

tant 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

ρdH

dt=

∂

∂z

({Ki(H)

K0

}∂H

∂z

)+Q. (84)

This equation is used in the PISM implementation.

4.3. Simplified relation among H, p, T, ω

As noted in section 2, the heat capacity Ci(T ) of ice is ap-

proximately constant in the range of temperatures relevant

to glacier and ice sheet modeling. Denote this constant heat

capacity of ice by ci so

Ci(T ) = ci +O(|∆T |)

where ∆T = T − T0. Here O(|∆T |) ≤ K|∆T | for a constant

K which is, for instance, a bound on |∂Ci/∂T | in the range

of relevant temperatures. Equations (5) and (6) become

Hi = ci(T − T0) +O(|∆T |2), (85)

Hw = ci(Tm(p)− T0) + L+O(|∆T |2). (86)

We now drop the terms “O(∆T 2)” above and recall that

the mixture enthalpy is H = (1−ω)Hi +ωHw and that ω = 0

if T < Tm(p). Thus

Hs(p) = ci(Tm(p)− T0). (87)

and

H(T, ω, p) =

{ci(T − T0), T < Tm,

Hs(p) + ωL, T = Tm and 0 ≤ ω < 1.(88)

In our simplified implementation, formula (88) replaces Equa-

tion (10). Instead of general forms (11) and (12), we have

these inverses:

T (H, p) =

{c−1i H + T0, H < Hs(p),

Tm(p), Hs(p) ≤ H,(89)

ω(H, p) =

{0, H < Hs(p),

L−1(H −Hs(p)), Hs(p) ≤ H.(90)

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 (subsec-

tion 5.1).

4.4. 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 formu-

lation, 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 D(ω)∆t

be the dimensionless mass fraction removed in time ∆t by

drainage of liquid water to the ice base. Note that D(ω)

is a rate, and the time step and drainage rate must sat-

isfy D(ω)∆t ≤ 1, so that no more than the whole mass is

drained in one timestep. The mass removed in the timestep

is ρwD(ω)∆t.

Though Greve (1997b) proposes D(ω) = +∞ for ω ≥ 0.01,

choosing D(ω) to be finite increases the regularity of the en-

thalpy 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 0.005 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 mod-

eling 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.

Aschwanden and others: An enthalpy formulation for glaciers and ice sheets 11

0 0.01 0.02 0.03 0.040.005

0.05

ω

D (

ω)

a−

1

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 sec-

tion 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 in-

complete experimental knowledge of drainage processes, and

not intrinsic in an enthalpy formulation.

The basal melt rate −Mb is increased by the drainage,

−Mb = −Mb +

h∫b

ρwD(ω) dz. (91)

We modify incompressibility equation (17) to globally con-

serve mass in the sense that the vertical velocity is computed

using the new basal melt rate,

w = −Mb +∇b · (u, v)∣∣z=b−

z∫b

(∂u

∂x+∂v

∂y

)dz′. (92)

We also modify equation (75), or its simplified version (76),

by using this new basal melt rate. Because incompressibil-

ity along with the surface kinematical equation are usually

combined in shallow ice sheet models to give a vertically-

integrated 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 mod-

ifications 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 im-

portance 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 ap-

proximately 1011 m2, and average surface elevation approx-

imately 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 km3 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 km3 (= 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 km3 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.

4.5. Simplified boundary conditions

MassIn all ice sheet and glacier models the surface process com-

ponents should report the runoff-modified net accumulation

Mh 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 veloc-

ity field and the surface elevation changes, while in shallow

models one may rewrite (92) to give a map-plane mass con-

tinuity equation (Bueler and Brown, 2009, for example). The

basal boundary condition for mass balance is the basal melt

rate (addressed next).

EnthalpyGenerally 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 deriva-

tive. In addition to the accumulation and snowfall rates al-

ready needed for the mass conservation boundary condition,

the surface process components should at least report a snow

temperature Tsnow. There must be a model to compute qatm,

the non-latent heat flux from the atmosphere, generally in-

cluding sensible, turbulent, and radiative fluxes. Modeling

these quantities is non-trivial, and simplified models are fre-

quently 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, ab-

lation zone, and equilibrium line. In general, regarding yearly-

average 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 ηh itself, if modeled in a surface pro-

cesses model, is needed primarily to determine the surface

mass balance Mh. 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 ηb may be

zero or positive and whereH ≥ Hs(p), either Equation (75) or

(76) allows us to compute a nonzero basal melt rate −Mb. In

doing so, however, we are left needing independent informa-

tion 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 condi-

tion” applicable to pure advection problems is not mathemat-

ically 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 K0∇H ·n = 0 in

the case of a positive-thickness layer of temperate ice, and

apply a Dirichlet boundary condition H = Hs(p) if the basal

temperate ice has zero thickness. In the discretized implemen-

tation, whether there is a positive-thickness layer is diagnosed

using a criterion like H(+∆z) > Hs(p(+∆z)).

MomentumEquation (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 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets

ηb > 0 &H < Hs(p) ?

H := Hs(p)

H < Hs(p) ?

yes

no

Eqn (74) is

Neumannb.c. for Eqn

(41); Mb = 0

yes (cold base)

positive thickness

of temperate iceat base?

H = Hs(p) is Dirichletb.c. for Eqn (41)

q = −Ki(H)∇Hat ice base

∇H ·n = 0 is Neumann

b.c. for Eqn (41)

q = −k(H, p)∇Tm(p)

at ice base

compute Mb from

Eqn (75) or (76)

no

no

yes

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 −Mb. The basal value of the ice mixture enthalpy is “H”.

4.6. 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 kilome-

ters. This is justified because the heat flux qlith = −klith∇T ,

which enters into the subglacial layer energy balance Equa-

tion (71), varies in response to changes in the insulating thick-

ness of overlain ice. Because the top few kilometers of the

lithosphere generally have a homologous temperature far be-

low 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 ne-

glected.

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 qlith 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 www.pism-docs.org,

we only outline enthalpy-related aspects here. The results be-

low used PISM trunk revision 0.3.1684.

Section 4 already identifies the significant continuum equa-

tions 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 time-

step 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 hydrol-

ogy 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.

5.2. Application to Greenland

Our goals in applying the enthalpy formulation to Greenland

are deliberately limited. We demonstrate differences in ther-

modynamical variables between the enthalpy formulation and

a temperature-based method. We examine the basal tempera-

ture, 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 col-

umn at all. For comparison, Aschwanden and Blatter (2009)

have modeled a Scandinavian thermal structure for Storglacia-

ren 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 sur-

face 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, Fb = 0 in Equation (76).

1The addition of a sliding model requires a membrane stress bal-

ance (Bueler and Brown, 2009). At minimum, this introduces anecessarily-parameterized subglacial process model relating the

availability of subglacial water to the basal shear stress. Inter-preting the consequences of such subglacial model choices, while

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 7.9× 10−8 K Pa−1

ci heat capacity of ice 2009 J (kg K)−1

cw . . . of liquid water 4170 J (kg K)−1

cr . . . of bedrock 1000 J (kg K)−1

ki conductivity of ice 2.1 J (m K s)−1

kr . . . of bedrock 3.0 J (m K s)−1

ρi bulk density of ice 910 kg m−3

ρw . . . of liquid water 1000 kg m−3

ρr . . . of bedrock 3300 kg m−3

L latent heat of fusion 3.34× 105 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 ac-

cording to Equation (39). The temperate ice diffusivity in

Equation (84) was chosen to beK0 = 1.045×10−4 kg (m s)−1,

which is 1/10 times the value Ki = ki/ci 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 K0. 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 simu-

lations 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 spac-

ing 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 ≥ Hs(p) while ice in COLD is temperate if its pressure-

adjusted temperature is within 0.001 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 temper-

ate 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 temper-

ate 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 ofthis paper about enthalpy formulations.

Table 2. Measurements at the end of each run.

CTRL COLDtotal ice volume [106 km3] 3.48 3.56temperate ice volume fraction 0.006 0.012temperate ice area fraction 0.24 0.43

total ice enthalpy [1023 J kg−1] 1.80 1.85

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 dissipa-

tion 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 defi-

ciency of cold-ice methods which was identified in the intro-

duction. Namely, at points within the three-dimensional ice

fluid where ice temperature has reached the pressure-melting

temperature, and where dissipation heating continues, gen-

erated 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 wa-

ter, 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 energy-

conserving way. Though an energy-conserving polythermal

method is obviously preferable, such methods nonetheless re-

quire 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 sub-

section 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 varia-

tions 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 ver-

sion to make it seem straightforward, but the possibility of

liquid water in the ice mixture raises modeling issues, in-

cluding choice of temperate ice rheology and how to model

intraglacial transport (drainage). One must interpret the en-

thalpy equation boundary conditions in the presence of mass

and energy sources and sinks at the boundaries, both supra-

glacial 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 con-

ditions follow from this technique. A more-complete basal

melt rate model appears, Equation (71). This view is well-

suited for coupling to surface mass and energy balance pro-

cesses, firn processes, runoff flow, subglacial hydrology, and

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 bound-

ary 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 knowl-

edge of the thermal structure nor a parameterization of the

cold-temperate transition surface (CTS) is required.

Our application to Greenland shows that an enthalpy for-

mulation can be used for continent-scale ice flow problems.

The magnitude and distribution of basal melt rate, a ther-

modynamical 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 Sci-

ence Foundation grant no. 200021-107480, by NASA grant

no. NNX09AJ38G, and by a grant of HPC resources from

the Arctic Region Supercomputing Center as part of the De-

partment of Defense High Performance Computing Modern-

ization Program.

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δ0σ

Σ+

w+

Σ−w−

V +

V 0

V −

∂V

Figure 9. A pill box V including a thin active layer volume V 0

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Appendix A. JUMP EQUATION WITHFLOW AND PRODUCTION

The “pillbox” argument in this appendix, which justifies Equa-

tion (44), is similar to the argument for jump equation (2.15)

in Liu (2002).

Consider a closed volume V with smooth boundary ∂V

enclosing a portion of the surface σ across which we wish

to compute the jump of a scalar field ψ in V . As shown in

Figure 9, Σ± are smooth oriented surfaces which separate V

into disjoint regions V −, V 0, and V +. Here V 0 is the region

between Σ− and Σ+ and it contains a portion of σ. Let δ0

be the maximum distance between the surfaces, and let w±

denote the normal speed of each surface Σ±. Decompose the

surface ∂V into disjoint components (∂V )± = V ± ∩ ∂V and

(∂V )0 = V 0∩∂V . Note that the boundary component (∂V )0

is a “band” with maximum height δ0.

The scalar field ψ is advected at velocity v, produced at

rate π, and experiences an additional non-advective flux φ

according to the general balance Equation (13). Assume that

ψ has a finite jump across each surface Σ±. Following pre-

cisely the logic and definitions used to justify equation (2.15)

in (Liu, 2002),∫V

∂ψ

∂tdv +

∫∂V

(ψv + φ) ·n da (A1)

−∫

Σ−

[[ψ]]−w− da−∫

Σ+

[[ψ]]+w+ da =

∫V

π dv.

We will consider the limit δ0 → 0 wherein all pillbox dimen-

sions normal to Σ± shrink to zero. In this limit, Σ± converge

to σ with normal vector nσ pointing from V − to V +. We

suppose that Σ± and (∂V )± are all within a normal distance

from σ which is some fixed multiple of δ0.

As δ0 → 0 we suppose that the integrals over V may not

disappear, though the volume of V goes to zero, because the

thin layer has much larger fluxes and active phase change

processes. Specifically we suppose:

1. there is a well-defined area density λσ defined in σ, with

dimensions of ψ per length, so that

limδ0→0

∫V

∂ψ

∂tdv =

∫σ

∂λσ∂t

da, (A2)

2. there is a density πσ defined on σ so that

limδ0→0

∫V

π dv =

∫σ

πσ da, (A3)

3. and the boundary component (∂V )0 becomes a smooth

curve L in σ, with length element ds, and there is a ve-

locity vσ, defined in σ and tangent to σ so that

limδ0→0

∫(∂V )0

(ψv + φ) ·n da =

∮L

λσvσ ·n ds. (A4)

The advective bulk flux (ψv) and non-advective bulk flux

(φ) combine in the limit (A4) to give an advective layer flux

(λσvσ). On the one hand, any flux qσ along the layer (units

[ψ] m2 s−1) can be factored qσ = λσvσ by introducing a layer

density λb (units [ψ] m) and a velocity vσ (units m s−1). On

the other hand, in the cases appearing in this paper such

layer fluxes are normally considered to be advective in the

glaciological literature.

Now for the pillbox argument: Split the flux integrals in

(A1) into V − and V + portions, and use Equations (A2),

(A3), and (A4). Equation (A1) becomes∫σ

∂λσ∂t

da+

∫(∂V )−

(ψv + φ) ·n da+

∫(∂V )+

(ψv + φ) ·n da

+

∮L

λσvσ ·n ds−∫

Σ−

[[ψ]]−w− da−∫

Σ+

[[ψ]]+w+ da

=

∫σ

πσ da+ o(δ0) (A5)

where o(δ0)→ 0 as δ0 → 0. In the limit:∫σ

∂λσ∂t

da+

∫σ

[[(ψv + φ) ·nσ]] da+

∮L

λσvσ ·n ds (A6)

−∫σ

([[ψ]]− + [[ψ]]+)wσ da =

∫σ

πσ da.

Compare Equation (2.14) in (Liu, 2002).

18 Aschwanden and others: An enthalpy formulation for glaciers and ice sheets

The divergence theorem allows us to rewrite the line inte-

gral as an integral over σ:∮L

λσvσ ·n ds =

∫σ

∇· (λσvσ) da. (A7)

Let [[ψ]] = [[ψ]]−+[[ψ]]+, which is the total jump across σ. The

result is an integral over σ only,∫σ

{[[(ψv + φ) ·nσ − ψwσ]] +

∂λσ∂t

+∇· (λσvσ)− πσ}

da = 0. (A8)

Because the pillbox argument leading to Equation (A8) can

be applied to any part of the surface, Equation (44) follows

by reordering the terms. If the area density λσ vanishes then

Equation (A8) is equivalent to Equation (2.15) in Liu (2002).