an enthalpy formulation for glaciers and ice sheets
TRANSCRIPT
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: [email protected] 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).