NASA Technical Reports Server (NTRS) 20120009085: Modeling of Firn Compaction for Estimating Ice-Sheet Mass Change from Observed Ice-Sheet Elevation Change

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AnnalsofGlaciology 52(59) 2011 1
Modeling of firn compaction for estimating ice-sheet mass change
from observed ice-sheet elevation change
Jun LI,1 H. Jay ZWALLY2
1SGT Inc., NASAGoddard Space Flight Center, Code 614.1, Greenbelt, MD 20771, USA
E-mail: [email protected]
2Cryospheric Sciences Branch, NASAGoddard Space Flight Center, Code 614.1, Greenbelt, MD 20771, USA
ABSTRACT. Changes in ice-sheet surface elevation are caused by a combination of ice-dynamic
imbalance,ablation,temporalvariationsinaccumulationrate,firncompactionandunderlyingbedrock
motion. Thus, deriving the rate of ice-sheet mass change from measured surface elevation change
requiresinformationontherateoffirncompactionandbedrockmotion,whichdonotinvolvechanges
inmass,andrequiresanappropriatefirndensitytoassociatewithelevationchangesinducedbyrecent
accumulation rate variability. We use a 25year record of surface temperature and a parameterization
foraccumulationchangeasafunctionoftemperaturetodriveafirncompactionmodel.Weapplythis
formulationtoICESatmeasurementsofsurfaceelevationchangeatthreelocationsontheGreenlandice
sheet in order to separate the accumulation-driven changes from the ice-dynamic/ablation-driven
changes, and thus to derive the corresponding mass change. Our calculated densities for the
accumulation-driven changes range from 410 to 610kgm–3, which along with 900kgm–3 for the
dynamic/ablation-drivenchangesgivesaveragedensitiesrangingfrom680to790kgm–3.Weshowthat
usinganaverage(or‘effective’) densitytoconvertelevationchangeto masschangeisnot validwhere
the accumulation and the dynamic elevation changes are of opposite sign.
INTRODUCTION larger temperature sensitivity due to their introduction of a
temperature-dependent activation energy and rate coeffi-
Knowledgeofthemassbalanceofpolaricesheetsisessential
cientbasedonlaboratoryexperiments.Theirresultsshowed
for understanding sea-level change. The recent assessment
that the seasonal variation in surface temperature caused
fromtheIntergovernmentalPanelonClimateChangeFourth
seasonal variations in surface elevation, as well as longer-
AssessmentReport(IPCC,2007)summarizesestimatesofthe
term changes from year-to-year trends in temperature. This
mass change from both the Greenland and Antarctic ice
strong dependence of firn compaction rate on temperature
sheets from 1993 to 2003 in the ranges from in balance
(0Gta–1) to a loss of 300Gta–1, equivalent to a rate of sea- has since been supported by field observations of compac-
level rise of 0–0.8mma–1. Other results from satellite and tion rates (Arthern and others, 2010).
aircraft measurements of surface elevation change (dH/dt), Prior to Zwally and others (2011), a constant effective
satellite measurements of changes in gravity, and the mass density, (cid:2)eff, with values between 300 and 900kgm–3 was
input–output method are reviewed(Alleyand others, 2007; generally usedto convert dH/dt todM/dt (e.g. 350 in Davis
ShepherdandWingham,2007)andtabulatedinDahl-Jensen andothers,2005;300forthe‘interior’and900to‘seaward’
andothers(2009).Aparticularissueofconcernforderiving inThomasandothers,2006;350and917inWinghamand
mass changes (dM/dt) from observed dH/dt has been the others, 2006). However, Figure 1 shows a case in which
appropriate density to use, because dH/dt is in general a using a constant density is clearly not valid. Consider a
combination of changes in firn thickness and solid-ice location where the accumulation rate, A(t), is increasing in
thickness, both of which have different densities. Recently timewhiletheicesheetisdynamicallythinning(i.e.thefirn
Zwally and others (2011) reported a net mass loss from the thickness change dIfirn/dt>0, and the underlying ice
Greenlandicesheetof171(cid:2)4Gta–1fortheperiod2003–07 thickness change dIice/dt<0, with example values of firn
andalossof7(cid:2)3Gta–1fortheperiod1992–2002basedon and ice thickness changes and associated densities given in
dH/dtobservationsoverthesetimeperiods.Themethodsfor Figure 1. The net mass change (per unit area and time) is
thesecalculationsaredescribedinmoredetailinthispaper. dM/dt=200–900=–700kgm–2a–1, and the net thickness
The relation between dH/dt and dM/dt is the result of change is dI/dt=0.4–1.0=–0.6ma–1. Thus, the effective
multiple processes occurring throughout the ice-sheet density using this dI/dt would be (cid:2)eff=(dM/dt)/(dI/dt)=
column. For example, variations in surface air temperature –700/–0.60=1167kgm–3, which is greater than the density
cause a change in the rate of compaction in the upper firn of glacier ice, (cid:2) =900kgm–3.
i
layers, which causes a surface elevation change, but this Ourexampleclearlyshowsthattheassumptionthatsome
temperature-driven change does not involve any mass value of (cid:2) between 300 and 900kgm–3 can be chosen is
eff
change (Zwally and Li, 2002). Arthern and Wingham notalwaysvalid.Infact,itwillbeinvalidwhereverthereisa
(1998) evaluated the impact of changes in accumulation combinationofaccumulation-driventhickeninganddynam-
rate, temperature and surface snow density by using a firn ic thinning, or vice versa. Furthermore, where dI /dt
firn
compaction model. Their results suggested that accumu- =dI /dt, the corresponding value of (cid:2) will become
ice eff
lation-rate induced changes were the most significant, and infinite. This example demonstrates the necessity of esti-
that temperature effects were relatively small. In contrast, mating changes in firn thickness and ice thickness sepa-
thefirncompactionmodelofZwallyandLi(2002)showeda rately, and of using the appropriate densities for each.
2 LiandZwally:Modelingfirncompaction
density, (cid:2)(z), and compaction rate, d(cid:2)(z)/dt, as
Z (cid:2) (cid:3)
z 1 d(cid:2)ðzÞ
V ðz,tÞ¼ dz, ð2Þ
fc (cid:2)ðzÞ dt
zi
wherez isthefirn–icetransitiondepthatwhichthedensity
i
is (cid:4)900kgm–3.
Essential to the compaction model is the constitutive
relation between the compaction rate, d(cid:2)/dt, and the
physical variables such as accumulation rate, A(t), and firn
temperature,T(t),thatdrivethedensitychange.Thisrelation
is given by
d(cid:2)ðzÞ (cid:4) (cid:5) (cid:4) (cid:5)
¼K TðzÞ A (cid:2) (cid:3)(cid:2)ðzÞ : ð3Þ
dt i
Equation (3) is a semi-empirical relation modified from that
initially proposed by Herron and Langway (1980) for the
steady-statecase,andA¼AðtÞistheconstantaccumulation
Fig.1. Example of upper firn and deep ice thicknesschanges and
associated mass changes, where 200kgm–2a–1 of firn is added to rate in their relation. As discussed by Zwallyand Li (2002),
thefirn–icecolumnbyanaccumulationincreaseand900kgm–2a–1 Equation(3)isbasedontheideathatthecompactionrateat
issubtractedbydynamicthinning.Inthiscase,theconventionally depthzisdeterminedbychangesinoverburdenpressure,firn
defined effective density, (cid:2) =(dM/dt)/(dI/dt), is 1167kgm–3, temperatureanddensity.However,Equation(3)isnottime-
eff
whichwouldbeunrealisticallygreaterthanthedensityofice. dependentwithrespecttoA(t)atthesurface.ThereforeAin
Equation (3) cannot simply be replaced by A(t) at z=0. For
example,whenA(t)atthesurfaceoftheicesheetiszero,the
In this paper, we present a firn compaction model for
compactionofdeeperfirnlayerscontinues,butEquation(3)
calculatingchangesintherateoffirncompactioncausedby
wouldindicatethatthecompactionrateiszeroatalldepths.
changes in both accumulation rate and temperature. The
Since A at least in part reflects the changes of overburden
model is used to separate the accumulation-driven and
pressure, we replace A by the integral from the surface to
ablation-dynamic-driven changes in surface elevation, and
depthzofA(t–(cid:2)t)/(cid:2)ttorepresenttheaveragechangeofthe
to calculate the appropriate firn density for accumulation-
overburdenpressureatdepthz,where(cid:2)tisthetimetakenfor
driven changes. It incorporates the processes of surface
thelayertopropagatefromthesurfacetodepthz.
melting,percolationandrefreezing.Italsoincludesacritical
The temperature dependence of the compaction rate,
improvement to accommodate a time-dependent accumu-
d(cid:2)(z)/dt, in Equation (3) has been conventionally taken as
lationrate,A(t).Weapplythemethodtothreeselectedfield
following an Arrhenius relation, i.e.
locations in Greenland.
KðTÞ¼K e(cid:3)E=ðRTÞ, ð4Þ
0
THE FIRN COMPACTION MODEL where K(T) is the rate factor due to the temperature T in
kelvin.K isaconstant,EistheactivationenergyandRisa
As described in Zwally and Li (2002), the dH/dt of an ice 0
gas constant. Experimental results of grain growth and ice
sheet is composed of several vertical-velocity components:
creepshowthatthetemperaturedependenceofK(T)ismore
dHðtÞ AðtÞ A ðtÞ dB sensitive than Equation (4) under constant K and E,
¼ (cid:3)V ðtÞ(cid:3) b (cid:3)V þ : ð1Þ 0
dt (cid:2) fc (cid:2) ice dt indicating that E is actually a function of temperature, E(T).
sf i
Its value increases with temperature (Jacka and Li, 1994).
Here accumulation rate, A(t), is the component that raises Since both grain growth and ice creep are involved in firn
thesurfaceatarateofA(t)/(cid:2)sf,where(cid:2)sfisthedensityofthe compaction,ZwallyandLi(2002)appliedtheexperimental
surface snow. In general, (cid:2)sf has a value of (cid:4)330kgm–3 in results to their firn compaction model. They indicated that
dry snow regions (Paterson, 1994). In areas with surface an increasing value of E(T) will decrease K(T) if K is kept
0
melt, the meltwater percolates downward in the firn layers constant, which is contrary to the experimental data.
and the densityand thickness of firn layers are modified by Therefore K must also be a function of temperature, K (T).
0 0
the meltwater and refreezing (Li and others, 2007; Reeh, They modified Equation (4) for firn compaction by introdu-
2008). The melting and refreezing process has been cingatemperature-dependentactivationenergy,E(T),anda
incorporated in our model and is described in detail by Li rate constant, K (T), with an adjustable parameter, (cid:3), i.e.
0
andothers(2007).A (t)istheablationrateoccurringonlyin
b KðTÞ¼(cid:3)K ðTÞ: ð5Þ
theablationzone,V istheverticalvelocityoftheiceatthe G
ice
firn/ice transition, and dB/dt is the vertical motion of the Thesecondtermontheright-handsideinEquation(5)isthe
underlyingbedrock.V (t)isthevelocityoffirncompaction grain growth rate,
fc
at the surface, which is the integral of the compressive
K ðTÞ¼K ðTÞe(cid:3)EðTÞ=ðRTÞ, ð6Þ
displacement of the firn layers over the length of the firn G 0G
column. Following the normal usage that surface elevation, given by the experiments (Fig. 2a).
H,ispositiveupwardsanddepth,z,ispositivedownwards, An error in the text of the Zwally and Li (2002) paper
dH/dt,AanddB/dtarepositiveupwardsandV ,A andV causedconfusionintheapplicationofthetheory.Byusinga
fc b ice
are positive downwards in Equation (1). According to best fit to the experimental data given by Jacka and Li
the mass conservation equation (Li and Zwally, 2002), the (1994), Zwally and Li (2002) derived the empirical expres-
velocity of firn compaction, V , at depth z is given by the sions of grain growth rate, K (T), and the activation energy,
fc G
LiandZwally:Modelingfirncompaction 3
E(T), as functions of temperature, T (Fig. 2a and b). Both
functionswereinitiallygivenbyZwallyandLi(2002,fig.3a
and b). However, K was mislabeled as K . Rearranging
G 0G
Equation (6) we have
K ðTÞ¼K ðTÞeEðTÞ=ðRTÞ: ð7Þ
0G G
Figure2cgivestheempiricalformofK (T)asafunctionof
0G
temperaturebysubstitutingtheexpressionsofE(T)andK (T)
G
showninFigure2aandbintoEquation(6),togetherwiththe
functionof E(T).AsshownbyFigure 2c, K increaseswith
0G
temperature apparently faster than E(T), indicating the
stronger temperature dependence of K (T) compared to
0G
E(T). Combining Equations (4–6), we then have
KðTÞ¼(cid:3)8:36ð273:2(cid:3)TÞ(cid:3)2:061, ð8Þ
where (cid:3) is an adjustable parameter determined by fitting
modeleddensityprofilestofieldmeasurements.Equation(8)
accounts for the temperature dependence of grain growth
and ice creep rates on firn compaction. It leads to much
higher temperature sensitivity than Equation (4) under
constant values of K and E (e.g. Herron and Langway,
0
1980), particularly for temperatures higher than –108C.
Equations (1–8) are coupled with a one-dimensional heat-
conductingequationandsolvedusingthemultilayersystem
described by Zwallyand Li (2002).
The empirical parameter, (cid:3), in Equation (5) is used to
calibratemodeleddensityprofilestofieldmeasurements.In
our previous version of the model, we presented (cid:3) as a
function of the annual mean temperature based on field
density profiles from Greenland (Li and others, 2003).
Fig. 2. (a) The temperature dependence of grain growth rate, K ;
G
Helsen and others (2008) extended this relation by using (b)activationenergy,E(T),forgraingrowth;and(c)thederivedrate
41 observed pore close-off depths (depth of the 830kgm–3 constant, K , for grain growth as a function of temperature,
0G
density) from Antarctica where firn temperatures are much according to Equation (6) (shown together with E(T) for com-
colder. They found a slightly different relation between (cid:3) parison). Note that the empirical functions in (a) and (b) were
and the annual mean temperature. However, for tempera- initiallygivenbyZwallyandLi(2002,figs3banda).However,KG
tures higher than about –308C (e.g. Greenland), the two wasmislabeledasK0Gintheirfigure3b.
relations are similar. To further improve the calibration, we
usetwocriticalpointsatdensitiesof550and830kgm–3as
the control, and tune the value of (cid:3) to force the modeled
ages at these two densities to match those given by Herron ELEVATION CHANGE COMPONENTS FROM
andLangway(1980)thatwerewellconstrainedbythefield ALTIMETRY dH=dt
density measurements. Although the previous calibrations
As described in Equation (1), the surface elevation change,
(Liandothers,2003;Helsenandothers,2008)showed that
(cid:3) is a function of annual mean temperature only, our tests dH/dt, is due to a combination of vertical velocity com-
ponentsfromdifferentprocesses.Fornon-steadystate,these
indicate that the accumulation rate also had an influence,
components can be represented by perturbations from
similar to the Herron and Langway (1980) density–age
steady state, assuming that dB/dt is constant during the
relation for which both temperature and accumulation rate
were involved. Our present values of (cid:3) as a function of measurements:
annualmeantemperature,Tm(8C),andlong-termaccumu- dH dHa dC dH dH dB
lation rate, hAi (ma–1), for Greenland are ¼ þ ATþ dþ bþ , ð11Þ
dt dt dt dt dt dt
(cid:3)¼(cid:3)1 wheredHa/dt=(A(t)–hAi)/(cid:2) isthedirectchangecausedby
sf
¼(cid:3)9:788þ8:996hAi(cid:3)0:6165Tm, ð9Þ A(t), dCAT/dt=–(Vfc(t)–hVfci) is the firn compaction-rate
(cid:2)(cid:5)550kgm(cid:3)3 caused elevation change driven by both A(t) and T(t),
dH /dt=–(A (t)–hA i)/(cid:2) is driven by changes in the ab-
b b b i
lation rate, and dH /dt=–(V (t)–hV i) is driven by
(cid:3)¼(cid:3)2 dynamic changes in dthe ice fliocew relatiicvee to hAi. The hi
¼(cid:3)1=ð(cid:3)2:0178þ8:4043hAi(cid:3)0:0932TmÞ, ð10Þ symbol indicates long-term averages of the various com-
(cid:2)>550kgm(cid:3)3: ponents, obtained during our model spin-up to a steady
state.Weconsiderthatablationiszerointheaccumulation
Equations(9)and(10)producemodeleddensity–depth(age) zonewherefirnexists.Intheablationzonewherethereisno
profiles that are in agreement with those from Herron and firn, accumulation is zero. In general, dC /dt depends on
AT
Langway (1980) within an error <(cid:2)1%. the history of both A(t) and T(t) as their effects propagate
4 LiandZwally:Modelingfirncompaction
into the firn. We separate their effects byassuming DERIVING DENSITY, (cid:2) , FOR dHa =dt
a ca
The density, (cid:2) , associated with dHa /dt shown in Equa-
dC dC dC a CA
AT ¼ Aþ T, tions (13) and (14) is the average density caused by
dt dt dt
perturbationstotheaccumulationrateoveraspecifiedtime
where dC /dt and dC /dt are changes driven by A(t) and period. Considering the surface elevation change driven by
A T
T(t), respectively. Thetotal elevation changefrom both A(t) variations in accumulation rate A(t) only, i.e. dH/dt =
and T(t) perturbations becomes dHa /dt, the corresponding mass change, (cid:2)M (t), over
CA a
(cid:2)t=t–t is given by
0
dHa dHa dC dC dHa dC Z
CAT (cid:6) þ Aþ T ¼ CAþ T, t(cid:4) (cid:5)
dt dt dt dt dt dt (cid:2)M ðtÞ¼ AðtÞ(cid:3)hAi dt:
a
wheredHa /dtisthetotalchangecausedbyA(t),including t0
CA Here A(t) and hAi are in units of kgm–2a–1. (cid:2)M (t) is in
both the direct change and that associated with the a
kgm–2.Thismasschangecausesasurfaceelevationchange
compaction rate. Equation (11) then gives
of
dI dH dC dB dHa dH Z (cid:8) (cid:9)
dt (cid:6) dt (cid:3) dtT(cid:3) dt ¼ dtCAþ dtbd, ð12Þ (cid:2)HaCAðtÞ¼ t dHaCA dt:
dt
t0
where dI/dt is the net thickness change in the firn/ice
Thus,
column,definedbytreatingdB/dtanddCT/dtascorrections Z Z (cid:8) (cid:9)
to dH/dt. dH /dt is the combined ablation and dynamic (cid:2)M t(cid:4) (cid:5) t dHa
term (as dHbbdd/dt(cid:6)dHd/dt+dHb/dt). In the ablation zone (cid:2)aðtÞ¼(cid:2)HaCaA ¼ t0 AðtÞ(cid:3)hAi dt= t0 dtCA dt: ð15Þ
where there is no firn, dH /dt is the mixed term of both
bd Equation (15) defines the density, (cid:2) , that is obtained using
dynamicsandmelting(ablation).Intheaccumulationzone, a
the firn compaction model. It represents the density of the
dH /dt is only the dynamic term. The two terms on the
bd
firnthathasbeenadded(removed)asaresultofanincrease
right-hand side in Equation(12) involve change in mass.To
obtain dHa /dt, our compaction model first calculates (decrease)inA(t)relativetothedensityprofileforthelong-
dHa /dt uCsAing both T(t) and A(t), and then calculates termaveragehAi.However,thecorrespondingchangeinthe
CAT
dC /dt using T(t) with constant hAi. The dHa /dt is then amount of firn and the compaction rate is distributed
T CA
throughout the firn column, so there is no associated
given by
physical layer in the firn with a density (cid:2) and thickness
dHaCA ¼ dHaCAT(cid:3) dCT, dHaCA/dt. a
dt dt dt
i.e. the total accumulation-driven change in surface eleva-
RESULTS AND DISCUSSION
tion is obtained by subtracting the temperature-driven
Theappropriatemethodtoderivethemasschange,dM/dt,is
compaction from that caused directly by the change in
described by Equation (14). Here we derive the values at
accumulation rate and by the change in compaction rate
threeselectedlocationsontheGreenlandicesheet(Fig.3).
driven by both accumulation and temperature variations.
SiteAisinthenorthernregionandisassociatedwithlower
accumulation rate and temperature. Site B is near the
MASS CHANGE FROM dH=dt COMPONENTS Summit of the ice sheet, and site C, in the south, is
ApplyingappropriatedensitiestoEquation(12)ontheright- associated with much higher accumulation rate and tem-
hand side, the mass change rate, dM/dt, of the ice sheet is perature. We use monthly surface temperatures, Ts(t), for
given by January1982–October2007asshowninFigure4afromthe
Advanced Very High Resolution Radiometer (AVHRR)
dM(cid:6)kgm(cid:3)2a(cid:3)1(cid:7)¼(cid:2)adHaCAþ900Hbd: ð13Þ (Comiso, 2003). We parameterize changes in the accumu-
dt dt dt lation rate, A(t), according to the annual mean surface
temperature at the rate of 5%K–1. The IPCC report (IPCC,
The appropriate density associated with the dynamic and
2007) summarized the sensitivity of accumulation rate to
ablation term, dH /dt (ma–1), is the density of glacier ice
bd temperature with a range 4.7–8.5%K–1. Data from Green-
(900kgm–3). The density, (cid:2) (kgm–3), associated with the
a land ice cores (Clausen and others, 1988) give a range
shorter-term change of dHaCA/dt (ma–1) is discussed in the 4–6%K–1 based on a sensitivity of d18O to temperature of
next section. In terms of dI/dt (ma–1), Equation (13) can be 0.69%d18OK–1 (Zwally and Giovinetto, 1997). For 1988–
written as
2005,thisparameterizationgivesanaverageA(t)increaseof
dM 900dI dHa 0.6%a–1. This is close to the 0.7%a–1 for 1988–2004
dt ¼ dt (cid:3)ð900(cid:3)(cid:2)aÞ dtCA: ð14Þ inferred from a regional-climate and surface mass-balance
modelforthepercolationanddry-snowzonesofGreenland
Equation(14)isusedtocalculatethemasschangeoftheice (Box and others, 2006). Although there is a good relation
sheet. It requires the altimetry measurements of dH/dt, between changes in accumulation and temperature for
surfacetemperatureT(t)andaccumulationrateA(t)histories annualmeanvalues,thecorrelationcouldbepoorforshort-
as inputs to the compaction model to obtain dC /dt, term changes (Kapsner and others, 1995). Therefore, in our
T
dHa /dt and the density, (cid:2) . Equation (14) indicates that A(t) parameterization, we use the annual mean tempera-
CA a
using dI/dt with the density of ice to derive dM/dt is an tures instead of monthly temperatures to avoid introducing
approximationthatneglectstheeffectsofrecentchangesin seasonal variations in A(t) due to strong seasonal variations
accumulation rate on the rate of firn compaction. in temperature. A more sophisticated approach that would
LiandZwally:Modelingfirncompaction 5
Fig. 3. A map showing three selected test locations: A (79.48N,
319.98E), B (72.98N, 321.28E) and C (61.98N, 315.98E) on the
Greenland ice sheet. The climatic characteristics for the three
locationsaregiveninTable1.
avoidsomeofthesecomplicationsinthesensitivitybetween
temperature and accumulation rate would be to use A(t)
from the climate models (e.g. Helsen and others, 2008).
Starting from the initial steady-state spin-up, we applied
the time series of T(t) and A(t) to our compaction model.
s
Figure 4c shows the modeled time variations in the three
components from firn compaction at site B, driven by the
variations in T(t) and A(t) shown in Figure 4a and b. The
s
profiles show significant interannual variations, especially
since 2000 due to the enhanced warming in that period.
Although the input temperatures are monthly mean values,
the annual cycles in the temperature profile still cause
seasonal variations in C (t) and Ha (t) profiles, but with
T CAT
smaller amplitudes ((cid:4)2cm) compared to the results from
model runsusing hourlyor daily timescales(Zwallyand Li,
2002).Incontrast,theHa (t)profiledoesnothaveseasonal Fig.4.(a)MeanmonthlysurfacetemperatureatsiteBinGreenland
CA (Fig.3)showingthegeneralwarmingduringthelasttwodecades.
variations, because our parameterization of A(t) does not
(b)Theassociatedincreaseinaccumulationrateasafunctionofthe
havea seasonalvariation. We use best fitsto the profiles in annualmeantemperatureusingarateof5%K–1.(c)Themodeled
Figure4cforsiteB(andthesimilarprofilesforsitesAandC), surface height changesin three components (asmarked) from firn
for the period October 2003–October 2007, to obtain the compaction driven by temperature and the accumulation rate
masschangerates,dM/dt,andassociatedcomponentsforthe historiesshownin(a)and(b).
three sites as summarized in Table 1. The associated dH/dt
are derived from Ice, Cloud and land Elevation Satellite
(ICESat)measurements for thesame period,and dB/dt were
Table1.Theclimaticcharacteristicsandthederivedratesofthemasschange,dM/dt,andassociatedvaluesofthecomponentsofelevation
change as described by Equation (14), together with other parameters for three selected locations (Fig. 3) on the Greenland ice sheet.
CalculationofdM/dtrequiresdeterminationoftheaccumulation-drivencomponent,dHa /dt,anditsassociateddensity,(cid:2) ,andtheice-
CA a
dynamic-andablation-drivencomponent,dH /dt,withdensityof900kgm–3,aswellascorrectionoftheobserveddH/dtfortemperature-
bd
drivenvariationsinfirncompaction,dC /dt,andbedrockmotion,dB/dt
T
Site hAi T dH/dt dC /dt dB/dt dI/dt dHa /dt dH /dt (cid:2) (cid:2) (cid:2) dM/dt
m T CA bd a avg eff
ma–1 8C ma–1 ma–1 ma–1 ma–1 ma–1 ma–1 kgm–3 kgm–3 kgm–3 kgm–2a–1
A 0.10 –29.3 0.068 –0.016 0.002 0.082 0.019 0.063 410 786 786 64.5
B 0.22 –29.7 0.046 –0.023 –0.003 0.072 0.036 0.036 460 680 680 49.0
C 0.90 –14.0 –0.083 –0.051 0.003 -0.035 0.104 –0.139 610 776 1762 –61.7
6 LiandZwally:Modelingfirncompaction
averagedfromthreemodelsasinZwallyandothers(2005). betransformedintomasschangesimplybyusingaconstant
AsshowninTable1,theimpactsfromthetemperature-and average (or effective) density (i.e. avalue between 300 and
accumulation-drivencomponents,dC /dtanddHa /dt,are 900kgm–3).
T CA
ofsimilarmagnitudes,butthetemperatureincreasecausesa
surfaceelevationdecreasethatisoffsetbytheincreaseinthe
accumulationrate.AtsiteC,themeasuredsurfaceelevation
ACKNOWLEDGEMENTS
change, dH/dt, is –0.083ma–1 and dHa /dt is 0.104ma–1,
CA
indicating that the obtained mass loss (–61.7kgm–2a–1) is This research was supported by NASA’s ICESat Project
duetotheice-dynamicimbalanceand/orablation.Also,we Science funding. We thank S.F. Price and two anonymous
note that for site C the values of dM/dt and dI/dt from referees for their valuable comments, which helped greatly
Table 1 give a conventionally defined effective density of to improve the paper.
1762kgm–3,whichisgreaterthanthedensityoficeasinthe
exampleofFigure1.
OncethecomponentsrequiredfordM/dtinEquation(14)
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