GEOPHYSICAL RESEARCH LETTERS, VOL. 24, NO.12, PAGES 1503-1506, JUNE 15, 1997
Spatial variations in the rate of sea level rise caused by
the present-day melting of glaciers and ice sheets
Clinton P. Conrad and Bradford H. Hager
Department of Earth, Atmosphericand Planetary Sciences,Massachusetts
Institute of Technology
Abstract.
The redistribution
of surface water mass associ-
ated with the melting of glacial ice causesuplift near areas
of mass depletion, depressionof the seafloors,and changes
in the earth's gravitational field which perturb the ocean
surface. As a result, local spatial variations exist in the
rate of sea level rise. Tide gaugeson continental coastlines
measurea sealevel rise 5% smaller than the global average.
trial Reference Frame, which can be tied to the center of
mass or center of figure of the solid earth [e.g., Aretern,
1995]. The melting of continentalice into the oceanscan
cause both of these points to move in space relative to an
inertial referenceframe defined by the center of mass of the
earth, ice and ocean system. In this referenceframe, movements of the center of mass of the solid earth compensate
Tide gaugesin the hemisphereoppositea sourceof continen- for the movement of water masson the surface. For simplictal massdepletionmeasuresealevelrise10 to 20% greater ity• we calculate crustal motions and changesin "absolute"
than the globalaverageproducedby that sourcewhile satel- sea level, as they would be measuredby satellites,in this
lites make measurements1'0%too low. Becausemost long inertial frame. This frame can be related to other geodetic
duration tide gaugesare in. the northern hemisphere,if the frames through a rigid translation of the solid earth.
sources of sea level rise are unbalanced
between
the two
hemispheres,estimates of global sea level rise could be in
error by 10 to 20%. Individual tide gaugescould be more
seriously unrepresentative if they are near regions of significant present-day mass depletion.
The expected rate of sea level rise can be calculated for
a given loading scenarioand elastic Earth model. The load,
which varies with latitude, O, and longitude, •, causesa
verticaldisplacement
of the earth'ssurface
s U(8, •), and a
change
in the earth'sgravitational
potential,•(8,•). Both
5r and ß can be calculatedby convolvingthe load with the
Green'sfunctionsfor the earth's elasticresponseto a point
Introduction
load[Farrell,1972].The Green'sfunctions
aresimply:
A century of tide gauge measurementsindicates that
globalsealevel is currentlyrisingbetweenI and 2 mm/yr
=
[Douglas,1991]. Of this, a significantportionmustinvolve
the meltingof continentalice massesinto the oceans[ Warrick and Oerlemans,1990]. The Earth's responseto conti- for vertical displacement and
nental water-mass wastageis the immediate elastic vertical
uplift of the crust around the depleted area. Vertical ve-
=
locitiesnear the Antarctic ice sheetwould be a few mm/yr
for a realisticscenarioof massdepletionthere [e.g., Hager,
ag
+
(co,
[1991];Jamesand Ivins, 1995; Wahr et al., 1995]. The for a changein gravitationalpotential [Farrell and Clark,
transfer of this mass to the oceans would similarly cause a 1976]. Here, • is angulardistance,and a, MB, and g are
smaller amplitude depressionof the sea floors. In addition, the earth's radius, mass, and surface gravitational acceleration. The elastic love numbers, h,, and k•, describe, for each
the gravitational equipotentialsurfacethat definessealevel
would be perturbed. As a result, the rate of sea level rise harmonic degree n, the vertical displacementand potential
should
varyspatially
asboththeseasurface
andtheseachange
which
arise
from
theimposition
ofa point
load
at
floors
move
vertically
atrates
determined
bythelocation
of • = 0. Included
in(2)isthepotential
change
duetothe
applied surface mass loads. Changesin sea level due to the
elastic responseof the earth to Quaternary climate change
havebeenstudiedpreviously[e.g.,Farrell and Clark, 1976];
0
(co,
Special attention must be paid to the degree one terms,
becausethey typically include a shift in the center of mass
our intent is to perform these calculationsfor plausiblesce- of the earth [e.g.,Farrell, 1972]. In the inertial frame,the
center of massof the earth plus load cannot move in space,
narios of present-day climate change.
sothe degreeonetermof the potentialin (2) is zero.Farrell
Elastic Models of Sea Level Change
[1972] giveselasticlove numbersin a referenceframe definedafterthe applicationof the load,sowe includea rigid
Changesin sealevel are currently measuredin two ways. translation of the solid earth such that the net change in
Relative sealevel is measuredby tide gaugesasthe difference the center of mass of the earth-load system is zero. Farrell
between the ocean-air and earth-ocean interfaces, both of [1972]givesa degreeone potentiallove numberof k• = 0;
which can move vertically as mass is redistributed. Geode- we require k• = -1 for the degreeone term of (2) to be
tic satellites, such as TOPEX/POSEIDON or the Global zero. This changemust be accompaniedby a correspondPositioningSystem(GPS), make measurements
relative to ing changeof h• = -0.290 to h• = -1.290. In this inertial
geodetic referenceframes such as the International Terres- frame, the elastic earth movesrigidly in spaceaway from a
positive load applied at the earth's surface. We have recal-
Copyright
1997bytheAmerican
Geophysical
Union.
Papernumber
97GL01338.
0094-8534/97/97GL-01338505.00
of
summationgivenby Farrell [1972].
The melting of ice into the oceanscauseslocal changesin
absolutesealevel, SA(8,A), which dependupon variations
1503
1504
CONRAD
AND HAGEl{: SPATIAL VARIATIONS
IN SEA LEVEL RISE
in
thelocal
gravitational
potential,
•(8,,•).Theaverage
(a)
changein sea level measuredin the inertial frame is given
by
the
average
height
ofof
sea
water
added,
Sv,
and
the
the seafloors,{U(8,A)}o, where0o indicatesan average
change
in
ocean
basin
volume
produced
by
depression
of"'"-,
'-•i•i:.-.-:::i;•?'•
"
i) =
+ i(., i) - (i(,. i))o + (v(,. i))o
a potenti• su•ace whi• conservesvolume
•elative sea level r•e, Sa, is absolute sea level •e measured relative to the su•ace of the sold earth, so we subtract
thelocal
vertical
displacement,
U(8,)•),from
S•t:
(b)
ß
Note
that{Sa)o
= Sv;the
&ange
inocean
b• volume
/
averagesof relative sea level. Be....
• not observed• •ob•
causeSa andSi differby U(8,1) thesequantities
exhibit
different
,pati•
ofsea
Because
bothpatterns
U(8 A) and
•(8level
A)•s,.
deter•e
•.•:.•:.-:::•?..::::•:•i•.•:•,
, ,
,:..:...........
..,..:,F•::•a•i•:•:::::•.•:•:•:•.•a......•?,
..•.;•.• .•,
•'•••••.•••••.••.,.•.
,........
a•:•
.....
• .........................................................
"•". :,•: ;:•.'-..
,•:•a•::•::•.•, ',--,•
ß
,•-,,:-•:•:•-':•.
'-
bothabso-
lute
and
relative
sea
level,
whi•
inturn
•volves
ared•
tribution
ofmass
witMntheoceans,
wesolve
(3) and(4)
by aniterativeprocess.
Themassre•t•bution wit• the
•'-L•:
•
:•J'-"•
oceans• c•culated for ea• step, as •e the result•g v•-
uesof U(8, A) and •(8, l). The sequence
converges
rapi•y;
we performed three iterations and determined sea level to
•th•
a few tenths of a percent. We used a load resolution
of •8 = •A = 0.5 de,tees, and an ocean s•d prodded by
the U.S. Navy GlobalElevationData [1984].
The Elastic Response to Sea Level Rise
As our first case,we add water to the oceanbasinsand let
the oceansurfacerede`tributeitself accordingto the resulting elastic surfacedeformation and gravitational potential.
In doing so, we ignore the continental sourcesof this water.
Thevalues
of U(8,•), S.t(8,•),andSa(8,•)(Fig.1),are Figure1. Calculations
fortheaddition
of waterto the
directlyproportional
to theaverage
heightof wateradded oceans,
withoutregardto the continental
sources
of this
to theoceans,
soweexpress
themaspercentages
ofSv.
water,givenasa percentage
of Sv. Shown
are(a) vertical
The geometry
of the oceanbasinson the earthis such displacement
ofthecrust,measured
in aninertialreference
that the centerof massof water addedto the oceansis frame,(b) absoluteand (c) relativesealevelchange.
located beneath the southern Pacific. To prevent a net shift
in the center of massof the earth-load system,the solid earth
moves
toward
a point
inAsia(28.9øN,
45.2øE)
by6.2%
of thanaverage
because
thesolid
earth
moves
away
fromthe
Sv.
This
degree
one
component
of
vertical
displacement
ocean
surface
there.
In
contrast,
the
North,
Mediterranean
causes the southern Pacific seafloor to sink and Asian crust
andB.ed seasdeepenup to 15%moreslowlythanthe global
to risein the inertial referenceframe (Fig. la). The higher average becausethe solid earth movestoward them.
order degreescontributesignificantlyas well, as shownby
the general depressionof the ocean basins. In this case,the
averagedisplacementof the solid earth over the oceanarea Mass Depletion
on the Continents
is -8.3% of Sv. As a result, the averageabsolutesea level
Continentalglacialmassesare probablythe majorsource
change,{S.,t}o,is only 91.7%of what it wouldbe on a rigid of present-day
sealevelrise. Because
glaciatedregionsare
earth. In fact, S.t • Sv everywhereon the earth (Fig. lb). lessuniformlydistributedthan are the oceans,we expect
Because absolute sea level is measured in an inertial refgreaterverticaluplift and significantlydepressed
potential
erenee
frame,it haszerodegree
onecomponent.
Variationsheightin areasofwidespread
deglaciation.
Wealsoexpect
a
in thehigher
orderdegrees
cause
theseasurface
toriseun- largetranslation
ofthesolid
earthtoward
thismelted
region.
evenly.The additionalseawaterincreases
the gravitationalThis combination
shouldproducedecreased
absoluteand
potential
in theocean
interiors,
thusraising
theirseasurfacerelativesealevelchange
nearthealeglaciated
area.
relative
to neighboring
coastal
areas.Thiseffectis dimin- Weinvestigated
theeffects
ofcontinental
mass
depletion
ishedin the Pacificwherethe potentialis decreased
dueto from threepossiblesources:the Antarcticand Greenland
themovement
ofthesolidearth.
icesheets,
andmountain
glaciers
worldwide.
In these
calcuRelative
sealevel(Fig.lc), isthedifference
between
ab- lations,
thespatial
patternofmass
depletion
isproportional
solutesealevel(Fig. lb) andvertical
displacement
(Fig. to theaverage
massbalance.
Although
thisassumption
is
la). In general,
coastlines
showlessthanaverage
sealevel unlikely
tobethecase,
it should
leadtothegeneral
patterns
rise,and openoceansmore. The Pacificdeepens
5% faster and magnitudes
of crustalmotionandsealevelrise.
CONl{AD
AND HAGEl{:
SPATIAL VARIATIONS
IN SEA LEVEL RISE
1505
is because the area over which mass depletion occurs in
Greenland is smaller, causing the changesin U and • to
be more severe. The earth moves toward
eastern Greenland
(70.4ø/7,26.0øE)by 49% of $v. Thus,displacements
in the
south are uniformly negative and only in the northern hemi-
sphereis $•t > $v (Fig. 3a). Because
the oceanareain the
northis small,{$•t)o = 91.6%of $v. l•elativesealevelrise
in the north is generally smaller than the global average,
balancedby largervaluesin the south(Fig. 3b). Parts of
North America and Europe experienceslowor negativerates
of relative sea level rise and crustal uplift rates of order $v.
Small
Mountain
Glaciers
The small mountain glaciersof the world are largely located in the northern hemisphere, so the results are similar
to those for Greenland. We used a compilation of the mass
balancesof 31 smallglaciersystemsgivenby Meier [1984]as
point sources. We spread the accumulation values for these
points over nearby mountainous areas given by the U.S.
Navy GlobalElevationData [1984].The rigid translationof
the solidearthtowardnorthernGreenland(85.30N,58.8•E)
is 30.5% of Sv. Again, vertical displacements
are positive
in the north, negative in the south and large near areas
of significant mass depletion. As before, the southern hemisphereexhibits below averageabsolutesealevel rise, causing
Figure 2. Calculations in which massdepletion in Antarctica is included
as the sole continental
source of sea level rise.
The minimumcontouraroundAntarcticain (a) is SA = 0.
Verticaldisplacement
is the difference
between(a) and (b).
(S•)o to be 9S.2%of Sv (Fig. 4a). Relativesealevelrise
(Fig. 4b) is higherthan averagein the south,exceptnear
the aleglaciatedsouthern Andes, and below averagein the
north, where it is extremely so at high latitudes.
Discussion
The volume
Antarctica
We used accumulation values of Giovinetto and Bentley
[1985]to calculatethe massimbalanceof Antarctica. For
and
Conclusions
of the ocean basins increases in all scenar-
ios, by up to 8% of the total volume of sea water added.
This volume changeleads to a discrepancybetweenglobal
averagesof absolutesealevel rise measuredby satellites and
the unloading of the Antarctic ice sheet and the subsequent
loading of the oceanbasins,the solid earth movesrigidly to-
wardEast Antarctica(81.6øS, 42.3øE) by about38%of Sv.
This translation results in negative vertical displacements
in the northern hemisphere. Close to Antarctica, vertical
displacements are positive and on the order of Sv.
The
southward
translation
of the earth causes absolute
in the north to be smaller than the
sea level rise measured
global average(Fig. 2a). The southernhemisphereexhibits correspondinglyhigher values of Sl, except close to
the Antarctic coast, where the potential is low due to mass
loss on the adjacent continent. Because the ocean area of
the southern hemisphereis larger than that of the northern,
the increasesin SA in the southcause{SA}o to increaseto
99.4%of Sv (compareto Fig. lb).
Relative sea level, which combinesdecreasesin sea surface
height and the uplift of the oceanfloor near the ice sheet,occurs at rates up to Sv around Antarctica, but is of opposite
sign(Fig. 2b). Decreases
in the southernhemisphere
are
balanced over much of the rest of the globe, where relative
sealevel changeis 10% to 20% greater than Sv.
Greenland
For the Greenland ice sheet, we used accumulationvalues
given by Giovinetto,perz. comm. [1996],obtainedusing
proceduresof Giovinettoand Zwally [1995]and data from
Ohmura and Reeh [1991]. The hemispheredependenceof Figure 3. Similarto Fig. 2, usingGreenlandice as the
SA and Sa for Greenland is oppositethat for Antarctica,
continental source of sea level rise. The minimum contours
and their values near the ice sheet are more extreme.
in Greenland
are(a) S• =-100 and(b) Sa =-300.
This
1506
CONRAD
ANDHAGER:
SPATIAL
VARIATIONS
IN SEALEVELRISE
ponent of sealevel rise probably originates from eachof these
sourcesand some combination of them would provide a more
realistic climate change scenario. In addition, other sources
of continentalmassdepletionmay be important [e.g., Gornitz et al., 1994],and the pattern of deglaciationis likely to
be different
than
assumed
here.
If we knew
the contribu-
tion to sea level rise from each source, it would be possible
to provide a correction factor for each tide gauge to relate
its measurement directly to the global average. The magnitudes of these contributions, however, are still uncertain
[e.g., Warrick and Oerleraans,1990],and it is currentlyimpossibleto do this for any tide gauge. It is possible,however,
that these correction terms could be estimated from discrepanciesamong the tide gauge recordsfrom different regionsof
the world. If this can be achieved, tide gaugescould provide
information about the sourcesof continental mass depletion
which have causedthe observedsea level rise of the past century. Additional spatial information could be gained from
satellite measurementsof sea surface height and vertical displacement, although these data are of limited duration.
Acknowledgments.
We thank M. Giovinetto for ice sheet
accumulation estimates and M. Fang, R. Nerem and two anonymous reviewers for comments which helped improve the paper.
This work was supported by NASA DOSE grant NAGS-1911 and
by a National ScienceFoundation Graduate Research Fellowship.
Figure 4. Similar to Fig. 2, using small mountain glaciers
as the continental
source of sea level rise.
=-00
The
minimum
= -200.
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10to 20%fasterthantheglobalaverage
(Fig. 2b).
These rates are correspondinglyslowfor massdepletionin
the north.
Some stations near sources of continental mass
C. P. Conrad,B. H. Hager,Dept. of Earth,Atmospheric,
and
Planetary
Sciences,
Mass.Inst. ofTech.,Cambridge,
MA 02139.
depletionmay seriouslyunderestimatethe globalaverage. (e-mail:
clint•}chandler.mit.edu,
brad•}chandler.mit.edu)
We have presentedresultsfor a few extrememodelsof
November
13,1996;revised
March20,1997;
present-dayclimate changein which all the sea level rise (Received
originatesfrom a singlecontinentalsource.In fact, acom- acceptedApril 18, 1997.)