t L +ical Journal (1989) 96, 497-517

Pleistocene and Holocene sea-level change in the Australian)n and mantle rheology

ro Nakada* and Kurt LambeckSchool of Earth Sciences, The Australian National Uniuersity, Canberra, ACT 2601, Australia

I i.988 September 14. Received 1988 September 9; in original form 1988 May 26.

SUMMARYSpatial and temporal variations in sea-level are produced by the melting of the LatePleistocene ice and by the Earth's response to the redistribution in surface loads. Byexamining different parts of the sealevel curves of the past 20000yr from geographicallywidely distributed regions it becomes possible to constrain models of the melting history ofthe ice sheets and of the Earth's rheology. Observations from sites away from the formerArctic ice sheets, such as the Australian and South Pacific region, are particularly importantfor constraining the total meltwater volumes added into the oceans in the past 20 000 yr andthe rates at which this occurred. These observations indicate that the Antarctic ice sheetsprovided a significant contribution to the sea-level rise at a rate that was approximatelysynchronous with the melting of the Laurentide ice sheet, except for the interval 9000-ó000 yrago when it may have lagged behind. Minor melting of the Antarctic ice sheet appears tohave continued throughout the Late Holocene. Differential observations of the Late Holocenesea-level change recorded at sites in the same region are particularly useful for estimatingparameters describing the Earth's non-elastic response to surface loading. The effectiveparameters used here are a lithospheric thickness, an upper mantle viscosity, and a lowermantle viscosity describing the response below 670km depth.'\ryith the observations usedhere, it is not possible to separate the lithospheric thickness 11 from the upper mantleviscosity and the viscosity results are based on the assumption that 50=FI s100km. Neitherdo these observations provide a resolution of the depth dependence of viscosity in the uppermantle and the resulting estimates are effective parameters only. Differential sea-levels alongcontinental margins and along the shores of large gulfs and bays constrain the effective uppermantle viscosity to be about (I-Z)x102oPas-1 while differential values from islands ofdifferent sizes are suggestive of a somewhat lower value. The lower mantle (taken to be below670 km depth) viscosity is about two orders of magnitude greater than this. The estimated iceand rheological models explain many of the Holocene sea-level observations throughout theAustralian and southern Paciflc region. The tilting of continental margins, as exemplified byobservations of variable amplitudes of the Holocene high-stands and the variable times atwhich sealevels first reached their present level along the north eueensland coast and GreatBarrier Reef, is well represented by these models. Differential Holocene sea-levels observedalong the narrow Spencers Gulf of South Australia are also well explained by the models andin neither case is it necessary to invoke tectonic motions. Predicted Late Holocene sea-levels atsmall- to medium-sized islands are characterized by small amplitude high-stands that reached theirmaximum values about 4000-2000 yr ago, consistent with observations from the Societv. Cookand Tuamotu Islands.

Key words: ice sheet melting models, mantle rheology, sea-level, vertical tectonics

)DUCTION

tions of Late Pleistocene and Holocene sea-leveln are important for studying the tectonic histories ofrtal margins and ocean islands, for evaluating thecal response of the Earth to surface loading, and forring the melting histories of the large polar ice

rt the Faculty of Science, Kumamoto University,to, Japan.

sheets. A separation of the parameters that define thesevarious processes is possible in principle by examiningrecords of sealevel change over different time periods fromgeographically widespread sites. For example, sea-levelchange at sites near the former ice load (locations in thenear-field) exhibit a quite different dependence on thevarious parameters defining the Earth's response and the iceload's geometry through space and time than do sealevelchanges at sites far from the ice sheets (the far-field). For

t t +Geophysical lournal (1989) 96, 497-577

Late Pleistocene and Holocene sea-level change in the AustralianregÍon and mantle rheology

Masao Nakada* and Kurt LambeckResearch School of Earth Sciences, The Australian National Uniuersity, Canberra, ACT 2601, Austalia

Accepted 1988 September 14. Received 1988 September 9; in original form 1988 May 26.

SUMMARYSpatial and temporal variations in sea-level are produced by the melting of the LatePleistocene ice and by the Earth's response to the redistribution in surface loads. Byexamining different parts of the sea-level curves of the past 20 000 yr from geographicallywidely distributed regions it becomes possible to constrain models of the melting history ofthe ice sheets and of the Earth's rheology. Observations from sites away from the formerArctic ice sheets, such as the Australian and South Pacific region, are particularly importantfor constraining the total meltwater volumes added into the oceans in the past 20 000 yr andthe rates at which this occurred. These observations indicate that the Antarctic ice sheetsprovided a significant contribution to the sealevel rise at a rate that was approximatelysynchronous with the melting of the Laurentide ice sheet, except for the interval 9000-6000 yrago when it may have lagged behind. Minor melting of the Antarctic ice sheet appears tohave continued throughout the Late Holocene. Differential observations of the Late Holocenesea-level change recorded at sites in the same region are particularly useful for estimatingparameters describing the Earth's non-elastic response to surface loading. The effectiveparameters used here are a lithospheric thickness, an upper mantle viscosity, and a lowermantle viscosity describing the response below 670km depth.'With the observations usedhere, it is not possible to separate the lithospheric thickness È1 from the upper mantleviscosity and the viscosity results are based on the assumption that 50 =,É1 s 100 km. Neitherdo these observations provide a resolution of the depth dependence of viscosity in the uppermantle and the resulting estimates are effective parameters only. Differential seaJevels alongcontinental margins and along the shores o_f large gulfs and bays constrain the effective uppermantle viscosity to be about (L-2)x102oPas-1 while differential values from islands ofdifferent sizes are suggestive of a somewhat lower value. The lower mantle (taken to be below670 km depth) viscosity is about two orders of magnitude greater than this. The estimated iceand rheological models explain many of the Holocene sealevel observations throughout theAustralian and southern Pacific region. The tilting of continental margins, as exemplified byobservations of variable amplitudes of the Holocene high-stands and the variable times atwhich sea-levels first reached their present level along the north eueensland coast and GreatBarrier Reef, is well represented by these models. Differential Holocene sealevels observedalong the narrow Spencers Gulf of South Australia are also well explained by the models andin neither case is it necessary to invoke tectonic motions. Predicted Late Holocene sea-levels atsmall- to medium-sized islands are characterized by small amplitude high-standsthatreached theirmaximum values about 4000-2000 yr ago, consistent with observations from the Societv. Cookand Tuamotu Islands.

Key words: ice sheet melting models, mantle rheology, sea-level, vertical tectonics

INTRODUCTION

Observations of Late Pleistocene and Holocene sea-levelvariations are important for studying the tectonic histories ofcontinental margins and ocean islands, for evaluating themechanical response of the Earth to surface loading, and forconstraining the melting histories of the large polar ice

* Now at the Faculty of Science, Kumamoto University,Kumamoto, Japan.

sheets. A separation of the parameters that define thesevarious processes is possible in principle by examiningrecords of sea-level change over different time periods fromgeographically widespread sites. For example, seaJevelchange at sites near the former ice load (locations in thenear-field) exhibit a quite different dependence on thevarious parameters defining the Earth's response and the iceload's geometry through space and time than do sealevelchanges at sites far from the ice sheets (the far-field). For

497

498 M. Nakada and K. Lambeck

the far-field sites, the sealevel variations before about6000yr ago are relatively insensitive to the Parametersdefining the Earth's response to time-dependent surfaceloading whereas the mid- to late-Holocene part of the curveis more sensitive to this response (see, for example, Farrell& Clark 1976; Peltier & Andrews 1976).

Sea-levels in the near-field, where the changes are mostdramatic, particularly at locations in eastern Canada andalong the Atlantic coast of North America, have beenextensively used to estimate the rheological structure of theEarth's mantle (e.g. Cathles 1975; Wu & Peltier 1983;Sabadini, Yuen & Gasperini 1985; Yuen et al. t986). Butthe changes at such sites, close to the margins of the LatePleistocene ice sheets, are about equally sensitive to therheology of the mantle as to the detailed description of theice load and generally the latter is inadequately known forthe purpose of estimating the Earth's viscosity (e.g. Nakada& Lambeck L987, 1988a). Furthermore, the sea-levelchanges here may also indicate tectonic movements causedby the sediment loading of the glacial outwash associatedwith the disintegration of the ice sheets (e.g. Newman e/ ø/.1980). We therefore avoid using these data here and insteadfocus on sea-level observations in the far-field, at sites thatlie at a considerable distance from both the Arctic andAntarctic ice sheets.

Sea-level change in the far-field is charaeterized by a rapidrise during the melting phase from about L8 000 yr beforepresent (yrer) to about 6000yrsp and a nearly constantlevel from about 6000yrBp to the present. These changesare predominantly indicative of the rates at which themeltwater has been added into the oceans. Regionalvariations from this general Pattern occur because of theresponse of the Earth to the time-dependent meltwater andice loads as the ice mass is transferred to the oceans. Thiswas recognized by Bloom (1967), Walcott (1972) andChappell (1974) who attempted to estimate mantle viscosityfrom observations of regional variations in the sea-levelcurve. The first global analyses of the far-field sea-levelswere by Cathles (1975), Farrell & Clark (1976) and Clark,Farrell & Peltier (1978) and these studies have indicatedthat such data for tectonically stable regions can be used toconstrain both the rheological structure of the mantle andthe gross melting characteristics of the ice sheets (see alsoNakiboglu, Lambeck & Aharon 1983; Nakada & Lambeck1988b).

FAR.F IELD OBSERVATIONS OFHOLOCENE SEA.LEYEL CHANGE

The sealevels examined in this paper are from theAustralian and New Zealand region and from a few selectedSW Pacific sites. This choice may seem parochial but it isguided by the Australian continent being tectonically stableover this time interval and by the careful fieldwork done inthis area. Few other far-ûeld regions meet these conditions.Following upon the pionecing paper by Fairbridge (1961),much detailed work has been carried out in the Australasianregion and the results for Australia have been summarizedby Hopley & Thom (1983), Hopley (1987) and Chappell(1987), while the New Zealand evidence has beensummarized by Gibb (1986). The Australian region,generally far from plate boundaries and free from significant

Quaternary tectonics, is believed to be relatively stable.New Zealand provides a less stable platform for measuringsea-level changes, although the eastern edge of the SouthIsland, south of about Christchurch, and the NW area, nearAuckland, is believed to be relatively stable (rù/ellman1979). Ocean islands, particularly young islands, are proneto tectonic subsidence but sea-levels from these sites areimportant in that, when compared with continental marginobservations, they may provide evidence for lateralvariations in the Earth's response to surface loading. Recentobservations include those by Pirazzoli et al. (1985, L987)and Pirazzoli & Montaggioni (198a) for the Tuamotu andSociety Islands.

The principal diagnostic characteristics of the sea-levelcurves observed at these far-field regions are (i) theamplitudes of sea-level, relative to the present level, atabout 6000 yr ee, (ii) the time when sea-level reached itspresent level or obtained its maximum Holocene level, (iii)the rates of sealevel rise from about 10 000 to 6000 yr re,and (iv) the seaJevel at about 18000-20000yrre. Therecords prior to about 6000yree in these regions areprimarily indicative of the gross melting characteristics ofthe ice sheets; of the total volume of meltwater that hasbeen added to the oceans and of the rates at which thismeltwater has been added (Nakada & Lambeck 1988b).

Figure 1 summarizes the Australian observations of theposition of seaJevel, relative to the present level, at about6000yree where these Holocene high-stands have beenidentified (see papers in Hopley 1983a, b; Chappell 1987).The occurrence of these high-stands is in the range5000-6000yrur (Table 1). Also indicated are locationswhere the high-stand has not been clearly identified butwhere this level is believed to be close to the Presentsealevel. The best information is from Karumba, in theGulf of Carpentaria, the Great Barrier Reef of northernQueensland (typified by the Halifax Bay site), the coast ofNew South Wales (typified by the Moruya site), andSpencer Gulf, South Australia. Information from the othersites are usually restricted to isolated observations whosereliabilities are difficult to assess. Several regional patternscan be seen in these observations. Well defined Holocenehigh-stands are found along the NE coast from the Gulf ofCarpentaria to as far south as central Queensland withamplitudes ranging from 0.5 to 3 m above present leveis.Along this coastline Holocene sealevels for offshore islandsare generally lower than those observed at the continentalmargin. The Holocene high-stand has not been clearlyidentified along the coast of New South Wales and Thom &Roy (1983, 1985) concluded that this level did not exceed1.0m above present sea-level (see also Hopley 1987). Norhave Holocene high-stands been found along the coast ofTasmania either. An extensive study of the South AlligatorRiver estuary in the Northern Territory failed to findevidence for a Holocene high-stand (Woodroffe et al. 1987)but tentative identiflcations have been made further to thewest within the estuaries of the Ord and Fitzroy Rivers ofWestern Australia (Brown 1983). The sea-level observationsalong the west coast of Australia have been reviewed byBrown (1983) and estimates of maximum Holocene levelsrange from about 0.5 to 3 m above their present levels. Thelower value corresponds to a site in the Swan River Estuarynear Perth (Kendrick 1977) whereas the higher value

Late Pleßtocene and Holocene sea-leuel change 499

Ord River- l m

-Cape York

All¡gator I Gu¡ of

!\uft (c"tp"nt",i"

Fitzroy River \- r/*" -) %n,*îïiÀ/ 1:å*-1 \

lHatifax Bay-ì e

- l m

I -1 t o""-

WesternAustralia

NorthernTerritory

Queensland

SouthAustralia

-l

It - - - -

Swan.River Port P¡rie New South'2.s-r3.8m WaþS0.5m

Rockingham- 2 . K Cape Spencer

<1mRottnest

- 3 m

-"""

l-.- -,- ./uoruya- -/ <1m

1150 lzso 1350 14so 1550Figure 1. Location map and amplitudes of some Late Holocene sea-level high-stands observed around the Australian coastline (see Table 1 fordetails).

20"

4oo

corresponds to sites on Rottnest Island, about 20 kmoffshore (Playford 1983), and to sites north and south ofPerth (Woods & Searle 1983; Searle & Woods 1986). Thedifferences between these and the Swan River result are, asemphasized by Chappell (L987), hard to reconcile unlessthere has been some local tectonic movement (Playford &Leech 1977). Clear Holocene high-stands have beenidentified in the Spencer Gulf, along the southern margin ofthe continent, where the highest levels of 2.5-3.8 m occur atthe head of the gulf near Port Pirie (Burne t982; Belperio etal. L984; Hails, Belperio & Gostin 1984) but there is noclear evidence for Holocene high-stands at the entrance tothe gulf (Belperio, Hails & Gostin 1983).

Table 1. Holocene high-stand amplitude and time of occurrence forselected sites in Australia and New Zealand (see Fig. 1).

The Holocene sea-level observations for New Zealand atthree different locations where tectonic movements arebelieved to have been negligible (Auckland, Christchurchand Blueskin Bay near Dunedin) are indicative of analmost constant sea-level from the time, at about 6500 yr ae,when the sea surface first reached its present level andHolocene maxima above present level have not exceeded1m (Gibb 1986). For the Pacific islands of the southernCook, Tuamotu and Society groups, this level is not alwaysevident but where it is, it was generally reached later than600 yr er and the maximum Holocene high-stands have beenvariously dated at between 4000 and 2000yrar (Fig. 2). Theevidence for this includes micro-atolls (Stoddart, Spencer &Scoffin 1985; Yonekura et al. 1984), the ages of cementationof reef debris, and the ages of chemical alteration of thecements (Pirazzoli et al. 1985).

THE SEA.LEVEL EQUATION

The equation describing the sea-level variation, (, caused bythe transfer of mass between the ice sheets and oceans on aviscoelastic Earth has been formulated by Farrell & Clark(1976). The solution for the relative sea-level (RSL),defined as

Al(0, A; t ) : C@, A; t ) - Ç(0, A; t ) (1)

at colatitude 0, longitude  and time f, can be written as

A((0, A; t ) : AE"(o, ) , ; t ) + AZr(0, A; t ) + AZ"(0, A; t ) (Z)

(Nakada & Lambeck L987), where /6 is the present time.

SiteKarumbaHalifax BayNew South Wales

Port Pirie

Cape SpencerRottnest IslandSwan River estuaryAlligator River

TasmaniaChristchurchAuckland

(Hauraki Gulf)

I max Time(*) (kyr nr)

-2.5 6.01.0-1.5 6.00.0-1.0 -6.0

2.5-3.8 -6.0

-0.03.0 5.5-6.00.5 -5.0

<1 .0

-0.00.9 -6.s

<1.0 -6.3

Reference

Chappell et al. (1982)Chappell et al. (1983)Thom & Roy (1983,

1e8s)Burne (1982), Belperio

et al. (1984), Hailset al. (7984)

Belperio et al. (1983)Playford (1983)Kenrick (1977)Woodroffe ef ¿/.

(1e87)Hopley (1987)Gibb (1986)

Gibb (1986)

EJahE

EJ3nIE

500 M. Nakada ønd K. Lambeck

Tahltl

4

T lmo (x1000 yo ¡ r r )BP

Figure 2. Schematic relative sea-level curves (RSL) rvith respect topresent-day values observed at selected sites in (a) Australia andNew Zealand and (b) the South Pacifrc.

Here Áf* is the relative seaJevel variation on a rigid Earthand includes the equivalent sea-level change defined as

(meltwater volume at time l) x (ice density)

(area of ocean surface) x 6;*"üi;;Ð ' (3)

and the gravitational terms defining the attraction betweenthe ice sheets and oceans. This term can be neglected for thepost-glacial phase; for the times when no further meltwateris added to the ocean. The other two terms in equation (2),AZ, and AZt, allow for the deformation of the Earth inresponse to the change in the ice and water load. AZtrepresents the sealevel change produced by the deforma-tion associated with the changes in ice load volume andincludes the modifications produced by self-attraction andthe Earth's viscoelastic deformation as a result of thissurface load. The term AZ2 reprèsents the sealevel changeassociated with the deformation produced by the meltwaterloading and includes the associated self-attraction and Earthdeformation. The two terms are defined such that mass isconserved and the ocean surface is an equipotential at alltimes. Details for the definition of these terms are given byNakada & Lambeck (1987). To obtain quantitative solutionsof the sealevel equation, the required inputs are (i) thespatial and temporal description of the ice model, (ii) ageometric description of the ocean surface, and (iii) arheological model of the Earth.

Ice model

The details of the melting histories of the polar ice sheets,both the geographical distribution and the rates of melting,are most important for understanding the sea-level changes

at sites near the margins of the ice sheets and theuncertainties in these ice models are generally such that it isnot possible to draw realistic conclusions about the Earth'sresponse from data collected at such sites. These details aremuch less important at the far-field sites considered herewhere sealevel change is primarily dependent upon thevolumes and rates at which the meltwater has been added tothe oceans. Three major ice sheets are considered here; (i)the Laurentide and western Cordilleran ice sheet, includingthe Innuitian and Greenland parts, (ii) the Fennoscandianand Barents-Kara shelf ice sheets, and (iii) the Antarctic icesheet. The ICE 1 model of Peltier & Andrews (L976),smoothed and defined here with a L" spatial resolution, hasbeen adopted for the first contribution although, for thesefar-field evaluations, coarser definitions of the spatialdistribution of the ice loads are adequate. Melting of theLaurentide ice sheet is assumed to have been complete by6000 yr ep. For the second contribution, the Fennoscandianpart of the above ICE 1 model has been adopted and this,plus the Laurentide ice model defines the ARC 1 model ofNakada & Lambeck (1987). To this has been added anapproximate model of the Barents-Kara ice sheet, based onthe maximum model of Denton & Hughes (1981), which hasan equivalent sea-level rise of about 12m and whose rate ofmeltlng has been assumed to be synchronous with theequivalent sea-level change of the Fennoscandian ice ofARC 1. The sum of the contributions (i) and (ii) is referredto here as ARC 3 and its sea-level equivalent as defined by(3) is 89 m. Because the Fennoscandian ice sheet vanishedby about 10 000 yr ae (De Geer 1954) the detailedtime-space distribution of this and the Barents-Kara loadsis unimportant in discussions of far-field sea-level change forthe past 8000-10 000 yr.

Several descriptions of the Antarctic contribution are usedhere. The first is the model ANT 1 of Nakada & Lambeck(1987) which is based on the ice sheet reconstruction atL8000yrnr by Denton & Hughes (1981, p. 269) and thepresent ice sheet thicknesses given by Drewry (1982). Eachice column in this sheet has been assumed to melt at thesame rate, synchronous with the bulk melting of the Arcticice sheet ARC 1 (Fig. 3), consistent with the hypothesis thatthe Antarctic ice sheet disintegrated, with minimal time-lag,in response to the rising sea-level produced by the meltingof the northern ice sheet. The contribution of the Antarcticmodels to the equivalent sea-level rise is about 37 m, greaterthan the 24m proposed by Denton & Hughes (1981). Asecond description, ANT 2, based on the Wu & Peltier(1983) model in which the assumption was made thatAntarctic melting preceded Arctic deglaciation, is rejectedon the basis of its inconsistency with far-field sea-levelobservations (Nakada & Lambeck 1987, 1988b; see alsoNakiboglu et al. t983).

The maximum sea-level drops in the far-field at about18 000 yr ne constrain the total meltwater, and observationsindicate that this level occurred at between 100 and 150 mbelow the present level (Table 2). Chappell (1987) hasreviewed the evidence for this low-stand in the Australasianregion and concludes that these depths are about 130-150mbelow present sealevel. Ota, Matsushima & Moriwaki(1981) concluded that around Japan these depths occurredat about 130 m below present sea-level. Some regionalvariation in this level can be expected because of the Earth's

Eo

gI6o6

teG

¿5c,t¡J

1 8 1 2 6 0Tlme(x1000 years)Bp

Figure 3. Equivalent sea-level rise (ESL) for t\üo Arctic ice modelsARC 1 and ARC 3 and three Antarctic ice models ANT 1. ANT 3.ANT 4.

deformation but this is relatively small for far-field sites andnot very sensitive to the choice of mantle model. The icemodels ARC 3 and ANT 1 produce low-stands of betweenabout 120 and 140 m for the sites included in Table 2 andare in reasonable agreement with the observed values exceptthat they do not reproduce the values near 150 m depthsuggested for some localities.

Nakada & Lambeck (1988b) noted that for the period10000 to 6000yr ago the predicted sea-levels, based on themodels ARC 3 + ANT 1, generally rose faster and earlierthan observed and they suggested that the Antarctic icesheet may have melted somewhat later than the Arctic icesheet. This forms the basis for the equivalent sea-levelmodel ANT 3 illustrated in Fig. 3 and in the followingdiscussions we use the ice models ARC 3 and ANT 3 as thereference ice-load model.

Because of the predicted regional variation in thesea-level response and because of the considerableuncertainty that is associated with the adopted ice model, it

Table 2. Estimates of the lowest stand of sea-level in the far-field atabout 20 000 to 18 000 yr ago. The predicted values are based on theice models ARC 3 + ANT 3 and on a rheological model of11 = 50 km, Iu =2 x 1ú0 Pa s-1, ?r*: 1022 Pa s-1.-Area Reference

North Queensland Carter &

Observed Predicteddepth (m) depth (m)

Johnson (1986) I74-L33Huon Peninsula Chappell &

Shackleton (1986) 130Central Japan Ota et al. (7987) -130Timor Sea van Andel &

Veevers (1967) 130Central Queensland Veeh & Veevers

(1970) Ls0-r7sArafura Sea Jongsma (1970) 150-175

Late Pleßtocene and Holocene sea-leuel chønge 501

is necessary to establish whether the sealevels in theAustralian and Paçific regions are sensitive to thedistribution of the load within the ice sheets. Sea-levels inthis region are insensitive to the geographical distribution ofthe northern hemisphere ice sheets (Nakada & Lambeck1988b) but because of their closer proximity to Antarctica itremains to be demonstrated the degree to which theobservations are sensitive to the details of the Antarcticload. Fig. 4 illustrates the thickness of the ice that isassumed to have been removed between 18 Ofr) and6000 yr sp. To examine the dependence of the Australianand New Zealand sea-levels on details of melting theAntarctic ice sheet has been divided into three sectors, aneastern sector (A) over which little change has occurred andtwo sectors (8, C), centred on the Ross and Weddell Seasrespectively, which are the major sources of meltwater. Theequivalent sea-level contributions to the global model ANT3 from each of these sectors are illustrated in Fig. 5. Eachice column of sector A is assumed to have meltedsynchronously with the equivalent sea-level model for ANT3 and two rather extreme hypothetical cases of differentialmelting times for the sectors B and C have been adopted forwhich the total equivalent sea-level curve remainsunchanged: in the first (Fig. 5b), the available ice in sector Bhas decayed before melting starts in sector C, and in thesecond (Fig. 5c) the sequence of hypothetical melting rateshas been reversed. The resulting sealevel variations at foursites are illustrated in Fig. 6 and in all cases the differencesare small, particularly for the sites furthest from Antarctica.These differences remain small over a range of realisticearth models and the sea-level observations used in thisstudy are not sensitive to details of the geographicdistribution of Antarctic ice load.

Ocean fr¡nction

The geometry of the ocean surface on to which themeltwater is added, is defined by the ocean function, afunction that equals 1 where there is ocean and 0 wherethere is land. The resolution required to describe thisfunction is variable but can be very high for those localitieswhere the mid- to late-Holocene sea-level variation iscontrolled by the distribution of the meltwater in the vicinityof the point of observation. Past studies of global variationsin sealevel used fairly coarse definitions of the oceanfunction. Nakiboglu et al. (1983) used a 10" resolution andCathles (1975, 1980) and Lambeck & Nakada (1985) used5". Wu & Peltier (1983) used a variable resolution of about5'in the near-field and along some continental margins, andas large as 20o in mid-ocean regions of the far-field. Thesecoarse definitions fail to model the regional response to localloading but, as already demonstrated by Chappell et ø/.(1982), such variations in the Earth's response can besignificant over distances much less than 500 km (see alsoNakada 1986). Nakada & Lambeck (1987) found that anocean function described with 10' spatial fesolution andexpanded into spherical harmonics up to degree 180provided an adequate description of the ocean function forall observation sites examined, provided that a lithosphericthickness of not less than about 50 km is adopted. The useof greater thickness lithospheres, by improving theconvergence of the solution of the sea-level equation,

t29

t26122

136

130138

ANT3, t - - - - - - - - - - -

- . 'a

ò,,.' ,,Þi/ ..'

// ,'-'/ ,."

.//ar" .."'t , ' t

t t '>.

502 M. Nakada ønd K. Lambeck

reduces the need for this high resolution but any estimatesof this thickness would then be geophysically meaningless(Nakada & Lambeck 1988a). We use here the 10' spatialdescription of the ocean geometry and adopt a lithosphericthickness of 50 km.

Earth models

The Earth's response to surface loading on characteristictime-scales of 103-L04 yr, is represented by a Maxwellviscoelastic body with a depth-dependent Newtonianviscosity. Most of the glacial rebound observations appear tobe adequately described by such a model (e.g. Kaula 1980)and there are no compelling observational reasons for

+o o

+

lgoo

Figure 4. Thickness of Antarctic ice removed between 18 000 and 6000 yr rr (in km).

adopting a non-linear rheology to describe the reboundphenomena. The rigidity, bulk modulus and derisity arethose of the preliminary earth reference model (PREM) ofDziewonski & Anderson (1981). The lithosphere is assumedto have a viscosity of 102s Pa s-1 so that it respondsessentially as an elastic layer to loads on time scales ofL03-104 yr. The mantle viscosity is assumed to betwo-valued, an upper mantle viscosity 4.- and a lowermantle viscosity 4,- where the boundary between the tworegions is taken to correspond to the 670km discontinuity.Rebound models with asthenospheric low viscosity channels(Walcott 1973; Cathles 1975) have not been explored here indetail but trial calculations with a low viscosity channel of170 km thickness immediately beneath the lithosphere

È= 2 03tt¡¡

A + B + C

c

. t ' B

. ' . / - - -

.*-;-,t"-"-..r"'

b

ANT3,

A + B + C

/ B

-//--

A + B + C

ANT3,,

/t ' B

/:' /:--tt

./" A

/:.. ---"r''' ï

1 2t 8

Tlme( r1000 yoars )BP

Figure 5. Three schematic models for equivalent sea-level rise resulting from differential rates of melting of the sectors A, B, C of Antarcticadefined in Fig. 4.

wEsT - + EAST

A N T 3 ' A N T 3

' \ o * r . "

Late Pleistocene and Holocene sea'leuel change 503

Tlme(x 1000 Years)BP

Figure ó. Late Holocene sea-Ievels relative to present-day levels predicted by the melt models ANT 3 and the two modifications ANT 3' and

A¡¡"f ¡" of the Antarctic contribution with (H, ?'-, ?r-: 50 km, 1020 Pa s-t, 10'2 Pa s-1)'

EO - ¿

gI6

8 2ID

goIE

indicate that the sea-level response at most of the far-freld

sites considered here is not very sensitive to the depth

dependence of the viscosity of the upper mantle and that

oniy the average upper mantle viscosity can be resolved.

Only at large island sites in the far-field, such as New

Z,ealand or Fiji, does the introduction of a low viscosity

channel have a measurable effect but here the separation of

viscosity parameters from lithospheric thickness estimates

does not appear to be readily possible (see Appendix 1).

The solutions for the response of the viscoelastic body use

the correspondence principle and they imply that the body is

non-adiabatic (Cathles 1975)' Fjeldskaar & Cathles (1984)

suggested that this assumption was inappropriate and that,

under certain conditions, this could lead to erroneous

conclusions being drawn from observations of the response

of the Earth to glacial unloading. Approximate calculations,

however, have indicated that for far-field sites and for

sea-level change during the late- and post-glacial phase the

question of whether the mantle is adiabatic or non-adiabatic

ii not important as can also be seen directly from fig. 9 in

Fjeldskaar & Cathles (1984).

TI IEORETICAL CHARACTERISTICS OF

THE HOLOCENE SEA.LEYEL CHANGES

The sea-level changes during the post-glacial period are

represented by the sum of the two tetms AZt and AQ

(equation 2) representing the deformation of the crust

próduced by the melting of the ice sheets and by the loading

of the ocean crust by the meltwater respectively. These two

terms, plus the total relative sea-level change defined by Q),are illustrated in Fig. 7 for the epoch 6000yrep for the

Australian, New Zealand and Pacific sites discussed

previously. In the first instance the lithospheric thickness has

been held constant at 50 km and the upper mantle viscosity

is assumed to be 1021 Pa s-1. The lower mantle viscosity is

allowed to vary through two orders of magnitude. The

results are characteristic for a range of plausible upper

mantle viscosities.In Fig. 7(a) AZt and AZt are given for the northern ice

models ARC 1 and ARC 3 although the results differ only

little from each other. The AZ, term does not produce

differential sealevels across the region due to the fact that

the sites are all located in the far-field of the Arctic ice

Hal l fax BaY

504 M. Nakada and K. Lambeck

( a ) 6

4

Chr is tchurch

RSL ( ARC 3)

2 3

î,- : 2õS oI(g.oa n ^

bE5ç, = 4x.gE

2ocot oor

---(.*:i4

H a l i f a x B a y

4--- ì:----- - - _ É - - - - + - _ - {

Cape Spencer

1- := 1-î :*=---+ - -:i:È-- :jr

-"'" - "'' "'-- ̂ 2ì-' - --- - -''

2 2

log l /m (Pa s )

Oahu

Vahitahi

L.r*'jJ-.'aÈ-r:_r

Cape York

{----ù-4:-:::=

2 3

2 2

Tahlt i(bl

5o

sIooo

E5Ex6

Eq¡

a,(,ooI

2

0

2 32 22 1

Rarotonga

2

2 2 2 3

log fl lm (Pa s)

Figure 7. Maximum Late Holocene seaJevels (RSL), the ice unloading term AZt and the meltwater loading term AZ, as a function of lowermántle viscosity. The upper mantle viscosity is 1021 Pa s-l and lithospheric thickness is 50 km. (a) For continental margin sites and Arctic icemodels, (b) for island sites of different surface areas and Arctic ice models, (c) for continental margin sites and the Antarctic ice model ANT 3.

2 12 32 22 1

sheets where the sealevel variations are not sensitive to thedetails of the ice models. The predicted meltwater loadingterms AQ are, however, variable from site to site withmaximum values ranging from less than 1m at Cape York tomore than 4 m at Port Pirie. In particular, these terms arequite different for sites within gulfs, such as at Port Pirie andCape Spencer, that are less than 300 km apart along themãrgin of Spencer Gulf, and for the Queensland sites at

Cape York, Halifax Bay and Karumba. These local andregional variations arise from the geometry of the shorelinein the vicinity of the point at which the RSL is evaluated; atPort Pirie and Karumba the sector of the crust subjected towater loading is restricted to relatively narrow azimuthranges whereas at Cape Spencer or,Cape York the crust isloaded over much larger azimuth ranges.

Figure 7(b) shows the same functions for the Pacific

fi:;::Por t P í r ¡e

\

- - f - _ - a

. - - - - - - - * - ( ' -ä¿- - - -À- ' - {

t{t '

' " '-"-.-:"-},

: t ' - \ \/ \ a \

_ \

Por t P l r ie

H a l ¡ f a x B a Y

- / ¿ F - - ^

, .2 - : - -_ .<-

- > - - - a - - - _ - - - - - + - _ _ {

Cape Spencer

R S L ( A R C 3 )

2 2log r¡,,n (Pa s)

Figure 7. (c)-see caption opposite.

Late Pleßtocene and Holocene sea-leuel change

R S L ( A R C 3 )M o r u y a

R S L ( A R C 1

- _-.-- - --+ " '- - '- Î i l --*- - -

t4 '_* +- . - -È<, \ 7- 2

Cape York

{-- -¡"s:J-:=

505

(c) 6

4

2 3

Êz

õo oIaúo, 6E5Ê. = 4xoE

2oco

3 oor

2 32 1

islands. Here the ice unloading terms AZ, for Fiji, Tahiti,Vahitahi (in the Tuamoto group) and Rarotonga are verysimilar but they differ from those predicted for Oahu,Hawaii, which lies closer to the former ice load. Themeltwater loading terms A\ are relatively insensitive to thelower mantle viscosity for all these islands but theamplitudes vary according to the size of the island; thelarger the surface area, the larger AQ (see also Nakada1986). Fig. 7(c) illustrates results similar to Fig. 7(a) but, inthis case, for the Antarctic ice load ANT 3. Thecharacteristics of the meltwater loading term AZ, isgenerally the same as for the Arctic ice loads although theamplitudes are much reduced. The Antarctic ice unloadingterm AZ, does exhibit a different dependence on the lowermantle viscosity than does the corresponding Arctic term.Compare, for example, the sites closest to Antarctica,Christchurch, with sites much further from the load, such asKarumba.

In the above analyses the lithospheric thickness ,EI hasbeen assumed to be 50 km but the above results remainvalid for 50=H= 100km. It has been argued by Peltier(1984) that the lithospheric thickness, as estimated from glacialrebound data along the Atlantic margin of North America,is of the order of 200km but Nakada & Lambeck (1988a)have shown that this is a consequence of inadequate spatialand temporal resolution of his adopted ice model. It mayalso be argued whether or not this parameter can beseparated from the upper mantle viscosity, particularly fromthe far-field data alone. Fig. 8(a) illustrates the Holocenehigh-stand for a 50km thick lithosphere and a 1021 Pas-1lower mantle viscosity as a function of upper mantleviscosity. The predicted high-stands at Karumba, Halifax

B"y, Cape Spencer and Rarotonga are not stronglydependent on the upper mantle viscosity within therelatively restricted range of 1020-1021 Pa s-1 but thehigh-stands for Port Pirie, Fiji and Christchurch are variableover this viscosity range. Fig. 8(b) illustrates the Holocenehigh-stands for a uniform mantle viscosity of 1021 Pa s-1 andfor a variable lithospheric thickness Il in the range of50-220 km. Here the sites that are sensitive to lithosphericthickness (e.g. Fiji) are also sensitive to upper mantleviscosity and sites that are insensitive to 11 are. alsoinsensitive to ?,- and for these continental margin andisland sites it will not be possible tÒ separate with confidencethe parameters ¡1 and 4.-. At sites such as Christchurch a50 km thick lithosphere with an upper mantle viscosity of2x!02oPas-1 gives about the same Holocene high-standamplitude as a lithospherc of 220 km thickness overlying aL021 Pa s-r viscosity upper mantle. Because of thisinterdependence of H and 4,- it is not possible to examinefrom this data alone whether the oceanic upper mantle has alow viscosity channel immediately below the lithosphere.For the present purpose a lithospheric thickness of 50kmhas been adopted throughout although it is important toestablish better bounds on this parameter. None of thefollowing results for continental margin sites is significantlymodified if this thickness is increased to 80 or even 100 km.

Figure 9 illustrates the dependence of the amplitude ofthe Holocene high-stand on both upper (4"j and lower(q,^) mantle viscosity in the respective ranges of5x101e-1.021 and L02r-1023 Pas-1 for a 50km thicklithosphere and the ice load ARC 3 + ANT 3. At sites suchas Auckland and Christchurch the Earth's response ispredominantly a function of the upper mantle viscosity,

Chr is tchurch

_ . _ - . - + - _ _ _ -- - - a " - - - 1 ' - ' -

\f -ql--=- p

P - C S

2 . 6

K - H

l o g f l u m ( p a s )

K - H

100

l l thospher lc

2 0 0

Figure E. (a) Relative sea-levels (RSL) as a function of upper mantle viscosity for a lower mantle viscosity of 1021 pa s-1 and a lithosphericthickness of 50km. (b) RSL as a function of lithospheric thickness for a uniform mantle viscosiry of ld2rpas-l. (c) Difierentiai relativesealevels (DRSL) as defined by equation (5), as a function of upper mantle viscosity, and (d) as a funcrion of lithospheric thickness.c = christchurch, cs : cape spencer, F: Fiji, H: Halifax Bay, K: Karumba, p: port pirie, Ra: Rarotonsa.

( u

o

EÊoo

\5

,NKarumba

23

22

21TaHtl

A \(-;ì'\'\

--\ \\

\"Ñ\

Rarotonga

212021202120212020

Figure 9. Predicted amplitudes 9fcontinental margin and island sites

log nr.(Pa ¡)Holocene high-stands as a function of upper mantle qum and lower mantlewith 11 :50 km.

h$rt i ì '¡ \ ' . '

Moruya

Fort Plrlo

I7

3

l./ 2

2r5

I

I

2.5

\. \2I

3is

-3 \

:-i_-\- - 1 , _

Cape York

Oahu Vahltald

4r* viscosities for selected

1.5

õ . ^à

, . t' a

Iq 0.5

Jo

- 0.0

while at sites such as Cape York or Karumba the response isprimarily controlled by the lower mantle viscosity and fromobservations at such sites it becomes possible to separateupper and lower mantle viscosities. Generally, thecomparison of observations with these predictions suggeststhat 4"-=102oPas-l and r¡r^=I021 Pas-1 for otherwisethe predicted Holocene high-stands become much greaterthan the observed values.

Mantle models with these viscosity values, however, areunsatisfactory in several respects. First, the observeddifferential values of Late Holocene maxima are not wellmatched by these models for sites such as Karumba andHalifax where these sea-stands are well determined, oralong the Spencer Gulf where large variations in themagnitude of the high-stand have been noted (Fig. 10).

Time (xl000 yers) BP

-7 -5 -3 -1Time (x1000 yeas) BP

Figure 10. Relative sea-levels for Late Holocene time for11=50km, ? . * :5x 101ePas-1 , 4 r - :1021 Pas- l . (a ) For twosites in northern Queensland, Australia; (b) for a small island inFrench Polynesia; and (c) for two sites in Spencér Gulf, SouthAustralia (cf. Figs 1 and 2).

Late Pleßtocene and Holocene sea-leuel change 507

Second, the models predict sea-level curves at small islandsites, such as the Society and Tuamotu Islands, that arecharacterized by short-duration high-stands at about6000 yr ee, in contrast with the observed values (Pirazzoli etal. t987) (compare Figs 2 and 10). Third, for continentalmargin sites, the sea-levels are predicted to drop rapidlybetween about 6000 and 4000 yr nr and this is generally notobserved (e.g. Chappell et al. 1983; Chappell 1987). Forthese reasons it is useful to explore the range of permissiblemantle models more fully, using observations of thedifferences in sealevel from sites in the same generalregion.

D IFFERENTIAL HOLOCENE H IGH-STANDS

The above-discussed characteristics of the Holocenehigh-stands suggests that useful observational constraints arethe differential sea-levels at locations in the same generalregion. The observed late Holocene high-stand (Áf,") at atectonically stable site i can be written as (c/. equation 2)

A€f = L€".t+ AZr.i+ A4,¡. (4)

For the ice models ARC 1to ARC 3 and ANT 1 to ANT 3.AEp..¡:0 for r< 6000yr but the following discussion is alsovalid when AgR.i+0 because this term is nearly constant forthe far-field sites discussed here. The differential high-standd(i,, between sites i and I is then defined by

dli,'= Azi - LE?: (AZ',¡ - A4ì + Ø4,¡ - AZr,') (s)

and is a function of the Earth's viscosity structure and of therates of melting during the time interval from 18 000 to6000 yr ne. The sealevels prior to 6000 yr nr have previouslybeen used to place gross constraints on the ice volumes andon the rates of melting of the Antarctic ice sheet and, withinthe range of melt models considered here, the differentialsea-levels (5) are primarily sensitive to the rheologicalstructure (Nakada & Lambeck 1988b). They do not,however, contribute greatly to separating the lithosphericthickness from upper mantle viscosity estimates (Figs 8c, d).

Differential Holocene sealevel maxima, as defined by eqn(5), are illustrated in Fig. 11(a) for selected pairs of sites ándas a function of upper and lower mantle viscosities. For thesites at Karumba and Halifax Bay in northem Queensland,the observed differential high-stand is =1.0 m and thisdefines a broad range of upper and lower mantle viscositiesthat yield predictions that are consistent with theobservations. For the sites Halifax Bay and Moruya alongthe eastern margin of Australia the observed differentialhigh-stand lies between 0.0 and 1.5m and this defines afurther range of permissible effective viscosities for the twoparts of the mantle. Likewise, the differential Holocenehigh-stand for the two nearby sites of Port Pirie and CapeSpencer in South Australia provides a further constraint andthe three sets of differential values along the Australianmargin deñne a relatively restricted range of values inrlt^-.Tu space that are consistent with the observations; of1020 s Tu- =2x IO2o Pa s-1 and Tr- à 5 x L02r Pa s-l 1Fig.11a). If Christchurch is considered to be representative of acontinental margin site then the differential heights betweenMoruya and Christchurch places a considerably tighterconstraint on both the average upper and lower mantleviscosities as (1020 1r¡u 12x 1020) Pas-1 and (5 x 1021 <

508 M. Nakada ønd K. Lømbeck

Karumba(l)Halifax Bayd(;r >r.o

- -0.5

Hallfar BaYMoruya(¡)

o.oSd(i , ,m<

Y Moruya(¡)Chr¡stchurch (¡)

-0.5< d( i j<1.0

2',t 20log nr. (Pa e)

20 21 20 21 20 21log ¡rr(Pa s)

Figure 11. Predicted differential sea-levels as a function of upper and lower mantle viscosities and for É1: 50 km. 'Ihe shaded regions for eachpair of sites indicates the range of permissible viscosity solutions. (a) For continental margin sites where (iv) illustrates that part of the viscosityrange satisfied by the individual solutions (Ð-(iiD, and (vi) illustrates the solution if Christchurch is included as a continental margin site. (b)For some Pacific island sites.

(a )

e 2 1

ë

c(tt-9

20

(b)

a , 2 3Gg

s

? z z 22

4r-(3 xl}22) Pas-1 respectively (Fig. 11a). To turtherconstrain this solution for effective viscosity, particularly toplace an upper bound on the lower mantle viscosity,requires further observations from pairs of sites whosedifferential response is particularly sensitive to the lowermantle structure. Differential sea-levels between the nearbyislands of Tahiti and Vahitahi illustrate the effect of islandsize (Fig. 11b). According to Pftazzoli et al. (1987) theHolocene high-stand at Vahitahi was about 0.7 m, similar tothat for Tahiti (Pirazzoli et al. 1985), and the smalldifferential value supports an upþer mantle viscosity of

=1020Pas-t. The difference Fiji-Tahiti, is likewise quitesensitive to upper mantle viscosity. The Holocenehigh-stand for Fiji has been estimated at up to 1.6m abovethe present level (Hopley 1987) and the differential valuewith Tahiti also indicates that 4,-s102oPas-1. Both theFiji and Tahiti observations, however, are not very reliableand the islands may be subject to vertical movements aswell, and while this suggestion is tempting, it is premature toconclude that upper mantle viscosity beneath the oceans isless than that beneath the continental margins. Differentialsea-levels between Tahiti and Oahu, islands of about equal

lll Port Plrle0)Gape SPencerlP

2.s s d(;,r< 4.0

l,V%2

\

IV

vt

Fii¡ - Tahit¡

I0

Tahiti - Oahu

il\..,o.u ".,l tj l (

1.6/ \=-__--

lt';;=l , ' - (l i - 1

Tahit¡ - Vahitahi

i | ", \\\\.

\ | .,' o.g ',

ì l i o o . uI \ ',.__-..,'

I " r \ /I i o . q \I o:L \ ò'1

size, are also illustrated in Fig. 11(b) and these areindicative of a particular sensitivity to the lower mantlerheology. Again the question of island stability needs to beexamined before conclusions can be drawn about mantlerheology but these examples illustrate clearly the impor-tance of island data for establishing whether lateralvariations occur in mantle rheology.

AN ADJUSTMENT OF THE MELT INGMODELS'With

the solution q, - (l-2) x 1020 Pa s-r and 4,*:L}nPas-1 that satisfies the differential sea-levels and withthe ice models ARC 3 + ANT 3, the predicted absoluteHolocene high-stand values are significantly greater than theobserved values (Fig. 11a). Increasing the volume ofmeltwater that has beeñ added into the oceans prior toabout 8000yree only increases this discrepancy while adecrease in the Antarctic contributions does not modify theresult greatly either. Fig. 12(b), for example, illustrates theresult for an Antarctic model (ANT 4) in which the totalmeltwater volume has been reduced to an equivalentsea-level rise of 23 m and in which melting did not start untilabout 12000yrre (see also Fig. 3).

Recourse to the argument of tectonic instability of thesites is not satisfactory in that it requires that all sites usedhere have subsided at rates of the order of about0.2-0.4mmyr-1 but there is no independent geological

Late Pleistocene and Holocene sea-leuel change 509

evidence to support such an interpretation; in particular, ifthis were so the inter-glacial maximum high-stand at120000yree would now be between 20 and 40m belowpresent sea-levels whereas it is usually observed to lie at afew metres above the present level (e.g. Chappell 1987). Analternate explanation is to assume that minor deglaciationcontinued throughout late-Holocene time. The Laurentideice sheet disappeared completely by about 6000yrne (e.g.Prest 1969) while the Fennoscandian ice sheet vanished at amuch earlier date (de Geer 1954), but there is no evidencethat the deglaciation of the Antarctic ice sheet also ceased at6000yree (Denton & Hughes 1981; Pickard 1986). Inparticular, if this latter deglaciation is controlled by therising sea-level, then minor melting may have continued wellinto mid- to late-Holocene time. Fig. 13 illustrates theequivalent sea-level curve for the past 8000 yr based on themodel ANT 3. This model can be adjusted such that thepredicted differential Holocene high-stand values arepreserved and the absolute values are consistent with theobsewational evidence. The amount by which the ice modelis adjusted is now a function of the mantle parametersalthough, within the range of mantle parameters thatproduce satisfactory differential Holocene levels, thisdependence is slight. Fig. 13 illustrates two models ANT 3Aand ANT 38 which are based on a lower mantle viscosity ofl022Pas-r, a lithospheric thickness of 50km, and uppermantle viscosities of 1020 and 2x 102oPas-1, respectively(Nakada & Lambeck 1988b). Both producê very similar

4 2

Tlme(x1000 ycar!) BP

E

ooI€oao

6oÊ,

EooI6ooo

ooG

6 4

Tlme(x1OOO Year!)BP

Figure 12. Predicted Holocene sea-levels for continental margin sites for the ice models (a) ARC 3 + NT 3 and (b) ARC 3 + ANT 4 (see Fig.3), and lithospheric thickness, upper mantle viscosity and lower mantle viscosity of 50 km, 1020 and 1.022 Pa s-1, respectively.

- -l-1.ÈÈ

Karunü¡

!

lt;

#ilt/

M. Nakada ønd K. Lambeck

4 0

3 5

ANT3AANT3

-/.-<:ã'ã'1-'-.-l-" \ nNtge

li

above the present level; the New Zealand sea-levels arepredicted to have had zero or very small Holocenehigh-stands and the Pacific island high-stands are also muchreduced and, particularly for small islands, these high-standsnow occur later, between about 4000 and 2000yr ago.

D ISCUSSION

Holocene high-stands along continental margins

Fig. 15 illustrates the predicted regional variability in thesea-level response in the Australian region for the followingcombination of ice and mantle models: (i) ARC 3 + ANT3A, H =50km, 4.* :1020Pas-1, ?r- = l022Pas-r , and( iÐ ARC 3+ANT 38 , H :50km, Tu :2x102oPas -1 ,q,.:1022 Pa s-l. As previously noted, values of .FI of up to100 km produce similar results. One feature of thesesolutions is the general increase in the amplitudes of theHolocene high-stands with decreasing latitude or withincreasing distance from Antarctica, in agreement with thegeneral trend observed from south to north along theeastern margin of Australia (cf. Fig. 1). In particular, noHolocene high-stand is predicted for Tasmania. This is aresult of the Antarctic ice unloading term AZ, because nomodification of the Arctic ice sheets can produce such aregional trend, and it supports the argu,ment that Antarcticmelting in Late Pleistocene time was significant and that thismelting occurred at approximately the same time as theArctic melting. A second feature of these solutions is the

0

and (i)

E

aDtu

8 6 4 2 0

. Tlme(x1000 years)BP

Figure 13. Equivalent sea-levels for the Antarctic ice models ANT3, ANT 3A and ANT 38 based on Holocene sealevel observations.

results for the Holocene sealevels as is illustrated in Fig. 14and now the predicted Holocene high-stands agree well withthe observed values at most of the sites examined. At PortPirie, for example, the maximum amplitude is about 3 mwhile the Cape Spencer sea-levels did not rise significantly

O E

T l ñ c ( ¡ 1 0 0 0 y o a r s ) B P

Figure 14. Relative sea-levels corresponding to the models with É/ :50 km, ?r-: 1022 Pa s-1?o-: 1020 Pa s-1, and (2) ARc 3 + ANT 3B for 4"- : 2 x 1020 Pa s-

ARC 3 + ANT 34, for

shorter wavelength variability in the sea-level response as aconsequence of the coastline geometry such as is predictedto occur in the Spencer Gulf and the Gulf of Carpentaria,where the high-stands are larger at sites within the gulfs thanat sites on neighbouring peninsulas. Even for relativelysmall coastal identations, such as occur along theQueensland coast, significant regional variation is predictedand the examination of sealevel indicators at such sites isparticularly important for studies of the lithospheric andupper mantle respoilse to surface loading.

Tilúing of continental margins and timing of high-stands

Another feature of the models is that they predict significanttilting of the continental shelves, the Holocene high-stands,for example, decreasing in amplitude with distance outwardsfrom the present shoreline: predicted high-stands for theouter part of the Great Barrier Reef are small or zero inamplitude and occur much later in time than at sites on theinner reefs or on the coast (Fig. 16). There is indeedobservational evidence for this. For example, Chappell et al.(1983) study of the age-height distribution of micro-atollsfrom North Queensland point to higher Holocene sea-levelsfrom samples on the mainland coast at Yule Point than fromthe samples from reefs on the inner shelf, possibly by asmuch as 0.7 m. Also, while a Holocene high-stand hasdeveloped along the mainland Queensland coast and at

Late Pleistocene and Holocene sea-Ieuel change 511

near-offshore islands and reefs, it is absent at outer reeflocations (see also Hopley 1983b). A potentially importantobservation is that the tilting of the continental shelf is quitesensitive to the upper mantle viscosity, as is illustrated inFig. 17.

Rigorous modelling of this tilting of the margin requiresconsideration of the migration of the shoreline with time assea-levels rises, an effect that may be particularly importantin shallow sea areas, such as the Gulf of Carpentaria, whereaverage present water depths are less than about 100 m.Approximate models of rebound with time-dependent oceanfunctions indicate that this effect on the relative sea-levelcurve may be of the order of 10 per cent for sites such asKarumba and along the Queensland coast.

A consequence of the tilting of the shelf is that itproduces a regional variation of the time at which theHolocene high-stand maxima first occurred or of the time atwhich sealevel first reached its present value, if this levelwas reached at all. Along the continental margins theselevels are predicted to have been reached at about6500-6000yrnr, with the earliest ages occurring at thesouthernmost sites. At island sites, however, these levels arepredicted to occur by as much as 2000yr later. This is also ingeneral agreement with observations along the northQueensland coast. For example, on the islands of Orpheusand Rattlesnake which lie about 10-15 km from themainland, the present sealevel was reached about

1 .1 (1 .0 )

0 .8 (0 .5 )

1 .2 (1 .21

0.7(0 .4

5 (3 .4 )

' /'-/-'1'2n'11

r0.s(1.5)(

ù-,;ä'i;"'\0.7(1.2)

. \ 1.6(1.1)

o.r1,.rf,o.stt.st /,---/| ---J0.7(0.8)-o.s(-0.4t .__áo(0.3)\-0 .4 ( -0 .4 )

1.2,1.g't -=¿

1.5r-.1/'

1.0(0 .7) .

0 .6 (0 .

r (3.9).4(2.91l . 7Q .2 l

\o.sn.sl

-0.+{0.6){\1

I y'-o.stt.ol\"do.nro.rl

-0 .7 ( -0 .9 )

4oo

1 1 5 0

1250 1350 1 4 S o l 5 5 o

Figure 15. Predicted Holocene seaJevels at 6000 yr BP along the Australian- margin for two ice-mantle models. The first number correspondstoìhe combination ARC 3 +ANT 3A and an upper mantle viscosity of 1020Pas-1. The second number, in parenthesis, corresponds to ARC3 + ANT 3B and an upper mantle viscosity of 2 x 1020 Pa s-1. In both, the lower mantle viscosity is 1022 Pa s-1 and the lithospheric thickness is50 km.

512 M. Nqkada and K. Lqmbeck

Tlme(x1000 yearr) BP

c t D C

Figure 16. (a) Predicted Holocene seaJevels (in metres) at 6000 yr BP, and (b) the times at which seaJevel first reached its present value (in1000 yr) along the margin and Great Barrier Reef of northern Queensland fo¡ the model ARC 3 + ANT 38, ?,- :2 x 1020 Pa s-1, ¡{: 50 kmand 4,-: 102 Pa s-r. The characteristic sea-level curves in the four zones indicated in (b) are illustrated in (c).

(nE

3 4!¡

g 2I6oato 0à6õ - 2E

5500-6000 yr np but at Grub and Wheeler Reefs, which lieabout 80 km from the mainland, sea-levels did not approachtheir present levels until about 2000yr later (Hopley 1983a).The sea-level indicators in these examples are based on reefcores and they assume that reef growth kept up with therising water level and the actual levels would lie above thesedata, but the significant observation is the differencebetween the two regions, a difference that is generallyconsistent with the predictions illustrated in Fig. 16. Thisdifference has usually been attributed to reef accretionlagging behind sea-level change by amounts that vary withdistance from the continental margin (e.g. Hopley 1983;Davies & Hopley 1983a) but these model calculationsillustrate that it may, at least in part, also reflect thedifferential sea-level change in the region, with quitesignificant variations in response occurring across the shelfover distances of the order of 100 km.

The predicted regional variabitity of the Holocenehigh-stand for northern Queensland is similar to that

15oo

Figure 17. Tilting of the continental edge along l8"S across thenorth Queensland coast for two models (1) ARC 3 + ANT 3A and?,r=1020p¿s-r, and (2) ARC 3+ANT 38 and Iu :2x1úo Pa s-l.

Longltude

predicted by Chappell et al. (1982) who computed thedeformation produced by the water loading only, equivalentto the AQ term, using a one-dimensional model alongseveral sections across the continental margin. This lattercalculation does not impose the condition that watersurfaces must be equipotential surfaces at all times but thiseffect will be small in the Late Holocene far-field and allmeltwater has been added into the oceans. The goodagreement between the two approaches suggests that forthese far-field sites it may be appropriate and convenient tocompute the Late Holocene sea-level changes in two parts inwhich the global spherical model is used to establish theregional contribution arising from the ice unloading effectand a more local and flat earth model is used to compute thecontributions resulting from the deformation of the crust inresponse to the change in water load in the vicinity of thesite. This approach may be particularly useful for areaswhere complex coastal geometry is not readily resolved withthe global models used here, as along, foi example, theupper reaches of Spencer Gulf.

Holocene sea-levels in the SW Pacific

Some characteristic sea-level responses predicted for islandsites in the SW Pacific are illustrated in Fig. L4. Throughoutthis region the ice-unloading term AZ, is quite constantalthough it does change for sites further to the north such asOahu. Differential sea-levels between Oahu and Tahiti, forexample, would be quite sensitive indicators of lower-mantleviscosity if tectonic subsidence of both islands could beexcluded (Fig. 11b). The amplitude of the meltwater term,AQ, throaghout the region is insensitive to the value of thelower mantle viscosity but it is a function of the island size,and differential sea-levels between small and large islands inthe same region (e.g. Fiji and Tahiti and to a lesser degreeTahiti and Vahitahi) are predominantly sensitive to theupper mantle viscosity (Fig. Ub). Simple regional patternsfor the sea-level change in the world's oceans do not,therefore, exist. Important for geophysics is that indicatorsof sea-level change from sites selected because of theirparticular sensitivity to mantle rheology parameters, provide.important information on mantle rheology, always assumingthat the sites themselves are not subject to tectonicmovements as may be the case for islands near plate marginsor for young volcanic islands. This warrants furtherexamination.

At many of the Pacific sites the time at which sea-levellast reached its present value is more recent than forcontinental margin sites and the maximum late Holocenesea-level often occurs in the interval 4000-2000yrne (see,for example, frg. I3.9 of Hopley 1987). Predictions based onice models in which all melting ceased at 6000 yr erinvariably produce high-stands at about this time whereasmodels in which some Antarctic melting is permitted in LateHolocene time predict Holocene sea-level maxima that aresmaller in amplitude, which occur later by about 2000 yr andwhich remain near their maximum level for about1000-2000yr (Fig. 18), in general agreement withobservations.

Antarctic melting

The conclusion that a small amount of meltwater(equivalent to a total of about 106 kml of grounded ice

Late Pleßtocene and Holocene sea-leuel change 513

- 1 0

8 6 4 2 0Tlme(x1000 years)BP

Figure lE. Predicted Holocene seaJevels for Tahiti for fourdifierent models. Observations indicate a maximum level of about0.5 m occurring at about 3000 yr rr, consistent with model 4 (ARC3 + ANT 3B, 4"-: 2 x 1020 Pa s-1).

above sealevel) continued to be added into the oceansbetween 6000 yr ee and the present is consistent withobservations of ice retreat in the Vestfold Hills of Antarcticaup to the present (Adamson & Pickard 1986). To effect achange in global seaJevel of 1m from Antarctic land ice, anaverage retreat by about 10 km of the Antarctic ice marginis required (Walcott 1974). Much of this margin is diffcult toexamine but according to Adamson & Pickard, the rate ofice retreat in the Vestfold Hills during late Holocene timehas averaged about 2-3ñyr-r, while Stuiver et al. (1987)have concluded that a slow Late Holocene retreat of the icemargins in the Ross and Weddell Seas cannot be excludedfrom the available data.

The conclusion concerning ongoing Antarctic melting intoLate Holocene time is consistent with our earlier suggestionthat the far-field sea-level observations in the time intervalof 10 000-6000 yr nr are indicative of Antarctic meltinghaving occurred somewhat later than the Arctic melting andlends support to the models in which the decay of theAntarctic ice sheet is triggered in whole or in part by the risein sea-level produced by the earlier melting of the Arctic icesheet (e.g. Stuiver et al. 1981; Budd & Smith 1982). V/iththe present sea-level data it is not possible to identify whichpart of the Antarctic contributes most to this latePleistocene and early flolocene meltwater and what isneeded to do this are sea-levels closer to Antarctica, or fromthe Antarctic margin itself.

Tectonic movements

The regional variations in sea-level resulting from the effectof meltwater loading can be sufficiently large in some

E!¡

9 _Iõ(,tt

o¡zgt,É.

rvroaer,f!tlfjå':, tili:ifJli'" n,ä, lí-

2 ANT3 50 2x1o2o 1022

3 ANT3A so 1o"o 1022

4 ANT3B so 2x1o2o 1022

51,4 M. Nakada qnd K. Lambeck

instances to be confused with tectonic movements. It has,for example, been proposed by Belperio et al. (1984) thatthe Spencer Gulf data is indicative of differential tectonicmovements but clearly this conclusion is not warranted asmuch of the regional variability can be explained in teims ofthe meltwater loading. Similarly, it has been suggested thatsome of the regional sea-level variations along the NorthQueensland coast are indicative of tectonic movementalthough Hopley (1983b) recognized that this could also be aconsequence of the Earth's response to water loading. Athird area where regional tectonics has been suggested isnear Perth, Western Australia, where the model predictionsfor the area range between about 0.7 and 1.5 m (Fig. 15).Holocene highs of about 3 m have been reported forRottnest Island (Playford 1987) and about 2.5 m to thesouth of Perth (Searle & Woods 1986) whereas Kendrick(L977) reported only about 0.5 m in the Swan River Estuarynear Perth. The region over which the observed variationsoccur is too small to be attributed to the variable responseof the Earth to meltwater loading and if all observations arecorrect then there indeed has been differential tectonic upliftin the area (Lambeck 1987). A wide range of models for theEarth's response and for the ice loads has been examinedbut no model explains the observations of Holocene maximaof 2.5-3 m without destroying the general agreement at allother Australian sites. The lower value observed in theSwan River estuary is more consistent with the models. Inview of the potential of this observation to constrain themantle rheology models (Fig. 1lb), these observationswarrant closer examination.

A further possible discrepancy between the predictions ofFig. 15 and observations (Fig. 1.) occurs along the NWmargin of Australia. Holocene high-stands of about 1mhave been reported for the Fitzroy and Ord River estuaries(Brown 1983) and of less than lm for the Alligator Riverestuary (Woodroffe et al. L987) compared with predictedvalues of 2-3m fo¡ these estuaries. Neither has evidencebeen found in this region for the last interglacial high-standwhich occurs elsewhere at a few metres above presentsealevel. Unless lateral variations in viscosity or litho-spheric thickness can explain these observations, one isforced to conclude that the NW region of Australia has beensubjected to subsidence at a rate of about 1. m in 6000yr (seealso Woodroffe et al. 1987).

Mantle and lithospheric parameters

While the observations examined here are stronglysuggestive of an increase in mantle viscosity with depth, theydo not suffce to give a unique determination of the depthdependence of viscosity. [n particular, it remains difficult toseparate the effects of lithospheric thickness and uppermantle viscosity, particularly if low viscosity channels areintroduced. The mantle models proposed here are therefore'effective' models which, for the continental margin sites,can be characterized by an average lithospheric thickness of50-100km, an upper mantle viscosity of (1-2)1020Pas-1down to 670 km, and a lower mantle viscosity of aboutl022Pas-1. The ocean island data are suggestive of asomewhat lower value for the upper mantle viscosity butthese observations are also questionable because of thepossible occurrence of vertical movements, and they warrant

further study. More important than these values themselvesis the demonstration that by examining differential sea-levelsin the far-field it becomes possible to separate the effectivemantle response parameters from the uncertainties as-sociated with the melting histories of the ice caps and thatby using differential values between continental margins andmid-ocean island sites it becomes possible to examinewhether lateral variations occur in these effective param-eters.

A C K N O W L E D G M E N T S

We thank Drs J. Chappell and D. Hopley for valuablediscussions on the Australasian evidence for sea-levelchange.

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516 M. Nakada and K. Lambeck

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APPENDIX 1 : DEPTH RESOLUTION OFUPPER MANTLE Y ISCOSITY PROFILES

The mantle models examined are defined by twoparameters, an average viscosity above and an average valuebelow the 670km discontinuity. Cathles (1975) has arguedthat a better representation of the structure is a thin(:75km), low viscosity (4x 101ePas-1) channel immedi-ately beneath the lithosphere and overlying a mantle of1021Pas-1 viscosity. This conclusion \ryas based on anexamination of rebound data from sites near the margins of

H¡llh¡ B¡y

the northern Late Pleistocene ice sheets and from LakeBonneville. Other studies (Passey 1981; Nakiboglu &Lambeck 1982, 1983) have supported this interpretation butin view of the anomalous tectonic setting of the LakeBonneville region it is debatable whether the upper mantlein this region is iepresentative of the mantle as a whole. Theability to resolve the detailed rheological structure fromobservations of rebound at ice-margin sites has previouslybeen questioned on the basis that a very accurate andhigh-resolution description of the evolution of the ice-load isrequired and this is generally not available (Nakada &Lambeck 1987, 1988a). Nevertheless, the question as towhether such a thin channel, low viscosity model isrepresentative of the mantle as a whole is important andwarrants further examination.

A four-layered mantle model is illustrated in the inset inFig. 41. The depth of the channel has been fixed by thebase of the lithosphere at 50 km and by the depth of 220 km,coresponding approximately to the base of the seismic lowvelocity layer observed beneath the oceans. Three differentvalues for the viscosity of this channel have been adopted(nw":L}re, 1020, 1021Pas-1). The predicted sea-lèvelvariations during the Late Holocene are generallyinsensitive to the introduction of the low-viscosity channel(Fig. Al), particularly at the continental margin sites where

, - | 00

Tlmc(x10OO yrtr!)BP

Figure À1. Relative sea-level curves for the ice models ARC 3 + ANT 3, lithospheric thickness fI = 50 km, lower mantle viscosity 1022 Pa s-1,uppermantleviscosityof 1021 Pas-1 andalowviscositychannelof (1)1olePas-1,(2) L0'?0Pas-t,(3) ld1 Pas-1.

the effective wavelength of the water load is of the order ofthe scale of the continent. Only at sites such as Port Piriewhere the length-scale of the Late Holocene meltwater loadis characterized by the dimensions of the gulf, does theresponse become dependent on the detailed viscositystructure by amounts of about 10-15 per cent for atwo-order-of-magnitude reduction in T,"..

At islands, the sensitivity of the crustal response to thepresence of a low viscosity channel is governed by the lengthscales of the island and the ocean basin. For very small

Late Pleßtocene and Holocene sea-leuel chønge 517

islands it is the latter that controls this response andsea-levels at these sites will be insensitive to the presence ofsuch a channel (Fig. A1). At large islands, such as Fiji orthe South Island of New Zealand, where the island size issimilar to the depth to the base of the channel, the seaJevelresponse is more sensitive to such a channel, provided thatthe viscosity reduction is by more than an order ofmagnitude and provided that the lithospheric thickness isknown.