Structure of the Earth from Glacio-Isostatic Rebound · 2007. 2. 22. · British Isles lies at...

Copyright 1973. All ri.qhts reserved

STRUCTURE OF THE EARTH FROMGLACIO-ISOSTATIC REBOUND’Richard I. WalcottEarth Physics Branch, Department of Energy, Mines and Resources, Ottawa, Canada

z lO001

INTRODUCTION

The deglaciated regions of Fennoscandia and North America have, since the dis-appearance of the ice sheet, experienced broad, domal uplifts of the earth’s surfacewhich are shown in the raised and tilted strandlines of the sea and lakes. Becausethey closely coincide in shape and position with the deglaciated area, these upliftsarc believed to be the result of the relaxation of the earth to the removal of the iceload. Their study will therefore given information on the dccp structure andtheology of the earth.

Apart from horizontal scale, in which they differ by a factor of about 2, theprincipal uplift of North America (the Laurentidc uplift) and the Fcnnoscandianuplift have many points of similarity. The central region of each is occupied by ashallow sea and surrounded by a Precambrian shield. In both regions the shieldmerges in one direction with a continental platform and in the other with heavilyglacicrized mountains that border the adjacent ocean. Both have a broadlyelliptical shape with an cllipticity of about 0.8, but thc major axis of the Laurcntidcuplift trends northwest, and that of the Fcnnoscandian uplift, northeast. To thenorth of each is a smaller region of rebound. In North America this is theInnuitian uplift (Blake 6, Walcott 65), centered over the Queen Elizabeth Islandsand contiguous with the Laurcntide to the south; in Fennoscandia, the island ofSpitzbergcn, once covered by ice, has bccn uplifted and the area of rebound mayextend southward over the area of present day shallow seas (the northern part of theBritish Islands is also dcglaciatcd and has undergone glacio-isostatic rebound). Thesimilarities may be understood as the result of the growth of ice caps in regions ofsimilar geology, physiography, history, and latitude. Because of these geologicalsimilarities we can infer that beneath both regions earth structure and rhcoldgy arealso closely similar. Comparativ~ studies of the glacio-isostatic rebound of NorthAmerica and Fennoscandia have a special geophysical significance, therefore, as wecan reasonably expect any differences in response to be directly attributable todifferences in scale.

Until recently the North American uplifts were inadequately known and theFcnnoscandian data have been the main body of information in this field of study,

1 Contribution of the Earth Physics Branch No. 426.

15

www.annualreviews.org/aronlineAnnual Reviews

Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.

http://www.annualreviews.org/aronline

16 WALCOTT

together with phenomena of different wavelength such as the net rotational accelera-tion of the earth (O’Connell 48) or the nonhydrostatic bulge (McKcnzic 38, Wang61 ; cf Goldreich & Toomre 19) which, however, are rather dubiously ascribed torelaxation processes. Another important source of relaxation data is the reboundof Lake Bonneville (Crittenden 11, 12), but this occurred in the highly anomalousregion of the Basin and Range province of the United States and there is everyreason to believe that the structure of the earth there is not comparable with thatof the shields.

The question of the relevance or compatibility of data from different sources is animportant one and is most difficult to answer. The general approach of manystudies has been to assume that the data are relevant and compatible, and todedfice the earth structure consistent with this assumption. As might be expected,very different models for the earth have been obtained, ranging from a uniform lowviscosity mantle of 102z P (O’Connell 48), to a very high viscosity mantle of 1027 P,

with a low viscosity channel in the upper few hundred kilometers (Takeuchi

60°

40°

t40° t00° 60°

/

subsidence(-13 )~l..~e~J / It [#.90-"’~.~.7/’-40 so°

90° 70° I0o 30°

Figure 1 Laurentide, Fennoscandian, and lnnuitian regions of glacio-isostatic rebounddrawn to the same scale. The contours of the North American regions are drawn on the6000 years B.P. sea level surface. The contours:of the Fennoscandian region are drawn onthe highest Littorina surface (ca 7000 years B.P.). Note the subsidence on the eastern UnitedStates and Northern European coasts.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


, GLACIO-ISOSTA~TIC REBOUND 17

Hasegawa 57, McKenzie 38, 39). If the objective is to deduce earth structure thenthis approach is clearly not very productive, and until such time as other evidenceis available to evaluate whether a particular phenomenon is indeed a relaxation effectlittle substantive progress will be made. By confining the study to glacio-isostaticrebound we can overcome this principal source of uncertainty to a very large extent.

OBSERVATIONS OF GLACIO-ISOSTATIC REBOUND

The Laurentide, Fennoscandian, and Innuitian areas of glacio-isostatic rebound areshown in Figure 1. The two North American examples are defined by contoursdrawn on the sea level surface of 6000 years B.P. (before present), based on datasummarized and discussed by Walcott (65). The accuracy of drawing the contoursis about _+~ contour interval when they are drawn as continuous lines. TheFennoscandian uplift is defined by contours drawn on the highest Littorina surface,from Sauramo (52) and Hyypp~i (27); the map shows the elevation of the level surface of about 7000 years B.P. (Hyypp~ 27) at the present time. The changein sea level in the central regions of each uplift as a function of time is given inFigure 2: the curves for the Laurentide uplift are based on data given by Webberet al (67) at a distance of about 400 km from the center of uplift at Richmond Gulf;the curve for the Innuitian uplift is largely based on data given by Mtiller (42)--the data of both areas are summarized and curves redrawn in walcott (65); the

140

1~0

IO0

4O

~0

- POSTGLACIAL UPLIFT ~

_-

t000 3000 5000 7000TIME {YEARS BP)

Fi~qure 2 Postglacial uplift in the center of the Laurentide, Fennoscandian, and Innuitianuplifts. The shaded envelope of the Laurentide and lnnuitian curves shows the confidencelimits of the data. The Laurentide curve is based on data from Cape Henrietta Maria about400 km from the center of uplift at Richmond Gulf.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


18 WALCOTT

data for Fennoscandia are given by Liden (33). The North American sea level curves

are based on a few radiocarbf~,.~_dated materials believed to have been depositednear the sea level of their time, and there are uncertainties in the measurements ofboth elevation and age. These uncertainties together with the sparseness of datacontribute to an overall imprecision in the curve. The error bounds are indicated,but the preferred curve, drawn as a heavy solid line, is the basis for later analysis.No information is available for the errors involved in the Fennoscandian data, andthe curve given by Liden (33) is the best available information at the present time.

Eustatic Changes in Sea Level

Sea level was depressed about lO0 m (Shepard & Curray 54) about 18,000 years B.P.,and with the melting of the i~e sheets rose to near its present position at about 6500years B.P. Since that time there have been no major changes in ice accumulation andthe sea level has remained within a few meters of its present level. Eustatic changesin sea level in postglacial time (6500 to 0 years B.P.) have been suggested, someinvolving a steady rise of about 3 m (Scholl, Craighead & Stuiver 53) and others irregular fall of about 3 m from a high point at about 5000 years B.P. (Fairbridge 14).The problem of determining eustatic changes is a very difficult one as the elastic andrelaxation responses of the earth to the changing loads of late glacial time (18,000 to6500 years B.P.) can be expected to result in differential vertical movements ofsignificant amplitude and to persist in postglacial time by continued relaxationeffects (Walcott 64). For comparisons of the isostatic response of the North Americanand Fennoscandian uplifts the precise eustatic changes are not, however, of crucialimportance, and the assumption that sea level has remained at the present level for thelast 6500 years is of sufficient precision. Accordingly, the curves of Figure 2 and mapsof Figure t are considered to represent actual movements of the land relative to thegeoid.

Peripheral Zones of Subsidence

In both Europe and North America, zones of subsidence peripheral to the region ofuplift are indicated by submerged terrestrial deposits. A maximum submergence of13 m in the last 6000 years is indicated at Brigantine NJ (Stuiver & Daddario 55) withsmaller amounts both to the north and south along the Atlantic seaboard. In Europe,Jelgersma (29) has shown that at 7000 years B.P. sea level was 10 m lower than presentin the vicinity of Zuid and Nordholland, corresponding to the time of the Littorinasea of Fennoscandia. Submerged peat 7340 years old on the Norfolk coast of theBritish Isles lies at about the same depth as material of similar age in the Netherlands,but in Somerset, further to the west, the submergence is less than that of the Nether-lands (Godwin, Suggate & Willis 18 ; cf Jelgersma 29), suggesting that the axis maximum submergence lies near the east coat of the British Isles and crosses thenorth European coast near the mouth of the Rhine.

Although evidence is sparse and inconclusive, the zones of submergence clearlydisplayed only in coastal sections of North America and Europe presumably extendentirely around the periphery of the uplifted area. Trowbridge (58) postulated subsidence of about 60 m in the central United States since the Wisconsin maximum,


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


GLACIO-ISOSTATIC REBOUND 19

and apparent changes in river gradient there are consistent with a regional tilting ofthe direction and magnitude to be expected for such a zone (Frye 15, Frye & Leonard16, McGinnis 37). Recent vertical movements from tidal trends on the Atlantic coastalmargin of North America show a high rate of subsidence near Chesapeake Bay,in the vicinity of the postglacial axis of maximum submergence, and lower rates to ttienorth and south (Hicks & Shofnos 25). The axis also appears in the releveling dataof the eastern United States (Meade 40, Walcott 65).

Thus the submergence displayed in the coastal section is believed to surround theregions of glacio-isostatic rebound with similar amplitude and wavelength and isas much a part of the pattern of glacio-isostatic rebound as the uplift itself.

Comparative Dimensions

The size, amount of uplift and subsidence, the gravity anomalies, and the presentrate of vertical movement in the center of the three areas of rebound are given inTable 1. The length and width are measured between the zero contours, and theradius is the distance from the center of the uplift to the axis of maximum sub-mergence (Figure 1). The position of the zero contour on the south side of the

Table 1 Comparison of rebound parameters

Parameter Laurentide Fennoscandia lnnuitian

Dimensionslength (km) 4000 2250 1100width (km) 2500 1350 900radius (km) 2200 1250 --

Maximum uplift (m)since 6000 B.P. 115a 86 45since 4000 B.P. 69 49 24since 2000 B.P. 34 21 10

Maximum subsidence (m)since 6000 B.P. 13 10

Mean free air gravityanomaly (mgal)

entire uplift -9 -- 1.6central area -- 35 -- 16

Present rate of upliftat center (cm/year) 1.7_+.5a .9b

~ At Cape Henrietta Maria, the uplift at Richmond Gulf since 6000 years B.P. is 138 m(Haselton 21), and rate of uplift is 2.0 + .5 cm/year (Walcott 65), but no data available therefor 4000 and 2000 years B.P.

b From K~i~iri~inen (30) and Lennon (32).


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


20 WALCOTT

Innuitian uplift and the presumed submergence associated with that uplift are notobservable; and hence, the cited width is uncertain and the radius unavailable.Otherwise, due to uncertainties in position of the contours, the accuracy of thevalues is about 10~.

Gravity Anomalies

Vening Meinesz (60) drew attention to the systematic decrease in free air gravityanomaly westward across Finland toward the center of uplift and suggested that theanomalies provided an estimate of the amount of remaining uplift. He estimatedthe anomaly due to undercompensation to be - 25 mgal, and, assuming the densityof the inflowing material to be 3.3 gcm 3, he estimated the remaining uplift to be180 m. However, this systematic decrease in anomaly across Finland is rather specialand is not as obvious on the Fennoscandian free air anomaly map as a whole(Honkasalo 26): indeed in Sweden positive anomalies occur very near to the center uplift. Nevertheless the mean free air anomaly in the center of uplift, defined by the80 m contour (Figure 1), is -16 mgal, whereas the mean anomaly over all o.fFennoscandia is - 1 mgal, indicating a systematic decrease toward the center. Kaula(31) has observed an isostatic anomaly over Fennoscandia of - 17 mgal obtainedfrom the spherical harmonic coefficients of the gravitational field of degrees 2 through16 given by Gaposchkin & Lambeck (17). Thus the anomaly related to the region the Fennoscandian uplift is probably about - (17 + 5) mgal.

In North America the free air anomalies associated with the deglaciated areaare much better defined. A large low corresponds in shape and position with thedeglaciated area and the area of uplift, and a zone of positive free air anomaliesoccurs around most of the periphery of the uplifted area (Walcott 65). The meanfree air anomaly in the center, defined by the 100 m contour of Figure 1, is - 35 mgal,a value that agrees very well with the -38 and -32 mgal free air and isostaticanomalies given by Kaula (31). The low is centered over the Hudson Bay sedimentarybasin, and a characteristic feature of the gravity field over similar basins, such as theMichigan and Williston basins, is a gravity high of between 10 and 20 mgal. Theuncompensated low in the Laurentide area may therefore be as much as -45 mgal.

Since there are examples of regions of positive anomaly that are rising and negativeanomaly that are sinking it is not certain that the negative anomaly does, in fact,indicate residual vertical movements. Nevertheless, as Kaula (31) has noted, negativefree air or isostatic anomalies do appear to be commonly associated with areas ofrebound, and if the models of rebound are correct, an anomaly due to the residualvertical movements should be expected. Thus, although the significance of theanomalies is open to question, they suggest that about 300 and 120 m of verticalmovement remain before equilibrium will be restored in the Laurentide andFennoscandian areas respectively.

Mode of Response of the Earth

There are a variety of concepts as to how the earth responds to glacio-isostaticrebound, and to some extent these influence the interpretation of field observationsand the drawing of shorelines and uplift curves used in the physical analyses. It is


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


GLAC10-ISOSTATIC REBOUND 21

~mportant to recogn!ze that we do not directly observe shorelines as continuousand easily identifia~ features. They are, rather, highly interpretative features con-structed.by corre.l~ting~sometimes indistinct strandline features from place to place,which involves consi~6,rable uncertainty that is seldom explicitly given or perhapseven recognized.

Three principal styles of deformation of the earth in rebound have been proposed :(a) There exist smooth, continuously, and slowly varying movements such as mightbe expected for the relaxation of an irregularity on the surface of a highly viscous fluid,a model which, because of its physical simplicity, is used in most physical analyses ;the other styles are more complex. (b) Vertical movements occur by tilting about series of horizontal hinges in the crust rather like the closing of parallel doors, anidea which results in the interpretation of shoreline profiles as involving successivestraight line segments with an angular discontinuity at the proposed hinges. (c) Move-ments occur in a mosaic of blocks with discontinuities both geographically andchronologically. The third idea results in straight line segments separated by steps inshoreline profiles and uplift curves that show spasmodic jumps and periods of nomovement. When we consider the proposal that sea level, the surface to which mostobservations are related, involved large and irregularly periodic oscillations, thedegree of complexity that could be anticipated from the last two hypotheses isappalling. Fortunately in those restricted regions where we have high quality, closelyspaced data this complexity is not evident in postglacial time.

Blake (6, 7) has given the ages and elevations of a large number of wood frag-ments buried in beach gravels that extend from 3200 to 6500 years B.P. in ageand from 11 to 40 m in elevation on southern Ellesmere Island in the CanadianArctic. The accuracy of age and relative elevation measurement is about + 140 yearsand 1 m respectively. In an elevation against age diagram all lie on a smooth curvewithin the errors of measurement and are sufficiently closely spaced that we can saythat no relative oscillations of sea level and land with an amplitude greater than 1 m,and wavelength longer than 500 years are apparent in the data, although smalleramplitudes or short period variations are still possible. It is this evidence that allowsus to draw smooth curves through discrete data points-where the data are sparser.

Also~ carefully measured elevations to the Algonquin strandline (12,000 years B.P.)and the Nipissing strandline (6000 years B.P.), both marked by particularly stronglydeveloped shoreline featurcs in the Great Lakes region of North America, showdepartures from a smooth surface by no more than 2 or 3 m, a value comparablewith the accuracy of identification of the strandline feature and the measurement ofits elevation. Furthermore, from high precision leveling (K~i~iri~inen 30) the mostconspicuous feature of the map of present day movements in Finland is its smooth-ness and regularity, particularly where, on the southern coast, the network is closelyspaced. Thus, although irregularities are present, they are small; block movements,if they occur, are subordinate in magnitude to the general trend.

It has been proposed that there are hinges in the Great Lakes, particularly forAlgonquin and earlier strandlines, but the term is used loosely and in many casesappears to mean no more than a change in gradient. The Algonquin hinge, probablythe best known and defined, occurs where that strandline merges with other nearly


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


’)2 WALCOTT

horizontal strandlines and where there is considerable difficulty in correlating thestrandline remnants, tndeed, the correlation is largely done on the assumption ofhorizontality. Alternative correlations are possible, including that of a smoothlydamped harmonic flexure such as suggested by the physics.of the bending of anelastic, lithospheric plate (Walcott 62).

The model dependence of shoreline profiles is most clearly shown by the contrast-ing interpretations of Sauramo (51, 52) and Hyypp~i (27) for the observations of Fennoscandian uplift in Finland. Sauramo postulated the existence of at least twomajor hinges on his profile and indicated considerable short wavelength irregularitiesin the shorelines. Hyypp~, using essentially the same data, fitted straight lines of tiltdecreasing with time to all observations. The short wavelength irregularities notedby Sauramo are an important part of the physical analysis of the data by McConnell(36) and Artyushkov (2, 3, 4) and the existence of the hinges is important in Lliboutry’s(34) study but it is clear from Hyypp~f’s work tha,t although they may be real, thesefeatures need not exist.

Late Glacial lsostatic Response

The vertical movements that occurred during the retreat of the ice sheets in lateglacial time (in North America, 18,000 to 6500 years B.P. ; in Fennoscandia, 18,000to 8500 years B.P.) are very much more complex than during postglacial time.This complexity is shown by very steep tilts of the strandlines of proglacial lakes inthe Great Lakes area of North America and very rapid rates of uplift and apparentsmoothing of short wavelength depressions in Vennoscandia. Such complexities areto be expected because of the major changes in surface load at that time causedby the melting of the ice and the resultant rise of sea level. One particular deforma-tional effect of considerable significance was a downward flexure of the earth’s surfaceadjacent to the ice edge. In North America this depression had a width of 300-400 kmand an amplitude of about 150m, and followed the retreat of the ice edge. The depres-sion is interpreted as the near-equilibrium flexure of the lithosphere by the ice load(Walcott 62, 64). Because of the flexure, the rate of uplift at any one point near theice edge will depend to a great extent on the rate of retreat of the ice, and a readvanceof the ice may even cause a reversal in vertical movement. The center of theFennoscandia uplift became ice-free before ~he ice had completely disappeared fromthe land to the west in Sweden and Norway (Hyypp~/27), and the ice load therecould well have had a significant effect on the rate of rebound in the center.

The detailed and well-controlled work of MOrner (41) in the Kattegat area of south-western Fennoscandia also shows very clearly the complexities of late glacial verticalmovements relative to sea level. There were large variations in the rate and even in thedirection of relative movement which arose partly from irregularities in the movementof the land and partly from irregularities in movements of sea level. From about7000 B.P. to the present, however, the variations were small and the uplift occurredas a smooth tilting throughout the region. For these reasons the interpretation oflate glacial data as a simple relaxation effect is erroneous and, to compare therelaxation of North American and Fennoscandian uplifts, only postglacial data areconsidered in this paper.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.



EARTH MODELS

The Fennoscandian shoreline profiles and the central uplift curve have been variouslyinterpreted as indicating a uniform mantle viscosity (Haskell 24), a low viscosityasthenospherc 100 km thick overlying a rigid or very high viscosity mesosphere(Van Bemmelen & Berlage 59), a mantle viscosity that decreases with depth by order of magnitude (Vening Meinesz 60), and a mantle viscosity that increases withdepth by an order of magnitude (McConnell 36). Evidently, inferences of earthstructure and theology are no less uncertain than the interpretation of the shorelineprofiles themselves.

The reason for the discrepancy is largely that the data are inadequate to resolvedifferent structural models. The early work, prior to the papers of McConnell (35)and Takeuchi (56), was mainly concerned with developing approximate solutions forthe uplift based on viscous half-space or thin-channel models and by two-dimensionalor radially symmetric approximations to the elliptical uplift. The later work, that ofthe last decade, is related to general solutions of differential equations of a viscousfluid with the pattern of rebound approximated by a series of surface harmonics.The current trend in analytical research is to construct increasingly complicatedmathematical models with more appropriate rheological prbperties than purelylinear (Newtonian) viscosity. However, the early work is of particular interest andvalue in that it shows the development of some ideas concerning earth structureand gives an insight into the overall time-varying pattern of rebound not as im-mediately evident in the later work.

Thin-Channel Models

The earliest analysis, that of Van Bemmelen & Berlage (59), assumed that flow withinthe earth in response to surface loading was confined to a thin channel aboutI00 km thick at a depth of about 100 km below the earth’s surface, a model con-sistent with the idea of the asthenosphere first given by Barrell (5). They showedthat the relaxation of an irregularity ( on the surface of a viscous fluid of viscosityq overlying a rigid substratum, where the linear dimensions of the irregularity werevery much greater than the thickness (h) of the fluid, was governed by the differentialequation

Ot - L~ ~x~ I.

irregularity. ’More generally this can be expressedfor a two-dimensional(Artyushkov 3)

c~ = DV~( 2.

where V = c~2/~xz+ ~2/~y2 with z vertical.They approximated the Fennoscandian uplift, by a two-dimensional profile of the

initial form

( = ~0 exp (- f12x2)


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


24 WALCOTT

where ~o is the displacement in the center and fl a constant with dimensions L-1,and derived the uplift as a function of time

~ = ~0[1 + 4Dfl2t]- t/2 exp [- f12x2/(1 q- 4Dfl2t)]

Equation 2 is a diffusion equation and the diffusion constant D is (Lliboutry 34)

D = pgh3/12~l 5.

where p is the density and r/the viscosity of the fluid, and g is gravity.Using Nansen’s (44) data for the uplift center, data that were later updated

Liden (33), they obtained a viscosity for a 100 km thick asthenosphere of 1020 Lliboutry (34) later indicated an arithmetic error in their work and when correctedthe viscosity becomes 3 × 1019p. As Lliboutry points out, however, the uplift datacan give a value only for D, the diffusion constant; the viscosity is derived from Dby the assumption (or some independent evidence) of a thickness of the channel.Lliboutry obtained a value for D of 48-35 km2/year, the difference arising fromthe assumption of different amounts of the remaining uplift in Fennoscandia. Walcott(66) obtained a value of 52 km2/year for the Laurentide uplift. Considering theapproximate nature of the differential Equation 2 and the possibilities of variationin q and h the similarity of values for D in the two areas is surprising, althoughperhaps fortuitous.

Viscous Half-Space Models

Daly (13) believed that flow due to surface loads occurred at "great depth" in zone of thickness comparable or greater than the width of the Fennoscandian uplift.This view was based on the supposition that flow confined to a thin, shallowchannel would cause a peripheral upbulge beyond the area loaded by ice of amagnitude very much greater than observed. No calculations were made and, aswe shall see, the magnitude of the upbulge is not as large as he supposed. His viewnevertheless profoundly influenced most of the early work on glacio-isostaticrebound.

Haskell (22) derived formal solutions for the motion of a highly viscous fluid a half-space subjected to a radially symmetric load and later extended the solutionto two-dimensional loads (Haskell 23). From these solutions he derived the displace-ment in the center as a function of the velocity for a radially symmetric uplift wherethe velocity profile was approximated by

v = v0 exp ( - f12r2) 6.

a form similar to that proposed by Van Bemme~en & Berlage (59) for the displace-ment profile. His equation was

~ ~ 2,/~fl,~o/p,q 7.The viscosity obtained, using Nansen’s data and assuming, according to Gutenberg

(20), 20 m of remaining uplift, was 3 × 1021 P. Vening Meinesz (60), however, inferredthe remaining uplift from gravity data to be 180 m and obtained a viscosity on thesame uplift data of 3 x 1022 P, an order of magnitude larger.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.



Vening Meinesz (58) extended the half-space model by assuming that the viscositychanged with depth according to an exponential law and, again using the uplift in thecenter, found a decrease from 3 × 1023p at 150 km to 3 × 10zzP at the base of themantle. Gutenberg (20), however, states that the accumulation of assumptions castssome doubt on this result.

The relaxation of a two-dimensional depression with a profile of Equation 3 wasalso studied by Niskanen (45), who obtained a viscosity of 3 × zz P.He,however,made an analytical error, pointed out by Burgers & Collette (9), and the equationshe developed were the same as Equation 4--those appropriate to a thin-channelmodel rather than those for a viscous half-space. Interestingly, Niskanen (45, 46) wasable to show by a variety of considerations that Equation 4 was widely consistent withobservation, but rather than indicating the validity of the half-space model as bothhe and Gutenberg supposed, his work supported the Van Bemmelen & Berlage (59)model.

Burgers & Collette (9) gave a most extensive discussion of the analytical solutionsfor the differential equations of viscosity in terms of elliptical, radially symmetric,and two-dimensional uplifts as well as considering both a viscous half-space and aviscous slab of finite depth. For the relaxation of a two-dimensional depression in aviscous half-space with the initial form of Equation 3, they obta!ned the equation

((x, t) = X/~ ,)1 exp ( - t/2 -- "C/U) COS ~U du 8.

where u is an integration variable, ¢ = 2fix and z = 9pt/4qfl, with other variablesas earlier defined. They give a numerical solution for the integral in Equation 8 fordifferent values of dimensionless parameters ¢ and z. Their solution is used later inFigure 5.

In summary, the early work on the viscous half-space and thin-channel modelsshow that both can be fitted to the data of the uplift of the center of Fennoscandia,and that if earth structure can be approximated by a viscous half-space its viscosity isabout 3 × 102z P. If a thin channel only 100 km thick is the appropriate model, thenthe viscosity, from the Fennoscandian uplift, is 3 x 1019p.

General Models

The importance of scale for evaluating relaxation phenomena was explicitly statedby Takeuchi (56) and McConnell (35). Takeuchi, referring to Jeffreys’ equation flow in a thin channel (Jeffreys 28, p. 322), pointed out that the nonhydrostatic bulge,assumed from a suggestion of Munk & Macdonald (43) to correspond to an equi-librium bulge 10 m.y. ago, and the Fennoscandian rebound could, in principle, bereconciled by a thin-channel model because the relaxation time would be inverselyproportional to the square of the dimensions of the area of rebound. Later, Takeuchi& Hasegawa (57) calculated relaxation time (zr) as a function of the degree (n) surface harmonics

2ha

[n(n+ 1)]


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


26 WALCOTT

where 2 is the wavelength and a the radius of the earth. Taking as data the relaxationtimes of the nonhydrostatic bulge (n = 2, z~ ~ 2 × 106 years), Fennoscandia (n = zr ~ 5 x l03 years) and Lake Bonneville (n = 200, rr "-- 4 x l03 years) rebound, theyshowed that the data could be explained by flow restricted to a channel 200 to400 km thick.

McConnell (35) extended the theory of relaxation times to a general modelcomposed of stratified viscous and elastic materials in which the surface load wastransformed into a series of surface harmonics. The general equation was solvedfor a number of specific models to illustrate the effect of changing variables ofviscosity and thickness of the layers. To make the solutions general, McConnell gavediagrams in which dimensionless wavenumber (~ was plotted against dimensionlessrelaxation time (O. The wavenumber u(= 2~/),) is defined for a radially symmetricdeformationf(r) on a plane surface, by the Hankel transform

= f~ f(r)J,(ur)rdr 10.F .(u)

and for a two-dimensional deformationf(x) by the Fourier transform

F(u)=f~o~f(_~;)exp(iux)dx 11.

The wavenumber has the dimensions of L- 1, and by introducing a length h which is aparameter of layer thickness defined in the specific models it can be written indimensionless form as

~= uh 12.

The dimensionless relaxation time is written

f = pghr/2~l 13,

where z is the relaxation time for a specific model and wavenumber. Equations 12 and13 are sufficient to obtain the actual relaxation time as a function of wavenumber.

Solutions for four models are given in Figure 3. The introduction of a rigid, incom-pressible half-space underlying a viscous layer of thickness h (Figure 3a) shows thatfor a short wavelength (5 > 4) the deformation "sees" only the viscous material andfor longer wavelengths (~ < 2) the slope of the curve becomes negative and therelaxation time increases with wavelength. Where the rigid half-space is replaced bya viscous half-space (Figure 3b) with a viscosity greater than that of the overlyinglayer, the effect of the lower layer starts to appear for wavelengths greater thantwice the thickness of the low viscosity layer. The wavelength at which only thelower layer is effective is strongly dependent on the viscosity ratio of the two layers.Where the viscosity of the half-space is lower than the surface layer, the relaxationcurve is shifted to the right of the homogeneous half-space.

If an elastic layer is added to a viscous half-space (Figure 3c), there is a pro-nounced decrease in relaxation times at shorter wavelength due to the energy storedin the strained elastic plate. With increasingly shorter wavelengths, however, abalance is set up between the retarding effects of the viscous material and the


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.



accelerating elastic effects, and the relaxation curves become asymptotic to the valuepgh/l~, where p is the rigidity.

In his analysis of Fennoscandia, McConnell (36) derived a relaxation timespectrum by the decomposition of Sauramo’s (52) shoreline profiles and by buildingincreasingly complex models combining different features of Figures 3a, b, and c,he showed that an adequate fit to the curve required an elastic lithosphere,about 100 km thick, overlying a mantle whose viscosity increased in depth from3 x 1021P at 100 km to 6x 1022p at 800 km (Figure 3d). To obtain sufficiently

t0t

1o0

10-t

- "~ = (pgh/2"r/)’r

I00 t02

t°°L- ~~-~ (el ’a~~.: tO0"N

_’~

t6t _ ~6~

~dt0-~ t0o

I I Ilitiilt0t

5 / t66

I I I IIIIII103 tO"2 tO’t 100 40t

- wove number u,(cm-t) ELASTIC .A

,.~,H. :6.5 x10lu

~ Nm-~Kin_ ,I Degree

: .vvr \

~ eoo~ I,,

- ~2OO1~ Reloxotion time T

109 40t0 t0 tt (sees) t2

Figure 3 Relaxation time curves of a variety of theological structures (after McConnell 35).

(a) A viscous layer overlying a rigid half-space. (b) A viscous layer overlying a viscous half-

space. (c) An elastic layer overlying a viscous half-space. The dimensionless rigidity

~ = k~/p,qh. (d) McConnell’s preferred model for the relaxation spectrum of the Fenno-scandian shoreline profiles, The solid line is that calculated by McConnel[ (36) for

planar model and the dashed line is that calculated by O’Connell (48) allowing for bodyforces, spherical geometry, and density stratification.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


28 WALCOTT

generalized data, the decomposition of shorelines into harmonics required a numberof assumptions about the pattern of rebound. McConnell assumed (a) that deformation occurred beyond 800 km from the center of rebound, (b) that the defor-mation of the shorelines prior to 8500 years B.P. was a simple relaxation effect,and (c) that a radially symmetric pattern of uplift was a sufficient approximation the elliptical uplift. All three assumptions will introduce errors into the analysis butthese errors are probably only significant at the extremes of his spectrum. Thespectrum of the relaxation curve ranges from a half wavelength of 350 km to3000 kin. The short wavelength relaxation times are strongly dependent on theaccuracy of Sauramo’s shorelines, questioned by Hyyppff. Spurious relaxation timesare probably introduced by assumption (b) and by neglecting the deformationaleffects of the ice, since the changes in ice load at these times would introduce verticalmovements that would tend to increase the apparent rate of uplift. This is especiallysignificant because the short wavelength part of the spectrum is derived largely fromthe central depression, that part nearest the residual ice. Nevertheless, a decrease inrelaxation times with wavelength is also shown in the later, postglacial shorelines, sothis criticism does not substantially affect McConnell’s conclusions.

A more serious criticism is that McConnelFs preferred model (number 62-12)indicates, as he himself noted, a remaining uplift and present rate of uplift in thecenter of less than 20 m and .5 cm/year respectively. Assuming that the observed- 17 mgal gravity anomaly noted earlier is due to undercompensation, a remaininguplift of 120 m is indicated; McConnell’s model can therefore explain only 3 mgalof the anomaly. It is possible to argue that the negative anomaly is caused by anunrelated and coincidental effect and can be dismissed as irrelevant; however, thepresent rate of uplift at the center is well determined as .9 cm/year, almost twicethat predicted, and the rate decreases systematically toward the periphery of theuplift (K~giri~inen 30, Lennon 32). Thus neither the gravity anomaly nor the presentrate uplift are satisfactorily explained by McConnell’s model. Relaxation timescalculated from gravity and the recent rate of uplift data are systematically greaterthan those derived from the shorelines, suggesting that there is an unresolved errorin the transformation procedure for the long wavelengths of the shoreline spectra.Combining the relaxation curves for the short and middle parts of the shorelinesystem with those determined for the gravity and rate of uplift, the model requiresmuch higher viscosities below the asthenosphere than in the model derived fromshorelines alone.

In McConnell’s work the elasticity of the mantle was not considered. McKenzie(39) has pointed out that the effect of treating the mantle as a viscoelastic, Maxwellbody rather than a simple Newtonian fluid is to increase the relaxation times forany given viscosity distribution. This increase occurs because the force producingviscous flow depends on the surface deformation which is reduced by elastic uplift.McKenzie gives the relaxation time (~,) for a surface harmonic of degree

r, = [2(n + 1)2 + l[tl/npga + q/lt 14.

where q is the viscosity, p the density, ~ the mean rigidity, and a the radius-of theearth. The value of the mean rigidity will vary according to the order number from


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


GLACIO-ISOSTAT1C REBOUND 29

101° Nm- 2 for the upper mantle (an appropriate value for n = 20) to 1.5 × 101 ~ Nm- for n = 2. For n = 20 the two terms on the right of Equation 14 are approximatelyequal and all estimates of viscosity for rebound of this size will be double the correctvalues. For orders very much less or greater than 20 the first term on the right is largerthan the second and the elastic effect can be neglected. Thus, McConnell’s viscositiesshould be reduced by a factor of 2.

Anderson & O’Connell (1) derived a mean viscosity for the mantle of 1023 P fromestimates of the anelasticity of the lower mantle obtained from seismology. Theyalso gave relaxation times for a variety of viscous models with spherical geometry asa function of the degree of the surface harmonic; O’Connell (48) later extended thiswork to include body forces and a density stratification within the sphere. In Figure 4the relaxation curves for a viscous model involving a 300 km layer of viscosity3 × 102~p overlying a mantle of uniform viscosity of 1023 P are given for both theAnderson & O’Connell (1) and O’Connell (48) models; for the low orders, relaxation times of the density stratified model are greatly increased. In his calcula-tions O’Connell used the density stratification of the Bullen model A, but a densitystratified model is valid only if the density gradient within the earth is nonadiabatic(Cathles 10), and a great part of the density structure of the Bullen model A is, in fact,due to elastic compression alone. Therefore the more acceptable relaxation curve

- O’Connell ~

Anderson .~I ~ ~and O’Connell f ’ ~\

’~:tOa3p

,5 10 ZO ,50 100DEGREE

\,oo \

~:1°2~P (b)

10 ZO ‘50 t00DEGREE

Figure 4 Relaxation time curves of two models involving elastic and viscous spherical shellsoverlying a viscous core of(a) 1023 P and (b) 1026 P. The continuous curves are those calcu-lated by Anderson & O’Connell (1) and the dashed curve in a is calculated by O’Connell (48).The area marked L shows the ranges in degree (wavelength) and relaxation time of theLaurentide uplift relative to those of the Fennoscandian uplift (F), the relaxation time which is fixed at 8 x 103 years.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


30 WALCOTT

for the viscous model will lie somewhere between the homogeneous and the densitystratified models. By taking body forces, spherical geometry, and density stratifica-tion into account, O’Connell showed that for models of similar viscosity structure,relaxation times of the lower order harmonics are increased compared to MeConnell’splanar models. Hence to fit the Fennoseandian relaxation curve, a lower viscositythan that given by McConnell (Figure 3d) is required in the lower mantle. O’Connellused the nontidal acceleration of the earth to estimate the relaxation time of thesecond order harmonic and shows that if that acceleration is a relaxation pheno-menon, then the mean viscosity of the mantle is 6× 1021P. By introducing finestructure into the upper mantle the higher order part of McConnell’s relaxation curvecan be reconciled with lower order relaxation. However, such a low viscosity in themantle fits with neither the present gravity field nor, more significantly, the presentrate of vertical movement of either the Laurentide or Fennoscandian uplifts. If thesedata are to have any weight in the analysis then O’Connell’s model is incorrect andwe must conclude that the nontidal acceleration cannot, as he proposes, be explainedas a relaxation effect.

RHEOLOGICAL STRUCTURE FROM GLACIO-ISOSTATICREBOUND

Viscosity of the Asthenosphere

A feature common to all models is the dependence of the rate of decay of a depres-sion on its size, a relationship expressed analytically by the exponential decay ofharmonics, where the decay constant for each harmonic is dependent on its wave-length. Thus to compare the rebound of Fennoscandian and Laurentide uplifts werequire estimates of the relative decay constants and the relative wavelengths of thetwo regions. As it is the relative rather than absolute value that is important, theprocedure to obtain these values can be simple. The decay in the center of the originaldepression is, to a first approximation, exponential and the decay constant can beobtained from a variety of different observations. The assumption is that the decay isof the form

h+R = ~ = ~0exp(-t/~) 15.

an equation involving five parameters : 1. the amount of vertical movement over agiven time interval (hI -h2) or, equivalently, the rate of vertical movement at a giventime (d~/dt); 2. the remaining uplift at the present time (R); 3. time or interval oftime (t~ - t2) ; 4. the decay constant (v); or 5. the amplitude constant (~0). The observa-tions of uplift (Table 1) can be combined in a number of ways to solve Equation but the resulting decay constants, for a variety of different combinations of data, showconsiderable uniformity. Table 2 gives four independent estimates of the decayconstant for both t.aurentide and Fennoscandian uplifts, and the decay constant ofthe Laurentide uplift is consistently larger than that of Fennoscandia. If the decayconstant of Fennoscandia is taken as .8 x 10’~ years then the best estimate for theIaurentide decay.constant is (1.7 +0.5)× 4 years. The decay constant correspondsto the relaxation time of the dominant harmonics of the uplift. In Fennoscandia,


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


GLACIO-ISOSTATIC REBOUND3 1

McConnell (36) has shown that the peak of the harmonic spectrum corresponds a wavelength-o~f 1950 km or of a degree-20 surface harmonic. Although the rangein his spectrum covered degrees 6 to 60, most of the power and the accurately deter-mined part of the spectrum is in the range of deggees 15 to 24. Because of thesimilarity in form of the Laurentide uplift, its comparable degree range is estimatedby a scaling factor of 1.8 derived from the ratio of comparable lengths of the twouplifts given in Table 1. Its degree range is therefore 8 to 13.

In Figures 4a and b the estimated decay constants of the two uplifts are plottedas the relaxation times of the harmonics in the estimated ranges. The model consistsof a 100 km thick elastic sheet (the lithosphere) overlying a thin channel of thicknessh and viscosity 3 × 1021 P (the asthenosphere), and in turn overlying a viscous half-space of viscosity 1023 P (Figure 4a) and 1026 P (Figure 4b). The Laurentide Fennoscandian data are consistent with either model : in Figure 4a they require theasthenosphere to be 300-400 km thick and in Figure 4b, 400-500 km thick. Althoughthe models do serve to indicate the probable viscosity structure within the earth,neither model is uniquely determined by the data. Because of the large increase inrelaxation time with increase in size, the thin-channel rather than half-space modelis the closest approximation to the earth for surface deformations in the range fromdegrees 8 to 25. Thus the existence of an asthenosphere of low viscosity is firmlyestablished but neither its viscosity nor its thickness are independently determined.As noted in the discussion of the thin-channel model, changes in the relaxation timeby changes in the viscosity of the channel can be offset by changes in its thickness.This is demonstrated by the diffusion constant (Equation 5) proportional to the cubeof the thickness of the channel and inversely proportional to the viscosity. Thus the500 km thick channel with a viscosity of 3 × 1021 P (the relaxation curve which passesthrough L and F in Model 4b), could be replaced by a 100 km thick channel witha viscosity of 4 × 1019 P without significantly affecting the relaxation curve.

The minimum viscosity of the lower mantle is 8 × 1022 P, given by the viscosity ofthat half-space, or uniform sphere which has the relaxation time of the Laurentideuplift. The slope of the line between Laurentide and Fennoscandian data is somewhatsteeper than the relaxation curves of Figure 4a and somewhat less steep than those of

Table 2 Exponential decay consonants for Laurentide and Fennoscandia

UpliftDecay constant (years)

Laurentide Fennoscandia

1. 0-2000 and 0-6000 years B.P.2. 0-4000 and 0-6000 years B.P.3. Present rate of uplift and uplift since 6000 B.P.

4. Present rate of uplift and remaining uplift (gravity)

1.2 x 104a .83 × 1041.8 × 10’*" .77 × 104

2.2 x 104"2.0 × 104b .78 × 1041.5×10’~b 1.1 ×104

Cape Henrietta Maria.Richmond Gulf.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


32 WALCOTT

Figure 4b. This suggests that the most likely viscosity for the subasthenospheremantle is between limits of 1023 and 1026 P.

Another analysis of both Laurentide and Fennoscandian rebound in terms ofearth structure is that of Cathles (10). He has extended O’Connell’s models to involvea linear visco-elastic mantle in which the elastic parameters are given by the Haddon& Bullen HB model earth. His preferred viscosity structure is 0-75 km deep into themantle, q = 4 x 1022 P; 75-1000 km, r/= 1022 P; 1000 km-core, r/= 8 x 1021 P; thisis a structure very similar to O’Connell’s in that it.possesses a relatively uniform,low viscosity mantle. However, many of the data and assumptions related to theLaurentide rebound differ from those obtained by other detailed studies. Cathlesassumes isostatic equilibrium and an ice thickness of 2.5 km at the last Wisconsinmaximum although isostatic equilibrium was probably not achieved and the moreprobable thickness of the ice is 3.6 km (Paterson 49). He dates the onset of deglacia-tion as 14,000 years B.P. rather than 18,000 years B.P. (Presto50). The pattern ice retreat is presumed to be that of Bryson et al (8) which shows significant Lauren-tide ice at 6500 years B.P., whereas the more recent study of Prest (50) indicates thatthe ice had disappeared at that time. It is difficult to estimate what effect changes inthese parameters would have on the viscosity structure, but it is probably small asthe changes tend to cancel each other out. Of considerable significance, however,are the values taken for the amount of postglacial uplift and the present rate ofvertical movement in the center of the Laurentide uplift. Cathles determines theamount of uplift since 6000 years B.P. to be 100 m rather than 138 m and thepresent rate of uplift to be .85 cm/year rather than the probable value of 2.0 cm/year.It is this very low rate of uplift that leads to low viscosities in his model since itrequires the rate of vertical movement to have decreased during the last 6000 yearsby a factor of 10, implying an exponential decay constant of about 2500 years, almostan order of magnitude smaller than the value determined above and smaller than thatof Fennoscandia. Cathles therefore obtains a relaxation time that apparentlydecreases with increase in size, the opposite of that obtained here.

The changes in the profile of rebound of a depression with time differ for differentviscosity structures and these differences may therefore be of use in evaluatingstructure. The calculated profiles of relaxation of two-dimensional or circular depres-sions are shown in Figure 5 for both half-space and thin-channel models. The verticalaxis shows vertical movement and the horizontal axis shows distance from center touplift, both in arbitrary units. Four curves are drawn in each diagram correspondingto the present horizontal surface (to) and to deformed horizontal surfaces of increas-ing age (t 1 to t3). Generally the shape of the thin-channel or half-space modelsare similar but they differ in the relative amplitude of subsidence in the outer zone touplift in the center. The relative amplitude varies from 18 to 31°/~, in two-dimensionalprofiles and from 8 (not shown) to 147, for models with circular symmetry. TheLaurentide and Fennoscandian rebound is most appropriately modelled by circularsymmetry along the line of section that includes the observed subsidence. Assumingthere has been no eustatic change in sea level, the relative values for Laurentideand Fennoscandian are 10 and 9~ respectively. If at 6000 years B.P. the possiblerange in eustatic sea level is _+ 5 m about present sea level, then the possible range in


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.



~ HALF-SPACEtj Two dimensionol

l~ ~ 18 %

~t~ Circular

----t,

DISTANCE FROM CENTER

Figure 5 Relaxation of viscous half-space and thin-channel models. The profiles show thevertical displacement with distance from the center of originally horizontal surfaces ofincreasing age for two-dimensional and radially symmetric uplifts. The units on vertical andhorizontal axes are arbitrary. Note the profiles migrate inwards in the half-space and outwardin the thin-channel models. Also note differences in amplitude of the maximum subsidencerelative to the maximum uplift expressed as a percentage.

relative amplitude is (10_+4)’~,,. Thus, with present uncertainties in the changes eustatic sea level, the best that can be said is that observations of subsidenceare consistent with either the thin-channel or half-space model.

Flexural Rigidity of the Lithosphere

To explain the apparently rapid decay of the short wavelength part of his spectrum,McConnell (36) introduced into his model an elastic lithosphere whose effect wasto, reduce the relaxation times for those wavelengths near the fundamental wave-length of flexure of the lithosphere. The fundamental wavelength is that of the dampedharmonic wave that will form beyond the edge of a surface load and is 2rt~ where~4 = 4D/(p,,,- P,.)9 (Walcott 63). The term ~ is the flexural parameter and is a functionof the flexural rigidity D [ ~ Eh3/(l - a2), where a is Poisson’s ratio] and the differencebetween the density of the displaced material below the lithosphere (Pro, the densityof the asthenosphere) and the density of material above the lithosphere (Pc) possiblywater, sediment, air, etc. For a 100 km thick lithosphere with a Young’s modulus(E) of 6.5 × ~° Nm2 andPoisson’s rati o of 0.25the valueof ~ i n air is 160 kmand the wavelength is about 1000 km. Referring to Figure 3c, the appropriate valueof/7 is 20 and t7 is 1.25, and the relaxation time is reduced by the introduction ofan elastic lithosphere from 1.25 to 025, a factor of 5.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


34 WALCOTT

A relaxation time of a few thousand years is therefore likely for the fundamentalwavelength. As the average rate of ice retreat in North America from about 14,000to 7000 B.P. was 1000 km in 5000 years, the marginal flexure due to bending of thelithosphere was never in complete equilibrium, although it was probably very closeto an equilibrium flexure in shape over all except the most rapid phases of retreat.Hence the distance between ice edge and crest of the marginal flexure will be a goodestimate of the flexural parameter and therefore of the flexural rigidity as well.Estimates of the flexural rigidities from the warped strandlines of the proglaciallakes of Algonquin and Agassiz lie between 6 and 9 x 1024 Nm (Walcott 63). For Young’s modulus of 6.5 x 101° Nm- 2 the thickness of the lithosphere is thus 110 km.If the departure from equilibrium flexure is large, this will be an overestimate andthe correct value of thickness will be smaller.

Apart from the marginal flexure and its effect on the late glacial behavior of therebound, the influence of the lithosphere on Fennoscandian and Laurentide upliftsis not very great as their wavelength is at least twice that of the fundamental wave-length. For the Innuitian uplift the effect, however, may be as large as the original iceload covering the Arctic Islands that had a width no greater than 600 km. If theflexural rigidity of the lithosphere is the same as in the Laurentide region then thewidth of the load is only one-half the fundamental wavelength. The effect is twofold.First, the amplitude of the isostatic depression beneath the center of the load willdiffer from that of local isostatic compensation; second, the width of the depressionwill be very much greater than the width of the load. This is demonstrated byJeffreys’ argument (28).

For harmonic loading hpg cos ~:x (h thickness, p density, 9 gravity, ~: wavenumber)on the surface of a thin elastic sheet of flexural rigidity D floating on a fluid of densityp,., the vertical displacement w is given by

hp9w = -- cos xx 16.

Dx’* + p,. 9

For long wavelengths D~:’~ ~ p,.9, the amplitude term in Equation 16 becomesequal to that for local coml~ensation hp/p,,. For short wavelengths D~’~ <~ p~,,q, theamplitude term decreases as the fourth power of the wavelength. The fundamentalwavelength is about 1200 km based on estimates of the flexural rigidity given abovefor the Laurentide area, and the amplitudes of isostatic depression arising from loadsof this wavelength are about one-half that of local compensation. For wavelengths of600 km the amplitude term is reduced by a factor of 17. Thus the lithosphere acts asa low pass filter in coupling the surface load to the depression 6n the surface ofthe asthenosphere, filtering out surface harmonics greater than about degree 35.Thus, in the Innuitian area, a substantial part of the ice load would have beensupported by the lithosphere which would have been bent beneath the load to adistance of about 300 km beyond the edge of the load.

CONCLUSIONS

The rheological structure of the earth consistent with Observations of glacio-isostatic


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


GLACIO-ISOSTATIC REBOUND35

rebound of the Laurentide, Fennoscandian, and Innuitian uplifts comprises: a 110 kmthick lithosphere which, although requiring a viscosity of 1025 P to explain somelong-term behavior (Walcott 63) behaves elastically in time scales of a few thousandyears; a thin, low viscosity channel solne 100-500 km thick with a viscosity dependent

,upon thickness of 1019 to 1021 P; and a lower mantle viscosity greater than 1023 P.The changes in rheological properties from one part to another are presumablygradational.

The relative changes of viscosity in this glacio-isostatic model are similar to thosededuced by Weertman (68) through studies of’the creep strength of the mantlefrom the standpoint of solid state physics. The actual values of effective viscosity(i.e. the viscosity defined for a specific strain rate) obtained by Weertman, however,are generally smaller, the mean viscosity of the lower mantle being about 4 × 1022 P.The value of effective viscosity is calculated from the ratio of melting temperature toambient temperature which is known only within wide limits. Thus the higherapparent viscosity deduced from glacio-isostatic rebound may indicate that themelting temperature increases faster with depth than assumed by Weertman.

Further improvements in the mathematical models of linear viscosity are unlikelyto change the present estimates of viscosity by more than a factor of 4 and, indeed, theestimate given here for the viscosity of the low viscosity channel does not differfrom that obtained by Van Bemmelen & Berlage (59) 37 years ago. Those areas whereconsiderable improvements in our knowledge may arise are in the structure of theasthenosphere and in the mechanism of strain within the earth. To this end studies ofrelaxation time relative to wavelength will continue to be important, and observationsof relaxation movements other than those of glacio-isostatic rebound are of greatinterest. As an example, a promising line of future geophysical research may involvethe relative changes in postglacial sea level in the ocean basins. Owing to the growthand decay of Pleistocene ice sheets, the ocean basins were subjected to periodicloading and unloading by about 100 m of water, and the last rise in eustatic sealevel ended about 6500 years ago. The possible isostatic movement in response tothis load is about 30 m. For the Pacific Ocean, an area of degree 3, the relaxationtimes from Figures 4a and b are about 104 and 105 years respectively. Accordingto these models, therefore, the present rate of vertical movement should be about.3 or .03 cm/year and the amount of vertical movement in the last 6500 years shouldbe about 15 or 2 m respectively. Thus the differences in isostatic behavior are largefor two models. Present data, however, are inadequate to choose between the two(Walcott 64), but with the further extension of radiocarbon dating to a worldwidestudy of sea levels the acquisition of suitable data can be anticipated.

Literature cited

1. Anderson, D. L., O’Connell, R. 1967.Viscosity of the earth. Geophys. J. Roy.Astron. Soc. 14 : 287-95

2. Artyushkov, Ye. V. 1966. On the isostaticequilibrium of the earth’s crust. Ann.Acad. Sci. Fenn. A III 90:455-66

3. Artyushkov, Ye. V. 1966. The nature of

viscosity variations of the upper mantlewith depth, lzv. Akad. Nauk SSSR Fiz.Zemli 8 : 484-91

4. Artyushkov, Ye. V. 1967. Establishmentof isostatic equilibrium of the earth’scrust. Izv, Akad. Nauk. SSSR Fiz. Zemli1:1,8


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


36 WALCOTT

5. Barrell, J. 1914. The strength of the earth’scrust; Part VI. Relations of isostaticmovements to a sphere of weakness--the Asthenosphere. J. Geol. 22 : 655-83

6. Blake, W. Jr. 1970. Studies of glacialhistory in Arctic Canada. I. Marineradiocarbon dates, and diflbrential post-glacial uplift in eastern Queen ElizabethIslands. Can. J. Earth Sci. 7 : 634-64

7. Blake, W. Jr. 1972. Climatic implicationsof radiocarbon-dated driftwood in theQueen Elizabeth Islands, Arctic Canada.Syrup. Clim. changes Arctic areas duringlast lO.O00 years. Onlanka-Kevo, Fin-land: Ibid A~ta Unit’. Ouluensis A. Geol..pp. 77-104

8. Bryson, R. A., Wendland, W. M., Ires,J. D., Andrews, J. T. 1969. Radiocarbonisochrones on the disintegration of theLaurentide Ice Sheet. Arctic Alp. Res.1 : 1-13

9. Burgers, J. M., Collette, B. J. 1968. Onthe problem of the postglacial uplift ofFennoscandia. Kon. NederL Akad.Wetensch. Proc. B 61 : 221-41

10. Cathles, L. M. lIl. 1971. The viscosity ofthe earth’s mantle. PhD dissertation.Princeton Univ., Princeton, NJ

11. Crittenden, M. D. Jr. 1963. Effectiveviscosity of the earth derived from iso-static loading of Pleistocene LakeBonneville. J. Geophys. Res. 68:5517-30

12. Crittenden, M. D. Jr. 1967. Viscosity andfinite strength of the mantle as deter-mined from water and ice loads. Geophys.J. Roy. Astron. Soc. 14:261-79

13. Daly, R. A. 1925. Pleistocene changes oflevel. Am. J. Sci. 5th Ser. 10:281-313

14. Fairbridge, R. W. 1961. Eustaticchanges in sea level. Phys. Chem. Earth4:99 185

15. Frye, J. C. 1963. Problems of interpretingthe bedrock surface of Illinois. Ill. Geol.Surv. Reprint Set. 11 pp,

16. Frye, J, C., Leonard, A. B. 1952. Pleisto-cene geology of Kansas. Kansas Geol.Surt’. Bull. 99. 230 pp.

17. Gaposchkin, E. M., Lambeck, K. 1971.Earth’s gravity field to the sixteenthdegree and station coordinates fromsatellite and terrestrial data. J. Geophys.Res. 76 : 4855-83

18. Godwin, H., Suggate, R. P., Willis, E. H.1958. Radiocarbon dating of the eustaticrise in ocean level. Nature 181 : 1518-19

19.Goldreich, P., Toomre, A, 1969. Someremarks on polar wandering. J. Geophys.Re~. 74 : 2555-67

20. Gutenberg, B. 1941. Changes in sea level,postglacial uplift and mobility of theearth’s interior. Bull. Geol. Soc. Am. 52:

721 7221. Haselton, G. M. Late Quaternary uplift

and glaciation in the Richmond Gulfarea, Quebec. Unpublished

22. Haskell, N. A. 1935. The motion of aviscous fluid under a surface load. Physics6: 265-69

23. Ibid 1936. 7:56 6124. Haskell, N. A. 1937. The viscosity of the

asthenosphere. Am. J. Sci. 38:22 2825. Hicks, S. D., Shofnos, W. 1965. Yearly

sea level variation for the United States.J. Hydraul. Div. Am. Soc. Cir. Eng.,91:23 33

26. Honkasalo, T. 1966. Gravity and landupheaval in Fennoscandia. Ann. Acad.Sci. Fenn. A III 90:139~.2

27. llyypp~i, E. 1966. The late-Quaternaryland uplift in the Baltic sphere and therelation diagram of the raised and tiltedshore levels. Ann. Acad. Sci. Fenn. A III90 : 153-68

28. Jeffreys, H. 1962. The Earth, its origin.history and physical constitution. Cam-bridge, England: Cambridge Univ.Press. 3rd ed.

29. Jelgersma, S. 1966. Sea level changesduring the last 10,000 years. Proc. Int.Syrup. Worm Clim., 8000-0 B.C. London:Roy. Meteorol. Soc. 229 pp.

30. Kg~riffinen, E. 1966. Land uplift in Fin-land as computed with the aid of preciselevellings. Ann. Acad. Sci. Fenn. A III90 : 187-90

31. Kaula, W. M. 1972. Global gravity andtectomcs. The Nature ¢~f the Solid Earth,ed. E. C. Robertson, 385-405

32. Lennon, G. 1966. An investigation ofsecular variations of sea level inEuropean waters. Ann. Acad. Sci. Fenn.A III 90:225 36

33. Liden, R. 1938. Den Senkvart~ra strand-f6rskjutningens f6rlopp och l~ronologiAngermanland. Geol. kb’ren. StockholmF6rh. 60:397 404

34. Lliboutry, L. A. 1971. Rheological pro-perties of the asthenosphere fromFennoscandian data. J. Geophys. Res.76:1433-46

35. McConnell, R. K. 1965. Isostatic adjust-ment in a layered earth. J. Geophys. Res.70:5171-88

36. McConnell, R. K. 1968. Viscosity of themantle from relaxation time spectra ofisostatic adjustment. J. Geophys. Res.73:7089-7105

37. McGinnis, L. D. 1968. Glacial crustalbending. Geol. Soc. Am. Bull. 79 : 769 76

38. McKenzie, D. P. 1966. The viscosity ofthe lower mantle. J. Geophys. Res. 71:3995-4010


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.



39. McKenzie, D. P. 1967. The viscosity ofthe mantle. Geophys. J. Roy. Astron. Soc.14 : 297-305

40. Meade, B. K. 1971. Report of the sub-commission on Recent Crustal Move-ments in North America. Recent CrustalMovements Symp. XV Gen. Assem. Int.Assoc. Geodesy, Moscow

41. Mt~rner, N.-A. 1969.The late Quaternaryhistory of the Kattegatt Sea arid theSwedish west coast. Sver. Geol. Unders.,C No. 640 (~lrsb. 63, No. 3.) 487 pp.

42. Mtiller, F. 1963. Radiocarbon dates andnotes on the climatic and morphologicalhistory. Axel Heiber9 Isl. Res. Rep.,Prelim. Rep. 1961 1962. McGill Univ.,169-72

43. Munk, W. H., Macdonald, G. J. F.1960. The Rotation of the Earth. Cam-bridge, England. Cambridge Univ. Press.323 pp.

44. Nansen, F. 1928. The earth’s crust, itssurface forms, and isostatic adjustment.Avh. Norske Vidensk. Akad. Oslo Mat.Naturvidensk. Kl, Vol. 12

45. Niskanen, E. 1939. On the upheaval ofland in Fennoscandia. Ann. Aead. Sci.Fenn. A 11I 10:3 30

46. Niskanen, E. 1943. On the deformationof the earth’s crust under the weight ofglacial ice-load and related phenomena.Ann. Acad. Sci. Fenn. A Ill, 7 : 1-59

47. Niskanen, E. 1948. On the viscosity of theearth’s interior and crust. Ann. Acad. Sci.Fenn. A 111, 15:1-22

48. O’Connell, R. J. 1971. Pleistoceneglacia-tion and the viscosity of the lower mantle.Geophys. J. Roy. Astron. Soc. 23:299-327

49. Paterson, W. S. B. 1972. Laurentide IceSheet: estimated volumes during late-Wisconsin. Rev. Geophys. Space Phys. Inpress

50. Prest, V. K. 1969. Retreat of Wisconsinand Recent Ice in North America. Geol.Surv. Can., Map 1257A

51. Sauramo, M. 1939. The mode of landupheaval in Fennoscandia in lateQuaternary time. Fennia 66: 2

52. Sauramo, M. 1958. Die Geschichte derOstsee. Ann. Acad. Sci. Fenn. A III,51 : 1-522

53. Scholl, D. W., Craighead, F. C. Sr.,Stuiver, M. 1969. Florida submergencecurve revised, its relation to coastal sedi-

mentation rates. Science 163 : 56~6454. Shepard, F. P., Curray, J, R. 1967.

Carbon-14 determination of sea levelchanges in stables areas. Pro(4r. Oceanogr.4:283-91

55. Stuiver, M., Daddario, J. L. 1963. Sub-mergence of the New Jersey Coast.Science 142:951

56. Takeuchi, H. 1963. Time scales of iso-static compensations. J. Geophys. Res.68 : 2357

57. Takeuchi, H., Hasegawa, Y. 1965. Vis-cosity distribution within the earth.Geophys. J. Roy. Astron. Soc. 9 : 503 8

58. Trowbridge, A. C. 1921. Erosional his-tory of the driftless area. Univ. Iowa Stud.Natur. Hist. 9:55-127

59. Van Bemmelen, R. W., Berlage, H. P.1935. Versuch einer mathematischenBehandlung geotektonischer Bewegungunter besonderer Berticksichtegung derUndationstheorie. Gerlands Beitr.Geophys. 43 : 19 55

60. Vening Meinesz, F. A. 1937. The deter-mination of the earth’s plasticity from thepostglacial uplift of Scandinavia; Iso-static adjustment. Proc. Kon. Ned. Akad.Wetenseh 40 : 654-62

61. Wang, C. Y. 1966. Earth’s zonal deforma-tions. J. Geophys. Res. 71 : 1713 20

62. Walcott, R. I. 1970. Isostatic response toloading of the crust in Canada. Can. d.Earth Sci. 7 : 716-27

63. Walcott, R. I. 1970. Flexural rigidity,thickness and viscosity of the lithosphere.J. Geophys. Res. 75 : 3941-54

64. Walcott, R. I. 1972. Past sea levels,eustasy and deformation of the earth.Quaternary Res. 2 : 1-14

65. Walcott, R. I. 1972. Late Quaternaryvertical movements in eastern NorthAmerica. Rev. Geophys. Space Phys. Inpress

66. Walcott, R. I. Inferences of the uppermantle rheology from Quaternarycrustal deformation. Syrup. RecentCrustal Movements, Moscow, In press

67. Webber, P. J., Richardson, S. W.,Andrews, J. T. 1970. Postglacial upliftand substrate age at Cape HenriettaMaria, southeastern Hudson Bay,Canada. Can. J. Earth Sci. 7:317-25

68. Weertman, J. 1970. The creep strengthof the earth’s mantle. Rev. Geophys.Space Phys. 8 : 145-68


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.


Ann

u. R

ev. E

arth

. Pla

net.

Sci.

1973

.1:1

5-37

. Dow

nloa

ded

from

arj

ourn

als.

annu

alre

view

s.or

gby

CA

LIF

OR

NIA

IN

STIT

UT

E O

F T

EC

HN

OL

OG

Y o

n 04

/14/

06. F

or p

erso

nal u

se o

nly.

Structure of the Earth from Glacio-Isostatic Rebound · 2007. 2. 22. · British Isles lies at...

Documents

Transcript of Structure of the Earth from Glacio-Isostatic Rebound · 2007. 2. 22. · British Isles lies at...