Geophys. J . Int. (1993) 115, 960-990

Glacial rebound of the British Isles-11. A high-resolution, high- precision model

Kurt Lambeck Research School of Earth Sciences, The Australian National University, Canberra ACT 0200, Australia

Accepted 1993 May 10. Received 1993 May 7; in original form 1992 September 11

S U M M A R Y Observations of ice movements across the British Isles and of sea-level changes around the shorelines during Late Devensian time (after about 25 000 yr BP) have been used to establish a high spatial and temporal resolution model for the rebound of Great Britain and associated sea-level change. The sea-level observations include sites within the margins of the former ice sheet as well as observations outside the glaciated regions such that it has been possible to separate unknown earth model parameters from some ice-sheet model parameters in the inversion of the glacio- hydro-isostatic equations. The mantle viscosity profile is approximated by a number of radially symmetric layers representing the lithosphere, the upper mantle as two layers from the base of the lithosphere to the phase transition boundary at 400 km, the transition zone down to 670 km depth, and the lower mantle. No evidence is found to support a strong layering in viscosity above 670 km other than the high-viscosity lithospheric layer. Models with a low-viscosity zone in the upper mantle or models with a marked higher viscosity in the transition zone are less satisfactory than models in which the viscosity is constant from the base of the lithosphere to the 670 km boundary. In contrast, a marked increase in viscosity is required across this latter boundary. The optimum effective parameters for the mantle beneath Great Britain are: a lithospheric thickness of about 65 km, a mantle viscosity above 670 km of about (4-5) lo2" Pa s, and a viscosity below 670 km greater than 4 x lo2' Pas .

Key words: British Isles, glacial rebound, mantle viscosity, sea-level.

1 INTRODUCTION

In an earlier paper (Lambeck 1993a, referred to hereafter as Part I) it was established that the complex pattern of sea-level indicators that formed around the British coastline during the past 15 000 years, provides an important data set for estimating the mantle response to changes in surface loading produced by the melting of the Late Devonsian ice sheet over the region, and by the concomitant rise in sea-level produced by the melting of the much more voluminous ice sheets of Fennoscandia and Laurentia. It was established that both the effective lithospheric thickness and a simple depth dependence of effective mantle viscosity can be inferred from the observational data. Furthermore it was shown that certain constraints on the areal extent and thickness of the ice sheet can also be inferred from the observations of past sea-levels. Simple models were developed that illustrated the principal characteristics and requirements for a model of the glacio-hydro-isostatic rebound that is consistent with the observational evidence.

The recognition that Scotland had been ice covered in the recent geological past goes back to A. Agassiz in 1840, and A. Geikie in 1865 produced a map of the ice-movement directions that is quite similar to the reconstruction by Sissons (1976) more than 100 years later (Price 1983). Despite the much greater information base now available, knowledge of the ice sheets at the time of, and subsequent to, the last glaciation remains limited. Denton & Hughes (1981, p. 311), for example, conclude in a more general context that 'our most important conclusion is that the distribution of Late Wisconsin-Weichselian ice sheets is not well known'. Sissons (1981) concludes that 'after 140 years of research it is surprising how little we really know about the extent of the last Scottish ice sheet and associated events'. With this state of affairs it is perhaps a bold or foolish action to propose any model for the last British ice sheet but this has been done here in the spirit of establishing, (1) what information is required for the glacial-rebound calculation, (2) what information about the ice sheet can be extracted from the observations of relative

960

Glacial rebound of the British Isles-I1 961

sea-level change, (3) what information can be extracted from these observations about the Earth’s response to surface loading, and (4) what information about coastal evolution can be extracted from these rebound models. The first two questions have been examined in Part I which established the need for a relatively high-resolution ice model in order to yield satisfactory predictions of the temporal and spatial changes in sea-level observed in the British Isles, particularly in Scotland.

In this paper a new detailed ice-sheet model for the Late Devensian ice sheet has been developed (Section 2) from the time of maximum glaciation to the end of the Loch Lomond Advance ice sheet soon after 10000yrsBP, together with an approximate model for the ice movements during Middle Devensian time after 50 000 yr BP. In Section 3 the observational evidence for past sea-levels in Scotland, England, Wales and the Channel coast of northern France has been reviewed, and a series of relative sea-level curves has been established for sites both within and without the former glaciated region. To address the third question raised above, this observational data base, together with the new ice-sheet model, provide the input for the inversion in Section 4 of the equations defining the glacio-hydro- isostatic sea-level change through time. Certain ice-model parameters are also determined in this inversion. In the inversion the earth model is defined by realistic depth- dependent elastic moduli and density as determined by seismological methods and by a depth-dependent radially symmetric linear viscosity structure for the lithosphere, upper mantle, transition zone and lower mantle. Section 5 discusses the resulting earth- and ice-model parameters. This section also includes some preliminary comparisons of the predicted sea-level changes since the last glacial maximum with the observed changes, but a more complete discussion of some of the geomorphological consequences will be given elsewhere.

2 A NEW ICE-SHEET MODEL

The predictions in Part I based on the ice model derived from the Boulton et al. (1977) and Andersen (1981) results indicated that major modifications to the ice sheet are required. Also, since these reconstructions new evidence of the ice movements across Great Britain and the surrounding shallow sea-floor has accrued and this has been used to develop a new model for the last maximum glaciation and subsequent retreat. The principal changes from the Andersen (1981) isochrone model and the Boulton et al. (1977) reconstruction concern the extent of the ice cover over the continental shelf and the North Sea, the initial rates of retreat of the ice and the ages attributed to some of the features defining the position of the retreating ice margin, and the initial thickness of the ice sheet at maximum glaciation.

Timing, areal extent and ice thickness of the Late Devensian maximum glaciation

The maximum extent of the ice sheet over the British Isles appears to have been reached between about 25000 and 20 000 yr BP (e.g. Huddart, Tooley & Carter 1977; Bowen & Sykes 1988; Eyles & McCabe 1989) although the precise

timing of this occurrence remains uncertain and may not have been synchronous along all fronts. Sissons (1981), for example, suggested that the ice sheet over northern Scotland was stationary or even retreating at a time when it was still advancing southwards over England. In East Yorkshire, the ice sheet may have continued to advance until about 18 000 yr BP, the Dimlington Advance, (Penny, Coope & Catt 1969; Rose 1985), whereas the advance from the Irish Sea across Cheshire and Shropshire probably occurred earlier, at perhaps 22 000 yr BP. A nominal date of 22000yrBP has been adopted here for the time of maximum glaciation of the entire British Isles region, with only minor changes in ice volume occurring up to about 18 000 r BP. Andersen’s (1981) estimate of the areal extent of the [ce sheet at this time does require some modification. Thus, the accumulating evidence that the North Sea was largely ice free during Late Devensian time (=26000- 10000yrBP) is accepted and the British ice sheet is assumed to have terminated at the Wee Bankie Moraine off the coast of eastern Scotland (Eden, Holmes & Fannin 1978; Sutherland 1984) (Fig. 1) and to have been stationary here between about 22000 and 18000yrBP. The Nor- wegian evidence also points to the Scandinavian ice sheet not having extended across the North Sea as far as Scotland and to the existence of an ice-free corridor between the two ice sheets during the last glacial maximum (e.g. Larsen & Sejrup 1990; Sejrup et al. 1987). The Shetland Islands appear to have been largely unaffected by the lastest Scottish and Scandinavian ice sheets and any evidence for glaciation and ice transport across them is interpreted as being the result of a local ice sheet or of an earlier glacial cycle (Sissons 1981; Sutherland 1991). In the proposed model (Fig. 2) the Orkney Islands and parts of Caithness and Buchan were ice free during the Late Devensian (Sutherland 1984; Hall & Connell 1991) and the nearby ice front is assumed to have been stationary between about 22000 and 18000yr BP (Figs 1 and 2a). The Outer Hebrides supported its own ice sheet (Peacock 1991) and the western limit does not extend as far across the shelf as it does in the models of Andersen (1981) or Boulton et al. (1977) in which the Outer Hebrides ice is an integral part of the main ice sheet and extends across the continental shelf as far as St Kilda (Sutherland, Ballantyne & Walker 1984). Instead the Outer Hebrides ice as assumed to have retreated to a local ice sheet and cleared the Minches after about 18 000 yr BP (Fig. 2a). Also, the extent of the ice into the Sea of the Hebrides and the Malin Sea is considered to have been the result mainly of outflows from the Little Minch and the Firth of Lorne (Davies, Dobson & Whiting 1984) and to have been less extensive than proposed in the Andersen or Boulton et al. models.

Furthermore, in this model the Scottish ice moved across the north Channel of the Irish Sea but did not penetrate far into Ireland during Late Devensian time (Stephens, Creighton & Hannon 1975; Hoare 1991). Instead, the ice flow was deflected southwards to form the Irish Sea Glacier which extended as far as the southern entrance to St George’s Channel (e.g. Garrard & Dobson 1974; Eyles & McCabe 1989), fed also from the east by an ice sheet centred over Wales and from the west by the Irish ice sheet. The ice front across southern Ireland has been taken to coincide with the South of Ireland Moraine and is given an

962 K . Lambeck

Figure 1. Location map of main geographic areas mentioned in text and the limit of the ice sheet at the last glacial maximum. The major terminal moraines are: (1) Wee Bankie Moraine; (2) South of Ireland Moraine; ( 3 ) Galtrim Moraine; (4) Drumlin Readvance Moraine and its extension into the Irish Sea.

age of 22000yr BP (Eyles & McCabe 1989). The initial northwards retreat of the ice across southern Ireland and the Irish Sea is assumed to have been rapid such that by 18000yr BP the ice margin stood north of the Galtrim Moraine and at the Isle of Man (Stephens & McCabe 1977) (Figs 1 and 2a), consistent with ice-free conditions having existed in northeastern Wales by this time (Bowen 1973). Along the western margin in Ireland the ice sheet -is assumed not to have extended much beyond the present shoreline such that the coastal fringe was ice free before

17000yr BP (McCabe, Haynes & MacMillan 1986; Bowen et al. 1986).

The southern limit of the ice sheet over Wales and England appears to be reasonably well constrained and the limits given by Andersen (1981) have been largely adopted here, with ice-free conditions existing locally in both Pembroke and Glamorgan (e.g. radiocarbon dates BM 374 and Birrn-340 of 18 460 and 22 350 yr BP respectively). The ice cover over Wales and dominated by a local ice sheet that formed over the high ground and coalesced to the west and

T = 22000 aBP

Glacial rebound of the British Isles-II 963

T=18000aBP

T = 16ooo aBP

Figure 2. Isochrones and ice-thickness estimates for the retreat of the ice sheet over the British Isles at selected epochs (a) from 22000 to 12 750 yr BP and (b) for the Loch Lomond Advance and subsequent retreat. Growth of this new ice cap is assumed to have started at 12 500 yr BP.

north with ice that advanced down the Irish sea basin from Scotland. The flow across Shropshire and Cheshire terminated at Wolverhampton (e.g. Thomas 1989) but there does appear to be a potential problem of timing here for if the ice had retreated from the southern part of the Irish Sea; by 18 000 yr B P the maximum advance into Shropshire, fed by ice flow from the Irish Sea, would have occurred earlier and a nominal date of 22 000 yr B P has been adopted for this advance. The ice front has been identified in the Peak District at the southern end of the Pennines and the more recent evidence is consistent with Andersen’s identification. The evidence for eastern Yorkshire suggests that a Late Devensian ice flow occurred over the Holderness Peninsula

of northeast England, through eastern Lincolnshire, to the northern edge of Norfolk and into the southern North Sea south of Dogger Bank (Boulton et af . 1985; Long et al. 1988). It is assumed here that this ice flow was a relatively short-lived phenomenon or surge and that it had vanished soon after 18 000 yr B P (see Fig. 2).

Two contrasting views of the last glacial maximum ice thickness over the British Isles are exemplified by the maximum and minimum ice sheet reconstructions of Boulton et al. (1977, 1985) (see, Fig. 7 of Part I). In their maximum model the ice thickness exceeds 1800 m and the ice sheet covers the peaks of the highest mountains. Others have argued for much thinner ice such that the higher

964 K. Lambeck

mountain tops were exposed as nunataks. Geomorphologi- cal evidence such as striae, erratics and trim lines can in some instances place constraints on maximum ice thickness (e.g. Ballantyne 1984) and they point to values of typically 1000-1200 m over Central Scotland. Thus Sutherland (1991) expresses a consensus view that the maximum ice thickness here is unlikely to have exceeded much more than 1300m. On the Outer Hebrides, estimates of ice thickness are of the order of 400m (Peacock 1980) while the evidence of an absence of erratics on the higher peaks in the Southern Uplands of Scotland suggest that here the ice thickness is unlikely to have exceeded about 1OOOm. Ice thickness estimates for the Central Plain of Ireland are about 400-500m according to Watts (1977) with the higher mountains standing out as nunataks. Such reduced ice-thickness estimates also appear to be consistent with the evidence from flow direction indicators that point to a polycentric ice sheet with at least two major ice foci, one over Rannoch Moor in the north and the other over the Southern Uplands in the south (Figs 1 and 2a), and a number of subsidiary ice centres over Scotland, northern England and Ireland.

Ice retreat during Late Devensian time

After 18000yrBP the ice retreat is assumed to have occurred without major standstills or readvances until about 12 500 yr BP when the ice sheets had vanished throughout Britain and Ireland. In eastern Scotland, the retreat of ice is marked by numerous Late Devensian shorelines formed at the ice margin, by the deposition of marine sediments and by glacial outflows, although the age control on these features is limited. Thus at time of the formation of the East Fife Shorelines, the ice margin had retreated across the low grounds of eastern Fife, Kincardine and eastern Lothian and locally may have retreated up the Forth and Tay valleys (see Fig. 3 of Part I and Fig. 4 later for locations). Of this series of shorelines, the youngest, (the East Fife 6 shoreline), has been associated with the deposition of the arctic-fauna- bearing marine Errol Beds with a nominal age of 14 750 yr BP (Paterson, Armstrong & Browne 1981) and this is adopted here. By the time of the formation of the Main Perth Shoreline, nominally at about 13 500 yr BP (Paterson et al. 198l), ice had retreated up the Forth and Tay River valleys and areas such as the upper Teith Valley west of Stirling were ice free by 12 750 yr BP (Gray & Lowe 1977). In contrast, in Andersen’s reconstruction the ice front at 14 000 yr BP was still to the east of the Wee Bankie Moraine and did not retreat to eastern Fife until about 13 500 yr BP (see Fig. 9 of Part I).

The oldest Late Devensian radiocarbon age recorded in northeastern Scotland is 16 630 f 650 yr BP for deposits offshore from Banff (Sissons 1981) and this area is assumed to have been ice free by this time. The Cromarty and Moray Firths may have been occupied by ice until about 13 500 yr BP (Peacock, Graham & Wilkinson 1980) and the ice front is considered to have stood near the upper Moray Firth at about this time. As emphasized by Firth (1989), the age of the ice movements across this region is poorly constrained, but the evidence does suggest an ice margin at 14 000 yr BP that lies considerably further to the west than assumed by Andersen (1981), consistent with other evidence

C

Figure 2. (Continued.)

Glacial rebound of the British Isles-Il 965

from the Spey Valley of rapid ice retreat from the eastern Highlands by about 13 500 yr BP (Sissons & Walker 1974). In southwestern Scotland the ice retreated to the north of the Solway Firth and most of the Southern Uplands are believed to have been ice free well before 13000yrBP (Lowe & Walker 1977; Bishop & Coope 1977). Also, evidence from western Scotland sites such as Cam Loch (Pennington 1977), southwest Mull (Walker & Lowe 1982) and Lochgilphead (Peacock et al. 1977), suggest that much of the coastal area of western Scotland was ice free before 13 000 yr BP. Taken together, the evidence suggests that a very large part of Scotland was ice free at or soon after about 13000yrBP (Sissons 1981; Price 1983) and by 12500yrBP, the Late Devensian ice sheet is assumed to have vanished everywhere over Scotland. Hence the late Glacial retreat over west and southern Scotland in this model occurs significantly earlier than suggested by Andersen's model in which much of the region was still ice covered at 13 000 yr BP (Fig. 9 of Part I).

In Ireland, the early retreat of the ice sheet from south to north is assumed to have been rapid, based on an age of 17 000 yr BP for the Main Drumlin or Kell Drumlin Moraine (McCabe 1987; McCabe et al. 1987) rather than the 14 000 yr BP adopted by Andersen (1981). The northern limits of the 'Drumlin event' moraines (Fig. 1) are taken to define the ice limit at 17000yrBP along the northern and north-western margins (McCabe et al. 1986) and by 14 000 yr BP Ireland appears to have been largely ice free (McCabe 1987; Carter 1982). Retreat of ice from the Irish Sea is also assumed to have been rapid, consistent with the 17 000 yr age for the Kells Drumlin Moraine extension on to the sea-floor (Stephens & McCabe 1977), so as to have been ice free by about 15 000 yr BP.

Few precise constraints appear to exist on the retreat of the ice across Wales and England during the Late Glacial. Ice-free conditions may have occurred in northwestern Yorkshire as early as 17 000 yr BP (Gascoyne, Schwarz & Ford 1983) and, apart from local residual ice sheets such as that over Cumbria, the main ice sheet appears to have retreated north of the Scottish border by 15000yrBP. Cumbria itself is assumed to have been ice free by 13500yr BP (Gale 1986) and all glacier ice over Wales vanished before 14 000-13 500 yr BP (Haynes et al. 1977; Musk 1986).

Figure 2(a) illustrates the proposed isochrones for the ice sheet over the British Isles from 22000yrBP to 12 750 yr BP. In addition to the arguments and observations summarized above, other information used to construct these isochrones have included the Late Devensian patterns of frontal retreat over northern England given by Boulton, Smith & Morland (1984) and the patterns of longitudinal and radial features, primarily drumlins and striae, left by the retreating ice (Boulton et al. 1985).

The Loch Lomond Readvance

The period of climatic improvement that started at about 14 000-13 500 yr BP and which led to the rapid collapse of the Late Devensian ice sheet came to an end by about 12 500 yr BP and was followed by a gradual climatic cooling, culminating with the Loch Lomond Stadia1 in Scotland between about 10 800-10 300 yr RP (Price 1983; Sissons

1974). The extent of the resulting Loch Lomond Advance ice sheet has been well documented for the Western Highlands with the ice again extending down to the Clyde estuary and into the upper valleys of the Forth and Tay rivers (Sissons 1974, 1981). Local ice sheets developed over the Inner Hebrides islands of Mull and Skye but no major glaciation appears to have occurred beyond the Western Highlands of Scotland. Ice thicknesses were never sufficient to cover the higher peaks and the morphological indicators suggest values of 400111, reaching 600m locally (Sissons 1974; Thorp 1986). Elsewhere in the British Isles only minor ice sheets formed as mountain glaciers at the time of this cold stage. Fig. 2(b) illustrates the adopted isochrones for this short-lived advance and retreat in which all glacier ice was finally gone by 9500 yr BP.

Ice model for 22 000-9500 yr BP

Ice heights through time have been established using eqs 4 and 5 in Part I, and the same procedure as discussed in Part I. At the time of maximum glaciation t,,, the heights hmax(tmax) at the ice foci, are based on the estimates discussed above and, with smax established from the maximum ice limits, this determines the coefficient di) (eq. 4b of Part I) for profiles ( i = 1 . . . 9) radiating out from the ice centres. The maximum ice heights at these foci for subsequent epochs t are given by

1 J i

h,,,(t) = - c "(i)[s:;x(t)]"'.

Ice heights along the profiles then follow from eq. 5 of Part I). Fig. 2 illustrates the resulting model at selected time intervals. Altogether, the model of Late Devensian ice retreat is defined at the time of maximum glaciation of 22000yrBP, then at 1OOOyr time steps from 20000 to 14000yr inclusive and thereafter at 500 year intervals until 9500 yr BP. The resulting ice model has been digitized with a spatial resolution of 0.25" in latitude by 0.5" in longitude, or approximately 25 km by 25 km.

Lower-middle Devensian glacial cycles

During Early Devensian time, before about 50 000 yr BP, much of England and Ireland appears to have been ice free (e.g. Shotton 1977; McCabe 1987). There also appears to be little evidence for early Devensian glaciation in Scotland before this time (e.g. Bowen et al. 1986) although some authors have suggested that the build-up of the ice sheet over Scotland may have started as early as 75000yrBP (Sutherland 1984). Price (1983) has drawn a tentative summary of environmental conditions in Scotland and suggests that glacial/periglacial environments existed here during 78 000-65 000 yr BP and again during 55 000- 50 000 yr BP, with interstadiai conditions between these two cold intervals as well as in the interval 50000-45000BP. Renewed ice-sheet development over Scotland possibly occurred in the interval of 45 000-32 000 yr BP. Palaeo- temperature records, particularly mean summer tempera- ture, provides an indicator of the likelihood of ice-sheet growth during Devensian time. Such temperature records have been proposed by Coope, Morgan & Osborne (1971)

966 K . Lambeck

l8

, I Western ' Scotland

'- I .. , r - - -

4 I t

0 -

3 s' 0.1 ; b

p 0.2 E -2 P

I

0 - .- 0.3 :

0.4 -

50 40 30 20 10 0 time ( xlo00 years BP)

Figure 3. Palaeotemperature records for central England and western Scotland (top) and the equivalent sea-level function for the past 50000 yr (bottom).

for lowland England, by Price (1983) for western Scotland, and by McCabe (1987) for Ireland (Fig. 3) and, within observational uncertainties, the results for the three regions exhibit similar fluctuations so that a simple relationship between temperature and ice-sheet extent can be estab- lished. For example, average summer temperatures in central lowland England in the interval 50 000-30 000 yr BP do not appear to have dropped below those at sites near the edge of the ice sheet during the last glacial maximum and it can be assumed that the ice sheet did not extend far south into England at any time in this interval. At 35 000 yr BP, as well as at 27 000 yr BP, temperatures were similar to those at 15000yrBP and the extent of the ice sheet at these earlier epochs is assumed to have been the same as that at 15 OOO yr BP. Fig. 3 illustrates the resulting equivalent sea-level curve for the British ice sheet for the past 50 000 yr BP and it compares well with the model previously derived from the evidence of ice movements across the northwest shelf of Scotland and across the North Sea (Eden et al. 1978) (Fig. 13 of Part I). Insofar as sea-levels are predicted only for the Late- and Postglacial stages such approximate models for the earlier glacial cycles suffice (see Part I).

3 A SUMMARY OF OBSERVATIONS OF SEA-LEVEL CHANGE

Quantitative evidence for sea-level change along the coast of Great Britain comes from a variety of different sources, the

particular problems of interpretation of which have been discussed in the volumes edited by van de Plassche (1986), Smith & Dawson (1983), Tooley & Shennan (1987) and Pirazzoli (1991). An important source of information, particularly for change during the latter part of the Holocene, is from the aquatic and herb flora of peats defining marine transgressions or regressions. Often these peats are intercalated with marine or freshwater sediments so that it becomes possible to deduce both the position of past shorelines and indicators of whether sea-levels were rising or falling at the time of deposition. In other examples, the past shorelines are inferred from the position of fossil intertidal or shallow-water marine molluscs, or from erosional features left when the sea retreated, such as the rock platforms and shingle beaches found in western Scotland. Generally, the height formation will not correspond to mean sea-level, the reference level used in all model calculations, and the successful interpretation of the observations in terms of sea-level change requires a knowledge of the relation of the stratigraphic boundaries or shoreline features to sea-level. Thus the peat sequences corresponding to the freshwater-to-marine transgressions, will usually correspond to a level near mean high-water spring (MHWS) tide although the actual formation range about this level may be quite large (e.g. Shennan 1986) as well as a function of tidal amplitude (e.g. Heyworth & Kidson 1982). Additional uncertainties result if the peats formed in a backswamp rather than in an open coastal environment, because in the former case the formation height may be near the middle of the tidal range rather than near the MHWS level. Compaction of the peats and consolidation of the underlying sediments add still further uncertainty to the inferred sea-levels. Other shoreline markers present a similar range of problems. For example, shingle beaches define a maximum level that generally lies above MHWS tide because of storm activity and the highest astronomical tide level may provide a more appropriate reference height. In contrast in situ molluscs define a lower part of the tidal range or a limiting depth below the tidal range, depending on species. Because of these different formation levels it is important that all shoreline markers are reduced to the same level, particularly in a region such as the Solway Firth where the tidal range presently exceeds 10m, or in the Bristol Channel where it exceeds 14 m. Mean tide level is the most appropriate level for this and all shoreline information will be reduced to this level, with the questionable assumption made that the tidal range has remained constant throughout a time interval in which major changes in sea-level have occurred in environments of indented shorelines and variable topography. Because the tidal range can vary significantly within a region, corrections have been applied individually to the data points at each location using the tidal amplitude information compiled by Shennan (1992). The height measurements of the shorelines are usually expressed relative to the British Ordinance Datum (OD) at Newlyn in the south of England. Zero-height OD does not generally coincide with mean sea-level (MSL) which lies at up to 0.35 m OD in northern England and up to 0.5m OD in Scotland and all heights have been reduced to MSL using the corrections given by Thompson (1980) except in a few instances where heights refer to a local datum in which case the reference is assumed to approximate MSL.

Glacial rebound of the British Isles-II 967

sequences of East Fife. Fig. 4(a) illustrates the shoreline isobases, contours of constant shoreline elevation above the present OD height datum, based primarily on the Cullingford & Smith analysis and on the elevations of the top of the Errol and St Abbs Beds in the Tay and Forth valleys. No reductions to MSL have been made at this stage.

Another conspicuous elevated beach and shoreline throughout southeast Scotland is the Main Perth Beach (Sissons & Smith 1967; Sissons, Smith & Cullingford 1967) and associated shoreline (MPS). Again, quantitative age information for this shoreline is lacking and its age is inferred to be about 13 500 yr BP from evidence of the ice front which stood near the shoreline during a temporary halt in the ice retreat (Paterson 1974; Sissons 1974). In East Fife the MPS has been identified at elevations below the EF-6 beaches (Smith, Sissons & Cullingford 1969) but the correlation is based on elevations and gradients rather than on quantitative evidence. The corresponding isobases of the MPS shorelines are illustrated in Fig. 4(b).

The best defined shoreline of all in the Forth valley is the Main Postglacial Shoreline defined by the abutment of the present flat valley floor, the ‘carse’ surface, against the higher adjacent ground. These carse clays were deposited in an estuarine environment during a time of relative sea-level rise and are overlain in some localities by peats carbon dated at 6500 yr BP. This determines the time of the end of the transgressive phase and the formation of the Main Postglacial (MPg) Shoreline (Sissons 1967a). It is also well defined in the Tay and Earn River valleys and along the Tay Firth (Cullingford, Caseldine & Gotts 1980; Morrison et al. 1981), in East Fife and along coastal areas as far north as the Ythan valley in Aberdeenshire (Smith et al. 1980, 1985). The precise age of the shoreline remains uncertain and the culmination is variously reported as between 6800 and 5700 yr BP although its formation may not be synchronous. Smith et al. (1985), for example, suggest that the shoreline decreases in age by as much as 500yr from the upper Tay and Earn River valleys to East Fife. A nominal age of 6000yr BP is adopted and Fig. 4(c) illustrates the corresponding isobases.

While the cane clays form a remarkably uniform deposit some layering and lamination does occur. One persistent feature near the top of the sequence is a thin silty-to- fine sand layer constrained in age by peats above and below it dated at about 7000 f 200 yr BP. It is found in the upper Forth valley (Sissons & Smith 1965), in East Fife (Morrison et al. 1981), near Montrose (Smith et al. 1980), and in the Ythan valley (Smith, Cullingford & Brooks 1983). Sissons (1983) and Smith et al. (1985) have suggested that this feature may have been produced during a major storm surge while Dawson, Long & Smith (1988a) and Long, Smith & Dawson (1989) have suggested that it may have been deposited by a tsunami generated by a major submarine landslide in the Norwegian Sea. Irrespective of its cause, the uniqueness of this layer makes it an ideal shoreline marker for a well-defined instant in time. Fig. 4(d) illustrates the ‘Tsunami’ isobase from localities where it has been identified and from elevations given by Long et al. (1989). The reference height for this shoreline is unknown but the MHWS tide level is adopted.

Cores taken through the carse clays in the Forth valley have revealed a number of buried beaches, the deepest and

The quoted precision of the height determinations based on the peat evidence is generally small, of the order of 0.1 m, but the accuracies of the inferred sea-levels will be considerably lower, of the order of f l m or more when some of the above-mentioned uncertainties are taken into consideration. These error estimates are comparable to the amplitudes of the minor oscillations in sea-level inferred, for example, by Tooley (1978) or Shennan (1986) and the latter are considered as local ‘noise’ superimposed on the more regional sea-level trends. The resulting sea-level trend is, therefore, represented by a smooth curve through the MSL estimates with upper and lower limits indicated by the spread of depths of the individual observations and by the accuracy estimates of the observations.

All ages used for the shorelines and sea-levels correspond to the conventional radiocarbon time scale because the history of the movements of the ice sheets are almost wholly defined within the same time scale. Fractionation and reservoir corrections will generally have been made or estimated. Age constraints for some of the major shorelines recorded in Scotland are poor and inferred rather than based on direct radiocarbon ages and the inferences of ages by correlating shoreline fragments from one area to another introduces further uncertainty, particularly because the formation of a shoreline need not be synchronous; diachronous formation being suggested by both the observational evidence (Morrison et al. 1981) and by rebound modelling (Lambeck 1991).

Southeast Scotland

This area encompasses the Forth and Tay valleys and adjacent coastal areas from the English border (near Berwick) to about Aberdeen (Fig. 4). Well-developed elevated shorelines occur up to elevations of about 40 m and these are believed to post-date the last ice retreat across the area from east to west. The main raised shorelines are the East Fife group, the Perth sequence and the postglacial features, some of which have been correlated across the regions using their morphological, stratigraphic and sedimentary characters, as well as their heights and gradients. Along the river valleys the evidence for change in sea-level is primarily in the form of marine deposits, peats, or erosional features, but only in a few instances has it been possible to establish time series for the sea-level change.

The East Fife Shorelines were initially mapped by Cullingford & Smith (1966). Six separate beaches have been identified in the sequence, each of which terminates in outwash plains with the break in slope between the beach or terrace and the back slope marking the shoreline. The lowermost feature East Fife 6 (EF-6) Shoreline extends furthest westward and it has been suggested that it correlates with the Errol Beds in the Tay estuary (Paterson et al. 1981) where they occur to the west of Perth, and the St Abbs Beds in the Forth Estuary where they have been recognized as far west as Inverkeithing (Paterson et al. 1981; Sissons & Rhind 1970). Based on age estimates of these clays, Browne (1980) tentatively associates an age of 14 750 yr BP with the EF-6 Shoreline, a suggestion that is adopted here. Further north, Cullingford & Smith (1980) have identified the same shoreline sequence on the basis of the correlation of their heights and gradients with the

968 K. Lambeck

I I 40 30 20 50

Figure 4. Isobases of the major shorelines identified in eastern Scotland. (a) The East-Fife 6 Shoreline, (b) the Main Perth Shoreline, (c) the Main Postglacial Shoreline, (d) the 7000 yr BP ‘tsunami’ Shoreline, (e) the Main Late Glacial Shoreline.

most extensive of which is a gravel layer backed by cliffs formed by marine erosion of older tills. This is the Buried Gravel Beach of Sissons (1967a) or the Bothekennar Gravel (Browne et al. 1984). The shoreline, defined by the base of these buried cliffs and referred to as the Main Late Glacial (MLg) Shoreline, has been identified from near Grange- mouth in the west where it occurs near present sea-level to near Berwick in the east where it occurs at -18mOD (Sissons 1974). It post-dates the Main Perth Shoreline and pre-dates the overlying sediment deposited at about 10 300 yr BP (Sissons 1967b). The extent of erosion suggests

that sea-level may have stood at this height for a considerable duration.

Figure 5 illustrates the time series of the change in shoreline heights for three locations with all heights reduced to MSL. The Arnprior results of Sissons & Brooks (1971) have been extended to late glacial time by a linear extrapola- tion of the MLg and MP shorelines of Sissons (1974) about 10 km and 20 km respectively beyond the western limits at which they have been observed. The upper Tay Forth result for the localities of Bridge of Earn and Glencarse is from Cullingford et al. (1980) and from the data points from

Glacial rebound of the British Isles-II 969

because no direct age measurements exist for the actual shorelines before about 9000 yr BP and the older ages are all inferred from relationships between the retreating ice front across the region and shoreline development. The results for Arnprior and Errol both indicate the occurrence of small oscillations in sea-level between about 10 000 and 8000 yr BP corresponding to the three Buried Shorelines of Sissons (1967a and b) (see also Paterson et al. 1981). The age of the first of these, the High Buried Shoreline, is constrained by pollen ages as having formed at about 10 200 f 100 yr BP, soon after the formation of the Main Late Glacial Shoreline. The second, the Main Buried Shoreline, is constrained in age by both pollen and radiocarbon age as about 9600 yr BP and the youngest, the Low Buried Shoreline, is constrained in age as 8700 yr BP (Sissons & Brooks 1971). Within the firths of eastern Scotland, at least, sea-level fluctuations exhibit a complex sequence of oscillations in earliest Holocene time. Fig. 5 also illustrates the envelopes of the adopted sea-level curves for the three localities, all referred to MSL, and the associated accuracy estimates. The standard deviations for the older shorelines (EF-6, MPS) are 5 m while for the MLg and Buried Shorelines accuracies of 3 m and for the MPg and Tsunami Shorelines values of between 1 to 2 m have been adopted depending on the quality of the original observations. Figs 4 and 5, with the former shoreline heights reduced to present MSL, summarize the eastern Scotland sea-level observation data that will be used below for estimating glacial-rebound parameters.

I

I MLg -10 ' I

15 10 5 0

(Earn, Abernethy, Glen Carse)

(b) 60

-20 15 10 5 0

(c) 50 . L I ) Errol

(Firth of Tay) . &

-20' . . . . . . . ' 1

15 10 5 0 time (~1000 years BP)

Figure 5. Time series of height-age relationships of former shorelines (relative to MSL) at (a) Amprior, (b) Bridge of Earn and Glencarse, (c) Errol. Also illustrated are the envelopes of the inferred sea-level change for these localities. In (a) and (c) HBS, MBS and LBS refer to the high-, main- and low-buried shorelines respectively.

Paterson et af. (1981) for the MP and the top of the marine clays identified with the EF-6 Shorelines. The Errol curve, to the east of Glencarse, is from Paterson et al. (1981) and includes the EF-6 Shoreline height taken from their height-distance diagram for the Tay Firth shorelines. It needs to be emphasized that these curves are less accurate than may be implied by the generally consistent pattern

Northeast Scotland

Most evidence for past sea-levels in this area comes from the shores of the Cromarty, Moray, Inverness and Beauly Firths (Fig. 6) and this has been reviewed by Firth (1989) for Late Devensian time and by Firth & Haggart (1989) for Holocene time. Within the inner Moray Firth a sequence of beaches of

58.0"

j

L 1 I I 40 50

Figure 6. Location map for northeast Scotland and (inset) the heights of the ILg-4A Shoreline. (1) Beauly Firth, (2) Firth of Inverness, (3) Cromarty Firth, (4) Dornoch Firth, (5) Munlochy Bay, (6) Beauly, (7) Inverness, (8) Nairn, (9) Banff, (10) Fort George, (11) Wick.

970 K. Lambeck

Beauly Firth

Late Devensian age reach altitudes of over 30m, and possibly as high as 42m (Synge 1977), associated with the westward retreat of the ice from Nairn to Inverness. Firth (1989) has identified at least 10 Late Glacial sequences, referred to collectively as the Inverness Late Glacial (ILg) Shorelines of which the best defined are the Ilg-4A and Ilg-5A surfaces with gradients of about 0.4 m km-' in an approximately northeasterly direction (Fig. 6). Direct age information for these shorelines is not available. They must post-date the retreat of the last ice cover and this places an age constraint on ILg-4A of about 13 500 yr BP correspond- ing to the deglaciation of Cromarty (Firth 1989), although Peacock (1974) suggests that this deglaciation may have occurred later. A nominal age of 13500yrBP is adopted here, equivalent to the Main Perth Shoreline of the Forth region (Firth 1989). Firth & Haggart (1989) have also identified a buried erosional surface covered by sediments dated at 9610f 130yrBP. This surface is terminated at some locations by the base of a cliff line and they suggest that it corresponds to the Main Late Glacial Shoreline of southeastern Scotland. This association is also adopted here.

A detailed examination of the Beauly evidence by Haggart (1986) and Firth & Haggart (1989) has indicated that after about 10 000 yr BP small sea-level oscillations occurred that are similar to those reported for the upper Forth valley (Fig. 7). The 'Tsunami Shoreline' has also been identified here as well as in Munlochy Bay, Dornoch Firth and Wick (Long et al. 1989) and its age is constrained to lie between 7430 f 120 and 7270 f 90 yr BP (Haggart 1986). The culmination of the early Holocene transgression forms the Inverness Flandrian Shoreline (IF1) and is defined by estuarine flats, terraces and shingle ridges along the margins of the firths of the region, attaining a height of about 9-10 m OD near Beauly and sloping in a north-northeast direction with a gradient of about 0.07mkm-'. The formation time is between 7100 and 5500 yr BP and Haggart (1986) suggests an age of about 6400 yr BP and correlates it with the Main Postglacial Shoreline of the Forth-Tay region. Further north, elevated shorelines of Late Holocene age have not been identified in Caithness or in the Orkney Islands while in Shetland, shorelines formed at 6000-

h

E 3 2 0 - e v

', ' t ' , '

8 ' . , ' \ '

Figure 7. Time series of height-age relationship in the upper Beauly Firth region and the inferred sea-level change envelope.

7000 yr BP, now appear to be well below present sea-level (Hoppe 1965; Flinn 1964). The observational data base used for the Moray region consists of the sea-level curve for the upper Beauly Firth (Fig. 7) and the elevations of the LIg-4A, MLg, Tsunami, and the IF-1 shorelines along the Inverness and Moray Firths.

Western Scotland

This area includes the Inner and Outer Hebrides and the mainland coast from Wester Ross in the north to the Firth of Clyde, Kintyre and Arran in the south. Evidence for sea-level change in this region is distinctly different from that in the southeastern region, reflecting the differences in geology and sediment supply and the fact that the west coast has been subjected to a greater degree of glacial erosion and scouring than the east coast. Raised shorelines and elevated marine sediment deposits have been recognized throughout the region at elevations up to 100m but no systematic studies for their origins and ages have yet been published. Some of the elevated deposits are believed to be of pre-Devensian age while others may hav been emplaced as a result of glacier action (Synge 1977; Dawson 1984). Well-developed rock platforms at up to 50 m OD also attest to higher shorelines at some time in the past but these also are of uncertain age (Sissons 1982, 1983; Dawson 1984).

One region where a variety of fossil-bearing Late Glacial sediments have been preserved is in the Clyde Firth and Clyde River. A useful data set is from the Ardyne locality (see Fig. 9b for location) where Peacock, Graham & Wilkinson (1978) have examined the heights and ages of in situ marine bivalves and their relationship with the maximum elevations of the same sedimentary units in nearby localities. They succeeded in estimating the sea-level curve from about 12000 to 10000yrBP and while uncertainties are large (Fig. 8a), this result indicates a rapid fall followed by a nearly constant level for about 1500yr. Other data points for the region include in situ marine molluscs dated at 13 780 f 124 yr BP (Browne, McMillan & Graham 1983) and a series of peat and mollusc dates between about 13 000 and 8000 yr BP. The in situ n~ollusc (points 1-3 in Fig. 8a) represent lower limits and, by analogy with the Ardyne results, could be up to 20 m below sea-level (see radiocarbon dates GU-12, Birm-122, SRR- 479). Point 4 is also based on marine shells but the height has been reduced to sea-level using nearby morphological indicators (Baxter, Ergin & Walton 1969). Points 5 and 6 (Bishop & Dickson 1970) represent lower limits whereas points 7 and 8 (Bishop & Coope 1977) define upper limits. The level at 6000yrBP is based on a number of observations throughout the firth and associated lochs and sounds, although nowhere is the shoreline defined with precision (e.g. Jardine 1986; Boyd 1986; Gemmell 1973). For a wide range of plausible ice and earth models, the predicted regional variability for the sites within the Firth of Clyde region are generally smaller than the uncertainties associated with the observations, and the data points can be combined into a single sea-level curve (Fig. 8a) that is representative of the region between Dumbarton and Ardyne. (See also Fig. 21 of Part I).

Other quantitative data for establishing part of a sea-level curve, come from the Lochgilphead region of Argyll

Glacial rebound of the British Isles-II 971

550fl (weslern Scotland)

(a) 40

30

E regional . - 2 20

P -

3 10 .

e

0 .

15 13 11 9 7 5

\Oban-Lome (western Scotland)

(b) 60

50

" 15 10 5 0

time (~1000 years BP)

Figure 8. (a) Height-age relationships for sea-level indicators in the Clyde Firth and the lower Clyde River. The data from Ardyne are illustrated by the solid circles with error bars and the numbered points are discussed in the text. All elevations are with respect to MSL. The estimated sea-level curve is also shown. (b) Sea-level evidence for the Firth of Lome and Loch Fyne regions. The Lower Fyne curve is given by Dawson er al. (1988b), the Oban-Lorne curve is from Synge (1977), the Lochgilphead observations (with error bars) are from Peacock et al. (1977) and the open circle corresponds to the rock barnacles at Oban (Gray 1974).

(Peacock et al. 1977) (see Fig. 9b for location), where the evidence is similar to that obtained by Peacock et al. (1978) for Ardyne in the Clyde Firth. Fig. 8(b) illustrates this result and here also the evidence points to a rapid fall in sea-level up to about 12 000 yr BP followed by a period of at least lo00 yr during which sea-level remained at an approximately constant level. Dawson et al. (198%) give a sea-level curve for the lower Fyne region that is presumed to incorporate the Lochlgilphead data and only the Holocene part of this curve is shown in Fig. 8(b). Near Oban, barnacles of a species that normally lives in the intertidal zone have been found at 20mOD (Fig. 8b) with an age of about 12 000 yr BP (Gray 1974).

Elsewhere, the only information is from the rock platforms and the shingle beach ridges. The latter are believed to correspond to the Main Postglacial shoreline of eastern Scotland and reach a maximum elevation of about

I I I I 1 70 6O s o

Figure 9. (a) Elevations (OD) of the Main Postglacial Shoreline for the Inner Hebrides and Western Highlands. Numbers in italics refer to locations as follow(1) Scarba, (2) Oronsay, (3) Oban, (4) Loch Fyne. (b) Same as (a) but for the Main Rock Platform. (1) Ardyne, (2) Dumbarton, (3) Lochgilphead, (4) Firth of Lome, (5) Ardnamurchan, (6) Moidart.

14 m OD near the heads of the inner sea lochs (Gray 1974). The appropriate reference level for these features is the height of modern deposits which in some instances may be up to 3 m above the MHWS tide level (e.g. Sissons & Dawson 1981). Fig. 9(a) illustrates the isobases for this

972 K . Lambeck

shoreline based on the work of Gray (1974) for the Firth of Lorn and Mull; of Dawson (1984) for Jura, Islay and Scarba; of Jardine (1978) for Oronsay; Synge & Stephens (1966) for Iona, Kintyre and Loch Etive; and the mean Clyde Firth result discussed above. In Wester Ross this shoreline feature is found at about 9 m near Applecross and at about 7 m OD between Loch Gairloch and Little Loch Brook (Sissons & Dawson 1981). All shingle beach deposits have been reduced to mean sea-level assuming a formation height at MHWS + 2 m for the exposed outer coast sites and MHWS + 1 m for the protected inner coast sites.

Of the rock platforms only the low rock platforms, the Main Rock Platform of Gray (1974), ranging in elevation from 0-10mOD are used. They are particularly well defined around the Firth of Lorne and the west coast of Jura but they have also been identified in Arran, Kintyre and the Firth of Clyde and as far north as Ardnamurchan and Moidart (Dawson 1984, 1988; Gray 1978). The spatial variation of the platform elevations forms a relatively simple pattern (Fig. 9b) with the highest values occurring along the inner sea lochs of the Highlands and decreasing with distance from the centre of maximum glaciation, and it is usually assumed that they correspond to the same erosion event(s). Different ages have been attributed to this feature at various times but the present consensus seems to be that it formed or was reshaped during the Loch Lomond Stadia1 by periglacial rock erosion (Dawson 1984, 1989). Within the Clyde Firth the platforms occur at between 5 and 10 m O D (Gray 1978), consistent with the minimum sea-level recorded at Ardyne in the time interval 11 500-10 000 yr BP. The association of the platform with the relative lowstand during the Loch Lomond Interstadial at 11 000-10 500 yr BP is therefore plausible and accepted here, either as a time of formation or as a time of re-occupation and re-shaping of an older feature. The High Rock Platforms form a less coherent pattern and the consensus is that they are a result of shaping by successive glacial events, possibly of pre-Devensian age, with the latest modifications having occurred at 13 500-13 000 yr BP when the ice front stood at the western edge of the Highlands. Because of the considerable uncertainty in their interpretation these shorelines are not considered here as appropriate for constraining models of Late Devensian rebound.

Southwest Scotland

This region includes Southern Ayr and the coastal regions of Wigtown, Kirkcudbright and Dumfries along the Solway Firth (see Fig. 5.3 for locations). Elevated shorelines have not been identified in the region (Jardine 1971) and as the coastal area was generally ice free soon after 14000yrBP (Fig. 2a), the absence of such features, if confirmed, places sea-level as being near or below the present level at that time. The oldest indicators of sea-level occur in the upper reaches of the firth and span a time interval of 12 290 f 250 to 10300f 185yrBP (Bishop & Coope 1977) and suggest that mean sea-level in this period was at -3 to -4m below the present level. The Holocene evidence has been examined by Jardine (1975) for the eastern Solway near Annan, and for Wigtown Bay and Luce Bay in the western Solway Firth. Fig. 10 illustrates the results, including the pre-Holocene result for the eastern Solway region. Four

1 1 (Redkirk, Annan) I 1 4 1 2 1 0 8 6 4 2 0

-8

West Solway Firth (Wigtown Bay & Luce Bay)

-8 1 4 1 2 1 0 8 6 4 2 0

time (~1000 years BP)

Figure 10. Age-height relations of sea-level indicators and estimated sea-level curves, all reduced to MSL, for southwest Scotland. (a) East Solway Firth near Annan and Redkirk point, (b) the west Solway Firth near Luce Bay and Wigtown.

data points are given by Jardine (1975) for the Girvan area on the south coast of Ayr and these indicate that mean sea-level here was at about 2 m between 9400 and 8400 yr BP.

England and Wales

The palaeo sea-level evidence for England and Wales is confined mainly to river estuaries, shallow bays and low-lying coastal plains where sediments and organic materials deposited in tidal flat or lagoonal environments have been preserved. The most complete examination of the evidence for northwestern England is by Tooley (1978). The preliminary model predictions in Fig. 21 of Part I, indicate that considerable spatial variation in sea-level response is predicted for this area and that it may not be appropriate to combine all data into a single sea-level curve. Sufficient information exists to construct sea-level curves for three localities within the region: Morecambe, the Ribble Estuary, and Formby. Fig. 11 illustrates the results. Following Tooley, all heights are assumed to correspond to the MHWS level and have been reduced to MSL in this figure. Height errors ranging from f 1 m to f 2 m have been adopted and the minor oscillations noted by Tooley have been ignored. A partial sea-level curve for a fourth region, the Duddon Estuary near Millom, has been inferred from a few data points in Tooley (1978) and from the mainly

Glaciai reborrnd of the British Isles-I1 973

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. 4

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' * Millom (northwest England) .

Formby (northwest England)

1

10 8 6 4 2 0 time (x1OOO years BP)

-25 '

0

-5

-10

-20 t Ribble Estuary (northwest England)

10 8 6 4 2 0 -25 ' .

' ' ' ' . ' . '

Figure 11. Sea-levels for northwest England. (a) Morecambe, (b) Ribble Estuary, (c) Formby, (d) Duddon Estuary, Millom. In (d) rn refers to molluscs which are considered not to be in their growth position but to define an upper limit to sea level. P refers to peat samples at a level near the MHWS level with P-T referring to results from Tooley. Wand C are wood and charcoal samples which also define an upper limit to MSL.

mollusc and wood fragment results by Andrews, King Lk Stuiver (1973). The shell data have come from dune and shingle ridge deposits and because they do not appear to be in their growth position they are assumed to define upper limits to sea-level. Fig. l l(d) illustrates these results but generally they do not provide useful constraints on past sea-levels. [The discovery of a spoon in the same strata as these shells (Tooley, private communication) highlights the uncertainty of this data.]

Tooley (1978) (see also Gaunt & Tooley 1974) discusses evidence from northeastern England, mainly from the Humber Estuary in an area between Spurn Head, Hessle and Brigg. Four points from Chapel Point in this compilation are also found in Shennan's (1986) data set for the Fenlands and are included below. The evidence is of several types; peats, wood fragments within estuarine and saltmarsh clays, and molluscs, and they generally constrain only the limits to sea-level height. Fig. 12(a) summarizes the

observed height-age relationships, with all heights reduced to MSL. Further south, in the Fenlands of East Anglia, unconsolidated sediments have been deposited in lagoonal, perimarine and tidal flat environments that extend over an area of nearly 5000 km'. The sea-level information, in the form of peats intercalated between lagoonal and tidal flat sands and clays, has been examined in detail most recently by Shennan (1986). It includes data from Devoy's (1982) analysis of sea-level in southern England as well as the Chapel Point data included in Tooley (1978). Shennan's estimates of MHWS have been reduced to MSL and his error bounds have been adopted (Fig. 12b). Predicted sea-levels through the region based on the preliminary model (Fig. 21 of Part I) exhibit some spatial variability across the region, but this generally lies within the range of the adopted error bounds and the single curve is assumed to be representative of a location near the geographic centre of the various data points.

974 K . Lambeck

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Humber Estuary "' (northeastern England) ,

-16 ' . I 8 6 4 2 0

4

-16

-20

, . . ~ ~ - .

(East Anglia)

6 4 2 0

Yare Valley

Waveney Valley

Norfolk

8 6 4 2 0 time (~1000 years BP)

Figure 12. Sea-levels for (a) the Humber Estuary in northeast England including the Chapel Point data, (b) the MSL curve for the Fenlands from Shennan (1986), (c) the Norfolk coast from three different locations.

The Norfolk coast has provided more localized environ- ments for the preservation of Holocene sea-level indicators. Evidence from the northern coast between Brancaster and Cley is discussed by Funnel1 & Pearson (1989), observations from the Yare estuary included in Devoy's (1982) compilation, and the data from the Waveney Valley has been examined by A. M. Alderton (Shennan 1987). Fig. 12(c) illustrates the results for the three localities, all reduced to MSL. The observations are consistent with each other within their errors and the range of spatial variability predicted across the region for the past 7000yr (cf. Fig. 21 of Part I).

An important source of information on sea-level change in southeastern England is provided by Devoy (1979, 1982).

His data includes observations from the Fenlands and Norfolk and these have been included in the above- discussed sea-level curves. The remaining data points fall into three regions within each of which the predicted spatial variability is smaller than the accuracy estimates for the reduced mean sea-levels. These are the Essex coast, mainly from the Blackwater and Colne River estuaries, the Thames Estuary as far upstream as London, and the eastern section of the Channel coast between Margate and Weymouth. The evidence is from a variety of peats that are indicative of changes from freshwater to marine-brackish environments. Fig. 13 illustrates the mean sea-level curves for the three localities. All heights have been reduced to mean sea-level and the corrections given by Devoy for consolidation and compaction of the peats and underlying sediments have been applied. The data from Devoy is in two parts. For the first (Devoy 1982, Table 3) the evidence is from peats that are assumed to have formed near the MHWS tide level. For the second (Devoy's Table 4) the dated horizons cannot be reliably related to this reference and they provide mainly upper limit estimates of the sea-level curve but they are useful for the interval 10000-8000yrBP for which no information would otherwise be available from this region.

Sea-level information for the western sector of the Channel coast is given by Heyworth & Kidson (1982) and is based on freshwater-estuarine transition peats and in situ tree roots and stumps. Fig. 13(d) illustrates the resulting mean sea-level curve for this part of the coastline. Another important collection of data points is given by Heyworth & Kidson (1982) for the Bristol Channel and Wales where the information is similar to that from the western Channel coast. From Fig. 21 of Part I it would appear inappropriate to group all Bristol Channel observations into a single sea-level curve and instead curves for three localities have been constructed, for Bridgwater Bay, Swansea Bay (Port Talbot, Fig. 3 of Part I), and the upper Bristol Channel (Fig. 14). For Wales, much of the evidence comes from the submerged forest beds near Borth (Heyworth & Kidson 1982) and this is the only locality in Wales for which it is possible to construct a sea-level curve for middle and late Holocene time (Fig. 14d). The tree-stump data is based on the assumption that the trees have been killed by rising groundwater associated with a rise of sea-level, although the relationship is not straightforward, particularly if there has also been significant coastal erosion (e.g. Tooley 1986). Despite this reservation, these observations are assumed here to be indicative of an upper limit to the sea-level curve.

French Atlantic Coast

To complement the observations from southern England, sites where sea-level is relatively insensitive to the changing British ice sheet but more strongly dependent on the changes in the global ice sheets through the equivalent sea-level function, observations from the French Atlantic coast are also considered. Ters (1986) has compiled observational evidence from locations along the entire Atlantic coast of France and estimated a single sea-level curve, but analogous to the predictions in Fig. 21 of Part I, some spatial variability in sea-level response can be expected through the spatial dependence of the water-load term and through the variable distances of the sites from the

Glacial rebound of the British Isles-II 975

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.

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Thames Estuary -50

10 8 6 4 2 0

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Figure 13. Sea-levels for (a) Mersea on the Essex coast, (b) Thames Estuary, (c) the eastern sector of the English Channel coast from Margate to Weymouth and (d) the western sector of the Channel coast from west of Weymouth to Cornwall.

-30

Swansea Bay . (Wales)

10 8 6 4 2 0

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0 .

-5

-10

-15 .

-20

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.

.

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10 8 6 4 2 0 time (x1ooO years BP)

Figure 14. Sea-levels for (a) Swansea Bay, south Wales, (b) the upper Britsol Channel near Avonmouth, (c) Bridgwater (Somerset), (d) Cardigan Bay, mainly from Borth Bog and the Dovey Estuary but including single data points from Newport and Newquay.

976 K . Lambeck

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m

10 8 6 4 2 0

1 Cotes-du-Nord

- 2 0 ' . ' . 1 10 8 6 4 2 0

time (~1000 years BP)

Figure 15. Sea-level curves for the France Atlantic coast at (a) Pas de Calais, (b) Manche & Calvados and (c) CBtes-du-Nord.

Fennoscandian ice sheet. The data has, therefore, been grouped to construct regional curves for three locations: Pas-de-Calais, Manche and Calvados, and CGtes-du-Nord. The observations in each area of a diverse nature and the relation of the sample positions to the tidal range are not always known with precision so that many of the observations represent lower limits only. Fig. 15 illustrates the three mean sea-level curves. The high-frequency oscillations in sea-level proposed by Ters have been ignored.

4 AN IMPROVED R E B O U N D MODEL

A strategy for parameter estimation

The parameters to be determined from the observations of sea-level include both the effective earth model parameters (the lithospheric thickness He and the mantle viscosity r]

assumed to be a function of depth) and certain ice model parameters. The local ice model is assumed known except for the possibility of a constant scale parameter 0"" to be applied in space and time to the height of the British ice sheet. The predicted sea-levels for most sites are not strongly dependent on the Fennoscandian ice sheet, and without including sea-level data from northern European sites it is not possible to solve for a similar parameter BFEN. An earlier study using such sites indicated that B""" = 1 (Lambeck, Johnston & Nakada 1990) and this value is adopted here. Also, a time-dependent equivalent sea-level correction S g ' ( t ) is introduced to allow for any inadequacies in the models of the distant ice sheets. The observation equation then becomes (cf. eq. 6 of Part I)

ACo(qpI A, t ) + 4% A, t )

+ B"" ACBR(q, A, t ) + ACFEN(q, I , t ) + ACf-'(q, I , t ) + SC"

= Ag'(t) + S g ' ( t )

= ACpredicted (1) where

A&, = observed sea-level reduced to MSL, at location (q, A) and at time t.

E ( ) = estimate of the observation error. Ag' = equivalent sea-level function for the totality of the

Sg' = correction term to Ag'. p"" = scale parameter for the British ice sheet.

ice sheets.

ACnR = predicted first-order contribution to sea-level change from the ice- and water-load. Terms of the British ice sheet for specified earth model parameters.

the ice- and water-load terms of the Fennoscandian ice sheet.

A CCf = predicted contribution to sea-level change from the water- and ice-load terms of the distant ice sheets of Laurentia, Barents Sea and Antarctica

S t w = second-order corrections to the total water-load terms arising from (1) the dependence of the water-load term on sea-level change itself, and (2) the time dependence of the ocean function (see Part I). The full formulation for this term as developed by Johnson (1993) is used here.

An iterative procedure has been adopted to solve for the various parameters, the He, r] defining the A < functions, B"" and S g ' ( t ) . In the first iteration predictions are made for the three sea-level terms AC"", AtFEN, AC'-' for earth model parameters covering a wide range of values. Predictions for each earth model k(k = 1 * * . K) within this range are made for each instant of time at every site where sea-level has been observed for a total of M ( m = 1 . . . M ) observations. The corrective terms Sg' and SC" are initially set to zero and for each model k the corresponding a,"" value is estimated that minimises the quantity

ACFEN = predicted contribution to sea-level change from

Glacial rebound of the British Isles-I1 977

where dm) is the standard deviation of the mth observation. For each earth model k the @"" value is established that gives the minimum solution variance and a search is then conducted through the earth model space to determine those parameters that give an overall minimum variance. Once this minimum variance solution is established, the second-iteration correction terms Sc" are computed only for those earth models in the neighbourhood of this first-iteration minimum because of the substantial computing requirements for these terms and because these corrections are relatively small. The solution of (1) for H,, q , p"" and the correction vector 6r(t) is then repeated for the subset of earth models and the new minimum variance solution is sought.

.I E

Earth model parameters

In the first series of calculations the earth model comprises a lithosphere (layer l ) , a combined upper mantle and transition zone down to a depth of 670km (layer 2) of viscosity q2 and a lower mantle (layer 3) of viscosity q,,. (This is the three-layered model of Table 1.) The viscoelastic lithosphere is assumed to have a high viscosity (lo2' Pa s) and its effective thickness He is to be determined from the comparison of the observations with the predicted sea-levels. The preliminary calculations (Part I) indicated that the predicted sea-levels are relatively insensitive to the choice of qIm and the initial search will be for He and q2 for qIm = 1.3 x Pa s (see below) with the search conducted within the range

(50 I He 5 100) km, (10'" 5 q2 5 5 X lo2') Pa s. (3) Figure 16 illustrates a typical result for the first-iteration

solutions for a subset of the K earth models and they indicate that while some trade off occurs between the earth- and ice-model parameters a separation of q2 and @"" appears feasible. This is further illustrated in Fig. 17 which summarizes the minimum variances and associated p"" for the parameters within the earth model space defined by (3). The results are for three observational data sets: (1) the observations from sites outside the former ice sheet margins in England, Wales and France; (2) the observations from sites within the formerly glaciated areas of Scotland and northwest England; and (3) the combined data set. The first data set (Fig. 17a) has no resolving power for @"" (see Part I) and this parameter has been set to unity in the first

Table 1. Definition of the multilayered earth models. CMB refers to the core-mantle boundary.

'three-layer Five-layer

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

0 - H Q H Q =80 krn

Hp -670 Hp -200

670-CMB 200-400

400-670

670-CMB

0.5 0.7 0.9 1.1 1.3 1.5 PBR

Figure 16. The variance function u: for three three-layer earth models with different q2 values and H, = 80 km as a function of the ice-height scale parameter, B"".

instance. Two local minima are identified in the q2 - He parameter space, one near q2 = 2-3 X lo2' Pa s , and the second near 4 X 102"Pa s. This example provides a good illustration of the frequently encountered existence of two solutions in viscous relaxation problems with one solution corresponding to an early stage of relaxation of a process with a long time-constant and the other corresponding to the late stage of relaxation of a short time-constant viscous model (e.g. O'Connell 1971). In the absence of any other information the higher viscosity solution would be the preferred one in this case. However, for the Scotland data, where the dominant contribution to sea-level change at the sites near the centre of loading comes from the local rebound, this higher viscosity mantle model does not predict well the considerable spatial and temporal variability observed and this solution is now excluded (Fig. 17b). The combined data set (Fig. 17c) also excludes the higher value and a well-defined optimum solution space is defined by the

= 3 contour and indicates earth parameters that lie in the range (60 5 He 5 75) km and (3 x lo2" 5 q2 5 5 x lo2") Pa s. The overall minimum variance is (1 .58)2 whereas the expected value from a perfect model and realistic estimates of observation variances would be unity and it appears that there is some room for model improvement unless the observational occurrences have been systematically underestimated.

The statistics of the minimum variance solution (He = 65 km, q2 = 4 x 102"Pa s) are summarized in Fig. 18 in terms of the degree of correlation between the M observed and predicted sea-levels (Fig. 18a) and in terms of the histogram of the residuals E ~ ) (eq. 1) (Fig. 18b) and although some of the individual observational errors are large, there does appear to be a high degree of consistency between the model predictions and the observations of sea-level change. At this stage no attempt has been made to solve for the correction 85;e(t) to the equivalent sea-level function and a first estimate of it is given as the weighted mean of the residuals E ~ , for each epoch t. This function is

978 K . Lambeck England &Wales

I I I , I I 1 I l(Pl 2 3 1 IOU 2 3 5

En+nd, Wales & Sro(l.nd

I 1 I I I , I I

l P 2 3 5 I l(P1 2 3 5

rli(P8 s)

Figure 17. The variance function u: as a function of earth parameters qz (the combined upper mantle and transition zone viscosity) and H , for the first iteration solution (a-c) and the three-layered model. (a) The England, Wales and northern France data set only, (b) the Scotland data set only, and (c) the combined data set. In (c) the B dependence on v 2 and H , is shown by the dashed lines. In (d) the results for the second-iteration solution are shown for the combined observational data set. All solutions correspond to the fully non-adiabatic models.

earth model dependent although this dependence is slight for solutions about the minimum variance solution.

In the second iteration the correction term cw is evaluated and included in the solution of (1) for parameters in the neighbourhood of the first-iteration solution. Also, the first-iteration estimated of bf'"(r) is now included, and the solution is illustrated in Figs 17(d) and 18(c, d). The resulting minimum variance occurs for

He = 65-70 km (4a)

r ] , = (4-4.5)102" Pa s. (4b)

and

Neither the inclusion of the second-order terms nor the introduction of the equivalent sea-level correction term significantly affects the solution for the earth model parameters, at least for the British rebound problem, and it appears reasonable to ignore these second-order terms in any further preliminary exploration of the full range of possible earth model parameters although these terms have been included in the five-layer solutions discussed below. Further iterations of the solution of (1) also are not required.

In the solutions of the sea-level equations discussed here the response of the mantle to loading has been assumed to be fully non-adiabatic in the sense that any particle displaced to a new depth in a medium of depth-dependent density retains its original density and thereby experiences a buoyancy force. It has been pointed out by Cathles (1975) and Fjeldskaar & Cathles (1984) that adiabatic models may be more appropriate when the mantle is chemically homogeneous or at phase transition boundaries. In these models the displaced particles blend with the material at their new depth and experience no buoyancy. Recently adiabatic solutions have been developed by P. Johnston so that it has been possible to estimate the effect of this assumption on the model parameters. Generally the differences between the two models as applied to the British ice load are relatively small, essentially because the surface load is composed largely of high wavenumber terms whose corresponding viscoelastic Love numbers differ little for either the adiabatic or non-adiabatic assumption. Further- more, the two models lead to results that are nearly linearly proportional to each other and, for this region at least, a direct trade off occurs between the degree of adiabaticity assumed and the scale factor /3 for the ice sheet. Both the adiabatic and non-adiabatic solutions lead to the same viscosities but not the same 0, with the latter leading to an underestimation of the ice height by about 10 per cent (Lambeck 1993b). Thus, in the following solutions, only the non-adiabatic models are considered in the discussion of the earth parameters.

It has previously been established (Part I) that the sea-level predictions for the British Isles are not strongly dependent on the properties of the lower mantle because of the relatively small dimensions of the ice load. However, the response to the water load, occurring with dimensions equal to those of the Atlantic Ocean, does stress the lower mantle to a certain degree and some dependence of the sea-level response to qI,,, can be anticipated. Fig. 19 illustrates the variance factor u', as a function of both r]* and qIm for He = 80 km and the result confirms the weak dependence of

Glacial rebound of the British Isles-11 979

' r e r.

p 40 .-'. 80 .

v - U [ 60

4 g -20 6 #7?- 0' 40

8 - B

2 20

B 0 :

. -

p =0.949

.

' ,a. .,' . 20 A$red = - 0.16 + 1.013 A{ob, - 4 0 . '

-60

, /

observed sea level (m) observed-predicted sea level (m)

3 - - 10

I I 1 I I

observed sea level (m) observed-predicted sea level (m)

Figure 18. Statistical evaluation of the minimum variance solutions for (a, b) the first iteration solution and (c, d) the second iteration solution for the three-layered earth model. In (a) and (c) the observed sea-levels are plotted as functions of the predicted values and the (unweighted) best fit linear trend is shown. In (b) and (d) the histograms are of the residuals E,, of eq. (1).

the rebound on qlm, with the smallest value occurring for

qIm = 1.3 x Pas. (4c)

Although the resolving power of this data set for the lower mantle viscosity is poor, models with qlm 5 4 x lo2' Pas appear to be excluded and there is strong support for models in which the mantle viscosity increases substantially across the 670 km boundary, consistent with previous studies of the crustal rebound in other areas (Lambeck & Nakada 1990; Lambeck el al. 1990). How this increase in viscosity is distributed with depth below 670 km cannot, however, be resolved with the present data.

The estimates (4) for the mantle response are effective parameters that describe the response of the Earth's surface- to-glacial unloading on a time scale and surface-load scale that is characteristic of the British ice sheet, but it remains to be demonstrated that this simple three-layered rheologi- cal model does in fact give the best representation of the upper mantle response. Therefore, an additional class of model is considered comprising five mantle layers (Table 1). Fig. 20 illustrates the first-iteration results for this model

980 K . Lambeck

7 -

7 q4=3x1020 Pa s

7(4=5x1020 Pas

7(3'7(4

I

8 0 x 4

W

2

1 2 3 5 7

10 r 1

7

5 - ? ' p 7 X l @ O Pa s

1 2 3 5 7 10

10 q4=10x1#* Pas

7 -

1 2 3 5 7 10

n 3 w 4

q3 ( ~ 1 0 2 0 Pa s) Figure 20. Minimum variance solutions for the second-iteration five-layered earth model as a function of the viscosities ( q 2 , 11) for the two upper mantle layers and the viscosity q4 for the transition zone from 400 to 670 km equal to (a) 3 X 10'" Pa s. (b) 5 X 102" Pas , (c) 7 X 10'" Pa S ,

(d) 10" Pas. The other earth model parameters are He = 80 km, qlm = 5 x 10" Pa s.

with H, = 80 km and in which the upper mantle comprises three layers; the lithosphere, a second layer which extends from the base of the lithosphere to 200 km with viscosity r]', and a third from 200 km to 400 km with viscosity q3. The transition zone down to 670 km forms the fourth layer of viscosity q4. The lower mantle viscosity in this example is equal t o 5 x 10" P a s for all models, near the lower limit permitted by the earlier solution (Fig. 19). Only the range of models in which the viscosity in any layer (other than the lithosphere) i s less than or equal to that of the layer below it (Table 1) are considered at this stage. The variances are plotted as functions of r]' and q3 for discrete values of v4 corresponding to the viscosity of the transition zone between 400 and 670 km depth, and for each q4 value the minimum variance occurs only when r ] , = q3. The overall minimum variance for the H , = 80 km solutions occurs for r ] , = q3 = 4.5 x 10'" Pa s and q4 = 7 x 102"Pa s suggesting that there may be a small increase in viscosity between the upper mantle and transition zone although solutions with

r ] , = q3 = q4 = 5 x lo2" Pa s yield only a slightly larger minimum variance. Fig. 2 l fa ) illustrates the solution variance for those models for which r ] , = q3 and it confirms that any increase in viscosity from the upper mantle to transition zone is either zero or very small. Similar results are illustrated in Fig. 21(b) for Q , ~ = 102'Pas and H, = 80 km and the conclusion is the same as before, that there is no compelling evidence f m substantial layering in the mantle viscosity above 670km. Nor d o the results support mantle models with a significant inversion of viscosity across the 400 km boundary (Fig. 21b).

The pattern of the isovariance curves does not suggest that models with low-viscosity channels (small r]' values) below the base of the lithosphere down t o a depth of 200km, except possibly if both q3 and q4 are substantially increased. However, even in this case, the results in Fig. 20(d) suggest that the smallest variance for such models will be greater than that found for solutions without such a channel. This is further illustrated in Fig. 21(c) for models in which q3 = v4

Glacial rebound of the British Isles-II 981

100

70

50

30

h 20

a" 2 s W x 10

m

N F

7

5

3

2

/

I I I I I I I 2 3 5 7 10 20 30

q3=(q4)(x1020 Pa s) 3

Figure 21. Minimum variance solutions for the second-iteration multilayered earth model with HI = 80 km and (a) q2 = q3 , qlm = 5 x lo2' P a s , (b) q2 = q3, q,,,, = 10'' P a s and (c) with q3 = q4, qlm = 5 x 10" Pa s.

and in which the viscosity q2 of the uppermost mantle layer from the base of the lithosphere to 200 km depth is allowed to be as low as 10" Pa s and the observations do not support mantle models with very low viscosities for the uppermost mantle.

Ice model parameters

While the solutions for earth model parameters, at least in the case of the British rebound, are not significantly affected by the inclusion of the second-iteration corrective term in eq. (l), the estimate of the scale factor p"" for the British ice sheet is, and the inclusion of these terms results in a lowering of the total ice thickness required (compare Figs 17c and d). The solution for the minimum variance three-layer earth model points to p"" = 1.02 f 0.04 (Fig. 17d) and indicates that the maximum ice thickness is unlikely to have exceeded about 1500m over central Scotland, wholly consistent with the geomorphological evidence (Section 2). Some trade off between this parameter and the earth-model parameters does occur (Fig. 17d), with increased lithospheric thickness estimates leading to greater p"" values being required in order to produce a comparable

rebound amplitude, although the minimum variance is increased. These @ estimates are for the so-called fully non-adiabatic models. Fully adiabatic models yield estimates that are about 5 per cent smaller and largely within the uncertainty estimate of this parameter.

The estimated correction 6c(t) to the equivalent sea-level function (eq. 1) from the second-iteration solution is illustrated in Fig. 22 and indicates that after about 6000yrBP, a rise of about 2 m in equivalent sea-level occurred; that sufficient glacial melting continued after this time over the next 1500 to 1000 years to raise sea-level globally by about 2m. This result is consistent with earlier analyses of Late Holocene sea-levels in both the Australian region and the northwestern European region (Nakada & Lambeck 1988; Lambeck et af. 1990) and lends support to glacial models in which the Antarctic ice sheet continued to add meltwater to the oceans after the complete melting of the northern ice sheets. Before 6000 yr BP the equivalent sea-level corrections change sign, indicating a more complex melting model than the relatively uniform melting rates adopted for the ARC and ANT ice models (see Part I). While the accuracy estimates of these corrective terms for earlier times are relatively low, the Sr(t) function points to

982 K. Lambeck

15 10 5 time (x1OOO years BP)

0

Figure 22. The corrective term 6T'(t) to the nominal equivalent sea-level function A r ( f ) . Intervals in which the gradient of this function is positive indicate times of more rapid global melting than assumed in the nominal melting model.

a faster global melting rate in the interval before about 11 000 yr BP than assumed (that part of the function for which the gradient of is positive) and to a slower melting rate at the time of the Loch Lomond Stadial, suggesting that the residual global ice sheets expanded in the interval 11 000 to 10500yrBP so as to lower sea-level by about 3-4m. However, the correction at 10500yrBP is based almost entirely on the nominal date given for the Late Interglacial Buried shoreline and Main Rock Platform of Scotland, and without a firmer basis for this age it would be premature to interpret this anomaly in the equivalent sea-level function in terms of a global sea-level response. A similar examination of the sea-level evidence along the coasts of Norway and southern Sweden is warranted.

Regional models

Generally, agreement between observations and predictions based on the above optimum models (4) with p"" = 1.02 is satisfactory, but some significant discrepancies occur, most notably for sites in the Beauly Firth of northeastern Scotland (Fig. 23c), which point to either a regional variation in the earth response or to local departures of the ice load from that assumed in the model. To examine these alternative interpretations, the observational data base for Scotland and northwestern England has been divided into four subregions: (1) southeastern Scotland, comprising primarily the Forth and Tay estuaries and valleys, (2) northeastern Scotland with the sites occurring in the Moray, Inverness and Beauly firths, (3) the Argyll region of western Scotland including the Clyde Estuary and Inner Hebrides, and (4) the Solway Firth of southwestern Scotland and the Morecambe Bay area of northwest England. In a first calculation, effective earth and ice parameters for the three-layer model are estimated independently for each subregion.

However, the resulting q2 and He all lie within the range of values defined by the overall solution (4) and a strong case for lateral variation in earth response cannot be made particularly as the solutions for both the Moray and Solway regions are not well constrained by the relatively small numbers of observations available for these regions. Hence,

in the second regional solution the earth parameters (4) have been adopted and local p values are estimated independently for each of the four subregions. Of these solutions only the Moray and Solway yield p"" estimates that are significantly different from the combined solution (Table 2). In particular, the Moray solution suggests a need to increase the ice load by about 17 per cent over northern Scotland, while the Solway solution points to a decrease in the ice heights by about 7 per cent over northwest England. These regional solutions are discussed further below.

5 DISCUSSION

A comparison of observations and predictions of sea-level change

The comparison of the predicted sea-levels with the observed levels can be affected in several ways, including the time series of sea-level change at characteristic sites (Fig. 23), as isobases of the mean sea-level for specific epochs (Fig. 24), and as height-distance diagrams of past mean sea-level surfaces relative to their present levels (Fig. 25). For the Forth and Tay firths of eastern Scotland the predicted isobases for mean sea-level at the time of formation of the principal shorelines (Fig. 24) can be compared with the observed isobases in Fig. 4 with the proviso that the latter correspond to the shoreline-formation heights (mainly the MHWS level which is typically at 2-3 m above MSL throughout the region). The generally satisfactory agreement between the two is also evident in the sea-level gradient plots for the major shorelines along the southern shore of the Firth of Forth (Fig. 25a), except that the predicted gradients for the older East Fife 6 and Main Perth Shorelines are somewhat steeper than observed. Along the northern shore of the Tay Firth the observed and predicted gradients are consistent (Fig. 25b) although the predicted heights for the East Fife 6 Shoreline lie below the observed heights. This, together with the Forth result, suggests that the ice load should be increased to the north or northwest of the Firth of Tay.

The regional solution for the Moray-Beauly region indicated a need to increase the ice load in this region (Fig. 23c) and Fig. 24(b) illustrates the sea-level isobases over northern Scotland based on the best-fitting local /3 value of j3 = 1.20. The predicted isobases in the region trend in a nearly easterly direction, whereas the observed trend is east of northeast (compare Fig. 24b with 6) and this points to a need to place the additional ice volume over the region of Ross & Cromarty to the northwest of Beauly (see Fig. 3 of I for location map). This conclusion is reinforced by the limited evidence from Wester Ross (in the area of Malvaig, Fig. 3 of Part I) where the model predicts Late Glacial shorelines that are near or below present mean sea-level (Fig. 24b), whereas the rock platforms attributed to a Late Glacial origin, Sissons & Dawson (1981), occurs at heights of 20-30 m. Thus, a considerably greater ice load is required in the northern region of Scotland unless the Wester Ross rock platform pre-dates the last glacial event. The predictions for the Main Postglacial event in this region are generally consistent with the observed levels reported by Sissons & Dawson (1981). Further north, in Orkney and the

Glacial rebound of the British Isles-11 983

-20

-25

Glencarse & Bridge of Earn Firth of Tay

-

-

- 10 16 14 12 10 8 6 4

Lochgilphead Loch Fyne

-10 1 1 15 10 5 0

(9) 0 1

Fenlands East Anglia 1

-15 1 8 6 4 2 0

time (x1ooO years BP)

Arnprior Forth River 40

15 10 S 0

s

-20 16 Morecambe Bay Lancashire

-25 - 10 8 6 4 2 0

T/ 1 I

Bridgwater Bay f Bristal Channel -30 -

1 0 8 6 4 2 0 time (x1ooO years BP)

Beauly Firth lnverness 30

.. 16 14 12 10 8 6 4 2

Thames Estuary

M .- 10 8 6 4 2 0

time (x1ooO years BP)

Figure 23. Predicted and observed sea-levels at selected sites in Scotland and England. The predictions are based on the earth model parameters defined by (4) and on the local /3 factors (Table 2). For the Beauly Firth the prediction based on the global /3 value is also shown (broken line).

Outer Hebrides, the model predicts sea-levels that are observed and predicted sea-level curves for Lochgilphead below the present level throughout Late Devensian and and the Clyde Firth is also satisfactory (Fig. 23d) and the Holocene time, consistent with an absence of observed association of the Main Rock Platform with sea-level at highstands in the region as well as with sea-levels at 10500yrBP leads to a consistent model (Fig. 25d). The 8000-5000 yr BP in the Outer Hebrides being several metres older and higher rock platforms of western Scotland at below present levels (Binns 1972; Ritchie 1985). heights of up to 50 m OD are predicted to occur only during

For the west coast of Scotland the agreement between the earliest stages of deglaciation, at about 16 000 yr BP

984 K . Lambeck

Table 2. Regional solutions for pBR and corresponding minimum variances for the earth model defined by (4). For the southern England, Wales and northern France solution, the PBR value has been set to unity.

Region P R Variance

Au data 1.03 2 .14

Forth-Tay 1 .oo 0.96

Moray Beady 1.20 0.74

Western Scotland 1.03 1.52

Solway and NW England 0.96 1.78

Southern England, Wales, France 1 .00 1.36

(e.g. Fig. 23d), when the region is believed to have still been ice covered and by the time the Inner Hebrides became ice free the predicted isobases are significantly lower than the observed heights. The model, therefore, supports the interpretation that these high-level platforms pre-date the last glacial cycle.

In neither eastern nor northeastern Scotland do the models predict the short-duration oscillations in sea-level as defined by the buried shorelines between about 10200 and 8700yrBP. If the fluctuations are the results of the Loch Lomond Advance then the ice volumes involved must have been greater than adopted and its duration and timing must be different from what is usually assumed. During the advance, renewed crustal loading will produce an apparent rise in sea-level at a site such as Arnprior near the advance's maximum limit (Fig. 17) but this would be expected to have occurred somewhat earlier than the age of 10200yrBP attributed to the highest of the buried beaches. Also, if the other buried beaches are attributed to the Loch Lomond Advance then the ice retreat would have occurred non-uniformly and more slowly until after 8700 yr BP, the age of the Low Buried Beach. Generally, neither the observational constraints nor the numerical resolution of the present rebound model are adequate to examine in detail the relative sea-level response to the Loch Lomond Advance.

Agreement between observations and predictions for the northern England sites is typified by the comparison for Morecambe (Fig. 23e). In this region the Main Postglacial zero contour isobase for mean sea-level is predicted to pass through Morecambe Bay and small highstands at about 6000 yr BP are predicted for sites such as the Esk or Duddon Estuaries to the north (Fig. 24a). The predictions for these localities suggest that sea-level should have reached its present level earlier than observed although the observa- tional data base (Andrews et al. 1973) is unsatisfactory for this region (see Fig. 1ld). To the south, at Southport and Formby, agreement between observations and predictions is again satisfactory. Also, the gradient of the Main Postglacial shoreline from the Solway Firth to north of Morecambe Bay is predicted to be about 0.04 m km-l which agrees well with the observed gradient of 0.03 m km-' for raised beaches reported by Walker (1966).

For the sites in Wales, southern England and the French Channel coast (Fig. 25), the predicted sea-levels are generally consistent with observations, although at a number of sites the observations lie above the predicted level. This is the case, for example, for the result at Borth Bog in Cardigan Bay (Fig. 25 ) where the tree stumps may define a level above MHWS tide as suggested by Tooley (1979). For the Thames Estuary the discrepancy is greatest for Devoy's (1982) Table 4 results which define mainly upper limits to the sea-level curve. Thus the discrepancies noted for southern England may reflect the different nature of the observational data as much as limitations of the model predictions, and no systematic pattern of the differences for the region as a whole occurs.

Mantle rheology

The effective parameters for the mantle below the Great Britain region are defined by the results (4) for the three-layer earth model as

He = 65-70 km

r ] , = (4-4.5)102" Pa s

4 x lo2' Pa s < qIm < 2 x lo2' Pa s

where r ] , represents the average viscosity for the mantle between the base of the lithosphere and the 670 km seismic discontinuity. These values are effective parameters that are representative of the mantle response to load cycles operating on the order of 104yr. The associated stress and strain-rate fields have not been evaluated at this stage, but simple order-of-magnitude calculations for the mantle above 670 km suggest deviatoric stresses and strain rates of the order of 1 MPa and (2-5)10-16 s-' respectively and considerably smaller values at greater depths.

Tests with multilayered models have shown that: (1) there is no significant change in viscosity from the upper mantle to the transition zone (Figs 20 and 21) and, (2) there is no strong evidence for viscosity layering in the upper mantle, including a low-viscosity layer immediately below the lithosphere (Fig. 20). The absence of a marked viscosity gradient above 670 km is perhaps not an obvious conclusion in view of the many variables that can be expected to affect the creep strength (i.e. viscosity) of the mantle materials. In particular, because garnet appears to have a considerable higher creep strength than either olivine or spinel (Karat0 1989), it has sometimes been argued that the transition zone could be expected to possess a significantly higher viscosity (Spada et al. 1992), but the result of these rebound inversions suggests instead that the creep strength of this layer as a whole is controlled by the weaker spinel phase such that the effective viscosity at ambient temperatures and pressures is not significantly different from that of the predominantly olivine mantle above 400 km depth. Another factor that could produce a strong depth dependence of viscosity is the mantle temperature, but the inferred near constancy of viscosity with depth suggests that, if this is an important mechanism, then the ratio of temperature to melting point temperature does not change significantly with depth in the mantle beneath the British Isles.

The solution for the earth model parameters based on the British Isles data agrees well with that previously

Glacial rebound of the British Isles-II 985

15 000 years BP

6 000 years BP

Figure 24. Contours of the predicted constant relative mean sea-level over northern Great Britain based on the earth model defined by (4) and (a) the global p"" value, and (b) for northern Scotland based on the p value from the northeastern Scotland regional solution.

determined from the Fennoscandian rebound using the same ARC model discussed in Part I). Because the Fennoscan- approach although in the latter case only three-layered dian ice sheet is considerably larger than the British ice mantle models were considered (Lambeck et al. 1990). In sheet the depth dependence of the mantle response to the that study relative sea-level observations were used from loading is different in the two cases so that the similar result sites either near the centre of rebound or well outside the for q2 lends further support to models of nearly constant former ice margin so that the model results were not viscosity profiles between the base of the lithosphere and the strongly dependent on limitations of the ice model used (the 670 km boundary.

986

0 -

15 OOO years BP

10 500 years BP

13 500 years BP \ \ \

-100

6 OOO years BP

- ' I \ c -

\ / I' -20

Figure 24. (Continued.) (b)

Amprior Kincardie Leith b b a r Stkling Aberlady

Firth of Forth (south side)

0 20 40 60 80 100 120 distance (km)

. - _ Moray & Beady firths

_- -5 0 5 10 15 20 25

distance (km)

lo t Tay Firth (north side) n [ I , , . 1

. .. " -20 0 20 40 60 80

distance (h) (d115 I I

Western Scotland T T

O h k 3 g upper Sound cyuy , (Jmf of Jura

-10 -20 0 20 40 60 80 1 0 0 120

distance (km) Figure 25. Predicted'(continu0us lines) and observed (broken lines and error bars) for some of the major shorelines (reduced to MSL) observed in Scotland.

Glacial rebound of the British Isles-I1 987

The effective viscosity q2 obtained for both the Fennoscandian and British regions is higher by a factor of about 2 than that obtained from analyses of sea-level change along the Australian margin. In the last example, the departures from the eustatic sea-level change is primarily a result of the response of the mantle to the water loading and to a transport of material between oceanic mantle and continental mantle. The value of (2-3)102"Pa s obtained (Lambeck & Nakada 1990) can, therefore, be interpreted as some average of the two mantle regions and the difference between this and the European value can be attributed to lateral variation in mantle structure (Nakada & Lambeck 1991). Such variations in upper mantle viscosity are generally consistent with observations of lateral variations in seismic shear wave velocities (e.g. Woodhouse & Dziewon- ski 1984; Tanimoto 1990a,b; Romanowitz 1990) and seismic wave attenuation.

Pa s for the lower mantle viscosity is constrained primarily by the mantle response to the water loading which occurs on length scales up to that of the dimension of the Atlantic Ocean, the resolution is not high (Fig. 19), but models with q l r n 5 4 X lo2' Pas are excluded. The result is consistent with previous studies (Lambeck & Nakada 1990; Nakada & Lambeck 1991) and the conclusion that the lower mantle has a significantly higher viscosity than the mantle above 670 km appears to be a robust one.

While the above estimate of

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