1 Rates of shoreline retreat have been startlingly rapid – locally over a metre a year – at a number of monitored sites on Atlantic coastlines of eastern Canada and elsewhere, notably the northeastern United States. At most such locales, retreat results from the rapid erosion of glacial material forming cliffs and bluffs (Wang and Piper 1982; Taylor et al. 1985; Boyd et al. 1987; Carter et al. 1990a, b; Shaw et al. 1993a; Manson 2002; Himmelstoss et al. 2006). Adjacent beaches reconfigure to keep pace. Many of us, from homeowners to planners, would like to have a basis for predicting patterns of retreat where sea-level continues to rise at rapid rates.

2 Significant parts of these shores expose remnants of glacial drumlins or other unconsolidated glacial material resting atop low shore platforms of bedrock extending out to sea (Fig. 1), commonly as undulating ‘whalebacks’, locally still showing glacial striae (Stea et al. 1992; Grant 1994). Indeed, the position of the shore generally depends on the protection provided by this bedrock in a stable state of temporary equilibrium relative to wave erosion, but rapid retreat of some adjacent shores show that even this apparent stability is fragile and transient.

3 This situation lends itself to analysis of successive configurations of these coastlines during expected future sea-level rise. Such analysis uses the late Holocene history of the same shores based on offshore features and of the shorefaces themselves, but is shown to depend on a variable that has not previously been incorporated into these analyses, namely the slope of the upper surface of bedrock, i.e. a subsurface feature of the adjacent land surface.

4 In coming decades, if shore stability is further disturbed by sea-level rise, better prediction of retreat rates and distribution could be valuable. The purpose of this article is to employ more information about the bedrock underpinning these cliffed shores, specifically the geometry of its upper surface, to enable such prediction.

5 The cliffed shores retreating most rapidly are clearly out of equilibrium with their hydrodynamic environments. The eroding glacial material can form overall slopes of about 70°, and erosional undercutting locally forms overhangs. Commonly, individual storms cut new profiles landward of the previous profiles, and/or trigger slump-block subsidence. Generally, a plume of muddy seawater, short-lived boulders of till in the surf, and slabs of overturned turf at the base of the cliff mark the location of the most severe erosion.

6 Similar shores with slopes of about 45°, generally somewhat vegetated, retreat much more slowly and seem to represent a state closer to equilibrium. Since this state is the one desired by most people, we need to understand the factors that allow it to occur. The more stable shore sectors generally consist of till resting on, or immediately adjacent to, shorefaces of bedrock, high enough to protect the till from rapid wave erosion and of course tough enough to resist its own erosion. The height of such shelves in these more-stable sectors varies from place to place, depending mostly on the local height of storm waves (W), the tidal range (T), and susceptibility to storm surges (S) (Fig. 2). The requisite height of bedrock shelves based on these three variables is shown as 0.5(T+W)+S, represented on Figure 2 as the difference between line 1 (mean sea-level) and line 2, assuming net sediment removal (i.e. requiring a headland-like situation), tough bedrock, and friable till. Note that coastal sectors with only moderate wave energy may still require high bedrock shorefaces if the tidal range is high, allowing smaller waves to reach unprotected till (as in ‘flower-pots’ along the Bay of Fundy). Where storm surge is appreciable and/or frequent there may be a similar result.

7 If in Figure 2 an arbitrary rise in sea-level disturbs this seemingly stable situation (new mean sea-level at line 3 on Figure 2), every aspect of the shore would have to adjust to reach a new state of stability, and the new stable bedrock shoreface height would be at line 4. As this new equilibrium height is projected into the cliff, the point at which it intersects bedrock is the new stable shore position. The position of this point depends on the slope of the surface between bedrock and overlying friable glacial till. Two contrasting scenarios of eventual shore retreat, based on different bedrock slopes, are presented (Fig. 2). If the seaward slope of the contact is steep, the point of intersection (the new stable shore position) is near the present shore, at point A. If instead the slope is very low, the point of intersection is much farther back, and the shore remains unstable until retreat has reached that point (point B). That is, the magnitude of retreat (in map view) varies with the cotangent of the bedrock slope angle. Note that local undulations (Fig. 2) in the bedrock surface could make shore retreat rates vary greatly.

8 Several complications to this simple model are common. First, where glacial material is resistant, erosion rates may be insufficient to prevent the hypothetical equilibrium. For example, where glacial till is exceedingly bouldery, displaced boulders may form a lag at the base of the eroding scarp that is resistant to further erosion (cf. Manson 2002). The resulting talus ramparts may slow shore retreat at constant sea-level, but as sea-level rises, stepover may occur to form bouldery retreat shoals just offshore of the new cliff position.

9 Second, sediment transported along the shore may be deposited in the plane of the cross-section (Fig. 2) and protect the shore from erosion. Locally, progradation may occur despite sea-level rise, especially between headlands and along their distal flanks (Wang and Piper 1982; Boyd et al. 1987; Shaw et al. 1993a). Thus the geometric model of Figure 2 is most applicable to headlands and their flanks, though beaches tangent to these headlands are also affected.

10 Third, bedrock toughness varies from place to place. Analysis here assumes that bedrock projects into the wave zone, preserving some semblance of its upper surface. This generally requires considerable induration, more than is shown by most post-Windsor sedimentary rocks of Nova Scotia, for example.

11 Fourth, bedrock slopes are seldom uniform. Whalebacks separated by bedrock lows are a common arrangement, and this morphology may form cores of the drumlins above. Consequently, coastal drumlin erosion tends to occur at varying rates, depending on the vagaries of nearby bedrock protection.

12 In the Holocene, the eustatic sea-level history in the Maritime Provinces of Canada, like that of other coastal areas worldwide, shows a rapid rise of about 1.2 m per century (or about 12 mm/yr) until about 6 Ka, followed by a slower rise (until the last few decades) of about 1.8 mm/yr. In Atlantic Canada, eustatic rise was complicated by post-glacial crustal rebound, represented regionally by a moving crustal fore-bulge and adjacent sag (Quinlan and Beaumont 1981). This non-eustatic but natural factor differs between localities, but in parts of Nova Scotia results in sea-level rise on the order of 2.5 mm/yr. Additional acceleration of sea-level rise beginning in the 20^th century, presumably spurred by anthropogenic activities (IPCC 2007), is on the order of 1.3 mm/yr. In some areas the sum of these factors, i.e. the measured modern rates of sea-level rise (about 5.6 mm/yr in the Canso Straits area of Nova Scotia) is about half the rapid early Holocene post-glacial rates – thus the urgency to better understand and predict coastal retreat rates (Boyd et al. 1987; Shaw et al. 1993a, b, 1998; Taylor et al. 2011).

13 The morphology of offshore platforms records the eventual consequences of sea-level rise along drum-lin-dominated coasts. Most commonly, the drumlin landforms, so prominent onshore, are obliterated or partially planed-off just offshore, even if Pleistocene material is still extensively exposed (Wang and Piper 1982; Piper et al. 1986; Stea 1997).

14 The geometric model of potential retreat on drumlin-dominated shores presented herein (Fig. 2) implies that any given rise of sea-level has different expected magnitudes of regional headland retreat depending on the slope of bedrock platforms underlying glacial material. Modern rates of sea-level rise in the region permit this rise to be converted to expected rates of coastal retreat, or elapsed times corresponding to a given magnitude of retreat.

15 Modern rates of coastal retreat on shores exposing glacial material are locally rapid in Atlantic Canada and adjacent northeastern United States. Erosion rates of more than one m/yr have been recorded at a number of localities (Taylor et al. 1985; Boyd et al. 1987; Carter et al. 1990a, b; Shaw et al. 1993a; Manson 2002; Himmelstoss et al. 2006). Some of these are in parts of Nova Scotia where the rate of modern sea-level rise is thought to be on the order of 0.5 m per century.

16 These figures can be directly related where bedrock slope controls erosion rate. For example, retreat rates of one m/yr at a sea-level rise of 0.5 m per century correspond to bedrock slopes of 0.5% given the conditions shown in Figure 2. Faster retreat rates would imply a lack of sufficient bedrock platform under the glacial material, slower ones steeper bedrock slopes, or any of the complicating features described above that prevent an approach to equilibrium.

17 To examine whether such conclusions are consistent with known bedrock configurations, an area in southernmost Cape Breton Island will serve as an example.

30 Generally, upper surfaces of bedrock slope seaward throughout the regions addressed here, as this slope basically controls the distribution of land and sea. The degree of seaward slope varies widely, however, and the northeastern Isle Madame example shows that regional bedrock slope may be quite small and surpassed by local variations. In such cases, shore retreat is potentially rapid, limited mostly by the kinetics of removal of the glacial material that forms the more obvious cliffs and bluffs.

31 Eventual shore positions depend on the configuration of the bedrock surface, as bedrock in the shoreface is required to slow or halt retreat. Along many vulnerable shores, systematic information on the bedrock surface is lacking. Mapping of this surface under the land along coasts fronted by glacial material is needed, in order to predict coastal retreat scenarios. A first step in such subsurface mapping can be comprehensive use of existing water-well logs, which in many coastal communities are sufficiently numerous to outline areas of greatest vulnerability.

32 Most coasts exposing glacial material are at best in fragile states of temporary equilibrium with respect to wave energy, tidal range, and storm surges; they are stable only if they are protected by platforms of tough bedrock on which glacial debris rests, by short-lived talus ramparts, or by locally abundant sediment supply. The height of the bedrock platform is an essential part of the equilibrium condition. As sea-level rises, the equilibrium position of the shoreline must move to maintain the same bedrock protection. How much it must move is a function of the slope of the upper bedrock surface under glacial material.

33 An example shows that some rapidly eroding coastal segments are significantly out of equilibrium already, but are temporarily relieved by slow kinetics; i.e., the bulk rates of removal of glacial material and eroded debris are limited. Rising sea-level in many areas could change coastal morphology significantly if the new equilibrium position is far different and kinetic rates are accelerated – and they can be accelerated by human activity. Systematic inventory of subsurface bedrock position is needed in evaluating these risks along drumlin-dominated and other glaciated shores.