Tors occur in many granitic landscapes and provide opportunities to better understand differential weathering. We assess tor formation in the Cairngorm Mountains, Scotland, by examining correlation of tor location and size with grain size and the spacing of steeply dipping joints. We infer a control on these relationships and explore its potential broader significance for differential weathering and tor formation. We also assess the relationship between the formation of subhorizontal joints in many tors and local topographic shape by evaluating principle surface curvatures from a digital elevation model of the Cairngorms. We then explore the implications of these joints for tor formation. We conclude that the Cairngorm tors have formed in kernels of relatively coarse grained granite. Tor volumes increase with grain size and the spacing of steeply dipping joints. We infer that the steeply dipping joints largely formed during pluton cooling and are more widely spaced in tor kernels because of slower cooling rates. Preferential tor formation in coarser granite with a wider joint spacing that is more easily grusified indicates that joint spacing is a dominant control on differential weathering. Sheet jointing is well developed in tors located on relatively high convex surfaces. This jointing formed after the gross topography of the Cairngorms was established and before tor emergence. The presence of closely spaced (tens of centimeters), subhorizontal sheeting joints in tors indicates that these tors, and similarly sheeted tors elsewhere, formed either after subaerial exposure of bedrock or have progressively emerged from a regolith only a few meters thick.
Tors are conspicuous rock masses that rise above the surrounding ground surfaces. They form in many rock types but are most commonly associated with granites [Migoń, 2006, p. 96]. Tor formation involves differential weathering, which is generally expressed where conditions within distinct pods of rock predispose these pods to slower weathering than the surrounding rock. If these resistant pods become subaerially exposed above a regolith cover, differential weathering is further promoted because weathering rates are lower on bare rock surfaces than surfaces that are covered by a thin regolith [Gilbert, 1909; Small et al., 1997, 1999]. Tors commonly develop where the joint spacing is wider, the crystal sizes are larger [Gibbons, 1981; Ehlen, 1992], and/or biotite [Eggler et al., 1969] or K-feldspar [Gibbons, 1981; Pye, 1986] is more abundant than in the surrounding bedrock. In some settings, where regolith thicknesses is no more than a few meters thick, such as in some arid environments, permafrost settings, and/or locales of especially rapid erosion, tors form where regolith is being most rapidly stripped [Anderson, 2002; Strudley et al., 2006]. Our primary objectives are to identify correlations between structural and petrological properties of granite associated with tors in the Cairngorm Mountains, Scotland, explore controls on these correlations, and highlight general implications of these controls for differential weathering and models of tor formation.
Two end-member models are commonly invoked to account for tors. In the first, tors form through a two-stage process, where they are initially sculpted by subsurface weathering within or beneath a regolith cover tens of meters thick, and then exposed subaerially by regolith stripping (Figure 1a) [Linton, 1955]. In this classic model, tors may reflect a change from an environment conducive to deep weathering to one where erosion dominates [Linton, 1955]. In the second end-member model, developed from observations in periglaciated landscapes, tors form in a single-stage process (Figure 1c) [Palmer and Nielsen, 1962]. In this model, bedrock is subaerially exposed by the downslope transport of mobile regolith and sculpted by subaerial processes into tors. Both end-member models imply climatic control on the processes of tor formation and resulting tor forms, with rounded and angular forms being used to infer dominance by chemical and physical weathering, respectively [see Ballantyne and Harris, 1994, p. 180 and Migoń, 2006, p. 109 for discussion]. However, because tors occur across a broad climatic range and similar tor morphologies may develop under differing climates and weathering processes, a climate control on tor formation remains uncertain [Ballantyne and Harris, 1994, pp. 180–181; Migoń, 2006, p. 104]. We use our findings to help distinguish which of these models, or a third, intermediate model (Figure 1b; discussed below), is best suited to explain the Cairngorm tors.
Tors in formerly glaciated landscapes, such as the Cairngorm Mountains (Figure 2), developed beneath surfaces that escaped deep glacial erosion (i.e., tens of meters of rock removal, vertically), either by being exposed as nunataks or by burial beneath cold-based ice during glacial periods [Linton, 1959; Sugden, 1968; Sugden and Watts, 1977; Ballantyne, 1994; Hättestrand and Stroeven, 2002; Briner et al., 2003; André, 2004; Phillips et al., 2006]. Partial glacial erosion of tors occurred in some places in the Cairngorms during comparatively short periods of warm-based ice coverage [Phillips et al., 2006]. Because of their spatial association with surfaces that have undergone little or no glacial erosion, tors in formerly glaciated landscapes have traditionally been interpreted as having pre-Quaternary origins. Accordingly, the two-stage model (Figure 1a) [Linton, 1955] was invoked, where tors were initiated within regoliths tens of meters thick that had developed under warmer Neogene climates and then subaerially exposed by periglacial regolith stripping during the colder Quaternary [Sugden, 1968; Bierman et al., 1999; Hättestrand and Stroeven, 2002; André, 2004; Thomas et al., 2004]. However, using a space-time transformation and cosmogenic nuclides to measure exposure ages of tor summits in the Cairngorm Mountains, Hall and Phillips  and Phillips et al.  developed a model of tor evolution through progressively more intense glacial erosion. They demonstrated that these tors had minimum exposure histories confined to the late Quaternary and invoked an intermediate formational model, where tors progressively emerged from a regolith only a few meters thick (Figure 1b). Their model implies higher rates of differential weathering, nonglacial erosion, and tor formation than had been previously recognized in this periglaciated landscape.
Here we assess controls on tor formation by investigating the origins of steeply dipping joints and subhorizontal joints that mirror surrounding ground surfaces. Steeply dipping joints occur in, and bound, all Cairngorm tors, and joint spacing has been suggested [Ballantyne, 1994] to control formation of these tors. Subhorizontal joints are prominent in many of the Cairngorms tors. However, the origins of these joints in tors have remained enigmatic [e.g., Migoń and Lidmar-Bergström, 2001, Figure 15; Migoń, 2006, pp. 30–33] and their potential relevance to tor formation is unexplored. We first assess whether tor location and size correlate with grain size and/or the spacing of steeply dipping joints. While studies of the Dartmoor tors and of granitic landforms elsewhere [Bateman and Wahrhaftig, 1966; Gibbons, 1981; Folk and Patton, 1982; Pye et al., 1984; Huber, 1987; Ehlen, 1992; Moore, 2000, pp. 334–335] have identified correlations between joint spacing, crystal size, and/or tor locations, none of these studies has explored why these correlations occur. We therefore attempt to identify the control(s) on these relationships and explore their potential broader importance for differential weathering and the formation of regolith and tors. Second, we assess a possible relationship between the formation of subhorizontal joints in tors and local topographic shape by evaluating principle surface curvatures from a digital elevation model (DEM) of the Cairngorm Mountains. We then explore the implications of these joints for tor formation models.
2 Geological Background
The Cairngorm Mountains comprise low-relief plateaus at altitudes of 900–1200 m above sea level (asl). Although inundated by ice during the Quaternary glaciations, many summit areas were subjected to cold-based subglacial conditions and display only minor glacial modification [Phillips et al., 2006]. Conversely, zones of structural weakness have been extensively eroded by fluvial and glacial processes, creating troughs as deep as 750 m between the plateaus [Glasser, 1995; Hall and Glasser, 2003; Thomas et al., 2004]. The Cairngorms are formed in a granitic pluton, which has a maximum exposed width of about 30 km in an E-W direction and extends a maximum of 17.5 km N-S. The granite is described by Thomas et al.  as relatively homogeneous, containing similar proportions of quartz, K-feldspar, and plagioclase, with smaller amounts of biotite. Crystal sizes generally range from <0.25 to 8 mm with a median of 5–6 mm. According to the Thomas et al.  crystal-size classification, most of the massif is underlain by medium-grained granite (4–7 mm; Figure 2). Fine-grained granite and microgranite (<4 mm) are centered on Sgòr Gaoith, Ben Macdui, and parts of Ben Avon, whereas coarse-grained granite (>7 mm) is centered on Glen Avon (Figure 2). The texture of the granite typically ranges from aphyric to moderately phyric. However, strongly phyric granite is restricted to Glen Avon and patches near Ben Macdui and Einich Cairn. Microgranites and pegmatites occur throughout the massif as patches or sheets. The dominant trends of major landscape features (e.g., valleys and ridges) and the strike of major structures (e.g., faults and master joints), visible in aerial photographs and satellite images, are NNE-SSW and ESE-WNW. Steeply dipping fractures with these strikes are also visible in cliff faces and in some of the tors.
Granite tors are numerous in the Cairngorm Mountains. They range in height from 1 to 23.5 m, with a mean of 4.3 m [Ballantyne, 1994; Mottram, 2001]. We examined petrologic and structural characteristics of 36 tors selected from five plateaus (Figure 2): (1) Ben Avon: 11 tors, nine of granite (BA1–9) and two of microgranite (μG1–2); (2) Bynack More: five tors (BB1–5); (3) Beinn Mheadhoin: seven tors (BM1–7); (4) Cairngorm: three tors (CG1–3); and (5) Sgòr Gaoith: 10 tors (SG1–10).
Most of these tors exhibit no more than moderate glacial modification and were selected on this basis (Table S1) as well as accessibility. According to the Hall and Phillips  classification scheme of tor glacial modification, these are stages 1–3 tors, where stage 1 tors maintain detached crowning boulders and display no glacial modification and stages 2–3 tors display glacial modification through the progressive loss of crowning boulders. More modified tors occur on Beinn Mheadhoin (BM2) and Cairngorm (CG1), where some tor bedrock was removed (stage 4). The most modified (stage 5) tors are on Sgòr Gaoith (SG2, 4, 5, 8–10) where tors are reduced to plinths and the surrounding regolith was glacially eroded. Two tors, BB5 and BA4, also appear as plinths but are interpreted as emerging tors because surrounding surfaces show no evidence of glacial erosion.
The visibility of the two sets of regional fractures varies among the five plateaus. Both the ESE-WNW and NNE-SSW fracture systems are well developed at the Sgòr Gaoith tors, which have equant shapes in plan view, although the ESE-WNW system generally dominates (Figure 2). On Beinn Mheadhoin, the ESE-WNW fracture system is better developed and elongated tors occur parallel to this fracture system (Figures 2 and 3b). On Cairngorm, the ESE-WNW fracture system is also present but it has less effect on tor shapes. The NNE-SSW fracture system is best developed on Bynack More and Ben Avon. However, its influence on tor shape and orientation is minimal with the exception of the Little Barns of Bynack (BB4), which form the ridge crest on the northern flank of Bynack More. Most other tors on Bynack More and Ben Avon have more equant shapes in plan view (Figure 2). Only the largest tors on Ben Avon (BA1 and BA8) are elongated, but the elongation direction is oblique to both fracture systems.
To evaluate controls on tor formation, we measured the size of tors, the spacing of steeply dipping and subhorizontal joints, and the lengths of feldspar and quartz crystals. We also estimated connected porosity and analyzed Sr/Zr ratios in granite samples to investigate possible controls on joint formation and differential weathering important to the development of tors. For comparative purposes, crystal lengths were also measured on boulders in surrounding autochthonous blockfields. We also analyzed how the formation of subhorizontal joints might be related to local topographic shape by calculating surface curvatures on a DEM of the Cairngorm Mountains.
We measured the length, width, and height of the 36 selected tors, using a tape measure and/or laser rangefinder. We used satellite images and aerial photographs (with a minimum of 0.25 m resolution) to correct for horizontal measurement errors during draping. We measured the spacing between steep joints, regardless of their trace lengths, along two profiles on each tor; one parallel to its length and the other parallel to its width. We also measured the vertical spacing of subhorizontal at each tor and calculated approximate tor volumes by multiplying the dimensions of the tors along the long horizontal, short horizontal, and vertical axes. These values overestimate volumes for tors, particularly those with irregular plan forms, such as BA9 (Figure 2), or those that taper toward their tops, such as BA8. We consider the tor volumes to be overestimated by up to 10–20%.
At each tor we measured the lengths of 25 quartz and 25 feldspar (plagioclase and K-feldspar) crystals. We began with a relatively large crystal on a fresh surface and then measured all crystals in its immediate vicinity. While there is a bias in the initial crystal selection, using a relatively large crystal for orientation eliminated the possibility of measuring crystals twice. Crystal lengths were also measured on boulders in surrounding autochthonous blockfields at distances tens of meters from tors on Bynack More and along transects originating from tors on Beinn Mheadhoin and on Cairngorm. We could reliably distinguish between quartz and feldspar in the field but not between plagioclase and K-feldspar. From each set of measurements, we calculated a mean crystal length and standard deviation. An initial field assessment of granite petrography at each tor was also undertaken using at least 450 point counts conducted with a wire mesh grid. One person (AS) performed all measurements to avoid interoperator variability.
We sampled relatively fine- and coarse-grained granite from eight locations on Bynack More for analysis of connected porosity (principally grain-scale fracture porosity). The relatively coarse-grained samples were removed from minimally weathered tor bedrock where this did not risk modifying the esthetic appearance of the tor, whereas relatively fine-grained granite was sampled from minimally weathered granite boulders in surrounding autochthonous blockfields. The samples were impregnated with a blue dye under vacuum, which fills connected voids in the granite, and thin sections were prepared. The National Institutes of Health public domain software ImageJ (http://rsb.info.nih.gov/ij) was then used to calculate the number of blue pixels on scanned images of each thin section to give an estimate of connected porosity. The connected porosities of the fine- and coarse-grained granite samples were then compared to determine if there was a grain-size control on weathering susceptibility that might be explained by differences in water access to the rock matrix.
To assess potential controls on crystal size, XRF-derived Sr/Zr ratios were analyzed in granite samples from 22 tors. Strontium (Sr) belongs to the large-ion lithophile (LIL) elements, whereas zirconium (Zr) belongs to the high field strength (HFS) elements. The LIL elements are generally considered to be more mobile than HFS elements in a fluid or a fluid-rich melt. The Sr/Zr ratio may therefore provide a first-order indication as to whether large crystals in tor granite reflect slow cooling and therefore more time for crystal growth or that the melt was fluid rich during crystallization, which allows for faster diffusion rates and therefore faster crystal growth.
Joints subparallel to surrounding ground surfaces occur in many of the Cairngorms tors. To evaluate how these joints are related to topographic shape at the length scale of ridge-valley spacing, we calculated the two principal normal curvatures of the ground surface on a grid with 10 m spacing from the NextMap DEM of the Cairngorm Mountains, which has a cell size of 5 m. At every point on a topographic surface, the surface bends up and down the most in two perpendicular directions, with the amount of bending in those directions being the magnitudes of the principle curvatures (k1 and k2). The DEM data first were filtered (smoothed) to retain features with wavelengths greater than 1000 m and to remove features with wavelengths less than 500 m [Perron et al., 2008]. The filtering process yielded artifacts restricted to within 300 m of the edges of the original DEM data set, so we evaluated the curvature within a slightly smaller central area devoid of those artifacts using techniques described by Martel . The resulting local topographic shapes were then classified according to their principal normal curvatures as either domes (doubly convex), ridges (singly convex), saddles (concavo-convex), planes, valleys (singly concave), or bowls (doubly concave). Principal curvatures with absolute values less than 0.0015 m−1 were rounded off to zero; without doing this, all the local topographic shapes would be identified as either domes, saddles, or bowls. As the tolerance level is reduced, the sensitivity is increased, and areas with ridges, valleys, or planes are depicted as increasingly intricate composites of smaller domes, basins, and saddles. Using these filter parameters and tolerances, we produce a curvature map that shows how the formation of joints subparallel to the ground surface relates to ridge-scale topographic form.
The following three subsections report findings on the following: (4.1) joint characteristics; (4.2) relationships between joint spacing, crystal sizes, and tors volumes; and (4.3) factors important to granular disintegration of granite on and around tors.
4.1 Jointing Observations and Measurements
Jointing varies widely among, and frequently within, individual tors (Figures 3 and 4 and Table S1), contributing to broad variations in tor volumes, shapes, and other morphological characteristics, such as whether tors appear composed of stacks of subhorizontal sheets where the spacing of steep joints is much larger than sheet thickness (Figure 3e) or of subrounded incipient boulders where the spacing of steep joints is similar to sheet thickness (Figure 3d). Generally, joint spacing perpendicular and parallel to the long axis of the tor is <4 m. The widest spacing mostly occurs between steeply dipping joints on Ben Avon tors (Figure 4). Joint spacing for each tor generally approximates a normal distribution, although outliers are common (Figure 4).
In Figure 5, joint spacing is plotted against the cube root of tor volume. The reason that we plot the cubic root of tor volume rather than tor volume is to allow for direct comparison of length scales, i.e., “tor length” versus joint spacing, both in units of meters. The data show a trend of the cube root of tor volume (and tor volume itself) increasing with joint spacing (Figure 5). A linear least squares best fit between the cube root of tor volume and the mean spacing of joints that cross the long horizontal, short horizontal, and vertical axes of tors provides an R2 value of 0.52 (Figure 5a). If subhorizontal joints are excluded under the assumption that they do not control the area of a tor in plan view, and the mean spacing of joints intersecting the long and short horizontal axes are weighted according to axis length to better reflect spacing of the most abundant joints in each tor, the linear correlation between the cube root of tor volume and joint spacing has a R2 value of 0.45 (Figure 5b). Least squares best fits for the cube root of tor volume versus joint spacing along each of the long and short horizontal axes provide R2 values of 0.32 and 0.28, respectively (data not shown). Because some of the tors are equant or have irregular shapes in plan view, we draw inferences from the data that combine joint spacing from both the long and short horizontal axes (Figure 5b).
Steep joints commonly extend through tor masses and control tor shapes in plan and profile view (Figures 2, 3a, 3b, and 3d). Many of these joints are open because of weathering along them (Figures 3a–3d) but some contain a quartz or hematite fill [Thomas et al., 2004]. Joints that crosscut or terminate against other joints have formed more recently. For example, the subhorizontal joints in Figure 3d terminate against the steep joints, indicating that the latter joints formed first.
Qualitative observations of block sizes in the surrounding autochthonous regoliths indicate that the spacing of steeply dipping joints away from the tors is smaller than that occurring in the tors. These observations are, however, based on surface blocks that frequently have their long axes partly buried in the unconsolidated granular (grusic) matrix of the blockfields. In addition, chemical weathering rates appear higher on block surfaces that are buried in the grus, as indicated by easier disintegration upon striking these buried surfaces with a hammer compared with striking subaerially exposed block surfaces. Because of partial burial in grus, block dimensions are, at best, difficult to measure and because of enhanced chemical weathering such measurements may underestimate steeply dipping joint spacing in the underlying bedrock. Conditions precluded excavating regolith and measuring joint spacing in the underlying bedrock, the transition to which is also indistinct in these types of autochthonous blockfields [Goodfellow et al., 2009]. From our qualitative observations, and our data indicating that tor sizes become smaller as joint spacing narrows (Figures 5a and 5b), we tentatively suggest that steeply dipping joint spacing is narrower in surrounding regoliths than in the tors, in agreement with previous observations in the Cairngorms [Ballantyne, 1994; Thomas et al., 2004].
The tors in most locations exhibit a prominent set of joints that are subparallel to the ground surface (Figures 3c, 3d and 6). These joints are gently curved, mirror the topographic surface, and cut microgranite and pegmatite dykes without offsetting them. In tors they are generally spaced <1 m apart, but their spacing increases with depth, as revealed in cliff exposures (Figure 7). They commonly cut, or terminate against, steeply dipping joints or dykes and are therefore the youngest structures in the tors. They thus exhibit the characteristics of sheeting joints [e.g., Gilbert, 1904; Jahns, 1943], and that is how we interpret them. In order to generate the tensile stresses perpendicular to the ground surface at shallow depth needed to produce sheeting joints, strong compressive stresses must parallel the topographic surface, and the topographic surface must be convex in at least one direction [Martel, 2006, 2011]. In most tors (e.g., BM4 in Figure 3e), the sheeting joints are slightly sinusoidal, with a wavelength of a few tens of centimeters. They commonly curve toward, and intersect, other subhorizontal joints. Some subhorizontal joints extend tens of centimeters into the surrounding granite from steeply dipping microgranite dykes (Figures 3c and 3d). Some of these joints might be associated with cooling of the dykes, but they also could be sheeting joints that either nucleated at, or terminated against, the dykes.
The presence, spacing, and sinuous form of sheeting joints vary among tors, even between some that are closely spaced, such as BA1N and BA1S, BA6 and BA8, and BB1 and BB4 (Figures 3, 4, and 6). Generally, the presence of sheet jointing correlates with surface convexity, consistent with factors that promote sheeting joint formation [Martel, 2006, 2011]. For example, sheet jointing occurs on BA7, which is located where topography is doubly convex (i.e., dome shaped) but is absent from BA8, which is located 650 m away on a planar ground surface (Figure 6). Even more marked is the presence of sheet jointing on BA1N, which straddles a convex summit, but absence from the adjoining BA1S, located on a low-convexity flank (Figure 6). Sheet jointing is also prominent on BB4, located on the convex crest of the Bynack More. Sheeting joints mirroring the surrounding topography are absent, however, from BB1, located 195 m to the east. There, joints have formed subparallel to the convex surface of BB1 and divide the rock into thin (<10 cm thick) sheets that are flaking off the tor (Figure 3f). Those joints might alternatively be considered spallation joints [Migoń, 2006, p. 32]. Tors with sheeting joints consistently develop on ridges, where the topography across the ridge is convex (Figure 6).
In summary, joint spacing is highly variable within and among tors. Steeply dipping joints control tor shapes in plan and profile view. Subhorizontal joints, interpreted as sheeting joints, occur in most tors, which are located where the topography is convex in at least one direction, usually along present ridge crests. Tor volumes increase with the spacing of joints, and the spacing of steeply dipping joints at the tors appears to be wider than in surrounding autochthonous blockfields.
4.2 Petrological Relationships
The Cairngorms tors have a generally homogenous, granitic composition (Table S1). They display mean abundances of 37% quartz, 33% plagioclase, mostly tending toward the albite end-member (29%), 26% K-feldspar, and 4% biotite, which is the only primary mafic phase observed (Table S2). Tor feldspar crystals are usually longer than quartz crystals, generally ranging between 4 and 15 mm compared with a range of 2 to 8 mm for quartz (Figure 8 and Table S2). The tors also typically contain feldspar phenocrysts, as indicated by large outliers (ovals in Figure 7). This is particularly the case for BA4, which is excluded from further analyses because it is an emerging, low-profile tor distinctly different in shape from other tors. Feldspar crystals are generally longest in the Ben Avon and Bynack More tors (mean = 9.4 mm) whereas quartz crystals are generally longer in Ben Avon, Bynack More, Beinn Mheadhoin, and Cairngorm tors (mean = 4.9 mm) than in the microgranite and Sgòr Gaoith tors (mean = 2.9 mm). The lengths of feldspar and quartz crystals have approximately normal distributions in most tors. Considering both feldspar and quartz crystal lengths, the Ben Avon and Bynack More tors are generally the coarsest grained.
Tor volume and joint spacing both increase with crystal length (Figure 5). Linear least squares best fits of the cube root of tor volume and joint spacing with crystal length have R2 values of 0.40 and 0.33, respectively (Figures 5c and 5d). In each case, crystal length is the mean of quartz and feldspar, weighted for mineral abundance. Some of the scatter in Figures 5a–5c can be attributed to glacial erosion, which reduces tor size, in particular in the Sgòr Gaoith group. If tors most modified by glacial erosion (i.e., stages 3–5 tors according to Hall and Phillips ) are excluded from the analyses, then R2 values in Figures 5a and 5b are increased to 0.59 and 0.56, respectively, whereas the R2 value in Figure 5c is 0.32. However, while glacial erosion of tors partly explains the observed data scatter, other factors may also contribute to this. These include the likelihood that tors are at different stages of unroofing, the presence of phenocrysts, which skews the crystal length data sets (Figure 8), and our simplified tor volume calculations. Applying estimated errors (Table S2) to our tor volume calculations changes the slopes of the regression lines in Figure 5 but R2 values vary by <5%. Our results are therefore insensitive to estimated errors in our tor volume calculations.
The linear least squares best fits of both tor volume and joint spacing are better for mean feldspar length than for mean quartz length (Figure 9). The fit between the cube root of tor volume and mean feldspar length has an R2 value of 0.42 (Figure 9a), compared with an R2 value of 0.31 for the cube root of tor volume plotted against mean quartz length (Figure 9b). In plots of joint spacing versus the mean lengths of feldspar and quartz crystals (Figures 9c and 9d), the linear least squares best fit is again better for feldspar (R2 = 0.28) than for quartz (R2 = 0.16), although in both cases it is lower than for the combined mean crystal lengths (R2 = 0.33; Figure 5d). In addition, these minerals are longer in tors than in the surrounding autochthonous blockfields. This is shown by spot measurements on Bynack More (Figure 10a) and by measurements along an inter-tor transect on Beinn Mheadhoin (Figure 10b) and along a cross-tor transect on Cairngorm (Figure 10c).
In summary, the volumes of these granitic tors generally increase with feldspar and quartz crystal lengths, and joint spacing also generally increases with crystal length. Coarser grain sizes occur in tors than in surrounding autochthonous blockfields.
4.3 Observations of Relationships Between Grain Size, Connected Porosity, and Grus Production on and Around Tors
Thin sections reveal that the connected porosity of the rock matrix is higher in fresh, coarser-grained granite (mean = 1.4%; 1σ = 0.8; n = 3) than in fresh, finer-grained granite (mean = 0.9%; 1σ = 0.5; n = 5). In examples from Bynack More (Figure 11), the connected porosity is largely attributable to grain-scale cracks and the connected porosity of the coarse-grained granite sample is 2.3% versus 0.4% for the comparatively fine-grained granite sample. In addition, qualitative field observations indicate that susceptibility to grusification (i.e., granular disintegration) through physical and chemical weathering increases with grain size. Microgranite outcrops maintain angular edges, whereas edges on coarser-grained tors, such as those on Bynack More, are well rounded by grusification. A similar grain-size control on edge rounding has been reported previously by Ballantyne and Harris [1994, pp. 168–171]. Furthermore, a particularly grusic regolith has developed in relatively porous, coarse-grained granite on Ben Avon (Figure 2 and the BA1 panel in Figure 6). This compares with a blockier regolith mantle on Beinn Mheadhoin, which is developed on granite with a less porous, finer-grained matrix (Figures 4c and 10b). Both regolith mantles are apparently little modified by glacial erosion, thereby negating a potentially confounding source of variability. This is evidenced by the preservation of tors and autochthonous blockfield mantles [Hättestrand and Stroeven, 2002; Briner et al., 2003; Phillips et al., 2006; Goodfellow, 2007] and a paucity of glacial erosion features. In summary, connected porosity in granite and its susceptibility to grusification increase with grain size.
Our results indicate that the tors of the Cairngorm Mountains have formed in spatially distinct kernels of granite that are relatively coarse grained and have a relatively wide spacing of steep joints compared with the surrounding granite from which regolith mantles have developed (Figures 4, 5, 9, and 10). Qualitative observations of block sizes indicate that the finer-grained granite from which the regoliths are derived is also more densely jointed, in agreement with conclusions of Ballantyne  and Thomas et al. . While the comparatively wide joint spacing of the tor kernels has also been previously recognized [Thomas et al., 2004], their large crystal sizes relative to the surrounding plateaus have not. Notably, relative differences in grain size and joint spacing between the kernels and surrounding granite, rather than absolute values, govern whether a tor forms. However, the size of the resulting tor increases with the grain size and joint spacing of the granite kernel (Figures 5a–5c, 9a, and 9b).
We see a few lines of evidence that indicate and support the interpretation that steeply dipping joints in many places in the Cairngorm Mountains developed during the cooling of the Cairngorm granite. We first present the evidence collected by Thomas et al.  that indicates that thermal stresses generated during the cooling of the granite controlled the strike of the steeply dipping joints at the scale of the Cairngorms as a whole. We then discuss some of our findings that indicate these same controls might have operated at scales relevant to the formation of the tors and are also important to the spacing of steeply dipping joints.
The major topographic lineaments (e.g., streams) that cross the intrusive contact between the Cairngorm granite and the older metasedimentary rocks [Thomas et al., 2004, Figure 1] (Figure 2) generally do so at angles between 70° and 90°. Examples include the following (proceeding clockwise from the north-central margin of the massif): (1) Strath Nethy; (2) the valley immediately east of Bynack More; (3) the two branches of Glen Quoich (on the southeast side of the massif); (4) the west branch of the River Dee; (5) Glen Dee; (6) the valley extending southeast from Carn Ban Mor; (7) the valley extending northwest from Carn Ban Mor; (8) the largest unnamed valley at the north side of the west end of the massif; (9) Glen Einich; (10) Lairig Ghru; and (11) the inlet stream to Glen More. In addition, three of the major streams and valleys (River Gairn, Glen Derry, and Glen Feshie) parallel the intrusive contact between the Cairngorm granite and the older metasedimentary rocks [Thomas et al., 2004, Figure 1]. The only major stream on Figure 1 of Thomas et al.  that clearly flows obliquely to this contact is the River Avon in the northeast part of the massif. The stream that flows SSE into Geldie Burn on the south side of the Cairngorm massif crosses the contact obliquely, but the contact turns to the NNW, parallel to the stream, less than a kilometer west of the stream. A large majority of the streams thus cross the external contact of the Cairngorm granite at angles greater than 70°. This relationship indicates that if the streams are controlled by preexisting fractures, the fractures might have formed in response to thermal stresses that arose during cooling of the plutons [e.g., Gerla, 1988; Bergbauer and Martel, 1999].
At many places within the massif, the valleys parallel fractures, especially quartz veins [e.g., Thomas et al., 2004, Figure 30]. We infer that many of the streams are eroding along quartz veins or associated fractures. Quartz veins typically are intruded in the late stages of the emplacement of plutons. Since these veins are opening mode fractures [e.g., Pollard and Segall, 1987], they tend to define surfaces within the massif that were perpendicular to the local most tensile stress when the host pluton cooled. The structural observations of Thomas et al.  thus both indicate and support a thermal stress control on at least some of the major fracture systems in the Cairngorm granite.
Thomas et al.  also mapped units of the coarse-grained, medium-grained, fine-grained granite and microgranite within the Cairngorm massif (reproduced here in Figure 2). Again, many stream valleys within the massif cross the contacts between these internal bodies at angles near 90°. Examples include the following (proceeding clockwise from the north-central margin of the massif): (A) River Avon at the north end of Beinn a' Bhuird; (B) the south branch of River Avon north of Ben Avon; (C) the eastern branch of the stream of Glen Quoich; (D) the two streams on opposing sides of Derry Cairngorm; (E) the two branches of the stream of Glen Dee that wrap around The Devil's Point; (F) the stream at the west end of the massif that flows into Glen Feshie; (G) Lairig Ghru; (H) the inlet stream to Glen More; and (I) possibly the headwaters of the River Avon. These relations also indicate and support the interpretation that major fracture systems within the Cairngorm granite developed during its cooling. Indeed, it would be difficult to account for these systematic relationships if they were not associated with cooling of bodies within the Cairngorm granite.
At the scale of the Cairngorm massif as a whole, the topographic lineament data and the fracture data are both indicative and supportive of thermal stresses associated with “subplutons” of different grain sizes controlling the fracturing. We infer that thermal stresses during the intrusion of the granite might also have controlled fracture formation at smaller scales relevant to the formation of the tors. For example, in the large, elongated tors on Ben Avon (BA1 and BA8), the strike of the dominant joint set is orthogonal to the long axis of each tor (Figure 2). Notably, the orientations of the long axes of these tors, which are located ~1200 m apart, are also (approximately) orthogonal and both are oriented obliquely to the regional ESE-WNW and NNE-SSW joint strikes. These latter observations indicate a local, rather than regional tectonic, control on the formation of these dominant joint sets, and together, the observations are consistent with radial joint formation in response to thermal stresses in elongated inclusions [Gerla, 1988; Bergbauer and Martel, 1999].
We now apply our data to the cooling hypothesis and consider whether kernels of coarse-grained granite with a wide joint spacing have formed from intrusions of fluid-rich melts and/or through slow cooling. Sr/Zr ratios plotted against weighted mean feldspar and quartz crystal lengths and against tor volumes permit an exploration of the potential contributions of these processes to kernel formation (Figures 12a and 12b). Two distinct groups of tors are apparent. The first of these includes CG2, BA1N, BA1S, BA8, and BA9, which possess high Sr/Zr ratios (>0.47), indicating that these tors had possible origins as fluid-rich melts. Granites derived from fluid-rich melts might display large feldspar and/or quartz crystal sizes if the cooling rate was rapid, because crystallization rates are also rapid, but not necessarily exhibit wide joint spacing and therefore large tors. Here, however, the tors with the highest Sr/Zr ratios account for four of the five largest tors in addition to displaying medium to large crystals. Furthermore, these tors have mean joint spacing that either exceed the joint spacing versus crystal-size trend line (BA8 and CG2) or lie only marginally below it (BA1N, BA1S, BA9; Figure 12c). Tors with moderate Sr/Zr ratios (0.35–0.47) also occur on either side of this trend line, rather than solely below it. It therefore appears that while some tors may have developed in granite derived from fluid-rich melts, cooling rate has likely been the key control on the spacing of steeply dipping joints and tor volumes. The contribution of fluid-rich melts to tor kernel formation adds to the observed scatter in correlations of tor volumes with joint spacing and crystal lengths (Figures 5 and 9).
The majority of tors in Figure 12 form a low-to-moderate Sr/Zr group (0–0.47), which is compatible with the host granites originating from slowly cooling melts. A slow cooling origin for many tor kernels is further indicated by the positive correlation of tor volume with both joint spacing and crystal length (Figures 5a–5c). This is because large bodies can generally be expected to take longer to cool than small bodies, thereby permitting formation of larger crystals and wider spaced joints. Had these tor kernels been generally derived from fluid-rich melts, in the absence of any volume control on cooling rate, these observed trends would likely not occur. We postulate that, locally, cooling of the granitic melts occurred slowly enough to yield large crystals and wide joint spacing at sites that eventually became tors but in a manner that did not result in large chemical variation within the granite. This hypothetical sequence is further supported by the absence of distinct compositional differences between the tor kernels and surrounding granite, which may have been expected with continued melt fractionation over longer periods. The observation that joint spacing shows a better linear least squares fit with feldspar crystal length than quartz crystal length (Figures 9c and 9d) could arise because K-feldspar crystals, which are generally larger and often more euhedral than quartz, crystallized first, leaving subsequent quartz crystal growth to occur interstitially. Feldspar size therefore provides a better indication of cooling rate, because other factors, including the availability of space, control quartz crystal size.
A cooling rate control on tor kernel formation is consistent with length scales of granite solidification in the Cairngorm pluton. Using the following empirically verified scaling relationship between crystal length, L (cm), and the time available for crystallization, t (s), from Zieg and Marsh 
where G° (effective growth rate) = 1.55 × 10−14 cm2 s−1 [Zieg and Marsh, 2002] and τ (constant to account for the time before solidification at the edge of a pluton) = 1.79 × 1012 s [Zieg and Marsh, 2002], with L = 0.5–0.6 cm, we calculate a timescale for crystallization, t, of 106 years. Entering this timescale into the equation of Jaeger  for the propagation distance, z (m), of a solidification front into a cooling magma
with λ (constant that depends on the latent heat of crystallization and the initial and final temperatures of the cooling magma) = 0.060, calculated from Jaeger  and κ (thermal diffusivity) = 0.01 cm2 s−1 [Zieg and Marsh, 2002] for a granitic melt cooling from 800°C to 650°C at a depth 12 km [Harrison, 1986] and country rock temperature of approximately 300°C, we calculate an approximate length scale for solidification of 700 m. This length scale is closer to those of the tor kernels (101–102 m) than the pluton length scale (104–105 m), confirming that heterogeneous solidification of the Cairngorm pluton was likely, as has been observed elsewhere [Coleman et al., 2004]. This inference is supported by the presence of zones of different crystal sizes (Figure 2) [Thomas et al., 2004], which indicates that the Cairngorm pluton did not simply cool from the outside inward but, rather, that cooling was complex, probably because of multiple pulses of magma and locally variable cooling rates within the pluton [Thomas et al., 2004]. Indeed, for the measured range of crystal sizes (1 to 40 mm), using equation ((1)), we calculate an order-of-magnitude range of solidification timescales from 0.3 to 2 Ma. The proposed cooling rate control on crystal size, joint spacing, and ultimately tor formation appears to be plausible.
The postulated role of cooling rate in establishing joint spacing remains little appreciated in the geomorphic literature but has been identified as a control on the spacing of columnar joints in basalt and the spacing of other joints in extrusive and intrusive rocks [Aydan and Kawamoto, 1990; Grossenbacher and McDuffie, 1995]. In each case, the joints result from contraction during cooling and their spacing is inversely related to local cooling rate at the place and time of fracture [Grossenbacher and McDuffie, 1995]. The strength of rocks, including granites, has been demonstrated to decline with increasing grain size [Brace, 1961; Onodera and Asoka Kumara, 1980; Martin et al., 1990; Tuğrul and Zarif, 1999; Vasconcelos et al., 2008], possibly because of a weaker crystal structure that reflects longer microcracks [Martin, 1990; Migoń, 2006, p. 42] and which is also indicated here by the higher connected porosities of coarse-grained granites (Figure 11). Fracturing through post cooling tectonic forces might then reasonably be expected to produce closely spaced joints in weaker, coarse-grained granites and more widely spaced joints in stronger, fine-grained granites. That the opposite is observed, for example, in Dartmoor, England [Ehlen, 1992], in the Sierra Nevada, California [Bateman and Wahrhaftig, 1966; Huber, 1987; Moore, 2000, pp. 334–335], and in the Cairngorm Mountains, indicates that cooling rate may offer an important, but frequently overlooked, control on joint spacing in intrusive rocks. A general decrease in the spacing of steeply dipping joints with an increasing cooling rate may provide an important control on the formation of different sized corestones in regolith, the erodibility of rock, and its susceptibility to mass failure. The potential importance of this relationship to these processes should therefore be assessed through investigations of grain sizes and jointing patterns in addition to more common considerations of rock strength and the contributions of tectonic and surficial weathering processes to rock fracturing [e.g., Molnar et al., 2007; Moore et al., 2009; Dühnforth et al., 2010; Fletcher and Brantley, 2010]. The postulated inverse relationship between joint spacing and cooling rate could thus have geomorphic effects many millions of years after pluton emplacement.
Joint spacing governs tor formation through its control on differential weathering. This is evidenced by the formation of tors in granite kernels that are widely jointed relative to the surrounding granite but which are also coarser grained and more susceptible to grusification, because of higher connected porosities in the rock matrix (Figure 11). In contrast, regoliths preferentially form in the surrounding, more densely jointed but comparatively fine-grained and less porous, granite (Figures 3c, 10, and 11). This contrast in landform development indicates that macroscopic fractures are a more dominant control than grain-scale cracks on tor formation through differential weathering, possibly because water gains less access to widely jointed granite. However, the grain-size control on the susceptibility of granite to grusification likely determines block angularity. We attribute pronounced edge rounding observed on blocks of more easily grusified, coarse-grained granite to this mechanism rather than to a climatic control on chemical weathering, as has been previously suggested [see Ballantyne and Harris, 1994, p. 180 and Migoń, 2006, p. 109 for discussion]. Together, these findings lead to a counterintuitive interpretation that these tors have preferentially formed in granite that more readily undergoes grain-scale disintegration than the surrounding granite.
The distribution of sheeting joints across different tors and their orientations relative to exposed tor surfaces are useful for inferring the surface-parallel stresses required for the joints to form and the time when they formed. Most of the tors in Figure 6 occur on ridges that trend roughly N-S. The ridges have a most concave curvature, k1, of approximately zero along the axis of the ridges. In contrast, the ridges in blue in Figure 6 have a least concave curvature (k2) value at least as negative as −0.0015; this is perpendicular to the ridges. For sheeting joints to open at the crest of a subhorizontal ridge, the combination of surface-parallel stress (σ22) across the ridge and surface curvature (k2) across the ridge must satisfy the following inequality:
where ρ = rock density (2.6 × 103 kg m−3) and g = gravitational acceleration (9.8 m s−2) [Martel, 2006, 2011]. Using a value for k2 of −0.0015, the inequality yields a value of σ22 of −17 MPa, with the negative sign denoting compression. More negative (i.e., more compressive) values of σ22 would also suffice. The required stress value is similar to the stresses at other sites with sheeting joints [Pahl et al., 1989; Martel, 2006, Table 1; Ziegler et al., 2013]. This compressive stress would be oriented approximately E-W in the Cairngorm Mountains. Sheeting joints generally cut through tors and are subparallel to the surrounding ground surfaces (Figure 3c, 3e, and 6), indicating that these subhorizontal joints formed after the gross topography of the area assumed its current configuration [Glasser, 1997] but before the steep faces of the tors were subaerially exposed (Scenario 1 in Figure 13). An exception occurs at tor BB1, where we interpret the “draping” of closely spaced (<10 cm) joints, which separate mechanically weak sheets, down the steep tor sides to largely indicate post emergence fracturing (Scenario 3 in Figure 13). In contrast, subhorizontal sheeting joints are unlikely to form in an emerged tor (Scenario 2 in Figure 13) for two reasons. First, a boxy tor shape will tend to diminish the high subhorizontal compressive stresses required for subhorizontal sheeting joints to form. Second, if the compressive stresses were large enough for sheeting joints to form, the joints would tend to become subparallel to the sides of a tor and lose their subhorizontal orientation (Scenario 3 in Figure 13). While the temporal constraints on the formation of sheeting joints appear robust, the required sources of high lateral compression [Martel, 2006, 2011] remain uncertain in general. Identifying the source of this compression in the Cairngorm Mountains could have implications for regional topographic development because of its effect on near-surface fracturing.
Sheet jointing provides important additional evidence for how tors have formed (Figure 13). With respect to the Cairngorms tors, the presence of closely spaced (tens of centimeters) sheeting joints subparallel to surrounding ground surfaces discounts the application of Linton's  end-member model of tor formation, where tors form in the subsurface due to weathering and emerge after surface erosion. This is because, in addition to forming near to the ground surface (tens of meters), sheet jointing forms only in massive unweathered rock of high compressive strength [Martel, 2006]. Tors displaying the sheet jointing described above will not form within a thick (up to tens of meters), intensely weathered regolith because high compressive stresses will not develop in the tor (Scenario 4 in Figure 13). If tors had formed at the base of a regolith tens of meters thick and were attached to underlying unweathered rock, then sheet jointing may be exhibited in the tors, if the other conditions necessary for the formation of these joints occur. However, in this case, sheeting joints should form subparallel to the tor surface rather than the surrounding ground surface and the spacing of sheeting joints would likely be many tens of centimeters (Figure 7 and Scenario 5 in Figure 13). The characteristics of sheeting joints in the Cairngorm tors support the application of either Palmer and Nielsen's  single-stage model or Hall and Phillips'  intermediate model to the formation of these tors. Although Linton's  two-stage model was developed from observations of the Dartmoor tors, the presence in those tors of closely spaced (few tens of centimeters) sheeting joints that mirror surrounding ground surfaces [Ehlen, 1992; Plate 3.7 in Migoń, 2006, p. 101; Gunnell et al., 2013] again supports the application of either of the alternative two models to the formation of these tors. These important implications of sheet jointing have been previously unrecognized in the tor literature and probably also apply to sheeted tors elsewhere.
The tors of the Cairngorm Mountains have formed in spatially distinct kernels of relatively coarse-grained granite surrounded by finer-grained granite, in which regolith mantles have developed. Tor volumes increase with grain size and the spacing of steeply dipping joints. Based on measurements of grain size and Sr and Zr concentrations, the spacing of steeply dipping joints seems likely to have been largely established by the cooling of the granite, with the wide spacing that generally characterizes the relatively coarse-grained tor kernels attributable to a slower cooling rate than occurred in the surrounding finer-grained granite. The role in surface processes of this potentially important control on igneous rock joint spacing has not been widely considered. While the spacing of steeply dipping joints and tor volumes increase with grain size, so does the susceptibility of the granite to grusification. This indicates that joint spacing is a more dominant control on differential weathering than grain-scale cracks and leads to a counterintuitive conclusion that these tors have preferentially formed in more easily grusified granite.
Sheet jointing is variably developed in the Cairngorms tors, with well-developed sheet jointing occurring on relatively high convex surfaces, in agreement with Martel's [2006, 2011] model. Although the source of compressive stress required to form sheeting joints remains uncertain, the sheeting joints visible in these tors formed after the gross topography of the Cairngorms was established and before tor emergence. The presence of closely spaced (tens of centimeters), subhorizontal sheeting joints discounts the application of Linton's  two-stage model to the formation of these tors. Rather, the Cairngorm tors have formed either according to Palmer and Nielsen's  single-stage model or have progressively emerged from a regolith only a few meters thick, in accordance with the Hall and Phillips  intermediate tor formation model. This important implication of sheet jointing has been previously unrecognized in the tor literature and probably also applies to subhorizontally sheeted tors elsewhere.
We thank Dylan Ward, anonymous reviewers of this manuscript and of an earlier version, and Journal Editors Simon Mudd and Alexander Densmore for insightful comments that improved the focus of this manuscript. This project was financed by grants to B.G. from Magn. Bergvalls Stiftelse, Carl Mannerfelts Fond, and Swedish Society for Anthropology and Geography Andreéfonden, and to A.P.S. from the Swedish Research Council. B.G. also gratefully acknowledges postdoctoral funding from Birgit and Helmuth Hertz's Foundation, through the Royal Physiographic Society in Lund, and the Wenner Gren Foundation in Stockholm. S.M. gratefully acknowledges support from NSF (grants EAR05-38334 and CMMI09-19584), the Yosemite Conservancy through the Hawaii-Pacific Islands Cooperative Ecosystems Studies Unit (agreement J8834110001), and JPL (agreement 1290138). We further acknowledge support of the Wallenberg Foundation for provision of analytical instrumentation used in this study. B.G. and A.S. were colead investigators and were involved in all aspects of this study. S.M. completed fieldwork and quantitative analyses of joint formation. A.P.S., K.N.J., C.H., and S.M. participated in fieldwork and manuscript reviewing.