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GJI Geodesy, potential field and applied geophysics Geophys. J. Int. (2009) 178, 1198–1219 doi: 10.1111/j.1365-246X.2009.04224.x An analysis of young ocean depth, gravity and global residual topography A. G. Crosby and D. McKenzie Bullard Laboratories of the Department of Earth Sciences, University of Cambridge, Cambridge CB3 0EZ, UK. E-mail: [email protected] Accepted 2009 April 23. Received 2009 April 22; in original form 2008 October 26 SUMMARY The variation of ocean depth with age in the absence of crustal thickening and dynamic support places valuable constraints on the thermal and rheological properties of the lithosphere and asthenosphere. We have attempted to estimate this variation using a global data set of shiptracks, with particular emphasis on young ocean floor. In this respect, this paper extends a previous study published in this journal by the same authors, which concentrated on the older parts of the ocean basins. We find that, prior to 80 Ma, subsidence patterns are reasonably consistent, with gradients of 325 ± 20 m Ma −1/2 and zero-age depths of 2600 ± 200 m. There is a strong inverse correlation between zero-age depth and the gradient of depth with the square root of age which is unrelated to local variations in dynamic support. Global depth-age trends to 160 Ma are not significantly different to those for the individual ocean basins. Within corridors of similar basement age, gravity–topography correlations are consistently 30 ± 5 mGal km−1 . Simple isostatic theory and numerical modelling of mantle plumes suggests that, if the minimum depth of convection is defined by the base of the mechanical boundary layer, the admittance should be a function of plate age. The observation that it is not implies that the active convective upwelling beneath young lithosphere ceases at the same depth as it does beneath old oceanic plates. This result is consistent with geochemical modelling of melts near mid-ocean ridges. We have examined the relationship between residual topography and gravity worldwide, and have found that good spatial correlations are restricted to the Atlantic, North Pacific and youngest Indian ocean basins. By contrast, residual topography and gravity are poorly or negatively correlated in the South and young North Pacific Ocean and in the older Indian Ocean. Away from regions of thick crust and flexure, histograms of residual topography and gravity have symmetric distributions about zero. We then use this residual topography to estimate the volume and buoyancy flux of seven major plume swells. In Hawaii, the clear correlation between melt and swell volumes in discrete age corridors is evidence that the horizontal velocity of the hot plume material far downstream from the plume is similar to the plate spreading velocity and that the plume pulses over time. Finally, comparison with seismic tomographic models suggests that the long-wavelength (>2000 km) residual topographic and gravity anomalies have an origin deeper than 250 km. This result is consistent with observations that the admittance is approximately constant at wavelengths longer than 800 km. Key words: Gravity anomalies and earth structure; Mid-ocean ridge processes; Intra-plate processes; Dynamics of lithosphere and mantle. 1 I N T RO D U C T I O N The isostatic subsidence of the ocean floor as it cools and moves away from a mid-ocean ridge depends on the thermal structure of the lithosphere and on the nature of the convective instability beneath the plates. Since this information is important and is difficult to determine by direct measurement, the relationship between ocean basement depth and age has been of interest since the development 1198 of plate tectonics, and numerous papers have covered both observational and theoretical aspects of the problem. The thermal plate model, in which the lithosphere is described as a rigid conducting layer with isothermal boundaries that starts at a constant temperature at the ridge, was first developed by McKenzie (1967), McKenzie & Sclater (1969) and Sclater & Francheteau (1970). Initially, ocean age data based on magnetic lineations only extended to ages younger than the lithospheric thermal time C 2009 The Authors C 2009 RAS Journal compilation Ocean depth, gravity and residual topography constant, so the first observational studies of this problem in the Pacific Ocean (Sclater et al. 1971; Davis & Lister 1974) observed a half-space subsidence trend, in which subsidence is proportional to the square root of age and the basal boundary condition is not apparent. However, as age constraints extended into Mesozoic time, Sclater et al. (1975) and Parsons & Sclater (1977) did observe a significant shallowing with respect to to the half-space model, implying that cooling did not extend below a certain depth. Using selected deep parts of the sea floor and ages outside the Cretaceous superchron (84–118 Ma), they found that the average sediment-corrected depth as a function of age could be fit well using a plate model with a thickness of approximately 125 km. Their predicted depth to unperturbed ocean floor at an age of 180 Ma was approximately 6.3 km, as opposed to 7.5 km for the half-space model. Since then, a number of physical explanations have been put forward for this flattening. These include (1) the addition of heat from convective instability underneath the rigid mechanical boundary layer (Parsons & McKenzie 1978); (2) radioactive heating in the upper part of the mantle (Forsyth 1977); (3) shear stress heating related to the large-scale flow (Schubert et al. 1978); (4) dynamic support associated with asthenospheric channel flow (Phipps Morgan & Smith 1992); (5) heating by numerous small upwelling plumes (Heestand & Crough 1981; Smith & Sandwell 1997); (6) large-scale convective processes of deep-seated origin (Davies & Pribac 1993) and (7) gradual changes in the average thickness of crust generated at ridges (Humler et al. 1999). Parsons & McKenzie (1978) argued that it is not possible to use subsidence alone to discriminate between competing models with equal total heat sources, and instead based their argument on an analysis of the mechanical stability of a cooling viscous layer. They found that instability will occur after ∼70 Ma if the viscosity of the thermal boundary layer is approximately 7.7 × 1019 Pa s, which is substantially less than the 1021 Pa s estimated for the whole upper mantle from postglacial uplift. Nevertheless, the predictions of a simple analytical model of the steady-state convective boundary layer agreed well with estimates of the asymptotic heat flow and subsidence in the Pacific and Atlantic Oceans. However, it is important to emphasize that there is no physical reason why simple plate models should account for the subsidence of cooling lithosphere between the onset of instability and the steady state. In early numerical experiments on the onset of convection beneath a cooling plate, the boundary layer was observed to cool by conduction, then to become unstable once its local Rayleigh number exceeded a critical value. The instabilities continued to grow as they cooled, but there was then a sudden increase in the average temperature of the boundary layer as the base of the layer detached and was replaced by hotter material from below. This new material then cooled by conduction, until it in turn became unstable, and so on. Houseman & McKenzie (1982) found that the mean temperature of the boundary layer, and hence the average topography, exhibited a series of decaying oscillations about the asymptotic plate model value. However, Fleitout & Yuen (1984) observed that the vigour and onset time of the convection varied strongly with the choice of the initial temperature distribution and the relationship between effective viscosity, temperature, pressure and stress. The form of the instabilities, and hence the average topography as a function of age, will also depend on whether the experiments are performed in 2-D or 3-D (e.g. Davaille & Jaupart 1994), and the presence or absence of large-scale circulation and a decoupling low-viscosity layer in the asthenosphere (e.g. Davies 1988; Robinson & Parsons 1988; Huang & Zhong 2005). Note that low-viscosity layers decouple both shear stresses (reducing the friction between plates and the C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation 1199 mantle) and vertical stresses, which attenuates the surface motion above underlying convection. It is this second kind of decoupling that is important here. Given these theoretical uncertainties, it is important that the observational data is selected as carefully as possible, and with full awareness of extraneous processes that affect ocean depths. Some authors (Sclater & Wixon 1986; Hayes 1988; Renkin & Sclater 1988; Marty & Cazanave 1989; Stein & Stein 1992) corrected for sediment loading alone; they observed significantly shallower depths than Parsons & Sclater (1977) but did not consider the bias from igneous crustal thickening or surface warping due to mantle convection. Others excluded seamounts, plateaus and prominent swells such as Hawaii from their data, and consequently estimated a deeper, more plate-like trend, but did not fully consider the full range of both positive and negative mantle dynamic topography (Shoberg et al. 1993; Carlson & Johnson 1994; Doin & Fleitout 1996; Smith & Sandwell 1997; Doin & Fleitout 2000; Hillier & Watts 2005; Zhong et al. 2007). Others simply excluded any data within 400– 600 km of a past or present hotspot (Heestand & Crough 1981; Schroeder 1984; Korenaga & Korenaga 2008) and consequently observed an even deeper, half-space trend, but again did not consider negative dynamic topography which tends to occur away from sites of mid-plate volcanism. Only a few authors (Colin & Fleitout 1990; Kido & Seno 1994; DeLaughter et al. 1999) attempted to correct for dynamic topography of both signs. Their studies had varying results, although all confirmed a shallowing with respect to the half-space trend. In a recent paper (Crosby et al. 2006), we argued that it is important to exclude prior to analysis topographic measurements both from areas of anomalously thick crust, and to take account of regions that are both elevated and depressed by mantle convection. It is especially important that dynamic topography is not thought of purely in terms of uplift. Our criterion for anomalous crust was visual, and the proxy we used for dynamic support was the intermediatewavelength gravity anomaly, which we found to be correlated with the topography by a factor of approximately 30 mGal km−1 within corridors of similar basement age away from regions of flexed and thickened crust. Such values for the admittance are consistent with those expected from simple fluid mechanical models of upwelling mantle plumes (McKenzie 1994; Crosby 2006; see also Appendix C). Once all perturbed regions had been excluded, we found that the average depth as a function of age in the North Pacific and Atlantic Oceans departed from the initial half-space conductive cooling trend more rapidly than predicted by the classical thermal plate model, and there is up to 250 m of transient shallowing between the ages of 80–130 Ma. We suggested that this shallowing is caused by the onset of convective instability in the thermal boundary layer as initially proposed by Parsons & McKenzie (1978). We also examined in more detail the relationship between residual topography and gravity in the wavenumber domain within large boxes in the Pacific and Atlantic Oceans, and found that there was no significant variation of the gravity–topography admittance with wavelength at length scales longer than approximately 1000 km. In this study, we expand on several aspects of the problem which we did not cover detail in our previous paper: a more extensive study of the relationship between depth, age and gravity on young ocean floor and its implications for the depth extent of mantle convection; a global study of depth versus age, including a convenient parametrization; and a global assessment of the relationship between long-wavelength gravity anomalies, residual topography and seismic tomography. We then conclude by using our improved 1200 A. G. Crosby and D. McKenzie maps of residual topography to estimate the buoyancy flux of several major plume swells. Finally, we emphasize that the principal aims of this study are observational rather than theoretical; in other words we are simply determining what information is contained in oceanic bathymetry and its related gravity field, rather than seeking to justify one physical model of the ocean lithosphere or mantle over another. Our explanations, where given, are less important than the underlying observations. 2 T H E R E L AT I O N S H I P B E T W E E N D E P T H , A G E A N D G R AV I T Y I N T H E YO U N G O C E A N S 2.1 Data and method In our previous study, we were for the most part concerned with the behaviour of the oceans after they depart from the half-space cooling trend, that is, where depth is proportional to the square root Figure 1. Map showing the gridded topography of Smith & Sandwell (1997, top panel) and the actual bathymetric measurements used in this study (bottom panel). Note that in the grid, depths with odd values, for example, 2001 m, are actual soundings, whereas depths with even values are predicted from gravity. Measurements from submarine continental crust, ocean crust covered by more than 1.5 km of sediment, anomalous igneous centres identified in Coffin & Eldholm (1994), and regions affected by lithospheric flexure as seen in Fig. 2 have been excluded. Regions selected for in-depth study of the depth of young ocean floor are bounded by thick black lines (a, Antarctic and adjacent African Plate; b, Australian Plate; c, Young Pacific Plate; d, Nazca Plate; e, West Atlantic Plate and f, East Atlantic Plate). C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography 1201 Figure 2. Marine free-air gravity anomalies, from Sandwell & Smith (1997). of basement age (Davis & Lister 1974), and for this reason our boxes contained a fairly small area of young ocean floor. We have now repeated our analysis for a much larger area. Fig. 1 shows the global database of shiptracks extracted from the grid of Smith & Sandwell (1997) together with the six regions of largely young ocean floor covering the Pacific, Nazca, Atlantic, Australian and Antarctic Plates. As before, we first visually excluded areas of thickened crust as described in Coffin & Eldholm (1994), and do not consider regions where the sediment is more than 1.5 km thick. The reason why we exclude thick sediments is because the standard National Geophysical Data Center (NGDC) global sediment thickness grid (Divins 2004) is of uncertain provenance and contains significant errors where sediment is thick near continental margins. To calculate unperturbed water-loaded depth in these areas requires careful examination of the primary seismic reflection and refraction data (Winterbourne et al. 2009) and is outside the scope of this study. However, prior experience has shown that backstripping errors using the NGDC grid are small in the deep oceans. We also exclude data in flexural moats and subduction zones visible in maps of sea surface gravity anomalies (Fig. 2). The new and updated basement age grid used here is described in Müller et al. (2008a). Within each region, we calculated water-loaded subsidence at the remaining locations using the standard backstripping method described in Sclater & Christie (1980) assuming a pure shale composition for sediment, and calculated the 0.5◦ block-median depth to ensure a more even geographical distribution of data. We then plotted gravity anomalies versus topography in age bins of 1 Ma1/2 to a maximum age of 81 Ma using the grid of Sandwell & Smith (1997), shown in Fig. 2. The estimated plate cooling contribution to the gravity field discussed in appendix B of Crosby et al. (2006) was subtracted beforehand. The reference depth within each age bin is then taken as the median depth of those blocks with residual gravity anomalies smaller than ±5 mGal, and the error is the interquartile range. Finally, we estimated the best-fit gravity–topography slope (the admittance) using the method of Marks & Sandwell (1991) C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation in which both data are presumed to contain errors. Upper and lower bounds are estimated by assuming that the errors are entirely confined to the topography and gravity, respectively. A simple linear correlation is used because there is no clear evidence that the admittance varies with wavelength once crustal-scale features have been removed (Crosby et al. 2006, Appendix C). This measure then has the advantage of allowing geographically discontinuous gravity and topography data to be used in our analysis. 2.2 Results Reference depth-age curves for the six regions shown in Fig. 1 to an age of 81 Ma are shown in Fig. 3, together with the weighted best√ fitting functions of the form d = a t + b, where d is depth in km and t is age in Ma. Corresponding gravity–topography correlations are shown in Fig. 5, and individual examples close to the ridge are given in Figs 6 and 7. The values shown in these plots are listed in tables at the end of this paper, where some minor typographical errors in Crosby et al. (2006) have been corrected. Fig. 3 shows that the reference depth-age trends are similar in different oceans. The depth increases by 307–347 m per Ma1/2 , and zero-age depths are between 2379 and 2750 m. Note that the extrapolated zero-age depth is not the same as the actual ridge depth. As expected, all six basins show an excellent correlation between depth and the square root of age. As Fig. 4 shows, there is also a clear inverse correlation between zero-age depth and basin √ slope with a best-fit line of b = −5736 − 9.62a where d = a t + b, as has been found by other authors (see, for instance, Fig. 6 in Korenaga & Korenaga 2008), although the variation we find is smaller than that in the range of studies they cite. We also find only slight asymmetry between either side of each mid-ocean ridge, in contrast to the more dramatic results of some earlier studies (e.g. Hayes 1988; Phipps Morgan & Smith 1992). Within these six regions, the gradient of depth with square root of age appears to decrease slightly with 1202 A. G. Crosby and D. McKenzie Figure 3. Median sediment-corrected depth as a function of the square root of median age within bins of 1 Ma1/2 in the regions shown in Fig. 1, calculated using only shiptracks in areas with gravity anomalies smaller than ±5 mGal and which do not cross areas of thickened crust. The grey bars show the interquartile range. The best-fitting straight lines were calculated by minimizing the weighted rms misfit to the data, where the weight is the inverse of the interquartile range. increasing spreading rate, but the range is larger than can be accounted for simply by differences in horizontal conduction (Sclater & Francheteau 1970; Davis & Lister 1974). One explanation is dynamic topography related to plate motions, such as that described by Phipps Morgan & Smith (1992). However, these viscous forces are uncompensated, so should be associated with significant gravity anomalies, which, by definition, our variations are not. Another possibility is a systematic variation in crustal thickness between ocean basins. The data in White et al. (1992) indicate that normal oceanic crust is on average 0.5–0.7 km thicker in the slow-spreading Atlantic Ocean than it is in the fast-spreading Pacific, although this variation is a factor of two too small to explain the ∼300 m difference in zero-age depth. It also does not explain why the zero-age depth is correlated with the average subsidence rate. We therefore do not have an explanation for the trend observed in Fig. 4, but suggest that the forces responsible are both dynamic and compensated. Figs 5–7 show that, although the slope of the best-fitting straight line relating gravity and topography within each age corridor is similar (approximately 25–35 mGal km−1 ), the correlation coefficients are very variable. In the Atlantic Ocean younger than 70 Ma and the Indian Ocean younger than 4 Ma, the correlations are consistently good (r s > 0.5 with a maximum of 0.84). These results compare favourably with the results in the North Central Pacific Ocean found by Crosby et al. (2006), where the best value of r s was 0.89 between the ages of 90–100 Ma. On the contrary, in the young Pacific, Nazca and older parts of the Indian and Antarctic Oceans, the correlations are much poorer and are not statistically significant. Nevertheless, there is no relationship between gravity–topography coherence and the linearity of the relationship between unperturbed depth and the square root of age in each region. 2.3 A simple model for the admittance over dynamic topography The gravity anomaly over dynamic topography is the sum of two components: the component arising from the deflected surface of the Earth, and the component arising from the buoyant mantle which supports it. Although numerical simulations (e.g. Ribe & Christensen 1994, 1999) indicate that dynamic swells are largely in isostatic equilibrium away from the plume itself, the overall gravity anomaly over a swell has the sign of the topography, because the surface component of the gravity is attenuated less than the more distant mantle component (e.g. Turcotte & Schubert 2002). Following this reasoning, we would thus expect the admittance to be related to the thickness of the lithosphere. It is therefore surprising that there is almost no difference between the admittance over C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography 1203 Appendix C. However, Z appears to be 25–40 mGal km−1 everywhere, even at ridges. The simplest interpretation is therefore that mantle convection at these spatial scales does not occur at depths shallower than approximately 100 km, regardless of the thermal structure of the lithosphere. Interestingly, this explanation is in agreement with inverse modelling of Rare Earth Element concentrations in oceanic island basalts, where McKenzie & O’Nions (1998) find there to be no relationship between melt composition and the age of the plate. Similar observations have been made more recently in Iceland (Maclennan et al. 2001), which suggests that active upwelling does not persist much shallower than the depth of the base of the old lithosphere, perhaps because of a change in viscosity due to dehydration in the initial stages of melting (Ito et al. 1999). 3 A G L O B A L S T U DY O F D E P T H VERSUS AGE Figure 4. Plot of zero-age elevation and the gradient of elevation versus the square root of age in Fig. 3. The linear correlation coefficient is −0.985. EA, East Atlantic (e in Fig. 1); WA, West Atlantic (f in Fig. 1); AU, Australian and adjacent Antarctic Plates (b in Fig. 1); AA, African and adjacent Antarctic Plates (a in Fig. 1); P, Pacific (c in Fig. 1) and N, Nazca and adjacent Antarctic (d in Fig. 1). mature ocean floor (see Crosby et al., Figs 6, 11 and C1) and the admittance found in this study near the mid-ocean ridge. A simple model for the expected variation of admittance with plate age is shown in Fig. 8. The gravity anomaly caused by the warping (h) of the surface and oceanic Moho is approximately gsurf ≈ 2πG [(ρc − ρw )h + (ρm − ρc )h] = 2πGh(ρm − ρw ) = 2π G M, (1) where ρc is the density of the crust, ρm is the density of the lithospheric mantle, ρw is the density of water and M is the anomalous mass per unit area. If isostatic equilibrium is assumed, then the mantle component of the gravity anomaly is approximately gsub ≈ −2πG Me−2π z/λ , (2) where z is the weighted midpoint depth and λ the lateral wavelength. The total gravity anomaly is the sum of these two contributions. Hence the admittance is Z≈ gsurf + gsub = 2πG(ρm − ρw )(1 − e−2π z/λ ). h (3) The shaded trends in Fig. 5 shows the expected variation of Z with lithospheric age, assuming that the anomalous mantle is bounded at its upper surface by the 1000 ◦ C isotherm, has a weighted midpoint depth ∼30 km below this boundary (e.g. Watson & McKenzie 1991), and has a lateral wavelength of 1200–2500 km. Using the plate model of McKenzie et al. (2005) with a thickness of 90 km, Z is expected to vary from 10 to 15 mGal km−1 at the ridge to 25–50 mGal km−1 over mature ocean floor. In the particular case of Hawaii, where the swell wavelength is 1800 km, the observed admittance of ∼28 mGal km−1 (Crosby et al. 2006) implies a sensible weighted midpoint depth of 100 km for the compensating mass, which is consistent with the numerical models discussed in C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation In Crosby et al. (2006), our focus was on constraining possible differences in the unperturbed subsidence between different ocean basins. We found, in fact, that the subsidence patterns were similar within the scatter of the data. It is therefore of interest to repeat that method using a worldwide data set, the distribution of which is shown in Fig. 1. The processing of the data was the same as outlined in the previous section. Figs 9(a) and (b) show the results of that analysis. As expected, the pattern of average unperturbed depth versus age is similar to figs 7–9 in Crosby et al. (2006), although the shallowing is slightly less pronounced when the North Pacific or East Atlantic Oceans are examined in isolation. However, there is still a clear deviation from the plate cooling model at an age of ∼80 Ma which has a maximum amplitude of ∼250 m between the ages of 100–125 Ma. At ages in excess of 140 Ma, the data set is dominated by the Pacific Ocean. A plate model using the parameters of McKenzie et al. (2005) and a thickness of 90 km fits the data well between the ages of 10– 80 Ma and 140–160 Ma, respectively, and predicts the heat flux within error in the mature Atlantic and Pacific Oceans (Louden et al. 1987; Crosby et al. 2006). We find a thinner equilibrium plate thickness than Parsons & Sclater (1977) for three reasons: we use different data selection criteria, we use a deeper zero-age depth (2652 m as compared to 2500 m), and we allow the thermal conductivity and expansivity to vary with temperature. Interestingly, and perhaps as a result of melting, the new 90 km plate model slightly underpredicts depths at ages younger than 10 Ma, with the difference reaching a maximum amplitude of 150 m between 1 and 2 Ma. At ages younger than 80 Ma, the depths are fit optimally √ by a function of the form d = 2652 + 324 t m. This function is similar to that found by Korenaga & Korenaga (2008), who did not explicitly exclude both positive and negative dynamic topography. Fig. 9(c) shows that within bins of similar age (width 1 Ma1/2 for ages under 81 Ma, 10 Ma for ages between 80 and 160 Ma) the overall correlations between gravity and topography are fairly poor, with correlation coefficients between 0.2 and 0.6. For this reason, we suspect that the shallowing between 80 and 130 Ma will be better resolved in individual basins such as the North Pacific where the correlations are much higher and there is less scatter in the data. However, the best-fit admittance values are almost all in the expected range 25–35 mGal km−1 and, again, there is no significant variation with age. There are a number of applications, such as plate reconstructions and the predictions of global sea level changes, for which it is 1204 A. G. Crosby and D. McKenzie Figure 5. Plots of gravity–topography correlations within age corridors of 1 Ma1/2 using the selected data in Figs 1 and 2 after correcting for sediment loading. The vertical axis is the best-fit slope using the method of Marks & Sandwell (1991) in which both data are presumed to contain errors and the colour denotes the Spearman’s Rank correlation coefficient. The error bars are determined by calculating the best-fit slope assuming the errors are confined to each variable in turn. The grey shading is the prediction of eq. (3) for wavelengths between 1250 and 2500 km. Note that the observed gravity–topography ratio shows no systematic variation with the age of the lithosphere. desirable to predict ocean depths given a predicted distribution of ocean basement age (e.g. Müller et al. 2008b). It is therefore useful to parametrize the data in Fig. 9(a) in a convenient form, such as the following. ⎧ √ m t ≤ 75 Ma ⎪ ⎨ 2652 + 324 t d = 5028 + 5.26t − 250 sin t−75 m 75 < t ≤ 160 Ma . 30 ⎪ ⎩ 5750 m t > 160 Ma (4) However, we agree with Müller et al. (2008b) that, for the purposes of whole-basin volume calculations, eq. (4) will overestimate the basin volume because it was calculated excluding anomalously thick and shallow crust such as the Ontong Java, Manihiki, Hess, Shatsky and Kerguelen Plateaus in the Pacific and Antarctic Oceans. A parametrization which includes all the data, such as that suggested by Stein & Stein (1992), may therefore be more useful for such purposes, even though it does not provide physical information about the lithosphere. 4 A G L O B A L E X A M I N AT I O N O F R E S I D UA L T O P O G R A P H Y Residual ocean topography is what remains after subtraction of the topography predicted using a depth-age model such as that given by eq. (4). It reflects changes in crustal thickness and mantle dynamic topography, and it is of interest to compare it to residual gravity. Both are shown in Figs 10–12. Note that the residual topography in these maps includes regions of thickened crust which were removed prior to the calculation of Fig. 9. Fig. 13 shows histograms of residual topography and gravity calculated using gridded data outside areas of thickened and flexed crust (see Fig. 1), and it can be seen that both data have an almost symmetric distribution about zero when these regions are excluded. As expected given the analysis in Crosby et al. (2006), the visual correlation between topography and gravity is good in the central North Pacific (Fig. 12), where both the Hawaiian and Line Island Swells and surrounding basins have residual topography and C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography 1205 Figure 6. Example plots of 0.5◦ block median sediment-corrected topography and gravity anomalies (see Figs 1 and 2) within age bins of width 1 Ma1/2 near the ridge in the Atlantic Ocean. r is the Spearman’s Rank Correlation Coefficient. The best-fitting straight line is shown in black. The grey bar shows the median and quartile depths of those points with gravity anomalies smaller than ±5 mGal (assumed to be unperturbed seafloor). gravity anomalies of the same sign. We believe these features result from convection in the mantle. The correlation is also good in the Atlantic Ocean (Fig. 10), where the Cape Verde, Azores, Bermuda, Cameroon and Iceland swells and surrounding basins also have residual topography and gravity anomalies of the same sign. Exceptions are off Southern Brazil (the Rio Grande Rise) and Namibia (the Walvis Ridge, Meteor Rise and Shona Ridge) where the crust has been thickened by plume-related melting during the breakup of the Atlantic Ocean (White & McKenzie 1989). Note, however, that not all of the lack of association off Southern Brazil can be explained by igneous crustal thickening alone (Winterbourne et al. 2009). The correlation in the oldest parts of the Pacific Ocean is also poor because of the isostatic crustal thickening associated with the Ontong Java Plateau, Mid-Pacific Mountains and pervasive seamount volcanism of Cretaceous age (Coffin & Eldholm 1994). Other large igneous provinces such as the Ninetyeast Ridge and Exmouth Plateau (Fig. 11) are also clearly visible in the topography but not the gravity. C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Interestingly, there is a negative correlation between residual gravity and topography in other parts of the South Pacific and young North Pacific Oceans (which is most pronounced west of Baja California and along the Pacific–Nazca ridge, including around the Galapagos volcanic centre), and over the South Pacific Superswell where the residual topography is elevated but the regional gravity anomaly is slightly negative (e.g. McNutt & Fisher 1987; Hillier & Watts 2004; Adam & Bonneville 2006). Clearly these features do not result from the same processes that produce the Hawaiian, Line Island, Cape Verde and Azores Swells and surrounding depressions; or if they do, there must be additional mass anomalies at depth, which dominate the gravity field but have restricted surface expression. Fig. 11 shows that there is a good positive correlation between residual topography and gravity along the coast of southern Africa (the African Superswell, Nyblade & Robinson 1994), along the Indian–Australian ridge (except near the Rodriguez Triple Junction), around the Crozet Swell, and in the South Australian Basin. 1206 A. G. Crosby and D. McKenzie Figure 7. Example correlation plots near the ridge in the Pacific and Indian oceans. Note how poor the correlations are in the youngest Pacific Ocean. There is some evidence that the crust in the South Australian Basin is anomalously thin (the area is a classic ‘cold spot’), although Gurnis et al. (1998) find that only ∼50 per cent of the topographic anomaly can be explained in this way. However, there is a negative correlation Southwest of India, where the residual topography is slightly elevated but there is a large negative gravity anomaly. The effect of this gravity anomaly on the estimation of depth versus age in the older parts of the Indian Ocean is discussed in Crosby et al. (2006). There is also a negative correlation in the Philippine Plate, where depths are anomalously deep but there is a slight positive gravity anomaly. We believe that this negative correlation arises because the crust here is anomalously thin (3.5–6.8 km, Louden 1980) but the dense subducted slabs underneath elevate the regional gravity field. Usually, the gravity anomaly associated with mantle motions has the same sign as the surface topography, which is negative over a sinking slab. However, an analysis of basin subsidence records shows that the change in elevation in the Philippine Plate due to subduction is much smaller than expected, which is why the net gravity anomaly is positive (Wheeler & White 2002). A possible reason why is a low-viscosity zone in the asthenosphere that attenuates the surface expression of vertical normal stresses in the deeper mantle (e.g. Robinson et al. 1987). In summary, we find that the strong positive correlation between residual topography and gravity, which is such a clear feature in the North Central Pacific and Atlantic Oceans, is by no means a globally consistent feature. These observations have implications for studies of the relationship between mantle convection and dynamic surface topography. 5 T H E R E L AT I O N S H I P B E T W E E N SEISMIC TOMOGRAPHY AND R E S I D UA L T O P O G R A P H Y A N D G R AV I T Y Although we have good physical models of some swells, for instance Hawaii (e.g. Watson & McKenzie 1991; Ribe & Christensen 1994, 1999), the origins of much of the Earth’s dynamic topography C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography 1207 Figure 8. A cartoon to illustrate eq. (3). In this simple model the anomalous mantle that gives rise to isostatic dynamic topography exists in a 60 km thick channel, the vertical extent of which is bounded by the 1000 ◦ C isotherm. The model predicts that the overall gravity per unit dynamic topography should be a function of z, the (weighted) midpoint depth of the anomalous mantle, and hence of the age of the lithosphere. The observation that it is not (see Figs 4–6) indicates that this model is incorrect. The simplest explanation is that mantle convection does not occur shallower than ∼100 km, regardless of the age of the lithosphere. remain poorly constrained in terms of mantle processes. Although past subduction may be a contributor (e.g. Lithgow-Bertelloni & Gurnis 1997; Spasojević et al. 2008), the predicted magnitudes and even signs of dynamic topography predictions based on subduction reconstructions and seismic tomography are often incorrect, which shows that our understanding is far from complete (e.g. Wheeler & White 2002; Winterbourne et al. 2009). In this section, we therefore elect to make a purely descriptive examination of the relationship between our maps of residual topography, gravity and mantle seismic velocity anomalies. Fig. 14 shows three slices through global S-wave tomographic model S20RTS (Ritsema et al. 1999, 2004) at depths of 250, 500 and 1500 km, respectively. This model has a lateral resolution of approximately 1000 km. At 250 km, there is a good visual correlation between seismic velocities and marine topography and gravity anomalies. In the Pacific Ocean, the Hawaiian Swell, the region of elevated seafloor west of Baja California and the South Pacifi Superswell all coincide with negative (slow) velocity anomalies, and the West Pacific and Young Pacific–Nazca depressions coincide with positive (fast) velocity anomalies. The same negative correlation between residual topography–gravity and seismic velocity is seen under the South Australian Basin, the depressed seafloor east of Brazil and the Bermuda Swell. These correlations are expected if positive (fast) seismic velocity anomalies reflect positive (heavy) density anomalies, and vice versa. Interestingly, the overall correlation between seismic velocity and residual topography is better than the correlation between seismic velocity and residual gravity. Indeed, certain features in the gravity, such as the Northeast Pacific low, appear to have no clear associated seismic features in the mantle. At 500 km depth, the correlation is weaker, although velocity anomalies coinciding with the Hawaiian Swell and surrounding basins are still visible, as are anomalies coinciding with the South Pacific Superswell and South Australian, Brazilian and Argentine Basins. In the lower mantle, at 1500 km depth, the correlation between S20RTS and the residual topography and gravity is poorer still, although the South Pacific and African Superswells and North West Pacific Basins still coincide with seismic velocity anomalies of opposite sign. The large negative gravity anomaly south of India, which does not coincide with a regional depth anomaly, does coincide with a large positive velocity anomaly at depths between 1000 and 2000 km. Finally, and not shown, the South Pacific Superswell coincides with a pronounced low-velocity anomaly at C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation the Core Mantle Boundary. A similar feature exists under the elevated southern part of Africa. The vertical resolution of S20RTS is poor at shallow levels. A better model for the asthenosphere in the Pacific Ocean is the surface wave-derived model of Priestley & McKenzie (2006), which has a vertical resolution of ∼25 km to a depth of ∼250 km. Using their velocity–temperature conversion, which is calibrated geologically using the thermal plate model of McKenzie et al. (2005), they found that at wavelengths longer than 800 km, the temperature and thus thermal density variation beneath the base of the plate is small, that is, less than 20 ◦ C or 3 kg m−3 . In other words, long-wavelength residual topography with correlated gravity anomalies must be caused either by deeper thermal variations, or by unmodelled compositional variations within or below the plate (Crosby 2006). Given that the observed admittance is approximately 30 mGal km−1 , eq. (3) suggests that these anomalies do indeed have a deeper rather than shallower origin: at a wavelength of 4000 km (appropriate for the Line Island Swell), a compensating depth of 125 km gives an admittance of 17 mGal km−1 , whereas a compensating depth of 300 km gives an admittance of 35 mGal km−1 . Furthermore, interpretation of the wavelength-invariance of longwavelength admittance in the Pacific (Crosby et al. 2006, Appendix C) using eq. (3) suggests that the depth, z, of the compensating mass in general increases with wavelength, where z ∼ 0.07λ. 6 T H E B U O YA N C Y F L U X O F M A J O R P LU M E S W E L L S In our discussion of residual topography, we now move from a global scale to the scale of individual swells. Although the physical origin of some of the Earth’s dynamic topography is unclear, many swells with active intraplate volcanism, such as Hawaii, are clearly a surface response to sublithospheric plumes. In these cases, the volume of a swell and the time over which it has grown provide information about the flux of material below the plate. We include this discussion here because estimating the volume and cross-sectional area of convective swells is a natural application of an improved reference depth-age trend. The volume of a swell depends on two processes: the accumulation of new hot mantle via the plume, which enlarges it; and the cooling of the anomalous mantle as it moves away from the plume, which diminishes it. 1208 A. G. Crosby and D. McKenzie Figure 9. (a) Depth versus age for the global data set shown in Fig. 1, using only those data with gravity anomalies smaller than ±5 mGal. The solid black line is the prediction of the thermal plate model in McKenzie et al. (2005) using a plate thickness of 90 km. The dashed line is the ad hoc parametrization given by eq. (4). As expected, there is a deviation from the plate model of up to ∼250 m between the ages of 80–140 Ma. Data older than 140 Ma are dominated by the Pacific Ocean. (b) Depth versus the square root of age for ocean floor younger than 90 Ma. The best-fit straight line is shown. Note that the plate model which best fits depths between 10 and 80 Ma and over 140 Ma underpredicts depths near the ridge by ∼150 m. (c) Gravity–topography correlations in bins of similar age (1 Ma1/2 under 80 Ma, 10 Ma thereafter). Grey shading as in Fig. 5. There is no systematic variation of admittance with age, but, globally, the correlations are not as good as they are locally in the North Pacific and Atlantic Oceans (see Fig. 5 and Crosby et al. 2006). For a young swell the bouyancy flux, B, is approximately B≈ V (ρm − ρw ) , t instability (e.g. Moore et al. 1998), the flux is approximately (5) where V is the volume and t is the age of the swell. Note that this calculation makes no assumptions about the extent to which the buoyancy is thermal or a compositional result of melt extraction (e.g. Phipps Morgan et al. 1995). For an old swell in steady state, where the gain in buoyancy due to the plume is balanced by the loss of buoyancy due to cooling and V (ρm − ρw ) , (6) τ where τ is the thermal time constant of the hot layer (∼30 Ma for 90 km thickness). This expression will be an underestimate of B when t is not much greater than τ . In this section, we use maps of residual topography to calculate swell volume and estimate buoyancy flux for seven swells around the world, and attempt to correlate these fluxes with estimates of melt production rate. Our results are shown in Table 1. We restrict B≈ C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography 1209 Figure 10. Maps of residual topography and gravity for the Atlantic Ocean, smoothed using a Gaussian filter with a 125 km radius. The contour interval is 500 m. Residual topography (left) was calculated by subtracting depths predicted using eq. (4) from sediment-unloaded observations of topography. The map illustrates regions of thickened crust and dynamic topography. Residual gravity (right) was calculated by subtracting the plate cooling gravity anomaly (Crosby et al. 2006, Appendix B) from the observations of free-air gravity. Note the following regions, which show a clear regional correlation between residual gravity and topography: B, Bermuda Swell; A, Azores Swell; CN, Canaries Swell; CV, Cape Verde Swell and CL, Cameroon Line. These swells are surrounded by basins, which also show a good correlation between residual depth and gravity. Yellow circles are sites of Holocene volcanism from the Smithsonian catalogue; note the strong association with positive residual topography and gravity, implying elevated temperatures in the asthenosphere. our analysis to plumes which are currently or recently volcanically active and also have coincident long-wavelength gravity anomalies, both of which we would expect if the swell was formed from hot material immediately beneath the base of the lithosphere. Iceland and Kerguelen are excluded because of the difficulty in disentangling support due to crustal thickening and mantle flow, and because in the case of Iceland there exist alternative methods for estimating plume flux using the velocities of propagating V-shaped ridges (Poore et al. 2008). 6.1 Hawaii Fig. 15(a) shows Hawaii, which in volcanic terms is the world’s most active intraplate swell. A particular problem with Hawaii is that the volcanic islands themselves occupy a significant fraction of the swell. We have therefore excised areas of thickened and flexed crust, and have attempted to estimate the full cross-sectional shape of the swell from its flanks. An example is shown in Fig. 15(b). Fig. 15(c) shows an estimate of the volcanic load, calculated from the topography using a thin elastic plate model assuming T e is between 20 and 36 km and that the load has a density of 2800 kg m−3 (e.g. Watts 1978). The age of each part of the load is also shown. Implicit is the assumption that the volume of igneous underplate is approximately equal to the volume of sediment infill of the flexural moat, which may be an underestimate. The total volume of the swell is estimated to be approximately 1.3 × 106 km3 between the ages of volcanism of 0–40 Ma. Fig. 12 shows that the swell has disappeared by the Hawaiian-Emperor bend, C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation which corresponds to an age of ∼43 Ma (Clouard & Bonneville 2005), which places constraints on the thermal time constant. In Fig. 15(d), we have used eq. (5) and reconstructed volumes within the individual age corridors marked in Fig. 15(a) to calculate the change in buoyancy flux over time. This simple calculation contains two assumptions: the hot asthenosphere has not cooled significantly, and it has moved with the same velocity as the overlying plate. Both are clearly not correct in general (e.g. in Iceland, velocities of material down the mid-ocean ridge are an order of magnitude faster than spreading velocities). Nevertheless, it is interesting to note a strong positive correlation between swell volume and melt production rate within individual corridors along the Hawaiian chain up to an age of 30 Ma. This correlation is consistent with hot asthenosphere far from the plume moving approximately with the plate with a thermal time constant not much shorter than ∼30 Ma, and with the plume pulsing in amplitude and temperature over time (see, for instance, Vidal & Bonneville 2004). Over the last 0–5 Ma, we estimate a buoyancy flux of 6–8 Mg s−1 , similar to Sleep (1990), and a melt production rate of 3–4 m3 s−1 , which is slightly less than the ∼5 m3 s−1 estimated by Watts & ten Brink (1989). If the excess plume temperature is 200 ◦ C (Watson & McKenzie 1991), the maximum swell amplitude of ∼1.5 km implies a maximum isostatic hot layer thickness of ∼150 km, although this estimate is an upper bound because it assumes no support by viscous stresses. The advective heat transport by a plume, F p is Fp = cp Bf , α (7) 1210 A. G. Crosby and D. McKenzie Figure 11. Maps of residual topography and gravity for the Indian Ocean. Key as Fig. 10. C, Crozet Swell; R, Réunion and Mascarene Plateau Swell; SA, South Australian Basin (Australian-Antarctic Discordance); NE, Ninetyeast Ridge; E, Exmouth Plateau and P, Philippine Plate. where B f is the buoyancy flux, cp is the specific heat capacity (1200 J kg−1 K−1 ) and α is the thermal expansivity (3.3 × 10−5 K−1 ). This can be related to the heat lost by melting, F m orders of magnitude smaller than the total heat supplied by the plume. Fm = ρ L Bm , 6.2 Cape Verde (8) where Bm is the melt generation rate by volume, L is the latent heat of fusion (320 kJ kg−1 ) and ρ is the density of the rock (2800 kg m−3 ). In the case of Hawaii over the last ∼2 Ma, we estimate F p and F m to be 3 × 1011 W and 3 × 109 W, respectively. In other words, even allowing for the considerable uncertainties in these calculations, the heat lost due to melting is about two Fig. 16(a) shows the Cape Verde swell off Northwest Africa. As in the previous section, Fig. 16(b) is an average and reconstructed cross-section over the active section of the swell. Visually, the swell appears to form two segments. The first, nearest the Mid-Atlantic Ridge, is not associated with substantial volcanism and is apparently older. The second, nearer Africa, is volcanic and reaches a maximum amplitude of 1.5–2 km. Volcanism has been active for at least C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography 1211 Figure 12. Maps of residual topography and gravity for the Pacific Ocean. Key as Fig. 10. OJ, Ontong Java (thickened crust); WP, West Pacific Basins (topography and gravity correlated); H, Hawaiian Swell; LI, Line Island Swell; EP, East Pacific (gravity and topography anticorrelated); S, South Pacific Superswell (gravity and topography anticorrelated) and LV, Louisville Swell. Figure 13. Histograms of water-loaded residual depth and residual gravity anomalies worldwide, calculated using the gridded data of Smith & Sandwell (1997) and Sandwell & Smith (1997). Regions affected by crustal thickening and flexure have been excluded, as have regions covered by more than 1.5 km of sediment. Note that the distributions are highly symmetrical about zero with an approximately Gaussian shape (topography: N = 1801 783, mean 0.037 km, skewness 0.403; gravity N = 1713 884, mean −2.32 mGal, skewness 0.082). 20 Ma, but uplift may have started as early as 50 Ma (Courtney & Recq 1986); therefore it is reasonable to assume that the swell is in approximate steady state. Because the plume is moving slowly with respect to the plate, it is important to consider conductive reheating of the lithosphere, which increases surface elevation without a corresponding increase in mantle buoyancy. Fig. 16(d) shows the predictions of two simple reheating models, details of which are given in Appendix B. Since Courtney & Recq (1986) observed a heat flow anomaly of 9–17 mW m−2 over the crest of the Cape Verde swell, we estimate that 30–50 per cent of the uplift of the Cape Verde has a lithospheric, rather than asthenospheric origin, and should be ignored when applying eq. (6). Further evidence for conductive reheating comes from observations of elastic thickness and admittance. In the oceans, T e coincides approximately with the 200–400 ◦ C isotherm (Crosby 2006), which the simple model shown in Fig. 16(d) predicts should be elevated C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation by 6–10 km. The Cape Verde and Hawaiian Islands have similar lithospheric loading ages, and it is interesting to note that T e estimates for Hawaii are generally in the range 28–40 km, whereas those for the Cape Verde are slightly smaller, in the range 20–30 km (Watts 2001). A second piece of evidence is the long-wavelength admittance, which Crosby et al. (2006) estimated to be 5–10 mGal km−1 smaller over the Cape Verde Swell than over other mid-ocean swells. This difference is expected if part of the compensating subsurface density anomaly is in the lithosphere rather than the deeper asthenosphere. The reduced volume of the active portion of the swell is approximately 3 × 105 km3 , and we estimate the volume of the volcanic load to be 2.75 × 1014 m3 , which equates to a melt generation rate of 0.2–0.45 m3 s−1 . For only the active part of the swell, the steady state buoyancy flux given a thermal time constant of 30 Ma (which implies a hot layer thickness of 80–90 km) is ∼0.7 Mg s−1 . 1212 A. G. Crosby and D. McKenzie Figure 14. Three slices through global tomographic model S20RTS (Ritsema et al. 1999), at depths indicated. Colourscale shows per cent variation in V s from PREM. Note correlations, or lack thereof, with Figs 10–12. 6.3 Other swells Fig. 17(a) shows the Réunion Swell in the Indian Ocean. At the present day, active volcanism is restricted to Réunion island itself, which is associated with only a minor topographic swell. Both the size of the swell and the volume of the volcanic island chain increase northwards through Mauritius and the Mascarene Plateau to a maximum age of age of volcanism of ∼64 Ma, demonstrating that the flux of the plume has decreased over time. Above a long-wavelength regional residual topography of ∼250 m, the volume of the reconstructed swell is ∼1.6 × 106 km3 , which is similar to Hawaii within error. This volume corresponds to a steady-state buoyancy flux of 3.6 Mg s−1 . We estimate the melt generation rate to have been 0.7 m3 s−1 between 34 and 7 Ma using an average T e of 15 km (Crosby 2006), which is consistent with the trend observed for Hawaii in Fig. 15(d). Fig. 17(b) shows the Azores Swell in the North Atlantic. It forms a fairly complicated structure, which includes the Azores islands and the Cruiser and Great Meteor seamounts to the south. Volcanism has occurred for at least the last ∼20 Ma, although there is considerable uncertainty about the overall age of the thickened crust (e.g. Tucholke & Smoot 1990; Cannat et al. 1999). The pattern of volcanism does not appear to be time progressive. Again, calculation of buoyancy flux is complicated because the swell itself is part of a much larger region of elevated topography extending south from Iceland. Above a base level of ∼500 m, we estimate the volume of the swell to be ∼7.6 × 105 km3 , which corresponds to a steady state buoyancy flux of 1.8 Mg s−1 . The C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography Table 1. Estimates of swell volume, buoyancy flux and melt generation rate (where known) for major offshore plume swells. Swell Icelanda Volume (km3 ) – Hawaii 1.3 × 106b Cape Verde 2.5 –3.5 × 105c Réunion 1.6 × 106 Azores 7.6 × 105 Crozet 3.5 –5 × 105c Louisville 2.0 × 105 Cameroon 3.7 × 105 Bermuda 2.0 × 105 Buoyancy flux (Mg s−1 ) Melt rate (m3 s−1 ) 20–40 6-8 0.7 3.6 1.8 0.8–1.1 0.8d 1.5 1.0e 8 3–4 0.2–0.45 0.7 0.5 – 0.1 0.3 0.05 Notes: Buoyancy fluxes for swells other than Hawaii, Iceland and the Line Islands are steady-state assuming a thermal time constant of 30 Ma for the hot layer. a From Poore et al. (2008) and Ito et al. (1996) (excess melt rate above normal mid-ocean ridge), shown for comparison. b Coincides with volcanism to an age of 40 Ma. c Excludes elevation due to conductive reheating. d Last 25 Ma. e Estimate for period of activity. estimated melt generation rate using the T e values of Crosby (2006) is ∼0.5 m3 s−1 , which is less than one would expect for a swell of this size. Fig. 17(c) shows the Crozet Swell in the southern Indian Ocean. Volcanic addition has formed two discrete structures: the Del Cano Rise to the west, and the Crozet Bank to the east (e.g. Recq et al. 1213 1998). The oldest dated volcanic rocks in the Crozet Bank are 8 Myr old, although low T e values of 8–15 km are consistent with a somewhat older age for the edifice, given that the underlying crust is 67 Myr old (Crosby 2006). The Del Cano Rise appears to be older with an even lower T e and is no longer growing. Like the Cape Verde, the crest of the swell has a 15–20 mW m−2 heat flux anomaly, implying similar steady-state conductive reheating of the lithosphere (Courtney & Recq 1986). As with the previous two swells, the whole region is elevated, this time as part of the so-called African Superswell (Nyblade & Robinson 1994). Above a regional base level of ∼500 m, the swell has a volume of approximately 7.0 × 105 km3 , of which 50–70 per cent is supported by buoyant asthenosphere by analogy with the Cape Verde. The corresponding steady state buoyancy flux is 0.8–1.1 Mg s−1 . Fig. 17(d) shows the Louisville Swell in the southern Pacific Ocean. The island chain extends for over 4000 km and has been created by time-progressive volcanism over the last 77 Ma (Clouard & Bonneville 2005). The swell persists for at least the last 45 Ma of volcanic activity. Volcano size has diminished dramatically in the last ∼20 Ma, indicating a reduction in flux, and the swell today is small. In the last 25 Ma, given a regional base level of 150 m, we estimate a buoyancy flux of 0.8 Mg s−1 and a melt generation rate of 0.1 m3 s−1 . Finally, Fig. 17(e) shows the Cameroon Swell off West Africa. Onland and immediately offshore, volcanism has occurred irregularly over the last 30 Ma (Fitton & Dunlop 1985) and does not appear to be time progressive. The age of volcanism further offshore is unknown, although the distribution of edifices is fairly irregular. We Figure 15. The Hawaiian Swell. Panel (a) shows residual topography with thickened crust and the flexural moat excised. The contour interval is 500 m. Panel (b) is an average cross-section within the box in (a) highlighted in yellow. The solid lines are super-Gaussian fits with p equal 2.0, 2.5 and 3.0, respectively (Watson & McKenzie 1991; Wessel 1993). Note the difficulty of reconstructing the crest of the swell when it is completely covered by volcanic loads and associated flexural moats. Panel (c) is an estimate of the total volcanic load, calculated from the topography assuming T e = 28 km, ρc = 2800 kg m−3 and that the volume of underplate is equal to the volume of sedimentary infill. Note the two pulses of volcanism, which correlate with swell amplitude. Panel (d) is a plot of estimated buoyancy flux versus melt generation rate for each of the boxes in (a) colour-coded by midpoint age. Error bounds are calculated by varying p between 2 and 3, and T e between 20 and 36 km, respectively. Note the good correlation, which is evidence that pulsing hot asthenosphere travels approximately with the plate, and its volume modulates surface melt production. C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation 1214 A. G. Crosby and D. McKenzie Figure 16. The Cape Verde Swell. Panel (a) shows residual topography with thickened crust and flexural moats excised. Panel (b) is an average cross-section within the swath in (a). The solid lines are super-Gaussian fits with p equal 2.0, 2.5 and 3.0, respectively. Panel swath (c) is an estimate of the total volcanic load, calculated from the topography assuming T e = 25 km (e.g. Ali et al. 2004) and that the volume of underplate is equal to the volume of sedimentary infill. Panel (d) shows the simple conductive reheating models discussed in Appendix B. The solid blue line is the predicted change in elevation when the thermal boundary layer is instantly heated and then allowed to cool (i.e. the plate is only briefly over the plume, and the spreading asthenosphere moves with the plate). The dashed blue line is the corresponding change in surface heat flux. This model is more appropriate for plumes under fast-moving plates such as Hawaii. The solid red line is the change in elevation when a plume is permanently applied to the base of the plate. The dashed red line is the predicted heat flux. The grey box shows the observed heat flux anomaly over the Cape Verde Swell (Courtney & Recq 1986). This observation is more consistent with the permanent rather than transient reheating model, which leads us to believe that ∼600 m of the elevation at the centre of the swell has a lithospheric, rather than asthenospheric origin. estimate a steady-state buoyancy flux and melt generation rate of ∼1.5 Mg s−1 and 0.3 m3 s−1 , respectively. 7 C O N C LU S I O N S We have extended the study of Crosby et al. (2006) to include a worldwide shiptrack data set, with particular emphasis on the variation of unperturbed depth with age on young ocean floor. We find that, prior to 80 Ma, subsidence patterns are reasonably consistent, with gradients of approximately 325 ± 20 m Ma−1/2 and zero-age depths of approximately 2600 ± 200 m. There is, however, a striking correlation within this range between unperturbed zero-age depth and subsidence rates. Global depth-age trends to 160 Ma are not significantly different to those for the individual ocean basins. Within regions of similar basement age, gravity–topography correlations are consistently 30 ± 5 mGal km−1 . Simple isostatic theory suggests that, if the minimum depth of convection is defined by the base of the mechanical boundary layer, the admittance should be a function of plate age. The observation that it is not implies that active convection under mid-ocean ridges persists no shallower than the depth of the old ocean plate, which is consistent with geochemical modelling of melts near mid-ocean ridges. We have examined the relationship between residual topography and gravity worldwide, and have found that good spatial correlations are restricted to the Atlantic, North Pacific and youngest Indian Oceans. By contrast, residual topography and gravity are poorly or negatively correlated in the South and young Pacific Ocean and in the older Indian Ocean. Overall, the distributions of both appear symmetric about zero. Above volcanic swells, maps of residual topography can be used to make estimates of buoyancy flux which are correlated with es- timates of melting rates. Finally, comparison with mantle density distributions predicted using surface wave tomography suggests that long-wavelength (>2000 km) residual gravity anomalies have an origin deeper than the asthenosphere. This conclusion is supported by admittance estimates and a qualitative comparison between the residual topography and gravity and global tomographic model S20RTS. AC K N OW L E D G M E N T S AC is grateful to BP Exploration for support, and we thank John Sclater, Barry Parsons, Bob White, Keith Priestley, Dietmar Müller, Jeff Winterbourne, Nicky White, Heather Poore, Steve Jones, John Maclennan and Laura Mackay for helpful discussions. We are also grateful to Jean Francheteau, Earl Davis and an anonymous reviewer for suggesting a number of important improvements to the manuscript. This paper forms a wider part of the BP-Cambridge Margins Research Project, and is Department of Earth Sciences Contribution ES931. REFERENCES Adam, C. & Bonneville, A., 2005. Extent of the South Pacific Superswell, J. geophys. Res., 110, B09408, doi:10.1029/2004JB003465. Ali, M.Y., Watts, A.B. & Hill, I., 2003. A seismic reflection profile study of lithospheric flexure in the vicinity of the Cape Verde islands, J. geophys. Res., 108, doi:10.1029/2002JB002155. Cannat, M. et al., 1999. Mid-Atlantic Ridge-Azores hotspot interactions: along-axis migration of a hotspot-derived event of enhanced magmatism 10 to 4 Ma ago, Earth planet. Sci. Lett., 173, 257–269. C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography 1215 Figure 17. Residual topography of other swells for which we have attempted to estimate buoyancy flux. Contours are 500 m. (a) is Réunion. The star shows the location of volcanism at the present-day, and the white boxes show ages of the older parts of the chain (e.g. Tiwari et al. 2007). The black lines denote the extent of thickened crust and the swell, respectively: (b) is the Azores. G, Great Meteor and Cruiser seamounts and A, the Azores proper; (c) is the Crozet swell; (d) is the Louisville Swell. Volume was estimated using the swath average cross-section shown; (e) is the Cameroon swell. Volume was estimated using the swath average cross-section shown. P denotes the island of Pagalu, which is the site of the furthest ridge-ward volcanic age measurement; the ages of the other islands are unknown; (f) is the now-inactive Bermuda Swell, shown here for comparison. The present-day volume of the swell is 2 × 105 km3 above a base level of −200 m residual topography. The melt flux given a ∼25 Ma history of volcanism is small, 0.05 m3 s−1 , given a T e of 30 km (Sheehan & McNutt 1989). 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Oceanic crustal thickness from seismic measurements and Rare Earth Element inversions, J. geophys. Res., 97, 19 683–19 715. C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation 1217 Winterbourne, J., Crosby, A. & White, N., 2009. Depth, age and dynamic topography of oceanic lithosphere beneath heavily sedimented Atlantic Margins, Earth planet. Sci. Lett, in press. Zhong, S., Ritzwoller, M., Shapiro, N., Landuyt, W., Huang, J. & Wessel, P., 2007. Bathymetry of the Pacific plate and its implictions for thermal evolution of lithosphere and mantle dynamics, J. geophys. Res., 112, B06412, doi:10.1029/2006JB004628. A P P E N D I X A : D ATA I N C L U D I N G C O R R E C T I O N S T O C RO S B Y E T A L . (2006), APPENDIX A Several of the tables in Crosby et al. (2006, appendix A), contained minor typographical errors. Here they are reproduced in their correct form, with the new results for young seafloor. Table A1. Unperturbed depth as a function of age for the world to 160 Ma. Age bin (Ma) 0–1 1–4 4–9 9–16 16–25 25–36 36–49 49–64 64–81 80–90 90–100 100–110 110–120 120–130 130–140 140–150 150–160 rs b LQ Median UQ 0.36 0.35 0.27 0.37 0.39 0.42 0.45 0.46 0.40 0.40 0.59 0.52 0.51 0.50 0.58 0.30 0.51 31.5 30.7 29.3 28.1 29.4 31.5 30.0 32.5 30.7 28.8 28.9 28.1 27.2 37.7 31.3 25.6 32.5 2740 2943 3265 3588 3844 4170 4596 4921 5114 5115 5189 5260 5255 5373 5353 5488 5475 2938 3152 3480 3753 4053 4442 4807 5141 5360 5392 5395 5464 5411 5513 5543 5808 5703 3092 3345 3678 3961 4269 4651 4987 5314 5529 5656 5631 5680 5640 5643 5703 6032 5862 Notes: r s is the Spearman’s Rank Correlation Coefficient, b is the bestfitting slope in mGal km−1 , and the last three columns are the three quartiles of the depth in metres. Table A2. Unperturbed depth as a function of age for the Pacific Plate (this study and Crosby et al. 2006). Age bin (Ma) 0–1 1–4 4–9 9–16 16–25 25–36 36–49 49–64 64–81 80–90 90–100 100–110 110–120 120–130 130–140 140–150 150–160 160–180 rs b LQ Median UQ −0.25 −0.18 −0.11 0.17 0.17 0.12 0.38 0.42 0.42 0.804 0.885 0.779 0.658 0.403 0.447 0.401 0.586 0.014 −31.9 −27.3 −22.1 −26.5 23.8 29.3 27.9 23.2 20.8 27.7 27.1 31.8 35.6 38.1 35.1 33.4 41.3 30.9 2820 2993 3361 3578 3791 4193 4660 4960 5188 5399 5410 5390 5333 5437 5668 5594 5703 5578 3035 3174 3542 3712 3997 4534 4792 5137 5407 5539 5579 5598 5472 5574 5811 5850 5886 5844 3111 3327 3808 3982 4291 4677 4926 5281 5488 5717 5718 5706 5610 5699 5958 6099 6030 6092 1218 A. G. Crosby and D. McKenzie Table A3. Unperturbed depth as a function of age for the whole West Atlantic Plate (0–80 Ma) and Northwest Atlantic (80–150 Ma). Table A6. Unperturbed depth as a function of age for the Southern Indian and Antarctic oceans to a maximum age of 64 Ma. Age bin (Ma) 0–1 1–4 4–9 9–16 16–25 25–36 36–49 49–64 64–81 80–90 90–100 100–110 110–120 120–130 130–140 140–150 rs b LQ Median UQ Age (Ma) 0.84 0.77 0.82 0.74 0.70 0.71 0.74 0.81 0.59 0.614 0.628 0.405 0.364 0.466 0.096 0.250 33.0 32.5 32.8 30.3 34.5 36.8 35.8 36.7 32.6 74.5 59.3 39.8 29.4 38.2 46.4 47.3 2598 2853 3108 3481 3810 4217 4576 4879 5063 5750 5760 5249 5169 5238 5649 5612 2789 3021 3297 3650 4005 4394 4794 5108 5247 5784 5820 5767 5460 5463 6046 5671 2916 3148 3513 3827 4157 4582 4966 5277 5459 5930 5889 5869 5842 5929 6223 6165 0–1 1–4 4–9 9–16 16–25 25–36 36–49 49–64 Table A4. Unperturbed depth as a function of age for the whole East Atlantic Plate (0–80 Ma) and Northeast Atlantic (80–140 Ma). Age bin (Ma) 0–1 1–4 4–9 9–16 16–25 25–36 36–49 49–64 64–81 80–90 90–100 100–110 110–120 120–130 130–140 rs b LQ Median UQ 0.77 0.73 0.81 0.76 0.69 0.59 0.54 0.64 0.48 0.696 0.607 0.440 0.639 0.704 0.255 34.3 33.3 32.7 27.8 25.6 26.4 27.4 26.3 29.8 32.9 31.5 34.7 30.0 39.6 27.0 2658 2601 2941 3237 3733 4072 4396 4721 5137 5274 5431 5453 5357 5328 5292 2822 2847 3152 3477 3917 4292 4713 4996 5369 5527 5602 5648 5561 5468 5468 2991 3102 3375 3672 4116 4521 4940 5290 5609 5811 5788 5819 5744 5686 5619 rs b LQ Median UQ 0.74 0.67 0.39 0.29 0.26 0.39 0.24 0.18 30.6 33.5 36.7 31.8 31.8 26.9 27.7 25.1 2796 2948 3182 3548 3914 4325 4405 4588 2883 3148 3420 3794 4170 4614 4842 4875 2992 3418 3601 3986 4452 4818 5112 5184 Table A7. Unperturbed depth as a function of age for the Nazca–South American Plate to a maximum age of 81 Ma Age (Ma) 0–1 1–4 4–9 9–16 16–25 25–36 36–49 49–64 64–81 rs b LQ Median UQ −0.25 −0.18 −0.11 0.17 0.17 0.12 0.38 0.42 0.42 −31.9 −27.3 −22.1 −26.5 23.8 29.3 27.9 23.2 20.8 2820 2993 3361 3578 3791 4193 4660 4960 5188 3035 3174 3542 3712 3997 4534 4792 5137 5407 3111 3327 3808 3982 4291 4677 4926 5281 5488 APPENDIX B: SIMPLE CONDUCTIVE M O D E L S F O R P L U M E R E H E AT I N G In one dimension, if ∂2T ∂T =κ 2 ∂t ∂z (B1) then T (z, t) = Tss + Tu , where Tu = ∞ An sin nπ z n=1 Table A5. Unperturbed depth as a function of age for the whole Indian– Australian (0–64 Ma) and North Indian Plate (60–120 Ma). See Crosby et al. (2006) for discussion of gravity data selection criterion. Age (Ma) rs b LQ Median UQ 0–1 1–4 4–9 9–16 16–25 25–36 36–49 49–64 60–70 70–80 80–90 90–100 100–110 110–120 0.80 0.74 0.54 0.56 0.52 0.63 0.56 0.55 0.598 0.621 0.671 0.707 0.901 0.895 31.7 31.7 25.1 29.3 26.4 23.7 33.6 48.4 42.2 40.0 44.2 35.3 41.6 45.7 2848 2928 3255 3686 3919 4261 4616 4565 5165 5192 5345 5454 5599 5710 2932 3033 3385 3795 4111 4400 4857 4925 5284 5273 5413 5837 5762 5867 3055 3166 3533 3897 4263 4542 5031 5109 5394 5431 5548 5959 5976 5958 (B2) An = 2 a a a Tu (z, 0) sin 0 −n 2 π 2 κt exp a2 nπ z a dz, (B3) (B4) T is temperature, t is time, z is depth, κ is the thermal diffusivity (7 × 10−7 m2 s−1 ), a is the thickness of the plate (100 km), T ss is the steady-state temperature after perturbation (i.e. T |t→∞ ), and T u is the unsteady component of temperature introduced by the perturbation. The geotherm prior to perturbation is linear with T = 0◦ C at the sea floor and T = 1330 ◦ C at the base. The heat flux at the surface is given as q = −k ∂ T , ∂z z=0 (B5) where k is conductivity (3.1 W m−1 K−1 ), and the elevation is calculated isostatically with respect to the unperturbed state L ρm α T (z, 0) − T (z, t) dz, (B6) e(t) = ρm − ρw 0 C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation Ocean depth, gravity and residual topography where ρm and ρw are the densities of the mantle and water, respectively, and α is the thermal expensivity (3.2 × 10−5 K−1 ). L is the depth of compensation. Tss = An = B1 Base of plate raised from Tm (1330 ◦ C) to Tp (1500 ◦ C) permanently. Depth of compensation at z = a. This model is most appropriate for plumes impacting slow plates such as the Cape Verde plume. Tss = Tp z a An = 2a π (B7) Tp Tm − L a (−1)n n (B8) B2 Thermal boundary layer (z = a to L) raised instantaneously to Tp at t = 0 and then allowed to cool This model is most appropriate for plumes impacting fast plates such as the Hawaiian plume. Tm z L (B9) nπa 2 Tm a (−1)n (Tm − Tp ) + Tp − cos nπ L L nπa Tm sin + nπ L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 APPENDIX C: NUMERICAL MODELS O F T H E A D M I T TA N C E O V E R U N D E RWAT E R S T E A D Y- S TAT E P L U M E S See McKenzie (1994) for experimental details. Ra Box dimensions (radial × vertical km) Lid (km) Admittance (mGal km−1 ) 3.94 × 106 3.94 × 106 3.94 × 106 5.26 × 106 7.89 × 106 7.89 × 106 7.89 × 106 1.05 × 107 1.05 × 107 1.32 × 107 1.32 × 107 1.44 × 107 1.44 × 107 1.58 × 107 1.58 × 107 1.58 × 107 1.78 × 107 2.30 × 107 2.30 × 107 6.58 × 107 832 × 555 575 × 383 440 × 293 828 × 552 827 × 551 569 × 379 434 × 289 607 × 405 536 × 357 591 × 394 528 × 352 569 × 379 436 × 291 832 × 555 574 × 383 436 × 291 602 × 401 509 × 339 469 × 313 832 × 555 69 72 73 69 69 71 72 71 71 70 70 71 73 69 72 73 72 73 73 69 32.5 36.3 38.8 32.0 31.2 35.1 37.8 33.7 35.0 33.5 34.8 33.9 36.6 29.7 33.6 36.5 33.0 34.3 35.2 27.0 Notes: Run 17 is best fit for Hawaiian melting and geoid anomaly (Watson & McKenzie 1991). Run 20 is from Crosby (2006). Note inverse correlation between admittance, convecting layer thickness, lid thickness and Rayleigh Number (Ra). Observed admittance in the oceans is 25–35 mGal km−1 . Average model admittance for boxes with vertical dimensions larger than 500 km is 30.5 mGal km−1 . Total vertical thickness is lid thickness plus box thickness. C 2009 The Authors, GJI, 178, 1198–1219 C 2009 RAS Journal compilation (B10) values for a and L are 80 and 120 km, respectively, and eq. (B6) is only evaluated to z = a. The boundary condition that z = L remains at T m is slightly unrealistic (we have effectively thickened the plate), but is used in order to achieve a simple analytical solution. Furthermore, the hot material at some point becomes convectively unstable, so this model is only appropriate until the local Rayleigh number is exceeded. Table C1. Convection runs, table modified from McKenzie (1994). Run 1219