Download An analysis of young ocean depth, gravity and global residual

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
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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
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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
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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
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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).
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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)
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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
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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
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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
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2009 The Authors, GJI, 178, 1198–1219
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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
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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
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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.
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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.
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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
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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
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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.
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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≈
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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,
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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)
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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
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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
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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 .
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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
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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.
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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.
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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.
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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
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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