Geoid height versus topography for a plume model of the Hawaiian swell

Earth and Planetary Science Letters 178 (2000) 29^38 www.elsevier.com/locate/epsl

Geoid height versus topography for a plume model of the Hawaiian swell L. Cserepes a; *, U.R. Christensen a , N.M. Ribe b b

a Institut fu«r Geophysik, Universita«t Go«ttingen, Herzberger Landstrasse 180, 37075 Go«ttingen, Germany Department of Geology and Geophysics, Yale University, P.O. Box 208109, New Haven, CT 06520-8109, USA

Received 2 July 1999; received in revised form 3 February 2000; accepted 19 February 2000

Abstract We use a three-dimensional variable-viscosity convection model of a stationary plume beneath a drifting lithosphere to study the factors that control the geoid-to-topography ratio (GTR) of the Hawaiian swell. The rate of melting in the plume is predicted using a batch melting parameterization, and the melt is assumed to migrate to the surface where it builds a volcanic edifice, equivalent to the Hawaiian island chain. Viscous stresses, elastic deformation of the lithosphere and (optionally) the volcanic material deposited on the ocean floor are included in the calculation of surface topography and the corresponding geoid. The derivation of the GTR from the model imitates methods that have previously been used to estimate the `observed' GTR for the Hawaiian swell. The first method we use here is that of Marks and Sandwell [J. Geophys. Res. 96 (1991) 8045^8055], which applies bandpass filters to retain only wavelengths from 400 to 4000 km as most characteristic of the swell topography and geoid, and the GTR is estimated from the slope of the regression line of geoid versus topography. Another group of methods analyzes the data along profiles drawn across the hotspot swell and eliminates the unwanted signals, e.g. the volcanic islands and the flexural moat around them, by cutting out parts of the sections. The GTR is then calculated from curves which best fit the topography and geoid profiles on the swell flanks only. In our plume model, when the effects of the volcanic surface loading are included, Marks and Sandwell's method yields 4.4 m/km for the GTR, while profile-fitting on the swell flanks gives 7^ 8.5 m/km. Ignoring the volcanic load leads to 7^8 m/km in all processing methods. The observed GTR for the Hawaiian swell has been reported to lie between 4 and 5 m/km. Analysis of the data processing methods shows that the applied bandpass filters cannot completely remove the signal due to the volcanic edifice and lithospheric flexure and this causes apparent GTRs around 4 m/km. The `pure' swell model containing no volcanic load on the surface demonstrates that the Hawaiian swell may have a proper GTR near 7 m/km. ß 2000 Elsevier Science B.V. All rights reserved. Keywords: geoid; topography; convection; models; mantle plumes

1. Introduction

* Corresponding author. Permanent address: Department of Geophysics, Eo«tvo«s University, Ludovika te¨r 2, 1083 Budapest, Hungary. Fax: +36-1-210-1089; E-mail: [email protected]

Among the most remarkable and largest topographic features on the ocean £oor are the hotspot swells. The best studied example is the Hawaiian swell. The close coincidence of these broad uplifts of the ocean £oor and the hotspots sug-

0012-821X / 00 / $ ^ see front matter ß 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 2 - 8 2 1 X ( 0 0 ) 0 0 0 6 5 - 0

EPSL 5424 17-4-00

30

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

gests that the swells are of thermal origin. The earliest models assumed that a swell of this kind is the consequence of some unspeci¢ed heat source which warms the base of the lithosphere and causes uplift through simple thermal expansion [1,2]. An increase of lithospheric temperatures means thinning of the lithosphere according to its usual thermal de¢nition. Several facts, however, many of them observed precisely in the case of the archetype Hawaii, are inconsistent with a pure thermal model. The uplift of the swell on the moving lithospheric plate is geologically fast: for Hawaii it takes only a few million years. This high speed cannot be explained by simple reheating because pure thermal conduction is far too slow for that. Besides, reheating should produce a signi¢cant heat £ow anomaly which cannot be detected over the Hawaiian swell [3]. Similarly, seismic wave propagation shows little if any change in the lithospheric structure under Hawaii compared to normal lithosphere of the same age [4,5]. Thus, in creating the hotspot swells, reheating must be accompanied by a more powerful agent, dynamic uplift, which quite evidently is the result of a local upwelling motion of the mantle. Today it is commonly assumed that the best theoretical model for the hotspot swells is the plume model which combines thermal and dynamic aspects. The plume, a cylindrical warm upwelling of mantle convection, induces a vertical stress ¢eld and temperature increase in the lithospheric plate which, together, can explain the topography of the swell, the rapidity of uplift and the heat £ow values as well. Convective swell models were calculated with two-dimensional numerical techniques, e.g. by Courtney and White [6], Robinson and Parsons [7] and Monnereau et al. [8], and in the full three dimensions by Ribe and Christensen [9]. These calculations yield no signi¢cant lithospheric thinning. With a more strongly temperature-dependent viscosity, Moore et al. [10,11] ¢nd some thinning in 3D models although they do not quantify how much this contributes to the total surface uplift. The large number of physical parameters in convective models allows a higher degree of freedom than the simple thermal thinning models. Adjustment

of a whole set of parameters gives more chance to ¢t all the relevant geophysical observations. A main aspiration of the convective models is to explain quantitatively two important characteristics of the hotspot swells : their topography and the corresponding geoid anomaly. In simple models of isostatic compensation, the geoid-to-topography ratio (GTR) is linearly related to the apparent compensation depth [12^14], therefore it is often regarded as a fundamental quantity in the physical theory of swell support. The topography and geoid (or gravity) ¢elds of the Hawaiian swell have been the target of many investigations [1,2,9,15^21]. Table 1 summarizes the estimated GTR values for Hawaii, obtained by processing the observed topography and geoid data of the swell. The values scatter between 3.5 and 6 m/km, a representative `mean' being 4.5 m/ km. The apparent compensation depth which corresponds to this mean GTR according to a formula given by Crough [2] is about 45 km. (Crough's formula identi¢es the compensation depth with the average depth of compensating density anomalies.) Using layered 2D convection models with a conductive lithosphere, a low-viscosity asthenosphere and a medium-viscosity deeper mantle, Robinson et al. [22], Robinson and Parsons [7] and Ceuleneer et al. [23] have shown that the above-mentioned GTR values and the corresponding compensation depths can be reproduced if the asthenosphere has a viscosity 10^100 times lower than the rest of the mantle. More realistic plume models with temperature- and pressure-dependent viscosity, however, did not have the same

Table 1 GTR values reported for Hawaii (m/km) Crough [2] McNutt and Shure [18] Cazenave et al. [27] Monnereau and Cazenave [13] Sandwell and Renkin [19] Sandwell and MacKenzie [12] Marks and Sandwell [28] Wessel [21] a

V6 5.6a 3.5 3.7^4.2 5.54 3.76 4.56 4^5 (varying along the swell)

This value is calculated from the depth and geoid anomalies given by the authors.

EPSL 5424 17-4-00

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

success. Either in 2D [8] or in 3D [9,10] these numerical models of the Hawaiian plume gave signi¢cantly higher GTRs than the observed values, by a factor of two at least. The aim of the present study is to reconsider the GTR derived from convective models of the Hawaiian swell. The starting point is Ribe and Christensen's work [9], which calculates the temperature ¢eld of the plume and the surface observables using a temperature- and pressure-dependent viscosity law and a moving lithospheric plate at the surface. This work is developed further in two respects. First, the partial melting which occurs in the central part of the plume head is taken into account explicitly. The melt is loaded onto the surface where it forms volcanic structures and bends the lithosphere, modifying the topography. This part of the calculations is discussed in detail by Ribe and Christensen [24] (hereafter : Paper 1), therefore Section 2 will touch this question but brie£y. Second, the synthetic data obtained for the topography and geoid of the Hawaiian swell will be processed in the same way as other authors treat raw observed data using ¢ltering, regression analysis or curve ¢tting. This will allow direct comparison between the theoretical and observed GTRs. 2. Model description This study uses the same numerical method for modelling the Hawaiian plume as Ribe and Christensen [9,24], completed with the explicit treatment of hotspot volcanism. Paper 1 has set up a `reference' model which describes the most recent volcanic history of the Hawaiian chain, matching the present-day volcanic mass production of 13.5U103 kg/s [25]. However, the Hawaiian swell is the product of a long period of time (40 million years) during which the melt production rate in the plume and the corresponding volcanic mass production on the surface have varied signi¢cantly. On average, the mass production rate has been about (7^8)U103 kg/s, as can be assessed from studies of the crustal structure in the neighborhood of the Hawaiian islands [25]. The present-day melting rate is thus much higher

31

Fig. 1. The computational domain of the convection model and the plume position.

than the average. Since this study analyzes the topography and geoid of the whole swell, we will slightly modify the reference case of Paper 1 and will use a theoretical model matching the mean melting rate observed on the Hawaiian swell. In the ¢rst part of the modelling work, we calculate 3D steady-state convection in a rectangular box using the Boussinesq approximation of the governing equations. The depth of the box is 400 km, its length in the x direction is 2304 km, while its width in the y direction is 1152 km. Rigid motion of the Paci¢c plate is simulated by prescribing a constant 8.6 cm/year velocity at the surface (Fig. 1). This generates a passive shear £ow; the stream ¢eld of the plume is superimposed on this background. The sides parallel to x are symmetry planes, while in£ow and out£ow corresponding to the background £uid motion are speci¢ed on the sides parallel to y. Since the bottom depth of 400 km is not a natural boundary of mantle convection, any choice of boundary condition here is to some extent arti¢cial. In our model, the horizontal velocity and the vertical normal stress are set to zero, but the boundary is permeable to vertical £ow. As to the thermal boundary conditions, the temperature is 0³C at the surface and 1300³C at

EPSL 5424 17-4-00

32

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

the bottom. The plume is generated by a circular Gaussian temperature anomaly on the bottom of the box with radius 58 km, centered at x = 648 km and y = 0. The maximum of this temperature anomaly was set after evaluating a series of models in order to ¢t the mean melt production rate of the Hawaiian hotspot for the last 35^40 million years. The plume excess temperature is then lower by 20³C than that used in the reference model of Paper 1. It is 280³C at 400 km depth and 273³C at the depth where melting starts. This is the only modi¢cation applied here with respect to the model of Paper 1. At the in£ow side of the model box (x = 0) the temperature varies according to an error function as is predicted by the boundary layer theory. The viscosity is a function of temperature and pressure [9] with an activation energy E = 3.3U105 J/mol and activation volume V = 4U1036 m3 /mol. The temperature in the plume head exceeds the solidus temperature above the depth of 145 km, and thus partial melting occurs. The degree of melting is calculated as in Paper 1 and it is assumed that the melt migrates to the surface vertically and instantaneously. The rate of volcanic mass accumulation on the surface is 7.6U103 kg/s. Melting reduces the density of the residue and thereby a¡ects the buoyancy force and the gravity ¢eld. The volcanic edi¢ce formed by the melt modi¢es the surface topography both by its mere presence and by bending the lithosphere elastically. If the plume £ow is in a steady state, a constant quantity of volcanic material will be accumulated at a point (x, y) on the surface. While the real volcanic structure of Hawaii consists of isolated islands forming a chain, the model produces a continuous volcanic `ridge'. Correcting for the fact that the real melt migration is not strictly vertical, the volcanic material at a given distance x in the model will be redistributed along the y direction according to a Gaussian pro¢le of half-width 166 km (i.e. width at half of the peak height). This value was chosen for conformity with the average lateral extent of volcanic bodies at Hawaii as revealed by the surface topography and deep crustal seismic studies [20,21,26]. The lithospheric topography h(x, y), i.e. the topography of the surface on which the volcanic

body sits, is calculated from the biharmonic equation:  2 2 D D2 ‡ h ‡ … 8 m 3 8 w †gh ˆ D D x2 D y2

c zz 3… 8 c 3 8 w †gd where D = 1.7U1023 Nm is the £exural rigidity (a good average for Hawaii, see [20]), 8m = 3300 kg/ m3 , 8c = 2700 kg/m3 , and 8w = 1000 kg/m3 are the densities of the mantle, crust and water, respectively, g is gravity, czz the vertical stress generated by the convective £ow at the surface, d(x, y) is the height of the volcanic mass on the surface. The total topography of the ocean £oor is: H ˆh‡d The geoid anomaly is calculated by summing the gravitational e¡ects of the internal density distribution, the lithospheric topography h and the volcanic mass. Because the latter has a lower density (8c ) than the mantle, the contributions of h and d to the geoid anomaly must be treated separately. 3. Evaluation of the topography and geoid data The ¢nal aim of this study is to interpret the topography and geoid anomalies of the swell model in terms of the GTR. The GTR can be rigorously de¢ned as N(k)/H(k) where N(k) and H(k), functions of the wavenumber k, are the power spectral densities of the geoid and topography, respectively. It is more common, however, to adopt a single characteristic value for GTR instead of a spectral function. This value is usually the output of some loosely de¢ned data evaluation procedure rather than the result of a rigorous mathematical formula. Most studies follow the methods described below. The ¢rst method uses linear regression of geoid versus topography crossplots [12,13,19,21,27,28]. The data are usually collected from the whole area of the hotspot swells, but they are also ¢ltered in some way to get rid of unwanted signals.

EPSL 5424 17-4-00

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

Experience shows that the wavelength range of the geoid and topography anomalies which is characteristic of the swell structure and compensation mechanism lies between 400 and 4000 km for Hawaii and many other hotspots. The wavelengths below 400 km are consequences of the elastic £exure of the lithosphere, variation of the crustal thickness and the presence of volcanic islands and seamounts, while the wavelengths longer than 4000 km have their source in the density heterogeneities of the deep mantle and in the cooling of the lithosphere. The bandpass-¢ltered data on the geoid versus topography crossplot lie nearly along a straight line whose slope gives the GTR. A precisely de¢ned representative of this method is that used by Marks and Sandwell [28]. This applies smooth (tapered) bandpass ¢lters to both the geoid and topography of the entire swell region, retaining the 400^4000 km wavelength range, and calculates linear regression from the ¢ltered data. The present study will concentrate on applying Marks and Sandwell's method on the synthetic swell data. Another group of methods analyzes data along pro¢les drawn across the hotspot swell and eliminates the unwanted components (lithospheric £exure, islands and seamounts) by simply cutting out parts of the pro¢les. Monnereau and Cazenave [14] mark out radial sections centered on the present-day hotspot site, stack the data of these sections to arrive at average pro¢les for the geoid and topography and ¢t Gaussian curves to these two pro¢les. The ratio of the amplitudes of the Gaussians is an estimate for the GTR. They exclude some central portion of the sections and ¢t the Gaussians to the £anks of the swell. After using bandpass ¢lters on the raw data as done by Marks and Sandwell [28], Wessel [21] draws pro¢les perpendicularly to the strike of the Hawaiian island chain and calculates linear regression on the geoid versus topography plot of individual pro¢les. The GTR is then the slope of the regression line. He also con¢nes his analysis to the £anks of the swell, determining by eye where the truncation of the sections is to be made. These latter pro¢le-cutting methods di¡er in two main respects from the technique of Marks and Sandwell [28]. First, they use pro¢les while

33

Marks and Sandwell's method calculates the GTR from data over the entire swell area. Second, Marks and Sandwell's method is rather straightforward and does not require ad hoc assumptions. In contrast, the pro¢le-cutting methods need to make assumptions as to what part of the pro¢le represents the swell proper and which parts are contaminated by the volcanic edi¢ce, the £exural bending and other ocean £oor features not related to the hotspot. The choice of which data to exclude is arbitrary to a certain extent ^ which may have an in£uence on the results ^ and is usually not documented in a quantitative and reproducible way. 4. Results Fig. 2 presents the ocean £oor topography and geoid map for the model described in Section 2. The upper two plots show the full model which includes the volcanic edi¢ce (Fig. 2a,b); the lower panels present the case where the volcanic surface loading is omitted from the calculation (Fig. 2c,d). The volcanic material forms a long continuous ridge in Fig. 2a. A trough appears beside the ridge due to the £exure of the lithosphere. Its distance from the ridge center, roughly 150 km, corresponds well to the position of the real Hawaiian `moat'. Far away from the ridge and the trough, for about y s 350 km, there is no perceptible di¡erence between the contours of the upper and lower panels in Fig. 2 (apart from a shift of the zero level which results from setting the mean elevation to 0 in each case). The central volcanic edi¢ce, the £exural moat and the unperturbed slope of the swell as well as the corresponding geoid anomalies are more clearly seen in Fig. 3 which shows sample pro¢les across the swell at x = 1152 km, middle of the box in the x direction (see the solid lines in the two panels). The maxima of the topography and geoid are found along the swell axis (x-axis), somewhat downstream of the plume root position (x = 648 km). If measured above the imaginary lithospheric surface free from the plume anomaly, this maximum amounts to 3689 m in Fig. 2a and 1115 m in Fig. 2c. Similarly the geoid maxima are 13.1 m and 8.5 m

EPSL 5424 17-4-00

34

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

Fig. 2. (a) Surface topography in the model including volcanic surface loading and (b) the corresponding geoid. (c) Surface topography in the model without volcanic load and (d) the corresponding geoid. Contouring is truncated at 1200 m topography and 10 m geoid height; this a¡ects only the area of the volcanic `ridge' in panels (a) and (b). Black arrowheads show the location of the plume center at the bottom of the model.

in the cases with and without volcanic load, respectively. It is to be noted that the maximum unperturbed swell height found here (1115 m) is less than the observed maximum uplift of the Hawaiian swell (V1350 m, see Paper 1), since this latter value corresponds to a recent high hotspot activity while the model used in this paper is a `time-averaged' model for the last 35^40 million years. The topography and geoid anomalies have been `processed' using a tapered bandpass ¢lter proposed by Sandwell and Renkin [19] and Marks and Sandwell [28], retaining only the 400^4000 km wavelength band. The resulting data are shown as geoid-to-topography crossplots in Fig. 4b,c. These plots contain 500 points chosen randomly from all over the model area. Adapting Marks and Sandwell's method [28], regression lines have been ¢tted to the data, and the slope

yields a geoid-to-topography ratio GTR = 4.37 m/ km with volcanic loading (Fig. 4B), whereas without loading the GTR is 7.74 m/km (Fig. 4c). Fig. 4a shows also 240 `observed' data from Marks and Sandwell [28], who processed the bathymetry and geoid of the Hawaiian swell by the same ¢ltering technique as used above. Marks and Sandwell's regression ¢t gave GTR = 4.56 m/km (see Table 1). Fig. 4d shows the overlap of the data clusters of panels a, b and c by delineating their envelopes. The coincidence of the two point clusters, the observed one in Fig. 4a and the model in Fig. 4b, is very good, with only one noticeable di¡erence: the scatter of the real data is greater than that of the calculated ones. The corresponding GTRs are very close to each other. It is to be noted that both data sets (Fig. 4a,b) bend into two sections around 200 m height, therefore the slope of the clusters is higher in the low-topogra-

EPSL 5424 17-4-00

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

Fig. 3. Representative cross-sections of our models taken at x = 1152 km. Solid lines: pro¢les from the model with volcanic surface loading. Dashed lines: pro¢les from the model with no volcanic load. Dotted lines: pro¢les from the model with volcanic load after applying Marks and Sandwell's [28] bandpass ¢lter.

phy range. Lower topography values come from the distant £anks of the swell and from the area of the lithospheric £exure. Consequently, if there were no volcanic peaks in the axial part of the swell, i.e. the high-topography region of the crossplots were absent, the GTR would be higher. This is just the case in Fig. 4c: the model without volcanic surface loading exhibits the `pure' e¡ect of the swell. The swell in itself seems to yield a GTR around 7^8 m/km. The second, `higher' part of C Fig. 4. Geoid versus topography crossplots. (a) Observed data of the Hawaiian swell from Marks and Sandwell [28]. (b) Model of Fig. 2a,b, with volcanic load. (c) Model of Fig. 2c,d, no volcanic load on the surface. (d) Envelopes of the data clusters: dashed line for panel (a), thick solid line for panel (b), thin solid line for panel (c). Regression lines in panels (a), (b) and (c) are labeled with their slope.

EPSL 5424 17-4-00

35

36

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

5. Discussion

Fig. 5. Application of Wessel's analysis [21] to the model with volcanic edi¢ce on the surface. The pro¢le is taken at x = 1152 km (as in Fig. 3). Crosses show the geoid-versustopography curve extracted from the ¢ltered data sets (dotted lines of Fig. 3). The solid line is obtained by regression over the swell slope for which the data lie between points A and B approximately. This particular section yields 8.2 m/km for the GTR.

the clusters in Fig. 4a,b is derived from the area of the volcanic ridge and has a lower slope. It is important to stress that this is seen in the `observed' data set as well as in the theoretical model. This shows that the applied ¢lter with a low-pass cuto¡ at 400 km wavelength cannot completely remove the e¡ect of the volcanic pile as has already been pointed out by Wessel [21]. If this e¡ect were perfectly eliminated, the result would be similar to Fig. 4c. We have also applied the pro¢ling methods of Monnereau and Cazenave [14] and Wessel [21]. The GTRs obtained in this way are between 7 and 8.5 m/km, depending signi¢cantly on where exactly we mark out the limits of the swell slopes. Nevertheless, these values scatter around the `pure swell GTR' found in Fig. 4c. An example of the application of Wessel's method is shown in Fig. 5. Here our model which includes the volcanic load has been processed by bandpass ¢ltering and by regression analysis along the swell cross-section used in Fig. 3. The dotted lines of Fig. 3 show the topography and geoid pro¢les after ¢ltering. These are reproduced as the geoid versus topography curve of Fig. 5. This latter ¢gure illustrates the ¢tting of a straight line to a portion of a curve which cannot be uniquely delimited.

A major de¢ciency of previous convective models of the Hawaiian and other hotspot swells is that they yield much higher GTRs than the observed ones [8^10]. An improvement here and in Paper 1 compared to earlier work has been achieved by taking into account the depletion buoyancy due to melting. As discussed in Paper 1, melt removal decreases the mean depth of the negative sublithospheric density anomaly, thereby reducing the GTR. Ignoring this e¡ect, Ribe and Christensen [9] obtained 10 m/km for the GTR, while a proper consideration of mantle depletion yields values in the range of 7^8 m/km for the swell itself (Paper 1 and this study, Fig. 4c). However, this is still too high relative to the observed 3.5^6 m/km (Table 1). The results of the previous section suggest that the remaining discrepancy could be due mostly to the e¡ect of volcanic loading and lithospheric £exure and to the way in which these e¡ects are handled by the data processing. All the studies that attempted to derive the swell GTR from observables [12^14,19,21,27,28] took some measure to eliminate these topographic signals and other `noise' (seamounts, oceanic plateaus not connected to the Hawaiian plume). Adapting Marks and Sandwell's method [28] for the analysis of our model data, we show that the applied bandpass ¢lter cannot completely eliminate the unwanted contribution of volcanic loading. This is clearly seen in the dotted pro¢les of Fig. 3. Applying the ¢lter to the model which includes the volcanic loading results in a cluster of data points ¢tting the observations very well (Fig. 4d). The calculated theoretical GTR closely reproduces the one derived from the observed data. On the other hand, the model which disregards the volcanic masses by de¢nition again gives a high GTR, over 7 m/km (Fig. 4c). The low GTR values found in Fig. 4a,b, either with measured or with theoretical data, result from the inability of the applied ¢lter to neutralize the in£uence of the volcanic load. This conclusion is con¢rmed by the pro¢ling methods [14,21] which use a su¤ciently distant part of the swell slopes so that the volcanic pile has no e¡ect on the results.

EPSL 5424 17-4-00

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

The model GTRs obtained in this way are high (7^8.5 m/km), comparable to that estimated for the `no load' case (Fig. 4c). Marks and Sandwell's [28] data processing method, selected as the main tool of our analysis by virtue of its well-de¢ned character, yields satisfactory agreement between the observed and synthetic GTRs for Hawaii which, in turn, can indicate the success of the self-consistent dynamic modelling of the underlying plume and its surface signatures. In the case of the pro¢ling methods, however, we found no agreement. Tests on our synthetic data yield a GTR of 7^8.5 m/km, close to the value obtained for the model swell una¡ected by surface loading, but in contrast to the values of 4^ 5 m/km derived from observations (Table 1). This suggests that the pro¢ling methods should in principle be able, despite the fuzzy de¢nition of cuto¡s, to give reliable estimates for the GTR of the swell proper. However, it is enigmatic why both kinds of methods give virtually identical results when applied to observed data [14,21,28] whereas they di¡er signi¢cantly for our synthetic data. The topography and the geoid anomaly of the Hawaiian swell can be `contaminated' by various signals that are not directly related to the plume or its e¡ect on the lithosphere. They could originate from long-wavelength variations in the crustal structure or from dynamic topography and related geoid height due to deep mantle sources. The GTR values associated with the former cause are low and those associated with the latter source are high. The long-wavelength part of the ¢lter could remove their contribution only partially. The way in which they a¡ect the overall GTR of the swell, as determined by the various methods, depends on how they correlate with the swell signal. Conceivably, this `noise' can a¡ect the various methods of data processing in di¡erent ways. Based on our comparison of processing methods with synthetic data, we conclude that the GTR of the Hawaiian swell proper is poorly constrained and might lie in the range of 4^8 m/km. The convection model presented here and in Paper 1 predicts a value near the upper end of this interval. However, the truncation of the physical

37

model at 400 km depth is likely to bias the model GTR towards the high side. The viscosity structure of the mantle below 400 km has an in£uence on the geoid amplitude, and especially an increasing viscosity can lower the GTR [7,22,23]. Our convection model represents an end-member in this respect, because the condition of vanishing normal stress at the bottom corresponds to an inviscid substratum below the model box. A ¢nite viscosity would support the plume head partially from below and would in particular reduce the geoid anomaly. 6. Conclusion The `true' (unperturbed) GTR of the Hawaiian swell is not well constrained. A convection model that takes volcanic surface loading and depletion buoyancy into account predicts geoid and topography data that compare favorably with observed data processed by certain methods, but shows discrepancies with respect to results of other methods. The GTR of the plume model, 7.7 m/km, might be compatible with the observations of the Hawaiian swell, but represents an upper limit because the model neglects a partial support of the plume head by viscous £ow in the transition zone and the deep mantle. Acknowledgements The authors are grateful to David Sandwell for granting permission to use his data set plotted in Fig. 4a. Thanks are due to Marc Monnereau and Garrett Ito for their helpful and constructive comments on the manuscript. Support from the German Science Foundation through Grant Ch77/8 and from the Hungarian National Grant OTKA T026630 is acknowledged.[FA]

References [1] R.S. Detrick, S.T. Crough, Island subsidence, hot spots, and lithospheric thinning, J. Geophys. Res. 83 (1978) 1236^1244.

EPSL 5424 17-4-00

38

L. Cserepes et al. / Earth and Planetary Science Letters 178 (2000) 29^38

[2] S.T. Crough, Thermal origin of mid-plate hot-spot swells, Geophys. J. R. Astron. Soc. 55 (1978) 451^469. [3] R.P. Von Herzen, M.J. Cordery, R.S. Detrick, C. Fang, Heat £ow and the thermal origin of hot spot swells: The Hawaiian swell revisited, J. Geophys. Res. 94 (1989) 13783^13799. [4] M.T. Woods, E.A. Okal, Rayleigh-wave dispersion along the Hawaiian Swell: a test of lithospheric thinning by thermal rejuvenation at a hotspot, Geophys. J. Int. 125 (1996) 325^339. [5] K. Priestley, F. Tilmann, Shear-wave structure of the lithosphere above the Hawaiian hot spot from two-station Rayleigh wave phase velocity measurements, Geophys. Res. Lett. 26 (1999) 1493^1496. [6] R.C. Courtney, R.S. White, Anomalous heat £ow and geoid across the Cape Verde Rise: Evidence for dynamic support from a thermal plume in the mantle, Geophys. J. R. Astron. Soc. 87 (1986) 815^867. [7] E.M. Robinson, B. Parsons, E¡ect of a shallow low-viscosity zone on the formation of midplate swells, J. Geophys. Res. 93 (1988) 3144^3156. [8] M. Monnereau, M. Rabinowicz, E. Arquis, Mechanical erosion and reheating of the lithosphere: A numerical model for hotspot swells, J. Geophys. Res. 98 (1993) 809^823. [9] N.M. Ribe, U.R. Christensen, Three-dimensional modeling of plume-lithosphere interaction, J. Geophys. Res. 99 (1994) 669^682. [10] W.B. Moore, G. Schubert, P. Tackley, Three-dimensional simulations of plume-lithosphere interaction at the Hawaiian swell, Science 279 (1998) 1008^1011. [11] W.B. Moore, G. Schubert, P.J. Tackley, The role of rheology in lithospheric thinning by mantle plumes, Geophys. Res. Lett. 26 (1999) 1073^1076. [12] D.T. Sandwell, K.R. MacKenzie, Geoid height versus topography for oceanic plateaus and swells, J. Geophys. Res. 94 (1989) 7403^7418. [13] M. Monnereau, A. Cazenave, Variation of the apparent compensation depth of hotspot swells with age of plate, Earth Planet. Sci. Lett. 91 (1988) 179^197. [14] M. Monnereau, A. Cazenave, Depth and geoid anomalies over oceanic hotspot swells: A global survey, J. Geophys. Res. 95 (1990) 15429^15438. [15] A.B. Watts, Gravity and bathymetry in the central Paci¢c Ocean, J. Geophys. Res. 81 (1976) 1533^1553.

[16] A.B. Watts, An analysis of isostasy in the world's oceans, 1. Hawaiian-Emperor seamount chain, J. Geophys. Res. 83 (1978) 5989^6004. [17] D.T. Sandwell, K.A. Poehls, A compensation mechanism for the central Paci¢c, J. Geophys. Res. 85 (1980) 3751^ 3758. [18] M. McNutt, L. Shure, Estimating the compensation depth of the Hawaiian swell with linear ¢lters, J. Geophys. Res. 91 (1986) 13915^13923. [19] D.T. Sandwell, M.L. Renkin, Compensation of swells and plateaus in the North Paci¢c: No direct evidence for mantle convection, J. Geophys. Res. 93 (1988) 2775^2783. [20] P. Wessel, A reexamination of the £exural deformation beneath the Hawaiian islands, J. Geophys. Res. 98 (1993) 12177^12190. [21] P. Wessel, Observational constraints on models of the Hawaiian hot spot swell, J. Geophys. Res. 98 (1993) 16095^16104. [22] E.M. Robinson, B. Parsons, S.F. Daly, The e¡ect of a shallow low viscosity zone on the apparent compensation of mid-plate swells, Earth Planet. Sci. Lett. 82 (1987) 335^ 348. [23] G. Ceuleneer, M. Rabinowicz, M. Monnereau, A. Cazenave, C. Rosemberg, Viscosity and thickness of the sublithospheric low viscosity zone: Constraints from geoid and depth over oceanic swells, Earth Planet. Sci. Lett. 89 (1988) 84^102. [24] N.M. Ribe, U.R. Christensen, The dynamical origin of Hawaiian volcanism, Earth Planet. Sci. Lett. 171 (1999) 517^531. [25] R.S. White, Melt production rates in mantle plumes, Phil. Trans. R. Soc. London A 342 (1993) 137^153. [26] A.B. Watts, U.S. ten Brink, Crustal structure, £exure, and subsidence history of the Hawaiian islands, J. Geophys. Res. 94 (1989) 10473^10500. [27] A. Cazenave, K. Dominh, M. Rabinowicz, G. Ceuleneer, Geoid and depth anomalies over ocean swells and troughs: Evidence of an increasing trend of the geoid to depth ratio with age of plate, J. Geophys. Res. 93 (1988) 8064^8077. [28] K. Marks, D.T. Sandwell, Analysis of geoid height versus topography for oceanic plateaus and swells using nonbiased linear regression, J. Geophys. Res. 96 (1991) 8045^8055.

EPSL 5424 17-4-00