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Produced by the NASA Center for Aerospace Information (CASI)
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On Cr.avity from SST, Ceoid from SEASKr, and Plate Age and fracture Zones

in the Pacific

by

,

B. D. Marsh

Earth and Planetary Sciences

.Johns Hopkins University

Baltimore, MD 21218 

J. G. ciarsh

Geodynamics Branch

Goddard Space Flight Center

Greenbelt, MD 20771

t. .

R. G. Williamson

EG&G Wash. Analytical Ser. Center Inc.

Riverdale, MD 20737

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(284— 10025) ON GRAVITY FROM SST, GEOID FROM N84- 1 1559

SEASATe AND PLATE AGE AND FRACTURE ZONES IN

THE PACIFIC (Johns Hopkins Univ. ); 39 p

HC A03/MF A01 CSCZ 08G Unclas

'r GS/43 000
OF pooR QUALITY,

Ab8tract

Our earlier measurements of the high degree and order (n,m>12)

gravity in the central Pacific using the method of satelli'Ce-td-satellite

tracking (SST) have been extended with an additional 50 passes of data.

The SST method utilizes line of sight Doppler tracking between the very

high geosynchronous ATS-6 spacecraft and the much lower (840 km) orbiting

GEOS-3 spacecraft. The observed changes in velocity with time are re- 

duced relative to the well-determined low degree and order (n,m<12) GEM-10

field model and accelerations are'found by analytical di.fferent.ati.on of

the range rates. This new map is essentially identical to the first

map and we have produced a composite map by combining all 90 passes

of SST data. The resolution of the map is at worst about 5 0 and much

better in most places. A comparison of this map with conventional. GEM

models shows very good agreement. A reddction of the SEASAT altimeter

data has also been carried out for an additional. comparison. Although

". tho SEASAT geoid contains much more high .frequency information, it agrees

very well with both the SST and GEM fields. The maps are dominated

(especially in the east) by a pattern of roughly east-west anomalies 

with a transverse wavelength of about 2000 km. A further comparison

with regional bathymetric data shows a remarkably close correlation with 

plate age. Each anomaly band is framed by those major fracture zones

having large offsets and the regular spacing ( =10 0 ) of these fractures

1 

seems to account for the fabric in the gravity field. There are other

anomalies that are accounted for by hot spots and altogether the immediate
w

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2

OF POOR QUAL-1 N

:source c:f at ].cast part of Chese Pacific anomalies is In the lithosphere

itself it therefore seems that; most of the anomalies in cite east half

of the Pacific could be partly caused simply by regional differences in

plate age. The amplitude of these geoid or gravity anomalies caused by

age differences should decrease with absolute plate ae-e, and large anomalies

(ti3 m) over old, smooth sea floor may indicate a further deeper source

within or perhaps below the lithosphere. We have also considered the possible

plume size and ascent velocity necessary to supply deep mantle material

to the upper mantle without complete thermal equilibration. A plume

emanating from a buoyant layer 100 km thick and 10 4 times less viscous than

the surrounding mangle will have a diameter of about 400 km. And it must

ascend at about 10 cm/yr to arrive still anomalously hot in the uppermost

mantle.
f¢ (ter r {ry.

, _ ^^ ;

OF POOR QQUALITV

j

Introduction

I In an earlier pr• per we. useu the method pioneered by W. L. S,jogren

of satellite-to-satellite ; tracki.ng (SST) to measure the gravity field

over the Pacific (Marsh et al., 1981). That study'tised 40 passes of

ATS-6/GEOS-3 data to construct a high degree and order (n,m > 1;=) gravity map. The

present study includes data from about 50 additional passes to produce

a more detailed gravity map.

In brief, this method measures the earth's gravity field by

Doppler tracking of a low orbiting satellite (GEOS-3 at 840 km) by a much

higher (40,000 km) geosynchronous satellite (ATS-6). Doppler tracking

furnishes the speed of the low satellite as a function of time or position. These

range-rates are then converted through differentiation to line-of-sight accelerations or gravity anomalies. Point values of these an.cmalies measured along each

track are then contoured into a map of gravity anc*nalies. The long wavelength components of the SSTs measurements are removed using a low degree and

order (n,m <12) field model, which is very well determined. The residual

anomalies thus represent harmonics of n,m > 12.

The great advantage of the SST method is that it is simple. No large

arrays of spherical harmonics need be found or manipulated, and there is

little chance for the data to become numerically adulterated during processing. The method is limited by the height of the low-orbiting satellite

and thus is sensitive to wavelengths longer than about 1500 km (i.e. n,m < = 2Q.

But as shall be shown,it is this range of the gravity Field (12 < n,m < 25)

it

that is geophysically particularly interesting. A disadvantage of the method

is that the map applies only at the altitude of 840 km and must be separately

downward-continued to give anomalies on the earth's surface, This method

also dues not strictly measure the radial component of gravity, but rattier
• OF POOR QUALITY

a ch,inging line-of-sight component of the field between ATS-G and C,80.4 -3.

Only within 30-40 degrees of the high satellites subsatellite point can these

an,)mal.lo^s he taken as f_,00d measurkis of the radial Fluid. Even with these. limitaLions we found in our previous study that with only 15 passes of data the map

derived of the gravity field over North America is essentially identical to that

produced by any previous field model, which involves data from 30 or more

satellites in addition to surface data.

The gravity data is not only geophysically useful, but it provides an

excellent opportunity to check one gravity field against another. For example, in

our previous study we compared the more conventional Goddard Earth Model (GEMIOB;

Lerch et al., 1981) and the SST gravity fields. tie also show for the first time

`f comparisons with the SEASAT altimeter-derived geoid over the Pacific, which is

highly accurate and useful for comparisons and geophysical interpretations. lie 

also show that there is often a good correlation between relatively young sea

floor and gravity and geoid anomalies, the major fracture zones of the Pacific,

because of their offsets, give the distinctive east-west fabric to the gravity and

geoid anomalies. Other anomalies correlate with the traces of hot spots, and

overall it is the interference of these two sources of anomalies that gives rise

to the distinctive gravity field of the Pacific area.

it

SST Map

A map of satellite tracks used here and in our previous study are shown ,

combined as Figure 1. The coverage is particularly dense over the central

Pacific. The gravity is determined at 70 km intervals along each track and

these values have been contoured into a map shown as Figure 2. This map shows

the same assemblage of anomalies as we previously noted, but now-there

is more detail. In particular: The positive Line Islands anomaly, in the

vicinity of the subsatelli.te point, is more rectangular and strikes

nearly east-west. The positive tail can the southeast corner of this anomaly
OF POOR QUpLf

is now smaller and e:tsentiully Wilgnificant from %vro. The positive

anomaly around Hawaii (see plotted islands) and northeast of Hawaii is much

the same as before except the contours within the anomaly show more detail.

There is also now a .stronger bridge of positive? VaIL S between the Gook-Austral

and Pitcairn anomalies. Otherwise the map is essentially the same as the previous

SST map of Marsh at al., (1981).

Comparison with SEASAT Geoid

Since the geometry of an essentially radial gravity field is the

same as its geoid, we show for comparison with the SST map a geoid over the same

area (Figure 3) . This geoid (mean sea surface) was computed using the radar altiMeter from the SEASAT satellite and it represents 70 days of data taken between

July and October 1978. The long wavelength field has again been removed

1

by subtracting a low degree and order (n,m < 12) field model (PGS-S4;

Lerch at al., 1982) from the original data. Each data point used in constructing this map represents an average of 1000 radar measurements taken

over a time of 1 sec or a ground distance of 7 km, and each point has a

precision of 3-4 cm (Tapley et al,., 1982). The accuracy of the altimeterderived mean sea surface with respect to the center of mass of the earth

is dominated by radial orbit error. The recent SEASAT ephemerides computed

at GSFC using laser, and Unified S-Band tracking datr, and the most accurate

earth gravity and geodetic models have a global rms radial accuracy of 70 cm.

This set of data therefore contains f&r more high frequency information over

the ocean than any previous GEM or SST gravity field. Through gridding and

contouring, however, these data have-been smoothed somew:,., t by using all

points within a cap wiCh a radius of five degrees. More details of the

procedure of computing this mean sea surface are described by Marsh and

Martin (1982).
ORIGINAL PAIR2 h

OF POOR QUALITY

Although the SEASAT geoid contains much more information than the

tilt! Lt:t) 111,1pq

se ame ankwiaLLos. It shoui.tl Ilts ,I, ow Qs, tilt- s

reci^lled, however, that the relative zero-level between the two maps may

be different and some features may appear larger or smaller on one or the

other of the maps. There are nevertheless some significant differences,

The Hawaiian anomalies (200*, 20 O N; 215 0 , 30 0 N) are hardly connected on the

SEASAT geoid. There is also a fairly strreig geoid anomaly all along the chain

of Hawaiian Islands, The SEASAT geoid shows almost a connection between

t!ie Hawaiian anomaly and the Line Islands anomaly to the south, Other

than a small break near 190* the Line Island geoid anomaly nearly connects

with the Gilbert-Marshall anomaly and forms a positive band nearly traversing

the entire Pacific. On both maps this anomaly forms the backbone of the

pattern of approximately east-west features with a transverse wavelength of

about 2000 km. This had been seen in our earlier work but is now particularly clear in these maps.

In sum, the SST and SEASAT ,raps show the same basic areas of positive

and negative anomalies and overall correlate closely. The SEASAT geoid

provides a more complete coverage and over the Pacific as a whole is

expected to be more accurate than the SST map.

Correlations with Fracture Zones and Hot Spot Traces

We noted in our recent (1981) paper that some of these gravcf ty anomalies

correlate well with residual depth anomalies. The Hawaiian and Cook-Austral

anomalies, for example, correlate to *various degrees with residual depth

anomalies delineated by, respectively, Watts (1976) and Crough (1978).
CA'^6^^wt

OF, PoOR QUALITY

And we found through a simple model that shallower-than-usual a,eas of

the sea floor can certainly cause the requisi te SST gravity anomaly

if they are compensated at a depth of about 30-100 km. Some positive

anomalies (e.g. northeast of Hawaii near 220 1 and 30") apparently do not

correlate with residual depth anomalies. The orderly pattern of eastwes,.t trending negative anomalies just south of Hawaii and the Line Islands

anomalies, which nearly span the Pacific, apparently do not correlate with

negative residual depth anomalies. Extensive residual depth data, however, is

largely wanting.

Across fracture zones with large offsets in lithosphere age (=10 m. y.)

there is a significant change in thickness of the lithosphere, which is

reflected by a sudden change in ocean lepth. This step change in topography

is well known to produce a distinctive gravity or geoid anomaly.

The change with age of these anomalies has been used to study 

the aging and growth of the lithosphere . (P. g. Detrick, 1981), and

these anomalies may persist for great distances. In their study of the 

Mendocino fracture zone, for example, Sandwell and Schuber t,, (1982) showed

that these effects may persist to ages of 135 m . y., which corresponds to a

distance near the bend in the Emperor-Hawaiian chain of seamounts. All of

these studies, however, have used a local array of tracks to study a single

anomaly or offset. Since the Pacific plate is cut by a regular - pattern

of fracture zones with large off ets, these may produce a certain fabric in

the gravity and geoid fields.

To investigate such correlations betwee n the gravity and geoid fields

and regional bathymetry we first made a map of the 1 0 x 1 0 depths as
ORIC"I,' IAL 1,2'r " ^ •

OF RDoR QUALITY

stippl.led by NOAA ( 1.980) For the north I)actf c. A comparison of Figure

and 3 with thiff map shows that major fracture zones essentially frame

the east-west pattern of geoi(l and gravity anomalies. The major positive

geoid anomaly northeast of Hawaii cor r elates with the relatively young

lithosphere bounded between the Murray and Molokai fracture zones. Because

the ocean floor magnetic anomalies closely record the age of the plate

and thus also its bachymetry, there' should also be a positive correlation

between geoid anomalies and the age of the ocean basin as .deduced from the

magnetic anomalies of the sea floor. The superposition of geoid on plate

age is shown as Figure 4. From this combined map there is a clear

correlation between the gravity anomalies and the age of the lithosphere.

The large fracture zones form a frame work around the gravit y anomalies.

These correlations suggest that gravity and geoid anomalies not

associated with active volcanism (i.e. so-called hot spots), especially in

the eastern Pacific, may result from differences in age of the sea floor.

The particular pattern of fracture zones where major offsets occur at

approximately equally-spaced intervals (=1000 kin) produces a fabric in the

If it were not for the seamount chains, the gravity and geoid

fields of the Pacific would appear as approximately east--west bands of

alternating sign. The topography and associated geoid fields of seamount

chains are superimposed on this regional plate fabric, which is set at the

ridges. Although the so-called hot spot seamount chains form at some

distance from the ridge, other chains (e.g. Line Islands) have apparently

formed near the ridge (Watts et al., 1980). Thus thu Pacific gravity 

geoid that dominates the entire Pacific.
OF pOOR

and geoid fields are principally the result of these two distinctive effects. BtIt

btacatiiae the .our'.'-v of the Pacific; piaw, for LIU-' 112 m,) • 11418 not been parE Llel to

the pattern of fracture zones, the twig- sets of anomalies interfere. Hence

we turn to n more detailed examination of the principal gravity and geoid

anomalies and their relation to fracture zones, plate age, and hot spot activity.

Hawaiian Anomaly: The Hawaiian chain of seamounts, for example, cuts

across the region bounded by the Murray and Molokai fracture zones which otherwise would be a positive band of gravity, and also across the negative region

to the north between the Mendocino and Murray fracture zones (see Figure 5).

This may have the resultant effect of producing the relative low in the geoid

along the Hawaiian seamounts near the Midway Islands. This could have important

implicationy for understanding the correlation of residual depth anomalies and

geoid anomalies.

tie found in our earlier (1981) study that the present gravity and geoid

fields show positive anomalies at the bend in the Hawaiian-Emperor seamount

chain. We also suggested that this anomaly may be an obstacle to Crough's

(1978) model of lithospheric thinning near Hawaii, through local reheating,

followed by a monotonic decay with cooling of both topc;raphy and geoid

anomaly. If this anomaly at the bend is actually a continuation of

the main Hawaiian chain anomaly, then the orginal Hawaiian anomaly may take

longer than expected to decay. The eastern portion of this anomaly

correlates with the shallow sea floor of the Hess Rise (e.g. Nemoto and

Kroenke, 1981) but the western portion does not. Following Crough's model.,

G

however, even an original irregularity in the sea floor should be erased

by the reheating and buoyant rise o¢ the lithosphere to a virtual age of about

;x

3 25 m.y. It is therefore not clear whether it is the positive anomaly at

•fi'

4)
©RIGINIA-L I?'AG"70 t

OF POOR QUAi.IV

the fiend or the necking of the Elturaii,an rnnomaly near :Midway that is

anomalous. For this model this differentiation could be important.

Line islands Anomal ; A'similar interference occurs between the

east-west trending, positive Line Islands geoid anomaly band (Marsh

et al., 1981) south of Hawaii and the trend of the Line Islands themselves. Watts et al. (1980) show that the gravity over the Line Islands

is consistent with their geologic age of about 97 m.y., implying that

they formed at a ridge. The continuation to the southeast of the trend

of the Line Islands, however, has been volcanically active in the last

1.3 m.y. (Duncan and McDougall, 074). Residual depth anomalies have been

computed for this area by Crough and Jarrard (1981), who found that this anomaly

decays roughly as predicted For a hot -spot trace tha includes the i.nconsistently older Tine Islands. This may imply that the Line Islands have

been reheated, perhaps as a result of the more recent volcanism of the

Cross Trend Lane or from much earlier, unnotir..ed,intrusive activity of

the Marquesas Islands.

This residual depth anomaly, the Marquesas-Line Swell (Crough and

Jarrard, 1981), strikes northwesterly but more northerly than that of the

trace of Pacific plate motion for the last 40 m.y. The associated geoid

anomaly shown by Crough and Jarrard was derived from the GEOS-3 geoid

of Brace (1977) by subtracting a regional geoid described by a seventh

degree polynomial. This geoid anomaly also shows a WNW trend, a trend

which is not evident in the more precise SEASAT geoid of Figure 3 nor in the

SST and GEM gravity fields, each ofrwh,ch have had a twelfth degree global

field removed. The trend of these latter fields is markedly east-west

which persists for about 4000 km. In fact, a principal 4 rp anomaly of
OF POOR QUALITY,

Ilr,ac(".r at -10 0 , 220 0 , which croii 11 .11 ► r1 rarr, ► rd found to 1-orrp late rLosel'%0

with the Marquesas depth anomaly, in absent in thc^. S1:ASAT data. This

descrepancy was described in our earlier (1981) SST study and was attributed

to the sparseness of GEOS-3 altimeter data in this region. The extensive

SEASAT data confirms this interpretation.

The WNW trend associated with the Tine Islands is evident on the

northwest corner of this regional Anomaly, but any correlation with the

Marquesas--Line Swell itself is less clear. The choice of removal, of the

regional, field is evidently crucial in producing a geoid anomaly that

correlates with this residual depth anomaly.

Regardleos of the age of the Tine Islands, they do produce a geoid

anomaly treading NW, which might be expected, on grounds of seamaunt

frequency, to continue through to the Hawaiian anomaly. These anomalies

nearly do connect, but they are separated by a distinctive regional

negative anomaly that spans the entire Pacific. This negative band has

its origin on the east between the Clipperton and Murray fracture zones,

which together frame relatively older lithosphere. The anomaly does not

carefully follow these bounds, which is probably due to interference with

the geoid anomaly of the Line Islands, and it cuts across moderately shallow

(<5000 m) sea floor! near 20 0 and 170-1800.

Cook-Austral Anomaly: This gravity and geoid anomaly centered near

-200 ,205 0 correlates roughly with the residual depth anomaly computed here

by Crough (1978). On close inspection, the principal part of the depth

anomaly (-800 m) is near -28 0 ,215 0 and trends northwes-'.:erly,which correlates

with a tail of the geoid anomaly to the southeast. The amplitude of this
OF POON Q V ,; M `4

parr of Cho geoid atum► k► l y H ntiver mor"' Orin 7 n► , wh it-h i,.±

sii;raificantly less than might be expected Judging from past: correlation8.

The much more regional nature of the geoid ano 1;ravity ranomal.i.e's may

in part be due to the proximity of the Tonga-Kermadec trench and its

associated outer rise. It is also clear in this region. that there is

no positive anomaly over the Manihiki Plateau, which is ,just northeast

of Samoa.

Early radiometric dating of lavas of the Cook-Austral Islands

showed a broad consistency with the hot spot hypothesis, but four of these

islands were updated and three others had only dates of unlocated

samples. Turner and Jarrard (1982) have recently reported ninety four

additlp, ,s!. dates that .supply this missing information. The ages of this

4.,*a6iii ;':o`a appear to be altogether inconsistent with those predicted

by a simple hot spot migration. They instead suggest that the results

give stronger support to a "hot line" hypotheses, which may stretch from

the Samoan chain through the Cook-Australs and on to Pitcairn Island. A

"hot line" could relate these volcanisms, but there seems to be little

evidence of it in the gravity or geoid of this area.

Pitcairn Anomaly: This anomaly is bounded on the east by the. east

Pacific rise and r,,n the north and south by two fracture zones with large

offsets. It does not form a continuous band westward to the Cook-Austral

anomaly, but is bounded on the west by a shallow negative anomaly (see more below).

Marshall-Gilbert Anomaly: This diffuse pesitive anomaly at 011,180°

is separated by a distinct but shallow loci from the Line Islands anomaly

to the east. The sea floor in this region is not highly populated with
ORIOINAL Wt"L

OF I-'OOR QUALITY

seamounts and is genernlLy 5000-6000 m deep. The tectortic history of

th1, rq,ion OL Olt, Nac.lftc h:, n^^t ^,1I^^':^r, but r w di..;, loc.tl,y I inear,

north-northwest trending facing edges of these anomalies (i,r,• GilbertMarshall and Line Islands) may suggest the presence of NN14 trending

fracture zones or other plate fabric. Some confirmztiou of this comes from

the correlations of magnetic anomalies in this region whose strike is

normal, to the strike of the inferred fracture zones (e.g. Pitman et al.,

1974). The chain of islands made up of the Marshall and Gilbert Islands,

although also striking in this (NNW) direction, does not have an associated

distinctive geoid anomaly. This general positive anomaly may therefore

be caused by a region of relatively young sea floor (see below).

Geoid and Gravity Anomalies Between Fracture Zones

The step-like change in geoid height across fracture zones separating

plates of different ages has been'used to study the aging of the lithosphere

• by Crough (1979), Detrick (1981), and Sandwell and Schubert (1982). The

amplitude of these anomalies is proportional to the offset in age,and the

history of the anomaly can be used to evaluate the cooling characteristics

and growth of the lithosphere. These results suggest deviations beginning

' at ages as early as 20-40 m.y. from the model of the lithosphere as a thermal 

boundary layer (Sandwell and Schubert, 1982). When modeled instead by the

cooling of a plate or constant thickness, as suggested by Parsons and 

. ai

f Sclater (1977), the closest fit to the change in slope of the geoid anomaly

4

y

with age is for a plate thickness of 125 km (Sandwell and Schubert, 1952). 

r ;4 Although in these studies the proper separation of the anomaly due to

the fracture zone Eromi the regional field is critical, in interpreting the 

...
• ORIGINAL PAGE: b`a

OF POOR QUALITY

residual SEASAT geoid of Figure 3 this is much less critical. This is so

L,ecause the principal cause of the anomalies bounded by fracture zones is the

r 11;1,onal diftureoc:e in plate: age. Lac h section of ISM CWC—:Nt1 ti , : major lracttiLo

zones is large enough (=1000 km) to be constantly near isostatic equilibrium. And

each section of plate is large enough such that it has a characteristic

geoid anomaly that, for the most part, depends critically on the absolute

age of the plate. These points have been treated in some detail by Haxby

and Turcotte (1978), Turcotte and McAdoo (1979), and Sandwell and Schubert

(1980). Under the assumption of isostatic equilibrium, the geoid or gravity

anomaly can be calculated from knowledge of the local vertical distribution

of density as long as the depth of compensation is much less than the wavelength of the nomal.y itself. For the plate regions between major fracture

zones, the anomaly wavelength is m1000 km. The geoid anomaly N(x,y) due

to a density distribution p(x,y,z) over a depth L is given by (e.g. Turcotte

and McAdoo, 1979)

L

N = - 2gG zAp(x,y,z)dz (11

o

where z is the vertical coordinate, g is gravity at the surface and G is

the universal gravitational constant. The application of (1) here is only an

approximation. To.model the details of the anomaly near fracture zones themselves the more exact techniques of, say, Sandwell and Schubert (1982) can be

employed. For the broad scale anomalies considered here, however, results

t

gained from (1) and also empirical relations (see below) should suffice.

F 1

Once a function for p is known.throughout L, it is routine to compute

N. For the present consideration, it is convenient to choose L as the

thickness of the lithosphere. Since the thermal regime of the lithosphere

changes with time, any equation of state will of necessity also be a

2

14-
. ORIGINAL PAGE B 

OF POOR QUALITY

function of time. *rho result is 110L especially settsitve to the equation 

of state or to the nssumed thermal distribution for timea less than about

50 tip . y .

For a thermal model of the lithosphere based on a thermal boundary layer,

the geoid height relative to Lire ocean ridges is (T urcotte and McAdoo, 1979)

N

1.17Gp2a(TL T 

0

)L2

1

2pLa(TCT0)

g n (pL-pw)

where p ,i.s the mantle density at a depth L at the base of the lithosphere

where the temperature is TL , To is the surface temperature, a is the

isothermal coefficent of thevial expansion, 

p 

is the density of sea water

and g is gravitational acceleration of the surface. A corresponding

expression for the plate model is given by Sandwell and Schubert (1930).

It is clear from (2) that the ;eoid anomaly is directly proportional to I.2,

the thickness of the lithosphe;P, which for the thermal boundary

layer model is L2 = (2.32) 2Kt, where K is thermal diffusivity and t is time.

With this substitution, (2) shows that the geoid anomaly is linearly dependent

on the absolute age (t) of the lithosphere. Here we are interested in the

anomaly caused because of the relative difference in age (At) of adjacent

regions of plate bounded by fracture zones with large offsets (i.e.. tens or

more of millions of years). This relative geoid anomaly (AN) is thus given

by

AN = - 6.30 gBK l + n (pB—pt,) AL (3)

L

(2)

where H = pLa(T

L

 -T 0 ). The last term in brackets is always of order one and
UFtGaNAL PA S."

OF POOR QUALITY

ror f , 1 = 4.:f0 )!,/cm 1, f; 980 om/^;2,

10-Z cin /s, JAN 1- 1.75 m per

10 m.y.) is reasonable for

e same order as those shown by

1

It is cleitr that AN 6.52 GRK At/g, which

'1' G-To = 1250°C, a= 3.3 x 10 5 deg-1 , and K a

10 m.y. of offset.. This time difference (At

many fracture zones and the anomaly is of th

Figure 3.

Although this result may apply when the lithosphere is young,

an unattractive feature of it is H,e fact that the relative geoid anomaly

is independent of the absolute age of the lithosphere itself. Whereas it

has been shown by Sandwell and Schubert (1980) that the rate of change of

the geoid anomaly across fracture zones decreases systematically with plate

age. Of the analytical models, they found that the plate model matches the

data better than the boundary layer model. But overall a good estimate of

the geoid anomaly can be obtain,-d from the observed anomaly decay. Their

actual data on the decay of the slope of the geoid anomaly with age for the

North Atlantic is of the form (see their Figure 4)

dN =2103 dt x. t-0.2 (4)

where N is the geoid height in meters and t is age in millions of years.

Integrating ( 4) and employing the condition that N ( t=0) = 0 and subtracting

two such formulas for adjacent regions of, respectively, age t 1 and 1:2

gives a result for the relative geoid.anomaly.

AN = (ti - t2) x 10-

3

 - 0.2(t l - t 2 ) (5)
s s 

OF pOQR 'Q lALIV

17 

Beginning at a ridge with a relative offset of 10 m.y. gives an anomaly 

of 1.9 m, which decays to about 1 m when the older plate has an age of 60 m.y., and

the anomaly vanishes at an age of about 100 m.y. (The actual sign of the anomaly

depends on the relative p lat" age.) Ftoi Lhiii result it is clear thatgeold

anomalies related to regional changes in plate age will only he significant over 

plates having an absolute age of less than about 80 m.y. The largest anoma).; es will

or course be closest to the ridge and they will have decayed to near nothing

over 80 m.y. In the Pacific significant anomalies can only be expected approximately east of the longitude of Hawaii. Overall it is useful, however, to apply

these results to the interpretation of the aerially large positive anomalies

that lie near the east Pacific rise; namely, the anomaly northeast of Hawaii,

the eastern and of the Line Islands anomaly, and the Pitcairn anomaly.

The positive anomaly northeast of Hawaii is bounded by the Murray and Molokai 

fracture zones. Across the fracture zones in this vicinity the average offset in

age is about 18 m.y. (Pitman et at., 1974). This would produce an anomaly of 

about 3 m when the .actual anomaly approaches 6 m. This is not necessarily 

a serious dissagreement, because the anomaly directly to the north, over the

older Murray-Mendocino area,.h,j also positive and the relative difference is

about 3 m. Why this positive spills northward to the Mendocino fracture zone

and beyond may be dupe to the proximity of North America. Nevertheless it seems

that this relative l 3 m anomaly decreases markedly toward Hawaii. This may be

due in part to a change in the offset age as has been noted for the Mendocino

fracture zone by Sandwell and Schubert (1982).

Moving south, the next section of plate with a major offset is bounded

by the Molokai and Clipperton fracture zones. This plate is relatively

older and the geoid anomalr is slightly negative. The next positive anomaly

is immediately to the south and is bounded, although not tightly, by the

Clipperton and Galapagos fracture zones. The anomaly itself seems too broad
k

M

R

f"+

i

• i

.t

j

t,

jj

ORIGINAL PAGE IS'

1 i;

OF POOR QUALITY

to be wholly the result of this relatively small offset ( ,! 5 m.y.), which

wotil d produce about 1 in anomaly when the actual anomaly has an average

amplitude of ubuut 2 m. The u11UI!Mlv noarly voachii;S tho M ai:quesais f rnettivo,

the next major fracture zone to the south. Moving southerly, normal to

the fracture zones along about 235 0 , each offset brings successively

younger plate further west. The eastern portion of this Line Islands anomaly is

probably the result of this cummulative effect it offset. This anomaly also

diminishes westward and then increases again, suggesting a change in the age of o

set, which along the Galapagos fracture zone may even change sign (Pitman et al., 1974)

Pitcairn anomaly is the only one of these positive anomalies that is bounded

by the east p acific rise itself. It is bounded by the Marquesas fracture

zone on the north and the Challenger on the south. The offset in age across

these fracture zones is largest nearer the ridge and decreases westward. This

has the effect of producing an anomaly nearer the ridge that attenuates

rapidly to the west, The anomaly itself lies almost wholly on sea floor formed

upon reorganization of the east Pacific 'rise beginning'about 60 m.y.b.p. (Herron,

1972). At this time the northwesterly trending ridge migrated north of the Eltanin

fracture zone forming a new, northeast trending ridge and reorienting fracture

zones to strike west-northwest. These newer fracture zones and young lithosphere

produce the Pitcairn anomaly.

Although these. interpretations in light of the presence of fracture

zones having significant offsets in age seem reasonable, they are only

qualitative. But they can be put on a firmer basis by computing a synthetic

geoid based strictly on the relative ages of the Pacific plate; which is now in

progress (Marsh and Hinojosa, 1983). This is quite feasible for the Pacific

plate because of the relative paucity of voluminous off-ridge volcanism

for large distances near the west flank of the east Pacific rise.
ORIGINAL PAGE

OF POOR QUALITY

19

is,

Possible Size. and Ascent Velocity of Hot Spat Plumes

Although the surUcial expross.iun of itch 4;l)OVS M; VoLc-11ILs111 and their

kinematic implications are by now clearly evident, there is much less

certainty about their subsurface structure anti ultimate origin. That is

little is known of the fluid mechanical conditions within the mantle that

gives rise to these features. Morgan (1971, 1973) has suggested that they

originate deep within the lower mantle as buoyant plumes, perhaps emanating

from the region of the core-mantle - boundary itself. Judging from the

wide nature of lithospheric swells (=1000 km), and using a "simple aspectratio arguement", Crough (1978) has also suggested that these plumes originate

from the lower mantle or at the core-mantle boundary itself. Sandwell

(1982) has attempted to place bounds on the diameter of the hot-spot below

the lithosphere by computing the gravity, geoid, topography, heat flow,

and subsurface temperature as a function of diameter and plate velocity

using an imposed temperature distribution. These results are not particularly diagnostic, but they suggest a sub•-lithospheric hot-spot diameter of

400-4000 km. It is the intent of this section to show some additional

relationships between plume size, mantle viscosity, and heat transfer during

ascent.

A necessary assumption in relating hot-spots to a source region for

plumes is that the plume must travel fast enough to the earth I s upper most

regions without cooling and loosing its buoyancy. The rate of cooling

depends critically on the diameter of the plume which is controlled by the

viscosity field in the source region. Plumes rise as a resu,lt . of a density

inversion or gravitational instability. It has been shown through analysis

and experiment that the characteristic size (a = radius) of a plume (of

viscosity u 2 ) emanating from such a low density layer (of viscosity u l) is

given by(Marsh, 1979)
ORIGINAL PAGE 19

OF POQR QUALITY

/r

2

where h2 is the initial thickness of the source layer. if ul 0

2 , the

plume has a diameter about equal to the thickness of its source layer,

which is approximately observed for salt domes within the earth's crust.

A buoyant layer within the mantle could be produced by anomalous

heating. The viscosity, which is otherwise nearly uniform (Cathles, 1973),

could also be lowered by this heating. The reduction in viscosity, howover,

is unlikely to be by more than a factor of 100 (poise) (e.g. Marsh, 1982))

unless it is partially melted, but there is no evidence that the

s

temperature of the deep mantle is near its solidus. From Figure 5,

n

computed from (4), for ul/μ2 - 100 the plume diameter is about three times

,f

the thickness of its source layer. Since a source layer much thicker than

about 100 km would be readily detectable by seismic methods, the plume

1

diameter would be about 300 km; even for a viscosity contrast of 104 , the

diameter is only about 400 km. The main lesson of equation (4) is that the

plume diameter will be of the same order as the thickness of its source.

Unless the scarce layer thickness is greater than about 5-10 times its

depth, there are naa results associated with the fluid mechanics of gravitational instabilities that give any information on the actual depth of the 

source (Marsh, 1975). The only region within the mantle where it is

dynamically reasonable to suspect plume growth is within but thermal

'boundary layers. Such a layer may occur at the core-mantle boundary

(Elsasser et al., 1978), f Aich may be associated with the V seismic layer, 

7
r. 21

OF POOR QUALITY

although this is controversial. If the mantle has two or more tiers of

convection cells, hot thermal boundary layers could also occur within the

body of the mantle itself, but their existence is highly controversial,,.

A small plume ascending from great depth may cool and loose its

buoyancy before. reaching the base of the lithosphere. Its final temperature 

depends critically on its rate of ascent, and even a small plume ascending suffi

ciently fast will hardly cool at all. This relationship between ascent velocity,

final temperature, and plume velocity may be investigated through a model of

heat transfer. These models and methods have been extensively developed by

Xarsh (1978; 1980 and Marsh and Kantha (1978), for understanding

the heat transfer of ascending bodies of magma. The method is parametric

and general and may be used for any geometry, and only a brief description

is given here. 

By conservation of energy, the mean temperature (T) in a plume changes

with time in response to the total flux of heat (Ql,) From the body.and that due to

adiabatic decompression (-YT),

PCpV1 d ^ ..QT y(T)/ pCpV' (S)

where P is density, Cp is specific heat, V is volume and y is the usual adiabatic

coefficient; all of which are considered constant (although this is not a necessaryt

assumption). A dimensionless number involving Q T can be conveniently defined as

Nu= QT

cd

where Nu is the well known Nusselt number, a pure number, oand Qcd is the

heat flux due to conduction (i.e. when the body is motionless). The flux

(6)
.y •I

ORIGINAL RAGE" 1,9

OF POOR QUALITY

AKc T-'I'ro(C)) /L, wliere A is the

he is tiiermaL conductivity, T M (t) is

from the plume, and L is the charact

duction; for a plume L = radius (a).

and rearranging gives (Marsh, 1918)

stirface area involved in the heat t ronsf or,

the. characteristic mincle geotherm Ear

wristic length Scale of heat conSubstituting Q cd and (6) into (5)

dTdt + JT ^ JTm (t) -Y T (7)

where J = Nu(AKc)/(LpCpV'). This has the general solution

T =Je 'f T. (t)ej d (8)

where J' = J + Y, when coupled with an intial condition this describes the

mean temperature of the body as a function of time or distance from its

source. Since the final temperature must be greater then T m (L), where L is

the total , ascent distance, this places a condition on the ascent velocity.

Although the ascent velocity has not yet appeared explicitly, it enters

through the relationship for Jt.

For magmatic transfer through the lithosphere, solutions to (8) are

given by Marsh (1978; 1982) and Marsh and Kantha (1978) where transfer by

diapirism, stoping, zone melting, and pipe flow are considered. Here we

consider an anomalous hot plume ascending from a buoyant region. The region

itself need not be globally continuous , but only locally continuous. Its mean temperature is Tp, which is the -temperature of the

itself at its source. The normaf mantle temperature at this depth is
R. ,u.w., M f'^'a a 'a1

OF rooR QUALITY

The, mantle temperature 1.s taken t• o be ndinba e l e. th rotighout the mantle, ant,

hence varies as rm (t)/To = I XLI(-yt) WWI (IL1 1)Gi1 (UV Lin) e) until nuar the

uppermost mantle. Substituting Tm (t) into (8) gives upon integration

T

= CXP(-(yto)t')-EXL'(-((j+y)to)t')4•(T Ito)EXY(-((J+y)to)t'), (9)

0

where the intial condition T(0) ffi Tp has been used. The mean temperature

of the plume as a function of t'( = t /to) and Jto as described by (9) is

given by Figure 6. Both the mantle and thr. plume have been assumed to

have the same adiabatic coe g ficient (y= agt o /Cp = 0.7, where Cp is

specific heat and other symbols as before).

The results of Figure? 6 suggest that if the plume is to remain hot

and buoyant and reach the upper mantle, it = 1, from which the ascent

velocity can be calculated. Recalling that J = (NuAKc)/(LpCpV'), where

for a plume A/V' = 2naZ/7ra 2 A = 2/a , where a is radius and R is a length,

Kc/pCp 10cm2Is, and L = a, we have J = Nu2x10-2/a2 . It is clear that

Nu must now be used and several choices are possible= The Nu for a plume

can be approximated by that for flow in a pipe,which for a fully developed

pipe flow is well known to be 4.36 (e.g. Kays, 1966). T'As result holds

only far from the thermal entry region of the pipe. The length of the

thermal entry region for most geophysical flows is large (Marsh, 1978) and

it is proportional to Vat/K, where V is velocity. If V = 10r7 cm/s, a = 107cm

and K = 10-2 cm2 Is, the thermal entry length is about 104 km, and for larger

velocities it increases. Thus Nu = '4.36 is a conservative approxiamtion for heat

ta--nsfer, more than likely it will be significantly larger, even by a factor of ten.

i
ORIGINAL PAGE b ^^

OF POOR QUALITY

TI-o- ii ,J ., 11.72x10-2 h a11d r.he tor ;1 ; Is, (t^nt time. i;t l;iveil I► y t0

115H:t2,

where: H is the nunmexLcal vaLiie of Jt o from the eurves of Vigure 6.

For JtoW 1 ,41 and, from Figure 5 for , say, n°200 km, the ascent time 1.s to-4x101S$cc)

which for an ascent di.staii,:;" of 2500 km, gives a typical ascent velocity of

about 2 cm/yr. That is, if a plume of a diameter of 400 kin ascends a distance

R

of 2500 kih at a velocity of about 2 cni /yr, it will still be anomalously

hot when it arrives in the upper mantle. It must be reminded, however, that

this is an absolute mimimum velocity. For a more reasonable Nu (-10x larger)

the velocity must be 20 cm/yr, and if the radius is 100 kin, V =80 cm/yr.

The main result of this calculation is that it is apparently possible even for

relatively small plumes to ascend a large distance through the mantle without

totally loosing their original anomalous temperature. It should also be

pointed out, however, that even an adiabatic plume cools more than an equivalent

"normal" mantle during its ascent. This is because the adiabatic temperature

change is proportional to the temperature of the body itself. Hence even for

identical thermal properties, a hotter body cools faster than an equivalent

coo,er body.

$1 Conclusions

The gravity field over the Pacific determined by Sjogren's method with

about 90 passes of CEOS-3/ATS-6 satellite-to-satellite tracking data. This

new determination is in good agreement with our previous (1981) determination

of the gravity field in this area using 40 Passes of SST data. A comparison

of this map with the more conventially-determined G D1 satellite gravity field
^, r

OgIGp4ALgJALITY

► haws goo d at;xeetr► ent . 14Q al.acr Show L11V a go ld ovor thf, l l avtf ; c dot ormtned

t tom SV'* 1SA'1 alGi1110-Ler r1,1W and it too aj ,, rovs well with both OWN a ►► d SST,

but this; geoid contains far more deL,111 than either of these rather maps.

It has been previously noted that areas with residual depth anomalies

associated with hot-spots correlate with geoid and gravity anomalies, These

so-called swells do often correlate with the prcoent map,,, , but there are

some clear exceptions and complications. The Marquesas-Line swell of Crough

and Jarrard (1981) was found by them to correlate well wish the GEOS-3 geoid

of Brace (1977). The positive geoid anomaly near the Marquesas Islands

of that work, however, was not found in our earlier SST study (Marsh at al.,

1981) and the present fields verity its absence. This errant positive

anomaly is apparently the result of poo r GEOS-3 coverage in this area.

Those anomalies not associated with thermal swells cirrelate well with

relatively young areas of the seafloor bounded by .fracture zones. Because

the major fracture zones of the Pacific are fairly evenly spaced and trend

approximately east-west, they produce a similar fabric in the gravity and geoid

fields. The anomalies are essentially framed by these VZacture zones. For

a typical offset in age of =15 m.y. the resulting geoid anomaly is about 2.5 m,

which is close to that observed near the east Pacific rise. Because the traces

of hot spots are not parallel to fracture zones, there is an interference in

anomalies from each source. The resulting anomaly field may thus be separable

using the known ages and history of the Pacific plate. Early modeling shows

that because the anomalies due to age ;offsets attenuate with absolute age,

large geoid anomalies over older sea floor (ti70 m.y. like NE of Hawaii) can not

be completely explained by this offset alone (Marsh and Hinojosa, 1983). A part 

of the cause may Also be below the lithosphere 
0 101NAL 1 AG"ho IS

OF POOR QUALITY

An examInat Ion of tho rl/(! • Intl n g vollt vol,ovit y noct?snary to brim, mantle'

phmivs Co LIW 1-Il>I)e1r 114lntiv withul ► r c ► ulli ► ti, show.p Ci ► .tt ltlen;lv diiu;leter L4 JI)t tv

be of the sail► ( order as the thickness of its sour o. Because low dt'llsity

regions within the mantle thickor than 7bOUt 1000 kn► , were they to exist, should

have been discovered through seismology, plume diameter is probably limited to less

than about 400 km. And the ascent velocity needed to prevent complete cooling

is at least 3 cm/yr; it could be ten times l,ari;er.

The explanation of this distinctive pattern of ,gravity and geoid

anomalies in the Pacific has been of interest since the first indication of

its presence (Marsh and Marsh, 1976). At that time we suggested they

might reflect the presence of a small scale form of convection in the uppermost

mantle. because several of these anomalies begirt very near the east Pacific

rise and because some time is necessary to initiate small scale convective rolls,

McKenzie et al. (1980) suggested that these anomalies must arise from small

.scale convective instabilities in the l.oVer limb of a larger cell confined to

the upper mantle. These secondary instabilities would then be already established as they reached the ridge itself. The close correlation between

sea-floor age, fracture zones and these near-ridge anomalies makes the interpretation of McKenzie et al. (1980) untenable. In fact as it now appears

the structure of the lithosphere itself may be a major factor in causing

these distinctive anomalies. Nevertheless, the ultimate origin of both hot

spots, the regularity of fracture zones with major offsets, and some geoid

anomalies over old, smooth sea floor is still unknown, and they could

conceivably be related to the effects of small convection.
OF POOR QUALITY

ACkPOWI0.1";ometst:a

F

i

?

B.M. thanks Ray Arvidson for an early image-analysis or the Pacific:

bathymetry. This work is supported by a grant (NAG5--32) to the Johns

Eiopkirts University from the National Aeronautics and Space Administration.

0
CMYiU*4L V 1ACt' 4r ;

OF POOR QUAL17Y

Rn f re nc os

Brace, K. L., 1977, Preliminary ocean-area geoid from GEOS-3 satellite

radar altimetry. St. Louis Air force Station, Nov. 1977.

Cath les, L. M., I11, The viscosity of the Earth's mantle.

Princeton Univ. Press, 389 p., 1975.

Cochran, J. R. and M. Tal.wani, Free-air gravity anomalies in the world's

oceans and their relationship to the residual elevation.

Geophys. J. R. Astr. Soc.,v.50, p. 495-552, 1977.

r^

} Crough, S. T. and R. D. Jarrard, The Marquesas-Line swell. Jour. Geophys

Res., 86, 11763-11771, 1981.

Crough, S. T., Thermal origin of mid-plate hot spot swells. Geophys. Jour.

R. Astronom. Soc., 55, p. 45-469, 1978.

Detrick, R. S., Jr., An analysis of geoid anomalies across the Mendocino

fracture :zone: Implications for thermal models of the lithosphere.

Jour. Geophys. Res., 86, 11,751-11,762, 1981.

Duncan, R. A. and I. McDougall, Migration of volcanism with time in the

Marquesas Islands, French Polynesia. Earth and Plant. Sci, Letts., 21,

414.-4201, 1974.

Elsasser, W. M., P; L. Olson, and B. D. Marsh, The depth of mantle convection.

Jeur. Geophys. Res., 84, 147 . 155, 1979.

Haxby, W. F. and D. L. Turcotte, On Isostatic Geoid Anomalies. J. Geophys.

Res., 83, p. 5473-5478, 1978., '

Herron, E. M., Sea floor spreading and the Cenozoic history of the eastcentral Pacific. Geol. Soc. Amer. Bull., 83, 167-1692, 1972.
p. 42-43, 1971. t ti '^w' ^++

OF POOR QUALITY

Lerch, F. .1., C. A. Wagner, S. M. Vo ko, R. P. t',e1ca^.t, It. l;. I,nubschev,

and W. A. Taylor, Gravity model improvement usin CEOS-3 altimetry

^k

(GEM-10A 4wd 1011). J., Mr. Gencl, (lit pre4s) .

Lerch, b'. J., J. G. Marsh, S. M. Kloskts, and It. G. 1411 iomsun, Gravity model

improvement for SEASAT. Jour. Geophys lies., 87, 3281-3296, 1982.

Marsh, 13. D., Plume spacing and source.

Nature, 256, 240, 1975

Marsh, B. D., On the cooling of ascending andesitic magma. Phil. Trans. Roy.

Soc. London. A., 288, 611-625, 1978.

Marsh, B. D., Island Arc Development: Some observations, experiments, and

speculation. Jour. Geol., 87, 687-713, 1979„

Marsh, B. D., On the mechanics of igneous diapirism, stoping, and zone melting.

Amer. Jour. Sci., 282, 808-855, 1982.

Marsh, B. D., Kantha, L. H., On the heat and mass transfer :from an ascending

magma. Earth and Planetary Sci. Letts., 39, 435-443, 1978.

Marsh, B., and J. Hinojosa, 1983, SEASAT Geoid Anomalies in the Pacific: Two Dimension

Spectra and Removal of Plate Sources'(abs tract) . AGU Spring Meeting 983.

Marsh, B. D. and J. G. Marsh, On Global gravity anomalies and two-scale mantle

convection. Jour. Geophys. Res., 81, 5267=5280.

Marsh, J. G,, B. D. Marsh, R. G. Williamson, and W. T. Wells, The gravity

field in the central Pacific from satellite-to-satellite tracking.

Jour. Geophys. Res., 86, p. 3979-3997, 1981.

Marsh, J. G. and T. V. Martin, The SEASAT altimeter mean so-a surface model.

Jour. Geophys. Res., 87, :3269-3280, 1982.

McKenzie, D., A. Watts, B. Parsons, and M. Roufosse, Planform of mantle

convection beneath the Pacific Ocean. Nature, v. 288, p. 442-446, 1980.

Morgan, W. J., Convection plumes in the lower mantle. Nature, v. 230,
ORIGINAL PAC-1- ;q

OF w00R QUALITY

Morgan, W. J., Plate motions and deep mantle convection. Qeo.l. Soc. Amer.

112 1 t9/21

Nemoto, K., and L. W. Kroenke, Marine geology of the Hess Rise 1. Bathymeri3y,

surface sediment distribution, and environment of deposition.

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Parsons, B., and J. G. Sclater, An analysis of the variation of ocean floor

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1977.

Pitman, W. C., 111, R. L. Larson, and E. M, Herron, Magnetic lineations of

the oceans. Geol. Soc. Amer, Map. 1974,

Sandwell, D. T., Thermal isostasy: Response of a moving lithosphere to a

distributed heat source. J. Geophys. Res., 87, 1001--1014, 1982.

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ridges. J. Geophys. Res., 85, 7235-7% ;1, 1980.

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87, 3949-3958, 1982.

Tapley, B. D., G. H. Born, M. E. Parke, The SEASAT altimeter data and its

accuracy assessment. Jour. Geophys. Res., 87, 3179-3188, 1982.

Turcotte, D. L. and D. C. McAdoo, Geoid anomalies and the thickness of the

lithosphere. J. Geophys. Res., v. 84, p. 2381-2387, 1979.

Turner, D. L. and R. D. Jarrard, K-Ar dating of the Cook-Austral island chain:

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Watts, A. B., Gravity and Bathymetry in the central Pacific Ocean,

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i.
or"N,,"NNIAL

OF POOR QUAL17Y

Writts, A. B., J. H. Bodine, and N. M. Ribe, Observations of flexure and the

guoLogi.ca.1 OvOlUtL011 Of Lho Ocoan

1980.
3

V'w9.1a Z :^ b i .:1

C9:?

VA Y'n'u

Figure Captions

Figure 1. Distribution of SST tracks used in this study.

Figure 2. SST gravity map for the central Pacific region. The circles

have radii of 30 0 and 40 0 about the subsatellite point, and within

this range the gravity is essential radial.

Figure 3. SEASAT altimeter geoid for the Pacific relative to the twelfth

degree and order GEM 10B geoid. The trace of the east Pacific

rise is also shown (far right) as well as the 20 and 65 m-y.

isochrons for the age of the Pacific plate.

Figure 4. The SEASAT geoid of Figure 3 superimposed on the map of the age

of the ocean basins of Pitman et al., (1974).

Figure 5. The relationship between the diameter (2a) of a plume rising from

• low density region of thickness h 2 and viscosity p 2 relative to

• surrounding !mantle of viscosity p I' The right axis gives diameter

for a source thickness of 100 km.

Figure E. The mean temperature (T/T 0 ),of a section of plume as it ascends

through mantle whose temperature is adiabatic and described by

Tm(t). Only for values of Jto = 1 does the plume reach the upper

mantle still anomalously hot.
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Figure S. The relationship between the diameter (2a) of a plume

rising from a loi , density region of thickness h2 and

viscosity 02 relative to a sourtounding mantle of

i viscosity ul. The right axis gives diameter for a

source thickness of '100 km.

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