~., :'~. -.~" .. ~. ;/-7

m"J}'~,l'l ELSEVIER Physics of the Earth and Planetary Interiors 103 (1997) 63-84

PHYSICS OFTHE EARTH

AN D PLANETARY INTERIORS

Density changes in upwelling mantle

F. Celia *, A. Rapolla Dipartimento di Geofisica e Vulcanologia, Universitb degli Studi 'Federico H' di Napoli, Napoli , Italy

Received 21 October 1996; accepted 3 April 1997

Abstract

Gravity modelling of the litho-asthenosphere system in regions involved in extensional tectonics requires an accurate estimation of the vertical density profile in the upwelling asthenosphere. To this end, the calculation of the variation of several parameters is needed, i.e., the degree of thermal anomaly in the asthenosphere, the amount of stretching, the thickness of the lithosphere before stretching, and the variation of melt fraction generated by adiabatic decompression. Therefore, knowledge of melt density and of the residual mantle rock composition is necessary in order to have the information to calculate the density of the uprising mantle. These calculations allow to partially solve the problem related to ambiguity of the gravity interpretation, and to provide information about the structural features of the upwelling mantle. © 1997 Elsevier Science B.V.

Keywords: Gravity modelling; Lithosphere; Upwelling mantle; Physical parameters; Petrological parameters

1. Introduction

An important aim of the interpretation of gravity data in regions involved in extensional tectonics (continental rifts, ocean ridges, etc.) is the modelling of the upwelling mantle. This implies not only the reconstruction of the lithosphere-asthenosphere boundary (L.A.B.) and Moho discontinuities, but also the vertical density distribution in the upwelling mantle.

When the asthenosphere rises, physical and petro- logical parameters (like density) of mantle rocks change depending on several factors:

Corresponding author.

(a) Generally, mantle uprising coincides with a thermal anomaly in the lithosphere-asthenosphere system. Therefore, a change in the vertical tempera- ture distribution has to be expected.

(b) If the temperature reaches the mantle solidus, partial melting sets in. The melt fraction and its composition are expected to change with depth.

(c) Consequently, the composition of the residual mantle rock will be modified after magma segrega- tion.

(d) The density of both liquid fraction and resid- ual mantle rock will change with depth depending not only on temperature and pressure, due to thermal expansion, but also on their gradual change in com- position.

0031-9201/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 3 1 - 9 2 0 1 ( 9 7 ) 0 0 0 2 1 - 6

64 F. Celia, A. Rapolla / Physics ~[the Earth and Planetary Interiors 103 (1997) 63 84

The aim of this study was to estimate vertical density distribution in the upper mantle for rifting at different degrees of evolution.

It must be remembered that knowledge of the above factors is important in gravity modelling in regions characterized by extensional tectonics. In fact, small density differences in the upwelling man- tle rocks can have a great influence on gravity anomaly because of the large volumes involved. Density of residual mantle upwelling can be esti- mated only by taking into account the above factors. Therefore, a quantitative model is needed, describing the evolution of the uprising mantle from both the physical and petrologic point of view. Among the few models elaborated to perform this, a significant one was suggested by McKenzie and Bickle (1988) and White and McKenzie (1988). It was here applied to solve the problems related to points (a) and (b). Point (c) was solved here by calculating the modal norm of residual mantle rock solving a linear system with a technique based on the 'minimum length with inequality constraints' method (Menke, 1984). Point (d) was solved according to the model proposed by Niu and Batiza (1991a).

2. The calculation of geotherms

The model proposed by McKenzie and Bickle (1988) and White and McKenzie (1988) describes the complex mechanism of upwelling mantle and related partial melting. In fact it is based both on experimental results and a theoretical approach to these problems from the thermal, rheologic and petrological points of view. The validity of this model is beyond the scope of this paper, but it will be shown because their model may be used fully extended to address points (a) and (b).

The first aim of this model is to estimate the vertical distribution of temperature. Geotherms are the results of several physical processes within the mantle.

Some authors did not take into account the tem- perature decrease due to the latent heat of melting and to the adiabatic gradient in the solid-liquid system (e.g., Green and Ringwood, 1967), thus ob- taining geotherms different from those calculated by

adopting the model by McKenzie and Bickle (1988). If a convecting fluid with constant viscosity is hy- pothesized beneath the solid lithosphere, the geotherms estimated based on the above mentioned model seem all to tend to the same adiabatic curve in the interior. The main problem solved by this model is the estimation of the heat content in the upwelling mantle. It has to be remembered that the advective heat transfer represents the main heat transport mechanism within the uprising asthenosphere (see Appendix A), whereas the heat transfer by conduc- tion may be neglected (Appendix A: Eq. (AI)).

The estimation of heat content presents several difficulties. In fact, it was already indicated that mantle uprising occurs under adiabatic decompres- sion at approximately constant entropy (Appendix A: Eq. (A2)). The problem is represented by the differ- ent values assumed by the adiabatic temperature gradient for the solid residual mantle rocks ( 0.6°C/km) and for the liquid fraction ( = 1.0°C/km) (McKenzie, 1984a), due to a different thermal ex- pansion coefficient (see Appendix A). The change in heat content of the residual mantle rock (low com- pressibility) is only proportional to the temperature variation. Consequently, it is relevant to compare temperatures in order to evaluate the heat content of the upwelling solid mantle at different depth. This is not possible in the case of a melt which is more compressible than the residual solid mantle. The compressibility of the mantle has to be taken into account because the temperature differences within the convecting mantle probably do not exceed 200°C and because the temperature gradient of the upper mantle is likely to differ little from the adiabatic gradient. Thus, the temperature of the mantle mate- rial will increase by 200°C on sinking to a depth of 300 kin. Hence, the temperature differences are not a good guide to differences in heat content. The model adopted overcomes this problem by considering the heat content of solid and liquid components of man- tle material as proportional to the temperature differ- ence between these components at a same reference pressure. Thus, the concept of 'potential tempera- ture' (Tp) is introduced, which, in terms of mantle dynamics, may be better defined as the temperature that the mantle material would have at the Earth surface if melting does not occur. This parameter allows to avoid the adoption of the entropy concept

F. Cella, A. Rapolla / Physics of the Earth and Planetary Interiors 103 (1997) 63-84 65

180

160

140

120

100

80

~ 60 o

~ 4 0

~ 2 0

-- - Temp. at the base ot TBL (*C) Tp = t 280°C

-Temp. at the base ol "rBL (oC) Tp = 1480°C

Li~hesphere thickneas (Km); Tp =1280~C ......... Lithosphere thickness (Kin); Tp =1480~C . . . . . I~L thickrmss (Krn); Tp =1280¢C = TBL thicRness (Km); Tp =1480~C

f . . . . . . . . .

7

'1 . . . .

! 1 -

, i . . . .

100 1

. . . . ! . . . . r . . . . LJ~ '-' i 1 ~ ' - ! ; J } _

l j J J J

1400

1350

o

t laO0 1250

1200

] 115o

i ] 1100 ~

; o l ° s °

MBL Thickness (Km)

Fig. 1. Lithospheric thicknesses, TBL thicknesses and tempera- tures at the base of TBL, as functions of MBL thickness.

(which is more difficult t o evaluate) in order t o

estimate the heat content of the upwelling mantle (Appendix A: Eq. (A3)). In other words Tp defines the degree of thermal perturbation in the mantle and controls the vertical temperature distribution within the upwelling mantle (see Appendix A: Eq. (A5)).

The adopted model attempts to figure out the main boundaries of the crust-mantle system by adopting the distinction between the mechanical boundary layer (MBL) and the thermal boundary layer (TBL) (White, 1988). The MBL represents the base of the upper Earth layer, only approximately coinciding with the geochemical definition of litho- sphere. The lower layer (TBL) is neither totally rigid, nor vigorously convecting, but it is in an intermediate state, thus accumulating only short term chemical anomalies. Therefore, the lower boundary of the lithosphere, in its thermal definition (the layer whose lower boundary coincides with the horizontal isotherm), is located within the TBL, and, as shown in Fig. 1, it may be related to the thickness of MBL (from Eqs. (A15), (A16), (A17), (A18) and (A19)). Similarly to Tp, the MBL thickness before the thin- ning process controls the profile of the geotherm (see Appendix A: Eq. (A5)).

A third parameter plays a significant role in the vertical distribution of mantle temperature, i.e., the stretching factor 13 (McKenzie, 1978). This is ex- pressed as the ratio of the final to initial surface area involved in the rifting process. An approximate 13 value can be inferred from geological and geophysi- cal data.

As far as the role influence of the viscosity on the temperature at the base of MBL and of the litho- sphere and TBL thicknesses are concerned (Fig. 2), it is summarized in Appendix A (Eq. (A8) and Eq. (A18)).

The model of McKenzie and Bickle (1988) was applied in this study to calculate geotherms as a function of Tp, 13, and MBL thickness. Calculations are based on analytical formulations described in Appendix A and were made for several values of the parameters previously mentioned.

As far as the choice of the potential temperature is concerned, a reliable Tp value can be estimated when the degree of stretching and the amount of subsi- dence are given (White and McKenzie, 1988). It has to be pointed out that, generally, a low T o value (1280 ~ 1320°C) indicates the absence of a thermal perturbation in the lower mantle and, consequently, a mantle uprising due only to plate separation caused by external driving forces is implied. This case, common to most parts of ocean ridges, could be considered as typical of 'passive rifting'. High Tp

140 . . . . . . . . . . . . . . . . . . . . 1350

12o ............... ~ - . - - - ~ ............ 1 ............................................ : 1 3 o o ~ : : ~ ] Temperature at I- . . . . . . . . . . . ~ ...... ~ the 5ase of MBL

100 ~- ................ ~ ................... J - - w Lithosphere thickness 1250 ~.

'~ -- 80 ............... ....................... ~ - - . . . . . TBL thickness .~

"-.. ,2oo L!

I~ 1150 ¢ 40 ..

20 ...... :-.~:..:.-:;--.:--.~- .~ ..................... 11oo

. . . . i . . . . . . . . . . . . . . . ,050 0 15 10 10 16 10 17

ViSCosity of melt (m 2 a "1 )

Fig. 2. Temperature at the base of MBL, lithospheric thickness and TBL thickness, as functions of melt viscosity. MBL Thick- ness = 100 km; Tp = 1380°C.

6 6 F. Celia, A. Rapolla / PhyMcs ¢~'the Earth and Planetat 3" Interior.s 103 (1997) 63-,~4

values (1460 ~ 1480°C) indicate the probable pres- ence of a 'hot rising jet ' (Oxburgh, 1980), or +hot plume' , located within the ocean or within plates and having a deep thermal origin (Courtney and White, 1986). Thus, values of Tp= 1280°C and 1480°C were chosen because they probably represent the extreme values for the most of anomalous thermal conditions.

Five 13 values were chosen in our calculations following the indications of White and McKenzie (1988). Two small values of 13 (13 = 2 and 13 = 3) were adopted because they are typical values for the intra-continental sedimentary basins which have not subsequently developed into ocean basins. A value of 13 = 5 was chosen because it roughly represents the point at which stretched continental crust is

Legend :

I 8=2 . . . . ~=3 ............ 8=5 .......... ~=10 - - ~=50 I 1600

L Solidus! curve ~ TP= 1480°C ~ 1soo . . . . . . . . . . . . . . . . . . .

_ i cl+-+o+ .e 1400 a

1100 ~ ' . ; n ~ - ; ~ = 3 = ..' I' ~ 0 50 100 150 200

Depth (kin) MBL thickness = 90 Km

1600

s o " +c 1500 ; Tp = 1380 °C

~ 140o

= i ; : Tp=1280°C b 1300

& - !: , c . . o , ~ , n , m : :: idNen¢~ o1

i. ~ 1200 : .~ thermal anomaly )

11oo Y~:,! ~" ,m, I/t, , , / / , / , ~ I , , , 50 100 150 200

Depth (kin) MBL thickness = 120 Km

~ 1500 I ~ 1 6 0 0 . . . . . . . . ~ -- + ~ ---

~, 1400 ¢= C ~ 1300

1200 ::' o1 thlWl~

0 50 100 150 200 Depth (kin)

MBL thickness = 160 Km

Fig. 3. Geotherms in uprising mantle as a function of Tp, f5, and undisturbed MBL thickness. Thick line indicates the curve of solidus. Diagrams (a), (b) and (c) show geotherms calculated for three different MBL thicknesses.

broken up to form igneous oceanic crust. Finally. two large B values (B = 10 and B = 50) were taken into account to represent an ocean basin at a young phase and a stable upwelling beneath oceanic spread- ing ridges, respectively.

The results of these calculations are presented in Fig. 3. Three different MBL thicknesses were chosen to cover the range that is likely encountered along rifting margins: 90 km (Fig. 3a), 120 km (Fig. 3b) and 160 km (Fig. 3c), respectively, corresponding to undisturbed thicknesses for continental lithosphere of about 108 km, 138 km and 180 km before stretching, respectively. Fig. 3 also shows the geotherm pre- dicted for unstretched lithosphere (13 = 1) and the mantle solidus. The latter is referred to as a garnet peridotite and was computed by taking into account experimental data from Ito and Kennedy (1967), Green and Ringwood (1967), Jaques and Green (1980), Stolper (1980), Harrison (1981), Takahashi and Kushiro (1983), Takahashi (1986), and McKen- zie and Bickle (1988).

Fig. 3 shows that as Tp increases the geotherms shift toward higher temperatures. Also for low 13, high temperature drops abruptly to normal values at greater depths than those at which 13 is higher. Thus, the higher the 13 values, the shallower the depths at which the thermal anomaly disappears. Moreover, for increased lithospheric thickness before stretching, the above variation occurs at progressively higher depths.

3. The est imation of fraction and composi t ion of melt

A complete model, capable of describing quantita- tively the melting process of a multiphase multi- component mantle system at different pressures and with a variable amount of solid solution between different phases, does not exist. Thus, a quantitative approach is adopted (see Appendix B), basing on a very large amount of empirical data (Ito and Kennedy, 1967; Green and Ringwood, 1967; Jaques and Green, 1980; Stolper, 1980; Harrison, 1981; Takahashi and Kushiro, 1983; Takahashi, 1986).

Significant results (Fig. 4) were obtained in the case of maximum lithospheric stretching (13 = 50). In

F. Celia, A. Rapolla / Physics of the Earth and Planetary Interiors 103 (1997) 63-84 67

4o 35

30

-~ 2s

~ 15 • 10

5 0

. . . . . . . , , . . . . . , , ,

0 20 40 60 80 100 120 Depth(km)

Fig. 4. Melt fraction versus depth generated by adiabatic decom-

pression in mantle uprising as a function of the potential tempera-

ture (Tp).

this case, partial melting continues up to the earth surface beneath the oceanic ridge. It can be observed that the melt fraction gradually becomes larger with decreasing depth. Moreover, the melt generated by adiabatic decompression is also strongly dependent on the degree of thermal anomaly in the astheno- sphere. Beneath the spreading ridges (13 = 50) the melt fraction can reach 23% (Fig. 5) when the asthenospheric temperature is 'normal' (Tp = 1280°C), and the process is controlled by external driving forces only. When the stretching factor is high (e.g., 13 = 50) but the thermal anomaly in the asthenosphere is higher (Tp = 1480°C), the melt frac- tion may be larger (about 36%). This case is less frequent and is typical of hot rising jets below spreading centres where lithospheric stretching is considerable (e.g., Iceland).

It must be observed that the lithospheric thickness before thinning has a great influence on the mantle partial melting. In fact, when 13 is small, the melt fraction does not reach the surface, and tends to zero at a certain depth. This depth coincides with the top of the mantle uprising, i.e., the base of the stretched lithosphere (Fig. 5a and b). Fig. 5a and b shows the results obtained for two different lithospheric thick- nesses (90 km and 160 km, respectively) and show how thickness controls the melt fraction. In fact, at constant values of 13 and Tp, the melt fraction occurs at shallower depths for smaller lithospheric thick- nesses.

Analogous considerations can be made by refer- ring to the total melt volume generated in the up- welling asthenosphere. Fig. 6 shows that total melt thickness becomes greater with increasing Tp, 13 and

40

35 g a0 = 25 o ~ 2o

. 1 5

5 0

0 20 40 60 80 1 O0 120

Depth (kin)

ThicRness of Mechanical Boundary Layer (MBL) = 160 Kin.

40

Z 30 :::::::::::::::::::::::::: ; . 25

20 b 15

10 5 0

0 20 40 60 80 100 Depth(km)

ThicRnessof MechanicalBoundaryLayer(MBL)= 90 Km.

Legend:

B=2 B=3 8=5 B=10 ~=50

Fig. 5. Melt fraction versus depth generated by adiabatic decom- pression in mantle uprising depending on stretching factor B), for

T o = 1280°C and Tp = 1480°C. Two different cases (MBL thick- ness = 90 and 160 km) are shown.

with decreasing lithospheric thickness before stretch- ing. The influence of stretching on melt thickness is

30

- ~"~P/?'~ -? " II .......... B i i ~ ................... i

0 5 10 15 20 25 30 35 40 45 50

Stretching factor (1~)

Fig. 6. Melt thickness versus stretching factor as a function of Tp (potential temperature) and of lithospheric thickness before stretching. Results for Tp = 1280°C, 1380°C, 1480°C and for MBL thicknesses = 70 krn, 100 km, 130 km are shown.

68 F. Celia, A. Rapolla / Physics qf the Earth and Planeta O' Interiors 103 (1997) 63 84

very high at low 13 values, but it abruptly decreases when 13 becomes higher. It is assumed that all the melt generated is extracted from the matrix and migrates upward, until it is added to the overlying crust, thus forming igneous underplating bodies with a thickness related to the amount of melt.

The largest melt thickness (26 km) is reached with a 70 km thick MBL, 13 >> 5 (13 = 50) and Tp = 1480°C. The reliability of this model is supported by the average crustal thickness predicted for oceanic plates, which coincides with the melt thickness pre- dicted by the model (6.5 kin) for T v = 1280°C and 13 = 50. White and McKenzie (1988) suggested that about half the melt generates cumulus gabbros form-

ing the lower oceanic crust, whereas the other half is erupted as pillow lavas or intruded as dykes.

The estimation of the amount of melt extracted from residual mantle, and migrated upward, is con- troversial (Foucher et al., 1982; McKenzie and Bickle, 1988). Nevertheless, most results confirm that the melt remaining in the residue is sufficiently small to be neglected, so that all magma produced by partial melting in the upwelling asthenosphere may be assumed to have migrated upward. Then, it either erupts at the surface or consolidates within the crust, forming underplating igneous bodies often detected by geophysical prospecting. The thickness of these underplating bodies can be so great that the amount

5 5 54 - 53 -

, ~ 5 2 - 51-

50" 49-

48- 47-

46

11 .

V . . . . .......

- % . . . . . . . . . . . . . .

" .',?~: i ....... ~ ........... ....... ~,',~ :: ...... i .......

i 2 0 4( ) 6 0 8 0 1 0 0

Depth (km)

9 -

O 8- M.

7 -

1 0 - ............. i . z ; i i -~/

0 2 0 40 6 0 8 0 1 0 0

Depth ( K r n ) 19

18

17

1 5

0 1 4

12 11 10

. . . . . . . L . ~ .

2 0 4 0 6 0 8 0 1 0 0

Depth (Kin)

1 7 ,

O ~ 1 2 - 1

11q ' / ! /

10 - { ' .~<.....-! .... ; ! ....

"f ~ . i , I , I , 0 2O 4O 6O 8 0 1 0 0

Depth ( K i n )

12 ~ ~ ~ E ~ . !

11 , ~ . ~,~,

0~10 ' ! ........ , ~: ~ ~ : $~ ,' ,. ~ !~.

! ! i 7 i

o ~o 4'o o'o 8'0 l oo Depth ( K r n )

3 , ! ~ :

~ 2.5

2 "

2'o ~b 6b ~b ~oo Depth (Kin)

2

/ i , ' i . ' , : , i / /

i' • ~ / /

8 •

i

0 , , ; ! 2 0 4 0 6 0 8 0 1 0 0

Depth (Kin)

0 . 3

o . 2 7 5

0 .2 ~.-~" ~ -

o.17.= 2~, 40 e6 80 too Depth (Km)

Legend :

T p = 1 2 8 0 ° C

T p = 1 3 2 0 ° C . . . .

T p = 1 3 6 0 ° C . . . .

T p = 1 4 0 0 ° C - -

T p = 1 4 4 0 ° C . . . . .

T p = 1480 °C . . . .

Fig. 7. Point a v e r a g e compos i t i on ( w e i g h t % o f main ox ides) o f the liquid fract ion, gene ra t ed by partial me l t ing o f a peridoti te in adiabat ic

upwel l ing mant le , ve r sus depth. Resul ts are g iven as a funct ion o f the potential t empera tu re (Tp).

F. Celia, A. Rapolla // Physics of the Earth and Planetary Interiors 103 (1997) 63-84 69

of extension could be underestimated if it is deter- mined only on the basis of the change of crustal thickness (McKenzie, 1984b).

In the estimation of the magma composition gen- erated by adiabatic decompression in upwelling as- thenosphere, a distinction must be made between instantaneous melt composition, i.e., the composition of the melt generated at any depth, and the 'average melt composition', i.e., the result of a complex weighted average of the melt composition produced.

Calculations in Appendix B (Fig. 7) show that the average melt composition changes significantly, mainly depending on the degree of the thermal anomaly within the asthenosphere (Tp).

With low asthenospheric temperatures and high degrees of stretching (as beneath oceanic spreading centers), melts are generated at shallow depths and have a composition crossing into the tholeiite field (Fig. 8). On the other hand, for lower degrees of stretching, as in many continental rifts, the melt

' . . . . ! . . . . i . . . . I . . . .

3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . .

3 " i i i

2 ........ ' ' ! ' r ~ . i ............. t ...........

1 .... - - ~ - ! ........... -~

/ o ?.)}:,. 0 5 10 15 20

Depth (kin)

55 . . . . . l ' " l ' " l " ' !

35 ~ /

30

25 i .. 0 20 40 60 80 100120

Depth (k in)

. , <

o

50~. ' , ' " ! " ' ! ' " ~ " "

~o i ~"i--~-i i 20 ii-~-i,~-! i i ......... 10 ~i,i~i i ~ " i

0 20 40 60 80 100120

Depth (k in)

23,

22

21

~2o 19

X a. 18

17

16

15

4

i

' i

, y ~ z '~, i ~-

/.)

/ ' i. 20 40 60 80 100 120

Depth (km)

~'i

z•'l .......... i ......... t . . . . . . . . . i

. . i . . ; t ' , ' , . :, . . . . . . 00; ' 20 40 60 80 100

Depth (km)

3.5

3

~ 2 . 5

0.15 ~ 0 20 40 60 80 100 120

Depth (km)

3O

25

~ 2o

0 20 40 60 80 100 120

Depth (k in)

A

- q i_/,~," I I

20 40 60 80 100 120

Depth (km)

Legend:

- - T p = 1 2 8 0 ° C

. . . . . . . . . Tp = 1320 °C

. . . . . Tp = 1360 °C

-- - - - T p = 1 4 0 0 ° C

- - - T p = 1 4 4 0 ° C

. . . . Tp = 1480 °C

Fig. 8. CIPW normative composit ion (weight% of main minerals) versus depth of igneous rocks derived from liquid fraction generated by partial melting of a peridotite in adiabatic upwelling mantle. Results are given as a function of the potential temperature (To).

7(/ F. Celia, A, Rapolla / Physics (#the Earth and Planetary Interiors 103 (1997) 63 84

composition is in the range of alkali basalts. Where the mantle thermal anomaly is very intense (Tp ~ 1480°C) and the melt fraction increases, as in the case of hot rising jets, the melt composition becomes more and more picritic. Furthermore, melts approach the composition of basaltic andesites at 13 ~ 50, whereas 13 ~ 2 defines the boundary separating alkali basalts from olivine tholeiites. These results are in good agreement with data from many authors (Clarke, 1970; Norry et al., 1980; Mohr, 1983).

Although the adopted model represents an useful way to perform a quantitative evaluation of these phenomena, nevertheless it presents some limits:

- - Because of the absence of a general theory of melting, it is necessary to adopt the experimental results. Most of these were carried out on peridotites and ridge basalts. Thus, as pointed out by the same authors, the estimation of the amount of melt and of its compositional changes are restricted only to the melting of garnet peridotite.

- - Some evidences from recent studies in the North Sea seem to indicate that the amount of melt generated could be underestimated by McKenzie and Bickle (1988) for smaller stretching factors (13 = 1.5-2) (Latin et al., 1990; Latin and Waters, 1991; Pedersen and Ro, 1992);

- - The McKenzie and Bickle (1988) model did not take into account the influence of the pressure exerted by the already formed basaltic crust on the amount of melting. This will lead to a reduced production of basaltic crust upon pressure-release melting (Vlaar et al., 1994).

- - Results obtained from Eq. (B4) (see Appendix B) are affected by a mean error of 3% that does not decrease by using more coefficients. However, this is expected since it is likely that the experimental data are affected by an error greater of 3%.

- - It must be pointed out that there could be significant differences between the amount of stretching related to the whole lithospheric thickness and that related to the single crustal layer. Where 13 is inferred from the change in crustal thickness, based on shallow geological and geophysical evidences, results could be underestimated, especially if the potential temperature is high and the MBL is thin. Nevertheless, to calculate the true lithospheric stretching, better results could be performed by de- ducing B from thermal data (heat flux) and from

subsidence measurements due to the fact that the vertical temperature distribution in the mantle de- pends on the thinning of the whole lithosphere.

4. Changes in the modal composition of residual mantle rocks

The knowledge of melt fraction and melt compo- sition allows to calculate the chemical composition of the residual mantle rock. First, a 'reasonable' average composition of a fertile mantle was chosen by adopting the MORB pyrolite with Fo90 (MPY90; Table 1) proposed by Fallon and Green (1987). This composition is similar to that of lherzolites and is considered by some authors (e.g., Niu and Batiza, 1991b) to be the best candidate for the MORB s o u r c e .

The calculation of the composition of residual mantle rock by taking into account the change of melt fraction and composition with depth was chal- lenging because it concerns a rock for which a standard norm calculation cannot be used.

Furthermore the pressure range of interest implies at least 2 different mineral phase changes:

(a) garnet-spinel peridotite transition (fixed at a depth of about 70 km):

olivine-orthopyroxene-clinopyroxe ne-spinel ;

olivine-orthopyroxene-clinopyroxene-garnet

(b) spinel-plagioclase peridotite phase transition (fixed at a depth of about 30 km):

olivine-orthopyroxene-clinopyroxene-plagioclase.

olivine-otthopyroxene-clinopyroxene-spinel

This problem was solved treating the calculation of the modal norm of the residual mantle rock as the solution of a linear system of equations. Thus, the

Table 1 Chemical composition of MPY90 (Fallon and Green, 1987)

SiO 2 MgO TiO 2 CaO A1203 Na20 FeO K 2 0 Sum

44.74 38.57 0.17 3.38 4.37 0.40 7.55 0.03 99.22

F. Celia, A. Rapolla / Physics of the Earth and Planetary Interiors 103 (1997) 63-84 71

following equation was used expressing the bulk rock composition (Y) in terms of its components:

k

Y = Y'~X i A i, (1) i= t

where A i is the composition of the ith component, and X i is the amount of A i. If the mode is esti- mated, Y and A i are known whereas X i must be calculated. In the present case Y is the weight per- centage of oxides in chemical analysis. Each compo- nent of the system represents the sum of their pro- portions in each phase component of the rock sam- ple. Thus, the Eq. (1) is expandable as:

Yl = X l l a l -}-x12a2 q- .... x i k a k

Y2 = x21al "[- x22a2 -[- .... x 2 k a k

Yn = Xnla l "]- x n 2 a 2 -[- .... X nka k

(2)

where Yn = amount of the nth system component in the rock (or the weight% composition of the nth oxide); xn~ = composition of the kth phase compo- nent; and a~ = proportion of the nth system compo- nent in the kth phase component.

The user can modify the input data set to compute the mode of the rock depending on its phase compo- nents. Therefore, the coefficients of the equations must be known. The coefficients are given by the ratio of the molecular weight of each oxide to the molecular weight of the mineral phase (Table 2).

The resulting linear system of equations was solved by adopting the IML technique ( 'minimum

length with inequality constraints'), based on the 'minimum length' solution for undetermined prob- lems (Menke, 1984) and adopted by Fedi and Rapolla (1995). The IML technique allows to define con- straints as 'a priori' information, defining a volume of feasible solutions. If the constraints are consistent, this volume is non-zero and there is at least one solution satisfying the system. It is, therefore, possi- ble to 'a priori' define the width of the maximum range in which the error of the results may vary. This is very useful in our case because, as already men- tioned, the known terms (Y) are given by the weight percentages of the oxides in the rock. Since these percentages are probably affected by analytical error if the modal norm is calculated as indicated in Eq. (1) and Eq. (2), it may be impossible to find the solution that satisfies the linear system. The problem was solved with the IML technique that allows to find the solutions of the system for which the known terms can differ from the oxides percentages by a small definable j quantity (I1, +J) . Besides, a set of data available with the bulk composition of the rock can be used as constraining data to obtain refined results. In our case, one constrain was given by the sum of all oxides weight percentage (100 + j), where j must be chosen depending on the uncertainties of the bulk composition of the rock.

The mode of MPY90 (Fallon and Green, 1987) mantle rock for three different phases: garnet, spinel and plagioclase phases was calculated. Then the same calculation was performed for the residual mantle rock by substituting coefficients in Table 2

Table 2 Coefficients used for Eq. (2)

Phase SiO 2 AI203 FeeO 3 FeO MgO CaO Na20

Olivine Forsterite 0.42700 0 0 0 0.57300 0 0 Olivine Fayalite 0.29500 0 0 0.70500 0 0 0 OPX Enstatite 0.59850 0 0 0 0.40151 0 0 OPX Ferrosilite 0.45543 0 0 0.54457 0 0 0 CPX Diopside 0.55492 0 0 0 0.18612 0.25896 0 CPX Hedenbergite 0.48437 0 0 0.28959 0 0.22604 0 CPX Jadeite 0.59448 0.25221 0 0 0 0 0.15331 Garnet Pirope 0.44713 0.25293 0 0 0.29994 0 0 Spinel Spinel 0 0.71670 0 0 0.28330 0 0 Spinel Hercynite 0 0.58663 0 0.41337 0 0 0 Plagioclase Albite 0.6874 0.19442 0 0 0 0 0.11818 Plagioclase Anortite 0.43194 0.36649 0 0 0 0.20157 0

7 2 F. Celia, A. Rapolla / Physics of the Earth and Planetary interiors 103 (1997) 63-84

starting from MPY90 after melt generation and its removal. Figs. 9 and l0 show that the mineral as- semblages of the residue change with increasing melting at different Tp. The minerals taken into account are olivine, orthopyroxene, clinopyroxene, garnet, spinel and plagioclase. Figs. 9 and 10 show how, independently from the T o value and the range of depth, olivine increases toward shallower depths, whereas orthopyroxene and clinopyroxene are de- creasing toward the surface, after an increase due to the garnet -spinel transition. The other mineral phases gradually increase with depth. With high astheno- spheric temperatures, spinel and plagioclase remain constant after an initial decrease toward shallower

100

=~ 90 . .? I eo

~e 70

60

. . ~ . i - " ~ [ / ~ i ' , i . . ................. i . . . . . .

..~.....~:.i . : . ., : : :,~ .. . . . . . . . . i .................. L

. . ~ . . . . . . . : .... . - . : : : . : : :~ . . . . . . ............

i ~ , , , ' i , , , i , = , 50

3 0

A 2 5

~c~ 20

15

X 10 Q. 0

5

10 30 50 70 90 110

Depth (kin)

. . . . ...z . : . < . i 2 "-.;, ............. .............. i : .

A

x Q. ¢J

0 20 40 60 80 100

Depth (km)

15 f ~ / ,

10i i T ;

20 40 60 80 1 O0

Depth (kin)

Legend :

Tp = 1280 °C Tp = 1320 °C Tp = 1380 °C Tp = 1440 °C Tp = 1480 °C

Fig. 9. Modal normative composition of residual mantle rock after total melt extraction versus depth, as a function of the potential temperature. Original mantle rock chosen is MPY-90 (Falloon and Green, 1987). Three different diagrams are shown for the follow- ing phases: (A) Olivine weight% versus depth; (B) Opx weight% versus depth; (C) Cpx weight% versus depth.

2 5

.~ 20

15 '~,

lO

0

7

~ 6 5

f~ 4 ~ 3 ~ 2 r , In I

0

: .......... ........................... i . . . . . . . . . . . . . . . . . . . . . . . . . ~ ................................. q - 1 1 - i

20 40 60 80 1 O0

Depth (Ken)

........ i ................ i ~', i .................. i

, ' i i .......... i , , , 20 40 60 80 1 O0

O~th (kin)

i ........ - ....... "

,, o . . . . . . . ; , , , i ,

Legend:

20 40 60 80 100

Depth (kin)

q Tp = 1280 °C Tp= . . . . . . . . . . . . . 1320°0 Tp = 1380 °C Tp = 1440 °C Tp=_ !487~CJ._

Fig. 10. Modal composition of residual mantle rock after total melt extraction versus depth, as a function of the potential temper- ature. Original mantle rock chosen is the MPY-90 (Falloon and Green, 1987). Three different diagrams are shown for the follow- ing phases: (A) Garnet weight% versus depth; (B) Spinel weight% versus depth; (C) Plagioclase weight% versus depth.

depths. It can be observed that modal composit ion of the mantle residue strongly depends on the astheno- spheric temperature (Tp). In fact, olivine increases with increasing Tp, whereas the opposite behaviour is observed for orthopyroxene and clinopyroxene.

5. Calculat ions o f me l t densi ty and res idual m a n - tle rocks

The calculation of the geotherms, melt fraction, its chemical composition and the mode of the resid- ual mantle rocks was essential for the calculation of density versus depth of melt and solid phases of the

F. Celia, A. Rapolla / Physics of the Earth and Planetary Interiors 103 (1997) 63-84 73

3,45 residual mantle. Theoretical studies have shown that the density differences between silicate melts and solid phases play a significant role for evaluating the style of the mantle upwelling, the melt segregation and the migration processes by porous flow (Mc- Kenzie, 1984a; Spiegelman and McKenzie, 1987; Sotin and Parmentier, 1989). These differences are here calculated as a function of temperature, pressure and composition following procedures suggested by Niu and Batiza (1991a,b,c). 3.45

The influence of the temperature on the melt ~ 3.4 density was estimated after Lange and Carmichael

3,35

3,3 (1987) by using the 'double-blob' Archimedean method for density measurement and considering the ~ 3,25

3,2 molar volume of a multi-component silicate having a

3,15 linear dependence on melt composition (see Ap- pendix C: Eqs. (C1), (C2) and (C3)).

As far as the estimation of the pressure effect on 3,45

melt density is concerned, the method by Lange and 3,4

Carmichael (1987) was preferred to that by Herzberg ~ 3,35 (1987a,b), because the former was confirmed by ~ 3,3

experimental results (Agee and Walker, 1988) (see ~" t 3,25

Appendix C: Eqs. (C4), (C5), (C6), (C7), (C8) and x 3,2 (C9)). 3,15

The density of melt (Fig. 11) was obtained for different values of Tp, B and lithosphere thickness, based on the assumption that all melt migrates up- ward from the solid matrix (see Section 3). At great depths (with very low degrees of partial melting) the

..... i .............. i

3,3

3,25 I with . . . . I temperature profile !

" - " ~--I- ' i ....................................... ~ ~ ' • " ~

¢~ 3,2 ~-~------~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3,1 ~ , I . . . . . ~ I i , , I , , 40 80 120 160 200

Dep th ( k m )

O r i g i n a r y M B L th i ckness = 9 0 Im~ ; T p = 1 2 8 0 °C

I i i i i i ~ i i : = t i J K i ~ L

40 80 120 160 200

Dep th (K in )

O r i g i n a r y M B L t h i c k n e s s = 120 k m ; T p = 1 2 8 0 °C

0 50 100 150 200 O~oth (K in )

O d g i n a r y M B L t h i c k n e s s = 160 km ; T p = 1 2 8 0 °C

Fig. 12. Density of residual mantle rock, after total melt extraction in adiabatic upwelling asthenosphere, as a function of 6 (13 = 2, 3, 5, 10, 50) and MBL thickness (MBL thickness = 90, 120, 160 km); Tp = 1280°C.

3.1 - - T p = 1 2 8 0 oC

- T p = 1 3 2 0 oC

3 - - - - " T p = 1 3 6 0 ° C .....

g ,p =1,oo oc TP = 1 4 4 0 ° C . ; ]

. . . . . . . . . . T p = 1 4 8 0 oC • v > , 2 .9 : j v

| i , " 2.8 • ,'4

--~

2 .7 L . . . . '- '- ~_.- ~

2 .6 . . . . . . . i i i i l l l l

10 1 0 0

Depth (km)

Fig. 11. Melt density versus depth in the uprising mantle as a function of the potential temperature (Tp).

density of silicate melts exceeds 3.05 g / c m 2. To- ward shallower depths, density values drop and grad- ually decrease.

It must be noted that density differences between melts generated in the presence of more or less intense thermal anomalies are negligible at great depths and with low melt fractions, but increase toward the earth surface. Density decrease at shal- lower depths is greater for lower than for higher Tp (Fig. 1 1). However, if the density difference between melts generated at great and shallow depths, respec- tively, is computed at the same pressure, it is smaller than observed in Fig. 11, but appears to be amplified because of the presence of strong temperature and pressure variations.

74 F. Celia. A. Rapolla / Physics of'the Earth and Phmetarv Interiors 103 (1997) 63 84

The main solid phases in the mantle are olivine, orthopyroxene, clinopyroxene, and in addition, gar- net or spinel or plagioclase. Because the phases are solid solutions, their end-member components are treated individually (Appendix C: Eqs. (C 10), (C 11), (C 12), (C 13), (C 14) and (C 15)).

The calculation was performed by adopting a technique suggested by Niu and Batiza (1991b) and by using information previously obtained from the geotherms, melt fraction and melt composition. The results have been computed for three different condi- tions, i.e., absence of thermal anomaly ( T v = 1280°C; Fig. 12), intermediate degree of thermal anomaly (Tp = 1380°C; Fig. 13) and high asthenospheric tem- perature ( T v = 1480°C; Fig. 14). Three different val- ues of lithospheric thicknesses (90 kin, 120 km and

3,45

3,4

E 3,35

3,3

3,25 o

3,2

3,15

; # t " ~ . . . . : : :

~i E;=IO

20 40 60 80 100 120 140 Dopth (Km)

Originary MBL thickness = 90 k m ; Tp = 1380 °C

160

3 ,45 i

3,2 ~ ................... ~ :

3,15 2 i i __..Cil. 8 - 0 3,1 ~ " , ~ = 1 ° , , I , , I ,

40 80 120 160 200 Depth (l~n)

Originary MBL thickness = 120 k m ; Tp = 1380 °C

3,45 ,

3,4 - ~ ~ i i . . . . -

3,35 .... , ....

. . . . . . . i ! !1 c 3,25 i: 3,2

40 80 120 160 Depth (kin)

Originary MBL thickness = 160 !ml ; Tp = 1380 °C

Fig. 13. Density of residual mantle rock, after total melt extraction in adiabatic upwelling asthenosphere, as a function of B (B = 2, 3, 5, 10, 50) and MBL thickness (MBL thickness= 90, 120, 160 km); Tp = 1380°C.

3,45 ~-

3,4

~" 3,35 o ~,~ 3,3

3,25

g 3,2 3,15

3,1

3,05

............ Y ~iil : Y ~!atPtur~e°~,i!ei - a

40 80 120 160 200 Depth (kin)

Originary MBL thickness = 90 km ; Tp = 1480 °C

3,45 : : 3 , .............. I ;

3,25 " b

~ 3,15 3,1

3,05 ~ , I , , i , L , I , , 4 40 80 120 160 200

Depth (km)

Originary MBL thid,,ness = 120 krn : Tp = 1480 °C

3,45

3,4 i . ~

3,3 ~ ~ C

3,1 40 80 120 160 200

Depth (km)

Originary MBL thickness = 160 km: Tp = 1480 °C

Fig. 14. Density of residual mantle rock. after total melt extraction in adiabatic upwelling asthenosphere, as a function of 6 (6 = 2, 3, 5, 10, 50) and MBL thickness (MBL thickness=90, 120, 160 kin), Tp = 1480°C.

160 km) and several stretching degrees (2, 3, 5, 10 and 50) were considered (seen Section 2).

Figs. 12-14 show: (a) the vertical density distri- bution of the upper mantle peridotite in absence of thermal anomaly. The only significant density varia- tion are represented by the two abrupt steps corre- sponding to the mineral phase transitions (garnet- spinel-plagioclase peridotite), whereas the density changes due only to the effect of the compressibility and of the thermal expansion coefficient of the solid phases are negligible; (b) the vertical density distri- bution when mantle uprises. In this case, the abrupt density decrease above mentioned, is accompanied by a gradual but significant decrease within any mineral phase range and is given by both the phase transitions and the partial melting, that is the sum of

F. Celia, A. Rapolla / Physics of the Earth and Planetary Interiors 103 (1997) 63-84 75

the effect represented in Figs. 9-11 and to the gradual variation in percentage and phase composi- (a) o lion of the solid and liquid fractions.

The results, obtained using Eq. (C7) and Eq. ~ -so (C10) (Appendix C) are expected to be subject to small errors, ranging from 5 × l 0 - 4 to 5 × 10 -3 ~ -100

(Niu and Batiza, 1991a,b,c). Possible uncertainties in ~ .~so the estimation of the composition of the solid phases

- 2 0 0

may occur but are rather difficult to evaluate. In a -2~ conservative hypothesis of a 10% error in the rela- ~ -40

tive modal proportion of the mantle minerals, the ~ -~o 5 -so predicted densities for solid mantle may be affected 9 -100 c~

by errors ranging from +0.01 to 0.03 g / c m 3. -120

6. Gravity effects

Fig. 15 shows the amplitude of gravity anomalies related to different kinds of asthenosphere upwelling through a 130 km thick lithosphere. The first dia- gram (Fig. 15a) illustrates a simplified model of rising mantle with a low Tp value (1280°C) and describes a typical case of passive rifting. The sec- ond one (Fig. 15b) shows a simplified model of mantle doming under very high thermal anomaly conditions (Tp = 1480°C) and describes an active rifting mechanism. The density vertical distribution of upwelling mantle rocks was calculated as de- scribed in previous sections and is shown in Fig. 12b and Fig. 14b.

Densities of garnet peridotite, spinel peridotite and plagioclase peridotite are respectively 3.4 g / cm 3, 3.37 g / c m 3 and 3.28 g / c m 3. One of the aims of this study was to estimate the gravity effect controlled only by the density distribution of the uprising man- tle and its shape, whereas other features of a rifting model (i.e., amount of crustal thinning, uplift or subsidence) were not considered. As far as the lateral extension of the uprising mantle rocks is concerned, the shape of the models in Fig. 15 were chosen based on numerical models by Keen (1985), who predicted very broad lithosphere thinning in the case of high Tp (active rifting). Where Tp is low, thus indicating a passive rifting mechanism, mantle is expected to uprise in a much smaller area.

The gravity effect for different lithospheric stretching degrees (13 = 2,3,5,10) is shown in Fig. 15. Where asthenosphere rises to crustal depths, the

(b)

o

- I O0

- 2 0 0

-3oo

- Z 8

- 4 0

- 6 0

~, - 8 8

-lOO

-120

: - . . . . . . . . . . . . . . . . . . . . ~ , . . . . . . . ~ ~ ; S = "

i ; ,L"-~"i~, ' -_ - - - : : :~ ;

~ ) ~ crUst-mantle= O.S ~ ~ e - 0.12

Tp - 1 2 8 0 *C

Crust

s ne' .i I . , F~=~51 0 " ~ ~ ' Garnet petidoUte

. • • , . • . , • . , , , • , • r ' • 0 200 4 0 0 600 8 0 0 1 0 0 0 1 2 0 0

D i s t a n c e ( k r n )

............. . ................ ) i~ "~"'/;,l'l - - ".:~ ,

) ) = - - :-,':. i " ~ i ......... :, "~ "-"?"" FI . . . . . . = 10 b

) i ~ • i I) Ap c rus t -man t le = 0 .5

:: I i ) ) Ap c rus t -man t le = 0 .12

Tp = 1 4 8 0 =C

- - ~ 1 0 Garnet per idoUte

4 0 0 8 0 0 1 200 1 600 2 0 0 0 2 4 0 0 2 8 0 0 3200

D i s t a n c e ( k r n )

(c)

o ~ ..--i i

i !i x..-~ ~/ ] ~ ! i - .... .=Sa - J i ~i / - - - .=.b . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . - " - = ~ . . . . . . . . . . . . . - - s = 1 0 a ~ -300 :: - - - 13= 10b

) i i L~ Ap Cfgst-mantle = 0.12

• L r_p - 14so "c -2o E ~ Cru~ _

_~ [_ p = 3.37 Q / c ~ . . / ~ \ \ \ Spi.e, ~ridotit~ - 8 g ~=1000000000~ Garnet periclotJte

-120 g=2 0 2 0 0 400 600 800 1 0 0 0 1 2 0 0

D i s t a n c e ( k i n )

Fig. 15. Gravity effect of upwelling asthenosphere for a passive rifting case (Tp= 1280°C) (a) and for an active rifting case (T o = 1480°C) (b), at different 13 values (13= 2; 13= 3; 13= 5; 13= 10); Lithospheric thickness = 130 km. Profiles 5b and 10b indicate the gravity effect due only to the mantle rising, in absence of a density contrast between the upper mantle and the crust. Fig. 15c is equivalent to Fig. 15b except the shape of the astheno- sphere rise. It is equal to that in Fig. 15a (see text).

76 F. Celia, A. Rapolla / Physics q(the Earth and Planetary Interiors 103 (1997) 63 84

presence of a short wavelength positive anomaly, whose amplitude increases for higher 13 values, can be observed. It is due to the positive density contrast between the crust and the rising mantle and overlaps the long wavelength negative anomaly due to the lower density values of the upwelling asthenosphere relative to those for the surrounding mantle.

Fig. 15 indicates that changes in vertical density distribution and vertical doming extension, depend- ing on the features of the lithospheric thinning pro- cess, cause very strong differences in gravity anomaly amplitude and shape. The maximum amplitude of the anomaly for T o = 1280°C and with a density contrast between upper mantle and crust (0.5 g /cm3) , ranges from 60 mGal for 13 = 2 to 20 mGal for 13 = 10 (Fig. 15a). For Tp = 1480°C (Fig. 15b) it ranges from 150 mGal for 13 = 2 to 205 mGal for 13 = 10.

With high stretching factors, and when the pres- ence of a crustal layer is not taken into account, these values increase reaching 175 reGal (Tp= 1280°C and 13 = 10) and 440 mGal (Tp = 1480°C and 13 = 10).

The strong difference in amplitude between grav- ity anomalies due to upwelling mantle processes

related to active (Tp = 1480°C) and passive (Tp = 1280°C) mechanisms is more noticeable. These dif- ferences range from 90 mGal, for 13 = 2, to 260 (185) reGal, for 13 = 10.

Obviously, differences in amplitude above re- ported are due not only to the different density distribution within the upwelling mantle, but also to the differences in the morphology of the rising man- tle. Fig. 15c shows the same model described in Fig. 15b (Tp = 1480°C), in which, however, the shape of the mantle rise is the same as in Fig. 15a (Tp = 1280°C). Basing on the results from Keen (1985), this model is unlikely from a structural point of view, but it allows to evaluate that the differences in gravity anomaly amplitude, due only to the different vertical density distribution in the mantle rising, remain rather noticeable.

7. T w o cases o f a p p l i c a t i o n

Two different cases of regional gravity modelling, based on the method proposed here, are presented as application.

50 . . . . . . . . . . . . . . . . . . 50 n ~ = Observed and filtered profile (low pass filter; [ ~

v _~_ ~-~ . . . . cut-off wavelenght = 200 Km)" I ~

-100 ~ ~ ' ~ Calculated profile; -~ =:--~ - ~ [ -100

-150 ~"~ ~ ,~ , -150 = -200 ~ . ~ ~ -200

\~ ..... ~ ~ -250

Somali plate Ethiopian Plateau NW~ SSE Sudan

:,:Continental crust ~,,'~,.'Z-,,.<~,~,,.,.,,,¢'..,.,¢'~,.,~',',.".~,~-'.'-'.','p = 2.85 g / c m c ~ , ~ ' -2o ~ ; . : : , : , ~ , , . ~ , ~ , , _ , , ~ , , ( ~ < . . i ; ~ ; ~ . ~ . , . " - . . . . . , , , , -~o

E~ -40 Spinel l : lec id~ t l~ ,~ : ~ ; ~ , ; ~ o ~ gneous underplatlng bodies -40

- ~ o - ~ ~ ' ~ .... ~ .o7 ~ m a l - ~ o

-100- -100 -120 -120

a

3.4 ~ - - . . . . . . . . .

b \ 3.2 ~ j l ~ I

50 100

Fig. 16. Gravity model (a) of the lithospheric structure along a profile crossing the Ethiopian plateau. The vertical density distribution in the upwelling mantle, calculated as in Fig. 13a, is shown in diagram (b).

F. Celia, A. Rapolla / Physics of the Earth and Planetary Interiors 103 (1997) 63-84 77

The first case refers to the Ethiopian rift and plateau, which are considered an interesting example of region undergone lithospheric thinning and LAB uprising. Fig. 16a shows the gravity 2D model of the lithospheric structure along a profile crossing the Ethiopian plateau and rift, located between 35°05'E - 12°15'N, 4°35'N-40°E, and 2050 ' N-46°10'E. Fig. 15b shows the resulting vertical density distribution assigned to the upwelling asthenosphere.

An estimated uplift of about 1000 m in the area (Le Bas, 1971; Burke and Whiteman, 1973; Spohn and Schubert, 1982; Mohr, 1983) was adopted. Moho depth was constrained by several seismic data sets (Rapolla et al., 1995); the density and thickness of the igneous body, generated by the consolidation of part of the melt produced in the upper mantle by adiabatic decompression and underplating the crust were constrained by choosing values predicted by White and McKenzie (1988). Therefore, only the lateral extension of the mantle dome and the under- plating body were modelled by fitting observed and calculated gravity data, thus reducing the interpreta- tion ambiguity due to the equivalent source problem.

Gravity modelling supports the hypothesis of a hot plume beneath the Afar Depression and the Ethiopian Rift (White and McKenzie, 1988) and indicates a very high potential temperature (Tp = 1480°).

The second case (Fig. 17a) concerns the Tyrrhe- nian Basin (Celia et al., 1997). A 2.5 D gravity modelling was performed along a profile from the Tunisian margin of the African plate (36°N-9°30'E) and crossing north-eastward the Sicily Channel, the Tyrrhenian Basin, the Central Apennines and the Adriatic Sea (46°N - 17°E). The high frequency component due to the crustal sources was suppressed by using a low-pass filter; a not very long-wave- length cut-off (100 km) was adopted to avoid the suppression of a significant part of the gravity signal due to the very shallow mantle rise beneath the Central Tyrrhenian Basin. The density profile in the upwelling mantle was estimated by adopting a stretching value of 13 = 5 based on thermal evidence (Hutchinson et al., 1985; Wang et al., 1989; Kastens et al., 1988). An unstretched lithosphere 110 km thick was chosen (Suhadolc and Panza, 1989). A

~ ~ ~ - - Calc. profile (density distrib, o f the upwelling asthenosphere calculated for Tp = 1 :>80 *C); ~

150 Observed and filtered profile (low pass ~ .

- 100 t - - - - Calc. profile (density distdb, of the upwelling asthenosphere calculated for Tp = 1480 °C);

! ~ " ~ -22 Tunisia ~ DistanCeTy nian Basin(Km) 1 rn ~ A p p e r m i n e s Adriat ic

-4 ss~w ~ . . . . . . 250 500 Oceanic crust 750 1000

0

3,4

3,3

3,2

3,1

' ! ' ' . . . . I ' ' ' I ' ' '

temperature , J , , ~ I ~ J J I , totofile 40 8O 120 160 2O0

Depth ~ )

A - 2 0 E -~ * 4 0

~. -6o ¢~ - 80

-100

2 5 0

100 ,~m=~

- 5 0 I

- 1 0 0 ~

2

0

0

- 20 - 40 -6o ~ ° 8 0

- 1 0 0

Fig. 17. Gravity model (a) of the lithospheric structure along a profile crossing the Tyrrhenian Basin. The vertical density distribution in the upwelling mantle, calculated as in Fig. 12, is shown in diagram (b).

78 F. Celia. A. Rapolla / Physics ~?/'the Earth and Planetary lnteriolw 103 (1997) 63-,~4

detailed description of gravity modelling in the Cen- tral Mediterranean is in Cella et al. (1997).

The gravity model was calculated by introducing several constraints.

The boundaries between continental and oceanic crust were located as suggested by Duschenes et al. (1986) in the Tyrrhenian Basin ( = 300 km) and by Pasquale et al., 1995) in the Ligurian-Provencal Basin ( = 250 kin). The Moho depth was constrained with results inferred by many geophysical prospects along the whole profiles (Him et al., 1977; Steinmetz et al., 1983; Recq et al., 1984; Le Douran et al., 1984; Duschenes et al., 1986: Nicolich, 1989; Pascal et al., 1993: Scarascia et al., 1994).

Calculated gravity profiles (Fig. 17a) clearly indi- cate a significant difference in amplitude among the long-wavelength gravity anomalies generated by the asthenospheric dome beneath the Tyrrhenian Basin, depending on the thermal regime chosen. The anomaly generated by calculating the density distri- bution at T~, = 1280°C (Fig. 17b) fits fairly well the observed anomaly, whereas the amplitude of the anomaly based on higher potential temperature (Tp = 1480°C) is much smaller, with a difference of about 120 mGal. These differences are so large that a slightly better agreement (even if not optimal as in the low Tp case) between observed and calculated anomalies, for adopting a density distribution at T v = 1480°C could be obtained only by assuming Moho depth values in wide disagreement with geophysical evidence, or by assuming a too high density contrast (=0 .35 g / c m 3) between continental and oceanic crust. This value becomes still less probable in the hypothesis that the Central Tyrrhenian Basin is com- posed only by thinned continental crust diffusely intruded by oceanic material (Thisseau et al., 1986). Thus, the absence of deep anomalous thermal regime in the lower mantle and the presence of a passive rifting process driven by an external regime of ten- sile stresses can be expected.

8 . C o n c l u s i o n s

The results of the method, to estimate density changes in the upwelling mantle, can be summarized as follows:

- - The mineral phase transitions cause a strong density decrease at depths of about 70 and 30 km.

Obviously, the last phase transition is reached where the crust is sufficiently stretched to allow the pres- ence of mantle material to reach shallower depths.

- - The modal composition of residual mantle rock gradually changes with increasing melt fraction and extraction. Mineral assemblages stable at shal- lower depths have lower densities.

- - Both temperature and pressure have to be taken into account, because they have opposite ef- fects on density. Thus, calculations performed in this study allow to estimate the resultant of these oppo- site influences. However, it must be remembered that the pressure influence is more important for melt, because solid phases are much less compressible.

- - The lithospheric stretching has a strong influ- ence: where B increases, anomalous high temperature conditions, high melt fraction values and, conse- quently, lower mantle densities, are present at shal- lower depths. The combined influence of these fac- tors is shown by differences between mantle density profiles in the presence or absence of thermal anomalies.

- - It is shown that the higher the anomalous asthenospheric temperatures, the larger the mantle density decrease.

- - Pre-stretching lithosphere has a noticeable in- fluence. For increasing thickness, the temperature drops toward normal temperature profiles at depths greater than those in a thin lithosphere (Fig. 3) and, consequently, a density stabilisation on higher values occurs.

- - Differences in vertical density distribution within upwelling asthenosphere, existing among sev- eral thermal conditions in the mantle, cause signifi- cant variations in the gravity effect due to this kind of sources. Therefore, it is important to characterise and to constrain gravity models in extensional re- gions in order to reduce the interpretation ambiguity.

This study demonstrates that with different ther- mal conditions and different degrees of lithospheric thinning, gravity anomalies will show very large differences in amplitude and shape (see Section 6) and allows to give a measure of these effects. It confirms that gravity modelling of litho-astheno- spheric structure in areas affected by extensional dynamics, can be performed not only by assigning densities as the result of a simple fitting existing between observed and calculated gravity anomalies

F. Celia, A. Rapolla / Physics of the Earth and Planetary. Interiors 103 (1997) 63-84 79

(Searle and Gouin, 1972; Darracott et al., 1972; Yungul, 1976; Fairhead, 1976; Bermingham et al., 1983; Fairhead and Okereke, 1987; Izzeldin, 1987; Makris et al., 1991) but, in addition, taking into account the physical and petrological effects of ther- mal anomalies in the asthenosphere on vertical den- sity distribution within the uprising mantle.

Therefore, a gravity modelling constrained by the informations obtained as above described can give a contribution to recognize the features of the mecha- nism causing rifting processes and to estimate the degree of thermal anomaly in the mantle.

where g = gravity acceleration; o/f = thermal expan- sion coefficient; T=absolute temperature; Cp= specific heat at constant pressure; and z = depth.

The 'potential temperature' (Tp) represents the temperature that a fluid mass would have if it were compressed or expanded at a constant reference pres- sure.

In other words Tp is given by integrating Eq. (Al):

T p : T e x p ( - g a z ] , --c7j (A3)

Acknowledgements

Our best thanks is mainly due to Prof. D. McKen- zie for the decisive contribution to the software to apply his model, and to Prof. M. Fedi for his signifi- cant helping. We are also grateful to Prof. A. Cun- dari, for carefully reviewing the paper and for pro- ductive discussions, and to Prof. E. Burov for its precious suggestions. This research was financially supported by the Italian Research Council sub-pro- ject: 'Somali continental margins and the evolution of the Gulf of Aden and the north-western Indian Ocean', contract: CNR 90-1104, CNR 92-00818 and CNR 92-04759.

Appendix A

The mechanism of heat transport is controlled by the Peclet number (Pet):

vl Pe t = - - , (A1)

k where v = material velocity; l = length scale; and k = thermal diffusivity.

If Pe t >> 1, the advective heat transport domi- nates; instead, if Pe t << 1, the conductive heat trans- port dominates; If an average velocity of 100 m m / y r for spreading ridges and a 100 km thick lithosphere are assumed, the resulting Pe t is 30, thus confirming the advective transfer as the prevailing heat transport mechanism (McKenzie and Bickle, 1988).

The adiabatic temperature is given by:

Cp '

where g = g r a v i t y acceleration; z=dep th ; at .= thermal expansion coefficient of the solid part of the mantle; T = absolute temperature; Cp = specific heat at constant pressure.

Because the reference depth at which T =Tp is defined as the Earth's surface, Tp does not change within the upwelling mantle, only depending on the entropy changes of the mantle material.

As far as the calculation of the geotherms is concerned, to generate geotherms for TBL beneath old plates the heat flux trough the MBL has to match that out of the top of the TBL. The Rayleigh number for fixed temperature boundaries Ra T is given by:

gad3AT Ra T - - - , (A4)

KU

where g = gravity acceleration; a = thermal expan- sion coefficient; d = d e p t h of the layer; A T = temperature difference between top and bottom; K = thermal diffusivity; v = viscosity of the fluid.

The knowledge of Tp and of the MBL thickness is needed to calculate the heat flux F through the MBL:

K(Tp + 273) e °~g')/cp - 273 F = , (A5)

t

where t = MBL thickness; cp = specific heat; and K = thermal conductivity of the fluid.

It is, therefore, possible to estimate the heat flux ( F c) in the convective region which is not carried by the adiabatic temperature gradient:

F c = F - [ K ( oeg/cp)(Tp + 273) e(~t)/cP]. (A6)

80 F. Celia, A. Rapolla / Physics o[" the Earth and Planetary Interiors 103 (1997) 63-84

When the heat flux ( F c) is known, the expression corresponding to Eq. (15) is:

g ozd4 F~. Ra F - - - (A7)

KKv

The Nusselt number is defined as the ratio of the total heat flux to the conductive heat flux out of the mantle:

Fc,d Nu K A T ' (A8)

and can be empirically estimated as:

Nu = 0.184Ra °281 . (A9)

Substituting, are obtained:

Ra F = 0.184Ra~: 281, (AI0)

Ra T = 3.75Ra °78j . (AI 1)

If the origin is fixed at the base of the MBL, the stetched variable 2 is given by:

where z = depth interval. Thus, the increase in tem- perature AT from the base of the MBL is obtained by:

[ Fd lRa_.O.2,9[ 4 ) A T = ~ - - ~ } v 1 1 ' 8 4 + e ~.=0E a , ~ - " ,

(A13)

where a 0 = -1 ,84; a~ = -1 .18; a 2 = -0 .22 ; a 3 =

0.15; and a 4 = 0.01. It is, therefore, needed to estimate AT(0) taking

into account the requirement that the heat transported by convection and conduction down adiabatic gradi- ent at the top of the TBL is the same as that conducted through the MBL when Tp and the MBL thickness are given. The calculation can be per- formed by iteration and the potential temperature ATp(Z) is determined fromn Eq. (A14) and converted to the actual temperature.

It must be remembered that the vertical length scale of Tp variations does not interests for ridge upwelling. However, beneath old lithosphere, the vertical length scale of Tp variations within the convecting region corresponds to the thickness of the

TBL which must be estimated if the temperature profiles has to be calculated. It can be demonstrated that the T o length scale variation (6) and the Tp magnitude (A0) variations depend only on the vis- cosity of the solid material of the upper mantle (v) and of the viscosity of TBL (v o) (McKenzie and Bickle, 1988):

( p )1/4 \/ /) ,1/4/y0 ] 6 = A ' - - A 0 = B ' [ - - ] , (A14) P0 where A' and B' are constant determined by numeri- cal experiments and do not depend on v.

The already mentioned relationship between the lithosphere and the MBL thicknesses is obtained (Fig. 1) by calculating a depth on the adiabatic geotherm with an area delimited by the geotherm and the adiabatic upwelling curve, that is:

ie = t / l O 0 0 z , and Df( i + ie)

= [ ( i - l)1000z] + t , (115)

where

(116)

Therefore, the following equations can be given:

Td( i + ie) = (Tp + 273)e {['~g Ol~i+i,,)l/cp}_ 27,

(A17)

and

Tf( i + ie) = e {t~e DF(i+ ie)l/cP}

× Tp + 273 - KRaO.2~9

× ( 1 " 8 4 + ~ a ~ z - n ) - 2 7 3 ' n = 0 (118)

Therefore

Td( i) = rd ( i) - Tf( i ) ,

where i = 1 to nt, nt = ie + it.

F. Celia, A. Rapolla / Physics of the Earth and Planetary Interiors 103 (1997) 63-84 81

Finally, the lithosphere thickness will be given by:

i=2 ( D f ( i - 1 ) ) Tl=

2 n t = 1000 ?000

×{[Td(i) + T d ( i - 1)]/2}/Td(1), (A19)

where Tl = lithosphere thickness. Finally, it must be also observed from Eq. (AS)

and Eq. (A18), that the dependence of the tempera- ture at the base of the MBL on the viscosity of fluid mantle component and, consequently, of the litho- sphere and TBL thickness, is also notable (Fig. 2). This must be taken into account because of some uncertainties where viscosity values are assigned to the melt (Craig and McKenzie, 1986).

Appendix B

The method used by McKenzie and Bickle (1988), to calculate melt fraction and composition, consists in three steps:

(1) Calculation of the analytic expression describ- ing changes in solidus temperatures with pressure in agreement with experimental data. The expressions used are:

P = (T s - 1100)/136 + 4.968E

- 4 e [12E- 2(rs- 1100) l ' (B 1)

T l = 1736.2 + 4.343P + 1 8 0 t a n - l ( P / 2 . 2 1 6 9 ) ,

(B2)

where P = pressure in GPa; T~ = Solidus tempera- ture in °C; and T 1 = liquidus temperature in °C.

(2) Calculation of the melt fraction X(T,P) as a function of pressure and temperature. To calculate X(T,P) directly is quite difficult; it is first to use the requirement T = 0 when T = T~ and X = 1 when T = Tj, by defining a dimensionless temperature T':

[ T - ( T s + TI)]I2 r' = (B3)

T,-rs Then X (T') has to pass through ( - 0 . 5 , 0) and (0.5, 1). The following polynomial satisfies these condi- tions:

X - 0.5 = (T ' + T '2 - 0.25)

X(ao+alT '+a2T '2 . . . ) . (B4)

Results are in good agreements with most authors by using only two coefficients (a0=0.4256; a I = 2.988).

(3) Estimation of the melt composition as a func- tion of melt fraction and pressure. The relationship between X and T' and the absence of the pressure effect on the partial melting suggest the melt compo- sition would be mainly controlled by the melt amount and that the abundance of oxides changes as function of T' and, consequently, of X. The estimation of the composition of the batch (or point average) melt generated from a garnet peridotite was based on the Rayleigh's law and is given by the following equa- tion:

b C(X) =a +

[ ( I / n ) + 1]X

d × ( ( 1 - X) (l/n)+1) + X' (a5)

where X- - melt fraction and a,b,d,n = coefficients experimentally determined for each oxide.

Appendix C

As far as the influence of the temperature on the melt density is concerned, the method suggested by Lange and Carmichael (1987) is here adopted. It uses the double-blob Archimedean method for density measurement and considers the molar volume of a multi-component silicate having a linear dependence on melt composition. Thus:

V(T,1) = E x , ( r , 1 ) v i ( r , 1 ), (C l )

where V = molar volume; X i = mole fraction in the melt of the ith oxide; T (°C)= temperature. V/ is the partial molar volume of the ith oxide and it is given by:

V/(T,1) = Vi,Tr "~- - ~ ( T - TR) , ( C 2 )

where V/.rr = partial volume of the ith oxide at a

dVi reference temperature T a = 1400°C;

dT temperature derivative of the partial molar value. Thus, the density of the silicate melt is given by:

melt molecular weight p (T ,1) = (C3)

V(T,1)

82 F. Celia, A. Rapolla / Physic.s' o['the Earth and Planetary Interiors 1(13 (1997) 6_¢...84

The estimation of the pressure effect on melt density was performed by following the method by Lange and Carmichael (1987), instead of that by Herzberg (1987a,b) because the first one has been confirmed by experimental results (Agee and Walker, 1988). In this model the isothermal compressibility of silicate (B T) linearly depends on the melt compo- sition:

f iT = E X i f i i ,T , (C4)

where X i is the mole fraction of the oxide i and 13 i is the isothermal compressibility of the oxide and is given by:

d fii ° fi,,T = fi0 + ( c 5 )

d T ( T - TR)

Values of Bi. T and d f i i ° / d T are given by Lange and Carmichael (1987). The melt compressibility (B T) is related to the bulk modulus:

K T = 1~f iT , (C6)

where K T is the isothermal bulk modulus at the temperature of interest and atmospheric pressure.

Thus, the pressure effect on the density of silicate melts is given using the third-order Birch- Murnaghan equation of state:

p = 1 . 5 K T [ R 7/3 - Rs/3 ]

×[1 + 0 . 7 5 ( 4 - K ' ) ( R 2 / 3 - l ) ] , (C7)

where

d K o K' - --~ 4 for silicate melt

d P (Lange and Carmichael, 1987), (C8)

and

R = V ( T , 1 ) / V ( T , P ) = p ( T , P ) / p ( T , l ) . (C9)

p(T,1) is calculated by Eq. (C3) and p ( T , P ) is evaluated by successive iterations of the pressure calculated by Eq. (C9) to the value of interests.

As far as the temperature effect on the density of a mineral solid is concerned, the density at a pressure of 1 Kb is given by (Niu and Batiza, 1991c):

p(T,1) = E X i p i ( T , 1 ) , (Cl0)

where X i is the end member i of the solid solution and pi(T,1) is the density of the end member at P = l atm and T = 25°C:

p i ( r , l ) = p°sx. + d p ° / d r ( T - 25)

+ d 2 p ° / d T 2 ( T - 25) 2. (C11)

o p25~c is referred to the densities of end-members at a pressure of 1 atm and T = 25°C: d~'p°/dT 2 and d p ° / d T ( T - 25) are the first- and the second-order temperature derivatives of the end-members densi- ties. Values for 0 d Z p ° / d T 2 and d p ° / d T ( T - P25°C , 25) are given in Niu and Batiza (1991b).

The pressure influence on the density of mineral solid phases is described by the Birch-Murnaghan third order equation of state (C8) in which the isothermal bulk modulus at the temperature of inter- est and atmospheric pressure (Kv) is given by:

dK~! K T = K~ + d T ( T - T,) ' (C12)

where K4 is the bulk modulus at the reference temperature and atmospheric pressure.

Since calculations were made with the bulk modu- lus K T of a solid solution, the use of the Voit-Re- uss-Hill mixing model was preferred rather than the ideal mixing model. In this case, K T for a solid solution is given by:

K v = ( K v + K R ) / 2 , (C13)

where

K v = E V i K i , v , (Ct4)

KR = [ S , ( V j K i , T ) ] - 2 , (C15)

and V i and Ki , T a re the molar volume and the bulk modulus of the and-member i, respectively.

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