Electronic appendix A
Article:
Pilet
et al.
(2011)
Monte
Carlo simulations of metasomatic enrichment in the lithosphere and implications for the source of alkaline basalts
1
Electronic Appendix A: A sample calculation of the trace
-
element evolution
of
metasomatic liquid and cumulates
during the differentiation of metasomatic
melt
within the lithosphere
Here we provide an example of a Monte Carlo simulation following the mode
l presented in Fig. 1 (here
, and throughout this Appendix,
references to specific
figures and tables are to those in the main article). As shown in Fig. 1, the model is divided into two parts:
(1) G
eneration of a metasomatic liquid (liq)
by low degrees of
melting of a source that varies in composition from E
-
DMM to DMM;
(2) Differentiation of this low
-
degree melt and the formation of anhydrous and hydrous cumulates.
In practice, the calculation follows the schematic model of vein formation illustrated in
Fig. 4 of the article. First, we calculate the composition of the initial
metasomatic liquid (L
0
), then the anhydrous cumulate composition as well as the composition of liquid L
1
, and finally the composition of hydrous cumulates
and the residual liquid L
2
.
In the following example of the calculation, the parameters that are allowed to vary are shown in yellow boxes, parameters that are held constant are not high
-
lighted, while the cal
culated values are listed in gra
y boxes.
Part 1
-
Composition of the i
nitial metasomatic liquid L
0
The composition of the initial metasomatic liquid L
0
depends on the source composition, the
degree of partial melting, the
solidus phase proportions
,
and the
proportions of phases that enter the melt.
The initial mantle sourc
e composition was calculated by varying the proportion of E
-
DMM to DMM in
the source along with the trace
-
element compositions
of these two end
members (Workman
and Hart, 2005):
Fraction of E
-
DMM in the source
(
can vary
from 0 to 1)
; in this example, th
e fraction of DMM is 0.481
Composition of E
-
DMM and DMM end
members (Workman
and Hart, 2005):
Calculated source:
The initial metasomatic liquid (shown in Fig.
5
a as L
0
) is calculated using the non
-
modal aggregate fractional
melting equation [C
l
/C
o
=
(
1/F
)
(1
–
(1
–
PF/D)
(1/P)
)]
where C
l
= the aggregated liquid, C
o
= the source composition, F = the melt fraction, P = the bulk distribution coefficient for those phases entering the melt,
and D = the bulk distribution coefficient at the beginning of melting;
the
source mineral
modes are from Table 1
and melting modes
are from Walter (1998)
Electronic appendix A
Article:
Pilet
et al.
(2011)
Monte
Carlo simulations of metasomatic enrichment in the lithosphere and implications for the source of alkaline basalts
2
and are
for partial melting at 3 GPa
(they have
been slightly modified to take into account the addition of minor sulphid
e in the source;
Table 1). The
D
min/liq
s
for ol
ivine
(ol)
,
orthopyroxene (
opx
)
,
clinopyroxene (
cpx
)
,
garnet (
gt
)
, and sulphide are reported in Table 2. The degree of partial melting is a free parameter
th
at can vary between 0.5% and 1.8
% according the distribution shown in Fig. 2b.
D
egree of partial mel
ting (can vary
from 0.005 to 0.01
8
)
.
Calculated bulk distribution coefficients D and P:
Composition of the initial metasomatic liquid (
corresponding to liquid
L
0
plotted
in Fig.
5a
) using F= 0.0077 and the mantle source composition calculated
above:
Part 2
–
Composition of
model
anhydrous and
model
hydrous cumulates
1) Composition of anhydrous cumulates and residual liquid L
1
The compositions of the anhydrous cumulates and the residual liquid (reported as L
1
in Fig. 5b) are calculated using the fr
actional crystallization equation, the
composition of the initial metasomatic liquid (L
0
), D
min/liq
s
reported in
Table 2, and proportions of clinopyroxene, garnet
, ol
ivine, orthopyroxene
, and trapped
liq
uid
that are allowed to vary within bounds that are c
onstrained by experimental petrology and the modes of xenolithic metasomatic vein material.
Mineralogical composition of the anhydrous fractionating assemblage:
Free to vary between 0
and 0.3
With garnet = 0.
122
, olivine can vary between 0.0
28
and 0.0
66
†
Free to vary between 0 and 0.0
8
Free to vary between 0 and 0.05
Fraction of cpx in the cumulate assemblage fixed by the expressi
on: 1
–
sum of all other phases
†
High
-
pressure expe
riments performed between 1.5 and
1 GPa suggest that the anhydrous cumul
ate assemblage formed at such
pressures is clinopyroxene + minor olivine
(Nekvasil
et al.
,
2004; Pilet
et al.
, 2010
). However, experiments at higher pressure
s indicate
that the cumulate assemblage
that crystallizes between ~3 and 2.5 GPa is clinopyroxene +
garnet
(Hauri
et al.
,
1994). Therefore,
as indicated in Fig. 4,
we include a link between garnet and olivine
abundances
in the anhydrous cumulate assemblage to mimic this pressure
effect.
Proportion of residual liquid
(ƒL
1
)
after anhydrous cumulate frac
tionation
: This parameter is free to vary between 0.65 to 0.45.
Bulk d
istribution coefficients (D) for the anhydrous cumulate ass
emblage. The individual mineral
-
melt
D
s are reported in Table 2
:
Electronic appendix A
Article:
Pilet
et al.
(2011)
Monte
Carlo simulations of metasomatic enrichment in the lithosphere and implications for the source of alkaline basalts
3
Composition of the residual liquid after
anhydrous cumulate fractionation (Liquid L
1
in Fig.
5b
). The trace
-
element
composition of this liquid is calculated using
the
fractional crystallization
equation,
C
l
/C
o
=F
(D
-
1)
:
Composition of the
model
anhydrous cumulates (corresponding to the
model
an
hydrous composition plotted in Fig.
5c
)
:
2) Composition of the
model
hydrous cumulate assemblage and the residual liquid L
2
While the fractionation of hydrous cumulates is carried out between ƒL
1
(the end point of anhydrous fractionation) and ƒL
2
, the
plotted composition of the
hydrous cumulates (e.g., Fig. 5c) is calculated between ƒmax and ƒL
2
; both ƒmax and ƒL
2
are free variables (with constraints).
f
max
Free to vary between
ƒ
after anhydrous cumulate formation (
i.e. ƒ
L
1
,
in this example 0.595) a
nd this value minus 0.3.
f
L
2
Free to vary between
(ƒ
max
–
0.1) and 0.2
with the condition that ƒ
ma
x
–
ƒL
2
>0.1
.
For
the formation of
the
hydrous cumulate
s, the fractionating
assemblage was separate
d
in
to
two parts
(#1 and #2)
in order
that
plagioclase a
nd zircon
crystallization be limited to the most differentiated and lowest temperature liquids. The transition between these two slightly different mineral assemblages
was fixed at ƒ = 0.4 (note that in some cases, ƒL
2
was higher that 0.4, so the second mi
neral assemblage was not used in the calculation).
Hydrous fractional assemblage (#1) between ƒL
1
and ƒ = 0.4 (see Fig. 4)
:
Free to vary between 0 and 0.5
F
ree to vary between 0
and
0.
03
F
ree to vary between 0 and 0.01; r
utile plus ilmenite is limited
to
a
max
imum of 0.01
I
lmenite i
s free to vary from 0 to (0.01
–
proportion rutile); i
n this example il
menite can
varied between 0
and
0.00
44
Free to vary between 0 and 0.02
Fr
ee to vary between 0 and 0.0005
Free to vary between 0 and 0.0
05
Free to vary bet
ween 0 and 0.0
01
Free to vary between 0 and 0.0
2
Free to vary between 0 and 0.
10
*
Fraction of amphibole in the cumulate assemblage fixed by the expressio
n: 1
–
sum of all other phases
Electronic appendix A
Article:
Pilet
et al.
(2011)
Monte
Carlo simulations of metasomatic enrichment in the lithosphere and implications for the source of alkaline basalts
4
*
As indicated in the main text, the model includes the addition of ph
logopite associated with trapped liquid to simulate the modal and cryptic metasomatism obs
erved at the
periphery
of
metasomatic veins (see Fig.
4
b). The proportion of phlogopite (representing veinlets in Fig.
4b) relative to
trap
ped
liquid (representing in
terstitial glass and metasomatism in
the
reaction
zone, Fig.
4b) i
s assumed to be 1: 3 (i.e.
,
the proportion of trap
ped liquid equal three times the
proportion of phlogopite in the hydrous vein).
Hydrous fractional assemblage (#2) between ƒ = 0.4 and ƒ =
L
2
:
Free to vary between 0 and 0.5
F
ree to vary between 0 and 0.01;
rutile plus ilmenite
is limited to a maximum of 0.01
Ilmenite i
s free to vary from 0 to (0.01
–
proportion rutile); in this example ilmenite can
varied between 0 and 0.0
073
Free to vary
between 0 and 0.02
F
ree to vary between 0 and 0.018
Fr
ee to vary between 0 and 0.0005
Fr
ee to vary between 0 and 0.0005
F
ree to vary between 0 and 0.006
Free to vary between 0 and 0.02
Same constraints on phlogopite as
in
the
hydrous fract
ionating assembl
e #1
Fraction of amphibole in the cumulate assemblage is fixed by the expressi
on: 1
–
sum of all other phases
a) Composition of the liquid at
ƒ
max
(f
rom
ƒ
= 0.
5
95 t
o ƒmax)
In the present example,
ƒmax
is hi
gher that 0.4 but lower than ƒ
after the fract
ionation of
the
anhydrous cumulate
s (ƒL
1
)
. So, the first step is to calculate the
composition of the residual liquid at
ƒmax
using hydrous assemblage #1.
The bulk Ds are calculated with the phase proportions from hydrous assemblage #1 and the D
min/liq
s
reported in Table 2
:
* For Ti, the bulk D corresponds to an assemblage without rutile, ilmenite, and sphene. For these three phases, stoichiometric TiO
2
contents are used to calculate the effects
of fractionation on liquid Ti contents (see discussion b
elow).
Composition of the residual liquid at ƒmax (ƒ = 0.409)
:
** The Ti content of the residual liquid was calculated in two steps. First, the Ti content of rutile, ilmenite and sphene weighted by their phase proportions was subtract from
the initial
liquid composition; second, the fractional crystallisation equation
was used along with
the remaining phases and their Ti
D
’s to calculate the
final
TiO
2
content of the
residual liquid.
Electronic appendix A
Article:
Pilet
et al.
(2011)
Monte
Carlo simulations of metasomatic enrichment in the lithosphere and implications for the source of alkaline basalts
5
b)
Model h
ydrous cumulate and residual liquid compositions produced
at ƒ
=
0.4
The bulk distribu
tion coefficients are the same as
those used for the calculation of the residual liquid at
ƒmax
(the fractionating assemblage is the same).
Composition of residual liquid at ƒ = 0.4:
Hydrous cumulate composition produced
between ƒmax and 0.4
(in this example ƒmax = 0.409)
:
T
he composition of amphibole present in the cumulate assemblage is calculated using the following equation: C
amph
= C
cumulate
/
[%amph +
(%min_1
×
D
min
_1
/liq
/
D
amph/liq
)
+...+
(%min_n
×
D
min_n/liq
/D
amph/liq
)
] where %min_1 to %min_n
are
the proportions of minerals 1 to n
in the cumulate
assemblage
and D
min
_1
/liq
to
D
min
_n
/liq
are the corresponding
mineral
-
liquid
distributio
n coefficients
.
Amphibole composition in the model hydrous cumulate assemblage produced
between ƒmax and ƒ = 0.4
:
#
Ti in amphibole is calcula
ted using the same equation as
described previously, but takin
g into account the presence of cumulate
Ti
-
rich phase
s assumed to have
stoichiometric TiO
2
contents
.
c)
Model h
ydrous cumulate and res
idual liquid compositions produced
at ƒ
L
2
For
ƒL
2
< 0.4, the second hydrous cumulate assemblage is used to calculate the distribution coefficients
.
The bulk Ds are calculated with hydrous assemblage #2 and the D
min/liq
s
reported in Table 2
:
* See pre
vious comment on Ti calculation.
+
For Zr and Hf, the
bulk
D
s correspond
to
the assemblage without zircon; stoichiometric Zr and estimated Hf contents of zircon
(see discussion in the main article) are used to estimate the effects of zircon fraction on the
Zr and Hf concentrations in the liquid.
Composition of residual liquid at ƒ
L
2
(liquid L
2
in Fig. 5b):
Electronic appendix A
Article:
Pilet
et al.
(2011)
Monte
Carlo simulations of metasomatic enrichment in the lithosphere and implications for the source of alkaline basalts
6
** See previous comment on Ti
liquid
calculation.
++
Zr and Hf contents of
the residual liquid are
calculated in two steps.
First, the Zr and Hf co
ntents in z
ircon
(Table 2)
are
subtracted from the initial liquid composition; second
,
the fracti
onal crystallisation equation i
s used to calculate the effect of fr
actionally removing the remaining solid phases
.
Model Hydrous cumulate composition produced
between ƒ = 0.4 and ƒ
L
2
(in this example ƒ
L
2
= 0.266):
Amphibole composition in the model hydrous cumulate assemblage produced between ƒ = 0.4 and ƒ
L
2
:
d) Mean
model
hydrous metasomatic v
ein composition produced between ƒmax
and
ƒ
L
2
The compositi
on of the hydrous cumulate assemblage produced between
ƒmax
and
ƒ
L
2
is calculated using the weighted averages of the cumulate phases
produce from
ƒmax
to
ƒ
=
0.4
and from
ƒ
=
0.4
to ƒ
L
2
.
Model hydrous cumulate composition produced between ƒmax and ƒ
L
2
(co
rresponding to hydrous cumulate assemblage plotted in Fig. 5c):
The composition of amphibole in the hydrous cumulate
s
produced between
ƒmax
and
ƒL
2
is
calculated using the
amphibole produce
d
from
ƒmax
to
ƒ
=
0.4
and
the
amphibole produced fro
m ƒ
=
0.4
to ƒL
2
weighted as function of
their proportions in each interval.
Amphibole composition in the model hydrous cumulates produced between ƒmax and ƒ
L
2
(corresponding to the amphibole composition plotted in Fig. 6):
Part 3
-
Composition of
model
metasom
atized hydrous lithosphere
(hydrous metasomatic veins + cryptic enrichment in
the peridotite
surrounding
the vein
s
)
Electronic appendix A
Article:
Pilet
et al.
(2011)
Monte
Carlo simulations of metasomatic enrichment in the lithosphere and implications for the source of alkaline basalts
7
As indicated in the main text, we model the potential cryptic and modal metasomatic enrichment in peridotite associated with metasomatic ve
in
s by including
phlogopite with
the hydrous cumulates and trapping residual liquid in
the peridotite surrounding the veins (see Fig. 4
). The addition of trap
ped
liquid to
the
hyd
rous cumulates yields
the composition of
metasomatized hydrous lithosphere
pl
otted in Fig.
5d
.
Fraction of residual liquid trapped in surrounding peridotite:
This p
arameter is free to vary from 0 to
0.3 but is linked to the proportion of phlogopite
(in this example: 0.06) in the hydrous vein by a ratio of phlogopi
te to trapped residual
liquid of 1 to 3.
The composition of
metasomatized hydrous lithosphere
is calculated using the proportion
s
of trap
ped
liquid
(L
2
) and
hydrous cumulate
s (those produced
between ƒmax and ƒL
2
).
Composition of the model
metasomatized
hydrous lithosphere
(plotted in Fig. 5d):