Geopkys. J. Int. (1995) 123,59-70
Numerical modelling of melting processes and induced diapirism in
the lower crust
D. Bittner' and H. Schmeling2
Jean-Paul-Str. 21, D-95444 Bayreutk, Germany
'lnstitut fur Meteorologie und Geophysik, Feldbergstr. 47, D-60323 Frankfurt am Main, Germany
Accepted 1995 March 27. Received 1995 January 31; in original form 1994 June 27
SUMMARY
The processes of partial melting and magmatic diapirism within the lower crust are
evaluated using a numerical underplating model. Fully molten basalt ( T = 1200°C) is
emplaced at the Moho beneath a solid granite ( T = 750OC)in order that a melt front
grows into the granite. If diapirism does not occur, this melt front in the granite reaches
a minimal depth in the crust before (like in the molten basalt) crystallization takes
place. The density contrast between the partially molten granite layer and the overlying
solid granite can lead to a Rayleigh-Taylor instability (RTI) which results in diapiric
rise of the partially molten granite. Assuming a binary eutectic system for both the
granite and the underplating basalt and a temperature- and stress-dependent rheology
for the granite, we numerically solve the governing equations and find (a) that diapirism
occurs only within a certain but possibly realistic range of parameters, and (b) that if
diapirs occur, they do not rise to levels shallower than 15 or perhaps 12km. The
growth rate depends on the degree of melting and the thickness of the partially molten
layer, as well as the viscosity of the solid and the partially molten granite. From a
comparison of the growth rate with the velocity of a Stefan front it is possible to
predict whether a melt front will become unstable and result in diapiric ascent or
whether a partially molten layer is created, which remains at depth. We carry out such
a comparison using our thermodynamically and thermomechanically consistent model
of melting and diapirism.
Key words: diapirism, granites, melting processes, phase-driven convection.
INTRODUCTION
While the occurrence of granitic intrusions at both shallow
(Pitcher & Cobbing 1985) and deep (Ocala & Myer 1972;
Schwartz et af. 1984) crustal levels has been described for
many regions, the physical processes of their formation, rise
and emplacement are not well understood. In principle, large
amounts of felsic melts in the lower crust can be produced by:
(1) a thickening of the crust in thrust regions (Buck &
Toksoz 1983) associated with heating due to thermal
relaxation;
(2) a regional increase of heat flow from the mantle
(De Yoreo, Lux & Guidotti 19891, due to upwelling of
asthenosphere (Bird 1978);
(3) basaltic underplating (Huppert & Sparks 1988);
(4) internal heating due to anomalous distribution of radioactive elements at mid-crustal depths (Lathop, Blum &
Chamberlain 1994).
The amount of granitic melt due to basaltic underplating
depends mainly on the volume of the underplating body. From
0 1995 RAS
a simple 1-D analysis one can derive that basaltic magma
intruding at Moho depths has the potential to partially melt
the overlying felsic crust up to a thickness comparable to that
of the basaltic layer (see also Bergantz 1989). In the case of
lithospheric root delamination (Bird 1979) the asthenosphere
might rise up to levels as shallow as the Moho, inducing
extensive melting of the lower crust (Lorenz & Nicholls 1984;
Matte & Nicolls 1986).
Different mechanisms of melt removal from the source region
and further ascent of felsic magma have been proposed.
Viscosities of felsic magmas are usually too high to allow for
significant distances of rise by segregation (McKenzie 1985;
Wickham 1986). However, segregation through vein-like networks might help to accumulate the melt within the upper
part of an initially partially molten granitic body, as observed
in the, Ivrea zone (Oncken 1993, private communication).
Models of diapiric ascent of fluid spheres (Stokes' flow) into
a crustal layer with temperature-dependent viscosity (Mahon
& Harrison 1988) or a non-Newtonian rheology (Weinberg &
Podladchikov 1994)show that the shallowest possible emplacement depth is about 15 km. Somewhat shallower emplacement
59
60
D. Bittner and H. Schmeling
depths were found by Reuther & Neugebauer (1987) and
Kukowski & Neugebauer (1990) who used a rather weak
temperature-dependent viscosity of the crust. Stoping may be
a viable mechanism, but it can explain only small distances of
magmatic rise (Oxburgh & McRae 1984). Dyke transport
has been shown to be a very efficient ascent mechanism by
Clemens & Mawer (1992), while Rubin (1993) argues that
dyke propagation might be inhibited due to high viscosity of
felsic magma.
Most of the above-mentioned approaches do not consider
melting/solidification processes during the rise. They also
assume simplified rheologies of the ductile granitic crust. In
the present study we focus on a thermodynamically and
thermomechanically consistent model of granitic plutonism
due to basaltic underplating. Of particular importance is the
rheology of the solid and partially molten granite. NonNewtonian rheology laws based on laboratory experiments
(Kirby & Kronenberg 1987; Shaocheng & Pinglao 1993)
are used.
If a granitic unmolten crust is underplated by a molten
basaltic layer, a melting front migrates into the crust. In the
absence of diapirism this front reaches a minimum crustal
depth before solidification of both the granitic and basaltic
layers occurs. During this process the density of the partially
molten granitic layer is less than that of the overlying crust,
giving rise to the possibility of a Rayleigh-Taylor instability
(RTI) with subsequent diapiric rise of the granitic magma
body. The growth rate of the RTI will depend on the degree
of melt, the thickness of the partially molten layer and the
viscosity of both the partially molten layer and the solid
overburden. These parameters may be estimated from 1-D
Stefan-problem-like analyses of the underplating problem
(Bergantz 1989) and be used to estimate the growth rate. This
growth rate can be compared with the velocity of the penetrating or retreating Stefan front. From such comparisons it should
be possible to predict whether a melting front penetrating the
base of the crust will become unstable and rise as diapirs, or
whether it will stay near the basaltic layer. We carry out such
a comparison using the full solutions of the thermodynamicalthermomechanical problem of melting and diapirism.
THEORETICAL F O R M U L A T I O N
In the formulation of our binary, two-phase, multirock model
we distinguish between N different rock types, each of which
may have different physical properties and may undergo
melting according to different binary phase diagrams. Two
different rock types will be used in our model. The basic
underplating rock will be referred to as basalt while the acidic
one above may be called granite. The governing equations
describe the conservation of mass, chemical components,
momentum and energy:
+ v (pu)= o
x= 1 ( S i + M ’ ) = l ,
ap/at
i = 1 ...N
o=-VP+-q
aT/at
ri
(1)
-+axj
a axj axi
+ (UV)T=
+pg,
(3)
&j)
+
“ ~ ~ - ( p ~ ~ , ~ / p C , ) ( a ~ /(a~tv ) M ) .
The definitions of variables are given in Table 1
(4)
Table 1. Variables.
E :=energy
a := thermal expansivity
T :=temperature
P :=pressure
p :=density
X :=mass in percent
v :=velocity
M :=liquid mass in percent
q :=viscosity
S :=solid mass in percent
:=1 ... N number of rock types
Tsol :=solidus temperature
I
Tliq :=liquidus temperature
g := gravitational acceleration
PO :=min(psolid.,Pliqud
L :=latent heat
k :=thermal diffusivity
ZII
:= 2nd invariant of deviatoric
stress tensor
In the momentum equation inertial terms are neglected
(high Prandtl number approximation). The Boussinesq
approximation is used, i.e. density is kept constant except for
the buoyancy term.
The equation of state is given as
=
C
i=I
(pi.solidSi + pi,liquidh,ii)
-p o c t ~ ,
(5)
...N
The viscosity for solid and partially molten rock with up to
10 per cent melt is assumed to be stress- and temperaturedependent:
(6)
q = Az:,-”exp(E/RT).
At higher melt fractions ( M > 10 per cent) an empirical
rheological law depending on the melt fraction is used
(Pinkerton & Stevenson 1992):typo: left paranthesis missing
q = ~ ~ e x p ( 2+. 5[(I
-M ) / M ] O . ~ ~ ) (
1- M ) )
(7)
The amount of melt M = M ( T , Kol,&,,L) is a function of
temperature, solidus and liquidus temperature and the latent
heat. It is calculated for a fully binary or a eutectic system
using analytical equations for the solidus and liquidus curves
(Paufler 1981) and the lever principle.
N U M E R I C A L FORMULATION
While the heat capacity and the thermal conductivity k
are different for granite and basalt, they are assumed to
be temperature-independent. In the model we assume no
multiphase flow or segregation.
A finite-difference (FD) code with uniform grid size is used.
For solid rock an explicit FD formulation of the heat eq. (4)
is used (forward in time, central in space). If melt is present,
an explicit formulation of the time derivatives is no longer
possible because of the implicit form of M ( T ) . An iteration
algorithm has to be used: first the heat equation is solved
without melting terms, to get a lower or upper bound for the
temperature; then eq. (4) and M = M ( T , KO,, L) from the
phase relation are solved iteratively to obtain the temperature
at the new time step.
The conservation of momentum (eq. 3) is solved by a streamfunction approach which fulfils also the conservation of mass
(eq. 1). The mass transport of different rock types is simulated
by markers. The movement of these markers is calculated by
a Runge-Kutta algorithm of fourth order.
A problem in the numerical calculation of this underplating
model arises from the large viscosity contrast between liquid
basalt (q < lo5Pas) and solid or partially molten granite
(q > 1014Pas). Accordingly, the Rayleigh numbers in the
basaltic and granitic regions might differ significantly, and
xi¶,
0 1995 RAS, GJI 123, 59-70
Modelling of melting processes and diapirism
61
viscosity of granite
23
n
F
21
n.
--
0
Q____*
._
15
.-3
-power-law
1 MPa
-power-law
10 MPa
(II
m 19
CI
>r 17
+.’
Haplogranite real
power-law 0.1 MPa
-melt
in model
> 13
11
9
520
640
760
880
temperature [“C]
Figure 1. Viscosity of granite as used in the models. In the first region where T < T,,,,, a power-law rheology is used (eq. 6 ) . Here the viscosity
calculated for stresses of 0.1 MPa, 1 MPa and 10MPa are shown. The stresses created in the model are less than 10MPa. In the second region
where T z T,,,,, the viscosity is calculated from eq. (7) for melt concentrations ( M ) greater than 10 per cent. Since within the present resolution,
the Rayleigh numbers larger than 105-106 lead to numerical instabilities, we artificially increase the viscosity of the highly molten regions and do
not use the data from Dingwell, Knoche & Webb (1992) shown as a dashed curve.
values of Ra > 10” are easily possible within a molten basaltic
layer of several kilometres thickness. Since within the present
resolution Ra numbers larger than 10s-106 lead to numerical
instabilities, we artificially increase the viscosity of the highly
molten regions.
The rheologies chosen for granite are shown in Fig. 1.
Below melt concentrations of 10 per cent the constants
A = 6 x 10’ MPa” s, n = 3.2 and E = 144kJ mol-’ are used for
granite (based on Kirby & Kronenberg 1987). Effective viscosities are shown for flow stresses of 0.1-10MPa. As can be
seen in the diagram, at stresses consistent with diapiric dynamics (- 10 MPa)the viscosity in the granite may d r i p to ialues
as low as 1OI6 Pa s. At melt concentrations above 10 per cent,
eq. (7) is used with qo = 5 x 1014Pasfor granite (Fig. 1, curve
‘model’) and qo = loi3 P a s for basalt (not shown). The values
of qo have been chosen arbitrarily in order to ensure numerical
stability in the liquid regions. This high viscosity implies that
the time-scale in the liquid rock is too long (i.e. the number of
overturns per time will be smaller than that in natural systems),
but we can simulate the mechanical behaviour of the ductile
granite (up to 10 per cent melt) and in particular the rise of
the diapirs with some confidence. For comparison, an experimentally derived curve for molten granite
( M > 30 per cent)
is shown (‘haplogranite’).
M-T relation Rapakivi-granite
+P
= 5 kbar
--S-..P = 2 kbar
0
20
40
60
80
1 00
melt proportion M (wt. %)
Figure 2. M-T relation of Rapakivi granite ( P = 2 kbar and P = 5 kbar with 2.8 per cent H,O) after Nekvasil (1991). The effects of the different
components of the system can be seen as curve segments with different slopes.
0 1995 RAS, GJI 123, 59-70
62
D. Bittner and H . Schmeling
PHASE DIAGRAMS A N D DENSITY
VARIATIONS U S E D I N THE MODEL
Lower crustal acidic or basic rocks may exhibit large variations
in melting behaviour, but can be generally characterized by
eutectic systems. The actual solidus and liquidus temperatures
and the eutectic point depend on pressure, water content, C 0 2
content, and composition. Since we do not intend to simulate
particular petrological systems, the actual choice of melting
parameters is not important, as long as the general distinction
between acidic (granite) and basic (basalt) rocks is made.
Melting may control the dynamic and thermal processes in
the lower crust by influencing density and by the release of
latent heat. Thus it is essential to know the melt fraction-temperature ( M - T ) relation. This relation can be derived from
experiments or phase diagrams of multicomponent systems.
Typical granite systems can be characterized by eutectic melting
phase diagram
0"
1100
2
1000
0
Y
3
c
,
900
Q)
800
Q)
c,
700
0
0,2
0,4
0,6
0,8
1
X (composition)
M-T relation
(i.e. increasing the melt fraction at constant temperature)
followed by an increase of melt with increasing temperature.
As an example, Fig. 2 shows the melting behaviour of Rapakivi
granite (Nekvasil 1991). The effect of the different components
of the system can be seen as curve segments with different
slopes.
To model the essential features of experimental eutectic
melting curves (i.e. horizontal branches of different lengths
followed by an increasing M-T curve) it is sufficient to use a
eutectic binary system. We assume that the kinks in the M-T
curves as seen in real multicomponent systems (Fig. 2) are of
minor importance to the dynamics of the model compared to
the effect of the eutectic part of the M-T relation.
The three phase diagrams used in the model to represent
the granite and the resulting M-T relationships are shown in
Fig. 3. There are two different types of M-T dependences seen
in the eutectic system. Near the eutectic temperature there is
an almost horizontal segment which is assumed to increase by
1"C along its entire length. This increase is sufficient to yield
an unequivocal relation between M and T , required by the
numerical formulation, Above the eutectic temperature the
curve increases in a convex manner. Variations in the length
of the horizontal part of the curve control the melting behaviour and therefore the dynamics of the model. This variation
in the M-T curve can be due either to different eutectic systems
as shown in the diagram or to different starting compositions
within the same eutectic system. Fixing the eutectic point and
varying the starting composition would lead to the same M-T
curves as those shown in Fig. 3(b). Only the eutectic point in
the granite phase diagram is varied.
The basalt is approximated using a eutectic phase diagram
of diopside-anorthite at 10kbar water pressure. Using a starting composition of X = 0.6 anorthite, the liquidus and solidus
temperatures are 1100"C and 950 "C,respectively.
An additional parameter which has a large influence on the
dynamics of the model is the density difference between the
solid and molten granite. The typical density contrast lies
between 50kgm-3 and 200kgm-3. The values used in the
model are listed in Table 2.
B O U N D A R Y A N D INITIAL CONDITIONS
a
1100
Y
As initial conditions (Fig. 4),the lowermost 5 km of the model
are occupied by a totally molten basalt with a temperature of
1200"C (100"C above liquidus temperature). In the overlying
solid granite a linear temperature profile with a gradient of
28 "Ckm-' is assumed, representing a heat flow of 70 mW m '.
At 27km depth the solid granite is 10°C below the eutectic
temperature. The thermal boundary conditions are constant
2 1000
a
c,
900
~
Q)
800
Q)
* 700
0
095
1
melt concentration M
Figure 3. Three different phase diagrams (Type 1, solid line; Type 2,
dashed line; Type 3, dotted line) are used for granite in the models.
The start composition of the granite at X = 0.8 is shown in Fig. 3(a).
A melt concentration-temperature (M-T) relationship (Fig. 3b) can
be calculated from these phase diagrams (Fig. 3a). Going from Type 1
to Type 3, the starting composition becomes closer to the eutectic
point. This results in a higher maximal melt concentration at the
eutectic temperature.
Table 2. Parameters used in the calculations (Johannes & Holtz 1990;
KTB Feldlabor 1994; Pribnow & Umsonst 1993).
density granite solid( 2450-2600ks/rn31
density basalt solid
2900 k@m3
latent heat granite
300 kJkg
manmum heat flow1
70 mW/rn2
heat capacity solid( loo0 J/(kg'K)
thermal conductivity granite
2.5 W/(m*K)
density granite liquid( 2400 k@m3
1
density basalt liquid 2800 k@m3
latent heat basalt
380 kJ/kg
I
minimum heat flow1 30 mW/W
I
I
heat capacity liquid/ 1500 Jl(k9.K)
I
thermal conductivity basalt 3,5 W/(m*K)
0 1995 RAS, GJI 123, 59-70
Modelling of melting processes and diapirism
63
Temperature at t=O
T=const=336"C
12 km
17km
22km
27km
32km
7OmW/W
u)mw/w
q (heatflow)
T
336°C
756°C
1200°C
Figure 4. Initial and thermal boundary conditions for the underplating model with solid granite and totally molten basalt.
temperature ( T = 336°C) at the top and a laterally varying
heat flow between 70 mW m-' and 30 mW m-' at the bottom.
Free slip at the top and bottom is chosen as the mechanical
boundary condition. Due to the high viscosity at the top, this
condition is equivalent to the no-slip condition.
RESULTS
Several calculations have been made, varying the density of
the solid granite (Table 2) and the eutectic composition of the
granite in the phase diagram (Fig. 3). First the typical time
evolution of the underplating model is described. For this a
model was chosen in which diapirs grow over a period of
c. 0.7Ma. The melt fraction M , temperature T , stream
function @ and the chemical components X are shown in
Fig. 5. An active marker-line (the basalt-granite boundary)
and a passive marker-line within the granite have been used
to represent X . The Type 1 phase diagram and a density
variation of 100kgm-3 between the solid and molten granite
is used in this model. During the first phase (here 0-0.2 Ma)
the cooling of the basalt leads to strong convection while the
growth of the melt front into the granite is purely conductive.
The first crystals begin to grow in the basalt. During the
second time phase (here from 0.2 Ma to 0.7 Ma; Fig. 5a) the
thickness of the partially molten granite is sufficiently high to
allow for convection within the granite. This results in perturbations of the interface betweens solid and molten granite
which can potentially form a RTI. The growth of diapirs is,
however, hindered by the dominating melting process. The
basalt begins to solidify during this phase. Diapirs begin to
rise out of the partially molten granite in the third phase (here
between 0.7Ma and 1.5Ma; Figs Sb, c and d). The diapirs
reach their maximum height after 1.5Ma. The dominating
process during phases 3 and 4 is the convection within the
granite driven by the solid-liquid phase transition. As a
characteristic feature of this phase-transition-driven convection, there exists a spatially stable partially molten region
(Figs 5c, d and e) that remains essentially undisturbed during
a large number of overturns. As a result, granite particles
0 1995 RAS, GJI 123, 59-70
involved in this flow undergo multiple cycles of melting and
crystallization. The crystallization of the basalt begins at the
contact to the granite and a crystallization front grows from
above into the basalt. Due to diapirism and the ensuing phasetransition-driven convection in the granite, the basalt solidifies
faster than for the purely conductive case as the heat transport
to the surface is more effective. Subsequently in the fourth
phase (Figs 5e and f), the cooling process begins also in the
granite. Similar time phases exist for other models, with the
time-scales of heating, ascent of diapirs (when existent) and
cooling being determined by convection of differing strengths.
In a first series of models the density contrast between solid
and liquid granite is varied (Ap = 50-200 kg m-3). Figs 6(a)-(c)
show three different models at the same instance of time, which
corresponds to time phase 4 as discussed above.
It can be seen that rising diapirs do not develop for all
cases. The occurrence of a RTI in our models depend on two
time-scales, namely the characteristic growth time of an instability and the time-scale associated with the existence of a
granitic melt layer. At low Ap the first time-scale is too large
compared with the second one, prohibiting the development
of diapirs.
These two time-scales do not only decide whether or not
diapirs ascend, they also control the time at which a diapir
starts to rise. The different density variations lead to different
growth rates of the RTI which results in the rise of diapirs at
different times. The time at which a diapir rises given a high
value of Ap is much earlier than for a small density difference.
For large values of Ap, therefore, the diapirs rise out of a
thin partially molten layer of granite (Fig. 6c). This results in
smaller volumes of melt, which rise by diapiric flow. Smaller
melt volumes cool and solidify more effectively and no overhangs are observed compared with cases where diapirs start
rising at later times and with larger volumes (Fig. 6b). Further
observations are pulsating flows for high Ap and near steadystate flows for a smaller density contrast. This behaviour can
be compared to thermal convection, which is time-dependent
at high Ra and steady state at low Ra.
Variations in the phase diagram as indicated in Fig. 3(a)
64
D. Bittner and H . Schmeling
amount of melt M
temperature T
stream function Q, chemical field X
12
(4
16
20
Time 190 Ka
-20
24
28
- 24
- 24
phase 2
- 28
32
-32
0
5
10
15
20
0
12
-12
16
-16
20
-20
24
-24
28
32
- 32
5
10
15
0
20
5
10
15
0
20
10
5
15
(bj
Time 730 Ka
-24
phase 3
-28
0
5
10
15
20
0
12
- 12
I8
- 16
20
-20
24
- 24
28
32
0
5
10
15
-28
-32
20
0
5
10
15
-32
20
0
- 32
5
10
15
20
0
5
10
15
(4
Time 1.2 Ma
-24
phase 3
5
10
15
-32
20
0
5
10
15
20
0
5
10
15
- 12
- 16
12
16
20
-20
24
-24
28
32
0
5
10
15
(4
Time 1.4 Ma
- 24
-28
-32
20
0
phase 3
- 28
5
10
15
20
0
5
10
15
20
0
5
.
10
.
15
12
- 12
16
-16
(el
Time 1.6 Ma
20
-20
24
-24
28
-28
32
0
5
10
15
-32
20
0
12
-12
16
-16
20
-20
- 24
28
-28
32
- 32
5
10
15
20
5
10
15
20
0
5
10
15
20
0
- 12
5
10
15
r----(f)
24
0
phase 4
0
Time 2.1 Ma
-24
phase 4
-28
-32
5
10
15
20
0
5
10
15
20
0
5
10
15
Figures. The development of a diapir in four phases. Parameters used are phase diagram Type 1, density contrast 100kgm-3, and the other
parameters listed in Table 2. The width and depth of the models are given in kilometres.
0 1995 RAS, GJI 123, 59-70
Modelling of melting processes and diapirism
amount of melt M
temperature T
chemical field X
- 12
- 16
- 12
-20
-20
-20
- 24
-24
-24
-28
-28
stream function qj
-16
32
-32
5
0
10
15
time steps
-32
0
20
5
4000.00
time in ycars
10
15
20
rnax I'crnp
- 12
-20
- 20
24
-24
- 28
- 32
-28
10
15
20
32
_I_c_____
5
0
10
r n n x . rnclt granitc
995 565
temperature T
- 12
- 16
5
0
0.0326730
rnax.
1.64631et-06
amount of melt M
-
65
riiin.
viscosily
20
15
0 2605
-5.30103
stream function @
chemical field X
1
1-
-16
- 32
time steps
4000.00
time in years
1.64996e+06
rnax. Streamf.
1.15537
r n a x melt g r a n i t e
mlBx ~ c m p .
807.731
rriitn. viscosity
-5.30103
Density Variation P s l i d granite - Pliquid granite = 200 kg/m3
(c)
amount of melt M
temperature T
chemical field X
stream function <p
_--~--
- 12
- 16
- 12
- 16
-20
-20
-20
-24
-24
-24
-28
-28
-32
-32
-12
0
0.2254
5
10
tirnc s t c p s
t i m e i n ycars
15
20
-28
-32
0
4000.00
1.64955e+06
5
10
15
20
1
0
5
10
15
20
0
5
max Slrearnf.
3.1 1698
n m x . rrielt g r a n i t e
r r , a x . T ~ : ~ ii~ , .
968.502
min. viscosity
10
__-_
15
20
0.2313
-5,:jOIO:j
Figure 6. Influence of density variation on diapirism. Phase diagram Type 1 and the parameters listed in Table 2 are used.
were used in further model calculations. The variations in
phase diagram resulted in changes in the melt concentrationtemperature ( M - T ) curves, which are shown in Fig. 3(b).
This causes a variation of the maximum melt concentrations
at the eutectic temperature. The buoyancy increases with
increasing melt concentration, and as a first result nearly the
same behaviour is observed as in the model series with the
0 1995 RAS, GJI 123, 59-70
density variations (Fig. 6). Thus a general rule can be formulated: the closer the starting composition of the granite is to
the eutectic point, the higher the potential diapiric rise.
From a comparison of the emplacement depths of the diapirs
shown in Fig. 7 it is possible to see that there is a small
increase in the ascent height for models with increasing density
differences. The influence of changes in the phase diagram
66
D. Bittner and H . Schmeling
comparison between diapir-models
phase diagrams
Figure 7. Different ascent heights are reached by the diapirs based on the variation of the density contrast between the solid and liquid granite
and the phase diagram. The 'wavenumber' (the number of diapirs across the width of the model, 40km) is indicated above each height bar. During
the ascent of the diapirs, the phase diagrams result in average melt concentrations of M 20-30 per cent (Type l), M 35-45 per cent (Type 2),
M 75-85 per cent (Type 3).
-
-
from a small melt fraction to a large melt fraction, however,
results in a slight decrease in the ascent height. (Only models
which resulted in diapirism are discussed here.) This is a
surprising result as a larger volume fraction of melt leads to
greater buoyancy. A closer analysis of the models, however,
shows that with increasing eutectic behaviour the melt concentration strongly increases, while the thickness of the partially
molten layer decreases. This leads to a decrease in the wavelength of the instability (see wavenumbers indicated in the
diagram). The resulting smaller characteristic wavelength leads
to smaller diapirs, which cool more efficiently and, if compared
to large diapirs, build up stress fields with smaller amplitudes.
The resulting higher viscosity of the surrounding rock impedes
high emplacement levels.
R I S I N G D I A P I R S OR HORIZONTAL
LAYERING
The density contrast between liquid and solid granite may
cause a RTI. The typical result of such a RTI are rising diapirs
(e.g. Ramberg 1981). Our results show that there are also
solutions in which diapirs do not form. The question arises,
which mechanism controls whether diapirism occurs or
whether the partially molten rock remains horizontally layered.
A RTI can only develop in the presence of granitic melt with
a density lower than that of solid granite. Thus the lifetime of
granitic melt can be used to formulate a first condition that
must be fulfilled before diapirs can rise. The amplitude A of a
RTI of two constant viscosity layers at the early stage is
A = A, exp(yt)
(8)
with A,=amplitude of the disturbance at time zero and
y = growth rate
y=-
1
APg
2%k
('1126,
+ 62)('112"1+ 4'
2khi
mi =.-7
sinh(2khi) - 2khi'
ai=
+
cosh(2khi) 1
.
slnh(2khi) - 2khi'
-
where h i = thickness of layer i; qi=viscosity of layer i; q I 2 =
q1/q2; k = wavenumber. Subscript i = 1 refers to solid granite;
i = 2 refers to partially molten granite (Woidt 1980).
The maximum time tmelmax
available for the ascent of diapirs
during the lifetime of the granite melt is dependent upon the
heat input Q,the depth and the temperature gradient. At any
time t < tmeltmaxthe remaining time tmelt= tmelmax- t corresponds to a critical growth rate of yc = l/tmeLt.Based on eq. (8)
this growth rate would result in an amplification of the original
amplitude by the factor e. Criterion 1 can be formulated as
follows: for diapirism to occur during the remaining time for
which the granite melt exists, the actual growth rate must be
larger than the critical growth rate, i.e. y > yc or t , > tmcltmax- t.
It is important to note that the critical growth rate is not
constant, but is dependent upon the remainder of time for
which a granitic melt exists, i.e. the critical growth rate increases
as a function of time. The values of the actual growth rate and
the critical growth rate at any time, based on criterion 1, are
shown in Fig. 8. The largest value for tmeltmaxin our models is
4Mio years. This results in an initial critical growth rate of
y c = 7.9 x 10-15 ijs.
However, as the creation of a melt front and the growth of
an RTI are competing with each other, a second criterion must
be fulfilled before the diapir grows. This criterion is related to
the speed of growth of both fronts.
At the early stage of a RTI the growth velocity of a
perturbation of the interface partial melt-solid having an
amplitude A is given as
udiapir = dA/dt = A,d(exp(yt))/dt = Ay ,
(9)
where A = amplitude of the RTI and y = growth rate.
The waviness of the instability is, however, smoothed and
the amplitude is outpaced by a growing melt front, if its
velocity is higher than that of the instability. The velocity of
the melting front is the time derivative of the height Hmeltof
the granite melting front,
Umelt = dHmeIt/dt.
(10)
0 1995 RAS, GJI 123, 59-70
Modelling of melting processes and diapirism
(a)
-1 1
10
.
'I 0- 1 6
0.0
growth
r a t e ,gamma
. , .
. ,
.
,
.
0.2
.
0.4
,
,
0.6
,
,
0.8
.
1.0
relative time
growth rate gamma
(b) l o - l l
-10s I
0.0
0.2
0.4
0.6
0.8
1.0
relative time
Figure8. Growth rate as a function of non-dimensional time. For
scaling, the time has to be multiplied by 2.16 x lo6 years. The actual
growth rate y (dashed line) must be greater than the critical growth
rate g, (solid line) in order that a diapir can rise during the existence
of granitic melt. Curves in (a) and (b) are from models Runl and Run2
shown in Figs 6(a) and (b).
We can summarize the two criteria to predict diapiric ascent:
(1) Y > Y ~ ; ~,=l/'(tme~tmax--),
(2) udiapir >
They are now tested with the help of the numerical model.
The parameters for the two velocities are obtained from the
numerical results of each model. Owing to the dynamical
behaviour of the models, average values are taken.
Owing to the variation in stress- and temperature-dependent
viscosities in our model there are a number of coupled parameters that can influence the growth rate curve. On one hand
the viscosity is dependent upon the strain rate and therefore
the strength of convection (i.e. diapirism), while on the other
hand the growth rate of the RTI is controlled by the viscosity.
The resulting interrelation between the strength of convection
and the growth rate of the RTI is responsible for the behaviour
of the y curve for stress-dependent rheology.
In the numerical calculations of the velocity from eqs (9)
0 1995 RAS, GJI 123, 59-70
67
and (lo), the amplitude A remains a model-dependent free
variable and can be adjusted to fulfil our two criteria. This
amplitude was chosen to be 150m for a granite layer of 15 km,
ensuring that the point of intersection of both velocity curves
and the start of diapiric ascent coincides in our models.
Figures 8 and 9 show the results when criteria 1 and 2 are
applied to the models. The first criterion y > y c is fulfilled in
Runl (Fig. 8a) for only 1 Ma. Diapirism is possible for more
than 2 Ma in the model Run2 shown in Fig. 8(b). In the model
without diapirs (Runl, and Run4, Figs 9a and d) there are no
time ranges in which both criteria are fulfilled at the same time.
In the models Run2, Run3, Run5 and Run6 (Figs 9b, c, e
and f), there are time regions in which both criteria are
simultaneously fulfilled and diapiric ascent occurs within this
time range.
There is a peak at early times for the RTI velocity to be
seen in the velocity curves. This is caused by the speed at
which convection starts in the basalt at the beginning of the
calculation. Owing to the coupling between the strength of
convection, the growth rate and the stress-dependent viscosity,
the growth rate increases very quickly. This is followed by a
decrease in both the growth rate and the strength of convection.
Shortly afterwards the velocity of the melt front reaches a
constant value for a short while. The increased heat transport
within the basalt during the early convection peak leads to an
increase in the heat flow into the granite. As a result, the
decrease in the growth velocity of the melt front is retarded
for a short time.
With the two criteria discussed above, we now have the
tools necessary to predict the occurrence of diapirism or the
growth of horizontal layers of partially molten granite. The
time-scale at which diapirs begin to ascend can also be
calculated.
DISCUSSION
As previously mentioned, there are a number of possible
scenarios for the heating and melting of the lower crust. We
wish to compare two examples and discuss their characteristic
heat balances. The first example is the present one of underplating by molten basalt. Based on the data used in the present
models, an internal energy of E = 5.5 x 1OI8kJ is stored in the
basalt underlying the granite in the lower crust at a temperature
of 760 "C. The model shows that this energy is transported to
the granite within maximally 1 to 1.5Ma. The second case
that we wish to discuss envisions the rise of hot unmolten
asthenosphere of 1300°C up to Moho level. As mentioned
above, this could possibly be the result of delamination of a
lithospheric root during orogeny. For reasons of comparison
we assume the asthenospheric layer of peridotite to have a
thickness of 5 km, as in the case of basaltic underplating. The
warm peridotite in this model has a heat content of E =
5.6 x 10l6kJ compared with granite at 760 "C. As no convection occurs in the solid peridotite, the time-scale on which the
energy is transported to the granite is estimated by a characteristic diffusion time to be 2 to 2.5Ma, thus it is only slightly
longer than in the case of basaltic underplating.
These calculations show that a 1300°C layer of crystalline
peridotite will create a similar amount of granite melt as will
underplating by basaltic magma. The realistic natural scenario
is probably a mixture of these two extremes.
An important variable in the ascent of granitic diapirs is the
D.Bittner and H. Schmeling
68
0.030.
-
'
'
'
'
'
'
.
'
'
'
'
'.' .
'
.'
i
E
-
[
\
-'
0.020
" '
E
0.010:
.
.,
..
" '
"
'
'
'
I
' '
'
'
'
' . .
.
'
'
'
'
"
'
'
'
0.030.'' ' *
'
.
.
'
'
'
.
1
.
. .
:
\
E
> 0.010:
'
"
'
' . ' .
'
'
'
'
' '
;
0.030.' '
0.020
::
'
I
-i
::
'
sh
> 0.010:
0.0307' '
0.020
L
;
+.
\
0.030r'
-
0.020 ;
Y
.
-'
\
"
'
::
:i
E
0.010:
' . I
'
-k
0.020
.
~
~
l
~
'
~ ' .l
-
::
.,
emplacement depth, which may range between a few kilometres
below the surface (Wilson 1989) to 10-15 km below the surface
(Barton et al. 1988; Miller, Watson & Harrison 1988). The
diapirs in our model reach a maximum height of 15 km (see
Fig. 7) below the surface. This is the depth of the critical
temperature of c. 500 "C below which the material can no
longer flow. The deviatoric stress created by diapiric ascent at
this depth is approximately 10MPa. This stress decreases as
the surface is approached. Tectonic stresses up to 100MPa
may be present in tectonic regimes like orogens or other plate
boundary environments which are not included in our model.
These stresses will decrease the modelled rheology by a factor
Aq = lo3, and based on an activation energy of granite of E =
150kJ mol-' the critical temperature for the ductile-brittle
transition will be decreased by c . 200 "C. Thus tectonic stresses
decrease the effective viscosity, allowing granitic intrusives to
be emplaced by diapiric ascent possibly to depths of up to
10-12 km.
In summary, our model shows that, based on realistic
rheological and thermodynamic assumptions, diapirism may
be an important mechanism within the ductile lower crust.
However, it is unlikely that any further ascent into shallower
crustal levels may occur by diapirism. To explain shallower
emplacement depths, other mechanisms like dyke propagation
(Clemens & Mawer 1992) or stoping (Marsh 1982) might
take over.
Diapirism is possible in the rheological setting employed in
the above calculations. This rheology is, however, the weakest
that has been observed in experimental studies of crystalline
granite (Kirby & Kronenberg 1987). At significantly higher
subsolidus viscosities and using the same realistic values of
density, etc., no diapiric ascent through solid host rock is
expected. As diapiric deformation is observed in both granitic
plutons and surrounding host rock (Brun & Pons 1981; Cruden
1988; Ramsay 1989) it may be concluded that the lower crust
might have a weak rheology (Miller et al. 1988).
None of the models presented here showed significant
entrainment of basaltic material into a rising granitic diapir.
Entrainment is controlled by both the density contrast and
viscosity contrast at the basalt-granite interface. The density
of both molten and unmolten basalt is more than 10 per cent
higher than that of granite with the presently chosen parameters. According to an empirical law based on isothermal
models (Cruden, Koyi & Schmeling 1995), this density
contrast would prohibit significant entrainment at any viscosity contrast. This prediction is confirmed now by our
thermodynamically consistent models.
Our models show that diapirism is convection-driven by the
0 1995 RAS, GJI 123, 59-70
~
'
'
Modelling of melting processes and diapirism
solid-liquid phase transition. Granite particles may experience
multiple overturns and melting-solidification events, which
should be detectable as zoned minerals. Another implication
here is that heterogeneous lower crust could be homogenized
by such a convective flow. Multiple overturns within diapirs
and their deformation patterns have been discussed by Cruden
(1988) and Schmeling, Cruden & Marquart ( 1988), being the
result of viscous drag acting on a diapir rising over long
distances. The present model suggests an alternative explanation leading to similar deformational structures but not
requiring large distances of rise.
Our diapirism model is able to predict the ascent of large
amounts of acidic melt (e.g. as has been inferred for the
Variscan Orogeny), and with the introduction of tectonic
stresses the ascent height of diapirs can be calculated in good
agreement with geological observations.
It should be emphasized that in our model melting occurs
as a consequence of heating from below. However, orogenic
processes associated with crustal thickening might induce
dehydration processes involving biotite or muscovite breakdown at lower crustal depths. The release of water reduces
solidus temperatures and melting might occur at isothermal
conditions (Johannes & Holtz 1991). Increasing the lithostatic
pressure within a thickening crust might be described by a
pressure front penetrating the crust from below in the
Lagrangian frame of reference. It has to be studied in future
models if the effect of a moving pressure front is comparable
to the effect of a temperature front as studied in this paper.
This numerical model is, however, only a predictive tool. In
order to successfully model the natural processess occurring
in the Earth it is necessary to better understand the relationships between experimental data, and modelled parameters
and mechanisms.
CONCLUSIONS
We have modelled melting processes and diapiric flow within
the lower crust in a fully self-consistent manner, i.e. by solving
the energy and flow equations coupled with phase relations
and rheological equations of state. The model leads us to the
following conclusions.
( 1) Assuming realistic thermodynamic and rheological parameters, depths of emplacement of granitic diapirs of 15 km
are possible, associated with diapiric stresses of up to 10 MPa.
In the presence of tectonic stresses of the order of 100MPa,
weakening of the stress-dependent rheology may result in
emplacement depths which are 3-5 km shallower.
(2) The model suggests a brittle-dutile transition temperature of about 500°C. Below this temperature no viscous flow
is observed.
(3) Convection driven by solid-liquid phase transformation
may result in multiple overturns of granitic material, leading
to multiple loops of the P T t path and multiple meltingsolidification events within a rock unit.
(4) Viscous flow is not confined to the partial molten
granite, but extends deeply into the unmolten surrounding
rock.
( 5 ) Above the 300 "C isotherm, lateral temperature variations are small (even in the models including the whole crust),
making it difficult to identify active diapirism within the lower
crust by heat-flow measurements at the surface.
Q 1995 RAS, G J I 123, 59-70
69
ACKNOWLEDGMENTS
This work profited significantly from stimulating discussions
with Volker von Seckendorf, Don Dingwell and Sharon Webb.
Reviews by Alexander Cruden and Giancarlo Serri have been
very helpful. We gratefully acknowledge financial support by
Deutsche Forschungsgemeinschaft (Grant Schm 872-1).
REFERENCES
Barton, M.D., Battles, D.A., Bebout, G.E., Capo, R.C., Christensen,
J.N., Davis, S.R., Hanson, R.B., Michelson, C.J. & Trim, H.E., 1988.
Mesozoic contact metamorphism in the western United States, in
Metamorphism and Crustal Evolution of the Western United States,
Vol. 7, pp. 110-178, eds Ernst, W.G. & Rubey, Prentice Hall,
Englewood Cliffs, NJ.
Bergantz, G.W., 1989. Underplating and partial melting: Implications
for melt generation and extraction, Science, 245, 1093-1095.
Bird, P., 1978. Initiation of intracontinental subduction in the
Himalaya, J. geophys. Rex, 83(B10), 4975-4987.
Brid, P., 1979. Continental delamination and the Colorado plateau,
J. geophys. Res., 84(B13), 7561-7571.
Brun, J.P. & Pons, J., 1981. Strain patterns of pluton emplacement
in a crust undergoing noncoaxial deformation, Sierra Morene,
Southern Spain, J . struct. Geol., 3(3), 219-229.
Buck, W.R. & Toksoz, M.N., 1983. Thermal effects of continental
collisions: thickening
a
variable
viscosity lithosphere,
Tectonophysics, 100, 53-69.
Clemens, J.D. & Mawer, C.K., 1992. Granite magma transport by
fracture propagation, Tectonophysics, 204, 339-360.
Cruden, A.R., 1988. Deformation around a rising diapir modeled by
creeping flow past a sphere, Tectonics, 7, 1092-1101.
Cruden, A.R., Koyi, H. & Schmeling, H., 1995. Diapiric basal
entrainment of mafic into felsic magma, Earth planet. Sci. Lett.,
in press.
De Yoreo, J.J., Lux, D.R. & Guidotti, C.V., 1989. The role of crustal
anatexis and magma migration in the thermal evolution of regions
of thickened continental crust, in Euolution of Metamorphic Belts,
pp. 187-202, eds Daly, J.S., Cliff, R.A. & Yardley, B.W.D., Geological
Society Special Publication No. 43.
Dingwell, D.B., Knoche, R. & Webb, S.L., 1992. The effect of P20, on
the viscosity of haplogranitic liquid, J. Mineral., 5, 133-140.
Huppert, H.E. & Sparks, R.S., 1988. The generation of granitic magmas
by intrusion of basalt into continental crust, J. Petrol., 29(3),
599-624.
Kirby, S.H. & Kronenberg, A.K., 1987. Rheology of the lithosphere:
selected topics, Reo. Geophys., 25, 1219-1244.
Kukowski, & Neugebauer, 1990. On the ascent and emplacement of
granitoid magma bodies-dynamic-thermal
numerical models,
Geol. Rundsch., 79( 1). 227-239.
Lathop, A., Blum, J. & Chamberlain, C., 1994. Isotopic evidence for
closed systems anatexis at midcrustal levels: An example from the
Acadian Appalachians of New England, J. geophys. Res., 99,
9453-9468.
Lorenz, V. & Nicolls, LA., 1984. Plate and interplate processes of
Hercynian Europe during the late Paleozoic, Tectonophysics, 107,
25-56.
Mahon, K.I. & Harrison, T.M., 1988. Ascent of a granitoid diapir in
a temperature varying medium, Geophys. Res., 93( 32), 1174-1 188.
Marsh, B.D., 1982. On the mechanics of igneous diapirism, stopping
and zone melting, Am. J . Sci., 282, 808-855.
Matte, P. 1986. Tectonics and platetectonics model for the variscan
belt of Europe, Tectonophysics, 126, 329-374.
McKenzie, D., 1985. The extraction of magma from the crust and
mantle, Earth planer. Sci. Lett., 75, 81-91.
Miller, C.F., Watson, M.E. & Harrison, T.M., 1988. Perspective on
70
D. Bittner and H.Schmeling
the source segregation and transport of granitoid magmas, Trans.
R. SOC.,Edinburgh, 79, 135-156.
Nekvasil, H., 1991. Ascent of felsic magmas and formation of Rapakivi,
Am. Mineral., 76, 1279-1290.
Ocala, L.C. & Myer, R.P., 1972. Crustal low velocity zone under the
Peru-Bolivian Altiplano, Geophys. J. R. astr. SOC.,30, 199-209.
Oxburgh, F.R.S. & McRae, T., 1984. Physical constraints on magma
contamination in the continental crust: an example, the Adamello
complex, Phil. Trans. R . SOC.Lond., A 310, 457-472.
Paufler, P., 1981. Phasendiagramme, Vieweg, Braunschweig.
Pinkerton, H. & Stevenson, R.J., 1992. Methods of determining the
rheological properties of magmas at subliquidus temperatures, Volc.
Geotherm. Rex, 53, 47-66.
Pitcher, W.S. & Cobbing, E.J., 1985. Phanerozoic plutonism in the
Peruvian Andes, in Magmatism at a Plate Edge, pp. 19-25, eds
Pitcher, W.S., Atherton, M.P., Cobbing, E.F. & Beckingsale, R.D.,
Blackie, London.
Pribnow, D. & Umsonst, T., 1993. Estimation of thermal conductivity
from the mineral composition: Influence of fabric and anisotropy,
Geophys. Res. Lett., 20(20), 2199-2202.
Ramberg, H., 1981. Graoity, Deformation and the Earth’s Crust in
Theory: Experiments and Geologic Applications, 2nd edn, Academic
Press, London.
Ramsay, J.G., 1989. Emplacement kinematics of a granite diapir: the
Chindamora batholith, Zimbabwe, Struct. Geol., 11(1/2), 191-209.
Reuther, C. & Neugebauer, H.J., 1987. Intrusion of igneous rocks:
physical aspects, Geol. Rundsch., 76( l ) , 89-99.
Rubin, A.M., 1993. On the thermal viability of dikes leaving magma
chambers, Geophys. Res. Lett., 20(4), 257-260.
Schmeling, H., Cruden, A.R. & Marquart, G., 1988. Finite deformation
in and around a fluid sphere moving through a viscouse medium:
Implications for diapiric ascent, Tectonophysics, 149, 17-34.
Schwarz, G., Haak, V., Martinez, E. & Bonnister, J., 1984. The
electrical conductivity of the Anden crust in Northern Chile and
Southern Bolivia as inferred from magnetotelluric measurements,
J. Geophys., 55, 169-178.
Shaocheng, Ji & Pinglao, Jhao, 1993. Flow laws of multiphase rocks
calculated from experimental data on the constituent phases, Earth
planet. Sci. Lett., 117, 181-187.
Weinberg, R.F. & Podladchikov, Y., 1994. Diapiric ascent of magmas
through power law crust and mantle, J. geophys. Res., 99(B5),
9543-9559.
Wickham, S.M., 1986. The segregation and emplacement of granitic
magmas, Geol. SOC.London, 144, 281-297.
Willson, M., 1989. Igneous Petrogenesis. A Global Tectonic Approach.
Unwin Hymann, London.
Woidt, W.D., 1980. Analytische und numerische rnodellexperimente
zur physik der Salzstockbildung, Dissertation, Techn. Uni.
Braunschweig, p. 128.
0 1995 RAS, GJI 123, 59-70