12.710 – Problem Set 4 solutions
1. What is “the geothermal gradient?” How has this relationship been established and
what are some of the anchor points? Why are the subcontinental and suboceanic
geothermal gradients postulated to be different?
“Geothermal gradient” refers to the temperature profile with depth in the Earth. Seismic
data, experimental work, and heat calculations (e.g. adiabatic behavior in the convecting
part of the mantle, heat conduction to the surface through the lithosphere) have helped to
establish this pressure-temperature relationship. Anchor points for the profile include the
temperature at the surface of the Earth; the core-mantle boundary (experiments and
seismic velocity profiles); the upper mantle-lower mantle boundary and the transition
zone (also experiments and velocity profiles), the crust-mantle boundary, and the base of
the lithosphere. Because continental crust and lithosphere are thicker than oceanic
lithosphere, their gradients must have different slopes.
2. What is “the adiabat?” What is the relationship between the adiabat and the Earth’s
geothermal gradient?
The term “adiabat” refers to the condition under which heat is not exchanged outside of
the system; this condition is such that with changes in pressure, the material behaves
isentropically. In the context of the Earth, “the adiabat” generally refers to adiabatic
conditions during up- or downwelling of material in the convecting mantle. Most
processes in the convecting Earth occur adiabatically, and the geotherm closely
approximates this adiabat. Likewise, materials upwelling or sinking in the mantle
generally follow the geotherm and, thus, behave adiabatically.
3. What is “the solidus” and how does it vary with pressure? How does the addition of
water into the mantle influence its solidus?
“The solidus” refers to the pressure and temperature conditions for a material at which
the very last bit of liquid crystallizes (e.g. with cooling), or the very first increment of
melt forms from a solid (e.g. with heating). A dry lherzolite solidus has a positive
pressure/temperature slope. The addition of water to the mantle weakens the structure of
the minerals in the rock and causes melting under higher pressure conditions – the wet
solidus is deeper in the Earth than the dry solidus. The wet solidus also has a negative P/T
slope.
4. Why does melting occur when solid mantle upwells?
Solid material upwelling adiabatically in the mantle at sufficiently high potential
temperatures intersects its solidus and thus begins to melt.
5. Consider the olivine phase diagram shown below... According to this phase diagram,
at what temperature will this particular melt begin to crystallize olivine? What will be the
Fo composition of the crystallizing olivine at T=1500ºC, and what will be the proportion
of crystals and liquid at this temperature?
In the olivine phase diagram shown, the melt at point O will begin to crystallize when it
reaches the liquidus at point L at temperature T1, ~1630ºC. When the temperature has
cooled to 1500ºC, the olivine crystallizing has a composition of Fo60 (60% forsterite, 40%
fayalite). Using the lever rule, there is 55% liquid and 45% crystalline olivine (all Fo60) at
this temperature.
6. The batch melting equation is... [Determine] the bulk partition coefficients for U and
Th (i.e. DTh and DU) for a “garnet lherzolite” mantle composed of garnet (12%), olivine
(59%), orthopyroxene (21%) and clinopyroxene (8%). Derive the batch melting
expression for the ratio of two elements (i.e. ClA/ClB = ?). For a Th/U source ratio
(CsTh/CsU) of 3.9, what is the Th/U in the melt (ClTh/ClU) if the total melt fraction (F) is
100%, 0%, 10%, 1%, 0.1%, and 0.001%? Make a plot of [Th/U] melt/[Th/U]mantle versus
total melt fraction (F). Considering the relationship shown on this plot, what is the
maximum melt fraction capable of fractionating Th from U during batch melting? Which
of the above minerals is responsible for the observed fractionation of Th from U during
melting? What would be different about the direction of Th/U fractionation (i.e. change
during melting) if the solid mantle undergoing melting was composed of only olivine,
orthopyroxene and clinopyroxene (i.e. no garnet)?
DThGt Lherz = 1.2e-3
DUGt Lherz = 2.4e-3
ClA/ClB = (CsA/CsB) * (F + (1-F)DB) / (F + (1-F)DA)
ClA/ClB = 3.9 (F=100%), 7.5 (0%), 3.9 (10%), 4.3 (1%), 5.9 (0.1%), 7.5 (0.001%)
According to the plot, F = 1% is the largest melt fraction capable of fractionating Th from
U during batch melting. Although Cpx is capable of causing a small amount of
fractionation, Gt dominates, since its DU and DTh values are very different. Without Gt
present, Cpx would dominate the fractionation, but it fractionates Th and U in the
opposite sense, such that [Th/U]melt/[Th/U]mantle would be <1 at small F.
7. The fractional crystallization equation is... For a magma which is crystallizing olivine
only, use the above mineral/melt partition coefficients for olivine and plot the normalized
Th concentration as a function of remaining melt fraction (i.e. ClTh/C0Th versus f), then on
another graph plot the change in Th/U as a function of remaining melt fraction (i.e.
[Th/U] remaining melt/[Th/U] original melt versus f). Now for a magma which is crystallizing
orthopyroxene only, use the above mineral/melt partition coefficients for
orthopyroxene and add to the first plot the change in Th concentration as a
function of remaining melt fraction (i.e. ClTh/C0Th versus f), and then add to the
second plot the change in Th/U as a function of remaining melt fraction (i.e.
[Th/U] remaining melt/[Th/U] original melt versus f) and label these lines accordingly. Now
do the same for clinopyroxene and again label these lines. Why does the relative
Th concentration change, but the Th/U ratio remain constant during
crystallization of olivine and orthopyroxene? What is different about
clinopyroxene?
Finally for a magma which is crystallizing olivine only, use the above
mineral/melt partition coefficients for olivine and plot normalized Ni
concentration as a function of remaining melt fraction (i.e. ClNi/C0Ni versus f). In
the context of the observed differences in the change of Th and Ni concentrations
in the remaining magma during olivine crystallization, discuss briefly what is
meant by the terms compatible and incompatible elements.
The relative Th concentrations change for Ol and Opx with increasing f, but the
Th/U ratios remain constant because DU and DTh are identical for both minerals.
Clinopyroxene, on the other hand, has different DU and DTh values, so both the Th
concentration and Th/U ratio changes with f.
Ni behaves compatibly in olivine – the DNiOl value is greater than 1, and with
decreasing f it leaves the melt and enters the crystals rapidly. Th, on the other
hand, is incompatible in olivine, and so Th concentrations increase with increasing
f. Th remains in the liquid until the last remnant of melt crystallizes.
8. Briefly describe and explain the succession of igneous rocks expected in
“normal” oceanic crust (i.e. give a description of ocean crust formed at midocean ridges).
“Normal” ocean crust has an upper layer of pillow basalts, underlain by a sheeted
dike complex, underlain by isotropic, foliated, and layered gabbros, underlain
finally by ultramafic materials, including layered and unlayered “tectonite”
peridotites. The pillow basalts are formed by extrusive flows on the ocean floor at
ridges, and the sheeted dikes are the conduits feeding the volcanics. The gabbros
represent the differentiating magma chamber beneath the ridge, and they are
infiltrated by and overlie pods of cumulate material. The deepest peridotitic and
harzburgitic rocks are the residuum of the uppermost part of the upwelling upper
mantle beneath the ridge.
9. What is the difference between rhyolite, andesite, and basalt?
Rhyolite, andesite, and basalt are three types of low-alkalinity, extrusive
(volcanic) igneous rocks, listed in order of decreasing silica content. Rhyolites are
very silica-rich (felsic), andesites and dacites (not listed) are intermediate in
composition, and basalt rocks are relatively silica-poor (mafic).
10. What is the fundamental difference between granite and rhyolite? Gabbro and
basalt?
Granite and rhyolite have the same chemical composition, but granites cool more
slowly beneath the Earth’s surface (intrusive or plutonic rocks), while rhyolites
cool quickly after eruption on the surface (extrusive or volcanic rocks). Similarly,
gabbro and basalt have the same composition; gabbros are intrusive, while basalts
are extrusive.
11. Describe the three primary tectonic settings where volcanism occurs and
discuss how differences in these settings result in different melting conditions,
styles of volcanism and resulting melt compositions.
Volcanism occurs at divergent margins (spreading ridges), convergent margins
(subduction zones or arcs), and intraplate settings (“hot spots”). Melts at spreading
centers form by decompression melting of upwelling solids in the convecting
mantle; most of the lavas erupted on earth erupt at ridges, the extent of melting is
relatively high, lavas are low in volatiles, and lavas undergo moderate
fractionation in magma chambers so that they are basaltic (tholeiitic) in
composition. The basalts have relatively low viscosity and cool rapidly under the
oceans. They show some isotopic and trace element variability, but not nearly as
much as ocean island basalts. Melts at convergent margins are produced by “wet
melting” of the mantle wedge due to the addition of water and other volatiles to
the wedge overlying the wet subducting slab of crust. They can experience a lot of
fractionation in the overlying crust, leading to differentiated melts ranging from
basalts to rhyolites, and they are high in volatiles that can exsolve suddenly and
generate explosive eruptions. Finally, intraplate or hot-spot volcanoes form
seemingly independent of plate tectonics; some of these are probably formed by
plumes of upwelling material from the mantle, while others may formed by some
other process (this is hotly debated). They are generally low in volatiles and erupt
as low-viscosity basalts. Their isotopic and trace element compositions are
extremely variable and help us fingerprint heterogeneous mantle reservoirs.
(Multiple answers will be accepted for this question since it is very open-ended.)
12. Please interpret this geologic cross-section in terms of the relative timing of
the different igneous, metamorphic and geologic processes.
Depositional order:
Layer B
Layer A
Layer D
Layer C
(Sill occurs after C and before first Unconformity)
Lava1
Folding event
Angular Unconformity
Layer E
(Overturn occurs after E and before Lava2)
Coarse-grained Granite
Fault
Dike
(Angular Unconformity occurs after Dike and before Lava2)
Fine-grained Granite
Lava2
A possible scenario: Layers B (a limestone), A (layered mudstone or sandstone), D
(shale), and C (finely-layered limestone) were deposited in a sedimentary basin as
water levels changed over time. A volcanic eruption produced the flow Lava1 over
Layer C, and either the same or another volcanic event injected the Sill between
Layers A and C. The region underwent compression, folding the rocks, and then
erosion created an angular unconformity above Lava1. Layer E (sandstone) was
deposited. A coarse-grained Granite was intruded, perhaps as a magma chamber.
The region was faulted, leading to what looks like extensional offset in this 2D
view (in 3D it might not be the case, though). A dike intruded along the fault. A
later, fine-grained Granite was then intruded across the other beds. Since Layer E
was deposited, the entire region underwent an overturning tilting event, either
between or concurrent with the granitic intrusions, faulting, and the dike intrusion.
Following the overturn and either before or after the second granitic intrusion,
another erosional event created a second angular unconformity. The lava flow
Lava2 was deposited above this unconformity.