Download Melting and Magma Generation

Why the Earth Melts
EAS 458 Volcanology
Lecture 7
How to Melt a Rock?
 Raise the temperature
 Heat production: radioactive heating, friction, impact heating
 Heat conduction or advection
 May occur in the crust,
crust, particularly where high-T magma intrudes
low solidus country rock, otherwise, conduction is not efficient
 Decompression
 If the adiabatic gradient and the solidus intersect, rising rock will
melt.
 This is the most important melting mechanism.
 Therefore, in a certain sense, volcanoes are like clouds.
 Lower the melting point
 Just as salt lowers the freezing point of water, addition of some
substances ( a flux) to rock can lower its melting point.
 Water is the most effective and most likely flux
 This is very likely important in subduction zones
1
 Volcanism occurs at:
 Divergent plate boundaries
 Convergent plate boundaries
 More rarely in plate interiors
Decompression Melting
 Decompression of rising mantle accounts
for most volcanism on Earth, in particular,
volcanism at divergent plate boundaries
and in intraplate settings.
 With the help of thermodynamics, we can
readily understand why this melting
occurs.
2
Fundamental Variables
 T: temperature
 always absolute temperature, or Kelvins,
Kelvins, in
thermodynamics
 P: pressure
 force per unit area; SI unit is the Pascal (P) =
1 Newton/m2 = 1 kg-m/s2; in geology, we use
MPa (106 Pa) or GPa (109 Pa). 1 atm ≈ 1 bar
= 0.1 MPa. In the mantle pressure increases
at a rate of ~ 1 GPa for each 35 km depth.
 V: volume
Fundamental Variables
 U: Energy
 SI unit is the Joule
 Q: Heat: a form of energy
 W: Work: a form of energy dW = -PdV
 S: Entropy - strictly defined as the ratio of heat
exchanged to temperature in a reversible
process: S = ∆Q/T
 Also: S = k ln Ω where Ω is the number of states
available to the system.
 a measure of the randomness of a system
 Units of J/K (Joules per Kelvin)
3
Derived Variables
 H: enthalpy:
enthalpy: heat content, units of Joules
 ∆H is the energy absorbed (or given up) by a system
as a consequence of heating, chemical reaction, or
phase change
 ∆Hfus is the latent heat of fusion (heat required to
transform a substance from solid to liquid at constant
temperature). ∆Sfus = ∆Hfus/T
 CP: Heat capacity at constant pressure
 Relation to H and T: Cp = (∂
(∂ H/
H/∂ T)
T)P
 α: coefficient of thermal expansion (fractional
expansion of a substance upon heating)
 κ: thermal diffusivity: the heat flux is the product
of κ times the thermal gradient.
Derived Variables
 (A: Helmholz Free Energy:
Energy: energy available for
work)
 G: Gibbs Free Energy - energy available for nonPV work (e.g., chemical work)
 ∆G = ∆H - T∆
T∆S
 dG = VdP-SdT
 Two important properties:
 Products are reactants are at equilibrium when their Gibb
Free Energies are equal.
 Chemical reactions always proceed in the direction if lower
Gibbs Free Energy
4
Laws
 First Law of Thermodynamics:
 Conservation of Energy
 Equivalence of Heat and Work
 dU = dQ - dW
 Second Law of Thermodynamics
 The entropy of the universe always increases
 dS ≥ dQ/T
Definitions
 System:
 That part of the universe under consideration
 Closed system: one which does not exchange mass
with its surroundings
 Isolated system: one which exchanges neither mass
nor energy with its surroundings.
 Adiabatic system:
system: on which exchanges neither heat or
mass with its surroundings, but which may do work or
have work done on it.
 An adiabatic system is one for which dQ = 0; therefore, since
dSrev = dQ/T,
dQ/T, an adiabatic system is also isoentropic in the
reversible case.
5
Definitions
 Solidus: temperature at which a substance
(solution) begins to melt
 Liquidus: temperature at which a
substance (solution) completely melts
Convection and the Adiabatic
Gradient
 We now know that the mantle convects (hot
parts rise, cool parts sink) and that this is closely
related to volcanism. Hot mantle rises beneath
mid-ocean ridges and also mantle plumes.
 Because of the scales involved and the low
thermal diffusivity of rock (10-6 m2-s-1), the rising
rock can be viewed as approximately adiabatic.
adiabatic.
 Question: How will the temperature of the rock
change as it rises? (This is the same as asking
what is the adiabatic gradient).
 Thermodynamically, we want to know (∂
(∂ T/
T/∂ P)
P)S
6
Derivation of the Adiabatic
Gradient
 “Adiabatic”
Adiabatic” implies dS ≈ 0.
 Express the entropy change as a ‘generic’
generic’
function of T and P
 dS = (∂
(∂ S/
S/∂ P)
P)TdP + (∂
(∂ S/
S/∂ T)
T)PdT
 Since dS = 0
 0 = (∂
(∂ S/
S/ ∂ P)
P)T+ (∂
(∂ S/
S/∂ T)
T)P (∂ T/
T/∂ P)
P)S
 Rearranging: (∂ T/
/
∂
P)
)
=
(∂
∂
S/
T P S ( S/∂ P)
P)T/(∂
/(∂ S/
S/∂ T)
T)P
 It can be shown that (∂ S/
S/∂ T)
T)P = CP/T
 and (∂ S/
S/∂ P)
P)T = -αV
 Thus: (∂ T/
T/∂ P)
P)S = αVT/CP
 Adiabatic gradients depends on properties of the
material (α
(α, CP, V) and temperature.
Clapeyron Equation
 Consider, for simplicity, a simple one-component




system that melts completely at a single
temperature. At the melting point (and only at
that temperature), the solid and liquid phases
are in equilibrium.
Equilibrium implies dG = 0.
Recall that
dG = VdP-SdT
For a reaction or phase change, such as
melting, this equation may be written as d∆G =
∆VdP - ∆SdT where ∆ designates the change in
the property upon reaction. Hence:
∆VdP - ∆SdT= 0; ∆VdP = ∆SdT
7
Clapeyron Equation
 From ∆VdP = ∆SdT we can derive:
dT/dP = ∆V/∆
V/∆S
 This is known as the Clapeyron Equation and
describes the slope of a phase boundary.
 Since ∆V and ∆S are functions of T and P, the
phase boundary will generally be curved.
 For all substances, ∆Sfus is positive; for most
substances ∆Vfus is positive. Hence the
Clapeyron slope will generally be positive. In
other words, the melting temperature will
increase with increasing pressure.
Decompression Melting
 Again, the adiabatic gradient is:
 (∂ T/
T/∂ P)
P)S = αVT/CP
 Rising parcels of mantle will follow this T-P path.
 For olivine V ≈ 43.8 cc/mol (=J/MPa-mol); Cp ≈ 193
J/K-mol (at 1650 K), α = 2.7 x 10-5 K-1.
 At 1650 K, the adiabatic gradient is ~10K/GPa
 Solidus slope is:
dT/dP = ∆V/∆
V/∆S
 ∆V ≈ 0.434 J/MPa/g; ∆S ≈ 0.362 J/K/g
 Slope of the solidus is ~120 K/GPa
 Solidus slope is much greater than adiabat, so
rising mantle can cross it when it is hot enough.
8
Effect of Lithosphere
Thick Lithosphere
Thin Lithosphere
9
Melt Productivity
 Once the solidus is reached, how much melting
occurs?
 We can again use thermodynamics to answer this
question (at least approximately).
 Recall the partial differential for S:
 dS = (∂
(∂ S/
S/∂ P)
P)TdP + (∂
(∂ S/
S/∂ T)
T)PdT
 and (∂ S/
S/∂ T)
T)P = CP/T and (∂ S/
S/∂ P)
P)T = -αV
 We ask, how will entropy of one phase (e.g., the
solid) change with pressure at the solidus?
 Thus:
dSs/dP = CPs/T (∂ T/
T/∂ P)
P)2φ -αsVs
Melt Productivity
 Let F be the fraction of melt.
 The entropy of the whole system, S0, is:
 S0 = FSl + (1-F)Ss
 Rearranging:
 F = (S0 - Ss)/(S
)/(Sl-Ss) = (S0 - Ss)/∆
)/∆Sfus.
 Differentiating with respect to pressure:
 (∂ F/
F/∂ P)
P)S = 1/∆
1/∆Sfus(CPs/T(
/T(dT/dP)
dT/dP)2φ - αsVs)
10
Melt Productivity
 Melt productivity is:
 (∂ F/
F/∂ P)
P)S = 1/∆
1/∆Sfus(CPS/T(dT/dP
/T(dT/dP))2φ - αSVS)
 Assuming (dT/dP
(dT/dP))2φ = ∆Vfus/∆Sfus
 Evaluating this for cpx at 1650 K:
 CP = 1.61 J/K/g
 ∆Sfus = 0.253 J/K/g
 ∆Sfus = 0.253 J/K/g
 Vs = 0.392 J/Mpa/g
 α = 3 x 10-5/K
 (∂ F/
F/∂ P)
P)S = 6%/GPa ≈ 6%/35 km
 Will increase as with decreasing P and T
Melt Productivity
 A little more math and we can derive the
following:
T )V " !T %
*$
# !P '& F
CP
" !F %
$# '& ( T
!P S
" !T %
+Sm + $
# !F '& P
Cp
 This is a more general relationship, and the term
(∂ T/
T/∂ F)
F)P is just the inverse of isobaric melt
productivity and can be determined from
experiments.
11
Melting in Subduction Zones
 In contrast to divergent plate boundaries,
convergent plate boundaries or subduction
zones are regions where sinking occurs.
So it is hard to see how decompression
melting can operate.
 While decompression may contribute to
melting in subduction zones, flux melting
may be more important.
Water & Plate Tectonics
 Interaction of seawater and hot
rock at mid-ocean ridges
results in metamorphic
reactions and formation of
hydrous minerals such as
chlorite, amphibole, and
serpentine. Water content of
“mature oceanic crust”
crust” may
reach several percent.
 Hydrous mineral such as these
are stable only at relatively low
temperatures. At high
temperature, the reactions
reverse and minerals break
down to form dry minerals +
water.
12
Effect of Water on the Solidus
Island Arc Magmatism
13
Water Content & Melting in the
Marianas
Mantle flow in Subduction Zones
14
Summary
 Decompression melting accounts for most
volcanism
 Rising mantle will intersect the solidus and melt,
provided it is hot enough (generally is) and the
lithosphere is thin enough (generally isn’
isn’t). Initial
temperature and lithospheric thickness dictate the
extent of melting.
 Accounts for volcanism at MOR’
MOR’s and mantle plumes
 “Flux melting”
melting” (addition of water) is important in
subduction zone melting.
 Melting by increasing T is rare (impact, crustal
melting).
15