Page 254 - Physical Principles of Sedimentary Basin Analysis

236 Subsidence

where X is the mass fraction of melt. The specific entropy for a partially melted system in
terms of only the mass fraction of melt is

s = Xs m + (1 − X)s s (7.144)
where s m and s s are the specific entropy of the melt and solid, respectively. The term
(s m − s s ) dX is therefore the increase in entropy when a mass fraction dX melts. The next
s
m
term (1 − X)c + Xc is the bulk specific heat capacity and (1 − X)(α s / s ) + X(α m / m )
p p
is in a similar way the bulk value of α/ . The mass fraction of melt X = X(p, T ) is a
function of pressure and temperature, which gives that

∂ X ∂ X
dX = dp + dT. (7.145)
∂p ∂T
T p
This change in melt fraction dX can be inserted into the entropy change (7.143), and setting
ds = 0 then gives

(s m − s s ) ∂ X − (1 − X) α s + X α m
dT ∂p T s m
=
. (7.146)
dp (s m − s s ) ∂ X − 1 (1 − X)c + Xc m
s
∂T p T p p
This version of the adiabat becomes the same as the previous one (6.338) in the case of zero
melt fraction. It has the form dT/dp = F(p, T ) since the melt fraction is a function of p
and T . This equation has to be integrated numerically in the case of a general melt fraction,
for instance with a simple Runge–Kutta scheme. McKenzie (1984) discusses melting and
magma generation and shows several examples of melt fractions and the adiabat (7.146).
Note 7.12 studies the adiabat (7.146) in the special case of linear solidus, liquidus and melt
fraction.

Note 7.12 Mantle adiabats and linear melt fraction: It is instructive to look at the
adiabat (7.146) in the case of linear solidus and liquidus
T s (p) = T s,0 + Bp and T l (p) = T l,0 + Bp (7.147)

and also a linear melt fraction
T − T s (p) T − T s,0 − Bp
X(p, T ) = = . (7.148)
T l (p) − T s (p) T l,0 − T s,0
The parameter B is B = A/( g), where A is the steepness of the solidus and liquidus as
in equation (7.127). The linear melt fraction (7.148)gives
∂ X 1 ∂ X B

= and =− . (7.149)
∂T p T l,0 − T s,0 ∂p T T l,0 − T s,0
We assume that the density, heat capacity and the thermal expansibility is the same for both
melt and solid. The adiabat (7.146) then becomes
dp a 1
= + a 2 (7.150)
dT T