Page 35 - Caldera Volcanism Analysis, Modelling and Response

10 Fidel Costa

One possibility is to use the zoning patterns of slowly diffusing elements to
constraint those that move faster and which we are interested in modelling (e.g.,
Costa et al., 2003). A similar approach was taken by Morgan and Blake (2006)
to model Ba and Sr zoning in sanidine of the Bishop Tuff. Other studies suggest
that an initial homogeneous concentration is a good assumption (e.g., Costa and
Chakraborty, 2004; Costa and Dungan, 2005) and was used by Bindeman and
Valley (2001) to model oxygen isotope zoning in zircon of Yellowstone magmas.
The boundary conditions refer to whether the crystal exchanges matter with its
surrounding matrix (open to flux) or not (isolating boundary). The boundary
conditions also include whether the concentration at the boundary remains
constant or not. In many situations related to magmatic processes a boundary open
to flux to an infinite reservoir is applicable. A constant composition at the boundary
is the simplest case but for example, Christensen and DePaolo (1993) presented a
model of Sr with changing boundary conditions due to in situ 87 Rb decay in
the case of the Bishop Tuff. The equilibration of glass inclusions by volume
diffusion can also be used to obtain time scales but we need to take into account
other boundary conditions, such as the solubility of the element in the host
(e.g., Qin et al., 1992).

2.3. Other methods of obtaining time scales of magmas related
to calderas
Thermal or crystallisation models, where viscosities and densities of liquids play
a prominent role can also be used to obtain time information (e.g, Wolff et al.,
1990; Bachmann and Bergantz, 2003; Michaut and Jaupart, 2006). For a first
approximation to the cooling times of magma reservoirs the following solution
of the heat diffusion equation can be used (Carslaw and Jaeger, 1986):

1 RR þ a RR a 2ðktÞ 0:5
T T 1
¼ erf 0:5 erf 0:5 0:5
T o T 1 2 2ðktÞ 2ðktÞ rp
2 2 (3)
ðRR aÞ ðRR þ aÞ
exp exp
4kt 4kt
which is for the case of a half sphere of at an initial temperature T o , and where k
is thermal diffusivity; T 1 the initial temperature of the host rock; r distance
measured from the centre of the intrusion and RR the radius of the intrusion.
This equation is used in Section 4 to compare magma solidification with crystal
residence times.
Crystal size distribution (CSD; Marsh, 1988) studies can also provide time
information. However, one should keep in mind that the times are directly
dependent on an arbitrarily chosen growth rate, which can be a variable (e.g.,
Cashman, 1991; Lasaga, 1998). The common syn-eruptive fragmentation of crystals
during explosive eruptions can also make the results difficult to interpret (e.g.,
Bindeman, 2005). First-order inferences about magma residence times are the
periodicity or time gaps between two successive eruptions from the same system
(Smith, 1979; Bacon, 1982).