Driving force for melt extraction at SMB
The increase of intensity of fabric with depth in planes illustrated in
Fig. 7, manifested as an increase in alignment factor, AF, with depth
(Fig. 7 and 8), suggests that the process controlling melt loss is grain
reorganization (repacking). The correlation between AF and paleodepth in
transect samples analyzed by the unmixing models provides evidence that
gravity is the driving force behind melt extraction. Under such
conditions, we perform a series of simple calculations to determine the
time required to develop a melt rich horizon of a given thickness atop a
compacting crystal matrix (Fig. 14). The melt rich horizon is formed by
the extraction and accumulation of interstitial melt in the crystal
matrix. The interstitial melt at the top of the crystal mush is constant
as there is no driver for melt extraction matrix (the pressure in
crystal matrix and melt at the top are equal). Instead, melt is
extracted from below and migrates upwards, lowering the height of the
crystal matrix and developing a melt rich layer. To solve for
FTL as a function of height in a compacting layer of
crystals with interstitial melt, we consider a combined statement of
momentum conservation between crystal matrix and melt (Florez et
al. , 2024),
\(\left(1-F_{\text{TL}}\right)\rho g=-\beta F_{\text{TL}}\left(V_{m}-V_{s}\right)-\left(\frac{\partial}{\partial z}\left[\left(1-F_{\text{TL}}\right)\left(\frac{\gamma}{F_{\text{TL}}}+1\right)\xi\frac{\partial}{\partial z}\ V_{s}\right]\right)\)(8a)
and statement of mass conservation,
\(\frac{\partial F_{\text{TL}}}{\partial t}\ +\ \frac{\partial}{\partial z}\left[F_{\text{TL}}\ V_{m}\right]=0\)(8b)
Here, β is proportional to \(\frac{\mu_{m}}{K}\), where\(\mu_{m}\) is the melt viscosity and is the permeability of the crystal
matrix, \(V_{s}\) and \(V_{m}\) are the crystal matrix and melt
velocity, respectively, \(\rho\) is the density difference between
crystal and melt, \(g\) is the gravitational constant, \(z\) is depth
within the compacting layer, \(\gamma\) is a geometric constant, and\(\xi\) is the viscous resistance to pore space closure in the crystal
matrix (effective matrix viscosity). Critically, \(\xi\) is a function
of \(F_{\text{TL}}\) that depends on the mechanism by which pore space
in the matrix is closed. Eq. (8a) is in essence a force balance where\(-\beta F_{\text{TL}}\left(V_{m}-V_{s}\right)\) is a drag force,\(\left(1-F_{\text{TL}}\right)\rho g\) is a buoyancy force, and\(\left(\frac{\partial}{\partial z}\left[\left(\frac{\pi}{F_{\text{TL}}}+1\right)\left(1-F_{\text{TL}}\right)\xi\frac{\partial}{\partial z}\ \left(F_{\text{TL}}\left[V_{m}-V_{s}\right]\right)\right]\right)\)is a compaction force. Here, melt and crystal matrix flow is driven by
the buoyancy force.
The compaction model solves for the mass and momentum conservation of
the two phases (melt and crystals) which provides us with the evolution
of FTL and mechanical phase velocities (velocity of melt
and crystal) in a compacting crystal column as a function of time. Mass
conservation is solved with an upwind and the momentum conservation with
a centered finite volume scheme. The results of these simulations are
strongly influenced by the effective matrix viscosity, \(\xi\). Here, we
use an expression for \(\xi\) assuming that pore closure in the matrix
is accommodated by repacking (particle rearrangements in the absence of
deformation of individual grains):
\(\xi=\xi_{0}\left(\frac{4}{3}\left[1+\frac{5}{2}\left(1-F_{\text{TL}}\right)\left(1-\frac{1-F_{\text{TL}}}{1-\ {F_{\text{TL}}}^{m}}\right)^{-1}+0.3\left(\frac{1-F_{\text{TL}}}{F_{\text{TL}}-\ {F_{\text{TL}}}^{m}\ }\right)^{2}\right]+\ \left[\frac{1-F_{\text{TL}}}{F_{\text{TL}}-\ {F_{\text{TL}}}^{m}}\right]^{2}\right)\)(9)
In this regime, \(\xi\) depends on
FTLm, which is the lowest
FTL of an aggregate that can be obtained by particle
rearrangements, and a reference viscosity, \(\xi_{0}\). Using
FTLm = 0.3 and \(\xi_{0}\) =
106 Pa s can explain the SMB trapped melt profiles
reasonably well in that the lowest FTL value calculated
using the unmixing model is 0.32 (Fig. 14). The
FTLm and \(\xi_{0}\) predicted in
Florez et al. (2024) by employing the numerical compaction model
to high temperature and pressure mechanical phase separation experiments
were ca. 0.3 and 105 – 106 Pa s,
respectively. Under such conditions, we expect that the timescales for
melt extraction and formation of the approximately 3.5 km thick region
of pure melt above the transect to be ca. 30 ka for melt with viscosity
of 105 Pa s and a grain diameter of 5 mm (Fig. 14).
The model is described in further detail in Florez et al. (2024).
Because of the longevity of the SMB, it is not surprising then that
significant melt loss has occurred throughout the majority of the SMG
samples.