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.