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\begin{document}

\title{Repacking-Driven Compaction in the Spirit Mountain Batholith,
Southern Nevada}



\author[1,2]{Darien Florez}%
\author[2]{Christian Huber}%
\author[3]{Olivier Bachmann}%
\author[4]{Alicia Sigworth}%
\author[4]{Lily Claiborne}%
\author[4]{Calvin Miller}%
\affil[1]{MIT}%
\affil[2]{Brown University}%
\affil[3]{ETH}%
\affil[4]{Vanderbilt University}%


\vspace{-1em}



  
  \date{May 28, 2024}


\begingroup
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\sloppy


\textbf{1. INTRODUCTION}

The processes that influence differentiation in magma chambers and the
rate at which the associated melt-crystal phases separate have important
ramifications for volcanic-plutonic connections among silicic igneous
rocks. Related to volcanic-plutonic connections among silicic igneous
rocks is the identification of cumulate signature in plutons. The
subtlety of crystal accumulation signals in silicic igneous rocks has
led to their interpretation as representing a true melt composition and
being genetically separated from volcanic rocks (Coleman \emph{et al.} ,
2004, Glazner \emph{et al.} , 2004). While the petrological signature of
the cumulate nature of silicic magmas is subtle, it is discernible
nonetheless (Bachmann \emph{et al.} , 2007, Deering \& Bachmann, 2010,
Gelman \emph{et al.} , 2014). Knowledge of how melt loss occurs at melt
fractions relevant to silicic magma chambers and the associated textural
and chemical indicators can facilitate identification of cumulates in
plutons. Furthermore, the rate at which the associated melt-crystal
phases separate have important ramifications for volcanic hazards. Here,
we investigate the Spirit Mountain Batholith (SMB) for chemical and
textural evidence of crystallization-differentiation and phase
separation by repacking-driven compaction (grain reorganizations).

The paper is organized such that we first introduce the geologic setting
of the region and of the SMB in particular and provide evidence from
previous studies supporting melt loss in the deeper parts of the SMB.
Then, results of geochemical analyses are provided, including
acquisition of major, minor, and trace elements of bulk rock SMB samples
and results from plagioclase composition analyses. Subsequently,
textural analyses of selected SMB samples are presented. We identify a
near linear unmixing trend in major, minor, and trace element
geochemistry defined by samples within a ca. 3 km transect at the base
of the exposed batholith and pooled leucogranites near the top of the
batholith. The plagioclase compositions suggest that the samples
crystallized from the same parental magma and that the magma was less
mafic than their bulk rock compositions. We then introduce a trace
element model that allows melt and crystal to be lost to estimate
relative melt loss (cumulate) or crystal loss (silicic cap) in the SMB.
The benefit of this model is that it doesn't assume a particular
separation mechanism; however, it is limited in that it doesn't provide
the range of crystallinities over which melt is lost and doesn't allow
calculation of trapped melt fractions. To accomplish this, we use an
unmixing model that treats the analyzed samples as combinations between
two different endmembers: melt and crystal at a certain crystallinity.
The trapped melt fraction profile is then compared to results from a
model of mush compaction based on a crystal repacking rheology to
provide order of magnitude timescale estimates for the growth of the
silicic cap (melt accumulation layer).

\textbf{2. GEOLOGIC SETTING}

The Colorado River extensional corridor in southern Nevada, California,
and Arizona, is a ca. 70 x 250 km region consisting of highly extended
crust (Fig. 1) (Faulds \emph{et al.} , 2001, Faulds \emph{et al.} ,
1995, Faulds \emph{et al.} , 1990, Howard \emph{et al.} , 1996, Howard
\& John, 1987, Howard \emph{et al.} , 1994). The corridor experienced
crustal thinning from ca. 16 to 11 Ma and a flare-up of magmatism from
ca. 18 to 8 Ma (Faulds \emph{et al.} , 1995, Howard \emph{et al.} ,
1996). Several Miocene age plutons of felsic composition are found in
the region, as well as extensive ash-flow tuffs, volcanic breccias, and
lava flows (Fig. 1). Crustal sections were exposed in the region by
tilting of fault blocks and subsequent erosion, the largest of which
include upper to mid-crustal portions of the magmatic plumbing systems
associated with region (Bachl \emph{et al.} , 2001, Walker Jr \emph{et
al.} , 2007). There are several intrusive units in the corridor, the
largest of which is Spirit Mountain batholith (Fig. 1).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/Figure-1}
\caption{{Regional geology of southern Nevada and parts of California and Arizona
relevant to the Spirit Mountain batholith. Locations of the Spirit
Mountain batholith and other Miocene plutons are highlighted.%
}}
\end{center}
\end{figure}

\textbf{Spirit Mountain batholith (SMB)}

The SMB is a ca.10 x 20 km granitic pluton offering kilometers of
largely uninterrupted exposure (Fig. 2). The pluton was emplaced in the
northern Colorado River extensional corridor over a period of
\textasciitilde{}2 My (ca. 17.5--15.3 Ma) (Claiborne \emph{et al} .,
2006, 2010, Walker Jr \emph{et al} ., 2007). Post-emplacement tilting
45\selectlanguage{ngerman}º W resulted in a cross-sectional window of the SMB exposed at the
surface today (Faulds \emph{et al.} , 1992, Haapala \emph{et al.} ,
2005, Howard\emph{et al.} , 1994) (Fig. 2).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/image2/image2}
\caption{\selectlanguage{ngerman}{A) Spirit Mountain geologic map modified from Walker et. al (2007). Bulk
rock geochemistry data is available for all samples in panel A. The
pluton has been tilted ca. 45º westward during exhumation. In later
sections, the distances above the deepest sample (red triangle in panel
A) are calculated by projecting the samples orthogonally onto the
dashed, red line in panel A. The distances are then adjusted for
post-emplacement tilting of the batholith and are intended to reflect
variations in paleo-depth. Panel B, C, and D show Spirit Mountain
geology zoomed in. B) Samples highlighted in yellow are those included
in transect analyzed using unmixing model developed in later sections.
C) Samples highlighted in pink are samples analyzed via Electron Probe
Microanalyzer (EPMA) in the present study. D) Samples highlighted in
blue are those that are investigated texturally in later sections. In
subsequent figures, samples from this study and Walker et. al (2007) are
no longer distinguished, for readability, and samples included in
transect appear yellow and all other samples are white.%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

The main units in the batholith, identified in Claiborne \emph{et
al.}(2006) and Walker Jr \emph{et al.} (2007), include the Spirit
Mountain granite (SMG), Spirit Mountain leucogranite (SMLG), the
fine-grained granite (FGG) suite, diorite sheets, dike swarms, and the
Mirage pluton. The SMG and SMLG were combined in Walker Jr \emph{et al.}
(2007) and referred to simply as ``SMG''. The SMG and SMLG make up the
vast majority of the pluton.

Walker Jr \emph{et al.} (2007) documented a continuous range in wt \%
SiO\textsubscript{2} of ca. 61 -- 79 between the SMG and SMLG (excluding
the more mafic diorite sheets). Leucogranite samples are typically fine-
to medium-grained and contain ca. 10\% plagioclase, 40 -- 50\% alkali
feldspar, 30 -- 40 \% quartz, and small amounts of biotite (Walker
Jr\emph{et al.} , 2007). The shallowest (western most) portions of the
SMLG include aplite sheets, granite porphyry, and miarolitic cavities.
SMG samples, on the otherhand, are coarse (millimeter scale) and contain
20 -- 50 \% to plagioclase, 30 -- 50 \% alkali feldspar, 5 -- 30 \%
quartz, and 3 -- 15\% biotite. The deepest (eastern most) and most
quartz-poor samples are enriched in major, minor, and trace elements
consistent with accumulation of early crystallized phases (i.e., Sr, Ba,
and Al\textsubscript{2}O\textsubscript{3} suggesting accumulation of
feldspars). This suggests mechanical phase separation may be important
in the SMG; however, the extent and mechanism responsible for separation
remain unconstrained.

The FGG, Mirage pluton, and dike swarms are similar in texture
(uniformally fine-grained), composition (71-74 wt \%
SiO\textsubscript{2}), and in general relative timing (late stage), as
they all intrude the SMG. The FGG and diorite units are found as
sill-like sheets, pods, and pillows mostly towards the base of the
exposed batholith, while the Mirage pluton intrudes the southern portion
of the SMG. Several magmatic units with similar traits (composition,
grain size, field relations) are found within other intrusive Miocene
complexes in the region, suggesting that such magmas were perhaps the
parental magmas from which different portions of the SMG differentiated
(Gualda \emph{et al.} , 2023). Here, we investigate the SMB (SMG in
particular) for records of phase separation to explore the mechanisms
that lead to chemical differentiation and the formation and accumulation
of evolved melt caps.

\textbf{3. ANALYTICAL METHODS}

\textbf{Geochemistry}

Forty-two samples were collected in the field, of which thirty were
analyzed for elemental compositions. Whole rock powders were prepared
from fresh samples using an alumina ceramic shatter box and analyzed for
major and trace elements at the Peter Hooper GeoAnalytical Lab at
Washington State University. Major and minor elements were obtained
using a Thermo-ARL automated X-ray fluorescence spectrometer (XRF).
Trace elements were obtained using an Agilent inductively coupled plasma
mass spectrometer (ICP-MS). The sample locations are shown in Fig. 2 and
represented as circles in panel A. A full table summarizing the location
and chemistry of the samples can be found in the supplementary Tables 1,
2, and 3. Example thin sections of samples are shown in Fig. 3.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\caption{{Thin sections of two SMB samples. A) XPL and B) WDS Si map scan of HC01
from Spirit Mountain granite (SMG) unit, near the bottom of the section.
Panel C and D are the same, for scans of BC02 sample from Spirit
Mountain granite unit from the middle of the section. The main phases
are quartz, biotite, plagioclase, and alkali feldspar (turquoise,
orange, purple, and blue phases in right column, respectively).%
}}
\end{center}
\end{figure}

\textbf{Plagioclase compositions and maps}

Eight thin section samples were analyzed using five
wavelength-dispersive X-ray spectrometers (WDS) to obtain precise,
quantitative compositional maps. The sample locations are shown in Fig.
2 and are represented by pink circles. Thin sections were coated with an
approximately 20 nm thick carbon layer. The maps were produced at the
Department of Earth Sciences, ETH Zurich, using a JEOL JXA-8230 Electron
Probe Microanalyzer (EPMA) instrument. The applied acceleration voltage
was 15 kV with a 100 nA beam current. A pixel size of 15 mm x 15 mm, a
dwell time of 35 ms, and a probe diameter of 10 mm was used for samples
BC306, BC304, BC11, BC02, DS12, and DS03. For samples HC01 and SMLG, a
pixel size of 25 mm x 25 mm, a dwell time of 50 ms, and a probe diameter
of 25 mm was used. The resulting scans allow for precise elemental maps
of the sections analyzed and the calculation of chemical formulas of
phases of interest. Here, we investigate plagioclase due to its
prolonged duration on the liquidus and because the composition records
changes in the melt with which it is in equilibrium.

\textbf{Texture analysis}

Four samples cut along planes striking NS and dipping 45\selectlanguage{ngerman}º E
(approximately paleovertical) were analyzed in thin section using a
petrographic microscope at the Nature Lab at the Rhode Island School of
Art and Design. The microscope renders mosaics of each sample consisting
of stitched together portions of an individual sample. The orientation
of the planes along which thin sections were made was chosen to
investigate whether magmatic folation due to gravity was detectable in
samples. Grain boundaries were identified using image processing,
following the technique implemented by Barraud (2006). This technique
makes use of the watershed segmentation function of the Insight Toolkit
(ITK). ITK is an open-source and cross-platform library that is
implemented in C++. The watershed segmentation technique is applied to
the images using the function \emph{WatershedSegmentation1.cxx} included
with the ITK examples build. The function uses an anisotropic diffusion
filter to smooth the image and then converts the image to greyscale. A
gradient function is then applied to the image which is then saturated
to allow for the segmentation of individual grains. The grain boundary
images are then thresholded and analyzed in ImageJ, approximating
individual grains as ellipses, to obtain information about the
orientation of each grain/ellipse.

\textbf{4. ANALYTICAL RESULTS}

\textbf{Major, minor, and trace element geochemistry}

The granitic samples obtained of the Spirit Mountain batholith record an
almost continuous chemical gradation from 61 to 79 wt\%
SiO\textsubscript{2} (Fig. 4 and 5). The variation of any particular
major element reaches a minimum at around 72 -- 74 wt\%
SiO\textsubscript{2} and increases as wt\%
SiO\textsubscript{2}decreases. At SiO\textsubscript{2} contents greater
than 72 -- 74 wt\%, the spread in oxide concentrations tends to either
cluster or spread again (Fig. 4). Similar trends occur in minor and
trace elements when plotted as a function of wt\% SiO\textsubscript{2}
(Fig. 4 and 5). The samples highlighted in yellow in Fig. 2b are plotted
in yellow in Fig. 4 and 5 and define a near linear trend in major,
minor, and trace element space.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/image4/image4}
\caption{{Harker diagrams for major element oxides for SMB samples of this study
and of Walker et. al (2007). Total Fe expressed as FeO. Samples in
yellow are analyzed in later sections using unmixing model. All other
samples are shown in white. Grey lines indicate the assumed parental
magma composition in later sections.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/image5/image5}
\caption{{Selected trace elements plotted as a function of wt \%
SiO\textsubscript{2}. Samples with low wt \% SiO\textsubscript{2} are
enriched in compatible trace elements such as Sr. Samples in yellow are
analyzed in later sections using unmixing model. All other samples are
shown in white. Grey lines indicate the assumed parental magma
composition in later sections.%
}}
\end{center}
\end{figure}

\textbf{Plagioclase compositions}

An example elemental plagioclase map is shown in Fig. 6a. The molar
anorthite number (An\#; mol CaO/{[}CaO + NaO\textsubscript{0.5} +
K\textsubscript{0.5}{]}) is calculated for each plagioclase pixel of the
scan and shown in the distributions of Fig. 6b. The statistical mode for
distributions in six of the seven samples are relatively consistent
(centered at around An \# 0.15 -- 0.25). This is especially surprising
considering the range in bulk wt \% SiO\textsubscript{2} (62 to 74 wt
\%) of these six samples (Fig. 6c). One sample, SMLG, however, has a
distinctly lower modal An \# (about 0.05) and a considerably higher wt\%
SiO\textsubscript{2} value (78 wt\%). Finally, for comparison, the
expected equilibrium range of molar anorthite numbers in plagioclase as
a function of bulk rock molar anorthite number is shown in Fig. 6d. The
range in equilibrium plagioclase compositions derives from the
equilibrim constant, K\textsubscript{eq}, from the exchange reaction
between plagioclase and melt (Drake, 1976, Dungan \emph{et al.} , 1978,
Putirka, 2008, Rhodes \emph{et al.} , 1979). Here, K\textsubscript{eq} =
0.10 \selectlanguage{ngerman}± 0.05 (Putirka, 2008), and depends on the molar ratio of
AlO\textsubscript{1.5}/ SiO\textsubscript{2}. The upper and lower bounds
for the equilibrium range plotted in Fig. 6c are obtained by taking the
lowest and highest AlO\textsubscript{1.5}/ SiO\textsubscript{2} ratios
amongst the yellow samples in Fig. 6c. The grey circles in Fig. 6c are
calculated using the bulk molar anorthite number and
AlO\textsubscript{1.5}/ SiO\textsubscript{2} ratio for each sample to
predict the equilibrium molar anorthite number in plagioclase. Fig. 6d
highlights that the analyzed samples are shifted off the equilibrium
line and have anorthite contents lower than would expect given that of
the bulk rock. SMLG, however, approaches equilibrium.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}
\caption{{A) quantified elemental scan of sample BC02 showing molar CaO/{[}CaO +
NaO\textsubscript{0.5} + K\textsubscript{0.5}{]} in plagioclase and
alkali feldspar (values between ca. 0 -- 55 \%). Plagioclase is light
blue, alkali feldspar is darker blue, and the grey areas are
non-feldspar phases. B) Frequency distribution showing percent of pixels
in plagioclase with a given An \#. C) Statistical mode of each
distribution plotted in panel B as a function of wt \%
SiO\textsubscript{2} of the bulk sample. D) Statistical mode of each
curve plotted in panel B as a function of CaO/{[}CaO +
NaO\textsubscript{0.5} + K\textsubscript{0.5}{]} in the bulk rock. For
comparison, expected mineral (plagioclase) and liquid equilibrium values
are superimposed using
K\textsubscript{eq}(An-Ab)\textsuperscript{pl-liq} = 0.10. Grey circles
are calculated using the bulk molar anorthite number and
AlO\textsubscript{1.5}/ SiO\textsubscript{2} ratio for each sample to
predict the equilibrium molar anorthite number in plagioclase. Results
suggest that SMG samples analyzed are out of equilibrium with their bulk
rock and are more evolved than expected given their, in cases, more
mafic bulk rock composition. Samples in yellow are analyzed in later
sections using unmixing model. All other samples are shown in white.%
}}
\end{center}
\end{figure}

\textbf{Texture and fabric}

The maps of identified grains and grain boundaries for DS03, BC11,
BC302, and HC01 (shallowest to deepest in terms of paleodepth, least to
most mafic, and highest to lowest Rb/Sr ratios) are shown in Fig. 7. The
thin sections were cut along planes that were originally perpendicular
to the surface (assuming a 45\selectlanguage{ngerman}º westward postemplacement tilt) and were
striking NS but that had since been tilted (Fig. 7a). Only grains away
from the edges were analyzed leading to some vacant space on certain
portions of the maps. Rose diagrams (last two column of Fig. 7) show the
number of particles oriented at a given angle (azimuth) with respect to
the bottom edge of the image. The tan rose diagrams include all
identified grains while the pink rose diagrams only include phenocryst
grains (euhedral plagioclase, alkali feldspar, and biotite) for each
sample. The intensity of fabric development can be described by the
alignment factor (Barraud, 2006). The alignment factor (AF) can be
calculated by first introducing an orientation tensor,
T\textsuperscript{n}, for each grain:

\(T^{n}=\ L_{n}\par
\begin{bmatrix}\operatorname{}\alpha_{n}&amp;\cos\alpha_{n}\sin\alpha_{n}\\
\cos\alpha_{n}\sin\alpha_{n}&amp;\operatorname{}\alpha_{n}\\
\end{bmatrix}\) (1)

Here, \emph{n} , refers to a grain of interest and L is the major axis
length. The bulk orientation tensor, M\textsubscript{ij}, can be related
to T\textsuperscript{n} as:

\(2.\ \ M_{\text{ij}}=\frac{1}{N}\sum_{1}^{N}T_{\text{ij}}^{n}\) (2)

Finally, AF is defined as:

\(alignment\ factor\ (AF)=100\ \times\ \frac{e_{1}-e_{2}}{e_{1}}\)(3)

where e\textsubscript{1} and e\textsubscript{2} are eigenvalues of
M\textsubscript{ij}. Conceptually speaking, AF represents the extent of
agreement between orientations of individual grains.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/image7/image7}
\caption{{A) Orientation for textural analysis and analysis results for B) DS03,
C) BC11, D) BC302, and E) HC01. A) Schematic showing the plane in which
thin sections were prepared relative to original orientation
(pre-tilting) of SMB. The left panel in B, C, D, and E includes
identified grains and grain boundaries, the identification of which is
described in the main text. Each color corresponds to an individual
grain and has no relation to mineralogy. The right panel in B, C, D, and
E includes identified grains and grain boundaries, but feldspar and
biotite phenocrysts have been highlighted with a pink-ish hue and all
other grains with tan-ish hue. The right panel in B, C, D, and E also
includes rose diagrams which display the number of grains whose long
axis is aligned at a given angle. The tan rose diagrams include all
grains (no mineralogic distinction), while pink rose diagrams only take
into account feldspar and biotite phenocrysts. The AF (alignment
factor), which describes the extent of foliation, of each grain
population is tabulated and displayed with the rose diagrams. The angles
are measured with respect to the bottom edge of the image.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}
\caption{{A) wt \% SiO\textsubscript{2}, B) Sr concentration, C) Rb concentration,
and D) distance above deepest sample of pluton as a function of
alignment factor (AF). An alignment factor (AF) of 100 corresponds to
populations of particles with identical orientations, while an alignment
factor of 0 corresponds to no preferred orientation. Here, the distance
above deepest sample is calculated by projecting the samples
orthogonally onto the dashed, red line in panel A of Fig. 1. The
distances are relative to the orthogonal projection of the deepest
sample, represented by a red triangle in panel A of Fig. 1. Data points
here correspond to samples shown in Fig. 7. Faded data points are AF
values that correspond to all grains and completely opaque data points
are AF values calculated for only feldspar and biotite phenocrysts.
Samples in yellow are analyzed in later sections using unmixing model.
All other samples are shown in white.%
}}
\end{center}
\end{figure}

Fig. 8 shows the AF for each sample that was analyzed texturally, for
both all grains (faded markers) and phenocryst grains (solid markers).
Fig. 7 and 8 illlustrate that the intensity of SPO (both for all grains
and phenocryst grains) observed in HC01 (the deepest sample) is the
strongest, while SPO in BC302 (shallower) is weaker in intensity. BC11
(shallower than BC302), on the other hand, exhibits the weakest SPO. The
shallowest sample, DS03, has the second most visible SPO. The SMG
samples exhibit AFs that are inversely correlated with wt \%
SiO\textsubscript{2.} and Rb and positively correlated to Sr (Fig. 8).
The AF of yellow samples in Fig. 8 (analyzed in later sections using
unmixing model) correlate inversely with distance above deepest sample.
The phenocryst population was identified in cross polarized light on the
basis of interference color, euhedrality, lack of undulose extinction,
and twinning (feldspars), and alteration and spottiness. Rose diagrams
and the AF for all grains are included for comparison to ensure that
bias introduced by manually assigning mineralogies to the identified
grains is minimized.

\textbf{5. TRACE ELEMENT MODELING AND RHYOLIT-MELTS CALIBRATION}

We implement a trace element (TE) model based on Gelman \emph{et
al.}(2014) in order to investigate if the range in trace elements
recorded by the SMB is consistent with crystallization and variable melt
loss from a single parental magma composition. The box model considers
the evolution of the concentration of a given trace element during
crystallization in three separate reservoirs: crystal, melt, and lost
melt (in alternate simulations we consider crystal, melt, and lost
crystals). The reason for this formulation is because accumulating
feldspars (melt loss) can explain SMB samples enriched in Sr, while
accumulating melt (crystal loss) can explain SMB samples depleted in Sr.
In the model, crystal and melt reservoirs are coupled or grouped
together and comprise the synthetic sample (Fig. 9). Lost melt, on the
other hand, is removed from the sample (Fig. 9 and 10). Strictly
speaking, however, the fundamental equations used to solve for the
evolution of the trace elements of interest (Sr and Rb) assume
fractional crystallization. This choice is governed by the diffusivity
of the trace element of interest and the longevity of the magma system
at elevated temperatures. Gelman \emph{et al.} (2014) demonstrated that
for Sr the results are relatively insensitive to this choice. In the box
model, the sample is initially considered entirely molten and
subsequently the mass of the trace element is partitioned between
crystals and melt as crystallization persists. Either reservoir (crystal
or melt) can be incrementally lost; however, only lost melt or crystals
are separated or ``lost'' from the sample (Fig. 9 and 10). If no mass is
lost, all of the melt will convert to crystal and the final sample
concentration (concentration of TE of interest in crystal reservoir once
all melt has crystallized) will be equivelant to the starting
concentration of the parental melt. However, if a portion of melt is
lost, only the melt that remains is converted to the crystal reservoir
that comprise the final sample (Fig. 10). Here, the final concentration
of the sample relative to the initial concentration of the sample will
change as total sample mass has decreased, while, depending on the
compatibility of the TE of interest, the mass of the TE in the crystals
potentially remains the same. Allowing crystal loss similarly decreases
the sample mass; however, the effect on the final concentration of the
sample will be the inverse of the case of melt loss (loss of melt for a
compatible TE increases TE concentration in the final sample relative to
initial whereas crystal loss decreases the final concentration).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}
\caption{{Schematic diagram of trace element box model developed in this study for
the case of melt loss. Four reservoirs exist: mass of melt,
M\textsubscript{M}, mass of melt lost, M\textsubscript{L}, mass of
crystallized solid, M\textsubscript{C}, and mass of crystal lost,
M\textsubscript{CL}. M\textsubscript{CL} is not shown in the
illustration but is similar to M\textsubscript{L} except that it takes
mass away from the solid. Simulations take into account either no mass
loss or either (not both) melt or crystal loss. The sum of mass in all
reservoirs is constant M\textsubscript{0}, the initial mass of melt.
Each reservoir has an associated concentration of the trace element of
interest which is solved for using eqs. 1-4. Here, S is mass of trace
element of interest, M is mass of reservoir, and C is concentration of
trace element. In the case of melt loss, the sample is comprised of only
retained melt (melt reservoir) and accumulated crystals (crystal
reservoir).%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}
\caption{{Example outputs of trace element model. A) Evolution of mass of each
reservoir as a function of total mass crystallized. B) Evolution of
concentration of Sr in each reservoir. In A) and B) the final mass
fraction of melt loss, M\textsubscript{L}\textsuperscript{F} = 0.7. C)
and D) show the same as panel A and panel B, respectively, except with
M\textsubscript{L}\textsuperscript{F} = 0.5. Here, the bulk partition
coefficient (included in later sections) is constrained using
rhyolite-MELTS simulations.%
}}
\end{center}
\end{figure}

\textbf{Table 1.} Symbols and definitions.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\textbf{Symbol} & \textbf{Definition}\tabularnewline
\midrule
\endhead
C & concentration\tabularnewline
M & mass of reservoir\tabularnewline
D & bulk partition coefficient\tabularnewline
K\textsubscript{D} & partition coefficient\tabularnewline
S & mass of trace element\tabularnewline
C (subscript) & crystal reservoir\tabularnewline
b (subscript) & bulk\tabularnewline
M (subscript) & melt reservoir\tabularnewline
L (subscript) & melt lost reservoir\tabularnewline
C\textsubscript{sample} & TE concentration in sample (crystals + melt in
\textbf{Figure 9})\tabularnewline
M\textsubscript{L}\textsuperscript{F} and
M\textsubscript{CL}\textsuperscript{F} & terminal amount of lost melt or
lost crystal, respectively\tabularnewline
\bottomrule
\end{longtable}

To develop the model, we solve for the change in concentration of a
trace element in the melt:

\(\frac{dC_{M}}{dM_{C}}=\ \frac{C_{M}}{M_{M}}\left(1-D\right)\)(4)

Here, C\textsubscript{M} is the concentration of a particular trace
element in the melt, M\textsubscript{C} is the mass of crystallized
solid, M\textsubscript{M} is the mass of melt, and D is the bulk
partition coefficient (see Table 1 for symbols and definitions). Eq. (1)
describes the change in concentration dC\textsubscript{M} in the melt
caused by an increment of crystallization, dM\textsubscript{C}.
Similarly we solve for the change in concentration of a trace element in
the solid, which, in the case where mass loss only occurs in the melt
phase, evolves as:

\(\frac{dC_{C,b}}{dM_{C}}=\ \frac{DC_{M}-C_{C,b}}{M_{C}}\) (5)

C\textsubscript{C,b} refers to the concentration in the bulk solid,
rather than that which is instantaneously crystallized. Finally, the
change in concentration in the lost melt phase is calculated as:

\(\frac{dC_{L}}{dM_{C}}=\ \frac{\left(C_{M}-C_{L}\right)}{M_{L}}\frac{dM_{L}}{dM_{C}}\)(6)

where C\textsubscript{L} is the concentration in the lost melt phase and
M\textsubscript{L} is the mass of melt lost. This model only accounts
for melt loss; however, many SMB samples are enriched in incompatible
trace elements, suggesting they may have accumulated melt (Fig. 4 and
5). When mass loss only occurs by removing mass from the solid, the
change in concentration of a particular element in the solid instead
evolves as:

\(\frac{dC_{C,b}}{dM_{C}}=\ \frac{DC_{M}}{M_{C}}-\ \frac{C_{C,b}}{M_{C}}\left[\frac{dM_{\text{CL}}}{dM_{C}}+1\right]\)(7)

where M\textsubscript{CL} is the mass of lost solid.

The group of ordinary differential equations above includes terms that
account for the change in concentration associated with a small amount
of crystallization (dM\textsubscript{C}), as well as melt
(dM\textsubscript{L}) and crystal (dM\textsubscript{CL}) loss. Here,
melt loss is relevant to crystal accumulation whereas crystal loss is
relevant to melt accumulation; however, the box model assumes no
physical process by which these occur. The full derivation is included
in the supplements. Crucially, the equations are derived with respect to
M\textsubscript{C}, which is the mass of crystallized solids. This means
that if the functional forms of the mass of melt (M\textsubscript{L})
and of lost crystals (M\textsubscript{CL}) as function of mass
crystallized are determined, then M\textsubscript{M} can be solved for
and the equations are completely resolved. We argue that in the limit
that D \textgreater{}\textgreater{} 1, the exact functional form of
M\textsubscript{L} and M\textsubscript{CL} is of little importance
(supplementary Fig. 1). The expressions for M\textsubscript{L} and
M\textsubscript{CL} used in trace element calculations in this
manuscript depend on M\textsubscript{L}\textsuperscript{F} and
M\textsubscript{CL}\textsuperscript{F}, which are the final amount of
melt or crystal loss as crystallization terminates. These parameters are
solved for each SMB sample assuming a parental magma composition
tabulated in Table 2.

\textbf{Table 2.} Composition of BC04 (assumed parental magma
composition). Oxides are reported as wt \% and elements as parts per
million (ppm). Total Fe expressed as FeO.\selectlanguage{english}
\begin{longtable}[]{@{}llllllllllll@{}}
\toprule
SiO\textsubscript{2} & TiO\textsubscript{2} &
Al\textsubscript{2}O\textsubscript{3} & FeO & MnO & MgO & CaO &
Na\textsubscript{2}O & K\textsubscript{2}O &
P\textsubscript{2}O\textsubscript{5} & Sr & Rb\tabularnewline
\midrule
\endhead
73.60 & 0.30 & 13.74 & 1.64 & 0.05 & 0.44 & 1.67 & 3.74 & 4.73 & 0.08 &
203 & 144\tabularnewline
\bottomrule
\end{longtable}

Finally, knowledge of the bulk partition coefficient, D, is required in
order to fully resolve the group of equations above. To accomplish this,
we perform thermodynamic simulations using rhyolite-MELTS 1.1.x (Gualda
\& Ghiorso, 2015). Simulations assume a parental magma composition
listed in Table 2, a fixed pressure of 200 MPa, ca. 5 wt \%
H\textsubscript{2}O and an QFM+1 for the oxygen fugacity. The choice of
parental magma composition is influenced by several observations . The
pressure of 200 MPa is chosen due to the clustering of SMB samples along
the 200 MPa minimum (Gualda \emph{et al.} , 2012, Gualda \emph{et al.} ,
2023) of the Qz-Ab-Or ternary (supplementary Fig. 2), while the water
content and oxygen fugacity are relatively unconstrained. However, the
concentration of water must be sufficiently high to precipitate biotite
as an early crystallizing phase (consistent with SMG mineralogy). The
sensitivity of the results to these choices is explored by altering the
wt \% H\textsubscript{2}O content to 2 wt \% and the pressure to 500 MPa
(supplementary Fig. 3 and 4). The starting temperature is set to assure
the system is entirely comprised of melt and equilibrium crystallization
proceeds in increments of 0.5\selectlanguage{ngerman}ºC until \textgreater{}95\% of the mass is
incorporated in the solid (usually ca. 730ºC). Equillibrium
crystallization is chosen because what is required for closure of the TE
model is D as a function of mass crystallized. For trace element
calculations, we model the evolution of both Sr and Rb using partition
coefficients, K\textsubscript{D}, reported in Table 3. The phase
abundances predicted by rhyolite-MELTS as a function of crystallinity
are shown in Fig. 11 as well as the associated evolution of the bulk
partition coefficients for Sr and Rb. Example model outputs are included
in Fig. 10.

\textbf{Table 3.} Table of partition coefficients used in trace element
calculations.\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
& \textbf{K\textsubscript{D}\textsuperscript{Sr}} &
\textbf{K\textsubscript{D}\textsuperscript{Rb}}\tabularnewline
\midrule
\endhead
\textbf{spinel} & 0\textsuperscript{d} & 0
\textsuperscript{d}\tabularnewline
\textbf{orthopyroxene} & 0.009\textsuperscript{c} &
0.003\textsuperscript{c}\tabularnewline
\textbf{orthoclase} & 5.6\textsuperscript{a} &
0.391\textsuperscript{a}\tabularnewline
\textbf{plagioclase} & 14\textsuperscript{a} &
0.018\textsuperscript{a}\tabularnewline
\textbf{biotite} & 0.307\textsuperscript{a} &
1.57\textsuperscript{a}\tabularnewline
\textbf{quartz} & 0 \textsuperscript{d} & 0
\textsuperscript{d}\tabularnewline
\textbf{oxide} & 0 \textsuperscript{d} & 0
\textsuperscript{d}\tabularnewline
\bottomrule
\end{longtable}

\textsuperscript{a}(Padilla \& Gualda, 2016)

\textsuperscript{b}Bachmann \emph{et al.} (2005)

\textsuperscript{c}Arth (1976)

\textsuperscript{d} Sr and Rb in quartz, oxides, and spinel are presumed
to be perfectly incompatible (partition coefficients = 0).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}
\caption{{A) Phase abundances as predicted by rhyolite-MELTS 1.1.x (Gualda \&
Ghiorso, 2015). Phase abundance here is in terms of mass. B) Bulk
partition coefficient, D, for Sr and Rb as a function of crystallinity.
Simulations performed at 200 MPa, ca. 5 wt \% H\textsubscript{2}O, a
starting composition listed in Table 2, and QFM+1 for the oxygen buffer.
Note eutectic is reached at ca. 20 \% crystallization.%
}}
\end{center}
\end{figure}

The results of the thermodynamic simulations (phase mass as a function
of temperature and composition of bulk liquid and bulk solids) are
included in the supplementary Fig. 3, 4, and 5. In most cases, the
majority of SMB data plot along tie-lines between solid and liquid
equilibrium end members. When considering equilibrium crystallization,
such as in these calculations, tie-lines connect liquid and solid lines
of descent in equilibrium, while the exact points at which they are
connected is a function of the mass crystallized (supplementary Fig. 5).

6. \textbf{TRACE ELEMENT MODELING RESULTS -- MELT AND CRYSTAL LOSS
CALCULATIONS}

Results from the trace element modeling suggest that the majority of
samples from the upper portion of the SMB have accumulated significant
amounts of Sr depleted melt. Samples from the lower portions of the
batholith, on the other hand, show evidence of melt loss and crystal
accumulation (Fig. 12). These results were obtained by solving for
M\textsubscript{L}\textsuperscript{F} in samples where the concentration
of Sr is greater than that the assumed parental magma composition (Table
2) and by solving for M\textsubscript{CL}\textsuperscript{F} in samples
where the concentration of Sr is depleted compared to the assumed
parental magma composition. The benefit of the trace element model is
that it does not assume a physical process controlling phase separation.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}
\caption{{Distance above the deepest sample plotted as a function of mass loss
calculated for each SMB sample. The vertical line indicating no melt
loss corresponds to samples whose Sr concentrations are equivalent to
that listed in Table 2 (assumed parental magma composition). The values
on the left hand side of the x-axis were calculated by solving for total
melt loss, M\textsubscript{L}\textsuperscript{F}, while values on the
right hand side of the x-axis were calculated by solving for total
crystal loss, M\textsubscript{CL}\textsuperscript{F}. Sample symbols
here are identical to Fig. 4 and 5.%
}}
\end{center}
\end{figure}

\textbf{7. DISCUSSION}

\textbf{Conceptual model}

The SMB is comprised of a patchwork of sills emplaced over ca. 2 Ma
(Claiborne \emph{et al.} , 2006, 2010, Miller \emph{et al.} , 2011,
Walker Jr \emph{et al.} , 2007). Field relations, regional geology, and
the plagioclase results presented in this study suggest that the sills
that comprise the SMB crystallized from similar parental magma
compositions. The conceptual model we invoke to explain the origins of
Spirit Mountain batholith granites is mechanical phase separation by
repacking-accommodated compaction (Bachmann \& Huber, 2019,
Boyer\emph{et al.} , 2011, Holness \emph{et al.} , 2017), a process
likely repeated over periods of as much as several million years
(emplacement of a sill like body of parental magma and subsequent
crystallization and then compaction). Numerical models of compaction
often consider the evolution of porosity or melt fraction of a
continuous crystal column with an initially uniform melt fraction
distribution (Bercovici \emph{et al.} , 2001, Huber \& Parmigiani, 2018,
McKenzie, 1984, McKenzie, 1985, McKenzie, 2011, Ricard \emph{et al.} ,
2001). Melt and crystals separate due to their density contrast, which
at melt fractions above the maximum packing fraction (minimum melt
fraction that can be obtained by particle reorganization) is likely
accommodated by a combination of hindered settling and the rotation and
translation of matrix crystals (repacking) (Bachmann \& Huber, 2019,
Boyer \emph{et al.} , 2011, Holness \emph{et al.} , 2017, Huber \emph{et
al.} , 2024). At any depth within the compacting layer, each sample can
therefore be thought of as the sum of different proportions of melt and
crystal components; proportions are controlled by melt-crystal
separation, here driven by gravity. Under this idealized model, at
depths nearest to the base within a single sill within the overall
batholith, the driving force for phase separation is greatest and a
layer depleted in trapped melt develops. At shallower depths the driving
force for separation diminishes and less melt is lost. Meanwhile, at the
top of the sill a melt rich layer develops. A consequence of this in the
context of the SMB is the enrichment of CaO,
Al\textsubscript{2}O\textsubscript{3}, Ba, and Sr, consistent with
accumulation of feldspars in samples nearest to the base of the sills
emplaced that comprise the pluton. Meanwhile, samples furthest from the
base of a given sill within the pluton are SiO\textsubscript{2} rich and
enriched in incompatible trace elements like Rb and Th, consistent with
melt accumulation. Samples at intermediate depths, meanwhile, display
chemical compositions intermediate to the two endmembers. Visually, this
is manifested by intermediate samples plotting along mixing (or more
precisely \emph{unmixing} ) tie-lines between melt and crystal
endmembers located on either side of the parental composition (Payac\selectlanguage{ngerman}án
\emph{et al.} , 2023). On the other hand, unmixing between melt and
crystal compositions by phase separation occurring over a range of
M\textsubscript{C}, crystallized mass, (different points along
equilibrium liquid and solid lines of descent) results in a wedge
pattern on either side of the parental magma composition in major and
trace element space and can explain the fact that not all samples
collapse onto the same straight line (Fig. 4 and 5).

The melt and crystal loss calculations using Sr modeling predicts that
the samples nearest to the base of the batholith, in general, have lost
Sr-depleted melt (i.e. accumulated feldspar) to become enriched in Sr,
while samples farthest from the base have lost crystals (i.e.
accumulated evolved melt) (Fig. 12). Because the plagioclase crystals at
all depths within a given sill have crystallized from the same magma,
plagioclase composition in the cumulate crystals is for the most part
invariant despite a broad range in major, minor, and trace element
composition. The most melt, and therefore SiO\textsubscript{2}, depleted
samples, would be more albitic than would be expected if they had
crystallized from a parental magma assumed to be compositionally
equivalent to their bulk chemistry (Cornet \emph{et al.} , 2022). Fig. 6
supports this interpretation for the SMB. Finally, melt loss is
accompanied by detectable foliation in the most melt depleted samples
nearest to the base of the pluton (Fig. 7 and 8).

\textbf{Unmixing model - calculation of trapped melt fraction}

While the TE model employed in earlier sections was used to calculate
melt and crystal loss using only Sr, we make use of an alternate
approach to calculate the trapped melt fraction and the range of
crystallinities over which melt is lost. When plotting the Rb against Sr
concentrations for the SMB (bulk) samples, the data collapses primarily
onto a straight line defined by the samples belonging to the transect in
Fig. 1b. We assume that the spread in Rb and Sr concentrations for the
analyzed samples (yellow samples in prior figures and in Fig. 13) is due
primarily to unmixing of fixed liquid and crystal end member
compositions for reasons outlined in the previous section. The choice of
parental magma composition is listed in Table 2 and is validated by
field observations, the fact that the spread in major, minor, trace
element with respect to wt \% SiO\textsubscript{2} is minimized at
compositions near the assumed parental magma composition, and the
ability of the starting composition to satisfy the observed major,
minor, and trace element data. The extent of crystallization (mass
fraction) before melt loss can be solved for by calculating the
evolution of Rb and Sr in the liquid and solid assuming fractional
crystallization where melt and crystals aren't separated from one
another and fitting the tie-lines between liquid and solid to the
samples composition (Fig. 13). In this case, the crystallized mass
fraction, M\textsubscript{C}, that minimizes the scatter about the
mixing line is 0.24. The trapped melt fraction of each sample is then
calculated as the relative contribution of each endmember -- the solid
and liquid at M\textsubscript{C} = 0.24 of a parental magma composition
listed in Table 2. Using rhyolite-MELTS, we convert mass to volume and
solve for the trapped melt volume (Fig. 13). The results are similar to
the melt and crystal loss results, except this technique allows us to
obtain the tie-line along which unmixing primarily occurs (i.e. how much
mass has crystallized before the onset of unmixing). This allows us to
quantify trapped melt fraction profiles from the trace element record of
the samples and compare to numerical models of (mechanical) compaction
by repacking (Florez \emph{et al.} , 2024).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image13/image13}
\caption{{Trace element unmixing model compared to SMB samples. Tie lines relate
the composition of liquid and solid in equilibrium with each other over
a range of mass fraction crystallized M\textsubscript{C}. These
compositions are calculated assuming fractional crystallization of a
parental magma where no melt loss occurs. The assumed parental magma
composition is listed in Table 2. The prior sections suggest that the
transect samples analyzed are mixtures of two chemically distinct
reservoirs along unmixing or tie lines and justifies this choice. The
tie line in red associated with M\textsubscript{C} = 0.24 is the tie
line that minimizes the least square error with respect to the analyzed
samples in yellow.%
}}
\end{center}
\end{figure}

\textbf{Minimum trapped melt fraction and maximum packing fraction}

An important result presented here is that the minimum trapped melt is
ca. 30\%. Lee and Morton (2015) analyzed plutons in the Peninsular
Ranges Batholith and similarly relied on trace element compositions to
estimate trapped melt fraction. They found similar results with 20-30\%
of residual melt trapped in cumulate at the root of high silica
granites, arguing on that basis that hindered settling was likely
responsible for melt-crystal separation. We propose that the lower limit
of ca. 30\% trapped melt is due to the transition in phase separation
mechanisms from a hydrodynamic (melt-crystals interactions) and friction
(crystal-crystal interactions) repacking dominated compaction regime to
a compaction regime dominated by viscous creep such as grain-boundary
diffusion-controlled creep. Repacking is efficient at intermediate melt
fractions, as individual grains do not need to be deformed to
accommodate pore closure (Boyer \emph{et al.} , 2011). However, as the
maximum packing fraction is approached, this is no longer the case and
the resistance to pore space closure associated with repacking diverges
(Boyer \emph{et al.} , 2011). At melt fractions lower than the maximum
packing fraction, pore space can only be closed by deformation of
grains. Pore closure at melt fractions lower than the maximum packing
fraction is much less efficient than pore space closure by repacking at
intermediate melt fractions. This common minimum trapped melt fraction
inferred for SMB and the Peninsular Ranges Batholith suggests that once
compaction by repacking reaches the maximum packing fraction (around 0.3
trapped melt), further compaction involves processes that are too slow
to be active before the thermal death of these plutons.

\textbf{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
F\textsubscript{TL} 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 \emph{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, \selectlanguage{greek}β\selectlanguage{ngerman} 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 F\textsubscript{TL} 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
F\textsubscript{TL}\textsuperscript{m}, which is the lowest
F\textsubscript{TL} of an aggregate that can be obtained by particle
rearrangements, and a reference viscosity, \(\xi_{0}\). Using
F\textsubscript{TL}\textsuperscript{m} = 0.3 and \(\xi_{0}\) =
10\textsuperscript{6} Pa s can explain the SMB trapped melt profiles
reasonably well in that the lowest F\textsubscript{TL} value calculated
using the unmixing model is 0.32 (Fig. 14). The
F\textsubscript{TL}\textsuperscript{m} and \(\xi_{0}\) predicted
in Florez \emph{et al.} (2024) by employing the numerical compaction
model to high temperature and pressure mechanical phase separation
experiments were ca. 0.3 and 10\textsuperscript{5} --
10\textsuperscript{6} 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 10\textsuperscript{5} Pa s and a
grain diameter of 5 mm (Fig. 14).

The model is described in further detail in Florez \emph{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.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/image14/image14}
\caption{{A) Volumetric trapped melt fraction (F\textsubscript{TL}) for each SMG
sample analyzed in transect (yellow) and all (white) SMG samples
calculated using the trace element unmixing model. Trapped melt
calculations are made by projecting sample data orthogonally onto the
M\textsubscript{C} = 0.24 tie line and using the lever rule to determine
relative proportions of both melt and cumulate end members. Range of
maximum packing fractions (in terms of melt fraction) determined from
analysis of centrifuge experiments in Florez \emph{et al.} (2024) are
superimposed. Here, a value of ca. 0.3 aligns well with the minimum
observed melt fraction estimates. B) F\textsubscript{TL} over entire
exposed batholith. C) Forward simulations to calculate the time required
to form an eruptible melt horizon of a given thickness using the forward
model from Florez \emph{et al.} (2024). Here, the calculations are
performed using different grain diameters.%
}}
\end{center}
\end{figure}

\textbf{8. CONCLUSIONS}

This work adds to the growing catalogue of research describing the
cumulate origins of intermediate and silicic plutons and geochemical and
textural evidence of phase separation in these systems (Bachl \emph{et
al.} , 2001, Barnes \emph{et al.} , 2019, Claiborne \emph{et al.} ,
2006, Claiborne \emph{et al.} , 2010, Cornet \emph{et al.} , 2022,
Fiedrich\emph{et al.} , 2017, Garibaldi \emph{et al.} , 2018, Gelman
\emph{et al.} , 2014, Hartung \emph{et al.} , 2017, Payac\selectlanguage{ngerman}án \emph{et
al.} , 2023, Schaen \emph{et al.} , 2018, Tavazzani \emph{et al.} ,
2020, Walker Jr\emph{et al.} , 2007). The textures and chemistry of the
samples suggest that the range in chemical compositions recorded by SMB
samples is consistent with phase separation by repacking-accommodated
compaction. Trace element modeling suggests that 24\% of the mass has
crystallized magma of uniform composition before unmixing occurs,
explaining the composition of the majority of SMB. We find in the
cumulates that the trapped melt fraction is correlated with wt \%
SiO\textsubscript{2} and incompatible trace elements and inversely
correlated with depth, fabric intensity, and compatible trace elements.
We argue that the batholith is the result of partial-crystallization and
subsequent mechanical phase separation by repacking-driven compaction of
repeated injections of similar magmas over the course of about two
million years as indicated by zircon age dates by previous researchers
(Claiborne \emph{et al} ., 2006, 2010, Walker Jr \emph{et al} ., 2007).

\textbf{ACKNOWLEDGEMENTS}

We appreciate the help of Julien Allaz and Alejandro Cortes Calderon in
the acquisition of EPMA data at the ETH electron microscopy lab. This
work was supported by the National Science Foundation grant EAR-2021328.

\textbf{DATA AVAILABILITY STATEMENT}

Code and scripts used for numerical model are available at
https://doi.org/10.24433/CO.9041342.v1. Geochemical datasets presented
are available at https://doi.org/10.60520/IEDA/113194.

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