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\begin{document}
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\title{Magma~ chamber formation by periodic dyke intrusions into the Earth's
crust~~~~~~~~}
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\authors{Oleg Melnik\affil{1}, Ivan Utkin\affil{2}}
\affiliation{1}{MSU Insitute of Mechanics}
\affiliation{2}{MSU Insitute of Mechanics}
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\begin{abstract}
We present a model of magma bodies formation by injection of hot
rhyolitic magmatic dikes leading to their incremental accumulation into
the plutons and magma chambers in the upper and middle crust. Our 2D
model simulates random or organized dike injection into a selected rock
volume, calculation of magma and rock displacement based on analytical
solution of an elastic problem of elliptical cavity expansion, a
realistic melt phase diagrams for country rock and magma. Lagrangian
particle transport is calculated in order to reduce numerical
dissipation and avoid unphysical mixing. Thermal histories in individual
batches of magma and country rocks are recorded. We further combine this
model with Bindeman and Melnik (2016) zircon crystallization/dissolution
software and compute zircon survival histories in each Lagrangian
particle.~
The model predicts shapes of realistic T-t histories, zircon age
distribution in different portions within a progressively growing
reservoir and generates output to estimate crustal vs mantle
contributions (e.g. Hf and O isotopes in zircons).
Simulations reveal that the rate of melt production is highly variable
in space and time, eruptible magma batches form in clusters, period of
initial magmatic incubation is followed by crustal rock melting and
formation of a large volume of eruptible magma with high melt fraction.
{Zircon survival and host rock eruptibility depends on magma injection
duration. After 700 y only \textasciitilde{}2 vol \% of molten rock can
be erupted, but due to slow dissolution most of zircon crystal are
preserved. After 1500 years eruptible rock amount reaches 8\% but
significant number of zircons looses their age information.~}%
\end{abstract}%
\section*{Introduction}
{\label{874460}}
The main mechanism of magma transport in the Earth's crust is the
formation of cracks (dykes) along which magma rises to the
surface\cite{Rubin_1995} .~ Basaltic magmas typically~ rise from depths
of several tens of kilometers~{\{REF\}}, for kimberlite magmas - up to
150-200 km~\citep{LENSKY_2006}. Dyke widths can vary from centimeters to
tens of meters, horizontal extend - from meters to kilometers. Magmaa~
ascent~ in dykes is controlled by the buoyancy forces and the tectonic
stress field. Most dikes do not reach the surface, but are blocked at
the level of neutral buoyancy {[}4{]}, or with structural barriers in
the form of stronger rock layers. As a result of repeated introduction
of dikes into the near-surface (first kilometers) region of the earth's
crust, it melts with the formation of magma foci, which can reach
thousands of cubic kilometers, although usually the volume of foci is
much smaller (kilometers-tens of kilometers). Foci of magmatic melt are
recorded by seismic tomography using shear wave attenuation. They may
have an irregular shape, but most often appear to be flattened bodies
with vertical or horizontal strike. Under active volcanoes there can be
several foci located at different depths {[}5{]}.
The formation of magma chambers is simulated both on global geodynamic
models {[}6{]} and in more detailed local models, where penetration and
heat transfer between individual dykes and host rocks are considered
{[}7, 8{]}. Models of the first type consider regions with a
characteristic size of tens of kilometers and a grid spacing of several
hundred meters. They cannot resolve the subtle heat exchange processes
that occur during the real transport of individual portions of magma,
but they allow one to estimate the size and position of magma chambers
based on the global distribution of temperatures, rheological properties
of rocks and stresses, as well as the consumption of magma between
individual chambers.
In the models of the second type, the region into which magma is
introduced, as well as the consumption of the latter, is set in advance
based on the geological structure of the rocks and estimates of the time
of formation of magmatic bodies. An example of the reconstruction of a
real magmatic system is {[}9{]}. The model assumes horizontal
introduction of dikes with lowering the underlying rock layer to the
width of the dike. The heat equation is solved taking into account the
heat of fusion of the rocks and the real temperature dependence of the
concentration of crystals. An explicit scheme for solving the heat
equation is used, which imposes a significant limitation on the time
step.
In the model {[}8{]}, the introduction of dikes can occur in an
arbitrary direction. To determine the field of displacements, the rocks
are considered a viscous fluid and the Navier-Stokes equation is solved.
This approach is not justified for low temperatures, at which the
behavior of the rocks is described by the relations of the theory of
elasticity. In {[}10{]}, the introduction of dikes is considered
vertical. Rock movement is determined solely by kinematic relationships.
\par\null
\section*{Mathematical model}
{\label{507127}}
Injection of individual dike leads to displacement of elastic host
rocks, heat transfer, rock melting and magma solidification.~ We model
individual dike as an ellipsoid with semi-axes a and b and use
analytical solution~\cite{Muskhelishvili_1977} in order to calculate host rock
displacement. Volume of the individual dike and frequency of emplacement
is controlled by the specified feeding rate of the magma
Q\textsubscript{in} (km\textsuperscript{3}/y).~ We assume that the~
emplacement occurs in 2D plain geometry and the third spacial dimension
is specified and constant. This situation is possible in the extensional
tectonic environment, where the local stress field leads to
preferentially parallel dyke orientation. We allow random anglo of an
individual dike emplacement or change in the dike orientation from
vertical at depth to horizontal near the surface.~ ~
\(\begin{equation}
\rho C\left( {\frac{{\partial T}}{{\partial t}} + \vec Vgrad\left( T \right)} \right) = div\left( {k\,grad\left( T \right)} \right) + \rho L\frac{{d\beta }}{{dt}}\\
\frac{{\partial \alpha }}{{\partial t}} + \vec Vgrad\left( \alpha \right) = 0\\
\rho C = {\rho _r}{C_r}\left( {1 - \alpha } \right) + {\rho _m}{C_m}\alpha \\
k = {k_r}\left( {1 - \alpha } \right) + {k_m}\alpha \\
{\beta _r} = {\beta _r}\left( T \right),\,{\beta _m} = {\beta _m}\left( T \right).
\end{equation}\)
Rock-magma temperature evolution~\(T\) is governed by heat
conduction equation~ ({\ref{eqn:eq1}}) that accounts
({\ref{eqn:eq1}}) for advection due to rock
displacement, latent heat of crystallization and heat conduction. Melt
fraction depends on temperature according to Annen et al 2006. Injection
rate is 0.015 km3/y (0.5 m3/s), initial magma temperature is 950 \selectlanguage{ngerman}°C,
typical for an island arc system.~ In order to minimize numerical
diffusion PIC/FLIP hybrid method which mixes the perspectives of solving
the system from a particle point of view (Lagrangian) and solving the
system from a grid point of view (Eulerian) is used. Particle
displacement is calculated after each individual dike injection, while
heat conduction is solved on a fixed grid.
\section*{Acknowledgements}
{\label{687807}}
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