Quantifying magma mixing with the Shannon entropy: application to simulations and experiments
📝 Abstract
We introduce a new quantity to petrology, the Shannon entropy, as a tool for quantifying mixing as well as the rate of production of hybrid compositions in the mixing system. The Shannon entropy approach is applied to time series numerical simulations and high-temperature experiments performed with natural melts. We note that in both cases the Shannon entropy increases linearly during the initial stages of mixing and then saturates toward constant values. Furthermore, chemical elements with different mobilities display different rates of increase of the Shannon entropy. This indicates that the hybrid composition for the different elements is attained at different times generating a wide range of spatio-compositional domains which further increase the apparent complexity of the mixing process. Results from the application of the Shannon entropy analysis are compared with the concept of Relaxation of Concentration Variance (RCV), a measure recently introduced in petrology to quantify chemical exchanges during magma mixing. We derive a linear expression relating the change of concentration variance during mixing and the Shannon entropy. We show that the combined use of Shannon entropy and RCV provides the most complete information about the space and time complexity of magma mixing. As a consequence, detailed information about this fundamental petrogenetic and volcanic process can be gathered. In particular, the Shannon entropy can be used as complement to the RCV method to quantify the mobility of chemical elements in magma mixing systems, to obtain information about the rate of production of compositional heterogeneities, and to derive empirical relationships linking the rate of chemical exchanges between interacting magmas and mixing time.
💡 Analysis
We introduce a new quantity to petrology, the Shannon entropy, as a tool for quantifying mixing as well as the rate of production of hybrid compositions in the mixing system. The Shannon entropy approach is applied to time series numerical simulations and high-temperature experiments performed with natural melts. We note that in both cases the Shannon entropy increases linearly during the initial stages of mixing and then saturates toward constant values. Furthermore, chemical elements with different mobilities display different rates of increase of the Shannon entropy. This indicates that the hybrid composition for the different elements is attained at different times generating a wide range of spatio-compositional domains which further increase the apparent complexity of the mixing process. Results from the application of the Shannon entropy analysis are compared with the concept of Relaxation of Concentration Variance (RCV), a measure recently introduced in petrology to quantify chemical exchanges during magma mixing. We derive a linear expression relating the change of concentration variance during mixing and the Shannon entropy. We show that the combined use of Shannon entropy and RCV provides the most complete information about the space and time complexity of magma mixing. As a consequence, detailed information about this fundamental petrogenetic and volcanic process can be gathered. In particular, the Shannon entropy can be used as complement to the RCV method to quantify the mobility of chemical elements in magma mixing systems, to obtain information about the rate of production of compositional heterogeneities, and to derive empirical relationships linking the rate of chemical exchanges between interacting magmas and mixing time.
📄 Content
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Quantifying magma mixing with the Shannon entropy: application to simulations and experiments
Perugini D.1*, De Campos C.P.2, Petrelli M.1, Morgavi D.1, Vetere F.P.1 & Dingwell D.B.2
1 Department of Physics and Geology, University of Perugia, Piazza Università, Perugia 06100, Italy 2 Department of Earth and Environmental Sciences, Ludwig-Maximilian-University, Theresienstrasse 41, 80333, Munich, Germany
- Corresponding author:
Diego Perugini (e-mail: diego.perugini@unipg.it)
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Abstract
We introduce a new quantity to petrology, the Shannon entropy, as a tool for
quantifying mixing as well as the rate of production of hybrid compositions in the mixing
system. The Shannon entropy approach is applied to time series numerical simulations and
high-temperature experiments performed with natural melts. We note that in both cases the
Shannon entropy increases linearly during the initial stages of mixing and than saturates
towards constant values. Furthermore, chemical elements with different mobilities display
different rates of increase of the Shannon entropy. This indicates that the hybrid composition
for the different elements is attained at different times generating a wide range of spatio-
compositional domains which further increase the apparent complexity of the mixing process.
Results from the application of the Shannon entropy analysis are compared with the
concept of Relaxation of Concentration Variance (RCV), a measure recently introduce in
petrology to quantify chemical exchanges during magma mixing. We derive a linear
expression relating the change of concentration variance during mixing and the Shannon
entropy.
We show that the combined use of Shannon entropy and RCV provides the most
complete information about the space and time complexity of magma mixing. As a
consequence, detailed information about this fundamental petrogenetic and volcanic process
can be gathered. In particular, the Shannon entropy can be used as complement to the RCV
method to quantify the mobility of chemical elements in magma mixing systems, to obtain
information about rate of production of compositional heterogeneities, and to derive empirical
relationships linking the rate of chemical exchanges between interacting magmas and mixing
time.
Keywords: magma mixing, chaotic dynamics, compositional variation, Shannon entropy, concentration variance
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- Introduction The mixing of magmas has been a subject of considerable scientific research in the last three decades (Anderson, 1982; Russell, 1990; Bateman, 1995; Abe, 1997; Jellinek and Kerr, 1999; Bergantz, 2000; Perugini et al., 2003; Slaby et al., 2011; Perugini and Poli, 2012; Pietruszka et al., 2015; Shorttle, 2015). It is considered by many as a major process generating extreme compositional variations in rock suites as well as one of the main processes responsible for triggering highly explosive volcanic eruptions (Sparks et al., 1977; Murphy et al., 1998; Leonard et al., 2002; Martin et al., 2008; Perugini et al., 2010; Tomiya et al., 2013). Magma mixing has been investigated using several different approaches: from classical geochemical studies (Langmuir et al., 1978; Poli et al., 1996; Xu et al., 2014; Hagen-Peter et al., 2015), through numerical simulations (Oldenburg et al., 1989; Folch and Martı́, 1998; Jellinek and Kerr, 1999; Petrelli et al., 2011) and experiments with both synthetic and natural compositions (Kouchi and Sunagawa, 1985; Morgavi et al., 2013a; Perugini et al., 2013; Laumonier et al., 2014). These studies highlight that the mixing of magmas is characterized by an extreme compositional variability in both space and time, resulting from the development of chaotic mixing processes between the interacting melts (Flinders and Clemens, 1996; Perugini et al., 2003; De Campos et al., 2011; Slaby et al., 2011). Deviations from classic linear mixing trends have been documented in natural samples, numerical simulations and high-temperature experiments (Perugini et al., 2006; De Campos et al., 2011; Morgavi et al., 2013a). These observations underline the non-linearity of this natural process. And this fundamental nonlinearity makes magma mingling/mixing one of the most complex petrogenetic processes on our planet.
4 Despite the considerable amount of literature on the subject, many fundamental aspects of magma mixing still remain unresolved. In particular, the time evolution of magma mixing still requires study in order to understand the impact of mixing upon the generation of extremely variable compositional domains in space and the rate of production of hybrid compositions with respect to mixing dynamics. This latter issue is of paramount importance because the rate at which hybrid compositions are generated is directly related to the disappearance of end-member compositions. The ability of a magmatic system to preserve information about the end-member com
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