On the utility of quantitative modeling to the interpretation of Ca isotopes
Introduction
Over the past twenty years, the use of calcium (Ca) isotopes has reached into almost every sub-discipline in the geosciences and is now gaining ground in archeological, medical, and forensic applications (Tacail et al., 2020). Calcium is a major element in the important sedimentary carbonate archive, it is correspondingly difficult to alter diagenetically (Fantle and Tipper, 2014; Fantle, 2015), and its chemistry in surface environments is relatively simple (e.g., not impacted by overly complex speciation or redox processes). We now have a reasonably robust understanding of how Ca isotopes fractionate when calcite precipitates and we are becoming increasingly sophisticated in the interpretation of this signal. Calcium isotopes are thus emerging as a robust tool both for probing the past (Gussone et al., this volume) and understanding the present (Griffith et al., 2020).
Though the potential is evident, the application of Ca isotopes to a host of questions remains relatively sparse. This may stem, at least in part, from a perception of analytical difficulty, or perhaps as a result of perceived complexity or ambiguity in the interpretation of Ca isotopes in natural systems. For example, the Ca isotopic composition of marine carbonates have been interpreted to reflect secular seawater evolution, sea surface temperature, local mineralogical and fractionation factor variability, and diagenetic alteration (e.g., De La Rocha and DePaolo, 2000; Fantle and DePaolo, 2005; Sime et al., 2007; Farkaš et al., 2007; Fantle, 2010; Fantle and Higgins, 2014; Du Vivier et al., 2015; Lau et al., 2017; Higgins et al., 2018; Bradbury & Turchyn 2018). Other questions remain to be answered, such as why the world's largest rivers show a surprisingly narrow range of Ca isotope values at high flow rates despite the complexity of water-rock interaction and riverine drainage networks (Tipper et al., 2010). Here, we argue that observations such as these are not a reason to avoid Ca isotopes as an analytical tool but are instead the motivation for delving more deeply into them. Ultimately, these are the critical clues to the underlying function and balance of the Earth system that Ca (and other metal) isotopes uniquely report.
Our ability to leverage the information contained in such complex signals and thus advance application of Ca isotopes across the geoscience community depends on the development of quantitative models capable of deconvolving complex signals. For our present discussion a “model” is considered any quantitative framework designed to combine external sources and internal processes to reproduce or even predict a dataset (or, more usefully, datasets). The primary objective of such work is to identify and even quantify the dominant controls on a system of interest. This distinction is important, as it focuses the remainder of our present discussion on process-based frameworks, thus excluding regression, component analysis, and interpolation methods. Such a modeling approach may be a simple mixing analysis, a box model with one or more reservoirs, a single- or multi-component reactive transport model, or an ab initio molecular simulation (Fig. 1). Models are a quantitative means of testing hypotheses across scales, thus serving a critically important service in deconvolving complex signals and developing confidence in proxies.
Combining models and stable isotopes, in addition to other geochemical parameters, removes degrees of freedom in poorly constrained problems. For example, the concentration of calcium in a given reservoir may appear to be stable through time, and thus we may suggest (multiple) simple steady state mixing model (s) of inputs and outputs that create this unique value. Yet variations in the δ44Ca values, if measured, could indicate that changes are indeed occurring, and could be used to either rule out certain combinations of input and output fluxes or constrain endmember compositional shifts through time. Thus, isotopes add important constraints to the models that integrate information, simplifying complex systems and isolating key processes.
The only perfect example of a system is the system itself. Models are fundamentally imperfect in their representations of reality, seeking to achieve reasonable approximations while simplifying the myriad of processes and underlying mechanisms. Becoming an experienced modeler is then largely a process of learning how to define what is a reasonable approximation for the datasets available and the hypotheses one wishes to test. To those unfamiliar with a given modeling tool, the learning curve for such model development can be steep. Difficulty commonly arises when new users wish to test hypotheses or explore behavior that requires a more elaborate modeling framework than an individual on their own could reasonably construct. Just as a researcher new to Ca isotopes would not be expected to purchase and operate a multi-collector mass spectrometer in order to produce their first dataset, many model applications require software that is already built by others, such as common open source and commercially available reactive transport packages (e.g., Steefel et al., 2015). For the inexperienced user, entering values into an input file or graphical user interface and then launching the simulation creates a ‘black box’ situation in which it becomes difficult to establish a clear understanding of what the model results have achieved or why a particular set of input parameters fails to run. Another common issue arises from the preconceived expectation that a model should reproduce observations or else it has failed. The frustration that results in such a situation can often cause a user to abandon the particular quantitative approach, when in fact the inability of a well-constructed modeling framework to reproduce expectations can be a direct test of underlying hypotheses.
Overcoming such barriers is crucial to the development of the Ca isotope proxy, given the value that models offer. Increasing familiarity with models will foster the ability to communicate in more certain terms about the capabilities of any given proxy. We suggest that such a collective proficiency can be initiated via a basic understanding of modeling language and techniques, which does not require years of (re)training. Just as we depend on specialists in field work, mass spectrometry, and laboratory experiments, the calcium isotope community will have to include and collaborate with specialists in model development. This does not mean that a ‘modeler’ is required in order to apply such quantitative techniques. Simply that a basic, shared language and understanding are necessary in order for the broader community to better integrate quantitative models into interpretive frameworks. For example, all models require some information to be input in order to generate a solution, and this information should be appropriate to the system of interest. Endmembers must be established for a mixing model, and these values should reflect realistic Ca mass and isotopic compositions. Rayleigh models require a fractionation factor that is appropriate to the pathway of interest, and a rationale that justifies applying a single fractionating pathway to a multi-process system. Numerical solutions for reactive transport equations require a starting point (initial conditions) and information regarding the mass and isotope ratio of reservoirs surrounding the domain (boundary conditions). Such values are the responsibility of the researchers to identify and constrain before a model can be applied.
To this end, and with our overall motivation to advance the development of the Ca isotope proxy, we present an overview of the general classes of models applicable to Ca isotope studies of the Earth system. To date, the majority of quantitative interpretations of Ca isotopic data have been accomplished using mixing, box model, and Rayleigh approaches, and we begin with these methods. Advancing from this more familiar foundation, we step through simple reactive transport frameworks to modern multi-component simulation approaches, finally turning to a new generation of highly-resolved mechanistic models that isolate the mass-dependent partitioning of calcium isotopes. In each section, we highlight areas in which these approaches have proven utility and success, as well as situations in which the relevant assumptions suggest that we require a more nuanced approach. An overall theme of the paper is that each modeling technique described offers the potential to impact interpretations and guide future research when they are appropriately developed, and the results carefully considered.
Before proceeding, we wish to clarify that this review focuses on Ca isotopes. However, in many instances, the descriptions and conclusions drawn are applicable to an array of stable isotope systems. Given the focus on Ca in this special issue, and for simplicity's sake, we will not discuss the diversity of metal isotopes. As a result, we may refer to the inapplicability of a given modeling technique, for example Rayleigh distillation describing precipitation of new mineral phases, that would be sound for a different isotope system. We ask the reader to keep this qualification in mind when utilizing the material herein.
Section snippets
Reservoir models
This classification refers to models that are applied to describe mass and isotopic distributions of reservoirs, such as ocean and crust, porous media and groundwater, and vegetation and soil, without explicitly resolving length scales or interfaces among and between these reservoirs. Mass transfer is thus accomplished through a process that is not physically resolved, such as an instantaneous or sequential mixing between endmembers, a coefficient of input, removal or exchange between
Reactive transport models
This classification is distinguished from the prior examples in that the spatial domain over which mass transfer and transformation occurs is explicitly represented by partial differential equations that describe changes in mass or concentration through space and time:where the term on the left-hand side describes the change in concentration of a compound, element or isotope (Ci) multiplied by the porosity (ϕ). The terms on the right-hand side include
Molecular and process-based models
As shown in the previous sections, the processes that dictate the Ca isotopic composition of solids and fluids in natural systems are complex. Yet all of the macroscopic quantitative modeling approaches described above are founded on a common assumption that a subset of critical controlling processes (referred to as “first-order processes” here) can be isolated that play a disproportionate role in determining Ca isotope systematics. As isotope geochemists, a key challenge we face is to identify
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
JLD acknowledges support for this manuscript from DOE Office of Science SBR program under award DE-SC0019198. LNL acknowledges support from DOE Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division, through Geosciences program at Lawerence Berkeley National Laboratory under Contract DE-AC0205CH11231.
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Calcium isotopes in deep time: Potential and limitations
2020, Chemical GeologyCitation Excerpt :Consequently, Ca concentrations and isotope ratios are likely homogenous throughout the ocean (Zhu and Macdougall, 1998), because carbonate precipitation in shallow water and dissolution at depth is not sufficiently large compared to the Ca inventory to create a significant isotope gradient in the ocean. Box modelling is addressed in greater detail by Druhan et al. (2020) in this issue. Using Eqs. (1) and (2), it is possible to evaluate the impact on the Ca seawater reservoir and isotope composition from changes in the Finputs through the weathering of carbonate and silicate rocks, and from changes in the Foutputs through transient perturbations to carbonate burial.