Technical Note: A generic law-of-the-minimum flux limiter for simulating substrate limitation in biogeochemical models

J Y Tang,W J Riley

doi:10.5194/bg-13-723-2016

Abstract

Abstract. We present a generic flux limiter to account for mass limitations from an arbitrary number of substrates in a biogeochemical reaction network. The flux limiter is based on the observation that substrate (e.g., nitrogen, phosphorus) limitation in biogeochemical models can be represented as to ensure mass conservative and non-negative numerical solutions to the governing ordinary differential equations. Application of the flux limiter includes two steps: (1) formulation of the biogeochemical processes with a matrix of stoichiometric coefficients and (2) application of Liebig's law of the minimum using the dynamic stoichiometric relationship of the reactants. This approach contrasts with the ad hoc down-regulation approaches that are implemented in many existing models (such as CLM4.5 and the ACME (Accelerated Climate Modeling for Energy) Land Model (ALM)) of carbon and nutrient interactions, which are error prone when adding new processes, even for experienced modelers. Through an example implementation with a CENTURY-like decomposition model that includes carbon, nitrogen, and phosphorus, we show that our approach (1) produced almost identical results to that from the ad hoc down-regulation approaches under non-limiting nutrient conditions, (2) properly resolved the negative solutions under substrate-limited conditions where the simple clipping approach failed, (3) successfully avoided the potential conceptual ambiguities that are implied by those ad hoc down-regulation approaches. We expect our approach will make future biogeochemical models easier to improve and more robust.

Highlights

Biogeochemical modeling has been one of the major themes in developing Earth system models (Hurrell et al, 2013), yet developing numerically robust and mathematically consistent biogeochemical (BGC) models has been challenging (Broekhuizen et al, 2008)
Many analyses indicate phosphorus is critical for improving carbon–climate feedback predictions (Vitousek et al, 2010; Yang et al, 2014; Wieder et al, 2015), and other nutrients may be important (Schmidt et al, 2013; Moro et al, 2014)
We show that, by ensuring mass conservation and non-negative solutions to the governing equations of a given biogeochemical model, it is possible to obtain a universal solution to the mass limitation for an arbitrary number of substrates

Summary

Introduction

Biogeochemical modeling has been one of the major themes in developing Earth system models (Hurrell et al, 2013), yet developing numerically robust and mathematically consistent biogeochemical (BGC) models has been challenging (Broekhuizen et al, 2008). Proper modeling of nutrient limitation is a prerequisite for credible predictions of carbon–climate feedbacks (Bouskill et al, 2014; Thomas et al, 2015). In the Earth system models (ESMs) joining phase 5 of the Coupled Model Intercomparison Project (CMIP5), only CLM-CN (Thornton et al, 2007) considered carbon and nitrogen interactions, observations indicate nitrogen has significantly limited the terrestrial carbon sink (Arora et al, 2013). Many analyses indicate phosphorus is critical for improving carbon–climate feedback predictions (Vitousek et al, 2010; Yang et al, 2014; Wieder et al, 2015), and other nutrients (e.g., sulfur, potassium, molybdenum) may be important (Schmidt et al, 2013; Moro et al, 2014). We expect that as more processes are included in future biogeochemical models, more substrates will limit different biogeochemical processes under different conditions

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Results

Conclusion