## 1. Introduction

[2] The long-term dynamics of soil and plant residue carbon (C) are receiving considerable interest because of the need to quantify long-term C sinks in soils under altered climatic conditions and following shifts in land use [*Lal*, 2008]. Two parameters are employed to define the soil effectiveness to retain C, transit time (sometimes referred to as residence time) and age of organic matter C (Figure 1). Transit time here is defined following *Eriksson* [1971] and *McGuire and McDonnell* [2006] as the time a soil organic matter (SOM) molecule remains in the soil since its deposition as plant residue, and until its release as mineralized products. The age is the time elapsed since the molecule entered the soil, and it is mathematically related to the transit time. According to this definition, the transit time is equivalent to the age at the time of exit.

[3] The transit time of SOM carbon depends on a number of processes acting at the yearly to decadal, or even longer timescales. In particular, the physical and chemical protection of SOM fractions, and variable microbial activity, may contribute to the different and possibly low decay rates that are observed [*Blanco-Canqui and Lal*, 2004; *von Luetzow et al.*, 2008]. Spatial heterogeneity also plays a role, since it mediates the relative importance of the different organic matter retention mechanisms. Hence, the transit time can be used as a single parameter that integrates a number of biochemical and physical processes characterizing SOM dynamics.

[4] Biogeochemical cycling is often described by means of mathematical models based on a number of interconnected compartments, each releasing C according to linear and nonlinear kinetic laws. Nonlinear kinetics may be preferred when describing short-term processes because the coupling of decomposer biomass and organic matter substrate is important [*Manzoni and Porporato*, 2007, 2009; *Schimel*, 2001]. However, for long-term studies (yearly timescale or longer), organic matter dynamics may be approximated by a system of compartments each exchanging C following first-order kinetics [*Baisden and Amundson*, 2003; *Manzoni and Porporato*, 2009]. At those timescales, the decomposer activity may be considered relatively constant, thereby making the amount of available substrate the predominant controlling factor of the C fluxes. The linear approximation is also often used to describe short-term dynamics where the decomposers' activity rapidly adapts to substrate availability and hence is never a limiting factor [*Manzoni and Porporato*, 2009]. Accordingly, soil is often regarded as a linear time-invariant input-output system, where the dynamics of the output (e.g., heterotrophic respiration) is computed as a function of the previous inputs of organic matter to the soil and a suitable “weighting function” that describes how much C remains in the soil of the past inputs at present time [*Bolin and Rodhe*, 1973; *Eriksson*, 1971]. Such a weighting function can be interpreted as the probability distribution of the times spent by each C molecule within the system, i.e., the transit time distribution, and can be used to transform a generic input into the output from the system [*Eriksson*, 1971; *Nir and Lewis*, 1975]. This overall transit time distribution results from the combination of the transit time distributions of the individual well mixed linear compartments making up the complex network of biogeochemical cycling, thereby carrying all (in the case of a linear system) the information on the system architecture.

[5] The mathematical theory behind linear input-output systems is well established, and has been used in climate studies [*Bolin*, 1981; *Enting*, 2007; *Thompson and Randerson*, 1999], hydrology [*Nash*, 1957; *Dingman*, 1994; *Rodriguez-Iturbe and Valdes*, 1979; *McGuire and McDonnell*, 2006], nutrient and contaminant transport [*Rinaldo et al.*, 2006; *Sardin et al.*, 1991], chemical engineering [*Aris*, 1989; *Nauman*, 2008], as well as in signal and electrical circuit theory [e.g., *Chen*, 2004]. Despite the fact that most long-term soil C models are linear [*Manzoni and Porporato*, 2009], there are only few applications of the theory to soil and ecosystem biogeochemistry. Notable exceptions are the linear analysis of organic matter cohorts [*Ågren and Bosatta*, 1996], of retention of C tracers in soils [*Balesdent*, 1987; *Ågren et al.*, 1996; *Bruun et al.*, 2004, 2005], and of the sensitivity of ecosystem C compartments to increased atmospheric carbon dioxide concentrations [*Emanuel et al.*, 1981; *Thompson and Randerson*, 1999]. In these studies, the theoretical relationships between the age and transit time distributions, as well as the sensitivity of C transit times to changes in model structure and kinetic constants, were not explicitly considered.

[6] In addition to the above applications, linear system theory is also useful to critically compare the variety of linear model structures that have been proposed and used in characterizing long-term SOM dynamics. The transit time distribution, which can be computed from the model compartmental organization, is especially suited to this goal. Directly computing the transit time distribution has also the advantage of retaining the relevant information on the long-term persistence of C in soils in an analytical compact formulation. For example, heavy-tailed transit time distributions hint at efficient C retention mechanisms resulting from low decomposition rates and the presence of physically and chemically protected zones.

[7] Along these lines, the primary objectives of this work are (1) to elucidate the controls of model structure on the system transit time and age distribution, and (2) to critically compare current soil biogeochemical models using these distributions. To address these two issues, we first present a range of models conceptually representing different biogeochemical processes and briefly review the theory of time-invariant linear impulse-response systems. Second, we compute transit times and age distributions for the different models, and describe the simulated responses to an external stochastic input. Finally, we focus on the mean transit time (which is proportional to the equilibrium soil organic matter content) and analyze how it is affected by changes in environmental conditions. Using a measure of the sensitivity of the mean transit time we discuss how different model formulations may lead to different responses to these environmental changes.