Periodic solutions for soil carbon dynamics equilibriums with time-varying forcing variables

Numerical models that simulate the dynamics of carbon in soil are increasingly used to improve our knowledge and help our management of the carbon cycle. Calculation of the long-term behavior of these models is necessary in many applications but encounters the difficulty of managing the periodic forcing variables, e.g. seasonal variations, such as carbon inputs and decomposition rates. This calculation is conventionally done by running the model over large time durations or by assuming constant forcing variables. Two methods, which make it possible to rapidly compute periodic solutions taking into account the time variations of these variables, are proposed. The first one works on discrete-time models and the second one on continuous-time models involving Fourier transforms. Both methods were tested on the Rothamsted carbon model (RothC), a discrete-time model which has also been given a continuous approximation, using realistic and unrealistic sets of time-varying forcing functions. Both methods provided an efficient way to compute the periodic solutions of the RothC model within the application domain of the model. Compared to running the discrete model to the equilibrium, reduction in the computational cost was of up to 95% at the expense of a maximum absolute error of 1% for the estimation of carbon stocks. For specific distributions of the forcing variables the use of Fourier transform of zero order, which was equivalent to assume constant forcing variables, led to a maximum absolute error of 55% in the estimation of the long-term behavior of the model. There, a Fourier transform of order higher than zero is required.