The MOPS model is based on phosphorus and simulates seven compartments. Phosphate, phytoplankton, zooplankton, DOP and detritus are calculated in units of millimoles of phosphorus per cubic metre (mmolPm-3). Oxygen is coupled to the P cycle with a constant stoichiometry given by R-O2:P. Aerobic remineralisation of organic matter follows a saturation curve, with half-saturation constant KO2. Aerobic remineralisation ceases when oxygen declines; at the same time, denitrification takes over, as long as nitrate is available above a defined threshold, DINmin. Like the oxic process, suboxic remineralisation follows a saturation curve for oxidant nitrate, with half-saturation constant KDIN. MOPS does not explicitly resolve the different oxidation states of inorganic nitrogen (nitrite, N2O, ammonium), but assumes immediate coupling of the different processes involved in nitrate reduction, the end product being dinitrogen see also. All organic components are characterised by a constant N : P stoichiometry of d=16. Loss of fixed nitrogen is balanced by a simple parameterisation of nitrogen fixation by cyanobacteria, which relaxes the nitrate-to-phosphate ratio to d with a time constant, μNFix*. In the long term, nitrogen fixation balances the simulated loss of fixed nitrogen via denitrification, although they may occur in distant areas (see , for more details).

Detritus sinks with a vertically increasing sinking speed: w=az. Assuming a constant degradation rate r, in equilibrium this would result in a particle flux curve given by F(z)∝z-b, with b=r/a. For better comparison with values of b derived from observations e.g., and with the simpler model RetroMOPS (see below), a is expressed in terms of b (assuming constant, nominal r=0.05 d-1). A fraction of detritus deposited at the sea floor (at the bottom of the deepest vertical box) is buried instantaneously in some hypothetical sediment. The fraction buried depends on the deposition rate onto the sediment. Non-buried detritus is resuspended into the last box of the water column, where it is treated as regular detritus. The phosphorus budget is closed on an annual timescale through resupply via river runoff. More details about the biogeochemical model and parameters and their effects on model behaviour can be found in and .

MOPS' structure has been simplified by skipping the explicit simulation of phytoplankton, zooplankton and detritus (see Fig. S1 in the Supplement). The remaining equations of, and functional relationships between, phosphate, nitrate, oxygen and DOP have been parameterised similar to MOPS. Because the downscaled model resembles so many features of earlier biogeochemical models simulated in a global context e.g., but keeps the oxidant dependency of MOPS, the model is named “RetroMOPS”.

All model simulations apply the Transport Matrix Method github.com/samarkhatiwala/tmm for tracer transport, with monthly mean transport matrices (TMs), wind, temperature and salinity (for air–sea gas exchange) derived from a 2.8∘ global configuration of the MIT ocean model, with 15 levels in the vertical, as described in and . The circulation model was forced with climatological annual cycles of wind, heat and freshwater fluxes, and subject to a weak restoring of surface temperature and salinity to observations. Its configuration is similar to that applied in the Ocean Carbon-cycle Model Intercomparison Project (OCMIP) , which has been assessed against observations of temperature, salinity and mixed-layer depth , CFCs , and radiocarbon . Overall, its performance is comparable to other global models.

Using this efficient offline approach, with a time step length of 1/2 day for tracer transport and 1/16 day for biogeochemical interactions, a simulation of 3000 years requires about 0.5–1.5 h on 4 nodes (24-core Intel Xeon Ivybridge) at a high-performance computing centre (www.hlrn.de). After 3000 years most tracers have approached steady state see alsofor long time trends of MOPS simulated in a different circulation, and the transient of the misfit function becomes very small (see Fig. S2). The last year is used for model analysis and evaluation of the misfit function.

Optimisation of parameters is carried out using an estimation of distribution algorithm, namely the covariance matrix adaption evolution strategy . The application of this algorithm to the coupled biogeochemistry–TMM framework has shown good performance with respect to quality and efficiency (in terms of function evaluations) and is described only briefly below. More details about the algorithm, its set-up and coupling to the global biogeochemical model can be found in .

Let n be the number of biogeochemical parameters to be estimated. In each iteration (“generation”) the algorithm defines a population of λ individuals (biogeochemical parameter vectors of length n), with λ=10 derived from the default parameter λ=4+3ln⁡(n),. The candidate vectors are sampled from a multi-variate normal distribution, which generalises the usual normal distribution, also known as Gaussian distribution, from R to the vector space Rn.

Following the simulation of these λ individual model set-ups to steady state (3000 years), the misfit function is evaluated, and information of the current as well as previous generations is used to update the probability distribution in Rn such that the likelihood to sample good solutions increases. Usually, the realisation of the probability distribution update ensures that information of former solutions fades out slowly, resisting for several iterations. Therefore, the population (the number of model simulations per generation) in CMA-ES is smaller, and of less computational demand, than in classical evolutionary algorithms. Nevertheless, CMA-ES can still, to a certain degree, perform well with misfit functions characterised by a rough topography .

As in the misfit to observations J is defined as the root mean square error (RMSE) between simulated and observed annual mean phosphate, nitrate and oxygen concentrations , mapped onto the three-dimensional model geometry. Although regridding the observations onto the coarser model geometry removes some of the variability, this method is computationally more efficient in an optimisation framework. Also, a sensitivity study with a similar coupled model showed that accounting for the variance inherent in the observational data and arising from regridding did not have a large influence on the misfit .

Deviations between model and observations are weighted by the volume of each individual grid box, Vi, expressed as fraction of total ocean volume, VT. The resulting sum of weighted deviations is then normalised by the global mean concentration of the respective observed tracer: J=∑j=13J(j)=∑j=131oj‾∑i=1N(mi,j-oi,j)2ViVT, where j=1,2,3 indicates the tracer type and i=1,…,N are the model locations for N=52749 model grid boxes; oj‾ is the global average observed concentration of the respective tracer. The terms mi,j and oi,j are model and observations, respectively. By weighting each individual misfit with volume, J serves more as a long-timescale geochemical estimator, in contrast to a misfit function that, for example, focuses on (rather fast) turnover in the surface layer, or resolves the seasonal cycle.

Based on a “hand-tuned”, a priori set-up of MOPS , which hereafter is referred to as MOPSr, presented an optimisation of mostly surface-related parameters (hereafter referred to as MOPS∘S). They chose a very wide range of parameter types, across all trophic levels and acting on different time and space scales. In that optimisation many of the surface parameters were difficult to constrain, because of a misfit function that consists mostly of observations in the deep ocean. Optimisation MOPS∘D presented here applies the same metric, but focuses on parameters in subsurface waters. The selection of parameters to be optimised is motivated by the large uncertainty regarding extent and expansion of oxygen minimum zones in models , and because little knowledge exists about their values, or even parameterisations.

Parameter KO2 determines the affinity of the aerobic remineralisation to oxygen, and the gradual transition from this process to denitrification see Eqs. 15 and 20 of. KDIN determines the affinity of denitrification to nitrate. Parameter DINmin defines the lower threshold for the onset of denitrification. MOPS∘D also optimises the maximum rate of nitrogen fixation, μNFix*, which balances fixed nitrogen loss through denitrification. The fifth and sixth parameter to be estimated are the oxygen requirement per mole of phosphorus remineralised, R-O2:P, and the flux (or remineralisation) length scale, b. Upper and lower boundaries of parameters to be optimised have been set to a rather wide range (Table ), to allow optimisation to explore a wide range of potential parameters. The optimal parameters of MOPS∘S for light and nutrient affinity of phytoplankton, zooplankton grazing and its mortality are retained in MOPS∘D (Table ). Therefore optimisation MOPS∘D builds upon a previous tuning of surface processes.

Most of the processes affected by the parameters to be optimised take place in suboxic waters, e.g. of the eastern equatorial Pacific (EEP). Given the coarse model geometry, it is possible that circulation dynamics are not represented well in the model. To investigate the influence of observations within this region on misfit function and parameter estimates, MOPS∘D is repeated with a reduced data set, that excludes the EEP (here: east of 140∘ W, between 10∘ S and 10∘ N) from the misfit function. This optimisation is named MOPS*∘D.

In the RetroMOPS model, processes such as grazing of phytoplankton, and its subsequent release of organic or inorganic phosphorus are parameterised via a single component, DOP. Because DOP production and decay regulate the partitioning between sinking and dissolved organic matter, optimisation RetroMOPS∘ targets these parameters, namely σ, λs and λ. While σ, as the parameter that regulates the export ratio, may be more or less well constrained, λs and λ both include a variety of processes, which may act on timescales of days to years. applied a multi-G model to incubations of DOP sampled in surface waters of the middle Atlantic Bight, and measured decay constants for the very labile fraction (32 % of total DOP) of ≈80 yr-1, with a range of 3–254 yr-1. Half of total DOP was in the labile fraction and characterised by a decay constant of ≈ 7 yr-1, ranging from 0.8 to 43 yr-1. However, these observations may not be directly transferable to globally simulated DOP, because most of the simulated ocean is far off the productive shelf areas; further, DOP in RetroMOPS is assumed to mimic a variety of biogeochemical components and processes. In a three-step optimisation study , who optimised a global model of semi-labile and refractory DOM against observations estimated rates of 0.016 yr-1 for semi-labile DOP at the surface, and 0.22 yr-1 for semi-labile DOP in the mesopelagic zone, i.e. much lower than suggested by . Summarising, the potential decay rate of the very labile to semi-labile fraction varies over several orders of magnitude, from O(0.01) to O(100) yr-1.

Optimisation of RetroMOPS focuses on the dominant labile to semi-labile fraction, but allows for some potential fast turnover rates of DOP at the sea surface towards the values observed by. To obtain a first impression on model sensitivity towards these parameters, a set of nine a priori experiments, that vary λ between 0.18 and 0.72 yr-1 and λs between 0 and 0.36 yr-1, has been carried out (Table ), which provides a guidance for upper and lower boundaries for optimisation of RetroMOPS. To nevertheless explore the full range of potential decay rates, the maximum possible rate (λ+λs) for optimisation is set to 7.2 yr-1, towards the average decay rate of the labile DOP observed by . Optimised RetroMOPS∘ will be compared to the sensitivity experiment with the lowest misfit (λs=0, λ=0.36), which is denoted as RetroMOPSr.

The explicit representation of detritus in MOPS may result in considerable numerical diffusion particularly on coarse vertical grids as used here; see also and thus in a different estimate of optimal b than when applying a direct flux curve, such as in RetroMOPS. Therefore, b is included as the fourth parameter to be optimised. The effect of explicit vs. implicit flux description on parameter estimate will be discussed in more detail below.

All other parameters (primary production, oxidant-dependent remineralisation, stoichiometry) have been fixed to those obtained in optimisations MOPS∘S and MOPS∘D (Table ). By doing so, optimisation RetroMOPS∘ builds upon previous optimisations of the more complex MOPS, and overlooks the faint possibility that a parameter that is insensitive in one model, might not be so in another. While it might be desirable to optimise all parameters of RetroMOPS at once, this study rather aims at investigating to what extent a simpler model can serve as a shortcut to the more complex one, given the applied misfit function and observations.

Both R-O2:P and b are constrained very well by the observations, as indicated by a well-defined minimum of the misfit function (Fig. S3) and a narrow, almost Gaussian distribution of the best 10–1 % of parameters (Fig. ). On the other hand, parameters related to the oxidant affinity of remineralisation or nitrogen fixation are determined with lower accuracy. This is also reflected in the rather wide range of candidate solutions within 1 ‰ of the best misfit, which vary between 10 and 20 % of their assigned a priori range (Table ). Thus, in the presence of noise inherent in the observations, some parameters could only be estimated within a quite wide range of uncertainty, a feature that has already been addressed in a one-dimensional model by . So far, the potential consequences of this parametric uncertainty for other metrics (such as extent of oxygen minimum zones, OMZs) and possibly transient scenarios (e.g. their impact on simulated future evolution of OMZ volume) are not known.

The good determination of b by dissolved inorganic tracers is in agreement with earlier studies that applied the same model . Its optimal value is very close to that obtained in MOPS∘S, i.e. higher than the value estimated by . Optimisation of maximum nitrogen fixation rate shows a slightly skewed distribution, but suggests an overall good estimate of this parameter. Optimal parameters for oxidant-dependent remineralisation also show wide, skewed distributions, with their mode near the lower (KO2) or upper (KDIN, DINmin) boundary.

The high thresholds for the limitation of denitrification protect nitrate from becoming depleted in the upwelling regions, particularly the eastern equatorial Pacific, and resemble results obtained by : To prevent their model from reproducing unrealistically low nitrate values in this region, they had to impose a threshold of 32 mmol NO3 m-3 for the occurrence of denitrification. An explanation for this requirement of a high nitrate threshold might be found in the representation of the equatorial intermediate current system in coarse-resolution models, which can result in spurious zonal oxygen gradients . It is possible that the optimisation of biogeochemical parameters attempts to ameliorate these effects, which are in fact caused by the parameterisation of physics.

To further investigate the impact of this region on the parameter estimate, an additional optimisation was carried out, that targets the same set of parameters, but omits the eastern equatorial Pacific from the calculation of the misfit function. This optimisation MOPS*∘D generates a lower threshold of nitrate for the onset of denitrification, and a higher maximum nitrogen fixation rate (Table ), resulting in slightly enhanced fixed nitrogen turnover, particularly in the eastern equatorial Pacific (Fig. ). Compared to MOPS∘D the estimates of KDIN and DINmin become more uncertain with respect to the best 10 % to 1 ‰ individuals, and even show a bimodal distribution (Fig. S4, Table ). The uncertainty in parameter estimates can be related to the missing data in regions of simulated denitrification. Because the misfit function excludes the EEP it is lower then when considering the entire ocean (Table ). A posteriori evaluation of misfit to the entire data set results in a misfit of 0.439, the same as for MOPS∘D. The only moderate effect of the eastern equatorial Pacific on optimisation is likely related to the small volume occupied by this region, compared to total ocean volume.

Global fixed nitrogen turnover depends on parameters for oxidant dependency of remineralisation: in MOPS∘S, both denitrification and nitrogen fixation are very high (Fig. ), and outside the observed range (Table ). Because of the reduced affinity to nitrate, in MOPS∘D pelagic fixed nitrogen loss is almost halved and now agrees with observed global estimates (Table ). Further, as a result of lower denitrification, the nitrate deficit in the eastern equatorial Pacific is smaller, but at the cost of a small underestimate of observed oxygen in this region (Fig. ). The latter is a consequence of the now very low half-saturation constant for oxygen uptake (Table ). In MOPS*∘D the constraint on nitrate affinity is again relaxed, resulting in an enhancement of fixed nitrogen turnover by about 20 %, towards the upper limit of observed estimates (Table ).

Overall, optimising parameters related to the oxidant affinity of oxic and suboxic remineralisation leads to a slightly improved fit to tracer concentrations, to J*=98 % of that of MOPS∘S (Table ), and to a better agreement with observed estimates of global biogeochemical fluxes (Table ). Although the eastern equatorial Pacific, and potential unresolved processes in simulated circulation, has no effect on global misfit, its effect on some parameter estimates results in an increase in global fixed nitrogen loss of about 20 %.

Given that parameters related to surface biology were difficult to constrain in MOPS∘S, and, within a certain range, exert only a small influence on the fit to global tracer distributions , this section examines if RetroMOPS, as a model that parameterises surface biology in a much simpler way, suffices to represent biogeochemical tracer fields. Starting from growth and decay parameters optimised in MOPS, sensitivity experiments and optimisation search for optimal parameters for DOP production and decay, that mimic the surface nutrient turnover of MOPS.

Current, state-of-the-art biogeochemical models address questions such as the future evolution of oxygen minimum zones, or uptake of anthropogenic carbon by the ocean e.g.. Compared to these models, MOPS and RetroMOPS, presented here, are of a rather low structural complexity. RetroMOPS is quite similar to early models addressing these tasks, among them the pioneering work of Ernst Maier-Reimer e.g., while MOPS resembles models of intermediate complexity such as HAMOCC e.g. or HadOCC . However, very simple models such as RetroMOPS are still being used, e.g. for inverse methods e.g. or to investigate specific processes, where their computational efficiency and structural simplicity facilitates model analysis e.g.. In contrast to these very simple model are models that simulate different plankton groups and size classes of detritus, e.g. PISCES , MEDUSA or PlankTOM .

Despite this large range of structural complexity, there have been only few studies which evaluate these models against a common data set, and with a common circulation. One example is the study by , who compared the output of six different global biogeochemical models, coupled to a common circulation model, and simulated over 118 years, against data sets of surface pCO2, DIC, alkalinity, DIN, Chl a and primary production. The models varied in complexity from 7 to 57 compartments, and thus also in their computational demand by almost a factor of 5. To assess model skill ranked the models with respect to spatial correlation between, and variance of, model and observations. In general, the more complex models performed better with respect to simulated variance, but the simpler models performed better with respect to spatial correlation. Although no model was superior across all metrics, they concluded that “Results suggest little evidence that higher biological complexity implies better model performance in reproducing observed global-scale bulk properties of ocean biogeochemistry” .

The lack of distinction between models and their ability to represent biogeochemical tracers is corroborated by the study by , who evaluated three different biogeochemical ocean models within a common framework for the Earth system. The models varied in complexity between 1 and 30 components. Following a spin-up over 100 years, analysed both a transient and pre-industrial scenario with respect to the model's representation of macronutrients, oxygen, DIC and export. All three models performed quite similarly with respect to the observed tracer fields, as well as with the transient evolution of carbon uptake and oxygen concentrations. Therefore, in the presence of noise inherent in observations, and given the sparsity of biological data sets, the question of whether more complexity is indeed beneficial seems unresolved so far – at least if the model is supposed to represent mostly biogeochemical processes, instead of biological interactions, and is compared against bulk biogeochemical properties.