We use the data-assimilated, steady (nonseasonal) circulation of , which has a horizontal resolution of 2∘×2∘ and 24 vertical levels with spacing that increases with depth. Temperature, salinity, radiocarbon, CFC-11, and PO4 have been used as constraints in the data assimilation. The circulation is also constrained dynamically, and the data assimilation used the wind-stress climatology of and specified horizontal and vertical viscosities of 5×104 and 10-4m2s-1, respectively. The circulation's advective–diffusive transport operator has fixed horizontal and vertical eddy diffusivities of 103 and 10-5m2s-1, respectively. We emphasize that the circulation effectively provides a ventilation-weighted transport because it has been optimized against PO4 and the ventilation tracers CFC-11 and radiocarbon. The steady circulation thus does not bias estimates of preformed nutrients in the way an annual-average circulation would.

Organic matter sinks as POP, dissolves, and remineralizes at depth. Inverse models of the phosphorus cycle suggest that dissolved organic phosphorus (DOP) represents a relatively small fraction of the total dissolved phosphorus that we neglect here (no DOP tracer) for simplicity and numerical efficiency. We note that our estimates of phosphorus export effectively capture the export due to DOP, despite DOP not being explicitly represented as a separate tracer. This is because the optimization of the biogeochemical parameters minimizes the mismatch with the observed PO4 distribution, which in the real ocean is determined by the remineralization of all organic phosphorus, including DOP. Because the particle transport is much faster than the fluid transport across a grid box, we approximate particle transport and remineralization, which acts as an interior source of nutrients, as instantaneous. We model this process for each phytoplankton functional class by the “source” operator, ScP, which reassigns a “detrital” fraction fc of the uptake rate to a remineralization rate throughout the water column, while a fraction 1-fc remineralizes in situ where the uptake occurred. We therefore express ScP in terms of a biogenic redistribution operator BP such that ScP=1-fc+BPfc, where it is understood that multiplication by the field fc precedes the action of BP. (The operator BP does not have a functional class subscript c because it redistributes a unit uptake with the same profile regardless of functional class.) We assume that the remineralization of organic matter releases dFe and phosphate in the same ratio with which they were taken up. Therefore, ScFe=ScP.

The values of fc, which set the export efficiency of each class, are not directly constrainable by the data used here. The nutrient concentrations constrain only the total export (i.e., summed over all classes), while the phytoplankton concentrations constrain the uptake, but not the export, of each class. We therefore use the optimized detrital fractions from the work of : the detrital fc fractions are modeled as decreasing with temperature T so that fc=fc0e-kfT, with kf=0.032 ∘C-1 independent of class, fsml0=0.14, flrg0=0.74, and we assign fdia=flrg. The large and diatom classes are thus ∼5 times more efficient at exporting organic matter than the small class. We acknowledge that suggested that small and large phytoplankton have similar export efficiencies. However, their very sparse data do not provide strong evidence that fsml=flrg, only that their values are uncertain. Indeed, our state estimates using the fc values of are consistent with the data presented by . Plots of the fractional uptake of each class versus its fractional export (not shown here) are broadly consistent with Fig. 1b of .

Following , we assume that the detrital production rate is fluxed as POP through the base of the euphotic zone at ze=73.4m with ϕcPOP(ze)=∫ze0fcUc dz, and that the POP flux attenuates with depth according to the Martin power law ϕcPOP(z)=ϕcPOP(ze)(z/ze)-b due to remineralization in the aphotic zone. The operator BP therefore injects PO4 with the divergence of ϕcPOP into the aphotic water column. The flux into the ocean bottom is remineralized in the lowest grid box as in the work of . The exponent b was determined to be b=0.82 using a restoring-type phosphate-only model. (Most parameters were optimized for the full coupled model – for details of our optimization strategy see Sect. .)

The redistribution operator BSi similarly injects silicic acid into the aphotic water column with the divergence of the opal flux, ϕbSi, which attenuates because of temperature-dependent opal dissolution following and . For each latitude and longitude, ϕbSi is computed as the solution to ∂zϕbSi(z)=-(κSimax/wSi)exp⁡(-TE/T(z))ϕbSi(z), with the boundary condition ϕbSi(ze)=∫ze0RSi:PfdiaUdia dz. We use TE=11,481K as and the same detrital fraction fdia for the opal export and diatom POP export. The parameter combination κSimax/wSi has nearly the same value as determined by , but was re-optimized here for a simple restoring-type model that takes subgrid topography into account (see Appendix ).

The scavenging operators Ss,POP and Ss,bSi act on JPOP and JbSi to redistribute a fraction of the iron scavenged at every layer throughout the water column below. In terms of the corresponding redistribution operators, we write Ss,POP=fPOPBs,POPandSs,bSi=fbSiBs,bSi, where the fractions fPOP and fbSi were both fixed at 0.9 (see Appendix ). The operators Bs,POP and Bs,bSi in effect “recycle” scavenged iron. They are very similar to BP and BSi but in addition to distributing scavenged iron from the euphotic zone to the aphotic zone, they also redistribute the scavenging rates of every aphotic layer to a source of redissolving iron with the divergence of the scavenging particle fluxes. The flux of scavenged iron into the bottom is assumed to be lost forever so that there would be iron loss even for 100 % efficient recycling of scavenged iron. (For details see Appendices  and .)

To compute accurate particle fluxes for constructing all S operators, we take subgrid topography into account as done by, using the high-resolution ETOPO2V2c data set . This is done by calculating for each grid box the fractional area occupied by the subgrid topography, which is also the fraction of the particle flux intercepted by the subgrid topography. For each grid box, the fraction of the flux intercepted is instantly remineralized and redissolved for Sci or buried in the sediments for Ss,POP and Ss,bSi (details in Appendix ).

The PO4 uptake rate at a given point is a function of the local temperature T, irradiance I, and nutrient concentrations. The uptake rate for functional class c is calculated as the product of its phytoplankton concentration, pc, and its specific growth rate, μc, as Uc=μcpc=pcmaxτceκTFI,cFN,c2, where τc is the timescale for growth, pcmax is the phytoplankton concentration under ideal conditions, and FI,c and FN,c are dimensionless factors in the interval [0,1) that represent light and nutrient limitation, respectively, as defined below. We derive Eq. () similarly to and as follows.

First, pc is calculated diagnostically by assuming steady state between growth and mortality, which avoids the need to carry explicit plankton concentration tracers. This is justified by the coarse resolution of our model, which implies transport timescales across a grid box that are much larger than the typical timescales for phytoplankton growth. Based on 's mortality formulation, pc can be modeled by a logistic equation ∂tpc=μcpc-λpcpc*pc, where the pc/pc* fraction scales the specific mortality rate λ, and pc* has been referred to as the “pivotal” population density e.g.,. Equation () has a nontrivial steady state, given by pc=μcλpc*. We assume that all phytoplankton classes share the same specific mortality rate λ, which depends only on temperature. For simplicity, we follow and approximate the T dependence of λ to be identical to that of the growth rate, which was determined by to be of the form eκT. Thus, λ=λ0eκT, where λ0 is a constant.

Our formulation differs from that of and , who raise the ratio pc/pc* to a power α=1 or α=1/3 to differentiate between small and large phytoplankton classes. Here, we instead differentiate between classes by assigning them different half-saturation rates and maximum uptake-rate constants similarly to the work of (details in Sect.  and ).

We model the specific growth rate μc as multiplicatively co-limited by temperature, light, and nutrients: μc=1τceκTFI,cFN,c, where τc is the growth timescale at 0∘C under ideal conditions and the temperature dependence eκT is identical to that used for the mortality rate e.g.,. To group parameters for more efficient optimization, we define pcmax=pc*/(λ0τc), so that diagnostic Eq. () for the phytoplankton concentration becomes pc=FI,cFN,cpcmax. Substituting Eq. () and Eq. () into Uc=μcpc gives Eq. (), which is similar to the uptake formulations of and .

We note that in the Sea of Japan the model's circulation produces unrealistic nutrient trapping, likely due to under-resolved currents. We therefore set the specific growth rate in the Sea of Japan to zero, effectively removing it from the computational domain of the biogeochemical model.

All three-dimensional fields (e.g., the concentrations χi) are discretized on our model grid and organized into column vectors (length n∼2×105 at our resolution). Linear operators such as T, Sci, and Ss,j are correspondingly organized into n×n sparse matrices. The steady-state tracer Eqs. ()–() then become a 3n×3n system of equations that are nonlinear because of the iron scavenging and the co-limitation of the PO4 uptake.

The 3n×3n system is solved efficiently using Newton's method e.g.,. Convergence of the Newton method depends on the initial guess for the solution and is not guaranteed. For the initial guess of χP and χSi we use the annual mean fields of the World Ocean Atlas WOA13, interpolated to our grid, and for the initial guess of χFe we use the dFe fields estimated by . The Newton solver typically converges to numerical precision in ∼10 iterations.

We optimize the model parameters by systematically minimizing a quadratic cost function of the mismatch between modeled and observed fields. For PO4 and Si(OH)4, for which gridded climatologies are available, we define the weights based on the grid box volumes, organized into vector v, as wP=v(χ‾Pobs)2V,andwSi=v(χ‾Siobs)2V, where we have normalized the weights by the total ocean volume V and the squared global mean observed concentrations. This nondimensionalizes the quadratic cost terms and scales them to the same order of magnitude. For dFe, for which only sparse observations are available, we also define weights wFe based on grid box volumes, but observations that are part of a vertical profile receive additional weight as detailed in Appendix .

With diagonal weight matrix Wi=diag(wi) for nutrient i, its cost for the mismatch with observations is then given by Ei=δχiTWiδχi, where δχi≡χi-χiobs. For χPobs and χSiobs we use WOA13 fields interpolated to our grid, and for χFeobs we used the GEOTRACES intermediate data product and the data set compiled by .

The cost terms for the nutrient mismatch do not provide a strong constraint on the relative sizes of the phytoplankton classes because the nutrients are determined by their combined export. We therefore include additional terms in our cost function that constrain the phytoplankton concentrations pc to the recent satellite derived estimates of . These estimates provide phytoplankton concentrations for picophytoplankton (0.5–2 µm in diameter), nanophytoplankton (2–20 µm), and microphytoplankton (20–50 µm), which we identify with our small, large, and diatom functional classes. We use the entire mission composite data set as the satellite climatology .

Because of the large dynamic range of the phytoplankton concentrations, we consider mismatches in the log of the concentrations; that is, δπc≡log⁡[(pc+pcobs‾)/p0]-log⁡[(pcobs+pcobs‾)/p0], where pcobs‾ is introduced to limit the logarithm where the phytoplankton concentration falls to zero, and p0=1mgCm-3 nondimensionalizes the argument of the logarithm. For each class, we construct normalized weight vectors: wc=v[log⁡(pcobs‾/p0)]2Veup, where Veup is the global euphotic volume.

Organizing mismatches into vectors and weights into diagonal matrices, we calculate the cost for the phytoplankton concentration mismatch as Eplk=∑cδπcTWcδπc, and combine the costs for the nutrient and plankton mismatches into the total cost: Etot=ωPEP+ωSiESi+ωFeEFe+ωplkEplk, which we minimize to constrain our model parameters by the available observations. In Eq. () the ω weights were chosen such that the four cost terms contribute roughly equally to the total cost for a typical member of our family of state estimates. This was achieved with (ωP,ωSi,ωFe,ωplk)=(1,0.47,0.044,0.30), the smaller weight for dFe reflecting its larger root mean square (rms) mismatch and hence much larger cost.

Our model has ∼50 biogeochemical parameters that can in principle be determined through objective optimization given appropriate observational data. However, even with perfect data, some parameters can compensate for others (e.g., two parameters appearing as a ratio) so that not all parameters are independent. Other parameters cannot be optimized because the mismatch with available nutrient and phytoplankton data is not sensitive to their value. In practice, it therefore is not possible to optimize all parameters, and care is needed to optimize only those parameters that independently shape the nutrient and phytoplankton concentrations.

The parameters associated with the remineralization of phosphate and dissolution of opal are well constrained by the high-quality climatologies of PO4 and Si(OH)4. However, the iron cycle is relatively poorly constrained because the dFe data are much more sparse in both time and space, and estimates of the iron sources range over 2 orders of magnitude e.g.,. Moreover, the ligand field that determines the scavengable free iron is highly uncertain. Given these challenges, the recent inverse model of the iron cycle by considered a family of state estimates for a range of external source strengths, an approach we will follow here for our coupled model.

Another key consideration is computational cost. Even with the numerically efficient Newton solver, optimization typically requires hundreds of solutions of Eqs. ()–() per optimized parameter. We therefore optimized no more than 13 parameters at a time. We acknowledge that the minimum attained by sequentially optimizing groups of independent parameters is generally different from jointly optimizing all independent parameters, but computational and practical considerations demanded a sequential approach. We justify this a posteriori by the fact that we are able to achieve fits to the observed nutrient concentration fields with rms mismatches similar to those of other recent data-constrained models e.g.,. Given these considerations, we adopted the following strategy: i.

Parameters that are measurable and considered well-known, as well as parameters that are unconstrainable by our cost function or with values that are not critical, because they are strongly compensated for by other parameters, were assigned values from the literature as collected in Table . The considerations that entered our choice of prescribed parameters are detailed in Appendix .

Figure  shows the quality of the fit to nutrient and phytoplankton data for all our optimized state estimates, which span a wide range of source strengths. For ease of presentation, state estimates are divided at σH=1GmolFeyr-1 into low and high hydrothermal cases, with σH ranging from 0.073 to 11.Gmolyr-1. For high σH, we focused on correspondingly higher aeolian and sedimentary source regimes. Source-parameter space was not explored uniformly because (i) the final step of our optimization adjusted our initial choice of sources, and because (ii) some source choices produced spurious numerical difficulties for the Newton solver.

All state estimates fit the macronutrient fields about equally well, but the overall quality of fit as quantified by the square root of the quadratic mismatch (“total cost”, top panels of Fig. ) gets systematically worse with increasing aeolian source strength, σA, especially for high hydrothermal sources. This worsening fit for high σA is reflected in the mismatch of all three nutrients. The state estimates with a total cost that exceeds the smallest misfit by ∼5 % (corresponding to a total cost above 15.4) are discarded from our family. This essentially eliminates state estimates with σA≳22Gmolyr-1 and σH≳5Gmolyr-1 (black crosses in Fig. ). (If we include the “crossed-out” states in plots below that show scatter across the family of state estimates, the visual impact is virtually imperceptible.) While it is clear from Fig.  that high-σA states are less likely, we hasten to add that the cost threshold for inclusion in the family is arbitrary as we do not have a formal error covariance to convert the cost into a likelihood.

For a small fraction of our state estimates, the optimization pushed the maximum possible Fe:P ratio R0Fe:P to near-zero values. These cases are unrealistic because zero R0Fe:P means significant P uptake and export are maintained without Fe uptake. We therefore also exclude cases for which the optimized R0Fe:P falls below 0.5mmolFe(molP)-1 from our family of state estimates. Removing these unphysical outliers has negligible visual impact on plots that show the entire family of state estimates.

In terms of total cost, there is little sensitivity to the strength of the sedimentary source – scavenging can be optimized for a sedimentary source ranging over 2 orders of magnitude for an overall similar quality of fit. For low σH, there are small opposing rms mismatches for PO4 and dFe, with a slightly better PO4 fit for higher sedimentary source and a slightly better dFe fit for lower sedimentary source, although the variation in the mismatch is less that 1 % of the global mean concentrations.

While the mismatch for dFe is substantial at ∼45 % of the global mean dFe concentration, the smallest dFe mismatch occurs when all three sources are low. The dFe mismatch rapidly increases with σA, consistent with the findings of the much simpler model of . The overall cost and the mismatch for each nutrient are insensitive to the strength of the hydrothermal source.

While Fig.  shows some variations with the source strengths in the overall quality of the fit, it is clear that the iron sources and scavenging sinks are poorly constrained by the available nutrient and phytoplankton observational data. Given the uncertainties in the sources and the small cost differential between family members, it is not appropriate to single out the state with the lowest numerical cost as being the most realistic. We therefore use the entire family of state estimates below to assess the robustness of our results and to elucidate the systematic variations of the carbon and opal exports with the fractional source of each iron type. The uncertainty in the value of any metric is assigned from its spread across the family.

As a typical representative of our family of state estimates, for which we plot results below, we selected the state with (σA,σS,σH)=(5.3,1.7,0.9)GmolFeyr-1. This state is typical in that it lies at the mode of the distribution of overall rms misfit values and, for most quantities, tends to lie in the middle of the range across the family.

We emphasize that the variations across the family of state estimates explored here are variations of the fully optimized biogeochemical states. These variations cannot be used to infer the system's response to dFe perturbations for which the other biogeochemical parameters would not change. Such perturbations, which are of great interest in themselves, are beyond the scope of this paper and will be examined in a separate publication.

The nutrient concentrations are well constrained for all members of our family of state estimates. We quantify the overall fit of the modeled nutrient concentrations in terms of the joint probability density function (pdf) of the modeled and observed concentrations. This joint pdf may be thought of as the binned scatter plot of the modeled versus observed values for all grid boxes. The binning for each nutrient was weighted using the associated cost-function weights, wi. These joint pdfs are shown in Fig.  for all three nutrients for our typical state estimate. Both the PO4 and Si(OH)4 pdfs fall close to the 1:1 line, showing high fidelity to observations. For PO4 the cost-weighted rms error is ∼5% of its global mean of 2.17µM. In comparison, achieved an rms mismatch of 3% by jointly optimizing the uptake rate of each grid box with the circulation. Silicic acid has a larger rms mismatch of ∼12% relative to its global mean of 89.1µM. This is similar to the 13% rms error reported by , who used the same circulation but a much simpler model of the silicon cycle.

The global mean dFe concentration is well constrained within the narrow range of 0.56–0.68nM across the family of state estimates. For iron, the joint probability is by necessity computed using only those grid boxes that contain dFe observations. The scatter from the 1:1 line is much larger than for the macronutrients with a substantial rms mismatch of 0.29nM or 44% of the mean. This mismatch is comparable to that of other models e.g.,. Compared to the simpler model of , the joint pdf shows that our dFe field has a wider, more realistic dynamic range. We note that, while report an rms mismatch of only 0.19nM, they also employed different weights for the model–observation mismatch. If we recompute the rms mismatch of the optimized dFe field of using the weights of this work, we also obtain a 0.29nM mismatch.

The relatively large mismatch for dFe not only quantifies model deficiencies, but to a large degree also reflects the fact that we are comparing snapshot observations against a steady-state coarse-resolution model. The dFe observations have difficult-to-quantify temporal and spatial sampling biases, and dFe being a trace element is sensitive to episodic events in the aeolian source e.g.,, and possibly to internal episodic events such as submarine volcanism e.g.,.

To quantify the spatial structure of the dFe mismatch, we examine vertical profiles for each basin. For both model and observations, we only use the grid boxes that contain observations and average horizontally over the basins using the cost weights wFe. The resulting profiles are shown in Fig. . The family of model profiles generally overlaps with the observational uncertainties. The estimates are particularly close to the observations near the surface. In the abyssal oceans, the spread in the family of profiles is larger. This spread is in part a reflection of the weights in our cost function. Most dFe observations are available in the upper ocean, implying a small variance of the mean concentration and hence large weights, while deep observations tend to be sparser with smaller weights (for details on the weights see Appendix ).

Figure  also shows systematic biases in the inferred dFe concentrations. Biases are particularly strong in the Pacific, where the observations tend to be underpredicted by as much as ∼ 0.3nM above ∼ 1500 m and overpredicted by ∼ 0.2 nM below ∼ 2000 m depth. The typical estimated Pacific profiles are too linear in the upper 1500m, with vertical gradients that are too weak above ∼ 300m and too strong below ∼ 1000m. In the Atlantic, a smaller low bias of ∼ 0.15nM can be seen between ∼ 500 and ∼ 1300m depth.

These biases could be due to deficiencies in our model such as oversimplified ligand parameterization, but one must also keep in mind that there are hard-to-quantify biases in the observations. The observations are too sparse to form a reliable climatology, and it is remarkable that we can fit the available observations as well as we do. The larger biases in the Pacific could well be due to the absence of Pacific transects in the GEOTRACES Intermediate Data Product 2014 , which means that mismatches in the Pacific incur a relatively smaller penalty in our cost function.

Figures d, , and  are appropriate quantitative comparisons between estimated and observed dFe, given that essentially raw bottle data are compared with a coarse-resolution steady-state model. For completeness, Appendix  also compares the main transects included in the GEOTRACES Intermediate Data Product 2014 with our typical state estimate.

The pattern of the aeolian source is identical for all family members because we only vary its global source strength, σA. The sediment source is keyed to export production, which is well constrained across the family of state estimates. Therefore, the sedimentary iron source patterns are very similar across all state estimates, with only the global strength σS varying among state estimates. The initial hydrothermal pattern is set by the OCMIP 3He source , but for total hydrothermal sources larger than ∼0.5Gmolyr-1, the optimized contributions from each basin changed substantially. Across our family of estimates the mean and standard deviations of the percentage contributions from each basin to the total hydrothermal source are (15±9) % for the Atlantic, (52±6) % for the Pacific, (16±2) % for the Indian Ocean, and (17±3) % for the Southern Ocean (south of 40∘ S). For reference, the vertically integrated iron sources of our typical state estimate are plotted in Appendix .

Because of the small variations in the source patterns, the patterns of the vertically integrated total sinks of dFe, ∫dz(1-SsP)JPOP, ∫dz(1-SsSi)JbSi, and ∫dzJdst, also vary little across the family of state estimates (see Appendix  for plots of these quantities for our typical state estimate). Note that these sinks balance the total source exactly in steady state. For our typical state, scavenging by POP and opal each account for about half of the total iron sink. The patterns of POP and opal scavenging are determined by the phosphorus and opal exports and by the concentration of free iron. Consequently, the POP scavenging sink is strongest in the tropics, and the opal scavenging sink is strongest in the Southern Ocean. The sink due to mineral dust scavenging reflects the pattern of the aeolian dust input modulated by the free-iron concentration. For our family of state estimates, the sink due to dust scavenging is essentially negligible, being about 3 orders of magnitude smaller than the POP and opal scavenging.

We note that the partition of scavenging among the different particle types cannot be inferred robustly from our inverse model. This is because the nutrient and phytoplankton data used do not provide separate constraints on the scavenging by each particle type, only on the total amount of scavenging. Moreover, scavenging by one particle type can be compensated for by another type because of overlap in their spatial patterns. However, the partition among particle types does vary systematically across our family of estimates. Scavenging by dust is negligible for all state estimates, while the fraction scavenged by POP ranges from ∼10 % for the lowest iron sources to saturation near 100 % for the highest iron sources. (The complementary fraction is due to opal scavenging.)

Figure  shows the typical state's zonally averaged dFe concentration for each basin and for the global ocean. For each zonal average, Fig.  also shows the corresponding profile of horizontally averaged dFe for each member of our family. The profiles are tightly clustered showing that the large-scale features of the dFe field are well constrained despite the large variations of the iron sources. The inverse model fits the observed dFe field for widely different sources by adjusting the corresponding scavenging. While these adjustments keep the total dFe field close to the observations, the relative contributions from the aeolian, sediment, and hydrothermal sources are unconstrained and can vary widely.

We calculate dFe concentrations due to each source following by replacing the dFe concentration tracer Eq. () by an equivalent linear diagnostic system that has the same solution. This linear system, corresponding to a given solution of the full nonlinear system, is obtained by replacing the iron uptake and scavenging by linear operators. Specifically, the dFe uptake RFe:PUc is replaced with LU,cχFe and the scavenging rate Jj with LJ,jχFe, where the linear operators, organized into matrix form, are simply specified from the uptake and scavenging rates of the nonlinear solution as LU,c≡diag(RFe:PUc/χFe) and LJ,j≡diag(Jj/χFe). The dFe concentration, χFek, due to source sk (with k∈{A,S,H}) is then computed by replacing the total source in the linear equivalent system with sk.