The ocean's nutrient cycles are important for the carbon balance of
the climate system and for shaping the ocean's distribution of dissolved
elements. Dissolved iron (

Here we build a steady-state model of the ocean's coupled phosphorus,
silicon, and iron cycles embedded in a data-assimilated steady-state global
ocean circulation. The model includes the redissolution of scavenged iron,
parameterization of subgrid topography, and small, large, and diatom
phytoplankton functional classes. Phytoplankton concentrations are implicitly
represented in the parameterization of biological nutrient utilization
through an equilibrium logistic model. Our formulation thus has only three
coupled nutrient tracers, the three-dimensional distributions of which are found
using a Newton solver. The very efficient numerics allow us to use the model
in inverse mode to objectively constrain many biogeochemical parameters by
minimizing the mismatch between modeled and observed nutrient and
phytoplankton concentrations. Iron source and sink parameters cannot jointly
be optimized because of local compensation between regeneration, recycling,
and scavenging. We therefore consider a family of possible state estimates
corresponding to a wide range of external iron source strengths. All state
estimates have a similar mismatch with the observed nutrient concentrations
and very similar large-scale

Both the magnitude and pattern of the phosphorus and opal exports are well
constrained, with global values of

The ocean's nutrient cycles control the primary
productivity of the global marine ecosystem and the ocean's biological carbon
pump, which are crucial components of the global carbon cycle that regulate
atmospheric

We focus on dissolved iron (

We model phosphate (

With a changing climate, we expect not only changes in the ocean circulation,
but also changes in the winds, hydrological cycle, and land use, and hence in
the aeolian iron supply. To understand how such changes impact the ocean's
nutrient cycles, it is necessary to model the coupling between the nutrients
mechanistically. While global biogeochemistry models have been used
extensively for this purpose

The intercomparison of iron models by

We build on the simple iron-only inverse model of

However, in the study of

We use our inverse-model estimates of the coupled

How well can the modeled

What are the nutrient limitation patterns that emerge from the data-constrained estimates of the coupled nutrient cycles, given that direct observational data on these patterns are very sparse?

How well constrained are the phosphorus and opal exports of optimized state estimates with widely different iron sources?

What fractions of the phosphorus and opal exports are supported by aeolian, hydrothermal, and sedimentary iron, and how do these fractions vary with the iron-source strengths?

In the following, we detail the model formulation in Sect.

We distinguish three phytoplankton functional groups, nondiatom small and
large phytoplankton as well as diatoms, with a nominal separation between
small and large at a cell diameter of

We consider the three nutrients

The terms proportional to

Equations (

We use the data-assimilated, steady (nonseasonal) circulation of

Organic matter sinks as POP, dissolves, and remineralizes at depth. Inverse
models of the phosphorus cycle

The values of

Following

The redistribution operator

The scavenging operators

To compute accurate particle fluxes for constructing all

The

First,

Our formulation differs from that of

We model the specific growth rate

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.

We model the limitation of functional class

Using a minimum over nutrient type

We prescribe the irradiance

Because we key all biological production to

Equation (

The Monod formulation (

Our representation of the

The aeolian source,

The sedimentary source,

To model the hydrothermal source,

Dissolved iron can be either chelated by ligands or “free”. We assume that
scavenging acts only on the concentration

To compute the concentration of the scavenging particles, we use the fact
that the flux divergences generated by the biogenic transport operators must
be balanced by local remineralization or dissolution rates; that is,

The key control on shaping the free-iron concentration, and hence the
scavenging, is the ligand concentration

As is the case for most iron models, there is no need to explicitly represent
the chemical precipitation of

All three-dimensional fields (e.g., the concentrations

The

We optimize the model parameters by systematically minimizing a quadratic
cost function of the mismatch between modeled and observed fields. For

With diagonal weight matrix

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

Because of the large dynamic range of the phytoplankton concentrations, we
consider mismatches in the log of the concentrations; that is,

Organizing mismatches into vectors and weights into diagonal matrices, we
calculate the cost for the phytoplankton concentration mismatch as

Our model has

The parameters associated with the remineralization of phosphate and
dissolution of opal are well constrained by the high-quality climatologies of

Another key consideration is computational cost. Even with the numerically
efficient Newton solver, optimization typically requires hundreds of
solutions of Eqs. (

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 parameters that set the phosphate remineralization and
opal dissolution profiles were optimized by minimizing the mismatch with

The remaining parameters were optimized using our coupled
model. We first assign initial values for all these parameters and then
sequentially update these initial values by optimizing subsets of parameters
as detailed in Appendix

Parameters that were prescribed from the literature or that were separately optimized in a submodel.

Optimized parameters and range across family of state estimates.

Total cost metric and rms mismatch of the nutrient concentrations as a function of the aeolian, hydrothermal, and sedimentary iron source
strengths (

Figure

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.

For a small fraction of our state estimates, the optimization pushed the
maximum possible

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

While the mismatch for

While Fig.

As a typical representative of our family of state estimates, for which we
plot results below, we selected the state with

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

We now examine in more detail how well our estimates match the observations
against which they were optimized. We focus here on the nutrient fields,
which contribute the bulk of the cost function. Each phytoplankton field
contributes only

Joint distribution of the cost-weighted observed and modeled concentrations of

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,

The global mean

The relatively large mismatch for

Basin-wide, cost-weighted average profiles of the (red) observed and (grey) modeled

To quantify the spatial structure of the

Figure

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

Figures

For a given phytoplankton functional class, different nutrients are known to
limit biological production in different parts of the ocean

Limiting nutrients can be determined observationally

To display the pattern of the nutrient limitations, we could use the fact
that we have three nutrients with which to define an RGB color as

The patterns of limiting nutrients for each phytoplankton functional class.
The color cube at the bottom right shows the eight possible limitation regimes of our inverse model:
red corresponds to

Figure

The small phytoplankton class shows a much simpler pattern. Limitation occurs
primarily in the subtropical oceans with small patches of iron limitation
also in the Southern Ocean and tropical Pacific. Iron limitation dominates
the subtropical South Pacific, while

The general features seen in Fig.

Our limitation patterns can also be compared to those calculated for summer
conditions in the BEC model of

A key metric of the nutrient cycles is their export production, which
determines the strength of the biological pump

We similarly calculate the opal export as

Local export production for each nutrient (maps on the left) and its zonal integral (curves on the right).
Maps are shown for our typical state estimate, while we plot the zonal integral of each family member (scaled for

Figure

The differences with the estimate of

Our estimates of export production compare well with the satellite-based
estimates of

Our estimates of phosphorus export in the subtropical gyres compare well to
the POP exports of

Figure

There is very little spread in the carbon and opal export productions across
our family of state estimates as can be seen by the tightly clustered zonal
integrals plotted in grey in Fig.

Figure

All export fields of Fig.

Here we document some of the key features of the iron cycle as constrained by
our inverse model. Certain features such as the

The pattern of the aeolian source is identical for all family members because
we only vary its global source strength,

Because of the small variations in the source patterns, the patterns of the
vertically integrated total sinks of

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

Estimates of the

Figure

We calculate

Figure

We quantify the contribution of each iron type to the export production as
follows. In our formulation, nonzero

Phosphorus export supported by each iron type [aeolian

While the total export production is well constrained regardless of the iron source strengths, the production supported by a given iron type varies substantially with the magnitude of the corresponding source. (Summing over the three iron types yields the well-constrained total.) However, regardless of the source amplitudes, the geographic patterns of the export supported by each iron type is similar across the entire family of state estimates.

Figure

Figure

Underscoring the similar source distribution of hydrothermal

Percent global phosphorus export (equivalently carbon export) supported by each iron type (aeolian, sedimentary, hydrothermal) versus the corresponding fractional source of that iron type. The superposed lines are least-squares fits to theoretical relationships with fixed relative export-support efficiencies. (See text for details.)

While the total phosphorus export is well constrained and varies little
across our family of state estimates, the magnitude of a given
iron-type-supported export production varies systematically with the
corresponding fractional source strength. To quantify these systematic
variations, Fig.

Figure

The key point of Fig.

The fact that the scatter plots of

The ability of aeolian iron to make disproportionately large contributions to
supporting organic-matter export, quantified here by a relative
export-support efficiency greater than unity, is presumably due to fresh
aeolian iron being directly injected into the euphotic zone. The
less-than-unity relative export-support efficiencies of sedimentary and
hydrothermal iron reflect the fact that iron from benthic sources is
generally subject to scavenging before it even reaches the euphotic zone.
Because most large sedimentary sources are relatively shallow, a typical
sedimentary

Opal export supported by each iron type [aeolian

Percent global opal export supported by each iron type (aeolian, sedimentary, hydrothermal) versus the corresponding fractional source of that iron type. Lines represent fits to theoretical curves with fixed relative export-support efficiencies. (See text for details.)

The opal export supported by each iron type can be calculated analogously,
and the corresponding geographic patterns are shown in Fig.

The amplitude of the opal-export patterns varies systematically with the iron
source strength as summarized in Fig.

Similar to our preceding analysis of phosphorus export, we define the
relative opal export-support efficiencies by

The lower efficiency of aeolian iron for supporting opal export is consistent
with the fact that opal export occurs primarily in the Southern Ocean, where
direct aeolian input is small. Similarly, the greater efficiency of
sedimentary and hydrothermal iron is consistent with the bulk of the opal
export occurring in the upwelling regions of the Southern Ocean where access
to deep iron sources is greatest. This is supported by the fact that plots of

Our approach has a number of limitations that should be kept in mind. Most
importantly, inverse-model estimates are only as good as the data used to
constrain them. The

Important nonnutrient observational fields for our inverse model are the
satellite-measured photosynthetically active radiation (PAR) and
ocean-color-derived estimates of the size-partitioned phytoplankton
concentrations. Small-scale features of the PAR field, e.g., in the Weddell
Sea where ice and cloud cover play a role, are uncertain with the PAR for
different time averages showing different features. The satellite-based
estimates of phytoplankton concentrations also carry unquantified
uncertainties due to a number of assumptions

Most biogeochemical parameters are determined through objective optimization
against available observations, but the construction of the cost function and
the choice of which parameters are optimized and which are prescribed are
necessarily subjective. For example, choosing a different set of weights

The uncertainty in key metrics (e.g., global phosphorus export) was
quantified in terms of their spread across our family of state estimates and
in terms of systematic variations with the iron source strengths. While our
efficient numerics allow us to easily determine the linear sensitivities of
any metric with respect to all parameters (from which one can also estimate
uncertainty), we did not do so here because the spread in the metric across
the family is more relevant. Given the large set of parameters

A key limitation of our approach is that seasonality is ignored and we use a
steady circulation. This circulation is constructed so that its transport
reproduces the annual-mean observed temperature, salinity, CFC-11,
radiocarbon, and

Our model of the macronutrient cycles makes a number of simplifying
approximations. We ignore external inputs of silicic acid and therefore also
neglect permanent burial of opal in sediments. While this approximation has
been shown to have a negligible impact on particle fluxes

Although our model of the iron cycle includes an explicit representation of
the redissolution of scavenged iron, effects of subgrid topography, and
dynamic coupling to the phosphorus and silicon cycles, and is thus much more
complex and mechanistic than the iron model of

Other uncertainties concern the phosphorus cycle to which the uptake of the
other elements is keyed. While the optimized phosphate fields have the
smallest misfit with observations, our model of the phosphorus cycle makes
several simplifying approximations. The Martin exponent is assumed to be
globally uniform although in reality it almost certainly varies spatially

We emphasize that the carbon export reported here was simply our estimate of
the phosphorus export converted to carbon units. No effort was made to
compute a more realistic carbon export such as could be achieved with an
explicit representation of the carbon cycle (which would require additional
tracers and was numerically too expensive) and the

We have formulated a steady-state model of the coupled phosphorus, silicon, and iron cycles that is embedded in a steady data-assimilated global circulation. The model is of intermediate complexity and couples the nutrient cycles through co-limitations on biological uptake and through the scavenging of iron by organic particles. The concentrations of the small, large, and diatom phytoplankton functional classes are calculated diagnostically, which avoids the need for plankton concentration tracers. We explicitly represent iron scavenging by POP, opal, and mineral-dust particles, and the redissolution of POP- and opal-scavenged iron. Subgrid topography is parameterized for the sedimentary iron sources and intercepts all vertical fluxes. The relative simplicity of the biogeochemical model and the matrix formulation of the steady-state advective–diffusive transport afford highly efficient numerics. Steady-state solutions are readily found using a Newton solver, which permits the model to be used in inverse mode to constrain many of the biogeochemical parameters through objective optimization. The optimization minimizes the mismatch with the observed nutrient concentrations and with satellite-derived estimates of phytoplankton concentrations.

Our estimates of the macronutrient concentrations closely match the
observational WOA13 climatology with volume-weighted rms errors of

Given that even the order of magnitude of the iron sources is uncertain, we
produced a family of state estimates with a wide range of iron source
strengths. Because different iron source strengths are compensated for by
optimally adjusting the scavenging parameters, each family member fits the
observations with roughly the same fidelity. This means that the available
observed

We partitioned the

Nutrient limitation patterns were defined by jointly considering whether the

The export productions of phosphorus and opal are well constrained across our
family of state estimates, in terms of both pattern and magnitude. Because we
model three phytoplankton functional classes with distinct optimized uptake
timescales, our phosphorus export (expressed in carbon units) of

We quantified the role of the iron cycle in shaping the phosphorus and opal
export productions. We find that each iron source type (aeolian, sedimentary,
hydrothermal) supports phosphorus and opal exports with a distinct geographic
pattern that is robust across the family of state estimates. The export
pattern supported by a given iron type reflects the nature of its source.
Sedimentary and hydrothermal iron support phosphorus export that is
dominantly shaped by the large-scale patterns of upwelling, which brings
these iron types to the surface. Aeolian iron supports export that is shaped
by both the pattern of direct aeolian input and by large-scale upwelling,
which brings regenerated as well as scavenged and redissolved aeolian

The fraction of the globally integrated export supported by a given iron type
varies systematically with its fractional global source. These variations
quantify the export-support efficiency of each iron type per source-injected
molecule. Aeolian iron is most efficient and supports a fraction of the
global export that is larger than its fractional source, while sedimentary
and hydrothermal iron are less efficient, supporting fractions of the global
export that are less than their fractional sources. This is because

Our optimized model is ideally suited for investigating the response of the global ocean ecosystem to a variety of biogeochemical perturbations. In the future, we will report on the model's response to perturbations in the iron supply and on a more comprehensive analysis of the detailed workings of the iron cycle.

The temperature, phosphate, and silicic acid data used in this study are
available from the World Ocean Atlas v2 2013
(

The recycling operator for POP-scavenged iron,

Similarly, the recycling operator for opal-scavenged iron,

We follow

The subgrid topography parameterization is implemented by applying Eqs. (

We use the

As in Eq. (

The following considerations determined which parameters were not optimized
and how their values were chosen. The recyclable fractions of POP and opal
scavenging,

We first chose an initial set of values for the remaining parameters as
collected in Table

The initial irradiance half-saturation constants were taken from the work of

The initial values of the parameters of the

We first optimized the hydrothermal iron source parameters,

We jointly optimized the three irradiance half-saturations

We jointly optimized the half-saturations

We were not able to optimize the

Because of compensation with the maximum

We then jointly re-optimized the

Only the parameters of the iron cycle remain to be optimized.
Iron source and sink parameters cannot jointly be optimized because of strong
local compensation. (Although the aeolian source injects

As a final step, we jointly optimized all source-strength
parameters

Figure

The inverse-model estimates capture the broad global patterns of the
phytoplankton concentrations reasonably well, although some biases are also
evident. The observation-based diatom and large concentration has a minimum
at

Comparison of modeled

The global mean phytoplankton concentration of each functional class was
stable across all members of our family of state estimates with ranges of

Local export production (maps on left) and its zonal integral (curves on the right) expressed in carbon units (using

Figure

Figure

We emphasize that a direct comparison with the GEOTRACES sections is subject
to a number of caveats. We use a coarse-resolution, steady-state inverse
model, while the GEOTRACES sections provide snapshots in space and time.
Therefore, our model cannot capture any transient plumes (e.g., from an
African dust event) that are highly localized and episodic. Our state
estimates can only capture the long-term average concentrations,
coarse-grained to

However, what matters for our inverse model, the biological production of which is
mechanistically driven by

Dissolved iron concentrations of the typical state estimate (contours) compared to the GEOTRACES Intermediate Data Product 2014 (dots). The abscissa runs south to north or west to east along the transects (map).

Vertically integrated sources of aeolian (top,

Maps of the vertically integrated iron sinks of our typical state estimate due to

Figure

Figure

The authors declare that they have no conflict of interest.

We thank François Primeau for making the data-assimilated circulation
available to us, Natalie Mahowald for providing the aeolian soluble iron and
dust flux estimates of