Iron fertilization is explored by tracking dissolved iron (DFe) through its
life cycle from injection by aeolian, sedimentary, and hydrothermal sources
(birth) to burial in the sediments (death). We develop new diagnostic
equations that count iron and phosphate regenerations with each passage
through the biological pump and partition the ocean's DFe concentration
according to the number of its past or future regenerations. We apply these diagnostics to a family of
data-constrained estimates of the iron cycle with sources

Iron is an essential micronutrient for phytoplankton photosynthesis

Many studies have been devoted to quantifying the extent to which biological
productivity and ocean carbon uptake can be influenced by altering the iron
supply through geoengineering

Here we develop new diagnostics for following an iron molecule's passages
through the biological pump from its “birth” by source injection to its
“death” by scavenging and burial in the sediments. We show that the number
of times that an iron molecule is biologically utilized and regenerated at
depth during its birth-to-death journey is a fundamental metric for
understanding the marine biosphere's iron fertilization efficiency. Using
Green-function techniques, we track DFe through past regenerations back to
its birth and through future regenerations forward to its death. We apply our
new diagnostics to a family of data-constrained state estimates of the iron
cycle and quantify the

Our analysis uses the 287 optimized state estimates of

All these optimized states fit the available observed DFe and nutrient
concentrations about equally well, with a total iron source that ranges from

While recent studies of the iron cycle have begun to quantify the
distribution and pathways of regenerated iron

Here, for the first time, the entire birth-to-death journey of DFe is considered.
Specifically, we aim to address the following questions.

What fraction of the DFe distribution in the current state of the ocean has
passed

How much does the DFe present at any given location in the current state of the ocean contribute to global export production per DFe molecule?

How do the mean number of past and future passages through the biological pump, and the closely related iron fertilization efficiency, depend on the uncertain iron source strengths?

We find that for states with global iron sources above

We link the mean number of future regenerations to the intrinsic iron fertilization efficiency, which we are able to quantify at any point in the ocean. At the surface, which is most relevant to geoengineering, we find that this fertilization efficiency is largest in the central tropical Pacific, with the Southern Ocean having comparable efficiency only for states with a high total iron source.

In Sect.

Following

Iron source strengths in units of

The uptake

By performing our analyses for a family of states spanning a wide range of
possible iron sources, we are able to establish features that are insensitive
to the iron-source scenario, while
also being able to quantify systematic variations of key metrics with the
uncertain iron source strengths. Below we will show spatial patterns for a
typical state estimate and for four other states that are representative of
the variations across our family of estimates. The source strengths and bulk
iron lifetimes of these five representative states are collected in
Table

In order to track iron from its birth at the source to its eventual death
(via the irreversible part of scavenging), we consider a labeling tracer
that we can think of as being attached to the nonlinearly evolving DFe. This
labeling tracer has the same concentration as DFe but satisfies a linear
evolution equation in which the nonlinear uptake and scavenging are replaced
by linear processes. These linear processes are diagnosed from the nonlinear
steady-state solution of Eq. (

To construct the linear processes for the iron labels, we diagnose the local
rate constants

The source

Schematic of the birth-to-death life cycle of a labeled DFe molecule (red
dot).
As captured by Eq. (

Because the superposition principle applies to the labeling tracers, we can
consider the DFe concentrations to be the sum of the concentrations of
different iron “source types”:

We caution the reader that the equivalent linear model
(Eq.

To establish how much a given iron molecule has contributed to organic matter export since its birth, we partition the DFe concentration according to the number of its past passages through the biological pump.

We first consider the concentration of DFe that has never been regenerated
since it was injected (born) by the source. This concentration, denoted by

We can now calculate the concentration of DFe from source

Global zonal averages of the concentration of DFe that was regenerated

Figure

As the number

How robust is the large-

The sedimentary and hydrothermal iron concentrations get their largest
contributions from iron that has never been regenerated (

Figure

Figure

Figure

To explore the variations of the asymptotic eigenmode pattern across our
different state estimates, Fig.

Normalized global zonal averages of the concentration of DFe that has passed

A key metric of how much the DFe field has contributed in the past to
organic matter export is the mean number of times,

A given iron molecule at

In other words, on average, one of the type-

Figure

The zonally averaged

Zonal averages of the mean number of past passages through the
biological pump (i.e., since birth),

We expect significant variations in the amplitudes of

We now ask how many times a given DFe molecule in the ocean will get
regenerated in the future before eventually being permanently scavenged out
of the system. The natural way to formulate the necessary equations is to
consider the time-reversed adjoint flow

To compute the number of future regenerations, it is useful to calculate the
death rate

One can show that the local fraction

In analogy with Eq. (

The mean number of regenerations until death,

Figure

Global zonal averages of the fraction of DFe that will be
regenerated

To quantify the approach of

We return to Fig.

Figure

We now return to the asymptotic eigenpatterns of

Global zonal averages of

Figure

Three-dimensional intrinsic fertilization efficiency metrics for our
typical state, zonally averaged across the global oceans.

Returning to Fig.

We begin by asking how much an arbitrary (“test”) source

Viewed another way, by construction

Figure

Figure

The surface patterns of

The key result of Fig.

Normalized patterns of our intrinsic fertilization efficiency
metrics at the surface.

How do our estimates of intrinsic fertilization efficiency compare to
previous estimates of differently defined fertilization efficiencies? Across
our entire family of estimates,

To compare the importance of the different iron source types in the current
state of the ocean,

It follows algebraically from Eq. (

Our analysis comes with a number of caveats that should be kept in mind. Some
of these were already identified in the work of

While the state estimates used were optimized against the observations
available at the time, we note that the sedimentary sources of the underlying
inverse model of

Other caveats relate to biogeochemical parameters that could not be
optimized. The recyclable fraction,

A key control on our model results is the Fe : P stoichiometry. The model
approximates the iron dependence of the Fe : P uptake ratio by a Monod
function with a half saturation constant that is the same for all
phytoplankton classes. We acknowledge that this may not be realistic. For
example,

The recent work of

Importantly, we would like to note that not every process thought to
influence DFe needs to be explicitly modeled for a useful representation of
the iron cycle. The model of

We have presented a new conceptual and mathematical framework for quantifying
the contribution of DFe to the biological pump during its journey from birth
by an external source to death by irreversible scavenging and burial. New
diagnostics were developed to partition the DFe concentration into the
fraction that was regenerated

We applied our new diagnostics to a family of optimized state estimates of
the global coupled iron, phosphorus, and silicon cycles, assuming steady
state. All states of the family match the observed nutrient concentrations
about equally well despite spanning a large range of external iron source
strengths. Performing our analyses across the family of states allowed us to
identify aspects of the birth-to-death journey of DFe that are robust and to
quantify systematic variations with iron source strength. Our key findings
are as follows.

A large portion of the global DFe inventory never participates in the
biological pump. For states with iron sources

The three-dimensional distribution of

The spatial distribution of DFe that was regenerated

The pattern of the large-

The fraction

The fraction of DFe that will be regenerated

We defined and quantified the intrinsic iron fertilization efficiency at
point

Intimately connected to the mean number of past and future regenerations are
the age and expected remaining lifetimes of DFe in the ocean. A full
exploration of these timescales and the associated transit-time distributions
is beyond the scope of this study. However, in future work, we plan to
explore the timescales of the iron cycle and their connection to setting the
efficiency with which iron fertilization achieves carbon sequestration

The state estimates used here were generated as documented by Pasquier and Holzer (2017) and are based on the following publicly available data sources. The temperature, phosphate, and silicic-acid data used are available from the World Ocean Database (Boyer et al., 2013). For the dissolved iron data, we used the GEOTRACES Intermediate Data Product 2014 (Mawji et al., 2015) as well as an earlier data compilation by Tagliabue et al. (2012). The satellite estimates of the concentrations of picophytoplankton, nanophytoplankton, and microphytoplankton are available from the PANGAEA data repository (Kostadinov et al., 2016). For annual mean irradiance, we used MODIS Aqua PAR data (NASA Ocean Biology Processing Group, 2015).

First we verify that

The defining equation for

Applying

To diagnose the local death rate with which iron is permanently removed from
the ocean, we first rewrite the reversible scavenging operator

Basin and global zonal averages of the death rate

For the interpretation of our diagnostics it is helpful to know how the local
iron death rate

To explore the sensitivity of our results to variations in

Figure

Figure

To derive the recursion relation for the fraction of iron at a given location
that is regenerated exactly

All adjoints here are defined in terms of the volume-weighted inner product
so that for linear operator

Now consider the concentration of iron

Similarly we can construct the concentration of DFe that has been regenerated
exactly zero times since being at

To construct the recursion equation for

Using similar techniques as in Appendix

Here we calculate the steady-state globally integrated export that results
from a steady injection with source

The authors declare that they have no conflict of interest.

Benoît Pasquier acknowledges support from the Government of Monaco, the Scientific Centre of Monaco, the Frères Louis et Max Principale Foundation, the Cuomo Foundation, Jefferson Keith Moore (DOE grant DE-SC0016539 and NSF grant 1658380), and François Primeau (NSF grant 1658380). Mark Holzer acknowledges a UNSW Goldstar award. Edited by: Jack Middelburg Reviewed by: Christoph Völker, Jonathan Lauderdale, and one anonymous referee

_{2}levels, Geophys. Res. Lett., 32, L09703,