BGBiogeosciencesBGBiogeosciences1726-4189Copernicus PublicationsGöttingen, Germany10.5194/bg-15-7177-2018The number of past and future regenerations of iron in the oceanand its intrinsic fertilization efficiencyPast and future regenerations of ironPasquierBenoîtpasquieb@uci.eduhttps://orcid.org/0000-0002-3838-5976HolzerMarkDepartment of Applied Mathematics, School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, Australianow at: Department of Earth System Science, University of California, Irvine, CA, USABenoît Pasquier (pasquieb@uci.edu)3December201815237177720314August201816August20182November20185November2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://bg.copernicus.org/articles/15/7177/2018/bg-15-7177-2018.htmlThe full text article is available as a PDF file from https://bg.copernicus.org/articles/15/7177/2018/bg-15-7177-2018.pdf
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
σtot in the range 1.9–41Gmolyr-1. We find
that for states with σtot>7Gmolyr-1,
50% or more of the DFe inventory has not been regenerated in the
past and 85% or more will not be regenerated in the future. The
globally averaged mean number of
past or future regenerations
scales with the bulk iron lifetime
τ∼σtot-1 and has a range of 0.05–2.2 for
past and 0.01–1.4 for future regenerations.
Memory of birth location fades rapidly with each regeneration, and DFe
regenerated more than approximately five times is found in a pattern shaped
by Southern Ocean nutrient trapping. We quantify the intrinsic fertilization
efficiency of the unperturbed system at any point r in the ocean as
the global export production resulting from the DFe at r per iron
molecule. We show that this efficiency is closely related to the mean number
of future regenerations that the iron will experience. At the surface, the
intrinsic fertilization efficiency has a global mean in the range
0.7–7molP(mmolFe)-1 across our family of state
estimates and is largest in the central tropical Pacific, with the Southern
Ocean having comparable importance only for high-iron-source scenarios.
Introduction
Iron is an essential micronutrient for phytoplankton photosynthesis
e.g.,. In the ocean, dissolved iron (DFe) is a trace
element that has been shown to limit biological production over vast
high-nutrient, low-chlorophyll (HNLC) regions
e.g.,. In HNLC regions, ample
phosphate and nitrate are available but phytoplankton are not able to
completely utilize these macronutrients because of a lack of sufficient DFe.
Iron thus exerts a major influence over the marine carbon cycle and hence
over the global climate system.
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
e.g.,. Relatedly, it is natural to
ask how efficient iron is in supporting the export of organic matter in the
current state of the system and what the relative efficiencies of the
different iron sources are (e.g., aeolian versus sedimentary).
suggested that artificial iron fertilization would
not sufficiently impact atmospheric CO2 to mitigate anthropogenic
global warming. Their argument relied, among other
things, on quantifying the efficiency of artificial iron fertilization, which
was inferred from the Southern Ocean Iron Experiment (SOFeX). Similarly,
used data from localized artificial iron
fertilization experiments to explore the per-iron-molecule efficiency of
supporting carbon export, defined by the change in local dissolved inorganic
carbon per unit added DFe. From this, estimated the
“natural” fertilization efficiency (i.e., the efficiency of perturbing DFe
only) by correcting for the lack of ligand protection in artificial iron
fertilization. used an adjoint technique and a
model of the coupled carbon, phosphorus, and iron cycles to quantify the
sensitivity of global biological production and air–sea carbon fluxes to
local perturbations in the aeolian iron source. They found that both
quantities were most sensitive to iron addition in the central and eastern
tropical Pacific.
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 intrinsic iron fertilization efficiency,
i.e., the fertilization efficiency of DFe molecules in the unperturbed system, in three dimensions throughout the global ocean. To the best of our
knowledge, this has never done before. Previous studies, like those of
and , estimated the
fertilization efficiency only for the specific surface regions where iron was
artificially injected into the euphotic zone.
Our analysis uses the 287 optimized state estimates of
, which were obtained from a steady-state inverse
model of the ocean's coupled iron, phosphorus, and silicon cycles, embedded
in the data-assimilated steady global ocean circulation of
. The members of our family of state
estimates correspond to different iron source strengths, given that the
actual values for the real ocean are still highly uncertain
e.g.,.
All these optimized states fit the available observed DFe and nutrient
concentrations about equally well, with a total iron source that ranges from
1.9 to 41GmolFeyr-1 across our family. (Atmospheric
models estimate an aeolian source of soluble iron to the ocean of
∼6GmolFeyr-1; e.g., .) Having states for a wide range of iron
source scenarios enables us to quantify systematic variations with source
strength.
While recent studies of the iron cycle have begun to quantify the
distribution and pathways of regenerated iron , to the best of our knowledge the
biological cycling of iron during its full birth-to-death lifetime has
previously not been quantified. In particular, the future passages of a given
DFe molecule through the biological pump have not been systematically
considered, thus missing a potentially important part of its contribution to
supporting carbon export. In fact, dismissed the
importance of iron regenerated at depth for subsequently fertilizing
production altogether based on the assumption that all DFe that is
regenerated at depth would be lost to scavenging. However, scavenging is
weaker at depth so that deep regenerated DFe molecules are not necessarily
scavenged and instead may be transported back to the surface where they may
support production multiple times before being scavenged out of the system.
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 n times through the biological pump in the past, and what fraction will
pass m times through the biological pump in the future for any given n and m?
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
7GmolFeyr-1, most of the DFe gets scavenged out of the
system without participating in the biological pump. Future passages through
the biological pump are less likely than past passages, but for sources
around 7GmolFeyr-1, roughly 10% of the iron
inventory will still participate in future biological production before
death. DFe that has been regenerated more than about five times in the past
can be found in a characteristic pattern (mathematically an eigenmode) that
bears a strong signature of Southern Ocean nutrient trapping. DFe that will
be regenerated many times in the future is similarly found in a
Southern-Ocean-intensified eigenmode.
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. , we briefly detail the salient features of the
iron model and introduce our diagnostic framework. The DFe distribution is
partitioned according to the number of past and future passages through the
biological pump in Sects. and ,
respectively. In Sect. , we develop the connection with
the iron fertilization efficiency. Caveats of our approach are discussed in
Sect. , and we present conclusions in
Sect. .
Iron modelNonlinear model
Following , we write the nonlinear tracer
equation for DFe concentration χ as
(∂t+T)χ=∑c(Sc-1)Uc+∑j(Sj-1)Jj+∑ksk,
where T is the advective–diffusive transport operator of the
data-assimilated steady ocean circulation of
. Uc is the iron uptake rate of
phytoplankton functional class c, Jj is the iron scavenging rate for
particle type j, and sk is the source of dissolved iron of type k,
with k=A, S, or H, for aeolian, sedimentary,
and hydrothermal iron, respectively. The subscript c ranges over small,
large, and diatom phytoplankton functional classes, and the subscript j
ranges over particulate organic phosphorus (POP), biogenic silica (bSi), and
dust particle types. (Note that we have simplified the notation
from the fully coupled nonlinear
model of , for clarity and readability
– see the table in
Appendix for symbol correspondence.)
Iron source strengths in units of GmolFeyr-1 for
the typical state and four other states sampling our family of solutions.
Source strengths are listed for aeolian (σA),
sedimentary (σS), hydrothermal (σH), and total (σtot) iron.
The corresponding bulk iron lifetime, τ, in units of years, is also tabulated
(τ is the ratio of the global DFe inventory to σtot).
The uptake Uc depends nonlinearly on the concentration of DFe, phosphate,
and silicic acid and hence couples the iron, phosphorus, and silicon cycles.
The remineralization rate of DFe taken up by phytoplankton class c is
modeled by ScUc, where the linear “source” operator
Sc accomplishes both the in situ remineralization in the
euphotic zone and the biogenic transport followed by remineralization at
depth. (Remineralization at depth is modeled as instantaneous with the
divergence of a Martin power-law POP flux profile; .)
Similarly, the redissolution rate of scavenged DFe is modeled by
SjJj, where the source operator Sj represents the
instantaneous transport to depth of iron scavenged by particles of type j.
The detailed model formulation of the coupled iron–phosphorus–silicon
cycles, including the construction of the operators Sc and
Sj, has been published by .
Family of optimal state estimates
coupled the iron cycle to the global phosphorus
and silicon cycles and determined the steady state of the system using an
efficient Newton solver. The biogeochemical model parameters were
systematically optimized by minimizing the quadratic mismatch between modeled
and observed nutrient and phytoplankton concentrations. This approach led to
a family of 287 possible state estimates, which correspond to widely
different iron-source scenarios and
a range in total iron source strength of 1.9 to
41GmolFeyr-1. (The true magnitude of the ocean's iron
sources is still highly uncertain; e.g., .) All
state estimates have very similar fidelity to the observational constraints.
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 . When emphasizing systematic variations of
a specific metric with iron source strength, we will plot the metric for all
287 states.
Equivalent linear model: iron labeling tracers
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 provide identical
uptake and scavenging rates and hence identical tracer solutions. It is
necessary to employ such iron labeling tracers because their linear
equations satisfy the superposition principle, which allows us to rigorously
partition the iron concentration according to source type, number of
regenerations, and so on. (The underlying parent model cannot directly be
used for this purpose because of its nonlinearities.)
To construct the linear processes for the iron labels, we diagnose the local
rate constants γc(r)=Uc(r)/χ(r) and
γj(r)=Jj(r)/χ(r) so that Uc and Jj
can be replaced by γcχ and γjχ, which are linear in
χ. Following , we write Sc=(1-fc)+Bfc to separate the remineralization rate into the
fraction (1-fc) that is remineralized in situ in the euphotic zone and
the detrital fraction fc that is exported to depth by the biogenic
transport and remineralization operator B. Substituting the
linear forms of Uc and Jj into Eq. () and
reorganizing terms, we obtain
(∂t+T)χ=Rχ-Lχ-Dχ+∑ksk,
where L≡∑cfcγc is the uptake operator for
DFe that gets exported, R≡BL is
the regeneration operator, and D≡∑j(1-Sj)γj is the reversible scavenging operator. More
precisely, D is the linear integral operator that, applied to the
DFe concentration field χ, gives the local rate of scavenging minus the
local rate with which scavenged iron is redissolved. Thus, D
provides both the transport of the “scavenging pump” (conservative
“reversible” scavenging) and the permanent iron sink due to burial
in the sediments (nonconservative “death”). The operator B
represents conservative biogenic particle transport and subsequent
regeneration. In Eq. (), DFe enters the biological
pump with the utilization rate Lχ and exits the biological
pump with the regeneration rate Rχ. Throughout,
“regeneration” refers to remineralization in the aphotic zone following a
passage through the biological pump. (We define the regeneration operator
R such that in the euphotic zone Rχ=0 so that
the biological pump's intake (Lχ) and output (Rχ) are separated.)
The source sk in Eq. () may be thought of as
injecting tracer labels attached to DFe. These labels are eventually
completely removed by D. The uptake L also removes
labels from iron in the euphotic zone, but the regeneration R=BL reinjects these labels throughout the water
column below without any losses (B is conservative). A
schematic of the transport and cycling of iron labels by these operators is
provided in Fig. . For convenience, we bring all the
operators to the left-hand side of the equation so that
Eq. () can be written compactly as (∂t+H)χ=∑ksk, where the complete linear system operator
is given by H≡T-R+L+D. Below it will be useful to consider the iron
cycle without regeneration, whose evolution is governed by F≡T+L+D. (Note that
F=H+R. The definitions and units of the
linear operators are collected in Appendix .)
Schematic of the birth-to-death life cycle of a labeled DFe molecule (red
dot).
As captured by Eq. (), the molecule makes several
passages through the biological pump (green, uptake L
and regeneration R) and through the scavenging pump (purple dotted,
D) while being transported by the ocean circulation (blue, T)
from its birth (red, sk, here the aeolian source) to its death by
eventual burial in sediments following scavenging.
This particular DFe molecule passed n=2 times through the biological
pump since its birth and will pass m=1 times through the biological pump until its death.
We consider the DFe concentration at location r and at present time t.
Our diagnostics partition DFe at (r,t) into the fractions that have
undergone n regenerations since birth and that will undergo m regenerations until death.
Computationally, we track DFe from its birth to the present using the usual
time-forward flow, while we track DFe backward in time from its death
to the present using the time-reversed adjoint flow.
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”: χ=∑kχk, where
χk is the concentration due to source sk. In steady state, we can
therefore calculate the concentration χk of each source type by solving
Hχk=sk.
We caution the reader that the equivalent linear model
(Eq. ) is not a linearization of the nonlinear parent
model (Eq. ) in the usual sense. The equivalent linear model is
constructed to partition DFe in the unperturbed system. By contrast,
linearization usually refers to the first-order Taylor expansion of the
nonlinear model around a base state, which captures the system behavior for
small perturbations about that base state.
Past contributions to export
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.
Iron concentration regenerated n times since birth
We first consider the concentration of DFe that has never been regenerated
since it was injected (born) by the source. This concentration, denoted by
χk0↓, can simply be computed from our equivalent linear
system by injecting iron labels with source sk, but not permitting them to
be regenerated. We therefore remove the R term from
Eq. (), which is equivalent to replacing H with
F, to obtain
Fχk0↓=sk.
We use arrow superscripts to indicate past processes that occurred since
injection into (↓) the ocean or future processes that will
occur until removal out of (↑) the ocean.
We can now calculate the concentration of DFe from source k that has been
regenerated exactly one time, χk1↓, as follows. The source
of labels for χk1↓ is the rate of first regeneration of
DFe, which is given by Rχk0↓. We simply allow
the system to cycle these labels but remove them on uptake using
F (no second regeneration). Thus, in steady state,
χk1↓ obeys Fχk1↓=Rχk0↓. Similarly, the source of DFe that has been regenerated
exactly n+1 times since birth is the rate of (n+1)st regeneration. This
gives the recursion relation
Fχk(n+1)↓=Rχkn↓,
with Eq. () providing the starting point of the recursion.
Note that the concentrations χkn↓ partition the DFe
concentration exactly, with ∑n=0∞χkn↓=χk, as shown in Appendix .
Global zonal averages of the concentration of DFe that was regenerated n
times since birth, normalized by its global mean, for n=0,1,2,3, and
10. The normalized concentrations of the three source types (aeolian,
sedimentary, hydrothermal) are shown, as is their sum, which
is the total concentration
regardless of source type. This figure is for our typical state.
Figure shows global zonal averages of the iron concentration
for our typical state, partitioned by source type (aeolian, sedimentary,
hydrothermal) and according to the number n of past regenerations since
birth. To emphasize the spatial patterns, each
χkn↓(r) field has been normalized by its global
volume-weighted average, 〈χkn↓〉. Iron that has
never been regenerated (n=0) carries a strong source signature, with peak
concentrations where the sources are largest. The patterns change
dramatically with one passage through the biological pump (n=1). Because
iron is regenerated at depth with a Martin power-law profile, the
concentration of aeolian DFe that has been regenerated n=1 times has a
subsurface maximum at roughly 1500m of depth. Similarly, sedimentary
iron that passed once through the biological pump has a mid-depth maximum and
is no longer concentrated near the seafloor. Sedimentary iron that has been
regenerated n=1 times is concentrated in the Arctic, presumably because the
model has already shallow sedimentary sources there. However, the model's
Arctic circulation is poorly constrained
and this particular feature may not be robust. Hydrothermal iron that has
been regenerated n=1 times shows the signature of Southern Ocean nutrient
trapping as expected
from the fact that hydrothermal iron is injected at seawater densities that
outcrop in the Southern Ocean.
As the number n of past regenerations increases, χkn↓
rapidly converges to a pattern that is independent of source type.
Physically, this is because the memory of birthplace quickly fades with each
passage through the biological pump. Mathematically, this can be seen from
recursion Eq. (),
which gives χkn↓=Anχk0↓,
where A≡F-1R. Thus, for
sufficiently large n, the concentration χkn↓ becomes
proportional to the eigenmode of A with the largest eigenvalue
λ, and the amplitude of the pattern decays exponentially like
λn=exp(-n/n*), where n*≡-1/log(λ). (Note that
λ<1, as must be the case for ∑n=0∞χkn↓ to converge to χk.) In other words, as A
is applied repeatedly, only the projection of χk0↓ onto
the eigenmode of A with the gravest eigenvalue survives. Because
A is independent of source type, all source types approach the
same large-n asymptotic eigenpattern. Convergence to this pattern is
remarkably rapid: for aeolian and sedimentary iron, the pattern has emerged
after about five regenerations (not shown) and for hydrothermal iron even
sooner. By n=10 the asymptotic pattern is well established and
indistinguishable for the different iron source types (bottom plots of
Fig. ).
How robust is the large-n eigenpattern of χkn↓ across
our family of solutions? To quantify the approach to the exponentially
decaying eigenmode, we plot in Fig. a the global
mean concentration that has been regenerated n times since birth as a
function of n for the five representative state estimates in
Table . Figure a shows
that the e-folding scale n* is independent of source type as expected
(same large-n slope for all source types on the semilog plots). For all
five
states the convergence to exponential decay is quickest for hydrothermal
iron. The value of the e-folding scale n* depends on the state of the
iron cycle. For our five representative states, the smallest n* values occur
for the high-source states (n*∼0.6 for HiA-LoS and 0.8 for
HiH), while the largest e-folding scales occur for the low-source states
(1.4 for both HiS-LoA and LoH), with n*∼1.2 for our typical
state. Across the entire family of state estimates, n* ranges from about
0.4 to 2.8. The dependence of n* on source scenario makes sense when
we consider that all these states are optimized against the observed DFe
concentrations: high aeolian input must be countered by vigorous removal from
the surface ocean, and hence a large proportion of the iron in the ocean is
accounted for by the first few regenerations since birth, leaving little DFe
for multiple regenerations. Conversely, low aeolian input corresponds to less
vigorous biological cycling so that a larger portion of the iron is available
to pass through the biological pump repeatedly. Similar considerations hold
for hydrothermal and sedimentary iron. For all states, asymptotic behavior is
well established for n≳5.
(a) Global mean DFe concentration regenerated n times in the past,
〈χkn↓〉, as a function of n for five
representative states of the iron cycle. (b) Mean concentration regenerated m times in the
future, 〈χkm↑〉, as a function of m.
The sedimentary and hydrothermal iron concentrations get their largest
contributions from iron that has never been regenerated (n=0), again
pointing to significant permanent removal of these iron types before they can
reach the euphotic zone following injection (birth) at depth. For aeolian
iron in the typical, LoH, and HiS-LoA states, the largest contribution comes
from iron that has been regenerated n=1 times as all freshly born aeolian
iron is immediately available for uptake and regeneration. However, for the
high-source states (HiA-LoS and HiH), the largest contribution still comes
from iron that was never regenerated because the increased scavenging
necessary to balance the high sources prevents most aeolian DFe from being
utilized even though it is deposited directly into the euphotic zone.
(a) Percentage of the total inventory of DFe that has not passed
through the biological pump in the past (i.e., since birth) as a function of
total source strength, σtot, for the family of state
estimates of . (b) Percentage of the total
inventory of DFe that will not pass through the biological pump in the future
(i.e., until death).
Note the logarithmic abscissa.
Shown are the fractional inventories of the individual source types of DFe
(color coded) and the total fractional inventory regardless of
source type (black).
Figure a shows that DFe that has not passed through
the biological pump since birth (n=0, unused iron) generally has the
highest global mean concentration, except for aeolian DFe under some source
scenarios. To quantify the amount of iron that was not regenerated in the
past, we now ask how the unused fraction of the global DFe inventory varies
with total iron source strength, σtot. Note that this
fraction can be considered to be the χk-weighted global average of the
local unused fraction fk0↓≡χk0↓/χk. This weighted average is defined as 〈fk0↓〉χk=〈fk0↓χk〉/〈χk〉, where we introduced the 〈⋅〉χk
notation, which will be used throughout. The unused fractional DFe inventory
regardless of source type is given by 〈f0↓〉χ, where f0↓≡∑kχk0↓/χ.
Figure a shows the fractional unused DFe inventories
〈fk0↓〉χk and 〈f0↓〉χ as a function of σtot for every member of
our family of state estimates. We emphasize that Fig.
does not show the response of the iron cycle to changes in
σtot, but instead shows the fraction of the unused DFe
inventory for distinct equilibrium states, each of which was optimized
against observations under a different prescribed iron source scenario
. Because the total iron inventory is well
constrained across the family, the bulk iron lifetime τ, given by the
ratio of inventory to source (see also Table ),
is inversely proportional to σtot, and hence the mean iron
age also scales with (σtot)-1.
The systematic increase in the fractional inventory of unused iron, and its
approach to saturation at 100% for the largest sources seen in
Fig. a, reflects the fact that the probability of past
regeneration decreases with the mean iron age until the age becomes so short
that very little iron has a chance to pass through the biological pump before
having to be scavenged out of the system to match the observed DFe
concentrations and inventory. For our smallest total source of
σtot∼2GmolFeyr-1, DFe molecules have
lived long enough to be regenerated at least once with 〈f0↓〉χ∼35%, while for our largest
total source of σtot∼40GmolFeyr-1, most
DFe molecules have not had sufficient time to be regenerated and 〈f0↓〉χ∼95%. For state estimates with
σtot>7GmolFeyr-1, more than half of the
DFe in the ocean has never passed through the biological pump, i.e., 〈f0↓〉χ>50%.
Figure a also shows that the fraction 〈fk0↓〉χk varies significantly with iron source type
k. This is because the probability of a past passage through the biological
pump is strongly dependent on birth location. The fraction that has not been
regenerated since birth is lower for aeolian DFe than for benthic DFe.
Benthic DFe must first be transported to the euphotic zone to participate in
biological production and is therefore more likely to be scavenged en route
compared to aeolian DFe, which is directly injected into the surface.
Hydrothermal DFe is the least likely to have passed through the biological
pump, with 〈fH0↓〉χH≳80% for all states and 〈fH0↓〉χH≳95% for
states with σtot>7GmolFeyr-1. For
sedimentary DFe, 〈fS0↓〉χS≳45% for all states, and 〈fS0↓〉χS≳60% for states with σtot>7GmolFeyr-1. As the mean age becomes ever smaller with
increasing σtot, the unused fractional benthic DFe inventory
saturates faster to 100% than the unused fractional aeolian DFe
inventory, which reaches only ∼90% even for
σtot∼40GmolFeyr-1, again reflecting
greater probability of biological utilization for surface-injected iron. We
will return to Fig. b below when we explore future passages through the biological pump.
To explore the variations of the asymptotic eigenmode pattern across our
different state estimates, Fig. shows the zonally
averaged patterns of χn↓=∑kχkn↓ for
n=10, which is well within the asymptotic regime, for our five representative
states. (Recall that individual source types all converge to the same
pattern, which is hence also the pattern of the total iron concentration.)
While there is a signature of Southern Ocean nutrient trapping for all
states, there is also significant variation across the state estimates.
Broadly, the Southern Ocean trapping has a stronger influence on the
eigenmodes of F-1R for the high-source states
than for the low-source states. This is likely due to the fact that
high-source states are able to match the observed DFe concentrations by
having both more active biological iron pumps and scavenging. The transport
to depth by both the biological pump and by scavenging particles (the
“scavenging pump”) enhances the Southern Ocean trapping. The scavenging is
largest at tropical and high latitudes so that stronger scavenging will tend
to remove iron from low and high latitudes (see Appendix
for zonally averaged death rates). At high latitudes this is counteracted by
stronger trapping, but at low latitudes the death rate dominates and the
concentration of iron that passes many times through the biological pump is
strongly diminished. Conversely, for the low-source states, reduced
scavenging allows iron to pass many times through the biological pump without
being scavenged out of the system at low latitudes. Consequently, the
low-source states have iron that has been regenerated many times flowing out
of the Southern Ocean with mode and intermediate waters well into the
Northern Hemisphere.
Normalized global zonal averages of the concentration of DFe that has passed
n=10 times through the biological pump since birth regardless of
source type, χ10↓, to show the eigenpatterns of our five
representative state estimates (Table ).
The zonal averages are normalized by the global mean.
Mean number of regenerations since birth
A key metric of how much the DFe field has contributed in the past to
organic matter export is the mean number of times, n‾k, that a
given iron molecule has been regenerated since its birth. From the local
fraction of the total DFe that was regenerated n times since birth,
fkn↓(r), the mean number of past regenerations is by
definition given by n‾k(r)≡∑n=0∞nfkn↓(r). Mathematically, as shown in
Appendix , it follows from Eq. ()
that n‾k obeys
H(n‾kχk)=Rχk.
Physically, Eq. () may be interpreted as the equation for a
labeling tracer n‾kχk that is cycled just like DFe by the
H operator, but whose numerical value accumulates
n‾k-fold with n‾k past regenerations.
Computationally, solving Eq. () provides an efficient means of
finding n‾k that avoids first finding and explicitly summing
fkn↓(r).
A given iron molecule at r is responsible, on average, for the export
of n‾k(r) iron molecules of source type k since its
birth. The corresponding organic matter export is quantified by the mean
number of phosphorus molecules that are exported along with the iron. The
regeneration operator for phosphorus is given by RP=∑cBfcγcP, where
γcP(r)≡UcP(r)/χ(r)
is diagnosed from the optimized phosphorus uptake UcP(r)
of the underlying model . The mean number of
phosphorus molecules, n‾kP(r), globally exported
and remineralized in the past per DFe molecule that is currently at
r obeys
H(n‾kPχk)=RPχk.
In other words, on average, one of the type-k DFe molecules currently at
r has supported the export of n‾kP(r)
phosphorus molecules since its birth. In analogy with
Eq. (), we interpret Eq. () as defining a
labeling tracer n‾kPχk that is cycled like DFe
by the H operator, but whose numerical value counts the number of
phosphate molecules that were regenerated in the past via the phosphate
regeneration operator RP.
Figure shows the global zonal averages of
n‾k(r) for our typical state estimate. Note that the
maximum values of n‾k are order unity and, on average, most of
the iron in the ocean has been regenerated less than once since its birth.
Iron with the largest n‾k is found near the Southern Ocean
surface and in mode and intermediate waters flowing out of the Southern
Ocean, which is presumably a signature of Southern Ocean nutrient trapping.
In terms of source types, aeolian iron has been most active in the biological
pump with n‾A exceeding 0.6 throughout most of the
ocean interior and exceeding 1.0 throughout much of the Southern Ocean
water column. In the surface waters of the low-production subtropical gyres
n‾A approaches zero. The mean number of regenerations
since birth is much smaller for sedimentary and hydrothermal iron, again
reflecting the greater scavenging hazard for benthic iron. For sedimentary
and hydrothermal iron, values of n‾k greater than 0.5 extend
from the subantarctic Southern Ocean surface into mode and intermediate
waters where scavenging is relatively weak (Appendix ).
The zonally averaged patterns of n‾k are remarkably
similar across our family of estimates. As shown in
Appendix , the spatial patterns are also insensitive to the
value of the recyclable fraction of scavenged iron, frec,
although their amplitude can vary by a factor of ∼2 between the
frec=0 and frec=1 extremes. The patterns of the
global zonal averages of n‾kP (not shown) are nearly
identical to those of n‾k.
Zonal averages of the mean number of past passages through the
biological pump (i.e., since birth), n‾k.
This figure is for our typical state.
We expect significant variations in the amplitudes of n‾k and
n‾ (the mean number regardless of source type) across the family
of state estimates. Because the number of regenerations that are possible
during the lifetime of an iron molecule should be proportional to the bulk
iron lifetime τ∝σtot-1, we expect
n‾∝σtot-1. To test this and to quantify
the range of possible variations, Fig. a shows the
global average 〈n‾〉 as a function of
σtot on a log–log plot and Fig. b
shows the corresponding behavior for 〈n‾P〉.
Both 〈n‾〉 and 〈n‾P〉
exhibit the expected approximate inverse relation with σtot.
Note that the 〈n‾〉∝σtot-1
scaling is not exact; instead, Fig. a suggests that
there are σtot-1 scaling regimes for a low-source cluster
and for a high-source cluster, with a transition for intermediate source
strengths. This approximate nature of the scaling reflects the fact that the
timescale for a passage through the biological pump is not merely set by the
prescribed circulation and (in our model) instant particle transport and
remineralization, but also depends on the spatial distribution of the
scavenging, which varies with the optimized source scenarios. The detailed
timescales that link circulation, scavenging, and pumping frequency will be
explored in a future publication. The numerical values of
〈n‾〉 range from 0.05 for σtot=41Gmolyr-1 to 2.2 for σtot=1.9Gmolyr-1, while 〈n‾P〉
correspondingly ranges from approximately 0.4 to
4.3molP(mmolFe)-1 (i.e., from 40 to
460molC(mmolFe)-1 using a simple uniform Redfield C : P
ratio of 106:1 for unit conversion).
(a) The globally averaged mean number of past and future DFe
passages through the biological pump per injected DFe molecule (regardless of
source type), 〈n‾〉 and 〈m‾〉,
as a function of the total iron source strength, σtot, for
the family of state estimates of . (b) The
corresponding globally averaged mean number of phosphorus molecules
exported per DFe molecule, 〈n‾P〉 and 〈m‾P〉.
Note the log–log axes to highlight the approximate inverse relationship with σtot.
To guide the eye, dashed grey lines indicate an exact σtot-1 power law.
Future contributions to export
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 , for which
the system is governed by the adjoint operators H̃,
F̃, R̃, and
D̃. These adjoints are defined in terms of the
volume-weighted inner product 〈x,y〉≡∫x(r)y(r)d3r, where the integral ranges over the entire
ocean volume. Thus, H̃, the adjoint of H,
is defined as usual so that 〈H̃x,y〉=〈x,Hy〉. In the time-reversed adjoint flow, the
death operator becomes a source of labels that we then track through
sequential regenerations backward in time to the present, analogously to what
we did in the previous section for the usual time-forward flow. The use of
adjoint operators allows for numerically efficient evaluation of our
diagnostics because a single tracer injected at death into the time-reversed
adjoint flow suffices to produce the full three-dimensional concentration
fields of interest at present. Here we provide the key equations –
their derivation in terms of Green functions is detailed in
Appendix .
Number of regenerations until death
To compute the number of future regenerations, it is useful to calculate the
death rate d(r) with which iron is permanently (nonreversibly)
removed at point r. This death rate can be diagnosed from the
scavenging operator D and the DFe concentration χ(r)
as detailed in Appendix , where we also show basin zonal
averages of d(r). The corresponding rate constant
γD for the linear equivalent model is defined such that
d(r)=γD(r)χ(r).
One can show that the local fraction f↑(r) that eventually dies obeys
H̃f↑=γD,
where f↑(r)=1 uniformly everywhere because all DFe must
eventually be scavenged out of the system.
In analogy with Eq. (), the fraction f0↑ that is
regenerated zero times until death obeys
F̃f0↑=γD.
By using the regeneration rate of DFe that will be regenerated m times
until death and going back in time by one regeneration, we obtain the
fraction that will be regenerated (m+1) times until death, giving us the
recursion relation
F̃f(m+1)↑=R̃fm↑.
Note that ∑m=0∞fm↑(r)=f↑=1 as can
be verified from Eq. () and Eq. ().
The mean number of regenerations until death, m‾=∑m=1∞mfm↑, can be shown to obey
H̃m‾=R̃f↑.
The mean number of phosphorus molecules, m‾P(r),
globally exported and remineralized in the future per DFe molecule that is
currently at r obeys
H̃m‾P=R̃Pf↑.
In other words, on average, a DFe molecule currently at r will
support the export of m‾P phosphorus molecules before
it is buried in the sediments.
Figure shows the zonally averaged fraction
fm↑ of DFe that will undergo m regenerations in the future
normalized by the global average, 〈fm↑〉, to
emphasize changes in the pattern with m. Note that this fraction is the
same for all source types because the future of a given DFe molecule is
independent of its past. Below the thermocline, the pattern of f0↑ is nearly uniform at a value just above its global mean but drops
to below 50% of the global mean near the surface where the
probability of further biological utilization and regeneration is largest
(careful inspection is required to see this in Fig. ).
The pattern of f1↑ has its largest amplitude at the surface and
in the Southern Hemisphere, where it spreads deepest from the surface,
reflecting the relatively low Southern Hemisphere death rates
(Appendix ). As m increases, fm↑ becomes
rapidly proportional to the gravest eigenmode of
F̃-1R̃ (see
Eq. ). As this eigenmode is approached, the
pattern of fm↑ contracts into the Southern Ocean, presumably
because iron that will be regenerated many times before death can only be
found where there is both a low death rate and efficient nutrient trapping.
Global zonal averages of the fraction of DFe that will be
regenerated m times, fm↑, normalized by its global mean value,
〈fm↑〉, for m=0,1,2,3, and 10.
This figure is for our typical state.
To quantify the approach of fm↑ to the gravest eigenmode of
F̃-1R̃, we return to
Fig. b, which shows the global averages of the
corresponding DFe concentrations χkm↑≡χkfm↑ as a function of m. Note that F̃-1R̃ and F-1R have the same
eigenvalues so that 〈χkm↑〉 and 〈χkn↓〉 approach the same exponential decay
(Fig. a and b).
We return to Fig. b to consider the fractional DFe
inventories that will not pass through the biological pump in the future for
all our state estimates. These inventories for source type k and regardless
of source type are given by 〈f0↑〉χk and
〈f0↑〉χ. We expect the probability of future
regenerations to increase with the remaining lifetime of DFe, which should
also scale like σtot-1 given a well-constrained global
DFe inventory. This is confirmed in Fig. b by the
systematic increase and approach to 100% saturation of 〈f0↑〉χk and 〈f0↑〉χ with
increasing σtot. (Note that in the theoretical
σtot→0 limit, we expect 〈f0↑〉χk→0 and 〈f0↑〉χ→0, although our lowest sources are
not nearly small enough to exhibit this limiting behavior.)
Figure shows striking asymmetries between unused past
and future inventories. The fractional inventory of iron that will not be
utilized in the future (Fig. b) is nearly independent
of iron type k, in sharp contrast with the fractional inventory that was
not utilized in the past (Fig. a). The insensitivity of
〈f0↑〉χk to source type is due to the
independence of f0↑ on source type so that the χk-weighted
global average is only sensitive to changes in the pattern of
χk, which varies little across the family of states
. Note that for a given σtot,
the inventory of total DFe unused in the future, 〈f0↑〉χ, is significantly larger than the inventory of
total DFe unused in the past, 〈f0↓〉χ. In
other words, total DFe is more likely to have been regenerated in the past
than it is to be regenerated in the future. This asymmetry stems from the
fact that f0↓=∑kχkχfk0↓ is
dominated by the relatively small unused aeolian fraction
fA0↓ (see Fig. a). In terms of
individual source types, aeolian and sedimentary DFe are also more likely to
have been regenerated in the past than in the future, but the reverse is true
for hydrothermal DFe for which the scavenging hazard following birth is
greatest. Thus, for a hypothetical state in which hydrothermal iron dominates
the total DFe inventory, it is possible that one could get 〈f0↓〉χ>〈f0↑〉χ;
however, none of our states fits this scenario.
We now return to the asymptotic eigenpatterns of fm↑. How robust
are these eigenpatterns across our family of state estimates?
Figure shows f10↑ normalized by its global
mean and zonally averaged over the global ocean for our five representative
states (for m=10, fm↑ is an excellent approximation to the
gravest eigenmode of F̃-1R̃).
While there is some variation across the family, the patterns are
qualitatively similar. Because the natural quantity that keeps track of
future regenerations is the source-type-independent fraction of DFe
that will undergo m regenerations, the spread in the large-m pattern of
fm↑ across the family of states is much smaller than the spread
in the large-n pattern of χkn↓ (χkn↓
being the natural quantity for keeping track of past regenerations).
Global zonal averages of f10↑ normalized by their global means,
〈f10↑〉, to show the approximate asymptotic eigenpatterns for five representative state estimates.
Figure a shows the global zonal average of the mean
number of future regenerations m‾(r), which like fm↑ is independent of source type. The pattern of m‾ is
similar to the pattern of f1↑, which dominates the rapidly
converging sum m‾=∑m=1∞mfm↑. The
m‾ pattern is surface intensified and largest in the Southern
Ocean. m‾ decays rapidly with depth, reflecting the fact that the
deeper the DFe, the harder it will be to escape death on the way to the
surface to participate in the biological pump. These qualitative features are
robust across the family of state estimates (not shown).
Figure b shows that the pattern of the global zonal
average of m‾P(r) is almost indistinguishable
from that of m‾(r). This is because the patterns of
m‾P and m‾ are dominated by the phosphorus
and iron export productions, respectively, which are similar despite the
substantial spatial variations of the Fe : P uptake ratio
.
Three-dimensional intrinsic fertilization efficiency metrics for our
typical state, zonally averaged across the global oceans. (a) The mean
number of iron molecules, m‾(r), that will be
regenerated per iron molecule at r during that molecule's lifetime. (b) The corresponding mean number of phosphorus molecules,
m‾P(r), that will be regenerated.
Returning to Fig. b, we see that both
〈m‾〉 and 〈m‾P〉 are
again approximately proportional to σtot-1, as expected.
The magnitude of 〈m‾〉 remains at or below order
unity and ranges from 0.01 to 1.4 as σtot varies from
41 to 1.9Gmolyr-1. Correspondingly,
〈m‾P〉 ranges from 0.07 to
2.8molP(mmolFe)-1 or from 7 to
290molC(mmolFe)-1 when converted using C : P =106:1.
Intrinsic iron fertilization efficienciesExport supported per unit DFe injection at r
We begin by asking how much an arbitrary (“test”) source sk(r)
contributes to the globally integrated export and remineralization of iron
and phosphate in organic matter. (In our model all exported phosphorus is
instantly respired with the divergence of a Martin POP flux profile – DOP
is not explicitly modeled.) We first consider the global export that will be
supported per unit injection of DFe at point r. The concentration
response to a unit injection is the Green function associated with operator
H, and the global export due to this response is simply
obtained as the global integral of the regeneration operator R
acting on this response. As shown in Appendix ,
a DFe injection into volume element d3r at rate
sk(r)d3r supports a globally integrated iron export
rate, Φk(r), given by
Φk(r)=m‾(r)sk(r),
while the corresponding phosphorus export, ΦkP(r), is given by
ΦkP(r)=m‾P(r)sk(r).
These equations have a straightforward physical interpretation: the mean
number of future regenerations, m‾(r), is simply the number
of DFe molecules that will be exported per DFe molecule injected at point
r, and m‾P(r) is the corresponding number
of phosphorus molecules that will be exported. The quantities
m‾(r) and m‾P(r) are hence
measures of the efficiency of iron fertilization at point r. Note
that this local efficiency is independent of source type; the efficiency is
determined by the biological pump and the transport of water, neither of
which depend on the iron source type. Furthermore, the local efficiency
m‾P(r) is defined regardless of whether an iron
source is actually present at r. The efficiency
m‾P(r) quantifies the global phosphorus export
rate per unit DFe source rate at r. At all points r,
even where there is no actual source in the system,
m‾P(r) can be considered to be the
“sensitivity” of the linear equivalent system to the insertion of the
arbitrary test source sk(r): Eq. () shows that
m‾P(r) is the proportionality between the export
response ΦkP(r) and the test source sk(r). In
this sense, the fertilization efficiency m‾P is a
close, but distinct, cousin of the sensitivity to small-amplitude
perturbations considered by .
Viewed another way, by construction m‾(r) and
m‾P(r) are the mean
numbers of DFe and phosphorus
molecules exported per molecule of iron that is present at r in the
current state of the ocean, before that molecule is buried. Thus,
m‾(r) and m‾P(r) again have the
natural interpretation of being fertilization efficiencies. We emphasize that
m‾(r) and m‾P(r) are defined
without perturbing the system and are hence noninvasive metrics of what we
call the intrinsic iron fertilization efficiency. The plots of the zonally
averaged m‾(r) and m‾P(r) in
Fig. may thus be interpreted as zonal averages of this
intrinsic, three-dimensional iron fertilization efficiency, which can be seen
to be highest at the surface and rapidly diminishes with depth. The zonal
averages of m‾ and m‾P are dominated by the
Southern Ocean. However, we will see that the Southern Ocean only plays a
secondary role for the fertilization efficiency at the surface.
Figure a shows that the surface patterns of
m‾ are qualitatively similar across the five representative states,
with a broad maximum in the central tropical Pacific and secondary
maxima in the subpolar oceans. The states with the largest global mean
surface fertilization efficiency, 〈m‾〉surf, (values provided in
Fig. a) are the low-source states as one would
expect: the less iron is in the system, the more the biological pump is
expected to benefit from the iron present. For the HiS-LoA state, a typical
molecule of DFe at the surface leads to a globally integrated export of
〈m‾〉surf∼1.4 molecules of iron.
For the high-source states (HiA-LoS and HiH), both the tropical Pacific and
the Southern Ocean are locally most prominent, but the surface mean
efficiencies are only 〈m‾〉surf∼0.2 (HiA-LoS) and 0.5 (HiH) molecules of iron exported per
typical surface molecule.
Figure b shows the surface patterns of the
efficiency for fertilizing organic matter export,
m‾P(r). The global surface average,
〈m‾P〉surf, is again highest for
the low-source states. However,
〈m‾P〉surf varies less across our
representative states than 〈m‾〉surf (a
range of ∼3 compared to ∼7). This is consistent with the fact
that all states have very similar phosphorus exports (well constrained by the
data used in the optimizations), but widely differing iron exports and Fe : P
uptake ratios .
The surface patterns of m‾P and m‾ are
similar, with very similar systematic variations across the different states.
However, because of the iron dependence of the Fe : P uptake ratio,
m‾P has sharper gradients than m‾, with a
more pronounced contrast between the low-efficiency subtropical gyres and the
high-efficiency tropical Pacific. The Fe : P uptake ratio in our model is
proportional to χ/(χ+k), which has its lowest values of around
0.1 in the central and eastern tropical Pacific for all our states. The
corresponding P : Fe uptake ratio (not shown) has a global pattern that is
broadly similar to the pattern of the intrinsic fertilization efficiency
m‾P because large P : Fe means that a relatively large
number of P molecules are taken up (and hence exported) per utilized DFe
molecule. The correspondence is not exact, but the P : Fe uptake ratio plays a
central role in shaping m‾P at the surface. At depth,
the effect of the P : Fe ratio on m‾P is less important
because, depending on where the DFe reemerges into the euphotic zone, deep
DFe will not necessarily be utilized in regions of high P : Fe.
The key result of Fig. is that regardless of
state, the tropical Pacific is where iron has its highest intrinsic
fertilization efficiency. For high-source states, the fertilization
efficiency of the Southern Ocean can be equally as large, although the global
surface mean efficiency is much lower than for low-source states. The
efficiency is not highest in the eastern tropical Pacific where production is
highest because of the associated large scavenging rate due to organic
particles. The sweet spot between upwelling-fertilized production and
relatively low scavenging lies in the central tropical Pacific to the west of
the highest productivity. The state-dependent relative importance of the
Southern Ocean compared to the tropical Pacific is a complicated function of
how our optimized states match the nutrient observations and reflects a
delicate balance between fertilizing biological production and the resulting
enhanced scavenging. Interestingly, found that
the sensitivity of their coupled model to aeolian iron addition is also
largest in the central tropical and eastern Pacific. This suggests that the
intrinsic fertilization efficiency of the unperturbed state is shaped by
similar processes as the sensitivity to small-amplitude perturbations.
Normalized patterns of our intrinsic fertilization efficiency
metrics at the surface. (a) The mean number of iron molecules,
m‾(r), that will be regenerated per iron molecule at
surface point r during that molecule's lifetime. (b) The
corresponding mean number of phosphorus molecules,
m‾P(r), that will be regenerated. To allow for a
meaningful comparison of the patterns of different states,
m‾(r) and m‾P(r) have been
normalized by their global surface averages,
〈m‾〉surf and
〈m‾P〉surf (values are given in
each plot).
How do our estimates of intrinsic fertilization efficiency compare to
previous estimates of differently defined fertilization efficiencies? Across
our entire family of estimates,
〈m‾P〉surf has a range of
0.7–7molP(mmolFe)-1, which converts to
73–750molC(mmolFe)-1 using a simple uniform C : P
ratio of 106:1. Thus, our estimate of the intrinsic fertilization
efficiency is roughly 1 to 2 orders of magnitude larger than the estimate of
about 3.3molC(mmolFe)-1 by , who
reported 900tC exported for 1.26t of iron added
during the SOFeX artificial fertilization experiment. There are multiple
reasons why our estimate of fertilization efficiency differs from that of
. First, the latter is a regional estimate based on a
localized artificial fertilization experiment, while the former is a global
mean estimate of the unperturbed system. Second, used
the data from the iron fertilization experiments without any corrections for
ligand protection in the natural unperturbed system, which leads to an
underestimate of the natural fertilization efficiency.
correct for this in their analysis, giving an
estimate of 2.6–100molC(mmolFe)-1 for the natural
fertilization efficiency. Despite being inferred from a finite-amplitude
perturbation experiment, the estimate of overlaps
with the low end of our estimates (which corresponds to the possibly more
realistic high-source states). The sensitivity of global production to
perturbations in the local aeolian source estimated by
has a spatial distribution with a range of about
20–180gCm-2yr-1 for a
0.02mmolFem-2yr-1 perturbation. The model of
exports one-third of its production as POP so that these sensitivities translate to
a POP export per added DFe molecule of about
28–250molC(mmolFe)-1, a lower bound on the full
carbon export and about a third of the intrinsic fertilization efficiency
estimated here. We emphasize that differences between these various estimates
are not only due to uncertainties in the iron cycle (as expressed by our
range of values), but also due to differences in the definition of
fertilization efficiency, not to mention due to differences among models and
methodologies.
Relation to relative export efficiency
To compare the importance of the different iron source types in the current
state of the ocean, defined the export-support
efficiency of iron type k as ϵ(sk)≡Φ^k/σ^k, where Φ^k is the fraction of the
global phosphorus export supported by source type k and σ^k
is the fractional global source of type k. We found that a key metric that
is robust across our family of state estimates is the relative
export-support efficiency, ekP=ϵ(sk)/ϵ(s̃k), which is the ratio of the
export-support efficiency of source type k to the export-support efficiency
of the other source types, whose combined source is the complement
s̃k≡stot-sk.
It follows algebraically from Eq. () that the relative export efficiency can be expressed as
ekP=〈m‾P〉sk〈m‾P〉s̃k,
where 〈⋅〉sk is the sk-weighted global mean,
defined so that for any field x(r), 〈x〉sk≡〈xsk〉/〈sk〉 and 〈⋅〉s̃k is the corresponding s̃k-weighted
global mean. Thus, the relative export-support efficiency is the ratio of the
mean intrinsic fertilization efficiency of the source to the mean intrinsic
fertilization efficiency of its complement.
found that all members of the family of state
estimates share approximately the same relative export efficiencies of
eAP=3.1±0.8, eSP=0.4±0.2, and eHP=0.3±0.1, where the uncertainties
represent the scatter across the family. What is new here is Eq. (),
which relates the global metric ekP to source-weighted averages
of the local intrinsic fertilization efficiency,
m‾P(r).
Discussion and caveats
Our analysis comes with a number of caveats that should be kept in mind. Some
of these were already identified in the work of ,
in which the design of our coupled Fe–P–Si inverse model is presented in
detail. In particular, the model used DFe data from the GEOTRACES
Intermediate Data Product (IDP) 2014 (in addition to an older compilation by
), which did not contain the newer data from the
Pacific and Southern Ocean made available only recently in the GEOTRACES IDP
2017. Although it is inevitable that assimilating the additional constraints
of the IDP 2017 would lead to some quantitative changes, especially in the
Pacific, we think that including the IDP 2017 data would not lead to
qualitative changes. The states used here do feature strong Pacific DFe
plumes of about the observed spatial extent to the west of the East Pacific
Rise (EPR), but unlike the observations these plumes also extend to the east
of the EPR. The unrealistic eastward plume would not be corrected by
assimilating the IDP 2017 data into the biogeochemical model because it is
due to small biases in the underlying circulation, which we hold fixed
throughout. The deep Pacific circulation can be corrected by assimilating
δ3He into a circulation inverse model, but this is beyond the scope
of the current study. Moreover, any quantitative differences due to
additional constraints would very likely be much smaller than the variations
across our family of state estimates, given its 2-order-of-magnitude range
in iron source strengths.
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 are keyed to the flux of
particulate organic matter into the bottom box where the organic matter is
completely oxidized in our model of the phosphorus cycle. This
parameterization of the sedimentary source is based on observations along the
California coast of a correlation between the DFe flux from sediments and the
flux of oxidized organic matter . All the model's
sediment sources are thus reductive as is the case in many current iron
models e.g., the PISCES and BEC
models;. However, the analysis of
δ56Fe by has highlighted the
importance of non-reductive sources, which our inverse model does not
include. Our state estimates feature sedimentary sources that are
realistically dominated by the continental shelves Fig. H1,, but
they also include sources that are
1 to 2 orders of magnitude smaller below highly productive regions in the eastern
tropical Pacific, where a sluggish abyssal circulation allows some DFe to
accumulate at depth, as can be seen in Fig. . Similarly, our
hydrothermal sources inject DFe into the ocean where it is then protected
from rapid scavenging by our enhanced ligand concentrations near hydrothermal
vents. However, the recent work of suggests
that this model of hydrothermal iron needs to be revised in the future to
include reversible exchange between DFe and particulate iron, which is
currently omitted in our scavenging parameterization. Regarding ligands, the
inverse model of prescribes an optimized
distribution of a single type of ligand that is enhanced in hydrothermal
plumes and in old waters, with optimized parameters. Ligands are not
dynamically transported and the effect of different ligand types on iron
bioavailability is neglected.
Other caveats relate to biogeochemical parameters that could not be
optimized. The recyclable fraction, frec, of DFe scavenged by
opal and POP was prescribed to be 90% for all scavenging
particles. frec is highly uncertain and in recent models its
value has spanned the entire 0–100% range
e.g.,. We
established the sensitivity of our results to the value of frec
by generating new state estimates from our typical state by prescribing
different values of frec ranging from 0 to 100% and
re-optimizing the scavenging and source parameters following the strategy of
(see Appendix for details). We
find that the patterns of n‾ and m‾ are robust to
changes in frec, although their global mean values, which are
order unity for the re-optimized typical state with frec=0,
systematically decrease by roughly a factor of 2 when frec is
increased to 100%. The mean numbers of past and future phosphorus
molecules exported per DFe molecule, n‾P and
m‾P, are robust to changes in frec in both
pattern and magnitude. This robustness reflects the fact that the
optimization can compensate for changes in the scavenging pump with changes in
the biological iron pump.
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, suggested that including the microbial
“ferrous wheel”, which operates with different stoichiometric ratios, could
affect iron budgets in the euphotic zone. Similarly,
explored variations in the iron regeneration rates
of different organisms and cautioned modelers to pay careful attention to the
details of the Fe:C stoichiometry. Relatedly, our model simply remineralizes
iron in the same Fe : P ratio with which it was utilized so that the
vertical profiles of iron and phosphate remineralization have identical
shapes. However, measurements by show that, at
least for some phytoplankton species, iron is remineralized more slowly than
phosphate, suggesting that our remineralization profile for iron could be too
shallow. Different remineralization length scales for iron and phosphate were
also emphasized by . Because our model is optimized to
fit the DFe observations with an emphasis on deep profiles relative to
surface measurements , a potentially too-shallow
remineralization of iron would be compensated for by an increased strength of the biological pump. Furthermore, the
relative amount of scavenging by opal and POP particles is optimizable in our
model so that deeper iron remineralization can be achieved by increasing the
scavenging by opal. We acknowledge, however, that when optimizing the match
to observed DFe, the model may produce biases in the relative contributions
of the biological and scavenging pumps, which would affect our estimates of
the number of passages through the biological pump.
The recent work of suggests that in surface waters
DFe must be preferentially recycled compared to nitrate in order to sustain
the observed nitrate consumption in the iron-limited equatorial Pacific.
While we do not model the nitrogen cycle, it is reasonable to assume that in
the euphotic zone DFe may also be preferentially recycled compared to
phosphate. At face value, this appears to contradict our assumption that the
detrital fractions, fc, are identical for Fe, P, and Si export. However,
the uptake and export of DFe in our model are proportional to the product
fcRFe:P, and RFe:P is optimized so that any
difference between the iron and phosphate detrital fractions is simply
absorbed into RFe:P. However, this does point to the need for
caution when interpreting the optimized values of RFe:P.
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 is of intermediate
complexity, and effects due to processes not explicitly modeled are captured
implicitly when parameters are optimized to fit the observed nutrient and
phytoplankton fields. Explicit modeling of all known processes in their full
complexity may be important for models that try to predict how the system
will change in the future, but this is neither necessary nor desirable for
constraining and diagnosing the large-scale cycling of DFe in the current
state of the ocean as we do here.
Conclusions
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 n times in the past or that will be
regenerated m times in the future. These diagnostics include new tracer
equations for the mean numbers of
regenerations n‾ and m‾, which afford numerically
efficient computation. The mean number m‾ of future regenerations
that iron at any point r will undergo is a measure of the intrinsic
fertilization efficiency of iron at r, giving the number of iron and
phosphorus molecules globally exported per DFe molecule at r.
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 σtot larger
than ∼7GmolFeyr-1, more than 50% of the
inventory has not passed through the pump since birth and more than 85%
will not pass through the pump before death.
Because of its direct injection into the euphotic zone, a larger portion of
aeolian iron passes through the biological pump than other source types. The
mean number of past passages and the mean number
of future passages through the biological pump,
n‾ and m‾, are both approximately proportional to the bulk iron lifetime τ∝σtot-1. For an increase in σtot
from 1.9 to 40GmolFeyr-1, the global average, 〈n‾〉, decreases from 2.2 to 0.05, while 〈m‾〉 decreases from 1.4 to 0.01.
The three-dimensional distribution of n‾k(r) has its
largest values in the Southern Ocean and a pattern that suggests nutrient
trapping for all iron source types k, but with different vertical structure
for different source types.
The precise patterns of n‾k(r) are shaped by the delicate
balance between regeneration and scavenging, which are processes with strong
spatial overlap near the surface.
For aeolian iron, the largest values of n‾k are found in surface,
mode, intermediate, and bottom waters, while for sedimentary and hydrothermal
iron n‾k is small in deep waters because of the greater scavenging
hazard during transit from benthic sources to first uptake in the euphotic zone.
The spatial distribution of DFe that was regenerated n times since birth
varies with n because DFe is reorganized in the water column with each
passage through the biological pump.
Unused DFe (n=0) is generally concentrated near the sources.
The concentration of unused aeolian DFe has a secondary maximum at ∼2km
of depth and extends to the seafloor because of the action of the scavenging
pump (scavenging and particle transport to depth followed by redissolution).
Aeolian DFe regenerated exactly once since birth (n=1) has its maximum
concentration well below the surface, and sedimentary DFe regenerated
exactly once has its maximum concentration well above the bottom, with both
maxima at ∼1.5 km of depth.
DFe that has undergone the largest number of regenerations
since birth is trapped in the Southern Ocean.
The pattern of the large-n DFe concentration is independent of iron source
type because the memory of birthplace quickly dissipates with successive
passages through the biological pump, although the rate of convergence to
the asymptotic large-n pattern does depend on source type.
The large-n pattern corresponds to the gravest eigenmode of
F-1R, which represents the combined transport
by the circulation and by sinking particles for one passage through the biological pump.
For n larger than ∼5, the concentration of DFe that was regenerated
n times since birth decays exponentially with n.
The e-folding scale n* ranges from 0.4–2.8 across our family, with
higher sources corresponding to smaller n* (faster decay consistent with shorter lifetimes).
The fraction fm↑(r) of DFe currently at r that
will pass m times through the biological pump in the future is independent of iron source type.
The local fraction that will not participate in biological production in the future,
f0↑(r), has a nearly spatially uniform distribution and
makes the largest contribution to the DFe inventory.
The local fraction that will be utilized exactly once before death, f1↑(r), is typically concentrated near the surface, where the likelihood of
passing through the biological pump before being scavenged is largest.
Correspondingly, the three-dimensional distribution of m‾(r)
has its maximum near the surface and is Southern Ocean intensified.
The fraction of DFe that will be regenerated m times in the future
again approaches an eigenmode for asymptotically large m.
The amplitude of this eigenmode decays with the same e-folding
scale n* as the eigenmode associated with past regenerations.
However, the eigenmodes associated with future and past regenerations
have different patterns, with the eigenmode for future regenerations
being much more surface intensified.
Both past and future eigenmodes are concentrated in the Southern Ocean
because multiple regenerations during an iron molecule's lifetime are more
likely where nutrients are effectively trapped .
We defined and quantified the intrinsic iron fertilization efficiency at
point r in terms of the number of globally exported phosphorus
molecules per DFe molecule at r.
This is the first time that a metric of iron fertilization efficiency
has been estimated in three dimensions, which may ultimately prove useful
for quantifying the importance of different iron reservoirs for supporting
the ocean's global export production.
At the surface, the intrinsic fertilization efficiency is largest in the
central tropical Pacific with secondary maxima in the subpolar oceans.
The relative importance of the Southern Ocean is greatest for high-source states.
Globally averaged, the surface fertilization efficiency ranges from 0.7
to 7molP(mmolFe)-1 across our family of state estimates,
with the low-source states having the largest efficiencies.
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
e.g.,. We also note that
the approach of using a linear equivalent linear model to partition iron and
diagnose its life cycle can also be applied to perturbed states (e.g., due to
iron addition or changes in circulation) to shed light on how the iron cycle
operates for various paleo or future climate scenarios. Finally, the concepts
and methods employed here can be applied to other nutrients for a more
complete picture of how the interaction between the biological pump and
physical transport shapes their distributions and cycling rates; we plan to
do so in future work.
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).
Glossary of mathematical symbols, their definitions, and SI units
(PH17 refers to Pasquier and Holzer, 2017).
SymbolDefinitionSI unitTAdvective eddy-diffusive transport operator: (Tχ)(r) is the flux divergence of χ at point rs-1LUptake operator: (Lχ)(r) is the rate of uptake of χ at point rs-1Sc, SjBiogenic transport and remineralization operators for the full water column (including the euphotic zone) for phytoplankton class c or particle type j (ScFe, Ss,j in PH17)dimensionlessBBiogenic transport and remineralization operator for export below the euphotic zonedimensionlessR, RPIron and phosphorus regeneration operatorss-1, molPmolFes-1DReversible scavenging operator: (Dχ)(r) is the rate of removal of χ by scavenging minus the rate with which scavenged iron is recycled at point rs-1H, FFull system operators H≡T-R+L+D and F≡T+L+Ds-1ARecursion operator A≡F-1R such that χk(n+1)↓=Aχkn↓dimensionlessλLargest eigenvalue of Adimensionlessn*e-folding scale for the decay of χkn↓ and χkm↑ with number or regenerations (independent of source type k)dimensionlessÕAdjoint of O for any operator O–rPosition vectormUc(r), UcP(r)Iron and phosphorus uptake rates for phytoplankton class c (RFe:PUc and Uc in PH17)molm-3s-1Jj(r)DFe scavenging rate for particle type jmolm-3s-1d(r)Death rate (rate of DFe scavenging that is not recycled and results in burial)molm-3s-1γc(r), γj(r), γD(r)Rate constants for iron uptake, scavenging, and death, respectivelys-1fcDetrital fraction for phytoplankton class cdimensionlessfrecFraction of scavenging that is recyclable (fbSi and fPOP in PH17)dimensionlessχ(r)Concentration of total DFe (χFe in PH17)molm-3χk(r)Concentration of DFe of source type k (aeolian: A, sedimentary: S, or hydrothermal: H)molm-3sk(r)DFe source of type k (global integral σk)molm-3s-1τBulk DFe lifetimesχn↓(r), χm↑(r)Concentration of DFe that passed n times, or that will pass m times, through the biological pumpmolm-3fn↓(r), fm↑(r)Fraction of DFe that passed n times, or that will pass m times, through the biological pumpdimensionlessn‾k(r), m‾(r)Mean number of past (n) or future (m) passages of DFe through the biological pumpdimensionlessn‾P(r), m‾P(r)Mean number of past (n) or future (m) moles of P regenerated per mole of FemolP(molFe)-1Φk(r), ΦkP(r)Globally integrated iron or phosphorus export rate due to iron source sk(r)molm-3s-1〈⋅〉, 〈⋅〉x, 〈⋅〉surfGlobal averages: volume-weighted, x-weighted, and area-weighted surface–Derivation of n‾(r)
First we verify that ∑n=0∞χkn↓=χk.
From Eqs. () and () we have
χkn↓=AnF-1sk,
where A≡F-1R. Because
A must have a maximum eigenvalue less than unity for
convergence, we can use the geometric operator sum ∑n=0∞An=(1-A)-1. Thus,
∑n=0∞χkn↓=(1-A)-1F-1sk.
Applying F(1-A) from the left, and recognizing that
F(1-A)=F-R=H, we
have
H∑n=0∞χkn↓=sk.
This shows that ∑n=0∞χkn↓ and χk obey
the same equation and hence that they are equal.
The defining equation for n‾k(r) is
χk(r)n‾k(r)=∑n=0∞nχkn↓(r).
From recursion
Eq. (), we
have χkn↓=Anχk0↓.
Substituting this, applying the
geometric operator sum ∑n=0∞nAn=(1-A)-1A(1-A)-1, and
using the fact that from
Eq. () χk0↓=F-1sk=F-1HH-1sk=(1-A)χk, we obtain
χkn‾k=(1-A)-1Aχk.
Applying RA-1(1-A)=F-R=H from the left yields Eq. ().
Local iron death rate
To diagnose the local death rate with which iron is permanently removed from
the ocean, we first rewrite the reversible scavenging operator
D in terms of a simplified local death rate. The operator
D is an integral operator so that (Dχ)(r)=∫d3r′KD(r|r′)χ(r′) with adjoint (D̃ϕ)(r′)=∫d3rϕ(r′)KD(r|r′). The term KD(r|r′)χ(r′) represents the
rate with which iron is scavenged at r′ minus the rate with which
this scavenged iron is redistributed to points r through particle
transport and redissolution. By integrating over all destination points
r, we obtain the deficit between the rate of iron scavenging at
r′ and the total water-column-integrated rate of redissolution of
that iron. Thus, d(r′)≡∫d3rKD(r|r′)χ(r′) is the rate of permanent
removal or death at r′. For the local rate constant we define
γD≡∫d3rKD(r|r′), which gives d(r′)=γD(r′)χ(r′).
Basin and global zonal averages of the death rate d(r′) for our
five representative states.
Note the logarithmic color scale.
For the interpretation of our diagnostics it is helpful to know how the local
iron death rate d(r′) is distributed through the ocean.
Figure shows basin and global zonal averages of the
death rate for our five representative states. Key features common to all
states are relatively low death rates in the subtropical gyres of both
hemispheres, where production and hence scavenging particle fluxes are low,
and relatively high death rates at high latitudes and in the tropics where
production is large. The highest death rates occur in the surface ocean where
particle fluxes are highest. The near-surface death rates are larger in the
Northern Hemisphere than in the Southern Hemisphere, presumably because the
DFe concentrations are higher in the Northern Hemisphere.
Variation of key diagnostics with recyclable fraction parameter
To explore the sensitivity of our results to variations in frec, we
changed the value of frec of our typical state (for which
frec=90%) and then re-optimized the biogeochemical
sink and source parameters following the optimization strategy described by
. This generated a set of eight optimized state
estimates with frec=0,10,30,50,70,80,90, and
100%. Except for the frec=100% case,
these states have similar iron source strengths (σA ranges
from 5.3 to 5.7Gmolyr-1, σS from 1.7 to
1.9Gmolyr-1, and σH is unchanged to two
significant figures at 0.88Gmolyr-1). For the extreme case of
frec=100%, when the only permanent iron sink in the
system is due to the flux of scavenged iron that reaches the ocean bottom
where we assume it is buried (in addition, iron scavenged by mineral dust is
always assumed not to be recyclable, but dust scavenging is very small for
our estimates), the optimized sedimentary iron source more than doubles and
the other sources also undergo significant adjustments during the
re-optimization (σA=5.1, σS=3.7, and
σH=0.85Gmolyr-1 for frec=100%).
Figure shows that the spatial pattern of the
zonally averaged n‾ is very insensitive to the value of
frec when the scavenging and source parameters are optimized. The
patterns of the zonally averaged m‾ have the same qualitative
features for all values of frec, but become more
surface intensified as frec increases to unity, presumably
because a greater recycling fraction increases the chance that DFe is
available near the euphotic zone to pass through the biological pump. The
patterns of n‾P and m‾P (not
shown) are similarly insensitive to frec.
(a) The
zonal means of n‾(r) normalized by the
corresponding global averages 〈n‾〉 for our typical
state estimate re-optimized with three very different recyclable fractions:
frec=0,50, and 100%. (b) The corresponding
normalized zonal means of m‾(r). The values of the global
averages, 〈n‾〉 and 〈m‾〉,
are given in the plot titles.
Figure shows how the volume-weighted global
means of n‾, m‾, n‾P, and
m‾P vary with the recyclable fraction,
frec. We find that 〈n‾〉 and 〈m‾〉 are order unity for frec=0% and
tend to decrease by roughly a factor of 2 to 3 when frec is
increased from 0 to 100%. We emphasize that this is not simple
sensitivity to frec because the scavenging and source parameters
have been re-optimized for each choice of frec. Broadly, a larger
value of the number of regenerations means that more iron is transported to
depth with the biological pump. This is consistent with the fact that for
small values of frec, less iron is transported to depth through
scavenging and redissolution. Given the similar iron sources of these
states, and hence similar global mean scavenging rates, reduced iron
transport through scavenging must be compensated for by increased transport
through biological uptake and regeneration, as measured here by the mean
number or iron regenerations. The mean numbers of phosphorus molecules that
are regenerated in the past or the future per iron molecule,
n‾P and m‾P, are much less
sensitive to the value of frec in these optimized states. The
reason for this insensitivity probably lies in the fact that the phosphorus
export is well constrained by the nutrient and phytoplankton data against
which we optimize , and any changes in iron
export are compensated for by different optimized values of the Fe : P uptake
ratio, which we also use as the Fe : P ratio for remineralization.
(a) The globally averaged mean number of past and future DFe
regenerations per injected DFe molecule, 〈n‾〉 and
〈m‾〉, as a function of the recyclable fraction
frec. The corresponding states were obtained by changing the
value of frec of the typical state and re-optimizing all other
source and scavenging parameters. The typical state has
frec=90% and is indicated by the vertical dashed
line. Panel (b) is the same as panel (a) but for the past and future number of phosphorus molecules
exported per injected DFe molecule, 〈n‾P〉
and 〈m‾P〉.
Recursion relation for fm↑Green functions
To derive the recursion relation for the fraction of iron at a given location
that is regenerated exactly m times before death, we use a Green function
approach. The Green function associated with the equation (∂t+H)χ=s is obtained by replacing the source s with a Dirac
delta function in space and time:
(∂t+Hr)GH(r,t|r′,t′)=δ(t-t′)δ3(r-r′),
where the subscript on Hr reminds us that H
acts on the field-point coordinates r. The Green function for the
time-reversed adjoint flow e.g., obeys
(-∂t′+H̃r′)GH̃(r′,t′|r,t)=δ(t-t′)δ3(r-r′),
where the adjoint Green function GH̃ obeys the
reciprocity relation GH̃(r′,t′|r,t)=GH(r,t|r′,t′). The Green functions associated
with (∂t+F)χ=s are defined in exactly the same
manner.
All adjoints here are defined in terms of the volume-weighted inner product
so that for linear operator H and any two fields x and y we
have ∫d3rx(r)(Hy)(r)=∫d3r(H̃x)(r)y(r).
For computation, all linear operators are discretized on a numerical grid and
organized into sparse matrices; e.g., H becomes matrix
H with adjoint H̃=V-1HTV, where V is a diagonal matrix of
the grid box volumes.
All DFe must die
Now consider the concentration of iron χ(r′,t′) in some volume
d3r′. As the system evolves the mass
χ(r′,t′)d3r′ results in concentration
X(r,t|r′,t′)d3r′ at (r,t), which is obtained by propagating with GH so that
X(r,t|r′,t′)d3r′=GH(r,t|r′,t′)χ(r′,t′)d3r′.
(Note that there is no integral as this is point-wise propagation.) The death
rate per unit volume incurred by X(r,t|r′,t′)d3r′ at (r,t) is given by
γD(r)X(r,t|r′,t′)d3r′. Integrating this death rate over all times t and
over the entire ocean for r, we must recover the initial mass χ(r′,t′)d3r′. Thus, we have
χ(r′,t′)d3r′≡∫dt∫d3rγD(r)GH(r,t|r′,t′)χ(r′,t′)d3r′,
where the initial volume d3r′ is not integrated
over. Because GH is just a function and not a differential
operator, and because we are integrating with respect to (r,t), we
can divide both sides by χ(r′,t′)d3r′, which
gives
f↑≡∫dt∫d3rγD(r)GH(r,t|r′,t′),
where f↑ is just a spatially uniform field of unit value, i.e.,
f↑=1. Applying the time-reversed adjoint operator
(-∂t′+H̃r′) from the left to
Eq. () and using the reciprocity relation of GH,
we obtain Eq. (). Note that H̃=T̃+(L̃-R̃)+D̃. Because T and L-R are mass-conserving operators, T̃f↑=0 and (L̃-R̃)f↑=0 so that Eq. () is
equivalent to D̃f↑=γD,
which reproduces the definition of γD.
Fraction not regenerated in the future
Similarly we can construct the concentration of DFe that has been regenerated
exactly zero times since being at (r′,t′). By propagating the mass
χ(r′,t′)d3r′ with F instead of
H, we obtain the resulting concentration at t that has not
passed through the biological pump since t′:
X0↑(r,t|r′,t′)d3r′=GF(r,t|r′,t′)χ(r′,t′)d3r′.
Calculating the death rate per unit volume by multiplying with
γD and integrating over all r and t must give the
mass in d3r′ that will not be regenerated in the future,
i.e., χ0↑(r′,t′)d3r′. Dividing both
sides by the starting mass χ(r′,t′)d3r′ and
defining the fraction f0↑(r′,t′)≡χ0↑(r′,t′)/χ(r′,t′) gives
f0↑(r′,t′)=∫dt∫d3rγD(r)GF(r,t|r′,t′).
Applying the time-reversed adjoint operator (∂t′+F̃r′) from the left gives Eq. ().
Recursion for the fraction regenerated m times in the future
To construct the recursion equation for fm↑, it is useful to
explicitly write the regeneration operator in terms of its integration
kernel:
(Rχ)(r)=∫d3r′′KR(r|r′′)χ(r′′,t′).
We again consider the concentration at (r,t) resulting from the mass
of DFe in d3r′ at t′ that was not regenerated since t′,
X0↑(r,t|r′,t′)d3r′, as
defined by Eq. (). The rate with which
X0↑(r,t|r′,t′)d3r′ is
regenerated (i.e., remineralized) in volume d3r′′ is given
by ∫d3rKR(r′′|r)X0↑(r,t|r′,t′)d3r′ (where
d3r′ is not integrated over) and the fraction of this rate
that will get regenerated a further m times in the future is given by
fm↑(r′′)∫d3rKR(r′′|r)GF(r,t|r′,t′)χ(r′,t′)d3r′.
Integrating this rate over all r′′ and t, we must recover the
fraction of the initial mass at t′ that will be regenerated exactly m+1
times in the future. Hence, we have
f(m+1)↑(r′)χ(r′,t′)d3r′=∫dt∫d3r∫d3r′′fm↑(r′′)×KR(r′′|r)GF(r,t|r′,t′)χ(r′,t′)d3r′.
Dividing both sides by the initial mass χ(r′,t′)d3r′ and with the adjoint regeneration operator
R̃ defined through (R̃f)(r)=∫d3r′′f(r′′)KR̃(r|r′′), where
KR̃(r|r′′)≡KR(r′′|r), we have
f(m+1)↑(r′)=∫dt∫d3r′′∫d3rGF̃(r′,t′|r,t),×KR̃(r|r′′)fm↑(r′′).
Note that Eq. () propagates the fraction to be
regenerated m times through one regeneration (modeled as instantaneous)
and through the adjoint flow backward in time to the fraction that will be
regenerated (m+1) times. Applying (-∂t′+F̃) from the left gives the recursion
Eq. (). Note that this recursion equation can be
written as fm↑=(F̃-1R̃)mf0↑. Using
Eq. () and regrouping the factors of F̃-1R̃, this recursion relation can be rewritten as
fm↑=F̃-1ÃmγD,
where A≡F-1R as in
Appendix , with adjoint Ã=R̃F̃-1.
Equation () is the analog of
Eq. () for the time-reversed adjoint problem.
Using similar techniques as in Appendix , it follows
readily that ∑m=0∞fm↑=f↑=1, as must be
the case. Equation () for m‾ is also derived
analogously to n‾ in Appendix .
Relation between export and m‾
Here we calculate the steady-state globally integrated export that results
from a steady injection with source s(r′) (in
molFem-3s-1) in some volume d3r′ during
dt′. Propagating the initial DFe mass
s(r′)d3r′dt′ with H results in
concentration GH(r,t|r′,t′)s(r′)d3r′dt′ at (r,t). This DFe is instantly
regenerated at r′′ with rate ∫d3rKR(r′′|r)GH(r,t|r′,t′)s(r′)d3r′dt′. Integrating over all
r′′ and t must give the globally integrated export Φ(r′)d3r′dt′ that is due to the initial injection of
the mass s(r′)d3r′dt′, and further
dividing by d3r′dt′ gives
Φ(r′)=∫d3r′′∫d3rKR(r′′|r)∫dtGH(r,t|r′,t′)s(r′).
The globally integrated export per unit source injection at r′ is
thus g(r′)≡Φ(r′)/s(r′). Dividing
Eq. () by s(r′) and applying (-∂t′+H̃) gives
H̃g=R̃f↑,
which is the Green function that propagates a unit source to globally
integrated export. Comparison with Eq. () shows that g=m‾ from which Eq. () of the main text
follows by replacing our generic source s(r′) with one of the sources
sk(r′). Equation () for the global
phosphorus export due to DFe injection at r′ can be derived in
exactly the same way by using regeneration operator RP
instead of R.
The authors declare that they have no conflict of
interest.
Acknowledgements
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
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