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**Research article**
24 Jan 2019

**Research article** | 24 Jan 2019

Modeling oceanic nitrate and nitrite isotopes

^{1}Department of Earth System Science, Stanford University, Stanford, CA, USA^{2}Department of Earth System Science, University of California, Irvine, Irvine, CA, USA

Abstract

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Nitrite (${\mathrm{NO}}_{\mathrm{2}}^{-}$) is a key intermediate in the marine nitrogen (N) cycle
and a substrate in nitrification, which produces nitrate (${\mathrm{NO}}_{\mathrm{3}}^{-}$),
as well as water column N loss processes denitrification and anammox. In
models of the marine N cycle, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ is often not considered as a
separate state variable, since ${\mathrm{NO}}_{\mathrm{3}}^{-}$ occurs in much higher
concentrations in the ocean. In oxygen deficient zones (ODZs), however,
${\mathrm{NO}}_{\mathrm{2}}^{-}$ represents a substantial fraction of the bioavailable N,
and modeling its production and consumption is important to understand the N
cycle processes occurring there, especially those where bioavailable N is
lost from or retained within the water column. Improving N cycle models by
including ${\mathrm{NO}}_{\mathrm{2}}^{-}$ is important in order to better quantify N
cycling rates in ODZs, particularly N loss rates. Here we present the
expansion of a global 3-D inverse N cycle model to include ${\mathrm{NO}}_{\mathrm{2}}^{-}$
as a reactive intermediate as well as the processes that produce and consume
${\mathrm{NO}}_{\mathrm{2}}^{-}$ in marine ODZs. ${\mathrm{NO}}_{\mathrm{2}}^{-}$ accumulation in ODZs is
accurately represented by the model involving ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction,
${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation, and anammox. We
model both ^{14}N and ^{15}N and use a compilation of
oceanographic measurements of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$
concentrations and isotopes to place a better constraint on the N cycle
processes occurring. The model is optimized using a range of isotope effects
for denitrification and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation, and we find that the
larger (more negative) inverse isotope effects for ${\mathrm{NO}}_{\mathrm{2}}^{-}$
oxidation, along with relatively high rates of ${\mathrm{NO}}_{\mathrm{2}}^{-}$, oxidation
give a better simulation of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$
concentrations and isotopes in marine ODZs.

How to cite

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How to cite.

Martin, T. S., Primeau, F., and Casciotti, K. L.: Modeling oceanic nitrate and nitrite concentrations and isotopes using a 3-D inverse N cycle model, Biogeosciences, 16, 347-367, https://doi.org/10.5194/bg-16-347-2019, 2019.

1 Introduction

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Nitrogen (N) is an important nutrient to consider when assessing biogeochemical cycling in the ocean. The N cycle is intrinsically tied to the carbon (C) cycle, whereby N can be the limiting nutrient for primary production and carbon dioxide uptake (Moore et al., 2004; Codispoti, 1989). Understanding the distribution and speciation of bioavailable N in the ocean allows us to make inferences about the effects on other nutrient cycles and potential roles that N may play in a regime of climate change (Gruber, 2008).

There are several chemical species in which N can be found in the ocean. The largest pool of bioavailable N is nitrate (${\mathrm{NO}}_{\mathrm{3}}^{-}$), a dissolved inorganic species, which can be taken up by microbes for use in assimilatory or dissimilatory processes. Another dissolved inorganic species, nitrite (${\mathrm{NO}}_{\mathrm{2}}^{-}$), accumulates in much lower concentrations but is a key intermediate in many N cycling processes. Models of the marine N cycle often include ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ together as a single dissolved inorganic N (DIN) pool, or exclude ${\mathrm{NO}}_{\mathrm{2}}^{-}$ entirely (DeVries et al., 2013; Deutsch et al., 2007; Brandes and Devol, 2002). However, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ does accumulate significantly in oxygen deficient zones (ODZs) in features known as secondary ${\mathrm{NO}}_{\mathrm{2}}^{-}$ maxima, and it is an intermediate or substrate in many important N cycle processes occurring there.

ODZs are hotspots for marine N loss (Codispoti et al., 2001; Deutsch et al., 2007), which
is driven by processes that result in conversion of bioavailable DIN to dinitrogen gas
(N_{2}). The two main water column N loss processes, denitrification and anammox,
use ${\mathrm{NO}}_{\mathrm{2}}^{-}$ as a substrate. Denitrification involves the stepwise reduction of
${\mathrm{NO}}_{\mathrm{3}}^{-}$ to ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and then to gaseous nitric oxide (NO), nitrous
oxide (N_{2}O), and N_{2}. Anammox consists of the anaerobic oxidation of
ammonium (${\mathrm{NH}}_{\mathrm{4}}^{+}$) to N_{2} using ${\mathrm{NO}}_{\mathrm{2}}^{-}$ as the electron
acceptor. ${\mathrm{NO}}_{\mathrm{2}}^{-}$ is also oxidized to ${\mathrm{NO}}_{\mathrm{3}}^{-}$ during anammox,
representing an alternative fate for ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in ODZs. Indeed, ${\mathrm{NO}}_{\mathrm{2}}^{-}$
oxidation appears to be prevalent in ODZs, with more ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation
occurring than can be explained by anammox alone (Gaye et al., 2013; Peters et al., 2016,
2018b; Babbin et al., 2017; Buchwald et al., 2015;
Casciotti et al., 2013; Martin and Casciotti, 2017). ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation results
in the regeneration of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ that would otherwise be converted to N_{2}
and lost from the system. The close coupling between ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction
to ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation back to ${\mathrm{NO}}_{\mathrm{3}}^{-}$, represents
a control valve on the marine N budget (Penn et al., 2016; Bristow et al., 2016). Where
${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation can outcompete ${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction via
denitrification and anammox, bioavailable N is retained. Water column N losses may
occur primarily where ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation rates are limited by oxygen availability. Thus,
understanding the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ dynamics in ODZ waters is critical to assess the N
loss occurring there.

The observed ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations alone do
not allow us to fully characterize the N cycling processes occurring in a
given region. Stable isotope measurements of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and
${\mathrm{NO}}_{\mathrm{2}}^{-}$ provide additional insight and constraints on marine N
cycle processes. There are two stable isotopes of N: ^{14}N and
^{15}N. The isotopic ratios for a given N species, usually expressed
in delta notation as *δ*^{15}N
(‰) = $\left(\right({}^{\mathrm{15}}\mathrm{N}/{}^{\mathrm{14}}\mathrm{N}{)}_{\mathrm{sample}}/({}^{\mathrm{15}}\mathrm{N}/{}^{\mathrm{14}}\mathrm{N}{)}_{\mathrm{standard}}-\mathrm{1})\times \mathrm{1000}$, are an
integrated measure of the processes that have produced and consumed that N
species. Each process imparts a unique isotope effect (*ε*
(‰) = ${(}^{\mathrm{14}}k{/}^{\mathrm{15}}k-\mathrm{1})\times \mathrm{1000}$, where ^{14}*k* and
^{15}*k* are the first-order rate constants for the ^{14}N and
^{15}N containing molecules, respectively) that impacts the isotopic
composition of the substrate and the product (Mariotti et al., 1981). In
particular, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ cycling processes have distinct isotope
effects, where ${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction occurs with normal isotopic
fractionation (Bryan et al., 1983; Martin and Casciotti, 2016; Brunner et
al., 2013) and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation occurs with an unusual inverse
kinetic isotope effect (Casciotti, 2009; Buchwald and Casciotti, 2010;
Brunner et al., 2013). Thus, the isotopes of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ are sensitive
to the relative importance of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation and
${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction in ${\mathrm{NO}}_{\mathrm{2}}^{-}$ consumption (Casciotti,
2009; Casciotti et al., 2013).

Models of the marine N cycle have employed isotopes and isotope effects in conjunction with N concentrations to elucidate N cycle processes (Brandes and Devol, 2002; Sigman et al., 2009; Somes et al., 2010; DeVries et al., 2013; Casciotti et al., 2013; Buchwald et al., 2015; Peters et al., 2016). A model can either assume a set of processes and infer the underlying isotope effects, or assume isotope effects and infer a set of processes. The latter isotope models are highly dependent on the chosen isotope effects used for given processes. Although there are estimates of isotope effects for processes based on both environmental measurements and laboratory studies, there is not always agreement between them. For example, laboratory cultures of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidizers indicate an N isotope effect of ${}^{\mathrm{15}}\mathit{\epsilon}=-\mathrm{10}$ to −20 ‰ (Casciotti, 2009; Buchwald and Casciotti, 2010), while measured concentrations and isotopes of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in ODZs indicate that isotope effects closer to −30 ‰ are needed to explain the observations (Buchwald et al., 2015; Casciotti et al., 2013; Peters et al., 2016).

Here we present an expansion of an existing global ocean 3-D inverse isotope-resolving N cycling model (DeVries et al., 2013) to investigate the isotopic constraints on N cycling in ODZs and the impact of these regions on global ocean N isotope patterns. An important step was to include ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and its isotopes as tracers. The addition of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ allows us to include additional internal N cycling processes, as well as a more nuanced and realistic version of the processes occurring in ODZs. We used a database of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ observations in order to assess the performance of the model as well as optimize the model N cycle parameters for which we do not have good prior estimates. In the model we employ a variety of isotope effect estimates for three important ODZ processes – ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction, and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation – to discern what isotope effects result in the best fit to the observations.

2 Methods

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The model used here is a steady-state inverse model that solves for the concentration and
*δ*^{15}N of ${\mathrm{NO}}_{\mathrm{3}}^{-}$, ${\mathrm{NO}}_{\mathrm{2}}^{-}$, particulate organic N (PON),
and dissolved organic N (DON) using a set of linear equations. Because the model assumes
that the system is in steady state, it is not able to capture time-dependent properties
of the system such as seasonality and anthropogenic change. However, on interannual
timescales the N cycle is thought to be approximately in balance (Gruber, 2004; Bianchi
et al., 2012). The residence time of N in the ocean, which is thought to be on the order
of 2000–3000 years (Gruber, 2008), is sufficiently long to preclude any detectable
changes in the global N inventory to date on timescales commensurate with the global
overturning circulation. An important advantage of the steady-state assumption for our
linear model is that it is possible to find solutions by direct matrix inversion without
the need for a spin-up period as required by forward models. The solution to the system
provides ^{14}N and ^{15}N concentrations of the N species of interest at
every grid point in the model system. Working with a linear system imposes some
restrictions on how complicated the rate equations can be, but there are improvements in
model performance and ease of use, allowing us to test hypotheses about the processes
that govern the marine N cycle and budget, particularly those occurring in and around
oceanic ODZs. We aimed to produce a realistic N cycle model that represented ODZ
processes accurately while limiting the number of free parameters. The description below
outlines the dependencies and simplifications employed in this version of the model.

The model's uncertain biological parameters were determined through an optimization
process that minimizes the difference between the modeled and observed ${\mathrm{NO}}_{\mathrm{3}}^{-}$
and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration and isotope data. Computational time limits the
number of parameters that we were able to optimize. We therefore focused our
investigation on parameters that are poorly constrained by literature values and to which
the model solution is most sensitive. In order to determine the parameters for
optimization, a sensitivity analysis was performed on each parameter, varying them
individually by ±10 % and computing the change in the modeled ^{14}N and
^{15}N. Those parameters that resulted in modeled ^{14}N and ^{15}N
variability of >5 % were chosen for optimization in the model. The sensitivity
analysis and the optimal values of the parameters contribute to an improved understanding
of the cycling of N in the ocean in general and in the ODZs in particular. The
optimization process is discussed in further detail in Sect. 2.6.

The sensitivity analysis revealed that the modeled distribution of ^{15}N was very
sensitive to chosen isotope effects, those parameters that control the relative rates of ^{15}N
and ^{14}N in chemical and biological processes. There are literature estimates
for each of the isotope effects of interest in this work, although there is often a
discrepancy between isotope effects estimated in laboratory studies and those expressed
in oceanographic measurements (Kritee et al., 2012; Casciotti et al., 2013; Bourbonnais
et al., 2015; Martin and Casciotti, 2017; Fuchsman et al., 2017; Marconi et al., 2017;
Peters et al., 2018b). Rather than optimizing the isotope effect values, we have chosen
to use multiple cases with different combinations of previously estimated isotope effects
in order to assess which values best fit the observations.

In addition to the optimized parameters and isotope effects, there were some nonsensitive parameters that were fixed prior to the optimization and whose values were chosen using literature estimates (Table 1). Some N cycle processes are also dependent on prescribed input fields that are not explicitly modeled, such as temperature, phosphate, oxygen, and net primary production. These external input fields will be discussed in detail in the relevant sections for each N cycle process.

The model uses a uniform 2^{∘} × 2^{∘} grid with 24 depth
levels. The thickness of each model layer increases with depth, from 36 m at
the top of the water column to 633 m near the bottom. Bottom topography was
determined using 2 min gridded bathymetry (ETOPO2v2) that was then
interpolated to the model grid. Our linear N cycle model relies on the
transport of dissolved N species (${\mathrm{NO}}_{\mathrm{3}}^{-}$, ${\mathrm{NO}}_{\mathrm{2}}^{-}$, and
DON) in the ocean. For this we use the annual averaged circulation as
captured by a tracer transport operator that governs the rate of transport of
dissolved species (${\mathrm{NO}}_{\mathrm{3}}^{-}$, ${\mathrm{NO}}_{\mathrm{2}}^{-}$, and DON) between
boxes. The original version of the tracer data-assimilation procedure used to
generate the transport operator for dissolved species (*T*_{f}) is described
by DeVries and Primeau (2011), and the higher resolution version used here is
described by DeVries et al. (2013).

In the N cycling portion of the model, we track four different N species (Fig. 1). There
are two organic N (ON) pools: dissolved (DON) and particulate (PON). There are also two
dissolved inorganic N (DIN) pools: ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$. We did not
explicitly model ammonium (${\mathrm{NH}}_{\mathrm{4}}^{+}$) because it typically occurs in low
concentrations throughout the ocean, and scarcity of data (especially *δ*^{15}N data)
would make model validation difficult. Although ${\mathrm{NH}}_{\mathrm{4}}^{+}$ has been observed to
accumulate to micromolar concentrations in some ODZs (Bristow et al., 2016; Hu et al.,
2016), this occurs largely in shallow, coastal shelf regions that are not resolved by the
model.

Because we used the concentrations of both ^{14}N and ^{15}N of
each N species to constrain the rate parameters, two sets of governing
equations were employed: one that depends on ^{14}N and another that
depends on ^{15}N. Generally, the rate for ^{15}N processes was
dependent on the rate of ^{14}N processes and an isotopic
fractionation factor ($\mathit{\alpha}{=}^{\mathrm{14}}k{/}^{\mathrm{15}}k$) that is specific to each process and
substrate. By solving for steady-state solutions to both ^{14}N and
^{15}N concentrations, we were able to model global distributions of
[${\mathrm{NO}}_{\mathrm{3}}^{-}$], [${\mathrm{NO}}_{\mathrm{2}}^{-}$], and their corresponding
*δ*^{15}N values.

We first describe the ^{14}N equations and the general format of
the N cycle in the model. Each equation is then broken down into its
component parts for further explanation of the biological processes and their
parameterization. The ^{15}N equations and isotope implementation will
be discussed in a later section.

The governing equations for the ^{14}N-containing DIN
(${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$) and organic N (DON and PON) state
variables can be written as follows:

$$\begin{array}{ll}{\displaystyle}\left({\displaystyle \frac{\partial}{\partial t}}+{T}_{\mathrm{f}}\right)\left[{}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{3}}^{-}\right]& {\displaystyle}={J}_{\mathrm{14}}^{\mathrm{dep}}-{J}_{\mathrm{14}}^{\mathrm{assim},{\mathrm{NO}}_{\mathrm{3}}}-{J}_{\mathrm{14}}^{\mathrm{NAR}}\\ \text{(1)}& {\displaystyle}& {\displaystyle}+{J}_{\mathrm{14}}^{\mathrm{NXR}}+\mathrm{0.3}{J}_{\mathrm{14}}^{\mathrm{AMX}}-{J}_{\mathrm{14}}^{\mathrm{sed}},{\displaystyle}\left({\displaystyle \frac{\partial}{\partial t}}+{T}_{\mathrm{f}}\right)\left[{}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}^{-}\right]& {\displaystyle}={J}_{\mathrm{14}}^{\mathrm{AMO}}-{J}_{\mathrm{14}}^{\mathrm{assim},{\mathrm{NO}}_{\mathrm{2}}}+{J}_{\mathrm{14}}^{\mathrm{NAR}}\\ \text{(2)}& {\displaystyle}& {\displaystyle}-{J}_{\mathrm{14}}^{\mathrm{NXR}}-{J}_{\mathrm{14}}^{\mathrm{NIR}}-\mathrm{1.3}{J}_{\mathrm{14}}^{\mathrm{AMX}},{\displaystyle}\left({\displaystyle \frac{\partial}{\partial t}}+{T}_{\mathrm{f}}\right)\left[\mathrm{DO}{}^{\mathrm{14}}\mathrm{N}\right]& {\displaystyle}=\mathit{\sigma}({J}_{\mathrm{14}}^{\mathrm{fix}}+{J}_{\mathrm{14}}^{\mathrm{assim},\mathrm{WOA}})\\ \text{(3)}& {\displaystyle}& {\displaystyle}+{J}_{\mathrm{14}}^{\mathrm{sol}}-{J}_{\mathrm{14}}^{\mathrm{remin}},\end{array}$$

$$\begin{array}{ll}{\displaystyle}\left({\displaystyle \frac{\partial}{\partial t}}+{T}_{\mathrm{p}}\right)\left[\mathrm{PO}{}^{\mathrm{14}}\mathrm{N}\right]& {\displaystyle}=(\mathrm{1}-\mathit{\sigma})({J}_{\mathrm{14}}^{\mathrm{fix}}+{J}_{\mathrm{14}}^{\mathrm{assim},\mathrm{WOA}})\\ \text{(4)}& {\displaystyle}& {\displaystyle}-{J}_{\mathrm{14}}^{\mathrm{sol}}.\end{array}$$

The model is designed to represent a steady state, thus the $\frac{\partial}{\partial t}$ term is 0. The *J* terms represent the source and sink processes for each state
variable, expressed in units of mmol m^{−3} yr^{−1}
and will be described in more detail below. Briefly, ${J}_{\mathrm{14}}^{\mathrm{dep}}$ is the
spatially variable deposition of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ from the atmosphere to the sea
surface. In the DIN model equations, ${J}_{\mathrm{14}}^{\mathrm{assim},{\mathrm{NO}}_{\mathrm{3}}}$ and
${J}_{\mathrm{14}}^{\mathrm{assim},{\mathrm{NO}}_{\mathrm{2}}}$ represent the assimilation of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and
${\mathrm{NO}}_{\mathrm{2}}^{-}$, respectively, by phytoplankton in the upper two box levels. This
assimilated ${\mathrm{NO}}_{\mathrm{3}}^{-}$ produces DON and PON, with proportions set by a spatially
variable term, *σ*. Assimilation in the DON and PON equations is represented by
${J}_{\mathrm{14}}^{\mathrm{assim},\mathrm{WOA}}$ and is dependent on 2013 World Ocean Atlas (WOA)
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] interpolated to the model grid. N_{2} fixation
(${J}_{\mathrm{14}}^{\mathrm{fix}}$) is split between DON and PON with the same *σ* term.
${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction (${J}_{\mathrm{14}}^{\mathrm{NAR}}$), ${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction
(${J}_{\mathrm{14}}^{\mathrm{NIR}}$), ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation (${J}_{\mathrm{14}}^{\mathrm{NXR}}$), and
anammox (${J}_{\mathrm{14}}^{\mathrm{AMX}}$) act on the ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$
pools. ${J}_{\mathrm{14}}^{\mathrm{sed}}$ represents the removal of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ via benthic
denitrification. ${J}_{\mathrm{14}}^{\mathrm{sol}}$ represents the dissolution of PON into DON.
${J}_{\mathrm{14}}^{\mathrm{remin}}$ represents the degradation of DON, which feeds into ammonia
oxidation (${J}_{\mathrm{14}}^{\mathrm{AMO}}$) and ${J}_{\mathrm{14}}^{\mathrm{AMX}}$ as described below.

Through the use of these *J* terms, the governing equations are all linear with
respect to the state variables. However, in order to introduce dependence of
rates on the concentrations of multiple state variables, for example allowing
heterotrophic ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction to be dependent on organic N as
well as ${\mathrm{NO}}_{\mathrm{3}}^{-}$, we run the organic N equations and the DIN
equations seperately. When [DON] is found in the [DIN] governing equations, that
[DON] value has already been determined for each grid box from the organic N
model. When [${\mathrm{NO}}_{\mathrm{3}}^{-}$] is found in the DON governing equations, it
is drawn from 2013 World Ocean Atlas annual data interpolated to the model
grid.

Atmospheric deposition and N_{2} fixation are the two largest sources of
N to the ocean (Gruber and Galloway, 2008) and the only sources of new
bioavailable N in the model. We do not consider the third largest source of
N, riverine fluxes, in the model due to a lack of coastal resolution and the
expectation that much of the river-derived N is denitrified in the shelf
sediments (Nixon et al., 1996; Seitzinger and Giblin, 1996). Representing
these processes may be possible in a future version of the model, but is
beyond the scope of the current model, given its coarse resolution near the
coasts.

N deposition is assumed to only occur in the top box of the model. We assume that most of the N deposited is as ${\mathrm{NO}}_{\mathrm{3}}^{-}$, and that the other species would be rapidly oxidized to ${\mathrm{NO}}_{\mathrm{3}}^{-}$ in the oxic surface waters.

$$\begin{array}{}\text{(5)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{dep}}={r}_{\mathrm{14}}^{\mathrm{dep}}{S}^{\mathrm{dep}}\end{array}$$

To calculate ${J}_{\mathrm{14}}^{\mathrm{dep}}$, the atmospheric deposition rate of
^{14}N, we use modeled total inorganic N deposition for 1993,
*S*^{dep} (Galloway et al., 2004; Dentener et al., 2006; data
available online at
https://daac.ornl.gov/CLIMATE/guides/global_N_deposition_maps.html,
last access: November 2017), which was
interpolated to our model grid. This term, *S*^{dep}, is then
multiplied by a prescribed fractional abundance of ^{14}N in the
deposited N (${r}_{\mathrm{14}}^{\mathrm{dep}}$), which is calculated from the isotopic
composition of deposited N (*δ*^{15}N_{dep}, −4 ‰;
Eq. 6), to yield the deposition of ^{14}N to the sea surface in each
box (${J}_{\mathrm{14}}^{\mathrm{dep}}$). To calculate ${r}_{\mathrm{14}}^{\mathrm{dep}}$ from
*δ*^{15}N_{dep}, we first calculate ${r}_{\mathrm{15}}^{\mathrm{dep}}$ using
${r}_{\mathrm{15}}^{\mathrm{air}}$, a standard with a value of 0.003676 (Eq. 6;
Mariotti, 1983).

$$\begin{array}{}\text{(6)}& {\displaystyle}{r}_{\mathrm{15}}^{\mathrm{dep}}=\left({\displaystyle \frac{{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{dep}}}{\mathrm{1000}}}+\mathrm{1}\right)\times {r}_{\mathrm{15}}^{\mathrm{air}}\end{array}$$

Then, using the approximation that ^{15}N∕^{14}N = ${}^{\mathrm{15}}\mathrm{N}/{(}^{\mathrm{15}}\mathrm{N}{+}^{\mathrm{14}}\mathrm{N})$, we calculate
${r}_{\mathrm{14}}^{\mathrm{dep}}$ as ($\mathrm{1}-{r}_{\mathrm{15}}^{\mathrm{dep}}$). The units of
*S*^{dep} are given in mg N m^{−2} yr^{−1}, which we convert to
mmol ${\mathrm{NO}}_{\mathrm{3}}^{-}$ m^{−3} yr^{−1} by dividing by the depth of the surface box. This source term of N to the model
is spatially variable but independent of the modeled N terms.

N_{2} fixation is the other source of new N to the model, and is assumed
to only occur in the top box of the model. It is parameterized similarly to
N_{2} fixation in the model of DeVries et al. (2013), with partial
inhibition by ${\mathrm{NO}}_{\mathrm{3}}^{-}$ (Holl and Montoya, 2005) and dependence on
iron (Fe) and phosphate (${\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}$) availability (Monteiro et al.,
2011).

$$\begin{array}{}\text{(7)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{fix}}={r}_{\mathrm{14}}^{\mathrm{fix}}{F}_{\mathrm{0}}{e}^{-{\mathrm{NO}}_{\mathrm{3},\mathrm{obs}}/\mathit{\lambda}}{e}^{\frac{{T}_{\mathrm{obs}}-{T}_{\mathrm{max}}}{{T}_{\mathrm{0}}}}{\displaystyle \frac{\mathrm{Fe}}{\mathrm{Fe}+{K}_{\mathrm{Fe}}}}{\displaystyle \frac{{\mathrm{PO}}_{\mathrm{4}}}{{\mathrm{PO}}_{\mathrm{4}}+{K}_{\mathrm{P}}}}\end{array}$$

*F*_{0} is the maximum rate of N_{2} fixation (1.5 mmol m^{−3} yr^{−1};
Table 1) and is calculated from the estimated areal rate of N_{2} fixation in the
western tropical Atlantic (Capone et al., 2005) divided by the depth of the top model
box. NO_{3,obs} is the 2013 World Ocean Atlas annually averaged surface
${\mathrm{NO}}_{\mathrm{3}}^{-}$ interpolated to the model grid (Garcia et al., 2013b).
The parameter *λ* is an inhibition constant for N_{2} fixation in the presence of
${\mathrm{NO}}_{\mathrm{3}}^{-}$ (Table 1).

The temperature (*T*) terms scale the rate of N_{2} fixation based on
the observed temperature (*T*_{obs}), maximum observed sea surface
temperature (*T*_{max}), and the minimum preferred growth temperature for
*Trichodesmium* (*T*_{0}; Capone et al., 2005). The temperature data
were taken from 2013 World Ocean Atlas annually averaged temperature
interpolated to the model grid (Locarnini et al., 2013). We recognize that
this will likely provide a conservative estimate of N_{2} fixation,
given the growing recognition of N_{2} fixation outside of the tropical
and subtropical ocean by organisms other than *Trichodesmium*
(Shiozaki et al., 2017; Harding et al., 2018; Landolfi et al., 2018).

Fe is the modeled deposition of soluble Fe interpolated to the model grid
(mmol Fe m^{−2} yr^{−1}; Chien et al., 2016) divided by the depth of
the top model grid box to give units of mmol Fe m^{−3} yr^{−1}. Fe and
${\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}$ are assumed to limit N_{2} fixation at low
concentrations via Michaelis–Menten kinetics. *K*_{Fe} and
*K*_{P} are their respective half-saturation constants. Additionally,
there is a term that allows us to set the isotopic ratio of newly fixed N,
${r}_{\mathrm{14}}^{\mathrm{fix}}$, which is the fractional abundance of ^{14}N
in newly fixed N and is calculated as in Eq. (6) from *δ*^{15}N_{fix} (−1 ‰; Table 1). All of the N_{2} fixation
parameters are fixed rather than optimized (Table 1). Due to the use of
non-optimized parameters and an input ${\mathrm{NO}}_{\mathrm{3}}^{-}$ field rather than
modeled ${\mathrm{NO}}_{\mathrm{3}}^{-}$, N_{2} fixation serves as an independent check
that our modeled N cycle produces reasonable N concentrations and overall N
loss rates. However, N_{2} fixation is not explicitly modeled here and
is instead taken as a fixed, though spatially variable, input field (Fig. S1
in the Supplement). The global rate of N_{2} fixation produced by this
parameterization is 131 Tg N yr^{−1}, which is in line with several
current estimates (Table S1 in the Supplement).

In the model, N_{2} fixation and ${\mathrm{NO}}_{\mathrm{3}}^{-}$ assimilation
(Sect. 2.3.3) are assumed to be the two processes that create exportable
organic N. A fraction, *σ*, of this organic N is partitioned into DON
rather than PON (Eqs. 3–4). In order to create spatial variability in this
constant, we assumed that 1−*σ*, the fraction of assimilated N
partitioned to PON, is equal to the particle export (*P*_{e}) ratio.
This *P*_{e} ratio is the ratio of particle export to primary
production, and is equivalent to the fraction of organic N that is exported
from the euphotic zone as particulate matter rather than recycled or
solubilized into DON. The *P*_{e} ratio is calculated for each model
grid square from the mixed layer temperature (*T*_{ml}) and net
primary production (NPP) as described by Dunne et al. (2005):

$$\begin{array}{}\text{(8)}& {\displaystyle}{P}_{e}=\mathit{\varphi}{T}_{\mathrm{ml}}+\mathrm{0.582}\mathrm{log}\left(\mathrm{NPP}\right)+\mathrm{0.419}.\end{array}$$

The constant Φ has a value of −0.0101 ^{∘}C^{−1} as determined
by Dunne et al. (2005). Net primary production estimates (in units of
mmol carbon m^{−2} yr^{−1})
were taken from a satellite-derived productivity model (Westberry et al.,
2008), annually averaged, and interpolated onto the model grid.
*T*_{ml} is calculated from the 2013 World Ocean Atlas annual average
(Locarnini et al., 2013), which has been interpolated to the model grid. The
temperature of the top two model boxes were averaged to give *T*_{ml}.
As temperature increases, the *P*_{e} ratio decreases and less PON is
exported, resulting in more DON recycling in the surface with several
possible explanatory mechanisms discussed in greater detail by Dunne et
al. (2005). As net primary production increases, the *P*_{e} ratio
increases and relatively more PON is exported;
net primary production explains 74 % of the observed variance in particle
export (Dunne et al., 2005).

Assimilation accounts for the uptake of DIN and its incorporation into organic matter in the shallowest two layers of the global model. Since assimilation affects both the organic and inorganic N pools, we must account for it in both sets of model runs. We will first address assimilation in the organic N model (Eqs. 9 and 10).

$$\begin{array}{}\text{(9)}& {\displaystyle}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{assim},\mathrm{WOA}}{=}^{\mathrm{14}}{k}_{\mathrm{assim}}[{\mathrm{NO}}_{\mathrm{3}}^{-}{]}_{\mathrm{obs}}\text{(10)}& {\displaystyle}& {}^{\mathrm{14}}{k}_{\mathrm{assim}}={\displaystyle \frac{\mathrm{NPP}}{{r}_{\mathrm{C}:\mathrm{N}}[{\mathrm{NO}}_{\mathrm{3}}^{-}{]}_{\mathrm{obs}}}}\end{array}$$

Since the organic N model is run first and the assimilation rates are
dependent on DIN concentrations, assumptions must be made about the DIN field
in order to account for assimilation prior to the DIN model runs. Here we
used observed [${\mathrm{NO}}_{\mathrm{3}}^{-}$] from the 2013 World Ocean Atlas annual
product interpolated to the model grid [${\mathrm{NO}}_{\mathrm{3}}^{-}$]_{obs}
(Garcia et al., 2013b) to calculate the assimilation rates for DON and PON
production (${J}_{\mathrm{14}}^{\mathrm{assim},\mathrm{WOA}}$). For this assumption to be valid,
our modeled surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] must be close to the observed values,
which we will test in Sect. 3.1. The rate constant for assimilation, ^{14}*k*_{assim}, varies spatially and is determined using observations of
surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] and satellite-derived net primary production (NPP; Westberry et al.,
2008). The rate constant is converted to N units using the ratio of carbon
(C) to N in organic matter (*r*_{C:N}), which we assume to be
106:16 (Redfield et al., 1963). The value of the rate constant is only
nonzero in the top two boxes of the model, where we assume primary production
to be occurring. The same rate constant is used in both the organic N and DIN
assimilation equations. We also assume from the perspective of organic N that
only ${\mathrm{NO}}_{\mathrm{3}}^{-}$ is being assimilated, since ${\mathrm{NO}}_{\mathrm{2}}^{-}$ is
present at relatively low concentrations in the surface ocean, and it may be
characterized as recycled production. Assimilated N is partitioned between
PON and DON using the *P*_{e} ratio as previously described and shown
in Eqs. (3) and (4).

The setup for assimilation in the DIN model (Eqs. 11 and 12) is similar, but
can use modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$] and [${\mathrm{NO}}_{\mathrm{2}}^{-}$] rather than the
World Ocean Atlas values. In order to appropriately reflect surface
${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations, both
${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ are assimilated. ^{14}*k*_{assim} is calculated as described above and is assumed to be the
same for both ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$. We justify using only
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] to parameterize ^{14}*k*_{assim} because
${\mathrm{NO}}_{\mathrm{3}}^{-}$ generally makes up the bulk of DIN available for
assimilation at the surface, but this assumption will be discussed in more
detail below.

$$\begin{array}{}\text{(11)}& {\displaystyle}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{assim},{\mathrm{NO}}_{\mathrm{3}}}{=}^{\mathrm{14}}{k}_{\mathrm{assim}}{[}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{3}}^{-}]\text{(12)}& {\displaystyle}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{assim},{\mathrm{NO}}_{\mathrm{2}}}{=}^{\mathrm{14}}{k}_{\mathrm{assim}}{[}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}^{-}]\end{array}$$

Solubilization is the transformation of PON to DON, and is dependent only on
[PON] and a solubilization rate constant (^{14}*k*_{sol}), which is
optimized (Table 2).

$$\begin{array}{}\text{(13)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{sol}}{=}^{\mathrm{14}}{k}_{\mathrm{sol}}\left[\mathrm{PO}{}^{\mathrm{14}}\mathrm{N}\right]\end{array}$$

The solubilization of PON, together with the particle transport operator
(*T*_{p}), produces a particle flux attenuation curve similar to a Martin curve
with exponent $b=-\mathrm{0.858}$ (Table 1; Martin et al., 1987). While in the real world, the
length scale for particle flux attenuation is somewhat longer in ODZs compared to
oxygenated portions of the water column, and also varies regionally (Berelson, 2002;
Buesseler et al., 2008; Buesseler and Boyd, 2009), our model uses a spatially invariant
^{14}*k*_{sol}. A spatially variable ^{14}*k*_{sol} that accounts for lower
apparent values in ODZs is a refinement that could be introduced in future model
versions.

Remineralization, or ammonification, is the release of DON into the DIN pool.
This is parameterized using the
concentration of DON and a remineralization rate constant
(^{14}*k*_{remin}), which is optimized (Table 2).

$$\begin{array}{}\text{(14)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{remin}}{=}^{\mathrm{14}}{k}_{\mathrm{remin}}\left[{\mathrm{DO}}^{\mathrm{14}}\mathrm{N}\right]\end{array}$$

The removal of this remineralized DON, since it does not accumulate as ${\mathrm{NH}}_{\mathrm{4}}^{+}$,
is either through ammonia oxidation (AMO) or anammox (AMX), depending on [O_{2}]
as described below and in Sect. 2.3.4. We use the same remineralization rate constant
regardless of the utilized electron acceptor (e.g., O_{2}, ${\mathrm{NO}}_{\mathrm{3}}^{-}$).
Since particle flux attenuation is observed to be somewhat weaker in ODZs compared with
oxygenated water (Van Mooy et al., 2002), this may slightly overestimate the rates of
heterotrophic remineralization occurring in ODZs.

AMO uses ammonia (NH_{3}) as a substrate. Since we do not include
NH_{3} or ${\mathrm{NH}}_{\mathrm{4}}^{+}$ in the model system, we treat
remineralized DON as the substrate for AMO. In order to maintain consistency
between the organic N and DIN model runs, remineralized DON is routed either
to AMO or AMX (lost from the system) based on the O_{2} dependencies of
AMO and AMX. Rather than using a strict O_{2} cutoff for AMO, it is
limited by O_{2} using Michaelis–Menten kinetics. The half-saturation
constant for O_{2}, ${K}_{m}^{\mathrm{AMO}}$ (Table 1), sets the O_{2}
concentration at which AMO reaches half of its maximal value.

$$\begin{array}{}\text{(15)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{AMO}}=(\mathrm{1}-{\mathit{\eta}}_{\mathrm{AMX}}){J}_{\mathrm{14}}^{\mathrm{remin}}+{\mathit{\eta}}_{\mathrm{AMX}}{\displaystyle \frac{\left[{\mathrm{O}}_{\mathrm{2}}\right]}{\left[{\mathrm{O}}_{\mathrm{2}}\right]+{K}_{m}^{\mathrm{AMO}}}}{J}_{\mathrm{14}}^{\mathrm{remin}}\end{array}$$

Recent studies have shown that AMO and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation (NXR),
both O_{2}-requiring processes, have very low O_{2} half saturation
constants and can occur down to nanomolar levels of [O_{2}] (Peng et
al., 2016; Bristow et al., 2016). In contrast, O_{2}-inhibited processes
such as AMX are only allowed to occur at O_{2} concentrations below a
given threshold. The handling of O_{2} thresholds for anaerobic
processes is discussed in more detail below (Sect. 2.3.4), though we describe
it briefly here due to the interplay between AMO and AMX in the model.
Briefly, the O_{2} dependence of AMX is represented by the parameter
*η*_{AMX}, which has a value between 0 and 1 for a given grid box
depending on the average number of months in a year its 2013 World Ocean
Atlas [O_{2}] falls below the [O_{2}] threshold for anammox
(${\mathrm{O}}_{\mathrm{2}}^{\mathrm{AMX}}$, Table 1). If, for example, the [O_{2}] in a given
grid box is always above the threshold for AMX, *η*_{AMX}=0 and
all of the remineralized DON (represented by ${J}_{\mathrm{14}}^{\mathrm{rem}}$) will be
oxidized via AMO. If [O_{2}] is less than ${\mathrm{O}}_{\mathrm{2}}^{\mathrm{AMX}}$,
*η*_{AMX} will be nonzero and a smaller fraction of the
remineralized DON will be oxidized via AMO. The fraction ultimately oxidized
by AMO is thus determined by the Michaelis–Menten parameterization of AMO,
as well as the O_{2} threshold for anammox.

The rates of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation (NXR) are dependent on the availability of
${\mathrm{NO}}_{\mathrm{2}}^{-}$ as well as O_{2}. Similar to AMO, we parameterize O_{2}
dependence using Michaelis–Menten kinetics and a fixed half-saturation constant for
O_{2} (${K}_{m}^{\mathrm{NXR}}$, Table 1). ${K}_{m}^{\mathrm{NXR}}$ was taken to be
0.8 µM O_{2}, based on kinetics experiments performed with natural
populations of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidizing bacteria (Bristow et al., 2016). Finally, we
employ an optimized rate constant (^{14}*k*_{NXR}, Table 2) to fit the available
data.

$$\begin{array}{}\text{(16)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{NXR}}{=}^{\mathrm{14}}{k}_{\mathrm{NXR}}\left[{}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}^{-}\right]{\displaystyle \frac{\left[{\mathrm{O}}_{\mathrm{2}}\right]}{\left[{\mathrm{O}}_{\mathrm{2}}\right]+{K}_{m}^{\mathrm{NXR}}}}\end{array}$$

${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction (NAR) and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction (NIR) are
two processes within the stepwise reductive pathway of canonical
denitrification. The end result of denitrification is the conversion of DIN
to N_{2} gas, rendering it bioavailable to only a restricted set of
marine organisms. Although there are intermediate gaseous products between
${\mathrm{NO}}_{\mathrm{2}}^{-}$ and N_{2}, we treat NIR as the rate-limiting step in
the denitrification pathway, where DIN is removed from the system.

For both NAR and NIR, we introduce a dependency on two state variables, their respective N substrates, and organic matter availability. Where NAR and NIR occur heterotrophically, they consume organic matter in addition to their main N substrates or electron receptors. When NAR occurs chemoautotrophically, it would be dependent primarily on the presence of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and an electron donor, such as hydrogen sulfide (Lavik et al., 2009). Since we do not model the production of reduced sulfur species in our model, our estimates of denitrification would not explicitly include the effects of this process. However, chemolithotrophic denitrification could be tacitly accounted for in the optimization process, since the rate constants that control the rates of NAR and NIR are optimized in order to best fit the observations, and the isotope effect for chemolithotrophic denitrification is thought to be similar to that of heterotrophic denitrification (Frey et al., 2014).

In order to maintain levels of heterotrophic NAR and NIR that are dependent on
both the available
${\mathrm{NO}}_{\mathrm{3}}^{-}$ or ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and the available organic matter in a
linear model, it was necessary to run organic N and DIN equations separately,
since it is not possible to include dependencies on two state variables
(e.g., DON and ${\mathrm{NO}}_{\mathrm{3}}^{-}$) in the linear system. Both NAR and NIR are
dependent on the remineralization rate (${J}_{\mathrm{14}}^{\mathrm{remin}}$) that is
calculated in the organic N model run. In model boxes where NAR and NIR are
occurring, some of the remineralization is carried out with electron
acceptors other than O_{2}. As mentioned above, we assume that ${J}_{\mathrm{14}}^{\mathrm{remin}}$ does not depend on
the choice of electron acceptor.

$$\begin{array}{}\text{(17)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{NAR}}& {\displaystyle}={\mathit{\eta}}_{\mathrm{NAR}}^{\mathrm{14}}{k}_{\mathrm{NAR}}\left[{}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{3}}^{-}\right]{J}_{\mathrm{14}}^{\mathrm{remin}}\text{(18)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{NIR}}& {\displaystyle}={\mathit{\eta}}_{\mathrm{NIR}}^{\mathrm{14}}{k}_{\mathrm{NIR}}\left[{}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}^{-}\right]{J}_{\mathrm{14}}^{\mathrm{remin}}\end{array}$$

The rate coefficients for NAR (^{14}*k*_{NAR}) and NIR (^{14}*k*_{NIR}) are
optimized rather than fixed (Table 2). Further, the dependence of ${J}_{\mathrm{14}}^{\mathrm{NAR}}$
and ${J}_{\mathrm{14}}^{\mathrm{NIR}}$ on ${J}_{\mathrm{14}}^{\mathrm{remin}}$ means that *k*_{NAR} and
*k*_{NIR} are not first-order rate constants and have different units than
*k*_{PON}, *k*_{DON}, and *k*_{NXR} (Table 2).

The inhibition of NAR and NIR by O_{2}, like AMX, is parameterized by a
parameter *η*, which inhibits these processes when [O_{2}] is above
their maximum threshold. Originally, we treated this term as a binary
operator that would be set to 0
if the empirically corrected 2013 World Ocean Atlas annually averaged
[O_{2}] was above the threshold for the process and 1 if [O_{2}]
was below the threshold. On further refinement, we wanted to account for the
possibility of seasonal shifts in [O_{2}] in ODZs. Thus, for each month,
we assigned a value of 0 or 1 to each model grid box. These values were then
averaged over the 12 months of the year to give a sliding value of *η*
between 0 and 1 for each grid box. The O_{2} thresholds used to
calculate *η*_{NAR} and *η*_{NIR} were fixed (7 and
5 µM, respectively; Table 1). Since we do not explicitly model
O_{2}, [O_{2}] was predetermined using the 2013 World Ocean Atlas
monthly O_{2} climatology (Garcia et al., 2013a) interpolated to the
model grid. We also applied an empirical correction that improves the fit of
World Ocean Atlas [O_{2}] data to observed suboxic measurements (Bianchi
et al., 2012).

Anammox (AMX) catalyzes the production of N_{2} from ${\mathrm{NH}}_{\mathrm{4}}^{+}$ and
${\mathrm{NO}}_{\mathrm{2}}^{-}$. Since we do not use ${\mathrm{NH}}_{\mathrm{4}}^{+}$ as a variable in our N
cycling equations, we substituted remineralized DON (${J}_{\mathrm{14}}^{\mathrm{remin}}$) as a
proxy for ${\mathrm{NH}}_{\mathrm{4}}^{+}$ availability. As described above in Sect. 2.3.3,
remineralized DON is routed through either AMO or AMX depending on [O_{2}]
and the O_{2} dependencies of AMO and AMX.

$$\begin{array}{}\text{(19)}& {\displaystyle}{J}_{\mathrm{14}}^{\mathrm{AMX}}={\mathit{\eta}}_{\mathrm{AMX}}\left(\mathrm{1}-{\displaystyle \frac{\left[{\mathrm{O}}_{\mathrm{2}}\right]}{\left[{\mathrm{O}}_{\mathrm{2}}\right]+{K}_{m}^{\mathrm{AMO}}}}\right){J}_{\mathrm{14}}^{\mathrm{remin}}\end{array}$$

The O_{2} threshold used to calculate *η*_{AMX} from monthly O_{2}
climatology is fixed (10 µM; Table 1). In order to maintain mass balance on
remineralized DON, we do not include dependence on [${\mathrm{NO}}_{\mathrm{2}}^{-}$] in Eq. (19),
although ${J}_{\mathrm{14}}^{\mathrm{AMX}}$ removes ${\mathrm{NO}}_{\mathrm{2}}^{-}$ (Eq. 2). This parameterization
inherently assumes that AMX is limited primarily by [${\mathrm{NH}}_{\mathrm{4}}^{+}$] supply and not
[NO_{2}], which may not always be correct (Bristow et al., 2016). Anammox also
produces 0.3 moles of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ via associated NXR for every 1 mole of N_{2}
gas produced (Strous et al., 1999). For this reason, anammox appears in the state
equation for ${\mathrm{NO}}_{\mathrm{3}}^{-}$ (Eq. 1).

Sedimentary (or benthic) denitrification (${J}_{\mathrm{14}}^{\mathrm{sed}}$) is an important
loss term for N in the marine environment, and in order to encapsulate it
within the model grid we assume that it is occurring within the bottom depth
box for any particular model water column. The parameterization for
sedimentary denitrification is based on a transfer function described by
Bohlen et al. (2012). The original transfer function was dependent on bottom
water [O_{2}], bottom water [${\mathrm{NO}}_{\mathrm{3}}^{-}$], and the rain rate of
particulate organic carbon (RRPOC). Here, RRPOC was calculated via a Martin
curve (Martin et al., 1987) using the *P*_{e} ratio, net primary production
(NPP), depth (*z*), euphotic zone depth (*z*_{eu}), and a Martin curve
exponent (*b*):

$$\begin{array}{}\text{(20)}& {\displaystyle}\mathrm{RRPOC}=\mathrm{NPP}\cdot {P}_{\mathrm{e}}\cdot {\left({\displaystyle \frac{z}{{z}_{\mathrm{eu}}}}\right)}^{b}.\end{array}$$

Net primary production is derived from the productivity modeling of Westberry et
al. (2008) as described in Sect. 2.3.2. The *P*_{e} ratio is calculated as
previously described in Sect. 2.3.2. The depth for any given model box is assumed to be
the depth at the bottom of the box. The euphotic zone depth is the bottom depth of the
second box (73 m), since all production is assumed to be occurring in the top two boxes.
As described above, the Martin curve exponent, *b*, is a fixed value in our model ($b=-\mathrm{0.858}$; Table 1), though this may result in underestimation of the particulate matter
reaching the seafloor below ODZs (Van Mooy et al., 2002).

The transfer function for sedimentary denitrification was originally
described using a nonlinear dependence of the rate on $\left[{\mathrm{O}}_{\mathrm{2}}\right]-\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]$. In order for sedimentary denitrification to be properly
implemented in our linear model, we broke the original nonlinear relationship
into three roughly linear segments to create a piecewise relationship between
$\left[{\mathrm{O}}_{\mathrm{2}}\right]-\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]$ and sedimentary denitrification rate
(Fig. S2). We obtained three linear relationships between $\left[{\mathrm{O}}_{\mathrm{2}}\right]-\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]$ and sedimentary denitrification rate, each applicable
across a given range of $\left[{\mathrm{O}}_{\mathrm{2}}\right]-\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]$ values. Due to
the nature of our linear model, we needed to express the interval cutoff
points that define the transition between the piecewise relationship segments
in terms of O_{2} rather than $\left[{\mathrm{O}}_{\mathrm{2}}\right]-\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]$.
Therefore, a linear relationship between O_{2} and $\left[{\mathrm{O}}_{\mathrm{2}}\right]-\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]$ was determined using the 2013 World Ocean Atlas annually
averaged data (Garcia et al., 2013a, b; Fig. S3). The cutoff points were
determined to be 75 and 175 µM O_{2}. The linear relationships
were then rearranged in order to estimate sedimentary denitrification rate as
a function of RRPOC, [O_{2}], and [${\mathrm{NO}}_{\mathrm{3}}^{-}$]. These equations
were then further broken down into a component that is dependent on
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] and a component that is dependent on [O_{2}] (see
Supplement).

An additional term is introduced that reduces the sedimentary denitrification
rate by 27 % if the depth of the bottom model box is less than 1000 m.
This term represents the potential for efflux of ${\mathrm{NH}}_{\mathrm{4}}^{+}$ into the
water column from shallow, organic rich shelf sediments (Bohlen et al.,
2012). This decreases overall sedimentary denitrification by approximately
6 Tg N yr^{−1}. This transfer function also assumes that all of the
${\mathrm{NH}}_{\mathrm{4}}^{+}$ efflux is immediately oxidized to ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and
does not alter its isotopic composition in bottom water. This is a
conservative estimate of the effects of benthic N loss on water column
${\mathrm{NO}}_{\mathrm{3}}^{-}$ isotopes, as several studies suggest that benthic N
processes may contribute to water column nitrate ^{15}N-enrichment
(Lehmann et al., 2007; Granger et al., 2011; Somes and Oschlies, 2015; Brown
et al., 2015). However, our current model parameterization does not require
enhanced fractionation during benthic N loss to fit deep ocean
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$. Additionally, our spatial resolution does not
well represent regions where this effect might be significant on bottom water
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$, such as the shallow shelves.

In our model, we are interested in using the isotopic composition of
${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ to constrain the rates of N cycling
and loss from the global ocean. As DON and PON are ultimate substrates for
${\mathrm{NO}}_{\mathrm{2}}^{-}$ and ${\mathrm{NO}}_{\mathrm{3}}^{-}$ production, it is essential to track
the ^{15}N in the organic N pools as well. The matrix setup for
^{15}N is similar to that for the ^{14}N species, but the rates
were changed as follows:

$$\begin{array}{}\text{(21)}& {\displaystyle}{J}_{\mathrm{15}}^{\mathrm{process}}=\mathrm{1}/{\mathit{\alpha}}_{\mathrm{process}}{\displaystyle \frac{\left[{}^{\mathrm{15}}{\mathrm{N}}_{\mathrm{substrate}}\right]}{\left[{}^{\mathrm{14}}{\mathrm{N}}_{\mathrm{substrate}}\right]}}{J}_{\mathrm{14}}^{\mathrm{process}}.\end{array}$$

${J}_{\mathrm{14}}^{\mathrm{process}}$ is the rate of each relevant ^{14}N process
as described above, and ${J}_{\mathrm{15}}^{\mathrm{process}}$ is the rate of each
^{15}N process. The *α*_{process} is the fractionation
factor for a given process, which is given by the ratio between the rate
constants for ^{14}N and ^{15}N ($\mathit{\alpha}{=}^{\mathrm{14}}k{/}^{\mathrm{15}}k$). A
fractionation factor greater than 1 indicates a normal isotope effect and a
fractionation factor less than 1 indicates an inverse isotope effect. Several
of these fractionation factors are well known, but others are more poorly
constrained, especially when values are calculated from in situ concentration
and isotope ratio measurements (Hu et al., 2016; Casciotti et al., 2013;
Ryabenko et al., 2012). For this reason, we ran several model cases with
different fractionation factors for NAR, NIR, and NXR during the optimization
process (Sect. 2.6, Table 3). The other fractionation factors were fixed
(Table 1). In order to produce the ^{15}N concentrations of N species
from our observations to constrain the model, we calculated
^{15}N∕^{14}N from measured *δ*^{15}N and multiplied by the
measured concentration of each modeled N species, assuming that
$\left[{}^{\mathrm{14}}\mathrm{N}\right]\sim \left[{}^{\mathrm{14}}\mathrm{N}\right]+\left[{}^{\mathrm{15}}\mathrm{N}\right]$.

This simple ^{15}N implementation was used with fixed fractionation factors for
remineralization (*α*_{remin}=1), solubilization (*α*_{sol}=1), assimilation (*α*_{assim}=1.004), sedimentary denitrification
(*α*_{sed}=1), and AMO (*α*_{AMO}=1) (Table 1). Isotope
effects for NAR (*ε*_{NAR}), NIR (*ε*_{NIR}), and NXR
(*ε*_{NXR}) were varied in different combinations during model
optimization (Table 3). Distinct isotopic parameterizations were also required for
atmospheric deposition, N_{2} fixation, and anammox, as described below.

For atmospheric deposition of N, we prescribe fixed *δ*^{15}N value of −4 ‰ (Table 1),
which can be related to the fractional abundance of ^{14}N
(${r}_{\mathrm{14}}^{\mathrm{dep}}$), previously described in Sect. 2.3.2, as well as the
fractional abundance of ^{15}N (${r}_{\mathrm{15}}^{\mathrm{dep}}$) in deposited N.
We multiply ${r}_{\mathrm{15}}^{\mathrm{dep}}$ by *S*^{dep}, the estimated rate of
total N deposition to obtain ${J}_{\mathrm{15}}^{\mathrm{dep}}$.

Similar to atmospheric deposition, newly fixed N has a *δ*^{15}N value
(−1 ‰; Table 1). In Sect. 2.3.2 we described
${r}_{\mathrm{14}}^{\mathrm{fix}}$, the fractional abundance of ^{14}N in
newly fixed N. Here we multiply the fractional abundance of ^{15}N,
${r}_{\mathrm{15}}^{\mathrm{fix}}$, by the other terms in the N_{2} fixation equation
(Eq. 6) to obtain the rate of ^{15}N fixation.

Anammox is the most complicated process to parameterize isotopically because
it has three different N isotope effects associated with it. There is an
isotope effect on both substrates that are converted to N_{2}
(${\mathrm{NO}}_{\mathrm{2}}^{-}$ and ${\mathrm{NH}}_{\mathrm{4}}^{+}$), as well as for the associated
${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation to ${\mathrm{NO}}_{\mathrm{3}}^{-}$. We assume that the
fractionation factor for ammonium oxidation via AMX
(${\mathit{\alpha}}_{\mathrm{AMX},{\mathrm{NH}}_{\mathrm{4}}}$) is 1, setting it to match the fractionation factor
for AMO (*α*_{AMO}; Table 1), both with no expressed
fractionation since ${\mathrm{NH}}_{\mathrm{4}}^{+}$ does not accumulate in the model. Since
all remineralized DON must be routed either through AMO or AMX, this
simplifies the mass balance and ensures that all remineralized ^{14}N
and ^{15}N is accounted for. ${}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}^{-}$ is removed with
the isotope effects of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction (*α*_{AMX,NIR}) and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation
(*α*_{AMX,NXR}) in the expected 1:0.3 proportion (Brunner
et al., 2013).

Once our N cycle equations were set up as described above, we input them into MATLAB in
block matrix form. The equations were of the general form **A**** x**=

In MATLAB, we used METIS ordering, which is part of the SuiteSparse
(http://faculty.cse.tamu.edu/davis/suitesparse.html, last access: December 2017) to order our large, sparse matrix **A**. We then used
the built-in function “umfpack” with METIS to factorize matrix **A**. The
built-in matrix solver “mldivide” was then used with the factorized components of
matrix **A** and matrix ** b** to solve for

There are many parameters in the model that control the rates of the different N cycle processes (Tables 1–3). Some of these parameters are well constrained by literature values (Table 1). Others, such as the rate constants, were objects of our investigation and were optimized against available observations (Table 2). For our optimization, we compared model output using different parameter values to a database of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations and isotopes. The database was originally compiled by Rafter et al. (2019) and has been expanded to include some additional unpublished data (Table S2). All of the database observations were binned and interpolated to the model grid. If multiple observations occurred within the same model grid box, the values were averaged and a standard deviation was calculated. The database was divided randomly into a training set, used for optimization, and a test set, used to assess model performance. The same number of grid points with observations was used in the training and test sets.

The optimization procedure used the MATLAB function “fminunc” to obtain
values for the nonfixed parameters that minimized a cost function (Eq. 22).
In each iteration of the optimization, the model system was solved by running
the ^{14}N-organic N model, ^{15}N-organic N model,
^{14}N-DIN model, and ^{15}N-DIN model. The modeled output
[${\mathrm{NO}}_{\mathrm{3}}^{-}$], [${\mathrm{NO}}_{\mathrm{2}}^{-}$], ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$, and
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ were compared to values from the database
training set. Though DON and PON observations were not used to optimize the
model, the open ocean and deep water ${\mathrm{NO}}_{\mathrm{3}}^{-}$ values were useful in
constraining the parameters that control PON solubilization and DON
remineralization. The entire model was run using a set of initial parameter
values (Table 2) and the optimization scheme continued to alter those
starting parameters until a minimum in the cost function was attained. We
optimized the logarithm of the parameter values rather than the original
parameters themselves so the unconstrained optimization returned positive
values. The transformed starting parameters and subsequent modified parameter
sets were then fed back into the model equations as *e*^{x}, where *x*
denotes the log-transformed parameter. The cost function in the optimization
procedure is as follows:

$$\begin{array}{ll}{\displaystyle}\mathrm{Cost}\phantom{\rule{0.125em}{0ex}}& {\displaystyle}={\displaystyle \frac{{w}_{{\mathrm{NO}}_{\mathrm{3}}}}{{n}_{{\mathrm{NO}}_{\mathrm{3}}}{\mathrm{sd}}_{{\mathrm{NO}}_{\mathrm{3}}}}}\sum \left(\right[{\mathrm{NO}}_{\mathrm{3}}^{-}{]}_{\mathrm{model}}-[{\mathrm{NO}}_{\mathrm{3}}^{-}{]}_{\mathrm{training}}{)}^{\mathrm{2}}\\ {\displaystyle}& {\displaystyle}+{\displaystyle \frac{{w}_{{\mathrm{NO}}_{\mathrm{2}}}}{{n}_{{\mathrm{NO}}_{\mathrm{2}}}{\mathrm{sd}}_{{\mathrm{NO}}_{\mathrm{2}}}}}\sum \left(\right[{\mathrm{NO}}_{\mathrm{2}}^{-}{]}_{\mathrm{model}}-[{\mathrm{NO}}_{\mathrm{2}}^{-}{]}_{\mathrm{training}}{)}^{\mathrm{2}}\\ {\displaystyle}& {\displaystyle}+{\displaystyle \frac{{w}_{\mathit{\delta}{\mathrm{NO}}_{\mathrm{3}}}}{{n}_{\mathit{\delta}{\mathrm{NO}}_{\mathrm{3}}}{\mathrm{sd}}_{\mathit{\delta}{\mathrm{NO}}_{\mathrm{3}}}}}\sum ({\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}},\mathrm{model}}-{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}},\mathrm{training}}{)}^{\mathrm{2}}\\ \text{(22)}& {\displaystyle}& {\displaystyle}+{\displaystyle \frac{{w}_{\mathit{\delta}{\mathrm{NO}}_{\mathrm{2}}}}{{n}_{\mathit{\delta}{\mathrm{NO}}_{\mathrm{2}}}{\mathrm{sd}}_{\mathit{\delta}{\mathrm{NO}}_{\mathrm{2}}}}}\sum ({\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}},\mathrm{model}}-{\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}},\mathrm{training}}{)}^{\mathrm{2}}.\end{array}$$

The *w* terms are weighting terms introduced to scale the contributions of the four
observed parameters to equalize their contributions to the cost function. The *n* terms
and standard deviation (sd) terms were used to normalize the contributions of each
measurement type to the cost function. Each *n* term is equal to the number of each type
of measurement in the training dataset (e.g., the number of [${\mathrm{NO}}_{\mathrm{3}}^{-}$] data
points $={n}_{{\mathrm{NO}}_{\mathrm{3}}}$). The sd term is equal to the standard deviation of all the
measurements of a given type (e.g., the standard deviation of all the [${\mathrm{NO}}_{\mathrm{3}}^{-}$]
data points within the training set).

In order to account for error in our model parameter estimates, we also
iterated over several possible values for three of the most important isotope
effects for processes in ODZs: *ε*_{NAR},
*ε*_{NIR}, and *ε*_{NXR} (Table 3). We
chose to iterate over these parameters rather than optimize them since there
is a large range of estimates for each of these
parameters. We assigned different possible values for each of these
parameters (Table 3), resulting in 12 possible combinations. The optimization
protocol was performed for each of those combinations and unique optimized
parameter sets were obtained. The parameter results were then averaged (final
values, Table 2) and their spread is categorized as the error (error,
Table 2).

3 Results

Back to toptop
The simulations of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration and isotopic composition are the most unique
features of this model in comparison to existing global models of the marine
N cycle. As such, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ accumulation in ODZs is a feature that
should be well represented by the model in order to use it to test hypotheses
about processes that control N cycling and loss in ODZs. Overall, we see
${\mathrm{NO}}_{\mathrm{2}}^{-}$ accumulating at 200 m in the major ODZs of the Eastern
Tropical North Pacific (ETNP), Eastern Tropical South Pacific (ETSP), and the
Arabian Sea (AS) (Fig. 2), which is consistent with observations and expected
based on the low O_{2} conditions found there. However, accumulation of
${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the model ETSP was lower than expected. The model also
accumulated ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the Bay of Bengal, which is a low-O_{2}
region off the east coast of India that does not generally accumulate
${\mathrm{NO}}_{\mathrm{2}}^{-}$ or support water column denitrification, but is thought to
be near the “tipping point” for allowing N loss to occur (Bristow et al.,
2017). Possible reasons for the underestimation of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the
ETSP and overestimation in the Bay of Bengal will be discussed further in
Sect. 4.2.

The model optimization described above yielded a set of isotope effects that
best fit the global dataset of [${\mathrm{NO}}_{\mathrm{3}}^{-}$], [${\mathrm{NO}}_{\mathrm{2}}^{-}$],
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ and ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$. The best fit was
achieved for isotope effects of 13 ‰ for ${\mathrm{NO}}_{\mathrm{3}}^{-}$
reduction (*ε*_{NAR}), 0 ‰ for ${\mathrm{NO}}_{\mathrm{2}}^{-}$
reduction (*ε*_{NIR}), and −13 ‰ for
${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation (*ε*_{NXR}). Figure 3 shows the
test set comparison for the global best-fit set of isotope effects overlaid
with a 1:1 line, which the data would follow if there was perfect
agreement between model results and observations. There is general agreement
between model and observations, with most of the data clustering near the
1:1 lines. Agreement between the observations and the training data
are similar (Fig. S4), indicating that we did not overfit the training data.

In the test set, there were some low [O_{2}] points where our model
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] exceeded observations (Fig. 3a, filled black circles);
these are largely within the ETSP. In contrast, the AS tended to show slightly lower modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$] than
expected. The [${\mathrm{NO}}_{\mathrm{2}}^{-}$] accumulation (Fig. 3b) and
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ signals (Fig. 3c) in the ETSP were also
generally too low compared with observations. These signals are likely tied
to insufficient ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction occurring in the model ETSP.
Another consideration is that there may be a mismatch in resolution between
the model and the time and space scales needed to resolve the high
${\mathrm{NO}}_{\mathrm{2}}^{-}$ accumulations observed sporadically (Anderson et al., 1982;
Codispoti et al., 1985, 1986).

Overall, the representation of ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ was fairly good (RMSE = 2.4 ‰), though there were a subset of points above ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}=\mathrm{10}$ ‰ where the modeled ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ exceeded the observed ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$, and others where modeled ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ was lower than observations (Fig. 3c). Many of the points with overestimated ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ were located within the AS ODZ, where there may be too much ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction occurring, leading to artificially elevated ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values. As indicated above, the underestimated ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ points largely fell within the ETSP where we believe the model is underestimating ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction. The representation of ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ was also fairly good (RMSE = 8.6 ‰), though the modeled ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ was generally not low enough (Fig. 3d), indicating an underestimated sink of “heavy” ${\mathrm{NO}}_{\mathrm{2}}^{-}$.

To further investigate the distribution of model N species within the three
main ODZs, we selected representative offshore grid boxes within each ODZ
that contained observations to directly compare with model results in station
profiles. Overall, the modeled ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$
concentration and isotope profiles in the AS and ETNP were consistent with
the observations, with [${\mathrm{NO}}_{\mathrm{3}}^{-}$] slightly underestimated in the AS
ODZ and overestimated in the ETSP (Fig. 4). As [O_{2}] goes to zero, the
O_{2}-intolerant processes NAR, NIR, and AMX are released from
inhibition. These processes result in a decrease in [${\mathrm{NO}}_{\mathrm{3}}^{-}$] (via
NAR) which corresponds to an increase in ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$,
since NAR has a normal isotope effect. ${\mathrm{NO}}_{\mathrm{2}}^{-}$ also starts to
accumulate in the secondary ${\mathrm{NO}}_{\mathrm{2}}^{-}$ maximum as a result of NAR. The
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ is lower than ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$
since light ${\mathrm{NO}}_{\mathrm{2}}^{-}$ is preferentially created via NAR, and this
fractionation is further reinforced by the inverse isotope effect of NXR
(Casciotti, 2009). These patterns are readily observed in the AS and ETNP,
but were less apparent in the ETSP, where [${\mathrm{NO}}_{\mathrm{3}}^{-}$] depletion and
[${\mathrm{NO}}_{\mathrm{2}}^{-}$] accumulation in the model were lower than observed. This
could be due in part to the time-independent nature of this steady-state
inverse model, which does not capture the effects of upwelling events in the
ETSP on N supply and cycling (Canfield, 2006; Chavez and Messié, 2009).

In order to gauge the model results for N loss, we also calculated
N^{*}, a measure of the availability of DIN relative to
${\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}$ compared to Redfield ratio stoichiometry (${\mathrm{N}}^{*}=\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]+\left[{\mathrm{NO}}_{\mathrm{2}}^{-}\right]-\mathrm{16}\cdot \left[{\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}\right]$; Deutsch et al., 2001). Negative N^{*} values
are associated with N loss due to AMX or NIR or release of ${\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}$
from anoxic sediments (Noffke et al., 2012), while positive N^{*} values
are associated with input of new N through N_{2} fixation (Gruber and
Sarmiento, 1997). Although we did not model ${\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}$, we used the
modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$] and [${\mathrm{NO}}_{\mathrm{2}}^{-}$] together with World Ocean
Atlas ${\mathrm{PO}}_{\mathrm{4}}^{\mathrm{3}-}$ data interpolated to the model grid to calculate
N^{*} resulting from the model. Both the AS and ETNP showed a decrease
in model N^{*} in the ODZ, as expected for water column N loss. Below
the ODZ, N^{*} increased again and returned to expected deep water
values. Modeled N^{*} in the ETSP, however, did not follow the observed
trend, consistent with an underestimate of N loss in the model ETSP.

Though the global best fit isotope effects for NAR, NIR, and NXR produced
good agreement to the data in general, the isotope effects that best fit
individual ODZ regions differed when the cost function was restricted to
observations from a given ODZ. For the ETSP, the best fit isotope effects
were the same as the previously stated global best fit. For the AS, the best
fit isotope effects were *ε*_{NAR}=13 ‰,
*ε*_{NIR}=0 ‰, and ${\mathit{\epsilon}}_{\mathrm{NXR}}=-\mathrm{32}$ ‰. For the ETNP, the best fit isotope effects were
*ε*_{NAR}=13 ‰, *ε*_{NIR}=15 ‰, and ${\mathit{\epsilon}}_{\mathrm{NXR}}=-\mathrm{32}$ ‰, though the
performance is only marginally better than with *ε*_{NIR}=0 ‰. The lower (more inverse) value for *ε*_{NXR}
resulted in higher ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ and lower ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$, which better fit the ODZ ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$
data compared to the global best fit ${\mathit{\epsilon}}_{\mathrm{NXR}}=-\mathrm{13}$ ‰. These results are consistent with earlier isotope modeling
studies in the ETSP (Casciotti et al., 2013; Peters et al., 2016, 2018b) and
in the AS (Martin and Casciotti, 2017). Although, in the AS, modeled
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values were too high, likely in part due to overpredicted
rates of NAR, which also resulted in lower modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$]
(Fig. 4).

We also investigated the agreement between global best fit model concentration and isotope distributions with data from two GEOTRACES cruise sections: GP16 in the South Pacific, and GA03 in the North Atlantic. For GP16, we see that [${\mathrm{NO}}_{\mathrm{3}}^{-}$] is low in surface waters and increases to a mid-depth maximum between 1000 and 2000 m. The highest [${\mathrm{NO}}_{\mathrm{3}}^{-}$] are found at mid-depth in the eastern boundary of the section. The model reproduces the general patterns, matching observations fairly well in the surface waters, but diverges below 500 m (Fig. 5). Although the patterns are generally correct, insufficient ${\mathrm{NO}}_{\mathrm{3}}^{-}$ is accumulated in the deep waters of the model Pacific. This could be due to an underestimate of preformed ${\mathrm{NO}}_{\mathrm{3}}^{-}$ (over estimate of assimilation in the Southern Ocean), or inadequate supply of organic matter to be remineralized at depth. In the Southern Ocean, model surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] are 5–10 µM lower than observations (Fig. S5), which could be enough to explain the lower-than-expected [${\mathrm{NO}}_{\mathrm{3}}^{-}$] in the deep Pacific, which is largely sourced from the Southern Ocean (Rafter et al., 2013; Sigman et al., 2009; Peters et al., 2018a, b).

In the GP16 section, we also see that there are elevated ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values in the model surface waters and in the ETSP ODZ (Fig. 5d), as expected from observations (Fig. 5c). However, we can also see that the insufficient depletion of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and increase in ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ in the ETSP ODZ (Fig. 5b and d) extends beyond the single grid box highlighted earlier (Fig. 4). The less-than-expected increase of ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ in the ETSP ODZ and the upper thermocline in the eastern part of the section is consistent with an underestimate of ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction. In GP16 we were also able to compare modeled and observed [${\mathrm{NO}}_{\mathrm{2}}^{-}$] and ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ (Fig. S6). Patterns of modeled [${\mathrm{NO}}_{\mathrm{2}}^{-}$] and ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ showed accumulation of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the ODZ, with an appropriate ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ value (Fig. S6). Although, generally lower modeled concentrations of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the ODZ also support an underestimate of NAR (Fig. S6).

Surface ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values were also not as high in the model
as in the observations (Fig. 5), which could result from insufficient
${\mathrm{NO}}_{\mathrm{3}}^{-}$ assimilation or too low supplied ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$
(Peters et al., 2018a). However, we do see a similar depth range for high
surface ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ and a local ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$
minimum between the surface and ODZ propagating westward in both the model
and observations, indicating that the physical and biogeochemical processes
affecting ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ are represented by the model.
Additionally, the model shows slightly elevated ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ in
the thermocline depths (200–500 m) west of the ODZ, which is consistent
with the observations (Fig. 5c), though not of the correct magnitude. This is
partly related to the muted ODZ signal as mentioned above and its lessened
impact on thermocline ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ across the basin. Peters et
al. (2018a) and Rafter et al. (2013) also postulated that these elevated
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values were in part driven by remineralization of
organic matter with high *δ*^{15}N. The *δ*^{15}N of
sinking PON in the model (6 ‰–10 ‰) was similar to those
observed in the South Pacific (Raimbault et al., 2008), as well as those
predicted from aforementioned N isotope studies (Rafter et al., 2013; Peters et al., 2018a). The model also shows
slightly elevated ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ in the intermediate depths
(500–1500 m), which is consistent with observations, again reflecting
remineralization of PON with *δ*^{15}N greater than mean ocean
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$. Overall, the patterns of ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ for the model GP16 are correct but the
magnitudes of isotopic variation are muted, largely due to the lack of N loss
in the ODZ and modeled surface ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values that are
lower than observations. The simplification of ${\mathrm{NH}}_{\mathrm{4}}^{+}$ dynamics in
the model could also contribute to underestimation of
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values if there was a large flux of
^{15}N-enriched ${\mathrm{NH}}_{\mathrm{4}}^{+}$ from sediments (Granger et al.,
2011), or if ^{15}N-depleted ${\mathrm{NH}}_{\mathrm{4}}^{+}$ was preferentially
transferred to the N_{2} pool via anammox. While the isotope effect on
${\mathrm{NH}}_{\mathrm{4}}^{+}$ during anammox (Brunner et al., 2013) is indeed higher than
that applied here, we chose to balance this with a low isotope effect during
aerobic ${\mathrm{NH}}_{\mathrm{4}}^{+}$ oxidation (Table 1).

In the North Atlantic along GEOTRACES section GA03, we see good agreement
between the observed and modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$] (Fig. 6). There is
generally low surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] with a distinct area of high
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] propagating from near the African coast. Deep water (>2000 m) [${\mathrm{NO}}_{\mathrm{3}}^{-}$] is lower than we see in the Pacific section,
and the model matches well with the Atlantic observations. Again, there is
not quite enough ${\mathrm{NO}}_{\mathrm{3}}^{-}$ present in Southern Ocean-sourced
intermediate waters (500–1500 m; Marconi et al., 2015). Modeled
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values at first glance appear higher than
observed values at the surface (Fig. 6). However, many of the surface
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] were below the operating limit for
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ analysis and were not determined. Focusing on
areas where both measurements and model results are present yields excellent
agreement. For example, we do see low ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ values in
upper thermocline waters in both the model and observations, likely
corresponding to low *δ*^{15}N contributions from N_{2}
fixation that is remineralized at depth and accumulated in North Atlantic
Central Water (Marconi et al., 2015; Knapp et al., 2008). The model input
includes significant rates of N_{2} fixation in the North Atlantic that
are consistent with this observation (Fig. S1). However, rates of N
deposition in the North Atlantic are also fairly high and can contribute to
the low *δ*^{15}N signal (Knapp et al., 2008). In our model,
atmospheric N deposition contributed between 0 % and 50 % of N input
along the cruise track.

4 Discussion

Back to toptop
As previously mentioned (Sect. 2.3), organic N and DIN were modeled separately in order to introduce dependence on both organic N and substrate availability for the heterotrophic processes NAR and NIR. These separate model runs required several assumptions to be made regarding the processes that impact both organic N and DIN, namely assimilation and remineralization.

The first assumption was that the
rates of N assimilation are equal between the organic N and DIN model runs.
The organic N model run uses World Ocean Atlas surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] to
estimate the contribution of DIN assimilation to the production of organic N,
whereas the DIN model uses modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$] and
[${\mathrm{NO}}_{\mathrm{2}}^{-}$] to estimate DIN removal via assimilation. Though these
two methods used the same rate constants for assimilation, differences in DIN
concentrations could cause some discrepancies between the overall rates.
Analysis of the results revealed that slightly more overall DIN assimilation
occurred in the DIN model run than organic N produced in the organic N model
(Fig. S7). This could be due in part to assimilation of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in
the top two boxes of the DIN model, since ${\mathrm{NO}}_{\mathrm{2}}^{-}$ assimilation is
unaccounted for in the organic N model. This is largely an issue in the
oligotrophic gyres, where surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] is very low and
${\mathrm{NO}}_{\mathrm{2}}^{-}$ accumulates to low but nonzero concentrations (Fig. 2).
Assimilation of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ accounts for a significant fraction of DIN
assimilation in these regions, but the overall assimilation rates there are
low and the resulting influence on the whole system is also low. Another
source of discrepancy would be where modeled surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] is
higher than the World Ocean Atlas surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] that is
supplied to the organic N model, which would result in higher assimilation
rates in the DIN model run. Indeed, points at which the DIN assimilation
rates are higher than the organic N production rates do tend to have modeled
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] that was higher than observed [${\mathrm{NO}}_{\mathrm{3}}^{-}$]
(Fig. S7). Likewise, points with relatively lower DIN assimilation had
modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$] less than observed [${\mathrm{NO}}_{\mathrm{3}}^{-}$]. However,
the majority of DIN assimilation estimates were within
10 µM yr^{−1} of the organic N production estimates, with an
average offset of approximately 3.5 % compared to DIN assimilation. The
total global assimilation rates were within 0.4 %, with some spatially
variable differences due to offset between surface [${\mathrm{NO}}_{\mathrm{3}}^{-}$] and
modeled [${\mathrm{NO}}_{\mathrm{3}}^{-}$]. However, we find that the World Ocean Atlas
surface ${\mathrm{NO}}_{\mathrm{3}}^{-}$ values are fairly well represented by our modeled
surface ${\mathrm{NO}}_{\mathrm{3}}^{-}$ (Fig. S5). We conclude that though the assimilation
rates are not identical in the organic N and DIN model runs, the discrepancy
in modeled DIN assimilation is less than 0.1 %, and there is unlikely to
be significant creation or loss of N as a result of the split model.

The modeled concentration and isotope profiles for the ETSP, unlike in the AS
and ETNP, reflected an underestimation of water column denitrification in the
best-fit model. In ETSP measurements, there is a clear
deficit in [${\mathrm{NO}}_{\mathrm{3}}^{-}$], coincident with the secondary
${\mathrm{NO}}_{\mathrm{2}}^{-}$ maximum and N^{*} minimum (Fig. 4). In our modeled
profiles, this ${\mathrm{NO}}_{\mathrm{3}}^{-}$ deficit is missing, and although a secondary
${\mathrm{NO}}_{\mathrm{2}}^{-}$ maximum is present, its magnitude is lower than observed
(Fig. 4). The model also does not capture the negative N^{*} excursion
(Fig. 4), which we think reflects a model underestimation of NAR and NIR in
the ETSP. The cause of this missing denitrification is likely to be poor
representation of the ETSP O_{2} conditions in the model grid space.
Since our model grid is fairly coarse ($\mathrm{2}{}^{\circ}\times \mathrm{2}{}^{\circ}$), only
a few boxes within the ETSP had averaged [O_{2}] below the thresholds
that would allow processes such as NAR and NIR to occur. The anoxic region of
the ETSP is adjacent to the coast and not as spatially extensive as in the AS
and ETNP (Fig. S8); therefore, this region in particular was less compatible
with the model grid. In order to test whether the parameterization of
O_{2} dependence was the cause of the low N loss, we ran the model using
the globally optimized parameters (Table 3) but with higher O_{2}
thresholds (15 µM) for NAR, NIR, and AMX (Table 1). This extended
the region over which ODZ processes could occur and resulted in an increase
in water column N loss from 6 to 32 Tg N yr^{−1} in the ETSP, which is
more consistent with previous estimates (DeVries et al., 2012; Deutsch et
al., 2001). This change also stimulated the development of a
${\mathrm{NO}}_{\mathrm{3}}^{-}$ deficit, larger secondary ${\mathrm{NO}}_{\mathrm{2}}^{-}$ maximum, and
N^{*} minimum within the ODZ (Fig. 7).

As previously mentioned (Sect. 3.1), modeled [${\mathrm{NO}}_{\mathrm{2}}^{-}$] in the Bay
of Bengal is higher than observations. The accumulation of ${\mathrm{NO}}_{\mathrm{2}}^{-}$
here in the model is likely due to O_{2} concentrations falling below
the set threshold for NAR but above the threshold for NIR, so
${\mathrm{NO}}_{\mathrm{2}}^{-}$ can accumulate via NAR but cannot be consumed via NIR.
Although AMX and NXR occur there, the modeled rates of their
${\mathrm{NO}}_{\mathrm{2}}^{-}$ consumption are rather low, which supports a higher
accumulation of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the steady-state model. This is in contrast to
observations that ${\mathrm{NO}}_{\mathrm{2}}^{-}$ production is tightly matched with
${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation in the Bay of Bengal, which limits
${\mathrm{NO}}_{\mathrm{2}}^{-}$ accumulation and N loss there (Bristow et al., 2017). The
fact that the model over predicts NAR in the AS may also be connected with
overprediction of NAR in the Bay of Bengal. It is possible that the oxygen
thresholds for ODZ processes are not the same in all ODZs, and further work
on oxygen sensitivities of N cycle processes will be addressed in a companion
study (Martin et al., 2019).

5 Conclusions

Back to toptop
A global inverse ocean model was modified to include ^{14}N and ^{15}N in
both ${\mathrm{NO}}_{\mathrm{3}}^{-}$ and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ as state variables. Adding the processes
required to describe the cycling of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the global ocean, including oxic
and anoxic processes, resulted in a globally representative distribution of
[${\mathrm{NO}}_{\mathrm{3}}^{-}$], [${\mathrm{NO}}_{\mathrm{2}}^{-}$], ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$, and
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$. In particular, the patterns of variation in both oxic and
anoxic waters are generally consistent with observations, though some magnitudes of
variation were somewhat muted by the model. This could be due to an underestimation of a
process rate, due to parameterization or model resolution, or an underestimation of the
isotope effect involved.

Importantly, we were able to generate a roughly balanced steady-state ocean N
budget without the need for an artificial restoring force. The
[${\mathrm{NO}}_{\mathrm{3}}^{-}$] and [${\mathrm{NO}}_{\mathrm{2}}^{-}$] distributions that were required
to achieve this roughly balanced budget are well within the range of observed
values. Some interesting take-home messages from this work are the following:
(1) a relatively low isotope effect for ${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction
(*ε*_{NAR}=13 ‰) gives a good fit to
${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{3}}}$ data, similar to that concluded in some recent
studies (Marconi et al., 2017; Bourbonnais et al., 2015; Casciotti et al.,
2013); (2) low O_{2} half-saturation constants for ${\mathrm{NO}}_{\mathrm{2}}^{-}$
oxidation allowing ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation to occur in parallel with
${\mathrm{NO}}_{\mathrm{3}}^{-}$ reduction, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ reduction, and anammox were
needed to achieve reasonable distributions of ${\mathrm{NO}}_{\mathrm{3}}^{-}$, ${\mathrm{NO}}_{\mathrm{2}}^{-}$, and their isotopes in
the ocean water column ODZs.

Though we have been able to adequately represent and assess N cycling in
ODZs, there are many areas in which this model could be improved in order to
expand its usefulness. Improving resolution of the model, particularly in
coastal regions where there are steep gradients in nutrient and O_{2}
concentrations, would improve the accuracy of the model in regions such as
the ETSP. Further, in regions that have high seasonal or interannual
variability, an annually averaged steady-state model may not represent some
important temporal dynamics. While we attempted to account for seasonal
variation in the strength of the ODZs through use of monthly O_{2}
climatologies, we did not simulate seasonal variations in net primary
production and the strength of the biological pump. Variations in these
parameters are likely to drive variations in N loss (Kalvelage et al., 2013;
Ward, 2013; Babbin et al., 2014).

In addition to the dependency on external static nutrient and parameter fields, this N cycle model is highly dependent on isotope effects for N cycle processes. Previous work has shown that the laboratory-derived isotope effects for some N cycle processes are not the same as their expressed isotope effects in environmental samples or under conditions relevant to environmental samples (Casciotti et al., 2013; Bourbonnais et al., 2015; Buchwald et al., 2015; Kritee et al., 2012; Marconi et al., 2017). Further probing the isotope effects using an inverse model such as this could provide insight into the expressed isotope effects that should be used in other modeling efforts involving field data. As presented in Sect. 3.1, the larger magnitude isotope effect for ${\mathrm{NO}}_{\mathrm{2}}^{-}$ oxidation best fit the ETNP and AS ODZs, where most of the ODZ volume resides. However, most model ${\mathit{\delta}}^{\mathrm{15}}{\mathrm{N}}_{{\mathrm{NO}}_{\mathrm{2}}}$ values still do not reach the lowest values observed on the edges of marine ODZs, indicating that further work is needed to understand the expression of these isotope effects.

The larger isotope effects resulted in better fits to observations of *δ*^{15}N
and DIN concentrations with lower rates of N cycling. This reinforces the importance of
obtaining realistic isotope effect estimates for each process that are relevant on an
environmental scale. Additionally, this highlights the need for critical consideration of
isotope effects used in N cycle models that use isotope balance to predict N cycling
rates. Though isotopes provide us with a useful tool to assess the relative contributions
of different processes, these estimates are highly subject to the isotope effects
employed. Also, as illustrated by the regional optimizations, the isotope effect for a
given process may vary, or be expressed differently, in different regions.

This model provides an excellent framework for further testing hypotheses
about controls on the marine N inventory and cycling of N on a global scale.
The distribution and sensitivities of N cycle rates resulting from this model
will be explored in a companion manuscript (Martin et al., 2019).
Incorporation of variable environmental input data, such as temperature,
productivity, and [O_{2}], could also help us predict how the N cycle
might be affected by past and future environmental changes.

Code availability

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Code availability.

Model code and model output from the three optimal ODZ isotope effect combinations, including the global best fit, are available in the Stanford Digital Repository (https://doi.org/10.25740/VA90-CT15; Martin et al., 2018).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/bg-16-347-2019-supplement.

Author contributions

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Author contributions.

KLC, FP, and TSM designed the study. TSM and FP constructed the model. TSM and KLC analyzed and interpreted the results. TSM, KLC, and FP wrote the manuscript.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

Thanks to Patrick Rafter for sharing a pre-publication version of his ${\mathrm{NO}}_{\mathrm{3}}^{-}$
isotope database. Thanks to Tim Davis for guidance on sparse matrix solvers. Thanks to
Tim DeVries for helpful discussions about earlier versions of the inverse model. Thanks
to Kevin Arrigo and Leif Thomas for comments on an earlier draft of this manuscript. This
work was partly supported by National Science Foundation (NSF) Chemical
Oceanography grant 1657868 to KLC.

Edited by: Jack
Middelburg

Reviewed by: Annie Bourbonnais and Itzel Ruvalcaba Baroni

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