The NitrOMZ model is based on the current understanding of the N cycle in OMZs as mediated by six major species: N2, NO3-, NO2-, N2O, NH4+, and organic nitrogen (OrgN) in either dissolved or particulate form. We only explicitly model NH4+ (the dominant dissolved form) and do not distinguish it from NH3. We also assume that organic nitrogen is linked to organic carbon by fixed stoichiometry , although variable stoichiometry can easily be accommodated.

A schematic of the model's tracers and transformation is shown in Fig. . Our approach represents a natural progression for current biogeochemical ocean models and takes a “system view” of the N cycle by focusing on the biogeochemistry of N transformation reactions rather than on microbial ecology . That is, the model explicitly resolves N chemical tracers and their transformations but not the populations of microbes that are responsible for these reactions.

The underlying assumption is that the occurrence and rates of N transformations are controlled by, and can be predicted from, the physical and chemical conditions of the oceanic environment. Implicitly, the model assumes that diverse populations of microbes are always present in the water column and that their activity (i.e., metabolic rate) is controlled by the abundance of substrates, analogous to chemical reactions, and dissolved O2, which inhibits anaerobic reactions . The focus on dissolved N forms and reaction rates bypasses poorly known aspects of microbial population dynamics, which are topics of ongoing research .

We assume that each reaction is implicitly mediated by specialized microorganism groups, each relying on a distinct metabolism . Thus, the model represents a “modular” N cycle, with individual reaction steps (i.e., individual redox reactions) represented separately and connected by the exchange of dissolved substrates . This premise is grounded on observations of high specialization and streamlined genomes for marine prokaryotes , including microorganisms carrying genes for N-based metabolic reactions .

These assumptions are sufficient for providing a broad representation of microbial N transformations and their environmental expressions in the ocean, while limiting model complexity and the proliferation of poorly constrained parameters. They are also grounding steps toward models that explicitly represent microbial populations, including their diversity and dynamics in OMZs .

The model focuses on microbial processes that take place below the euphotic zone, as driven by the flux of organic matter produced near the surface and exported into the ocean interior by the biological pump . We include heterotrophic and chemolithotrophic pathways that are commonly observed in the open ocean and require N species as substrates (Fig. ). Additional pathways, for example, involving sulfur or iron, could also be represented following a similar approach.

Heterotrophic reactions resolved by the model (Fig. ) consist of aerobic organic matter respiration (Rrem, pathway 1), which relies on O2 as the oxidant, and the three main steps of denitrification: dissimilatory NO3- reduction to NO2- (Rden1, pathway 4), NO2- reduction to N2O (Rden2, pathway 5), and N2O reduction to N2 (Rden3, pathway 6). Chemolithotrophic processes consist of aerobic oxidation of NH4+ to both NO2- (Raono2, pathway 2a) and N2O (Raon2o, pathway 2b, via both hydroxylamine oxidation and nitrifier denitrification); aerobic oxidation of NO2- to NO3- (Rno, pathway 3); and anammox, the anaerobic oxidation of NH4+ with NO2- to produce N2 gas (Rax, pathway 7). Reactions are parameterized as functions of substrates (i.e., model tracer concentrations) and environmental parameters such as dissolved O2 and organic matter. Tracers are expressed as concentrations, with units of mmol m-3.

We do not include an explicit representation of nitric oxide, NO, because of the poor understanding of its cycle in the marine environment . NO is thought to be an obligate intermediate or a byproduct of N cycle reactions, including nitrification and denitrification . However, it is a very reactive chemical with extremely low concentrations (on the order of pmol m-3) and rapid turnover in seawater . As a consequence, in situ NO observations are limited , and rate measurements targeting NO reactions are missing. Implicitly, we assume that NO cycles so rapidly that accumulation and transport by the oceanic circulation are negligible and that its dynamics can be folded into the cycle of other N tracers.

There are also several notable processes that are not represented in the current model formulation but could be introduced in future releases. Some of these processes (e.g., dissimilatory NO2- reduction to NH4+, DNRA) are not thought to be quantitatively relevant in oceanic oxygen minimum zones. Others, while relevant, require further measurements to constrain their significance and response to environmental variability.

Production of N2O via NH4+ oxidation in NitrOMZ is represented as a single O2-dependent function designed to model the transition in bacterial metabolisms from predominantly hydroxylamine oxidation to nitrifier denitrification at low O2 . However, growing evidence suggests that ammonia-oxidizing archaea (AOA, which greatly outnumber their bacterial counterparts) can also produce N2O via a hybrid mechanism . Production of N2O via AOA appears to be similarly enhanced at low O2 , although evidence from argues otherwise.

DNRA, which can be dominant in anoxic sediment, has been sporadically observed in the water column of oxygen-deficient zones, where it may provide an additional source of NH4+ to anammox bacteria . However, DNRA is commonly undetectable in OMZ waters , and its importance to the N cycle of OMZ is still debated .

Recent tracer incubation studies show substantial and often dominant formation of N2O from NO3- rather than NO2- . This suggests that denitrifying bacteria capable of direct production of N2O from NO3- reduction (as NO2- reduction proceeds entirely within the cell) could be a major source of N2O. This idea, which contrasts with the model assumption of a fully modular N cycle, is further supported by isotopic evidence . However, observations needed to constrain the proportion of N2O from NO3- and NO2- and its environmental sensitivity remain limited .

Other work suggests the occurrence of NO2- oxidation in apparently O2-deficient waters . This may involve NO2- oxidation coupled to iodate reduction or NO2- disproportionation – two poorly characterized processes. It may also reflect the high affinity to O2 of nitrite-oxidizing bacteria in regions where vanishing O2 concentrations are maintained by infrequent lateral intrusions .

Finally, the model could easily accommodate missing processes that couple the N cycle with other elemental cycles, in particular carbon and sulfur. These include formation of organic matter by chemolithotrophy; changes in inorganic carbon chemistry (e.g., pH) by anaerobic reactions ; and additional metabolic pathways such as anaerobic oxidation of sulfide with NO3- and anaerobic oxidation of methane with NO2- , both chemolithotrophic denitrification reactions.

Heterotrophic reactions (i.e., organic matter remineralization) are parameterized as a function of the respective oxidants and organic matter concentration and expressed in carbon units per unit volume and time. Heterotrophic reaction rates are assumed to be on the first order in the concentration of organic matter and limited by the oxidant, following a Michaelis–Menten formulation . Anaerobic reactions are inhibited by the presence of O2, based on an exponential limitation term . The resulting equation for a general heterotrophic reaction is 1RH=kH⋅[X][X]+KHX⋅e-O2KHo2⋅POC. Here, H indicates the heterotrophic process considered (e.g., dissimilatory reduction of NO3- to NO2-), RH the heterotrophic reaction rate (mmol C m-3 s-1), kH the specific first-order reaction rate (s-1), [X] the concentration of the oxidant (i.e., O2, NO3-, NO2-, or N2O), KHX the half-saturation constant for oxidant uptake (mmol m-3), KHo2 the scale for inhibition of the reaction by O2 (mmol m-3), and POC the concentration of particulate organic matter in units of mmol C m-3. No O2 inhibition is applied to aerobic respiration (i.e., KHo2 can be thought of as arbitrarily large).

Chemolithotrophic reactions are proportional to the respective substrates. A maximum reaction rate is modulated by the concentration of oxidants and reductants, following Michelis–Menten dynamics. For anaerobic reactions (here, anammox), an O2-dependent inhibition term limits the reactions when O2 is present. The resulting equation for a general chemolithotrophic reaction is 2RA=kA⋅[X][X]+KAX⋅[Y][Y]+KAY⋅e-O2KAo2⋅ Here, A indicates the chemolithotrophic process considered (e.g., anammox); RA the reaction rate (mmol N m-3 s-1); kA the maximum reaction rate when the process is not limited (mmol N m-3 s-1); [X] and [Y] the concentrations of the oxidant and reductant, respectively (e.g., NO2- and NH4+ for anammox); KAX and KAY the half-saturation constants for oxidant and reductant uptake, respectively (mmol m-3); and KAo2 the scale for inhibition of the reaction by O2 (mmol m-3). For aerobic reactions, KAo2 is set to infinite, removing O2 inhibition.

Equations for each of the heterotrophic and chemolithotrophic reactions are presented in Appendix  and , respectively; parameter names, units, and suggested values from the literature are presented in Table .

In the model, we assume that heterotrophic reactions are first-order to the concentration of organic matter; thus all organic matter can be utilized by microorganisms without saturation at high concentrations. Because of the low abundance of organic matter in seawater and extensive colonization of particles by heterotrophic bacteria, this is a reasonable first-order assumption. However, see for a discussion of microbial–particle interactions in ocean biogeochemical models and more complex aspects of their dynamics. For simplicity, we represent organic carbon by a single component. This assumption is easily relaxed to include multiple carbon species, for example, separate particulate or dissolved forms.

We do not explicitly model conversion of dissolved CO2 to organic matter by chemolithotrophy because of the small rates compared to the remineralization of organic matter in the upper ocean. This assumption can also be relaxed in future implementations of the model, allowing a more complete integration between chemolithotrophy and the carbon cycle.

The use of an exponential inhibition term for anaerobic reactions by O2 is based on the observation that they are limited at O2 concentrations to a few mmol m-3 or smaller . However, coexistence of anaerobic and aerobic reactions at O2 concentrations of 10–20 mmol m-3 or higher is also observed , perhaps related to the presence of redox microenvironments within organic particles , which are not explicitly considered here. The exponential inhibition formulation has the advantage of being controlled by a single parameter, allows anaerobic reactions at concentrations of finite O2, and approximates empirical rates from incubation experiments reasonably well .

Parameter values for maximum reaction rates, half-saturation constants, and O2 inhibition terms (Eqs.  and ) are informed by analysis of previous work and further optimized against in situ observations of tracers and rates (Sect. ). Table  presents a list of the model parameters and measured values based on a review of the literature. Note that these studies are based on shipboard and laboratory incubations that differ in the setup, conditions, and microbial populations tested. Despite these caveats, experimental results provide valuable starting points to further constrain parameter values in the model.

We implement the model for a one-dimensional water column that includes physical transport by vertical advection and turbulent diffusion and, if required, parameterized lateral transport by horizontal currents and eddies . The model is configured to represent the typical weak upwelling conditions that characterize open ocean oxygen minimum zones, following previous work .

In the one-dimensional framework, the conservation equation for the concentration [C] of a generic dissolved tracer can be written as 3∂[C]∂t=-∂wu⋅[C]∂z+∂∂zKv∂[C]∂z+∑i=1NHrC,Hi⋅RHi+∑i=1NArC,Ai⋅RAi+LT, where wu is the vertical upwelling velocity (m s-1) and Kv is the vertical turbulent diffusion coefficient (m2 s-1, distinct from molecular diffusion, which is much smaller), both of which can be a function of depth. The first and second summations are, respectively, over the NH heterotrophic and NA chemolithotrophic processes that involve the tracer (Eqs.  and ), with rC,Hi and rC,Ai being the corresponding stoichiometric ratios (Appendix A4). LT represents any parameterized lateral transport process. The explicit equations for each of the model tracers are detailed in Appendix .

The lateral transport term LT can be included to parameterize horizontal circulation by advection and diffusion in the one-dimensional framework. Typically, these terms are simplified by a linear restoring to far-field tracer concentration profiles , [C]far, with a relaxation timescale τC (s): 4LT=-1τC⋅[C]-[C]far. For typical open ocean conditions, τC can be estimated as the minimum of an advective timescale LU and a diffusive timescale, L2KH, where L, U, and KH are, respectively, the horizontal spatial scale, the horizontal velocity scale, and the horizontal eddy diffusion. Assuming L on the order of 1000 km, U on the order of 0.01 m s-1, and KH on the order of 1000 m2 s-1 results in a timescale τC=108 s, i.e., on the order of 3 years and in agreement with recent estimates of the residence time of water within the eastern tropical South Pacific (ETSP) .

In the one-dimensional model implementation, we represent organic matter (OrgC and OrgN) as a single particulate organic carbon (POC) class that sinks through the water column. We assume that this sinking is rapid compared to advection and diffusion, leading to a steady-state distribution of POC that is only controlled by sinking and remineralization . Since remineralization rates are proportional to the concentration of organic matter, the resulting steady-state one-dimensional equation for POC is 5∂ws⋅POC∂z=-∑i=1NHRHi=∑i=1NHkHeff,i⋅POC, where ws is the depth-dependent sinking speed of POC in the water column, and kHeff,i (s-1) is the effective rate constants for each heterotrophic process, i.e., the maximum rate constants multiplied by the respective substrate limitation and O2 inhibition terms (Eq. ).

Considering the flux of sinking POC, ΦPOC (mmol C m-2 s-1), 6ΦPOC=ws⋅POC. Equation () can be written as 7∂ΦPOC∂z=-∑i=1NHRHi=-∑i=1NHkHeff,i⋅POC or, equivalently, 8∂ΦPOC∂z=-∑i=1NHkHeff,iws⋅ΦPOC. Equation () can be recast to relate the concentration of POC in the water column to the remineralization of the POC flux with depth: 9POC=-1∑i=1NHkHeff,i⋅∂ΦPOC∂z.

The advantage of Eq. () is that it allows us to diagnose sinking POC concentrations when the POC flux and remineralization rate constants are known. In the one-dimensional implementation of the model, we parameterize the POC flux following a typical depth-dependent power-law function or Martin curve : 10ΦPOC=ΦPOCz0⋅zz0-b, where z0 is the upper boundary of the model and b the power-law or Martin coefficient. A plot of the model POC is shown in Fig. . Another advantage of this formulation is that it allows coupling NitrOMZ to more complex parameterizations for the remineralization of organic matter in ocean biogeochemical models, some of which rely on explicit representation of sinking organic particles and some of which only represent sinking organic particle fluxes in the water column . Because NitrOMZ's equation can be cast as a function of prescribed vertical organic matter flux or remineralization profiles, the model can be coupled to existing biogeochemical models with minimal interference in their formulation of organic matter cycles.

For the purpose of testing and illustration, we implement NitrOMZ in a one-dimensional representation of the water column below the mixed layer, following previous work . Model tracers are discretized on a one-dimensional vertical grid, with equal spacing Δz=10m, where z is depth. Boundary conditions are set at the top (z0) and bottom grid (zbot) cells, as Dirichlet (or fixed concentration) boundary conditions, with values taken from observations (Tables B2–B3). The conservation equation for each tracer (following Eq. ; see Appendix  for full equations) is then solved using a forward-in-time, centered-in-space numerical scheme, with a constant vertical grid spacing, and the option for a variable or constant time step. In the baseline simulations (Fig. ), we adopt a time step of 5 d for the initial 650-year spinup and decrease it to 3 h for the final 2 years of the simulation (years 698 and 699) to increase accuracy.

As in , NitrOMZ does not represent primary production in the surface layer and is instead forced at the uppermost boundary by a flux of sinking POC, ΦPOCz0=wsz0⋅POCz0, where POCz0 provides the boundary condition for POC. The flux ΦPOC remineralizes in the water column based on a Martin curve profile (Eq. ). At each depth, the steady-state conservation equation for POC (Eq. ) is solved with a forward-in-space method, using a depth-dependent sinking speed ws chosen to produce, together with the maximum aerobic remineralization rate constant, krem, a POC flux profile matching a Martin curve with exponent b appropriate for the oxygenated ocean . To this end, the sinking speed is calculated at each depth as 11ws=kRem⋅zb. The concentration of POC in the water column is then diagnosed using Eq. () and used to calculate the heterotrophic remineralization rates RH in Eq. () (see Appendix ).

Under constant forcings and boundary conditions, the model tracers evolve towards steady state (∂[C]∂t≈0, Fig. ) with a timescale τSS that can be estimated from the advection velocity wu, the turbulent vertical diffusion Kv, and the vertical scale H as the minimum between Hwu and H2Kv. For wu on the order of 10 m y-1, Kv on the order of 10-5 m2 s-1, and a vertical scale of 1000 m, the timescale to approach steady state is τSS=3×1010 s or about 100 years.

Figure  shows an example of model spinup to steady state in NitrOMZ, with parameters taken from an optimal solution discussed in Sect.  and uniform initial tracer concentrations in the water column. At the start of the simulation, high water column O2 leads the aerobic remineralization (Rrem) to dominate total POC consumption. As the simulation proceeds, an O2 minimum develops in subsurface waters, reaching suboxic (<10 mmol O2) concentrations around year 100. NO3- reduction rates (Rden1) are relieved of O2 inhibition and begin to take up a larger fraction of total POC remineralization, as revealed by the depletion of N*, signaling NO3- consumption in the water column. Reduction of NO3- also leads to a subsurface peak in NO2- within the O2 minimum (Fig. ). With newly available NO2- substrate and low-O2 conditions, NO2- reduction (Rden2) begins, resulting first in a subsurface spike in N2O. With further decrease in O2 concentrations, N2O is reduced to N2, leading to a layer of low N2O concentrations within the OMZ that persists to the end of the simulation. Anammox (Rax) is similarly relieved of O2 inhibition as the O2 minimum is established, reaching maximum values near the upper oxycline, reflecting a relatively high supply of both NO2- and NH4+.

The CMA-ES is a stochastic, population-based algorithm that seeks to minimize an objective cost function . The CMA-ES falls within the broader class of evolutionary optimization algorithms, where the search for an optimal solution proceeds by an iterative improvement of a population of parameters, with each iteration including a stochastic “evolutionary” element, in loose analogy with biological processes of mutation, recombination, and selection (illustrated in Fig. ). In contrast with typical evolutionary computation algorithms such as genetic algorithms, in the CMA-ES the mutation and recombination operations are substituted by sampling from a multivariate normal distribution in which parameters (the covariance matrix) are deterministically updated based on previous iteration steps .

The CMA-ES has been shown to be more efficient (i.e., requiring fewer objective function evaluations), accurate (i.e., able to approximate the global optimum when it is known to exist), and robust (i.e., not overly sensitive to the initial choice of parameters) compared to other optimization algorithms, when applied to multi-dimensional, non-linear optimization problems . These properties make it suitable for optimization of ocean biogeochemical models . A detailed description of the algorithm procedure can be found in ; an overview of the main steps of the algorithm and its application to ocean biogeochemistry are presented in .

As an illustration of NitrOMZ, we perform a series of optimizations against ETSP OMZ observations. For this configuration, we set a constant upwelling velocity (wup) but impose a variable vertical diffusion (Kv) profile, with lower diffusion in upper stratified layers, and a transition to higher diffusion in deeper layers (Fig. , left panel). This is a simplifying assumption that allows us to control the vertical scale for advective–diffusive transport (given by the ratio between vertical diffusivity and upwelling velocity, Kvwup), without requiring vertical divergence terms in the conservation equation for tracers associated with variable vertical velocities. Since this simulation targets the core of the OMZ, generally characterized by sluggish horizontal circulation , we turn off far-field tracer restoring. This simplifies analysis of model balances between transport and reaction rates, while resulting in realistic tracer distributions. The top and bottom boundary conditions are listed in Table  and are extracted from observations.

As a first step, we select parameters that control aerobic remineralization processes (Rrem) and lead to a realistic vertical O2 profile relative to ETSP observations, including the vertical position and thickness of oxygen-deficient waters (O2<5 mmol m-3) (Fig. ). These consist of the vertical diffusion and upwelling magnitude, the Martin curve coefficient (b), and the upper-ocean POC flux (Φpoctop), based on values consistent with observations (Table  and Fig. ). For simplicity, we also set the maximum aerobic remineralization rate (krem) and the O2 half-saturation constants for NH4+ and NO2- oxidation (Kaoo2 and Knoo2, respectively) to reported values in the literature (see Table ). We then employ the CMA-ES algorithm in NitrOMZ to optimize the remaining 20 parameters that control heterotrophic and chemolithotrophic reactions in Fig. , using the range of parameter values listed in Table .

To optimize more uncertain parameters that control the anaerobic N cycle, we then conduct four sets of optimizations, with cost functions devised to match desired characteristics of tracer and rate profiles in the ETSP OMZ. Briefly, the cost function is calculated as the mean square of the difference between observations and model output profiles for a series of variables that include tracers and N transformation rates (listed in Table ). Before each optimization, a random error of up to 20 % is assigned to each observation to increase the variability in observational constraints and improve the robustness of the optimization ensemble by preventing it from always converging in the neighborhood of a specific local minimum controlled by non-relevant features of the observations. Three additional constraints are imposed to improve the fit to observations for N cycle processes occurring within the core of the OMZ. First, all rates are weighted equally, whereas different weights are assigned to each tracer, giving higher weight to N2O and NO2-, which are central to the anaerobic N cycle. Because of possible influence from horizontal advection in observations, discrepancies exist between modeled and observed NO3- and PO43-. To compensate for this, we also assign lower weights to NO3- and PO43- and higher weight to N*. Second, a depth-dependent weighting scheme is included to emphasize the match to observations in the OMZ interior. This vertical weight is shaped as a Gaussian curve centered at the core of the observed OMZ, where the bulk of anaerobic transformations targeted by our model occurs so that values within the core of the OMZ are weighted up to twice as much as values outside the OMZ. Finally, N cycle transformation rates are shifted vertically to match their depth relative to the oxycline (here defined as O2=1 mmol m-3) in both model and observations and rescaled by a factor proportional to observed vs. modeled POC flux in the upper ocean. The only difference between the four sets of optimization is the relative weights assigned to each tracer, listed in Table . In total, we obtain 382 optimized parameter sets for further analysis.

The distributions of the parameter values from the 382 sets of optimizations (see Sect.  and Table ) are shown in Fig. . Rather than always converging to the same set of parameters, the optimization shows some variability for specific parameters. This reflects the stochastic nature of the CMA-ES algorithm, the inclusion of random variations in the observations, and the highly non-linear nature of the optimization problem, which may allow for non-unique optimal solutions. Optimized maximum rates (such as kao, kno, kden1, and kden3) and exponential O2 inhibition parameters for step-wise denitrification (Kden2o2 and Kden3o2) reveal more variability than half-saturation concentration coefficients (K terms), which often settle to the minimum or maximum allowed value (Table ).

Pairwise correlations in Fig.  reveal several parameter pairs which exhibit strong relationships, reflecting the fact that in a significantly non-linear optimization, similar results can be obtained by trade-offs between different parameters and processes. Notably, the exponential O2 inhibition constants for NO2- and N2O reduction (Kden2o2 and Kden3o2, respectively) are strongly correlated with each other (R=0.73) and with other parameters controlling the denitrification steps. These include positive correlations with the maximum rate parameters for NO3- and NO2- reduction (kden1 and kden2, respectively) and negative correlations with the half-saturation constants for NO2- and N2O reduction (Kden2no2 and Kden3n2o, respectively). These correlations suggests tight couplings between modeled denitrification steps, wherein high/low maximum denitrification rates can be compensated by lower/higher half-saturation coefficients, respectively.

Considering the variability in the optimal parameter sets and the complexity of the cost function, which depends on observations for multiple variables at different depths, the resulting N cycle profiles show similar features across all optimal solutions (Fig. , top panels; see also Fig.  for macronutrient profiles). When compared to observations, the majority of parameter sets are able to skillfully model (1) the vertical distribution of O2, including the oxygen-deficient layer between roughly 100 to 400 m; (2) the subsurface maximum in NO2-; (3) the rapid attenuation of NH4+ with depth; and (4) the subsurface minimum in N*.

N cycle transformation rates also show similar consistency in their vertical profiles, albeit with more notable discrepancies with observations, possibly reflecting the higher variability and more complex nature of these measurements. Lower rates than observed may also reflect the fact that incubation experiments provide potential rates rather than in situ rates. In general, the yield of N2O from NH4+ oxidation (Raon2o) is O(100) times less than for NO2- (Raono2), following Eq. (A8) and (A9), consistent with observations . The step-wise denitrification rates (Rden1, Rden2, and Rden3) show remarkably similar vertical profiles, with higher NO3- reduction rates (Rden1) and nearly identical magnitudes between Rden2 and Rden3. Anammox (Rax) shows a similar profile as denitrification, albeit with enhanced local maxima near the upper- and lower-oxycline depths surrounding the OMZ core, consistent with observations .

Several robust features emerge from the optimized parameter solutions, suggesting underlying mechanisms that need to be captured for a faithful representation of the OMZ N cycle. In particular, the differences in the exponential O2 inhibition parameters for denitrification, shown in Fig.  (left panel), reveal the existence of progressively lower O2 tolerance for step-wise denitrification (Kden3o2<Kden2o2<Kden1o2) from all optimized parameter sets. As a result, denitrification can stop at either N2O or NO2- as O2 increases above anoxic levels, leading to “incomplete” denitrification .

Within the anoxic core of the OMZ (∼100 to 350 m depth), O2 is low enough in all optimizations to allow each of the steps to proceed unimpeded (Fig. ). The large differences between NO3- and NO2- reduction (Rden1-Rden2, middle panel of Fig. ) allow accumulation of a characteristic subsurface peak in NO2- near the OMZ core. Conversely, N2O produced via NO2- reduction (Rden2) is quickly consumed via N2O reduction (Rden3), leading to a pronounced N2O deficit near the OMZ core. The progressive O2 inhibition of the three steps of denitrification results in a decoupling between these reactions that is particularly evident in the oxycline layers above and below the OMZ, where N2O accumulation dominates, as N2O reduction (i.e., consumption) is more strongly inhibited by O2 than NO2- reduction (i.e., N2O production, right panel of Fig. ). Thus, the O2 range defined by Kden2o2 and Kden3o2 can be thought of as a N2O production “window” that allows net N2O accumulation in the water column . This O2-driven decoupling of anaerobic reactions is consistent with the observed sequential inhibition of N2O and N2 production in incubation experiments , although we find O2 inhibition thresholds that are somewhat higher than suggested by those experimental studies. Conversely, other studies have suggested much higher O2 inhibition thresholds for anaerobic processes, on the order of several mmol m-3 .

The vertical profile of the step-wise denitrification rates (Rden1, Rden2, and Rden3) shows remarkable agreement across solutions, with only a small subset of parameter sets that behave as outliers (Fig. ). As a consequence, the fraction of POC remineralized by each heterotrophic reaction remains consistent across optimizations (Fig. , top panels). Near the base of the euphotic zone, at around 30 m depth, aerobic remineralization (Rrem) far exceeds denitrification, reflecting O2 inhibition of the latter. However, as O2 decreases to suboxic levels around 100 m depth, NO3- reduction becomes the dominant remineralization pathway (up to 60 % of total remineralization). As O2 drops further within the OMZ core (∼100 to 350 m depth), NO2- and N2O reduction rapidly take up the remaining fraction (∼25 % and 15 %, respectively), albeit with more variability than near the euphotic zone. Below the OMZ, as the water column reverts to oxic conditions, aerobic remineralization dominates, and by 500 m depth, all solutions show essentially no denitrification.

The processes responsible for fixed N loss (anammox, NO2- reduction, and N2O production from NH4+ oxidation) are also consistent across optimizations (Fig. , bottom panels). Within oxygenated waters, N2O production from NH4+ oxidation (Raon2o) is by far the dominant fixed N loss term, as all other sources are inhibited by O2. Anammox (Rax) becomes the dominant term within the upper and lower oxycline due to increased availability of both NO2- (from denitrification and nitrification) and NH4+ (from the decomposition of sinking POC), consistent with observations . In the anoxic OMZ core, relief from O2 inhibition allows NO2- reduction to outcompete anammox for NO2- and contributes up to 60 % of the total N loss, with anammox making up the remaining 40 % (also see Fig. ). This is somewhat higher than expected from purely stoichiometric constraints , likely reflecting vertical transport of NO2- and NH4+, co-occurrence of aerobic and anaerobic processes, and the higher O2 threshold for anammox inhibition in oxygenated waters. The resulting profile of total N loss thus reveals subsurface maxima predominantly driven by anammox, with denitrification leading total OMZ losses.

Among tracers, N2O profiles show significant variability between optimizations. While all optimizations generate two peaks in N2O surrounding the oxygen-deficient core, only a subset is able to reproduce the observed magnitude of the secondary peak at the lower oxycline (roughly 500 m depth; see Fig. ). This subset forms a “cluster”' of optimizations that share common features that facilitate the formation of a realistic deep N2O peak, including higher O2 inhibition thresholds (between 1.0 and 2.0 mmol m-3 for NO2- reduction and between 0.5 and 1.0 mmol m-3 for N2O reduction) and a wider O2 window where net N2O production is favored (between 0.5 and 1.0 mmol m-3 width). Additionally, while most optimizations are able to reproduce the OMZ peak in NO2-, significant variability in its magnitude exists. Given the central roles of N2O and NO2- in both nitrification and denitrification pathways (Fig. ) and the importance of oceanic N2O emissions to the atmosphere, we assign high priority to optimizations that reproduce realistic features in the distribution of these tracers, in particular a higher magnitude for the secondary N2O maximum. To this end, we select a parameter set (hereafter Optsel), which results in N2O and NO2- profiles closer to observations (bold red curves in Fig. , with parameter values reported in Table ). We use this Optsel parameter set for further analysis of the model sensitivity.

Compared to the other parameter sets, Optsel is characterized by weaker maximum NH4+ and NO2- oxidation rates (kao and kno, respectively) and smaller half-saturation constants for reductant uptake (Kaonh4 and Knono2, respectively) (Fig. ). In surface oxygenated waters, this results in relatively higher NH4+ and NO2- (Fig. ). In contrast, maximum denitrification rates (kden1, kden2, and kden3) are close to the median values from all optimizations. Rates of NO2- and N2O reduction (Rden2 and Rden3, respectively) are generally larger than other solutions, in particular near the lower oxycline (Fig. ). This increases POC consumption within this depth range via denitrification compared to other solutions (Fig. ). As a consequence, the residual between the NO3- and NO2- reduction (Rden1-Rden2; see Fig. ) leads to higher NO2- accumulation at these depths, providing the necessary NO2- substrate to fuel either NO2- reduction (i.e., N2O production) or anammox. Since the parameterization scheme in Optsel also results in reduced NO2- oxidation (Rno) and anammox (Rax) rates (see Fig. ), likely because of higher anammox half-saturation constants for substrate uptake (Kaxnh4 and Kaxno2), more NO2- is available for reduction by denitrification, leading to a surplus in production (Rden2) relative to consumption (Rden3) and high concentrations of N2O at the lower oxycline.

As shown in Sect.  and Fig. , strong correlations exist between parameter pairs in the optimization ensemble. Since Optsel demonstrates good comparisons with ETSP tracer and rate observations, we perform a series of sensitivity tests around parameters (P) most responsible for controlling specific features (F) of the tracer distributions. These include concentrations of NH4+ and NO2- at 50 m depth, the peak NO2- concentration in the OMZ, the N2O concentrations at the primary and secondary N2O maxima, and the minimum in the OMZ NO3- deficit (i.e., N*). Additionally, we evaluate which parameters govern total N loss, including the fractional contribution of anammox; the partitioning of POC consumption via NO3-, NO2-, and N2O reduction; and total N2O production and air–sea flux (here approximated by the vertical transport at the upper model boundary). To this end, we calculated the sensitivity coefficient (ϕij) for each P and F pairing by evaluating the impact of varying each Optsel P value by ±5% of its range in Table  and recording the resulting relative change in the F: 12ϕij=PiFj⋅∂Fj∂Pi.

The results demonstrate high sensitivity to changes in the maximum rates for all reactions (Fig. ). Specifically, higher maximum rates correlate negatively with the concentrations of their substrates and positively with the concentrations of their products. For example, increasing kden1 results in an increase in OMZ NO2- and a decrease in OMZ N*. Similarly, increasing kden2 decreases OMZ NO2- and increases N2O concentrations in the upper and lower oxycline and its flux to the atmosphere. These impacts are further modulated by the half-saturation and O2 inhibition constants.

Figures  and  further summarize the sensitivities to the maximum denitrification rates and their inhibition by O2, detailing the resulting changes to O2, N2O, NO2-, and N* profiles. As expected, changes in maximum rates affect reaction substrates and products in opposite ways. For example, a positive perturbation of kden1 (top panels) stimulates NO3- reduction, causing an increase in OMZ NO2- and a decrease in N* as expected. Similarly, a positive perturbation of kden2 increases N2O and decreases NO2- nearly everywhere. However, these sensitivities also have specific depth-dependent signatures. While changes in NO2- are more pronounced within the OMZ core, in particular the upper section, changes in N2O are stronger at the upper and lower oxyclines, i.e., within the N2O production window defined by Kden2o2 and Kden3o2 (see Sect. ).

Notably, by increasing kden1 (top panels in Fig. 9) or kden2 (middle panels) from Optsel values, the vertical extent of oxygen-deficient waters is reduced as a result of increased POC consumption via denitrification (not shown). This enhances aerobic remineralization and nitrification below the OMZ, providing an enhanced source of NO3- that partly offsets the OMZ losses seen via kden1 enhancement. This may indicate a potential negative feedback: if denitrification is locally enhanced (i.e., via increased competition for POC by denitrifying heterotrophs), a resulting reduction in the vertical extent of the OMZ would inhibit further N loss.

Figures  and  highlight significant sensitivities to the O2 inhibition constants, which control O2-dependent modulation of the maximum reaction rates. These effects are particularly evident at the boundaries of the OMZ. For example, an increase in Kden2o2 allows for more NO2- reduction at higher O2, leading to a slight depletion in OMZ NO2- and, as a consequence, an increase in suboxic N2O concentrations (Fig. , middle panels), consistent with observations of these processes in the Peruvian oxygen-deficient zone . In a similar manner, an increase in Kden3o2 leads to more N2O reduction, reducing the magnitude of both the primary and secondary N2O peaks, while leaving other OMZ tracers (NO2-, N*) relatively unaffected.

The main features of the OMZ simulated by the model are strongly dependent on environmental parameters such as upwelling and mixing; organic matter fluxes; and the model boundary conditions, including mixed-layer depth and O2 concentrations. Critically, these parameters are likely to vary over time under the effects of natural climate variability (e.g., ) and anthropogenic climate change . While each of these parameters control OMZ tracer profiles and N cycle reactions in complex ways, the main responses can be ascribed to changes in the position, thickness, and strength of the anoxic OMZ layer. Perturbations that replenish O2 above the thresholds for anoxic processes – such as those predicted under climate warming scenarios – thus have cascading impacts on anaerobic N cycle intermediates, such as NO2- and N2O, and on the fixed N removal and NO3- deficit of the oxygen-deficient zone.

Figure  shows the sensitivity of the optimal solution Optsel to the magnitudes of vertical upwelling (wup) and turbulent diffusion (Kv). Increasing wup results in higher O2 supply from below the OMZ, leading to increasing O2 concentrations, and an upward shift and thinning of the anoxic layers. At high upwelling, the anoxic layer is effectively wiped out and is replaced by a suboxic layer. Similar results are obtained with higher Kv values, with an increase in diffusive O2 supply from both above and below the OMZ, resulting in a progressive shrinking of the anoxic layer. As this layer vanishes, anaerobic processes cease, drastically reducing the concentration of NO2- and the N deficit in the OMZ core. Notably, as the OMZ reaches the brink of anoxia, i.e., as the minimum O2 concentration falls within the N2O production window, the upper and lower N2O maxima merge into a single N2O spike with particularly high N2O concentrations, reflecting the largest imbalance between production and consumption.

Opposite changes are observed for a reduction in both wup and Kv, which result in an expansion of the OMZ layer; increased NO3-, NO2-, and N2O reduction; a larger OMZ NO2- peak; and a broader separation of the upper and lower N2O maxima. The interplay between the position of the oxygen-deficient layer, sinking particle fluxes, and transport processes further modulates the response of tracer profiles. For example, as anoxic waters expand upwards following a reduction in Kv, they intercept a higher concentration of sinking organic matter, which in turn fuels higher remineralization rates. Together with reduction in diffusive fluxes, this likely favors the strengthening of the upper N2O maximum at low Kv observed in Fig. .

Because the supply of POC to the OMZ controls the overall magnitude of remineralization reactions, including O2 consumption and denitrification, the model is particularly sensitive to the sinking POC flux at the upper model boundary (Φpoctop, Table ; Fig. , top panel). Increasing Φpoctop causes a greater remineralization rate, which reduces available O2, and drives a progressive thickening of the OMZ, with a series of cascading impacts on tracers similar to the ones discussed above. In contrast, decreasing Φpoctop reduces the Rrem rates and increases O2 to the point that anoxic conditions and their signature disappear.

Similar changes can also be driven by variations in the bottom-boundary O2 concentration, which directly controls upward O2 supply by upwelling (Fig. , bottom panel). Increasing bottom O2 progressively decreases the thickness of the OMZ, shifting it upwards and eventually eroding the anoxic layer. Conversely, decreasing bottom O2 leads to a downward expansion of the OMZ and an intensification of anoxic conditions and the resulting anaerobic reactions.