A one-dimensional diffusion-reaction model was developed to simulate the concentrations of inorganic N compounds (NO3-, NO2-, NH4+, N2, N2O), distinguishing between 14N and 15N isotopes (14NO3-, 15NO3-, 14NO2-, 15NO2-, 14NH4+, 15NH4+, 14N2, 14N15N, 15N2, 14N2O, 14N15NO, 15N2O), as well as for O2 and sulfate (SO42-) concentrations. Their production and consumption rates are described by incorporating key processes of the canonical N cycle: aerobic mineralization, denitrification, nitrification, anammox, DNRA, mineralization by SO42- reduction, and anaerobic mineralization (other than SO42- driven) (Fig. ). All reactions (Table ) are described using the general formula: 1rate=kmax⋅limitation⋅inhibition where kmax represents the maximum conversion rate under ideal conditions (in µMd-1). The terms for limitation by substrate X and inhibition by substance Y for the process i are defined following Michaelis–Menten kinetics (Martin et al., 2019): 2limitation=[X]KX,i+[X]3inhibition=KY,iKY,i+[Y] where [X] and [Y] are the concentrations (in µM) of substances X and inhibitor Y, respectively, while KX,i and KY,i are their respective half-saturation and inhibition constants (in µM) for process i, respectively. While the model supports exponential equations for limitation and inhibition terms, Michaelis–Menten kinetics were chosen for this study, as they are more commonly employed in N models (Rooze and Meile, 2016). The specific reaction rate equations are implemented taking into account the concentrations of 14N, 15N, 14N14N, 14N15N, and 15N15N species separately for the limitation term. For 15N-containing species, specific reaction rates are reduced by (1-ε/1000) relative to 14N-containing species, reflecting the isotope effect associated with a given reaction (detailed descriptions of the model processes are provided in Appendix : Model processes and stoichiometry).

Molecular diffusion is modelled taking into account the reduced solute movement due to tortuosity (Burdige, 2007). Additionally, bioturbation is included as a transport term enhancing diffusion, with its influence exponentially decreasing with depth. Boundary conditions are set based on observed concentrations of N compounds, O2, SO42- at the upper boundary, and by zero fluxes at the lower boundary, except for NH4+. The NH4+ flux (and its δ15NFNH4) was jointly estimated with the model parameters, as the field data display a clear NH4+ concentration gradient at 5 cm. Total N, 14N and 15N concentrations, along with their fluxes, are used for model parameterization (see Appendix : Reaction-diffusion model for details).

The model is formulated as a dynamic model, but simulated to steady-state for comparison with observational data. Concentrations of 14N- and 15N-containing compounds are converted to total concentrations and δ15N.

This section outlines the modelled processes for 14N and 14N14N compounds (Table ). A comprehensive overview of the transformation processes for all isotopologues, and stoichiometric relations is provided in Appendix : Model processes and stoichiometry.

Mineralization of OM, the sole external N source, is differentiated in the model according to the specific electron acceptor involved: aerobic mineralization (O2), denitrification and DNRA (NO3-), SO42- reduction, and anaerobic mineralization. The latter encompasses all remaining redox species (i.e., other than O2, NO3-, and SO42-) below the nitracline (e.g., manganese oxides, iron oxides, carbon dioxide).

Denitrification is modelled as a three-step process: (1) NO3- to NO2-; (2) NO2- to N2O; and (3) N2O to N2. The first step, typically regarded as the rate-limiting step (Kampschreur et al., 2012), is the primary control on the overall expression of the N isotope effect (Kessler et al., 2014; Rooze and Meile, 2016). To prevent unrealistic rates, subsequent steps are constrained by setting kDen2=fDen2×kDen1 and kDen3=fDen3×kDen1, and specifying priors for fDen2 and fDen3. The re-parameterization of the second and third steps using the fDen2Den1 and fDen3Den1 factors corresponds to exactly the same model without any approximation or simplification. It serves solely to facilitate the specification of priors, as more knowledge is typically available about ratios of maximum rates (i.e., fDen2Den1=kDen2/kDen1) than about the absolute maximum rates themselves. The NO3- N isotope effect during benthic denitrification is known to be suppressed in the overlying water due to diffusion limitation (Dale et al., 2022; Kessler et al., 2014; Lehmann et al., 2003), though its expression at the porewater level remains less well constrained (Wankel et al., 2015). Transiently accumulating intermediates, such as N2O, that can escape to the overlying water and alter benthic N fluxes (Rooze and Meile, 2016), are also considered. Lastly, to ensure mass balance, the model accounts for clumped (doubly substituted; e.g., 15N15NO and 15N15N) isotopocules, but does not distinguish between isotopomers (i.e., 14N15NO and 15N14NO) due to lack of N2O isotope data needed for model validation. For the purpose of comparison with previous N models, a simplified one-step denitrification pathway (NO3- to N2 with no release of NO2- or N2O into the environment) approach is also implemented in the model code.

Nitrification is modelled as a two-step process: (1a) NH4+ to NO2-; (1b) NH4+ to N2O; (2) NO2- to NO3-. As for denitrification, the second step of nitrification is constrained to prevent unrealistic rates: kNit2=fNit2×kNit1, with specifying a prior for fNit2. N2O production yield during the first step is O2-dependent, and is modelled accordingly: 4fN2O_Nit1=ba[O2]+a where b and a are empirical parameters derived from Ji et al. (2018). N2O production also occurs via nitrification-denitrification, implicitly modelled by allowing reaction coupling via the intermediate NO2-. The expression of isotope effects depends on substrate availability and reaction completion. For instance, incomplete nitrification has been shown to result in isotopically heavy NH4+ efflux from the sediments (Dale et al., 2022; Lehmann et al., 2004; Rooze and Meile, 2016). However, similar phenomena for N2O and NO2- remain poorly understood.

The limited understanding of porewater N isotope dynamics, especially for processes other than denitrification, hinges on the scarcity of isotope data for crucial N species like NH4+ and NO2- in natural settings (Martin et al., 2019; Wankel et al., 2015). In the present model, we investigated the importance of these solutes, and how N-turnover processes like DNRA and anammox shape the distribution of their N isotopes. DNRA is modelled as a two-step process: (1) NO3- to NO2-; and (2) NO2- to NH4+. This approach separates the impact of NO2- reduction on NH4+, and allows comparison of NO2- isotopic signatures induced by denitrification, DNRA, and anammox. Anammox is modelled to include both the comproportionation of NH4+ and NO2- to N2 (main reaction, “m”), and the NO3- production via NO2- oxidation (side reaction, “s”) (0.3 molNO3- produced per 1 molNH4+ and 1.3 molNO2-) (Tables  and ) (Martin et al., 2019), which imparts a strong inverse isotope fractionation (Brunner et al., 2013; Magyar et al., 2021).

The relative importance of reductive NO3- pathways is constrained by altering maximum conversion rates, k, as: kDNRA1=fDNRA1,Den1×kDen1; kDNRA2=fDNRA2,Den2×kDen2; kAnam=fAnam,Den2×kDen2, where prior information on f factors was obtained from experimental rate measurements (see below). Altogether these reactions provide a comprehensive overview of N isotope dynamics in porewater and enable the assessment of influential environmental conditions in shaping them.

The model builds on the following considerations and assumptions: i.

The inputs of sinking OM and associated advective transport relative to the sediment surface are not explicitly modelled, as the dissolved O2 and N-compound profiles tend to reach quasi-steady state on short timescales (days to weeks). This simplification may not be valid for continental shelf sediments, where advection dominates solute movement due to high sediment permeability (Rooze and Meile, 2016). Therefore, in our model, porewater profiles are shaped primarily by molecular diffusion and bioturbation (the latter approximated as enhanced diffusion), along with reaction processes.

The model incorporates deliberate simplifications to reduce complexity, while remaining adaptable to new data or insights; however, it is acknowledged that these assumptions may significantly influence model outcomes and should be carefully considered when interpreting results.

Model parameter values were derived from an extensive literature review, and formulated as prior distributions, as detailed and referenced in Appendix : Prior values for inference. Positive parameters were parameterized as Lognormal priors, while priors of positive or negative parameters were parameterized as Normal distributions. Mean values were derived from the provided references, standard deviations were assigned either as absolute values or as percentages of the mean, depending on the class of variables. For parameters that are lake-specific (see model assumption iii.) and expected to be well identifiable from data, such as the maximum conversion rates of various processes (i.e., aerobic mineralization, the first step of nitrification, the first step of denitrification, mineralization by SO42- reduction, anaerobic mineralization) and the NH4+ flux from deeper sediment layers, only limited prior knowledge is available, making the use of uniform priors preferable. As their interpretability can be questionable, uniform priors were applied only to parameters expected to be well-identifiable, ensuring that prior variations within the marginal posterior range would remain small, even with alternative broad priors. This approach avoids specifying typical expected values, while maintaining robust identifiability. The maximum conversion rates for anammox, DNRA, as well as the second step of nitrification and the second and third steps of denitrification (Anam, DNRA1, DNRA2, Nit2, Den2 and Den3) were more challenging to identify from data, as the sensitivity of model results to these parameters becomes very low when the concentration of the converted substance becomes small. Additionally, prior specification for these rates was difficult, due to the expected variability among different lakes, similar to other maximum conversion rate parameters. Therefore, their priors were formulated as ratios relative to the better-constrained maximum conversion rate of the first nitrification (i.e., kNit1) or denitrification step (i.e., kDen1). This approach allowed for the characterization of the relative importance of each process without requiring absolute rate values. The joint prior for all parameters was assumed to be an independent combination of their respective marginal prior distributions.

To partially reduce structural uncertainty of the model and to account for parameter non-identifiability, Bayesian inference was applied, considering all uncertain parameters listed in Appendix : Prior values for inference. Some parameters were excluded from this analysis, including molecular diffusion coefficients, compound concentrations at the sediment surface, zero fluxes from deeper sediment layers (except for the NH4+ flux, which was inferred jointly with other parameters) and bioturbation. These values are considerably less uncertain than the other model parameters, except for bioturbation, which was addressed separately through a scenario analysis, following Bayesian inference under the Base scenario.

The posterior distribution (probability density) of the model parameters, fpost, is expressed as 5fpost(θ)=fL(C|θ)fpri(θ)∫fL(C|θ′)fpri(θ′)dθ′ where fpri is the prior distribution (probability density) of the model parameters, fL(C|θ) is the likelihood function of the model, C represents the observed compound concentrations, or δ15N values, and θ denotes the model parameters. The likelihood function fL(C|θ) is defined as a multivariate, uncorrelated Normal distribution with constant variances (standard deviation, σδ) for δ15N values, and variances increasing linearly with concentration, leading to a standard deviation σC=σC,aC+σC,b2 for O2, SO42-, and N compound concentrations. This formulation incorporates the combined uncertainties in model structure, sampling, and concentration measurements. To account for the unknown magnitude of these uncertainties, the coefficients of these relationships, σC,a, σC,b, and σδ, were inferred alongside the model parameters.

The marginal posteriors of individual parameters were compared with their priors to evaluate whether observational data provided information about these parameters, and whether this information was in conflict with the priors. In addition, two-dimensional marginals were examined to identify potential identifiability issues. Finally, uncertainty in the model results was calculated by propagating parameter uncertainty to the model results under consideration of their uncertainty for given parameter values as formulated in the likelihood function: 6fpost(C)=∫fL(C|θ)fpost(θ)dθ

For the parameters with marginal posteriors in conflict with prior information, we conducted additional scenario analyses, fixing parameters, and narrowing or widening prior distributions. These analyses evaluated the model's compatibility with observational data if parameters better aligned with prior information and assessed changes in posterior distribution with weaker priors. These scenario analyses complemented the assessment of bioturbation uncertainty mentioned above.

The partial differential equations outlined in Appendix : Reaction-diffusion model were solved using the Method of Lines. For spatial discretization, a grid was employed with cell thickness increasing progressively from the sediment surface toward deeper layers. This adaptive grid design reduced the total number of cells required, while still maintaining high resolution near the sediment-water interface, where steep concentration gradients typically occur (Appendix : Model discretization). The resulting system of ordinary differential equations (ODE) was solved by a standard ODE solver. Parameter inference was conducted using two advanced Bayesian inference algorithms: Metropolis (Andrieu et al., 2003; Vihola, 2012) and Hamiltonian Monte Carlo (Betancourt, 2017; Neal, 2011) algorithms.

The model was implemented in Julia (Bezanson et al., 2017) (https://julialang.org, last access: 11 July 2024) to achieve high-performance and facilitate automatic differentiation. The DifferentialEquations.jl package (Rackauckas and Nie, 2017) was used to solve the system of ODEs; performance testing of several ODE solvers identified the FBDF solver (adaptive order and adaptive time-step backward-differencing solver) as the most suitable for handling the stiffness of the ODE system. The ForwardDiff.jl package (Revels et al., 2016) was used for automatic differentiation; Bayesian inference was conducted using the adaptive Metropolis sampler from the AdaptiveMCMC package (Vihola, 2020), and the Hamiltonian Monte Carlo algorithm implemented in the AdvancedHMC.jl package (Xu et al., 2020). Further implementation details are provided in Appendix : Model implementation. Simulations were performed at sciCORE (https://scicore.unibas.ch, last access: 26 February 2025), the scientific computing centre at the University of Basel.

Sediment cores were retrieved at the deepest location of the Kreuztrichter basin in Lake Lucerne, a large oligotrophic lake in Switzerland (Baumann et al., 2024), in April 2021 using a gravity corer with PVC liners. The sediment cores were stored at 4 °C and processed using two porewater-sampling methods: whole-core squeezing (WCS; Bender et al., 1987) for NO3- samples, and Rhizon samplers (Rhizosphere research products, Wageningen, NL) for NH4+ samples. The WCS technique provides a high depth resolution near the sediment-water interface (0–5 cm, resolution: ∼ 0.7–1 mm), where NO3- is present in porewaters, while the Rhizon sampling method allows collecting samples at greater sediment depths (> 5 cm, resolution: ≥ 0.5 cm). NO3- and NH4+ concentrations were measured using ion chromatography (940 Professional IC Vario, Metrohm). δ15N-NO3- and δ15N-NH4+ were determined using the denitrifier method (Casciotti et al., 2002; Sigman et al., 2001), and the hypobromite-azide method (Zhang et al., 2007), respectively. In both methods, sample N from NO3- or NH4+ is converted into N2O, which is then purified and analysed by isotope ratio mass spectrometry (Delta V Plus, Thermo Fisher Scientific). The typical analytical precision is ∼ 0.25 ‰ (McIlvin and Casciotti, 2010).

For model parameterization, reaction rates for denitrification, DNRA, and anammox were determined using established protocols for 15N-tracer incubations (Holtappels et al., 2011). After recovery and sectioning of the core into 1 cm intervals, 1 g of sediment was placed into 12 mL gas-tight glass vials (Exetainers^®, Labo, UK). These Exetainers were then filled with anoxic, sterilized bottom water, amended with the following tracers: (Exp1) 15NO3-, (Exp2) 15NH4++14NO2-. Exetainers were incubated at 6 °C in the dark, and terminated at designated time points (0, 6, 12, 24, and 36 h) by adding ZnCl2. Gas headspace samples were analysed for the production of 14N15N and 15N15N using gas-chromatography isotope ratio mass spectrometry (GC-IRMS; Isoprime, Manchester, UK). Linear regression of 14N15N and 15N15N production over time was used to calculate N2 production rates, with standard errors derived from deviations in the regression slopes across the five-time points. For the determination of 15NH4+ production from 15NO3- additions, 15NH4+ was chemically converted to N2 gas using the alkaline-hypobromite method (Jensen et al., 2011). The resulting 14N15N was quantified by GC-IRMS. Linear regression of 14N15N production over time was used to calculate potential rates of 29N2 (i.e., 15NH4+) production. Rates of denitrification, DNRA, and anammox were calculated according to Holtappels et al. (2011) and Risgaard-Petersen et al. (2003). Only data from the upper 1 cm were used to parameterize the model, as the investigated sediments displayed a shallow nitracline and the highest anammox contribution at 0–0.5 cm depth.

The model implementation was highly efficient, achieving simulation times of about 12 s on a 13th Gen Intel^® Core™ i9–13 900 K processor with 3.00 GHz and 64 GB of memory (of which only a small fraction was needed) for a 100 d simulation starting from constant concentration profiles. This efficiency enabled the execution of Markov chains of 20 000 iterations within a few days on the scientific computing centre at the University of Basel (https://scicore.unibas.ch, last access: 26 February 2025). By combining these chains, samples of 100 000 iterations were generated. The Hamiltonian Monte Carlo algorithm outperformed the adaptive Metropolis algorithm during burn-in to the core of the posterior distribution. However, for final posterior sampling with about 60 parameters, adaptive Metropolis sampling proved more efficient in terms of effective sample size per unit of simulation time. Despite these efforts in getting computational efficiency, and the use of advanced MCMC algorithms, reaching convergence of the Markov chains remained challenging. We got five consistent Markov chains without discernible trends for each scenario; however, some widening of the chains and the resulting effective sample size on the order of 500 indicate that we were not able to get a good coverage of the tails of the posterior distribution. This outcome demonstrates that incorporating so many uncertain model parameters pushes the limits of Bayesian inference in terms of numerical tractability. However, the resulting uncertainty estimates are certainly more realistic than those obtained by fixing many poorly constrained parameters to unique values to reduce the dimension of the parameter space.

The simulation results of solute concentration and δ15N profiles in the most plausible Base scenario (Fig. ) integrate prior knowledge (Appendix : Prior values for inference) with observational data through Bayesian inference. The profiles closely reproduce the available, albeit limited, data, and conform to expected depth-related trends: oxidants (i.e., O2, NO3- and SO42-) are readily consumed via aerobic mineralization and nitrification (O2), denitrification (NO3-), and SO42- reduction. While mineralization is assumed to involve negligible N isotopic fractionation, the first step of nitrification causes significant enrichment in 15N of the residual NH4+ pool, yielding δ15N-NH4+ values up to 11.2 ‰ at 0.15 cm, due to strong N isotope fractionation, estimated at εNit1 = 12.0 ‰ (to NO2-) and 36.4 ‰ (to N2O). Unfortunately, extremely low NH4+ concentrations measured in the top 2 cm hindered the determination and verification of the modelled δ15N-NH4+ in this zone with field data. Both NO2- and N2O accumulate in the upper 0.5 cm, reaching up to 0.4 and 2 µM, respectively. Below 0.3 cm, denitrification leads to the progressive 15N enrichment of NO3-, NO2- and N2O, while N2-producing mechanisms (i.e., denitrification and anammox) cause only minimal changes to the modelled δ15N-N2 profile, due to the dominance of a large pre-existing N2 pool. For concentrations, the 95 % credibility intervals of parametric uncertainty are rather narrow, whereas the much broader total uncertainty is dominated by the lumped uncertainty term in the likelihood function, which primarily reflects the model's structural uncertainty. The error, beyond the parameter error, is parameterized using the two sigma values (σC,a and σC,b; see Sect. ), and exceeds what would arise from measurement and sampling alone. This suggests that the larger error is attributable to the model's structural limitations. Conversely, δ15N profiles exhibit small total uncertainty, as model results for δ15N closely match observational data, with minimal random and systematic deviations (parameterized using the sigma value σδ, see Sect. ).

The model provides insights into the underlying process rates (Fig. ) that shape the simulated profiles (Fig. ). Vertical profiles of transformation rates for NH4+, NO3-, NO2- and N2O clearly illustrate the sequential dominance of different N-transformation processes with increasing sediment depth and decreasing O2 availability. Aerobic processes, namely aerobic mineralization and nitrification, primarily control NH4+ transformation rates, peaking at 450 and 350 µMd-1, respectively (Fig. a). Nitrification sustains denitrification by producing both NO2- (up to 350 µMd-1) and NO3- (up to 275 µMd-1) in the upper 0.4 cm (Fig. b and c). A strong spatial overlap of nitrification and denitrification emerges in the depth distribution of processes affecting the NO2- pool, suggesting a potential interplay between these pathways (Fig. c).

A key strength of this model is the incorporation of N2O as a state variable. Our model results reveal that, although N2O production via nitrification is minimal (not visible in Fig. d), the strong isotopic fractionation associated with this reaction (εNit1,N2O = 36.4 ‰) generates N2O with δ15N values of -1.2 ‰ to -2.2 ‰ in the top 0.2 cm (Fig. c). At a depth of approximately 0.35 cm, up to 2.1 µM of N2O accumulate, coinciding with the highest rates of N2O production through denitrification. Conversely, N2O consumption by the last denitrification step peaks at 0.5 cm, leading to a progressive increase in δ15N-N2O with depth. This zonation likely reflects the O2 sensitivity of the distinct N2O-producing and -consuming processes. Specifically, N2O reductases are known to be strongly inhibited by O2, and therefore exhibit greater activity below the oxycline (Wenk et al., 2016). Although the model does not explicitly include the enzymes responsible for N-transformation pathways, the chosen and estimated kinetic parameters reflect substrate affinity and inhibition strength. Consequently, inhibition constants like KO2,Den2 and KO2,Den3 provide indirect insights into the O2 dependency of these enzyme-mediated reactions, effectively shaping the modelled redox zonation.

The model adequately captures the concentration and isotopic composition of the state variables, in agreement with field measurement and the expected patterns of underlying N-transformation processes and reaction coupling (Figs.  and ). One key strength of the step-wise model is its ability to quantify reaction coupling, which is challenging to infer directly from state variable pools (i.e., reactive intermediates), if they are rapidly turned over.

To address the variable ranges for the model parameters found in the literature, and to reduce structural uncertainty imposed by fixed parameter values, we estimated a large set of parameters using Bayesian inference. The obtained joint posterior distribution of model parameters enabled us to assess the knowledge acquired from data. Marginal posterior distributions of individual parameters, and two-dimensional marginal distributions of parameter pairs, were particularly useful in this context (Fig.  shows examples for the four categories defined below; Fig. S1 in the Supplement provides an overview of all marginal prior and posterior parameter distributions). By comparing marginal posterior distributions with their corresponding priors, parameters were classified as well identifiable or poorly identifiable. While this classification involves some subjectivity in determining how much narrower a posterior distribution should be compared to its prior distribution to classify such parameter as well identifiable, some clear patterns emerged:

1.

Well identifiable parameters: The marginal posterior distribution is clearly narrower than the prior, indicating that data provide meaningful information about the parameter's value. Two cases were observed: a.

The marginal posterior distribution is within the prior range, suggesting that the information from the data is in agreement with prior knowledge (Fig. a). Examples include: f factors for anammox (fAnam,Den2 = 0.2) and both DNRA steps (fDNRA1,Den1 = 0.005, fDNRA2,Den2 = 0.005), estimated using 15N-tracer incubation experiments for the investigated system, and parameters such as KNO3,Den1 and KO2,MinOx, constrained from clearly defined oxidant declines. Maximum conversion rates for aerobic mineralization, denitrification, SO42- reduction, and anaerobic mineralization, as well as the NH4+ flux from deeper sediment layers, also belong to this category, although we approximated very wide priors by uniform priors (see Sect. ), making it less visible in the plot.

The comparison of marginal priors and posteriors of the parameters (Fig. S1) demonstrates that excellent agreement between model outputs and observational data (Fig. ) can be achieved for 54 of the 58 estimated parameters compatible with their priors. Exceptions include: the higher-than-expected rate for the second denitrification step relative to the first (expressed by the factor fDen2,Den1), the large half-saturation constant for SO42- reduction (KSO4,MinSulfRed), and smaller-than-expected N isotope effects for the first steps of denitrification and nitrification (εDen1 and εNit1,NO2-, respectively). The largest deviation is observed for εDen1, which is further examined in the next subsection.

Notably, the seven parameters, for which a uniform prior was chosen to approximate a very wide prior (kMinOx, kDen1, kMinSulfRed, kMinAnae, kNit1, FNH4, δ15N, FNH4), were identifiable, indicating that highly system-specific prior knowledge is not crucial for these estimates. Most of the other model parameters showed limited narrowing of the marginal posterior relative to the prior, reflecting the rather limited information gain that can be obtained from data. The three model error parameters (σC,a, σC,b, σδ) were well identifiable and will be used in the following sections to compare the fit quality across different modelling scenarios.

Building on the findings discussed in the previous section, we explored the apparent prior-data conflict regarding εDen1 in greater detail. Additionally, we assessed whether the estimated process rates overlooked potential reaction coupling, which might go undetected through 15N-tracer incubation experiments, by exploring the variability in contributions of anammox and DNRA (i.e., fAnam, fDNRA1 and fDNRA2). Lastly, given the uncertainty regarding solute-diffusion enhancement by bioturbation, we investigated a scenario with increased bioturbation. These considerations led to four key scenarios: A.

Narrow priors for ε. This scenario investigated the effects of restricting ε variability to a narrower range (prior standard deviation of 1 ‰ instead of 5 ‰). The aim was to test whether the marked reduction in the marginal posterior of εDen1 persisted under stricter prior assumptions, and whether this decreased flexibility significantly impacted the quality of the model fit.

The results demonstrate a strong dependence of the estimated parameters on the chosen prior assumptions (Fig. ). Across all scenarios, marginal posterior distributions for the selected parameters are generally narrower than the prior distributions, though results vary substantially. In Scenario (A) (Narrow priors for ε), restricting the prior range significantly constrained εDen1, limiting its deviation from the prior (Fig. m; note that the prior for Scenario (A) is 5 times narrower than the one shown, which represents the prior for all other scenarios). These results closely resemble those from Scenario B (Fixed ε), where no deviation was possible (Figs.  and S2 in the Supplement). Both scenarios exhibit lower denitrification rates than the Base scenario (Fig. b), but comparable fit quality for total (14N+15N) concentration, quantified by σC,a (i.e., the dominant term of standard deviation of the model error for concentrations, see Sect. ) (Fig. x). On the other hand, scenarios (A) and (B) display poorer fit quality for δ15N profiles, indicated by a large value of σδ (Fig. z), suggesting that the model structure cannot adequately reproduce the δ15N-NO3- profiles without adapting the εDen1 value. While biological isotope effects of 15 ‰–30 ‰ are typical for NO3- reduction (Lehmann et al., 2007), lower values under almost-complete NO3- consumption have been reported (Thunell et al., 2004; Wenk et al., 2014). This finding is further confirmed by comparable marginal posteriors for εDen1 across all scenarios considered in this study, besides scenarios (A) and (B). To test the robustness of our model, we ran a base scenario simulation for marine sediments in the Bering Sea (station MC16) (Lehmann et al., 2007) (data not shown). Moreover, a manuscript currently in preparation presents an extensive comparison of model application across different sites and demonstrates a much wider range of 15εDen1 values, exceeding 20 ‰.

In Scenario (C) (Wider f), allowing greater variability in anammox and DNRA contributions results in the lowest fAnam,Den2 values, although such deviation is not substantial compared to the Base scenario output (Fig. i). The estimated fDNRA1,Den1 and fDNRA2,Den2 values in Scenario (C) mostly align with those of the Base scenario, corroborating the marginal role of DNRA in Lake Lucerne. Such findings confirm the accuracy of the rate measurements performed with 15N tracer incubations.

Scenario (D) (Enhanced bioturbation) stands out with the highest conversion rates (i.e., kMinOx, kMinSulfRed, and kNit1) (Fig. a, e, and g) to ensure sufficient oxidant consumption at higher supply/flux rates (reproducing the observed gradient despite higher diffusivity). Despite these changes, bioturbation had negligible effects on porewater N isotope dynamics, with estimated isotope effects and fit quality for δ15N profiles (σδ) comparable to those of the Base scenario.

The obtained concentration depth profiles for the four scenarios are generally comparable, as newly estimated parameters ensured good fitting of the data (Fig. S2). However, in Scenarios A and B, stricter constraints on prior knowledge for parameter estimation result in little to no suppression of all isotope effects (i.e., relatively strong N isotopic fractionation), leading to great variability in the δ15N profiles. Poor fits to the δ15N data are observed under these conditions, as evidenced by the greater 15N enrichment of the NO3- pool compared to the measured-data profiles (Fig. S2). Similarly, the δ15N-N2O profiles exhibit sharp declines to approximately -15 ‰ in the upper 0.5 cm under scenarios (A) and (B), driven by the strong expression of εNit1,N2O (40.1 ‰ and 40.0 ‰, respectively). In contrast, Scenarios (C) and (D) closely resemble the Base scenario, with only minor δ15N-N2O variations.

The importance of modelled processes and their impact on N isotope signatures were investigated by selectively deactivating individual processes and comparing the model outputs to the Base scenario. Aerobic mineralization, denitrification, and SO42- reduction were considered essential to preserve redox zonation (e.g., sequential decline of O2, NO3-, and SO42-) and N dynamics. The following processes were individually turned off: (a) nitrification (“NitOff”); (b) anammox (“AnamOff”); and (c) DNRA (“DNRAOff”). Initially, each process was simply inactivated to assess its impact on model outputs (Fig. ). Subsequently, inference was conducted after deactivating each process, to investigate their importance for model performance, parameter and flux estimation, and for the identifiability of rate parameters by evaluating the quality of the fit to the data, especially on the δ15N profiles (Fig. , Figs. S3 and S4 in the Supplement).

Switching off nitrification significantly alters the model output compared to the Base scenario (Fig. a, b, e, and f), indicating its central role in the benthic N dynamics. Key effects include NH4+ accumulation throughout the investigated depths, with a flattening of the δ15N-NH4+ profile (i.e., less curvature towards higher δ15N values) in the upper 0.5 cm, as the only other source of 15N-enriched NH4+ besides nitrification would be anammox, which is inhibited under oxic conditions. Furthermore, nitrification-denitrification coupling via NO2- weakens in this scenario, resulting in lower overall N2 production (as indicated by the lower maximum N2 concentration of 734 µM compared to 745 µM in the Base scenario). These results suggest that partially reducing, or fully eliminating, nitrification lowers the system's capacity to act as an efficient N sink. In other words, the findings confirm that nitrification is a critical process that, when closely coupled to denitrification, helps to enhance the ecosystem's potential to remove fixed N. All other N-isotopic state variables also show a flatter δ15N profile, with only a progressive enrichment in 15N below 0.5 cm, primarily driven by denitrification (NO3-, NO2-, and N2O). The impact of disabling nitrification is clearly reflected in the δ15N-N2O profile across the upper 0.3 cm, where the typical nitrification-induced dip is absent, and δ15N-N2O values remain relatively constant (∼ 7 ‰–8 ‰). In contrast, the effects of turning off anammox or DNRA are more subtle, owing to their generally lower reaction rates in Lake Lucerne (Fig. c, d, g, and h). Notably, in the absence of anammox, N2O exhibits lower δ15N values in the upper 0.3 cm compared to the Base scenario, likely due to higher N2O yields via nitrification, as reduced competition for NH4+ with anammox provides more substrate for nitrification.

Upon running inference for each case, concentration and N isotope profiles for the NitOff, AnamOff, and DNRAOff scenarios are generally similar to those of the Base scenario (Fig. S3), with notable exceptions in the NitOff case. In the absence of nitrification, NH4+ accumulates and the δ15N-NH4+ profile remains largely flat, since anammox, the only other NH4+-consuming process, is minimal under oxic conditions. No δ15N-NH4+ measurements are available for the top 1 cm, so the model output could not be verified with field data. The N2O pool systematics also diverge between the NitOff and Base scenarios. Specifically, in the NitOff case, no nitrification-derived N2O accumulates in the upper 0.4 cm, and consequently, the δ15N-N2O profiles lacks the typical nitrification-associated decline in this layer. Instead, N2O becomes progressively enriched in 15N below 0.4 cm. While most estimated parameters and fluxes are consistent across the four scenarios, the NitOff scenario stands out again, exhibiting strong effects on the anammox rates and associated isotope effects (e.g., fAnam,Den2, εAnam,NH4) (Fig. S4), as well as on benthic fluxes of NH4+, NO2-, NO3- and N2O (Fig. ). Nonetheless, the NH4+ concentration profile is well-captured, as indicated by a low σC,a, reflecting a good match between model and concentration data even in the absence of nitrification. This finding implies that the model cannot resolve the relative contributions of nitrification versus anammox to NH4+ consumption based on the concentration and isotope data, highlighting the importance of prior knowledge regarding fAnam,Den2.

The comparison of process rates across these four scenarios provides insights, unveiling the extent of process coupling and competition (Fig. S5 in the Supplement) (Hines et al., 2012). For instance, anammox and nitrification compete for both NH4+ and NO2- as substrates, causing the rate of one process to be enhanced, when the other is switched off. The NH4+ oxidation and NO2- production rates via nitrification (Nit1) are higher (∼ 0.2 cm depth) in the AnamOff scenario than in the Base scenario. Even more obviously, enhanced rates of NH4+ oxidation, NO2- consumption, and NO3- production via anammox are observed in the NitOff scenario than in the Base scenario. Process coupling, specifically nitrification-denitrification, is further confirmed by lower rates for NO2- reduction via denitrification (Den2) in the absence of nitrification. In general, the influence of DNRA on production and consumption rates of the considered state variable appears minimal, owing to the limited environmental relevance of DNRA in Lake Lucerne. Overall, the similarly good fits obtained across these three scenarios and the Base scenario reflect the poor identifiability of the switched off processes. This suggests that the data can be well-fitted even without these three processes, emphasizing the importance of prior knowledge about their environmental relevance.

Previous models of benthic N isotope dynamics have focused on individual reactions or overlooked the role of intermediate species, such as NO2- (Kessler et al., 2014; Lehmann et al., 2007). Our study confirms that NO2- plays a critical role in coupling multiple N-transformation processes and shaping benthic N isotope dynamics, including that of δ15N-NO3-. While such process coupling has been examined in the water column (Frey et al., 2014), it remains, to our knowledge, largely unexplored in sedimentary environments.

To assess the significance of this coupling, we implemented a one-step denitrification approach that bypasses NO2- as an intermediate, replacing the three-step pathway used throughout this paper (Fig. ). In this simplified model, NO2- concentrations and isotopic signatures are shaped solely by nitrification (and to a marginal extent, DNRA and anammox), as denitrification no longer contributes to NO2- production. This modification leads to significantly reduced NO2- accumulation, restricted to the upper 0.3 cm, and lower anammox activity, due to a lack of NO2- substrate below the oxycline. The absence of denitrification-derived NO2- has profound effects on the N isotope dynamics. First, a consistent ∼ 15 ‰ offset between δ15N-NO3- and δ15N-NO2- is evident across all modelled depths (Fig. c). This offset is ascribed to the isotope effect of the second nitrification step (εNit2 = -13.7 ‰), and the lack of 15N enrichment in the NO2- pool from denitrification. Second, the estimated isotope effect for NO3- reduction (εDen) increases to 5.5 ± 0.9 ‰, nearly double than in the Base scenario, indicating that elevated δ15N-NO3- values in the field data may, to some extent, reflect NO2- isotope dynamics, rather than solely the effect of NO3- reduction (Fig. ).

These findings emphasise the importance of both NO2--producing and -consuming processes in modulating δ15N-NO3-, and consequently, estimates of εDen1. Although nitrification is typically aerobic and denitrification anaerobic, evidence exists that indicates spatial overlap of these two processes at the bottom of oxyclines in natural aquatic environments (Frey et al., 2014; Granger and Wankel, 2016). In this transition zone, NO2- produced by either pathway can be oxidised to NO3- or reduced to N2O, NH4+ or N2 (Fig. ), significantly affecting its δ15N signature (depending on the N-branching). For instance, NO2- reduction to N2O enriches the residual NO2- pool in 15N. If this 15N-enriched NO2- is subsequently oxidized to NO3- (a reaction that exhibits an inverse kinetic isotope effect), the resulting NO3- will be markedly enriched in 15N (Fig. ). Such interactions have been shown to influence apparent isotope effects for NO3- in the water column (Frey et al., 2014), and likely exert similar effects in sediments, where sharp redox gradients create overlapping zones of nitrification and denitrification. This coupling may explain the discrepancy in estimated εDen1 values between the Base scenario (2.8 ± 1.1 ‰) and the one-step denitrification model approach (5.5 ± 0.9 ‰).

Anammox further complicates these dynamics, as it depends on NO2- excreted into the environment. Without denitrification, which releases NO2- (Sun et al., 2024), anammox is substrate limited (Fig. ). Thus, while previous benthic studies estimated denitrification isotope effects using one-step denitrification approaches (Lehmann et al., 2007), our findings call for the adoption of a stepwise modelling approach (Sun et al., 2024) that better captures the interdependence of N-transformation pathways, and their integrated effects on NO3- isotope dynamics. A more detailed examination of these interactions is essential for refining our understanding and quantification of isotope effects associated with NO3- reduction in sedimentary systems.