Introduction

Numerical simulations of coupled hydrodynamic and biogeochemical processes are increasingly being used to manage water quality for beneficial uses and to achieve ecological goals in estuaries (Staehr et al. 2012; Ganju et al. 2016; Liu et al. 2018). These models assess responses of estuaries to prospective management actions and to other factors that impact aquatic ecosystems (Hopkinson and Vallino 1995; Howarth et al. 2011; Li et al. 2016; Testa et al. 2017). However, modeling biogeochemical processes with the combination of generality, precision, and reliability needed to support management needs is difficult. One challenge is representing all important processes without introducing more parameters than can be constrained in calibration with available observational data (Ganju et al. 2016). Even as observational data sets become larger and richer, modeling is complicated by potentially important processes with unknown rate parameters that may vary across time and space. We chose to simulate nitrogen cycling in the Sacramento-San Joaquin Delta (Delta) in the northern San Francisco Estuary to illustrate the power of Lagrangian modeling approaches in a complex system with ample monitoring data available to calibrate and validate the model.

Quantitatively predicting nitrogen transport, cycling, and fate in aquatic systems is complicated by the numerous biologically mediated transformations that nitrogen species undergo during transport. These processes include nitrification: conversion of ammonium to nitrate by nitrifying microbes; photosynthesis: nitrate and ammonium uptake and incorporation into organic molecules, for example by photosynthesizing phytoplankton and aquatic macrophytes; mineralization: recycling of organic nitrogen back to ammonium (NH4) or nitrate (NO3) by microbial degradation or respiration; denitrification: conversion of nitrate to nitrogen gas (N2) by heterotrophic microbes; annamox: conversion of ammonium to N2 gas by autotrophic microbes; N2 fixation: microbial conversion of N2 gas to ammonium; and, dissimilatory nitrate reduction (DNRA): the microbial transformation of nitrate to ammonium (Damashek and Francis 2018; Chen et al. 2021; Testa et al. 2022). While facets of microbial transformation processes are well understood, their rates are site-specific and can vary with habitat characteristics such as substrate, temperature, dissolved oxygen concentration, and the availability of labile organic carbon. Biological uptake of dissolved inorganic nitrogen (DIN) by primary producers such as vascular plants and phytoplankton are also affected by environmental conditions such as temperature, water depth, light and turbulence, which typically vary widely across an estuary (Bergamaschi et al. 2020; Ganju et al. 2020). In most cases there is insufficient a priori information for assessing rates and understanding which processes dominate (Damashek and Francis 2018; O’Meara et al. 2020; Chen et al. 2021; Testa et al. 2022). Furthermore, the fluxes between organic and inorganic pools of nitrogen are bidirectional, driven by autotrophic and heterotrophic metabolism (Creed et al. 2015; Paerl 2018; Gold et al. 2021).

Biogeochemical models represent biological processes and their responses to physical conditions by systems of equations with levels of complexity that vary widely among models. Some models track nutrient storage and transformations within multiple compartments including dissolved ions in the water column, organic nitrogen retained in organisms (phytoplankton, macrophytes, zooplankton), and detrital nitrogen in the water column and in sediments (Testa et al. 2013; Liu et al. 2018). The concentrations and transformation processes among these pools are represented by systems of equations including zero-order (concentration independent), first-order (reactant concentration dependent), and second-order (dependent on concentrations of two reactants) reaction kinetics, with rates that can be influenced by abiotic factors such as temperature, salinity, or dissolved oxygen concentrations (Smits and van Beek 2013). For example, benthic fluxes have been represented with a range of sophistication from relatively complex approaches representing diffusive fluxes of nutrients in an explicitly modeled sediment bed with oxic and anoxic layers, to a simple approach of applying a constant flux from or to the water column at the bed (Soetaert et al. 2000). While more complex nutrient modeling approaches hold the promise of improved generality and accuracy, they also require determination of additional site-specific model parameters (e.g., sediment characteristics) and inputs (e.g., boundary and initial conditions for additional analytes and compartments), which are often unavailable or highly uncertain. Furthermore, the computational cost of some coupled hydrodynamic-biogeochemical models is on the order of one week to predict nutrient concentrations over a year (Nuss et al. 2018), placing practical limits on exploration of the parameter space.

Researchers have applied various approaches to estimate the rates used in modeling studies. Nutrient transformation rates are most commonly quantified through field and laboratory studies that apply mass balance or 15N tracer approaches (Cornwell et al. 2014; Damashek et al. 2016; Kraus et al. 2017). Other studies capitalize on differential biological fractionation of natural-abundance isotopes to discriminate cycling processes (Dähnke et al. 2008; Wankel et al. 2009; Oakes et al. 2010; Glibert et al. 2019; Fackrell et al. 2022). In some cases, whole-stream amendments of isotopically distinct nutrients are used (Mulholland et al. 2004; Cohen et al. 2012; Glibert et al. 2019; Kim et al. 2021; Ledford et al. 2021). These studies provide constraints on net rates and sometimes on individual process rates. However, the limited spatial and temporal scales limited the ability to assess process rates over a diversity of aquatic landscapes and environmental conditions.

The goal of our study was to identify predominant biogeochemical processes affecting the distribution and speciation of dissolved inorganic nitrogen (DIN) between its two predominant forms, nitrate and ammonium. As described above, identifying these processes and estimating associated rates is challenging in modeling and field studies. Our approach overcomes the computational challenge associated with multi-dimensional coupled hydrodynamic-biogeochemical models by using transport information from a hydrodynamic model in a Lagrangian biogeochemical model to predict nitrate and ammonium concentrations. The approach combines the power of a well-calibrated three-dimensional hydrodynamic model to quantify transport processes with a computationally efficient Lagrangian DIN model. Our DIN model used tracer-based water age, shallow vegetated area (SVA) exposure time and depth exposure predicted by the hydrodynamic model across time and space, and contained rate parameters for (i) net transformation of ammonium to nitrate (e.g., nitrification), (ii) nitrate loss at the bed due to denitrification and other processes, (iii) ammonium flux from the bed to the water column, (iv) loss of ammonium in SVAs, and (v) loss of nitrate in SVAs. These SVAs are widely distributed in the Delta and contain submerged aquatic vegetation species, such as the invasive Brazilian waterweed, Egeria densa, and floating aquatic vegetation species including the invasive water hyacinth, Pontederia crassipes (Ta et al. 2017; Khanna et al. 2022). The quantitative effect of SVAs on DIN cycling in the Delta is unknown, but SVAs are biogeochemically complex and can be highly productive (Boyer et al. 2023).

The rate parameters in our DIN model were determined by an optimization algorithm and were based on high-frequency nitrate observations from a fixed-station monitoring network. Parameter values were validated to an independent dataset of broad-area synoptic surveys of ammonium and nitrate concentrations. Our study builds on Kraus et al. (2017) which estimated whole-system transformation rates using flow measurements to estimate travel times between two fixed stations along a channelized tidal reach of the Sacramento River. By incorporating a hydrodynamic model and tracer-based analysis, our approach can be applied to complex, branching tidal systems, and we demonstrate its use in the Sacramento-San Joaquin Delta.

Methods

Site Description

Our study area was the Sacramento-San Joaquin Delta (Fig. 1), the landward portion of San Francisco Estuary. The regional climate is Mediterranean with highly variable precipitation in winter and spring and consistently dry conditions during summer and autumn. The Sacramento River entering the northern boundary of the Delta contributes most of the inflow to the estuary. Numerous water-diversion facilities in the Delta remove some of that inflow, with the largest diversions at facilities of California’s State Water Project (SWP) and the Federal Central Valley Project (CVP) in the south Delta (Fig. 1). The difference between freshwater inflow to and precipitation within the Delta and the combined flow in all diversions is termed “net Delta outflow,” an index of the flow to the rest of the estuary from Suisun Bay to San Francisco Bay. This outflow is typically small during summer and fall. Flows throughout the study area are influenced by mixed semidiurnal tides with a mean range of 1.25 m at the mouth of the estuary. Salinity in summer and fall ranges from brackish water in Suisun Bay and the confluence of the Sacramento River and the San Joaquin River to fresh water in most of the Delta.

Fig. 1
figure 1

Study area and locations of US Geological Survey continuous nitrate monitoring stations, Sacramento Regional Wastewater Treatment Plant (SRWTP), Stockton Regional Wastewater Control Facility (Stockton RWCF) discharge, diversion points of the Central Valley Project (CVP) and State Water Project (SWP), Sacramento River Deep Water Ship Channel (DWSC), and other locations of interest. USGS stations included FPT (11447650), DWS (11455142), LCT (11455146), WLD (382006121401601), PCT (381424121405601), FRI (381614121415301,) LIB (11455315), DEC (11455478), and JPT(11337190) (U.S. Geological Survey 2023). Bathymetry data from Wang et al. (2018)

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The July–August 2018 simulation period was characterized by low river flows and relatively long residence times, with biological activity elevated by high temperatures and solar radiation (Kimmerer 2004). During summer, the Sacramento River contributed over 90% of the freshwater input to the study area. The largest summertime source of DIN to the Delta in 2018 was the Sacramento Regional Wastewater Treatment Plant (SRWTP; Fig. 1; Dahm et al. 2016), which discharges to the Sacramento River immediately below the U.S. Geological Survey (USGS) monitoring station number 11447650 at Freeport (U.S. Geological Survey 2023). Loads from other point discharges to the Delta (Fig. 1) were excluded from the simulation. In 2018, most of the nitrogen loading from the SRWTP took the form of ammonium (Kraus et al. 2017). Despite elevated levels of DIN and other nutrients from anthropogenic inputs to the study area, high turbidity and grazing rates have limited the occurrence of algal blooms and hypoxia (Jassby 2008; Kimmerer and Thompson 2014).

Shallow subtidal areas containing aquatic vegetation (SVAs) are widely distributed across the Delta. During summer, abundant submerged aquatic vegetation species in the Delta include the invasive species Egeria densa (Brazilian waterweed) and Hydrilla, and abundant floating aquatic vegetation species include invasive Pontederia crassipes (water hyacinth), and Ludwigia spp. (primrose-willow) (Ta et al. 2017). Other non-native and native vegetation species were present during this period (Ta et al. 2017). The observed relative abundance of species has also varied through time; during the 2012–2016 drought, invasive submerged aquatic vegetation extent in the Delta expanded substantially (Kimmerer et al. 2019). Recent aerial hyperspectral data show that submerged aquatic vegetation was more extensive than floating aquatic vegetation in surveys bracketing our study period (Khanna et al. 2022).

Observational Data

During the study period, nitrate plus nitrite (hereafter, nitrate because it is the quantitatively dominant species) concentrations were measured by the USGS every 15 min at a network of fixed location continuous monitoring stations (Fig. 1) using automated submersible ultraviolet nitrate analyzers (SUNA, Version 2; Satlantic, NS, Canada). Eight fixed stations were used for model calibration while the station located on the Sacramento River near Freeport was used for boundary conditions for our study (Fig. 1). USGS also collected discrete water samples approximately monthly at these stations which were sent in to the USGS National Water Quality Laboratory for determination of nitrate, nitrite and ammonium concentrations (Fishman 1993; Patton and Kryskalla 2011). To account for instrument offsets relative to laboratory-derived data, a correction was applied to the raw sensor data (Pellerin et al. 2016). The data are available in the USGS National Water Information System (US Geological Survey 2023).

During 24–26 July 2018, the USGS collected continuous data for surface ammonium and nitrate concentrations across a broad spatial extent of the Delta. Using a boat operating at speeds up to 13 m s−1, samples were taken at 1-s intervals using a flow-through system equipped with a SUNA nitrate analyzer and a Timberline ammonium analyzer (TL-2800, Timberline Instruments, Boulder, Colorado) (Bergamaschi et al. 2020). This dataset is referred to hereafter as a high-resolution mapping survey.

Overview of Modeling Approach

Our modeling approach was designed to allow rapid estimation of rates of key DIN transformation processes by decoupling the hydrodynamic simulations from the biogeochemical modeling. The Lagrangian biogeochemical modeling approach uses tracer-based information extracted from the hydrodynamic model. In this way a single computationally expensive hydrodynamic simulation provides the physical transport information shared by thousands of highly efficient biogeochemical simulations performed during optimization of rate parameters.

The modeling proceeded in five distinct steps. The first step was application of a three-dimensional hydrodynamic model to simulate a suite of tracers associated with water passing the location of the SRWTP discharge at Freeport, California (Fig. 1). These three-dimensional and time-varying tracer fields track the transport and exposure history of this water from the tracer insertion cross-section at Freeport. This cross-section serves as the “origin” for transport time scales and is the location of the boundary condition for tracer concentrations. The second step was to calculate the following information from the tracer fields: mean water age, representing transport time from Freeport (Fig. 1); fraction of time spent in SVAs since passing Freeport; and mean depth exposure, representing the mean water depth the tracer has encountered since passing Freeport. Depth-averaged values of each of these properties were calculated from the tracer fields at the times and locations of nitrate observations in the estuary. The third step was to use these tracer-derived values associated with mean age and property experience of the tracer in a biogeochemical model to predict nitrate and ammonium concentrations at times and locations corresponding to observations. Model parameters were iteratively adjusted to maximize agreement with a calibration subset of the nitrate data. The fourth step was validation of nitrate and ammonium predictions with an independent dataset. The fifth step was analysis of model predictions to quantify relative contributions of each term in the biogeochemical model to nitrate and ammonium concentrations as freshwater travels through the study area.

The biogeochemical model applied in the third and fourth steps included two sources of ammonium, the SRWTP discharge to the Sacramento River (Fig. 1), and a spatially diffuse constant source at the bed which represented the net effect of remineralization and other benthic processes. A constant, first-order rate coefficient parameterized the net conversion of dissolved ammonium to nitrate via nitrification or other pathways. An additional loss of ammonium was applied in SVAs. Nitrate loss terms represented losses to the bed and additional losses in SVAs. All rate coefficients estimated in this study are net coefficients which may include contributions from several biogeochemical processes.

The model calibration step used an optimization algorithm for estimating biogeochemical rate parameters to provide nitrate predictions most consistent with observations at fixed stations. In the validation step, the model was applied with the optimization-derived parameters and predicted nitrate and ammonium concentrations were compared to data from the July 2018 high-resolution mapping surveys (Bergamaschi et al. 2020).

Hydrodynamic Model

Hydrodynamics were simulated using the three-dimensional Resource Management Associates (RMA) UnTRIM San Francisco Estuary Model (Andrews et al. 2017; Gross et al. 2019). UnTRIM solves the discretized Reynolds-averaged shallow water equations on an unstructured grid and allows for wetting and drying of computation cells and a sub-grid-scale representation of bathymetry (Casulli and Stelling 2011). Vertical turbulent mixing in the model was parameterized using a k-ε closure, which solves one equation for turbulent kinetic energy (k) and another for turbulent dissipation (ε) using published parameter values (Warner et al. 2005). Bed friction was parameterized using a quadratic stress formula and bed roughness height, zo.

The model domain extended from the coastal ocean through the estuary, including the Delta (Fig. 2). Cell side lengths ranged from less than 5 m to more than 1000 m, and 1-m layer spacing was used in the vertical. Bathymetry was specified using digital elevation models and sounding data from several bathymetric surveys. The hydrodynamic simulation period was April 1, 2018, to September 1, 2018, a period of low Delta outflow during a below-normal water year in the Sacramento and San Joaquin Valleys (California Department of Water Resources 2024). The model included a number of inflows and diversions (Fig. 2) and accurately predicted water level, flow, salinity and temperature at over 50 stations through the estuary during the DIN species simulation period (Resource Management Associates 2021).

Fig. 2
figure 2

RMA UnTRIM San Francisco Estuary model grid bathymetry and locations of primary flow inputs and diversions. East Bay Municipal Utility District (EBMUD), East Bay Discharge Authority (EBDA), San Jose-Santa Clara (SJSC)

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Classification of Shallow Vegetated Areas

The exposure of Sacramento River water to SVAs during transport was calculated by the three-dimensional hydrodynamic model. Identification of SVAs in the model was derived from estimated distributions of submerged and floating aquatic vegetation based on airborne hyperspectral imagery (Khanna et al. 2022). Imagery data were from a 2018 survey when available, but two major swaths missing from the 2018 data were filled using 2020 classification data. Each pixel (approximately 2 m by 2 m) was assigned a classification indicating whether it contained vegetation or not. Most of the cells of the computational grid were larger than the pixel size, so we assigned an SVA fraction to each cell by calculating the fraction of the cell covered by submerged aquatic vegetation or floating aquatic vegetation pixels (Fig. 3). Drag associated with this vegetation was not modeled in the hydrodynamic simulation.

Fig. 3
figure 3

Shallow vegetated areas (SVAs) inferred from the coverage of submerged and floating aquatic vegetation, interpolated onto the model grid. Outlines denote which data came from 2018 aerial surveys versus 2020 surveys

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Water Age

To estimate the mean age of a source of water, we simulated the transport of two conservative tracers in the hydrodynamic model following the widely used Constituent-oriented Age and Residence Time (CART) theory and approach (Deleersnijder et al. 2001). The first tracer marked (“tagged”) water entering the study area as it passed Freeport (Fig. 1) and was calculated using a three-dimensional advection diffusion equation

$$\frac{\partial C}{\partial t}+\nabla \cdot ({\varvec{u}}C)=\frac{\partial }{\partial z}({K}_{T}\frac{\partial C}{\partial z})$$
(1)

where \(C\) is the tracer concentration with dimensions of mass per volume, \({\varvec{u}}\) is a three-dimensional velocity vector with dimensions length per time, and \({K}_{T}\) is the vertical eddy diffusivity with dimensions length squared per time. Horizontal eddy diffusion was neglected because horizontal numerical diffusion is likely at least as large as turbulent eddy diffusivity.

This equation was discretized with a conservative finite-volume approach (Casulli and Zanolli 2005). The discretized equation can be summarized as

$${C}^{n+1}=\mathcal{A}\left({C}^{n}\right)$$
(2)

where \(\mathcal{A}\) represents a discrete advection–diffusion operator, and superscripts denote time steps.

The second tracer that we tracked was “age-concentration” (Deleersnijder et al. 2001). The governing equation of age-concentration is

$$\frac{\partial \alpha }{\partial t}+\nabla \cdot \left({\varvec{u}}\alpha \right)=\frac{\partial }{\partial z}\left({K}_{T}\frac{\partial \alpha }{\partial z}\right)+C$$
(3)

where \(\alpha\) is the age-concentration with dimensions time-mass per volume. Its discretized form is

$${\alpha }^{n+1}=\mathcal{A}\left({\alpha }^{n}\right)+\Delta t{C}^{n}$$
(4)

where \(\Delta t\) is the computational time step. The mean water age was then estimated as the ratio of the age-concentration to the tracer concentration,

$$a= \frac{\alpha }{C}$$
(5)

where a has dimensions of time.

In our application, the concentration \(C\) in each cell and time step represents the portion of water at that location that passed Freeport on the Sacramento River during the simulation period, and a is the mean age of that water, quantifying mean time elapsed since passing Freeport. This is referred to as the “mean age” because the age calculated at any cell and any time step represents the algebraic mean of the age of source water parcels present at that time and location (Deleersnijder et al. 2001). The initial conditions of the scalar transport equations were zero concentration of each tracer throughout the domain. The boundary conditions of \(C\) were zero at all boundaries. The tracer concentration in water passing Freeport (at the SRWTP discharge location) was set to 1 at each time step (effectively as an internal boundary condition). The boundary conditions of \(\alpha\) were zero at all boundaries and it was internally reset to zero at Freeport. The tracer simulation was performed from April 1, 2018 through September 1, 2018, with April–June serving as a “spin-up” period during which the tracer introduced at Freeport spread through the northern estuary, prior to the DIN species prediction period of July–August 2018. After concentration and age-concentration were predicted throughout the domain and simulation period, the mean age (\(a\)) was then calculated in a post-processing analysis.

The exposure time tracer, referred to as partial age-concentration in Mouchet et al. (2016), for region or spatial compartment j was calculated as

$${\alpha }_{j}^{n+1}=\mathcal{A}\left({\alpha }_{j}^{n}\right)+{\delta }_{j}\Delta t{C}^{n}$$
(6)

where \({\delta }_{j}\) was the proportion of each cell that is in compartment j. The exposure time (partial age) for compartment j at time step n is then

$${a}_{j}=\frac{{\alpha }_{j}}{{C}}$$
(7)

In our application, the exposure time tracer tracked the accumulated time spent in SVAs (Fig. 3) since being introduced at Freeport. The same boundary conditions described above for age-concentration were applied for the exposure time tracer. Many cells were partially in SVAs, so the approach accounted for the proportion of the cell area in SVAs using a fractional \({\delta }_{j}\).

Property Exposure

We also used a “property tracking” approach (Gross et al. 2019, 2024; Deleersnijder 2001) to estimate the mean depth exposure of the tracer. In this approach, the property-age-concentration is governed by the equation

$$\frac{\partial \beta }{\partial t}+\nabla \cdot \left({\varvec{u}}\beta \right)=\frac{\partial }{\partial z}\left({K}_{T}\frac{\partial \beta }{\partial z}\right)+ \psi C$$
(8)

where \(\beta\) is the property-age-concentration and \(\psi\) is the instantaneous value of the property, which is water depth in this application. The discretized form of property-age-concentration is

$${\beta }^{n+1}=\mathcal{A}\left({\beta }^{n}\right)+\Delta t{\psi }^{n}{C}^{n}.$$
(9)

Then the mean property exposure by the tracer was estimated as the ratio of the property-age-concentration to the age-concentration.

$$b= \frac{\beta }{\alpha }$$
(10)

where b is the mean property exposure by the tracer. The mean property estimated by Eq. 10 in our application is mean depth exposure (\(H\)) of the tracer from entry at Freeport to a subsequent location. The initial condition of the depth-age-concentration (\(\beta )\) was zero throughout the domain. The boundary conditions of \(\beta\) were zero. We note that this method cannot recover the full history of property exposure, only aggregate measures such as the mean property exposure.

Biogeochemical Approach

We used a Lagrangian biogeochemical model driven by tracer results from the hydrodynamic model to simulate the evolution of nitrate and ammonium concentrations as Sacramento River water advects and disperses downstream of Freeport (Fig. 4). The specific processes included in our model were selected by preliminary exploration of alternative biogeochemical formulations with a range of complexity. We concluded that a parsimonious formulation for this period should include the following terms:

  1. 1.

    First-order nitrification

  2. 2.

    Zero-order ammonium source at the bed due to mineralization and other processes

  3. 3.

    First-order loss of nitrate at the bed due to denitrification and other processes

  4. 4.

    First-order loss of ammonium in SVAs

  5. 5.

    First-order loss of nitrate in SVAs

Fig. 4
figure 4

Model inputs. A) observed flow in the Sacramento River at Freeport and SRWTP discharge flow; B) observed NO3 in discharge, calculated flow weighted cross-sectional averaged NO3 downstream of discharge, and low-pass filtered NO3 used as model input data; C) observed NH4 in discharge, calculated flow weighted cross-sectional averaged NH4 downstream of discharge, and low-pass filtered NH4 used as model input data. Values below the y-axis in panels A and B correspond to periods of zero flow from the SRWTP (discharge “holds”)

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Each of these terms could include contributions from multiple biogeochemical processes. Furthermore, processes which we represent only as bed terms may also be active in the water column. However, including both a bed and water column source of ammonium from mineralization was not found to influence accuracy in preliminary modeling work.

Changes to ammonium concentration via these processes were represented by the equation

$$\frac{d[{{\text{NH}}_{4}]}_{t,{\varvec{x}}}}{da}=-\left(k+{{f}_{sva}k}_{sva,am}\right)[{{\text{NH}}_{4}]}_{t,{\varvec{x}}} +\frac{{k}_{bed, am}}{H}$$
(11)

where \([{{\text{NH}}_{4}]}_{t,{\varvec{x}}}\) is the depth-averaged molar concentration of ammonium, \(t\) is time, \({\varvec{x}}\) is a horizontal position, \(a\) is mean water age, \({f}_{sva}\) (\({a}_{j}/a\)) is the portion of time the tracer spent in SVAs, \(k\) is the first-order nitrification rate, \({k}_{bed,am}\) is a zero-order net bed source rate of ammonium, \({k}_{sva,am}\) is a first-order ammonium loss rate applied to the time that the tracer spent in SVAs. The ordinary differential equation (Eq. 11) expresses change with mean water age, not with time at a fixed location, and therefore applies to transformations in a moving water mass, not at a fixed location.

The ammonium that was nitrified was assumed to be converted entirely to nitrate. Loss terms for nitrate were a first-order loss at the bed and an additional first-order loss in SVAs

$$\frac{d[{{\text{NO}}_{3}]}_{t,{\varvec{x}}}}{da}=k[{{\text{NH}}_{4}]}_{t,{\varvec{x}}} - \left(\frac{{k}_{bed,ni}}{H}+ {{f}_{sva}k}_{sva,ni}\right)[{{\text{NO}}_{3}]}_{t,{\varvec{x}}}$$
(12)

where \([{\text{NO}}_{3}]\) is the depth-averaged molar concentration of nitrate, \({k}_{sva,ni}\) is a first-order nitrate loss rate applied to the time that the tracer spent in SVAs (Fig. 3) and \({k}_{bed,ni}\) is the loss rate of nitrate at the bed.

All coefficients represent the net effect of multiple processes. For example, \({k}_{bed,am}\) and \({k}_{bed,ni}\) represent the net effect of remineralization processes, denitrification, dissimilatory nitrate reduction to ammonium (DNRA), and anammox processes that take place at and near the sediment–water interface (Damashek and Francis 2018). These bed terms are formulated as a mass flux per unit area, whereas a water column process would be a mass flux per unit volume. However, because we do not have separate terms in Eqs. 11 and 12 to represent biogeochemical processes within the water column, the coefficients \({k}_{bed,am}\) and \({k}_{bed,ni}\) also represent contributions from remineralization and other processes that occur within the water column. Furthermore, we make a substantial simplifying assumption that the parameters \({f}_{sva}\) and \(H\) can be replaced by their respective average values along the Lagrangian trajectory and considered as constants during the time integration of the ordinary differential equations.

The system of linear ordinary differential equations constituted by Eqs. 11 and 12 then permits an analytical solution by first solving for ammonium in Eq. 11, then substituting the solution into Eq. 12 and solving for nitrate. The resulting solution, confirmed by multiple computer algebra systems, is

$$[{{\text{NH}}_{4}]}_{t,{\varvec{x}}}={\left({A}_{0}-\frac{{s}_{a}}{{D}_{1}}\right)e}^{-{D}_{1}a}+ \frac{{s}_{a}}{{D}_{1}}$$
(13)
$$[{{\text{NO}}_{3}]}_{t,{\varvec{x}}}= \frac{k\left({A}_{0}- \frac{{s}_{a}}{{D}_{1}}\right)}{{D}_{2}-{D}_{1}}{e}^{-{D}_{1}a}+ \frac{{A}_{0}{D}_{2}k+{N}_{0}{D}_{1}{D}_{2}-{N}_{0}{D}_{2}^{2}-k{s}_{a}}{{D}_{2}\left({D}_{1}-{D}_{2}\right)}{e}^{-{D}_{2}a}+\frac{{ks}_{a}}{{D}_{1}{D}_{2}}$$
(14)
$${D}_{1}= k+{{f}_{sva}k}_{sva,am}$$
(15)
$${D}_{2}={{\frac{{k}_{bed, ni}}{H}+f}_{sva}k}_{sva,ni}$$
(16)
$${s}_{a}=\frac{{k}_{bed, am}}{H}$$
(17)

where \({A}_{0}=[{{\text{NH}}_{4}]}_{t-a,{{\varvec{x}}}_{0}}\) and \({N}_{0}=[{{\text{NO}}_{3}]}_{t-a,{{\varvec{x}}}_{0}}\) are ammonium and nitrate concentrations at time \(t-a\) and location \({{\varvec{x}}}_{0}\) which represents Freeport. \({D}_{1}\), \({D}_{2}\), and \({s}_{a}\) are temporary variables used to simplify Eqs. 13 and 14. Conceptually these equations represent transformations of nitrate and ammonium in a water parcel over the travel time (mean water age) from Freeport to an observation time and location.

The inputs to the analytical solution included; 1) estimated cross-sectionally averaged low-pass filtered nitrate (\({N}_{0}\)) and ammonium (\({A}_{0}\)) concentrations at Freeport (Fig. 1); 2) depth-averaged, temporally and spatially varying mean water age (\(a\)), fractional exposure to SVAs (\({f}_{sva}\)) and mean depth exposure (\(H\)) in the tracer simulations; 3) five constant biogeochemical rate parameters (\(k, {k}_{bed, am}, {k}_{bed, ni},{k}_{sva,am},{k}_{sva,ni}\)) that were estimated by fitting to the calibration data.

\({N}_{0}\) was calculated by first performing a flow-weighted average of nitrate in the Sacramento River upstream of the SRWTP discharge and nitrate in the SRWTP discharge (Fig. 4C). This flow-weighted average was then filtered by a 4th order Butterworth filter with a 7-day cutoff period to remove variations in concentration due to tides and short-term changes in operations at the SRWTP (Fig. 4C), This filtering approximated the effect of small-scale longitudinal dispersion which was otherwise not accounted for in Eqs. 11 and 12. We assumed uniform nitrate concentration across the channel at Freeport. The ammonium concentration at Freeport (\({A}_{0})\) was calculated in the same manner as nitrate (Fig. 4B).

To quantify the contributions of terms in Eqs. 11 and 12, the contribution from each individual term was integrated and stored separately using the rate coefficients found in the fitting approach. The numerical integration was carried out with the odeint method from the scipy python module (Virtanen et al. 2020). For the closed-form solution (Eqs. 13–17) and integration of the ordinary differential equation, each predicted nitrate value is an independent calculation and is only computed at times and locations of observations.

Fitting Approach

The five rate parameters in Eqs. 13–17 were simultaneously fit to the continuous USGS fixed-station monitoring data at 8 locations (Fig. 1). The fitting procedure was formulated as a optimization problem to minimize Root Mean Square Deviation (RMSD) in comparison to the fixed station observations, using the scipy implementation of differential evolution (Virtanen et al. 2020) for robust global optimization. Note that there were no station-specific parameters, only global parameters. The model performance metrics used were bias, RMSD, correlation coefficient and a Willmott Skill score (Willmott 1981), for which a maximum value of 1 indicates perfect agreement.

After the rate parameters were estimated by optimization to the station data, the model (Eqs. 13–17) was validated by comparing predicted constituent concentrations to the high-resolution mapping data collected in July 2018. Data used for validation were taken only from locations where at least 50% of the water had passed Freeport on the Sacramento River during the simulation period. This screening did not remove many data from the mapping dataset because the Sacramento River provided over 90% of the inflow to the Delta during these dry summer conditions, compared to ~ 3% from the San Joaquin River. Tracer-derived predictions at the exact time and location of each high-resolution mapping observation were used in the analytical solution (Eqs. 13–17) with parameters previously determined through the fixed station model fitting to predict DIN concentration and speciation at those times and locations.

Results

Tracer Age

The hydrodynamic simulation generated three-dimensional time-varying fields of the four tracer-based properties across the hydrodynamic model domain (Fig. 2) and the July–August 2018 simulation period. As an example for visualization, depth-averaged values of each of these fields were calculated for noon on July 25th, 115 days after the start of the hydrodynamic simulation (Fig. 5). As expected, throughout much of the northern and central Delta, the predicted concentration (\(C\) in Eq. 1) of tracer for the fraction of water that had passed Freeport (“Sacramento River fraction”) had values very near 1 (i.e., 100% Sacramento River water). Values decreased gradually with distance from Freeport, particularly in portions of the south Delta closer to inflows from the San Joaquin River. In the long dead-end Sacramento River Deep Water Ship Channel, a portion of the water present prior to the model run start time (April 1, 2018) had not yet been replaced by “new” Sacramento River water that had flowed past Freeport before July 25th, resulting in Sacramento River fraction values of less than 0.5 (Fig. 5A), though water in that channel is primarily supplied by the Sacramento River.

Fig. 5
figure 5

Snapshot on July 25, 2018, at noon local standard time, of depth-averaged tracer-based properties. A The fraction of water that has passed Freeport during the simulation period (“Sacramento River fraction”). B The mean time since the Sacramento River tracer entered the model at Freeport (“mean water age”). C Fraction of time that the Sacramento River tracer has spent in shallow vegetated areas (“fraction of time in SVAs”). D mean depth exposure of Sacramento River tracer since entering at Freeport (“mean depth exposure”)

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The mean water age distribution visualized in Fig. 5b represents the average transport time required for water passing Freeport to reach any location in the model domain on July 25th at noon. The mean water age ranged from less than a day along a portion of the Sacramento River to over 60 days in portions of the Cache Slough Complex (Fig. 5b). The predicted mean water age cannot exceed the simulation period.

The fraction of time in SVAs (Fig. 5c) is the mean accumulated proportion of time that tracer has spent in shallow vegetated areas since entering at the SRWTP (Fig. 3). Generally, the fraction of time spent in SVAs is < 0.1 except for notably higher values of 0.3 to 0.4 in and near SVAs (Fig. 3) and with slow flushing, including portions of the Cache Slough Complex and Franks Tract State Recreation Area (Fig. 1).

The mean depth exposure (Fig. 5d) represents the mean depth encountered by the tracer during its transit from Freeport. The mean depth exposure (Fig. 5d) was often quite different from local depth (Fig. 2), particularly in shallow areas because the tracer present in those areas spent substantial time in deep channels during transit from Freeport.

All four of these estimated tracer-based quantities are time variable because the tracers are advected by mean and tidal flows. Thus, while Fig. 5 shows model output for a specific time (July 25, 2018, at noon), the model generated these data at a one-minute time step and saved output to a netCDF format file at a 30-min time interval. The ranges of values of each tracer-based property during the simulation period are summarized for fixed stations in Fig. 6. The Sacramento River fraction is not shown because it is near 1 at all eight stations. The fraction of time in SVAs is small at most stations but higher and tidally variable in some Cache Slough Complex stations, including LCT, PCT, FRI and LIB (Fig. 6b). The mean depth exposure was typically higher at deeper stations such as DWS in the Sacramento River Deep Water Ship Channel (Fig. 1) and lower at stations adjacent to shallow and marsh areas, such as LCT in the (Fig. 6c) Cache Slough Complex (Fig. 1) but varied among stations and times only by a factor of 2.

Fig. 6
figure 6

Box plots showing tracer-based quantities generated for fixed stations throughout the July–August simulation period. Station locations in Fig. 1. A Mean water age. B Fraction of time in SVAs. C Mean depth exposure. Boxes indicate interquartile range, horizontal line indicates median, and whiskers indicate range of 95% of predictions. For station locations see Fig. 1

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Nitrate and Ammonium Observations and Model Calibration

Using the predicted tracer-based properties from the hydrodynamic model and nitrate concentration data from fixed stations, we derived the best-fit values for the five biogeochemical rate parameters listed in Table 1. Loss of nitrate associated with SVA exposure was effectively weighted by the time the tracer spent in SVAs (Fig. 5c). In SVAs, this loss term was the dominant (fastest) process. Although only nitrate data were used in calibration, the ammonium loss in SVAs affects nitrate predictions by reducing the pool of ammonium available for nitrification.

Table 1 Best-fit rates for calibration to fixed station nitrate observations
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Model predictions of nitrate using the best-fit rates (Table 1) were compared with data from the eight USGS fixed stations (Fig. 7, Table 2). The correlation coefficient (R) of a linear regression of simulated and observed nitrate was 0.957. Observed nitrate concentrations ranged from consistently below 2 µmol L−1 at LCT and WLD in the northern Cache Slough Complex (Fig. 1) to typically above 10 µmol L−1 at DEC in the western Delta (Fig. 1). These spatial patterns were predicted well by the model (Fig. 7; Table 2). The range of predicted nitrate at each station is generally consistent with observations (Fig. 7). Some of the observed tidal variability was predicted (Fig. 8). The model predicted the observed tidal variability of nitrate best at stations PCT, DEC, and LIB (Fig. 8; Table 2) spanning a large geographical extent (Fig. 1). Nitrate was generally overpredicted at FRI and LCT (Fig. 8), both located in the Cache Slough Complex. All of the observed data during the simulation period at LCT were below the method detection limit of 1.0 µmol L−1, which was set to half the instrument detection limit (Fig. 7).

Fig. 7
figure 7

Median, interquartile range (boxes) and 5th and 95th percentiles (whiskers) of observed and predicted nitrate at 8 USGS monitoring stations (see Fig. 1) during the simulation period of July and August of 2018

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Table 2 Model calibration metrics for observed versus predicted nitrate concentrations at specific USGS continuous monitoring stations and nitrate, ammonium, and DIN from high-resolution mapping measurements, including bias, root-mean-square deviation (RMSD), correlation coefficient and a Willmott skill score. Correlation coefficient and Willmott skill score could not be computed at LCT because all observations at that location were below the method detection limit
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Fig. 8
figure 8

Observed (black) and predicted (green) nitrate concentrations at the seven USGS continuous monitoring stations in the study area with data in the last two weeks of the simulation period, corresponding to the period of maximal data availability. For station locations see Fig. 1. Data were not available at WLD during the period shown

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Model Validation

The observed spatial patterns of ammonium, nitrate, and DIN were all represented well by the model (Fig. 9). Some of the largest errors in nitrate prediction were in regions with other water sources, including the San Joaquin River. Nitrate was substantially underpredicted near Stockton, likely due to our omission of the substantial nitrate loading from the Stockton RWCF, which discharges DIN mostly in the form of nitrate (Dahm et al. 2016). The model (Eqs. 13–17) reproduced key spatial patterns in the mapping data, including the strong decreases in nitrate from south to north in the Cache Slough Complex and from north to south across Franks Tract State Recreation Area (Fig. 9C).

Fig. 9
figure 9

Observed (outer circles) and predicted (inner circles) ammonium (A), nitrate (B), and DIN (C) at the locations and times of high-resolution mapping measurements in July 2018. Observations and predictions are not compared for data associated with Sacramento River fraction less than 0.5

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Ammonium data were not used in the parameter fitting by differential evolution (Storn and Price 1997) during model calibration but were included in the validation (Fig. 9a), and ammonium prediction performance metrics were as good as those for nitrate (Table 2). Because the SRWTP was the only source of ammonium in the model, predicted ammonium decreased with mean water age due to nitrification and with fraction of time in SVAs. Broad spatial patterns in observed ammonium (Fig. 9a) were predicted well including gradients in predicted ammonium in Franks Tract State Recreation Area which resulted primarily from the net loss of ammonium in SVAs due to the higher rate of loss in SVAs (0.40 d−1) relative to the nitrification rate (0.13 d−1). Predicted DIN concentrations, which are not affected by nitrification, also decreased across Franks Tract State Recreation Area and other SVAs because of the ammonium and nitrate loss terms in SVAs (Fig. 9), and generally agreed well with observations. The most notable errors in predicted ammonium and DIN were close to the SRWTP discharge where strong tidal (low-pass) filtering of ammonium to calculate the ammonium “boundary condition” (\({A}_{0}\)) at this location (Fig. 4b), removed the effect of the tidally variable loading from the SRWTP (Fig. 4).

Reaction Terms Influencing Nitrate and Ammonium Concentrations

The solutions of the ordinary differential Eqs. 11 and 12 by numerical integration resulted in the same predicted nitrate and ammonium concentrations as the closed-form solution and allowed quantification of contributions of individual terms to these predicted concentrations. While closed-form solutions for the individual contributions exist, a numerical approach was used here to verify the results from the analytical integration and simplify the diagnostic analysis for which computational expense was negligible. We characterized the relative importance of transformation processes at fixed monitoring sites (Fig. 10) and at the times and locations of the high-resolution mapping survey (Fig. 9). Because the ammonium bed flux rate was estimated to be zero (Table 1), the SRWTP served as the only source of ammonium. Ammonium losses at fixed stations occurred primarily due to nitrification, with smaller losses in SVAs (Fig. 10a). Nitrate concentrations entering at the SRWTP were less than 3 µmol L−1 (Fig. 4) and nitrification acted as a substantial nitrate source (Fig. 10b). Nitrate losses occurred in SVAs and due to bed flux (e.g., denitrification), with time spent in SVAs accounting for the largest share of the loss at most of the fixed station sites (Fig. 10b). Throughout large portions of the system, time spent in SVAs accounted for most DIN loss (Fig. 11c) with a relatively minor contribution from bed flux (Fig. 11d).

Fig. 10
figure 10

Contributions to predicted change from incoming concentrations on the Sacramento River downstream of the SRWTP due to individual processes for ammonium (A) and nitrate (B). Positive values indicate a gain and negative values a loss. Colored bars show change associated with individual processes and horizontal black lines are net change. For station locations see Fig. 1

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Fig. 11
figure 11

Model predictions at the time of the high-resolution mapping. A Predicted DIN concentration. B Nitrification of ammonium to nitrate. C Loss of DIN in shallow vegetated areas. D Flux of nitrate to the bed

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Discussion

Evaluation

Lagrangian modeling approaches such as the Streeter-Phelps equation have been extensively applied in rivers (Schnoor 1996) where the relevant age is the advective travel time from an upstream location to a downstream location. In contrast, estuaries, with their complex geometry and oscillatory flow, have multiple pathways and transport times from a starting location to a seaward (“downstream”) location. The calculated mean water age estimates the arithmetic mean of the transport times of water particles (Lucas and Deleersnijder 2020). While only the mean transport time is represented by mean water age, age-based Lagrangian approaches enable highly efficient, though approximate, modeling of biogeochemical processes in estuaries (Wang et al. 2019). Like the Streeter-Phelps equation, the concept of these approaches is to apply biogeochemical transformations from a known starting concentration over a known travel time. We have extended these simpler Lagrangian approaches that only track transport time (age) by also tracking mean depth exposure and mean proportion of time spent in a defined areas. When expressed in Lagrangian form and using a single representative depth and average fraction of time in shallow vegetated areas (SVAs), the biogeochemical equations of DIN cycling permit a closed-form solution that can be solved efficiently for any time and location in the model domain using the tracer-based properties. Our Lagrangian model is evaluated only at the times and locations where observations are available, which is different from frequently used Eulerian models in which concentrations are determined at all computational points in the domain at each time step.

The Lagrangian biogeochemical model has a computational time of 12 ms on a laptop computer. This computational efficiency enables rapid exploration of multiple model formulations and optimization of model parameters. In our application, the computational efficiency stems not only from the closed-form solution but also from computing nitrate and ammonium concentrations only at the times and locations of observations instead of on a fixed grid and time step. More complex formulations involving feedbacks among several biogeochemical constituents can also be explored in the same Lagrangian framework using an ordinary differential equation solver at a large time step resulting in a moderate increase in computational cost relative to the closed-form solution.

The application of the tracer-based Lagrangian approach to predict DIN dynamics in the Delta targets a summer and fall study period in 2018 when the dominant freshwater source to the system was the Sacramento River. During this period, the SRWTP was the dominant source of DIN, primarily in the form of ammonium. The model indicates that a large fraction of the ammonium was converted to nitrate by nitrification during transport, consistent with results of a previous study (Kraus et al. 2017). The tracer-based mean water age representing transport time from Freeport was generally highest in the northern portion of the Cache Slough Complex (over 50 days in the SDWSC, Fig. 1). Although the tracer moved through the Sacramento River primarily through advection by net flows, tidal dispersion was responsible for most of the tracer transport to the north into the Cache Slough Complex, where net flows were minimal during the study period.

Our tracer-driven biogeochemical model accurately predicted the observed spatial distribution of nitrate concentration (Fig. 7). The model also captured a substantial part of the observed tidal variability of nitrate at fixed stations (Fig. 8) with better agreement at stations having higher average nitrate concentrations. The limited accuracy of predicted tidal variability at some stations may be partially due to model assumptions. For example, predictive ability may be limited by applying the same loss rate in any shallow vegetated area (SVA), independent of the vegetation type and density, and neglecting other environmental drivers, such as temperature effects, on reaction rates. In addition, use of a single mean depth exposure instead of a time history of depth exposure is a likely source of error. Nevertheless, the model predictions matched broad patterns observed in high-resolution mapping for nitrate concentration, as well as for ammonium concentration, which was not used to estimate the model parameters (Table 1; Fig. 9). In addition, the predictions match observed spatial patterns of DIN constituents far from stations used to calibrated the model, demonstrating broad utility of estimated transformation rates.

Nitrogen Sources, Transformation and Sinks

The findings of our study aligned well with existing literature. First, as expected, the Sacramento River was the dominant source of water to most of the Delta throughout our simulation period of July and August 2018 (Fig. 5A). The model accurately predicted observed data without considering additional point sources (the model allowed for a bed-source term, but the term was not active due to a best-fit value of zero for ammonium, Table 1). This demonstration of model accuracy supports model results and previous nutrient loading estimates indicating that the SRWTP and upstream sources on the Sacramento River are the dominant sources of ammonium to the Delta (Jassby 2008; Novick et al. 2015; Saleh and Domagalski 2021).

Our modeling results also indicated that the primary source of nitrate for most of the study domain was nitrification of ammonium from the SRWTP discharge. This is consistent with the conclusions of studies conducted in the northern portion of the study area (Kraus et al. 2017; Fackrell et al. 2022) but the modeling results extend this conclusion through most of the study area. One exception is a portion of the Delta where a smaller waste water treatment plant (WWTP) (Stockton RWCF; Fig. 1) was a clear source of nitrate not represented by the model, resulting in locally underpredicted concentrations (Fig. 9). However, the influence of the Stockton RWCF was spatially limited by small net flows from the San Joaquin River (Fig. 1) during the study period.

The estimated rate constants (Table 1) were also generally consistent with prior estimates. The model determined a first-order nitrification rate constant (k) of 0.13 d−1 (Table 1). Assuming ammonium concentrations of 10 to 50 µ mol L−1 (Fig. 3), the 0.13 d−1 model-derived rate equates to 1.3 to 6.5 µmol L−1 d−1, which aligns with zero-order nitrification rates previously measured for the Sacramento River that ranged from 1.5 to 6.4 µmol L−1 d−1 (Parker et al. 2012; Kraus et al. 2017).

The estimated DIN bed fluxes were small relative to other terms in the model (Fig. 10) and within the range of prior estimates (Cornwell et al. 2014). These terms represent processes such as remineralization, DNRA, denitrification and algal uptake, which can be active at the bed and in the water column. Because our formulation did not have a distinct term representing these processes in the water column, the modeled bed terms represent the net effect of benthic and pelagic processes. When analogous water column transformation terms were included in the parameter fitting, they were estimated to be zero (shown in supplement). The estimated \({k}_{bed, am}\) of zero (Table 1) corresponds to no net contribution of ammonium from the bed during the study period. This term was constrained to be non-negative in our parameter fitting. Cornwell et al. (2014) reported variable bed fluxes of ammonium, with mostly positive fluxes, but negative fluxes reported at some locations and light conditions. In contrast, the estimated \({k}_{bed, ni}\) was 0.068 m d−1, indicating a net loss of nitrate to the bed. For a representative nitrate concentration range of 10 to 20 µmol L−1, this corresponds to a loss rate ranging from 0.68 to 1.36 mmol m−2 d−1, consistent with reported estimates of denitrification in the San Francisco Estuary ranging from 0.6 to 1.0 mmol m−2 d−1 (Cornwell et al. 2014). Studies that measured benthic nutrient fluxes in the estuary report a wide range of values, suggesting the benthos can be either a source or sink for ammonium and nitrate, with differences in rates of remineralization, DNRA, uptake and denitrification among locations and seasons likely contributing to this variability (Kuwabara et al. 2009; Cornwell et al. 2014).

The estimated DIN losses associated with SVAs were much greater than estimated bed fluxes (Fig. 10), despite tracer residing in SVAs for less than 25% of the transit time from Freeport to each fixed station (Fig. 6). Rate constants for ammonium (\({k}_{sva,am}\)= 0.40 d−1) and nitrate (\({k}_{sva,ni}\)= 0.41 d−1) losses in SVAs were higher than the nitrification rate (Table 1) (Fig. 11). The dominant role of DIN loss in SVAs is consistent with observed DIN in the high-resolution mapping (Fig. 9c). Observations show sharp decreases in DIN at Franks Tract State Recreation Area and with distance north in the Cache Slough Complex, both of which contain SVAs (Fig. 3).

SVAs are complex environments that facilitate several physical and biogeochemical processes influencing DIN species concentrations. The presence of aquatic vegetation in these regions can impede flow and decrease velocities, which, in turn, increases residence time, settling of sediment (Lacy et al. 2021), and light penetration. Because phytoplankton production in the Delta is generally light-limited (Jassby 2008), increased water clarity permits greater algal productivity and associated DIN demand. Aquatic vegetation can also support populations of periphyton and macroalgae serving as additional sinks for DIN. Furthermore, this elevated productivity may provide the labile carbon needed to fuel bacteria responsible for denitrification (Cornwell et al. 2014; Damashek and Francis 2018).

Productivity of the aquatic vegetation may also place a demand on DIN. While rooted vegetation may take up nutrients from sediment (Novick et al. 2015; Boyer and Sutula 2016), Egeria densa, a species of submerged aquatic vegetation that is widely distributed through the Delta (Ta et al. 2017; Khanna et al. 2022), has been shown to take up most of its nitrogen from the water column (Feijóo 2002). Carbon uptake by Egeria Densa is estimated to be 464 g C m−2 y−1 (Boyer et al. 2023). Assuming a carbon to nitrate ratio of 11.1 (Cloern et al. 2002), this corresponds to a nitrate uptake of 41.8 g N m−2 y−1. We further assume a representative water depth in SVAs of 2 m, a representative DIN concentration in SVAs of 20 µmol L−1 (Fig. 9c), and a growing season of 5 months. The first-order loss formulation in Eqs. 11 and 12 then imply an annual DIN uptake of 34 g N m−2 y−1, roughly consistent with the carbon uptake estimate of Boyer et al. (2023). This rough calculation indicates that uptake by vegetation growth alone could plausibly account for the estimated DIN loss in the model results. While the complexity of processes that occur within SVAs warrants further study, our results indicate that nutrient cycling models should allow for loss rates in SVAs that are distinct from loss rates in other regions. This distinction is often omitted even in state-of-the-art models (Liu et al. 2018).

General Applicability of Lagrangian Biogeochemical Modeling Approach

The tracer-based Lagrangian modeling approach presented here can be used in a range of biogeochemical applications including exploration of alternative formulations and rate estimation. This Lagrangian modeling approach can be applied to other locations, seasons, and constituents and generalized for application to more complex biogeochemistry, including interactions among several constituents. More complex biogeochemical models might not permit an analytical solution (e.g., Eqs. 13–17) but can be integrated numerically with standard methods for coupled ordinary differential equations. This Lagrangian modeling approach allows multiple point sources or distributed sources of constituents by introducing additional tracers to “fingerprint” contributions from each source. For example, the Lagrangian modeling approach could readily be applied to track the evolution of DIN from multiple WWTPs and other point source inputs; used to track mass of chemical constituents in suspended sediment, particulates, and pelagic organisms; and applied with temporally variable biogeochemical rates.

In our approach, we represent the age distribution of “water particles” making up a water parcel, following the terminology of Lucas and Deleersnijder (2020), by a single mean age in the governing equations (Eqs. 11 and 12). While the CART framework (Deleersnijder et al. 2001) is based upon the distribution of age within the “concentration distribution function,” typical tracer-based modeling predicts only the first moment of this distribution, the mean age. In some cases the second moment of age variance is calculated by a related tracer-based approach (Delhez and Deleersnijder 2002). In highly dispersive environments, the variance could be large, and the use of mean age in lieu of the full age distribution may incur substantial error.

A limitation of the method is the inherently Lagrangian perspective and limited ability to incorporate Eulerian information (Gross et al. 2023). To the extent that distinct regions or spatial compartments (habitat types or geographic features) can be identified a priori, exposure to these regions can be tracked with exposure time tracers and regionally distinct biogeochemical rates can be applied. In addition, the average conditions, such as depth and temperature, encountered by the tracer in these discrete regions can be calculated by a property-tracking approach (Eqs. 8–10). However, a complete time history of the property is not retained. Instead, the tracer-based estimate integrates the exposure of many water particles that would each have a different time history of a property. This lack of a time history of the conditions encountered by the tracers makes the approach inappropriate for some situations described below. Furthermore, the Lagrangian approach has only a single set of state variables, without a direct way to represent storage and subsequent release in Eulerian compartments such as SVAs or the benthos.

Because phytoplankton move with the water, cycling of nutrients through phytoplankton only influences DIN species concentrations when there is a net change in the mass of nitrogen species in the phytoplankton compartment of a moving water parcel. During the study period, measured phytoplankton concentrations were typically low (< 4 µg L−1) at fixed stations (US Geological Survey 2023) and in high-resolution mapping during July (Bergamaschi et al. 2020); substantially higher concentrations (> 10 µg L−1) were measured in small portions of the east Delta and in the western Delta on the Sacramento River. Therefore, net exchange of DIN species with phytoplankton biomass in a moving water parcel was expected to be small during the study period but may be greater in periods and locations with large algal blooms. For study periods with blooms, a likely improvement to the DIN cycling model presented here would be inclusion of phytoplankton biomass as a compartment for storage, transformation, and release of nitrogen species. DIN uptake by phytoplankton in estuaries can be substantial, particularly during blooms when chlorophyll concentrations exceed 20 µg L−1.

Building on the DIN modeling approach presented here, phytoplankton and zooplankton equations could be added to develop an NPZ (nutrient, phytoplankton, and zooplankton) model. These components could be easily added to the nitrogen model as a coupled set of ordinary differential equations. The time history of coupled DIN, phytoplankton and zooplankton in a moving water parcel can be integrated in the Lagrangian model using an ordinary differential equation (ODE) solver. However, following an approach analogous to our nitrogen model, the time history of primary production would be unknown. In cases in which the variability in production is primarily temporal, this variability can be incorporated into the Lagrangian model using time-varying insolation, temperature, and turbidity data (Gross et al. 2023). The spatial component of variability in phytoplankton production due to spatially variable turbidity and other factors is more difficult to address in a Lagrangian framework (Gross et al. 2023). These limitations can be particularly acute when the time scale of biogeochemical processes is shorter than transport time scales. In this case, a Lagrangian modeling approach may not be appropriate.

Using tracer information to build a Lagrangian biogeochemical model is feasible for many water bodies if simulations identify the provenance of most of the water at times and locations of biogeochemical predictions. Biogeochemical rates can be estimated most reliably when incoming constituent concentrations are measured or can reasonably be assumed to be constant and fit in the parameter optimization. In our study, biogeochemical rate estimation was aided by the strong spatial variability in DIN concentrations (Fig. 9c) and speciation resulting from a large point source of DIN, primarily in the form of ammonium, at the SRWTP.

In the context of reactive transport modeling, the computational expense of typical coupled hydrodynamic-biogeochemical models prevents broad exploration of the parameter space, while box models and reduced-dimension models often cannot account for important transport effects. The use of water age, partial age, and property exposure in modeling reactive transport provides a middle ground for modeling reactive transport by retaining much of the computational advantage of box models while accounting for important aspect of transport processes.