Episodic events of flood deposit in coastal environments
are characterized by deposition of large quantities of sediment containing
reactive organic matter within short periods of time. While steady-state
modelling is common in sediment biogeochemical modelling, the inclusion of
these events in current early diagenesis models has yet to be demonstrated.
We adapted an existing model of early diagenetic processes to include the
ability to mimic an immediate organic carbon deposition. The new model
version (FESDIA) written in Fortran and R programming language was able to
reproduce the basic trends from field sediment porewater data affected by
the November 2008 flood event in the Rhône River prodelta. Simulation
experiments on two end-member scenarios of sediment characteristics dictated
by field observation (1–high thickness deposit, with low TOC (total organic carbon) and 2–low
thickness, with high TOC), reveal contrasting evolutions of
post-depositional profiles. A first-order approximation of the differences
between subsequent profiles was used to characterize the timing of recovery
(i.e. relaxation time) from this alteration. Our results indicate a longer
relaxation time of approximately 4 months for
SO42- and 5 months for DIC (dissolved inorganic
carbon) in
the first scenario, and less than 3 months for the second scenario which
agreed with timescale observed in the field. A sensitivity analysis across a
spectrum of these end-member cases for the organic carbon content (described
as the enrichment factor α) and for sediment thickness indicates
that the relaxation time for oxygen, sulfate, and DIC decreases with
increasing organic enrichment for a sediment deposition that is less than 5 cm.
However, for larger deposits (>14 cm), the relaxation time for
oxygen, sulfate, and DIC increases with α. This can be related to the
depth-dependent availability of oxidant and the diffusion of species. This
study emphasizes the significance of these sediment characteristics in
determining the sediment's short-term response in the presence of an
episodic event. Furthermore, the model described here provides a useful tool
to better understand the magnitude and dynamics of flooding event on
biogeochemical reactions on the seafloor.
Introduction
Coastal margins play a crucial role in the global marine systems in terms of
carbon and nutrient cycling (Wollast, 1993; Rabouille et al., 2001b; Cai,
2011; Regnier et al., 2013; Bauer et al., 2013; Gruber, 2015). Due to their
relatively shallow depth, sedimentary early diagenetic processes are
critical for the recycling of a variety of biogeochemical elements, which
are influenced by organic matter (OM) inputs, particularly carbon
(Middelburg et al., 1993; Arndt et al., 2013). Furthermore, these processes
have the potential to contribute to the nutrient source that fuels the
primary productivity of the marine system. In river-dominated ocean margins
(RiOmar; McKee et al., 2004), organic matter input can also be enhanced by
flood events which provide a significant fraction of the particulate carbon
(POC) delivered to depocenters (Antonelli et al., 2008). Organic matter
derived from riverine input to sediment has biogeochemical significance in
coastal marine systems (Cai, 2011). As a result, the coastal environment
serves as both a sink for particulate organic carbon and nutrients, and an
active site of carbon and nutrient remineralization (Burdige, 2005; McKee et
al., 2004; Sundby, 2006).
In the context of early diagenetic modelling, numerical models with
time-dependent capability are well-established (Lasaga and Holland, 1976;
Rabouille and Gaillard, 1991; Boudreau, 1996; Soetaert et al., 1996a, b;
Rabouille et al., 2001a; Archer et al., 2002; Couture et al., 2010; Yakushev
et al., 2017; Munhoven et al., 2021; Sulpis et al., 2022), and they are used
in many coastal and deep-sea studies. However, because of the scarcity of
observations and their unpredictability, the role of massive deposit of
sediment in these early diagenesis models has frequently been overlooked
(Tesi et al., 2012). As these rare extreme events are being currently
documented in various locations, there is a growing appreciation for their
impact on the coastal margin (Deflandre et al., 2002; Cathalot et al., 2010;
Tesi et al., 2012).
Attempts to use mathematical models to understand perturbation-induced
events such as sudden erosion/resuspension event, bottom trawling, and
turbidity-driven sediment deposition on early diagenetic processes have
resulted in a variety of approaches that incorporate this type of
phenomenon. As an example, Katsev et al. (2006) demonstrated that the
position of the redox boundary (depth zone beneath the sediment–water
interface that separates the stability fields of the oxidized and reduced
species of a given redox couple) in organic-poor marine sediment can undergo
massive shifts due to the flux of new organic matter on a seasonal basis,
whereas on a longer timescale (e.g. decadal), redox fluctuation linked to
organic matter deposition can induce the redistribution of solid-phase
manganese with multiple peaks (due to depth-wise oxidation reduction of Mn).
Another study in a coastal system revealed that coastal sediments change as
a result of an anthropogenic perturbation in the context of bottom dredging
and trawling (van de Velde et al., 2018). More recently, using a similar
model, De Borger et al. (2021) highlighted that perturbation events such as
trawling can possibly decrease total OM mineralization.
In river-dominated ocean margins, episodic flood events can deliver sediment
with varying characteristics depending on its source origin, frequency, and
intensity (Cathalot et al., 2013). Therefore, the flood characteristics have
direct impact on the deposited sediment's characteristics such as
scale/thickness of the deposited layer, composition (mineralogy and
grain size), OM content, and so on. For example, In the Rhône prodelta, a
single flash flood can deliver up to 30 cm of new sediment material in a
matter of days (Cathalot et al., 2010; Pastor et al., 2018). Despite the
large amount of sediment introduced by this episodic loading, vertical
distribution of porewater species like oxygen (O2) can be restored
after a few days (Cathalot et al., 2010). It has also been noticed (Rassmann
et al., 2020) that spring and summer porewater compositions measured for
several years following fall and winter floods show quasi-steady-state
profiles for sulfate and DIC (dissolved inorganic
carbon). Similar massive deposition was also reported
in the Saguenay fjord (Quebec, Canada) (Deflandre et al., 2002; Mucci and
Edenborn, 1992). The recovery timescale from this perturbation has only been
roughly estimated for species with short relaxation time such as oxygen, but
this is not always the case for sulfate
(SO42-), dissolved inorganic
carbon (DIC), or other redox species. Furthermore, due to the limitation in
temporal resolution of the observations, the short-term post-depositional
dynamics in the aftermath of this flood deposition event are scarcely
described, making it difficult to discern how the system responds after the
event. While experimental approaches (Chaillou et al., 2007) can provide
useful insight into how they work, they lack the ability to provide
continuous system dynamics and are often difficult to set up. A modelling
approach can assist in addressing these issues, providing useful feedback in
terms of the scale and response of the sediment to this type of event.
The goal of this study is to better understand episodic events in the
context of flood-driven sediment deposition and their impact on benthic
biogeochemistry, post-flood evolution dynamics, and relaxation timescale. As
the relaxation dynamics represent a gap in our understanding of how coastal
systems respond to external drivers, we characterize the timescale of the
recovery of sediment porewater profiles using a first-order approximation.
To accomplish this, we developed an early diagenetic model called FESDIA.
The ability to explicitly simulate non-steady early diagenesis processes in
systems subject to perturbation events such as massive flood or storm
deposition is a novel contribution of FESDIA to early diagenetic models. In
the following ways, FESDIA differs therefore from the OMEXDIA model
(Soetaert et al., 1996a) by implementing:
an explicit description of the anoxic diagenesis including (i) iron and
sulfur dynamics, and (ii) methane production and consumption. In comparison,
OMEXDIA has a single state variable (ODU: oxygen demand unit) to describe
reduced species;
the possibility to include sediment perturbation events such abrupt deposition
of sediment.
In this paper, we only discuss part of the FESDIA model concerned with
implementation of a perturbation event as it relates to some biogeochemical
indicators. The model is implemented in Fortran (for speed) and linked to R
(for flexibility). We demonstrate the model's utility in describing data
collected from a flood event in November/December 2008 (Pastor et al., 2018)
and numerically investigate the impact of varying degrees of flood
type characteristics on the system's relaxation dynamics. This work is a
foundation for a more in-depth investigation of the model–data
biogeochemistry of the porewater and solid phase components of core samples
from Pastor et al. (2018), and it provides a useful baseline for
understanding the spatiotemporal dynamics of coastal marine systems subject
to event-driven organic matter pulses.
Materials and methodsSite and events description
The Rhône prodelta serves as a case study for the development of the
model used to evaluate sediment perturbation dynamics. This particular
coastal area acts as the transitory zone between the inland river channel
and the continental shelf (Gulf of Lion) of the Mediterranean Sea. The
Rhône River with a drainage basin of 97 800 km2 and mean water
discharge of 1700 m3s-1 delivers
up to 1.6×1010 moles C of
particulate carbon (POC) annually (Sempéré et al., 2000) to the
pro-deltaic part (i.e. where the river meets the sea). The Rhône prodelta
covers an area of approximately 65 km2 with depth ranging from 2 to 60 m (Lansard et al., 2009) and is characterized by high sedimentation rates
reaching up to 41 cm yr-1 in the proximal
zone (Rassmann et al., 2016; lat: 43∘18.680′ N, long:
4∘51.038′ E and average depth of 21 m) (Radakovitch et
al., 1999; Miralles et al., 2005). The organic matter delivered to the
depocenter typically reflects the different compositional materials derived
from the terrestrial domain (Pastor et al., 2018), whereas the magnitude of
material transported and the quantity of organic carbon transferred
laterally varies according to seasons and the period of massive
instantaneous deposition (Lansard et al., 2008; Cathalot et al., 2013).
Map showing the locations of sampling sites off the Rhône
River mouth.
Relating to the episodic pulse of organic matter, numerous studies have
documented instances of flood-driven deposition from the Rhône River
from a hydrographic perspective (Boudet et al., 2017; Hensel et al., 1998;
Pont et al., 2017). Pastor et al. (2018) go beyond sedimentology and
hydrographic characteristics to provide a concise description of the various
flood types, their diagenetic signatures, and biogeochemical implications.
Furthermore, published porewater chemistry and solid-phase data have
highlighted sediment characteristics following such an event (Cathalot et
al., 2010; Toussaint et al., 2013; Cathalot et al., 2013; Pastor et al.,
2018).
Model development and implementation
Following the description of the Rhône River flood types and the
composition of the flood deposit (mainly in terms of organic carbon) at the
proximal station A (Pastor et al., 2018), we proceed to describe the model
developed to explore the observed data and their diagenetic implications in
terms of relaxation times and their evolution following this transient
perturbation.
Description of notations, phrases, acronyms, and abbreviations, as
used in this paper.
SymbolsDescriptionTOCrefAsymptotic carbon content. This is equal to the refractory component of sedimentary carbon not modelled explicitly by FESDIA.ΔzThickness of vertical layer. Unequal in each layer of the modelled domain.Ncur, NNumber of grid layer. Equal to default modelled layer (N=100).NpertZpertDepth of sediment deposition. This corresponded to the observed depth of sediment deposited due to flood input.TOCzpertOrganic carbon content in the deposited layer. This corresponds to the TOC (total organic carbon) content introduced by the flood layer and differs from the ancient layer depending on its concentration of carbon.TOColdOrganic carbon content in the ancient layer. This corresponds to the TOC content in the previous layer prior to the flood deposition.t-Time index prior to the flood event deposition.αCarbon enrichment factor. This is a multiplicative coefficient for which the solid component of the sediment in the newly deposited layer can be increased dynamically during the simulation. As such, a “new initial condition” for the deposition can be realized without stopping the simulation using this factor.CorgConcentration of fast and slow degradable organic matter. This is the sum of both model variables (Corgfast and Corgslow).CorgfloodThis variable symbolically specifies concentration of fast and slow degradable organic matter immediately after the event.fluxorgDaily flux of organic carbon flux derived from the annual average flux.φtTime-dependent, differential operator of successive depth integrated over the modelled domain.τtRelaxation time derived from φt.
Our model combines the development in the OMEXDIA model (Soetaert et al.,
1996a), applied in the Rhône prodelta area (Ait Ballagh et al., 2021;
Pastor et al., 2011) and which has recently been equipped with event-driven
processes (De Borger et al., 2021). In De Borger et al. (2021), the authors
specifically addressed the issue of bottom trawling as a mixing and an
erosional process that removes an upper layer of sediment and mixes a
certain layer below. In addition, the model considers a bulk categorization
of reduced substance in a single state variable, ODU (oxidative oxygen
unit). For our approach, the event is defined by an addition of a new layer
on top of the former sediment–water interface (Table 1). Furthermore, we
explicitly modelled pathways involving sulfur and iron. Following this
preamble, the following sections go over aspects of the model description
and parameterization. Table 1 provide some key glossary of mathematical
notations used in the model.
Model state variables
The complete model describes the concentration of labile
(Corgfast) and semi-labile
(Corgslow) decaying organic matter,
oxygen (O2), nitrate
(NO3-), ammonium
(NH4+), and dissolved inorganic
carbon (DIC), following the classic early diagenetic equation of Berner (1980) and Boudreau (1997). In addition to the model from De Borger et al. (2021), our model includes sulfate
(SO42-), hydrogen sulfide
(H2S), methane
(CH4), and iron species
(Fe2+ and Fe(OH)3) (Table 2).
State variables described in the model.
State variableModel notationUnitsDescriptionSolid CorgfastFDETmmol C m-3Fast decaying detritusCorgslowSDETmmol C m-3Slow decaying detritusFeOH3FEOOHmmol Fe m-3Oxidized ferric ironLiquid O2O2mmol O2 m-3OxygenNO3-NO3mmol N m-3NitrateNH4+NH3mmol N m-3AmmoniumSO42-SO4mmol S m-3SulfateH2SH2Smmol S m-3Hydrogen sulfideFe2+Femmol Fe m-3Reduced ferrous ironDICDICmmol N m-3Dissolved inorganic carbonCH4CH4mmol N m-3Methane
In some coastal settings, oxidation via sulfate reduction has been
highlighted as the primary pathway for organic carbon (OC) mineralization,
with minor contributions from manganese and iron oxidation (Burdige and
Komada, 2011). In addition, the flux of integrated remineralization products
such as DIC has previously been estimated to contribute up to 8 times that
of diffusive oxygen uptake (Rassmann et al., 2020) – thus highlighting its
importance in describing the amplitude of benthic recycling in coastal
water. As such in this paper, we focus our analysis on these proxy variables
(O2,
SO42-, DIC) because they serve
as indicators of the integrated effect of the main diagenetic processes.
Biogeochemical reaction
Early diagenesis processes on the seafloor are driven by organic matter
deposition. For areas such as the Rhône prodelta, continental organic
carbon input is dominant, and it is difficult to identify the fraction of
labile fraction responsible for fast OM pool consumption (Pastor et al.,
2011). Moreover, observations show that some organic compounds are
preferentially degraded and become selectively oxidized (Middelburg et al.,
1997; Pozzato et al., 2018). As a result, the model assumed solid phase
organic carbon with two reactive modelled fractions with different
reactivities and C/N ratios (Westrich and Berner, 1984; Soetaert et al.,
1996a). The mineralization of OM occurs sequentially, with the labile
fraction mineralizing faster than the slow decaying carbon. During the
timescales considered here, the refractory organic matter class is not
reactive. To compare with the observation, we consider an asymptotic OC
constant (TOCref) for the inert fraction that scales the model-calculated TOC output to the observation (Pastor et al., 2011) (see Sect. 2.2.8). This organic carbon degradation requires oxidants, and the
depth dependency in sequential utilization of terminal electron acceptors
assumption first proposed by Froelich (1979) is used here. Oxygen is
consumed first, followed by nitrate, iron oxides, sulfate, and finally
methanogenesis occurs (Eq. 3). Because the quantity of organic matter and
the relative proportions of fast and slow degrading materials decrease with
depth, the overall organic matter degradation rate decreases accordingly. In
the formulation of the individual biogeochemical processes, we use a similar
paradigm as Soetaert et al. (1996a) (Eq. 2).
This rate of carbon mineralization of organic matter (mmol m-3 d-1) can be expressed as
Cprod=rFast×Corgfast+rSlow×Corgslow×1-ϕϕ,
where the rFast and rSlow are the decay rate constant
(d-1) for fast and slow detritus component. ϕ and
1-ϕ are the volume fraction for both solutes
and solid respectively. This process is mediated by microorganisms and
oxidant availability. The primary redox reaction includes (1) oxic
respiration, (2) denitrification, (3) Fe (III) reduction, (4) sulfate reduction,
and (5) methane production:
OM+O2→CO2+1C:NNH3+H2OOM+0.8NO3-+0.8H+→CO2+1C:NNH3+0.4N2+1.4H2OOM+4FeOOH+8H+→CO2+1C:NNH3+4Fe2++7H2OOM+0.5SO42-+H+→CO2+1C:NNH3+0.5H2S+H2OOM→0.5CO2+1C:NNH3+0.5CH4.
This reaction can be modelled using a Monod type relationship with each
oxidant having a half-saturation constant (ksC)
represented as ks* in the model code. The inhibition of mineralization by the
presence of other oxidants is also modelled with a hyperbolic term
(subtracted from 1), where kinC is concentration at which
the rate drops to half of its maximal value. Using these limitation and
inhibition functions, a single equation for each component across the
model–depth domain can be realized (Rabouille and Gaillard, 1991; Soetaert
et al., 1996a; Wang and Van Cappellen, 1996), together with some possible
overlap (Froelich et al., 1979; Soetaert et al., 1996a). For a generic
species, this can be described mathematically as
lim=CksC+C∏1-CkinC+C,
where C is one oxidant. Formulation for individual pathways as well as
values of half-saturation and inhibition constants for each oxidant can be
found in Appendix A1. With this limitation term, mineralization rate per
solute can be estimated using potential carbon produced via OM degradation
in (Eq. 1),
ratemin=Cprod×lim×1∑lim,
with the ∑lim the sum of all limitation terms which
normalizes the term in order to always achieve the maximum degradation rate.
See Soetaert et al. (1996a) for more details on the derivative of this
equation.
Secondary redox reaction includes reoxidation of reduced substances
(nitrification, Fe oxidation, H2S oxidation,
methane oxidation) (Eq. 5) and the precipitation of FeS. Anaerobic oxidation
of methane occurs in the absence of O2 following
upward diffusion of methane to the sulfate–methane transition zone (SMTZ)
(Jørgensen et al., 2019):
NH4++2O2→NO3-+H2O+2H+4Fe2++O2+6H2O→4FeOOH+8H+CH4+O2→CO2+2H2OH2S+O2→SO42-+2H+CH4+SO42-→HCO3-+HS-+H2OFe2++H2S→FeS+2H+.
These reactions are mathematically described using a coupled reaction
formulation. Nitrification is limited by the availability of oxygen and the
other reactions are described with a first-order term.
Nitri=Rnit×NH4×O2O2+ksnitri(Nitrification)Feoxid=RFeOH3×Fe×O2(Iron oxidation)H2Soxid=RH2S×H2S×O2(sulfide oxidation)CH4oxid=RCH4×CH4×O2(Methane oxidation)AOM=RCH4×CH4×SO4(Anaerobic oxidation of methane),
where Rnit is the rate of nitrification (d-1),
RFeOH3, RH2S, and RCH4 are the maximum rate of oxidation
of iron, sulfide, and methane via oxygen, respectively
(mmol-1 m3 d-1). Because
sulfide precipitation can occur in some coastal sediments, we accounted for
this sink process by removing produced sulfide from sulfate reduction as a
first-order FeS formation.
FeSprod=RFeSprod×Fe×H2S(FeS production),
with RFeSprod the rate of production of FeS
(mmol-1 m3 d-1).
Transport processes
Transport processes in the model are described by molecular diffusion and
bio-irrigation for dissolved species whereas bioturbation is the main
process for mixing the solid phase. In addition, advection occurs in both
the solid and dissolved species. The model dynamics described as a partial
differential equation (PDE) is the general reaction–transport equation
(Berner, 1980). We use a similar paradigm and formulations to that of
Soetaert et al. (1996a). For substances that are dissolved:
∂ϕC∂t=-∂∂z-ϕ×Dsed×∂C∂z+w∞×ϕ∞×C+∑ϕ×REAC.
With special consideration of ammonium adsorption to sediment particles, the
governing equation is given by
∂ϕC∂t=-∂∂z-ϕ×Dsed1+kads×∂C∂z+w∞×ϕ∞×C+∑ϕ×REAC1+kads,
where we assumed that the immobilization of NH4+ is in
instantaneous, local equilibrium (i.e. any changes caused by the slow
NH4+ removal process results in an immediate adjustment of
the NH4+ equilibrium; so, can be modelled with a simple
chemical species) and kads is the adsorption coefficient. The inclusion
of this formulation for the diffusion and reaction term has the effect of
slowing down ammonium migration in sediment. Derivation of this formulation
is given in Berner (1980) and Soetaert and Herman (2009).
For the solid phase:
∂1-ϕS∂t=-∂∂z-1-ϕ×Db×∂S∂z+w∞×1-ϕ∞×S+∑1-ϕ×REAC,
where C is the concentration of porewater (unit of mmol m-3 liquid) for Eq. (8) and S for solid (unit
of mmol m-3 solid) Eq. (10). w
(cm d-1) and Dsed
(cm2 d-1)
represent the burial/advection and molecular diffusion coefficient in the
sediment respectively, and REAC is the source/sink processes linked to
biogeochemical reactions in the sediment. This term includes both biological
and chemical reaction within the sediment column and non-local
bio-irrigation transport term (see next section). Db is the
bioturbation term for solid driven by the activities of benthic organisms.
For dynamic simulation, w can change as a function of time but in most
cases we assumed a constant value.
Diffusive fluxes of solutes across the sediment–water interface are driven
by the concentration gradients between the overlying seawater and the
sediment column. Fick's first law is used to describe the solute flux due to
molecular diffusion,
Jd=-ϕDsed∂C∂z,
where the Dsed (cm d-1) is the
effective diffusion coefficient corrected for tortuosity and given as
Dsed=Dswθ2, with Dsw the molecular
diffusion coefficient of the solute in free solution of sea water, and
θ is the tortuosity derived from the formation factor (F) and
porosity (ϕ) of a sediment matrix (Berner, 1980; Boudreau, 1997). This
molecular diffusion coefficient is calculated as function of temperature and
salinity using compiled relation of Boudreau (1997), implemented in the R
package Marelac (Soetaert and Petzoldt, 2020).
As a simplifying assumption, material accumulation has no effect on
porosity. We further assumed the porosity profile decreased with depth but
invariant with time. Although, this assumption is a restrictive as the site of
flood deposition can undergo variation in grain size which might affect
their porosity (Cathalot et al., 2010), we proceed noting that the fixed
parameters which define the porosity curve can be changed when necessary.
Thus, using optimized parameters fitted with data in the proximal sites of
Rhône prodelta (Ait Ballagh et al., 2021), porosity (ϕz) in Eqs. (10)–(8) is prescribed as an exponential decay,
ϕz=ϕ∞+ϕ0-ϕ∞e-z-zswiδ,
where ϕ0 and ϕ∞ is the porosity surface and
at deeper layer, respectively, while Zswi is the depth of the SWI (sediment–water interface), and
δ (cm) is the porosity exponential decay coefficient with depth.
Bioturbation and bio-irrigation
Bioturbation in the model is characterized by the movement and mixing of
particles by benthic organisms. This is parameterized as a diffusivity
function in space (Dz) and acts on the concentrations of
the different solid species in the sediment. In our model, this bioturbation
flux is assumed to be interphase, with porosity ϕz
remaining constant over time. Thus, this process is prescribed as
Dbz=Db0if Z≤ZLD∞+Db0-D∞e-Z-ZLbiotattif Z>ZL,
where Db0 is the bio-diffusivity coefficient
(cm2 d-1) at the SWI and
in the mixed layer, ZL is the depth of the mixed layer (cm),
and biotatt is the attenuation coefficient
(cm-1) of bioturbation below the mixed
layer. D∞ is the diffusivity at the deeper layer as
usually specified as zero. In the model, we did not account for mortality of
benthic fauna following the deposition as in De Borger et al. (2021) where
they focus on habitat recolonization after trawling.
Bio-irrigation is modelled in an identical manner to that of biodiffusion,
and acts as a non-local exchange process between the porewater parcels and
the overlying bottom water.
Irrz=Irr0if Z≤ZLIrr∞+Irr0+Irr∞e-Z-ZLIrrattif Z>ZL,
for which Irr0 is the bio-irrigation rate (d-1)
and Irratt is the attenuation of irrigation (cm) below the depth of
the irrigated layer Zirr (cm). At depth, the bio-irrigation
rate (Irr∞) is generally set to zero.
Model vertical grid
The model is vertically resolved with grid divided into 100 layers (N), of
thickness (Δz) increasing geometrically from 0.01 cm at the
sediment–water interface to 6 cm at the lower boundary. The result is a 100 cm model domain comprising of a full grid with non-uniform spacing and
maximum resolution near the SWI. Depth units are in centimetres. This choice
of modelled depth allows for complete carbon degradation. This modelled grid
is generated by the grid generation routine of the ReacTran R package
(Soetaert and Meysman, 2012), which implements many grid types used in
early diagenesis modelling. During the time instance of the event
specification, the added grid of new layers (Npert≈Zpert) and the current grid (Ncur≈N) is rescaled to the
model's common grid of N layer by linear interpolation (see Sect. 2.2.6
and Fig. S1). The concentration of state variables is defined at the layer
midpoints, whereas diffusivities, advection (sinking/burial velocities), and
resulting transport fluxes are defined at the layer interfaces.
Schematic of model implementation for the deposition event scenario. Profile from previous time step (left) and after addition of new layer over a predefined depth layer (right). For the solid (a), the new layer can be enriched (blue) or depleted (red) relative to the old (average) (black). The dissolved substances (b) are set equal to the bottom water concentration during the deposition. Thereafter, the profile is integrated forward with time. The whole sequence of step occurs dynamically with time capitalizing on the integrator ability to simulate dynamic event process. α is the carbon enrichment factor applied over depth Zpert (see text for detail).
Deposition event
The inclusion of the deposition event as a separate external routine to
modify the sediment properties (i.e. porewater species,
Corg) is a fundamental difference between our
approach and the other previous early diagenesis model applied in the Rhone
Delta, but it bears similarity with De Borger et al. (2021). We assume the
event occurred as an instantaneous deposition of organic carbon
(Corgfast and
Corgslow) over a depositional layer,
Zpert (Fig. 2).
The event calculation was carried out dynamically within the same simulation
time. For the solid species, following the flux of organic carbon via the
boundary condition (see Sect. 2.2.7), the portion of organic carbon is
split between the fast and slow decaying component using a proportionality
constant (pfast) as in Ait Ballagh et al. (2021).
pfast varies from 0 to 1 and it is expressed in percentage of
carbon flux deposited associated to either fraction (fast and slow).
However, at the time when the event is prescribed, the integrated profile of
the solid species Corgfast and
Corgslow from previous time step, defined
as (t-), was used to create a virtual composite of
the deposited layer. This integral calculation was performed over a
specified sediment thickness (Zpert), which
corresponded to the vertical extent of the depositional event. This average
concentration for the solid, which we define exclusively for the time of
deposition as (Corgflood), is scaled with
an enrichment factor (α) (see below) and then nudged on top
of the old layer which is supposed to be buried beneath after the event. To
avoid numerical issues caused by the discontinuity of both layers with
different properties, an interpolation of the composite profile was
performed over the modelling domain. This smoothes the interface between the
deposited layer's base and the current model grid's upper layer. This
algorithmic procedure is schematically shown in Fig. 2 and we summarized
this process mathematically as
Corgflood≈αCorgit|0zpert=α∫0zpertCorgit-dzZpertα<1, if TOCzpert<TOColdα>1, if TOCzpert>TOCold;i=1fast,2slow,
where TOCzpert corresponds to the TOC content introduced by the
flood layer and TOCold represents the TOC in the previous layer prior
to the flood deposition. The carbon enrichment factor (α)
in the model (confac in the model code) is introduced here in order to scale
the deposited OC with those observed from field data. This helps in
calibrating the deposited organic matter concentration
(Corgfast and
Corgslow) in the new layer relative to
the previous sediment fraction, simulating the wide range of TOC content
observed in the field. For instance, when the newly deposited organic matter
is similar to the former sediment topmost layer (average pre-flood layer
concentration over an equivalent Zpert depth), an
(α) value of 1:1 is used. If the new material is lower in
organic carbon content compared to what is near the sediment–water
interface, then (α)<1, while if the newly
deposited material is higher in carbon content than the sediment surface,
(α)>1. This flexibility can be used to
constrain the simulation to match the corresponding TOC profile from field
observation. In the modelling application, this parameter is generally
specified by using different value for the magnitude of OC in each fraction
depending on the empirical observation of the TOC data. This quantity is
therefore tunable and the upper bound of this parameter is dictated by the
maximum TOC in the sediment sample.
It is important to note that this parameter differs from pfast.
This OC flux partitioning by pfast occurs regardless of the event
and it is related to the carbon flux received at the boundary, but the
carbon enrichment factor occurs only during the event. The carbon enrichment
factor (α) can be viewed as a method of imposing a new
initial condition only at the time of the event by using the integral
concentration from the previous time. However, using the approach described
here, all calculations can be done dynamically without stopping the model.
For the solutes (O2,
NO3-,
NH4+, DIC,
SO42-, H2S, and CH4),
the bottom water concentration is imposed through the perturbed layer at the
time of event by assuming this new layer is homogenously mixed.
Boundary Conditions
The boundary conditions for the model are of three types:
At the sediment–water interface, a Dirichlet concentration condition for
most solutes equalling the bottom water concentration was used,C|z=0=Cbw.Both pore water and solid have a zero-flux boundary condition at the bottom
of the model,dCdz|z=z∞=0.For solid, an imposed flux at the upper boundary for most of the year is
used,fluxorg|z=0=-1-ϕ0Db0dCdz|z=0+1-ϕ∞w∞C|z=0.
The model also includes the ability to include time-varying organic carbon
flux with user-specific time series or a functional representation such as
sinusoidal pattern. In the case study presented here, this carbon flux
varies over the annual carbon flux (flux‾org) in the region
in question. This was expressed mathematically as
fluxorgt|z=0=flux‾org×1+sin2⋅π⋅t365.
At the time of the instantaneous deposition, this deposited carbon is
treated as described in Sect. 2.2.6.
Model parameterization and verification
The model parameters in Table 2 (for full model parameter, see Table S1 in
Supplement) were derived from previously published model in the Rhône
Delta (Pastor et al., 2011; Ait Ballagh et al., 2021). The organic matter
stoichiometry for both fractions is represented here by the NC ratio
(NCrFdet and NCrSdet) with values of 0.14 and 0.1, respectively. The flux of carbon in the
upper boundary of the model was defined using a yearly mean flux (flux‾org) of 150 mmol m-2 d-1 in Rhône prodelta (Pastor et al., 2011). TOC
(in % dw) is estimated from both carbon fractions
(Corgfast and
Corgslow) assuming a sediment density
(ρ) of 2.5 g cm-3 and conversion from the model unit for
detrital carbon fraction of mmol m-3 to a unit in percent mass. The
model diagnostics TOC value is then computed as follows:
TOC=Corgfast+Corgslow×1200×10-92.5+TOCref,
with TOCref the asymptotic TOC value at deeper layer of the sediment,
thus representing concentration of refractory carbon not explicitly
modelled. The sedimentation rate used in this modelling study was kept
constant at 0.027 cm d-1 (Pastor et al.,
2011). The decay rate constant for the labile and semi-labile detritus
matter is set as 0.1 and 0.0031 d-1, respectively, with
both fractions split equally with a proportionality constant (pfast) of
0.5. Using parameters fitted by the model of Ait Ballagh et al. (2021) to
data observed in the Rhône prodelta area, the rate of bioturbation and
bio-irrigation is fixed as 0.01 cm2 d-1 and 0.23 d-1 with these fauna-induced activities occurring
down to a depth of 5 and 7 cm, respectively.
The bottom water temperature was fixed at 20 ∘C. The bottom water
salinity is nearly constant below the Rhône River plume, ranging from
37.8 to 38.2. In the model, the average temperature and salinity is used to
calculate the diffusion coefficient for the solute chemical species (Sect. 2.2.3). Bottom water solute concentrations were constrained using previously
reported values in previous modelling efforts (Ait Ballagh et al., 2021) and
adapted with new data for the time corresponding to the flood deposit event
(see Table 3). Porosity decreases exponentially with depth from 0.9 at the
sediment water interface to 0.5 at deeper layer with a decay coefficient of
0.3 cm (Lansard et al., 2009).
Core parameters used in the model.
ModelModelValuesUnitsDescriptionReferencesparametersnotationflux‾orgCFlux150mmol m-2 d-1total organic C depositionPastor et al. (2011)pfastpFast0.5–part FDET in carbon fluxPastor et al. (2011)flux‾FeOO3FeOH3flux0.01mmol m-2 d-1deposition rate of FeOH3AssumedrFastrFast0.1d-1decay rate FDETAit Ballagh et al. (2021)rSlowrSlow0.0031d-1decay rate SDETAit Ballagh et al. (2021)NCrFdetNCrFdet0.14molN / molCNC ratio FDETPastor et al. (2011)NCrSdetNCrSdet0.1molN / molCNC ratio SDETPastor et al. (2011)O2bwO2bw197mmol m-3upper boundary O2Ait Ballagh et al. (2021)NO3bwNO3bw0.0mmol m-3upper boundary NO3Ait Ballagh et al. (2021)NH3bwNH3bw0.0mmol m-3upper boundary NH3Ait Ballagh et al. (2021)CH4bwCH4bw0.0mmol m-3upper boundary CH4Rassmann et al. (2016)DICDICbw2360mmol m-3upper boundary DICPastor et al. (2018)Fe2+bwFebw0.0mmol m-3upper boundary Fe2+Pastor et al. (2018)H2SbwH2Sbw0.0mmol m-3upper boundary H2SPastor et al. (2018)SO4bwSO4bw30 246mmol m-3upper boundary SO4Pastor et al. (2018)ww0.027cm d-1advection ratePastor et al. (2011)D0biot0.01cm2 d-1bioturbation coefficientAit Ballagh et al. (2021)ZLbiotdepth5cmdepth of mixed layerAit Ballagh et al. (2021)biotattbiotatt1.0cm-1attenuation coeff below biotdepthAit Ballagh et al. (2021)Irr0irr0.2d-1bio-irrigation rateAit Ballagh et al. (2021)Zirrirrdepth7cmdepth of irrigated layerAit Ballagh et al. (2021)Irrattirratt1.0cmattenuation coeff below irrdepthAit Ballagh et al. (2021)temptemperature16∘temperatureAit Ballagh et al. (2021)salsalinity38psusalinityAit Ballagh et al. (2021)TOCrefTOC01.1%refractory carbon concPastor et al. (2018)∅0por00.8–surface porosityAit Ballagh et al. (2021)∅∞pordeep0.6–deep porosityAit Ballagh et al. (2021)δporcoeff2cmporosity decay coefficientAit Ballagh et al. (2021)kadsKads1.3–adsorption coefficientSoetaert et al. (1996a)
For the verification of the model output, data from Pastor et al. (2018)
corresponding to the diagenetic situation 26 d after an organic-rich flood
were used. We restricted our benchmark to data from the proximal station
(station A) near the river mouth, where the impact of this flood discharge
is more visible (Fig. 1).
Numerical integration, application, and implementation
Because the procedure is based on OMEXDIA, complete details of the
derivation can be found in that paper and is referenced therein (Soetaert et
al., 1996a). Here we recap the mathematical formulation of the
method of lines (MOLs) algorithm used by FESDIA. Direct differencing of Eqs. (8)–(10) results to
∂Ci∂t=Φi,i+1DΦi,i+1Ci+1-CiΦiΔzi,i+1Δzi-w∞Φ∞λi,i+1Ci+1-λi,i+1Ci+1ΦiΔzi-Φi-1,iDΦi-1,iCi-Ci-1ΦiΔzi-1,iΔzi+w∞Φ∞λi-1,iCi-1+1-λi-1,iCiΦiΔzi,
for a generic tracer C with a phase properties index Φ and DΦ denoting porosity and dispersive mixing term, respectively, for solid or
liquid. This equation is calculated such that the variables and parameters
are defined both at the centre of each layer zi and at the interface
between layers zi-1,izi,i+1. The
position at the centre of the grid is then given as
zi=zi-1,i+zi+1,i2.
Δzi represents the thickness of the i layer and Δzi,i+1 is the distance between two consecutive grid
layers. A Fiadeiro scheme (Fiadeiro and Veronis, 1977) based on the model's
Peclet number (a dimensionless ratio expressing the relative importance of
advective over dispersive processes) is used to set λi,i+1, thus providing a weighted difference of the
transport terms which reduces numerical dispersion.
Equations (8)–(10) implemented as Eq. (22) are integrated in time using an
implicit solver, called lsodes, that is part of the ODEPACK solvers
(Hindmarsh, 1983). This solver uses a backward differentiation method (BDF);
it has an adaptive time step, and is designed for solving systems of
ordinary differential equations where the Jacobian matrix has an arbitrary
sparse structure. The model output time and its time step (dt) is set by
the user and is generally problem-specific. Because of the aforementioned
challenge in observability of the massive flood event deposition, daily
resolution is most often used for user dt. However, there is the
possibility of obtaining higher resolution by decreasing dt.
The model application starts by estimating the steady-state condition of the
model using the high level command FESDIAperturb(). This steady-state condition is
calculated using iterative Newton–Raphson method (Press et al., 1992) and is
then used as a starting point for a dynamic simulation, with perturbation
times as in perturbTimes and depth of perturbation given as perturbDepth in the model function call.
As the event can be given as a deposit and mixing process, further
specification of the perturbation type (deposit or mix) is provided as an argument to
the simulation routine. In our case, we used only the deposit mode. The
event algorithm is used at the stated time point to estimate the model
porewater and solid properties driven by the instantaneous change in the
boundary condition. The concentrations are successively updated by their
diagenetic contributions during this time step. Afterward, this modified
profile is integrated forward in time. The model is written in Fortran for
speed and integrated using the R programming language (R Core Team, 2021)
via the “method of lines” approach (Boudreau, 1996). In addition, the
model made use of the event-handling capabilities' specific numerical solvers
written in the R deSolve package (Soetaert et al., 2010b). The R programming
language is used in the preprocessing routine for model grid generation
(Soetaert and Meysman, 2012), porewater chemistry parameter (Soetaert et
al., 2010a), steady-state calculation (Soetaert, 2014), and time integration
(Soetaert et al., 2010b). Further information about the model usage can be
found in the model-user vignette found in R-forge page (https://r-forge.r-project.org/R/?group_id=2422, last access: 2 August 2022).
Quantification of sediment diagenetic relaxation timescale
Quasi-relaxation timescale.
Given the strong non-linearity and coupled nature of the biogeochemical
system in question, we used an approximate approach to define the timescale
of relaxation. Recognizing that in a nonlinear system, a perturbed
trajectory is frequently arbitrarily divided into a fast, transient phase
and a slow, asymptotic stage that closes in on the attractor (i.e. steady-state concentration; Kittel et al., 2017), we proceeded to estimate the
relaxation timescale by using the time for which the memory of the perturbed
signature disappears. We estimate the relaxation timescale by first
calculating the absolute difference (φt) between
successive model output after the event, assuming that a slowly evolving
state will eventually converge to the pre-perturbed state as time after the
disturbance approaches infinity. This point-by-point concentration
difference between two successive discretized profiles is then terminated at
the point where the sum of absolute differences at each time point is less
than the threshold (i.e. given by the median over the entire time duration).
The relaxation time, τ, for each porewater profile species is then
defined as the first time this threshold is crossed. A similar technique was
employed by Rabouille and Gaillard (1990).
φt=1N∑i=1N||Xt+1i-Xti||In the limit of time (t):τt⇒φt≤φtthresholdwhere φtthreshold=φt‾≈seasonal background,
where N is the total number of grid point (i) used to discretize the
depth profile (Xt), and Xt+1 is the depth profile at
t+1 after the event.
This relaxation timescale calculation based on the disappearance of the
perturbed signal (via successive profile similarity) may differ from an
approach in which the profile returns to a pre-defined “old profile”.
Because the exactness of pre-flood and post-flood profiles is difficult to
quantify numerically (Wheatcroft, 1990), and the return to the former is
frequently driven by slow dynamics, the approach used here can provide a
window of estimate for which a particular signal fades toward the background
of a theoretically pre-perturbed signal.
Uncertainty in relaxation timescale estimate.
The uncertainty introduced by this technique is quantified using a
non-parametric bootstrap of the τ statistics. The objective of
bootstrapping is to estimate a parameter based on the data, such as a mean,
median, or any scalar or vector statistics but with less restrictive
assumptions about the form of the distribution that the observed data came
from (Efron, 1992).
The bootstrapping technique used to calculate the uncertainty in
the relaxation timescale. The resampled median about a reference provides a
replicate over which the standard error estimate is defined. The solid red
represents the expected value of the quantity estimated while the vertical
red line is the deviation from this expected value.
In this case, we employ a modified bootstrapping technique to estimate the
uncertainty in the relaxation timescale by resampling on the cutoff point
introduced in Eq. (23) (i.e. median, φt‾
of a given reference simulation). This calculation takes advantage of the
fact that the time series will be dominated by the slowly varying seasonal
cycle over a long time period away from the point of perturbation, with the
influence of the perturbation fading to the background. The variation of
this reference time series over time reflects the uncertainty in this median
threshold point. This variance, along with the reference cutoff value, can
be used to generate n random perturbations varying about the normative
threshold value. We can proceed to create a histogram of the replicate
threshold(s) distribution. The histogram of this distribution is
depicted schematically in the left margin of Fig. 3. The relaxation time in
each realization of the threshold is calculated (τi^). The
median absolute deviation from this ensemble of relaxation times is then
used to calculate the level of uncertainty in the statistics of interest
(timescale of relaxation – (τ^)). Figure 3 depicts this concept
schematically. It should be noted that this method eliminates the need to
rerun the deterministic model for each iteration, reducing the computational
burden of this technique.
The 95 % confidence intervals (σ^τlevel/2 and σ^τ1-level/2) are reported in this paper by
calculating the quantiles of this empirical distribution of σ^τ.
Model simulation
The model is initialized as explained in Sect. 2.2.9. Thereafter, for the dynamic
simulation, the model is spin-up for a sufficiently long time to attain
dynamic equilibrium (≥5 years). A 2 yr run is carried out
for the respective model application. The time step (dt) for dynamic
simulation is daily in order to match the frequency for which observation of
field data is possible. For specific numerical experiment, model
configuration required for the simulation will be detailed in Sect. 2.2.11, “end-member type numerical experiment”.
End-member type numerical experiment
For the numerical model experiment, we investigate the sediment's response
to two end-member types of deposition that can represent actual field
observations in the Rhône prodelta (Pastor et al., 2018).
Low OC content with high sediment thickness scenario (EM1). In this
scenario, we assume that a 30 cm new layer of sediment of degraded sediment
was deposited. This scenario can describe old terrestrial material and is
similar to the extreme case of flood event of May/June 2008 in the proximal
outlet of Rhône River where lateral transfer of low TOC sediment (around
1 %) was deposited on top of the previously deposited sediment (OC around
1.5 %–3 %) (Cathalot et al., 2010). Using the partitioning of the carbon as
explained in Sect. 2.2.7, an α value of 0.5 and 0.7 for
Corgfast and
Corgslow, respectively, was used to scale
the TOC profile in order to mimic this type of trend.
High OC with low sediment thickness scenario (EM2). For this, we
assumed a moderate 10 cm deposition of a new layer enriched in carbon during
a flood discharge event. This scenario can correspond to the end-member case
of November 2008 flood type with high TOC around 2.5 %, reaching more than
6 % in some sediment cores from the prodelta (Pastor et al., 2018), (most
likely composed of freshwater phytoplankton detritus, debris, and freshly
dead organisms) overlain on a less labile layer. In order to simulate this
type of pattern, an α value of 20 and 10 for
Corgfast and
Corgslow, respectively, was used to adjust
the TOC profile to such high-deposit OC scenario.
Except for the α and the thickness of the flood deposit, all other
parameters were held constant in all numerical experiments. The time of the
event occurrence in both scenarios were initialized at a period corresponding
to published dates for May and November 2008 flood deposition as reported in
Pastor et al. (2018). This helps to provide some realism to this
hypothetical case study and appropriate context to the environmental
regime when these events occur.
Sensitivity analysis
Lastly, we conducted a sensitivity analysis of the relaxation timescale for
oxygen, sulfate, and DIC concentrations in terms of their variation to the
thickness of the new sediment layer and the quantity of organic
carbon introduced by the deposition.
We assumed a 15 cm average deposit thickness and conducted simulations with
a thickness variation ranging from 1 to 30 cm. A 5 cm thickness increment
was used for the sensitivity analysis. The α value is calculated in
the same way: assuming a 1:1 ratio in the fast and slow OC fractions, and
because deposited sediment can be highly refractory in nature, we
geometrically conducted simulations with values ranging from 0.3 to 35. This
was done only by changing the α corresponding to
Corgfast with the slow fraction fixed as
1. We also made sure that both series are equilateral in length, and that
the values were chosen to span the range of values in EM1 and EM2, thus
bracketing the normative value for the end-member case. This range
encompasses the large spectrum of flood deposits such as those experienced
in the Saguenay fjord, Canada (Deflandre et al., 2002; Mucci and Edenborn,
1992), the Rhône prodelta, France (Pastor et al., 2018), and in the Po
River, Italy (Tesi et al., 2012).
ResultsQualitative model performance: Cevenol flood in the Rhône prodelta
In order to compare the model evolution to field data, we made a comparison
between the simulated profiles 26 d after a flood layer deposition and
data collected in the Rhône prodelta in December 2008 (observed data
collected 26 d after a Cevenol flood). During this flooding period,
riverine discharge delivered 0.4×106t of sediment which amounted to
approximately 10 cm of sediment deposited in the site A of depocenter (Fig. 4).
The general pattern of the simulated profile agrees well with the observed
data (Fig. 4). The newly introduced organic carbon-rich sediment resulted in
rapid oxygen consumption. The data for total organic carbon (TOC) shown in
Fig. 4 suggest a good agreement with the model, with high TOC (2.5 wt %–2.0 wt %) deposited at the upper 10 cm. Twenty-six days after the flood, the oxygen
concentration dropped from 250 µM at the new sediment
interface to nearly zero at 0.2 cm depth, and oxygen may have already
returned to pre-flood levels; the simulated porewater profile was within the
data's range (Fig. 4). The model diffusive flux of oxygen at this period was
18 mmol m-2 d-1 while the measured DOU (diffusive oxygen uptake) flux was 16.6±2.9 mmol m-2 d-1.
Overall, the model–data trend was satisfactory with observed depth
distribution of sulfate (SO42-)
26 d after the flood event fitted well, without much parameter
fine-tuning. Only the sedimentation rate of the sediment was changed from
0.027 to 0.06 cm d-1 in order to match the observed
distribution at depth. Sulfate reduction was high in the new layer. However,
below the flood layer, the SO42-
concentration in the data seem to asymptote to a value of 10 mM at 25 cm,
while the model simulates complete sulfate depletion below 20 cm (Fig. 4).
The DIC profile shows a similar trend to the data collected after the flash
flood. Within the depth interval of data, the model tends to follow the
data. It drifts at lower depths, on the other hand, by overestimating the
concentration of DIC observed at deeper layers. Similarly, the modelled
NH4+ shows a gradual increase
with depth, and the model overestimates the production of
NH4+ below 15 cm (Fig. 4).
Model and observation depth profile of TOC (%), SO42-,
DIC,NH4+, and O2 for November/December event in station A
(Rhône prodelta). The green and red dashed lines depict the vertical
depth profile of the model before (3 d) and after (1 d) the flood
deposit. The blue solid line represents the model result on the day the
observations were collected (26 d after the flood, as indicated by the
black circle).
Numerical experiment on end-member scenariosLow carbon, high thickness scenario (EM1)
With a test case of 30 cm of new material deposited during the event (EM1)
in the spring, the sediment changes as thus: prior to the event, the oxygen
penetration depth (OPD) was about 0.17 cm. The OPD increases to 1.17 cm
after the deposition of these low OC materials. The model showed a gradual
return to its previous profile within days, with the OPD shoaling linearly
with time (Table 4). By day 5, oxygen has returned to the pre-flood profile
with similar gradient to the pre-flood state.
Against a background OM flux following the introduction of the flood layer,
the sediment responded quickly. As a result, the perturbation has a
significant effect on sulfate penetration depth, with concentration
remaining nearly constant within the perturbed depth (≈20 cm). This corresponds to the bottom water concentration (30 mM)
trapped within the flood deposit. Within that layer, sulfate reduction rate
was low with an estimated integrated rate of 2.14 mmol C m-2 d-1 from the
surface to 30 cm.
Model vs. data comparison for oxygen penetration depth (OPD), flux
of oxygen, sulfate, and DIC (26 d after deposition).
Scenario 1 (EM1): model evolution for sulfate following
deposition. Relative deviation of successive profile with time shown below.
Dashed vertical line signifies cutoff point by the median (dashed horizontal
line). Inset: histogram of bootstrap estimate of sulfate relaxation
timescale for EM1 with 95 % confidence interval.
Scenario 1 (EM1): model evolution for DIC following deposition.
Relative deviation of successive profile with time shown below. Dashed
vertical line signifies cutoff point by the median (dashed horizontal line).
Inset: histogram of bootstrap estimate of DIC relaxation timescale for EM1
with 95 % confidence interval.
Below the interface with the newly deposited layer, the sediment is enriched
in OM whose mineralization results in a higher sulfate reduction rate (SRR)
at the boundary that delineates the newly deposited layer and the former
sediment–water interface. The simulated SRR falls from 437 mmol C m-3 d-1 at the former
sediment–water interface (now re-located at 26 cm) to 24 mmol C m-3 solid d-1. This
high interior sulfate consumption at the boundary correlates well with the
higher proportion of reactive organic material buried by the new layer
containing less reactive material. From day 10, the consumption of this OM
stock by sulfate controls the shape of the profile (Fig. 5). This anoxic
mineralization via sulfate reduction will continue until the entire stock of
carbon is depleted 50 d after deposition. Following that, OM
mineralization via sulfate reduction shift becomes more intense at the top
layer by day 60 (2 months after the event), when it begins to gradually
evolve to the typical depth-decreasing sulfate profile. By day 115
(∼4 months), the profile had almost completely returned to
its pre-flood state. We estimate that it took approximately 4 months for
sulfate to relax back to within the range of background variability (with
lower and upper bootstrap estimate between 92–139 d).
Correspondingly, OC mineralization products (such as DIC) were significantly
lower in the upper newly deposited layer, as a consequence of the reduced
quantity of OC brought by the flood. This concentration increased with depth
to about 80 mM. Starting from the deposition, higher production of DIC
below the former SWI led to a distinct boundary in the sediment: a DIC-depleted layer above an increasing DIC with concentrations up to 75 mM trapped in the region below the new–old sediment horizon 20 d after deposition (Fig. 6). This increased DIC production continued
despite complete exhaustion of buried labile fraction with mineralization
driven by the slow decaying component. The depth gradient caused by the
increased DIC production enhances diffusive DIC flux. Following that initial
period, DIC began to revert to its previous state. This slow
re-organization, mostly driven by diffusion continues, with an estimated
recovery time of 5 months (with a 95 % bootstrap confidence interval of
137–147 d respectively), as it temporarily lags behind
SO42- in its return to the
previous pre-flood state.
High carbon, low thickness scenario (EM2)
A flood deposition scenario of 10 cm thick material with enhanced OC content
was used for the other end-member case experiment (EM2) in autumn. In this
scenario, the modelled sediment exhibits a variety of response
characteristics. The newly introduced sediment resulted in rapid oxygen
consumption. The OPD decreased to 0.74 cm shortly after the event, according
to the model, and stabilized there for days. There was no visible
deformation in the shape of oxygen during its recovery trajectory, and total
oxygen consumption for organic matter mineralization decreased by 8 %
during the first 2 d after the event, from 12 to 11 mmol O2 m-2 d-1.
The SO42- concentration that
developed as a result of the deposition showed two gradients: a
concentration gradient from 30 mM at the “new” sediment water interface to
26 mM in the newly deposited layer (Fig. 7). Accordingly, the DIC in the
corresponding depth layer gradually increased up to 20 mM (Fig. 8). An
intermittent increase in SO42-
was simulated below the new interface, at the boundary with the “old”
sediment–water interface (SWI), reaching up to 29 mM from 9 to
12 cm (Fig. 7). This layer, which corresponded to the depth horizon where
the new layer gradually mixed with the old layer, resulted in less sulfate
reduction and DIC production in comparison to the new layer. Porewater
SO42- concentrations decreased
monotonically with depth from this interface, with a corresponding increase
in DIC. Within 26 d of the event, the sulfate profile appears to be
returning to its original shape. By then, 75 % of the newly introduced
fraction of OM had been depleted, with OM remineralization in the upper
layer fuelled by the small amount of remaining detrital materials. As the
temporal memory of the deposition fades, the profile continues to gradually
evolve towards the background, fed by the slow decaying OM, up to day 90,
when the sulfate profile appears to have reached a similar pre-flood state.
In this scenario, the estimated
SO42- and DIC relaxation
timescales were around 3 months (91 d for
SO42- and 102 d for DIC)
(Fig. 7).
Scenario 2 (EM2): model evolution for sulfate following
deposition. Relative deviation of successive profile with time shown below.
Dashed vertical line signifies cutoff point by the median (dashed horizontal
line). Inset: histogram of bootstrap estimate of sulfate relaxation
timescale for EM2 with 95 % confidence interval.
Scenario 2 (EM2): model evolution for DIC following deposition.
Relative deviation of successive profile with time shown below. Dashed
vertical line signify cutoff point by the median (dashed horizontal line).
Inset: histogram of bootstrap estimate of DIC relaxation timescale for EM2
with 95 % confidence interval.
Relaxation timescale (τ) in days as function of deposited
sediment thickness and enrichment factor (α) for degradable OM.
Sensitivity of relaxation time to variation in enrichment factor
(<inline-formula><mml:math id="M354" display="inline"><mml:mi mathvariant="italic">α</mml:mi></mml:math></inline-formula>) and sediment thickness (<inline-formula><mml:math id="M355" display="inline"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant="normal">pert</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>)
We then examine the sensitivity analysis of the relaxation timescale (τ) for oxygen, sulfate, and DIC for variation in sediment deposit thickness
(Zpert) and the concentration factor for
Corgfast enrichment (α) covering
values ranging between the two EM scenarios.
Over all runs varying the enrichment factor (α) and the thickness of
the flood input layer, relaxation time for oxygen varied from 2 d for a
flood-deposited layer consisting of a thin layer of high concentration of
labile OC to 9 d for a thicker deposited layer with low concentration of
labile OC. In contrast, the relaxation timescales for
SO42- and DIC were significantly
longer than those for O2 (3 to 4 months). In
addition, the relaxation timescale surface structure for
SO42- appears complex with
divergence gradient at mid-depth of 15 cm. For deposited depth layers above
5 cm and at low α value, the relaxation time for
SO42- varied between 75–100 d (2–3 months). Below 5 cm (bioturbated depth imposed in the model),
relaxation time was constant across all α variations (100 d). As
organic enrichment (α) and thickness increase, the model estimates a
longer relaxation time with a maximum time span of 105 d.
Similar variation of relaxation time for DIC was simulated for different
α and sediment deposit thickness. However, unlike
SO42-, relaxation time for DIC
varies smoothly across the range of α and thickness combinations
with relatively constant relaxation time (100 d) at low thickness and
α combinations. The relaxation time increased exponentially as
sediment deposit thickness and labile OC concentration increased (α), with maximum recovery time (171 d/6 months) simulated at the
extremes of both combinations (Fig. 9).
Discussion
In highly dynamic coastal ecosystems, such as RiOmar (river-dominated ocean margins) systems, driven by
seasonal variability and meteorologically extreme events, the response of
early diagenetic processes to time varying deposition of organic matter is
generally non-stationary (Tesi et al., 2012). While dynamic equilibrium as a
steady-state condition may be reasonable in the case of seasonal
variability, such an assumption may fail in cases of instantaneously
event-driven deposition. An intermittent supply of sediment and OC, like
those presented here, can cause a change in the system's properties on a
short- or long-term basis. Previous works have highlighted excursions in
sediment redox boundary (Katsev et al., 2006), flux of solutes at the
sediment–water interface (Rabouille and Gaillard, 1990), and
modification of other system properties due to depositional flux of organic
matter. Thus, the premise of steady-state conditions in early diagenetic
processes which often depends on the temporal resolution of the observation
might need revisiting especially in areas of episodic sedimentation
(Wheatcroft, 1990; Tesi et al., 2012). Here, we discuss the evolution and
dynamics of a non-stationary sedimentary system following a singular
perturbation.
Model representation and utility
Non-steady-state models are increasingly being applied in dynamic coastal
environments, but they are still primarily based on forcing from smooth
varying boundaries that mimic seasonal forcing or long-term variability
(Soetaert et al., 1996b; Rabouille et al., 2001a; Zindorf et al., 2021).
Explicit consideration of abrupt changes in the upper boundary of the model
caused by events such as landslides, flash flooding, turbiditic transfer of
materials on a continental slope, and trawling is still relatively
uncaptured by these models (but see De Borger et al., 2021, for inclusion of
erosion events). In this paper, we adapt OMEXDIA (Soetaert et al., 1996a), a
well-known reaction transport model, to investigate the changes in the solid
and liquid phases during massive deposition event. Our efforts highlight the
algorithm's utility in incorporating this process with minimal numerical
issues. The model represented the basic characteristics of the data derived
from the November/December 2008 flood event at station A in the Rhône
Delta's depocenter (Fig. 4). The simulated flux was also in agreement with
the estimate from field data, as diffusive oxygen uptake (DOU) rate sampled
26 d after the event (8 December 2008) was 16.6±2.9 mmol m-2 d-1
(Cathalot et al., 2010) while the estimate from the model was 18 mmol m-2 d-1.
As the inclusion of such discontinuity in PDE(s) presents numerical
challenges in classic solvers, the implementation utilized by our model
ensures such difficulties are overcome. This is the result of improved
development of solvers adapted to such a problem (Soetaert et al., 2010b).
This difference in the approach employed here distinguishes ours from other
published models (e.g. Berg et al., 2003; van de Velde et al., 2018) with similar
scientific motivation for time-dependent simulation. Overall, the validation
of the model output with field observations lends some confidence in using
the model in scenarios involving abrupt changes in boundary conditions and
investigating biogeochemical changes in the sediment as a result of such an
event. This is despite the model underestimation of the amplitude of
sulfate and DIC at depth which can be improved with better optimization of
some parameters, especially those derived from previous studies that might not
be suited for such a flooding regime or with better process resolution
relating to these pathways. Nonetheless, there are advantages to this model
especially in the case of episodic flood deposit events, where only a
snapshot of data is available at any given time. Modelling tools capable of
simulating this event with high fidelity can provide continuous information
of the system state and help fill in data gaps needed to understand the
sediment's response on different timescales.
Role of end-member flood input OM in the diagenetic relaxation dynamics
Flooding events can transport large amounts of material through the river to
transitional coastal environments such as deltas and estuaries. River floods
can account for up to 80 % of terrigenous particle inputs (Antonelli et
al., 2008; Zebracki et al., 2015), and they can have a significant impact on
geomorphology (Meybeck et al., 2007), ecosystem response, and biogeochemical
cycles (Mermex Group et al., 2011). If the source materials have a different
organic matter composition (Dezzeo et al., 2000; Cathalot et al., 2013), the
rapid deposition of these flood materials can alter diagenetic reactions and
resulting fluxes.
Furthermore, the relaxation timescale associated with the sediment recovery
following this external perturbation can be important in terms of the process
affecting the biogeochemistry of solid and solutes species. With a series of
numerical experiments ranging in between two end-members of the input
spectrum for flood events such as those in the Rhône prodelta (Pastor et
al., 2018), our study revealed contrasting sedimentary responses and
associated typical timescales at which porewater profiles relax back to
undisturbed state. Using a simple metric for estimating relaxation timescale
of the perturbation, our calculations for the first end-member scenario (EM1)
show that the upper bound of the timescale of relaxation for oxygen is 5±3 d, whereas it was approximately 2±2 d for the second end-member scenario (EM2). This reflects the property of
oxygen, which quickly approaches a steady-state situation after an event
(Aller, 1998). This viewpoint is supported by an ex situ controlled
laboratory setup. In their studies, Chaillou et al. (2007) demonstrated that
after gravity-levelled sediment was introduced, oxygen consumption quickly
recovered to its first-day level, with a sharp response time of 50 min
and gradual shoaling of OPD within 5 d. We conclude that the tiny
difference in oxygen relaxation and diagenetic response between the two
scenarios can be attributed to the slow kinetic degradability of the
refractory carbon deposited in the first scenario versus the labile nature
of the deposit in the second scenario. This kinetically driven OM
degradation has been extensively studied and provides the basis for the
reactive continuum in early diagenesis models (Middelburg, 1989;
Jørgensen and Revsbech, 1985; Burdige, 1991).
Other terminal electron acceptors (TEAs), such as
SO42-, relax toward natural
variation over a longer timescale than oxygen. For EM1, our simulation
predicts a sulfate relaxation time of 117 d with a 95 % confidence
interval (CI) estimate between 92 d (lower CI) and 139 d (upper CI) while in the case of EM2, we estimate a sulfate relaxation time of 91 d with comparatively low temporal variability (lower CI–80 and upper CI–103 d). This difference in relaxation time is caused by the differences
in sediment characteristics and how their mineralization occurs over the
sediment layer. In the first scenario, organic-rich sediment is buried by
less reactive new material. The buried sulfate fraction is reduced faster
than in the new layer above and controls the short-term recovery. As the
buried carbon stock depletes and the physical imprint of the flood
deposition fades, the profile begins to revert to its pre-flood shape. The
post-flood evolution for the second scenario (EM2), on the other hand,
differs in that the OM is consumed in the classical manner, with decreasing
sulfate consumption with depth, caused by top-down control of the OM flux
that adds OM to the sediment surface.
Such a long time lapse for the recovery of an element with a complex
pathway, such as SO42-, has been
reported in the literature (Anschutz et al., 2002; Stumm and Morgan, 2012;
Chaillou et al., 2007). Similarly, estimates from our simulation for each
end-member scenario indicate that mineralization products such as DIC have a
longer relaxation time. This is especially true for the first scenario as
opposed to the second, with evidence of slow convergence at depth within the
simulation timescale for the first scenario. We estimate that DIC will
recover to its pre-deposition state in 5 months for EM1 and in a
comparatively shorter time for EM2 (3 months). This lag in DIC recovery
could be attributed to the fact that its post-flood dynamics is governed by
the slow decaying detrital material that contribute to the already buried
refractory carbon. This long-term quasi-static behaviour of the porewater
concentration despite such dynamic introduction of flood input can be
understood by introducing the concept of a biogeochemical attractor effect – a similar analogy
to the Lorenz attractor (Lorenz, 1963). This idea is derived from the
mathematical theory that describes chaos in the real world (Strogatz, 2018;
Ghil, 2019). The existence of a “biogeochemical attractor” may explain why
multiple temporal data sets in the Rhône River prodelta show a similar
diagenetic signature from spring to summer (Rassmann et al., 2016; Dumoulin
et al., 2018). Our timescale analysis estimates that such rapid system
restoration is indeed plausible and of the correct order of magnitude, based
on the range of uncertainty reported here.
In addition, our calculations show that the timescale of return to the
previous “pre-flood” profile is bracketed by the range of recovery due to
purely molecular diffusion, putting an upper bound on our estimate. For
example, using the Einstein's approximation, a species such as oxygen with a
sediment diffusion coefficient (Ds) of 1.52 cm2 d-1 takes
approximately 300 d to be transported solely by diffusion through a 30 cm
sediment column and approximately 30 d for a 10 cm sediment column.
Similar scaling argument could be made for species such as
SO42- (Ds=0.86 cm2 d-1) with
>500 d to be transported through 30 cm and ≈60 d for 10 cm. Because our estimates are less than these values, it
suggests that processes other than diffusion (thickness effect) may
contribute to relaxation control. It emphasizes the importance of
biogeochemistry (OM kinetic) in modulating the response after the event.
Besides that, any long-term recovery timescale is governed by the solid
deposited. In comparison to the timescale of relaxation roughly estimated
from field data (Cathalot et al., 2010), our estimate shows the right order
of magnitude.
The relaxation time may also vary depending on the diagenetic interaction,
and the characteristics of the organic matter available for degradation.
This difference in characteristics was partially imposed in our study by
assuming variations of α in the new deposit. The empirical
observation of sediment characteristics associated with flood input dictates
this parametric turning to match the TOC characteristics (Pastor et al.,
2018; Deflandre et al., 2002; Mucci and Edenborn, 1992; Tesi et al., 2012;
Bourgeois et al., 2011). However, more data from the field and laboratory
experiments that resolve the OM composition of flood deposits are required
to constrain the choice of this numerical parameter.
Control of relaxation time by sediment deposit properties
With the sensitivity analysis, we further explore the variation of
relaxation timescale under variation of the thickness of layer and
enrichment factor of input material given by α in our model. The
model's sensitivity analysis reveals that the thickness and concentration of
the reactive fraction of TOC control the relaxation time across a wide range
of deposited sediment perturbation characteristics (Fig. 9).
In terms of the recovery time as a function of the availability of labile
OC, our results revealed a contrasting pattern for oxygen and sulfate.
Several factors related to how different oxidants react with sediment matrix
disturbances can explain these differences:
With oxygen that has a high molecular diffusion coefficient, variations in
relaxation time depend on the levels of labile OC, with thin sediments
containing a high level of labile OC showing a shorter recovery time than
thicker sediments with a low OC content. This pattern can be attributed to
the higher relative importance of oxygen consumption in OM-poor sediment
relative to the OM-rich sediment.
For low thickness deposits, sulfate and DIC relaxation times were more or
less constant. However, a longer relaxation time was simulated for larger
deposits and higher labile OC. This can be attributed to the increased
distance required for solutes to migrate back after the event. This is
clearly the case for sediment thicknesses greater than 14 cm. Such two-way
dynamics could be explained by the fact that biological reworking and
physical mixing within the surface mixing layer (SML) can improve OC
degradation by promoting the replenishment of electron acceptors (i.e.
oxygen, sulfate, nitrate, and metal oxyhydroxides) (Aller and Aller, 1992;
Aller, 2004), resulting in a shorter recovery time for the porewater profile
to reorganize within the SML.
This critical depth could also be the distance horizon at which the slow
diffusion of the profile when retracting back to its pre-flood profile
becomes an important factor in controlling the relaxation timescale. This is
especially true for DIC, where the connection is more obvious. It has been
proposed that when flood deposits extend beyond the sediment bio-mixing
depth, the relaxation time for the constituent species is determined by the
concentration gradient between the historical and newly deposited layers
(Wheatcroft, 1990). In our sensitivity analysis, higher α
corresponds to higher Corgfast
concentrations at depth, resulting in a case of enhanced OC degradation
(both at the surface and within the sediment matrix). This depletes electron
acceptors such as sulfate, which are required for OM mineralization at this
depth. The slow diffusion across the displaced distance, on the other hand,
cannot quickly compensate for its demands, which may explain the longer
relaxation time. In other words, a higher concentration of OC in a region
where all oxidants are nearly consumed results in a profile that takes a
relatively longer time to recover to its previous state due to the
constraints imposed by oxidant availability. This viewpoint is consistent
with previous research from the Rhône prodelta area, where a minimum
transport distance of 20 cm is suggested for efficient connection with the
SWI; above which several processes are decoupled (Rassmann et al., 2020) and other eutrophic systems, where evidence of large accumulation of
organic matter in subsurface sediments serves as a constraint on system
restoration (Mayer, 1994; Pusceddu et al., 2009). Indeed, more observational
and experimental studies are needed to better understand these processes.
Model limitations and future development
Because it is based on the well-established OMEXDIA model, FESDIA has
several capabilities that make it suitable for a wide range of application
domains for non-steady-state early diagenetic simulation. However, due to
assumptions made during model development, some limitations in model usage
must be considered:
First, we assumed that porosity is time independent. This may not be the
case in some coastal systems that receive sediment materials from regions
with distributary channels, which contribute particles of varying origin and
grain size (Grenz et al., 2003; Cathalot et al., 2010). The composite
sediment that is eventually transported to the depocenter by a flood event
may differ in porosity, and thus vary temporarily depending on when and where
the source materials are derived during the flood event. In this case, model
estimates of fluxes in dissolved species may be over/underestimated. The
resulting porosity in the new layer is barely predictable and could range
between 0.65 and 0.85 in the proximal zone of the prodelta (Grenz et al.,
2003; Cathalot et al., 2010), allowing us to justify our assumption.
Second, in our examples, we assumed that the burial rate and bioturbation
were constant. With the introduction of these flood events, such assumptions
may be called into question (Tesi et al., 2012). In addition, benthic
animals respond to other perturbation event such as trawling in ways that
may warrant explicit description of their recovery, which is linked to
bioturbation (De Borger et al., 2021; Sciberras et al., 2018). While some
coastal sediment burial rates have been shown to vary seasonally (Soetaert
et al., 1996b; Boudreau, 1994), in the proximal zone of the Rhône
prodelta, approximately 75 % of sediment deposition can occur during the
flood (e.g. 30 cm d-1), with the remaining 25 % distributed throughout the
year at a low range daily constant rate (0.03 cm d-1). The dominance of flood
deposition over non-flood sedimentation, and the low bioturbation
rate observed in the Rhône prodelta (Pastor et al., 2011), prompted the
use of constant rate in the application shown here. Moreover, we designed
the FESDIA model to allow for the use of a temporarily varying rate constant
and coefficient for these processes, and the possibility of imposing
an observational time series in cases where such data exist.
The current FESDIA version does not include a diffusive boundary layer,
which can be important for material exchange between the overlying bottom
water and the sediment. This is critical for calculating fluxes of species
such as O2, where the depth extent of the DBL (diffusive boundary layer) zone is comparable to the
depth at which oxygen consumption occurs (Boudreau and Jorgensen, 2001). As
a result, the current version of FESDIA may overestimate the flux of
O2. However, because the primary focus of this paper is on the
relaxation dynamics of species (SO42- and DIC), where the
DBL has negligible impact on the relaxation time and overall diagenetic
processes (Boudreau and Jorgensen, 2001), the simplification presented here
is justified. Even for oxygen, the inclusion of DBL which might result to
corresponding change in the concentration at the SWI only have a marginal
effect on its relaxation time (<2 d – within the range of
uncertainty reported here), so the conclusions drawn in the case studies
discussed here are still valid.
In terms of future development, we hope to improve the model's diagenetic
pathways, particularly for the iron and sulfur cycles. Furthermore,
processes such as calcite formation have been shown to affect DIC profile by
10 %–15 % in the proximal sites of Rhône prodelta (Rassmann et al.,
2020), thus might necessitate inclusion in future versions of the model. This
will enable FESDIA to account for carbonate system dynamics in marine
sediment which can play an important role in the coast carbon cycle (Krumins et
al., 2013).
Relaxation time metric: limitation and perspective
While one main focus of this study is on providing a quantitative estimate
of relaxation time, the difficulty of objectively defining what
relaxation means necessitates some commentary. This difficulty is not unique to
marine biogeochemistry, as accurate quantification of recovery time is an
open research question in other fields. In the context of a sedimentary
system, Wheatcroft (1990) proposed that determining “dissipation time”
(analogous to our “relaxation time”) can be subjective when it comes to
signal preservation after sediment event layer deposition. The difficulties
are exacerbated by previous work on episodic pulse on sediment
biogeochemistry (Rabouille and Gaillard, 1990), in which two metrics for
estimating relaxation timescale for silica were proposed. Outside of benthic
early diagenesis, Kittel et al. (2017) proposed two generic metrics for
systems with well-defined asymptotic properties that can be applied to a
distance function from a given target (subject to certain mathematical
assumptions). Because porewater profiles are inherently nonlinear, and
biogeochemical pathways in sediment are tightly coupled, the mathematical
suggestion of asymptoticity using such a distance metric for an evolving
profile converging toward the target proposed in that paper is frequently not
met. This is the case for our investigation. Overall, while we provide a
first-order approximation of relaxation time following perturbation for some
model-state variables, these studies highlight also some of the challenges
associated with defining the timescale at which a signal can be validly
assumed to have returned to its prior state. However, our method allows a
full discussion of relaxation times for the main biogeochemical pathways.
Conclusion
The need to comprehend extreme events and their relationship to marine
biogeochemistry prompted the development of novel methods for diagnosing
flood-driven organic matter pulses in coastal environments. In this paper,
we propose a new model (FESDIA) for characterizing flood deposition events
and the biogeochemical changes that result from them. This type of event can
have an impact on the benthic communities and the response of the whole
ecosystem (Smith et al., 2018; Bissett et al., 2007; Gooday, 2002). Our
modelling study shows that the post-depositional sediment response varies
depending on the input characteristics of the layer deposit. For instance,
we tested the combined effect of enrichment of labile organic carbon and
deposit thickness on space–time distribution and relaxation time of key
dissolved species (oxygen, sulfate, DIC). This integral timescale of
relaxation is constrained by the intrinsic properties of the solutes
(diffusion) and the characteristics of the flood input (thickness and
concentration of labile organic carbon). In essence, the findings from this
study highlight the importance of the quantity and quality of organic carbon
in modulating the sediment response following such a singular perturbation,
and the role of flood events with heterogeneous quantitative
contributions in the coastal ocean.
Biogeochemical reaction
The full model equation explained in Sect. 2.3.2 is described fully below.
Organic matter is composed of three fractions: fast degradable organic
matter, slow degradable organic matter, and refractory organic matter. Given
the long timescale for the degradability of the refractory OM, it is
parameterized using TOCref as the asymptotic value. For the two other
fractions, five mineralization pathways are included: aerobic respiration
(AP), denitrification (DE), dissimilatory iron reduction (DIR), sulfate
reduction, and methanogenesis (MG).
Degradation of organic matter:
Cprod=rFast×Corgfast+rSlow×Corgfast×1-ϕϕNprod=rFast×Corgfast×NCratioCfast+rSlow×Corgfast×NCratioCslow×1-ϕϕ.
Limitation terms: the limitation of a mineralization pathway by the availability of the
oxidant is modelled by a MOND-type hyperbolic limitation function with
inhibition of a pathway represented by a reciprocal hyperbolic function.
Oxicminlim=O2O2+kO2×1limDenitrificlim=NO3NO3+kNO3×1-O2O2+kinO2den×1limFeredminlim=FeOH3FeOH3+kFeOH3×1-NO3NO3+kinNO3ano×1-O2O2+kinO2ano×1limBSRminlim=SO4SO4+kSO4×1-FeOH3FeOH3+kinFeOH3ano×1-NO3NO3+kinNO3×1-O2O2+kinO2ano×1limMethminlim=1-SO4SO4+kinSO4ano×1-FeOH3FeOH3+kinFeOH3ano×1-NO3NO3+kinNO3ano×1-O2O2+kinO2ano×1limlim=1Oxicminlim+Denitrificlim+Feredminlim+BSRminlim+Methminlim.
Depth-dependent kinetic reaction: this limitation is used to reconstruct the vertical distribution of the
successive mineralization pathways with a rescaling term “lim” to ensure
that the sum of the individual pathway equal the total degradation rate.
Oxicmin=Cprod×Oxicminlim×limDenitrific=Cprod×Denitrificlim×limFeredmin=Cprod×Feredlim×limBSRmin=Cprod×BSRlim×limMethmin=Cprod×Methminlim×lim.
Secondary reaction: the re-oxidation of reduced substance and other secondary reactions are
modelled with a first-order reaction term.
Nitri=Rnit×NH4×O2O2+ksnitri(Nitrification)Feoxid=RFeOH3×Fe×O2(Ironoxidation)H2Soxid=RH2S×H2S×O2(sulfideoxidation)CH4oxid=RCH4×CH4×O2(Methaneoxidation)AOM=RAOM×CH4×SO4(AnaerobicoxidationofmethaneFeSprod=RFeSprod×Fe×H2S(FeSproduction)DICprodCH4=-0.5×Methmin+CH4oxid+AOM(DICproductionfrommethane).
Removal of sulfide via FeS production and oxidation with oxygen:
FeSprod=RFeSprod×Fe×H2S(FeSproduction)H2Soxid=rH2Soxid×O2×H2S(Sulfideoxidation).
Rate of change in state variable:
∂Corgfast∂t=transport+rFast×Corgfast∂Corgslow∂t=transport+sFast×Corgslow∂O2∂t=transport-Oxicmin-1.5Nitri-0.25FeOxid-2H2Soxid-2CH4oxid∂NH3∂t=transport+Nprod-Nitri1+NH3ads∂NO3∂t=transport-0.8Denitrific+Nitri∂CH4∂t=transport-DICprodCH4∂DIC∂t=transport+Cprod+DICprodCH4∂Fe∂t=transport+4×Feredmin-Feoxid-FeSprod∂FeOH3∂t=transport+Feoxid-4Feredmin×ϕ1-ϕ∂H2S∂t=transport+0.5BSRmin-H2Soxid-FeSprod+AOM∂SO4∂t=transport-0.5BSRmin+H2Soxid-AOM.
Code availability
As a whole, the model is bundled as an R package for easy accessibility and
can be downloaded from R-forge (https://r-forge.r-project.org/R/?group_id=2422, last access: 2 August 2022). The most recent version of the model, and its evolution, can be found on the project development page
(https://r-forge.r-project.org/projects/diagenesis/, last access: 2 August 2022)
with subsequent expected release in CRAN. Full R vignette illustrating the
capabilities of the model can be found on the model doc folder. The version
used to produce the results used in this paper is archived on Zenodo
(10.5281/zenodo.6369288; Nmor and Soetaert, 2022), along with
the input data and scripts to recreate the simulation presented in this
paper. FESDIA users should cite both this publication and the relevant
Zenodo reference.
Data availability
The data and paper used to evaluate the model (Pastor et al., 2018) can be
found at 10.5281/zenodo.6369288 (Nmor and Soetaert, 2022). Users of the data should cite Pastor et al. (2018)
and Ait-Ballagh et al. (2021).
The supplement related to this article is available online at: https://doi.org/10.5194/gmd-15-7325-2022-supplement.
Author contributions
All authors contributed to the paper in several capacities. The project was
supervised by CR and EV. SIN, CR, and EV conceptualized the method design, result
interpretation, and assist in the initial draft of the paper. Model
development was jointly design by KS and SIN. LP and BL provided insight on
the data used in the model.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Financial support
This research has been supported by the Université de Versailles Saint-Quentin-en-Yvelines (Ecolé doctorale des science de l'environment, Ile de France (grant no. ED129)) and the Centre National de la Recherche Scientifique (INSU EC2CO DELTARHONE).
Review statement
This paper was edited by Sandra Arndt and reviewed by two anonymous referees.
References
Ait Ballagh, F. E., Rabouille, C., Andrieux-Loyer, F., Soetaert, K.,
Lansard, B., Bombled, B., Monvoisin, G., Elkalay, K., and Khalil, K.:
Spatial variability of organic matter and phosphorus cycling in rhône
river prodelta sediments (NW mediterranean sea, france): A model-data
approach, Estuaries Coasts, 44, 1765–1789, 2021.
Aller, R. C.: Mobile deltaic and continental shelf muds as suboxic,
fluidized bed reactors, Marine Chem., 61, 143–155, 1998.
Aller, R. C.: Conceptual models of early diagenetic processes: The muddy
seafloor as an unsteady, batch reactor, J. Marine Res., 62,
815–835, 2004.
Aller, R. C. and Aller, J. Y.: Meiofauna and solute transport in marine
muds, Limnol. Oceanogr., 37, 1018–1033, 1992.
Anschutz, P., Jorissen, F., Chaillou, G., Abu-Zied, R., and Fontanier, C.:
Recent turbidite deposition in the eastern Atlantic: early diagenesis and
biotic recovery, J. Marine Res., 60, 835–854, 2002.Antonelli, C., Eyrolle, F., Rolland, B., Provansal, M., and Sabatier,F.: Suspended sediment and 137Cs fluxes during the exceptional December
2003 flood in the Rhône River, southeast France, Geomorphology, 95,
350–360, 2008.
Archer, D., Morford, J., and Emerson, S.: A model of suboxic sedimentary
diagenesis suitable for automatic tuning and gridded global domains, Global
Biogeochem. Cycles, 16, 17-1–17-21, 2002.
Arndt, S., Jørgensen, B. B., LaRowe, D. E., Middelburg, J., Pancost, R.,
and Regnier, P.: Quantifying the degradation of organic matter in marine
sediments: a review and synthesis, Earth-Sci. Rev., 123, 53–86, 2013.
Bauer, J. E., Cai, W.-J., Raymond, P. A., Bianchi, T. S., Hopkinson, C. S.,
and Regnier, P. A.: The changing carbon cycle of the coastal ocean, Nature,
504, 61–70, 2013.
Berg, P., Rysgaard, S., and Thamdrup, B.: Dynamic modeling of early
diagenesis and nutrient cycling. A case study in an artic marine sediment,
Am. J. Sci., 303, 905–955, 2003.Berner, R. A.: Early Diagenesis: A Theoretical Approach, Princeton
University Press, 10.1515/9780691209401, 1980.
Bissett, A., Burke, C., Cook, P. L., and Bowman, J. P.: Bacterial community
shifts in organically perturbed sediments, Environ. Microbiol., 9,
46–60, 2007.Boudet, L., Sabatier, F., and Radakovitch, O.: Modelling of sediment
transport pattern in the mouth of the Rhône delta: Role of storm and
flood events, Estuarine, Coast. Shelf Sci., 198, 568–582,
10.1016/j.ecss.2016.10.004, 2017.
Boudreau, B. P.: Is burial velocity a master parameter for bioturbation?, Geochim. Cosmochim. Ac., 58, 1243–1249, 1994.
Boudreau, B. P.: A method-of-lines code for carbon and nutrient diagenesis
in aquatic sediments, Comput. Geosci., 22, 479–496, 1996.Boudreau, B. P.: Diagenetic models and their implementation, vol. 410,
Springer, Berlin, 10.1007/97S-3-642-60421-S, 1997.
Boudreau, B. P. and Jorgensen, B. B.: The benthic boundary layer: Transport
processes and biogeochemistry, Oxford University Press, ISBN 0-19-511881-2, 2001.Bourgeois, S., Pruski, A. M., Sun, M.-Y., Buscail, R., Lantoine, F., Kerhervé, P., Vétion, G., Rivière, B., and Charles, F.: Distribution and lability of land-derived organic matter in the surface sediments of the Rhône prodelta and the adjacent shelf (Mediterranean Sea, France): a multi proxy study, Biogeosciences, 8, 3107–3125, 10.5194/bg-8-3107-2011, 2011.
Burdige, D. J.: The kinetics of organic matter mineralization in anoxic
marine sediments, J. Marine Res., 49, 727–761, 1991.Burdige, D. J.: Burial of terrestrial organic matter in marine sediments: A
re-assessment, Global Biogeochem. Cycles, 19, GB4011, 10.1029/2004GB002368, 2005.
Burdige, D. J. and Komada, T.: Anaerobic oxidation of methane and the stoichiometry of remineralization processes in continental margin sediments, Limnol. Oceanogr., 56, 1781–1796, 2011.Cai, W.-J.: Estuarine and coastal ocean carbon paradox: CO2 sinks or sites
of terrestrial carbon incineration?, Annu. Rev. Marine Sci., 3,
123–145, 2011.Cathalot, C., Rabouille, C., Pastor, L., Deflandre, B., Viollier, E., Buscail, R., Grémare, A., Treignier, C., and Pruski, A.: Temporal variability of carbon recycling in coastal sediments influenced by rivers: assessing the impact of flood inputs in the Rhône River prodelta, Biogeosciences, 7, 1187–1205, 10.5194/bg-7-1187-2010, 2010.Cathalot, C., Rabouille, C., Tisnérat-Laborde, N., Toussaint, F.,
Kerhervé, P., Buscail, R., Loftis, K., Sun, M.-Y., Tronczynski, J.,
Azoury, S., and Lansard, B.: The fate of river organic carbon in coastal areas: A
study in the Rhône River delta using multiple isotopic (δ13C,
Δ14C) and organic tracers, Geochim. Cosmochim. Ac., 118,
33–55, 2013.
Chaillou, G., Anschutz, P., Dubrulle, C., and Lecroart, P.: Transient states
in diagenesis following the deposition of a gravity layer: dynamics of O 2,
Mn, Fe and N-species in experimental units, Aquatic Geochem., 13,
157–172, 2007.
Couture, R.-M., Shafei, B., Van Cappellen, P., Tessier, A., and Gobeil, C.:
Non-steady state modeling of arsenic diagenesis in lake sediments,
Environ. Sci. Technol., 44, 197–203, 2010.De Borger, E., Tiano, J., Braeckman, U., Rijnsdorp, A. D., and Soetaert, K.: Impact of bottom trawling on sediment biogeochemistry: a modelling approach, Biogeosciences, 18, 2539–2557, 10.5194/bg-18-2539-2021, 2021.
Deflandre, B., Mucci, A., Gagné, J.-P., Guignard, C., and Jørn
Sundby, B.: Early diagenetic processes in coastal marine sediments disturbed
by a catastrophic sedimentation event, Geochim. Cosmochim. Ac., 66,
2547–2558, 2002.
Dezzeo, N., Herrera, R., Escalante, G., and Chacón, N.: Deposition of
sediments during a flood event on seasonally flooded forests of the lower
Orinoco River and two of its black-water tributaries, Venezuela,
Biogeochemistry, 49, 241–257, 2000.Dumoulin, J., Pozzato, L., Rassman, J., Toussaint, F., Fontugne, M.,
Tisnérat-Laborde, N., Beck, L., Caffy, I., Delqué-Koli, C. E.,
Moreau, C., and Rabouille, C.: Isotopic Signature (δ13C,14C) of DIC in Sediment
Pore Waters: An Example from the Rhône River Delta, Radiocarbon, 60,
1465–1481, 2018.Efron, B.: Bootstrap methods: another look at the jackknife, in:
Breakthroughs in statistics, Springer, 569–593, 10.1007/978-1-4612-4380-9_41, 1992.
Fiadeiro, M. E. and Veronis, G.: On weighted-mean schemes for the finite-difference approximation to the advection-diffusion equation, Tellus, 29, 512–522, 1977.
Froelich, P., Klinkhammer, G., Bender, M. L., Luedtke, N., Heath, G. R.,
Cullen, D., Dauphin, P., Hammond, D., Hartman, B., and Maynard, V.: Early
oxidation of organic matter in pelagic sediments of the eastern equatorial
Atlantic: suboxic diagenesis, Geochim. Cosmochim. Ac., 43,
1075–1090, 1979.
Ghil, M.: A century of nonlinearity in the geosciences, Earth Space
Sci., 6, 1007–1042, 2019.
Gooday, A. J.: Biological responses to seasonally varying fluxes of organic
matter to the ocean floor: a review, J. Oceanogr., 58, 305–332,
2002.
Grenz, C., Denis, L., Boucher, G., Chauvaud, L., Clavier, J., Fichez, R.,
and Pringault, O.: Spatial variability in sediment oxygen consumption under
winter conditions in a lagoonal system in new caledonia (south pacific), Elsevier,
285, 33–47, 2003.
Gruber, N.: Carbon at the coastal interface, Nature, 517, 148–149, 2015.Hensel, P. F., Day, J. W., Pont, D., and Day, J. N.: Short-Term
Sedimentation Dynamics in the Rhône River Delta, France: The Importance
of Riverine Pulsing, Estuaries, 21, 52, 10.2307/1352546,
1998.
Hindmarsh, A. C.: ODEPACK: A systemized collection of ODE solvers, 55–64,
1983.
Jørgensen, B. B. and Revsbech, N. P.: Diffusive boundary layers and the
oxygen uptake of sediments and detritus 1, Limnol. Oceanogr., 30,
111–122, 1985.Jørgensen, B. B., Findlay, A. J., and Pellerin, A.: The biogeochemical
sulfur cycle of marine sediments, Front. Microbiol., 10, 849, 10.3389/fmicb.2019.00849, 2019.Katsev, S., Sundby, B., and Mucci, A.: Modeling vertical excursions of the
redox boundary in sediments: Application to deep basins of the Arctic Ocean,
Limnol. Oceanogr., 51, 1581–1593,
10.4319/lo.2006.51.4.1581, 2006.Kittel, T., Heitzig, J., Webster, K., and Kurths, J.: Timing of transients:
quantifying reaching times and transient behavior in complex systems, New
Journal of Physics, 19, 083005, 10.1088/1367-2630/aa7b61, 2017.Krumins, V., Gehlen, M., Arndt, S., Van Cappellen, P., and Regnier, P.: Dissolved inorganic carbon and alkalinity fluxes from coastal marine sediments: model estimates for different shelf environments and sensitivity to global change, Biogeosciences, 10, 371–398, 10.5194/bg-10-371-2013, 2013.Lansard, B., Rabouille, C., Denis, L., and Grenz, C.: In situ oxygen uptake
rates by coastal sediments under the influence of the Rhône River (NW
Mediterranean Sea), Cont. Shelf Res., 28, 1501–1510,
10.1016/j.csr.2007.10.010, 2008.
Lansard, B., Rabouille, C., Denis, L., and Grenz, C.: Benthic
remineralization at the land–ocean interface: A case study of the Rhône
River (NW Mediterranean Sea), Estuarine, Coast. Shelf Sci., 81,
544–554, 2009.
Lasaga, A. and Holland, H.: Mathematical aspects of non-steady-state
diagenesis, Geochim. Cosmochim. Ac., 40, 257–266, 1976.
Lorenz, E. N.: Deterministic nonperiodic flow, J. Atmos.
Sci., 20, 130–141, 1963.
Mayer, L. M.: Surface area control of organic carbon accumulation in
continental shelf sediments, Geochim. Cosmochim. Ac., 58,
1271–1284, 1994.
McKee, B. A., Aller, R., Allison, M., Bianchi, T., and Kineke, G.: Transport
and transformation of dissolved and particulate materials on continental
margins influenced by major rivers: benthic boundary layer and seabed
processes, Cont. Shelf Res., 24, 899–926, 2004.
Mermex Group, T. M., de Madron, X. D., Guieu, C., Sempéré, R.,
Conan, P., Cossa, D., D'Ortenzio, F., Estournel, C., Gazeau, F., Rabouille,
C., and Stemmann, L.: Marine ecosystems' responses to climatic and anthropogenic
forcings in the Mediterranean, Prog. Oceanogr., 91, 97–166, 2011.
Meybeck, M., Dürr, H., Roussennac, S., and Ludwig, W.: Regional seas and
their interception of riverine fluxes to oceans, Marine Chem., 106,
301–325, 2007.
Middelburg, J. J.: A simple rate model for organic matter decomposition in
marine sediments, Geochim. Cosmochim. Ac., 53, 1577–1581, 1989.
Middelburg, J. J., Vlug, T., Jaco, F., and Van der Nat, W.: Organic matter
mineralization in marine systems, Global Planet. Change, 8, 47–58,
1993.
Middelburg, J. J., Soetaert, K., and Herman, P. M.: Empirical relationships
for use in global diagenetic models, Deep-Sea Res. Pt. I, 44, 327–344, 1997.Miralles, J., Radakovitch, O., and Aloisi, J.-C.: 210Pb sedimentation rates
from the Northwestern Mediterranean margin, Marine Geol., 216, 155–167,
10.1016/j.margeo.2005.02.020, 2005.
Mucci, A. and Edenborn, H. M.: Influence of an organic-poor landslide
deposit on the early diagenesis of iron and manganese in a coastal marine
sediment, Geochim. Cosmochim. Ac., 56, 3909–3921, 1992.Munhoven, G.: Model of Early Diagenesis in the Upper Sediment with Adaptable complexity – MEDUSA (v. 2): a time-dependent biogeochemical sediment module for Earth system models, process analysis and teaching, Geosci. Model Dev., 14, 3603–3631, 10.5194/gmd-14-3603-2021, 2021.Nmor, S. and Soetaert, K.: FESDIA: An early diagenesis model including Iron (Fe), Sulfur (S) and Methane (CH4) dynamics, Zenodo [code], 10.5281/zenodo.6369288, 2022.Pastor, L., Cathalot, C., Deflandre, B., Viollier, E., Soetaert, K., Meysman, F. J. R., Ulses, C., Metzger, E., and Rabouille, C.: Modeling biogeochemical processes in sediments from the Rhône River prodelta area (NW Mediterranean Sea), Biogeosciences, 8, 1351–1366, 10.5194/bg-8-1351-2011, 2011.Pastor, L., Rabouille, C., Metzger, E., Thibault de Chanvalon, A., Viollier, E., and Deflandre, B.: Transient
early diagenetic processes in Rhône prodelta sediments revealed in
contrasting flood events, Cont. Shelf Res., 166, 65–76,
10.1016/j.csr.2018.07.005, 2018.Pont, D., Day, J. W., and Ibáñez, C.: The impact of two large floods
(1993–1994) on sediment deposition in the Rhône delta: Implications for
sustainable management, Sci. Total Environ., 609, 251–262,
10.1016/j.scitotenv.2017.07.155, 2017.Pozzato, L., Rassmann, J., Lansard, B., Dumoulin, J.-P., Breugel, P. van, and Rabouille, C.: Origin of remineralized organic matter in sediments from the rhone river prodelta (NW mediterranean) traced by Δ14C and δ13C signatures of pore water DIC, Prog. Oceanogr., 163, 112–122, 2018.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T.: Numeric recipes in c: The art of scientific computing, Cambridge university press, 3rd edn., 1992.
Pusceddu, A., Dell'Anno, A., Fabiano, M., and Danovaro, R.: Quantity and
bioavailability of sediment organic matter as signatures of benthic trophic
status, Marine Ecol. Prog. Ser., 375, 41–52, 2009.
Rabouille, C. and Gaillard, J.-F.: The validity of steady-state flux
calculations in early diagenesis: a computer simulation of deep-sea silica
diagenesis, Deep-Sea Res. Pt. A, 37,
625–646, 1990.Rabouille, C. and Gaillard, J.-F.: Towards the EDGE: Early diagenetic global
explanation. A model depicting the early diagenesis of organic matter, O2,
NO3, Mn, and PO4, Geochim. Cosmochim. Ac., 55, 2511–2525, 1991.
Rabouille, C., Mackenzie, F. T., and Ver, L. M.: Influence of the human
perturbation on carbon, nitrogen, and oxygen biogeochemical cycles in the
global coastal ocean, Geochim. Cosmochim. Ac., 65, 3615–3641,
2001a.
Rabouille, C., Witbaard, R., and Duineveld, G.: Annual and interannual
variability of sedimentary recycling studied with a non-steady-state model:
application to the North Atlantic Ocean (BENGAL site), Prog.
Oceanogr., 50, 147–170, 2001b.Radakovitch, O., Charmasson, S., Arnaud, M., and Bouisset, P.: 210Pb and
Caesium Accumulation in the Rhône Delta Sediments, Estuarine, Coast. Shelf Sci., 48, 77–92, 10.1006/ecss.1998.0405, 1999.Rassmann, J., Lansard, B., Pozzato, L., and Rabouille, C.: Carbonate chemistry in sediment porewaters of the Rhône River delta driven by early diagenesis (northwestern Mediterranean), Biogeosciences, 13, 5379–5394, 10.5194/bg-13-5379-2016, 2016.Rassmann, J., Eitel, E. M., Lansard, B., Cathalot, C., Brandily, C., Taillefert, M., and Rabouille, C.: Benthic alkalinity and dissolved inorganic carbon fluxes in the Rhône River prodelta generated by decoupled aerobic and anaerobic processes, Biogeosciences, 17, 13–33, 10.5194/bg-17-13-2020, 2020.R Core Team: R: A Language and Environment for Statistical Computing, R
Foundation for Statistical Computing, Vienna, Austria, https://www.R-project.org/ (last access: 2 August 2022), 2021.
Regnier, P., Friedlingstein, P., Ciais, P., Mackenzie, F. T., Gruber, N.,
Janssens, I. A., Laruelle, G. G., Lauerwald, R., Luyssaert, S., Andersson,
A. J., and Arndt, S.: Anthropogenic perturbation of the carbon fluxes from land to
ocean, Nat. Geosci., 6, 597–607, 2013.
Sciberras, M., Hiddink, J. G., Jennings, S., Szostek, C. L., Hughes, K. M., Kneafsey, B., Clarke, L. J., Ellis, N., Rijnsdorp, A. D., McConnaughey, R. A., et al.: Response of benthic fauna to experimental bottom fishing: A global meta-analysis, Fish Fish., 19, 698–715, 2018.Sempéré, R., Charrière, B., Van Wambeke, F., and Cauwet, G.:
Carbon inputs of the Rhône River to the Mediterranean Sea:
Biogeochemical implications, Global Biogeochem. Cycles, 14, 669–681,
10.1029/1999GB900069, 2000.
Smith, K. L., Ruhl, H. A., Huffard, C. L., Messié, M., and Kahru, M.:
Episodic organic carbon fluxes from surface ocean to abyssal depths during
long-term monitoring in NE Pacific, P. Natl. Acad.
Sci. USA, 115, 12235–12240, 2018.Soetaert, K.: Package rootSolve: roots, gradients and steady-states in R,
Google Scholar [code], https://cran.r-project.org/web/packages/rootSolve/index.html (last access: 2 August 2022), 2014.Soetaert, K. and Herman, P. M.: A practical guide to ecological modelling:
Using r as a simulation platform, Springer, 10.1007/978-1-4020-8624-3, ISBN 978-1-4020-8623-6, 2009.
Soetaert, K. and Meysman, F.: Reactive transport in aquatic ecosystems:
Rapid model prototyping in the open source software R, Environ.
Model. Softw., 32, 49–60, 2012.Soetaert, K. and Petzoldt, T.: marelac: Tools for Aquatic Sciences,
https://CRAN.R-project.org/package=marelac (last access: 2 August 2022), r package version 2.1.10,
2020.
Soetaert, K., Herman, P. M., and Middelburg, J. J.: A model of early
diagenetic processes from the shelf to abyssal depths, Geochim. Cosmochim.
Ac., 60, 1019–1040, 1996a.
Soetaert, K., Herman, P. M., and Middelburg, J. J.: Dynamic response of
deep-sea sediments to seasonal variations: A model, Limnol.
Oceanogr., 41, 1651–1668, 1996b.Soetaert, K., Petzoldt, T., and Meysman, F.: Marelac: Tools for aquatic
sciences, [code], https://cran.r-project.org/web/packages/marelac/index.html (last access: 2 August 2022), 2010a.Soetaert, K., Petzoldt, T., and Setzer, R. W.: Solving Differential
Equations in R: Package deSolve, J. Statist. Softw., 33, 1–25,
10.18637/jss.v033.i09, 2010b.
Strogatz, S. H.: Nonlinear dynamics and chaos: with applications to physics,
biology, chemistry, and engineering, CRC press, ISBN 13 978-0-8133-4910-7, 2018.
Stumm, W. and Morgan, J. J.: Aquatic chemistry: chemical equilibria and
rates in natural waters, John Wiley & Sons, 126, ISBN 0-471-51184-6, 2012.Sulpis, O., Humphreys, M. P., Wilhelmus, M. M., Carroll, D., Berelson, W. M., Menemenlis, D., Middelburg, J. J., and Adkins, J. F.: RADIv1: a non-steady-state early diagenetic model for ocean sediments in Julia and MATLAB/GNU Octave, Geosci. Model Dev., 15, 2105–2131, 10.5194/gmd-15-2105-2022, 2022.
Sundby, B.: Transient state diagenesis in continental margin muds, Marine
Chem., 102, 2–12, 2006.
Tesi, T., Langone, L., Goñi, M., Wheatcroft, R., Miserocchi, S., and
Bertotti, L.: Early diagenesis of recently deposited organic matter: A 9-yr
time-series study of a flood deposit, Geochim. Cosmochim. Ac., 83,
19–36, 2012.Toussaint, F., Tisnérat-Laborde, N., Cathalot, C., Buscail, R.,
Kerhervé, P., and Rabouille, C.: Depositional processes of organic
matter in the Rhône River Delta (Gulf of Lions, France) traced by
density fractionation coupled with Δ14C and δ13C,
Radiocarbon, 55, 920–931, 2013.van de Velde, S., Van Lancker, V., Hidalgo-Martinez, S., Berelson, W. M.,
and Meysman, F. J. R.: Anthropogenic disturbance keeps the coastal seafloor
biogeochemistry in a transient state, Sci. Rep.-UK, 8, 5582,
10.1038/s41598-018-23925-y, 2018.Wang, Y. and Van Cappellen, P.: A multicomponent reactive transport model of
early diagenesis: Application to redox cycling in coastal marine sediments,
Geochim. Cosmochim. Ac., 60, 2993–3014, 1996.
Westrich, J. T. and Berner, R. A.: The role of sedimentary organic matter in
bacterial sulfate reduction: The G model tested 1, Limnol.
Oceanogr., 29, 236–249, 1984.
Wheatcroft, R. A.: Preservation potential of sedimentary event layers,
Geology, 18, 843–845, 1990.Wollast, R.: Interactions of carbon and nitrogen cycles in the coastal zone,
in: Interactions of C, N, P and S biogeochemical cycles and global change, Springer,
195–210, 10.1007/978-3-642-76064-8, 1993.Yakushev, E. V., Protsenko, E. A., Bruggeman, J., Wallhead, P., Pakhomova, S. V., Yakubov, S. Kh., Bellerby, R. G. J., and Couture, R.-M.: Bottom RedOx Model (BROM v.1.1): a coupled benthic–pelagic model for simulation of water and sediment biogeochemistry, Geosci. Model Dev., 10, 453–482, 10.5194/gmd-10-453-2017, 2017.
Zebracki, M., Eyrolle-Boyer, F., Evrard, O., Claval, D., Mourier, B.,
Gairoard, S., Cagnat, X., and Antonelli, C.: Tracing the origin of suspended
sediment in a large Mediterranean river by combining continuous river
monitoring and measurement of artificial and natural radionuclides, Sci. Total Environ., 502, 122–132, 2015.
Zindorf, M., Rooze, J., Meile, C., März, C., Jouet, G., Newton, R.,
Brandily, C., and Pastor, L.: The evolution of early diagenetic processes at
the Mozambique margin during the last glacial-interglacial transition,
Geochim. Cosmochim. Ac., 300, 79–94, 2021.