We introduce a time-dependent, one-dimensional model of
early diagenesis that we term RADI, an acronym accounting for the main
processes included in the model: chemical reactions, advection, molecular
and bio-diffusion, and bio-irrigation. RADI is targeted for study of
deep-sea sediments, in particular those containing calcium carbonates
(CaCO3). RADI combines CaCO3 dissolution driven by organic matter
degradation with a diffusive boundary layer and integrates state-of-the-art
parameterizations of CaCO3 dissolution kinetics in seawater, thus
serving as a link between mechanistic surface reaction modeling and
global-scale biogeochemical models. RADI also includes CaCO3
precipitation, providing a continuum between CaCO3 dissolution and
precipitation. RADI integrates components rather than individual chemical
species for accessibility and is straightforward to compare against
measurements. RADI is the first diagenetic model implemented in Julia, a
high-performance programming language that is free and open source, and it
is also available in MATLAB/GNU Octave. Here, we first describe the
scientific background behind RADI and its implementations. Following this, we evaluate
its performance in three selected locations and explore other potential
applications, such as the influence of tides and seasonality on early
diagenesis in the deep ocean. RADI is a powerful tool to study the
time-transient and steady-state response of the sedimentary system to
environmental perturbation, such as deep-sea mining, deoxygenation, or
acidification events.
Introduction
The seafloor, which covers ∼70 % of the surface of the
planet and modulates the transfer of materials and energy from the biosphere
to the geosphere, remains for the vast majority unexplored. Today, this
rich, poorly understood ecosystem is threatened locally by deep-sea mining
activities (e.g., plowing of the seabed) because it contains abundant
valuable minerals and metals essential for the energy transition (Thompson
et al., 2018). The deep ocean is also being perturbed globally by climate
change, including seawater acidification caused by the uptake of
∼10×109 t of anthropogenic carbon dioxide (CO2)
into the ocean each year (Perez et al., 2018; Gruber et al., 2019), roughly
a quarter of our total annual emissions (Friedlingstein et al., 2020). In
this context, it is important to improve our understanding of the seafloor's
response to environmental change.
Accumulation of sinking biogenic aggregates and lithogenic particles at the
seafloor provides reactive material that regulates the chemical composition
of sediment porewaters. Whereas biogenic particles typically sink through
the water column at rates from a few meters to hundreds of meters per day
(Riley et al., 2012), the same particles accumulate in sediments much more
slowly, typically a few centimeters per thousand years (Jahnke, 1996). The
residence time of solid particles in the top centimeter of sediments is
therefore very long (a few hundred or thousand years) compared to their
residence time in the water column (a few weeks). Additionally, while
solutes are dispersed by advection in the water column, molecular diffusion
dominates in porewaters, which is slower. The long residence time of
reactive solid material in surface sediments, coupled with the slow
diffusive transport of dissolved species, can lead to large gradients in
chemical composition between sediment porewaters and the overlying seawater,
inducing solute fluxes between the two (Hammond et al., 1996). Thus, the top
few millimeters of the seafloor play a significant role in many major marine
biogeochemical cycles.
The overall rate of biogeochemical reactions is determined by the slowest,
“rate-limiting” step, which can be (i) transport to or from the reaction
site or (ii) the reaction kinetics of the particle at the mineral–water
interface. At the seafloor, the rate-limiting step for many biogeochemical
reactions is solute transport via molecular diffusion through the sediment
porewaters or through the diffusive boundary layer (DBL). The DBL is a thin
film of water extending up to a few millimeters above the sediment–water
interface in which molecular diffusion is the dominant mode of solute
transport. The presence of a DBL above the sediment–water interface (Fig. 1)
has been reported by several investigators (Morse, 1974; Archer et al.,
1989b; Gundersen and Jørgensen, 1990; Santschi et al., 1991; Glud et al.,
1994) and its thickness depends on the composition and roughness of the
substrate, as well as on the flow speed of the overlying seawater (Chriss
and Caldwell, 1982; Dade, 1993; Røy et al., 2002; Han et al., 2018).
Diffusive fluxes of solutes across the sediment–water interface are driven
by concentration gradients between the overlying seawater and the sediment
column being considered. If most of the concentration gradient for a given
solute occurs within the porewaters, rather than within the DBL, then the
diffusive flux of this solute is termed “internal” or “sediment-side
controlled” (Boudreau and Guinasso, 1982). Conversely, if the majority of
the concentration gradient for a given solute is within the DBL, the
chemical flux across the sediment–water interface is termed “external” or
“water-side transport-controlled”. In practice, the chemical exchange of
most solutes is controlled by a combination of both regimes termed
“mixed-control”, such as dissolved oxygen (Jørgensen and Revsbech, 1985;
Hondzo, 1998), radon (Homoky et al., 2016; Cook et al., 2018), and the
products of calcium carbonate dissolution (Sulpis et al., 2018; Boudreau et
al., 2020), which have concentration gradients on both sides of the
sediment–water interface. Despite the importance of the DBL in controlling
diffusive fluxes across the sediment–water interface, DBLs are not
explicitly included in most models that simulate early diagenesis in the
deep ocean.
Schematic of RADI's vertical structure alongside steady-state
depth profiles of porosity φz (see Eqs. 3 and 4), porewater (u,
solid light blue line) and solid (w, solid brown line) burial velocities at
in situ conditions taken at station 7 of Sayles et al. (2001). Burial
velocity varies with depth due to porosity, as described in Sect. 2.3.
The open circles in the porosity profile are porosity measurements from
Sayles et al. (2001).
Multiple numerical models simulating early diagenesis have previously been
published (Burdige and Gieskes, 1983; Rabouille and Gaillard, 1991;
Boudreau, 1996b; Van Cappellen and Wang, 1996; Soetaert et al., 1996b;
Archer et al., 2002; Munhoven, 2007, 2021; Couture et al., 2010; Yakushev et al.,
2017; Hülse et al., 2018), each with its own assumptions
and best area of application (Paraska et al., 2014). For instance, most
existing models are limited to a steady state and are thus unable to predict
the transient sediment response to time-dependent phenomena such as tides,
seasonal change, ocean deoxygenation, or acidification. Moreover, most of
these models do not take the presence of a DBL into account, even though
diffusion through the DBL may control the overall rate of many
biogeochemical reactions. Finally, as the landscape of computing software
and programming languages evolves and improves computing efficiency and code
accessibility, it is important to leverage emerging developments to
implement new biogeochemical models. Here, we describe a new sediment
porewater model built upon earlier work termed RADI, an acronym accounting
for the main processes included in the model that control the vertical
distribution of solutes and solids: chemical reactions, advection, molecular
and bio-diffusion, and bio-irrigation. The novelty of RADI is that it
combines degradation-driven organic matter CaCO3 dissolution (Archer et
al., 2002) with a diffusive boundary layer (Boudreau, 1996b) and integrates
the state-of-the art parameterization of CaCO3 dissolution kinetics in
seawater (Dong et al., 2019; Naviaux et al., 2019a). RADI thus links
mechanistic surface reaction modeling to global-scale biogeochemical models
(Carroll et al., 2020). By integrating components (e.g., total alkalinity)
rather than individual chemical species (e.g., carbonate and bicarbonate
ions), RADI is easy to compare to observations. RADI is implemented in two
popular scientific programming languages: Julia and MATLAB/GNU Octave. To
our knowledge, this is the first diagenetic model implemented in Julia
(https://julialang.org, last access: 28 January 2022), a high-level, high-performance, and
cross-platform programming language that is free and open source (Bezanson
et al., 2017). Here, we first describe the scientific background behind RADI
and its implementations. Following this, we evaluate its performance in three selected
locations and explore other potential applications, such as the influence of
tides and seasonality on early diagenesis in the deep ocean.
Model description
In the following section, we describe how reactions, advection, diffusion,
and irrigation are implemented in RADIv1. Model variables are italicized and their
names as coded in the model are shown in monospaced font. Tables 1 and 2
include an inventory of model variables and parameters and a list of
nomenclature for chemical species, respectively.
Nomenclature of model parameters and variables.
VariableModel notationDescriptionEquation no.GeneralZz_maxTotal height of the sediment columndzz_resDepth resolutionzdepthsArray of modeled depths within the sedimentTstoptimeTotal simulation timedtintervalTime stepsttimestepsArray of modeled time pointsφzphiPorewater porosity3βphiBetaPorosity attenuation coefficient3φs,zphiSSolid volume fraction4θ2tort2Squared tortuosity24FvFvarSolid deposition fluxvwvar_wBottom waters solute concentrationδdblDiffusive boundary layer thicknessTwTTemperatureAdvectionuuPorewater burial velocity17wwSolid burial velocity15, 16, 18Peh,zPehHalf of the cell Péclet number22σzsigmaNumber from Fiadero and Veronis (1977)21Reactionsc/pRCRedfield ratio for carbonn/pRNRedfield ratio for nitrogenp/pRPRedfield ratio for phosphorusKvKvarHalf-saturation constant for a given electron acceptorKv′KvariInhibition constant for a given electron acceptorkreactionkvarRate constant for a given chemical reactionfv,zfvarFractions of organic matter degraded by a given oxidant7, 8ηdiss. ca.order_diss_caReaction order for calcite dissolution12ηdiss. ar.order_diss_arReaction order for aragonite dissolution13ηprec. ca.order_prec_caReaction order for calcite precipitation14ΩcaOmegaCaSeawater saturation state with respect to calcite12, 14ΩarOmegaArSeawater saturation state with respect to aragonite13Diffusiondz(v)D_var_tort2Effective molecular diffusion coefficient23, 27dz∘(v)D_varFree-solution molecular diffusion coefficient23bzD_bioBioturbation coefficient25, 26λblambda_bCharacteristic bioturbation depth26IrrigationαzalphaIrrigation coefficient30, 31λilambda_iCharacteristic depth for irrigation31
Nomenclature of modeled chemical species. All variables are
concentrations, expressed in mol per cubic meter of solid for solid
species and mol per cubic meter of water for solute species.
* We consider all clay minerals to be montmorillonite (Al2H2O12Si4; molar mass is equal to 360.31 gmol-1).
Model structure and fundamental equation
RADI uses the same set of reactive-transport partial differential equations
as implemented in CANDI (Boudreau, 1996b), i.e., for each solute component,
∂ν∂t=1φ∂∂zφd∂ν∂z-φuν+α(vw-v)+∑R,
and for each solid component,
∂ν∂t=1φs∂∂zφsb∂ν∂z-φswν+∑R,
where v is the concentration of a given component, t is time, φ is
sediment porosity, φs is the solid–volume fraction, d is the
effective molecular diffusion coefficient, b is the bioturbation
coefficient, z is depth, u is the porewater burial velocity, w is the solid
burial velocity, α is the irrigation coefficient, vw is the
concentration of a solute in the bottom waters, and ΣR is the net
production rate from all biogeochemical reactions for a given component.
Each of these terms will be described in detail later in this section. These
partial differential equations are solved numerically using the method of
lines described in Boudreau (1996b). Instead of searching for steady-state
solutions directly, RADI computes the concentrations of a set of solids and
solutes at each depth and time step following a time vector set by the
user. The user determines the simulation time depending on the objectives,
e.g., multimillennial to predict a steady state, or a few days to study the
response of the sedimentary system to high-frequency cyclic phenomena such
as tides. For initial conditions, the user can choose between predefined
uniform values (e.g., set all concentrations to zero) or a set of saved
concentrations (e.g., from a previous simulation that has reached steady
state). T is the total simulation time, dt is the temporal resolution, i.e.,
the interval between each time step, and t refers to the array of modeled
time points. All time units are in years (a). The interface between the
surface sediment and overlying seawater, conventionally set at a sediment
depth z=0, represents the top layer of RADI's vertical axis (Fig. 1). The
bottom layer of the model is at a sediment depth Z. Between these limits, n
layers are present, each being separated by a constant vertical gap dz. Depth
units are in meters. The values assigned to dz and dt depend on the nature of
the problem and on the kinetics of the chemical reactions. In the present
study, all cases use dz=2 mm and dt=1/128000 a, i.e., ∼4 min. If a lower dz is used, dt needs to be lowered as well to preserve
numerical stability. In general, the ratio dz/dt should be kept below a value
of 256 ma-1. If dz is divided by two, dt needs to be divided by two as well,
and the speed at which RADI runs will be reduced by a factor of four.
RADI operates on a static, user-defined porosity profile. Sediment porosity,
φz in Fig. 1, refers to the porewater volume fraction in the
sediment (dimensionless) and typically decreases exponentially with sediment
depth due to steady-state compaction. The sediment porosity profile is
parameterized following Boudreau (1996b) as follows:
φz=φ∞+(φ0-φ∞)e-βz,
where φ∞ is the porosity at great depth, φ0 is
the porosity at the sediment–water interface, and β is an attenuation
coefficient (in m-1). A typical deep-sea sediment porosity
profile is shown in Fig. 1. Here the measured porosity profile at station 7
of cruise NBP98-2 (Sayles et al., 2001) is fit using φ∞=0.87, φ0=0.915, and β=33m-1. The solid
volume fraction (φs, dimensionless) is defined as follows:
φs,z=1-φz,
and increases with sediment depth (as compaction forces squeeze porewaters
out).
Within this grid and for each time step, RADI computes the concentrations of
11 solute variables (TAlk, ΣCO2, O2, Ca2+, ΣNO3, ΣSO4, ΣPO4, ΣNH4, ΣH2S, Fe2+, and Mn2+) and 8 solid variables (Calcite, Aragonite,
Fe(OH)3, MnO2, Clay, and three kinds of particulate organic carbon,
collectively termed POC). Note that clay is simply modeled as a
non-reactive solid that is included because the clay accumulation flux to
the sediment–water interface participates in the calculation of the solid
burial velocity; see Sect. 2.3. Concentration units
are in mol per square meter of water for solutes and in mol per square meter of solid for solid
species. For each modeled solute or solid concentration v at time t and
sediment depth z, the following equation applies:
v(t+dt),z=vt,z+[R(vt,z)+A(vt,z)+D(vt,z)+I(vt,z)]⋅dt,
where R(vt,z) quantifies the rate of change of vt,z due to
chemical reactions, A(vt,z) quantifies the rate of change of
vt,z due to advection, D(vt,z) quantifies the rate of
change of vt,z due to molecular and bio-diffusion, and
I(vt,z) quantifies the rate of change of vt,z due to
bio-irrigation. In general, only the subscript z variables are explicitly written out
in this paper for variables and parameters that vary with depth. The
t variables are implicit but excluded for clarity.
Reactions
In RADI, biogeochemical reactions operate on solutes and solids throughout
the entire sediment column, including the very top and bottom layers.
R(vz) is the net rate at which vz is being consumed (negative R) or
produced (positive R) by these reactions. Biogeochemical reactions in RADI
(Table 3) are grouped into three categories: (i) organic matter degradation,
(ii) oxidation of reduced metabolites (organic matter degradation
byproducts), and (iii) dissolution or precipitation of calcium carbonate
minerals. RADI has been designed for early diagenesis in deep-sea sediments,
and thus formation and re-oxidation of metal sulfide minerals are not considered.
Diagenetic reactions, reaction rates, and reaction contributions to porewater.
Organic carbon deposited on the seafloor originates mainly from primary
production in the upper ocean or on land and (to a lesser extent) from the
ocean interior via chemoautotrophy. Despite the differences in origin,
detrital organic matter found in marine sediments typically has the same
composition: ∼60 % proteins, ∼20 % lipids,
∼20 % carbohydrates, and a fraction of other compounds
(Hedges et al., 2002; Burdige, 2007; Middelburg, 2019). Here, the
stoichiometry of organic matter is represented by the coefficients c (for
carbon), n (for nitrogen), and p (for phosphorus). By default, the c:p ratio is
set to 106:1 and the n:p ratio set to 16:1, following the Redfield ratio that
describes the average composition of phytoplankton biomass (Redfield, 1958),
but these values can easily be adjusted. In RADI, c/p is denoted RC, n/p is
denoted RN, and p/p is denoted RP, which is unity. Organic matter is also
simplified here as an elementary carbohydrate (CH2O). In reality, loss
of H and O during biosynthesis of proteins, lipids, and polysaccharides
occurs (Anderson, 1995; Hedges et al., 2002; Middelburg, 2019), which
results in an effective molar ratio of O2 consumed to C degraded of
∼1.2 during aerobic respiration (Anderson and Sarmiento,
1994) instead of 1 as assumed here (Table 3).
Observations show that some organic compounds are preferentially degraded
and become selectively depleted (Cowie and Hedges, 1994; Lee et al., 2000).
As a result, the bulk reactivity of organic matter decreases with increasing
age (Middelburg, 1989). Degradation of organic matter deposited at the
seafloor typically follows a sequential utilization of available oxidants,
O2, NO3-, MnO2, Fe(OH)3, and SO42-,
followed by methanogenesis (Froelich et al., 1979; Berner, 1980; Arndt et
al., 2013). All organic matter degradation reactions implemented in RADI are
shown in Table 3.
To account for the decrease in organic matter degradation rate with sediment
depth, we separate organic matter into fractions of different reactivity,
and we assign a rate constant to each of the degradable fractions. Following
Jørgensen (1978), Westrich and Berner (1984), and Soetaert et al.
(1996b), three different classes of organic matter are considered:
refractory, slow-decay, and fast-decay organic matter. The refractory organic matter class
is not reactive during the timescales considered here. The fast- and
slow-decay organic matter fractions each have a depth-dependent,
oxidant-independent reactivity. The overall organic matter degradation rate
decreases with depth because the quantity of organic matter and the relative
proportions of fast- and slow-decay materials decline with depth. Organic
matter is degraded following the sequential utilization of available
oxidants. The oxidant limitation is represented by a Michaelis–Menten-type
(also termed “Monod”) function, in which each oxidant has an associated
half-saturation constant (Koxidant in molm-3) that symbolizes the
oxidant concentration at which the process proceeds at half its maximal
speed (Soetaert et al., 1996b). The presence of some oxidants may also
inhibit other metabolic pathways; this is represented by an inhibition
constant (Koxidant′ in molm-3) that is specific to each oxidant.
These limiting and inhibitory functions have been widely used (Boudreau,
1996b; Van Cappellen and Wang, 1996; Soetaert et al., 1996b; Couture et al.,
2010), they allow a single equation to be used for each component across
the entire model depth range, and they also permit some overlap between the
different pathways, as observed in nature (Froelich et al., 1979). In RADI,
the overall degradation of fast- or slow-decay organic carbon occurs at the following
rate:
RPOCfast or slow,z=foxidant,z⋅kPOCfast or slow⋅[POCfast or slow]z
where kPOC is the rate constant for the degradation of a given
type of organic carbon (fast- or slow-decay types, expressed in a-1),
[POCfast or slow] is the concentration of organic carbon (fast-
or slow-decay) in sediments, and fOx. is the sum of the fractions of
organic carbon degraded by each oxidant (dimensionless, always equal to
one), given by
foxidant,z=fO2,z+fΣNO3,z+fMnO2,z+fFe(OH)3,z7+fΣSO4,z+fCH4,z,
where
8afO2,z=[O2]zKO2+[O2]z,fΣNO3,z=[ΣNO3]zKΣNO3+[ΣNO3]z8b×KO2′KO2′+[O2]z,fMnO2,z=[MnO2]zKMnO2+[MnO2]zKΣNO3′KΣNO3′+[ΣNO3]z8c×KO2′KO2′+[O2]z,fFe(OH)3,z=[Fe(OH)3]zKFe(OH)3+[Fe(OH)3]zKMnO2′KMnO2′+[MnO2]z8d×KΣNO3′KΣNO3′+[ΣNO3]zKO2′KO2′+[O2]z,fΣSO4,z=[ΣSO4]zKΣSO4+[ΣSO4]zKFe(OH)3′KFe(OH)3′+[Fe(OH)3]z×KMnO2′KMnO2′+[MnO2]zKΣNO3′KΣNO3′+[ΣNO3]z8e×KO2′KO2′+[O2]z,fCH4,z=KΣSO4′KΣSO4′+[ΣSO4]zKFe(OH)3′KFe(OH)3′+[Fe(OH)3]z×KMnO2′KMnO2′+[MnO2]zKΣNO3′KΣNO3′+[ΣNO3]z8f×KO2′KO2′+[O2]z.
Half-saturation and inhibition constants for each oxidant used in RADI are
given in Table 4. The degradation rate constant of organic carbon,
kPOCfast or slow, is computed as a function of the
flux of organic carbon reaching the seafloor and is sediment depth dependent
(Archer et al., 2002):
9akPOCfast=(1.5×10-1)(FPOC⋅102)0.85,9bkPOCslow=(1.3×10-4)(FPOC⋅102)0.85,
where FPOC is the total flux of organic carbon reaching the seafloor
(i.e., fast, slow, and refractory, in molm-2a-1). The numbers
1.3×10-4 and 1.5×10-1 have been tuned to
best fit observations of both a Southern Ocean station and a North Atlantic station;
see Sect. 3.
Suggested values for model parameters.
ParameterModel notationValueUnitSourceKO2/KO2′KdO2/KdO2i3/10µMSoetaert et al. (1996b)KΣNO3/KΣNO3′KdtNO3/KdtNO3i30/5µMSoetaert et al. (1996b)KMnO2=KMnO2′KpMnO2/KpMnO2i42.4mMVan Cappellen and Wang (1996)1KFe(OH)3=KFe(OH)3′KpFeOH3/KpFeOH3i265mMVan Cappellen and Wang (1996)1KΣSO4=KΣSO4′KdtSO4/KdtSO4i1.6mMVan Cappellen and Wang (1996)1kFe oxkFeox106mM-1a-1Boudreau (1996b)2kMn oxkMnox106mM-1a-1Boudreau (1996b)2kS oxkSox3 ×105mM-1a-1Boudreau (1996b)2kNH oxkNHox104mM-1a-1Boudreau (1996b)2βphiBeta33m-1Tunedλblambda_b0.08mArcher et al. (2002)λilambda_i0.08mArcher et al. (2002)
1 Assuming a solid density of 2.65 gcm-3. 2 Values for the “deep sea”.
Oxidation of organic matter degradation by-products
Organic matter degradation reactions primarily change oxidants (e.g.,
O2, NO3-, MnO2, Fe(OH)3, SO42-) into
their reduced forms (e.g., H2O, N2, Mn2+, Fe2+,
H2S; Table 3). If oxygen is introduced into the system or the
reduced metabolites diffuse upwards in oxic porewaters, then these reduced
byproducts are converted back into their oxidized form and the energy
contained in them becomes available to the microbial community, though these
energetics are not considered in RADI.
Here, four redox reactions involving organic matter degradation byproducts
are implemented (Table 3): oxidation of Fe2+, Mn2+,
ΣH2S, and ΣNH3, respectively. These four reactions consume
porewater total alkalinity (TAlk) but do not alter porewater ΣCO2 (Table 3), thus locally acidifying porewaters. Here, we use the
TAlk definition of Dickson (1981), in which TAlk is defined as
“the number of moles of hydrogen ion equivalent to the excess
of proton acceptors (bases formed from weak acids with a
dissociation constant K≤10-4.5 and zero ionic strength) over proton donors (acids with K>10-4.5) in one kilogram of sample.”
This scheme should be sufficient for all open-ocean applications but may not
be suitable for coastal and anoxic environments with extensive metal sulfide
mineral turnover, which require a more complete set of redox reactions such
as that from the CANDI model of Boudreau (1996b). Additional components and
reactions can easily be implemented in future versions (see Sect. 5). The
rate constants for these four redox reactions are taken from Boudreau
(1996b) and reported in Table 4.
CaCO3 dissolution and precipitation
RADI includes two CaCO3 polymorphs: low Mg calcite and aragonite, but
more could be added in future versions, e.g., high Mg calcite and/or
vaterite. Calcite and aragonite both have different dissolution kinetics, in
which their dissolution rates increase as the undersaturation state of
seawater with respect to calcite (1-Ωca,z) or aragonite (1-Ωar,z) increases (Keir, 1980; Walter and Morse, 1985; Sulpis
et al., 2017; Dong et al., 2019; Naviaux et al., 2019b). Here, Ωz is the sediment-depth-dependent saturation state of seawater with
respect to calcite or aragonite, computed as [Ca2+]z⋅[CO32-]z/Ksp∗, where Ksp∗ is the
stoichiometric solubility constant of calcite or aragonite at in situ
temperature, pressure, and salinity, as given in Mucci (1983) and Millero
(1995). At each time step, Ωz is computed using porewater
[Ca2+]z and [CO32-]z from the previous time step,
the latter being calculated as a function of TAlk and the proton
concentration [H+]. At each model time step, the total hydrogen ion
concentration [H+] is computed from TAlk and ∑CO2 using a
single Newton–Raphson iteration from the previous time step (Humphreys et
al., 2022):
[H+]t=[H+]t-110-[TAlk]([H+]t-1,[∑CO2])-[TAlk]d[TAlk]([H+]t-1,[∑CO2])/d[H+]t-1,
where [H+]t is the new [H+] value and [H+]t-1 is
the [H+] from the previous time step. TAlk([H+]t-1,
∑CO2) is the total alkalinity computed from user-specified total
dissolved silicate, [∑PO4] and total borate calculated from
salinity (Uppström, 1974), plus equilibrium constants for silicic acid
(Sillén et al., 1964) and phosphoric acid (Yao and Millero, 1995). Its
derivative is computed following the approach of CO2SYS; see Humphreys et
al. (2022). The carbonate ion concentration is then computed as follows:
[CO32-]=[∑CO2]×K1∗×K2∗K1∗×K2∗+K1∗×[H+]t+[H+]t2,
where K1∗ and K2∗ are the first and second
dissociation constants for carbonic acid, respectively, taken from Lueker et
al. (2000).
The dissolution rates (Rdiss, in molm-3a-1) of calcite
(solid blue line in Fig. 2a) and of aragonite (solid red line in Fig. 2a) as
a function of (1-Ωca) are empirically defined as follows:
12Rdiss. ca.,z=[Calcite]⋅kdiss. ca.⋅(1-Ωca)ηdiss. ca.,13Rdiss. ar.,z=[Aragonite]⋅kdiss. ar.⋅(1-Ωar)ηdiss. ar..
In these expressions, the dissolution rate constant (kdiss, in
a-1) and the reaction order (ηdiss, unitless) are
mineral specific. The dissolution rate constants implicitly account for each
mineral's specific surface area. Similar formulations have previously been
implemented to describe calcite dissolution rates (e.g., Archer et al., 2002)
but in most cases used a high reaction order and a tuned rate
constant independent of solution chemistry (Fig. 2). Such discretizations
are convenient but lack a mechanistic description of the controls on calcite
dissolution in seawater (Adkins et al., 2021).
(a) Dissolution rate of calcite as computed using Eq. (12) and
[Calcite]=104molm-3 (solid blue line), and dissolution rate of
aragonite as computed using Eq. (13) and [Aragonite]=104molm-3 (solid
red line). Note that for each dissolution rate profile, two different rate
constants (kdiss) and reaction orders (ηdiss) are used,
depending on the seawater saturation state, with each accounting for a separate
dissolution mechanism, i.e., step-edge retreat or homogeneous-edge pit
formation. The dashed black line stands for a “traditional” dissolution
rate profile obtained using [Calcite]=104molm-3, a single
dissolution rate constant for the entire (1-Ωca) range
kdiss=10%d-1, and a reaction order ηdiss of 4.5.
(b) Precipitation rate of calcite as computed from Eq. (14). Note that
dissolution rates are normalized here per total solid sediment volume and not
per CaCO3 surface area as in traditional kinetics studies.
The latest advances using isotope-labeling approaches to study carbonate
dissolution kinetics show abrupt changes in dissolution mechanism depending
on solution saturation state with either calcite or aragonite (Subhas et
al., 2017; Dong et al., 2019; Naviaux et al., 2019a, b).
Close to equilibrium, dissolution occurs primarily at sites on the crystal
surfaces that are most exposed to the solution, e.g., steps and kinks.
Further away from equilibrium, dissolution etch pits are activated at
surface sites associated with defects and impurity atoms. Far away from
equilibrium, there is enough free energy for dissolution etch pits to occur
anywhere on the mineral surface, without the aid of crystal defects (Adkins
et al., 2021). However, at temperatures most relevant to the deep oceans,
∼5∘C or less, the defect-assisted dissolution
mechanism is skipped (Naviaux et al., 2019b) and only the step-edge retreat
(close to equilibrium) and homogeneous etch-pit formation (far away from
equilibrium) dissolution regimes remain (Naviaux et al., 2019b) (Fig. 2).
For both aragonite and calcite, while homogeneous etch-pit formation is
indeed associated with a high-order dependency on the solution saturation
state, step-edge retreat dissolution rates dominating near equilibrium show
very little dependence on seawater saturation (Dong et al., 2019; Naviaux et
al., 2019a). This could have significant consequences for the predicted
carbonate dissolution rate near equilibrium: saturation-state-independent
step-edge retreat dissolution will always be predicted to be faster close to
equilibrium than dissolution associated with a high reaction order because a
high reaction order forces the dissolution rate to converge to zero as the
solution gets closer to equilibrium (Fig. 2).
Naviaux et al. (2019a) derived reaction orders for two separate regions of
the (1-Ωca) spectrum: the Ωca threshold value
dividing these two regions was Ωca. critical≈0.8. Here, based on the results of Naviaux et al. (2019a), we set ηdiss. ca.=0.11 for 0.828<Ωca<1
and ηdiss. ca.=4.7 for Ωca≤0.828. The Ωca critical value used here is slightly higher
than the ∼0.8 value given in Naviaux et al. (2019a) in order
to have a smooth transition between defect-assisted and homogeneous
dissolution. For aragonite, based on the results of Dong et al. (2019), we
set ηdiss. ar.=0.13 for 0.835<Ωar<1 and ηdiss. ar.=1.46 for Ωar≤0.835. The rate constants are tuned to best fit the observations in the two
stations presented in Sect. 3. We use kdiss. ca.=6.3×10-3a-1 for 0.828<Ωca<1,
kdiss. ca.=20a-1 for Ωca≤0.828,
kdiss. ar.=3.8×10-3a-1 for 0.835<Ωar<1, and kdiss. ar.=4.2×10-2a-1 for Ωar≤0.835. Both calcite and aragonite
dissolution rate constants are lower than the values reported in the
original publications. We suspect that (i) the reactive surface area of
grains in sediments is much smaller than their specific surface area
measured using adsorption isotherms via the Brunauer–Emmett–Teller (BET) method and (ii) unaccounted
dissolution inhibitors are present in sediments, such as dissolved organic
carbon (Naviaux et al., 2019a). A comparison of the steady-state
[CO32-] and [Calcite] porewater profiles predicted by RADI using the
tuned rate constants implemented in RADIv1 and the original rate constants
is shown in Fig. S1 in the Supplement.
Calcite precipitation is also included in the model and its rate (solid blue
line in Fig. 2b) is parameterized with the following function:
Rprec. ca.,z=kprec. ca.⋅(Ωca-1)ηprec. ca.,
where kprec. ca is the precipitation rate constant set to 0.4 molm-3a-1 and η is equal to 1.76. The precipitation reaction
order is taken from Zuddas and Mucci (1998), corrected for a seawater-like
ionic strength of 0.7 molkg-1. The precipitation and dissolution rate
continuum implemented in RADI (see Fig. 2) is very different from what a
classic model with only calcite dissolution following high reaction order
kinetics would display. For comparison, the dissolution rate of calcite
using a dissolution rate constant kdiss of 10 %d-1 and a
reaction order η of 4.5, as implemented in most diagenetic models,
including Archer (1991), is shown in Fig. 2a. The value of 10 %d-1 for the rate constant was chosen because it makes the
“traditional” calcite dissolution law overlap with the RADI dissolution
law so that any differences between the two can be attributed to enhanced
dissolution caused by step-edge retreat close to equilibrium. Mechanistic
interpretations of the “kinks” in the dissolution rate profiles and of a
non-zero dissolution rate near equilibrium still require more research, but
the implications of these features for our understanding of marine
CaCO3 cycles can be explored with the present model.
Advection
The solid burial velocity at the sediment–water interface, w0 in ma-1, is given by
w0=∑Fv⋅Mvρv/φs,0,
where Fv is the flux of a solid species at the sediment–water interface
(molm-2a-1), Mv is the molar mass of that solid (gmol-1),
and ρv is its solid density (gm-3). The solid and
porewater burial velocity at greater depth are assumed to be equal and are
computed as follows:
w∞=u∞=w0φs,0/φs,∞.
Thus, the porewater burial velocity, u, at all depths is
uz=u∞φ∞/φz,
and the solid burial velocity, w, is
wz=w∞φs,∞/φs,z.
Depth profiles of u and w are shown in Fig. 1, computed from the solid fluxes
at station 7 of cruise NBP98-2 (Sayles et al., 2001); see Sect. 3.2.
In Fig. 1, the sharp porosity decline in the top centimeters of the
sediments causes the solid fraction at ∼5 cm depth to be
roughly 50 % higher than just below the interface. This leads to a solid
burial velocity decrease of about the same magnitude (Fig. 1).
Advection is implemented following Boudreau (1996b), where the advection
rate (Az, in molm-3a-1) for solutes is given by
Az(v)=-uz-dz(v)φz⋅dφzdz-d∘(v)⋅d(1/θz2)dz19⋅v(z+dz)-v(z-dz)2dz,
where dz is the effective diffusion coefficient for a given solute at a
given depth (in m2a-1), d∘ is the “free-solution” molecular
diffusion coefficient for a given solute (in m2a-1), and
θz is the depth-dependent tortuosity (unitless) defined in Sect. 2.4. For solids, a more sophisticated weighted-difference scheme is employed
(Fiadeiro and Veronis, 1977; Boudreau, 1996b):
Az(v)=-wz-dbzdz-bzφs,z⋅dφs,zdz20⋅(1-σz)v(z+dz)+2σzvz-(1+σz)v(z-dz)2dz,
where bz is the depth-dependent bioturbation coefficient (m2a-1) and
σz(v)=1/tanh(Peh,z)-1/Peh,z,
where
Peh,z=wz⋅dz/2bz.
The parameter Peh is half of the cell Péclet number, which expresses
the influence of advection relative to bioturbation across a distance
separating two points of the grid, centered at the depth z. If bioturbation
dominates (Peh≪1), e.g., near the sediment–water
interface, σz tends toward zero and a centered-difference
discretization is implemented. If advection dominates (Peh≫1), e.g., deeper in sediments, σz tends toward
unity and backward-difference discretization prevails; see Eq. (20). This
differencing scheme, originally developed by Fiadeiro and Veronis (1977),
maintains stability in the entire sediment column (Boudreau, 1996b).
Diffusion
The diffusion flux of any species depends on its effective diffusion
coefficient, dz(v), which varies with depth within the sediment.
For each solute, free-solution diffusion coefficients, denoted
dz∘(v), were computed at in situ temperatures (Li and Gregory,
1974; Boudreau, 1997; Schulz, 2006). For solute variables representing
several individual species (e.g., ΣPO4, ΣCO2), the
diffusion coefficient of the dominant species was considered. Given the high
proportion of HCO3- relative to CO32- and CO2 (aq) in seawater and porewaters (see Fig. S2 in the Supplement), the diffusion coefficient
of HCO3- was adopted for both TAlk and ΣCO2. However,
this approach may not be suited for sedimentary environments in which pH is
lower than 7 because a greater proportion of dissolved inorganic species
would then be under the form of carbonic acid, i.e., CO2 (aq), which has
a higher diffusion coefficient than HCO3- (Fig. S2). Free-solution
diffusion coefficients, their temperature dependencies, and their sources are
reported in Table 5. The diffusion of solutes in the porewaters is slower
than in an equivalent volume of water as a result of the physical barriers
caused by the presence of solid grains in a sediment. To correct for this
effect, we follow Boudreau (1996b) and compute the effective diffusion
coefficient for a given solute as follows:
dz(v)=d∘(v)/(θz2),
where so-called tortuosity (θ) is defined as follows (Boudreau, 1996a):
θz=1-2lnφz.
Diffusion coefficientValueSourcedz∘ (TAlk)0.015179+0.000795×TwBoudreau (1997), Schulz (2006)1dz∘ (ΣCO2)0.015179+0.000795×TwBoudreau (1997), Schulz (2006)1dz∘ (Ca2+)0.011771+0.000529×TwLi and Gregory (1974)dz∘ (O2)0.031558+0.001428×TwBoudreau (1997), Schulz (2006)dz∘ (ΣNO3)0.030863+0.001153×TwLi and Gregory (1974)2dz∘ (ΣSO4)0.015779+0.000712×TwLi and Gregory (1974)3dz∘ (ΣPO4)0.009783+0.000513×TwBoudreau (1997), Schulz (2006)4dz∘ (ΣNH4)0.030926+0.001225×TwLi and Gregory (1974)5dz∘ (ΣH2S)0.028938+0.001314×TwBoudreau (1997), Schulz (2006)dz∘ (Fe2+)0.001076+0.000466×TwLi and Gregory (1974)dz∘ (Mn2+)0.009625+0.000481×TwLi and Gregory (1974)
1 value for HCO3- ion,
2 Value for NO3- ion.
3 Value for SO42- ion.
4 Value for HPO42- ion.
5 Value for NH4+ ion.
For each solid, effective diffusion occurs through the mixing action of
burrowing microorganisms, quantified using a bioturbation coefficient that
decreases with depth. Archer et al. (2002) used a dataset including 53
sediment sites ranging in depth from 47 to 5668 m to derive an optimal
bioturbation rate profile, in which the rate of bioturbation increases with
increasing flux of total organic carbon reaching the seafloor
(FPOC) and attenuates in low-oxygen conditions. This pattern
was also observed by Smith et al. (1997) and Smith and Rabouille (2002). As
in Archer et al. (2002), we couple both bioturbation and irrigation to the
incoming carbon deposition flux (Fig. 3) rather than water depth or sediment
accumulation rate (Boudreau, 1994; Middelburg et al., 1997; Soetaert et al.,
1996c), although all these quantities are related to each other. From an
ecological perspective, more carbon to the seafloor represents more food
available to benthic communities, hence more biological transport. Linking
bioturbation activity to carbon deposition flux also allows for a direct
coupling with Earth system models simulating carbon sinking fluxes in the
ocean. Following Archer et al. (2002), we express the surficial bioturbation
mixing rate (b0, in m2a-1) as follows:
b0=(2.32×10-6)(FPOC×102)0.85,
where FPOC is expressed in molm-2a-1. The
bioturbation mixing rate at all depths (bz, in m2a-1) is
bz=b0e-(z/λb)2[O2]w[O2]w+0.02,
where the characteristic depth λb=8 cm, following Archer et
al. (2002), and [O2]w is the oxygen
concentration in the bottom waters. This depth-dependent bioturbation mixing
rate is common to all solids, and its depth distribution is shown in Fig. 3
as a function of in situ [O2]w and
FPOC. The effective diffusion coefficient for solids is then
set as follows:
dz(v)=bz.
Bioturbation mixing rate bz and irrigation coefficients
αz as a function of sediment depth z, organic carbon deposition
flux FPOC, and dissolved oxygen concentration in the bottom waters
[O2]w.
The (bio)diffusion is implemented in RADI following the centered difference
discretization scheme from Boudreau (1996b). At sediment depth z, where 0<z<Z, for both solutes and solids:
Dz(v)=dz(v)⋅(v(z-dz)-2vz+v(z+dz))/(dz)2,
where dz(v) is the relevant effective diffusion coefficient.
Irrigation
The mixing of solutes caused by burrow flushing or ventilation occurs
through an ensemble of processes collectively termed irrigation. Macroscopic burrows
are often present in the seafloor sediment, with a complex three-dimensional
structure and filled with oxygenated water that is ventilated for aerobic
respiration. In a one-dimensional framework, this causes apparent internal
sources or sinks of porewater solutes at particular depths (Boudreau, 1984;
Emerson et al., 1984; Aller, 2001). Mathematically, this is parameterized as
a non-local exchange function, i.e., a first-order kinetic reaction:
It,z(v)=αz(vw-vz),
where αz is an irrigation coefficient common to all solutes
(expressed in a-1). Following Archer et al. (2002), who used a dataset of
53 sediment sites comprised of microelectrode oxygen profiles and chamber
oxygen fluxes across the sediment–water interface to derive an
irrigation rate profile, we express the surficial irrigation coefficient as
a function of the organic carbon deposition flux and the oxygen
concentration of the overlaying waters:
α0=11tan-15FPOC×102-400400π+0.5-0.930+20[O2]w[O2]w+0.01⋅FPOC×102FPOC×102+30⋅e-[O2]w0.01
and the irrigation coefficient at all depths as follows:
αz=α0e-(z/λi)2,
where the characteristic depth λi is 5 cm (Archer et al., 2002).
The depth distribution of the irrigation coefficient is shown in Fig. 3 as a
function of in situ [O2]w and
FPOC.
Boundary conditions
Modeling of advection and diffusion processes requires appropriate boundary
conditions in the layers above and below (z-dz and z+dz, respectively).
Effective values of each variable immediately adjacent to the modeled depth
domain are calculated following Boudreau (1996b) and used to compute the
effects of advection and diffusion in the top and bottom layers using the
same equations as within the sediment itself.
At the sediment–water interface, RADI enables prescribed solid fluxes and a
diffusive boundary layer control for solutes. Following Boudreau (1996b), we
calculate advection and diffusion at z=0 for solutes and solids as follows:
v(-dz)=vdz+2θz2dzδ(vw-v0),
and
v(-dz)=vdz+2dzb0Fvφs,0z-w0v0,
respectively. Here, θ is the tortuosity, δ is the boundary
layer thickness (expressed in m; see Fig. 1), and vw is the solute
concentration above the diffusive boundary layer, i.e., in the bottom
waters. At the sediment depth z=Z, v(Z+dz) falls
outside the depth range of the model. The bottom boundary condition demands
that concentration gradient disappear, which can be translated by the
following for both solutes and solids:
v(Z+dz)=v(Z-dz).
This “no-flux” bottom boundary condition should be appropriate here because
we set Z so that all action occurs at shallower depth. However, if anaerobic
methane oxidation or subsurface weathering are included in future versions,
a “constant” flux boundary condition might need to be included.
Julia and MATLAB/GNU Octave implementations
We have implemented RADI both in Julia (Humphreys and Sulpis, 2021) and in
MATLAB/GNU Octave (Sulpis et al., 2021). Both implementations use similar
nomenclature and provide identical results. Documentation for both is
available from https://radi-model.github.io (last access: March 2022). The Julia
implementation is available from https://github.com/RADI-model/Radi.jl (last access: March 2022), and the MATLAB/GNU Octave
implementation is available from https://github.com/RADI-model/Radi.m (last access: March 2022).
Julia (https://julialang.org, last access: March 2022) is a high-level,
high-performance, and cross-platform programming language that is free and
open source (Bezanson et al., 2017). Its high performance stems primarily
from just-in-time (JIT) compilation of code before execution, which has been
built-in since its origin. RADI uses Julia's multiple-dispatch paradigm, a
core feature of the language, which improves the readability of the code and
reduces the scope for errors. Specifically, each modeled component of the
sediment column is either a porewater solute or a solid. These components
are initialized in the model as variables either of Solute or Solid type.
Advection and diffusion are governed by different equations for porewater
solutes than for solids, but the same top-level functions (advect! and
diffuse!) can be used within RADI to calculate the effects of these
processes for both component types; the multiple-dispatch paradigm ensures
that the correct equations are automatically used on the basis of the type
of the input variable. While the model has been designed to solve a single
profile at a time, Julia's support for parallelized computation (across
multiple processors) would also support efficient computations across a
series or grid of vertical profiles.
As of version R2015b, MATLAB also features JIT compilation with a
corresponding execution speed-up. However, MATLAB is an expensive,
proprietary software, which limits how widely it can be used. The MATLAB
implementation also runs in GNU Octave (https://www.gnu.org/software/octave/, last access: March 2022), which is a free and open-source
clone of MATLAB. However, GNU Octave executes more slowly than MATLAB for a
variety of reasons, including a lack of JIT compilation.
For a model that necessarily includes long simulations with relatively short
time steps, computational speed is an important consideration. Our testing
indicates that the Julia implementation runs ∼3 times faster
than the MATLAB (R2020a) implementation and ∼70 times faster
than the GNU-Octave implementation.
Simultaneously developing the model in two languages allowed us to quickly
identify and remedy bugs and typographical errors in both implementations.
Each was coded independently, with equations and parameterizations
written out from the original sources, thus avoiding code copy-and-paste
errors. Frequent comparisons were made throughout this process to ensure
that the results were consistent. For a typographical error to survive to
the final models would therefore require an identical mistake to have been
made independently in both implementations. The risk of such errors is thus
substantially reduced by our dual-language approach. Where errors were
identified, in some cases they were subtle enough that they may otherwise
not have been noticed, while still causing meaningful errors in final model
results.
Model evaluation
To evaluate the performance of RADI, we used in situ data obtained at three
different locations and compared our predictions to the measured porewater
and sediment solid-phase composition profiles. We used these comparisons to
tune the CaCO3 dissolution and POC degradation rate constants; all
other parameters were assigned a priori using values from the literature.
Thus, we did not aim to reproduce observations as accurately as possible by tuning
a wide selection of parameters. Instead, we evaluated whether a generic
approach using measured deposition fluxes and bottom-water conditions could
explain observations while tuning only the inorganic and organic reactivity
constants.
Northwestern Atlantic Ocean
First, RADI was compared to the porewater and sediment composition
measurements of station no. 9 described in Hales et al. (1994), located in
the northwestern Atlantic Ocean (24.33∘ N, 70.35∘ W)
at a 5210 m depth. The bottom-water TAlk and ΣCO2
were 2342 and 2186 µmolkg-1, respectively, bottom-water
in situ temperature was 2.2 ∘C, salinity was 34.9, and oxygen
concentration was 266.6 µmolkg-1 (Hales et al., 1994). The
computed bottom-water saturation state with respect to calcite was 0.88. The
only CaCO3 polymorph reaching the seafloor was assumed to be calcite.
The calcite flux to the seafloor was set to 0.20 molm-2a-1
(20.02 gm-2a-1) and the POC flux to 0.18 molm-2a-1,
which correspond to the low end of fluxes measured by sediment traps on the
continental slope (Hales et al., 1994). The clay flux was set to a value of
26 gm-2a-1 to fit the calcite sediment surface concentration
measured by Hales et al. (1994). The porosity at the sediment–water
interface was set to that measured by Sayles et al. (2001) in the Southern
Pacific Ocean station; see Fig. 1. Following the diffusive boundary layer
distribution from Sulpis et al. (2018), δ at the station location
was set to 938 µm. This value represents an annual-mean estimate
derived using a number of assumptions, e.g., considering the sediment–water
interface to be a horizontal surface and neglecting sediment roughness. A
complete description of the environmental parameters for this North Atlantic
station, along with their sources, is available in Table S1 in the Supplement.
RADI was run using the environmental conditions described above and the
steady-state concentration profiles of O2, ΣNO3, calcite,
and POC were compared with observations. Complete methods for solute and
solids measurements are described in Hales et al. (1994). Briefly, porewater
O2 concentration was measured both in situ using microelectrodes and
on board (along with ΣNO3) from the retrieved box core (Hales
et al., 1994). The steady-state calcite, TAlk and ΣCO2 profiles
were compared to those obtained from a RADI simulation with “traditional”,
4.5-order calcite dissolution kinetics (see Fig. 2) with all other variables
being unchanged.
RADI predicts a porewater O2 concentration decreasing from the
bottom-water value to zero at ∼20 cm depth (Fig. 4). In the
top 2 cm, the RADI porewater O2 predictions near the surface are in
good agreement with the in situ microelectrode measurements. The
RADI-predicted [O2] is lower than that measured on board, but [ΣNO3] is well reproduced by RADI. RADI predicts that organic matter
respiration is mainly aerobic (see Table 3a) until about 20 cm depth.
Between 20 and 35 cm depth, ΣNO3 is the preferred oxidant for
organic matter degradation (see Table 3b), which leads to a decrease in
porewater [ΣNO3] in both RADI predictions and observations,
followed by ΣSO4 deeper than 35 cm depth. The calcite profile
is relatively well reproduced by RADI in the top 20 cm, but the measured
calcite concentration drop below 20 cm depth is not well reproduced. When
“traditional” 4.5-order calcite dissolution kinetics are implemented,
calcite concentrations are similar to those predicted by RADI, but the
predicted [TAlk] and [∑CO2] are slightly different, being lower
(≲10µmolkg-1) than RADI's in the top
15 cm, and higher in the deeper part of the sediment column. The observed
lower calcite concentrations below 20 cm depth may be attributed to a lower
calcite accumulation rate to the seafloor in the past, whereas the model
considers accumulation of solids to be unchanged through time. Calcite
concentration predicted by RADI does increase again below 25 cm depth due to
porewater supersaturation (Ωca at 40 cm depth is about 1.05),
but this increase is too small to be noticed in the figure.
Comparison of RADI with measurements from the northwestern
Atlantic station no. 9 described in Hales et al. (1994). The lower panels
represent (f) the computed CO32- concentrations in porewaters
(solid black line) and at equilibrium with respect to calcite (dashed blue
line) and (g) the fractions of organic matter degraded by a given
oxidant.
Southern Pacific Ocean
RADI was also compared with data collected at the station no. 7 mooring
no. 3 described in Sayles et al. (2001), located in the southern Pacific
Ocean (60.15∘ S, 170.11∘ W), where the seafloor lies at
a 3860 m depth. This dataset (http://usjgofs.whoi.edu/jg/dir/jgofs/southern/nbp98_2/,
last access: 28 January 2022)
constrains the sedimentary system well: sediment trap CaCO3 and POC
fluxes, CaCO3, and POC sediment composition, sediment porosity, and
porewater solute depth profiles are all available from the same cruise and
location.
The bottom-water chemical composition was taken from the GLODAPv2
1∘×1∘ climatologies (Lauvset et al., 2016), linearly interpolated
over depth, latitude, and longitude to match the station location and
seafloor depth. The bottom-water in situ temperature was 0.84 ∘C,
salinity was 34.696, oxygen concentration was 215.7 µmolkg-1, and
calculated saturation state with respect to calcite was 0.85. Solid fluxes
at this station were measured by Sayles et al. (2001) using sediment traps
collecting sinking particles between the months of November and December
1997. Their deepest sediment trap available was at a depth of 3257 m, i.e.,
600 m a.s.f., which we assume to be representative of
sinking fluxes to the seafloor, although the loss of material after
collection usually causes sediment traps to underestimate the real sinking
fluxes (Buesseler et al., 2007). The only CaCO3 polymorph reaching the
seafloor was assumed to be calcite, and its flux was set to 0.25 molm-2a-1 (25.02 gm-2a-1), rather than using the sediment trap
value, in order to fit the calcite sediment surface concentration measured
by Sayles et al. (2001). This CaCO3 flux to the seafloor is slightly
higher than the measured CaCO3 flux at 600 m above the seafloor in
mid-January 1997 (0.19 molm-2a-1; Sayles et al., 2001). The POC
flux was set to 0.14 molm-2a-1 (4.62 gOMm-2a-1),
which is also slightly higher than the measured POC flux averaged between
the months of November and December 600 m above the seafloor (0.11 molm-2a-1). The clay flux, which we considered to be the total
measured sediment flux minus the assumed POC and calcite fluxes, was 32 gm-2a-1. The porosity profile was tuned to best fit the porosity
measurements at this station (Sayles et al., 2001, see Fig. 1). Finally,
using the diffusive boundary layer world map computed in Sulpis et al.
(2018) based on bottom current speeds at in situ temperature and pressure
measurements, the diffusive boundary layer thickness (δ) at the
station location was set to 715 µm. A complete description of the
environmental parameters for this station, along with their sources, is
available in Table S2 in the Supplement.
RADI was run using the environmental conditions described above to compare
the steady-state concentration profiles of TAlk, ΣCO2, O2,
ΣNO3, calcite, and POC to in-situ measurements. Methods for
solutes and solids concentration analyses are described in Sayles et al.
(2001). Briefly, TAlk, ΣCO2, and ΣNO3 were sampled
in situ using the Woods Hole Interstitial Marine Probe (Sayles, 1979), while
O2 was sampled at a higher depth resolution but in the ship laboratory
using whole-core squeezing (Bender et al., 1987). We also compare the RADI
steady-state concentration profiles with those obtained from a simulation
with “traditional” 4.5-order calcite dissolution kinetics, all other
variables being the same.
RADI predicts porewater O2 concentrations that are slightly higher than
observed (Fig. 5). Because RADI does not predict porewater O2 to go to
zero until about the 30 cm depth, the dominant organic matter degradation
pathway switches from mainly aerobic to ΣNO3 at about the 30 cm
depth. Nevertheless, the RADI-predicted ΣNO3 profile is lower
than observed values, particularly toward the bottom of the resolved depth.
The TAlk and ΣCO2 porewater profiles predicted by a RADI
simulation using 4.5-order calcite dissolution kinetics are slightly lower
(≲40µmolkg-1) than those using the
new calcite dissolution kinetics scheme, and the predicted calcite
concentrations are slightly higher (≲2 %).
Comparison of RADI with measurements from the station no. 7
mooring no. 3 MC-1 described in Sayles et al. (2001). The lower panels
represent (f) the computed CO32- concentrations in porewaters
(solid black line) and at equilibrium with respect to calcite (dashed blue
line) and (g) the fractions of organic matter degraded by a given
oxidant.
Central equatorial Pacific Ocean
As a third case study to evaluate the performance of RADI, solute fluxes
through the sediment–water interface computed from model steady-state runs
were compared to fluxes measured using benthic chambers. The comparison took
place at station no. W-2 described in Berelson et al. (1994) and Hammond et
al. (1996), located in the central equatorial Pacific Ocean (0∘ N, 139.9∘ W) at a depth of 4370 m. Bottom-water in situ temperature
was set to 1.40 ∘C and salinity was set to 34.69 (Lauvset et al., 2016).
Bottom-water oxygen concentration was set to 159.7 µmolkg-1 and
the bottom-water saturation state with respect to calcite computed using the
carbonate system solver within RADI was 0.78. For the purposes of this
evaluation, the CaCO3 flux to the seafloor was assumed to be entirely
calcite. The calcite flux was set to 0.22 molm-2a-1, which
represents 22.02 g of calcite m-2a-1, the POC flux was 0.20 molm-2a-1, that is, 6.6 g of organic matter m-2a-1, and
the clay flux was set to 2.0 gm-2a-1. The steady-state calcite
content within the top centimeter was 61 drywt%, in line with CaCO3
contents observed in this area (Archer, 1996; Hammond et al., 1996). The
porosity profile was built using an attenuation coefficient β=33m-1, φ0=0.85, which is the measured surface porosity
(Hammond et al., 1996), and φ∞=0.74, which is the
measured porosity at depth (Berelson et al., 1994); see Eq. (3). The DBL
thickness, δ, was fixed to a value of 1 mm. A complete description
of the environmental parameters for this station, along with their sources,
is available in Table S3 in the Supplement.
The diffusive fluxes for a given solute (Jv) between the sediment–water
interface and the bottom waters occur as a response to the concentration
gradient within the DBL and can be expressed by
Jv=φ0Dv×v0-vwδ,
where v0 and vw are solute concentrations at the sediment–water
interface and in bottom waters, respectively. In this definition, a positive
Jv indicates a solute release from the sediment porewaters to the bottom
waters, while a negative Jv represents a solute flux towards the
sediment.
The predicted TAlk, ΣCO2, ΣPO4, and O2 fluxes
(0.30 molm-2a-1, 0.32 molm-2a-1,
1.9 mmolm-2a-1, and
-0.23molm-2a-1, respectively) are all within the
uncertainty bounds of the fluxes measured by benthic chambers at the same
location (0.28±0.09molm-2a-1, 0.24±0.09molm-2a-1, 1.4±0.5mmolm-2a-1, and -0.26±0.03molm-2a-1, respectively; see Fig. 6). Nevertheless, the predicted ΣNO3 flux (9.0 mmolm-2a-1) is lower than
its measured value (18±5mmolm-2a-1).
Comparison of fluxes computed from RADI with benthic-chamber
measurements from the station no. W-2 described in Hammond et al. (1996).
Error bars are included for all measured fluxes but are not always visible.
Discussion of model performance
These three model evaluation examples allowed us to determine a set of
organic carbon degradation rate constants, CaCO3 dissolution rate
constants and organic carbon flux composition (fast-decay, slow-decay, or
refractory) that can best reproduce sediment and porewater measurements in
all stations while keeping all other model parameters to values from the
literature. In each station, bottom-water composition was fixed from
observations. Due to a lack of adequate data, solid deposition fluxes were
tuned to best fit observed CaCO3 and POC contents in sediments, except
in the northwestern Atlantic station, where the CaCO3 and POC fluxes
were taken from measurements, and in the southern Pacific station, where the
clay flux was inferred from observations. The POC composition that allows to
best fit porewater and sediment data in the three stations was as follows:
70 % of fast-decay POC, 27 % of slow-decay POC, and 3 % of refractory
POC. The tuned fast- and slow-decay POC degradation rate constants are
reported in Sect. 2.2.1 and are of similar orders of magnitude as in most
other models (Arndt et al., 2013). The tuned CaCO3 dissolution rate
constants are reported in Sect. 2.2.3 and are 2 orders of
magnitude lower than their laboratory-based values (Fig. S1), which we
attribute to the presence of dissolution inhibitors (e.g., dissolved organic
carbon, Naviaux et al., 2019a) or to the reactive surface area of natural
grains in situ being lower than in laboratory experimental settings. Thus,
with its current settings, RADI should be able to accurately predict
porewater chemistry and sediment composition in deep-sea environments,
provided that the POC and CaCO3 deposition fluxes are known.
In the central equatorial Pacific, all RADI diffusive fluxes are within the
uncertainty range of observations except the ΣNO3 flux, which
is underestimated by RADI. The low ΣNO3 flux could be
attributed, for example, to the presence of organic matter with a
stoichiometry different than the Redfield ratio used in the current version
of RADI or to errors in the nitrification parameters.
In addition, we note that the choice of calcite dissolution kinetics
implemented in RADI does not seem to have a large impact on TAlk and ΣCO2 porewater profiles or on the predicted calcite concentrations.
RADI's step-edge retreat dissolution regime and its low reaction order
induce calcite dissolution rates near equilibrium that are orders of
magnitude higher than what is predicted in a high-order rate law (Fig. 2),
but if the rate constant of a high-order rate law is tuned so that it
overlaps the homogeneous dissolution rate law far from equilibrium,
differences are limited (Fig. 2). Thus, we conclude that using a 4.5-order
rate law with a 10 %d-1 rate constant or using the new,
mechanistic calcite dissolution rate scheme implemented in RADI should lead
to similar predictions.
Potential model applications
In the following section, we continue to analyze the results obtained using
the environmental conditions of the equatorial Pacific station no. W-2 and
present a few examples of relevant model applications. RADI can be used to
study both steady-state and transient conditions, but in the following
subsections we focus on time-dependent problems, since transient diagenetic
models are underrepresented in the literature.
Seasonal variability
At the seafloor, the fluxes of sinking material regulating the chemical
composition of sediment porewaters are patchy in both space and time.
Seafloor microbes and macrofauna respond quickly to pulses of organic matter
delivery to the seafloor (Smith et al., 1992), causing short-term
variability of sediment oxygen consumption (Smith and Baldwin, 1984; Smith
et al., 1994). In addition, both the POC and CaCO3 fluxes to the deep
seafloor are strongly affected by seasonal flux variability (Billett et al.,
1983; Smith and Baldwin, 1984; Lampitt, 1985; Lampitt et al., 1993, 2010). In the northeastern Atlantic Ocean at 3000 m depth, Lampitt et
al. (2010) have shown that the summer POC and CaCO3 fluxes can be
∼10 and ∼4 times higher, respectively, than
the wintertime minima. The seasonal coupling between organic matter and
CaCO3 fluxes to the seafloor and the state of upper-ocean ecosystem is
the result of rapid vertical transport of these materials (Sayles et al.,
1994). If the fluxes of reactive material reaching the seafloor are affected
by seasons, early diagenesis could display a seasonal signal, and this should
be taken into account when interpreting sedimentary data (Martin and Bender,
1988; Sayles et al., 1994; Soetaert et al., 1996a). Here we use RADI to
assess how the porewater chemistry and solid composition of deep-sea
sediments may be impacted by seasonally varying fluxes.
Seasonally time-varying solid fluxes to the seafloor (F, in molm-2a-1)
can be represented with the following sinusoidal function:
F(t)=Faverage+ΔFsin2πtΔt,
where t is time in years, Δt is the time period separating two maxima
(here set to 1 year), and ΔF is the amplitude. We assume that all
CaCO3 settling at the seafloor is calcite and set the mean
FCalcite to 0.22 molm-2a-1, its amplitude change ΔFCalcite to 0.11 molm-2a-1, the mean FPOC to 0.20 molm-2a-1, and its amplitude change ΔFPOC to 0.10 molm-2a-1. All other parameters correspond to values from the
central equatorial Pacific stations described in Sect. 3.3.
Seasonal variations in the calcite and POC fluxes reaching the seafloor are
visible in sediment profiles of both solids and solutes (Fig. 7).
Nonetheless, the amplitude of concentration changes separating an annual
minimum from an annual maximum is very small, barely if (at all) detectable
by observations. The annual amplitude is about 0.5 wt % for calcite, 0.07 wt % for POC, 2 µmolkg-1 for [O2], and 7 µmolkg-1 for [ΣCO2]. While concentrations at the
sediment–water interface respond quickly to seasonal flux changes, there is
a phase lag that increases with depth between the concentrations of solids
and porewater solutes and the seasonally changing fluxes to the seafloor.
Thus, it is possible that porewaters and solid particles at the top millimeter-thick
sediment layers are never really at a steady state but always lagging behind
seasonal changes, even if these are minimal. This is in agreement with
earlier modeling (Martin and Bender, 1988) and observational (Sayles et al.,
1994) studies, indicating that biogeochemical reactions and bioturbation at
the deep seafloor are too slow to show a discernible seasonal signal.
However, this might not be the case for sites receiving more reactive
organic matter (Soetaert et al., 1996a).
Response of porewater O2, ΣCO2, calcite, and POC
concentrations to seasonal fluctuations in the calcite and POC fluxes
reaching the seafloor.
Tidal cycles
This section explores the applicability of RADI for studying the response of
sediments to higher-frequency phenomena such as tides. The DBL thickness is
dependent on the overlying current speed (Levich, 1962; Santschi et al.,
1983; Lorke et al., 2003): slower currents generate thicker DBLs, whereas
faster currents cause the DBL to thin (Larkum et al., 2003; Lorke et al.,
2003; Higashino and Stefan, 2004). In the deep sea, tidal forces are an important contributor
to benthic current speeds, which means that tidal currents are a
potentially important contributor to biogeochemical exchanges across the
sediment–water interface (Egbert and Erofeeva, 2002; Sulpis et al., 2019).
If tidal current speed fluctuations induce DBL thickness fluctuations, they
may induce solute concentration fluctuations at the sediment–water
interface, thus affecting early diagenesis. The strongest tidal currents
occur during the transition from high to low tides. For semidiurnal tides,
the time period separating a low from a high tide is ∼6 h
(Pugh, 1987). Setting the average DBL thickness to δ=1 mm and
assuming that tides generate δ fluctuations with an amplitude
Δδ=0.5 mm, the time-dependent δ can be expressed
as follows:
δ(t)=δaverage+Δδsin2πtΔt,
where t is time in years and Δt is set to 1/1461 a (∼6 h). RADIv1 was run using the steady-state solutes and solids depth
profiles from the equatorial Pacific station no. W-2 as initial conditions,
with a DBL thickness fluctuating in response to tidal currents computed
using Eq. (37).
While none of the solids seem to respond to tidal velocity fluctuation due
to their slow accumulation rate, solutes show a clear response (Fig. 8). At
the sediment–water interface, the simulated porewater [ΣCO2]
variation amplitude within a single tidal cycle is about 25 µmolkg-1,
while [O2] oscillates with an amplitude of about
8 µmolkg-1. Deeper than a few millimeters below the sediment–water interface,
the amplitude of both [ΣCO2] and [O2] changes become very
small and tidal cycles are not measurable in the concentration profiles.
Response of porewater O2, ΣCO2, calcite, and POC
concentrations to fluctuations in DBL thickness (δ) driven by tidal
currents.
The implications are potentially important for our interpretation of
porewater microprofiles. Profiles of pH (Archer et al., 1989b; Cai and Reimers, 1993;
Zhao and Cai, 1999; Cai et al., 2000); pCO2 (Cai et al., 2000; Zhao and
Cai, 1997); CO32- (de Beer et al., 2008; Han et al., 2014; Cai et
al., 2016); O2 (Revsbech et al., 1980; Reimers, 1987; Archer et al.,
1989a; Sosna et al., 2007); and even dissolved Fe, Mn, or S(-II) (Brendel and
Luther, 1995) microelectrodes have been developed during the past decades.
According to the results presented here, microprofiles, which capture
instantaneous snapshots of porewater chemistry, should show appreciable
differences depending on when they are carried out during a tidal cycle.
That organic matter degradation rates inferred from oxygen microprofiles
span a wide range (Archer et al., 1989a; Arndt et al., 2013; Wenzhöfer
et al., 2016) may be, among other factors, due to the dependency on tidal
and other ocean bottom current fluctuations. To adequately capture O2
consumption rate in sediments, O2 fluxes should be measured and
integrated over a period of time longer than a tidal cycle (Berg et al.,
2022).
Benthic chambers
RADI can also be used in the calibration of sensors and the optimization of
sampling protocols and experimental designs. In the DBL, molecular diffusion
is the dominant mode of solute transport, and laboratory experiments of
CaCO3 dissolution in seawater suggest that diffusion through the DBL is
the rate-limiting step for CaCO3 dissolution at the seafloor in the
absence of organic matter respiration (Sulpis et al., 2017; Boudreau et al.,
2020). Nevertheless, earlier assessments of in situ CaCO3 dissolution
at the sediment–water interface in the central equatorial Pacific indicated
that DBL thickness does not impact overall dissolution rates (Berelson et
al., 1994).
In their study, Berelson et al. (1994) deployed a set of free-sinking
benthic chambers onto the seafloor. In each chamber, the portion of the
chamber exposed above the sediment–water interface was sealed and isolated
from external bottom waters, and water samples were drawn during the
incubation period. Each incubation lasted between 80 and 120 h. Chambers
were stirred with a paddle at various rates to quantify the dependency of
the measured diffusive fluxes across the sediment–water interface on the DBL
thickness, which were calibrated via anhydrite dissolution to the
300–600 µm range. Seeing no influence of the stirring rate on the measured
diffusive fluxes, Berelson et al. (1994) discarded the hypothesis of fast,
surficial carbonate dissolution and instead argued for slow,
high reaction order calcite dissolution kinetics at the seafloor, as
subsequently implemented in most models. To better interpret the results
from a benthic-chamber study such as that of Berelson et al. (1994), RADI
can be used to predict the time response of diffusive fluxes across the DBL
following an instantaneous change of DBL thickness due to, for instance, a
change in paddle stirring rate within a chamber.
Response of TAlk, ΣCO2, and O2 diffusive fluxes
through the DBL to instantaneous changes in the DBL thickness (δ) as caused by,
for instance, a stirred benthic chamber. Positive values
represent solute fluxes toward the bottom waters, while negative values
represent solute fluxes toward the sediments.
RADI was run using the steady-state solutes and solids depth profiles from
the equatorial Pacific station no. W-2 as initial conditions. In the initial
run, the DBL thickness was set to 1 mm. We simulated the response of this
sediment to an instantaneous change in the DBL thickness, with one model run
representing a situation where δ increases from 1 to 5 mm (e.g., a
slow stirring rate) and one model run representing a δ drop from 1 mm to 200 µm (e.g., a fast stirring rate). Following a 5-fold increase
in δ, diffusive fluxes of TAlk, ΣCO2, and O2
initially decrease by a factor ∼5 but then increase back as
the solute concentrations at the interface adapt to the new DBL (Fig. 9).
A total of 2 h after the δ increase, diffusive fluxes converge toward a
new steady state. Following a 5-fold decrease in δ, diffusive fluxes
immediately increase by the same magnitude but go back to close to their
initial value within an hour as the interfacial porewater concentrations
adjust to the new DBL. These results suggest that the incubation periods of
the Berelson et al. (1994) benthic-chamber experiments were long enough to
let porewaters adjust to the changes caused by the paddle stirrers and
confirm that, in sediment rich in organic matter and CaCO3 such as that of
the Equatorial Pacific, the influence of a DBL on diffusive fluxes across
the sediment–water interface should indeed be limited. Additionally, these
results confirm the observation by Berelson et al. (1994) that
changing the stirring rate in a benthic chamber does not alter steady-state
diffusive fluxes by much under the
CaCO3 and POC deposition fluxes encountered in the equatorial Pacific. Part of the reason may be the quick adjustment of
porewater concentrations to the new diffusive boundary layer (see Fig. S3 in the Supplement).
Additional applications
The time-dependent problems presented above focus on relatively short
timescales (from minutes to months). A non-steady-state model such as RADI
can also be used to project the sediment response to perturbations over
longer periods of time. Examples include estimating the effect of negative
emission technologies, such as coastal enhanced weathering with olivine on
early diagenesis (Meysman and Montserrat, 2017; Montserrat et al., 2017),
deep-sea mining (Haffert et al., 2020), or bottom trawling (Trimmer et al.,
2005; van de Velde et al., 2018; De Borger et al., 2021); the impacts of a
decadal bottom-water deoxygenation event such as in the Saint Lawrence
estuary (Jutras et al., 2020); or the present anthropogenic CO2
transient. However, the current version of RADI cannot deal with long-term
major dissolution (erosion) events because the burial velocity calculation
scheme currently implemented does not account for solid mass gain or loss
within the sediment. To study the response of sediments to a global-scale,
long-duration ocean acidification event such as the Paleocene–Eocene
Thermal Maximum (Zachos et al., 2005; Cui et al., 2011), a different burial
velocity calculation scheme would have to be implemented, such as that
adopted by Munhoven (2021).
Future developments
One advantage of RADI is that it is easily tunable by the user: adding new
components is straightforward as long as the chemical reactions are known.
In future releases, we plan to add oxygen, carbon, and calcium isotopes as
individual components in order to predict the diagenetic response of
isotopic signals. Additionally, adsorption and desorption reactions on clay
surfaces could be a critically important advance, especially regarding the
prediction of sedimentary pH profiles (Meysman et al., 2003), as RADI
currently treats clay minerals as non-reactive.
The representation of organic matter in the current version of the model is
oversimplified. All reactive organic matter in RADI is associated with a
Redfield stoichiometry, but marine organic matter can considerably deviate
from this ideal (Martiny et al., 2013; Teng et al., 2014).
Finally, a model is only as good as its assumptions. RADI is targeted to
study deep-sea, carbonate sediments. To be used in coastal environments,
additional biogeochemical reactions would be necessary, particularly those
involving methane and iron sulfide. Close to the shore, sediments become
more permeable, and the assumption of molecular diffusion as the dominant
mode of solute transport in porewaters does not hold. In very shallow
environments that are subject to high wave energy, pressure-induced
advection in the sediment porewaters also needs to be included (Huettel et
al., 2014). Moreover, coastal sediments typically have lower pH than
open-ocean sediments, which may render our assumption of both ΣCO2 and TAlk diffusing with a fixed diffusion coefficient set to that
of the HCO3- ion inaccurate; see Fig. S2. Other chemical
species (e.g., dissolved sulfide, ammonium) that also contribute to the
measured pore water alkalinity may also invalidate this assumption.
Code availability
The current versions of RADI in both Julia and MATLAB/GNU Octave are freely
available from GitHub (https://github.com/RADI-model, last access: March 2022) under the
GNU General Public License v3. The exact version of the model used to
produce the results used in this paper is archived on Zenodo (RADI.jl v0.3;
10.5281/zenodo.5005650 (Humphreys and Sulpis, 2021); v1 will be released
after review), along with input data and scripts to run the model for all
the simulations presented in this paper. RADI users should cite both this
publication and the relevant Zenodo reference (10.5281/zenodo.5005650, Humphreys and Sulpis, 2021; 10.5281/zenodo.4739205,
Sulpis et al., 2021).
Data availability
Sediment and porewater composition, porosity, and solid fluxes data for the
southern Pacific Ocean station described in Sayles et al. (2001) are
available at http://usjgofs.whoi.edu/jg/dir/jgofs/southern/nbp98_2/.
Sediment and porewater composition for the northwestern Atlantic Ocean
station described in Hales et al. (1994) are available at
https://doi.pangaea.de/10.1594/PANGAEA.730420. The GLODAPv2 dataset used
in this study is available at https://www.glodap.info/index.php/mapped-data-product/ (last access: March 2022, https://doi.org/10.5194/essd-8-325-2016, Lauvset el al., 2016).
The supplement related to this article is available online at: https://doi.org/10.5194/gmd-15-2105-2022-supplement.
Author contributions
OS was responsible for conceptualization, methodology, software, validation, formal
analysis, investigation, writing of the original draft, and visualization.
MPH was responsible for conceptualization, methodology, software, formal analysis, investigation, and writing of the original draft.
MMW was responsible for conceptualization, methodology, software, formal analysis, and reviewing and editing the manuscript.
DC was responsible for the methodology, software, formal analysis, and reviewing and editing the manuscript.
WMB was responsible for validation and reviewing and editing the manuscript.
DM was responsible for reviewing and editing the manuscript and supervision of the project.
JJM was responsible for the methodology, validation, reviewing and editing the manuscript, and supervision of the project.
JFA was responsible for conceptualization, methodology, validation, resources, reviewing and editing the manuscript, and supervision of the project.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
Thanks are due to Bernard P. Boudreau, whose CANDI model (Boudreau, 1996b)
was a large source of inspiration during the creation of the present RADI
model, and to Daniel L. Johnson for fruitful discussions. We thank David Burdige and one anonymous reviewer for their constructive feedback. We also
thank Lukas van de Wiel for assistance with the Utrecht Geoscience computer
cluster. Olivier Sulpis also
acknowledges the Department of Earth and Planetary Sciences at McGill University
for financial support during his residency in the graduate program and the
Faculty of Science at McGill University for a graduate mobility award.
Monica M. Wilhelmus, Dustin Carroll, and Dimitris Menemenlis carried out research at the Jet Propulsion
Laboratory, California Institute of Technology, under a contract with NASA,
with support from the Biological Diversity, Carbon Cycle, Physical
Oceanography, and Modeling, Analysis, and Prediction Programs.
Financial support
This research has been supported by the Netherlands Earth System Science Centre (grant no. 024.002.001), the Department of Earth and Planetary Sciences at McGill University, and NASA.
Review statement
This paper was edited by Andrew Yool and reviewed by David Burdige and one anonymous referee.
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