RADI uses the same set of reactive-transport partial differential equations as implemented in CANDI (Boudreau, 1996b), i.e., for each solute component, 1∂ν∂t=1φ∂∂zφd∂ν∂z-φuν+α(vw-v)+∑R, and for each solid component, 2∂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: 3φz=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=33 m-1. The solid volume fraction (φs, dimensionless) is defined as follows: 4φs,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: 5v(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.

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.

The solid burial velocity at the sediment–water interface, w0 in ma-1, is given by 15w0=∑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: 16w∞=u∞=w0φs,0/φs,∞. Thus, the porewater burial velocity, u, at all depths is 17uz=u∞φ∞/φz, and the solid burial velocity, w, is 18wz=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 21σz(v)=1/tanh⁡(Peh,z)-1/Peh,z, where 22Peh,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).

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: 23dz(v)=d∘(v)/(θz2), where so-called tortuosity (θ) is defined as follows (Boudreau, 1996a): 24θz=1-2ln⁡φz.

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: 25b0=(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 26bz=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: 27dz(v)=bz.

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: 28Dz(v)=dz(v)⋅(v(z-dz)-2vz+v(z+dz))/(dz)2, where dz(v) is the relevant effective diffusion coefficient.

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: 29It,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: 31αz=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.

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: 32v(-dz)=vdz+2θz2dzδ(vw-v0), and 33v(-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: 34v(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.

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.

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 (≲10 µmolkg-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.