Following , IMP considers degradation of organic matter and dissolution of CaCO3 as the main biogeochemical reactions occurring in marine sediments. (In this version of IMP, we omit the role and geochemistry of opal and its dissolved pore-water phase, silicic acid; see however, e.g., , for a summary of the sedimentary system of opal).

The reaction term for organic matter is given by 3ROM=(1-ϕ)mOMkOM, where kOM is the first-order degradation rate constant for organic matter (yr-1). To account for anaerobic degradation of organic matter by SO4, IMP simulates an anoxic pathway below the dynamically calculated oxygen penetration depth (zox). Different rate constants for oxic (kox) and anoxic (kanox) degradation can be adopted: 4kOM=kox(z≤zox)kanox(z>zox). Following , both rate constants are considered the same for the initial validation of our model in this study. While clearly an oversimplification, it serves as a first approximation of the importance of OM degradation on calcite dissolution and is also a requirement in order to be able to benchmark IMP to the model of . Although other pathways are used to degrade organic matter in marine sediments, such as nitrate and metal oxides, these have been shown to be quantitatively of less importance on a global scale combined likely <20 %;. It is, however, possible to artificially add DIC and ALK fluxes at a given depth, thereby simulating the production of ALK and DIC from a pathway that is not explicitly simulated (Supplement).

The reaction term for any class ℓ of CaCO3 particles is given by 5Rℓ=(1-ϕ)mℓkcc,ℓ(1-Ωcc)ηccH(1-Ωcc), where kcc,ℓ is the rate constant (yr-1); Ωcc is the saturation degree; ηcc is the reaction order for CaCO3 dissolution; and the Heaviside function H guarantees that net CaCO3 precipitation does not occur . Note that the model allows assignment of different dissolution rate constants to different classes of CaCO3 particles e.g.,. For this study, however, unless otherwise described, we assume a dissolution rate of kcc,ℓ=365.25 yr-1 for all classes, which is a value determined by .

The clay species is assumed to be nonreactive. Hence, 6Rclay=0.

The reaction terms for aqueous species O2, DIC and ALK are correspondingly given by 7RO2=-γO2-OM(1-ϕ)mOMkox,8RDIC=ROM+∑ℓ=1nccRℓ,9RALK=(1-ϕ)mOMkanox+2∑ℓ=1nccRℓ. Here, γO2-OM in Eq. (7) is the mole ratio of oxygen to organic matter consumed upon oxic degradation of organic matter. We assume that the aqueous carbonate system is always at equilibrium, and we calculate the partitioning of the aqueous carbonate species (H2CO3, HCO3- and CO32-) based on alkalinity and DIC concentrations in conjunction with the apparent equilibrium dissociation constants adjusted for pressure, salinity and temperature (Tables , ). Other options to utilize published routines for the calculation of the aqueous carbonate system, mocsy 2.0 and CO2SYS , are presented in the Supplement.

Bio-mixing of solid-phase species in the model is simulated by means of a transition matrix. A wide range of bio-mixing styles can be captured by the transition matrix because a transport probability of solid particles from one sediment layer to another can be specified with the value of a cell whose row and column numbers correspond to the two layers between which particles are transported. Thus, the use of the transition matrix facilitates the implementation of user-defined/biology-based particle mixing, whether local or nonlocal e.g.,. In this section, we elaborate upon how the bioturbation term in Eq. (1) can be derived from the transition matrix.

The rate at which particles of solid species θ are transported from layer i to layer j, Pθ,ij (yr-1), is given by 10Pθ,ij=Nθ,ij∑j=1nmlNθ,ij1τ, where Nθ,ij is the number of particles of species θ moved from layer i to layer j, nml is the total number of layers within the bioturbated zone and τ is the time (years) required for the displacements. Note that Pθ,ij×τ represents the particle transport probability and corresponds to components at (i,j) of the transition matrix . When bioturbation causes mixing of sediment particles based on the above transport rate, the number of particles of species θ in layer i changes with time according to the following equation: 11dNθ,idt=-Nθ,i∑j=1nmlPθ,ij+∑j=1nmlNθ,jPθ,ji, where Nθ,i is the total number of particles of species θ in layer i compare Eq. 11 with Eq. 3.117 of.

The concentration of species θ in layer i, mθ,i (molcm-3), can be given by cf., 12(1-ϕi)mθ,i≡αθNθ,iAδzi, where ϕi and δzi are the porosity and the thickness (cm) of layer i, respectively; αθ represents the moles of species θ (mol) included in one particle; and A is the cross-sectional area in the model (cm2). One can then deduce the following from Eqs. (11) and (12): d(1-ϕi)mθ,idt=-(1-ϕi)mθ,i∑j=1nmlPθ,ij13+∑j=1nml(1-ϕj)δzjδzimθ,jPθ,ji compare Eq. 13 with Eq. 3.118 of. Equation (13) can be simplified with a modified transition matrix for species θ, with components at (i,j) denoted as Kθ,ij and calculated based on the particle transport rate Pθ,ij: 14Kθ,ij=δziPθ,ij/δzj(i≠j)-∑j≠inmlPθ,ij(i=j). Using Eq. (14), we can rewrite Eq. (13) as a function of Kθ,ij: 15d(1-ϕi)mθ,idt=∑jnml(1-ϕj)mθ,jKθ,ji.

Formulation of bioturbation in a continuum system needs a corresponding continuous function. We define a continuous exchange function Eθ (cm-1yr-1) as follows cf.,: 16Eθ(zi,zj)≡lim⁡δzj→0(Pθ,ij/δzj), where zi and zj denote the depths of sediment layer i and j, respectively. With Eq. (16), we can write a continuous form of Eq. (13) in the limits of zero thicknesses for discretized sediment layers: ∂(1-ϕ)mθ∂t=-(1-ϕ)mθ∫0zmlEθ(z,z′)dz′17+∫0zml{1-ϕ(z′)}mθ(z′)Eθ(z′,z)dz′. Here, z′ denotes any depth except at z, and zml is the thickness of the mixed layer. Eq. (17) is the same as Eq. (3.121) of and the two bioturbation terms in Eq. (1). Note that Eq. (15) is a finite difference version of Eq. (17), and the transition matrix corrected for porosity therein (i.e., (1-ϕi)Kθ,ij representing components at (i,j)) corresponds to the bioturbational transport part of the Jacobian matrix for species θ, which is used for solving the governing equations (Sect. 2.3).

Three different transition matrices were created for the present study to illustrate different styles of bio-mixing (Fig. 1): Fickian mixing, homogeneous mixing and the more mechanistic automaton-based mixing simulated by the particle-tracking bioturbation simulator LABS e.g.,.

The transition matrix that assumes Fickian diffusion for bioturbation parameterized with Db,θ,, can be expressed by 18Kθ,ij=-Kθ,ij(j=i+1)(i=j=1)-Kθ,ij(j=i+1)-Kθ,ij(j=i-1)(1<i=j<nml)-Kθ,ij(j=i-1)(i=j=nml)(1-ϕi)Db,θ,i+(1-ϕj)Db,θ,j/{δzi(1-ϕi)⋅(δzi+δzj)(2≤j=i+1≤nmlor1≤j=i-1≤nml-1)0(else), where Db,θ,i represents the biodiffusion coefficient for solid species θ at sediment layer i. As a default biodiffusion parameterization, a depth-independent value of 0.15 cm2yr-1 is assumed Table 1. Note that the biodiffusion considered in this study is only intraphase biodiffusion and does not include interphase biodiffusion e.g.,. The implementation of interphase biodiffusion requires a different transition matrix.

The transition matrix for homogeneous mixing can be given by 19Kθ,ij=δziPh,θ/δzj(i≠jand1≤i,j≤nml)-(nml-1)Ph,θ(1≤i=j≤nml)0(else), where Ph,θ (yr-1) is the homogeneous transport rate for solid species θ. A value of 10-3 yr-1 is assumed for the default homogeneous mixing (Table 1).

To obtain the mechanistic automaton-based transition matrix, we utilized the eLABS v.0.2 code, the latest release of lattice-automaton bioturbation simulator (LABS) by , with which a transition matrix can be extracted based on Eqs. (10) and (14). The new features of LABS added by , i.e., 2-D porewater flow and diagenesis, were disabled to extract mixing controlled dominantly by benthos biology as in . A 200-year LABS simulation was run with a deposit feeder with a body size of 0.25×0.25×1.65 cm3, a locomotion speed of 10 cmd-1 and a maximum ingestion rate of 1 g of sediment per gram of organism per day in a 0.25×12×15 cm3 3-D sediment system. Transition matrices were created every 10 model days cf., and the averaged transition matrix over 200 model years multiplied by a factor of 1/10 was adopted to represent the transition matrix derived from the above LABS simulation. The factor of 1/10 was introduced above because the LABS mixing would otherwise have a relatively high mixing intensity equivalent to a biodiffusion coefficient of 0.1–10 cm2yr-1; cf. and also to facilitate the numerical solution of the model with the LABS mixing (see below).

The default transition matrices, corrected for porosity, are shown in Fig. . Fickian mixing is a local mixing, allowing translocation of particles only between adjacent sediment layers, resulting in a tridiagonal matrix (Fig. 1a). On the other hand, nonlocal mixing (homogeneous and LABS mixing) allows the transportation of particles between remote layers and, thus, is characterized with the spread of nonzero components away from the main diagonal in the transition matrix (Fig. b, c). As defined in Eq. (19), the transition matrix for homogeneous mixing has components that systematically change with rows and columns (Fig. b) compared with the transition matrix for LABS mixing that has more randomly spread noncontinuous values (Fig. c). The porosity-corrected transition matrix corresponds to the bioturbational transport part of the Jacobian matrix used for solving the governing equations (Sect. 2.3). Therefore, the difficulty to achieve a numerical solution of the model differs between chosen mixing styles reflecting corresponding transition matrices: in general, this is the least difficult with Fickian mixing and the most difficult with LABS mixing (Fig. ). Note that the transition matrices for Fickian and homogeneous mixing change with assumed mixed-layer depth (related to nml) and/or parameters that define mixing intensity (Db,θ and Ph,θ) as in Eqs. (18) and (19) (cf. Table ). Additional LABS simulations, with variations related to deposit-feeder behavior and/or modified sediment grid dimensions, are necessary to generate a new LABS-based transition matrix.

The burial velocity in IMP changes according to the volume change of solid material caused by biogeochemical reactions and bio-mixing because a constant, time-independent porosity profile is assumed (Eq. 23). This section describes how the change in burial rate is calculated in the model.

Multiplying the governing equation (Eq. 1) by the molar volume Vθ (cm3mol-1) for solid species θ leads to 20∂(1-ϕ)Vθmθ∂t=-∂(1-ϕ)wVθmθ∂z-VθRθ+Vθ-(1-ϕ)mθ∫0zmlEθ(z,z′)dz′+∫0zml{1-ϕ(z′)}mθ(z′)Eθ(z′,z)dz′. Note that the molar volume Vθ can be obtained from the density, ρθ (gcm-3), and the molar mass, Mθ (gmol-1), of species θ as Vθ=Mθ/ρθ. Summing Eq. (20) for all solid-phase species gives ∂(1-ϕ)w∂z=-∑θVθRθ+∑θVθ-(1-ϕ)mθ∫0zmlEθ(z,z′)dz′21+∫0zml{1-ϕ(z′)}mθ(z′)Eθ(z′,z)dz′. For the derivation of Eq. (21), the following relations are enforced: 22∑θVθmθ=1,23∂ϕ∂t=0. Equations (22) and (23) express the constraint that the volume fractions of all solid species sum to 1  cm3cm-3 and the assumption of time independency of porosity, respectively. Unless bio-mixing is Fickian (intraphase) biodiffusion with the same intensity and the same mixed-layer depth for all solid species (see below), the burial velocity is calculated based on Eq. (21).

If bio-mixing of solid species θ is Fickian biodiffusion with a coefficient Db,θ (cm2yr-1), Eq. (20) can be expressed as ∂(1-ϕ)Vθmθ∂t=-∂(1-ϕ)wVθmθ∂z-VθRθ24+∂∂z(1-ϕ)Db,θ∂Vθmθ∂z. Further, if bio-mixing of all solid species occurs as Fickian biodiffusion with the same mixing intensity (Db) and depth (zml), Eqs. (22) and (23) lead to a simpler burial velocity equation: 25∂(1-ϕ)w∂z=-∑θVθRθ. Therefore, when the transition matrix is specified to represent intraphase biodiffusion (e.g., Fig. a) and the same matrix is applied to all solid species, Eq. (25) is used to calculate burial velocity, otherwise Eq. (21) is used. In either case, the model generally satisfies Eq. (22).

At the beginning of the calculation, we must define both initial (e.g., solid and pore-water composition) and boundary conditions as well as the structure of the grid.

In the default setting of IMP, the calculation domain represents a ztot=500 cm sediment column and is discretized into N=100 layers whose thickness increases with depth from less than 10-2 to more than 102 cm following a logarithmic function (Table 2). Furthermore, a time-independent exponential porosity profile is imposed (Table 2). One may modify the grid structure and porosity profile by changing the associated parameter values (Table 2) defined in the code (Supplement).

As initial conditions for the sediment grid, the model assumes almost nonexistent concentrations of 10-8 molcm-3 for all solid species (carbonate, organic matter and clay), and the volume deficiency relative to the solid space prescribed by the assumed porosity is filled by the later time-integration (see below). Ambient ocean concentrations at the seawater–sediment interface are adopted as the initial concentrations for all aqueous species at all depths. These initial values, however, do not have an impact on our results, as the model is run to steady state before an experiment is started (e.g., a proxy signal change event is simulated).

The upper boundary conditions at the seawater–sediment interface are given by mass fluxes of simulated solid species and concentrations for simulated aqueous species (Tables , ). The lower boundary conditions at ztot for all aqueous species are given by zero concentration-gradients. If oxygen is consumed within the simulated sediment column (i.e., zox<ztot), the dynamically calculated oxygen penetration depth marks a lower boundary for oxygen (i.e., cO2=0 at z=zox). As boundary conditions can change with model time (e.g., in the proxy signal change experiments), they are specified before each time integration.

Solutions for the temporal and spatial evolution of individual solid and aqueous species are obtained by solving the governing equations with the finite difference method e.g.,. Figure  summarizes the structure of the code to solve the governing equations, and the calculation at a given time is conducted by the model in the following four main steps.

First, organic matter and oxygen concentration profiles are calculated using Eqs. (1) and (2) (for θ=“OM” and σ=“O2”). As both calculations depend on the oxygen penetration depth zox, they are conducted iteratively by the following steps cf.: a.

zox is calculated based on the O2 profile from the previous iteration or time instance;

The concentration profiles of individual species are solved based on the difference equations of Eqs. (1) and (2), which are obtained by the finite difference method. The second-order and first-order spatial differential terms are discretized by the second-order central and the first-order upwind differencing schemes, respectively e.g.,. The finite difference form of the bioturbation term in Eq. (1) is formulated with a transition matrix (Eq. 15). The difference equations for all the solid and aqueous species are solved time-implicitly e.g.,. For the solution of the difference equations that are nonlinear, as is the case for the carbonate system (multiple CaCO3 classes, DIC and ALK), the Newton–Raphson method is utilized (Fig. ) where the solution is iteratively updated along with the Jacobian matrix until its relative difference from the previous iteration becomes ≤10-6 e.g.,. Note that the porosity-corrected transition matrix corresponds to the bioturbational transport part of the Jacobian matrix (Fig. ).

The time step taken for the time integration of the governing equations can vary between and within simulations, and can be specified by the user (cf. Sect. 3.1). In the default setting, the time step increases with model time from 100 to 105 years to reach steady state (e.g., a spin-up phase of simulation prior to imposing a signal change event; Sects. 2.4, 3), and a smaller and fixed time step is taken when simulating a signal change event (5 or 10 years for a 10 or 50 kyr signal change event, respectively) as well as its aftermath (Sects. 2.4, 3).

By default, the model monitors and records depth-integrated fluxes of individual rate terms in Eq. (1) or (2) (fluxes caused by amount change in sediment, sediment rain, biogeochemical reactions, advection, bio-mixing and so on) for each solid/aqueous species, as well as the residual flux as a sum of all the fluxes, which is ideally zero, to confirm the mass balance of the species. The residual fluxes for all the solid and aqueous species are negligible (e.g., ≤10-6 times the rain fluxes) for all the simulations presented in this paper (Sect. 3).

Tracking of proxy signals in carbonates is conducted by assigning different numerical values to the simulated CaCO3 classes and by scaling their input fluxes to reflect the overall change in proxy signal with time. Thus, proxy signal changes are reflected as changes in the boundary conditions (i.e., rain fluxes of different CaCO3 classes) in the model (see Sect. 2.3). Assignment of proxy signals and fluxes to CaCO3 classes can be realized by three methods (Fig. ).

In the first method (a “time-stepping” method), any change in proxy signal is approximated by a step function, i.e., a continuously varying analogue signal is (digitally) discretized (see, e.g., dotted curve in the top panel of Fig. ). Each step is represented by a separate and unique CaCO3 class, characterized by the approximate proxy value (Fig. a). For example, if a signal change event is discretized into 10 steps, 10 different CaCO3 classes with unique proxy values are simulated. Any change in a proxy signal during each discretized time interval is thus muted. Accordingly, the accuracy of the proxy signal approximation is increased by increasing the number of steps and thus the number of simulated CaCO3 classes, which, however, results in an increased computation cost (Supplement). As an advantage, one can track any number of proxies as long as the signal changes of all tracked proxies occur within a simulated event (Supplement).

The second method to assign proxy signals (an interpolating method) simulates only the end-member CaCO3 classes, each of which possesses a unique combination of the maximum and/or minimum input-signal values. As an example, one proxy can be tracked with two CaCO3 classes, with the first possessing the maximum and the second the minimum proxy value. Intermediate values of an input proxy are realized by assigning varying fluxes to the two end-member classes such that the sum of their flux-weighted values results in the input-signal value at each time step (Fig. b). Accordingly, the input proxy signal is always accurately represented regardless of the resolution of the time discretization. As a disadvantage of method 2, the number of simulated CaCO3 classes increases with the number of proxies to be tracked. In general, 2np classes of CaCO3 particles are necessary when tracking np proxies because the number of unique combinations of the maximum and/or minimum signal values is increased by a factor of 2 for every additional proxy to be tracked: 262proxy 1×2proxy 2×⋯×2proxynp︸np=2np. Nonetheless, the computational demand is lower compared with method 1 in most cases because we are interested in a limited number of proxies and, thus, fewer CaCO3 classes are simulated (2np in method 2<time steps in method 1).

The third method (a direct tracking method) separates bulk CaCO3 into multiple classes based on how the simulated proxies are determined. For example, when the tracked proxy is δ13C, which is determined by the 13C/12C ratio (X in Fig. ), method 3 simulates classes of Ca13CO3 and Ca12CO3 (Y and G, respectively; Fig. c). The rain fluxes of individual classes at a given time step are directly calculated based on the definition of the proxy and the contemporaneous proxy value (see boxes in Fig. c). Thus, one can regard method 3 as a derivative of method 2 that defines the end-member CaCO3 classes based on the definition of the tracked proxy. Because the flux calculation must change with the simulated proxy signal, method 3 is not as flexible as methods 1 and 2, but the computational effort can be further reduced because a certain class can be used to define multiple proxies (e.g., Ca12CO3 is related to the definition of both 14C age and δ13C). Method 3 has the unique advantage of enabling additional biogeochemical reaction terms for any specific CaCO3 class if necessary. For instance, when tracking 14C age, one needs to account for the radioactive decay of Ca14CO3 and the accompanied generation of alkalinity, which can be implemented with method 3. Currently method 3 tracks four proxies including 14C age with five CaCO3 classes (Supplement).

After the signal and flux assignment by any of the three methods, the model is spun up to steady state with only the CaCO3 class(es) with pre-event proxy values being deposited to sediment (Fig. ). After the spin-up, a proxy signal change event is simulated by changing the rain fluxes of different CaCO3 classes with different proxy values (i.e., the boundary conditions) with model time (Fig. ). After the signal change event, the model is run until a new steady state is reached.

Note that the methods and procedures described above can be applied not only to track proxy signals but also any other property of CaCO3 particles such as particle size and deposition time (cf. Sect. 2.4.2). In this case, methods 2 and 3 track the property in the same way (Sect. 2.4.2; Supplement).

After input signals are reflected in rain fluxes by any of the three methods in Sect. 2.4.1, they are modified within the sediment by bioturbation and chemical erosion. Caution needs to be taken with respect to numerical diffusion, which is inevitably introduced to the difference form of the advection term (first term on the right-hand side of Eq. 1) in a finite difference approach e.g.,. For an accumulating column of sediment in a fixed grid, numerical diffusion artificially mixes the deposited and buried sediment particles along with their proxy signals, especially at depths where grid cells are relatively coarse (Fig. ). An alternative is to allow for a partial surface layer and to accrete or remove complete layers depending on the growth or erosion at the surface, such as in . However, such an approach is impractical if the depth-dependent diagenetic reactions are to be solved rather than just recording historical accumulation (or erosion).

Here, to minimize the effect of numerical diffusion, we read out the proxy signal as a function of time, from just below the mixed layer and before the start of the “historical” layer (zml, see arrow in Figs.  and ). Accordingly, signal values are not plotted against the depth of the sediment domain but against a sediment stack composed of the sediment layers that were used to record the proxy signal (i.e., at depth zml) during the course of the simulation. The depth of this sediment stack is called diagnosed depth (zdiag, Fig. ) and can be calculated as follows: 27zdiag=zml+∫tttot(1-ϕml)wmldt, where ϕml and wml (cmyr-1) denote the porosity and burial velocity at the mixed-layer depth (z=zml), respectively, and ttot is the total duration of a simulation (years). While reading proxy signals at the bottom of the mixed layer is likely effective in most cases (cf. Supplement), it is also possible to specify a different depth point to read proxy signals. In such a case, the definition of diagnosed depth needs to be modified by replacing zml, ϕml and wml in Eq. (27) with the corresponding parameter values at the specified depth.

To convert the signal profiles plotted against diagnosed depth to profiles plotted against model time, an age model is required, which can be obtained by tracking model time as a proxy. The application of the three methods explained in Sect. 2.4.1 (i.e., to assign numerical values to multiple classes of CaCO3 particles and calculate their rain fluxes from the input values) is not limited to tracking proxy signals but can also be applied to any other characteristic including the model time at which particles are deposited. In method 1, individual classes of CaCO3 particles are defined based on the time steps discretized from a signal change event (Fig. a) and, thus, already have their own model time to be assigned with. Note, however, that tracking model time with method 1 is computationally more expensive because a larger number of explicit CaCO3 classes is needed to represent the continuously changing model time. When using method 2 or 3 to track model time in addition to paleoceanographic proxies, the number of CaCO3 classes must be doubled (cf. Eq. 26). For example, when using method 2, one proxy signal can be simulated with two (or a pair of) CaCO3 classes representing the maximum and minimum proxy value. Additionally tracking model time requires an extra pair of CaCO3 classes, whereas the start and end of model time is assigned to the two pairs, respectively (cf. Eq. 26). In either method, model time tracked in bulk CaCO3 can be plotted against diagnosed depth, which is the age model of IMP, and can be used to plot the other tracked proxy signals against model time. Examples of obtaining and using IMP's age model are provided in the Supplement.

To illustrate the initial evolution of the model, a spin-up experiment was run until a steady-state sediment composition was achieved. For this, we assumed Fickian mixing using the default conditions given in Table  (Fig. a). Model output includes depth profiles of density and volume fraction of solid sediment (Fig. a, c), burial velocity (Fig. b), concentrations of solid and aqueous species (Fig. d–k), and rates of biogeochemical reactions (Fig. l–n) for five time instances during the spin-up experiment (1, 10 and 100 kyr, and 1 and 3 Myr).

A second experiment illustrates how a change in the boundary conditions affects the temporal evolution of the depth profiles in IMP. This experiment starts from the end of the first spin-up experiment and artificially imposes significant carbonate dissolution by changing the water depth from 3.5 to 5.0 km between 5 and 45 kyr (Fig. ). Because of the longer timescale to achieve steady state (see the first experiment), the second experiment run for 50 kyr is in transient states except for the initial steady state at 0 kyr (Fig. ).

Finally, IMP was run to steady state assuming various carbonate rain fluxes (ranging from 6 to 60 µmolcm-2yr-1, in increments of 6 µmolcm-2yr-1), ratios of organic matter to carbonate (0, 0.5, 0.67, 1 and 1.5) and water depths (ranging from 0.24 to 6.00 km, in increments of 0.24 km) cf.. These lysocline experiments were performed for both the oxic-only OM degradation model and the oxic–anoxic model (Figs. , ). To facilitate comparison of our results with , IMP assumes a single class of CaCO3 particles, Fickian mixing for bioturbation and a sediment column depth of 50 cm. All other boundary conditions are as described in Table .

One can use the IMP code of any of the three programming languages (i.e., Fortran90, MATLAB or Python) to conduct the simulations presented in this paper. The model code for each language is stored in the respective directory (i.e., “Fortran”, “MATLAB” and “Python”), and a language-specific readme file provides instructions on how to run the simulations (e.g., \iMP\Fortran\readme_Fortran.txt for the Fortran version). The boundary conditions can be specified with time-invariant values at run time (e.g., the third experiment above; see the readme file for the chosen version of the code) but can also be changed as a function of time (as in the second experiment above). The temporal changes in the boundary conditions must be prescribed in the input files that are stored in a directory “input” and can be modified by the user (see the readme file therein, \iMP\input\readme_input.txt, for the details). We also provide Python scripts to plot concentrations of solid and aqueous species (e.g., Figs. –) as well as tracked proxy signals (Sect. 3.2), stored in a directory “plot” (see a readme file therein, \iMP\plot\readme_plot.txt, for more details).

In the spin-up to steady state, spaces for solid sediment defined by assumed porosity (1-ϕ) are initially empty (not filled) because of the low initial concentrations of solid species (∑θVθmθ≅0; Sect. 2.3) but soon become filled with clay (as a “dilatant”) and OM, and later with CaCO3 as Eq. (22) is enforced and steady state is approached (∑θVθmθ=1; Fig. a, c). In contrast, pore spaces are assumed to be always filled with pore water and pore-water chemistry achieves steady state much faster (Fig. g–k) e.g.,. The steady-state results for bulk phases (Fig. ) are not affected by changing the number of CaCO3 classes or the time step of each time integration (cf. Sect. 2.3.2).

The second experiment demonstrates that once steady state is achieved, a change in boundary conditions does not generate significant void spaces (∑θVθmθ≪1) and/or expansions (∑θVθmθ≫1) in solid sediment (Fig. c), thus generally satisfying Eq. (22). In other words, prescribed spaces for solid sediment by assumed porosity are almost perfectly matched with the sums of volumes of all solid-phase species (∑θVθmθ=1; Fig. c) even when the concentrations of solid species dynamically change with time, leaving steady state (e.g., Fig. d). Absence of significant void spaces or expansions in solid sediment provides a convergence diagnostic (adapted from one of the convergence diagnostics in the steady-state diagenesis model of Archer et al., 2002).

Finally, we compare steady-state lysoclines simulated with IMP to results from the CaCO3 diagenesis model of , who showed that the lysocline is sensitive to rain rates of carbonate and organic matter to the seafloor and, in particular, to the ratio of these fluxes. The simulated lysocline and carbonate burial rates for the oxic-only OM degradation model are presented in Fig. a and b. The results for the oxic–anoxic model are shown in Fig. a and b.

In general, our predicted mixed-layer CaCO3 wt % and the CaCO3 burial fluxes match the steady-state estimates by (compare with Figs. 5 and 6 from ). For instance, as in , increasing the carbon rain to the sediments for lower OM/CaCO3 rain ratios (i.e., ≤0.67) enhances carbonate preservation and causes the lysocline to deepen for both the oxic-only and the oxic–anoxic OM degradation model (Figs. , ). The only notable difference occurs for the oxic-only OM degradation model under the most extreme carbon rain fluxes (i.e., rain ratio=1.5; CaCO3 rain>40 µmolcm-2yr-1). Here, IMP simulates higher CaCO3 preservation than the model of (Fig. , right panels). This difference can be explained by a burial velocity enhancement caused by high organic matter preservation in the oxic-only model, which is not considered by (see the lysocline experiment with VOM=0 in the Supplement). For the same high OM/CaCO3 rain ratio (1.5), the oxic–anoxic OM degradation model simulates an enhancement in the carbonate accumulation rate and a deepening of the lysocline for an increase in the CaCO3 rain, which is in line with the results of .

The effects of three different styles of bioturbation on the recorded proxy signals are considered: (i) Fickian local mixing with a biodiffusion coefficient of Db,θ=0.15 cm2yr-1, (ii) homogeneous nonlocal mixing to represent random mixing as simulated by, e.g., TURBO2 and (iii) process-based nonlocal mixing simulated by deposit-feeder automata from the LABS model e.g.,. Because the LABS-derived transition matrix contains less continuous and more irregular transport probability than the other two styles of bio-mixing (Fig. ), it is susceptible to convergence problems cf.,Sect. 2.2.2. When convergence was not achieved, model results with bio-mixing from LABS are not shown in the following subsections (Sects. 3.2.1–3.2.3).

The input proxy values of δ13C and δ18O in CaCO3 either experience a step change over 5 kyr or a 5 kyr duration impulse event, respectively (Fig. a). Four end-member classes of CaCO3 particles are used for signal tracking (Fig. c), and simulated proxy signals are recorded just below the sediment mixed layer and plotted against diagnosed depth to minimize the effect of numerical diffusion (Sect. 2.4.2). A first set of experiments is conducted with dissolution disabled for all CaCO3 classes (kcc,ℓ=0) in order to solely consider the effect of different styles of bioturbation. In a second set of experiments, the default CaCO3 dissolution rate constant is used for all classes.

To visualize signal distortions by comparison, the input signals as a function of time (Fig. a) are plotted against diagnosed depth in Fig. , using the age model for the no bioturbation case (Supplement). Slight deviations of the recorded signals (pink curves in Fig. a and b) from the input signals (dotted black lines) in the “no bioturbation” case can be attributed to numerical diffusion but are minor compared with signal distortions exhibited by bioturbated sediments (blue, yellow and green curves). More specifically, dispersion of the recorded signals occurs over a larger depth interval and, for the impulse event in δ18O, the signal magnitude is significantly reduced with bioturbation (Fig. a, b). Fickian and homogeneous mixing distorts the input signals similarly (blue and yellow curves, respectively, which are almost completely superimposed in Fig. a and b), but LABS mixing results in slightly different signal shifts that extend to deeper depths (green curves). This difference may be explained by defecating/pushing of particles by deposit-feeder automata resulting in rare occasions where particle displacements propagate to depths even below the mixed layer Fig. c; e.g.,. Note that bio-mixing in LABS can vary with assumed physicochemical and ecological conditions and animal types e.g.,; thus, our results should not be regarded as the exclusive results with a LABS transition matrix (cf. Sect. 2.2.2).

Results for the second set of experiments with CaCO3 dissolution enabled are presented in Fig. d–f. Different modes of bioturbation result in variations in the extent of CaCO3 dissolution (Fig. f): no bioturbation leads to the lowest degree of dissolution, and efficient homogeneous mixing causes the highest degree of dissolution (Fig. f). Correspondingly, sediment accumulation rates and, thus, age models differ between different styles of bioturbation (Supplement), and one observes signal change events at shallower depths with a more enhanced dissolution (Fig. d, e). By enabling dissolution, proxy signals are slightly lost along with CaCO3 particles, especially when bio-mixing is not efficient. This can be recognized by a reduction in the magnitude of the δ18O impulse for the no bioturbation case by enabling dissolution (slightly smaller peak of the pink curve in Fig. e than in Fig. b). We examine the dissolution effect in more detail in the next subsection.

While evidence for significant dissolution of sedimentary carbonates provides information about ocean chemistry e.g.,, it also distorts proxy signals recorded in these carbonates. In this subsection, we examine how and to what extent dissolution distorts proxy signals.

We consider a negative δ13C excursion over 40 kyr with a relatively rapid onset and recovery of the isotope signal (over 5 kyr). At the same time, a more gradual ramp down and up change of the δ18O signal over 50 kyr is simulated (Fig. a). The signal shifts for the two proxies are intentionally made decoupled in time and should not be associated with any “real” geological event. These signal changes are accompanied by water depth changes from the background depth of 3.5 to 4.5 and 5.0 km over 5 kyr in order to cause different extents of dissolution (Fig. c) through destabilizing CaCO3 by increasing pressure (Millero, 1995). These imposed changes in water depths are not intended to be “realistic”; rather, they drive conditions of enhanced CaCO3 dissolution as might have been caused by environmental changes such as ocean acidification e.g., see: , but without the additional interpretative complications of actually changing the ocean chemistry at the sediment surface in the model. (Note that it is also possible to drive IMP with changing upper geochemical boundary conditions to explicitly simulate, e.g., ocean acidification.) The water depth and related dissolution changes are assumed to be synchronous with the proxy signal changes (Fig. a, c).

Signal tracking is conducted by simulating the same four classes of CaCO3 as in the previous subsection (Fig. d; cf. Fig. c), with enabling Fickian or homogeneous bio-mixing (Fig. a, b) or without bioturbation. An additional set of experiments was run without changing the water depth as a “no dissolution” control (dotted line in Fig. c). Simulated signals against sediment depth (Fig. ) are compared with input signals (dotted black curves in Fig. ) which are obtained from their temporal changes (Fig. a) and the age model for the no bioturbation case (cf. Supplement) as in the previous subsection.

When dissolution is intensified by changing the water depth from 3.5 to 4.5 km (experiment 1; solid line in Fig. c), the total amount of CaCO3 is reduced from ∼90 wt % to ∼50 wt % for all cases with and without bioturbation (Fig. f). As described in Sect. 3.2.1, dissolution is enhanced by bio-mixing, and signal change events are correspondingly observed at different depths between different modes of bioturbation (Fig. d–f; cf. Supplement). Apparent durations of the signal change events become shorter compared with the control experiment (Fig. a–c) because less sediment accumulates during the events with more enhanced dissolution (Fig. c, f). However, because imposed dissolution is still moderate (Fig. f) and relatively long-term signal change events are considered (e.g., compare Fig. a with Fig. a), no significant reduction in the magnitude of signal peaks is observed in experiment 1.

Further increasing the dissolution rate by changing the water depth to 5.0 km during the isotope excursion (experiment 2; dashed line in Fig. c) causes CaCO3 to completely disappear for all cases with and without bioturbation (Fig. i). Note that a concentration of absolute zero is not allowed for solid species in the model. Simulated concentrations are truncated at a threshold of 10-300 molcm-3. As for dissolution experiment 1 (Fig. f), different styles of bioturbation cause different CaCO3 dissolution rates (Fig. i). Under this more intense dissolution scenario, simulated proxy signals are considerably distorted and reduced for all styles of bioturbation (Fig. g, h). Simulated excursions of proxy signals are observed for considerably shorter apparent duration or sediment depth interval, as described in the paragraph above.

It is noted that the carbon and alkalinity fluxes from dissolved CaCO3 in sediments under any destabilization can vary with the mode of bioturbation (Figs. , ). This indicates the potential role of benthic ecosystems to determine the feedback of sedimentary CaCO3 to a climate perturbation e.g.,.

It has been suggested that carbonates of different sizes can be differently bioturbated and dissolved in marine sediments e.g.,. IMP is well suited for examining the effect of differential mixing and/or dissolution rate among CaCO3 size classes on the signal distortion.

Here, we consider eight CaCO3 classes, consisting of two sets of the same four CaCO3 classes as in the previous subsections (Table ). We assign two distinctive sizes to these two sets (Fig. c, d). CaCO3 particles in the first set are assumed to be of “fine” grain size and are consequently bioturbated (by Fickian and in a second experiment by homogeneous mixing) to deeper depths 20 cm; cf., with the correspondingly modified transition matrices (Eqs. 18 and 19; Sect. 2.2.2). They are also dissolved at a faster rate by adopting a dissolution rate constant increased by a factor of 10 cf., (classes 1–4 in Fig.  and Table ). CaCO3 particles in the second set are of “coarse” grain size and adopt the default particle characteristics (Table , classes 5–8 in Fig.  and Table ) and transition matrices (Fig. a, b). The total mass flux and isotope signal input are the same as in Sect. 3.2.2, and the water depth remains unaltered at 3.5 km. In concert with the δ18O decrease, the coarse species becomes more dominant over the fine species the rain fraction of the coarse species increases from 50 % to 90 %; Fig. c; cf.,.

The differences in dissolution and mixing properties of fine and coarse CaCO3 species have a prominent effect on their relative preservation (Fig. c). In general, the coarse species shows higher preservation due to its lower dissolution rate. The more efficient the adopted mixing mode (e.g., homogeneous mixing), the better the preservation of the coarse species and the more obscured the preservation of the imposed CaCO3 input flux changes. Correspondingly accumulation rates are different for fine and coarse CaCO3 species; thus, excursions of proxy signals as well as peaks in coarse vs. fine species abundance are offset by ∼10 cm between the two species (compare solid and dotted curves in Fig. ). Observed apparent offsets of peaks in proxy signals and species abundance can be mostly removed by applying individual age models to the two species, although the reduction in the magnitude of abundance shifts cannot be recovered (Supplement).

Although the above experiment is not designed to simulate any specific surface-environment change event in the past, signal offsets among CaCO3 species have been observed in, e.g., hyperthermal events e.g.,. The application of IMP to such events can be useful, as it might lead to an insight into population shifts among calcifiers associated with environmental changes in the past (cf. Figs.  and c).

To track radiocarbon age, the direct tracking method (method 3 in Sect. 2.4.1) is utilized. The method simulates five CaCO3 classes corresponding to five isotopologues (Ca12C16O3, Ca12C18O16O2, Ca13C16O3, Ca13C18O16O2 and Ca14CO3) to track four associated isotopic signals (δ13C, δ18O, Δ47 and 14C age) recorded in CaCO3 particles that are of the same size (see the Supplement for the details). Because we further track the “size” of CaCO3 particles by simulating two distinct CaCO3 species of “fine” and “coarse” sizes, two sets of the above five classes (i.e., 10 classes in total) are necessary (cf. Eq. 26; Table ). The first set of five classes (classes 1–5) possesses the dissolution and bio-mixing properties for the fine species defined in Sect. 3.2.3, whereas the second set (classes 6–10) represents the coarse species (Table ).

Steady-state simulations were run with the above 10 CaCO3 classes adopting Fickian mixing, for three water depths (3.7, 3.9 and 4.1 km), three total sediment fluxes (12 (default), 6 and 3 µmol total CaCO3  cm-2yr-1 with the fixed default OM/CaCO3 and clay/CaCO3 rain ratios; Tables , ) and for different rain fractions of the fine CaCO3 species (10 % , 50 % , 90 % and 99 %). The rain fluxes of individual CaCO3 classes are calculated from the total CaCO3 rain, the rain fraction for fine species, and assuming that δ13C=δ18O=Δ47=0‰ and the 14C/12C ratio is 1.2×10-12 (cf. Supplement). The mixed-layer depth (and, thus, the transition matrix) and the dissolution rate constant are defined differently between the fine (classes 1–5) and coarse (classes 6–10) species (cf. Sect. 3.2.3). All the other parameters were set at the default values (Table ).

Because dissolution and transport is fully coupled in IMP (i.e., dissolution is “homogeneous”), a decrease in CaCO3 wt % caused by increasing water depth generally leads to a younger radiocarbon age (e.g., see Fig. c where 14C ages are highest for blue curves and lowest for green curves). However, when the decrease in the CaCO3 concentration is caused not by increasing the water depth but by increasing the rain fraction of the fine species that dissolves faster (trajectories depicted with curves in Fig. ), the trend of 14C age vs. CaCO3 wt % differs. The trend for the coarse species is especially counterintuitive, where an older 14C age is observed for lower CaCO3 wt % (Fig. a). The opposite trend is recognized for the fine species (Fig. b). Bulk CaCO3 shows a combination of the above two contrasting aging trends, and whether bulk 14C age increases or decreases with bulk CaCO3 wt % depends on the contribution of fine vs. coarse species (Fig. c). The magnitude of the aging effect (whether by changes in the rain fraction of the fine species or the water depth) can be amplified when the total sediment rain is decreased because both CaCO3 species are buried at a slower rate (dashed and dotted curves in Fig. ).

Note that it is not our intention to perfectly reproduce the observations with the parameterization adopted in this experiment, given that a large number of parameters would need to be constrained and/or modified e.g.,. Nonetheless, the 14C age sensitivity to the rain fraction of fine species shown above illustrates the utility of the model to interpret proxy signals in an extended and more realistic environmental parameter space.