NPP occurs in the top two vertical levels of the ocean domain above the fixed compensation depth (zc) of 100 m. Key parameter values are given in Table 2a. Nutrient uptake by phytoplankton type i (Γi) depends on the optimal nutrient uptake timescale (τi), nutrients, temperature (T), irradiance (I), and mixed-layer depth (zml): 1Γi=1τi⋅FN,i⋅FT⋅FI⋅max⁡1,zczml. Subscript i refers to PFT (i=1: eukaryotes; i=2: cyanobacteria; i=3: diazotrophs). The nutrient dependence FN,i is given by Liebig's law of minimum combined with Michaelis–Menten uptake kinetics of limiting nutrients: PO4, NO3, CO2 (aq), total dissolved iron (sum of free iron and ligand-bound iron: FeT = Fe′+FeL), and Si(OH)4: 2FN,i=min⁡PO4PO4+KPO4,i⋅PO4,NO3NO3+KNO3,i⋅NO3⋅QN,i-1,CO2aqCO2aq+KCO2,i⋅CO2aq⋅QC,i-1,FeTFeT+KFeT,i⋅FeT⋅QFe,i-1,SiOH4SiOH4+KSiOH4⋅SiOH4⋅QSi-1, where KX,i is the half-saturation concentration of nutrient X for PFT i. Only eukaryotes (i=1) are limited by Si(OH)4. Diazotrophs (i=3) are not limited by NO3. Nutrient uptake Γ is based on the master nutrient variable P, and all other nutrient uptake is related to Γ by the uptake stoichiometry QX,i, where X is N, Fe, Si, or C. For example, QC,i=1P:Ci for PFT i. Thus, QC,i is numerically equivalent to C:P for PFT i, but we write the equations in terms of P:C for numerical stability and convenience. The QX,i ratios represent the flexible phytoplankton uptake stoichiometry and are described more fully in Sect. 2.2.

The temperature dependence FT of Eq. (1) is given by 3FT=T∘C+2T∘C+10, which is analogous to the commonly used Q10=2 relationship. Light limitation FI of Eq. (1) is described by a hyperbolic function: 4FT=II+20, where I is the seasonally variable solar short-wave irradiance in W m-2. Light is attenuated exponentially from the ocean surface with a 20 m depth scale.

Nutrient uptake in Eq. (1) has a dependence on zml, which is diagnosed using the σt density gradient criterion (Levitus, 1982). Following the Sverdrup (1953) model of the spring bloom, Eq. (1) allows for the shoaling of zml relative to zc to enhance nutrient uptake.

As noted above, all nutrients and O2 are related to the main model currency P by QX,i. We describe three different, mutually exclusive formulations in this section. The standard formulation is the power law model (Matsumoto et al., 2020; Tanioka and Matsumoto, 2017). The other two (linear model and optimality-based model of stoichiometry) are alternative formulations that have been coded, and the user can activate them (one at a time) in place of the power law formulation. However, the alternative formulations are not calibrated. Key parameter values are given in Table 2b for the power law formulation.

In the top 100 m of the model domain, where phytoplankton P uptake occurs (i.e., Γi>0, see Sect. 2.1), NPP is immediately routed to POM and DOM pools (Fig. 1). The production fluxes of POM, DOMsl, and DOMr from NPP are given as Jprod. Here we write the equations in terms of the master nutrient variable P: 16JprodPOPi=1-fDOM⋅Γi,17JprodDOPsl=∑i(1-fDOMr)⋅fDOM⋅Γi,18JprodDOPr=∑ifDOMr⋅fDOM⋅Γi.

The term fDOM denotes the fraction of NPP that is routed to DOM as opposed to POM. Likewise, fDOMr is the fraction of DOM that is routed to DOMr as opposed to DOMsl. The value of fDOMr is not well known but estimated to be ∼ 1 % (Hansell, 2013), which we tentatively adopt in MESMO 3. If DOMr is not selected in the model run, fDOMr=0. In previous versions of MESMO, fDOM was assigned a constant value of 0.67. In reality, a large variability is observed locally for this ratio, ranging from 0.01–0.2 in temperate waters to 0.1–0.7 in the Southern Ocean (Dunne et al., 2005; Henson et al., 2011; Laws et al., 2000). In MESMO 3, fDOM is calculated as a function of the ambient temperature following Laws et al. (2000): 19fDOM=1.0-min⁡0.72,max⁡0.04,0.62-0.02⋅T∘C.

This formulation gives low export efficiency (i.e., high fDOM) in the warmer regions compared to the colder high-latitude regions. Locally, we impose fixed fDOM upper and lower bounds of 0.96 and 0.28, respectively, as estimated from a previous study (Dunne et al., 2005).

In MESMO 3, a new DOM production pathway below the production layer is available as an option. In previous MESMO versions, sinking POM was respired in the water column with the loss of O2 directly to the dissolved inorganic forms (i.e., POC → DIC, POP → PO4, and PON → NO3). In the new “deep POC split” pathway, sinking POM is simply broken down into DOM without the loss of O2 as in the production layer (Fig. 1). If DOMr is selected in the model, the broken-down POM is further routed to both DOMsl and DOMr according to fDOMr. If not, all of the broken down POM is converted to DOMsl. Thus, when the deep POC split is activated, the presence of DOM in the deep ocean can be accounted for by in situ production of DOM and DOMr in addition to DOM transport from the surface. Thus, the deep POC split pathway offers an alternative means to control deep ocean DOM distribution.

In MESMO 2, opal production was associated with the “large” PFT, and CaCO3 production was associated with the “small” PFT. We recognize that coccolithophorids and diatoms, which are the producers of these biogenic tests, are both eukaryotes. Therefore, in MESMO 3, we associate both CaCO3 and opal production with the POP production by the same eukaryote PFT (JprodPOP1): 20JprodCaCO3=rCaCO3:POC⋅JprodPOP1⋅QC,1,21Jprodopal=JprodPOP1⋅QSi.

The concept of the residual nitrate potential growth (RNPG) (Balch et al., 2016) is useful in allowing competition between diatoms and non-siliceous phytoplankton within the same PFT (Matsumoto et al., 2020). Typically, in the real ocean, non-Si phytoplankton are able to grow faster and dominate the community if Si concentration is low and diatom growth is Si limited. Otherwise, diatoms are more competitive, as they have higher intrinsic growth rates. The RNPG index recasts the ambient concentrations of NO3 and Si(OH)4 into potential algal growth rates: 22RNPG=NO3NO3+KNO3,1-SiOH4SiOH4+KSiOH4.

If RNPG is more positive, the index indicates that nitrate-dependent growth exceeds silica-dependent growth. Thus, non-Si phytoplankton are more competitive, and this leads to higher CaCO3 production. On the other hand, a more negative RNPG implies that silica limitation for diatoms is relieved, leading to enhanced diatom growth and reduced CaCO3 production. The RNPG index is incorporated in the calculation of the rain ratio rCaCO3:POC presented in Eq. (20) as follows: 23rCaCO3:POC=r0CaCO3:POC⋅Ω-1η⋅min⁡1,max⁡0.1,RNPG⋅kT,CaCO3.

Equation (23) indicates the base rain ratio r0CaCO3:POC (set to 0.30) is also modified by the carbonate ion saturation state Ω by η (set to 1.28) and temperature (see Ridgwell et al., 2007, and references therein): 24Ω=Ca2+CO32-Ksp,25kT,CaCO3=min⁡1.0,T∘C+2T∘C+8.

Ksp is the solubility product of CaCO3. The temperature dependency of CaCO3 formation (kT,CaCO3) is similar to that of Moore et al. (2004) where warmer temperatures favor the growth of carbonate-bearing phytoplankton.

Once produced, both POM and DOM undergo remineralization throughout the water column. Key remineralization parameter values are given in Table 2d. Previously, POM remineralization had a temperature dependence and decayed exponentially with depth (Yamanaka et al., 2004). In MESMO 3, we incorporate an additional dependency on dissolved oxygen following Laufkötter et al. (2017): 26RPOMi=VPOM⋅ekR⋅T⋅O2[O2]+KO2⋅POMi.

VPOM is the base remineralization rate, kR expresses the temperature sensitivity of remineralization, and KO2 is half-saturation constant for oxygen-dependent remineralization. When the sediment model is not coupled, any POM that reaches the seafloor dissolves completely to its inorganic form and is returned to the overlying water.

In MESMO 3, all forms of semi-labile DOM remineralize at the same rate. It is represented by τsl, the inverse of the timescale of DOMsl decay, which has been estimated previously to be ∼ 1.5 years (Hansell, 2013): 27RDOMsl=τsl⋅DOMsl.

All forms of DOMr also remineralize at the same rate in MESMO 3. In total, there are three optional, additive sinks of DOMr in the model: slow background decay, photodegradation, and degradation via hydrothermal vents (Fig. 1). Observations clearly indicate that the 14C age of deep-ocean DOCr is 103 years (e.g., Druffel et al., 1992), much older than DI14C. In addition, the deep ocean DOCr concentration decreases modestly along the path of the deep water from the deep North Atlantic to the deep North Pacific (Hansell and Carlson, 1998). Thus, it is understood that there is a slow DOMr background decay in the deep ocean. We represent this ubiquitous process with τbg, which is the inverse of the background decay timescale, estimated to be ∼ 16 000 years (Hansell, 2013).

Observations to date indicate that photodegradation is a major sink of DOMr (e.g., Mopper et al., 1991). This process is believed to convert DOMr that is upwelled from the ocean interior into the euphotic zone into more labile forms of DOM. We represent photodegradation with τphoto, the inverse of the decay timescale, estimated to be ∼ 70 years (Yamanaka and Tajika, 1997). This occurs only in the surface.

Finally, observations of DOM emanating from different types of hydrothermal vents indicate that they have variable impacts on the deep-sea DOMr (Lang et al., 2006). However, the off-axis vents circulate far more seawater through the fractured oceanic crust than the high-temperature and diffuse vents and are thus believed to determine the overall impact of the vents on the deep-sea DOMr as a net sink (Lang et al., 2006). Here we assume simply that seawater that circulates through the vents loses all DOMr (i.e., 1/τvent < Δt, where Δt is the biogeochemical model time step of 0.05 year). This means that the more seawater circulates through the vents, the more DOMr is removed: the total removal rate depends on the vent flux of seawater Hflux. We implement the vent degradation of DOMr in MESMO 3 by first identifying the wet grid boxes located immediately above known mid-ocean ridges. We then distribute the annual global Hflux of 4.8×1016 kg yr-1 (Lang et al., 2006) equally among those ridge-associated grid boxes. The grid cells contain a mass of seawater much greater than the mass that circulates through vents in Δt (1021 kg vs. 1013 kg). Therefore, the seawater mass in the vent grid cells that does not circulate through the vents in Δt is subject only to background degradation in MESMO 3.

The three DOMr sinks are not mutually exclusive. They can thus be combined to yield the total DOMr remineralization rate: 28RDOMr=τbg+τphoto+τvent⋅SWflux_localSWgrid⋅DOMr, where SWflux_local is the mass of seawater that circulates through the vents in each grid box in Δt, and SWgrid is the total mass of seawater in the same grid box.

The amount of O2 respired as a result of these POM and DOM remineralization processes is related to the organic carbon pools by the respiratory quotients of POC and DOC, r-O2:POC and r-O2:DOC, respectively. These are molar ratios of O2 consumed per unit organic carbon respired. They are variable and calculated from the ambient POM and DOM concentration (Tanioka and Matsumoto, 2020b): 29r-O2:POC=1.1+2PONPOC,30r-O2:DOC=1.1+2DONDOC.

Remineralization of CaCO3 and opal particles occurs as they sink through the water column and remains the same as in MESMO 2. Key parameter values are given in Table 2d. Remineralization of CaCO3 is a function of temperature similar to that of particulate organic matter remineralization but without oxygen dependency. The temperature-dependence term kR modifies the base remineralization rate VCaCO3: 31RCaCO3=VCaCO3⋅ekR⋅T⋅CaCO3. Opal remineralization in the water column follows Ridgwell et al. (2002). The rate of opal remineralization Ropal is given by the product of normalized dissolution rate (ropal), base opal dissolution rate (kopal), and opal concentration [opal]. 32Ropal=ropal⋅kopal⋅opal33ropal=0.16⋅1+T∘C15⋅uopal+0.55⋅1+T∘C4004⋅uopal9.2534uopal=SiOH4eq-SiOH4SiOH4eq ropal is a function of temperature (T) and the degree of under-saturation (uopal), which in turn is calculated from the ambient SiOH4 and SiOH4 at equilibrium. The equilibrium concentration is a function of ambient temperature: 35log⁡10SiOH4eq=6.44-968TK.

Without the sediment module of MESMO activated, both CaCO3 and opal particles that reach the seafloor are completely dissolved back to inorganic forms.

The time rate of change of the biogenic organic matter and tests are given by the sum of the production terms (i.e., sources) and the remineralization terms (i.e., sinks). The circulation-related transport terms are omitted as noted above, but the vertical transport due to particle sinking is included here. The sinking speed w is the same for all particles. The sum of POMi of all the PFTs give the total POM concentrations. 36∂POPi∂t=JprodPOPi-∂∂zwPOPi-RPOP,i37∂POCi∂t=JprodPOPi⋅QC,i-∂∂zwPOCi-RPOC,i38∂PONi∂t=JprodPOPi⋅QN,i-∂∂zwPONi-RPON,i39∂POFei∂t=JprodPOPi⋅QFe,i-∂∂zwPOFei-RPOFe,i40[POM]=∑iPOMi The time rate of change of CaCO3 and opal is expressed in much the same way as POM. 41∂CaCO3∂t=JprodCaCO3-∂∂zwCaCO3-RCaCO342∂opal∂t=Jprodopal-∂∂zwopal-Ropal

The DOM pools have the production and remineralization terms without the particle sinking term. 43∂DOPsl∂t=JprodDOPsl-RDOPsl44∂DONsl∂t=JprodDONsl-RDONsl45∂DOCsl∂t=JprodDOCsl-RDOCsl46∂DOFesl∂t=JprodDOFesl-RDOFesl47∂DOPr∂t=JprodDOPr-RDOPr48∂DONr∂t=JprodDONr-RDONr49∂DOCr∂t=JprodDOCr-RDOCr

The time rate of change of the inorganic nutrients have organic carbon production as sink terms and remineralization as source terms. The production terms (Jprod) are zero below the upper-ocean production layer. Nutrients have a unit of mol element kg-1 in the model. 50∂PO4∂t=-∑iΓi+∑iRPOP,i+RDOPsl+RDOPr51∂NO3∂t=-∑iΓi⋅QN,i+∑iRPON,i+RDONsl+RDONr+FixN-DenN52∂DIC∂t=-∑iΓiQC,i+JprodCaCO3+∑iRPOC,i+RDOCsl+RDOCr+RCaCO3+Fgas,CO253∂ALK∂t=-2⋅JprodCaCO3-∑iΓiQN,i-∑iRPON,i-RDONsl-RDONr-FixN+DenN+2⋅RCaCO354∂FeT∂t=-∑iΓiQFe,i+∑iRPOFe,i+RDOFesl+RPOMFe+AeolianFe55∂SiOH4∂t=-Jprodopal+Ropal56∂O2∂t=∑iΓi⋅Q-O2,i-(r-O2:DOC⋅RDOCsl+RDOCr+∑ir-O2:POC,i⋅RPOC,i)+1.25DenN+Fgas,O2

In Eq. (51), FixN is the N fixation carried out by diazotrophs, and DenN is the water column denitrification. There is an air–sea gas exchange term Fgas in Eqs. (52) and (56) for gaseous CO2 and O2, respectively. In Eq. (53), alkalinity increases with decreasing nitrate concentrations and increasing CaCO3 dissolution. Equation (54) contains RPOMFe, which is an iron source that represents remineralization of the Fe′ scavenged by sinking particles. These terms are explained in the following sections.

Biological production by diazotrophs is stimulated when the ambient NO3 is low. Nitrogen fixed by diazotrophs during their growth is added to the marine NO3 pool. The prognostic nitrogen fixation model employed here is similar to that used in the HAMOCC biogeochemical module (Paulsen et al., 2017): 57FixN=Γ3⋅QN,3⋅INO3,58INO3=1.0-NO32KNO3_Nfix2+NO32, where FixN is the nitrogen fixation rate and INO3 is the nitrate dependency term in quadratic Michaelis–Menten kinetics form with the half-saturation constant KNO3_Nfix. See Table 2e for the values related to the N cycle.

Water column denitrification is formulated in an approach similar to that of the original GENIE model (Ridgwell et al., 2007), in which 2 mol of NO3 are converted to 1 mol of N2 and liberating 2.5 mol of O2 as a byproduct: 592NO3-+2H+→2.5O2+N2+H2O.

Denitrification takes place in grid boxes, in which O2 concentration is below a threshold concentration (O2,def) and is stimulated if the total global inventory of NO3 relative to PO4 is high. In other words, denitrification can effectively act as negative feedback to nitrogen fixation. The threshold O2 concentration (O2,def) takes the minimum of the hard-bound O2 threshold concentration (O2,crit) and the NO3 to PO4 ratio, scaled by a parameter kD. The parameters O2,crit and kD are calibrated to give the global denitrification rate of roughly 100 Tg N yr-1, which balances the total nitrogen fixation rate in the model. 60DenN=0.8yr-1⋅max⁡O2def-O2,061O2def=min⁡O2,crit,kD⋅NO3inventoryPO4inventory

The iron cycle in MESMO 3 remains the same as in MESMO 2. Key parameter values are given in Table 2e. The two species of dissolved iron (Fe′ and FeL) are partitioned according to the following equilibrium relationship: 62Kligand=FeLFe′⋅L, where [L] is the ligand concentration and Kligand is the conditional stability constant. The sum of ligand and FeL is set at a constant value of 1 nmol kg-1 everywhere. Iron is introduced into the model domain by a constant fraction (3.5 wt %) of aeolian dust deposition at the surface (Fin) following the prescribed modern flux pattern (Mahowald et al., 2006) with constant solubility (β): 63SFe=β⋅Fin.

Particle-scavenged iron POMFe (note the difference from POFe) is produced below the productive layer when sinking POM scavenges Fe′ to sinking POM: 64JFe=-τsc⋅Ko⋅POC0.58⋅Fe′, where τsc and Ko are empirical parameters that determine the strength of scavenging. Remineralization of Fe scavenged to POM (POMFe) is identical in form to that of POM remineralization: 65RPOMFe=VPOM⋅ekRT⋅O2[O2]+KO2⋅POMFe.

The conservation equation of the particle scavenged iron is thus expressed as follows: 66∂POMFe∂t=JFe-∂∂zwPOMFe-RPOMFe.

Any scavenged iron that escapes remineralization in the water column reaching the seafloor is removed from the model domain in order to keep the total Fe inventory at a steady state.

The air–sea gas exchange formulation remains the same as in MESMO 2 and follows Ridgwell et al. (2007). It is the function of gas transfer velocity, the ambient dissolved gas concentration, and saturation gas concentration. The flux of CO2 and O2 gases across the air–sea interface is given by 67Fgas,CO2=k⋅CO2sat-CO2⋅1-A,68Fgas,O2=k⋅O2sat-O2⋅1-A, where k is the gas transfer velocity, CO2sat and O2sat are saturation concentrations, and A is the fractional ice-covered area that is calculated by the physical model. Gas transfer velocity k is a function of wind speed (u) following Wanninkhof (1992) where Sc is the Schmidt number for a specific gas: 69k=0.31⋅u2⋅Sc660-0.5.