Studying the temporal bias of the steady-state approximation of 234Th-derived carbon export during phytoplankton blooms

The ²³⁴Th–²³⁸U pair technique is widely used in estimating particulate organic carbon (POC) flux, typically with a steady-state (SS) assumption. There is often a temporal bias between the SS-derived and actual POC fluxes caused by neglecting the temporal change in ²³⁴Th. However, this temporal bias has not been fully investigated due to the lack of continuous sampling of ²³⁴Th profiles and sediment traps. Here, we develop a radioactive trace model of ²³⁴Th built on a physical–biogeochemical model to simulate the scavenging of ²³⁴Th in the water column by POC sinking processes at the South East Asia time series (SEATS) site. The seasonal patterns of the ²³⁴Th profiles simulated by the model generally compared well with in situ observations. Analysis based on the model simulation suggests that the temporal bias can be depicted and reproduced in a simplified ²³⁴Th continuity equation. By obtaining an analytical solution for the SS-derived POC flux from the simplified ²³⁴Th equation, we found that the temporal bias results from the phase difference in time between the SS-derived and direct sinking POC fluxes. To provide a method that does not need repeated samplings to reduce this temporal bias for in situ observations, a modification term was constructed for the SS-derived POC flux from the analytical solution. Applying this term to the data obtained at the Bermuda Atlantic time series and SEATS reduced the bias by up to 67% and 34%, respectively. This study provides a feasible way to improve ²³⁴Th-derived POC flux under the SS assumption.

1 Introduction

The ocean’s biological pumps are a primary pathway for sequestering atmospheric carbon dioxide (CO₂) through the gravitational sinking of particulate organic carbon (POC), significantly mitigating global warming (DeVries et al., 2017; Terhaar et al., 2021; Boyd et al., 2019; DeVries et al., 2019; Friedlingstein et al., 2020). However, there is a challenge in quantifying the magnitude of POC export flux, as discrepancies in magnitudes exist among different assessments (DeVries et al., 2019; Gruber et al., 2019; Hauck et al., 2020; Iida et al., 2021; Ceballos-Romero et al., 2022).

Currently, POC flux is widely estimated using sediment traps, which directly collect sinking particles from the water column to quantify flux (Buesseler et al., 2007; Lamborg et al., 2008; Owens et al., 2013; Bourne et al., 2019; Baker et al., 2020; Estapa et al., 2020).

Radioactive pairs (mainly the ²³⁴Th–²³⁸U pair) are also used for POC flux estimation (Buesseler et al., 1992; Stewart et al., 2011; Ceballos-Romero et al., 2016; Umhau et al., 2019; Alkalay et al., 2020; Zhou et al., 2020). The ²³⁴Th (half-life of 24.1 days) produced from the radioactive decay of ²³⁸U (half-life of 4.47×10⁹ years) is expected to be in secular equilibrium with ²³⁸U in seawater. While the particle-adsorptive ²³⁴Th is scavenged by the downward flux of particles and redistributed by the transport of physical processes, the equilibrium is broken in the upper layer of the water column. With the steady-state assumption (SS) that the change in ²³⁴Th over time and physical transport is negligible, the downward ²³⁴Th flux at a depth of interest is equal to the deficit of ²³⁴Th–²³⁸U integrated over depth multiplied by a thorium decay constant λ (0.02876 d⁻¹; e.g., Buesseler et al., 1992; Ceballos-Romero et al., 2018). Once the ²³⁴Th profile is obtained from the field measurements, the ²³⁴Th flux can be estimated using the SS model, and the POC flux is thus obtained by multiplying the ²³⁴Th flux by the POC:²³⁴Th ratio.

The ²³⁴Th–²³⁸U pair is used frequently because of its advantage in obtaining spatially and temporally resolved POC flux profiles (Buesseler et al., 1992; Umhau et al., 2019). However, inconsistencies have often been observed between SS-derived and sediment trap-estimated POC fluxes (e.g., Stewart et al., 2011; Le Moigne et al., 2013). Such discrepancies have been attributed to variable factors, such as the influence of physical processes (Buesseler et al., 2008; Resplandy et al., 2012; Stukel et al., 2017), uncertainties in trapping efficiency (Buesseler et al., 2007; Baker et al., 2020), and the POC:²³⁴Th ratio in the radioactive pair technique (Buesseler et al., 2006; Umhau et al., 2019). In addition, neglecting the temporal change in ²³⁴Th under the SS assumption may introduce biases compared with the actual sinking fluxes (Buesseler et al., 1992; Savoye et al., 2006; Buesseler et al., 2008; Ceballos-Romero et al., 2016, 2018). Previous studies showed that SS-derived POC fluxes might be underestimated during the pre-bloom period and overestimated during the post-bloom period (Buesseler et al., 1992; Ceballos-Romero et al., 2018). To reduce the temporal bias, the non-steady state (NSS) method for the radioactive pair technique has been used, which resolves the changes in ²³⁴Th over time by repeated sampling of the same water mass (Savoye et al., 2006; Resplandy et al., 2012; Ceballos-Romero et al., 2018). However, the NSS model often has large uncertainties, as resampling the same water mass is not always applied (Savoye et al., 2006; Resplandy et al., 2012; Ceballos-Romero et al., 2018).

The temporal bias of the SS-derived POC flux has not been fully investigated due to the lack of continuous sampling of ²³⁴Th profiles and sediment traps. This study presents a model of radioactive isotopes ²³⁸U and ²³⁴Th that are included in a physical–biogeochemical model (ROMS-CoSiNE-²³⁴Th; described in Section 2) to simulate the scavenging of 234Th by sinking POC at the South East Asia Time-Series (SEATS; 116°E, 18°N) site. Using the model simulation, we diagnosed the continuity equation of ²³⁴Th to investigate the source of temporal bias between the SS-derived and direct sinking POC fluxes. The diagnosis analysis suggests that the temporal bias between the SS-derived and direct sinking POC flux can be depicted and reproduced in the simplified ²³⁴Th equation. We solved the simplified ²³⁴Th equation and obtained an analytical solution for the SS-derived POC flux. To provide a method that does not need repeated samplings to reduce this temporal bias for in situ observations, a modification term was then constructed for the SS-derived POC flux from the analytical solution and applied to field observations at the SEATS and the Bermuda Atlantic time series study (BATS; 31°50′N, 64°10′W) site by using satellite data.

2 Data and methods

2.1 The physical–biogeochemical model

A physical–biogeochemical model (ROMS-CoSiNE-²³⁴Th) that includes a ²³⁴Th module was constructed to compute the evolution of vertical ²³⁴Th profiles. The physical model was based on the Regional Ocean Modeling System (ROMS) and set up at SEATS with a vertical grid of 200 layers covering the depth from the surface to 2000 m. The Mellor and Yamada Level 2.5 (MY-2.5) turbulence closure scheme (Mellor and Yamada, 1982) was used for vertical mixing. Atmospheric forcing was applied using bulk formulas with six-hourly data of net surface shortwave radiation, net surface longwave radiation, 10 m height wind, 2 m height air temperature, mean sea level pressure, and specific humidity from the National Centers for Environmental Prediction (NCEP) Reanalysis II. The biogeochemical model was based on a modified version of the carbon, silicate, and nitrogen ecosystem (CoSiNE; Chai et al., 2002) model that includes two functional groups of phytoplankton [small phytoplankton (P1), diatoms (P2)], two chlorophyll groups [chlorophyll of P1 (Chl1) and P2 (Chl2)], two zooplankton groups [microzooplankton (Z1) and mesozooplankton (Z2)], small and large organic nitrogen detritus (PON_S and PON_L), biogenic silica (bSi), nitrate (NO₃), ammonium (NH₄), phosphate (PO₄), silicate (Si(OH)₄), dissolved oxygen (DO), dissolved inorganic carbon (DIC), and total alkalinity (TALK). The POC concentration in the CoSiNE model was quantified as the concentration of detritus and phytoplankton. Particles with slow and fast sinking rates in the observations (Villa-Alfageme et al., 2016; Tréguer et al., 2018; Boyd et al., 2019) were simulated as the groups of P1, P2, and PON_S (sinking velocity<2 m/d) and the groups of PON_L and bSi (sinking velocity >10 m/d), respectively (Supplementary Table S1). The CoSiNE model considers the processes of nutrient uptake, mortality, and grazing for phytoplankton, and fecal pellet production, predation, remineralization, aggregation, and sinking for detritus. The detailed equations for the CoSiNE model used in this study are listed in Ma et al. (2019). Initial conditions for nutrients and physical fields were derived from the World Ocean Atlas 2013 (WOA13), and the model was spun up with climatological monthly forcing of atmospheric conditions from the NCEP for 5 years. The adjusted biogeochemical and ²³⁴Th fields at the end of this spin-up were used as initial conditions for the realistic simulation, which was then integrated from 2004 to 2014, driven by real-time six-hourly NCEP atmospheric forcing.

As shown in the schematic diagram of the model (Figure 1), the ²³⁴Th module includes tracers of dissolved ²³⁴Th (²³⁴Th_w) and particulate ²³⁴Th (²³⁴Th_p; unit: dpm m⁻³) adsorbed onto P1, P2, Z1, Z2, PON_S, PON_L, and bSi (denoted ²³⁴Th_P1, ²³⁴Th_P2, ²³⁴Th_Z1, ²³⁴Th_Z2, ²³⁴Th_PONL, ²³⁴Th_PONS, and ²³⁴Th_bSi). The continuity equation describing the total ²³⁴Th behaviors in the model includes the change in ²³⁴Th (total ²³⁴Th in the water) over time (Rate), the radioactive production decay of ²³⁴Th (Production-Decay), the direct ²³⁴Th flux due to sinking particles (sinking), and the ²³⁴Th change due to physical processes (PHY):

Figure 1

Figure 1. Schematic diagram of the CoSiNE-²³⁴Th model.

$\begin{array}{ll}\underset{\text{Rate }}{\underbrace{\frac{{\partial }^{234}Th}{\partial t}}}=\underset{\text{Production}-\text{Decay}}{\underbrace{\lambda {(}^{238}U{-}^{234}Th)}}-\underset{\text{Sinking}}{\underbrace{W_p·\frac{{\partial }^{234}T{h}_p}{\partial z}}}+\underset{\text{PHY}}{\underbrace{PHY{(}^{234}Th)}}& (1)\end{array}$

where λ and W_p are the decay constant (0.02876 d⁻¹) and particle sinking rate (m d⁻¹), respectively. In a one-dimensional framework, vertical diffusion (Vdiff) is the dominant term for the PHY term. Total ²³⁴Th is produced by the decay of ²³⁸U. Here, total 234Th represents the sum of ²³⁴Th_w and ²³⁴Th_p. The distributions of ²³⁴Th_p are determined by the balance of adsorption and desorption processes (Dunne et al., 1997). Detailed equations and parameters for all ²³⁴Th processes are listed in Supplementary Data Sheet 1 - Text S1.

Since the time series of the ²³⁴Th profiles at SEATS were reproduced by the realistic simulation of the ROMS-CoSiNE-²³⁴Th model, the ²³⁴Th flux at 100 m that generally represents those directly measured by sediment traps (hereafter noted as Sinking) was calculated using the model outputs at 100 m (Equation 2):

$\begin{array}{ll}\text{Sinking}={({\text{W}}_{\text{p}}{·}^{234}{\text{Th}}_{\text{p}})}_{100\text{m}}& (2)\end{array}$

where W_p is the particle sinking velocity (Supplementary Table S1). Following the approach in Buesseler et al. (1992), the SS-derived ²³⁴Th flux at 100 m (hereafter noted as SS) was calculated using the ²³⁸U-²³⁴Th deficit from the model outputs of the ²³⁴Th profiles (Equation 3):

$\begin{array}{ll}\text{SS}=\text{λ}{\displaystyle {\int }_0^{100\text{m}}{[}^{238}\text{U}{-}^{234}\text{Th]dz}}& (3)\end{array}$

Similar to the calculation of SS-derived ²³⁴Th flux, the NSS-derived ²³⁴Th flux (hereafter noted as NSS) accounts for the rate of ²³⁴Th and was obtained by Savoye et al. (2006) (Equation 4):

$\begin{array}{ll}\begin{array}{c}\text{NSS}=\frac{\text{λ}}{1-{\text{e}}^{-\text{λΔt}}}\hfill \\ ·{\displaystyle {\int }_0^{100\text{m}}{[}^{238}\text{U}·(1-{\text{e}}^{-\lambda \Delta \text{t}}+{\text{e}}^{-\lambda \Delta \text{t}}{·}^{234}{\text{Th}}_1{-}^{234}{\text{Th}}_2)]\text{dz}}\hfill \end{array}& (4)\end{array}$

where ²³⁴Th₁ and ²³⁴Th₂ are the total ²³⁴Th concentrations at two different times from the model outputs, and Δt is their time interval. In this study, the temporal patterns of NSS with Δt=10 days [NSS (10d)] and Δt=60 days [NSS (60d)] were calculated. To avoid introducing the uncertainty of the POC: ²³⁴Th ratio to the temporal bias between the SS-derived and direct sinking POC fluxes simulated at SEATS, the direct sinking, SS-derived and NSS-derived ²³⁴Th fluxes calculated from the model outputs were converted to the POC flux by multiplying the same POC:²³⁴Th ratio of 4.9×10⁻³ mmol C/dpm. This POC:²³⁴Th ratio was estimated as the average over POC:²³⁴Th ratio at 100 m collected from 11 cruises between 2004 and 2014 at SEATS (Zhou et al., 2020).

2.2 Deriving analytical solutions for the ²³⁴Th-derived POC flux

In the ROMS-CoSiNE-²³⁴Th model, the total ²³⁴Th behaviors are described by the continuity equation of ²³⁴Th (Equation 1). To derive the analytical solutions for the ²³⁴Th-derived POC flux, we used a simplified continuity ²³⁴Th equation by ignoring the advection and diffusion terms in Equation 1 (e.g., Ceballos-Romero et al., 2018; de Soto et al., 2018):

$\begin{array}{ll}\frac{{\partial }^{234}Th}{\partial t}={\lambda }^{238}U-{\lambda }^{234}Th-W_p·\frac{{\partial }^{234}T{h}_p}{\partial z}& (5)\end{array}$

By integrating each term of Equation 5 from the surface to a depth of Z, we obtained:

$\begin{array}{ll}\frac{\partial {\displaystyle {\int }_0^{\text{Z}234}Thdz}}{\partial t}=\lambda {\displaystyle {\int }_0^{Z238}Udz-\lambda }{\displaystyle {\int }_0^{Z234}Thdz-W_p·}{\displaystyle {\int }_0^Z\frac{{\partial }^{234}T{h}_p}{\partial z}}dz& (6)\end{array}$

Let $TH={\displaystyle {\int }_0^{\text{Z}234}Thdz}$, $U={\displaystyle {\int }_0^{\text{Z}238}Udz}$, and P= $W_p·{\displaystyle {\int }_0^Z\frac{{\partial }^{234}T{h}_p}{\partial z}}dz$; then, Equation 6 is simplified as a differential equation with each term relevant to variable t:

$\begin{array}{ll}\frac{dTH}{dt}+\lambda TH=\lambda U-P& (7)\end{array}$

The function of the particle direct sinking flux can be represented with a Fourier series:

$$P=a_0+{\displaystyle \sum _{i=1}^I(a_i\cos i\omega t+b_i\sin i\omega t)}$$

where a₀, a_i, and b_i are amplitudes, and ω is the angular frequency. Here, U is regarded as a constant. The analytical solution for TH is given by solving Equation 7:

$\begin{array}{ll}\begin{array}{c}TH(t)=c{e}^{-\lambda t}\\ +{\displaystyle {\sum }_{i=1}^I\left\{\left[\frac{b_{i}i\omega -a_i\lambda }{{\lambda }^2+{(i\omega )}^2}\right]\cos i\omega t-\left[\frac{b_i\lambda +a_{i}i\omega }{{\lambda }^2+{(i\omega )}^2}\right]\sin i\omega t\right\}}\hfill \\ +U-\frac{a_0}{\lambda }\hfill \end{array}& (8)\end{array}$

where the constant c can be determined once an initial condition for TH is given. Hence, the analytical solution for the SS- and NSS-derived ²³⁴Th flux is given by:

$\begin{array}{ll}SS(t)=\lambda (U-TH)& (9)\end{array}$

$\begin{array}{ll}NSS(t)=\lambda U+\frac{\lambda \alpha }{1-\alpha }T{H}_1-\frac{\lambda }{1-\alpha }T{H}_2& (10)\end{array}$

where $\alpha =e^{-\lambda \text{Δ}t}$. SS(t) and NSS(t) represent the analytical solutions for the SS and NSS-derived ²³⁴Th fluxes. TH₁ and TH₂ are integrals of ²³⁴Th₁ and ²³⁴Th₂ over depth.

2.3 Data

In situ data at SEATS collected from 11 cruises between 2004 and 2014 were used to validate the model simulation, including the temperature, salinity, chlorophyll, particulate ^234Th, and total ²³⁴Th collected by 12-L Niskin bottles at multiple depths (Supplementary Table S2; Zhou et al., 2020). During this period, two sediment trap-estimated POC fluxes at 100 m at SEATS were obtained from Ho et al. (2010) and Wei et al. (2011). The sediment trap-estimated POC fluxes were used to evaluate the corrected results of the SS-derived POC flux at SEATS.

Satellite-derived POC flux as described in Supplementary Data Sheet 1 - Text S2 and micro-sized phytoplankton chlorophyll concentration [CHL(Micro)] obtained from the Copernicus Marine Environment Monitoring Service (CMEMS) were used to estimate the duration of the phytoplankton blooms at both SEATS and BATS.

To verify the developed correction method at BATS, the time series of POC flux estimated using sediment traps at 150 m was obtained from the BATS website. The SS-derived POC fluxes at 150 m near BATS (distance to BATS ≤ 150 km) were obtained from Buesseler et al. (2008) and Stewart et al. (2011). With this dataset, we tried to reduce the bias of SS-derived POC flux compared with the sediment trap-estimated POC flux.

2.4 Model validation

To evaluate the ROMS-CoSiNE-²³⁴Th model’s ability to represent seasonal ²³⁴Th dynamics at SEATS, we compared realistic simulations (2004–2014) with in situ data from 11 cruises conducted between 2004 and 2014 (Supplementary Table S2; Zhou et al., 2020). These cruises, spanning a decade, represent the most extensive dataset available for SEATS to our knowledge. Comparisons were made over winter (December to February) and summer (June to August) seasons, focusing on temperature, salinity, chlorophyll, and ²³⁴Th profiles (Figure 2). In winter, in situ measurements showed small changes in the mixed layer, with temperature ranging from 23.2°C to 24.9°C, salinity from 33.7 to 33.9 PSU, chlorophyll from 0.21 to 0.46 mg/m³, and particulate ²³⁴Th from 280 to 450 dpm/m³. The model reproduced these homogeneous distributions well, yielding climatological averages of 25.1°C for temperature, 33.9 PSU for salinity, 0.25 mg/m³ for chlorophyll, and 505 dpm/m³ for particulate ²³⁴Th. Total ²³⁴Th from in situ data ranged from 1,500 to 2,400 dpm/m³, while the model overestimated this slightly, lacking a minimum observed at ~50 m depth, suggesting an underestimation of particle sinking in the setup. In summer, observations indicated a temperature decrease and salinity increase below a shallow mixed layer, with subsurface maxima in chlorophyll and particulate ²³⁴Th. The model reasonably captured these vertical distribution patterns and the large ²³⁴Th-²³⁸U deficits beneath the mixed layer depth (MLD), consistent with particle sinking effects (Zhou et al., 2020; Figure 2j). While the model’s performance supports its use for simulating ²³⁴Th dynamics, validation of the POC flux correction was limited to two concurrent sediment trap measurements at 100 m (Ho et al., 2010; Wei et al., 2011), reflecting the scarcity of such paired observations at SEATS.

Figure 2

Figure 2. In situ and model-simulated vertical profiles of (a, b) temperature, (c, d) salinity, (e, f) chlorophyll, (g, h) particulate ^234Th, and (i, j) total ²³⁴Th in (a, c, e, g, i) winter and (b, d, f, h, j) summer at the SEATS site. Black and gray solid lines represent the seasonal mean and daily vertical profiles of the model outputs, respectively. Blue solid lines are the in situ profiles with standard deviations. The horizontal lines indicate the mixed layer depth.

3 Results

3.1 Temporal evolution of POC flux at SEATS

The seasonal cycle of POC flux was calculated from the 10-year realistic simulation at SEATS (Figure 3a). The POC sinking flux increased from late autumn to winter and was generally in phase with the primary production of P2 [PP (P2)] and surface chlorophyll of P2 [CHL (P2)] (Figure 3b), which was associated with the winter phytoplankton peak induced by the northeast monsoon (Palacz et al., 2011; Zhou et al., 2020). Linear regression analysis revealed that the POC flux was strongly correlated with the sum of PON_L (r=0.96, p<0.01), PP (P2) (r=0.79, p<0.01), and CHL (P2) (r=0.8, p<0.01), whereas it was weakly correlated with small phytoplankton chlorophyll and total phytoplankton chlorophyll. The weak correlation between total chlorophyll and POC flux could be attributed to the variability in carbon export efficiency induced by different phytoplankton compositions (Henson et al., 2015).

Figure 3

Figure 3. (a) Seasonal POC export flux at 100 m for (blue) Sinking, (red) SS, (green) NSS (10 days), and (orange) NSS (60 days) from simulations at SEATS. The shaded area is the 10% error for each curve. (b) Seasonal distributions of (blue) surface chlorophyll for large phytoplankton, (orange) 0–100 m integrated primary production for large phytoplankton from model outputs, and (cyan) satellite-derived surface chlorophyll of micro-sized phytoplankton. (c) Correlation coefficients of the Sinking and NSS-derived POC flux at 100 m as a function of the time interval (Δt) for estimating the NSS-derived flux. The vertical dotted line represents the half-life of ²³⁴Th.

The model showed a seasonal difference between SS and Sinking, while NSS (10 d) generally followed the change in Sinking (Figure 3a), which was consistent with the results of Ceballos-Romero et al. (2018). Considering the annual time series, there was up to 80% of the time when the NSS (10 d) could predict the Sinking within a 10% uncertainty, while it was only 37.8% for SS. However, the accuracy of NSS is determined by the sampling interval (Δt) (Savoye et al., 2006; Ceballos-Romero et al., 2018). As an example, the peak of NSS (60 d) was temporally mismatched with that of Sinking at SEATS. Given a large Δt, the result of NSS can approach that of SS as e^−λΔt approaches 0 (Savoye et al., 2006; Resplandy et al., 2012). The correlation coefficient (R) between NSS and Sinking decreased as Δt increased, particularly with a dramatic decline after Δt> the half-life of ²³⁴Th (24.1 d) (Figure 3c).

3.2 Dynamics of ²³⁴Th in temporal evolution

We diagnosed each term of Equation 1 to quantify the dynamics controlling ²³⁴Th activity in the water column. The change rate of ²³⁴Th over time (Rate) was negative within the mixed layer due to the negative contribution of vertical diffusion (Vdiff) when winter mixing developed (Figures 4a, b). The temporal patterns of the particle sinking term (Sinking) and the Production-Decay term were similar (Figures 4c, d), suggesting a positive contribution of particle sinking to the ²³⁴Th change above 100 m (Buesseler et al., 1992; Savoye et al., 2006) and a negative contribution by remineralization near 100 m (Zhou et al., 2020; Xiu and Chai, 2020). The combined Production-Decay and Sinking terms contributed positively to the Rate term after the Sinking term reached its maximum, which was responsible for the time lag between the peaks of the SS-derived flux and the sinking flux.

Figure 4

Figure 4. The seasonal cycle of each term in Equation 1 from simulations at SEATS. Vertical distributions of (a) the change rate over time (Rate), (b) vertical diffusion term (Vdiff), (c) radioactive production-decay of ²³⁴Th (Production-Decay), and (d) the particle sinking term (Sinking). The black lines in parts (a–d) are the mixed layer depths. (e) 0–100 m integrations of (blue) the Rate, (orange) Vdiff, (purple) radioactive production-decay (SS), and (green) the Sinking terms. (f) 0–100 m integrations of (blue) the Sinking term, (red) the radioactive production-decay term (SS), and (orange) the term of (SS-Rate).

The upper 100 m integration of the Rate term was negative during the pre-bloom period, which was attributable to Vdiff and particle sinking, while the Rate term became positive during the post-bloom period, largely due to the Production-Decay term (Figure 4e). Combining SS with Rate is a feasible way to reduce the time lag between SS and Sinking (Figure 4f).

The evolution of ²³⁴Th shows the influence of the winter phytoplankton peak on the ²³⁴Th distribution (Supplementary Figure S1). During the period of phytoplankton development, the dissolved ²³⁴Th (²³⁴Th_w) was adsorbed onto the particles (²³⁴Th_p) and reached its minimum when the particle sinking was largest (Supplementary Figures S1a, b, d). At this time, the total ²³⁴Th was not at its minimum, as most ²³⁴Th_p was not removed (Supplementary Figures S1c, d). After the phytoplankton peak, the concentration of ²³⁴Th_w was compensated by the decay of ²³⁸U (Supplementary Figures S1a, d). ²³⁴Th_p accumulated at the base of the mixed layer after particle sinking reached its peak and was retained at the subsurface when the mixed layer became shallow (Supplementary Figure S1b).

3.3 Analytical solutions for the ²³⁴Th-derived POC flux

The diagnosis analysis in Section 3.2 suggests that the combination of the Rate, Production-Decay, and Sinking terms, which is equivalent to the simplified continuity equation of ²³⁴Th, can reproduce the temporal bias between the SS-derived and direct sinking POC fluxes. It is thus applicable to derive the SS(t) and NSS(t) using the simplified equation of total ²³⁴Th (Equation 5).

We tested the SS(t) and NSS(t) using the model outputs at SEATS. The temporal pattern of Sinking curve was best fitted (r=0.98, p<0.01) using a Fourier series that comprised eight trigonometric functions. The SS(t) and NSS(t) were obtained from Equations 9 and 10 (Figures 5a–c; Supplementary Data Sheet 1 - Text S3). Both the SS(t) and NSS(t) were close to the realistic simulations [r>0.98, p<0.01, root-mean square error (RMSE)<5.8 mg/m²d], demonstrating the feasibility of the analytical solution in producing the temporal patterns of SS and NSS.

Figure 5

Figure 5. Analytical solution of the seasonal POC flux at the SEATS site. (a) Comparison of the curve-fitted Sinking using the Fourier series function with the modeled sinking term. (b) Comparison of the analytical solution of SS from the fitted sinking curve (Fitted SS) with the SS calculated directly from the model results. (c) Comparison of the analytical solution of NSS from the fitted sinking curve (Fitted NSS) with the NSS calculated directly from the model results. The root-mean square error (RMSE; unit: mg/m²d) and the correlation coefficient R are shown in parts (a–c).

For applications of the analytical solutions [SS(t) and NSS(t)] in the real ocean, we can simplify the Fourier series of the particle sinking flux by considering periodic export events. For a regular phytoplankton bloom that has obvious periodic variations, P increases gradually and then decreases after it reaches a peak, which can be assumed to follow a cosine perturbation during this period:

$\begin{array}{ll}\text{P}={\text{a}}_0-{\text{a}}_1\cos \text{ωt}& (11)\end{array}$

where a₁ is the amplitude of the variation of POC flux, and ω is determined by the duration (T) of the POC export flux that is connected with the phytoplankton bloom in the euphotic zone as $\text{ω}=\frac{2\text{π}}{\text{T}}$. The assumption that the seasonal variation of POC flux follows a pattern of cosine functions is reasonable for most oceans except for coastal regions, tropical oceans, and high-latitude areas (Supplementary Data Sheet 1 - Text S4). By combining Equations 8, 9, and 11, SS can be obtained as:

$\begin{array}{ll}\text{SS}(\text{t})={\text{a}}_0-{\text{λce}}^{-\text{λt}}-{\text{a}}_1\text{A}\sin (\text{ωt}+\text{φ})& (12)\end{array}$

where c is a constant determined by the initial condition of ²³⁴Th, $\text{φ}={\tan }^{-1}\frac{\text{λ}}{\text{ω}}$, and $\text{A}=\frac{\text{λ}}{\sqrt{{\text{λ}}^2+{\text{ω}}^2}}$. The second term on the right-hand side of Equation 12 represents the decay of the initial ²³⁴Th, which varies slightly during the bloom period compared with its initial state. The variation in SS is mainly controlled by the third term, which specifies the influence of particle sinking on the ²³⁴Th flux.

The duration of phytoplankton blooms varies from subtropical to subpolar gyres (160–180 days and 120–125 days in the subtropical and subpolar gyres, respectively; Sapiano et al., 2012). a₀ and a₁ generally change due to the variation in primary production in global oceans (Le Moigne et al., 2013). Using observational data collected by Le Moigne et al. (2013), we constructed the idealized patterns of SS and Sinking during phytoplankton blooms for subtropical gyres (case 1 in Figure 6a: T=180 days, a₀ = 28.5 mg C/m²d and a₁ = 22.5 mg C/m²d) and subpolar gyres (case 2 in Figure 6b: T=120 days, a₀ = 73.95 mg C/m²d and a₁ = 48.25 mg C/m²d). Detailed mathematical formulas for cases 1 and 2 are shown in Supplementary Data Sheet 1 - Text S5. The amplitude of the seasonal cycle in SS was a₁A, attenuated by a factor of A compared with the amplitude in Sinking (a₁) of Equation 11. The time lag between SS and Sinking was quantified as ${\text{ΔT}}_{\text{lag}}=\frac{\frac{\text{π}}{2}-\text{φ}}{\text{ω}}$, with ΔT_lag=25.3 days for case 1 and ΔT_lag=20.4 days for case 2. The time lag calculation suggested that a longer duration of phytoplankton blooms yielded a larger time lag, consistent with the simulations in Ceballos-Romero et al. (2018).

Figure 6

Figure 6. (a, b) Temporal variations in POC export flux derived from (blue) Sinking and (red) SS, (cyan) modified SS (SS_cor), and (orange) △SS between SS and SS_cor for cases 1 and 2. The shaded error bars show the 10% uncertainty for each curve. (c, d) The relative errors of (blue) SS, (green) SS_cor, and (red) △SS to Sinking for cases 1 and 2, respectively.

3.4 Reducing the bias in SS

The analysis of the phytoplankton blooms in cases 1 and 2 suggests that the bias between Sinking and the SS-derived flux was largely attributed to their phase difference over time. If we assume that there is a modified SS (SS_cor), which is in phase with Sinking (Equation 11), its bias with Sinking would be reduced (Equation 13).

$\begin{array}{ll}{\text{SS}}_{\text{cor}}={\text{a}}_0-{\text{λce}}^{-\text{λt}}-{\text{a}}_1\text{A}\cos \text{ωt}& (13)\end{array}$

Then, the difference between SS and SS_cor is given by:

$\begin{array}{ll}\text{ΔSS}=\text{SS}-{\text{SS}}_{\text{cor}}={\text{a}}_1\text{A}[\cos \text{ωt}-\sin (\text{ωt}+\text{φ})]& (14)\end{array}$

At time t₁= $\frac{\frac{\text{π}}{2}-\text{φ}}{2\text{ω}}$. and t₂= $\frac{\frac{5\text{π}}{2}-\text{φ}}{2\text{ω}}$, ΔSS was zero (t₁ = 12.6 and t₂ = 102.6 in case 1; t₁ = 10 and t₂ = 70 in case 2; Figure 6). From the initial to t₁ and from t₂ to the time that SS reached its peak, the relative error of SS was smaller than that of SS_cor. Except for these two periods, the relative errors were reduced for SS_cor compared to SS (Figures 6c, d). The reduction in relative errors (ΔSS/P) was smaller in the pre-bloom period than in the post-bloom period (on average 25% vs. 64% in case 1 and 17.4% vs. 42.8% in case 2). We hereby suggest a modification for SS in each sampling by subtracting ΔSS (hereafter noted as the modification term).

According to Equation 14, a modification term can be estimated for each in situ sampling given that the export duration (T) and the amplitude of the seasonal POC flux (a₁) are obtainable. To obtain applicable timing information for the export, we utilized satellite data. Time series of satellite-derived total chlorophyll concentrations have mostly been used to estimate phytoplankton blooms (Stange et al., 2017; Ceballos-Romero et al., 2018), while a time lag between the peak in total chlorophyll and POC export has also been reported (Henson et al., 2015; Stange et al., 2017; Ceballos-Romero et al., 2018). The time lag between phytoplankton blooms and POC export was attributed to multiple factors (Henson et al., 2015, 2019). One reason for the time lag was that POC sinking was more related to large phytoplankton, which was not in phase with total phytoplankton (Villa-Alfageme et al., 2016; Giering et al., 2020). We used different satellite data to estimate the timing information of export, including satellite-derived chlorophyll for micro-sized phytoplankton and satellite-derived export fluxes from Henson et al. (2011) and Laws et al. (2011) (Supplementary Data Sheet- Text S2). We derived the amplitude of the POC export flux (a₁) from the historical range of the maximum and minimum POC fluxes from local sediment traps.

The method was first tested based on realistic simulation results at SEATS. The duration of the phytoplankton bloom estimated from the satellite-derived micro-sized phytoplankton chlorophyll was ~150 days (Supplementary Figure S2a; Palacz et al., 2011; Zhou et al., 2020). The time series of SS_cor was generally in phase with Sinking (Supplementary Figure S2a) and was better fitted to Sinking than SS (Supplementary Figure S2b). At times t₁ = 11.5 days and t₂ = 86.5 days, ΔSS≈0. Consistent with cases 1 and 2, the relative errors of SS_cor were higher than those of the SS from the initial time to t₁ and were reduced during the pre- and post-bloom periods, with average reductions of 29% and 48.7%, respectively (Supplementary Figure S2b).

3.5 Applications of the method with field measurements

We applied the method to a different location (BATS) for verification, as BATS has a long-term record of sediment trap deployment and a ²³⁴Th dataset (Le Moigne et al., 2013; Puigcorbé et al., 2020). The SS-derived POC fluxes from Buesseler et al. (2008) and Stewart et al. (2011), were corrected in different cruises and compared with trap-derived fluxes at 150 m (Table 1). Time differences between cruise and trap collection dates range from 1 to 22 days for 5 points where direct trap data were available nearby and require linear interpolation (23 February to 16 August 2004) for 4 points lacking concurrent data (see Supplementary Table S3 and Supplementary Data Sheet 1 - Text S6 in Supplementary Material for details). The POC flux from the sediment trap was generally in phase with satellite-derived micro-sized phytoplankton chlorophyll during 2003–2007 (Figure 7), suggesting a well-established yearly cycle of phytoplankton blooms (Martin and Pondaven, 2006) and carbon export (Brix et al., 2006). For the data at BATS (Table 1; Buesseler et al., 2008; Stewart et al., 2011), those samples during the post-bloom period were mainly overestimated due to the temporal bias between the sediment trap and SS (Savoye et al., 2006; Ceballos-Romero et al., 2018). Estimating the $\frac{{\partial }^{234}\text{Th}}{\partial \text{t}}$ term shows that $\frac{{\partial }^{234}\text{Th}}{\partial \text{t}}$ is ~1,993 dpm/m²d (Supplementary Table S4), suggesting a potentially large temporal bias for the SS-derived POC flux. Applying the modification term to the SS-derived POC flux during the post-bloom periods can reduce the relative errors except for one data point sampled on 2 August 2004 (Figure 7; Table 1). For 25 and 28 June 2004, where trap fluxes were interpolated over the 174-day span, improvements should be interpreted cautiously, as the interpolation may not capture short-term POC flux variations. There were two data points available in the pre-bloom period (sampled on 8 November 2006 and 29 January 2007), and the relative error of one data point was reduced, while the other one was not. Unlike an underestimation predicted by the SS bias, the failed bias had already been overestimated, which was probably due to an increase in thorium scavenging by the increase in inorganic particles from the sea surface (Kim et al., 1999; Tian et al., 2008) or influence from episodic events—such as eddy-driven particle pulses (Buesseler et al., 2008). Using the satellite-derived micro-sized phytoplankton and satellite-derived export flux for timing estimation produced similar results, both were capable of reducing SS bias. The reduction ranged from 8.21% to 67.17%, with an average of 40.43% (Table 1).

Table 1

Table 1. The ²³⁴Th-derived POC flux and comparison to the sediment trap-estimated POC flux at BATS.

Figure 7

Figure 7. POC export flux derived from (purple) SS and (cyan) SS_cor for data sampled using the ²³⁴Th technique at BATS during 2003–2007. Black and orange lines are POC flux from sediment traps collected at BATS and satellite-derived chlorophyll concentration of micro-sized phytoplankton during September 2003–December 2007, respectively.

This modification term was also applied at SEATS with two simultaneous ²³⁴Th and trap data. For the data collected on 8 January 2007, the correction had minimal impact, as the SS-derived flux already closely matched the sediment trap data (Supplementary Table S5). For the data from 12 September 2009, although the modified SS still underestimated the flux, the method reduced the relative bias in the SS-derived POC flux by 34.19%, with an average improvement across the 2 points of 25.20% (Supplementary Table S5).

4 Discussion and conclusions

The scarcity of continuous ²³⁴Th profiles and sediment trap data has limited investigation into seasonal discrepancies between SS-derived and direct sinking POC fluxes. To address this, we developed the ROMS-CoSiNE-²³⁴Th model to simulate ²³⁴Th profiles and POC flux dynamics at SEATS. The simulations reveal that only 37.8% of SS-derived POC fluxes align with direct sinking fluxes within a 10% margin due to inherent temporal biases. Our model outputs, analyzed with the simplified ²³⁴Th continuity equation, suggest that this bias stems from phase differences between SS-derived and direct fluxes. Solving the simplified ²³⁴Th continuity equation analytically, we propose a correction method that mitigates bias without requiring repeated sampling. This approach, based on a cosine perturbation reflective of phytoplankton bloom cycles, is well-suited to regions with stable bloom patterns, such as subtropical gyres. However, it requires caution in areas with complex particle flux dynamics, like coastal, frontal, or high-latitude regions (Supplementary Data Sheet 1 - Text S4).

Our correction method effectively reduces systematic seasonal biases in SS-derived POC fluxes in mid- to low-latitude regions like SEATS and BATS, where POC export follows a regular, periodic pattern approximated by a cosine function. This assumption is supported by our analysis of satellite-derived POC flux data from 2003 to 2020 (Supplementary Data Sheet 1 - Text S4), which shows a mean relative difference (MRD) of <20% between the cosine fit and observed fluxes in these regions (Supplementary Figures S5a–c). However, the method’s performance is constrained under several conditions. In high-latitude, coastal, or frontal zones, POC export deviates from the assumed regular seasonal cycle—for instance, due to abrupt spring blooms or ice-edge upwelling (Ardyna et al., 2020)—increasing the MRD in the cosine fit (Supplementary Figure S5) and undermining the method’s foundational assumption. Likewise, in regions with strong currents, such as the equatorial Pacific and Southern Oceans, lateral advection can become a dominant term in the ²³⁴Th continuity equation, overshadowing the vertical sinking and decay processes central to our method. Moreover, mesoscale and submesoscale processes introduce significant spatial and temporal variability in POC flux (Buesseler et al., 2008; Stukel et al., 2017; Buesseler et al., 2020). For example, observations near the Hawaii Ocean Time-Series (HOTS) site revealed spatial variability exceeding seasonal variability (Benitez-Nelson et al., 2001), and Resplandy et al. (2012) found that small-scale spatial variability in ²³⁴Th activity (270–550 dpm m⁻³ over ~100 km) can lead to errors in SS-derived fluxes, particularly when physical transport by eddies introduces variability that may exceed seasonal biases in dynamically active regions. Furthermore, measurement errors in the POC:²³⁴Th ratio, which varies widely due to factors like particle size, composition, or aggregation–disaggregation processes (Le Moigne et al., 2013; Puigcorbé et al., 2020), can dominate over the seasonal biases our method corrects. For instance, differences in sampling techniques can yield POC:²³⁴Th ratios varying by a factor of 2–4 (Buesseler et al., 2006), potentially masking the systematic errors we address. Although our model does not require a constant POC:²³⁴Th ratio—applying a variable, accurate ratio to both direct sinking and SS-derived fluxes—such measurement inaccuracies often render the seasonal correction less meaningful when they overshadow the targeted biases. As noted by Resplandy et al. (2012) and Ceballos-Romero et al. (2018), uncertainties from small-scale dynamics cannot be easily separated from SS or NSS biases, further complicating flux estimates in these scenarios. Consequently, while our method excels in stable, periodic mid- to low-latitude environments, its utility is limited where irregular export patterns, dominant physical transport, or substantial POC:²³⁴Th measurement errors prevail. Future enhancements could include three-dimensional modeling to account for lateral advection and improved POC:²³⁴Th ratio measurement techniques to reduce uncertainties in variable conditions.

The correction relies on estimates of seasonal POC flux amplitude and bloom duration, obtainable from satellite chlorophyll or in situ data, and proves most effective post-bloom, where it reduces errors more significantly than in pre-bloom periods. In summary, this study demonstrates that the ROMS-CoSiNE-²³⁴Th model accurately simulates ²³⁴Th dynamics at SEATS, identifying phase differences as the primary source of temporal bias in SS-derived POC fluxes. Our analytical correction method successfully reduces this bias by up to 67% at BATS and 34% at SEATS, offering a practical tool for regions with regular bloom cycles without the need for extensive sampling. These findings enhance our understanding of biases in ²³⁴Th-based POC flux estimates, a key component of the ocean’s biological carbon pump. By improving flux accuracy in stable mid- to low-latitude environments, this work contributes to refining global carbon cycle models and supports climate change research through better quantification of carbon sequestration, although its application requires careful consideration of regional dynamics and measurement precision.

Data availability statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/Supplementary Material.

Author contributions

MG: Formal Analysis, Methodology, Writing – original draft, Writing – review & editing. PX: Funding acquisition, Methodology, Project administration, Writing – original draft, Writing – review & editing. KZ: Data curation, Investigation, Writing – review & editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. This study is supported by the National Natural Science Foundation of China (92258302), the National Key Research and Development Program of China (2022YFE0136600 and 2022YFC3105303), and the National Natural Science Foundation of China (42430401 and 42376154).

Acknowledgments

The numerical simulation is supported by the High Performance Computing Division of the South China Sea Institute of Oceanology.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fmars.2025.1554932/full#supplementary-material

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