# An Inverse Modelling Approach to Constrain Be Cycling in the Subpolar North Atlantic

Paul Lerner, Mélanie Grenier, Olivier Marchal, Pieter van Beek

► To cite this version:

Paul Lerner, Mélanie Grenier, Olivier Marchal, Pieter van Beek. An Inverse Modelling Approach to Constrain Be Cycling in the Subpolar North Atlantic. Deep Sea Research Part I: Oceanographic Research Papers, 2025, pp.104465. 10.1016/j.dsr.2025.104465. hal-04956188

HAL Id: hal-04956188

https://hal.science/hal-04956188v1

Submitted on 19 Feb 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Journal Pre-proof

An Inverse Modelling Approach to Constrain 7 Be Cycling in the Subpolar North Atlantic

Paul Lerner, Mélanie Grenier, Olivier Marchal, Pieter van Beek

PII: S0967-0637(25)00024-X

DOI: https://doi.org/10.1016/j.dsr.2025.104465

Reference: DSRI 104465

To appear in: *Deep-Sea Research Part I*

Received Date: 4 September 2024

Revised Date: 9 February 2025

Accepted Date: 12 February 2025

<figure><img src="image_2.png" /></figure>

Please cite this article as: Lerner, P., Grenier, M., Marchal, O., van Beek, P., An Inverse Modelling Approach to Constrain 7 Be Cycling in the Subpolar North Atlantic, *Deep-Sea Research Part I*, https://doi.org/10.1016/j.dsr.2025.104465.

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2025 Published by Elsevier Ltd.
1
2

# An Inverse Modelling Approach to Constrain ⁷Be Cycling in the Subpolar North Atlantic

3

Paul Lernerª

4

Mélanie Grenierᵇ

5

Olivier Marchalᶜ

6

Pieter van Beekᵇ

7

a. Department of Applied Physics and Applied Mathematics, Columbia University, New York City, NY, 10027, USA

8

b. LEGOS, University of Toulouse, CNRS, CNES, IRD, Toulouse, 31400, France

9

c. Woods Hole Oceanographic Institution, Woods Hole, MA, 02543, USA

10

## Abstract

Beryllium-7 is a short-lived cosmogenic radionuclide that has been used as a tracer of atmospheric deposition at the sea surface and of physical processes in the upper ocean. These applications generally assume that (i) the fraction of marine ⁷Be in particulate form is negligible, and/or (ii) the interactions between the particulate and dissolved forms of ⁷Be in seawater can be neglected. In this study, we test different steady-state models of upper ocean ⁷Be cycling from measurements of total ⁷Be and particulate ⁷Be activities collected at two stations of the GEOVIDE cruise in the subpolar North Atlantic (May-June 2014). The most complete model includes vertical advection, vertical diffusion, gravitational settling, radioactive decay, atmospheric deposition, and the reversible exchange between dissolved ⁷Be (⁷Be_d) and particulate ⁷Be (⁷Be_p). This model reproduces the measured ⁷Be activities at both stations to within their uncertainties (± 1 standard deviation). In the East Greenland-Irminger Current (station 51/60), models that do not consider adsorption and/or desorption can still reproduce the measured activities, while in the southern Labrador Sea (station 69) models that neglect reversible exchange or desorption poorly fit the data. Thus, ⁷Be_d at station 51/60 could have been supplied entirely by surface deposition, whereas ⁷Be_d at station 69 originated at least partly from the release of particulate ⁷Be into solution. The subsurface ⁷Be_p maxima, present at both stations, seem to require a flux of ⁷Be between particles and solution at station 69 but not at station 51/60. At both stations, most of the total ⁷Be deposited at the ocean surface was removed by radioactive decay, with at most 5% removed by sinking of ⁷Be_p, suggesting that reversible exchange can be neglected in applications of total ⁷Be as a tracer for atmospheric deposition. However, reversible exchange is important to consider in applications of ⁷Be_d as a deposition tracer, with a maximum bias of 40% when reversible exchange is neglected. The steady state assumption does not alter our results regarding the solid-solution exchange of ⁷Be, but it does result in a significant bias when deriving atmospheric ⁷Be_{tot} deposition fluxes from water column ⁷Be_{tot} inventories. Overall, our findings
suggest that reversible exchange could significantly influence the oceanic cycling of ⁷Be at some locations, and should not be systematically neglected when using ⁷Be_d as an oceanic tracer.

## 1. Introduction

Beryllium-7 is a cosmogenic radionuclide with a half-life ($\tau_{1/2}$) of 53.2 days (Nir-El et al., 2007). It is produced by cosmic ray spallation of oxygen and nitrogen nuclei in the stratosphere and troposphere (Lal and Peters 1967). ⁷Be is subsequently scavenged by submicron particles, with median aerodynamic diameters ranging from 0.58 to 1.22 µm (Papastefanou and Ioannidou 1996; Ioannidou 2011). ⁷Be is then transported by these aerosols in the atmosphere and ultimately enters the ocean through a combination of wet and dry deposition processes, though wet deposition is thought to be the dominant oceanic source (Brost et al., 1991).

The half-life of ⁷Be of ~ 2 months, in combination with its atmospheric source, has resulted in the use of ⁷Be as a tracer of surface atmospheric deposition of other trace elements in the open ocean (Kadko et al., 2015; Shelley et al., 2017; Kadko et al., 2019). Such an approach assumes that ⁷Be activities measured in the water column are at steady state with atmospheric fluxes, and that ⁷Be is soluble in seawater, so that ⁷Be losses due to particle scavenging and sinking are negligible. These assumptions are appropriate for low-particle environments (Kadko et al., 2015), but not for coastal and estuarine environments where ⁷Be has been shown to be highly particle reactive (Aaboe et al., 1981). Other applications of ⁷Be, such as the estimation of vertical turbulent diffusivity (Silker 1972b) or upwelling velocity in the thermocline (Kadko and Johns, 2011; Kadko 2017), rely on similar assumptions.

The assumption that ⁷Be is strongly soluble in open-ocean environments is based on evidence that ⁷Be bound to particulate matter represents less than 10% of total (dissolved + particulate) ⁷Be in these environments (Silker 1972a; Andrews et al., 2008). A common measure of the partitioning of a particle-reactive radionuclide between the particulate and dissolved phases is the distribution coefficient, $K_d$, defined as the activity of the radionuclide per mass of particles (specific activity, in dpm g⁻¹, dpm: disintegrations per minute) divided by the dissolved radionuclide activity per volume of water (dpm m⁻³). The distribution coefficients for beryllium radioisotopes estimated in previous studies correspond to $logK_d$ ranging from 3 to 6-7, where log is the logarithm in base 10, and are generally smaller than estimates for thorium radioisotopes, with $logK_d$ ranging from 4.17 to 7.08 (Chase et al., 2002; Luo and Ku, 2004; Chuang et al., 2013, 2015). Based on these studies, it is often assumed that ⁷Be scavenging by reversible exchange (⁷Be adsorption onto particles, ⁷Be desorption from particles, and the sinking of ⁷Be_p on particles) does not significantly influence the activities of total or dissolved ⁷Be in the upper ocean, so that it can be neglected in applications of ⁷Be as a tracer of atmospheric deposition (e.g., Kadko et al., 2015) or oceanic vertical transport (Silker 1972b; Kadko and Johns, 2011; Kadko 2017).

Prior studies on the importance of chemical scavenging for Be relied either on (i) measurements of radionuclide activity on particles caught by sediment traps and co-located with (but temporally separated from) measurements of total radionuclide activity (Chase et al., 2002; Luo and Ku, 2004; Chuang et al., 2013), or (ii) laboratory experiments where ⁷Be was added to seawater
samples collected in the Gulf of Mexico (Chuang et al., 2015). However, simultaneous
measurements of total (or dissolved) ⁷Be, particulate ⁷Be, and suspended particle matter
concentration in the open ocean are rare with only a few exceptions (Kremenchutskii et al., 2021;
Grenier et al., 2023). Size-fractionated measurements are needed to better constrain the ⁷Be
cycle in the ocean and to address the following questions: what is the fate of ⁷Be deposited at
the ocean surface? To what extent is reversible exchange responsible for the partitioning of ⁷Be
between the dissolved and particulate fractions in the upper water column? How best can ⁷Be
be used as a tracer for atmospheric deposition, and when used as such a tracer, can ⁷Be bound
to particles be neglected? The analysis of ⁷Be in different particulate fractions collected at the
same site and at the same time would allow one to address these important questions.

In a companion paper, we showed from paired measurements of total (dissolved + particulate) ⁷Be activities (⁷Be<sub>tot</sub>) and particulate ⁷Be activities (⁷Be<sub>p</sub>) that the ⁷Be<sub>p</sub> inventory range from 5 to 19% of the ⁷Be<sub>tot</sub> inventory at two stations of the GEOVIDE cruise in the subpolar North Atlantic (Grenier et al., 2023). The distribution coefficient for ⁷Be was found to increase with depth, with logK<sub>d</sub> averaging 5.2 ± 0.1 at 5 m and 6.1 ± 0.1 between 70 and 150 m, suggesting that particle scavenging of dissolved ⁷Be (⁷Be<sub>d</sub>) is more intense in thermocline waters than in the mixed layer when differences in particle concentration are considered.

In this study, we use the paired measurements of ⁷Be<sub>tot</sub> and ⁷Be<sub>p</sub> activities at the two GEOVIDE
stations (Grenier et al., 2023) to constrain aspects of the ⁷Be cycle in the upper water column in
the subpolar North Atlantic. The rate parameters of a 1-dimensional (vertical) steady-state model
describing the cycling of dissolved and particulate ⁷Be are estimated from a fit of the model to
the paired ⁷Be activity data from these stations. Different versions of the model, in combination
with an examination of the ⁷Be<sub>d,p</sub> budgets and ancillary particle composition data at the same
stations, are then used to test the significance of particle scavenging for ⁷Be cycling and to
evaluate the use of ⁷Be in the estimation of atmospheric deposition rates of trace elements and
isotopes (TEIs) at the sea surface.

This paper is organized as follows. In section 2, we describe the data used in this study, the
hydrographic setting and mean circulation at both GEOVIDE stations, the ⁷Be cycling model, and
the inverse method used to fit the ⁷Be cycling model to the ⁷Be data. Multiple inversions are
performed to cover a broad range of assumptions regarding the hydrographic setting and the
potential biases associated with the ⁷Be measurements. The rate parameters and deposition
fluxes of ⁷Be estimated from the inversion of ⁷Be data are reported in section 3. In section 4, we
compare our estimates of the rate parameters and of the deposition fluxes to estimates from
prior studies (section 4.1). We determine (i) whether reversible exchange is necessary to explain
the paired measurements of ⁷Be<sub>tot</sub> and ⁷Be<sub>p</sub> at the two GEOVIDE stations, and (ii) whether the
total and dissolved ⁷Be inventories could be used to estimate the deposition of ⁷Be at the sea
surface. Conclusions follow in section 5.
## 2. Methods

In this section, we detail (i) the data used in this study, (ii) the oceanic conditions at the two investigated stations, (iii) the ⁷Be cycling model, and (iv) the inverse method used to combine the data with the model

### 2.1 Data and Prior Estimates

We use data from two stations occupied in the subpolar North Atlantic in the framework of the GEOVIDE section (GA01) that was completed between May 15, 2014 and June 30, 2014 as part of the GEOTRACES program (Sarthou et al., 2018): station 51/60 along the continental slope of southeast Greenland, and station 69 along the southern edge of the Labrador Sea (Fig. 1). Station 51/60 combines station 51, where samples dedicated to particulate ⁷Be analysis were collected, and station 60, where samples for total ⁷Be analysis were collected two days after the station 51 occupation and at the same location (Shelley et al., 2017; Grenier et al., 2023).

**Total ⁷Be:** The collection and analysis of ⁷Be$_{tot}$ samples are described in detail by Shelley et al. (2017). Briefly, ⁷Be$_{tot}$ was measured by collecting seawater (50-100 L) using standard Niskin bottles at three to four depths between 0 and 150 m. The unfiltered seawater samples were acidified to a pH of 1, spiked with a stable Be tracer, introduced with an Fe³⁺ carrier, and allowed to equilibrate for 12 hrs. Beryllium was co-precipitated with iron-hydroxides by adjusting the pH to ~ 8.5. Gamma counting of co-precipitated ⁷Be was performed at the Universitat Autònoma de Barcelona or at the LAFARA underground laboratory in the French Pyrénées.

**Particulate ⁷Be:** The collection and analysis of ⁷Be$_p$ samples are detailed by Grenier et al. (2023). The ⁷Be$_p$ samples were collected using different filter types: (i) 1-µm pore size QMA filters or 0.8-µm pore size Supor filters, mounted on McLane large volume filtration pumps and (ii) 1-µm pore size pocket filters (sock filters) used to filter surface waters. Filter samples were analyzed two weeks to four months after arrival in the laboratory using low-background gamma-ray spectrometers at LAFARA and at Modane (another underground laboratory) in the French Alps. Differences in ⁷Be$_p$ activities associated with the use of different filter types were investigated from particulate ⁷Be$_p$ measurements on surface samples collected at a station offshore of the French coast of the Mediterranean Sea (Grenier et al., 2023). ⁷Be$_p$ activity was found to be 1.4 times lower on sock filters than on Supor filters, and 2.1 times higher on QMA filters than on Supor filters. Using Supor filters as a reference, Grenier et al. (2023) multiplied all ⁷Be$_p$ activities measured on sock filters by 1.4 and divided all ⁷Be$_p$ activities measured on QMA filters by 2.1. In this study, we rely on the “corrected” activities of ⁷Be$_p$. Finally, ⁷Be activities in the dissolved phase (⁷Be$_d$) are calculated by subtracting particulate activities (⁷Be$_p$) from total activities (⁷Be$_{tot}$).

**Hydrographic Data:** We use temperature, salinity, and pressure data from Conductivity-Temperature-Depth measurements (CTD SBE-911) at both stations (Garcia-Ibanez et al., 2018). Data from two different CTD casts at station 51/60 are used to examine the sensitivity of our results to hydrographic conditions (see section 2.6).

**Particle concentration and composition:** The collection and analysis of “suspended” particulate matter (PM) are described in Grenier et al. (2023). Total PM concentration was calculated as the
sum of the concentrations of particulate organic carbon (POC), calcium carbonate, biogenic opal, lithogenic material, and iron and manganese (oxy)hydroxides, following the method developed by Lam et al. (2015). POC and biogenic opal analyses are described in Sarthou et al. (2018). Lithogenic material concentration was estimated based on the concentration of particulate aluminum, calcium carbonate was determined from particulate calcium (Grenier et al., 2023), and iron and manganese (oxy)hydroxides were estimated based on the concentrations of particulate Fe and Mn (Gourain et al., 2019). Samples for each component of PM were collected from the same Niskin/Go-Flo bottles.

**Adsorption and Desorption:** We use prior estimates of the second-order rate constant for ⁷Be adsorption ($k_{1,c}$) and of the first-order rate constant for ⁷Be desorption ($k_{-1}$) from sorption experiments on sediment from San Clement Basin (off San Diego, California, USA), Narragansett Bay (off the coast of Rhode Island, USA), and MANOP Site H (eastern equatorial Pacific; Nyffeler et al., 1984). The averages of $k_{1,c}$ and $k_{-1}$ from these experiments are used in this study. We set the uncertainties in these rate constants equal to the range in their respective experimental values (Table 1).

**Particulate ⁷Be Sinking Speed:** We obtain a prior estimate of the sinking speed of particles carrying ⁷Be ($w_p$) from two sources. First, Kremenchutskii et al. (2021) estimated $w_p$ of a ⁷Be$_{tot}$ cycling model that considers vertical advection, vertical diffusion, particle sinking, and radioactive decay. They applied this model to estimate $w_p$ from ⁷Be$_p$ data at several stations in the Black Sea. The $w_p$ estimates reported by these authors have the advantage of being constrained by ⁷Be$_p$ data but the disadvantage of not being based on data from the subpolar North Atlantic. Second, we consider $w_p$ estimates obtained from measurements of dissolved oxygen concentration and of particulate optical backscatter from BGC-Argo floats deployed in the Irminger Sea and the southern Labrador Sea from 2013 to 2016 (Wang et al., 2022). Results from these two studies are used to derive an estimate of $w_p$ for stations 51/60 and 69. Specifically eleven estimates from the Black Sea (Kremenchutskii et al., 2021) and five estimates from the subpolar North Atlantic (Wang et al., 2022) are used. We use the mean (2.3 m d⁻¹; Table 1) and standard deviation (2.1 m d⁻¹; Table 1) of these sixteen estimates. For comparison, the mean and standard deviation of $w_p$ from Kremenchutskii et al. (2021) are 1.1 ± 0.6 m d⁻¹, while those from Wang et al. (2022) are 4.9 ± 1.9 m d⁻¹.

**Surface Deposition of ⁷Be:** We use prior estimates of total ⁷Be deposition flux ($FBe_{tot}$) and of the fraction of total ⁷Be deposition flux that is in the dissolved phase (f). Our estimates of $FBe_{tot}$ are obtained from total deposition estimates derived from the measured inventory of ⁷Be$_{tot}$ in the upper 150 m at station 51/60 and in the upper 100 m at station 69 (Shelley et al., 2017). To obtain a prior value of f, we use an estimate of dry deposition from Shelley et al. (2017) obtained by multiplying the ⁷Be activity measured on an aerosol sample collected at station 69 by an assumed dry deposition velocity of 0.3 cm s⁻¹, which is at the upper limit of dry deposition velocities reported by Duce et al. (1991) for aerosols < 1 µm. Since aerosols were not collected at station 51/60, and because station 69 is the closest GEOVIDE station for which an estimate of dry ⁷Be deposition is available, we apply the dry deposition flux of Shelley et al. (2017) at station 69 to
obtain an estimate of *f* for both stations (see section 2.3). The uncertainties in the estimates of *FBe*<sub>tot</sub> and *f* are obtained from, respectively, (i) the interquartile range of <sup>7</sup>Be<sub>tot</sub> deposition fluxes in the Atlantic Ocean between 50°N and 60°N from the compilation by Zhang et al. (2021), and (ii) the interquartile range of the fractions of dry to total fallout of <sup>7</sup>Be in the same region, also compiled by Zhang et al. (2021). In our study, we assume that dry deposition is responsible for the deposition of particulate <sup>7</sup>Be in the size fraction sampled by the filters and that wet deposition is responsible for the deposition of dissolved <sup>7</sup>Be.

*Vertical Velocities:* We consider prior estimates of the vertical velocity (*w*) near stations 51/60 and 69 obtained from the fit of an ocean general circulation model to oceanic and atmospheric observations (ECCO v4 non-linear inverse solution; Forget et al., 2015). We linearly interpolate the *w* estimates from ECCO v4.4 for the month and year of the station occupations (June 2014) onto our model grid (see section 2.3). The uncertainties in *w* estimates at each grid level are set equal to the standard deviation of the ECCO v4 *w* estimates for the upper 200 m at each station.

## 2.2 Hydrographic Setting and Circulation

Station 51/60 (59.80°N, 42.01°E, 1722 m water depth; GEOVIDE Cruise Report 2014) is located along the continental slope southeast of Cape Farewell. It is situated within the western boundary current of the Irminger Gyre, where near-surface waters are advected by the East Greenland-Irminger Current (Fig. 1), a strong southward current with a speed of up to 20 cm/s (Våge et al., 2011; Daniault et al., 2016). The water masses at station 51/60, as evaluated from water property data, are dominated primarily by Irminger Subpolar Mode Water (IrSPMW; 0-1500 m depth) and secondarily by Subarctic Intermediate Water (SAIW, 0-300 m depth; García-Ibáñez et al., 2018).

Station 69 (55.84°N, 48.09°E, 3687 m water depth; GEOVIDE Cruise Report 2014) is situated at the southern boundary of the Labrador Sea. The station is located offshore of the Labrador Current. Shipboard profiles from a Lowered Acoustic Doppler Current Profiler deployed at the same longitude as station 69 but at 53°N between May and July from 1996 to 2001 have shown the presence of a near-surface southward-flowing current with velocities ~ 5 cm/s (Fischer et al., 2004). Near-surface waters at station 69 are composed almost entirely of Labrador Sea Water (LSW; García-Ibáñez et al., 2018). The LSW is formed by wintertime deep convection in both the Labrador Sea and, as supported by mooring and Argo-float hydrographic data, the Irminger Sea (Pickart et al., 2003; Jong and Steur 2016; Piron et al., 2016).

## 2.3 <sup>7</sup>Be Cycling Model

The domain of the <sup>7</sup>Be cycling model considered in this study extends from the sea surface (z = 0 m) to the depth of the deepest <sup>7</sup>Be<sub>p</sub> (or <sup>7</sup>Be<sub>tot</sub>) sample (z = 150 m at station 51/60 and z = 100 m at station 69). The water column in this depth interval is divided into different layers, with the layer boundaries positioned at the sea surface and at the depths of the <sup>7</sup>Be<sub>p</sub> and <sup>7</sup>Be<sub>tot</sub> samples. The number of layers is thus defined by the number of <sup>7</sup>Be<sub>p</sub> and <sup>7</sup>Be<sub>tot</sub> measurements at each station. For example, if <sup>7</sup>Be<sub>tot</sub> is measured at 5, 20, and 70 m, while <sup>7</sup>Be<sub>p</sub> is measured at 5, 20, 30,
and 70 m, then the layer boundaries are set at 5, 20, 30, and 70 m. Linear interpolation is used when activities are missing at layer boundaries (in the above example to provide $^{7}$Be$_{tot}$ at 30 m). Similarly, measurement uncertainties are linearly interpolated to obtain uncertainties for interpolated values. Moreover, it is assumed that $^{7}$Be$_{d,p}$ activities at the surface are equal to those at 5 m, the shallowest depth where $^{7}$Be measurements are available (Grenier et al., 2023).

Our model of $^{7}$Be cycling is based on, but extends, the model of Kadko and Johns (2011). In the model of Kadko and Johns (2011), the balance equation for total $^{7}$Be ($^{7}$Be$_{tot}$) includes vertical advection, vertical diffusion, and radioactive decay (atmospheric deposition is represented in the boundary conditions). We extend this model by considering separately the $^{7}$Be activities in the dissolved phase and in the particulate phase, and by including reversible exchange between these two phases, as for thorium radioisotopes (e.g., Nozaki et al., 1981; Bacon and Anderson 1982; Murnane 1994a; Murnane et al., 1994b; Lerner et al., 2017). The governing equations for $^{7}$Be$_{d}$ and $^{7}$Be$_{p}$ are, respectively,

$$ \frac{\partial^{7} \mathrm{Be}_{\mathrm{d}}}{\partial t}=-\lambda^{7} \mathrm{Be}_{\mathrm{d}}-\frac{\partial}{\partial z}\left(w^{7} \mathrm{Be}_{\mathrm{d}}\right)+\frac{\partial}{\partial z}\left(K \frac{\partial^{7} \mathrm{Be}_{\mathrm{d}}}{\partial z}\right)-k_{1}^{7} \mathrm{Be}_{\mathrm{d}}+k_{-1}^{7} \mathrm{Be}_{\mathrm{p}} \quad (1a) $$

$$ \frac{\partial^{7} \mathrm{Be}_{\mathrm{p}}}{\partial t}=-\lambda^{7} \mathrm{Be}_{\mathrm{p}}-\frac{\partial}{\partial z}\left([w+w_{\mathrm{p}}]^{7} \mathrm{Be}_{\mathrm{p}}\right)+\frac{\partial}{\partial z}\left(K \frac{\partial^{7} \mathrm{Be}_{\mathrm{p}}}{\partial z}\right)+k_{1}^{7} \mathrm{Be}_{\mathrm{d}}-k_{-1}^{7} \mathrm{Be}_{\mathrm{p}} \quad (1b) $$

where $\lambda$ is the radioactive decay constant for $^{7}$Be (d$^{-1}$), $w$ the vertical velocity (m d$^{-1}$), $K$ the vertical turbulent diffusivity (m$^2$ d$^{-1}$), $k_1$ the first-order adsorption rate constant (d$^{-1}$), $k_{-1}$ the first-order desorption rate constant (d$^{-1}$), and $w_p$ the sinking speed of particles carrying $^{7}$Be$_p$ (m d$^{-1}$). Clearly $t$ is time and $z$ is depth. In equations (1a)-(1b), the terms on the right-hand side represent the effects of, respectively, radioactive decay, vertical advection (plus particle sinking for $^{7}$Be$_p$), vertical diffusion, adsorption onto particles, and desorption from particles. The effects from horizontal transport are neglected. Unless stated otherwise, we further assume that the $^{7}$Be activities are at steady state, i.e., the left-hand side of equations (1a)-(1b) are omitted. Our model is thus a steady state, 1-dimensional (vertical) model. On the other hand, the values of $K$, $k_{1,c}$, $k_{-1}$, $w_p$, and $w$ are all allowed to vary with depth in our model.

The first-order rate constant for $^{7}$Be$_d$ adsorption onto particles is set to vary with total PM concentration (C, see section 2.1):

$$ k_1 = k_{1,c} C \quad (2) $$

Here, $k_{1,c}$ is a second-order adsorption rate constant (L µg$^{-1}$ d$^{-1}$), i.e., $k_{1,c}$ is the rate of adsorption of $^{7}$Be onto particles per unit PM concentration and per unit $^{7}$Be$_d$ activity. The parameterization (2) is consistent with the notion that the attachment of trace metals to particles increases with the number of surface sites available for adsorption (e.g., Honeyman et al., 1988). It has been
adopted in models describing the cycling of thorium and protactinium isotopes in the water column (Lerner et al., 2020; Chen et al., 2021), and is a particular case of the power-law formulation of Honeyman et al. (1988) when the exponent for C is set to 1 (for further discussion, see Lerner et al., 2017).

For this study, the steady-state forms of equations (1a)-(1b) are integrated from the top to the bottom of each layer, and the diffusive fluxes at the layers boundaries are approximated using finite differences:

$$ 
\begin{aligned}
269 \quad 0 &= -(\lambda + k_{1,i-\frac{1}{2}})^7 \mathrm{Be}_{d,i-\frac{1}{2}}(z_i - z_{i-1}) - w_i^7 \mathrm{Be}_{d,i} + w_{i-1}^7 \mathrm{Be}_{d,i-1} \\
270 \quad &\quad + (k_{-1,i-\frac{1}{2}})^7 \mathrm{Be}_{p,i-\frac{1}{2}}(z_i - z_{i-1}) - K_{i-1}(a_{i-1}^7 \mathrm{Be}_{d,i-2} + b_{i-1}^7 \mathrm{Be}_{d,i-1} + c_{i-1}^7 \mathrm{Be}_{d,i}) \\
271 \quad &\quad + K_i(a_i^7 \mathrm{Be}_{d,i-1} + b_i^7 \mathrm{Be}_{d,i} + c_i^7 \mathrm{Be}_{d,i+1}) \quad (3a)
\end{aligned}
$$

$$ 
\begin{aligned}
273 \quad 0 &= -(\lambda + k_{-1,i-\frac{1}{2}})^7 \mathrm{Be}_{p,i-\frac{1}{2}}(z_i - z_{i-1}) - (w_i + w_{p,i})^7 \mathrm{Be}_{p,i} + (w_{i-1} + w_{p,i-1})^7 \mathrm{Be}_{p,i-1} \\
274 \quad &\quad + (k_{1,i-\frac{1}{2}})^7 \mathrm{Be}_{d,i-\frac{1}{2}}(z_i - z_{i-1}) \\
275 \quad &\quad - K_{i-1}(a_{i-1}^7 \mathrm{Be}_{p,i-2} + b_{i-1}^7 \mathrm{Be}_{p,i-1} + c_{i-1}^7 \mathrm{Be}_{p,i})
\end{aligned}
$$

$$ 
\begin{aligned}
276 \quad &\quad + K_i(a_i^7 \mathrm{Be}_{p,i-1} + b_i^7 \mathrm{Be}_{p,i} + c_i^7 \mathrm{Be}_{p,i+1}) \quad (3b)
\end{aligned}
$$

where

$$ 
\begin{aligned}
279 \quad a_i &= - \frac{\Delta z_{i+1}}{(\Delta z_i + \Delta z_{i+1})\Delta z_i} \quad (4a)
\end{aligned}
$$

$$ 
\begin{aligned}
280 \quad b_i &= - \frac{\Delta z_i - \Delta z_{i+1}}{\Delta z_i \Delta z_{i+1}} \quad (4b)
\end{aligned}
$$

$$ 
\begin{aligned}
281 \quad c_i &= \frac{\Delta z_i}{(\Delta z_i + \Delta z_{i+1})\Delta z_{i+1}} \quad (4c)
\end{aligned}
$$

Here i is an index for the layer boundaries (i = 1, ..., N) and half-integer indices represent layer centers. The $^7\mathrm{Be}$ activities at a layer center are taken as the averages of the activities at the top and bottom boundaries of the layer, e.g., $^7\mathrm{Be}_{d,p,i-1/2} = (^7\mathrm{Be}_{d,p,i} + ^7\mathrm{Be}_{d,p,i-1})/2$. Note also that the rate constants $k_1$ and $k_{-1}$ are specified at the center of each layer, while parameters $w$ and $w_p$ are specified at layer boundaries. Figure 2 illustrates the $^7\mathrm{Be}_{tot,p}$ measurements and their interpolated values at stations 51/60 and 69.
The model is subjected to boundary conditions specified (i) at the upper boundary of the uppermost layer (i.e., at the sea surface) and (ii) below the lower boundary of the bottom-most layer (i.e., below the depth of the deepest available sample). At the sea surface, atmospheric deposition fluxes of dissolved and particulate $^7$Be are imposed. Respectively,

$$w \ ^7\text{Be}_d - K \frac{\partial \ ^7\text{Be}_d}{\partial z} = f F \text{Be}_{\text{tot}} \quad (5a)$$

$$(w + w_p) \ ^7\text{Be}_p - K \frac{\partial \ ^7\text{Be}_p}{\partial z} = (1 - f) F \text{Be}_{\text{tot}} \quad (5a)$$

Below the lower boundary of the bottom-most layer, the $^7$Be activities are specified:

$$\ ^7\text{Be}_d = \ ^7\text{Be}_{d,B} \quad (6a)$$

$$\ ^7\text{Be}_p = \ ^7\text{Be}_{p,B} \quad (6b)$$

Here $^7\text{Be}_{d,B}$ and $^7\text{Be}_{p,B}$ are, respectively, the dissolved and particulate activities at the depth $z_{N+1} = z_N + (z_N - z_{N-1}) = 2z_N - z_{N-1}$.

## 2.4 Vertical Turbulent Diffusion

The vertical turbulent diffusivities in the $^7$Be balance equations (3a)-(3b) are estimated using hydrographic data at stations 51/60 and 69. Specifically, $K$ is derived from the relationship suggested by Osborn (1980):

$$K = \Gamma \epsilon / N^2 \quad (7).$$

Here $N^2 = (-\frac{g}{\rho_0}) \frac{\partial \rho}{\partial z}$ is the square of the buoyancy frequency, $g = 9.81 \text{ m s}^{-2}$ is the acceleration due to gravity, $\rho$ is the in-situ density, $\rho_0$ is a reference density set equal to the density of surface water ($\rho_0 = 1027.44 \text{ kg/m}^3$ at station 51/60 and $1027.23 \text{ kg/m}^3$ at station 69), $\epsilon$ is the turbulent kinetic energy (TKE) dissipation rate ($\text{W kg}^{-1}$), and $\Gamma$ is the mixing efficiency (dimensionless). For $\Gamma$, we follow previous studies and take the upper bound of 0.2 suggested by Osborn (1980) (Forryan et al., 2012; Sun et al., 2013; Painter et al., 2014). A compilation of in-situ microstructure measurements and tracer-release experiments support a range of $0.1 < \Gamma < 0.3$, suggesting that $\Gamma = 0.2$ is a reasonable approximation (but subject to considerable variability; Gregg et al., 2018). In-situ density ($\rho$) is computed from CTD temperature, salinity, and pressure data at each station using the international thermodynamic equation of state for seawater (IOC, SCOR, and IAPSO, 2010; see their section 2.8). To reduce the influence of variability between CTD casts at a given station (due, e.g., to the passage of internal waves), we first average in-situ density values in 5-m depth increments (Fig. 3), and then use the density averages to compute $N^2$ with central differences (Fig. 4).

To our knowledge, microstructure velocity data along the GEOVIDE section are not available to estimate $\epsilon$. Instead, we use estimates of $\epsilon$ obtained from microstructure data in the Iceland and Irminger basins reported by Painter et al. (2014). To obtain these estimates, we approximate
323 visually the averages in the composite $\epsilon$ profile of their Figure 4. Estimates of $\epsilon$ are obtained in 324 three layers: the mixed layer, the transition layer (defined below), and the “deep layer”, defined 325 as the depth interval between the transition layer and the depth of the deepest $^7$Be 326 measurement. These three layers are chosen because between them the $\epsilon$ values, averaged 327 across all vertical profiles measured in the subpolar North Atlantic, display the greatest 328 differences in both order of magnitude and vertical structure (Painter et al., 2014; their Fig. 4).

329 The TKE dissipation rates for the mixed layer, the transition layer, and the deep layer are set to 330 the following values. The mixed layer depth is defined using the variable density threshold 331 algorithm of Boyer Montégut et al. (2004). The rate of TKE dissipation in the mixed layer is set to 332 $\epsilon = 10^{-7}$ W/kg (Painter et al., 2014; their Fig. 4).

333 The transition layer is defined from the $N^2$ profile (Johnston and Rudnick 2009; their Eq. 4). The 334 upper boundary of the transition layer is the depth where the following two criteria are met:

335 (i) the depth is shallower than the depth where $N^2$ is maximum.

336 (ii) the depth is the depth where $N^2 = 0.5[\max(N^2) + \text{mean}(N^2)]$ above the $N^2$ maximum.

337 where $\text{mean}(N^2)$ is the arithmetic average of $N^2$ within 20 m of the base of mixed layer. The lower 338 boundary of the transition layer is defined similarly, but criterion (i) is that the depth of the lower 339 boundary is greater than the depth where $N^2$ is maximum and criterion (ii) applies below the $N^2$ 340 maximum. We set $\epsilon$ in the transition layer to $10^{-8}$ W kg$^{-1}$ (Painter et al., 2014; their Fig. 4).

341 Finally, for the deep layer, $\epsilon$ is set to $10^{-9}$ W kg$^{-1}$ (Painter et al., 2014; their Fig. 4). Thus, in our 342 model, the rate of TKE dissipation decreases by two orders of magnitude from the mixed layer to 343 the deep layer, which is qualitatively consistent with the notion that the source of turbulence is 344 primarily located near the sea surface. The layer boundaries for each cast are shown in Figures 345 3-5.

346 The $K$ profiles resulting from the values of $\Gamma^-$, $N^2$, and $\epsilon$ which are described above are shown in 347 Figure 5. For all inversions (see section 2.6), unless stipulated otherwise, we use the $K$ profile 348 obtained from CTD data collected during cast 106 at station 51/60 and cast 142 at station 69. At 349 station 51/60, hydrographic data are available from cast 106, the cast closest in time to the time 350 of $^7$Be$_p$ sample collection, and cast 119, during which $^7$Be$_{tot}$ samples were collected (see Figs. S1- 351 S3 for differences in potential density, squared buoyancy frequency, and $K$ between these two 352 casts). We use hydrographic data from cast 106 and explore the sensitivity of our results to the 353 choice of cast in section 3. At station 69, cast 142 is selected because it was (i) closest in time to 354 the collection of $^7$Be$_p$ samples, and (ii) the cast during which $^7$Be$_{tot}$ samples were collected.

355 Note that when the model domain extends below the greatest depth at which an estimate of $K$ 356 is available from hydrographic data, we assume that $K$ is uniform between this depth and the 357 deepest layer boundary. For example, for cast 142 at station 69, the model domain extends to 358 100 m, but estimates of $K$ only extend to 62 m. For this cast, the values of $K$ at 70 and 100 m are 359 set equal to the deepest $K$ estimate derived from the CTD data, at 62 m.
## 2.5 Inverse Method

The $^{7}$Be balance equations (3a)-(3b) are fit to the $^{7}$Be$_{d,p}$ activities at the layer boundaries and at the depth $z_{N+1}$ at stations 51/60 and 69 using an inverse method. The $^{7}$Be$_{d,p}$ activities at these depths are constrained by activity measurements and are called “measured” values or “data” below. The fit is obtained by adjusting the rate constants $k_{1,c}$ and $k_{-1}$, the velocities $w$ and $w_p$, the atmospheric deposition of total $^{7}$Be ($FBe_{tot}$), the fraction of atmospheric $^{7}$Be deposition in the dissolved phase ($f$), and the measured $^{7}$Be$_{d,p}$ activities, from the prior estimates of all these variables and given the uncertainties of these estimates. Note that the vertical turbulent diffusivity, $K$, which is estimated from hydrographic data, is not adjusted (section 2.4), unless stated otherwise.

The inverse method used to fit the $^{7}$Be$_{d,p}$ balance equations (3a)-(3b) to the $^{7}$Be$_{d,p}$ data is briefly described below (for details see Lerner et al., 2017). Let $\mathbf{x}$ be a vector including the variables describing the state of the $^{7}$Be cycle at station 51/60 or 69 according to the model. The elements of $\mathbf{x}$ are the variables to be adjusted to fit the model equations to the data. They include (i) the $^{7}$Be activities in dissolved and particulate forms at the boundaries of each model layer and at $z_{N+1}$, (ii) the total $^{7}$Be deposition flux ($FBe_{tot}$) and its fraction in the dissolved phase ($f$), (iii) the rate constants $k_{1,c}$ and $k_{-1}$ in each layer, and (iv) the velocities $w$ and $w_p$ at the layer boundaries, at the investigated station. Note that $\mathbf{x}$ does not contain the $^{7}$Be$_{d,p}$ activities at the center of each model layer (Eqs. 3a-b): these are computed by averaging the activities at the upper and lower layer boundaries (see section 2.3). We seek a vector $\mathbf{x}$ that is consistent with the prior estimates of all the variables (i)-(iv), given their respective error statistics, while satisfying equations (3a)-(3b) perfectly. This vector is found at a minimum of the objective function

$$J(\mathbf{x}) = (\mathbf{x} - \mathbf{x}_0)'\mathbf{C}_0^{-1}(\mathbf{x} - \mathbf{x}_0) \quad (8)$$

subject to the constraint $f(\mathbf{x}) = 0$, where $f(\mathbf{x})$ is a vector that represents the right-hand side of equations (3a)-(3b). Here, $\mathbf{x}_0$ is a vector including the prior estimates of the elements in $\mathbf{x}$, $\mathbf{C}_0$ is the error covariance matrix for the prior estimates (the diagonal elements of $\mathbf{C}_0$ are the squared errors in the estimates and the off-diagonal elements of $\mathbf{C}_0$ are the covariances between the errors), and the prime designates the transpose. We impose that the variables in $\mathbf{x}$, which are to be estimated, should be positive, with one exception: no positivity constraint is placed on $w$, so that the solution allows for either downwelling or upwelling velocities (note that positive $w$ means downwelling according to our sign convention). Moreover, $f$ is constrained such that $0 \le f \le 1$.

The prior estimates in $\mathbf{x}_0$ and their error covariances in $\mathbf{C}_0$ are obtained as follows. The prior estimates of $^{7}$Be$_{d,p}$ are derived from the $^{7}$Be$_{tot}$ and $^{7}$Be$_p$ measurements at station 51/60 and station 69 (Figure 2). Prior estimates $k_{1,c}$, $k_{-1}$, $w_p$, $f$, and $FBe_{tot}$ are previous observational estimates of these variables in different oceanic environments (Table 1). Finally, the prior estimates of $w$ come from the data-constrained ocean circulation simulation, ECCOv4 (section 2.1; Table 1). The matrix $\mathbf{C}_0$ is taken as diagonal, given the difficulty to constrain error covariances. The diagonal elements of $\mathbf{C}_0$ are based on the uncertainties (errors) in the prior estimates, which are shown in Figure 2 and listed in Table 1.
Note that while measurements of $^{7}$Be$_{tot}$ and $^{7}$Be$_{p}$ are available at stations 51/60 and 69, the vector $\mathbf{x}_0$ and the matrix $\mathbf{C}_0$ require prior estimates of $^{7}$Be$_{d}$ and their errors, respectively. The prior estimate of $^{7}$Be$_{d}$ at a given depth is taken as the difference between $^{7}$Be$_{tot}$ and $^{7}$Be$_{p}$ at that depth, while the uncertainty in this estimate is determined by error propagation,

$$ \sigma(^{7}Be_{d}) = \sqrt{\sigma(^{7}Be_{tot})^{2} + \sigma(^{7}Be_{p})^{2}} \quad (9), $$

where $\sigma(^{7}Be_{tot})$ and $\sigma(^{7}Be_{p})$ are the errors in, respectively, the $^{7}$Be$_{tot}$ activity and the $^{7}$Be$_{p}$ activity at that depth.

## 2.6 Alternative Models

The model described in section 2.3 (called model BASE below) is relatively complete as it includes the processes of vertical advection, vertical mixing, particle sinking, radioactive decay, adsorption, and desorption. It can be used to derive estimates of rate parameters ($k_{1,c}$, $k_{-1}$, $w$, $w_p$) and deposition variables ($FBe_{tot}$ and $f$) consistent with the $^{7}$Be$_{tot,p}$ data and given their errors, but it cannot identify the processes that are necessary to describe these data. It is possible that simpler models, for example a model that neglects adsorption and desorption, is just as capable at fitting the data as the model BASE. To determine the extent to which sorption processes and particle sinking are necessary to explain the observed $^{7}$Be$_{tot,p}$ activities, we consider the following alternative models:

*Model NO\_ADS neglects the adsorption of $^{7}$Be$_{d}$ onto particles and thus considers dry deposition as the only source of $^{7}$Be$_{p}$ in the water column.*

*Model NO\_DES omits the desorption of $^{7}$Be$_{p}$ from particles and thus considers wet deposition as the only source of $^{7}$Be$_{d}$ in the water column.*

*Model NO\_REX omits both $^{7}$Be$_{d}$ adsorption and $^{7}$Be$_{p}$ desorption and thus considers that surface deposition is the only source of the radionuclide in the water column.*

*Model NO\_SINK neglects the sinking of $^{7}$Be$_{p}$, so that vertical transport of $^{7}$Be$_{p}$ proceeds only through advection and diffusion.*

## 3 Results

In this section, we present the results from the five different models described in sections 2.3 and 2.6. Two measures of the fit of the modeled $^{7}$Be$_{tot,p}$ activities to the measured $^{7}$Be$_{tot,p}$ activities are considered: the root mean square error and the bias. The root mean square error is defined as $RMSE = \sqrt{\sum(^{7}Be_{tot,p} - ^{7}Be_{tot,p,o})^{2}}/N$, where $^{7}$Be$_{tot,p,o}$ ($^{7}$Be$_{tot,p}$) is the prior (posterior) estimate of $^{7}$Be$_{tot,p}$ and $N$ is the number of estimates of total or particulate $^{7}$Be activities. The bias is defined as $B = \sum(^{7}Be_{tot,p} - ^{7}Be_{tot,p,o})/N$. The values of RMSE and B for model BASE and alternative models are listed in Table 3.
433 **3.1 Model BASE**

434 We find that model BASE, where all processes in Eqs. (3a)-(3b) are considered, can reproduce all
435 the observed ⁷Be<sub>tot,p</sub> activities within their respective uncertainties (± 1 standard deviation, STD)
436 both at station 51/60 and at station 69 (Fig. 6). The RMSE and the bias for both ⁷Be<sub>tot</sub> and ⁷Be<sub>p</sub>
437 are small (Table 2). In particular, the RMSE and bias for ⁷Be<sub>p</sub> are close in magnitude to the deepest
438 measured ⁷Be<sub>p</sub> activities, which are ~ 0 dpm m<sup>-3</sup> (Fig. 2b).

439 At station 51/60, the posterior estimates of k₁,c varies between 0.25 x 10⁻⁴ and 0.29 x 10⁻⁴ L µg⁻¹ d⁻¹ (Fig. 7a), with a maximum at 45 m (Fig. 2a). They exceed the prior estimate of (0.19 ± 0.36) x 10⁻⁴ L µg⁻¹ d⁻¹, but not significantly so. The desorption rate constant k₁ shows much larger values at 2.5 and 12.5 m (0.12 d⁻¹) than at 45 and 110 m (< 0.05 d⁻¹). The posterior values of k₁ are lower than the prior value of 0.17 ± 0.20 d⁻¹ but remain consistent with it given the prior uncertainty. The particle sinking speed wₚ is roughly uniform with depth, with values between 2.1 and 2.3 m d⁻¹, and is also consistent with the prior value of 2.3 ± 2.1 m d⁻¹. The vertical velocity w ranges from -1.5 to +0.3 m d⁻¹ and at each vertical level is consistent with its prior estimate (to within ± 1 STD of the prior value). The atmospheric deposition flux of ⁷Be<sub>tot</sub>, FBe<sub>tot</sub>, is 127 ± 49 dpm m⁻² d⁻¹ (estimate ± 1 STD), which is smaller than the prior estimate of 143 ± 82 dpm m⁻² d⁻¹ (Shelley et al., 2016), although these two values are not different at the 1 STD level. Finally, 80% of the total ⁷Be deposition at the surface ocean occurs in the dissolved phase (f = 0.8), consistent with prior knowledge (f = 0.87 ± 0.20).

452 Note that the results presented above for station 51/60 use diffusivity estimates based on
453 hydrographic data from cast 106. Using *K* estimates based on cast 119 data produces a similar fit
454 to the observed ⁷Be<sub>tot,p</sub> activities but some differences in the estimates of rate parameters and
455 of ⁷Be<sub>tot</sub> deposition (Figs. S4 and S5; Table S1): the estimate of k<sub>1,c</sub> at 12.5 m is larger by ~ 30%,
456 the estimate of k<sub>1</sub> is substantially reduced (by over a factor of 2) at 12.5 m and increased at 45
457 m, and the estimate of *w* at 20 m is positive and much smaller in magnitude (*w* = 0.09 m d<sup>-1</sup> for
458 cast 119 whereas *w* = -1.5 m d<sup>-1</sup> for cast 106). Finally, the deposition flux of ⁷Be<sub>tot</sub> is reduced by
459 13 dpm m<sup>-2</sup> d<sup>-1</sup>, while the fraction of this flux in the dissolved phase is the same. For this inversion,
460 as for BASE, the posterior estimates of the rate parameters and of the deposition fluxes are
461 indistinguishable from their prior estimates to within ± 1 STD.

462 At station 69, the posterior estimates of k₁,c range from 0.0 to 0.42 x 10⁻⁴ L µg⁻¹ d⁻¹. At this station,  
463 a maximum in k₁,c occurs at 65 m, 20 m deeper than at station 51/60 (Fig. 8a). The desorption  
464 rate constant k₁, in contrast, displays a maximum of 0.24 d⁻¹ at 30 m at station 69 (Fig. 8b).  
465 Contrary to station 51/60, the posterior estimates of w are all positive (downwelling) and  
466 confined to a narrower range of 0.25-0.5 m d⁻¹ (Fig. 8c). Finally, wₚ seems to present a bimodal  
467 distribution, with values of ~ 2.3 m d⁻¹ at 5, 30, 60, and 100 m, and values ~ 1 m d⁻¹ at 20 and 70  
468 m. The deposition flux FBe<sub>tot</sub> = 130 dpm m⁻² d⁻¹ is similar to the estimate at station 51/60, whereas  
469 the fraction of it that occurs in the dissolved phase f = 0.7 is smaller. For all rate parameters and  
470 deposition fluxes, the posterior estimates are within ± 1 STD of the prior values.

471 As described in section 2.4, our diffusivity estimates rely on ε averages in the Iceland and Irminger
472 basins (Painter et al., 2014). Based on the spread in the ε profiles shown in their Fig. 4, the
uncertainty of the $\epsilon$ average could be as large as an order of magnitude relative to the mean in all three layers (the mixed layer, the transition layer, and the deep layer). To test the influence of this uncertainty on our results, we perform two additional inversions at each station using again the BASE model, but now treating the diffusivities $K$ as unknowns in $x$. In these inversions, the prior estimates of $K$ are based on the layer-averages of $\epsilon$ as described in section 2.4, and the errors in these estimates are set by assuming a relative uncertainty of 100% or 1000%. We find that our results are largely unaltered: the posterior estimates of $K$ are similar to the prior estimates (Fig. S6), and the posterior estimates of the rate parameters and fluxes are within ± 1 STD of the values obtained from the inversion where the diffusivity values were imposed (not adjusted) (Table S2, Figs. S7 and S8).

## 3.2 Alternative Models

### 3.2.1 Station 51/60

At station 51/60, models NO_ADS, NO_DES, and NO_REX are all able to reproduce the $^7Be_{tot,p}$ data within their uncertainties (Table 2, Fig. 6a,b). This finding suggests that at this station, sorption processes are not needed to simultaneously explain the measured activities of $^7Be_{tot}$ and $^7Be_p$. Similarly, model NO_SINK can reproduce the $^7Be_{tot,p}$ data within their uncertainties. This other result suggests that particle sinking is also not necessary to explain the measured activities (Fig. 6a,b).

The $^7Be$ cycling parameters inferred from the fit of models NO_ADS, NO_DES, NO_REX, and NO_SINK are compared to those from the fit of model BASE. When adsorption is absent (NO_ADS), the $k_{-1}$ estimates are strongly reduced at all depths (~ 0 d⁻¹ below 5 m; Fig. 7b), and the estimates of $w$ and $w_p$ at 70 m increase and decrease, respectively, by a factor of ~ 2 (Fig. 7c,d). When desorption is removed (NO_DES), $k_{1,c}$ is estimated to be lower than for model BASE and range from 0.00 x 10⁻⁴ to 0.14 x 10⁻⁴ L µg⁻¹ d⁻¹, with the vanishing value at 12.5 and 110 m (Fig. 7a). In this case, the velocities $w$ and $w_p$ show comparatively smaller changes (Fig. 7c,d), with the exception of $w_p$ at 70 m, which is roughly half the value inferred from BASE (Fig. 7d). When both adsorption and desorption are removed (NO_REX), estimates of $w_p$ and $w$ are similar to model BASE estimates, except that $w_p$ at 70 m is less than half the value inferred from BASE, while $w$ at 70 m is almost double the value inferred from BASE (Fig. 7c,d). When particle sinking is omitted (NO_SINK), we find at 2.5 m a substantial decrease and increase of $k_{1,c}$ and $k_{-1}$, respectively, compared to BASE (Fig. 7a,b). We also find a substantial increase in $k_{1,c}$ at 45 m compared to BASE (Fig. 7a), and that the $w$ estimate at 20 m is roughly half of the estimate from BASE (Fig. 7c).

Models NO_ADS and NO_REX result in an increase of 11 dpm m⁻² d⁻¹ in the $^7Be_{tot}$ deposition flux, and in a decrease of 0.1 in the fraction of this flux in the dissolved phase, compared to BASE (Table 3). These results could be understood from the fact that in BASE, adsorption of $^7Be_d$ provides a source of $^7Be_p$ and a sink of $^7Be_d$ in the water column, so that when $^7Be_d$ adsorption is removed the flux of $^7Be_p$ from the atmosphere is increased and its dissolved fraction is decreased. Interestingly, we do not see a corresponding increase in the flux of $^7Be_d$ from the atmosphere
512 when desorption is removed (models NO_DES and NO_REX), likely because in BASE the
513 predominant source of $^7Be_d$ is already from the atmosphere.

514 In summary, the changes in the estimated values of the rate parameters generally act to
515 compensate the removal of specific process(es) in the alternative models. For example, when
516 removing adsorption, the desorption rate constant $k_{-1}$ correspondingly decreases, presumably to
517 compensate for the lack of a dissolved $^7Be$ sink and a particulate $^7Be$ source. Despite these
518 changes, the posterior estimates of the rate parameters in NO_ADS, NO_DES, and NO_REX are
519 all consistent (within ± 1 STD) with their prior estimates.

520 3.2.2 Station 69

521 At station 69, models NO_ADS and NO_SINK are able to fit the observed $^7Be_{tot,p}$ activities to within
522 ± 1 STD, as for BASE (Fig. 6c,d). However, models NO_DES and NO_REX are not able to fit these
523 observations, e.g., the $^7Be_{tot,p}$ activities at 5 and 20 m estimated by inversion are significantly
524 lower than the observed activities (Fig. 6c,d). These misfits result in a much larger RMSE for
525 $^7Be_{tot,p}$ compared to models NO_ADS, NO_SINK, and BASE (Table 2). They suggest that at station
526 69, desorption is necessary to simultaneously explain the measured $^7Be_{tot}$ and $^7Be_p$ activities.

527 We consider the rate parameters estimated from the two models that reproduce the $^7Be_{tot,p}$ data
528 (NO_ADS and NO_SINK). When adsorption is removed (NO_ADS), we find a substantial decrease
529 in $k_{-1}$ at 2.5, 12.5, and 25 m. Interestingly, and in contrast to the results at station 51/60, there
530 are three depths, 25, 45, and 85 m, where removing adsorption does not substantially change $k_-$
531, and at 65 m $k_{-1}$ is increased, rather than reduced, compared to BASE (Fig. 8b). Removing
532 adsorption also results in a large decrease in $w_p$ at 20 m, and a large increase in $w_p$ at 70 m (Fig.
533 8d). Removing particulate $^7Be$ sinking (NO_SINK) results in a dramatic reduction in $k_{1,c}$ at 2.5 m
534 and 65 m, and a small increase (< 0.1 x 10⁻⁴ L µg⁻¹ d⁻¹) in $k_{1,c}$ at 25 and 45 m (Fig. 8a). Posterior
535 estimates of $k_{-1}$ from NO_SINK are similar to those from BASE, with the exception of the estimate
536 at 2.5 m, which is ~ 25% larger, and the estimate at 85 m, which is ~ 50% smaller, compared to
537 BASE (Fig. 8b). Estimates of $w$ from NO_SINK are either similar to, or slightly larger than, the
538 corresponding estimates from BASE. Interestingly, although model NO_ADS assumes that all $^7Be_p$
539 comes from atmospheric deposition, the flux of $^7Be_{tot}$ from the atmosphere and the fraction of
540 this flux in the dissolved phase are nearly identical in this model and in BASE (Table 4). This finding
541 suggests that at station 69, even when adsorption occurs, the main source of $^7Be_p$ in the water
542 column was atmospheric deposition.

543 In summary, we find that at both stations 51/60 and station 69, $^7Be_d$ adsorption and $^7Be_p$ sinking
544 do not appear to be necessary to explain the measured $^7Be_{tot,p}$ activities. In models where
545 adsorption or sinking is omitted (NO_ADS and NO_SINK), the observations are reproduced to
546 within ± 1 STD at both stations. At station 51/60, desorption also appears unnecessary to explain
547 the measured $^7Be_{tot,p}$ activities, while this process is required to explain the activities at station
548 69. Interestingly, at both stations, all models that reproduce the measured $^7Be_{tot,p}$ activities yield
549 similar posterior estimates of $FBe_{tot}$ and $f$ (Tables 3 and 4), suggesting that these estimates are
550 robust and do not strongly depend on the $^7Be$ cycling model used for their inference.
551 **4. Discussion**

552 **4.1 Comparison to Results from Previous Studies**

553 In this section, we compare our posterior estimates of ⁷Be cycling rate parameters and deposition
554 variables at GEOVIDE stations 51/60 and 69 in the subpolar North Atlantic, to estimates obtained
555 from prior studies. We only consider our estimates from models that reproduce the ⁷Be<sub>tot,p</sub>
556 activities. At station 51/60, these models include BASE, NO_ADS, NO_DES, NO_REX, and
557 NO_SINK. At station 69, they include BASE, NO_ADS, and NO_SINK. For conciseness, we will refer
558 to these models as the “best models”.

559 **4.1.1 Atmospheric deposition**

560 Total ⁷Be deposition fluxes estimated by our best models range from 119 to 138 dpm m⁻² d⁻¹ at
561 station 51/60 and from 122 to 130 dpm m⁻² d⁻¹ at station 69 (Tables 3 and 4). Using the total ⁷Be
562 inventory method (Kadko et al., 2015), Shelley et al. (2017) estimated that the atmospheric
563 deposition flux of ⁷Be<sub>tot</sub> amounted to 143 dpm m⁻² d⁻¹ at station 51/60 and to 126 dpm m⁻² d⁻¹ at
564 station 69. Posterior estimates of ⁷Be<sub>tot</sub> deposition from our best models therefore encompass
565 the estimate reported by Shelley et al. (2017) at station 51/60, and are 5 to 24 dpm/m³ smaller
566 than the estimated reported by these authors at station 69 (Tables 3 and 4).

567 Our estimates of total ⁷Be deposition at stations 51/60 and 69 are within the range found from
568 high latitude (50-70°N) observations in the North Atlantic (50-350 dpm m⁻² d⁻¹; Zhang et al.,
569 2021) and are generally lower than those found at lower latitudes in the North Atlantic (137-785
570 dpm m⁻² d⁻¹; Young and Silker, 1980; Aaboe et al., 1981; Kadko and Johns, 2011; Kadko et al.,
571 2015; Zhang et al., 2021). The latter result is consistent with the meridional pattern of surface air
572 ⁷Be activities, with subpolar minima in both hemispheres (Villareal et al., 2022). The subpolar
573 minima in surface air ⁷Be activity were attributed by these authors to the rising of tropospheric
574 air masses due to the convergence of the poleward flow in the Ferrel Cell and the equatorward
575 flow in the Polar Cell. Interestingly, at both stations 51/60 and 69, the largest posterior estimates
576 of the ⁷Be<sub>tot</sub> deposition flux are still below the range of flux estimates found from precipitation
577 samples within the same area (255-400 dpm m⁻² d⁻¹ for precipitation Area 1; Shelley et al., 2017).
578 Thus, deposition fluxes obtained by inversion broadly agree with fluxes estimated using the ⁷Be
579 inventory method, but not with fluxes obtained from precipitation samples.

580 We also find that, at both stations and for all best models, atmospheric ⁷Be enters the ocean
581 predominantly in the dissolved phase (f ≥ 0.7; Tables 3 and 4). It is difficult to constrain with prior
582 observations the fractions of ⁷Be deposition in each of the operationally defined phases
583 (dissolved and particulate). ⁷Be deposition fluxes are usually separated as wet deposition and dry
584 deposition fluxes (Brown et al., 1989; Laguionie et al., 2014; Shelley et al., 2017; Zhang et al.,
585 2021), or assumed to be dominated by precipitation (Lal, 1963; Kadko and Johns, 2011). The latter
586 assumption is supported by the observation that the dry deposition of ⁷Be rarely exceeds 20% of
587 the total deposition of ⁷Be (Zhang et al., 2021).

588 Even in cases where deposition fluxes of ⁷Be are dominated by precipitation, there are two modes
589 of wet deposition that need to be considered. Aerosols serve as cloud condensation nuclei and
are removed from the cloud by rainwater (rainout), or are scavenged by raindrops below the cloud (washout), both processes resulting in deposition of ⁷Be by precipitation (Heikkilä, 2007). Laguionie et al. (2014) measured the wet deposition of ⁷Be in Nantes, western France, and used a semi-empirical model to derive the fraction of deposition contributed to by washout, computing rainout as the difference between wet deposition and the modelled washout. They found that most of the wet deposition flux originated from rainout, with washout contributing ≤ 10% of the total flux. Rainout was similarly found to dominate wet ⁷Be deposition from precipitation measurements collected in Onagawa, Japan (Ishikawa et al., 1995). Karsi et al. (2018) reported that 90% of aerosols deposited by rainout had a diameter ≤ 0.8 µm. Importantly, measurements by Laguionie et al. (2014) and Karsi et al. (2018) were taken in urban and semi-urban environments, respectively, so the applicability of their results to open ocean wet deposition is questionable. Nevertheless, if rainout is the dominant mode of wet deposition of ⁷Be, then the findings by Karsi et al. (2018) suggest that nearly the entire flux of 119 to 138 dpm m⁻² d⁻¹ at station 51/60 and 122 to 130 dpm m⁻² d⁻¹ at station 69 would be in the operationally defined dissolved phase (≤ 0.8 µm).

However, other factors need to be considered in the interpretation of ⁷Be deposition fluxes. Nieto et al. (1994) and Chate et al. (2003) compiled measurements of aerosol particle size distribution outside of urban areas, and simulated the response of these distributions to below-cloud scavenging. They found that the effect of below-cloud scavenging is to mainly wash out aerosols of diameter > 2.5 µm, with washout coefficients (defined as the removal rate of aerosol number concentration) that were much larger for coarse (> 2.5 µm) than fine (0.1-1 µm) aerosols. Numerical and theoretical models (Foret et al., 2006) as well as a compilation of laboratory and field measurements (Santachiara et al., 2012) of below-cloud scavenging, show that the washout coefficient increases with particle size above 1 µm. However, since the surface area to volume ratio of an aerosol is expected to decrease with aerosol size, ⁷Be aerosols impacted by washout may still be predominantly in the fine mode (diameters between 0.1 and 1 µm) or in the accumulation mode plus fine mode, where the accumulation mode represents aerosols with diameters between 0.1 and 2.5 µm (Laguionie et al., 2014). Moreover, ⁷Be in aerosols has been found to be predominantly in the fine mode (< 1 µm) at ground level, supporting the notion that ⁷Be is deposited in this mode regardless of whether ⁷Be deposition occurs via rainout or washout (Ioannidou & Paatero, 2014.; Narazaki et al., 2021).

The aforementioned studies collectively suggest that the wet deposition of ⁷Be is predominately associated with ⁷Be in the dissolved phase (< 0.8 µm), whether that deposition is dominated by in-cloud or below-cloud scavenging. Then, the large values of f (> 0.7) estimated at stations 51/60 and 69 in the subpolar North Atlantic are consistent with wet deposition being the dominant mode of ⁷Be delivery to the surface of the ocean. However, we cannot conclude with confidence whether wet deposition dominated over dry deposition at either station, given the lack of studies on the size distribution of ⁷Be-bearing aerosols transported to the surface by dry deposition.

### 4.1.2 ⁷Be sorption

Consider first our posterior estimates of the second-order rate constant for ⁷Be adsorption ($k_{1,c}$ in Figs. 7,8a). At both stations, $k_{1,c}$ estimates range from 0.00 to 0.50 x10⁻⁴ L µg⁻¹ d⁻¹, depending
on depth and the best model considered, with the average from all depths and all models equaling 0.18 ± 0.30 x10⁻⁴ L µg⁻¹ d⁻¹ (mean ± 1 STD). The average of the posterior estimates of k₁,c is nearly equal to the prior estimate of 0.19 x 10⁻⁶ L µg⁻¹ d⁻¹. Its magnitude varies between an order of magnitude lower than, and the same order of magnitude as, k₁,c estimates for Th isotopes from previous studies. For example, Nyffeler et al. (1984) found k₁,c for ²²⁸Th to be 0.44 x 10⁻⁴ L µg⁻¹ d⁻¹ for fecal pellets and 2.65 x 10⁻⁴ L µg⁻¹ d⁻¹ for sinking particles, both collected from sediment traps deployed in the North Atlantic. While their value for sinking particles is about 15 times higher than our average k₁,c estimate, their value for fecal pellets is within 1 STD of this average. The k₁,c estimates constrained from multiple Th isotope and particle concentration data from GEOTRACES section GA03 in the subtropical North Atlantic were found to range between 0.82 x 10⁻⁴ and 1.09 x 10⁻⁴ L µg⁻¹ d⁻¹ (Lerner et al., 2017). This range is 4.5 to 5.9 times higher than the average estimate of k₁,c for ⁷Be in our best models. Thus the average of k₁,c estimates for ⁷Be at both stations differs from previous k₁,c estimates for Th by a factor of 2.5 to 5.8, with the exception of the k₁,c estimate for sinking particles collected by sediment traps (Nyffeler et al., 1984), which is ~ 15 times this average value.

Consider then our posterior estimates of the rate constant for ⁷Be desorption, k⁻¹. At station 51/60, our k⁻¹ estimates are the largest at 2.5 and 12.5 m for both model BASE and model NO_SINK. For BASE, these estimates are slightly smaller than the prior estimate of 0.17 d⁻¹, while for NO_SINK k⁻¹ at 2.5 m is slightly larger than the prior estimate and k⁻¹ at 12.5 m is about equal to the prior estimate. In contrast, model NO_ADS produces a much smaller estimate of k⁻¹ at both of these depths. At the remaining depths, k⁻¹ is largely reduced in all of the best models, generally being < 0.05 d⁻¹ at depths below 12.5 m. In contrast, at station 69, k⁻¹ has roughly the same magnitude at all depths in most of the best models. Again, an exception is model NO_ADS, for which k⁻¹ is reduced by ~ 40% compared to BASE at 2.5 m and vanishes at 12.5 m. However, below 12.5 m, k⁻¹ remains larger than 0.1 d⁻¹ in all best models, with the exception of NO_SINK at 85 m (Fig. 8b). Another contrast between stations 51/60 and 69 is that at station 51/60, desorption is not required to fit the observed ⁷Beₜot,p activities, while it is at station 69 (Fig. 6c,d). These results collectively show that desorption was more important for the cycling of ⁷Be at station 69 than at station 51/60, suggesting that different particle types may have controlled sorption kinetics of ⁷Be at these stations.

We explore this possibility by examining measurements of the chemical composition of particulate samples collected by in situ filtration at stations 51/60 and 69 (Sarthou et al., 2018). Particle composition data below 40 m show that the fraction of the total particle mass accounted for by opal was more than double at station 51/60 than at station 69 (Fig. 9a). Above 40 m, the mass fraction of POC was relatively similar at stations 51/60 and 69, while below 40 m, it was 20 to 70% higher at station 69 than at station 51/60 (Fig. 9b). Calcium carbonate accounted for only a small fraction of total particle mass: < 10% at all depths at both stations (Fig. 9c). The largest discrepancy in the CaCO₃ mass fraction between both stations is found at 20 m, where CaCO₃ made up 3% of total particle mass at station 51/60 and 9% of total particle mass at station 69. Lithogenic material and Mn and Fe (oxy)hydroxides had mass fractions < 8% and < 1%, respectively, at all depths at both stations (not shown). Overall, these results indicate some noticeable differences in particle composition at stations 51/60 and 69.
Both sediment trap data (Chase et al., 2002; Luo and Ku, 2004) and laboratory experiments using synthetic microparticles (Yang et al., 2013) have shown that particles comprising opal have higher distribution coefficients for ⁷Be than particles comprising organic material (i.e., POC) or calcium carbonate. While CaCO₃ contributions to total particle mass at stations 51/60 and 69 were similar (Fig. 9c), CaCO₃ contributed to a much smaller fraction than either POC or opal (Fig. 9a,b), suggesting only a small influence of CaCO₃ on ⁷Be adsorption and desorption at both stations. Assuming a stronger affinity of Be for opal vs. POC (Chase et al., 2002; Luo and Ku, 2004; Yang et al., 2013), the higher mass fraction of opal at station 51/60 may at least partly explain our lower estimates of the desorption rate constant at this station compared to station 69. It is interesting to note, however, that despite these differences in particle composition, our estimates of the adsorption rate constant k₁,ₐ are similar in magnitude at both stations (Figs. 7a and 8a). It appears that in our analysis the influence of particle composition is captured through differences in desorption, rather than adsorption, between both stations.

### 4.1.3 Particle sinking speed

Estimates of the sinking speed of large particles (> 53 µm) were obtained by analyzing ¹³⁷Cs and other artificial radionuclides released by the Chernobyl nuclear accident and present in sediment trap material collected at the DYFAMED station in the Western Mediterranean Sea. Fowler et al. (1987) thus reported estimates of ca. 30 m d⁻¹ for these large particles, a relatively large value due to zooplankton grazing in the upper water column. The settling speed of smaller particles (< 53 µm; which we refer to in this study as PM) - as investigated here - has often been inferred from the vertical gradient of particulate ²³⁰Th activity in the water column (e.g. Krishnaswami et al., 1981). When applying a reversible exchange model to various data collected worldwide, settling speeds for PM have been estimated to be in the range 200-1000 m yr⁻¹ (e.g. Bacon and Anderson, 1982; Roy-Barman et al., 1996; Roy-Barman et al., 2002).

The range in posterior estimates of wₚ for ⁷Be at stations 51/60 and 69 is broadly consistent with sinking speeds constrained from Th radioisotope data in the Atlantic Ocean. For example, Scholten et al. (2008) applied a reversible exchange model to interpret particulate ²³⁰Th samples collected in the Angola and Cape basins between 0 and 5000 m. They estimated an “apparent” sinking speed of 2.2 m d⁻¹ (800 m yr⁻¹), which is within the range of our wₚ estimates for ⁷Beₚ at both stations (Figs. 7d and 8d). More recently, Lerner et al. (2017) found that the sinking speed for particles with size > 0.8 µm, estimated using ²²⁸,²³⁰,²³⁴Th and particle concentration data, was between 0.27 and 5.48 m d⁻¹ (100-2000 m yr⁻¹) above 200 m at several stations along GA03. The range of wₚ estimates reported by these authors is comparable to the overall range of wₚ estimates for ⁷Beₚ at stations 51/60 and 69 (0.8-4 m d⁻¹ or 290-1460 m d⁻¹; Figs. 7d and 8d).

To our knowledge, no previous study has attempted to estimate the sinking speed of particles carrying ⁷Be except for Kremenchutskii et al. (2021). These authors used particulate activities of samples collected in the Black Sea to estimate particle sinking speeds in the range from 0.66 to 2.71 m d⁻¹ (240-990 m yr⁻¹). From Stokes formula, they inferred that the sinking particles were organic particles with diameters of 13 to 26 µm, or lithogenic particles with diameters of 3 to 6 µm. While our estimates of wₚ (> 2 m d⁻¹) are of the same order of magnitude, most are at the high end of the range, or above the range, found by these authors. This difference may be due
to differences in the contributions of lithogenic and/or biogenic particles to PM, and/or to differences in phytoplankton community composition. Since Kremenchutskii et al. (2021) did not characterize the composition of PM, we cannot determine whether differences in the major particle phases (POC, opal, CaCO₃) are responsible for differences in wₚ estimates between the GEOVIDE stations and the stations occupied in the Black Sea.

719

720

## 4.2 $^7Be$ Budgets at stations 51/60 and 69

In this section, we assess the relative importance of different processes in the cycling of $^7Be_{d,p}$ at stations 51/60 and 69. This task is addressed by estimating the $^7Be_{d,p}$ fluxes for all processes in equations (3a)-(3b) at both stations from the fit of the BASE model to $^7Be_{tot,p}$ data (Fig. 10). We use the results from the BASE model since among all models considered in this study BASE is the most complete and thus the less prone to potential biases, e.g., BASE includes both sorption processes (e.g., Nyffler et al., 1984) and gravitational sinking (Kremenchutskii et al., 2021); the budgets for the other models are included in the supplementary material (Figs. S9-S12). For each station, volumetric fluxes (in units of dpm m⁻³ d⁻¹) are derived by dividing the terms on the right-hand side of (3a)-(3b) by the thickness of the corresponding layer (Fig. 2). We also derive error estimates (STDs) for each volumetric flux by propagating the uncertainties in the estimated $^7Be_{d,p}$ activities and rate parameters (section 2.5), accounting for error covariances.

722

723

724

725

726

727

728

729

730

731

732

First, we consider station 51/60. In the top layer (0 – 5 m), we find that atmospheric deposition is the most important source of both $^7Be_d$ and $^7Be_p$. Below this layer, sorption processes play a minor role in the $^7Be_d$ budget but a significant role in the $^7Be_p$ budget: adsorption is close to the largest source of $^7Be_p$ in three out of four layers, and desorption is the largest sink of $^7Be_p$ in half of the layers (Fig. 10b). However, for both phases, vertical advection and diffusion are often the dominant sources and sinks, depending on the phase and layer considered (Fig. 10c). For $^7Be_p$, particle sinking is also a dominant term in every layer, either as a source (third and fourth layer) or a sink (first and second layer; Fig. 10c).

733

734

735

736

737

738

739

740

Next, we consider station 69. Atmospheric deposition is again the most important source of both $^7Be_d$ and $^7Be_p$ in the top layer (0 – 5 m). Generally, however, desorption is an important term in the budgets of $^7Be_{d,p}$: it is the dominant sink of $^7Be_p$ in most layers, and it is the dominant (or almost dominant) source of $^7Be_d$ in half of the layers (Fig. 10c,d). Adsorption is also the major source of $^7Be_p$ in two out of six layers. As at station 51/60, vertical advection is a significant term in the budget of $^7Be_d$, being a dominant source or sink in five out of six layers (Fig. 10c). However, vertical advection is less important for the budget of $^7Be_p$, never being the main source or sink for the particulate phase (Fig. 10d). Unlike at station 51/60, where particle sinking is a dominant term in every vertical layer, particle sinking at station 69 is only a dominant term in the $^7Be_p$ budget in the top layer and in the three bottom-most layers (Fig. 10d). Notice that at station 51/60, the particle sinking speed is roughly the same at 5 and 20 m. Since $^7Be_p$ activities are smaller at 5 m than at 20 m (Fig. 2a), particle sinking is a sink of $^7Be_p$ into the layer centered at 12.5 m (between 5 and 20 m) at station 51/60 (Fig. 10b). In contrast, at station 69, the particle

741

742

743

744

745

746

747

748

749

750

751

752

753
754 sinking speed drastically decreases from 5 to 20 m (Figs. 7d and 8d), resulting in particle sinking
755 being a source of ⁷Beₚ in this layer, despite ⁷Beₚ activities still being smaller at 5 m than at 20 m
756 (as at station 51/60; Fig. 2b).

757 Notably, both stations 51/60 and 69 present subsurface (20 m) maxima in ⁷Beₚ activity, which
758 may indicate enhanced ⁷Be adsorption onto particles and/or reduced ⁷Be desorption from
759 particles, at this depth. Yet, despite their apparent importance to the budgets in model BASE,
760 sorption processes are not required to fit any of the ⁷Beₜot,p data at station 51/60, including the
761 subsurface ⁷Beₚ maximum. Adsorption is also not required to fit the ⁷Beₜot,p data, including the
762 ⁷Beₚ subsurface maximum, at station 69 (e.g., Fig. 6). The fact that neglecting adsorption yields a
763 reasonable fit to ⁷Beₜot,p observations at both stations, while adsorption, when it is not neglected,
764 is an important term in the ⁷Beₚ budgets, is seemingly contradictory. Indeed, the contribution is
765 only apparent: the fact that a model omitting adsorption can fit the ⁷Beₜot,p data does not imply
766 that adsorption is not important, since the omission of adsorption can be accommodated through
767 variations in the rate parameters for other processes in order to achieve a best fit (cf., e.g., the
768 large reduction in the desorption rate constant at most depths in model NO_ADS; Fig. 7,8b). This
769 consideration suggests that the significance of individual sorption processes (adsorption and
770 desorption) in the ⁷Be₉ₚ budgets at stations 51/60 and 69 cannot be determined with great
771 confidence.

772 However, we highlight one key finding from our analysis: desorption is necessary to explain the
773 ⁷Beₜot,p data at station 69, but not at station 51/60 (section 4.1.3). Thus, at station 69, a fraction
774 of ⁷Be found in the dissolved phase should have originated from the release of ⁷Be from particles
775 in the water column, while the entirety of ⁷Be₉ₚ at station 51/60 could have been supplied from
776 surface deposition. In particular, the subsurface ⁷Beₚ maximum at station 69 requires a flux of
777 ⁷Be from the solid phase to the dissolved phase, in contrast to station 51/60. ⁷Be present in
778 particulate form in the water column could represent ⁷Be adsorbed in situ or ⁷Be bounded to
779 aerosols deposited at the sea surface. Speculatively, aerosol particles may be chemically distinct
780 at stations 51/60 and 69, allowing for more rapid desorption at station 69 than at station 51/60.
781 At station 69, Shelley et al. (2017) reported that aerosols originated from a mineral dust source,
782 while at station 51/60, they found a mixed source of dust, with V and Ni enriched aerosols,
783 suggesting a source from shipping emissions. It is unclear, however, whether this difference in
784 aerosol source could lead to differences in ⁷Be desorption rates in seawater, as the affinity of Be
785 for aerosols of different chemical compositions has to our knowledge not yet been determined.

## 4.3 Estimating Atmospheric Deposition from ⁷Be Inventory

787 Kadko et al. (2015) developed a method for estimating the atmospheric deposition of a trace
788 element (TE) at the sea surface from (i) measurements of total ⁷Be activity in the upper water
789 column and (ii) measurements of the TE/⁷Be activity ratio of aerosol samples:

$$F_{TE} = \left(\frac{C_{TE}}{C_{Be}}\right) \int_{z=0}^{z_{max}} \lambda \ ⁷Be_{tot}(z) dz \ (10).$$

791 Here, F<sub>TE</sub> is the deposition flux of the trace element at the surface (e.g., in dpm m<sup>-2</sup> d<sup>-1</sup>), z<sub>max</sub> is the
792 maximum depth of the water column over which the inventory is calculated, C<sub>TE</sub> is the
concentration of the trace element in aerosols (µg g⁻¹), and $C_{Be}$ is the activity of ⁷Be in the aerosols (µg g⁻¹). This method assumes that the depth-integrated decay rate of ⁷Be$_{tot}$ (the water column inventory of ⁷Be$_{tot}$ multiplied by the radioactive decay constant) is balanced by the atmospheric deposition flux of ⁷Be$_{tot}$ at the sea surface. It could thus be applied in situations where (i) lateral transport of ⁷Be$_{tot}$ is negligible, (ii) water column measurements of ⁷Be$_{tot}$ are collected down to the depth at which ⁷Be$_{tot}$ activities are negligible, (iii) the measured activity profile represents a system at steady state, and (iv) there is no ⁷Be removal by sinking particles. This method was applied by Shelley et al. (2017) to estimate the deposition fluxes of different trace elements along the GEOVIDE section. Some investigators used Eq. 10 with ⁷Be$_{tot}$ substituted by ⁷Be$_{d}$ to estimate the atmospheric deposition of ⁷Be$_{d}$ (e.g., Kadko et al., 2015), which requires the additional assumption that adsorption of ⁷Be$_{d}$ onto, and desorption of ⁷Be$_{p}$ from, particles are negligible.

In this subsection, we assess the extent to which sorption processes could produce deviations of the depth-integrated radioactive decay rate of ⁷Be$_{tot}$ and ⁷Be$_{d}$ from their corresponding deposition fluxes. To this end, the integrated decay rates of ⁷Be$_{tot}$ and ⁷Be$_{d}$ at stations 51/60 and 69 are compared to the surface deposition rates of ⁷Be$_{tot}$ and ⁷Be$_{d}$ for each of our best models (Fig. 11). For ⁷Be$_{d}$, the depth-integrated decay rates are also compared to the integrated net adsorption (adsorption minus desorption) rates.

Consider first the results for station 51/60. We find that the integrated ⁷Be$_{tot,d}$ decay rates slightly overestimate the deposition fluxes of ⁷Be$_{tot,d}$ in all models, but by no more than 20% (Fig. 11a,b). Losses of ⁷Be$_{tot}$ by ⁷Be$_{p}$ sinking are small, and balance at most 5% of the atmospheric source. These results suggest that biases in ⁷Be$_{d}$ (⁷Be$_{tot}$) deposition estimates derived only from water column ⁷Be$_{d}$ (⁷Be$_{tot}$) data would be relatively small. Notice that at station 51/60, for some models, the activities of ⁷Be$_{tot,p}$ at the lower boundary of the deepest model layer ($z_{N}$) exceed the detection limits, resulting in non-negligible advective and sinking fluxes there. Accordingly, we are limited in our ability to determine the accuracy of ⁷Be deposition estimates derived from ⁷Be inventories, and the above assessment of the biases in these estimates should be viewed with caution.

Consider then the results at station 69. Here the integrated ⁷Be$_{tot}$ decay reproduces the surface flux of ⁷Be$_{tot}$ in all best models with almost no bias (Fig. 11c). As at station 51/60, ⁷Be that enters the water column is almost completely removed by radioactive decay. However, unlike at station 51/60, some models display larger discrepancies between the depth-integrated ⁷Be$_{d}$ decay and the surface deposition of ⁷Be$_{d}$ (by up to ~40% in NO_ADS; Fig. 11d). Depth-integrated rate of net adsorption is a more substantial term in the integrated upper-water column budget of ⁷Be$_{d}$ at station 69 than at station 51/60. Importantly, unlike at station 51/60, the deepest ⁷Be$_{tot,p}$ measurements at station 69 are below the detection limit, so that at station 69 inventories are computed to the depth at which the ⁷Be$_{tot,p}$ activities (and by extension the ⁷Be$_{d}$ activities) are ~0 dpm m⁻³. Thus, the use of water column inventory to estimate the atmospheric flux of either phase can be more reliably assessed at station 69.

## 4.4 Sensitivity to Steady-State Assumption
The assumption of steady-state used in our analysis of ⁷Be<sub>tot,p</sub> data may be questionable. For example, mixed layer depths in the Labrador Sea can vary from values of over 1000 m in winter to values less than 50 m in summer (Körtzinger et al., 2008). In the Irminger Sea, they can vary from over 500 m in winter to less than 50 m in summer (Sterl and Jong, 2022). Since ⁷Be has a mean-life of 77 days, the activity of ⁷Be of samples from the seasonal thermocline may include a significant component that is a remnant from deeper mixed layers that existed several weeks to months in the past (Kadko and Johns, 1996). In this subsection, we examine the influence of the steady-state assumption on our results by performing inversions where this assumption is relaxed and the time-dependency terms ∂ ⁷Be<sub>d</sub>/∂t and ⁷Be<sub>p</sub>/∂t are treated as unknowns in the state vector **x**. We use results from these inversions to determine whether the following conclusions hold: (i) reversible exchange processes are not required to fit the ⁷Be<sub>d,p</sub> data at station 51/60, but are required to fit the ⁷Be<sub>d,p</sub> data at station 69, and (ii) atmospheric supply of ⁷Be<sub>tot</sub> is largely balanced by radioactive decay, indicating the reversible exchange can be neglected when using ⁷Be<sub>tot</sub> as a tracer of atmospheric deposition (Fig. 11).

The prior estimates of the terms ∂ ⁷Be_d /∂t and ∂ ⁷Be_p /∂t and their uncertainty are described in the supplementary material (Text S1). We perform inversions of ⁷Be_{tot,p} data using the model BASE and the alternative models (section 2.3-2.6) with the time-dependency terms included as unknowns (Fig. S14). We find that including the time-dependency terms does not significantly alter the rate parameter estimates from the BASE model (Fig. S15, S16). We further find that the fit of each of the models to the ⁷Be_{tot,p} activity profiles is largely unaltered by the inclusion of the time-dependency terms (Figs. 13 and 14), with two exceptions: the fits of models NO_DES and NO_REX to ⁷Be_{tot,p} data at station 69 change significantly when these models are time-dependent (Fig. 14). The steady-state version of models NO_DES and NO_REX show large negative biases at 5 and 20 m in both ⁷Be_{tot} and ⁷Be_p (Fig. 14a,c). In contrast, the unsteady version of these models shows large positive biases in ⁷Be_{tot} at 5 m, and smaller but still significant negative biases in ⁷Be_p at 70 m (Fig. 14b,d). Additionally, the biases in ⁷Be_p at 5 and 20 m are, while still significant, smaller in the time-dependent models than in the steady-state models. Nonetheless, the results obtained with the time-dependent models are still consistent with our conclusion that at station 69, desorption is required to explain the ⁷Be_{tot,p} activities: both with and without the time-dependency terms, models without desorption cannot fit the activity profiles at this station.

As for the rate parameters, at station 51/60, including the time-dependency terms does not significantly alter the atmospheric deposition fluxes of ⁷Be<sub>tot,p</sub> estimated by inversion. In contrast, these fluxes are significantly altered, increasing by up to 40%, at station 69 (Table S3). As a result, including the time-dependency terms in our ⁷Be<sub>tot,p</sub> data inversions only slightly changes the vertically integrated sources and sinks of ⁷Be<sub>tot</sub> at station 51/60, but significantly changes them at station 69 (Fig. 15). At the latter station, the atmospheric deposition flux of ⁷Be<sub>tot</sub> is roughly 40% greater than the integrated ⁷Be<sub>tot</sub> decay rate. As explained in section 4.3, testing the use of water column inventory to assess atmospheric deposition fluxes is more reliable at station 69 than at station 51/60. Therefore, our conclusion that reversible exchange can be neglected in applications of total ⁷Be as a tracer for atmospheric deposition is dependent on the steady state assumption.
In summary, except for two alternative models at station 69 (NO-DES and NO_REX), we find that relaxing the steady-state assumption does not significantly alter our conclusion that ⁷Be fluxes between solution and particles are necessary to explain the activity profiles at station 69. Moreover, neither the rate parameters estimated by inversion, nor the fit to the ⁷Be<sub>d,p</sub> activities, are significantly changed. However, we do find that, at least at station 69, atmospheric deposition is no longer balanced by radioactive decay in all of our best models. Therefore, our finding that the water column inventory ⁷Be<sub>tot</sub> can be used to accurately estimate the atmospheric deposition flux of ⁷Be<sub>tot</sub> relies on assuming that the ⁷Be<sub>d,p</sub> activities are at steady-state.

## 5. Conclusions

In this study, different models of ⁷Be<sub>d,p</sub> cycling in the upper water column are fitted to ⁷Be<sub>tot,p</sub> measurements collected at two stations of the GEOVIDE section in the subpolar North Atlantic: station 51/60, near the southern tip of Greenland, and station 69, in the southern Labrador Sea. Consideration of multiple models allows us to test the influence of adsorption, desorption, and particle sinking in the ⁷Be<sub>d,p</sub> budgets at each station.

We find that the measured ⁷Be<sub>tot,p</sub> activities can be closely reproduced by the model which considers all processes listed above with in addition vertical advection, vertical diffusion, and radioactive decay. They can also be reproduced at both stations by models that omit adsorption or particle sinking. At station 51/60, model versions that neglect desorption can also explain the ⁷Be<sub>tot,p</sub> data, while this process is required to fit the ⁷Be<sub>tot,p</sub> data at station 69. These results point to a greater importance of ⁷Be desorption from particles at station 69 than at station 51/60. They indicate that dissolved ⁷Be found at station 69 originated at least partly from the release of particulate ⁷Be into solution, whilst dissolved ⁷Be at station 51/60 could have been supplied entirely from surface deposition. They also indicate that the ⁷Be<sub>p</sub> subsurface maximum at station 69 requires a flux of ⁷Be between particles and solution, which is not required at station 51/60. Moreover, estimates of the first-order desorption rate constant are generally larger at station 69 than at station 51/60, particularly beneath 12.5 m, perhaps as a result of differences in particle chemical composition. The estimated ⁷Be<sub>d,p</sub> budgets suggest that reversible exchange of ⁷Be occurred at both stations.

We also find that estimates of surface deposition of ⁷Be<sub>tot</sub> are biased by at most 20% when derived from water column ⁷Be<sub>tot</sub> inventories and assuming steady-state. This finding suggests that ⁷Be deposited at the sea surface is mainly removed by radioactive decay in the water column. However, estimates of surface ⁷Be<sub>d</sub> deposition fluxes based on water column ⁷Be<sub>d</sub> inventories are subjected to biases of up to 40% if reversible exchange processes are not considered. Therefore, we recommend that solid-solution exchange be not systematically disregarded in applications of ⁷Be as a tracer of atmospheric deposition, particularly when dissolved ⁷Be, as opposed to total ⁷Be, is considered as the tracer. Additionally, we find that relaxing the steady-state assumption increases the bias in the surface deposition of ⁷Be<sub>tot</sub> up to 40% when derived from water column ⁷Be<sub>tot</sub> inventories. The latter finding suggests that the steady-state assumption must be assessed before estimating atmospheric deposition fluxes of trace metals from ⁷Be<sub>tot</sub> inventories, ideally by utilizing a time-series of ⁷Be activity measurements.
Additional measurements and experiments are needed to further our understanding of the behavior of ⁷Be in the upper ocean. Given potential departures from steady state conditions, future sampling efforts should constrain the seasonal variations in ⁷Be<sub>d,p</sub> activities via, e.g., the repeated occupation of a few stations representative of a range of oceanic environments, particularly since mixed layer depth, surface deposition, and chemical scavenging could each display large seasonal variability. The distribution of ⁷Be activities in the upper water column should be determined with higher vertical resolution, especially in the upper part of the seasonal thermocline, where our preliminary data suggest the presence of ⁷Be<sub>p</sub> activity maxima. Paired measurements of ⁷Be activity in the dissolved phase (or the total fraction) and the particulate phase are needed to better constrain the interaction of ⁷Be with marine particles. To aid in the interpretation of these ⁷Be activities, ancillary data such as bulk particle concentration, particle composition, and phytoplankton community composition should be collected, ideally at the same depths and at the same times as the ⁷Be data. A better understanding of surface ⁷Be deposition, and of its partitioning between the dissolved and particulate fractions, is also desirable. Future laboratory studies should focus on determining the rates of ⁷Be adsorption and desorption for different particle types in order to better assess the significance of sorption processes for ⁷Be cycling in the ocean.

## 931 Acknowledgements

Mélanie Grenier and Pieter van Beek acknowledge support by the French National program LEFE (Les Enveloppes Fluides et l'Environnement) funded by CNRS-INSU (BE-7-FLUX). Olivier Marchal is supported by grant OCE-1829790 from the US National Science Foundation. The data were acquired in the framework of the GEOVIDE cruise (GEOTRACES GA01). Therefore, we thank the chief scientists (Géraldine Sarthou and Pascale Lherminier), the captain and crew of RV *Pourquoi Pas?*, and all team members who contributed to collect the samples and generate the dataset used here (see Shelley et al., 2017). The GEOVIDE project was co-funded by the French national program LEFE/INSU (GEOVIDE), ANR Blanc (GEOVIDE, ANR-13-BS06-0014) and RPDOC (ANR-12-PDOC-0025-01), LabEX MER (ANR-10-LABX-19) and IFREMER. The International GEOTRACES Programme is possible in part thanks to the support from the U.S. National Science Foundation (Grant OCE-2140395) to the Scientific Committee on Oceanic Research (SCOR).

## 947 References

Aaboe, E, E P Dion, and K K Turekian. 1981. "⁷Be in Sargasso Sea and Long Island Sound waters." *J. Geophys. Res. Oceans* 86 (C4): 3255–57.
950 Anderson, R F, Y Lao, W S Broecker, S E Trumbore, H J Hofmann, and W Wolfli. 1990. "Boundary scavenging in the Pacific Ocean: a comparison of <sup>10</sup>Be and <sup>231</sup>Pa." *Earth and Planetary Science Letters* 96 (3-4): 287-304.

953 Andrews, J, C Hartin, and K Buesseler. 2008. "$^{7}$Be analyses in seawater by low background gamma-spectroscopy." *Journal of Radioanalytical and Nuclear Chemistry* 277 (1): 253-59.

955 Bacon, Michael P, and Robert F Anderson. 1982. "Distribution of thorium isotopes between dissolved and particulate forms in the deep sea." *J. Geophys. Res. Oceans* 87 (C3): 2045-56.

957 Benitez-Nelson, C, K O Buesseler, D M Karl, and J Andrews. 2001. "A time-series study of particulate matter export in the North Pacific Subtropical Gyre based on <sup>234</sup>Th:<sup>238</sup>U disequilibrium." *Deep Sea Res. Part I: Oceanogr. Res. Pap.* 48 (12): 2595-2611.

960 Boyer Montégut, C de, G Madec, A S Fischer, A Lazar, and D Iudicone. 2004. "Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology." *Journal of Geophysical Research: Oceans* 109 (C12): 1-20.

963 Brost, R A, J Feichter, and M Heimann. 1991. "Three-dimensional simulation of 7Be in a global climate model." *Journal of Geophysical Research: Atmospheres* 96 (D12): 22423-45.

965 Brown, L, G J Stensland, J Klein, and R Middleton. 1989. "Atmospheric deposition of $^{7}$Be and $^{10}$Be." *Geochimica Et Cosmochimica Acta* 53 (1): 135-42.

967 Buesseler, K, L Ball, J Andrews, C Benitez-Nelson, R Belastock, F Chai, and Y Chao. 1998. "Upper ocean export of particulate organic carbon in the Arabian Sea derived from thorium-234." *Deep Sea Research Part II: Topical Studies in Oceanography* 45 (10-11): 2461-87.

970 Chanut, J, B Barnier, W Large, L Debreu, T Penduff, J M Molines, and P Mathiot. 2008. "Mesoscale eddies in the Labrador Sea and their contribution to convection and restratification." *Journal of Physical Oceanography* 38 (8): 1617-43.

973 Chase, Z, R F Anderson, M Q Fleisher, and P W Kubik. 2002. "The influence of particle composition and particle flux on scavenging of Th, Pa and Be in the ocean." *Earth Planet. Sci. Lett.* 204 (1): 215-29.

976 Chate, D M, P S P Rao, M S Naik, G A Momin, P D Safai, and K Ali. 2003. "Scavenging of aerosols and their chemical species by rain." *Atmospheric Environment* 37 (18): 2477-84.

978 Chen, S Y S, O Marchal, P E Lerner, D C McCorkle, and M M R van der Loeff. 2021. "On the cycling of <sup>231</sup>Pa and <sup>230</sup>Th in benthic nepheloid layers." *Deep Sea Research Part I: Oceanographic Research Papers* 177 (103627).

981 Chuang, C Y, P H Santschi, Y F Ho, M H Conte, L Guo, D Schumann, M Aryanov, and Y H Li. 2013. "Role of biopolymers as major carrier phases of Th, Pa, Pb, Po, and Be radionuclides in settling particles from the Atlantic Ocean." *Mar. Chem.* 157: 131-43.
984 Chuang, C Y, P H Santschi, L S Wen, L Guo, C Xu, S Zhang, Y Jiang, et al. 2015. "Binding of Th, Pa, 985 Pb, Po and Be radionuclides to marine colloidal macromolecular organic matter." *Mar. Chem.* 986 173: 320–29.

987 Daniault, N, H Mercier, P Lherminier, A Sarafanov, A Falina, P Zunino, F F Pérez, et al. 2016. "The 988 northern North Atlantic Ocean mean circulation in the early 21st century." *Deep Sea Research Part I: Oceanographic Research Papers* 146: 142–58.

990 Duyck, E., and M.F. De Jong. (2021). Circulation over the South-East Greenland Shelf and 991 potential for liquid freshwater export: a drifter study. *Geophysical Research Letters*, *48*(5), 992 e2020JB020886.

993 Duyck, E., R. Gelderloos, M.F. de Jong (2022). Wind-Driven Freshwater Export at Cape 994 Farewell. *Journal of Geophysical Research: Oceans*, 127(5), e2021JC018309.

995 Fan, X, U Send, P Testor, J Karstensen, and P Lherminier. 2013. "Observations of Irminger Sea 996 anticyclonic eddies." *Journal of Physical Oceanography* 43 (4): 805–23.

997 Fischer, J, F A Schott, and M Dengler. 2004. "Boundary circulation at the exit of the Labrador 998 Sea." *Journal of Physical Oceanography* 34 (7): 1548–70.

999 Foret, G, G Bergametti, F Dulac, and L Menut. 2006. "An optimized particle size bin scheme for 1000 modeling mineral dust aerosol." *Journal of Geophysical Research: Atmospheres* 111 (D17): 1–1001 15.

1002 Forget, G, J M Campin, P Heimbach, C N Hill, R M Ponte, and C Wunsch. 2015. "ECCO version 4: 1003 an integrated framework for non-linear inverse modeling and global ocean state estimation." *1004 Geosci. Model Dev.* 8: 3071–3104.

1005 Forryan, A, A P Martin, M A Srokosz, E E Popova, S C Painter, and M C Stinchcombe. 2012. 1006 "Estimates of the local rate of vertical diffusion from dissipation measurements." *Deep Sea 1007 Research Part I: Oceanographic Research Papers* 63: 20–35.

1008 Fowler, S. W., Buat-Menard, P., Yokoyama, Y., Ballestra, S., Holm, E., & Nguyen, H. V. (1987). 1009 Rapid removal of Chernobyl fallout from Mediterranean surface waters by biological activity. *Nature*, 329(6134): 56-58.

1011 Garcia H.E., T.P. Boyer, O.K. Baranova, R.A. Locarnini, A.V. Mishonov, A. Grodsky, C.R. Paver, 1012 K.W. Weathers, I.V. Smolyar, J.R. Reagan, D. Seidov, and M.M. Zweng (2019). World Ocean 1013 Atlas 2018: Product Documentation. A. Mishonov, Technical Editor.

1014 García-Ibáñez, M I, F F Pérez, P Lherminier, P Zunino, H Mercier, and P Tréguer. 2018. "Water 1015 mass distributions and transports for the 2014 GEOVIDE cruise in the North Atlantic." *Biogeosciences* 15 (7): 2075–90.

1017 Gehlen, M, L Beck, G Calas, A M Flank, A J Van Bennekom, and J E E Van Beusekom. 2002. 1018 "Unraveling the atomic structure of biogenic silica: evidence of the structural association of Al 1019 and Si in diatom frustules." *Geochimica Et Cosmochimica Acta* 66 (9): 1601–9.
1020 Gourain, A, H Planquette, M Cheize, L Lemaitre, J M Barraqueta, R Shelley, P Lherminier, and G
1021 Sarthou. 2019. "Inputs and processes affecting the distribution of particulate iron in the North
1022 Atlantic along the GEOVIDE (GEOTRACES GA01)." *Biogeosciences* 16 (7): 1563-82.

1023 Grenier, M, P van Beek, P Lerner, V Sanial, M Souhaut, M Lagarde, O Marchal, and J L Reyss.
1024 (2023). "New insights on the 7Be cycle in the ocean." *Deep Sea Research Part I: Oceanographic
1025 Research Papers*, 194, 103967.

1026 Gregg, M. C., E. A., D'Asaro, J.J. Riley, and E. Kunze (2018). Mixing efficiency in the
1027 ocean. *Annual review of marine science*, *10*, 443-473.

1028 Heikkilä, U. 2007. "Modeling of the Atmospheric Transport of the Cosmogenic Radionuclides
1029 10Be and 7Be Using the ECHAM5-HAM General Circulation Model." PhD thesis, ETH Zurich.

1030 Honeyman, B D, L S Balistrieri, and J W Murray. 1988. "Oceanic trace metal scavenging: the
1031 importance of particle concentration." *Deep Sea Res. A* 35 (2): 227-46.

1032 IOC, SCOR and IAPSO, 2010: *The international thermodynamic equation of seawater – 2010:*
1033 *Calculation and use of thermodynamic properties*. Intergovernmental Oceanographic
1034 Commission, Manuals and Guides No. 56, UNESCO (English), 196 pp.

1035 Ioannidou, A. 2011. "Activity size distribution of 7Be in association with trace metals in the
1036 urban area of the city of Thessaloniki, Greece." *Atmospheric Environment* 45 (6): 1286-90.

1037 Ioannidou, A. and Paatero, J (2014)." Activity size distribution of 7Be in association with trace
1038 metals in the urban area of the city of Thessaloniki, Greece." *Atmospheric Environment*, 45(6),
1039 1286-1290.

1040 Ishikawa, Y, H Murakami, T Sekine, and K Yoshihara. 1995. "Precipitation scavenging studies of
1041 radionuclides in air using cosmogenic 7Be." *Journal of Environmental Radioactivity* 26 (1): 19-
1042 36.

1043 Johnston, T S, and D L Rudnick. 2009. "Observations of the transition layer." *Journal of Physical
1044 Oceanography* 39 (3): 780-97.

1045 Jong, M F de, and L de Steur. 2016. "Strong winter cooling over the Irminger Sea in winter
1046 2014-2015, exceptional deep convection, and the emergence of anomalously low SST." *Geophysical Research Letters* 43 (13): 7106-13.

1048 Kadko, D. 2017. "Upwelling and primary production during the US GEOTRACES East Pacific
1049 Zonal Transect." *Global Biogeochemical Cycles* 31 (2): 218-32.

1050 Kadko, D, A Aguilar-Islas, C Bolt, C S Buck, J N Fitzsimmons, L T Jensen, W M Landing, et al. 2019.
1051 "The residence times of trace elements determined in the surface Arctic Ocean during the 2015
1052 US Arctic GEOTRACES expedition." *Marine Chemistry* 208: 56-69.

1053 Kadko, D, and W Johns. 2011. "Inferring upwelling rates in the equatorial Atlantic using 7Be
1054 measurements in the upper ocean." *Deep Sea Res. Part I Oceanogr. Res. Pap.* 58 (6): 647-57.
1055 Kakdo, D, W Landing, and R U Shelley. 2015. "A novel tracer technique to quantify the atmospheric flux of trace elements to remote ocean regions." *J. Geophys. Res. Oceans* 120 (2): 848-58.

1058 Karsi, M B B, S Yenisoy-Karakas, and D Karakas. 2018. "Investigation of washout and rainout processes in sequential rain samples." *Journal of Radioanalytical and Nuclear Chemistry* 190: 53-64.

1061 Katsman, C A, S S Drijfhout, H A Dijkstra, and M A Spall. 2018. "Sinking of dense North Atlantic waters in a global ocean model: Location and controls." *Journal of Geophysical Research: Oceans* 123 (5): 3563-76.

1064 Kremenchutskii, D A, G F Batrakov, I I Dovhyi, and Y A Sapozhnikov. 2021. "Role of suspended matter in controlling beryllium-7 ($^7$Be) in the Black Sea surface layer." *Journal of Marine Systems* 217 (103513).

1067 Laguionie, P, P Roupsard, D Maro, L Solier, M Rozet, D Hébert, and O Connan. 2014. "Simultaneous quantification of the contributions of dry, washout and rainout deposition to the total deposition of particle-bound $^7$Be and $^{210}$Pb on an urban catchment area on a monthly scale." *Journal of Aerosol Science* 77: 67-84.

1071 Lal, D. 1963. "On the investigations of geophysical processes using cosmic ray produced radioactivity." *Earth Science and Meteoritics*, 115-42.

1073 Lal, D, and B Peters. 1967. "Cosmic ray produced radioactivity on the Earth." In *Kosmische Strahlung II*, edited by K Sitte, 551-612. Berlin: Spring.

1075 Lam, P. J., Ohnemus, D. C., & Auro, M. E. (2015). Size-fractionated major particle composition and concentrations from the US GEOTRACES North Atlantic Zonal Transect. Deep Sea Research Part II: Topical Studies in Oceanography, 116, 303-320.

1078 Lerner, Paul, O Marchal, P J Lam, W Gardner, M J Richardson, and A Mishonov. 2020. "A model study of the relative influences of scavenging and circulation on $^{230}$Th and $^{231}$Pain the western North Atlantic." *Deep Sea Research Part I: Oceanographic Research Papers* 155 (103159).

1081 Lerner, P, O Marchal, P J Lam, K Buesseler, and M Charette. 2017. "Kinetics of thorium and particle cycling along the US GEOTRACES North Atlantic Transect." *Deep Sea Res. Part I Oceanogr. Res. Pap.* 25: 106-28.

1084 Lilly, J M, P B Rhines, F Schott, K Lavender, J Lazier, U Send, and E D'Asaro. 2003. "Observations of the Labrador Sea eddy field." *Progress in Oceanography* 59 (1): 75-176.

1086 Luo, S, and T L Ku. 2004. "On the importance of opal, carbonate, and lithogenic clays in scavenging and fractionating $^{230}$Th, $^{230}$Pa and $^{10}$Be in the ocean." *Earth Planet. Sci. Lett.* 220 (1): 201-11.
1089 Maiti, Kanchan, Ken O Buesseler, Steven M Pike, Claudia Benitez-Nelson, Pinghe Cai, Weifang
1090 Chen, Kirk Cochran, et al. 2012. "Intercalibration studies of short-lived thorium-234 in the water
column and marine particles." *Limnol. Oceanogr. Methods* 10 (9): 631–44.

1092 Mikaelyan, A S, A G Zatsepin, and V K Chasovnikov. 2013. "Long-term changes in nutrient supply
1093 of phytoplankton growth in the Black Sea." *Journal of Marine Systems* 117: 53–64.

1094 Moran, S B, and R M Moore. 1988. "Evidence from mesocosm studies for biological removal of
dissolved aluminium from sea water." *Nature* 335 (6192): 706–8.

1096 Moran, S B, and R M Moore. 1992. "Kinetics of the removal of dissolved aluminum by diatoms
in seawater: A comparison with thorium." *Geochimica Et Cosmochimica Acta* 56 (9): 3365–74.

1098 Murnane, R J, J K Cochran, and J L Sarmiento. 1994b. "Estimates of particle and thorium cycling
rates in the northwest Atlantic Ocean." *J. Geophys. Res. Oceans* 99 (C2): 3373–92.

1100 Murnane, Richard J. 1994a. "Determination of thorium and particulate matter cycling
parameters at station P: A reanalysis and comparison of least squares techniques." *J. Geophys.
Res. Oceans* 99 (C2): 3393–3405.

1103 Narazaki, Y., Sakoda, A., Takahashi, S., & Momoshima, N. (2021). "Cosmogenic ⁷Be: Particle size
distribution and chemical composition of ⁷Be-carrying aerosols in the atmosphere in Japan.
"*Journal of Environmental Radioactivity*, 237, 106690.

1106 New, A L, R Bleck, Y Jia, R Marsh, M Huddleston, and S Barnard. 1995. "An isopycnic model
study of the North Atlantic. Part I: Model experiment." *Journal of Physical Oceanography* 25
(11): 2667–99.

1109 Nieto, P G, B A Garcia, F Diaz, and M R Brana. 1994. "Parametric study of selective removal of
atmospheric aerosol by below-cloud scavenging." *Atmospheric Environment* 28 (14): 2335–42.

1111 Nir-El, Y., G. Haquin, Z. Yungreiss, M. Hass, G. Goldring, S.K. Chamoli, B. S. Nara Singh, S.
1112 Lakshmi, U. Köster, N. Champault, A. Dorsival, G. Georgiev, V. N Fedoseyev, B. A. Marsh, D.
1113 Schumann, G. Heidenreich, and Teichmann, S. (2007). Precision measurement of the decay rate
of Be 7 in host materials. *Physical Review C*, 75(1), 012801.

1115 Nozaki, Y, Y Horibe, and H Tsubota. 1981. "The water column distributions of thorium isotopes
in the western North Pacific." *Earth Planet. Sci. Lett.* 54 (2): 203–16.

1117 Nyffeler, U P, Y H Li, and P H Santschi. 1984. "A kinetic approach to describe trace-element
distribution between particles and solution in natural aquatic systems." *Geochim. Cosmochim. Acta* 48 (7): 1513–22.

1120 Osborn, T R. 1980. "Estimates of the local rate of vertical diffusion from dissipation
measurements." *Journal of Physical Oceanography* 10 (1): 83–89.

1122 Painter, S C, S A Henson, A Forryan, S Steigenberger, J Klar, M C Stinchcombe, N Rogan, A R
1123 Baker, E P Achterberg, and C M Moore. 2014. "An assessment of the vertical diffusive flux of
1124 iron and other nutrients to the surface waters of the subpolar North Atlantic Ocean.”
1125 *Biogeosciences* 11 (8): 2113–30.

1126 Papastefanou, C, and A Ioannidou. 1996. "Beryllium-7 aerosols in ambient air. Environment
1127 International." *Environment International* 22: 125–30.

1128 Pedlosky, J. 2003. "Thermally driven circulations in small oceanic basins." *Journal of Physical
1129 Oceanography* 33 (11): 2333–40.

1130 Pedlosky, J, and M A Spall. 2005. "Boundary intensification of vertical velocity in a β-plane
1131 basin." *Journal of Physical Oceanography* 35 (12): 2487–2500.

1132 Pickart, R S, F Straneo, and G W K Moore. 2003. "Is Labrador Sea water formed in the Irminger
1133 basin?" *Deep Sea Research Part I: Oceanographic Research Papers* 50 (1): 23–52.

1134 Piron, A, V Thierry, H Mercier, and G Caniaux. 2016. "Argo float observations of basin-scale
1135 deep convection in the Irminger sea during winter 2011–2012." *Deep Sea Research Part I:*
1136 *Oceanographic Research Papers* 109: 76–90.

1137 Roy-Barman, M., Chen, J. H., and Wasserburg, G. J. 1996. <sup>230</sup>Th-<sup>232</sup>Th systematics in the central
1138 Pacific Ocean: The sources and the fates of thorium. *Earth and Planetary Science Letters*, 139(3-
1139 4): 351–363.

1140 Roy-Barman, M., Coppola, L., and Souhaut, M. 2002. Thorium isotopes in the western
1141 Mediterranean Sea: an insight into the marine particle dynamics. *Earth and Planetary Science
1142 Letters*, 196(3-4): 161-174.

1143 Santachiara, G., F. Prodi, F. Belosi (2012). "A review of termo-and diffusio-phoresis in the
1144 atmospheric aerosol scavenging process. Part 1: Drop scavenging." *Atmospheric and Climate
1145 Sciences*, Vol. 2 No. 2, 2012, pp. 148-158.

1146 Sarthou, G, P Lherminier, E P Achterberg, F Alonso-Pérez, E Bucciarelli, J Boutorh, V Bouvier, et
1147 al. 2018. "Introduction to the French GEOTRACES North Atlantic transect (GA01): GEOVIDE
1148 cruise." *Biogeosciences* 15 (23): 7097–7109.

1149 Shelley, R U, M Roca-Martí, M Castrillejo, V Sanial, P Masqué, W M Landing, Pieter van Beek,
1150 Héléne Planquette, and Geraldine Sarthous. 2017. "Quantification of trace element
1151 atmospheric deposition fluxes to the Atlantic Ocean (> 40 N; GEOVIDE, GEOTRACES GA01)
1152 during spring 2014." *Deep Sea Res. Part I Oceanogr. Res. Pap.* 119: 34–49.

1153 Silker, W B. 1972a. "Beryllium-7 and fission products in the GEOSECS II water column and
1154 applications of their oceanic distributions." *Earth Planet. Sci. Lett.* 16 (1): 131–37.

1155 Silker, W.B. 1972b. "Horizontal and vertical distributions of radionuclides in the North Pacific
1156 Ocean." *Journal of Geophysical Research* 77 (8): 1061–70.

1157 Sun, O M, S R Jayne, K L Polzin, B A Rahter, and L C St. Laurent. 2013. "Scaling turbulent
1158 dissipation in the transition layer." *Journal of Physical Oceanography* 43 (11): 2475–89.
1159 Tonnard, M. Etude du cycle biogéochimique du fer : distribution et spéciation dans l'Océan Atlantique Nord (GA01) et l'Océan Austral (GIpr05) (GEOTRACES). Université de Bretagne occidentale - Brest; Université de Tasmanie (Australie), 2018.

1162 Våge, K, R S Pickart, A Sarafanov, Ø Knutsen, H Mercier, P Lherminier, H M van Akene, J
1163 Meincke, D Quadfasel, and S Bacon. 2011. "The Irminger Gyre: Circulation, convection, and
1164 interannual variability." *Deep Sea Research Part I: Oceanographic Research Papers* 58 (5): 590-
1165 614.

1166 Villarreal, R E, A Arazi, and J F Niello. 2022. "Correlation between the latitudinal profile of the
1167 ⁷Be air concentration and the Hadley cell extent in the Southern Hemisphere." *Journal of
1168 Environmental Radioactivity* 244 (106760).

1169 Wang, B., and Fennel, K. (2022). Biogeochemical-Argo data suggest significant contributions of
1170 small particles to the vertical carbon flux in the subpolar North Atlantic. *Limnology and
1171 Oceanography, 67*(11), 2405-2417.

1172 Yang, W, L Guo, C. Y Chuang, D Schumann, M Ayranov, and P H Santschi. 2013. "Adsorption
1173 characteristics of ²¹⁰Pb, ²¹⁰Po and ⁷Be onto micro-particle surfaces and the effects of
1174 macromolecular organic compounds." *Geochim. Cosmochim. Acta* 107: 47-64.

1175 Yashayaev, I., and Loder, J. W. (2017). Further intensification of deep convection in the
1176 Labrador Sea in 2016. *Geophysical Research Letters*, 44(3): 1429-1438.

1177 Young, J A, and W B Silker. 1980. "Aerosol deposition velocities on the Pacific and Atlantic
1178 oceans calculated from ⁷Be measurements." *Earth Planet. Sci. Lett.* 50 (1): 92-104.

1179 Zhang, F, J Wang, M Baskaran, Q Zhong, Y Wang, J Paatero, and J Du. 2021. "A comprehensive
1180 global dataset of atmospheric ⁷Be and ²¹⁰Pb measurements: air concentration and depositional
1181 flux." *Earth System Science Data* 13 (6): 2963-94.

1182 Zunino, P, P Lherminier, H Mercier, N Daniault, M I García-Ibáñez, and F F Pérez. 2017. "The
1183 GEOVIDE cruise in May-June 2014 reveals an intense Meridional Overturning Circulation over a
1184 cold and fresh subpolar North Atlantic." *Biogeosciences* 14 (23): 5323-42.

1185

1186

1187

1188

1189

1190

1191
1192

1193

1194

1195

1196

1197

1198

<table><caption>Table 1: Prior estimates of rate parameters and deposition variables of the ⁷Be cycling model*</caption><thead><tr><td>parameter or flux</td><td>prior estimate</td><td>uncertainty</td><td>source</td></tr></thead><tbody><tr><td>k<sub>1,c</sub> (L µg<sup>-1</sup> d<sup>-1</sup>)</td><td>0.19 x 10<sup>-4</sup></td><td>0.36 x 10<sup>-4</sup></td><td>Nyffeler, Li, and Santschi (1984)</td></tr><tr><td>k<sub>-1</sub> (d<sup>-1</sup>)</td><td>0.17</td><td>0.2</td><td>Nyffeler, Li, and Santschi (1984)</td></tr><tr><td>w<sub>p</sub> (m d<sup>-1</sup>)</td><td>2.3</td><td>2.1</td><td>Kremenchutskii et al. (2021); Wang et al. (2022).</td></tr><tr><td>FBe<sub>tpt</sub> (dpm m<sup>-2</sup> d<sup>-1</sup>)</td><td>143 (S60), 126 (S69)</td><td>82 (S60 and S69)</td><td>Shelley et al., 2017<br>Zhang et al. (2021)</td></tr><tr><td>f</td><td>0.87 (S60), 0.80 (S69)</td><td>0.20</td><td>Shelley et al., 2017<br>Zhang et al. (2021)</td></tr><tr><td>w (m d<sup>-1</sup>)</td><td>variable (see Fig. 7).</td><td>1.94 (S60), 0.89 (S69)</td><td>Forget et al.<br>(2015)</td></tr></tbody></table>

* S60: station 51/60. S69: station 69.

1201

1202 Table 2: Root mean square error (RMSE) and bias (B) (both in dpm m⁻³) for ⁷Be<sub>tot,p</sub> at stations

1203 51/60 and 69 for different ⁷Be cycling models

<table><thead><tr><td>model version</td><td>RMSE(⁷Be<sub>tot</sub>) S60/S69</td><td>B(⁷Be<sub>tot</sub>) S60/S69</td><td>RMSE(⁷Be<sub>p</sub>) S60/S69</td><td>B(⁷Be<sub>p</sub>) S60/S69</td></tr></thead><tbody><tr><td>BASE</td><td>2.05/1.15</td><td>0.51/0.49</td><td>0.09/0.06</td><td>-0.04/-0.03</td></tr></tbody></table>
<table><tr><td>NO_ADS</td><td>3.76/1.63</td><td>0.75/-0.04</td><td>0.11/1.15</td><td>-0.08/-0.51</td></tr><tr><td>NO_DES</td><td>3.31/11.22</td><td>0.38/-6.93</td><td>0.08/11.83</td><td>0.01/-7.32</td></tr><tr><td>NO_REX</td><td>3.80/11.41</td><td>0.68/-7.32</td><td>0.09/11.88</td><td>-0.03/-7.57</td></tr><tr><td>NO_SINK</td><td>3.33/0.97</td><td>0.67/0.31</td><td>0.32/0.08</td><td>-0.06/-0.00</td></tr></table>

1204

1205

<table><caption>Table 3: Posterior estimates of FBe<sub>tot</sub> and f for the different <sup>7</sup>Be cycling models: Station 51/60.</caption><thead><tr><td>model version</td><td>FBe<sub>tot</sub> (dpm m<sup>-2</sup> d<sup>-1</sup>)</td><td>f</td></tr></thead><tbody><tr><td>BASE</td><td>127±49</td><td>0.8 ±0.2</td></tr><tr><td>NO-ADS</td><td>138±50</td><td>0.7 ±0.2</td></tr><tr><td>NO-DES</td><td>122±49</td><td>0.9 ± 0.2</td></tr><tr><td>NO-REX</td><td>131±48</td><td>0.8 ±0.1</td></tr><tr><td>NO-SINK</td><td>119±48</td><td>0.9 ± 0.2</td></tr></tbody></table>

1206

1207

<table><caption>Table 4: Posterior estimates of FBe<sub>tot</sub> and f for the different <sup>7</sup>Be cycling models: Station 69.</caption><thead><tr><td>model version</td><td>FBe<sub>tot</sub> (dpm m<sup>-2</sup> d<sup>-1</sup>)</td><td>f</td></tr></thead><tbody><tr><td>BASE</td><td>130±8</td><td>0.7 ±0.2</td></tr><tr><td>NO-ADS</td><td>129±7</td><td>0.7 ±0.2</td></tr><tr><td>NO-DES</td><td>122±14</td><td>1.0 ±0.1</td></tr><tr><td>NO-REX</td><td>122±14</td><td>1.0 ±0.0</td></tr><tr><td>NO-SINK</td><td>130±5</td><td>0.9 ±0.1</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Fig. 1 Map showing the locations of stations (black circles) for which ⁷Be samples are used in this study. Arrows provide a schematic of surface circulation elements, including the North Atlantic Current (NAC; red), the East-Greenland-Irminger Current (EGIC; magenta), the West-Greenland Current (WGC; magenta), and the Labrador Current (LC; green) (García-Ibáñez et al., 2018). The crossed black circle depicts a region of deep convection in the Labrador Sea, where surface buoyancy loss and intermediate and deep-water formation occur (Yashayaev and Loder 2016). The 100 m, 250 m, 1000 m, and 3000 m isobaths are highlighted in black.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2 Activities of total ⁷Be (blue) and particulate ⁷Be (black) at stations (a) 51/60 and (b) 69. Filled (open) markers show measured (interpolated) values, and triangles (squares) show corrected (uncorrected) ⁷Beₚ activities (see section 2.1). Error bars for uncorrected ⁷Beₚ activities represent ± 1 standard deviation (counting statistics). Error bars for corrected ⁷Beₚ activities are calculated by propagating the errors in the uncorrected activities and the errors in the filter corrections. The horizontal dashed grey lines show the boundaries of the layers of the ⁷Be cycling model.</figcaption></figure>

1208

1209

1210

1211

1212

1213

1214

1215

1216

1217

1218

1219

1220
<figure><img src="image_5.png" /><figcaption>Fig. 3: Profiles of potential density referenced to 0 dbar at (a) station 51/60, cast 106, and (b) station 69, cast 142. The red curves show potential density derived from CTD data, and the blue curves show potential density averaged in 5-m bins. The horizontal lines show, from top to bottom, the mixed-layer depth (MLD;), the upper boundary of the transition layer (TLupand the lower boundary of the transition layer (TLlw). Note that MLD = TLup for cast 106.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Profiles of squared buoyancy frequency at stations 51/60 (a) and 69 (b) derived from the 5-m bin-averaged potential density profiles (Fig. 3). The horizontal lines show, from top to bottom, the mixed-layer depth (MLD), the upper boundary of the transition layer (TLup), and the lower boundary of the transition layer (TLlw). Note that MLD = TLup for cast 106.</figcaption></figure>

<figure><img src="image_7.png" /></figure>

1221

1222

1223

1224

1225

1226

1227

1228

1229

1230

1231
Fig. 5: Profiles of K at stations 51/60 (a) and 69 (b) derived from the N² profiles (Fig. 4). The horizontal lines show, from top to bottom, the mixed-layer depth (MLD), the upper boundary of the transition layer (TLup), and the lower boundary of the transition layer (TLlw). Note that MLD = TLup for cast 106. The black dots show the K values at the layer boundaries of the ⁷Be cycling model and used in the inversions of ⁷Be<sub>tot,p</sub> data.

<figure><img src="image_8.png" /></figure>

Fig. 6. Profiles of the biases in ⁷Be<sub>tot</sub> (a,c) and ⁷Be<sub>p</sub> (b,d) activities. The top panels are for station 51/60, and the bottom panels are for station 69. The colored bars show the results for different models of ⁷Be cycling. The error bars are the uncertainties in the biases (±1 standard deviation).

1236

1237

1238

1239
<figure><img src="image_9.png" /><figcaption>Fig. 7. Profiles of k₁,c (a), k⁻¹ (b), w (c), and wₚ (d) estimates at station 51/60. The colored circles and colored error bars show posterior estimates and their uncertainties (± 1 STD) for different models of ⁷Be cycling. The grey circles and grey error bars show the prior estimates and their uncertainties (±1 STD). For clarity, the different estimates for the same depth are offset vertically and dashed horizontal grey lines separate estimates for different depths.</figcaption></figure>

1240

1241

1242

1243

1244
<figure><img src="image_10.png" /><figcaption>Fig. 8. Same as figure 7 but for station 69. Note that in panel (d), the posterior estimate for model NO-DES at 60 m is 8.45 ± 2.1 m d⁻¹.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9: Profiles of the measured weight percentage of (a) opal, (b) particulate organic carbon (POC), and (c) calcium carbonate (CaCO₃) in total particulate matter (Grenier et al., 2023). The red dots are for station 51/60, and the blue dots are for station 69.</figcaption></figure>

1245

1246

1247

1248

1249

1250

1251
<figure><img src="image_12.png" /><figcaption>Fig. 10: Profiles of the volumetric fluxes of (a,c) $^{7}$Be$_{d}$ and (b,d) $^{7}$Be$_{p}$ at station 51/60 (a,b) and station 69 (c,d) for the model BASE: atmospheric deposition (red), radioactive decay (blue), vertical advection (green), vertical diffusion (magenta), adsorption (purple), desorption (black), and particle sinking (brown). For clarity, fluxes for the same depth are offset vertically and dashed horizontal grey lines separate fluxes for different depths. The error bars show the uncertainties (±1 STD).</figcaption></figure>

1252

1253

1254

1255

1256

1257
<figure><img src="image_13.png" /><figcaption>Fig. 11. Vertical integral of volumetric $^{7}$Be$_{tot,d}$ fluxes over the upper 150 m at station 51/60 (a,b) and the upper 100 m at station 69 (c,d). The left panels are for $^{7}$Be$_{tot}$, and the right panels are for $^{7}$Be$_{d}$. Color convention is: red for atmospheric deposition, blue for radioactive decay, green for vertical advection, magenta for vertical diffusion, purple for net adsorption (adsorption minus desorption), and brown for particle sinking. The error bars show the uncertainties (±1 STD).</figcaption></figure>

1258

1259

1260

1261

1262

1263
<figure><img src="image_14.png" /><figcaption>Fig.12: Vertical profiles of ⁷Be<sub>tot</sub> (a,c) and ⁷Be<sub>p</sub> (b,d) at stations 51/60 and 69, as simulated by our time-dependent model of ⁷Be cycling (Eqs. S1-3a,b). The black profiles are the initial conditions obtained by integrating the model to steady state using a fixed mixed layer depth from March 1st, 2014, derived from the WOA2018 0.25 x 0.25 gridded climatology (see Text S1; Garcia et al., 2019). The colored lines show the profiles in increments of 10 days. The dashed grey curve are the profiles at the time when the mixed-layer depth reaches the value observed during GEOVICE occupations (42 m for station 51/60, 24 m for station 69).</figcaption></figure>

1264
1265
1266
1267
1268
1269
1270
<figure><img src="image_15.png" /><figcaption>Fig. 13: Profiles of the biases for $^{7}$Be$_{tot}$ (a,b) and $^{7}$Be$_{p}$ (c,d) activities for station 51/60. The left panels are for the steady-state models (section 2.3-2.6), and the right panels are for the time-dependent models (section 4.4. and supplementary text S1). The error bars show the uncertainties ± 1 STD).</figcaption></figure>

1271

1272

1273

1274
<figure><img src="image_16.png" /><figcaption>Fig. 14: Same as Fig. 13 but for station 69.</figcaption></figure>

1275

1276
<figure><img src="image_17.png" /><figcaption>Fig. 15: Vertical integral of volumetric $^{7}$Be$_{tot}$ fluxes over the upper 150 m at station 51/60 (a,b) and the upper 100 m at station 69 (c,d). The left panels are for the steady-state models and the right panels are for the time-dependent models. Color convention is: red for atmospheric deposition, dark blue for radioactive decay, green for vertical advection, magenta for vertical diffusion, brown for particle sinking, and light blue for the time-dependency term. The error bars show the uncertainties (± 1 STD).</figcaption></figure>

1277

1278

1279

1280

1281

1282

1283

1284

1285

1286

1287

1288

1289

1290

1291
<figure><img src="image_18.png" /></figure>
Highlights

* An inverse model is used to investigate processes controlling ⁷Be activities

* The influence of reversible exchange processes on ⁷Be activities is regionally variable.

* Reversible exchange cannot always be neglected when using ⁷Be as a tracer in the open ocean.
**Declaration of interests**

☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☒ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Melanie Grenier reports financial support was provided by National Centre for Scientific Research.
Olivier Marchal reports was provided by National Science Foundation. Pieter van-Beek reports was provided by Centre National de la Recherche Scientifique. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.