By substitution in Eq. (B16), the following holds.

$$
\begin{align*}
& \sum_{j=1}^{n} (\phi_{ij} \tilde{W}_{ij} - \phi_{ji} \tilde{W}_{ji}) = 0 \\
\Leftrightarrow & \sum_{j=1}^{n} \left[ \left(1 + \frac{\lambda_j - \lambda_i}{2} \tilde{W}_{ij}\right) \tilde{W}_{ij} \right. \\
& \left. - \left(1 + \frac{\lambda_i - \lambda_j}{2} \tilde{W}_{ji}\right) \tilde{W}_{ji} \right] = 0 \\
\Leftrightarrow & \sum_{j=1}^{n} \left[ \frac{\lambda_j - \lambda_i}{2} \tilde{W}_{ij}^2 - \frac{\lambda_i - \lambda_j}{2} \tilde{W}_{ji}^2 + \tilde{W}_{ij} - \tilde{W}_{ji} \right] = 0 \\
\Leftrightarrow & \sum_{j=1}^{n} \left[ \lambda_j \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} - \lambda_i \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} + \tilde{W}_{ij} - \tilde{W}_{ji} \right] = 0 \\
\Leftrightarrow & \sum_{j=1}^{n} \left[ (\lambda_j - \lambda_i) \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} + \tilde{W}_{ij} - \tilde{W}_{ji} \right] = 0 \\
\Leftrightarrow & \sum_{j=1}^{n} \left[ (\lambda_j - \lambda_i) \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} \right] = -y_i \tag{B19}
\end{align*}
$$

With the same definition of the divergence vector **y** as in Sect. B1,

$$y_i = \sum_j (\tilde{W}_{ij} - \tilde{W}_{ji}). \quad (B20)$$

Here again, one must notice that the system of equations (B19) only has, at most, $n-1$ linearly independent equations: applying $\sum_i$ to Eq. (B19) gives the following.

$$
\begin{align*}
& \sum_{i=1}^{n} \sum_{j=1}^{n} (\lambda_j - \lambda_i) \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} = - \sum_{i=1}^{n} y_i \\
\Leftrightarrow & \sum_{i=1}^{n} \sum_{j=1}^{n} \lambda_j \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} \\
& - \sum_{i=1}^{n} \sum_{j=1}^{n} \lambda_i \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} = - \sum_{i=1}^{n} y_i \\
\Leftrightarrow & \sum_{i=1}^{n} \sum_{j=1}^{n} \lambda_j \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} \\
& - \sum_{j=1}^{n} \sum_{i=1}^{n} \lambda_i \frac{\tilde{W}_{ji}^2 + \tilde{W}_{ij}^2}{2} = - \sum_{i=1}^{n} y_i \\
\Leftrightarrow & 0 = - \sum_{i=1}^{n} y_i \tag{B21}
\end{align*}
$$

And we show in Appendix B1 that we indeed have $\sum_i y_i = 0$. In addition, one may notice that, as in Appendix B1, if $\lambda$ is a solution of Eq. (B19), then $\lambda + c$ (for any constant $c$) is also a solution and that adding $c$ does not change the value of $\phi$ (Eq. B18). We therefore choose to impose $\sum_i \lambda_i = 0$. To

do so, we use the following change of variables to substitute vector $\lambda$ by vector **x**.

$$
\left\{
\begin{array}{l}
x_1 = \lambda_1 \\
x_2 = \lambda_2 \\
\vdots \\
x_n = \sum_{k=1}^{n} \lambda_k
\end{array}
\right.
\quad \Leftrightarrow \quad
\left\{
\begin{array}{l}
\lambda_1 = x_1 \\
\lambda_2 = x_2 \\
\vdots \\
\lambda_n = x_n - \sum_{k=1}^{n-1} x_k
\end{array}
\right.
\qquad (B22)
$$

Equation (B19) is only solved for $i \neq n$ and $i = n$. Indeed, the last equation ($i = n$) can be deduced from the first $n-1$ ones. Hence, $\forall i \neq n$.

$$
\begin{align*}
& \sum_{j=1}^{n} \left[ (\lambda_j - \lambda_i) \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} \right] = -y_i \\
\Leftrightarrow & \sum_{j=1}^{n-1} \left[ (x_j - x_i) \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} \right] \\
& + \left[ \left( x_n - \sum_{k=1}^{n-1} x_k - x_i \right) \frac{\tilde{W}_{in}^2 + \tilde{W}_{ni}^2}{2} \right] = -y_i
\end{align*}
$$

Using the fact that $x_n = 0$ (corollary of $\sum_i \lambda_i = 0$), the following holds.

$$
\begin{align*}
\Leftrightarrow & \sum_{j=1}^{n-1} x_j \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} - \sum_{j=1}^{n-1} x_i \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} \\
& - \left( \sum_{k=1}^{n-1} x_k \right) \frac{\tilde{W}_{in}^2 + \tilde{W}_{ni}^2}{2} - x_i \frac{\tilde{W}_{in}^2 + \tilde{W}_{ni}^2}{2} = -y_i \\
\Leftrightarrow & \sum_{j=1}^{n-1} x_j \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} - \sum_{k=1}^{n-1} x_k \frac{\tilde{W}_{in}^2 + \tilde{W}_{ni}^2}{2} \\
& - \sum_{j=1}^{n} x_i \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} = -y_i
\end{align*}
$$

Then, substituting $k$ for $i$ in the sum of the second term gives the following.

$$
\begin{align}
\Leftrightarrow & \sum_{j=1}^{n-1} \frac{\tilde{W}_{in}^2 + \tilde{W}_{ni}^2 - \tilde{W}_{ij}^2 - \tilde{W}_{ji}^2}{2} x_j \nonumber \\
& + \left( \sum_{j=1}^{n} \frac{\tilde{W}_{ij}^2 + \tilde{W}_{ji}^2}{2} \right) x_i = y_i \tag{B23}
\end{align}
$$

Equation (B23) is a linear system $\mathbf{M} \cdot \mathbf{x} = \mathbf{y}$ (of $n-1$ equations and $n-1$ unknown $x_i$), with $\mathbf{M}$ a size $(n-1)$ square matrix defined by

$$
M_{ij} = \begin{cases} \frac{1}{2}(\tilde{W}_{in}^2 + \tilde{W}_{ni}^2 - \tilde{W}_{ij}^2 - \tilde{W}_{ji}^2) & \text{if } i \neq j \\ \frac{1}{2}(\tilde{W}_{in}^2 + \tilde{W}_{ni}^2 - \tilde{W}_{ii}^2 - \tilde{W}_{ii}^2) + \sum_{k=1}^{n} [\tilde{W}_{ik}^2 + \tilde{W}_{ki}^2]) & \text{if } i = j. \end{cases} \quad (B24)
$$
This system of $n-1$ equations is solved by numerically inverting the matrix **M**, i.e., $\mathbf{x} = \mathbf{M}^{-1} \mathbf{y}$. We note that there is no guarantee that **M** is invertible, so this method may not always be numerically stable, in contrast to the additive correction method. Once **M** is inverted and vector **x** is computed (with the additional term $x_n = 0$), $\lambda$ is computed using Eq. (B22), and finally, $\phi$ is computed using Eq. (B18).

## Appendix C: Flux matrices used by GEOCLIM

Figure C1 shows the default flux matrix **W** that was used in previous GEOCLIM versions and the one computed from IPSL-CM5A2 pre-industrial climate fields with the method presented in the article (Sect. 3.2 and Appendix B). The “historical” flux matrix significantly underestimated the horizontal exchanges between coastal and open-ocean boxes and did not represent the connection between coastal boxes and high-latitude ones. When estimated from IPSL-CM5A2 fields, the horizontal exchanges between southern high latitudes and midlatitudes also appears to be larger than “historical” exchanges. The deepwater formation estimated from IPSL-CM5A2 is 10.3 Sv in northern high latitudes and 8.9 Sv in southern high latitudes, which is consistent with the intensity diagnosed from IPSL-CM5A2 simulations (Sepulchre et al., 2020) though somewhat smaller for the southern high latitudes. This underestimation is likely due to the summing of upward and downward velocities within the area of definition of GEOCLIM boxes #8 and 9.

## Appendix D: Fourier analysis of oceanic oxygen time series in GEOCLIM Turonian experiment

This Appendix presents the Fourier transform analysis of the experiments presented in Sect. 4 (more specifically, Sect. 4.5). We computed the Fourier transforms of the 95–85 Ma time series of the three orbital parameters (Fig. D1a–c) and of the Arctic and North Atlantic open-ocean oxygen concentration of the corresponding GEOCLIM experiment (Fig. D1d–i). For all time series, the time average was removed before computing the fast Fourier transform. To identify the contribution of the three orbital parameters in the oxygen time series, we simply selected non-overlapping frequency ranges (colored in Fig. D1) and looked at the peaks within those frequency ranges in the oxygen Fourier transform.

<figure><img src="image_17.png" /></figure>

**Figure C1.** Flux matrices **W** representing oceanic circulation between GEOCLIM boxes in pre-industrial default nine-box GEOCLIM configuration. Fluxes are oriented from column *i* to line *j*. The numbering of boxes is the default GEOCLIM one (#1–2: N Hemisphere high latitudes (surface and deep), #3–4–5: midlatitude (surface, intermediate, and deep), #6–7: coastal oceans (surface and deep), #8–9: S Hemisphere high latitudes). However, the positions of the box pairs (6, 7) and (8, 9) have been swapped in the figure to better illustrate the distinction between “open-ocean” and “coastal” boxes.
<figure><img src="image_18.png" /></figure>

**Figure D1.** Amplitude spectrum of the Fourier transforms of the 95–85 Ma time series of orbital parameters and of open-ocean oceanic oxygen concentration from the 95–85 Ma GEOCLIM transient simulation (see Sect. 4.5). (a) Eccentricity, (b) cosine of precession angle ω, (c) obliquity, (d) O₂ in the surface (0–100 m) Arctic, (e) O₂ in the intermediate (100–900 m) Arctic, and (f) O₂ in the deep (900 m–bottom) Arctic; (g–i) same as (d–f) for the North Atlantic. Small encapsulated panels are close-ups of the 0–0.02 kyr⁻¹ frequency range for the corresponding bigger panels. Minor ticks on the “period” axis are every 1 kyr up to 20 kyr, every 10 kyr up to 100 kyr (200 kyr in the close-up panel), and every 100 kyr from 200 kyr to 1 Myr in the close-up panel. The non-overlapping frequency ranges associated with each orbital parameter are colored in red (eccentricity), violet (precession), and green (obliquity).

*Code and data availability.* The code and instructions for running the model are available on the GEOCLIM GitHub repository, which is archived on Zenodo (all-version “concept”, DOI: https://doi.org/10.5281/zenodo.5246621, Maffre, 2025a). The specific version 7.0.1 of GEOCLIM can be found at https://doi.org/10.5281/zenodo.15808864 (Maffre, 2025b), and the modified version dedicated to the Cenomanian–Turonian simulations can be found at https://doi.org/10.5281/zenodo.15808899 (branch “Mil-90Ma” of the GitHub repository, Maffre, 2025c). The outputs of climate simulations (used as inputs by GEOCLIM) and the outputs of GEOCLIM simulations presented in this article can be found on a separate Zenodo archive at https://doi.org/10.5281/zenodo.14228131 (Maffre, 2024).

*Author contributions.* YG, YD, GLH, EN, and PM conceptualized the model and developed its code. PM developed the methods to generate GEOCLIM boundary conditions and wrote the associated code. PM also formatted the different codes for the distribution of GEOCLIM. ACS and YD conducted the climate simulations. PM conducted the GEOCLIM simulations, and PM, YD, and ACS analyzed their results. PM wrote Sects. 1 to 3, and PM and YD wrote Sect. 4. All authors contributed to reviewing and editing.

*Competing interests.* The contact author has declared that none of the authors has any competing interests.
*Disclaimer.* Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

*Acknowledgements.* This work used computing hours on the HPC resources of the TGCC under GENCI allocations A0110102212 and A0130102212.

*Financial support.* This research has been supported by the Agence Nationale de la Recherche (grant no. ANR-20-CE49-0014 RISE).

*Review statement.* This paper was edited by Tatiana Egorova and reviewed by two anonymous referees.

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