Supplementary material to "Particle flux dynamics amplify marine carbon cycle differences between climate states"

Implementation of the implicit version of MSPACMAMMSPACMAM (Dinauer et al., 2022) calculates vertical profiles of organic (POC), calcite, aragonite, and opal (OPAL) particles through depth-by-depth temporal integration of particle losses and downward fluxes until a dynamic equilibrium is achieved.This approach requires much shorter time steps than those of Bern3D (∼3 days), making its direct implementation costly.To reduce computational cost and increase numerical stability, we devised an implicit scheme which estimates the dynamic 5 equilibrium based on primary production, particle sinking speed and environmental conditions in the water column.Our model operates on a 3D grid with 𝑖 and 𝑗 indicating longitude and latitude of the center of a grid cell.𝑘 ′ indicates the depth of the upper boundary of the grid cell 𝑘, and 𝑘 ′ +1 indicates the lower boundary of the grid cell .𝑑𝑧(𝑘) = 𝑧(𝑘 ′ +1)-𝑧(𝑘 ′ ) is the thickness of cell 𝑘.In the following 𝑖, 𝑗 are omitted to ease notation.Variables are specific for depth (𝑘), size class (𝑙) and particle component (𝑚; POC,calcite,aragonite,opal): 10 𝑓 0(0, 𝑙, 𝑚): export flux out of euphotic zone (mol/(m 2 /s)) 𝑣(𝑘, 𝑙): mean sinking speed in box 𝑘 (m/s) (approximated by 𝑣(𝑘 ′ , 𝑙)) 𝑑𝑡 𝑃 (𝑘, 𝑙) = 𝑑𝑧(𝑘)/𝑣(𝑘, 𝑙) : average time particle is spending in box 𝑘 (s) (approximated by 𝑑𝑧(𝑘)/𝑣(𝑘 ′ , 𝑙) 𝑟 𝑃 (𝑘 ′ , 𝑙, 𝑚): particle remineralization/dissolution rate (1/s) at depth 𝑘 ′ 15 𝑑𝑡: model time step (s)The 1-dimensional equation for the divergence of a particle flux can be written as 𝜕𝑐 𝜕𝑡 + 𝜕𝑓 𝜕𝑧 = 0, with particle concentration 𝑐, particle velocity 𝑣, and particle flux 𝑓 = 𝑐•𝑣.For particles dissolving with rate 𝑟 𝑃 , the temporal change in particle concentrationWhere 𝜕𝑧/𝑣 is the time, 𝑑𝑡 𝑃 , the particles require for travelling 20 distance 𝜕𝑧.We get 𝜕𝑓 = 𝑟 𝑃 • 𝑓 • 𝑑𝑡 𝑃 and this equation is transferred to our model grid.We can then calculate the degradation rate in box k between depth k' and k'+1 either relative to the input f(k',l,m), or output f(k'+1,l,m), of the box, or a mix of the two, resulting in three different approaches to derive f(k'+1,l,m) from f(k',l,m):