# Theory

## Mass-Dependent Decay

CarbonDrift has two mass decays implemented. The first is a mass-dependent decay $$ \frac{dm}{dt} = -km, $$ where $k = k(T(z(t)))$ is a first order decay rate. There exist two models for the decay rate (Lebrato et al, 2019): a linear model $$ k_{\mathrm{lin}} = 0.064\,\mathrm{^\circ C^{-1}\,d^{-1}}\, T(z) + 0.02\,\mathrm{d^{-1}}, $$ and an exponential model $$ k_{\mathrm{exp}} = 0.140\,\mathrm{d^{-1}}\,\exp\big(0.125\,\mathrm{^\circ C^{-1}\,}T(z)\big). $$

## Surface-Dependent Decay

The second is a surface-dependent decay $$ \frac{dm}{dt} = -\kappa S, $$ where $S$ is the surface area and $\kappa$ is now a different decay rate. Assuming a spherical-body approxiamtion, we can write $$ \frac{dm}{dt} = -km^{2/3}m_0^{1/3}, $$ where $m_0 = m(t=0)$ is the initial mass and $k$ is again the Lebrato et al, (2019) decay rate.

## Vertical Sinking Velocity

There are two models for the vertical sinking velocity. A constant sinking velocity $$ w = w_0, $$ and a variable mass-dependent sining velocity $$ w(t) = w_0 \sqrt[6]{\frac{m(t)}{m_0}}. $$

## Initial GZ Mass

The initial mass conditions can be given in a number of ways, e.g as a relative value, i.e. one unit of mass per each grid cell. The other option is to asign a different mass to each grid cell using biome speciffic values reported in Luo et al, (2020). Here, the ocean is split into 4 biomes, based on chlorophyll, stratification and ocean-depth data. The initial mass is then distributed in such a way, that the intial mass flux is constantant across each biome.

## Horizontal Advection

Horizontal dynamics are governed by ocean advection. OpenDrift - a Lagrangian Particle Tracking tool - is used to advect the particles at each time step using 3D Eulerian velocity fields. This is achieved with the OceanDrift.advect_with_ocean(), which is inherited by CarbonDrift.