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Basics

The mass conservation of a passive tracer \(C\) in a fluid can be summarized as follows:

\[ \frac{\partial C}{\partial t} = \underbrace{-\vec{v}\nabla C}_{Advection} + \underbrace{\nabla_{h}.(K_{h}\nabla_{h}C) + \frac{\partial}{\partial z}K_{z} \frac{\partial C}{\partial z}C}_{\substack{\text{Diffusion}\\ \text{(unresolved processes)}}} + \underbrace{SMS(C)}_{\substack{\text{Biogeochemical}\\ \text{Sources minus Sinks}}} \]

The first term on the right hand side of the equation represents the advection of the tracer \(C\) in the three dimensions. It can be interpreted as the budget between the incoming and outgoing tracer fluxes of a volume cell \(V_t= e_{1t}\cdot e_{2t}\cdot e_{3t}\) where \(e_{1t}\), \(e_{2t}\), and \(e_{3t}\) are the horizontal and vertical scaling factors of each model grid cell.

The second term represents the change due to horizontal diffusion (left side) and vertical diffusion (right side), the latter parameterized as eddy diffusion to represent vertical turbulent fluxes.

Those two terms, representing the physical transport, are computed along the dynamics. They will not be described in this document as they are set in the configuration of the hydrodynamic model driving PISCES. Documentation dedicated to each of these two models is available below:

  • NEMO Ocean Engine Reference manual: DOI
  • CROCO Technical and Numerical Documentation: DOI

\(SMS(C)\), the last term on the right hand side of the equation, is the Sources Minus Sinks inherent to the tracer \(C\). The \(SMS(C)\) term includes all biogeochemical processes, as well as gas exchange, river inputs, sediment fluxes, and gravitational sinking. For example, in the case of phytoplankton, \(SMS(C)\) is the balance between phytoplankton growth and its loss through mortality and grazing.