# Eulerian simulation of drift trajectories

## Convection

Convection consists of advection and diffusion.

Diffusion is given by:

\[\frac{\partial U}{\partial t} = D \left( \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} \right)\]

The convection equation is (wiki):

\[\frac{\partial c}{\partial t} = ...\]

with the assumptions that:

the diffusion constant D is constant for the field,

and that the flow u is incompressible (i.e. has

no divergence).

the equation simplifies to:

\[\frac{\partial c}{\partial t} = D \nabla^2 c - \mathbf{v} \cdot \nabla T\]

where \(\nabla^2 = \triangle\) is the Laplacian.

## Diffusion

Diffusivity (\(m^2/s\)). E.g. between 0.01 and 0.1 for oil on the surface of the ocean (Matsuzakia et. al., 2017).

Decreasing diffusivity places stricter stability criteria on time step.

### Porosity

Porosity, rate of liquid volume to total volume (fraction of flux)

## Numerical schemes

### Explicit simulation

A simple explicit scheme for integrating the convection-equation.

Forward difference in time

ndimage.laplace and np.gradient for spatial differences.