# 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.

https://en.wikipedia.org/wiki/Numerical_solution_of_the_convection%E2%80%93diffusion_equation#Solving_the_convection%E2%80%93diffusion_equation_using_the_finite_difference_method