Eulerian simulation of drift trajectories


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.


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, 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.