Drawing from Larson-Bengzon FEM book, I wanted to experiment with transport instabilities. It looks there might be an instability in my ocean-ice model but before being able to address that, I wanted to wrap my head around the 1D simplest example one could find. And that is the Convection-Diffusion Equation:

(1)

The first term is responsible of the smearing of the velocity field proportionally to , and is thus called the *diffusion term*. Intuitively, it controls how much the neighbors of a given point are influenced by the behavior of ; how much velocities (or temperatures: think of whatever you want!) *diffuse* in the domain.

The second term controls how much the velocities are transported in the direction of the vector field , and is thus called the *convective term*. A requirement for this problem to be well-posed is that — otherwise it would mean that we allow velocities to vanish or to come into existence.

The instability is particularly likely to happen close to a boundary where a Dirichlet boundary condition is enforced. The problem is not localized only close to the boundary though, as fluctuations can travel throughout the whole domain and lead the whole solution astray.

## Transport instability in 1D

The simplest convection-diffusion equation in 1D has the following form:

(2)

whose solution, for small is approximately just . This goes well with the boundary condition at 0, but not with the condition at 1, where the solution needs to go down to 0 quite abruptly to satisfy the boundary condition.

It’s easy to simulate the scenario with FEniCS and get this result (with and the unit interval divided in 10 nodes):

in which we can infer two different trends: one with odd points and one with even points! In fact, if we discretize the 1D equation with finite elements we obtain:

(3)

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