Some Pitfalls in Numerical Modelling of Vertical Turbulent Mixing

In applied oceanography, Lagrangian particle methods are commonly used to model advection-diffusion problems, where the diffusion part is modelled by a random walk. Using the theory of stochastic differential equations (SDEs), it can be shown that a correctly formulated random walk is equivalent with the diffusion equation, but this requires certain conditions to be satisfied, among them that the diffusivity is a reasonably well-behaved function.

We consider vertical turbulent diffusion in the water column with a reflecting boundary at the surface, and study three different cases: Step function diffusivity, linearly interpolated diffusivity, and diffusivity that goes linearly to zero at the surface boundary. In each of these cases, extra care must be taken to obtain results that are consistent with the diffusion equation. We use the so-called well-mixed condition (WMC) as a check for Eulerian consistency, and consider different strategies for achieving consistency. These strategies include different numerical schemes for SDEs, different reflection schemes, and small adjustments to make the diffusivity more well-behaved. The different approaches are evaluated with respect to numerical efficiency, order of convergence and practical applicability.