Numerical peculiarities in solving the planetary boundary layer equations

The numerical experiments performed with all possible combinations of approximations for the equations of a one-dimensional planetary boundary layer model employing a one-equation turbulence closure scheme and a staggered, homogenous, finite-difference grid show that numerical peculiarities in the form of spikes and jittering can be detected in the vertical profiles of the predicted variables. It is found that these peculiarities are caused by an inappropriate numerical description of the turbulent kinetic energy equation. Spikes occur for large time steps only and have a life time of several days. There exist two types of jittering: transient and permanent. The former has a lifetime of about 1 week and occurs with large time steps, whereas the latter has a lifetime of more than a 1000 days and is observed when using moderate time steps. It is found that transition to instability of the solution can either be a sudden or a gradual one. In the first case, no numerical peculiarities are observed, whereas in the second case, jittering is present during the entire transition period. In this sense, jittering may be regarded as a herald of instability. The only way of avoiding these numerical artefacts is to employ implicit approximations consistently for all the terms in the model equations.