Unsteadiness in the Prandtl model of the katabatic flow due to Coriolis effects

Katabatic flow on glaciers is a most pronounced boundary layer phenomonon in the polar regions. It also plays an important part in the atmospheric general ciruculation at high latitudes. Unlike the strongly-stable ABL on flat terrain, where turbulence is surface induced, the katabatic flow jet is the main source of turbulence production in the ABL on sloping surfaces. Still, our knowledge on the interactions between turbulence and large-scale geophysical processes remains insufficient, e.g. how strongly-stratified turbulence may impact the large-scale flow. This question was tackled analytically by means of the Prandtl model for a simple, idealized, environment: a long cold slope where Coriolis force acts to deflect the katabatic flow.

If the Coriolis force is added to the classic Prandtl model of the katabatic flow, the cross-slope wind component does not approach a true steady state, but rather diffuses upwards in time without a well defined time scale. Thus straightforward analytical solution to the steady Prandtl model does not satisfy the boundary conditions for any finite time and is not a good representation of the katabatic flow. On the other hand, the down-slope component and the potential temperature perturbations do reach stationarity on the same time scale as in the classic Prandtl model, so that neglecting the Coriolis term and in that way going back to the classic Prandtl model solutions represents a valid approximation. The cross-slope component can thus be decoupled from the rest of the flow but must be time-dependent. An asymptotic solution for unsteady cross-slope component was derived and was shown to be an excellent approximation to the time-dependent system.

We conclude that the steady Prandtl model is not equivalent to its time dependent counterpart, even after long time periods and that the simplest Prandtl model which includes the Coriolis effect should include time-variations of the cross-slope wind component as well. This is an important issue for parameterizations which often assume the existence of a steady state. Moreover the asymptotic solution indicates that the cross-slope component may affect the circum-polar stratospheric vortex after a few months of polar night.