Linearization of a simplified planetary boundary layer parameterization

This study examines the linearization properties of a simplified planetary boundary layer parameterization based on the vertical diffusion equations, in which the exchange coefficients are a function of the local Richardson number and wind shear. Spurious noise, associated with this parameterization, develops near the surface in the tangent linear integrations. The origin of this problem is investigated by examining the accuracy of the linearization and the numerical stability of the scheme used to discretize the vertical diffusion equations. The noise is primarily due to the linearization of the exchange coefficients when the atmospheric state is near neutral static stability and when a long time step is employed. A regularization procedure based on the linearization error and a criterion for the numerical stability is proposed and tested. This regularization is compared with those recently adopted by Mahfouf, who neglects the perturbations of the exchange coefficients, and by Janiskova et al., who reduce the amplitude of those perturbations when the Richardson number is in the vicinity of zero. When the sizes of the atmospheric state perturbations are 1 m/s for the winds and 1 K for the temperature, which is the typical size of analysis increments, regularizations proposed here and by Janiskova et al. perform similarly and are slightly better than neglecting the perturbations of the exchange coefficients. On the other hand, when the state perturbations are much smaller (e.g., 3 orders of magnitude smaller), the linearization becomes accurate and a regularization is no longer necessary, as long as the time step is short enough to avoid numerical instability. In this case, the regularization proposed here becomes inactive while the others introduce unnecessary errors.