PBL Schemes

Planetary Boundary Layer (PBL) schemes are used to model unresolved transport in the vertical direction within the planetary boundary layer when mesh resolutions are too coarse to resolve any of the turbulent eddies responsible for this transport (1 km grid resolution or larger). The PBL scheme is used to provide closure for vertical turbulent fluxes (i.e., \(\widetilde{w'\phi'} = \widetilde{w\phi} - \widetilde{w}\widetilde{\phi}\), for any quantity \(\phi\)). PBL schemes may be used in conjunction with an LES model that specifies horizontal turbulent transport, in which case the vertical component of the LES model is ignored.

Right now, the only PBL scheme supported in ERF is the Mellor-Yamada-Nakanishi-Niino Level 2.5 model.

MYNN Level 2.5 PBL Model

In this model, the vertical turbulent diffusivities are computed in a local manner based on a transported turbulent kinetic energy value. The model was proposed by Nakanishi and Niino, building on the work of Mellor and Yamada

The prognostic equation for \(q^2 = \widetilde{u_i u_i} - \widetilde{u}_i\widetilde{u}_i\) is

\[\frac{\partial \bar{\rho} q^2}{\partial t} + \left[ \frac{\partial \bar{\rho} \widetilde{u}_i q^2}{\partial x_i} \right] = \frac{\partial}{\partial z} \left(K_{q,v} \frac{\partial q^2}{\partial z} \right) + 2\bar{\rho} \left(-\widetilde{u'w'} \frac{\partial \widetilde{u}}{\partial z} - \widetilde{v'w'}\frac{\partial \widetilde{v}}{\partial z} + \beta g \widetilde{w'\theta'} - \frac{q^3}{B_1 l} \right)\]

where \(B_1\) is a model parameter, \(\beta\) is the thermal expansion coefficient and l is a lengthscale. The vertical turbulent transport coefficients are then computed:

\[K_{m,v} = l q S_m, K_{q,v} = 3 l q S_m, K_{\theta, v} = l q S_\theta\]

where \(S_m\) and \(S_\theta\) are stability parameters thaat account for bouyancy effects. These coefficients are then applied in evaluating the vertical component of turbulent fluxes in a similar manner as is described for the Smagorinsky LES model. Computation of the stability parameters and lengthscale depend on the Obukhov length and surface heat flux, which are obtained from the MOST module. Further detail on these computations can be found in the cited works. Several model coefficients are required, with default values in ERF taken from the work of Nakanishi and Niino.