# 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, largely matching the original forumulation proposed by Nakanishi and Niino in a series of papers from 2001 to 2009.

## MYNN Level 2.5 PBL Model¶

In this model, the vertical turbulent diffusivities are computed in a local manner based on a gradient diffusion approach with coefficients computed based on a transported turbulent kinetic energy value. The implementation and description here largely follows Nakanishi and Niino, Journal of Meteorological Society of Japan, 2009, but has also been influenced by the full series of papers that led to the development of this model and by a few documents published since then, as listed in the Useful References section below.

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 Boundaries. 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.

### Useful References¶

The following references have informed the implementation of the MYNN PBL model in ERF:

Discussions with Branko Kosovic (NCAR) and Joseph B. Olson (NOAA) have also played a major role in informing the implementation of MYNN PBL models in ERF.

## MYNN-EDMF Level 2.5 PBL Model¶

More recent advancements that add significant complexity to the MYNN scheme have been incorporated into WRF, as described in Olson et al. 2019. These advancements are not included in ERF, but may be in the future.

## YSU PBL Model¶

The Yonsei University (YSU) PBL model is another commonly use scheme in WRF. It is not yet supported in ERF, but is under development.