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

Warning

Implementation is in progress, this option is not yet supported

The Yonsei University (YSU) PBL model is another commonly use scheme in WRF. It includes nonlocal mixing with contergradient diffusion within the PBL, and a local mixing treatment outside the PBL.

Turbulent diffusion for prognostic variables (\(C, u, v, \theta, q_k\)), where \(q_k\) includes all moisture variables and \(C\) any additional scalars (other terms in the equations omitted for brevity):

\[\frac{\partial C}{\partial t} = \frac{\partial}{\partial z} \left[ K_c \left( \frac{\partial C}{\partial z} - \gamma_c \right) - \overline{\left(w'c' \right)_h} \left( \frac{z}{h} \right)^3 \right]\]

Note

Not applied for vertical velocity?

Where for each variable the turbulent diffusion coefficient \(K_c\), countergradient correction \(\gamma_c\), and entrainment flux at the PBL top \(\overline{\left(w'c' \right)_h}\) must be diagnosed for each variable. The main controlling parameter is the PBL height \(h\). Notably, a nonlocal model for turbulent diffusion is used for \(z \leq h\), but a local model is used for \(z \ge h\).

The first step is to determine the PBL height \(h\). This is defined as the smallest value of \(z\) where the bulk Richardson number equals the critical value, which is taken to be 0:

\[{\rm Rib}(z) = \frac{ g \left[ \theta_m(z) - \theta_s\right] }{\theta_{ma} U(z)^2}z\]
\[{\rm Rib}(h) = {\rm Rib_{cr}} = 0\]

where

  • \(\theta_m\) is the moist potential temperature,

  • \(\theta_{ma}\) is the value at the lowest vertical cell in a column,

  • \(U = \sqrt{u^2 + v^2}\) is the horizontal wind speed,

  • \(\theta_s = \theta_{ma} + \theta_T\) is the virtual temperature near the surface,

  • \(\theta_T = a\frac{\overline{\left(w'\theta_m' \right)_0}}{w_{s0}}\) is the excess virtual temperature near the surface,

  • \(a\) is a constant taken to be 6.8 per HND06 (matching the \(b\) constant that appears elsehwere in the YSU model)

  • \(\overline{\left(w'\theta_m' \right)_0}\) is the surface virtual heat flux (determined from the MOST surface layer model),

  • \(w_{s}(z) = \left(u_*^3 + 8 k w_{*b}^3z/h \right)^{1/3}\) is a representative velocity scale in the mixed layer, with \(w_{s0} = w_s(h/2)\) (note this equation matches the WRF implementation and description in H10, but differs from HND06, where \(\phi_m\) appears in place of the constant 8),

  • \(u_*\) is the surface frictional velocity scale determined from the MOST surface layer model,

  • \(k = 0.4\) is the von Karman constant

  • \(w_{*b} = \left[ g/\theta_{ma} \overline{\left(w'\theta_m' \right)_0} h \right]^{1/3}\) for \(\overline{\left(w'\theta_m' \right)_0} > 0\), \(w_{*b} = 0\) otherwise, is a convective velcoity scale for moist air

In practice, an approximate value of \(h\) is determined through a two-step process. First, \(\theta_T\) is set to be zero and a provisional value of \(h\) is estimated. Then this provisional value of \(h\) is used to compute \(\theta_T\), which is in turn used to provide an improved estimate of \(h\), which is the value used in subsequent calculations.

Note

This two step-process matches the WRF implementation, but this could be extended iteratively to reach convergence.

Countergradient corrections are computed as follows:

\[\gamma_\theta =\]
\[\gamma_u =\]
\[\gamma_v =\]
\[\gamma_{q_k} = \gamma_C = 0\]

Entrainment fluxes are computed:

\[\overline{\left(w'c' \right)_h} =\]
\[\overline{\left(w'c' \right)_h} =\]

Within the PBL (\(z \leq h\)),

Useful References

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