Parameterization of Convective Boundary Layer Turbulence and Clouds
115
resolution; (2) the initialization is directly based on the TKE budget equation, as
the eddy-diffusivity formulation; (3) the MF profi le avoids the use of an empirical
formula for σ w (Holtslag and Moeng 1991) and requires almost no tuning; and, additionally; and (4) the MF profi le can be directly interpreted in the light of the standard
conceptual picture of turbulence in the CBL.
For the shallow cumulus BL, Soares et al. (2004) followed the bulk mass fl ux
scheme, where the mass fl ux vertical profi le is given by the cloud-core continuity
Equation 4.48 where M c is the cloudy updraft MF. To integrate it, one needs boundary conditions for M c (the cloud base MF) and the fractional entrainment and detrainment rates. The thermodynamic structure of the cloud is given by Equation 4.51.
The cloud base closure needs particular care since it prescribes the initial cloudy
plume properties and represents the ventilation of the subcloud layer (Tiedtke et al.
1988). Betts (1976) introduced a closure of the MF at cloud base to describe the coupling between the two layers. Neggers et al. (2004) examined three different closures
for the MF at the cloud base for a diurnal cycle of shallow cumulus convection. They
concluded that the convective sub-cloud velocity scale closure (Grant 2001) captures
the coupling between the two layers at cloud base, reproducing the timing of both
the maximum and the fi nal decrease in the cloud base MF in LES results. This type
of closure relies on the relationship between the cloud base MF and the TKE in the
subcloud layer. For these reasons, the cloud base mass fl ux was taken as the product
of the core fraction by the updraft vertical velocity (M c = a co w u ). The core fraction
was estimated as 10% of the cloud fraction, a co = 0.1a c , where the cloud fraction a c is
given by the sub-grid condensation scheme, all values computed at cloud base. The
values of entrainment and detrainment rates chosen were Equation 4.47.
The diagnostic of cloud cover and cloud water mixing ratio is a crucial component
of any NWP or mesoscale model due to its potential impacts on the radiation budget.
MesoNH has a statistical sub-grid condensation scheme, based on the distributions
of the grid scale values of θ l and q t , and their variances, which are supplied by the
general turbulence scheme (Sommeria and Deardorff 1977; Cuijpers and Bechtold
1994). Consistent with the EDMF, the evaluation of those variances have both the
ED and MF contributions. Following Lenderink and Siebesma (2000), the variance
of a conserved variable φ was computed according to
2
2
2
2
(
)
u
K
M
z
z
φ
φ
⎛ ⎞
∂φ
∂φ
φ ≅ τ
− τ
φ −φ
′
⎜ ⎟
∂
∂
⎝ ⎠
(4.84)
where the two terms in the r.h.s. account, respectively, for the ED and MF contributions, assuming for simplicity that τ φ = 600 s is a typical eddy-turnover time (Cheinet
and Teixeira 2003).
4.6.2 EDMF RESULTS
4.6.2.1 Dry BL
The idealized case presented here differs slightly from the previous case, once
humidity is taken into account. The surface force in this case corresponds to the
© 2010 by Taylor and Francis Group, LLC
115
resolution; (2) the initialization is directly based on the TKE budget equation, as
the eddy-diffusivity formulation; (3) the MF profi le avoids the use of an empirical
formula for σ w (Holtslag and Moeng 1991) and requires almost no tuning; and, additionally; and (4) the MF profi le can be directly interpreted in the light of the standard
conceptual picture of turbulence in the CBL.
For the shallow cumulus BL, Soares et al. (2004) followed the bulk mass fl ux
scheme, where the mass fl ux vertical profi le is given by the cloud-core continuity
Equation 4.48 where M c is the cloudy updraft MF. To integrate it, one needs boundary conditions for M c (the cloud base MF) and the fractional entrainment and detrainment rates. The thermodynamic structure of the cloud is given by Equation 4.51.
The cloud base closure needs particular care since it prescribes the initial cloudy
plume properties and represents the ventilation of the subcloud layer (Tiedtke et al.
1988). Betts (1976) introduced a closure of the MF at cloud base to describe the coupling between the two layers. Neggers et al. (2004) examined three different closures
for the MF at the cloud base for a diurnal cycle of shallow cumulus convection. They
concluded that the convective sub-cloud velocity scale closure (Grant 2001) captures
the coupling between the two layers at cloud base, reproducing the timing of both
the maximum and the fi nal decrease in the cloud base MF in LES results. This type
of closure relies on the relationship between the cloud base MF and the TKE in the
subcloud layer. For these reasons, the cloud base mass fl ux was taken as the product
of the core fraction by the updraft vertical velocity (M c = a co w u ). The core fraction
was estimated as 10% of the cloud fraction, a co = 0.1a c , where the cloud fraction a c is
given by the sub-grid condensation scheme, all values computed at cloud base. The
values of entrainment and detrainment rates chosen were Equation 4.47.
The diagnostic of cloud cover and cloud water mixing ratio is a crucial component
of any NWP or mesoscale model due to its potential impacts on the radiation budget.
MesoNH has a statistical sub-grid condensation scheme, based on the distributions
of the grid scale values of θ l and q t , and their variances, which are supplied by the
general turbulence scheme (Sommeria and Deardorff 1977; Cuijpers and Bechtold
1994). Consistent with the EDMF, the evaluation of those variances have both the
ED and MF contributions. Following Lenderink and Siebesma (2000), the variance
of a conserved variable φ was computed according to
2
2
2
2
(
)
u
K
M
z
z
φ
φ
⎛ ⎞
∂φ
∂φ
φ ≅ τ
− τ
φ −φ
′
⎜ ⎟
∂
∂
⎝ ⎠
(4.84)
where the two terms in the r.h.s. account, respectively, for the ED and MF contributions, assuming for simplicity that τ φ = 600 s is a typical eddy-turnover time (Cheinet
and Teixeira 2003).
4.6.2 EDMF RESULTS
4.6.2.1 Dry BL
The idealized case presented here differs slightly from the previous case, once
humidity is taken into account. The surface force in this case corresponds to the
© 2010 by Taylor and Francis Group, LLC