Parameterization of Convective Boundary Layer Turbulence and Clouds
89
Expression 4.40 is equivalent to Equations 4.5 or 4.6, in the MF approach, where the
source terms are gathered in F, with the exception of the term of vertical divergence
of the turbulent fl ux (fi rst term in the r.h.s. member). Thus, the tendency of a property
φ
– , that can be any meteorological variable in a grid point of a numerical weather
prediction model, depends on knowing M c and φ c .
4.3.5 TRANSFER OF PROPERTIES AT INTERFACES
4.3.5.1 Lateral Mixing
Entrainment and detrainment were previously defi ned. They can be both scaled by
the total MF (Tiedtke 1989):
,
,
c
c
E
M
D
M
= ε
= δ
(4.45)
defi ning ε and δ, respectively, the fractional rate of lateral mixing of entrainment and
detrainment, and can be interpreted as inverse mixing length scales (m −1 ).
The studies about lateral mixing processes in isolated thermals and plumes
(Squires and Turner 1962; Simpson and Wiggert 1969; Simpson 1971) suggested a
lateral mixing rate, ε, inversely proportional to the plume radius:
ϑ
ε ≈
radius
(4.46)
with ϑ ≈ 0.2. If one considers a typical cumulus radius, say 500 m, the mixing rate
has the order of magnitude 10 −4 m −1 . This approximation was widely used for representing the lateral mixing associated with an ensemble of cumulus, namely, in the
MF scheme of the ECMWF model (Tiedtke 1989), where these rates were assumed
as ε = δ = 3 × 10 −4 m −1 .
Siebesma and Cuijpers (1995) constructed a LES case study, from observations of
the BOMEX fi eld campaign, with the objective of better understanding the processes
of lateral mixing that occurs in an ensemble of shallow cumulus. These authors concluded that the lateral mixing rates would have to be 10 times higher than those that
were until then considered:
3
1
3
1
2 10 m ;
3 10 m .
−
−
−
−
ε ≈ ×
δ ≈ ×
(4.47)
These values were tested in a 1D model, improving the results for the thermodynamic properties of the cumulus ensemble (Siebesma and Holtslag 1996).
It is worthwhile to mention that this description of the lateral mixing refers to a
cumulus ensemble and not to an individual cumulus. Unlike a dry thermal, where the
lateral mixing seems to be the dominant mixing process, in cumulus this process has
less importance when compared with the cloud-top entrainment (Stull 1988). Hence,
© 2010 by Taylor and Francis Group, LLC
89
Expression 4.40 is equivalent to Equations 4.5 or 4.6, in the MF approach, where the
source terms are gathered in F, with the exception of the term of vertical divergence
of the turbulent fl ux (fi rst term in the r.h.s. member). Thus, the tendency of a property
φ
– , that can be any meteorological variable in a grid point of a numerical weather
prediction model, depends on knowing M c and φ c .
4.3.5 TRANSFER OF PROPERTIES AT INTERFACES
4.3.5.1 Lateral Mixing
Entrainment and detrainment were previously defi ned. They can be both scaled by
the total MF (Tiedtke 1989):
,
,
c
c
E
M
D
M
= ε
= δ
(4.45)
defi ning ε and δ, respectively, the fractional rate of lateral mixing of entrainment and
detrainment, and can be interpreted as inverse mixing length scales (m −1 ).
The studies about lateral mixing processes in isolated thermals and plumes
(Squires and Turner 1962; Simpson and Wiggert 1969; Simpson 1971) suggested a
lateral mixing rate, ε, inversely proportional to the plume radius:
ϑ
ε ≈
radius
(4.46)
with ϑ ≈ 0.2. If one considers a typical cumulus radius, say 500 m, the mixing rate
has the order of magnitude 10 −4 m −1 . This approximation was widely used for representing the lateral mixing associated with an ensemble of cumulus, namely, in the
MF scheme of the ECMWF model (Tiedtke 1989), where these rates were assumed
as ε = δ = 3 × 10 −4 m −1 .
Siebesma and Cuijpers (1995) constructed a LES case study, from observations of
the BOMEX fi eld campaign, with the objective of better understanding the processes
of lateral mixing that occurs in an ensemble of shallow cumulus. These authors concluded that the lateral mixing rates would have to be 10 times higher than those that
were until then considered:
3
1
3
1
2 10 m ;
3 10 m .
−
−
−
−
ε ≈ ×
δ ≈ ×
(4.47)
These values were tested in a 1D model, improving the results for the thermodynamic properties of the cumulus ensemble (Siebesma and Holtslag 1996).
It is worthwhile to mention that this description of the lateral mixing refers to a
cumulus ensemble and not to an individual cumulus. Unlike a dry thermal, where the
lateral mixing seems to be the dominant mixing process, in cumulus this process has
less importance when compared with the cloud-top entrainment (Stull 1988). Hence,
© 2010 by Taylor and Francis Group, LLC