Parameterization of Convective Boundary Layer Turbulence and Clouds
79
Different fi rst-order theories developed by Prandtl, Taylor, and von Karman are
presented by Monin and Yaglom (1971). Those approaches showed some skill when
tested with measurements in wind tunnels and laboratorial tanks, and even with
some atmospheric observations, that at the time were limited to the surface layer.
Consequently, they have been widely used.
The K-diffusion theories are essentially theories of small eddies that represent
well the turbulent mixing in neutral and steady BLs (Stull 1984, 1993). These are
appropriate when the mixing length is smaller than the resolution of the model,
and belong to the inertial subrange of the spectra of energy of the turbulence.
However, they give a defi cient representation of the nonlocal turbulent fl uxes associated with thermals, of vertical extension larger than the vertical resolution of the
BL model.
Lilly (1962) considered a formulation to take into account the effect of buoyancy,
proposing that eddy diffusivities were made dependent on the Richardson number
(4.13). The Prandtl approach can be derived from a steady-state equation of e – , equaling the production by shear with the dissipation. If buoyancy is included in the production term, it becomes
∂
= ∂
2
( ),
u
K l
F Ri
z
(4.12)
where Ri is the gradient Richardson number. Ri is based on the K-diffusion approach,
more precisely in Equations 4.7 and 4.8 allowing to write,
2
2
,
v
v
g
z
Ri
u
v
z
z
∂θ
∂
θ
= ⎡
⎤
∂
∂
⎛ ⎞ ⎛ ⎞
+
⎢
⎥
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∂
∂
⎢
⎥
⎣
⎦
(4.13)
To solve Equation 4.12 one has to specify l and F(Ri). Usually, it is assumed that
l → kz in the surface layer and F(Ri) → 1 in a neutral stratifi cation, where k is the
von Karman constant.
Both Expressions 4.11 and 4.12 depend critically on l. On the basis of these observations, Blackadar (1962) built an empirical formulation for l:
1 1 1 ,
l kz
=
+ λ
(4.14)
where λ is an adjustable parameter representing an asymptotic mixing length. This
expression consists in an interpolation between the two limits: l → kz when z → 0
and l → λ when z → + ∞. Other expressions for the CBL are based on empirical prescription of profi les for K, making use of empirical expressions (Holtslag and Moeng
1991). Often, these expressions depend on the computation of the BL height, z i , and
other scale variables of the surface layer.
© 2010 by Taylor and Francis Group, LLC
79
Different fi rst-order theories developed by Prandtl, Taylor, and von Karman are
presented by Monin and Yaglom (1971). Those approaches showed some skill when
tested with measurements in wind tunnels and laboratorial tanks, and even with
some atmospheric observations, that at the time were limited to the surface layer.
Consequently, they have been widely used.
The K-diffusion theories are essentially theories of small eddies that represent
well the turbulent mixing in neutral and steady BLs (Stull 1984, 1993). These are
appropriate when the mixing length is smaller than the resolution of the model,
and belong to the inertial subrange of the spectra of energy of the turbulence.
However, they give a defi cient representation of the nonlocal turbulent fl uxes associated with thermals, of vertical extension larger than the vertical resolution of the
BL model.
Lilly (1962) considered a formulation to take into account the effect of buoyancy,
proposing that eddy diffusivities were made dependent on the Richardson number
(4.13). The Prandtl approach can be derived from a steady-state equation of e – , equaling the production by shear with the dissipation. If buoyancy is included in the production term, it becomes
∂
= ∂
2
( ),
u
K l
F Ri
z
(4.12)
where Ri is the gradient Richardson number. Ri is based on the K-diffusion approach,
more precisely in Equations 4.7 and 4.8 allowing to write,
2
2
,
v
v
g
z
Ri
u
v
z
z
∂θ
∂
θ
= ⎡
⎤
∂
∂
⎛ ⎞ ⎛ ⎞
+
⎢
⎥
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∂
∂
⎢
⎥
⎣
⎦
(4.13)
To solve Equation 4.12 one has to specify l and F(Ri). Usually, it is assumed that
l → kz in the surface layer and F(Ri) → 1 in a neutral stratifi cation, where k is the
von Karman constant.
Both Expressions 4.11 and 4.12 depend critically on l. On the basis of these observations, Blackadar (1962) built an empirical formulation for l:
1 1 1 ,
l kz
=
+ λ
(4.14)
where λ is an adjustable parameter representing an asymptotic mixing length. This
expression consists in an interpolation between the two limits: l → kz when z → 0
and l → λ when z → + ∞. Other expressions for the CBL are based on empirical prescription of profi les for K, making use of empirical expressions (Holtslag and Moeng
1991). Often, these expressions depend on the computation of the BL height, z i , and
other scale variables of the surface layer.
© 2010 by Taylor and Francis Group, LLC