Parameterization of Convective Boundary Layer Turbulence and Clouds
95
justifying to scale Δθ vu with the ratio between (
)
v s
w θ
′ ′ and σ w , for any level z 1 of the
surface BL:
θ
′ ′
θ
=θ
+ σ
1
1
1
(
)
( )
( )
.
( )
v s
v
vu
i
w
w
z
z
b
z
(4.54)
The coeffi cient value b i of Equation 4.54 was adjusted, through diagnostics of LES
results, to 0.3. Similar expressions could be applied for the other variables, like θ lu
and q tu . This initialization requires the knowledge of σ w .
The vertical velocity of the thermals ensemble, w u , is then computed through a
modifi ed version of the Simpson and Wiggert (1969) equation, adding a buoyancy
term B = g(θ vu − φ
–
v )/φ
–
v :
2
,
u
u
u
w
w
b w
a B
z
∂ = −ε
+
∂
(4.55)
where ε represents, as before, the lateral mixing rate. The presence of the coeffi cients a and b is discussed in several papers (e.g., Siebesma et al. 2004) and is to
account approximately for the effects of the pressure perturbations and sub-plume
turbulence. The values of these coeffi cients, still in discussion, were diagnosed from
LES runs and were considered as: a = 2.0 and b = 1.0. The BL height, z i , corresponds
to the level where w u is nil.
Following Siebesma and Cuijpers (1995), the lateral mixing rates for the dry BL
were diagnosed. Expression 4.51 was applied to potential temperature, to compute
the vertical profi les of the lateral mixing for each of the three referred decompositions, corresponding to the fractions 1%, 3%, and 5%. The obtained profi les can be
observed in Figure 4.8, suggesting a strong relation between the lateral mixing rate,
the height, and the BL height z i , despite some dispersion. The same type of scaling
was, previously, suggested by Siebesma (1998) for the lateral mixing of cumulus
cores, proposing that the rate should be inversely proportional to the distance to the
cloud base (ε ≈ 1/ z − z b ).
The LES results, depicted in Figure 4.8, allowed to propose an empirical expression for the solid curve, to compute the rate of lateral mixing, present in Equations
4.51 and 4.55, given by
1
1 ,
i
c
z z z
ε
⎛
⎞
ε =
+
⎜
⎟
⎝
⎠
−
(4.56)
where c ε ≈ 0.4.
The last expression showed some unwanted dependency on the vertical resolution, thus a modifi ed expression was proposed to deal with coarser resolutions:
ε
ε
⎧
⎫
⎡
⎤
⎛
⎞
⎪
⎪
ε =
+
⎨
⎬
⎢
⎥
⎜
⎟
⎝
⎠
Δ
−
⎪
⎪
⎣
⎦
⎩
⎭
1
1
1
max 0,min
,
.
i
c
c
z
z z z
(4.57)
© 2010 by Taylor and Francis Group, LLC
95
justifying to scale Δθ vu with the ratio between (
)
v s
w θ
′ ′ and σ w , for any level z 1 of the
surface BL:
θ
′ ′
θ
=θ
+ σ
1
1
1
(
)
( )
( )
.
( )
v s
v
vu
i
w
w
z
z
b
z
(4.54)
The coeffi cient value b i of Equation 4.54 was adjusted, through diagnostics of LES
results, to 0.3. Similar expressions could be applied for the other variables, like θ lu
and q tu . This initialization requires the knowledge of σ w .
The vertical velocity of the thermals ensemble, w u , is then computed through a
modifi ed version of the Simpson and Wiggert (1969) equation, adding a buoyancy
term B = g(θ vu − φ
–
v )/φ
–
v :
2
,
u
u
u
w
w
b w
a B
z
∂ = −ε
+
∂
(4.55)
where ε represents, as before, the lateral mixing rate. The presence of the coeffi cients a and b is discussed in several papers (e.g., Siebesma et al. 2004) and is to
account approximately for the effects of the pressure perturbations and sub-plume
turbulence. The values of these coeffi cients, still in discussion, were diagnosed from
LES runs and were considered as: a = 2.0 and b = 1.0. The BL height, z i , corresponds
to the level where w u is nil.
Following Siebesma and Cuijpers (1995), the lateral mixing rates for the dry BL
were diagnosed. Expression 4.51 was applied to potential temperature, to compute
the vertical profi les of the lateral mixing for each of the three referred decompositions, corresponding to the fractions 1%, 3%, and 5%. The obtained profi les can be
observed in Figure 4.8, suggesting a strong relation between the lateral mixing rate,
the height, and the BL height z i , despite some dispersion. The same type of scaling
was, previously, suggested by Siebesma (1998) for the lateral mixing of cumulus
cores, proposing that the rate should be inversely proportional to the distance to the
cloud base (ε ≈ 1/ z − z b ).
The LES results, depicted in Figure 4.8, allowed to propose an empirical expression for the solid curve, to compute the rate of lateral mixing, present in Equations
4.51 and 4.55, given by
1
1 ,
i
c
z z z
ε
⎛
⎞
ε =
+
⎜
⎟
⎝
⎠
−
(4.56)
where c ε ≈ 0.4.
The last expression showed some unwanted dependency on the vertical resolution, thus a modifi ed expression was proposed to deal with coarser resolutions:
ε
ε
⎧
⎫
⎡
⎤
⎛
⎞
⎪
⎪
ε =
+
⎨
⎬
⎢
⎥
⎜
⎟
⎝
⎠
Δ
−
⎪
⎪
⎣
⎦
⎩
⎭
1
1
1
max 0,min
,
.
i
c
c
z
z z z
(4.57)
© 2010 by Taylor and Francis Group, LLC