82
Air Pollution and Turbulence: Modeling and Applications
and
1
2
, ,
, ,
,
m h e
m h e
K
e
= Λ
(4.20)
where different Λ are length scales associated with dissipation and mixing of the
different properties (all other). The closure is complete with the establishment of
functional relations for those different length scales.
In general, the performance of K-diffusion schemes in the CBL is very much
dependent on the mixing length in the region of the inversion. One advantage of
schemes that incorporate the TKE equation is that they are less sensitive, since they
have to satisfy the equation of TKE. In the upper half of a CBL, the gradients of
the horizontal components of the wind are practically zero, and the TKE balance is
due mainly to buoyancy, transport, and dissipation. In the surface layer, the vertical
transport of TKE is negative and considerable, changing the signal in the interior of
the BL, which implies a positive transport of TKE for the BL upper region. When the
surface heat fl ux is ascending, the surface layer is unstable.
Higher-order closures, incorporating prognostic equations for part or all secondorder moments and even for third-order moments, may be easily deduced, eventually leading to a large set of simultaneous equations (e.g., Mellor and Yamada 1974;
André et al. 1978). Some of these sets have been successfully tested in idealized
cases, but they have not been generally incorporated in numerical weather prediction
models, even at mesoscale, due to their high computational cost.
4.3.4 NONLOCAL APPROACHES
Nonlocal approaches were inspired by the observation of many thermals that make
an ascent almost without lateral mixing, transporting air at signifi cant distances in
the boundary layer, a picture of the fl ow that is consistent with the analysis of cloud
images (Lenschow and Stephens 1980; Agee 1984; Stull and Driedonks 1987; Ebert
et al. 1989). Because turbulent terms arise in the Reynolds equations as a consequence of the nonlinearity of advection, it is easy to argue that at least part of its
effect in the mean fl ow should not behave as if it were an extra diffusion term. As a
consequence, nonlocal closures relate the turbulent terms with variables known in
all the BL, and not just in the vicinity of the grid point.
The simplest nonlocal approach, frequently used in atmospheric models, consists in the ad hoc introduction of a “counter-gradient” term in the balance equations (Deardorff 1966), permitting upward heat fl ow in the upper half of the mixed
layer, and in the inclusion of a “mass-fl ux” (MF) term in the parameterization of the
transport in cloud updrafts (Betts 1973). Other, more complex, nonlocal closures
include the transilient turbulent theory (Stull 1984) and the spectral diffusive theory
(Berkowicz and Prahm 1979).
4.3.4.1 Counter-Gradient
The inconsistency of K-diffusion theory with observations was made clear in the
convective BL, when an upward heat fl ux is maintained through a neutral or slightly
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
and
1
2
, ,
, ,
,
m h e
m h e
K
e
= Λ
(4.20)
where different Λ are length scales associated with dissipation and mixing of the
different properties (all other). The closure is complete with the establishment of
functional relations for those different length scales.
In general, the performance of K-diffusion schemes in the CBL is very much
dependent on the mixing length in the region of the inversion. One advantage of
schemes that incorporate the TKE equation is that they are less sensitive, since they
have to satisfy the equation of TKE. In the upper half of a CBL, the gradients of
the horizontal components of the wind are practically zero, and the TKE balance is
due mainly to buoyancy, transport, and dissipation. In the surface layer, the vertical
transport of TKE is negative and considerable, changing the signal in the interior of
the BL, which implies a positive transport of TKE for the BL upper region. When the
surface heat fl ux is ascending, the surface layer is unstable.
Higher-order closures, incorporating prognostic equations for part or all secondorder moments and even for third-order moments, may be easily deduced, eventually leading to a large set of simultaneous equations (e.g., Mellor and Yamada 1974;
André et al. 1978). Some of these sets have been successfully tested in idealized
cases, but they have not been generally incorporated in numerical weather prediction
models, even at mesoscale, due to their high computational cost.
4.3.4 NONLOCAL APPROACHES
Nonlocal approaches were inspired by the observation of many thermals that make
an ascent almost without lateral mixing, transporting air at signifi cant distances in
the boundary layer, a picture of the fl ow that is consistent with the analysis of cloud
images (Lenschow and Stephens 1980; Agee 1984; Stull and Driedonks 1987; Ebert
et al. 1989). Because turbulent terms arise in the Reynolds equations as a consequence of the nonlinearity of advection, it is easy to argue that at least part of its
effect in the mean fl ow should not behave as if it were an extra diffusion term. As a
consequence, nonlocal closures relate the turbulent terms with variables known in
all the BL, and not just in the vicinity of the grid point.
The simplest nonlocal approach, frequently used in atmospheric models, consists in the ad hoc introduction of a “counter-gradient” term in the balance equations (Deardorff 1966), permitting upward heat fl ow in the upper half of the mixed
layer, and in the inclusion of a “mass-fl ux” (MF) term in the parameterization of the
transport in cloud updrafts (Betts 1973). Other, more complex, nonlocal closures
include the transilient turbulent theory (Stull 1984) and the spectral diffusive theory
(Berkowicz and Prahm 1979).
4.3.4.1 Counter-Gradient
The inconsistency of K-diffusion theory with observations was made clear in the
convective BL, when an upward heat fl ux is maintained through a neutral or slightly
© 2010 by Taylor and Francis Group, LLC