## 1. Introduction

The description of the atmospheric boundary layer (ABL) processes, understanding of complex boundary layer interactions, and their proper parameterization are important for air quality as well as many other environmental models. In that sense single-column vertical mixing models are comprehensive enough to describe processes in ABL. Therefore, they can be employed to illustrate the basic concepts on boundary layer processes and represent serviceable tools in boundary layer investigation. When coupled to 3D models, single-column models can provide detailed and accurate simulations of the ABL structure as well as mixing processes.

Description of the ABL during convective conditions has long been a major source of uncertainty in the air quality models and chemical transport models. There exist two approaches, local and nonlocal, for solving the turbulence closure problem. While the local closure assumes that turbulence is analogous to molecular diffusion in the nonlocal-closure, the unknown quantity at one point is parameterized by values of known quantities at many points in space. The simplest, most popular local closure method in Eulerian air quality and chemical transport models is the K-Scheme used both in the boundary layer and the free troposphere. Since it uses local gradients in one point of model grid, K-Scheme can be used only when the scale of turbulent motion is much smaller than the scale of mean flow (Stull, 1988), such as in the case of stable and neutral conditions in the atmosphere in which this scheme is consistent. However, it can not: (a) describe the effects of large scale eddies that are dominant in the convective boundary layer (CBL) and (b) simulate counter-gradient flows where a turbulent flux flows up to the gradient. Thus, K-Scheme is not recommended in the CBL (Stull, 1988). Recently, in order to avoid the K-scheme drawbacks, Alapaty (Alapaty, 2003; Alapaty & Alapaty, 2001) suggested a “nonlocal” turbulent kinetic energy (TKE) scheme based on the K-Scheme that was intensively tested using the EMEP chemical transport model (Mihailovic & Jonson, 2005; Mihailovic & Alapaty, 2007). In order to quantify the transport of a passive tracer field in three-dimensional simulations of turbulent convection, the nonlocal and non-diffusive behavior can be described by a transilient matrix whose elements contain the fractional tracer concentrations moving from one subvolume to another as a function of time. The approach was originally developed for and applied to geophysical flows known as turbulent transilient theory (T3) (Stull, 1988; Stull & Driedonks, 1987; Alapaty et al., 1997), but this formalism was extended and applied in an astrophysical context to three-dimensional simulations of turbulent compressible convection with overshoot into convectively stable bounding regions (Miesch et al., 2000). The most frequently used nonlocal-closure method is the asymmetric convective model (ACM) suggested by Pleim & Chang (1992). The design of this model is based on the Blackadar’s scheme (Blackadar, 1976), but takes into account the important fact that, in the CBL, the vertical transport is asymmetrical (Wyngaard & Brost, 1984). Namely, the buoyant plumbs are rather fast and narrow, while downward streams are wide and slow. Accordingly, transport by upward streams should be simulated as nonlocal and transport by downward streams as local. The concept of this model is that buoyant plumbs rise from the surface layer and transfer air and its properties directly into all layers above. Downward mixing occurs only between adjacent layers in the form of a slow subsidence. The ACM can be used only during convective conditions in the ABL, while stable or neutral regimes for the K-Scheme are considered. Although this approach results in a more realistic simulation of vertical transport within the CBL, it has some drawbacks that can be elaborated in condensed form: (i) since this method mixes the same amount of mass to every vertical layer in the boundary layer, it has the potential to remove mass much too quickly out of the surface layer and (ii) this method fails to account for the upward mixing in layers higher than the surface layer (Tonnesen et al., 1998). Wang (Wang, 1998) has compared three different vertical transport methods: a semi-implicit K-Scheme (SIK) with local closure and the ACM and T3 schemes with nonlocal-closure. Of the three schemes, the ACM scheme moved mass more rapidly out of surface layer into other layers than the other two schemes in terms of the rate at which mass was mixed between different layers. Recently, this scheme was modified with varying upward mixing rates (VUR), where the upward mixing rate changes with the height, providing slower mixing (Mihailović et al., 2008).

The aim of this chapter is to give a short overview of nonlocal-closure TKE and ABL mixing schemes developed to describe vertical mixing during convective conditions in the ABL. The overview is supported with simulations performed by the chemical EMEP Uniﬁed model (version UNI-ACID, rv2.0) where schemes were incorporated.

## 2. Description of nonlocal-closure schemes

### 2.1. Turbulent kinetic energy scheme (TKE)

As we mentioned above the well-known issues regarding local-closure ABL schemes is their inability to produce well-mixed layers in the ABL during convective conditions. Holtslag & Boville (1993) using the NCAR Community Climate Model (CCM2) studied a classic example of artifacts resulting from the deficiencies in the first-order closure schemes. To alleviate problems associated with the general first-order eddy-diffusivity

The starting point of approach is to consider the general form of the vertical eddy diffusivity equation. For momentum, this equation can be written as

where

According to Moeng & Sullivan (1994), a linear combination of the turbulent kinetic energy dissipation rates associated with shear and buoyancy can adequately approximate the vertical distribution of the turbulent kinetic energy,

where

Following LES (Large Eddy Simulation) works of Zhang et al. (1996) and Moeng & Sullivan (1994), Alapaty (2003) suggested how to estimate the vertically integrated mean turbulent velocity scale

where

The formulation of eddy-diffusivity by Eq. (1) depends on

where

where

and

Using

In the free atmosphere, turbulent mixing is parameterized using the formulation suggested by Blackadar (1979) in which vertical eddy diffusivities are functions of the Richardson number and wind shear in the vertical. This formulation can be written as

where
^{2} s^{-1}),

The critical Richardson number in Eq. (9) is determined as

where

### 2.2. Nonlocal vertical mixing schemes

The nonlocal vertical mixing schemes were designed to describe the effects of large scale eddies, that are dominant in the CBL and to simulate counter-gradient flows where a turbulent flux flows up to the gradient. During convective conditions in the atmosphere, both small-scale subgrid and large-scale super grid eddies are important for vertical transport. In this section, we will consider three different nonlocal mixing schemes: the Blackadar’s scheme (Blackadar, 1976), the asymmetrical convective model (Pleim & Chang, 1992) and the scheme with varying upward mixing rates (Mihailovic et al., 2008).

Transilient turbulence theory (Stull, 1988) (the Latin word *transilient* means to jump over) is a general representation of the turbulent flux exchange processes. In transilient mixing schemes, elements of flux exchange are defined in an
*N* is the number of vertical layers and mixing occurs not only between adjacent model layers, but also between layers not adjacent to each other. That means that all of the matrix elements are nonzero and that the turbulent mixing in the convective boundary layer can be written as

where *c* is the concentration of passive tracer, the elements in the mixing matrix *M* represent mass mixing rates, and *i* and *j* refer to two different grid cells in a column of atmosphere. Some models specify mixing concepts with the idea of reducing the number of nonzero elements because of the cost of computational time during integration.

*The Blackadar’s scheme* (Blackadar, 1976) is a simple nonlocal-closure scheme, that is designed to describe convective vertical transport by eddies of varying sizes. The effect of convective plumes is simulated by mixing material directly from the surface layer with every other layer in the convective layer. The schematic representation of vertical mixing simulated by the Blackadar’s scheme is given in Figure 1. The mixing algorithm can be written for the surface and every other layer as

and

respectively, where *Mu* represents the mixing rate,

*The asymmetrical convective model* (Pleim & Chang, 1992) is a nonlocal vertical mixing scheme based on the assumption of the vertical asymmetry of buoyancy-driven turbulence. The concept of this model is that buoyant plumes, according to the Blackadar’s scheme, rise from the surface layer to all levels in the convective boundary layer, but downward mixing occurs between adjacent levels only in a cascading manner. The schematic representation of vertical mixing simulated by the ACM is presented in Fig. 2a. The mixing algorithm is driven by equations

and

where *Mu* and *Md* are the upward and downward mixing rates, respectively. The downward mixing rate from level *k* to level *k* −1 is calculated as

The mixing matrix controlling this model is non-zero only for the leftmost column, the diagonal and superdiagonal.

*The scheme with varying upward mixing rates* (VUR sheme), sugested by Mihailović et al. (2008) is a modified version of the ACM, where the upward mixing rate changes with the height, providing slower mixing. The schematic representation of vertical mixing simulated by this scheme is shown in Fig. 2b. The upward mixing rates are scaled with the amount of turbulent kinetic energy in the layer as

where *Mu _{1}
* is the upward mixing rate from surface layer to layer above and

*e*denotes the turbulent kinetic energy in the considered layer. The upward mixing rate from surface

_{k}layer to layer above is parameterized as

where

The algorithm for the other layers is very similar to the ACM algorithm [Eqs. (16) and (17)], with the upward mixing rate *Mu* substituted with varying upward mixing rates *Mu _{k}
*.

## 3. Numerical simulations with nonlocal-closure schemes in the Unified EMEP chemical model

In the EMEP Unified model the diffusion scheme remarkably improved the vertical mixing in the ABL, particularly under stable conditions and conditions approaching free convection, compared with the scheme previously used in the EMEP Unified model. The improvement was particularly pronounced for NO_{2} (Fagerli & Eliassen, 2002). However, with reducing the horizontal grid size and increasing the heterogeneity of the underlying surface in the EMEP Unified model, there is a need for eddy-diffusivity scheme having a higher level of sophistication in the simulation of turbulence in the ABL. It seems that the nonlocal eddy-diffusivity schemes have good performance for that. Zhang et al. (2001) demonstrated some advantages of nonlocal over local eddy-diffusivity schemes. The vertical sub grid turbulent transport in the EMEP Unified model is modeled as a diffusivity effect. The local eddy-diffusivity scheme is designed following O’Brien (1970). (In further text this scheme will be referred to the OLD one). In the unstable case,

where

To compare performances of the proposed nonlocal-closure schemes TKE scheme (Eqs.(1)-(4)) and local OLD scheme (Eq. (22)), both based on the vertical eddy diffusivity formulation, in reproducing the vertical transport of pollutants in the ABL, a test was performed with the Unified EMEP chemical model (UNIT-ACID, rv2_0_9).

### 3.1. Short model description and experimental set up

The basic physical formulation of the EMEP model is unchanged from that of Berge & Jacobsen (1998). A polar-stereographic projection, true at 60ºN and with the grid size of
^{2} was used. The model domain used in simulation had (101, 91) points covering the area of whole Europe and North Africa. The

### 3.2. Comparison with the observations

The comparison of the TKE and VUR schemes with OLD eddy diffusion scheme has been performed, using simulated and measured concentrations of the pollutant NO_{2} since it is one of the most affected ones by the processes in the ABL layer. The simulations were done for the years (i) 1999, 2001 and 2002 (TKE scheme) and (ii) 2002 (for VUR scheme) in the months when the convective processes are dominant in the ABL (April-September). The station recording NO_{2} in air (µg(N) m^{-3}) concentration was considered for comparison when measurements were available for at least 75% of days in a year [1999 (80 stations), 2001 (78) and 2002 (82)]. We have calculated the bias on the monthly basis as (M-O)/O*100% where M and O denote the modeled and observed values, respectively. The comparison of the modeled and observed NO_{2} in air (µg(N) m^{-3}) concentrations and corresponding biases for both schemes (TKE and OLD) are shown in Figure 3. The values used in calculations were averaged over the whole domain of integration. It can be seen that both schemes underestimate the observations. However, for all considered months, NO_{2} concentrations calculated with the TKE scheme are in general higher and closer to the observations than those obtained by the OLD scheme (of the order of 10%). Correspondingly, the bias of the TKE scheme is lower than the OLD scheme. The comparison of the modeled and observed NO_{2} in air (µg(N) m^{-3}) concentrations between VUR and OLD schemes is shown in Figure 4. The values used in the calculations were also averaged over the whole domain of integration. It can be seen that both schemes underestimate the observations. However, for all considered months, NO_{2} concentrations calculated with the VUR scheme are in general higher and closer to the observations than those obtained using the eddy diffusion scheme (of the order of 15-20%). Accordingly, the bias of the VUR scheme is lower than the OLD eddy diffusion scheme.

To quantify the simulated values of the both schemes we have performed an error analysis of the NO_{2} concentration outputs NO_{2} based on a method discussed in Pielke (2002). Following that study, we computed several statistical quantities as follows

Here,

The statistics gave the following values: (1) TKE (
^{-3},
^{-3},
^{-3}
^{-3}) and OLD (

## 4. Conclusions

In the ABL during convective conditions, when much of the vertical mixing is driven by buoyant plumes, we cannot properly describe mixing processes using local approach and eddy diffusion schemes. Nonlocal-closure schemes simulate much better vertical mixing than local ones. In this chapter, two nonlocal schemes (the TKE scheme and the VUR scheme) for applications in air quality and environmental models are described. The comparison of the TKE scheme and the VUR one with an eddy diffusion scheme (OLD) commonly used in chemical transport models was done. These comparisons were performed with the EMEP Unified chemical model using simulated and measured concentrations of the pollutant NO_{2} since it is one of the most affected ones by the processes in the ABL layer. Nonlocal shemes gave better results than local one.