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 K -schemes, they proposed a nonlocal K -scheme. Hong & Pan (1996) presented an enhanced version of the Holtslag & Boville (1993) scheme. In this scheme the friction velocity scale ( u∗ ) is used as a closure in their formulation. However, for moderate to strong convective conditions, u∗ is not a representative scale (Alapaty & Alapaty, 2001). Rather, the convective velocity ( w∗ ) scale is suitable as used by Hass et al. (1991) in simulation of a wet deposition case in Europe by the European Acid Deposition Model (EURAD). Depending on the magnitude of the scaling parameter h/L ( h is height of the ABL, and L is Monin-Obukhov length), either u∗ or w∗ is used in many other formulations. Notice that this approach may not guarantee continuity between the alternate usage of u∗ and w∗ in estimating K - eddy diffusivity. Also, in most of the local-closure schemes the coefficient of vertical eddy diffusivity for moisture is assumed to be equal to that for heat. Sometimes this assumption leads to vertical gradients in the simulated moisture fields, even during moderate to strong convective conditions in the ABL. Also, the nonlocal scheme considers the horizontal advection of turbulence that may be important over heterogeneous landscapes (Alapaty & Alapaty, 2001; Mihailovic et al. 2005).

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 Km is the vertical eddy diffusivity, e¯∗ is the mean turbulent velocity scale within the ABL to be determined (closure problem), k is the von Karman constant ( k=0.41 ), z is the vertical coordinate, p is the profile shape exponent coming from the similarity theory (Troen & Mahrt, 1986; usually taken as 2), and Фm is the nondimensional function of momentum. According to Zhang et al. (1996), we use the square root of the vertically averaged turbulent kinetic energy in the ABL as a velocity scale, in place of the mean wind speed, the closure to Eq. (1). Instead of using a prognostic approach to determine TKE, we make use of a diagnostic method. It is then logical to consider the diagnostic TKE to be a function of both u∗ and w∗ . Thus, the square root of diagnosed TKE near the surface serves as a closure to this problem (Alapaty & Alapaty, 2001). However, it is more suitable to estimate e¯∗ from the profile of the TKE through the whole ABL.

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, e(z) , in a variety of boundary layers ranging from near neutral to free convection conditions. Following Zhang et al. (1996) the TKE profile can be expressed as

where LE characterizes the integral length scale of the dissipation rate. Here, Фm=(1−15z/L)−1/4 is an empirical function for the unstable atmospheric surface layer (Businger et al., 1971), which is applied to both the surface and mixed layer. We used LE=2.6h which is in the range 2.5h−3.0h suggested by Moeng & Sullivan (1994). For the stable atmospheric boundary layer we modeled the TKE profile using an empirical function proposed by Lenschow et al. (1988), based on aircraft observations

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 e¯∗ that within the ABL can be written as

where Ψ(z) is the vertical profile function for turbulent kinetic energy as obtained by Zhang et al. (1996) based on LES studies, later modified by Alapaty (personal communication), and dz is layer thickness.

The formulation of eddy-diffusivity by Eq. (1) depends on h . We follow Troen & Mahrt (1986) for determination of h using

where Ric is a critical bulk Richardson number for the ABL, u(h) and v(h) are the horizontal velocity components at hg/θ0 , is the buoyancy parameter, θ0 is the appropriate virtual potential temperature, and θv(h) is the virtual potential temperature of air near the surface at h , respectively. For unstable conditions (L0)θs is given by (Troen & Mahrt (1986))

where C0=8.5 (Holtslag et al., 1990), ws is the velocity while wθv0¯ is the kinematics surface heat flux. The velocity ws is parameterized as

and

Using c1=0.6 . In Eq. (6), θv(z1) is the virtual temperature at the first model level. The second term on the right-hand side of Eq. (6) represents a temperature excess, which is a measure in the lower part of the ABL. For stable conditions we use θs=θv(z1) with z1=2 m. On the basis of Eq. (5) the height of the ABL can be calculated by iteration for all stability conditions, when the surface fluxes and profiles of θv u and v are known. The computation starts with calculating the bulk Richardson number Ri between the level θs and subsequent higher levels of the model. Once Ri exceeds the critical value, the value of h is derived with linear interpolation between the level with RiRic and the level underneath. We use a minimum of 100 m for h . In Eq. (5), Ric is the value of the critical bulk Richardson number used to be 0.25 in this study.

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 K0 is the background value (1 m s), S is the vertical wind shear, l is the characteristic turbulent length scale (100 m), Rc is the critical Richardson number, and Ri is the Richardson number defined as

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

where Δz is the layer thickness (Zhang & Anthes, 1982).

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×N transilient matrix, where 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, ξ is the vertical coordinate, and Δξ denotes the layer thickness. The mixing matrix which controls this model is nonzero only for the top row, the left most column, and the diagonal.

Figure 1.

A schematic representation of vertical mixing in a one dimensional column as simulated by the Blackadar’s scheme.

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 Mu₁ is the upward mixing rate from surface layer to layer above and e_k denotes the turbulent kinetic energy in the considered layer. The upward mixing rate from surface

Figure 2.

Schematic representation of vertical mixing in a one-dimensional column as simulated by the (a) ACM and (b) VUR scheme.

layer to layer above is parameterized as

where ρ is the air density while H represents the sensible heat flux. Using the VUR scheme, the mixing algorithm for the lowest layer can be written in the form

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 .

3.1. 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 50×50 km was used. The model domain used in simulation had (101, 91) points covering the area of whole Europe and North Africa. The σ terrain-following coordinate was used with 20 levels in the vertical- from the surface to 100 hPa and with the lowest level located nearly at 92 m. The horizontal grid of the model is the Arakawa C grid. All other details can be found in Simpson et al. (2003). The Unified EMEP model uses 3-hourly resolution meteorological data from the dedicated version of the HIRLAM (HIgh Resolution Limited Area Model) numerical weather prediction model with a parallel architecture (Bjorge & Skalin, 1995). The horizontal and vertical wind components are given on a staggered grid. All other variables are given in the centre of the grid. Linear interpolation between the 3-hourly values is used to calculate values of the meteorological input data at each advection step. The time step used in the simulation was 600 s.

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 NO₂ 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 NO₂ in air (µg(N) m) 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 NO₂ in air (µg(N) m) 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, NO₂ 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 NO₂ in air (µg(N) m) 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, NO₂ 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 NO₂ concentration outputs NO₂ based on a method discussed in Pielke (2002). Following that study, we computed several statistical quantities as follows

Here, Γ is the variable of interest (aforementioned variables in this study) while N is the total number of data. An overbar indicates the arithmetic average, while a caret refers to an observation. The absence of a caret indicates a simulated value; ν is the rmse, while νBR is rmse after a bias is removed. Root-mean-square errors (rmse) give a good overview of a dataset, with large errors weighted more than many small errors. The standard deviations in the simulations and the observations are given by η and η^ . A rmse that is less than the standard deviation of the observed value indicates skill in the simulation. Moreover, the values of η and η^ should be close if the prediction is to be considered realistic.

Figure 3.

The eddy diffusion (OLD) versus TKE scheme. Comparison of: the modeled and observed NO₂ in air (µg(N) m) concentrations (left panels) and their biases (right panels) in the period April-September for the years 1999, 2001 and 2002. M and O denotes modeled and observed value, respectively.

The statistics gave the following values: (1) TKE ( ν=0.548 , νBR=0.293 η=0.211 η^=0.147 ) and OLD ( ν=0.802 , νBR=0.433 η=0.303 , η^=0.147 ) and (2) VUR ( ν=0.571 µg(N) m, νBR=0.056 µg(N) m, η=0.219 µg(N) m η^=0.211 µg(N) m) and OLD ( ν=0.802 , νBR=0.159 η =0.303, η^ =0.211). A comparison of η and η^ , for (1) and (2), shows that difference between them, is evidently smaller with the TKE and VUR scheme schemes versus the OLD one.

Figure 4.

The eddy diffusion (OLD) versus the VUR scheme. Comparison of: (a) the modelled and observed NO₂ in air (µg(N) m) concentrations and (b) their biases in the period April-September for the year 2002. M and O denotes modelled and observed value, respectively.