By Alexander Baklanov, Branko Grisogono, A. Baklanov, B. Grisogono

Most of practically-used turbulence closure types are according to the idea that of downgra- ent delivery. consequently the types convey turbulent uxes of momentum and scalars as items of the suggest gradient of the transported estate and the corresponding turbulent delivery coef cient (eddy viscosity, ok , warmth conductivity, ok , or diffusivity, okay ). Fol- M H D lowing Kolmogorov (1941), turbulent delivery coef cients are taken to be proportional to the turbulent pace scale, u , and size scale, l : T T okay ? okay ? ok ? u l . (1) M H D T T 2 frequently u is identi ed with the turbulent kinetic power (TKE) according to unit mass, E ,and ok T is calculated from the TKE finances equation utilizing the Kolmogorov closure for the TKE dissipation expense: ? ? E /t , (2) okay okay T the place t ? l /u is the turbulent dissipation time scale. This technique is justi ed whilst it T T T is utilized to impartial balance ows, the place l may be taken to be proportional to the space T from the closest wall. although, this system encounters dif culties in strati ed ows (both good and uns- ble). The turbulent Prandtl quantity Pr = ok /K indicates crucial dependence at the T M H strati cation and can't be regarded as constant.

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**Example text**

Empirical constants that appear in the above formulations are given in Table 1. The proposed method can be applied, in particular, to the shallow ABL, when the lowest computational level is close to h, and standard approaches completely fail. But it has advantages also in situations when the ABL (the height interval 0 < z < h) contains several computational levels. In such cases, it provides several independent estimates of h, u 2∗ and F∗ , and by this means makes available a kind of data assimilation, namely, more reliable determination of h, u 2∗ and F∗ through averaging over all estimates.

Zilitinkevich et al. Fig. 6 Dimensionless vertical gradients of (a) mean velocity, k T zτ 1/2 H = −Fz kz M = τ 1/2 ∂U , and (b) potential tempera∂z ∂ ture, ∂z , versus z/L, based on our local closure model [solid lines plotted after Eq. 13a, b)] compared to the same LES data as in Fig. 5 [from Figs. 1 and 2 of Zilitinkevich and Esau (2007)] z, does not appear in the set of parameters that characterise the vertical turbulent length scale in sufficiently strong static stability (z/L 1). Without this assumption the linear asymptote for H loses ground while for M it holds true.

In Eq. 17, h E is taken after Eq. 16, wh is the mean vertical velocity at the height z = h (available in numerical models), the combination Ct u ∗ h −1 E is the inverse ABL relaxation time scale, Ct ≈ 1 is an empirical dimensionless constant, and K h is the horizontal turbulent diffusivity (the same as in other prognostic equations of the model under consideration). Finally, given h, the free-flow Brunt-Väisälä frequency, N , is determined through the root-mean-square value of the potential temperature gradient over the layer h < z < 2h: N4 = 1 h 2h h β ∂ ∂z 2 dz (18) and substituted into Eq.