Identifying Counter-Gradient term in atmospheric convective boundary layer

Figures & data

FIGURE 1 Different parameterizations for the nondimensional counter-gradient.

FIGURE 2 New counter-gradient estimation by second-order maximum entropy with given in [18]: (a) noise 1%, with λ = 2.5×10¹⁰; (b) noise 2.5%, with λ = 4.5× 10¹⁰; and (c) noise 5%, with λ = 4.0×10¹⁰.

FIGURE 3 Influence of the regularization parameter to 1% of noise: (a) without regularization; (b) λ = 1.0 × 10⁷ (c) λ = 2.5 × 10¹⁰; (d) λ = 1.29 × 10¹¹.

FIGURE 4 L-curve for reconstruction of counter-gradient with 1% of noise.

FIGURE 5 Concentration profiles at three times t = 0 s, t = 400 s and t = 2000 s: the solid line represents the diffusion without counter-gradient term, the dashed line represents the use of counter-gradient term given by Eqs. (2), (4) and (A.3).

FIGURE 6 Concentration profiles for three time periods: the solid line with counter-gradient term given by Eqs. (2), (4) and (A.3); the dashed line the counter-gradient is given in [4] (Eqs. (2), (3) and (A.2)).

TABLE I Difference between several formulations for the counter-gradient term

γModel	ρModel
γCH98	ρCH98=3.5 × 10^−4
γHB93	ρHB93=6.4 × 10^−3

TABLE II Square error between exact and estimated counter-gradient terms for several levels of noise

1% noise	2.5% noise	5 % noise
3.1 × 10^-5	7.5 × 10^-5	2.6 × 10^-4

Table

b	=	constant in Eq. (2)
	=	mean concentration (g m^−3)
C i,n	=	approximated solution for concentration
h	=	boundary layer height (m)
J(γ)	=	objective function
K zz	=	vertical eddy diffusivity (m^2 s^−1)
R(γ)	=	square difference term
S(γ)	=	entropy regularization function
u *	=	friction velocity (m s^−1)
w	=	wind vertical component (m s^−1)
w *	=	convective scale for velocity (m s^−1)
	=	vertical turbulent flux of quantity
γ	=	counter-gradient term (g m^−2 s^−1)
λ	=	regularization parameter
	=	variance of the vertical velocity (m^2 s^−1)
χ *	=	scale for mean quantity (g m^−3)
Δ t	=	time discretization (s)
Δ z	=	vertical space discretization (m)
Ψ	=	nondimensional dissipation function
Ω (γ)	=	general regularization function

Sorbjan Z 1989 Structure of the Atmospheric Boundary Layer, Prentice Hall

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Cuijpers , JWM and Holtslag , AAM . 1998 . Impact of skewness and nonlocal effects on scalar and boundancy fluxes in convective boundary layers . J. Atmos. Sci , 55 : 151

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