Vertical Profiles of Turbulence Parameters in the Thermal Internal Boundary Layer

Abstract

The correct simulation of pollutant dispersion in coastal regions demands understanding the turbulence structure of the thermal internal boundary layer (TIBL), which typically occurs in daytime when maritime air is advected over the continent. Such a structure is investigated using 10 levels of turbulence observations made at a 140-m micrometeorological mast installed at 3500 m from the shoreline in south-eastern Brazil, with TIBL dimensionless vertical profiles of the turbulence parameters commonly used in Lagrangian and Eulerian dispersion models determined. To accomplish that, the TIBL height $z_i$ is estimated using a vertical-flux-convergence approach, developed here. The values experimentally obtained for $z_i$ agree with those predicted by a widely-used model for TIBL growth. In general, the normalized turbulent profiles evaluated for the TIBL differ from those previously obtained in the convective boundary layer (CBL). The horizontal eddy diffusivities evaluated for the TIBL are larger than those typically observed in the CBL, while the vertical ones are similar for both boundary layers. Finally, it is shown that CBL similarity relationships can be used to describe turbulent parameters as long as the proper empirical constants are used.

Appendices

Appendix 1: Uncertainty on the Linear Fit

The linear adjustment performance is evaluated using two goodness-of-fit measures: R-squared ($R^2$) and the standard error of the regression (S) defined respectively as

$$\begin{aligned} S = \sqrt{\frac{1}{(N-2)}\sum _{i=1}^{N}\left( y_i-\hat{y_i}\right) ^2} \end{aligned}$$

(30)

and

$$\begin{aligned} R^2=1-\frac{\sum _{i=1}^{N}\left( y_i-\hat{y_i} \right) ^2}{\sum _{i=1}^{N}\left( y_i-\overline{y} \right) ^2}, \end{aligned}$$

(31)

where y are the original measurements, $\hat{y}$ are the estimated values of y, and N is the total number of measurements.

The percentage relative uncertainty of the linear ($dA=100\sigma _A/A$) and angular ($dB=100\sigma _B/B$) estimated coefficients using the linear fit method have been evaluated as (Bevington and Robinson 2003)

$$\begin{aligned} \sigma _A=S\sqrt{\left( \frac{\sum _{i=1}^Nx_i^2}{\varDelta }\right) } \end{aligned}$$

(32)

and

$$\begin{aligned} \sigma _B=S\sqrt{\frac{N}{\varDelta }}, \end{aligned}$$

(33)

where

$$\begin{aligned} \varDelta =N\sum _{i=1}^Nx_i^2-\left( \sum _{i=1}^Nx_i\right) ^2. \end{aligned}$$

(34)

The absolute and percentage relative uncertainty on $z_i$ due the linear fit is obtained, respectively, as

$$\begin{aligned} \sigma _{z_i}=\sqrt{\left( \frac{\partial z_i}{\partial A} \right) ^2\sigma _A^2 + \left( \frac{\partial z_i}{\partial B} \right) ^2\sigma _B^2} \end{aligned}$$

(35)

and

$$\begin{aligned} dz_i=100\frac{\sigma _{z_i}}{z_i}. \end{aligned}$$

(36)

Appendix 2: Uncertainty on the Adjusted Turbulence Parameters Profiles

Vertical profiles of the turbulence parameters are used to adjust analytical relationships found in the literature. To reduce the uncertainty in the adjusted parameters due the outliers, we present the following methodology: a first adjustment is obtained by the nonlinear least-squares method; assuming that the error between experimental data and the modelled curve is normally distributed, error values larger than three standard errors with respect to the mean are marked as outliers; a second fit is performed using the retained measurements.

The uncertainty on the adjusted parameter can be obtained directly from the covariance matrix $\mathbf{C} = \mathbf{A} ^{-1}$ as (Vetterling et al. 2002)

$$\begin{aligned} \sigma _{\alpha _j}=\sqrt{C_{jj}}, \end{aligned}$$

(37)

where

$$\begin{aligned} A_{jk} =\sum _i\frac{1}{\sigma _i^2}\left( \frac{\partial f(x_i;\alpha _0...\alpha _{m-1})}{\partial \alpha _j} \right) \left( \frac{\partial f(x_i;\alpha _0...\alpha _{m-1})}{\partial \alpha _k} \right) \end{aligned}$$

(38)

are the elements of the matrix A$_{M\times M}$, and where m is the number of unknown parameters. In Eq. 38, $x_i$ are the measured dependent variables, $\sigma _i$ is the measurement error of the ith data point, $\alpha _j=\alpha _0...\alpha _{m-1}$ are the parameters to be evaluated, and f is the model function.