An evaluation of the dissimilarity in heat and momentum transport through quadrant analysis for an unstable atmospheric surface layer flow

Abstract

To elucidate the role of each fluid motion in the transport of momentum and heat fluxes in an unstable atmospheric surface layer (ASL) flow from single point measurements on a micrometeorological tower, we develop a novel method based on quadrant analysis where the contour maps of the turbulent statistics (fluxes, temperature variances and triple order moments between vertical velocity and temperature) are plotted on the quadrant planes between streamwise (u)–vertical (w) velocities, and vertical velocity (w)–temperature (T). We find that the dissimilarities in the heat and momentum transport with atmospheric stability are closely linked to the non-Gaussian nature of the joint probability density function (JPDF) between w and T. To highlight the changes in the fluid motions which cause this dissimilarity, we plot the contour maps of the third order moments between w and T on the u − w quadrant plane, and also of the streamwise momentum flux conditioned on every quadrant of u − w plane onto the T − w plane, referred to as octant analysis. The results indicate that in a highly-convective ASL, the cold downdrafts interspersed with strong ejections of hot fluid, carry a significant amount of both down-gradient and counter-gradient momentum flux, thus making the momentum transport inefficient. However, in a near-neutral ASL, the heat and momentum both are carried by the ejection and sweep quadrants of u − w quadrant plane, which indicates the temperature fluctuations are highly correlated with the high-speed and low-speed streaks commonly found in pure shear flows in the laboratory experiments.

Appendices

Appendix 1: Construction of the contour maps of the turbulent statistics on the quadrant plane

We will explain briefly in this section, how we can construct the contour maps of the turbulent statistics such as momentum and heat fluxes, triple order moments between w and T, or the temperature variance, on the quadrant planes of $\hat{u} - \hat{w}$ or $\hat{T} - \hat{w}$, from the half-hourly high frequency 3-D sonic anemometer data collected simultaneously in the ASL flow, at a single point on a micrometeorological tower. A step-by-step implementation is given below:

• 1. Select the simultaneously measured flow variables w and x (where x can be u or T) from the half-hourly high frequency sonic anemometer data collected in a convective ASL flow at a single height above the surface.

• 2. Remove the half-hourly linear trend from the flow variables, to calculate the fluctuations w′ and x′.

• 3. Normalize the fluctuations by their respective standard deviations and denote these normalized values as $\hat{w}$ and $\hat{x}$.

• 4. Grid these $\hat{w}$ and $\hat{x}$ into N uniform bins which are linearly spaced. The binned $\hat{w}$ and $\hat{x}$ values are denoted as $\hat{w}_{\text{bin}}$ and $\hat{x}_{\text{bin}}$ respectively, and the bin-width is defined as $d\hat{w} = \frac{{\hat{w}_{\max} - \hat{w}_{ \min } }}{N}$ and $d\hat{x} = \frac{{\hat{x}_{ \max } - \hat{x}_{ \min} }}{N}$.

• 5. Locate the indices for which $\{ \hat{x}_{\text{bin}} (i) < \hat{x}(t) < \hat{x}_{\text{bin}} (i) + d\hat{x}, \hat{w}_{\text{bin}} (j) < \hat{w}(t) < \hat{w}_{\text{bin}} (j) + d\hat{w}\}$, where 1 $\le i \le N$ and 1 $\le j \le N$.

• 6. For these indices, calculate the fractional contribution to the turbulent statistic $w^{m} x^{n}$, where $\{ 0 \le m \le 3, 0 \le n \le 3\}$ (e.g., for m =  1, n =  1, and x = u, the concerned statistic wu is the streamwise momentum flux), and divide that by the area of each grid that is $d\hat{w} \times d\hat{x}$, so when summed over all i and j, the result is 1. Assign this area-divided fractional contribution to the grid value $\{ \hat{x}_{\text{bin}} (i), \hat{w}_{\text{bin}} (j)\}$.

• 7. Repeat the steps 5–6 for all i and $j$, to construct a two-dimensional matrix of the turbulent statistic $w^{m} x^{n}$.

After completing the steps 1–7, we get a two-dimensional matrix of the turbulent statistic where each matrix element corresponds to the fractional contribution to this statistic from each grid point $\{ \hat{u}_{\text{bin}} (i), \hat{w}_{\text{bin}} (j)\}$ or $\{ \hat{T}_{\text{bin}} (i), \hat{w}_{\text{bin}} (j)\}$ of the quadrant plane $\hat{u} - \hat{w}$ or $\hat{T} - \hat{w}$. When this two-dimensional matrix is plotted against $\{ \hat{u}_{\text{bin}} , \hat{w}_{\text{bin}} \}$ or $\{ \hat{T}_{\text{bin}} , \hat{w}_{\text{bin}} \}$, we get the contour map of the turbulent statistic on the quadrant planes of $\hat{u} - \hat{w}$ or $\hat{T} - \hat{w}$, which allows us to visually identify how and how much each fluid motion from each quadrant plane contributed to the transport of this turbulent statistic in the ASL flow.

Appendix 2: The general form of the JPDF between two variables

Following Nakagawa and Nezu [27], the general form of the JPDF between two variables $\hat{x}$ ($\frac{{x^{{\prime }} }}{{\sigma_{x} }}$) and $s$ ($\frac{{y^{{\prime }} }}{{\sigma_{y} }}$), can be written out in the form of a Gram–Charlier probability distribution, expressed as,

$$ P\left( {\hat{x},\hat{y}} \right) = G\left( {\hat{x},\hat{y}} \right) \left[ {1 + \mathop \sum \limits_{j + k = 3}^{4} \frac{{Q_{j,k} }}{j!k!} H_{j,k} \left( {\hat{x},\hat{y}} \right)} \right], $$

(15)

where the terms $j + k > 4$ have been ignored. In Eq. 15, $G(\hat{x},\hat{y})$ is the equivalent gaussian JPDF defined as,

$$ G\left( {\hat{x}, \hat{y}} \right) = \frac{1}{{2\pi \sqrt {1 - R^{2} } }} {\text{exp}}\left[ { - \frac{{\hat{x}^{2} - 2R\hat{x}\hat{y} + \hat{y}^{2} }}{{2\left( {1 - R^{2} } \right)}}} \right], $$

(16)

with $R$ being the correlation coefficient between x and y, $H_{j,k} (\hat{x},\hat{y})$ is the two-dimensional Hermite polynomial, and each $Q_{j,k}$ is defined as, $Q_{0,0} = 1$, $Q_{1,0} = 0$, $Q_{0,1} = 0$, $Q_{2,0} = 1$, $Q_{0,2} = 1$, $Q_{1,1} = - R$, $Q_{1,2} = \overline{{\hat{x}\hat{y}^{2} }}$, $Q_{2,1} = \overline{{\hat{x}^{2} \hat{y}}}$, $Q_{3,0} = \overline{{\hat{x}^{3} }}$, $Q_{0,3} = \overline{{\hat{y}^{3} }}$, $Q_{1,3} = \overline{{\hat{x}\hat{y}^{3} }} + 3R$, $Q_{3,1} = \overline{{\hat{x}^{3} \hat{y}}} + 3R$, $Q_{2,2} = \overline{{\hat{x}^{2} \hat{y}^{2} }} - 2R^{2} - 1$, $Q_{4,0} = \overline{{\hat{x}^{4} }} - 3$, and $Q_{0,4} = \overline{{\hat{y}^{4} }} - 3$.

We can rewrite Eq. 15 as,

$$ P\left( {\hat{x}, \hat{y}} \right) = G\left( {\hat{x}, \hat{y}} \right) [1 + D_{1} \left( {\hat{x}, \hat{y}} \right) + D_{2} \left( {\hat{x}, \hat{y}} \right)], $$

(17)

where,

$$ D_{1} \left( {\hat{x}, \hat{y}} \right) = \frac{{Q_{1,2} }}{2} H_{1,2} \left( {\hat{x}, \hat{y}} \right) + \frac{{Q_{2,1} }}{2} H_{2,1} \left( {\hat{x}, \hat{y}} \right) + \frac{{Q_{3,0} }}{6} H_{3,0} \left( {\hat{x}, \hat{y}} \right) + \frac{{Q_{0,3} }}{6} H_{0,3} \left( {\hat{x}, \hat{y}} \right), $$

(18)

$$ D_{2} \left( {\hat{x}, \hat{y}} \right) = \frac{{Q_{1,3} }}{6} H_{1,3} \left( {\hat{x}, \hat{y}} \right) + \frac{{Q_{3,1} }}{6} H_{3,1} \left( {\hat{x}, \hat{y}} \right) + \frac{{Q_{2,2} }}{4} H_{2,2} \left( {\hat{x}, \hat{y}} \right) + \frac{{Q_{4,0} }}{24} H_{4,0} \left( {\hat{x}, \hat{y}} \right) + \frac{{Q_{0,4} }}{24} H_{0,4} \left( {\hat{x}, \hat{y}} \right). $$

(19)

By expanding the two-dimensional Hermite polynomials as given by Feŕiet [8] and using the expressions for $Q_{j,k}$, we can rewrite $D_{1} \left( {\hat{x}, \hat{y}} \right)$ and $D_{2} \left( {\hat{x}, \hat{y}} \right)$ as,

$$ D_{1} \left( {\hat{x}, \hat{y}} \right) = \frac{{\overline{{\hat{x} \hat{y}^{2} }} }}{2} \left( {\xi \eta^{2} - 2b\eta - c\xi } \right) + \frac{{\overline{{\hat{x} \hat{y}^{2} }} }}{2} \left( {\xi^{2} \eta - 2b\xi - a\eta } \right) + \frac{{\overline{{\hat{x}^{3} }} }}{6} \left( {\xi^{3} - 3a\xi } \right) + \frac{{\overline{{\hat{y}^{3} }} }}{6} (\eta^{3} - 3c\eta ), $$

(20)

$$ \begin{aligned} D_{2} \left( {\hat{x}, \hat{y}} \right) & = \frac{{\overline{{\hat{x} \hat{y}^{3} }} + 3R}}{6} \left( {\xi \eta^{3} - 3b\eta^{2} - 3c\xi \eta + 3cb} \right) + \frac{{\overline{{\hat{x}^{3} \hat{y}}} + 3R}}{6} \left( {\xi^{3} \eta - 3b\xi^{2} - 3a\xi \eta + 3ab} \right) \\ & \quad + \frac{{\overline{{\hat{x}^{2} \hat{y}^{2} }} }}{4} \left( {\xi^{2} \eta^{2} - c\xi^{2} - 2b\xi \eta - a\eta^{2} + ac + 2b^{2} } \right) + \frac{{\overline{{\hat{x}^{4} }} - 3}}{24} \left( {\xi^{4} - 6a\xi^{2} + 3a^{2} } \right) \\ & \quad + \frac{{\overline{{\hat{y}^{4} }} - 3}}{24} (\eta^{4} - 6c\eta^{2} + 3c^{2} ), \\ \end{aligned} $$

(21)

where,

$$ \xi = a\hat{x} + b\hat{y}, $$

(22)

$$ \eta = b\hat{x} + c\hat{y} $$

(23)

and $a = \frac{1}{{1 - R^{2} }}$, $b = - \frac{R}{{1 - R^{2} }}$, $c = \frac{1}{{1 - R^{2} }} .$

We can thus plugin Eqs. 20 and 21 in Eq. 17 and obtain a general expression for the JPDF between $\hat{x}$ and $\hat{y}$. As shown by Raupach [34], it is sufficient to include only up to third order terms, i.e., $j + k = 3$ in Eq. 17 to model the JPDFs between $\hat{x}$ and $\hat{y}$.