The role of coherent turbulent structures in explaining scalar dissimilarity within the canopy sublayer

Abstract

Scalar similarity is widely assumed in models and interpretation of micro-meteorological measurements. However, in the air space within and just above the canopy (the so-called canopy sublayer, CSL) scalar similarity is generally violated. The scalar dissimilarity has been mainly attributed to differences in the distribution of sources and sinks throughout the canopy. Since large-scale coherent structures in the CSL (e.g. double roller and sweep/ejection) arise from the instabilities generated by the interaction between the mean flow and the canopy, they may encode key dynamical features about the production term responsible for the source–sink dissimilarity of scalars. Therefore, it is reasonable to assume that the geometric attributes of coherent structures are tightly coupled to the onset and the vertical extent of scalar dissimilarity within the CSL. Large-eddy simulation (LES) runs were used to investigate the role of coherent structures in explaining scalar dissimilarity among three scalars (potential air temperature, water vapour and $\text{ CO }_2$ concentration) within the CSL under near-neutral conditions for horizontally uniform but vertically varying vegetation leaf area density. It was shown that coherent structures, when identified from the first mode of a novel proper orthogonal decomposition (POD) approach, were able to capture some features of the scalar dissimilarity in the original LES field. This skill was quantified by calculating scalar–scalar correlation coefficients and turbulent Schmidt numbers of the original field and the coherent structures, respectively. However, coherent structures tend to magnify the magnitude of scalar–scalar correlation, particularly in cases where this correlation is already strong. The ability of coherent structures to describe more complex features such as the scalar sweep-ejection cycle was also explored. It was shown that the first mode of the POD does not capture the relative importance of sweeps to ejections in the original LES field. However, the superposition of few secondary coherent structures, derived from higher order POD modes, largely diminish the discrepancies between the original field and the POD expansion.

Appendix 1: A general POD approach in the wavenumber space

For continuous applications, Eq. (9) has a countable infinity of solutions [27], each including an eigenvalue $\lambda ^{(n)}$ and an associated eigenfunction $\hat{\phi }_i^{(n)}$, where the index $n$ is added to distinguish between different solutions. However, for the application of this paper, the approach of discretization needs to be conducted such that the solutions are finite (see e.g. [70]). For convenience, we maintain the notation of continuous conditions. The eigenfunctions are orthogonal and can be normalized such that

$$\begin{aligned} \int _H\hat{\phi }_i^{(m)}(k_x,k_y,z)\hat{\phi }_i^{(n)*}(k_x,k_y,z){\mathrm{d}}z=\delta _{mn}. \end{aligned}$$

(17)

We may sort the series of solutions by decreasing magnitude of the modulus of $\lambda ^{(n)}$ such that $\lambda ^{(1)}>\lambda ^{(2)}>\cdots$, and define eigenmode as $\hat{\psi }_i^{(n)}=\sqrt{\lambda ^{(n)}}\hat{\phi }_i^{(n)}$ such that $\hat{\psi }_i^{(n)}$ carry the information of both the spatial shape and its importance toward describing the variance [19]. The POD of $\hat{{\mathrm{V}}}$ can now be expressed based on the eigenfunctions,

$$\begin{aligned} \hat{{\mathrm{V}}}_i(k_x,k_y,z,t)=\sum _{n=1}^\infty \hat{a}^{(n)}(k_x,k_y,t)\hat{\phi }_i^{(n)}(k_x,k_y,z), \end{aligned}$$

(18)

where $\hat{a}^{(n)}$ is the coefficient corresponding to $\hat{\phi }_i^{(n)}$. Multiply both sides of Eq. (18) by $\hat{\phi }_i^{(m)*}$, integrate in $z$ over $H$ and then substitute Eq. (17) into the resulting equation, we obtain the expression of the coefficient $\hat{a}^{(n)}$ as

$$\begin{aligned} \hat{a}^{(n)}(k_x,k_y,t)=\int _{H}\hat{{\mathrm{V}}}_i(k_x,k_y,z,t)\hat{\phi }_i^{(n)*}(k_x,k_y,z){\mathrm{d}}z. \end{aligned}$$

(19)

$\hat{a}^{(n)}$ is also orthogonal across different solutions in the sense that

$$\begin{aligned} \langle \hat{a}^{(m)}(k_x,k_y,t)\hat{a}^{(n)*}(k_x,k_y,t)\rangle _t=\delta _{mn}\lambda (k_x,k_y). \end{aligned}$$

(20)

Combining Eq. (10) and Eq. (18), we can obtain the reconstruction formula of $\varPhi _{ij}$ from $\hat{\phi }_i^{(n)}$ as

$$\begin{aligned} \varPhi _{ij}(k_x,k_y,z,\tilde{z})=\sum _{n=1}^\infty \lambda ^{(n)}(k_x,k_y)\hat{\phi }_i^{(n)}(k_x,k_y,z)\hat{\phi }_j^{(n)*}(k_x,k_y,\tilde{z}). \end{aligned}$$

(21)

The two-point correlation tensor is the inverse Fourier transform of Eq. (21)

$$\begin{aligned}&R_{ij}(r_x,r_y,z,\tilde{z})\nonumber \\&\quad =\frac{1}{4\pi ^2}\sum _{n=1}^\infty \int \int \lambda ^{(n)}(k_x,k_y)\hat{\phi }_i^{(n)}(k_x,k_y,z)\hat{\phi }_j^{(n)*}(k_x,k_y,\tilde{z})e^{ik_xr_x+ik_yr_y}{\mathrm{d}}k_x{\mathrm{d}}k_y, \end{aligned}$$

(22)

where $r_x$ and $r_y$ are separation distances in $x$ and $y$, respectively. Letting $r_x = r_y = 0$ and $z = \tilde{z}$ leads to the one-point second-order statistics

$$\begin{aligned} \langle {\mathrm{V}}_i{\mathrm{V}}_j\rangle (z)=\frac{1}{4\pi ^2}\sum _{n=1}^\infty \int \int \lambda ^{(n)}(k_x,k_y)\hat{\phi }_i^{(n)}(k_x,k_y,z)\hat{\phi }_j^{(n)*}(k_x,k_y,z)e^{ik_xr_x+ik_yr_y}{\mathrm{d}}k_x{\mathrm{d}}k_y.\nonumber \\ \end{aligned}$$

(23)

Furthermore, letting $i = j$ and integrating $z$ over $H$, the conservation of the overall variance is given by

$$\begin{aligned} E=\int _H\langle {\mathrm{V}}_i{\mathrm{V}}_i\rangle (z){\mathrm{d}}z=\frac{1}{4\pi ^2}\sum _{n=1}^\infty \int \int \lambda ^{(n)}(k_x,k_y){\mathrm{d}}k_x{\mathrm{d}}k_y. \end{aligned}$$

(24)

If we write $\varLambda ^{(n)}=\frac{1}{4\pi ^2}\sum _{n=1}^\infty \int \int \lambda ^{(n)}(k_x,k_y){\mathrm{d}}k_x{\mathrm{d}}k_y$, then

$$\begin{aligned} E=\sum _{n=1}^\infty \varLambda ^{(n)}. \end{aligned}$$

(25)

The original field of ${\mathrm{V}}_i$ can be approximated by a truncated reconstruction using the first $p$ eigenvalue/eigenfunction solutions [conf. Eq. (18)]:

$$\begin{aligned} {\mathrm{V}}_i^{(p)}(x,y,z,t)=\frac{1}{4\pi ^2}\sum _{n=1}^p\int \int \hat{a}^{(n)}(k_x,k_y,t)\hat{\phi }_i^{(n)}(k_x,k_y,z)e^{ik_xr_x+ik_yr_y}{\mathrm{d}}k_x{\mathrm{d}}k_y, \end{aligned}$$

(26)

and the contribution of ${\mathrm{V}}_i^{(p)}$ to the second-order statistics is given by,

$$\begin{aligned} \langle {\mathrm{V}}_i^{(p)}{\mathrm{V}}_j^{(p)}\rangle (z)\!=\!\frac{1}{4\pi ^2}\sum _{n=1}^p\!\int \int \!\lambda ^{(n)}(k_x,k_y)\hat{\phi }_i^{(n)}(k_x,k_y,z)\hat{\phi }_j^{(n)*}(k_x,k_y,z)e^{ik_xr_x+ik_yr_y}{\mathrm{d}}k_x{\mathrm{d}}k_y.\nonumber \\ \end{aligned}$$

(27)

The 3D coherent structure has been referred to here as the first eigenmode $\psi _i^{(1)}$ since $\varLambda ^{(1)}$ represents the greatest percentage of $E$ out of all other non-Fourier-mode choices of the coherent structure [19, 27, 33, 34, 53]. However, the POD framework does not provide the phase angles for $\phi _i^{(1)}$, which are critical in determining the spatial shape of the coherent structure in the physical space. This issue is commonly tackled through the shot-effect expansion theory [50] in conjunction with an extra assumption regarding the physical property of the coherent structure. For example, Lumley [50] proposed the bi-spectrum or three-point correlation criterion, which states that the coherent structure should conserve as much as possible the three-point correlation of the original velocity field. The second method, termed the ‘compactness criterion’, was originally proposed by [26], and assumes that the coherent structure is spatially compact. The third method is termed the ‘wavenumber continuity’ or the ‘spectral smoothness’ criterion, which implies that the phase angle of the coherent structure is continuous in wavenumber space [53]. In this paper, we apply the compactness criterion considering that the coherent structures in the CSL revealed by flow visualization experiments are typically compact [55]. Finnigan et al. [20] compared the coherent structures derived from the POD together with the compactness criterion and from a conditional average method using local maxima of static pressure at the canopy top as a trigger, and found that unlike the latter, the former does not reveal that a sweep motion is often followed by an ejection motion downstream. However, this does not affect our use of the POD approach as far as the topic of this paper is concerned because scalar dissimilarity is dominant in the vertical direction. Additionally, the orientation of the coherent structure is forced to be consistent with a sweep motion owing to the known fact that sweep is the dominant contributor to the Reynolds stress within the canopy [19]. Denoting the phase angles as $\eta (k_x,k_y)$, the coherent structure can now be written as

$$\begin{aligned} \psi _i^{(1)}(r_x,r_y,z)=\frac{1}{4\pi ^2}\int \int \sqrt{\lambda ^{(1)}(k_x,k_y)}\hat{\phi }_i^{(1)}(k_x,k_y,z)e^{ik_xr_x+ik_yr_y+i\eta (k_x,k_y)}{\mathrm{d}}k_x{\mathrm{d}}k_y. \end{aligned}$$

(28)

$\psi _i^{(1)}$ connects to ${\mathrm{V}}_i^{(1)}$ by

$$\begin{aligned} {\mathrm{V}}_i^{(1)}(x,y,z,t)=\int \int \psi _i^{(1)}(x-\tilde{x},y-\tilde{y},z)\beta (\tilde{x},\tilde{y},t){\mathrm{d}}\tilde{x}{\mathrm{d}}\tilde{y}, \end{aligned}$$

(29)

where

$$\begin{aligned} \beta (x,y,t)=\frac{1}{4\pi ^2}\int \int \hat{a}^{(1)}(k_x,k_y,t)e^{-i\eta {k_x,k_y}} \Bigg / \sqrt{\lambda ^{(1)}(k_x,k_y)}{\mathrm{d}}k_x{\mathrm{d}}k_y, \end{aligned}$$

(30)

and $\beta (x,y,t)$ satisfies

$$\begin{aligned} \langle \beta (x,y,t)\beta (\tilde{x},\tilde{y},t)\rangle _t=\delta (x-\tilde{x},y-\tilde{y}). \end{aligned}$$

(31)

The variance $\varLambda ^{(1)}$ is conserved in ${\mathrm{V}}_i^{(1)}$ as well as in $\psi _i^{(1)}$, given by

$$\begin{aligned} \int \!\!\int \!\!\int \langle {\mathrm{V}}_i^{(1)}(x,y,z,t)^2\rangle _t{\mathrm{d}}x{\mathrm{d}}y{\mathrm{d}}z=\int \!\!\int \!\!\int \psi _i^{(1)}(r_x,r_y,z)^2{\mathrm{d}}r_x{\mathrm{d}}r_y{\mathrm{d}}z=\varLambda ^{(1)}. \end{aligned}$$

(32)