Mean and Turbulent Flow Statistics in a Trellised Agricultural Canopy

Abstract

Flow physics is investigated in a two-dimensional trellised agricultural canopy to examine that architecture’s unique signature on turbulent transport. Analysis of meteorological data from an Oregon vineyard demonstrates that the canopy strongly influences the flow by channelling the mean flow into the vine-row direction regardless of the above-canopy wind direction. Additionally, other flow statistics in the canopy sub-layer show a dependance on the difference between the above-canopy wind direction and the vine-row direction. This includes an increase in the canopy displacement height and a decrease in the canopy-top shear length scale as the above-canopy flow rotates from row-parallel towards row-orthogonal. Distinct wind-direction-based variations are also observed in the components of the stress tensor, turbulent kinetic energy budget, and the energy spectra. Although spectral results suggest that sonic anemometry is insufficient for resolving all of the important scales of motion within the canopy, the energy spectra peaks still exhibit dependencies on the canopy and the wind direction. These variations demonstrate that the trellised-canopy’s effect on the flow during periods when the flow is row-aligned is similar to that seen by sparse canopies, and during periods when the flow is row-orthogonal, the effect is similar to that seen by dense canopies.

Appendix: Derivation of Turbulent Kinetic Energy Budget

Following Raupach and Shaw (1982), the budget of $\overline{e}$ in a canopy can be written as,

(4)

where primes represent deviations from temporal averages that are represented as overbars, double primes represent deviations from spacial averages (taken over volumes of very small $\varDelta z$ with $\varDelta x$ and $\varDelta y$ each larger than the local scales of the canopy architecture) represented by angled brackets, g is the acceleration due to gravity, $\epsilon$ is the dissipation rate of $\langle \overline{e}\rangle$, $\mathcal {R}1$ is the budget residual that represents the pressure transport term that cannot be explicitly calculated, and index notation ($u_i = \{u_\perp , v_\parallel , w\}$ and $x_i = \{x_\perp , y_\parallel , z\}$ for this coordinate system) is used for simplicity (Brunet et al. 1994; Kaimal and Finnigan 1994). The labelled terms represent mean advection of $\langle \overline{e}\rangle$ into the averaging volume, shear production, turbulent transport, dispersive transport, wake production, and buoyancy production of $\langle \overline{e}\rangle$ inside the averaging volume, respectively.

The budget equation can be considerably simplified through standard assumptions used for canopy flows. However, due to the coordinate system used here and the discrete nature of the vine architecture some terms that are often neglected in other studies, e.g., for streamwise aligned coordinate systems, had to be retained. Additionally, although Taylor’s hypothesis would suggest that our anemometer signal could be interpreted as a spatial signal, such an approach is problematic in a canopy, especially considering the vineyard’s heterogeneity. Still, it is common to assume that the temporal average of the anemometer data also approximates a spatial average (e.g., Meyers and Baldocchi 1991), thus making use of a temporally-spatially averaged budget appropriate. If instead the budget of $\overline{e}$ could be investigated at multiple spatial locations within the vineyard canopy, using a temporally-averaged budget may be more useful in understanding transport.

First, out of lack of interest for temporal variations at time scales larger than 30 min and because only quasi-steady 30-min periods of data were used, the unsteady term was set to zero.

The advection term is the summation of three individual components, all of which are zero by definition. Although $\partial \overline{e}/ \partial x_\perp$ may be non-zero locally within the wakes of individual vines, when spatial averaging is used, such spatial variations disappear from the advection term and are instead accounted for in the dispersive transport term. The spatial averaging causes $\langle \overline{e}\rangle$ to be a constant within the averaging volume and thus $\partial \langle \overline{e}\rangle / \partial x_\perp = \partial \langle \overline{e}\rangle / \partial y_\parallel = 0$. This same rule can be applied to any term wherein the slope of a spatially-averaged variable is to be determined in either the $x_\perp$ or $y_\parallel$ direction. By definition, a planar-averaged variable has no slope along any axis on the plane. Likewise, although $\overline{w}$ may be persistently non-zero at some points within the canopy, $\langle \overline{w}\rangle$ must be zero (assuming an accurate tilt correction and neglecting subsidence at spatial scales larger than that used for $\langle \rangle$). We therefore assumed that the anemometers were placed within the aisle in such a way that no persistent features in w caused $\overline{w}$ to not equal $\langle \overline{w}\rangle$.

The shear production term is the summation of nine individual terms each including a component of the stress tensor. The six terms containing $\partial \langle \rangle / \partial y_\parallel$ and $\partial \langle \rangle / \partial x_\perp$ are zero as described above. Additionally, any term containing $\partial / \partial y_\parallel$ can be set to zero due to the continuity of the canopy in the row-parallel direction. The two remaining terms which contain $\partial \langle \overline{w}\rangle / \partial x_i$ are zero due to the spatial averaging of $\overline{w}$ but would likely be of interest when using only local ensemble averaging of spatially resolved data. This leaves only the two terms which contain the components of the vertical momentum flux paired with the vertical gradients of their respective velocity components, i.e., $\langle \overline{u_\perp ^\prime w^\prime }\rangle \partial \langle \overline{u_\perp }\rangle /\partial z + \langle \overline{v_\parallel ^\prime w^\prime }\rangle \partial \langle \overline{v_\parallel }\rangle /\partial z$.

The turbulent transport term can also be expanded into nine components, six of which have either $\partial \langle \rangle / \partial x_\perp$ or $\partial \langle \rangle / \partial y_\parallel$ and are therefore zero. This leaves only the three terms with $\partial / \partial z$. Based on the observed behaviour of the cross-vine momentum flux in Sect. 3 and the discontinuity of the canopy in the row-perpendicular direction, it is likely that terms like $\partial (\overline{u_\perp ^\prime v_\parallel ^{\prime 2}}) / \partial x_\perp$ would be of some significance in a local budget of e even if such cannot be resolved with this approach and data.

The dispersive transport term in Eq. 4 appears due to the spatial averaging procedures performed on the advection term in the temporally averaged budget of e. Again six of the nine components of the dispersive transport term are zero because they contain $\partial \langle \rangle / \partial x_\perp$ or $\partial \langle \rangle / \partial y_\parallel$. The final three terms are $\partial /\partial z [\langle \overline{u_\perp ^{\prime 2}}^{\prime \prime }\overline{w}^{\prime \prime }\rangle +\langle \overline{v_\parallel ^{\prime 2}}^{\prime \prime }\overline{w}^{\prime \prime }\rangle + \langle \overline{w^{\prime 2}}^{\prime \prime }\overline{w}^{\prime \prime }\rangle ]$. Because it was assumed that the data from the anemometers accurately represents a spatial average and deviations in the data can only be treated as temporal deviations, we are unable to resolve any terms that contain dispersive components that arise from the spatial decomposition of the temporally averaged budget. Fortunately, the sum of these three terms should be small compared to the other terms of the budget and will be accounted for in a budget residual, $\mathcal {R}2$. Böhm et al. (2013) showed that the dispersive transport term was only non-zero in a small portion of the upper canopy and never had a magnitude greater than $\approx 25\%$ of the wake production term at those heights.

The wake production term only exists as a byproduct of the spatial averaging procedures performed on the shear production term. It is within the wake production term that the energy associated with spatial gradients at scales smaller than the averaging scale of $\langle \rangle$ is accounted for. Following the work of Raupach and Shaw (1982), many other studies (e.g., Brunet et al. 1994; Raupach et al. 1996; Böhm et al. 2013) have used $\langle \overline{u_i}\rangle \partial /\partial x_j\left[ \langle \overline{u_i^\prime u_j^\prime }\rangle + \langle \overline{u_i}^{\prime \prime } \overline{u_j}^{\prime \prime }\rangle \right]$ as an equivalent for the wake production term. This approach can then be expanded into 18 individual terms, all but four of which are zero for the reasons discussed above, i.e., $\langle \overline{w}\rangle = \partial \langle \rangle / \partial y_\parallel = \partial \langle \rangle / \partial x_\perp = 0$. Of the remaining four terms, two contain the dispersive form of the Reynolds stress based on $u_\perp$ and $v_\parallel$, and cannot therefore be determined with the anemometer data, and must be neglected and will appear in $\mathcal {R}2$. Fortunately, Bailey and Stoll (2013) showed that $\overline{u_\perp }^{\prime \prime }\overline{w}^{\prime \prime }$ was never more than $\approx 20\%$ of the total vertical flux in a vineyard-like canopy under perpendicular conditions and was only non-zero in a small span of heights just below the mid-canopy. Poggi et al. (2004) also showed that the ratio of $\overline{u_\perp }^{\prime \prime }\overline{w}^{\prime \prime }/\overline{u_\perp ^\prime w^\prime }$ varies as a function of canopy density and that for even the most extreme canopy case the ratio never exceeded $\approx 40\%$.

All of these assumptions and simplifications reduced the full budget of e (Eq. 4) to the version shown as Eq. 2, all of the terms of which, could be solved for with the anemometer data in the vineyard. The two residual terms as defined here were simply combined into one term $\mathcal {R}$ because the contributions cannot be individually partitioned. It is important to understand however that some portion of the residual arose from a physical process (pressure transport) while some other portion is attributed to spatial variations which cannot be explicitly solved for with a single tower of anemometers.