Impact of different within-canopy wind attenuation formulations on modelling sensible heat flux using TSEB

Abstract

The unique vertical canopy structure and clumped plant distribution/row structure of vineyards and orchards creates an environment that is likely to cause the wind profile inside the canopy air space to deviate from how it is typically modelled for most crops. This in turn affects the efficiency of turbulent flux exchange and energy transport as well as their partitioning between the plant canopy and soil/substrate layers. The objective of this study was to evaluate a new wind profile formulation in the canopy air space that explicitly considers the unique vertical variation in plant biomass of vineyards. The validity of the new wind profile formulation was compared to a simpler wind attenuation profile that assumes attenuation through a homogeneous canopy. We evaluated both attenuation models using measurements of wind speed in a vineyard interrow, as well as turbulent flux estimates retrieved from a two-source energy balance model, which uses land surface temperature as the key boundary condition for flux estimation. This is relevant in developing a robust remote sensing-based energy balance modelling system for accurately monitoring vineyard water use or evapotranspiration that can be applied using satellite and airborne imagery for field-to-regional scale applications. These tools are needed in intensive agricultural production regions with arid climates such as the Central Valley of California, which experiences water shortages during extended drought periods requiring an effective water management policy based on robust water use estimates for allocating water resources. Results showed that the new wind profile model improved sensible heat flux estimates (RMSE reduction from 42 to 35 $\text{W}\,\text{m}^{-2}$) when the vine canopy is in early growth stage resulting in a strongly clumped canopy.

Appendix: TSEB model

The basic equation of the energy balance at the surface can be expressed following Eq. 4.

$$\begin{aligned}&R_{n}\approx H+\lambda {}E+G \end{aligned}$$

(4a)

$$\begin{aligned}&R_{n,{\mathrm{S}}}\approx H_{{\mathrm{S}}}+\lambda {}E_{{\mathrm{S}}}+G \end{aligned}$$

(4b)

$$\begin{aligned}&R_{n,{\mathrm{C}}}\approx H_{{\mathrm{C}}}+\lambda {}E_{{\mathrm{C}}} \end{aligned}$$

(4c)

with $R_n$ being the net radiation, H the sensible heat flux, $\lambda {}E$ the latent heat flux or evapotranspiration, and G the soil heat flux. “C” and “S” subscripts refer to canopy and soil layers, respectively. The symbol “$\approx$” appears since there are additional components of the energy balance that are usually neglected, such as heat advection, storage of energy in the canopy layer or energy for the fixation of CO$_2$ (Hillel 1998).

The key in TSEB models is the partition of sensible heat flux into the canopy and soil layers, which depends on the soil and canopy temperatures ($T_{\mathrm{S}}$ and $T_{\mathrm{C}}$, respectively). If we assume that there is an interaction between the fluxes of canopy and soil, due to an expected heating of the in-canopy air by heat transport coming from the soil, the resistances network in TSEB can be considered to be in series. In that case H can be estimated as in Eq. 5 (Norman et al. 1995, Eqs. A1–A3)

$$\begin{aligned} \begin{aligned} H=H_{{\mathrm{C}}}+H_{{\mathrm{S}}}=&\rho _{\mathrm{air}} C_{\mathrm{p}}\frac{T_{\mathrm{AC}}-T_{\mathrm{A}}}{R_{\mathrm{a}}}\\ =&\rho _{\mathrm{air}} C_{\mathrm{p}}\left[ \frac{T_{{\mathrm{C}}}-T_{\mathrm{AC}}}{R_{x}}+\frac{T_{{\mathrm{S}}}-T_{\mathrm{AC}}}{R_{\mathrm{s}}}\right] \end{aligned} \end{aligned}$$

(5)

where $\rho _{\mathrm{air}}$ is the density of air ($\text{kg m}^{-3}$), $C_{\mathrm{p}}$ is the heat capacity of air at constant pressure ($\text{J kg}^{-1}$$\text{K}^{-1}$), $T_{\mathrm{AC}}$ is the air temperature at the canopy interface, equivalent to the aerodynamic temperature $T_0$, computed with Eq. 6 (Norman et al. 1995, Eq. 4).

$$\begin{aligned} T_{\mathrm{AC}}=\frac{\frac{T_\mathrm{A}}{R_{\mathrm{a}}}+\frac{T_{\mathrm{C}}}{R_x}+\frac{T_{\mathrm{S}}}{R_{\mathrm{s}}}}{\frac{1}{R_{\mathrm{a}}} +\frac{1}{R_x}+\frac{1}{R_{\mathrm{s}}}}. \end{aligned}$$

(6)

Here $R_{\mathrm{a}}$ is the aerodynamic resistance to heat transport ($\text{s}\,\text{m}^{-1}$), $R_{\mathrm{s}}$ is the resistance to heat flow in the boundary layer immediately above the soil surface ($\text{s}\,\text{m}^{-1}$), and $R_{x}$ is the boundary layer resistance of the canopy of leaves ($\text{s}\,\text{m}^{-1}$). The mathematical expressions of these resistances are detailed in Eq. 7 and in Norman et al. (1995) and Kustas and Norman (2000) and discussed in Kustas et al. (2016).

$$\begin{aligned} R_{\mathrm{a}}&=\frac{\ln \left( \frac{z_{T}-d_0}{z_{0H}}\right) -\Psi _{h}\left( \frac{z_{T}-d_0}{L}\right) +\Psi _{h}\left( \frac{z_{0H}}{L}\right) }{\kappa '\,u_{*}} \end{aligned}$$

(7a)

$$\begin{aligned} R_{\mathrm{s}}&=\frac{1}{c\left( T_{{\mathrm{S}}}-T_{\mathrm{A}}\right) ^{1/3}+b\,u_{\mathrm{s}}} \end{aligned}$$

(7b)

$$\begin{aligned} R_{x}&=\frac{C'}{\mathrm {LAI}}\left( \frac{l_{\mathrm{w}}}{U_{d_0+z_{0{\mathrm{M}}}}}\right) ^{1/2} \end{aligned}$$

(7c)

where $u_{*}$ is the friction velocity ($\text{m s}^{-1}$) computed as:

$$\begin{aligned} u_{*}=\frac{\kappa '\,u}{\left[ \ln \left( \frac{z_{u}-d_0}{z_{0{\mathrm{M}}}}\right) -\Psi _{m} \left( \frac{z_{u}-d_0}{L}\right) + \Psi _{\mathrm{m}}\left( \frac{z_{0{\mathrm{M}}}}{L}\right) \right] }. \end{aligned}$$

(8)

In Eq. 8$z_{u}$ and $z_{T}$ are the measurement heights for wind speed u ($\text{m s}^{-1}$) and air temperature $T_{\mathrm{A}}$ (K), respectively. $d_0$ is the zero-plane displacement height, $z_{0{\mathrm{M}}}$ and $z_{0H}$ are the roughness length for momentum and heat transport, respectively (all those magnitudes expressed in m), with $z_{0H}=z_{0{\mathrm{M}}}\exp ({-kB^{-1}})$. In the series version of TSEB $z_{0H}$ is assumed equal to $z_{0{\mathrm{M}}}$ since the term $R_x$ already accounts for the different efficiency between heat and momentum transport (Norman et al. 1995), and therefore $kB^{-1}=0$. The value of $\kappa '=0.4$ is the von Karman’s constant. The $\Psi _{m}\left( \zeta \right)$ terms in Eqs. 7a and 8 are the adiabatic correction factors for momentum. The formulations of these two factors are described in Brutsaert (1999, 2005). These corrections depend on the atmospheric stability, which is expressed using the Monin–Obukhov length L (m):

$$\begin{aligned} L=\frac{-u_{*}^{3}\rho _{\mathrm{air}}}{\text{kg}\left[ ^H/_{\left( T_{\mathrm{A}}C_{\mathrm{p}}\right) } +0.61E\right] } \end{aligned}$$

(9)

where H is the bulk sensible heat flux ($\text{W m}^{-2}$), E is the rate of surface evaporation ($\text{kg s}^{-1}$), and g the acceleration of gravity ($\text{m s}^{-2}$).

The coefficients b and c in Eq. 7b depend on turbulent length scale in the canopy, soil-surface roughness and turbulence intensity in the canopy, which are discussed in Sauer et al. (1995), Kondo and Ishida (1997) and Kustas et al. (2016). $C'$ is assumed to be 90 $\text{s}^{^1/_2}\,\text{m}^{-1}$ and $l_{\mathrm{w}}$ is the average leaf width (m).

Wind speed at the heat source–sink ($z_{0{\mathrm{M}}}+d_0$) and near the soil surface was originally estimated using Goudriaan (1977) wind attenuation model (Eq. 10)

$$\begin{aligned} U\left( z\right) =U_{\mathrm{C}}\exp \left[ -a_G\left( 1-z/h_{\mathrm{c}}\right) \right] \end{aligned}$$

(10)

$$\begin{aligned} a_G=0.28\text{LAI}^{2/3}h_{\mathrm{c}}^{1/3}l_{\mathrm{w}}^{-1/3}. \end{aligned}$$

(11)

Since Eqs. 5–9 are interrelated, an iterative scheme is performed until the convergence of L and $u_*$ is reached. The iterative process is as follows: neutral conditions are firstly assumed ($L\rightarrow {}\infty$, $\Psi _{M}\left( \zeta \right) =0$ and $\Psi _{H}\left( \zeta \right) =0$) and an initial estimate of H is calculated using Eqs. 8 to 5, and E with Eq. 4. An initial value of L is then obtained from Eq. 9 and the stability functions are then calculated, which gives a new friction velocity (Eq. 8) and resistance set (Eq. 7) and new estimates of H and E (Eqs. 6, 5 and 4). L is recalculated again and the process continues (Eqs. 9–5) until the change in L and $u_*$ between two successive iterations is lower than a certain threshold.

When only a single observation of $T_{\mathrm{rad}}$ is available (i.e., measurement at a single angle), partitioning of $T_{\mathrm{rad}}$ requires some assumptions to help to define $T_{\mathrm{C}}$ or $T_{\mathrm{S}}$. One approach developed for TSEB (Norman et al. 1995) starts with an initial estimate that assumes plants are transpiring at a potential rate, as defined by the Priestley and Taylor (1972) relationship, applied to the canopy divergence of net radiation ($R_{n,{\mathrm{C}}}$)

$$\begin{aligned} \lambda E_{\mathrm{C}}=\alpha _{\mathrm{PT}}\,f_{\mathrm{g}}\frac{\Delta }{\Delta +\gamma }R_{n,{\mathrm{C}}} \end{aligned}$$

(12)

where $\alpha _{\mathrm{PT}}$ is the Priestley–Taylor coefficient, initially set to 1.26, $f_{\mathrm{g}}$ is the fraction of vegetation that is green and hence capable of transpiring, $\Delta$ is the slope of the saturation vapor pressure versus temperature curve, and $\gamma$ is the psychrometric constant. This allows the canopy-sensible heat flux to be calculated using the energy-balance at the canopy layer ($H_{\mathrm{c}}=R_{n,{\mathrm{C}}}-\lambda E_{\mathrm{C}}$) and hence an estimate of $T_{\mathrm{C}}$ to be obtained by inverting Eq. 5 (Norman et al. 1995, Eqs. A7, A11 and A12). Then $T_{\mathrm{S}}$ is the derived from Eq. 13 having both $T_{\mathrm{rad}}$ and $T_{\mathrm{C}}$ and an estimate of $f_{\mathrm{c}}\left( \theta \right)$ the fraction of vegetation observed by the sensor view zenith angle $\theta$.

$$\begin{aligned} T_{\mathrm{rad}}^4\left( \theta \right) =f_{\mathrm{c}}\left( \theta \right) T_{{\mathrm{C}}}^4+\left[ 1-f_{\mathrm{c}}\left( \theta \right) \right] T_{{\mathrm{S}}}^4. \end{aligned}$$

(13)

The value of $f_{\mathrm{c}}\left( \theta \right)$ is typically estimated as an exponential function of the leaf area index, which includes a clumping factor or index $\Omega$ for row crops and canopies where the LAI is concentrated for plants sparsely distributed or are organized such as row crops (Kustas and Norman 1999; Anderson et al. 2005) and has the following form.

$$\begin{aligned} f_{\mathrm{c}}\left( \theta \right) =1-\exp \left( \frac{-0.5\Omega LAI}{\cos \theta }\right) . \end{aligned}$$

(14)

If the initial $T_{\mathrm{C}}$ implied by this approximation is unusually low in comparison with the observed $T_{\mathrm{rad}}$, $T_{\mathrm{S}}$ will likely be overestimated and therefore produce unrealistic estimates of soil latent heat flux (negative values during daytime). In this case, the $\alpha _{\mathrm{PT}}$ coefficient is iteratively reduced assuming the canopy is stressed and transpiring at sub-potential levels until soil latent heat flux becomes zero or positive.