Simple analytical model of evapotranspiration in the presence of roots

Evaporation of water out of a soil involves complicated and well-debated mechanisms. When plant roots are added into the soil, water transfer between the soil and the outside environment is even more complicated. Indeed, plants provide an additional process of water transfer. Water is pumped by the roots, channeled to the leaf surface, and released into the surrounding air by a process called transpiration. Prediction of the evapotranspiration of water over time in the presence of roots helps keep track of the amount of water that remains in the soil. Using a controlled visual setup of a two-dimensional model soil consisting of monodisperse glass beads, we perform experiments on actual roots grown under different relative humidity conditions. We record the total water mass loss in the medium and the position of the evaporating front that forms within the medium. We then develop a simple analytical model that predicts the position of the evaporating front as a function of time as well as the total amount of water that is lost from the medium due to the combined effects of evaporation and transpiration. The model is based on fundamental principles of evaporation fluxes and includes empirical assumptions on the quantity of open stomata in the leaves, where water transpiration occurs. Comparison between the model and experimental results shows excellent prediction of the position of the evaporating front as well as the total mass loss from evapotranspiration in the presence of roots. The model also provides a way to predict the lifetime of a plant.

Figure 1

A sample image of a lentil root grown inside a 2D Hele-Shaw cell of 10 cm $\times$ 15 cm. The effective dimensions of the cell are 7.5 cm $\times$ 13.5 cm $\times$ 1.2 mm.

Figure 2

(a) Typical evaporation curve in a 2D Hele-Shaw cell in the absence of roots: water mass loss is plotted as a function of time. In this figure, both regimes are represented by an image and the transition is indicated by the broken vertical line. (b) Comparison of water loss as a function of time in in the presence $(◯)$ and absence $(-)$ of roots. The solid arrows $(\searrow )$ point to the transition point between the first and second regimes. (c) Measured evaporating and percolating fronts denoted as $z_{\sup }$ and $z_{\inf }$, respectively, as a function of time. (d) Measured height of the PSZ, $h$, where $h=z_{\inf }-z_{\sup }$, as a function of time. The inset figures depict the evolution of the PSZ during regime 2, showing that its height $h$ roughly remains constant. These curves were obtained under conditions of $T=32\pm 2{}^{\circ }\mathrm{C}$ and $H_R=20.0\pm 2.0\%$.

Figure 3

(a) Schematic of the evapotranspiration process in the presence of roots. The saturation of water in the medium is defined by $\Phi$. The system starts out fully saturated, $\Phi =1$. Water is lost from the system through evaporation and transpiration, leading to the appearance of a partially saturated zone (PSZ) of height $h=z_{\inf }-z_{\sup }$, which is a mixture of air and water $\left(\Phi <1\right)$. As water is lost during the first regime, a portion of the root${}^{\prime }\mathrm{s}$ length becomes exposed to the PSZ. The vertical distance between the tip of the root and the cell surface is $L_v$. During the second regime, a dry region $\left(\Phi =0\right)$ develops above the receding evaporating front, $z_{\sup }$, which slows down evaporation. A depletion zone also develops, as illustrated by the small white area in the PSZ around the root. (b) Experimental images showing the evolution of the front during evapotranspiration in the presence of roots. The roots absorb water to replenish the amount of water lost by transpiration; this is clearly manifested by the appearance of a depletion zone around the roots. The depletion zone is considered part of the PSZ because thin films of water are still present in this zone.

Figure 4

Illustration of a stoma at the leaf surface (adapted from Ref. [29]). Water vapor diffuses through the characteristic height ${\delta }_{\mathrm{st}}$ of the stomata and across the external boundary layer $\delta$. $C_{\mathrm{sat}}$ is the concentration of water inside the leaf, $C_s$ refers to the concentration of water at the surface of the leaf, and $C_e$ is the water concentration in the surrounding atmosphere. $C_e$ depends on the relative humidity.

Figure 5

Plot of root length, $L_T$, as a function of distance between the root tip and cell surface, $L_v$. These results reflect an average of the experiments performed for the characterization of lentil roots under fully saturated conditions. Experiments show a linear proportionality between root length and tip distance, as seen from the broken line.

Figure 6

(a) Position of the evaporating front $z_{\sup }$ as a function of time in the presence of roots under various relative humidity conditions. The solid curves are calculated from the implicit expression for $z_{\sup }$ shown in Eq. (16). The solid arrows point to the transition point between regime 1 and regime 2 when the evaporating front, $z_{\sup }$, begins to develop. We see that the model remarkably fits the experimental results. (b) Total water mass loss curves in the porous medium in the presence of roots under different relative humidity conditions. The solid curves are generated using Eqs. (13), (15), and (16). The change in inflection of each curve, indicated by the solid arrows, denotes the transition point between regime 1 and regime 2. Again, our model fits experimental results.