Freely available online through the author-supported open access option.

Abstract

In this study, we analyzed the potential of distributed soil temperature and soil moisture observations for identifying the spatiotemporal variability of near-surface water and energy fluxes. We studied the soil energy balance using soil moisture and temperature data collected during the Second Microwave Water and Energy Balance Experiment (MicroWEX-2) in Florida. We found that heat transfer in the shallow subsurface could not be explained by conduction. Sinks and sources of energy in each soil layer were quantified using an inversion approach to the heat diffusion equation. We investigated the extent to which the sinks and sources could be explained by advection and phase change. From our analysis, it seems that, for dry days, advection is a comparatively minor contributor to heat transfer and that phase change plays a more significant role. Yet vapor diffusion rates, required for sustaining phase changes and thus evaporation in the soil large enough to explain the sinks and sources, were beyond the plausible range. We concluded that soil moisture and temperature observations can yield quantitative information on the surface energy balance and heat partitioning. There is a lack of understanding of heat transfer in the shallow subsurface, however, that hampers the translation of soil temperature and moisture observations to water and energy fluxes.

Temperature measurements in the topsoil revealed an unexpected heat sink at 4 cm. We explored relevant explanations for this sink, the prime candidate being evaporation, yet water vapor diffusion is one order of magnitude too small to account for the heat sink. The findings have direct importance for satellite observations and soil distributed temperature sensing.

The motivation for this study was to develop a new technique based on distributed temperature sensing (DTS) to observe the spatiotemporal variability of land surface–atmosphere exchange processes of water and energy from the micro- to the mesoscale. Understanding land–atmosphere exchanges is of great importance in the study of the global energy and water cycles. Land–atmosphere exchanges of water and energy are mainly controlled by soil moisture and temperature, vegetation type, and meteorologic forcing. The dependence on soil characteristics, vegetation, and meteorology means that land–atmosphere exchanges are characterized by patterns that vary at different spatial and temporal scales.

Heat transfer in the shallow subsurface is particularly crucial for understanding land– atmosphere exchanges; the difference between the surface temperature and the air temperature largely determines the sensible heat flux (Gao et al., 2008). Furthermore, recent and imminent satellite missions to observe land surface characteristics rely on radiative transfer schemes to relate satellite observations to surface characteristics such as soil moisture and vegetation. These schemes are particularly sensitive to skin temperature estimation (Holmes et al., 2008).

To validate and improve land surface models and radiative transfer schemes, observations of heat-transfer-related variables across various spatial and temporal scales are essential. Distributed temperature sensing (e.g., Selker et al., 2006; Tyler et al., 2009) is a promising technique to monitor soil temperature and soil moisture across scales between 1 m and 10 km, with the potential to bridge the gap between point measurements and the scale of a remote sensing pixel or land surface model cell. In DTS, fiber optic cables are used as thermal sensors, providing thousand of simultaneous temperature readings. These cables are easy to install in the shallow subsurface. Distributed temperature sensing can yield information on both temperature itself and soil moisture at high spatial and temporal resolution across long distances. Steele-Dunne et al. (2010) demonstrated that soil temperature observations at multiple depths could be used to infer soil moisture. This type of soil moisture sensing is called passive soil DTS. Passive soil DTS measures the temperature response in buried or submerged cables to the diurnal radiation cycle. In contrast, active soil DTS applies a heat pulse to the soil, and the resultant temperature change is used to determine soil moisture. Sayde et al. (2010) demonstrated that soil moisture in a laboratory column in the range of 0.05 to 0.41 m3 m−3 could be measured with a precision of 0.004 to 0.046 m3 m−3 using an active DTS approach. If we can use temperature and soil moisture data to study heat transfer, we will be able to use DTS to study land–atmosphere exchanges at a high resolution across large areas.

To infer fluxes from temperature and moisture observations, heat transfer models are necessary. Energy partitioning and heat transfer in the shallow subsurface is poorly understood. Commonly used models based on full partitioning at the surface and heat conduction (e.g., heat flow equations by van Wijk and de Vries [1963] or the force restore method [Dickinson, 1988]) perform well when applied to deep soil layers but result in large errors when applied to the shallow subsurface (Holmes et al., 2008). In the shallow subsurface, phase change and advection can be more significant than conduction. Recently, Holmes et al. (2008) and Gao et al. (2008) proposed improved models including these modes. The approach of Holmes et al. (2008) is based on the assumption that heat partitioning occurs within a shallow soil layer of up to 4 cm rather than at the surface, so that phase change plays a dominant role in the heat transfer in his zone. Gao et al. (2008) included advective heat transfer due to water movement in the soil.

The objective of this study was to determine what we can learn about the energy balance in the shallow subsurface from soil moisture and temperature observations. We aimed to quantify fluxes and evaluate the relative contributions of the heat transfer modes of conduction, advection, and phase change by comparing an estimated source–sink term to the physically reasonable limits of these transfer modes.

The data used in this experiment were from the Second Microwave Water and Energy Balance Experiment (MicroWEX-2). The relative importance of several heat transfer modes was evaluated by comparison with observations and physically plausible ranges of these transfer modes. Our ability to use soil moisture and temperature data to quantify the shallow subsurface energy balance (demonstrated here with point data) means that DTS could be used to yield new insight into the variability of land–atmosphere interactions at a range of scales.

Background and Theory

Soil Heat Transfer

Temperature changes in the soil are driven by five modes of heat transfer: radiation, conduction, convection, advection, and phase changes, commonly referred to as latent heat transfer or evaporation. We use the term convection for sensible heat transport, driven by density gradients, and advection for passive scalar transport by, for example, water or air.

Radiative heat transfer is dominant at the soil–air interface where the soil receives direct solar shortwave radiation, Rs,down (W m−2). Depending on the albedo of the soil, part of this shortwave radiation is reflected, Rs,up (W m−2), and does not contribute to heating. The albedo of (vegetated) soil can vary between 0.05 and 0.30. In addition to shortwave radiation, the soil receives longwave radiation RL,down (W m−2) from the atmosphere but also from objects such as trees and scrub and overlying vegetation. The soil emits longwave radiation, RL,up (W m−2), as a function of its temperature. Soil heat transfer models are typically based on the assumption that there is full heat partitioning at the surface into sensible heat H (W m−2), latent heat LE (W m−2), and ground heat flux G (W m−2). The latter can, in this case, be approximated as 
\[G{=}R_{\mathrm{s},\mathrm{down}}{-}R_{\mathrm{s},\mathrm{up}}{+}R_{1,\mathrm{down}}{-}R_{1,\mathrm{up}}{-}H{-}\mathrm{LE}\]
[1]
Holmes et al. (2008) suggested, however, that energy partitioning, including latent heat formation, occurs not at the surface but in the top few centimeters of soil. Therefore, we also included an explicit latent heat term in our model of soil heat transfer below the surface.

Below the surface, radiative heat transfer becomes negligible and temperature changes are induced by heat conduction, convection, advection, and phase changes. Additional sources or sinks of energy may arise due to biochemical activity (e.g., Wadsö, 2009), but these were not considered in our study.

Assuming thermal equilibrium between the solid, liquid, and gas phases, the one-dimensional energy conservation equation may be written as 
\begin{eqnarray*}&&C(z)\frac{{\partial}T}{{\partial}t}{=}\mathrm{{\kappa}}(z)\frac{{\partial}^{2}T}{{\partial}z^{2}}{-}{\upsilon}(z)c_{\mathrm{w}}\mathrm{{\rho}}_{\mathrm{w}}\mathrm{{\theta}}\frac{{\partial}T}{{\partial}z}\\&&{+}L(n{-}\mathrm{{\theta}})D_{\mathrm{v}}\frac{{\partial}^{2}\mathrm{{\rho}}_{\mathrm{v}}}{{\partial}z^{2}}{-}\frac{S}{{\partial}z}\end{eqnarray*}
[2]
where T (K) is temperature, C (J m−3 K−1) is the volumetric heat capacity of the soil, κ (W m−1 K−1) is its thermal conductivity, v (m s−1) is the liquid flow velocity, cw (J kg−1 K−1) is the gravimetric heat capacity of water, ρw (kg m−3) is the liquid water density, θ (m3 m−3) is the volumetric water content, L (J kg−1) is the latent heat of vaporization, n (m3 m−3) is the porosity, Dv is the water vapor diffusivity (m2 s−1), ρv (kg m−3) is the vapor density, S (J m−2) is a sink term, and z (m) and t (s) are the depth and time coordinates, respectively. The soil thermal properties C and κ are further discussed below.

The terms on the right-hand side represent conductive heat transfer, advective heat transfer due to liquid water flow, phase changes, and a sink term and are illustrated in Fig. 1 . Advective heat transfer due to gas flow is neglected because the volumetric heat capacity of gas is three orders of magnitude smaller than the volumetric heat capacity of the solid or liquid. In the formulation of the phase change term, we assumed that latent heat formation is not moisture limited and that phase changes are limited by the diffusion of water vapor. This assumption was valid because we were considering an irrigated corn (Zea mays L.) field. In addition, we assumed that the water vapor transport was purely diffusive and that forced convection (heat advection by air) was neglectable.

Inversion Approach

The inversion approach of Steele-Dunne et al. (2010) was used to: (i) estimate thermal properties assuming that conduction was the dominant heat transfer mode; and (ii) estimate the total contribution of the remaining terms in Eq. [2]—advection, phase change, and other sources or sinks.

First, it was assumed that conduction was the dominant heat transfer mode and that heat transfer in a soil column can be described by the diffusion equation: 
\[\frac{{\partial}T}{{\partial}t}{=}D(\mathrm{{\theta}})\frac{{\partial}^{2}T}{{\partial}z^{2}}{=}\frac{\mathrm{{\kappa}}(\mathrm{{\theta}})}{C(\mathrm{{\theta}})}\frac{{\partial}^{2}T}{{\partial}z^{2}}\]
[3]
An implicit finite difference scheme was used to solve Eq. [3] for the diffusion coefficient D (m2 s−1). For sets of three temperature sensors, the upper and lower sensors provided the boundary conditions. The diffusivity was obtained using the observed temperature at the middle sensor and the Johansen model (Johansen, 1975; see below). For a window of 60 min and a spatial resolution of 5 mm, the MATLAB function fminsearch (MathWorks, Natick, MA) was used to find the diffusivity term that minimized the root mean squared error between the simulated and observed temperatures at the depth of the middle temperature sensor. The fminsearch function is a multidimensional unconstrained nonlinear minimization algorithm that uses the Nelder–Mead direct search method. For the first time step, linear interpolation between the three temperature measurements was used to give the initial temperature profile.

Second, we specified the diffusivity from the observed soil moisture at the middle sensor and used the optimization scheme to estimate a source–sink term representing the remaining terms in Eq. [2]. This is a somewhat simplified approach: A single value of soil moisture and hence diffusivity was used for the whole soil layer, so any gradient in soil moisture was neglected; however, estimation of the optimal diffusivity described above and comparison with a plausible range (discussed below) enabled us to distinguish between errors due to variable diffusivity and errors due to significant influence of heat transfer modes other than conduction. We also estimated a single, uniform source–sink term for the whole soil layer, which neglects the potentially steep temperature gradient, hence the source–sink, at the surface for example.

Soil Thermal Properties

The volumetric heat capacity of the soil, C (J m−3 K−1), is a simple, well-understood linear function of soil moisture: 
\begin{eqnarray*}&&C{=}(\mathrm{{\rho}}c)_{\mathrm{bulk}}{=}\frac{V_{\mathrm{a}}}{V_{\mathrm{t}}}\mathrm{{\rho}}_{\mathrm{a}}c_{\mathrm{a}}{+}\frac{V_{\mathrm{w}}}{V_{\mathrm{t}}}{\rho}_{\mathrm{w}}c_{\mathrm{w}}{+}\frac{V_{\mathrm{s}}}{V_{\mathrm{t}}}\mathrm{{\rho}}_{\mathrm{s}}c_{\mathrm{s}}\\&&C{=}n(1{-}S_{\mathrm{r}})\mathrm{{\rho}}_{\mathrm{a}}C_{\mathrm{a}}{+}nS_{\mathrm{r}}\mathrm{{\rho}}_{\mathrm{w}}c_{\mathrm{w}}{+}(1{-}n)\mathrm{{\rho}}_{\mathrm{s}}c_{\mathrm{s}}\end{eqnarray*}
[4]
where the subscripts m, a, w, and s denote the bulk soil, air, water, and soil solids, respectively, V is volume (m3), ρ is density (kg m−3), c is the specific heat capacity (J kg−1 K−1), Sr is the relative saturation (dimensionless), and n is the porosity (dimensionless).
Thermal conductivity is considerably more complicated, and there are many models available (Peters-Lidard et al., 1998). We used the model of Johansen (1975) in which the thermal conductivity is calculated as a linear combination of the dry and saturated thermal conductivities: 
\[{\kappa}{=}K_{\mathrm{e}}(\mathrm{{\kappa}}_{\mathrm{sat}}{-}\mathrm{{\kappa}}_{\mathrm{dry}}){+}\mathrm{{\kappa}}_{\mathrm{dry}}\]
[5]
where Ke is the Kersten number (Kersten (1949), given by 
\[K_{\mathrm{e}}{=}\left\{\begin{array}{ll}0.7\mathrm{log}S_{\mathrm{r}}{+}1.0&\mathrm{for}{\,}\mathrm{coarse}{\,}\mathrm{soil},{\,}S_{\mathrm{r}}{>}0.05\\\mathrm{log}{\,}S_{\mathrm{r}}{+}1.0&\mathrm{for}{\,}\mathrm{fine}{\,}\mathrm{soil},{\,}S_{\mathrm{r}}{>}0.1\end{array}\right.\]
[6]
The dry thermal conductivity, κdry, is given by a semi-empirical model: 
\[\mathrm{{\kappa}}_{\mathrm{dry}}{=}\frac{0.135\mathrm{{\gamma}}_{\mathrm{d}}{+}64.7}{2700{-}0.947\mathrm{{\lambda}}_{\mathrm{d}}}\]
[7]
where the dry density, γd, can be approximated as γd = (1 − n)2700. The saturated thermal conductivity, κsat, is given by 
\[\mathrm{{\kappa}}_{\mathrm{sat}}{=}\mathrm{{\kappa}}_{\mathrm{s}}^{1{-}n}\mathrm{{\kappa}}_{\mathrm{w}}^{n}\]
[8]
where the thermal conductivities of water (κw) and the solids (κs) are combined according to their respective volume fractions. The thermal conductivity of the solids is determined by the quartz content q: 
\[\mathrm{{\kappa}}_{\mathrm{s}}{=}\mathrm{{\kappa}}_{\mathrm{q}}^{q}\mathrm{{\kappa}}_{\mathrm{o}}^{1{-}q}\]
[9]
The thermal conductivity of quartz is κq = 7.7 W m−1 K−1 and that of other minerals is given by 
\[\left\{\begin{array}{ll}\mathrm{{\kappa}}_{\mathrm{o}}{=}2.0{\,}\mathrm{\mathbb{W}m}^{{-}1}{\,}\mathrm{K}^{{-}1},&i\mathrm{f}{\,}q{>}0.2\\\mathrm{{\kappa}}_{\mathrm{o}}{=}3.0{\,}\mathrm{\mathbb{W}m}^{{-}1}{\,}\mathrm{K}^{{-}1},&\mathrm{if}{\,}q{>}0.2\end{array}\right.\]
[10]

MicroWEX-2

The MicroWEX-2 (Judge et al., 2005) was conducted by the Center for Remote Sensing, Agricultural and Biological Engineering Department, at the Plant Science Research and Education Unit of the University of Florida, Gainesville, from 17 Mar. through 3 June 2004. The goal was to understand the land–atmosphere interactions during the growing season of sweet corn and their effect on the observed microwave brightness temperatures. The experimental site was 183 by 183 m. We briefly describe the observations used in the analysis. Judge et al. (2005) provided a detailed description of the experiment and the complete data set.

Meteorologic Data

Surface Flux Data

A Campbell Scientific (Logan, UT) eddy covariance system (ECS) was located at the center of the field. The system included a CSAT3 anemometer and KH20 hygrometer. The CSAT3 is a three-dimensional sonic anemometer, which measures wind speed and air temperature along three non-orthogonal axes. The KH2O measures the water vapor in the atmosphere. Its output voltage is proportional to the water vapor density. Latent and sensible heat fluxes were measured every 15 min. The height of the ECS was 2.1 m from the ground and the orientation of the system was 232° toward southwest. Data collected by the ECS were processed for coordinate rotation, O2, and sonic temperature corrections (see Judge et al., 2005).

A Kipp and Zonen (Delft, the Netherlands) CNR-1 four-component net radiometer was located at the center of the field to measure up- and down-welling short- and longwave infrared radiation. The sensor consisted of two pyranometers (CM-3) and two pyrgeometers (CG-3). The sensor was installed 2.66 m above the ground.

Soil Moisture, Temperature, and Heat Flux

Soil moisture and soil temperature were each measured at two locations. We used the observations closest to the ECS and CNR sensors. Ten Campbell Scientific water content time-domain reflectometers were used to measure the soil volumetric water content and temperature at depths of 2, 4, 8, 32, 64, and 100 cm every 15 min. The observations of soil moisture were duplicated at the 2-cm depth.

Two Campbell Scientific soil heat flux plates (HFT-3) were used to measure the soil heat flux at depths of 2 and 5 cm in the row and near the root area.

Soil and Vegetation Characteristics

Vegetation Characteristics

Figure 2 shows the leaf area index (LAI), plant height, and root length density observed at the study site. The LAI was measured weekly with a Li-Cor LAI-2000 (Li-Cor Biosciences, Lincoln, NE) in the interrow region, with four cross-row measurements. The LAI-2000 was set to average two locations into a single value for each vegetation sampling area so one observation was taken above the canopy and four beneath the canopy: in the row, one-quarter of the way across the row, half of the way across the row, and three-quarters of the way across the row. This provided a spatial average for row crops of partial cover.

Crop height was measured weekly by placing a measuring stick at the soil surface to average the height of the crop. The heights inside the vegetation sampling areas were taken for each vegetation sampling. While the LAI increased steadily during the growth period, the corn height increased very rapidly from the end of April to mid-May.

At tasseling, root samples were taken with a soil coring tool between rows and between plants at depths of 0 to 15, 15 to 30, 30 to 60, 60 to 90, and 90 to 120 cm. After cleaning the samples, the root length density was determined using a scanner and the WinRhizo software (Regent Instruments, Quebec, QC, Canada). The values shown are root length density at the midpoints of the specified depths. From Fig. 2, it can be seen that most of the roots are in the top 20 to 30 cm of the soil column.

Soil Properties

Because the goal of MicroWEX-2 was related to soil moisture retrieval algorithms, particular attention was paid to soil characterization. Table 1 lists the properties of the sandy soil measured at the study site.

Using the parameter values from Table 1, together with the models presented above, the soil volumetric heat capacity, thermal conductivity, and thermal diffusivity for the soil at the study area were calculated. The results are shown in Fig. 3 .

Results and Discussion

Impact of Vegetation

Figure 2 shows the increase in the LAI and height of the corn crop during the study period. To examine the impact of vegetation, the study period was divided into two halves, with the period before 1 May being considered as the low-vegetation period and the period after as the high-vegetation period. On this date, the vegetation height was beginning to increase rapidly, while the LAI was at two-thirds of its maximum value.

Figure 4 shows how each of the components of net radiation was influenced by the vegetation. Downward longwave radiation increased by 50 W m−2 in the second half of the study period. This was due to the atmospheric state (temperature and cloud cover), however, and was unrelated to vegetation. The diurnal cycle of upward longwave radiation was reduced during the latter half of the experiment. This might have been due to a change in either surface emissivity or temperature. The maximum was reduced by about 25 W m−2 and the minimum increased by the same amount, suggesting that the difference was dominated by the change in temperature rather than emissivity. The presence of a canopy insulates the surface at night, preventing temperatures from falling, while during the day it shields the surface, reducing the temperature. Moreover, emissivity was found to be positively correlated with vegetation growth rather than negatively. Shortwave net radiation increased in the second period as insolation approached its summer maximum. The increase was larger in the morning (up to 70 W m−2). At night, the magnitude of the upward net radiation was reduced by about 20 W m−2, while the maximum net radiation increased by about 100 Wm−2. The maximum also occurred earlier in the second period. The variability in net radiation appeared to increase slightly after 1 May, but its temporal distribution was essentially unchanged. No discernible change in albedo was detected.

The top row in Fig. 5 shows the resultant change in temperature at 2 cm. Recall, however, that the 2-cm temperature was in the soil below the canopy while the net radiometer was above the canopy. The soil temperature was also influenced by the orientation of the crop rows relative to the sun. There was a 5 K increase in nighttime temperatures due to the insulation provided by the canopy and a decrease of up to 5 K in the afternoon. We expect that this was due to evaporative cooling and shading from the canopy. Variability was more than halved, with a notable exception between 0900 and 1200 h when the temperature was increasing. Again, this can be attributed to the orientation of the canopy. Crops were in rows from northwest to southeast, oriented at 30° from the north.

The remaining rows in Fig. 5 show how the diurnal temperature cycle propagated with depth into the soil column. The amplitude of the mean diurnal cycle was damped with depth, and the variability decreased with depth. This was expected because the surface layers respond more quickly to variations in radiative forcing.

As the summer progressed, the mean temperature in all soil layers was seen to rise. The vegetation growth coincided with a dampening of the diurnal cycle, however, particularly in the top two layers. Increased vegetation cover slowed cooling of the surface at night and enhanced the latent heat flux during the day, thus reducing the amplitude of the diurnal cycle. Vegetation growth also led to a reduction in the standard deviation at all depths. In the top two layers, the minimum standard deviation associated with the rising limb of the diurnal temperature cycle disappeared in the latter half of the experiment.

Figure 5 also shows the temperature at each soil layer as simulated using an implicit finite difference solution to the diffusion equation (the inversion approach). For example, to estimate temperatures at 4 cm, the diffusivity was specified using soil moisture at 4 cm, and the temperatures at 2 and 8 cm provided the boundary conditions. If the model and observations matched, it suggests that conduction was the dominant heat transfer process. In the top two soil layers (2–8 and 4–32 cm), this was clearly not the case. Before 1 May, the diffusion model underestimated the nighttime temperature and warmed too quickly and too early during the day. After 1 May, the nighttime temperatures improved somewhat, but again the daytime temperatures were overestimated by the diffusion model. In contrast to the observed changes, the diffusion model predicted an increase in variability during the day in both of the top two layers.

Part of the disagreement between the diffusion model and the observations may be attributed to two sources of uncertainty in the diffusion coefficient. First, we assumed that diffusivity was constant throughout a soil layer, while soil moisture actually varied with depth. Second, there are uncertainties in the Johansen model that was used to estimate the soil thermal properties.

Optimal Diffusivity Estimates

Using the inversion approach of Steele-Dunne et al. (2010) and assuming that conduction was the only heat transfer process, the optimum diffusivity was determined for each soil layer as the diffusivity that gave the best match with the temperature at the middle observation point. For example, in the 2- to 8-cm layer, temperatures at 2 and 8 cm provided the boundary conditions. The optimal diffusivity was defined as that which gave the best agreement between the simulated and observed temperatures at 4 cm. The results for each soil layer are shown in Fig. 6 . The dashed lines indicate the upper and lower limits for the thermal diffusivity for this soil assuming the Johansen model. Figure 6 shows that the layer from 8 to 64 cm was the only one in which the diffusivity remained within the physically reasonable range. The physically reasonable range, as determined using a Monte Carlo approach to account for the uncertainty in soil parameters, is illustrated in Fig. 8 (dashed lines). In the lower layer (32–100 cm), the temperature variations were too small to reliably determine the diffusivity with this approach. Figure 6 suggests that in both these layers, diffusion was the dominant heat transfer process.

Closer to the surface, it is clear that this was not the case; neither of the upper two estimation layers yielded diffusivity within the plausible range, indicating that processes other than conduction were also driving heat transfer. Estimates were sensitive to the sensor depth, as discussed in Steele-Dunne et al. (2010). Taking into account an uncertainty on the order of 1 mm for the sensor depth, diffusivity in the two top layers was still out of range.

Estimates of the Source–Sink Term

The inversion approach discussed above was used to estimate the sink every hour in each estimation layer. In Fig. 7 , these results are averaged to show which layers were net sources or sinks and how this varied with time. The first thing to note is that the order of magnitude of the source–sink term decreased with depth. In the two lower layers shown, the source–sink term was three orders of magnitude smaller than that at the surface and was therefore negligible. Given uncertainty in the thermal diffusivity (see Fig. 8 ), the source–sink term for the lower two layers is not significant. This is consistent with the modeling results shown above in which it was demonstrated that conduction alone accounted for heat transfer in the lower layers of the soil column.

In the top layer shown (2–8 cm), the term is primarily a sink term, particularly in the latter half of the experiment. The days on which the sink was very large (e.g., 5−10 May) were those on which the amplitude of the diurnal temperature cycle was largest. The second layer (4–32 cm) is visually a mirror image of the top layer and was primarily a source of energy, particularly in the latter half of the experiment. The magnitude of this source, however, like that of the lower two layers, was orders of magnitude smaller than that of the sink at the surface.

Each line in Fig. 9 represents the diurnal cycle of the estimated source–sink term for 1 day. The source–sink term seems to have been largely determined by the 2-cm temperature. The increase in the 2-cm temperature peak and the shift in its timing resulted in a similar increase and shift in the peak of the sink term. The reduced variability in the 2-cm temperature after 1 May correlates with a reduced variability in the sink term between 2 and 8 cm. The increased variability in the 2-cm temperature after 1 May coincides with an increased variability in the magnitude of the sink term at 2 to 8 cm. It is interesting that the maximum sink in the top layer occurred between 0900 and 1200 h, and that this was followed by the maximum source in the second layer after 1200 h. This suggests that there might have been an exchange between the two layers, with some of the energy lost in the top layer providing an energy source in the lower layer.

The temperature profiles averaged for the pre- and post-vegetation periods shown in Fig. 10 give more insight into why the diffusion model failed. If we averaged profile temperatures for a full year, we would expect no change in temperature with depth. In our case, a decrease in temperature with depth was expected because we were looking at a spring temperature series. For both periods, however, the profiles showed a local temperature minimum at the 4-cm depth. Such minima cannot be modeled with a diffusion equation. If the temperature at 4 cm is modeled using the temperatures at 2 and 8 cm as boundary conditions, a large sink is needed to obtain a minimum at 4 cm like that shown in Fig. 11 . The opposite is the case for the temperature at 8 cm.

Based on the analysis of the profiles in Fig. 10, we suppose that the small source–sink in the second layer may primarily be an artifact of the chosen boundary conditions and that possibly biochemical activity may also play a role. In the first layer, we expect that other heat transfer mechanisms than conduction may explain the source–sink term and this is investigated below.

Explaining the Source–Sink Term

The origin of the source–sink term was further explored for a dry day during the observation period, 27 May. The upper row in Fig. 11 shows the diurnal temperature cycle at the 4-cm depth for this day. The amplitude of the observed temperature was smaller than the amplitude of the temperature modeled with a diffusion model. Therefore we found a source term at night and a sink term during the day. In the lower rows of Fig. 11 we investigated the origin of the source–sink term. The explanations offered are advection and latent heat.

First, we considered advection as the dominant heat transfer mechanism next to conduction. If we assume that there were no phase changes or sinks or sources related to biochemical activity, we can estimate a liquid flow velocity necessary to transport the source–sink using Eq. [2]. The results are presented in the third row in Fig. 11. The estimates go to infinity when the temperature gradients are close to zero, so these were left out of the plot. We found that a capillary flow velocity of approximately 0.4 × 10−4 m s−1 would be necessary to transport the source–sink. This implies a total capillary rise of 35 m d−1, which is not plausible and advection can therefore not explain a significant part of the source–sink. In addition, advection could not explain the average profiles found in Fig. 10 because it is not likely that the temperature at 4 cm was lower than the temperature at 8 cm under conditions of upward advective heat transfer.

Second, we considered latent heat as the dominant heat transfer mechanism. By dividing the sink term by the latent heat of vaporization (L = 2260 kJ kg−1), we find a water flux. If we compare this flux to the latent heat observations from the flux tower, it is plausible that the source–sink contributes to the latent heat flux measured at the flux tower and that the remaining latent heat stems from evaporation from the first 2 cm of the soil column and vegetation. For larger parts of the observation period, the sink term in the upper layer was compared with the observed surface fluxes in Fig. 12 . The 2 d (8–9 May 2004) when reasonable agreement was found were days at the end of a drying period. The dry surface led to a large temperature variation during the day, and the large gradients led our model to estimate a significant sink term. On 8 and 9 April, the sinks were smaller than usual but the turbulent fluxes were still quite high. This suggests that the latent heat flux may include contributions from interception, or evaporation at or close to the surface (<2 cm), which is plausible because there was precipitation or irrigation on 8 April. Conversely on 8to 9 May, the absence of intercepted or surface water could mean that the surface fluxes were drawn entirely from the 2- to 8-cm layer. This is consistent with the suggestion of Holmes et al. (2008) that the thermally active layer is smaller under wet surface conditions. If we compare the order of magnitude of the sinks with surface fluxes, it seems plausible that latent heat contributes significantly to heat transfer in the shallow subsurface.

To sustain evaporation, the water flux has to leave the soil. As written in Eq. [2], we assumed that the dominant process in this was vapor diffusion. We estimated vapor diffusion across a layer with thickness zD as (Philip and De Vries, 1957) 
\[Q_{vap}{=}D_{\mathrm{v}}(n{-}\mathrm{{\theta}})\frac{(1{-}\mathrm{RH}_{0})\mathrm{{\rho}}_{\mathrm{v},sat}}{z_{D}}\]
[11]
where RH0 (dimensionless) is the relative humidity at the surface and ρv,sat (kg m−3) is the saturated vapor density, which depends on the temperature. It was assumed that air is saturated with vapor at z = zD and the tortuosity and mass flow factors were omitted from the equation. Figure 13 shows typical values of the vapor diffusive flux for various temperatures. Comparing Fig. 13 to the third plot in Fig. 11, we find that the maximum vapor diffusive flux is an order of magnitude smaller than the phase change equivalent of the sink.

Enhanced vapor diffusion is a known phenomenon. Three explanations offered for enhanced vapor diffusion in porous media are thermal vapor diffusion (e.g., Philip and de Vries, 1957), capillary action (Shokri et al., 2009), and turbulent diffusion due to surface winds (Ishihara et al., 1992). Enhanced vapor diffusion due to thermal effects cannot explain our results because thermal vapor diffusion is a density-driven process, enhancing vapor diffusion from higher to lower temperatures, and we found enhanced vapor diffusion from lower to higher temperatures. Capillary action may explain enhanced vapor transport from a mass balance point of view but does not solve our energy balance problem. For bare soil, Ishihara et al. (1992) proposed that turbulent diffusion significantly enhances vapor transport, leading to local temperature minima at a depth of a few centimeters—the evaporation front depth. Turbulent diffusion may explain our results, but this needs to be further investigated. Fully coupled simulation of the heat and mass balance equations of the individual phases at the pore scale may increase our understanding of the physics of enhanced vapor transport.

We did not evaluate the effects of plant activity and other biochemical processes on heat transfer in the soil. Root water uptake will remove water and therefore heat from the soil and respiration will add heat to the soil. We expected that these contributions would be relatively small compared with the source–sink in the top layer, but the contributions may be more significant at deeper depths. Future work could include an analysis on the contribution of plant activity and other biochemical processes.

To take into account sinks and sources in the shallow subsurface for practical applications, such as radiative transfer schemes, empirical models may be used. Holmes et al. (2008) developed an empirical model for bare soil applications. This model was not directly applicable to our case with vegetation. An important parameter in the Holmes model is β, the ratio between the soil heat flux at 5 cm and the net radiation. Holmes assumed that β has a constant value of 0.25. Using radiation and ground heat flux measurements, we found that for our experiment β was highly variable and significantly lower than 0.25. For the low-vegetation period, we found an average value of 0.20 and for the high-vegetation period a value of 0.06. An extension of the Holmes model with a vegetation-related parameter may make the model applicable to vegetated surfaces. Distributed temperature sensing observations can help in deriving and validating such a model for a wide range of surface characteristics and vegetation covers.

Conclusions

We analyzed profile measurements of soil moisture and soil temperature to investigate the potential of distributed soil temperature and soil moisture observations for identifying the spatiotemporal variability of water and energy fluxes. We found that soil moisture and temperature observations can yield quantitative information on the energy balance and its variability under changing vegetation conditions.

Our analysis demonstrated that conduction is dominant at greater depths and that other heat transfer processes play an important role in the shallow subsurface (<8 cm). An inversion approach identified a significant source–sink term in the upper soil layer. Not surprisingly, its magnitude and diurnal cycle proved to be strongly correlated with the temperature, which in turn depends on radiation and vegetation conditions.

It proved impossible, however, using temperature and soil moisture data from 2 to 100 cm, to explain the source–sink term in terms of advection due to liquid water flow or phase change. From our analysis, it seems that, for dry days, advection is a comparatively minor contributor to heat transfer. While phase change appears to play a more significant role, it alone could not explain the source–sink term. When we consider vapor diffusion out of the soil as the limiting process for phase change, maximum values of phase change are an order of magnitude smaller than the observed source–sink term.

While temperature and soil moisture data can yield useful information on whether or not heat advection or phase changes are occurring in the soil, our lack of understanding of heat transfer in the very shallow subsurface inhibits our ability to directly relate this to land–atmosphere exchanges. Distributed temperature sensing could yield valuable information on the spatiotemporal variability in the magnitude of the sources and sinks; however, directly relating this to surface fluxes requires further research into the physics of heat transfer in the shallow subsurface.

We would like to thank Dr. Scott Tyler for his valuable comments on an earlier version of this manuscript. Part of this research was performed at Laboratory of Environmental Fluid Mechanics and Hydrology at Ecole Polytechnique Fédérale de Lausanne. Support for the MicroWEXs was obtained from the NSF Earth Science Directorate (EAR-0337277), the NASA New Investigator Program (NASA-NIP-00050655), and internal funds from the University of Florida. Open access to this article was supported by the NWO Incentive Fund for Open Access Publication as part of the project “Assimilation of time-lapse backscatter data to estimate root-zone soil moisture” (Veni-ALW 016.101.012).