An empirical expression to relate aerodynamic and surface temperatures for use within single-source energy balance models

Abstract

Single-source energy balance models are simple and particularly suited to assimilate mixed pixel remote sensing data. Mixed pixels are made up of a combination of two main elements, the soil and the vegetation. The use of single-source models implies that the reference temperature for the estimation of convective fluxes, the aerodynamic temperature, is linked to the available remotely sensed surface temperature. There are many relationships relating both temperatures in the literature, but few that try to find objective constraints on this link. These relationships accounts for the difference between both temperatures by dividing the roughness length for thermal turbulent transport by an expression known as “radiometric kB⁻¹”, which depends mostly on Leaf Area Index (LAI). Acknowledging that the two temperatures should be similar for bare soil and high LAI conditions, we propose an empirical relationship between LAI and the ratio of the difference between the aerodynamic and the air temperatures and the difference between the surface and the air temperatures, also known as “β function”. Nine datasets obtained in agricultural areas (four in south western France near Toulouse, four in south eastern France near Avignon, one in Morocco near Marrakech) are used to evaluate this new relationship. They all span the entire cropping season, and LAI values range from 0 to about 5. This new expression of the β function is then compared to the β function retrieved from measured sensible heat flux and in situ radiometric measurements as well as the β function simulated by a two-source SVAT model (ICARE). Its performance in estimating the sensible heat compares well to other empirical or semi-empirical functions, either based on a β function or a radiometric kB⁻¹.

Introduction

Assessing the turbulent fluxes of latent and sensible heat at the land surface is a crucial issue for both water resource management (computation of evapotranspiration) and meteorological forecasting (evolution of the planetary boundary layer). In order to compute these fluxes at a suitable spatial scale, estimation methods based on the use of remote sensing data are favoured. There is a large array of evapotranspiration estimation methods that use as input or constraint remotely sensed variables such as the NDVI (Courault et al., 2005, Gowda et al., 2008, Olioso et al., 2005). Evapotranspiration in potential conditions (i.e. not water limited) can be assessed with relatively good precision from in situ (Cleugh et al., 2007) or remote sensing derived (Venturini et al., 2008) meteorological data, and, very often, NDVI. But when water stress occurs the latent and sensible heat fluxes are more difficult to assess. In those cases, there is a tight coupling between the evapotranspiration and the radiative surface temperature and, consequently, methods based on remote sensing data in the Thermal Infra Red (TIR) domain are favoured (Kalma et al., 2008). Those methods often compute the instantaneous latent heat flux as the residual of the energy budget and, in most cases, an expression of each individual term of the energy budget is proposed (Boulet et al., 2007):

$$\text{Rn}-G=H+\text{LE}$$

While net radiation Rn and, to a lesser extent, soil heat flux G can be expressed directly as a function of the radiative surface temperature, the turbulent fluxes H (sensible heat) and LE (latent heat) depend on a mixed-surface (soil, air and vegetation) temperature source T_aero:

$$H=\rho c_p\frac{T_{\text{aero}}-T_{\text{air}}}{r_{a,h}}$$

where ρ is the air density, c_p the specific heat of air at constant pressure, T_air is the air temperature at a reference level above the canopy and r_a,h the aerodynamic resistance for heat exchange. The temperature T_aero called aerodynamic temperature represents the average temperature of the air in the vicinity of the vegetation elements within the canopy, at the height of the aerodynamic level (defined as the sum of the displacement height and the roughness length for momentum). There is no measuring device for this temperature, which is usually inverted from turbulent flux measurements. Moreover, it can be significantly different from the ensemble directional radiometric surface temperature T_surf (as defined by Norman and Becker, 1995) which is usually derived from brightness temperature measurements made by a thermo-radiometer or infrared thermometer at nadir or at a specific view angle (Kustas and Anderson, 2009, Kustas et al., 2007, Stewart et al., 1994) and which is used to assess energy balance from remote sensing data. In general, the relationship between aerodynamic and surface temperatures is obtained through the use of a dual-source energy balance when either the vegetation or the soil bulk skin temperatures are known (Lhomme et al., 2000). Retrieving surface temperature for each of the different components of the surface (mostly soil and plants) is difficult, especially with current remote sensing platforms (Jia et al., 2003). The use of single-source models is therefore favoured over dual-source models to estimate pixel average turbulent fluxes from a mixed-pixel radiometric temperature. For those models, there is a need to develop robust yet simple methods to relate the aerodynamic temperature to the surface temperature. This has been subject of debate for a long time (e.g. see Carlson et al., 1995, Kalma and Jupp, 1990). Many formulations exist in the literature, and a comprehensive terminology and conversion formulae are proposed by Matsushima (2005). Historically, most expressions governing the relationship between the aerodynamic and the surface temperatures have been built on an analogy between wind and temperature profiles within the canopy. However, while the bottom boundary condition for wind (a null value at the height of the aerodynamic level) allows defining a roughness length for momentum exchange (z_om), the bottom value of the temperature profile, the aerodynamic temperature, is generally unknown. One assumes usually that a roughness length z_or (improperly named roughness length for thermal exchange) can be defined so that, at a height corresponding to the displacement height plus z_or, the surface temperature of the vegetation can be considered as representative of the aerodynamic temperature. The relationship between both roughness lengths translates into what Matsushima names the “radiometric kB⁻¹” $\text{k}{\text{B}}_{\text{radio}}^{-1}$ which is written as $z_{\text{or}}=z_{\text{om}}/e^{\text{k}{\text{B}}_{\text{radio}}^{-1}}$. In that case, the difference between the aerodynamic temperature and the surface temperature in the sensible heat flux formulation is expressed by an additional resistance (Lhomme et al., 1997):

$$H=\rho c_p\frac{T_{\text{surf}}-T_{\text{air}}}{r_{a,\text{hm}}+(k{B}_{\text{radio}}^{-1}/k{u}_{\ast })}\iff H=\frac{\rho c_{p}k{u}_{\ast }(T_{\text{surf}}-T_{\text{air}})}{\text{ln}((z-d)/z_{\text{or}})-{\Psi }_h((z-d)/L)+{\Psi }_h(z_{\text{or}}/L)}$$

where $r_{a,\text{hm}}=\left[\text{ln}((z-d)/z_{\text{om}})-{\Psi }_h((z-d)/L)+{\Psi }_h(z_{\text{om}}/L)\right]/k{u}_{*}$ is the aerodynamic resistance for heat exchange before kB⁻¹ correction, u_* the friction velocity, k the von Karman constant, z the measurement height of the atmospheric forcing, d the displacement height, L the Monin Obhukhov length and ψ_h the stability correction function for heat transfer given by Paulson (1970). $\text{k}{\text{B}}_{\text{radio}}^{-1}$ is derived according to the expected air temperature profile within the canopy and expressed as a function of meteorological data, LAI and plant height. Amongst the well known formulae, one can cite those from Blümel (1998), Massman (1999) revisited by Su et al. (2001) and Lhomme et al. (2000).

Other authors have proposed a somewhat simpler, and easier to interpret, formulation of the relationship between T_surf and T_aero, called the “β function”, originally proposed by Chehbouni et al. (1997). β is expressed solely in terms of the temperatures, independently from wind speed:

$$H=\rho c_p\beta \frac{T_{\text{surf}}-T_{\text{air}}}{r_{a,h}}$$

i.e.

$$\beta =\frac{T_{\text{aero}}-T_{\text{air}}}{T_{\text{surf}}-T_{\text{air}}}$$

Even for isothermal surfaces, usually bare soils or very dense canopies, the aerodynamic temperature can be slightly different from the surface temperature, because the diffusion process for heat transfer adds to the convective exchange of air. There is therefore a difference between the effective eddy diffusivities for momentum and heat exchange, which can be again translated into an excess resistance function of an “aerodynamic kB⁻¹” or $\text{k}{\text{B}}_{\text{aero}}^{-1}$. The available kB⁻¹ formulae, derived either empirically or from scalar and flux theoretical profiles in the canopy, account for both aspects: the difference between the aerodynamic and the radiometric temperatures (radiometric kB⁻¹) and the difference between momentum and heat exchange diffusion processes (aerodynamic kB⁻¹). For isothermal surfaces, one can expect that there is no radiometric component within the combined (radiometric and aerodynamic) kB⁻¹. The combined kB⁻¹ retrieved for those surfaces from observations by solving for $\text{k}{\text{B}}_{\text{radio}}^{-1}$ in Eq. (3) is fairly low (within the range 0–5 according to Verhoef et al. (1997), Massman (1999) and Yang et al. (2008)). For strongly non-isothermal situations, which correspond in general to intermediate LAI values (LAI in the range 0.5–2), the combined kB⁻¹ is much higher (in the range 10–30 according to the same authors). One can thus assume that $\text{k}{\text{B}}_{\text{aero}}^{-1}$ is usually smaller than $\text{k}{\text{B}}_{\text{radio}}^{-1}$ for all LAI values, or that the difference between the surface temperature and the aerodynamic temperature will have on average a much larger impact on the sensible heat flux than the difference between the diffusion processes for heat and momentum at the vicinity of the canopy. While the radiometric kB⁻¹ do not discriminate between both differences (the difference between the aerodynamic and surface temperatures and the difference between the roughness lengths for momentum z_om and heat exchange z_oh), the β function allows us to separate both aspects and keep the difference between z_oh and z_om in the formulation of the aerodynamic resistance: $z_{\text{oh}}=z_{\text{om}}/e^{\text{k}{\text{B}}_{\text{aero}}^{-1}}$ and $r_{a,h}=r_{a,\text{hm}}+\text{k}{\text{B}}_{\text{aero}}^{-1}/k{u}_{*}$. Consequently, Eq. (4) can be rewritten as:

$$H=\rho c_p\beta \frac{T_{\text{surf}}-T_{\text{air}}}{r_{a,\text{hm}}+(k{B}_{\text{aero}}^{-1}/k{u}_{\ast })}\iff H=\frac{\rho c_p\beta k{u}_{\ast }(T_{\text{surf}}-T_{\text{air}})}{\ln ((z-d)/z_{\text{oh}})-{\Psi }_h((z-d)/L)+{\Psi }_h(z_{\text{oh}}/L)}$$

One must also note that the published values of the radiometric kB⁻¹ $\text{k}{\text{B}}_{\text{radio}}^{-1}$ (Matsushima, 2005) according to Lhomme, Blümel and Massman/Su (see formulations in Table 1) can be converted into β values by combining Eqs. , (6):

$$\beta =\frac{r_{a,\text{hm}}+(\text{k}{\text{B}}_{\text{aero}}^{-1}/k{u}_{\ast })}{r_{a,\text{hm}}+(\text{k}{\text{B}}_{\text{radio}}^{-1}/k{u}_{\ast })}$$

The Blümel and Massman/Su formulations depend on Leaf Area Index LAI through, mostly, the fraction cover (f_c) of the canopy and on two parameters difficult to assess, the component aerodynamic kB⁻¹ for bare soil and for vegetation canopy, respectively (See Table 1). Since for bare soil and full cover conditions there is no large difference between both temperatures, the β function is fairly easily interpreted: β values are close to 1 for those bare soil and full cover conditions, that is, more generally, for all homogeneous isothermal surfaces, while for sparse vegetation β decreases. In those conditions, the soil temperature has a large impact on the radiative surface temperature whereas the aerodynamic temperature remains closer to a mix of air and vegetation temperatures and is less influenced by the soil temperature. Since the soil temperature around midday is generally higher than the vegetation temperature, the observed radiative surface temperature is often larger than the aerodynamic temperature around that time. Factors influencing β and $\text{k}{\text{B}}_{\text{radio}}^{-1}$ include LAI and other plant geometrical features such as height and fraction cover, friction velocity, time of the day, solar radiation, etc. However, most studies agree on the fact that LAI is by far the main driving factor, at least for agricultural canopies for which the turbid medium (random leaf dispersion, Myneni et al., 1989) and permeable-rough transfer hypotheses are valid (Kustas et al., 2007, Verhoef et al., 1997). This is further confirmed by dual-source land surface models which predict the aerodynamic temperature through the classical dual-source approach (Shuttleworth and Wallace, 1985). The evolution of β as a function of LAI is presented in Fig. 1a as obtained from an uncalibrated run of the ICARE SVAT model (Gentine et al., 2007) for the B124 wheat site in Morocco (see below). One can observe that the simulated shape of the β(LAI) relationship decreases sharply from 1 when LAI increases to about 1, and increases more slowly for higher LAI, tending again to 1 for LAI well above 3. The lognormal distribution function is therefore a good candidate to represent the evolution of 1 − β(LAI) for the whole range of LAI values. Consequently, we propose the following empirical relationship for β:

$$\beta =1-\frac{a}{\text{LAI}*b*\sqrt{2\pi }}{e}^{-{\left(\text{ln}(\text{LAI})-c\right)}^2/(2*b^2)}$$

where a, b and c are empirical coefficients that need to be calibrated.

The objectives of the present paper are threefold:

• to retrieve β variations with LAI from observations for nine experimental cultural cycles where seasonal evolution of factors governing β is available and LAI values range from 0 to well above 2, and by doing so, assess the variability in shapes and scales of the β(LAI) relationship

• to compare several formulae of β(LAI), including the new one (Eq. (8); hereafter referred to as the Boulet et al. expression), against observed trends and

• to compare the performances of the various formulae in computing sensible heat flux from observed in situ radiometric surface temperature.

The new Boulet et al., 2012 expression (Eq. (8)) of the β function will be first calibrated over the values of β derived from experimental data acquired on the B124 wheat site in Morocco (i.e. values will be obtained for a, b and c in Eq. (8)). Next, it will be tested on data acquired over eight other crop cycles in South of France.