**The Relationship Between Incoming Solar Radiation and Land Surface Energy Fluxes**

Edgar G. Pavia

*Centro de Investigación Científica y de Educación Superior de Ensenada Mexico* 

## **1. Introduction**

Incoming solar radiation (*R*) is the driver of the land surface energy fluxes: latent heat (*E*) or soil evaporation (i.e. the natural transfer of water from the topsoil to the atmosphere, although it might include also condensation), sensible heat (*H*) so-called because it can be "felt" (i.e. it is related to temperature differences between the surface and the atmosphere), and the ground heat flux (*G*) so-called because it is restricted to the interior of the ground (i.e. it is related to temperature differences between ground layers). All this seems rather obvious during the daytime, when *R* provides the energy input and apparently the output of *E*, *H* and *G* balance it. However the situation is less clear at night when *R* is nil but *E*, *H* and *G* may not vanish, while the energy balance must be kept. To understand this simple idea lets consider the following example: under certain conditions, like wet soil in low- and mid-latitudes, *E* may be considered almost as proportional to *R*; that is *E ~* (*a*1 *× R*) *+ OT*, where *a*1 is a proportionality factor (not necessarily a constant) and *OT* are other terms (in this case: *H* and *G*, which are usually smaller than *E* and (*a*1 *× R*), but there could be other terms). If we are able to estimate *E*, *R*, *H* and *G*, or these terms are somehow known, we can tentatively solve for *a*1, which could characterize the relationship between *R* and these fluxes, and the term (*a1 × R)* is called the net radiation (*Rn*); i.e. the part of *R* which is actually balanced by *E*, *H* and *G*. The problem, however, is not trivial because even if we restrict ourselves to this simplified case, and we could measure *R* and *E*, the smaller terms *H* and *G* would have to be assessed as well. Nevertheless we believe that this difficulty may be partially overcome by empirically modeling *E*. Recall that calculating *a*1 with observations of total and net solar radiation: *a*1 = (*Rn*)*obs*/*Robs* may not be appropriate for our purposes, because it would not consider *E*, *H* and *G*, and *a*1 is but an element of a vector **a** yetunknown. Therefore our goal is to develop a full energy flux model to show that indeed the relationship between *R* and surface fluxes may be achieved in this empirical way. That is, in this work we will attempt to approximate these surface energy fluxes by simultaneously modeling them based on a simplified energy balance. Although, due to their importance in many environmental issues (from crop-field irrigation (Brisson & Perrier, 1991), (Allen et al., 1998), to the study of the global water cycle (Huntington, 2006)) one usually first looks for *Rn* in order to evaluate surface fluxes; here we will attempt to model *E*, *H* and *G* in order to estimate *Rn* = (*a*1 *× R*). However even if our model is successful, that is even if in general it correlates well with observations, we will examine the situations in which the model fails, the energy balance does not hold, and a simple relationship between *R* and surface fluxes cannot be established. Thus the limitations of this study should serve as a motivation for future work.

#### **1.1 Modeling approach**

The simplest way to determine if a model is appropriate is to compare it with observations. Even though assessing *E* in general is difficult because it depends not only on the ambient conditions, but also on soil composition and moisture content, here we use observations of soil evaporation (*Eobs*) obtained through micro-lysimetry (Figure 1). That is, we will try to model *E* from *E*o*bs* and calculate their correlation coefficient r(*Emod*, *Eobs*) to evaluate the appropriateness of our model: *Emod* = (*Rn* - *H* - *G*)*mod*. Although diverse efforts have been devoted to model *E* for different applications (Penman, 1948), (Priestley & Taylor, 1972), (Twine et al., 2000), (Brutsaert, 2006), (Agam et al., 2010), all these efforts possess different limitations and degrees of difficulty. In other words, there is no general way to model *E* which is practicable for all situations (Crago & Brutsaert, 1992), and thus we must develop an *ad hoc E*-model to estimate *a*1 for our particular case. In this sense we will focus on a relatively simple case, the diurnal variation of bare soil evaporation when water is not a limiting factor (for example wet sand with substantially more than 5% of water (Pavia & Velázquez, 2010)). That is, when the main diurnal surface energy balance is between *Rn* and *E*: *Rn* ~ *E.* Previous works in cases similar to the present one have confirmed that daytime *E* is highly correlated with *R* (Pavia, 2008); therefore we should expect our model to reflect daytime better than nighttime conditions. We will perform an experiment with an evaporating tray containing a small amount of wet sand (~35 Kg maximum), so that *Eobs* should be easier to measure throughout the day than *Rn*. Our hypothesis is that we can obtain *Emod* from a small number of standard meteorological observations and experimentally-obtained variables, which are chosen by their assumed relationship to energy terms; namely *R*, air temperature (*Ta*), surface temperature (*To*), soil temperature (*Ts*) and observed soil evaporation (*Eobs*). Therefore we will try to fit *Eobs* to a linear combination of terms derived from the above variables. Specifically *Emod* = *Eobs* ~ L(*R*, Δ*Ta*, Δ*Ts*), where, as it will be explained in the next section, the model *E* (*Emod*) is achieved from *R*, Δ*Ta = To - Ta*, Δ*Ts = To - Ts* and *Eobs*, through a multiple regression procedure yielding a vector **a** which includes a1 among other parameters. This approach is physically-motivated by the primary land surface energy balance:

$$R\_n = E + H + G\_\prime \tag{1}$$

Where *Rn* would be approximated by the *R* term (Gay, 1971) and the sensible heat flux *H* and the ground heat flux *G* would be similarly approximated by the Δ*Ta* and Δ*Ts* terms, respectively. Therefore it is anticipated that the multiple-regression parameter-vector **a** resulting from our model may give a preliminary assessments of the relative importance of *R* on each of these surface energy flux densities.

#### **2. Methods**

In this section we describe the original technique to find the relationship between *R* and the surface energy fluxes. This includes the experimental evaluation of *E*, the approximation made of *H* ~ Δ*Ta* and *G* ~ Δ*Ts* from the observed temperatures, and the multiple regression method to optimize these approximations.
