## Estimating the temperature at the surface

 E. Linacre 10/'97

problem

The surface temperature (or skin temperature) Ts can be estimated from satellite IR measurements, but it is not straightforward to derive the air temperature (at screen height) T from Ts. Alternatively, a florist who is concerned about a possible frost and has access only to predicted screen-level temperatures, needs to estimate Ts from T. The skin temperature Ts governs upwards longwave radiation and the rate of photosynthesis in the case of a surface of leaves. And the difference (T - Ts) determines the flow of sensible heat from the surface.

theory

The basic information is as follows:

1) surface energy-balance equation: H = Rn - G - L.E where H is the sensible heat flux from surface to air, Rn is the net radiation to the surface, G is the heat absorbed into the soil, L is the latent heat of evaporation, and E is the evaporation rate. This is the energy-balance equation at the surface.

2) surface heat flux equation: H = r.c (Ts - T) / r, where r is the density of air, c its specific heat, and r is the diffusion resistance between the surface and the air.

3) soil heat flux (empirical expression) G = 0.2 Rn in many circumstances, it is found empirically (see Linacre 1992: 338).

4) Penman’s formula for the rate of evaporation from a wet surface: L.E = s.Rn/(s + d) + r.c.D / r.(s + d) where s is the slope of the psychometric curve between T and Td (the dewpoint temperature), d is the psychrometric constant, D is the saturation deficit of the air.

5) vapour pressure deficit: D = s (T – Td), from the definition of s.

6) slope of the psychometric curve: s / (s + d) = 0.42 + 0.012 T, approximately (Linacre 1993a: 244). Call this expression J.

Combination of all this information yields the following

 (Ts - T) = r.Rn (0.8 - J) / r.c - J (T - Td) (i)

discussion

T is approximately the mean of the daily extreme temperatures, and the dewpoint Td can also be inferred from the daily extremes (Linacre 1992: 86) or (less accurately, especially in dry climates) assumed to be the same as the daily minimum.

The product r.c has a value of about 1,200 J/m3.K near sea level. As an example, we can estimate (Ts - T) in a typical case, assuming that r has a value of 100s/m (Linacre 1993b: 46), Rn is maybe 150W/m2, T is 25° C and Td is 10° C. The temperature difference (Ts - T) is then given as - 9.8K, i.e. the wet surface is 9.8K cooler than the air.

However, eqn (i) becomes the following if the same surface is so dry that there is no evaporation (ie L.E = 0),

 (Ts - T) = 0.8 r.Rn / r.c (ii)

The temperature difference in the environment described above would then be + 10K, ie the surface would be ten degrees hotter than the air. In other words, the temperature difference depends very much on the wetness of the surface.

It also depends on the aerodynamic resistance r (Linacre 1964), which in turn is governed by the surface roughness and the wind speed. The value of 100s/m used above applies to grassland, but for vine leaves in the open it is 36s/m (Linacre 1972: 378), and for a pine forest around 5s/m (Linacre 1993b: 42). In the case of a dense forest, it seems likely that the heat absorption G is negligible, in view of the tenuous nature of the foliage compared with the ground. The temperature difference for a wet forest in the environment described above then is:

 (Ts - T) = r.Rn (1 - J) / r.c - J (T - Td) (iii)

In the above example (Ts - T) is then given as - 10.6K, i.e. almost the same as for grassland after rain. But the difference would be + 0.8K for a non-evaporating forest (however improbable). This difference is much less than for a dry grassland (ie + 10K). Thus the effect of differences of r is related to surface wetness.

#### References

Linacre,E.T. 1964a. Calculations of the transpiration rate and temperature of a leaf. Archiv. Meteor. Geophys. & Bioklim. B13, 391-9.

______ 1972. Leaf temperatures, diffusion resistances and transpiration. Agric. Meteor. 10, 365-82.

______ 1992. Climate Data & Resources. (Routledge, London) 366pp.

______ 1993a. Data-sparse estimation of lake evaporation using a simplified Penman equation. Agric. & Forestry Meteor. 64, 237-56.

______ 1993b. A three-resistance model of crop and forest evaporation. Theoret. Appl. Climatol. 48, 41-8.