Agriculturalists, foresters and ecologists are interested in surface heat balance because it is a major factor in the productivity of vegetation and the irrigation of crops. In hydrology, heat balance is important for the water balance of surfaces and water resources. In climatology, the surface heat balance determines surface temperature and for individual organisms, heat balance determines the energy needs of homoiothermic animals and the water requirements of terrestrial plants and animals. (See Physiology and Heat Transfer.)

*Heat Balance* is the key concept in *environmental heat transfer*. The heat balance equation is simply the statement of the * First Law of Thermodynamics* for a particular system. The usual convention adopted in this field is that fluxes toward the surface are positive on the left hand side of the equation:

For uniform, extensive surface, Heat Transfer through the boundary layer may be treated as quasi one-dimensional, with transport being driven by gradients of concentration, temperature and velocity normal to the surface. Rates of heat transfer are expressed in terms of the SI unit of energy flux density, W m^{–2}. The alternative for climatological time scales is MJ m^{–2} d^{–1}. Surface conditions in nature are rarely uniform, but the solution to the heat balance can be recognized as the equilibrium temperature and vapor pressure at the surface. For example, for a dry surface, the difference between surface temperature and air temperature is proportional to the supply of heat by radiation and inversely proportional to the resistance to transport across the Boundary Layer above the surface.

For natural surfaces, the main driving term of the heat balance is *Net Radiation*, R_{n},—the sum of four terms representing the components of the radiation microclimate of the surface:

where S and ρS are the incident and reflected *solar radiation*, respectively; ρ is the *reflection coefficient* for solar radiation; L_{d}*downward thermal radiation*; and L_{u} the *upward thermal radiation* emitted by the surface. (See also Solar Energy.) L_{u} may be estimated from *Stefan’s law*, provided the surface temperature T_{s} is known:

where σ (= 56.7 × 10^{−9} W m^{−2} K^{−4}) is the *Stefan-Boltzmann* constant and ε is the Emissivity of the surface. (See Radiative Heat Transfer.)

Natural surfaces may be approximated to black bodies in the thermal infrared (ε = 1), since measured values of ε * for plant leaves* are in the range 0.94–0.99 [Monteith and Unsworth (1990)]. Similar values have been measured for animal coats [Hammel (1955)]. Net radiation can therefore be approximated to:

Solar irradiance is determined by illumination geometry and atmospheric attenuation, i.e., by season and weather. Obviously, S = 0 at night while daytime peak insolation is about 1,000 and 850 W m^{–2} in the tropics and in temperate latitudes, respectively. L_{d} is also determined by weather, but ground surface and vegetation cover have a strong influence on net radiation, via ρ and T_{s}. Taking an extreme example, ρ is about 20% for bare tundra but 80 to 85% for fresh snow [Lewis and Callaghan (1975)]. Even in strong sunshine, the high reflectivity of snow makes net solar radiation S(l – σ) smaller than the (negative) net thermal radiation (L_{d} – L_{u}), and therefore R_{n} is negative. The resulting negative feedback in the microclimate tends to preserve snow-cover in spring, despite high insolation.

Strictly, we are concerned with the total heat (enthalpy) transfer from surfaces, but conventionally the components are treated separately, ignnoring the effects of height changes within the turbulent boundary layer. The dissipation terms in the heat balance are usually regarded as the *sensible heat flux* to the air above the surface, g_{s}; the *latent heat flux* carried by water evaporating from the surface g_{E} (equal to the product of the evaporation mass flux E with the specific enthalpy of evaporation h_{lg}, i.e., g_{E} = h_{lg}E); the *ground heat flux, g _{G};* and heat storage in the surface layer CdT/dt:

where C and dT/dt are the thermal capacity and the instantaneous rate of change of temperature of the surface layer (air plus a vegetation canopy), respectively. Over periods of several days, the storage term is negligible [Monteith and Unsworth (1990)] and is omitted from the remaining analysis.

Ground heat flux depends on the **thermal conductivity**, λ, and volumetric specific heat of the substrate (density × specific heat, ρ'C_{s} ). The diurnal and annual cycles of solar heating result in the propagation into the ground of temperature waves, with damping depth δ given by:

where κ is the thermal diffusivity of the soil, equal to λ/ ρ'c_{s}, and ω is the angular frequency of the temperature signal. (See Thermal Diffusion.) The *thermal properties of soil*depend on composition and water content, with values of δ for the daily cycle ranging from about 5 cm for peat soils; 7–14 cm for clay; and 7–18 cm for sandy soils [Van Wijk & De Vries (1964); Monteith & Unsworth (1990)].

The other determinant of ground heat flux is net radiation. Fluxes below dark surfaces, such as a tarmac, are high because they absorb most solar radiation; but under vegetation, g_{G} is moderated by shade and becomes a relatively constant fraction of R_{n} above the canopy. The heat balance equation can then be simplified to:

Sensible and latent heat fluxes are usually considered together because they are both transported by atmospheric Turbulence and because they ‘compete’ to dissipate the radiant heat input. The simplest case is for a dry surface—a tin roof, road surface or desert—where all the radiant heat must be used to heat the air. In this case,

where ρ_{a}c_{p} (J m^{−3} K^{−1}) is the volumetric specific heat of air; T_{a} is the air temperature; and r_{a} (s m^{−1}) is the *aerodynamic resistance* for turbulent transport between the surface and the air, r_{a} depends mainly on wind speed and surface roughness, but it is also influenced by atmospheric stability [Monteith & Unsworth (1990)].

For wet surfaces, the evaporative heat flux g_{E} may be expressed as:

where r_{c} is the surface (canopy) resistance and c_{s}(T_{s}) and c_{a} are the saturated concentration of water vapor at the surface temperature and the atmospheric concentration, respectively. For surfaces with free water, such as lakes, reservoirs and vegetation covered by dew or recent rain, r_{c} = 0; but for vegetated surfaces, r_{c} is usually larger than r_{a} and therefore limits the flux of water vapor and the associated latent heat flux.

For dry surfaces, it is easy to solve the heat balance equation to estimate surface temperature. In contrast, the solution for wet surfaces is complicated by *stomatal control of water loss from plants* and the approximately exponential temperature dependence of the saturation vapor pressure over water.

A number of algorithms have been developed to predict water loss from natural surfaces, but most employ empirical relations which are invalid when used away from their original context. The scientific basis for the prediction of evaporation was first proposed by Penman (1948), who manipulated the heat balance equation to eliminate the unknown term, surface temperature.

The *Penman Equation* estimates *Potential Evaporation* E_{p} from a surface as the combination of two terms, which represent radiative energy supply and the evaporative demand of the atmosphere;

where c_{s}(T_{a}) is the saturation vapor pressure at the air temperature and f(u) is a function of wind speed with appropriate units. The weighting terms Δ/(Δ + γ) and γ/(Δ + γ) depend on the *psychrometer ‘constant’ γ*, equal to 0.066 kPa K^{−1} at sea level, and Δ, the slope of saturation vapor pressure versus temperature relation at air temperature. Paw U. (1992) has pointed out that Penman’s original derivation involved significant errors because of the approximations necessary to permit analytical solution and hand calculation. However, McArthur (1990: 1992) argues that the principles of Penman’s work remain valid, and that approximations may be avoided when computing E_{p}.

The Penman equation remains the standard basis for the estimation of potential evaporation from weather measurements, both for irrigation scheduling and as a measure of climate. Monteith (1965) [Monteith and Unsworth (1990); Jones (1992)] has subsequently adapted the Penman equation to include stomatal resistance, while Jarvis and McNaughton (1986) have shown that the atmosphere interacts with surface extent, roughness and canopy resistance to modify heat balance and evaporation at the landscape scale. Jarvis and McNaughton have identified coupling between the surface and the atmosphere as a key factor in the heat balance and evaporation in natural environments.

At one extreme are well-coupled conditions; when wind speed is high and/or the vegetation tall and rough, when the boundary layer resistance is small and the canopy temperature is close to air temperature. Well-coupled surfaces lose water at the imposed evaporation rate, determined by canopy resistance and atmospheric saturation deficit, with little effect of net radiation. At the other extreme are poorly-coupled conditions, over smooth surfaces and/or at low wind speeds. Then water is lost at the equilibrium evaporation rate, which is driven by net radiation.

There are three basic methods for measurement of surface heat balance: the *profile* and *Bowen Ratio* methods, both of which are valid only over extensive uniform surfaces, and *eddy correlation*.

The *profile method* depends on an understanding of the vertical structure of the turbulent boundary layer. The vertical profiles of wind speed, atmospheric temperature and water vapor concentration are measured in the constant flux layer [e.g., Biscoe et al. (1975)], then the turbulent diffusivity is estimated from the wind profile, with adjustment for atmospheric stability. The sensible and latent heat fluxes from the gradients of temperature and humidity and that of turbulent diffusivity.

The *Bowen Ratio* β is the ratio of sensible to latent heat fluxes, g_{S}/g_{E}. In principle, the Bowen ratio method only requires measurements of net radiation, g_{G} and the temperature and humidity at two heights [Monteith and Unsworth (1990)], but may also be applied to analyze profile measurements. The method depends on rearrangement of the heat balance equation so that there is only one unknown. Expansion of the Bowen Ratio by substitution for g_{S} and g_{E} from Eqs. (7) and (8) gives:

Substitution for g_{S} in the heat balance equation therefore gives:

*Eddy correlation* is simple in principle [e.g., Leuning et al. (1982)], but its routine application to measurements in the constant flux layer has been possible only since the development of portable computers, sonic anemometers [e.g., Kaimal et al. (1974)] and rapid response sensors. The net flux of sensible heat is computed as the time integral of the instantaneous vertical velocity, u_{z}, multiplied by the volumetric specific heat and temperature:

Similarly, the flux of latent heat is given by:

Eddy correlation therefore allows measurement of both sensible and latent heat fluxes, provided the sensors are fast enough to follow the full spectrum of turbulence.

Within the constraints of this article, it is impossible to cover both heat transfer from extensive surfaces and that from individual organisms. Those interested in the heat balances of leaves, fruits and animals can refer to the texts of Monteith and Unsworth (1990) and Jones (1992).

#### REFERENCES

Biscoe, P. V., Clark, J. A., Gregson, K., McGowan, M., Monteith, J. L., and Scott, R. K. (1975) Barley and its environment. I. Theory and practice. *Journal of Applied Ecology*, 12: 227–257.

Hammel, H. T. (1955) Thermal properties of fur. *American Journal of Physiology*, 182: 369–376.

Jarvis, P. G. and McNaughton, Stomatal control of transpiration: scaling up from leaf to region. *Advances in Ecological Research*, 15: 1–49. DOI: 10.1016/S0065-2504(08)60119-1

Jones, H.G. (1992) *Plants and Microclimate*. 2^{nd} edn. Cambridge University Press, Cambridge.

Kaimal, J. C., Newman, J. T., Bisberg, A., and Cole, K. (1974) An improved three component sonic anemometer for investigation of atmospheric turbulence. Flow, its Measurement and Control in Science and Industry. *Instrument Society of America*, 1: 349–345.

Leuning, R., Denmead, O. T., and Lang, A. G. R. (1982) Effects of heat and water vapor transport on eddy covariance measurements of CO2 fluxes. *Boundary-layer Meteorology*, 23: 209–222.

Lewis, M. C. and Callaghan, T. V. (1975) Tundra. Chapter 13, *Vegetation and the Atmosphere* (Ed. J. L. Monteith) Academic Press, London.

McArthur, A. J. (1990) An accurate solution to the Penman equation. *Agricultural and Forest Meteorology*, 51: 87–92. DOI: 10.1016/0168-1923(90)90043-6

McArthur, A. J. (1992) The Penman form equations and the value of Delta: a small difference of opinion or a matter of fact? *Agricultural and Forest Meteorology*, 57: 305–308. DOI: 10.1016/0168-1923(92)90126-O

Monteith, J. L. (1965) Evaporation and environment. *Symposia of the Society for Experimental Biology*, 19: 205–234.

Monteith, J. L. and Unsworth, M. H. (1990) *Principles of Environmental Physics*, 2^{nd} edn. Arnold, London.

Penman, H. L. (1948) Evaporation from open water, bare soil and grass. *Proceedings of the Royal Society of London*, A. 194: 120–145.

Paw, U. K. T. (1992) A discussion of the Penman form equations and comparisons of some equations to estimate latent energy flux density. *Agricultural and Forest Meteorology*, 57: 297–304. DOI: 10.1016/0168-1923(92)90125-N

Van Wijk, W. R. and De Vries, D. A. (1963) Periodic temperature variations. In *Physics of Plant Environment* (Ed. W. R. Van Wijk) North Holland, Amsterdam.

#### References

- Biscoe, P. V., Clark, J. A., Gregson, K., McGowan, M., Monteith, J. L., and Scott, R. K. (1975) Barley and its environment. I. Theory and practice.
*Journal of Applied Ecology*, 12: 227â€“257. DOI: 10.2307/2401731 - Hammel, H. T. (1955) Thermal properties of fur.
*American Journal of Physiology*, 182: 369â€“376. - Jarvis, P. G. and McNaughton, Stomatal control of transpiration: scaling up from leaf to region.
*Advances in Ecological Research*, 15: 1â€“49. DOI: 10.1016/S0065-2504(08)60119-1 - Jones, H.G. (1992)
*Plants and Microclimate*. 2^{nd}edn. Cambridge University Press, Cambridge. - Kaimal, J. C., Newman, J. T., Bisberg, A., and Cole, K. (1974) An improved three component sonic anemometer for investigation of atmospheric turbulence. Flow, its Measurement and Control in Science and Industry.
*Instrument Society of America*, 1: 349â€“345. - Leuning, R., Denmead, O. T., and Lang, A. G. R. (1982) Effects of heat and water vapor transport on eddy covariance measurements of CO2 fluxes.
*Boundary-layer Meteorology*, 23: 209â€“222. DOI: 10.1007/BF00123298 - Lewis, M. C. and Callaghan, T. V. (1975) Tundra. Chapter 13,
*Vegetation and the Atmosphere*(Ed. J. L. Monteith) Academic Press, London. - McArthur, A. J. (1990) An accurate solution to the Penman equation.
*Agricultural and Forest Meteorology*, 51: 87â€“92. DOI: 10.1016/0168-1923(90)90043-6 - McArthur, A. J. (1992) The Penman form equations and the value of Delta: a small difference of opinion or a matter of fact?
*Agricultural and Forest Meteorology*, 57: 305â€“308. DOI: 10.1016/0168-1923(92)90126-O - Monteith, J. L. (1965) Evaporation and environment.
*Symposia of the Society for Experimental Biology*, 19: 205â€“234. - Monteith, J. L. and Unsworth, M. H. (1990)
*Principles of Environmental Physics*, 2^{nd}edn. Arnold, London. - Penman, H. L. (1948) Evaporation from open water, bare soil and grass.
*Proceedings of the Royal Society of London*, A. 194: 120â€“145. DOI: 10.1098/rspa.1948.0037 - Paw, U. K. T. (1992) A discussion of the Penman form equations and comparisons of some equations to estimate latent energy flux density.
*Agricultural and Forest Meteorology*, 57: 297â€“304. DOI: 10.1016/0168-1923(92)90125-N Van Wijk, W. R. and De Vries, D. A. (1963) Periodic temperature variations. In

*Physics of Plant Environment*(Ed. W. R. Van Wijk) North Holland, Amsterdam.