T 1

Log height, z

Log height, z

Fig. 10.5 Resistance analogy and logarithmic wind profile in a tall vegetation canopy for the big leaf model. Canopy resistance, rc, is intended to represent the bulk effect of the soil, xylem and stomatal resistances at canopy scale.

where ¡3 = C/1Et is known as the Bowen ratio. The value of the Bowen ratio indicates how the available energy is partitioned into that used to heat the atmosphere and that used in evapotranspiration. Experimental studies have shown how, for a vegetated surface, the Bowen ratio can stay fairly constant (at least for clear-sky conditions with no water limitations) (e.g. Crago and Brutsaert, 1996, see also Fig. 10.6).

As for open water, the fluxes C and 1Et are represented by gradient equations, here in the form:


where To and eo are temperature and vapour pressure at the canopy surface, Ta and ea are air temperature and vapour pressure at the reference height, Cp is the specific heat of air («1.005kJkg-1°C-1), and raH and ra v are the aerodynamic resistances to the transport of heat and vapour from the surface to the atmosphere. The problem in applying these equations is that the surface temperature and vapour pressure of the 'big leaf' surface approximation cannot be easily measured. Penman and Schofield (1951) and later Monteith (1965, 1973) came up with the idea of using an additional

Fig. 10.6 Hourly net radiation (diamonds) and Penman-Monteith évapotranspiration estimate (squares) in Wm-2 for dry, short, grass canopy on a clear day in May. Values of the Bowen ratio (fi = C/XEt) are shown as open circles.

Fig. 10.6 Hourly net radiation (diamonds) and Penman-Monteith évapotranspiration estimate (squares) in Wm-2 for dry, short, grass canopy on a clear day in May. Values of the Bowen ratio (fi = C/XEt) are shown as open circles.

resistance for the transfer of vapour from within the leaf stomata (where the air can be assumed saturated) to the surface to eliminate these variables from the equations. Thus the vapour flux may then also be written:

where rc is called the canopy resistance, and es(To) is the saturated vapour pressure at the canopy temperature. An important feature of the canopy resistance is that it should be zero when the canopy is wet, i.e. when there is intercepted water on the canopy during and after rainfall. It will be shown below how this can make a significant difference to rates of water loss from the canopy, particularly for rough canopies with low aerodynamic resistance. Under dry conditions, we expect the canopy resistance to be related to the stomatal resistance within individual leaves, but under the big leaf assumption it is used here as an effective parameter for the canopy as a whole. It will increase as the plant starts to restrict the loss of water by closing the stomatal openings under dry conditions.

The aerodynamic resistance is often approximated by assuming that both ra H and ra are equal to the equivalent resistance for momentum flux in a neutrally stable atmospheric boundary layer above the canopy. For the resulting logarithmic windvelocity profile, ra is given by:

a K2uz where uz is the measured wind speed at the reference heightz, d is called the zero plane displacement, zo is called the roughness height and k is the von Karman constant (= 0.41). The parameters d and zo are introduced to allow for the fact that over a tall, rough, vegetated surface the effective zero wind velocity will be displaced above the ground. We should expect that the resistance for a rough tree canopy will be much lower than that of a smooth grass sward. For a range of vegetation canopies, d and zo can be very approximately estimated from the canopy height h as d = 0.67 h and zo = 0.1 h.

Then under the assumption that ra = ra,V = ra,H, we can use the same approximation as in (10.12) to (10.14) above to eliminate To from the equations by using the gradient of the saturation vapour pressure curve at the measured air temperature, A. The result, after some rearranging, is the Penman-Monteith equation in the form:

Thus, in applying the Penman-Monteith equation, we can proceed in a similar way to the calculation of the Penman Eo, outlined above. To do so, the user must specify:

the energy available for evapotranspiration, H; wind speed at the reference height, uz; zero plane displacement, d; roughness height, zo;

dry-bulb air temperature at the reference height, Ta;

wet-bulb air temperature at the reference height (to give vapour pressure, ea); the canopy resistance, rc.

Fig. 10.6 shows these calculations applied on an hourly basis for a clear day in May for a dry grass canopy (with ra = 50 s m-1; Table 10.1), together with the Bowen ratio (P = C/1Et). The estimates in energy units (Wm-2) can be converted to depth of water (ms-1) by dividing by the latent heat of vaporisation, 1 (2.470 x 10-6 Jkg-1). With such short time steps, storage of energy in the ground and canopy starts to become more important. In this case, the data indicate that there is a source of sensible heat from the atmosphere and ground at night, leading to evapotranspiration rates that are higher than the net radiation, although over the day the energy balance is driven by the net radiation and dominated by the split between sensible and latent heat transfers to the atmosphere. Fig. 10.7 shows the application of the Penman-Monteith equation at a daily time step using the same August 2003 Hazelrigg data as above to estimate daily potential evapotranspiration rates, assuming a dry grass canopy without water limitation. In this case, the actual evapotranspiration total will be much less because

Table 10.1 Typical values of resistances in the Penman-Monteith equation for different surface and wetness conditions

Aerodynamic resistance (sm ')

Canopy resistance (sm 1 )

ra ,

ra ,

rc (dry rc

rc (soil water

(u = 2 ms 1 )

(u = 5ms ')

canopy surface) (wet surface)


Open water








50 0


Mature wheat



70 0


Pine trees



100 0


August 2003

August 2003

Fig. 10.7 Daily estimates of Penman-Monteith evapotranspiration (in mm day-1) for August 2003 at the Hazelrigg climate station at Lancaster University assuming a short, dry, grass canopy without water limitation. Dotted line is mean daily potential evapotranspiration for the month.

the hot dry conditions will mean that the grass evapotranspiration will be water limited and canopy resistance will increase.

Beven (1979) looked at the sensitivity of evapotranspiration estimates to changes in the different parameters for typical UK summer conditions. His analysis showed quite clearly how the predicted rates increase with low canopy resistance (wet canopies) and low aerodynamic resistance (rough canopies) and that rates can be very high for a rough wet canopy (Fig. 10.8). This is one reason why frequently wetted forest canopies in the UK uplands tend to show greater cumulative volumes of evapotranspiration than grassland (e.g. Fig. 10.1). Typical values of these resistances for different canopies are shown in Table 10.1. Values of canopy resistance are known to exhibit a diurnal variation (e.g. Szeicz and Long, 1969; Stewart and Thom, 1973) and many canopies will also show a seasonal variation (Calder, 1977). We can examine the sensitivities rc(sm-1)

Fig. 10.8 Sensitivity of Penman-Monteith actual évapotranspiration estimates to changes in aerodynamic (ra) and canopy (rc) resistance parameters for mean August midday conditions in central England (from Beven, 1979, with kind permission of Elsevier).

Fig. 10.8 Sensitivity of Penman-Monteith actual évapotranspiration estimates to changes in aerodynamic (ra) and canopy (rc) resistance parameters for mean August midday conditions in central England (from Beven, 1979, with kind permission of Elsevier).

directly by calculating the estimates of evapotranspiration for different values of the canopy and aerodynamic characteristics (see example).

Example: Comparison of estimated evapotranspiration for different canopies For the midday conditions for the plot in Fig. 10.6, the net radiation value is 227Wm-2, the air temperature is 16° C (289.2 K), the wet-bulb temperature is 12.1° C, the wind speed is 2.5 m s-1. At this air temperature, the density of air, p , is 1.22 kgm-3 (Appendix Table A4), the specific heat of air, cp, is 1.005 kJ kg-1 K-1 (Appendix Table A4), the saturation vapour pressure is 18.17 hPa (Appendix Table A4), and from the wet-bulb temperature the vapour pressure deficit is 4.16 hPa. The slope of the saturated vapour pressure v. temperature curve at this air temperature (A) is 1.61 hPa K-1 and the psychometric constant, y, 0.67 hPa K-1.

For a dry, short, grass canopy, assume an aerodynamic resistance at this wind speed of 57 s m-1 and a canopy resistance of 50 s m-1.

Substituting into the Penman-Monteith equation (noting that 1 W = 1J s-1 )

_ 1.61 x 227 + 1.22 x 1005 x (4.16/57) = 1.61 + 0.67 x (1 + 50/57)

To convert from energy to an equivalent depth of water, we need to divide by the latent heat of vaporisation l (2.470 x 10-6 J kg-1 ) and convert to the required depth and time units, so assuming a density of water of 1000kgm-3 (so that 1 kgm-2 = 1 mmm-2 water)

Repeating the calculations for a tree canopy with aerodynamic resistance under the same wind conditions of 5.7 s m-1 and canopy resistance of 50 s m-1 gives

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