6. Penman¶

6.1. The combination equation (Penman, 1948)¶

Given the difficulty in directly observing evaporative fluxes – and the need to estimate evaporation to determine irrigation of crops Penman (1948) developed the combination equation. Only standard meteorological observations at some height above either water or a surface covered with well-watered short grass, were required.

This equation “combines” two physical controls on evaporation: the supply of available energy \(Q^*-Q_G\) and the turbulent diffusion of water vapour from the surface. Firstly, the sensible and latent heat fluxes should be written in resistance notation

(6.1)¶\[ Q_{H} = - \rho c_{p}\frac{T - T_{0}}{r_{h}}\]

(6.2)¶\[ L_{v}E = - \frac{L_{v}}{R_{v}T}\frac{e - e_{0}}{r_{v}}\]

We can rewrite Eq.6.2 in terms of \(r_h\) and the effective psychrometer constant \(\gamma^*\):

(6.3)¶\[L_{v}E = - \rho c_{p}\frac{e - e_{0}}{{\gamma^*}r_{h}},\]

where the “thermodynamic” psychrometer constant is given by \(\gamma = \rho c_{p}/L_{v}\varepsilon\) and has a value of 0.66 hPa K^-1 near sea level, the ratio of gas constants is \(\varepsilon=R_a/R_v\), and the effective psychrometric constant is \(\gamma^*=\gamma r_v/r_h\).

The key to estimating the exchange of sensible and latent heat with the air above a surface is to establish the surface temperature \(T_0\). If this is not known, then it can be eliminated from the equations by assuming that the saturation vapour pressure is a linear function of temperature for small temperature differences. Additionally, it is assumed that the surface vapour pressure is at saturation, i.e.

(6.4)¶\[e_{0}=e_{s}\left(T_{0}\right)=e_{s}(T)-s\left(T-T_{0}\right)\]

By substitution of Eq.6.4 into Eq.6.2 we get \(Q_E\) in terms of \(T-T_0\) which is not known. But by combining the new equation for \(Q_E\), \(T-T_0\) may be eliminated from Eq.6.1 for \(Q_H\).

The final assumption is that of energy balance closure (EBC), i.e. that \(Q^*-Q_G=Q_H+Q_E\). By substitution of the new equation for \(Q_H\), we obtain the Penman equation

(6.5)¶\[L_{v}E = \frac {s\left( Q^* - Q_{G} \right) + \rho c_{p}\left( e_{s}(T) - e \right)/r_{H}} {s + \gamma_{*}}\]

giving latent heat flux as a function of variables measured at one height only.

6.2. Resistance notation¶

In the combination equation, fluxes are recast in terms of aerodynamic resistances. Thus, the momentum flux across a finite layer \(\Delta z = \left( z_{2} - z_{1} \right)\) can be expressed as

(6.6)¶\[\tau = \frac{\rho\left( u_{2} - u_{1} \right)}{r_{a}}\]

where \(r_a\) is the aerodynamic resistance. If \(u_1\) is taken to be the \(u(z_0)=0\), then the resistance can be related to the log law

(6.7)¶\[r_{a} = \frac{u(z)}{u_{*}^{2}} = \frac{ln\lbrack\frac{z - d}{z_{0}}\rbrack}{ku_{*}} = \frac{\left\lbrack ln\lbrack\frac{z - d}{z_{0}}\rbrack \right\rbrack^{2}}{k^{2}u(z)}\]

Notice that the quantity in square brackets depends only on the nature of the surface: if that is known, then \(r_{a}\) is determined only by the single measurement of mean wind speed. For fluxes this leads to an expression predicting a flux of some variable \(\alpha\), \(F_{\alpha}\) in the form

(6.8)¶\[F_{\alpha} = u_{*}^{2} \left( \frac{\phi_{m}}{\phi_{\alpha}} \right) \frac{\Delta\alpha}{\Delta u} \equiv \frac{\Delta\alpha}{r_{\alpha}},\]

in which \(r_{\alpha} = \left( \frac{\Delta u}{u_{*}^{2}} \right) \times \left( \frac{\phi_{\alpha}}{\phi_{m}} \right)\) is the aerodynamic resistance of the layer (some authors use conductance, which is the reciprocal of \(r_{\alpha}\)). In the case of momentum, \(\phi_{\alpha}=\phi_m\). For non-neutral conditions:

(6.9)¶\[r_{a} = \frac{\left\lbrack \ln\left( \frac{z - d}{z_{0}} \right) - \ \psi_{m}\ \left( z^{'}/L \right) \right\rbrack^{2}} {k^{2}u}.\]

Under stable conditions, the aerodynamic resistance for sensible heat transfer (\(r_h\)) is usually taken as equal to \(r_a\). Under unstable conditions, the assumption \(\phi_{h}=\phi_{m}^{2}\) means that we need to multiply \(r_a\) by \(\phi_{m}\) to get \(r_h\). In both cases, the aerodynamic resistance for water vapour and other scalar fluxes is generally assumed to be the same as that for sensible heat. Note that, in general, \(z\) should be replaced by \(z-d\) if the sampling height is not much greater than the zero-plane displacement height \(d\).

6.3. The Penman-Monteith equation¶

Evaporation from a vegetated surface is also impacted by plant physiology. Stomata in the leaves open to allow transfer of CO₂ during photosynthesis, but can close when the plant is water stressed to avoid undue loss of water vapour. Monteith (1965) adapted Penman’s equation to allow for this effect, giving what we now know as the Penman-Monteith equation where the symbols have the same meaning, except that \(\gamma^*\) is an apparent psychrometer ‘constant’. Over crops the resistance to evaporation is larger than the resistance to heat transfer, due to canopy resistance. To allow for this, the effective psychrometer constant is usually assumed to be of the form

(6.10)¶\[\gamma_{*} = \gamma\left( \frac{r_{h} + r_{s}}{r_{h}} \right) = \gamma\left( 1 + \frac{r_{s}}{r_{h}} \right),\]

where \(r_s\) is an effective surface resistance. The latter depends in a complicated way on soil moisture, type of vegetation, fractional cover, and time of year. It is usual to consider it as the result of a canopy (or crop) resistance (\(r_{sc}\)) and a bare soil resistance (\(r_{ss}\)) acting in parallel, so that

(6.11)¶\[\frac{1}{r_{s}} = \frac{\left( 1 - A \right)}{r_{\text{sc}}} + \frac{A}{r_{\text{ss}}}\]

where \(A\) is an effective fraction of bare soil area. Appropriate values of crop resistance are known for various types of vegetation. For ‘moist’ surface conditions during the day, \(r_{ss}\) is usually taken to be 100 s m^-1.

Finally, \(r_h\) can be approximated by the aerodynamic resistance \(r_a\) as it is more readily measurable. For observations close to the ground (e.g. below 3 m) the stability correction can be neglected, and \(r_a\) therefore estimated using (4). The Penman approach does not require ‘special’ equipment but assumptions about the transfer processes at the Earth’s surface as well as the nature of the surface itself are needed.

The P-M equation is used practically by the FAO. Evapotranspiration is computed for a grass reference crop (\(ET_0\)), then multiplied by a crop coefficient (\(K_c\)) to give an estimate of actual evapotranspiration. For instance, \(K_c \sim 1 \textrm{ to } 1.2\) for cabbage, but 1.1 to 1.5 for sugar cane. To aid calculation of irrigation requirements, \(ET_0\) is expressed in \(\textrm{mm day}^{-1}\) and can range from \(1–3\textrm{ mm day}^{-1}\) in cool, arid regions to \(5–6\textrm{ mm day}^{-1}\) in warm tropical regions.

6.4. Penman Monteith Method Measurements¶

Meteorological variables measured at one height can be used to estimate the evaporative flux from a surface using the Monteith (1965) adaptation for vegetated surface of Penman (1948):

(6.12)¶\[Q_{E} = \frac{s\left( Q^{*} - Q_{G} \right) + \rho c_{p}(e_{s} - e)/r_{H}} {s + \gamma\ (1 + r_{s}/r_{a})}\]

For the practical, some assumptions are made in addition to those made in deriving the equation. The aerodynamic resistance for heat transfer is assumed to be equal to that for momentum transfer, i.e. \(r_h \sim r_a\).

6.5. Aerodynamic Resistance¶

To calculate \(r_a\), one could assume neutral conditions are applicable or apply stability correction. Surface characteristics influence the surface (\(r_s\)) or canopy resistance \(r_c\). Around the site the surface characteristics vary with grass in the near fetch but arrange of other land covers further away from the sensors.

6.6. Canopy or surface resistance¶

By re-arranging the PM equation, with EC and other observations you can determine the surface resistance (\(r_s\)) or its inverse surface conductance (\(g_s\)) (Ward et al. 2016):

(6.13)¶\[\frac{1}{g_{s}} = r_{s} = \left( \frac{s}{\gamma}\frac{Q_{H}}{Q_{E}} - 1\ \right) r_{\text{av}} + \frac{\rho c_{p}\text{VPD}}{\gamma Q_{E}}\]

where \(\text{VPD}\) (\(=e_s-e\)) is the vapour pressure deficit. For our purposes we will assume \(r_h \sim r_a \sim r_{av}\).