Derive PET expression by Penman's method

Penman's method combines the energy-balance method and the aerodynamic (mass-transfer) method to compute evaporation (or PET) from standard weather data.

Basic relations

Let $e_w$ = saturation vapour pressure at the surface temperature $T_s$, $e_s$ = saturation vapour pressure at the air temperature $T_a$, and $e_a$ = actual vapour pressure of the air.

1. Energy balance (heat stored in the ground is ignored): net radiation $H_n$ is used for evaporation $E$ and for heating the air $H_a$ (all in mm of water/day):

$$H_n = E + H_a$$

2. Bowen's ratio for sensible heat:

$$\frac{H_a}{E} = \gamma\,\frac{T_s - T_a}{e_w - e_a}$$

3. Aerodynamic (Dalton type) equations:

$$E = f(u)(e_w - e_a), \qquad E_a = f(u)(e_s - e_a)$$

$E_a$ is the evaporation if the surface were at air temperature. 4. Slope of the saturation vapour pressure curve:

$$A = \frac{e_w - e_s}{T_s - T_a}$$

Derivation

From (3): $e_w - e_a = E/f(u)$ and $e_w - e_s = (E - E_a)/f(u)$.

From (4): $T_s - T_a = \dfrac{e_w - e_s}{A} = \dfrac{E-E_a}{A f(u)}$.

Put these in the Bowen ratio:

$$H_a = \gamma E\,\frac{(E-E_a)/(A f(u))}{E/f(u)} = \frac{\gamma\,(E - E_a)}{A}$$

Substitute in $H_n = E + H_a$:

$$H_n = E + \frac{\gamma(E - E_a)}{A} \;\Rightarrow\; A H_n = (A+\gamma)E - \gamma E_a$$ $$E = PET = \frac{A H_n + E_a\gamma}{A+\gamma}$$

This is Penman's equation. The unknown surface temperature has been eliminated, so only standard weather data are needed.

Terms

• $H_n = H_o(1-r)(0.18+0.55n/N) - \sigma T_a^4(0.56-0.092\sqrt{e_a})(0.1+0.9n/N)$ mm/day, where $H_o$ is the solar radiation reaching the top of the atmosphere (not the same as the heating of air $H_a$ above), $r$ = reflection coefficient, $n/N$ = ratio of actual to possible sunshine hours and $\sigma$ = Stefan-Boltzmann constant.
• $E_a = 0.35\left(1+\dfrac{u_2}{160}\right)(e_s - e_a)$ mm/day, with $u_2$ = wind speed at 2 m height (km/day) and vapour pressures in mm Hg.
• $A$ = slope of the saturation vapour pressure curve at $T_a$ (mm Hg/°C); $\gamma = 0.49$ mm Hg/°C.

When $A$ is large (hot weather) the radiation term dominates; when $A$ is small (cold weather) the aerodynamic term dominates.