4.1 Derivation of the Budyko Relationship

The general water balance of a catchment can be written as

\[ Q = P - E + \Delta S \tag{4.1} \]

where $P$ is precipitation in mm, $E$ is evaporation in mm, $\Delta S$ is net storage in mm and $Q$ is specific discharge in mm. Evaporation is the phenomenon by which a substance is converted from its liquid into its vapor phase, independently of where it lies in nature (Miralles et al. 2020). This definition of evaporation encompasses evaporation from inside leaves (transpiration), evaporation from bare soils, evaporation from intercepted precipitation (interception loss), evaporation from open water surfaces, and finally, evaporation over ice- and snow-covered surfaces (often referred to as sublimation).

Over the period of hydrological years and longer time scales, we expect $\Delta S$ to be 0 since neither water storage nor destorage happen over these periods. This would of course not be true for catchments where for example man-made storage infrastructure was built over the period under consideration. If $\Delta S = 0$, the above Equation (4.1) can be rewritten as

\[ Q = P - E \tag{4.2} \]

Dividing by $P$, we get

\[ \frac{Q}{P} = 1 - \frac{E}{P} \tag{4.3} \]

where $Q/P$ can be called the runoff index and $E/P$ is the evaporation index or evaporative fraction.

For a catchment, annual mean $E$ and $Q$ are governed by total water supply $P$ and the total available energy which is normally expressed as potential evaporation $E_{pot}$ and denotes the (atmospheric) water demand. If $E_{pot}$ is small, the discharge $Q$ is normally bigger than evaporation $E$. Similarly, if the available radiative energy is very high, the water demand $E_{pot}$ is very large and $Q<<E$ (Arora 2002). $E_{pot}$ and $P$ are thus the key determinants of annual or longer time-scale runoff and evaporation rates. Michael Budyko has termed the ratio $E_{pot} / P$ aridity index (Budyko 1974).

As explained above, water demand is determined by energy. Solar radiation is the primary energy source for the earth-atmosphere system and the key driver of the hydrological cycle. At the earth’s surface, the net radiative flux $R_N$ is the energy that is available for

heating and cooling of the soil (ground heat flux),
changing the phase of water (latent heat flux), and
heating or cooling air in the boundary layer thus causing atmospheric dynamics (sensible heat flux).

This can be formalized with the following relationship

\[ R_{N} = H_{S} + H_{L} + \Delta H_{G} \tag{4.4} \]

where $R_{N}$ is the net radiation [in W/m² = kg/s³], $H_{S}$ is the upward sensible heat flux, $H_{L}$ is the latent heat flux and $\Delta H_{G}$ the net ground heat flux. The latent heat flux is directly proportional to evaporation $E$. Thus, $H_{L} = L \cdot E$ where $L = 2.5 \cdot 10^{6}$ J/kg [= m²/s²] is the latent heat of vaporization and $E$ is the actual evaporation in [m/s]. As in the case of the water balance, at the annual or longer time scales, we can neglect the heat storage effect in the ground and get

\[ R_{N} = H_{S} + L \cdot E \tag{4.5} \]

With the Bowen ratio defined as the fraction of the sensible heat flux divided by the latent heat flux, i.e.

\[ \gamma = \frac{H_{S}}{H_{L}} = \frac{H_{S}}{L \cdot E } \tag{4.6} \]

and by rearranging the terms, the long-term energy balance in Equation (4.5) can simply be rewritten as

\[ R_{N} = (1 + \gamma)L E \tag{4.7} \]

Using the fact that $R_{N} = L E_{pot}$, where $E_{pot}$ is the potential evaporation, and dividing by precipitation, we can rewrite the above Equation (4.7) as

\[ \frac{E_{pot}}{P} = (1 + \gamma) \frac{E}{P} \tag{4.8} \]

where the left-hand side is called the aridity index, i.e. $\phi = E_{pot}/P$ and $E/P$ is called the evaporative fraction or evaporation index. With this, Equation (4.3) from above can be written as a function of the Bowen ratio and the aridity index, i.e.

\[ \frac{E}{P} = 1 - \frac{Q}{P} = \frac{\phi}{(1 + \gamma)} \tag{4.9} \]

$Q/P$ is again the runoff index. Since the Bowen ratio is also water supply and energy demand limited, it too is a function of the aridity index and we can thus rewrite Equation (4.9) to

\[ \frac{E}{P} = 1 - \frac{Q}{P} = F[\phi] \tag{4.10} \]

The Budyko relationship thus allows for a simple parameterization of how the aridity index $\phi$ controls the long-term mean partitioning of precipitation into streamflow and evapotranspiration and it is capable of capturing the behavior of thousands of catchments around the world. This explains its growing popularity over recent years (Berghuijs, Gnann, and Woods 2020).

$Basics of the Budyko framework. The x-axis contains the aridity index ($E_{pot}/P$) and the y-axis the evaporative fraction ($E/P$), which often is approximated by one minus the runoff ratio ($E/P = 1-Q/P$) because storage changes are assumed to be negligible at multi-year timescales. Together, these two axes form the two-dimensional Budyko space. Many catchments around the world fall around the Budyko curve (black solid line, see Equation below), including 410 US MOPEX catchments which are indicated by blue markers [@berghuijs_2020].$

Figure 4.2: Basics of the Budyko framework. The x-axis contains the aridity index ($E_{pot}/P$) and the y-axis the evaporative fraction ($E/P$), which often is approximated by one minus the runoff ratio ($E/P = 1-Q/P$) because storage changes are assumed to be negligible at multi-year timescales. Together, these two axes form the two-dimensional Budyko space. Many catchments around the world fall around the Budyko curve (black solid line, see Equation below), including 410 US MOPEX catchments which are indicated by blue markers (Berghuijs, Gnann, and Woods 2020).

Figure 4.2 shows a plot of data from catchments in the US for which consistent long-term hydro-climatological data records are available. Individual catchments’ aridity indices are plotted against evaporative fractions, averaged over many years. The catchment data plots along the Budyko curve in the two-dimensional Budyko space as indicated in the Figure where the Budyko curve is defined as

\[\begin{equation} \frac{E}{P} = \left[ \frac{E_{pot}}{P} \text{tanh} \left( \frac{P}{E_{pot}} \right) \left( 1 - \text{exp} \left( - \frac{E_{pot}}{P} \right) \right) \right]^{1/2} \tag{4.11} \end{equation}\]

This non-parametric relationship between the aridity index and the evaporative fraction was developed by M. Budyko (Budyko 1951).

The Budyko space is delineated by the demand and supply limits. Catchments within the space should theoretically fall below the supply limit ($E/P = 1$) and the demand limit ($E/E_{pot} = 1$), but tend to approach these limits under very arid or very wet conditions (Berghuijs, Gnann, and Woods 2020). The data from the US shows that a large percentage of in-between catchment variability can be explained by the Budyko curve. After the seminal work Budyko in the last century, the evidence for a strong universal relationship between aridity and evaporative fraction via the Budyko curve has since grown. As catchment hydrology still lacks a comprehensive theory that could explain this simple behavior across diverse catchments Gentine et al. (2012), the ongoing debate about the the underlying reasons for this relationship continues (see e.g. (Padron et al. 2017; Berghuijs, Gnann, and Woods 2020)).

While almost all catchments plot within a small envelope of the original Budyko curve, systematic deviations are nevertheless observed from the original Budyko curve. Several new expressions for $F[\phi]$ were therefore developed to describe the long-term catchment water balance with one parameter (see e.g. (Budyko 1974; Sposito 2017; Choudhury 1999). One popular equation using only 1 parameter is the Choudhury equation which relates the aridity index $\phi$ to the evaporative fraction $E/P$ in the following way

\[\begin{equation} \frac{E}{P} = \left[ 1 + \left( \frac{E_{pot}}{P} \right) ^{-n} \right]^{1/n} \tag{4.12} \end{equation}\]

where $n$ is a catchment-specific parameter which accounts for factors such as vegetation type and coverage, soil type and topography, etc. (see e.g. (Zhang et al. 2015) for more information). In other words, $n$ integrates the net effects of all controls of of the evaporative fraction other than aridity. The Figure 4.3 shows the control of $n$ over the shape of the Budyko Curve.

$$n$ is determining the shape of the Budyko Curve. Note how large values of $n$ favour evaporation over discharge. Note that $E_{pot}$ is denoted with $E_{0}$. Figure taken from [@yang_2008].$

Figure 4.3: $n$ is determining the shape of the Budyko Curve. Note how large values of $n$ favour evaporation over discharge. Note that $E_{pot}$ is denoted with $E_{0}$. Figure taken from (Yang et al. 2008).