2.3.1 Exploring the drivers of observed Ewc losses using the Penman-Monteith equation
To explore the meteorological conditions required for consistency with observed Ewc losses, a broad range of Ewc estimates were made using the Penman-Monteith equation (Monteith, 1965 ;Eqn. 1); estimates were made using stratified samples from ranges of relative humidity (85% to 100%; expressed via\(\text{\ e}_{z}\) where Tz is assumed to be 10oC) and aerodynamic resistances (0.5 s m-1 to 12 s m-1). This analysis creates a response surface where different combinations of meteorological variables that lead to similar Ewc losses can easily be visualised. The Penman-Monteith equation takes the form:
\(\lambda E_{\text{PM}}=\ \frac{\Delta_{e}H{\ +\ \rho}_{a}c_{p}\ (e_{s}\left(\text{Tz}\right)-e_{z})/r_{a}\_s\ }{\Delta_{e}+\ \Upsilon\left(1+r_{g}/r_{a}\_s\right)}\)(1)
where \(\lambda\) is the latent heat of vaporisation (J kg-1), \(E_{\text{PM}}\ \)is the evapotranspiration rate (kg m-2 s-1), \(\Delta_{e}\)is the slope of the saturation vapour pressure curve versus temperature relationship (Pa K-1) at temperature,\(\left.\ \text{Tz}\right.\ ,\) where z is the observation height in metres, \(H\) is the total energy available for evaporation (J m-2), \(\rho_{a}\) is the density of air at Tz (kg m-3), \(c_{p}\ \)is the specific heat capacity of air (J kg-1 K-1),\(e_{s}\left(\text{Tz}\right)\) is the saturation vapour pressure at Tz, \(e_{z}\ \)is the actual vapour pressure at z ,\(\gamma\) is the psychrometric constant (\(\approx\) 66 x 10-3 Pa K-1) and \(r_{a}\_s\) and\(r_{g}\) are the resistance to the aerodynamic exchange for scalars (sensible heat and vapour) and surface resistances respectively (s m-1). Note that in all calculations made here, the canopy is assumed to be wet and \(r_{g}\) is assumed to be zero such that the term \(\left(1+r_{g}/r_{a}\_s\ \right)\) disappears (Stewart, 1977; VanDijk et al., 2015). The total energy available (\(H\)) was assumed to be the approximate net radiation (\(R_{n}\)) for a cloudy day during winter in Northern England: nominally 2.5 MJ m-2 d-1.
2.3.2 Estimates of aerodynamic exchange and\(\mathbf{E}_{\mathbf{\text{PM}}}\) using meteorological observations
To explore the potential for Ewc across mountainous regions of the UK during large and extreme rainfall events, \(E_{\text{PM}}\) was calculated using meteorological data for 17 sites (Figure 1a and Table Supp. 2) using Eqn. 1. The aerodynamic resistance for momentum (\(r_{a}\_m\)) was estimated using Eqn. 2:
\(r_{a}\_m=\frac{\ln\left(\frac{z-d}{z0\_m}\right)^{2}}{\kappa^{2}U_{\text{z\ }}}\)(2)
where \(z\) is the wind speed observation height, \(d\) is the zero-plane displacement, \(z0\_m\) is the roughness length for momentum (all in metres), \(U_{\text{z\ }}\)is the wind speed (m s-1) at \(z\) and \(\kappa\) is the dimensionless von Karman constant (\(\approx\ \)0.41). The canopy height (Zc) was arbitrarily assumed to be 20 m, \(d\) to be 0.75(Zc) and\(z0\_m\) as 0.1(Zc) in accordance with Szeicz, Endrödi and Tajhman (1969) and Rutter, Robins, Morton and Kershaw (1972). However, studies have shown an enhancement of exchange compared to estimates assuming these approximations for \(z0\_m\) (e.g. Holwerda et al ., 2012). Enhancement of momentum exchange has been observed both for tall canopies and in complex terrain owing to breakdown of theoretical vertical logarithmic wind profiles (Cellier and Brunet, 1992; Raupach, 1979; Simpson, Thurtell, Neumann, Den Hartog, & Edwards, 1998). For the indicative calculations made here, where z < Zc, wind speed was extrapolated to Zc using a logarithmic wind profile relationship. Wind speed observations used here are taken over short grass surfaces and extrapolated to hypothetical canopy height as if the logarithmic profile assumption is valid. It is recognised that this may not be the case in complex terrain but, as the degree of enhancement of momentum exchange is not easily estimated and because the calculations made here are purely indicative no enhancements have been made for \(r_{a}\_m\).
It is often assumed that \(r_{a}\_s\) is equal to \(r_{a}\_m\), but this assumption can to lead to considerable error (Brutstaert, 1982, p62) owing to so-called excess resistance for scalars. Excess resistance occurs because pressure forces associated with form drag increase momentum exchange, but not scalar exchange and because of differences in source and sink distributions for these entities (Brutstaert, 1982; Moors, 2012; Simpson, et al ., 1998; Stewart & Thom, 1973). Although there can be differences between the magnitude of exchange for different scalars, we assume that the exchange of heat and vapour are equal for the purposes of this study and hence only explore differences between the magnitude of scalar exchange compared to the exchange of momentum. Aerodynamic exchange estimated using Eqn. 2 is more sensitive to the value of \(z0\) than it is to the value of \(d\)(Gash, Wright & Lloyd, 1980). The value of \(z0\) has been shown to vary significantly with wind speed for forest canopies, whilst \(d\)tends to remain relatively constant (Bosvelt, 1999; Szeicz, & Endrödi, 1969). Consequently, \(d\) is fixed as specified above for all calculations made here and it is assumed that the primary differences between \(r_{a}\_s\) and \(r_{a}\_m\) are driven by differences in\(z0\_m\) and \(z0\_s\). It is worth noting that \(d\) may vary significantly for very sparse canopies or for deciduous canopies during the leafless period (Brutstaert, 1982, p116; Dolman, 1986).
The ratio of \(z0\_s\)/\(z0\_m\) used in previous studies varies over approximately an order of magnitude as it is influenced by canopy roughness, canopy density, atmospheric stability and wind speed (Bosvelt, 1999; Brutstaert, 1982, p114; Lalic, Mihailovic, Rajkovic, Arsenic, & Radlovic, 2003; Raupach, 1979; Thom, Stewart, Oliver & Gash, 1975). The sensitivity of \(r_{a}\_s\) and \(E_{\text{PM}}\) to the ratio \(z0\_s\)/\(z0\_m\) is explored here using three scenarios:
Scenario 1 - \(z0\_s/z0\_m\text{\ \ }=\ \ 1.0\ \); i.e. \(\ z0\_s=\) \(0.1(\text{Zc})\);
Scenario 2 -\(\text{\ \ }z0\_s/z0\_m\text{\ \ }=\ \ 0.5\); i.e. \(\ z0\_s=\)\(0.05(\text{Zc})\);
Scenario 3 - \(z0\_s/z0\_m\text{\ \ }=\ \ 0.1\); i.e. \(\ z0\_s=\) \(0.01(\text{Zc})\).
It is likely that vapour pressure deficit (also expressed as relative humidity, RH ) observations over grassland meteorological observation sites are likely to be lower than those over an adjacent forested area (e.g. see Pearce, Gash, & Stewart 1980). No attempt has been made to correct RH observations for this study owing to the complexities associated with such a correction and the indicative nature of our calculations; this is also the case for Tz which is likely to be lower above a forest canopy (Rutter, 1967). Additionally, Eqn. 2 is strictly only valid for neutral atmospheric conditions (Szeicz, and Endrödi, 1969) but corrections for non-neutral conditions are often assumed to be insignificant during rainfall (e.g. Morton, 1984; van Dijk et al ., 2015) and we assume they are negligible here.