An important scaling consideration is introduced into the formulation of the complementary relationship (CR) of land surface evapotranspiration (ET) by specifying the maximum possible evaporation rate (Epmax) of a small water body (or wet patch) as a result of adiabatic drying from the prevailing near‐neutral atmospheric conditions. In dimensionless form the CR therefore becomes yB = f( Epmax−EpEpmax−EwxB) = f(X) = 2X2 − X3, where yB = ET/Ep, xB = Ew/Ep. Ew is the wet‐environment evaporation rate as given by the Priestley‐Taylor equation, Ep is the evaporation rate of the same small wet surface for which Epmax is specified and estimated by the Penman equation. With the help of North American Regional Reanalysis data, the CR this way yields better continental‐scale performance than earlier, calibrated versions of it and is on par with current land surface model results, the latter requiring vegetation, soil information and soil moisture bookkeeping. Validation has been performed by Parameter‐Elevation Regressions on Independent Slopes Model precipitation and United States Geological Survey runoff data. A novel approach is also introduced to calculate the value of the Priestley‐Taylor parameter to be used with continental‐scale data, making the new formulation of the CR completely calibration free.

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Recent research into the complementary relationship (CR) between actual and apparent potential evaporation has resulted in numerous alternative forms for the CR. Inspired by Brutsaert (2015), who derived a general CR in the form y = function (x), where x is the ratio of potential evaporation to apparent potential evaporation and y is the ratio of actual to apparent potential evaporation, an equation is proposed to calculate the value of x at which y goes to zero, denoted xmin. The value of xmin varies even at an individual observation site, but can be calculated using only the data required for the Penman (1948) equation as expressed here, so no calibration of xmin is required. It is shown that the scatter in x‐y plots using experimental data is reduced when x is replaced by X = (x − xmin)/(1 − xmin). This rescaling results in data falling along the line y = X, which is proposed as a new version of the CR. While a reinterpretation of the fundamental boundary conditions proposed by Brutsaert (2015) is required, the physical constraints behind them are still met. An alternative formulation relating y to X is also discussed.

A rigorous approach to the complementary relationship (CR) in land surface evaporation was introduced by Brutsaert (2015, https://doi.org/10.1002/2015WR017720) in which the problem was cast in nondimensional form for generality, boundary conditions (BCs) were established from physical constraints, and a suitable mathematical solution was formulated. Building on Brutsaert's insightful foundation, Crago et al. (2016, https://doi.org/10.1002/2016WR019753) showed the need, for rational reasons, to modify Brutsaert's BCs by introducing Emax, an upper limit to the apparent potential evaporation corresponding to an actual evaporation of 0. Following the rigorous approach of Brutsaert, this paper presents the derivation and resulting CR. The BC associated with Emax requires a solution that is rescaled with regard to Brutsaert's dimensionless framework and represents a fundamentally different solution from that of Brutsaert. In essence, this formulation acknowledges a pattern of organized variability, which exists within the data when scaled in Brutsaert's dimensionless framework, and our rescaled CR reorganizes the data, collapsing it toward a more universal representation. Our rescaled CR, implemented with two versions of Emax, one based on mass transfer and another on the Penman equation, is evaluated alongside Brutsaert's original formulation. Multiyear data sets from seven Fluxnet sites in Australia, ranging from a sparsely vegetated ephemeral tropical wetland to a temperate forest with a 75‐m‐tall canopy were used to test the formulations on a weekly basis. All three formulations performed adequately. Overall, the rescaled model with Emax based on the Penman equation performed best; it extracts more information with no additional observational data requirements.

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