Advection Effect and Boundary Condition in the Formulation of Generalized Complementary Relationship for Evapotranspiration

Land-surface evapotranspiration (ET) is a major component of the hydrologic cycle. It is a very attractive approach to estimate land surface ET by means of complementary relationship (CR). After 60 years of continuous exploration, the CR has developed from linear relationships to the present nonlinear ones. There are usually four boundary conditions (BCs) for the nonlinear CR, among which the first-order one in completely wet environments (dy/dx_x=1) has been a debatable issue, including both the difference in values of dy/dx_x=1, and the divergence in definitions of the independent variable x. It has always been a problem how to consider the advection effect in CR. The effect degree of advection from outside the region varies in the ET process at different spatial scales. In this paper, x denotes the ratio of equilibrium ET (ET_e) to apparent potential ET (ET_pa), y denotes the ratio of ET to ET_pa, and x=1 is set as the benchmark with ET_e as the lower limit of ET_pa. According to the characteristics of ET processes at different spatial scales, we extend the value range of dy/dx_x=1, and take dy/dx_x=1=k (k≥0) to establish the generalized BC. The generalized CR model for ET is then proposed by using an exponential function, expressed as y=EXP[k/d(1-1/x^d)] (denoted by GCR-EXP; d>0), where k and d are model parameters. k is equal to 2 in the absence of advection, which is the most complementary case. When k < 2, warm advection plays a role, and the value of k gradually decreases as the advection influence increases. Brutsaert (2015) considered the effect of minimal advection, and used the potential ET (Priestley and Taylor,1972) as ET’s constraint to determine the first-order BC in completely wet environments for the polynomial model of CR, which is a case that fits quite well with a large number of observed data. When k = 0, the CR is no longer valid, and the ET is always equal to ET_pa, which reflects the ET of a small wet surface. When 0≤x≤x_min, y is equal or approximately equal to 0. x_min and Priestley-Taylor coefficient α can be determined by the values of x at y close to 0 and to 1 in GCR-EXP model, respectively. For instance, the value of x at y=0.001 can be taken as the value of x_min. k reflects advection effects and the corresponding degrees of CR. Moreover, the GCR models, which satisfy the four BCs including dy/dx_x=1=k, can be also expressed as a power-exponential function form and other ones besides the proposed exponential one (Supported by Project 41971049 of NSFC).