The Budyko functions under non-steady-state conditions

Abstract

Abstract. The Budyko functions relate the evaporation ratio E / P (E is evaporation and P precipitation) to the aridity index = E p / P (E p is potential evaporation) and are valid on long timescales under steady-state conditions. A new physically based formulation (noted as Moussa-Lhomme, ML) is proposed to extend the Budyko framework under non-steadystate conditions taking into account the change in terrestrial water storage S. The variation in storage amount S is taken as negative when withdrawn from the area at stake and used for evaporation and positive otherwise, when removed from the precipitation and stored in the area. The ML formulation introduces a dimensionless parameter H E = − S / E p and can be applied with any Budyko function. It represents a generic framework, easy to use at various time steps (year, season or month), with the only data required being E p , P and S. For the particular case where the Fu-Zhang equation is used, the ML formulation with S ≤ 0 is similar to the analytical solution of Greve et al. (2016) in the standard Budyko space (E p / P , E / P ), a simple relationship existing between their respective parameters. The ML formulation is extended to the space [E p / (P − S), E / (P − S)] and compared to the formulations of Chen et al. (2013) and Du et al. (2016). The ML (or Greve et al., 2016) feasible domain has a similar upper limit to that of Chen et al. (2013) and Du et al. (2016), but its lower boundary is different. Moreover, the domain of variation of E p / (P − S) differs: for S ≤ 0, it is bounded by an upper limit 1 / H E in the ML formulation, while it is only bounded by a lower limit in Chen et al.'s (2013) and Du et al.'s (2016) formulations. The ML formulation can also be conducted using the dimensionless parameter H P = − S / P instead of H E , which yields another form of the equations.

Dominant effect of increasing forest biomass on evapotranspiration: interpretations of movement in Budyko space

“…dS/dt = 0) in these basins, affecting the calculations of E (Eq. 1) (Bring et al, 2015;Budyko, 1974;Moussa and Lhomme, 2016). We therefore applied the same bootstrapping and regression analysis to the 17 values and 10 mean values resulting from averaging the annual data of the period 1961-2012 into 3-and 5-year means, respectively.…”

Incorporating the Vadose Zone into the Budyko Framework

“…Equation (6) constitutes the "Budyko picture" of ET in the "Budyko space", i.e., ET/P as a function of ET 0 /P and ∆S/P, while Equation (7) constitutes the "Turc picture" of ET in the "Turc space", i.e., ET/ET 0 as a function of P/ET 0 and ∆S/ET 0 [1,8]. In the present paper, only the Budyko picture will be utilized.…”

“…The present paper is a response to this challenge. It will be shown that, by making a single physically-relevant hypothesis about the functional relationship in Equation (4), the Budkyo framework can be extended to include changes in vadose-zone water storage in a manner broad enough to include all extant models [1,3,6] as special cases, while offering a rigorous approach to the specific modeling of Equation (4) that does not require adjustable parameters.…”

“…For example, the boundary condition in Equation (3) can be violated by annual ET from an arid sub-catchment if ET exceeds P because of contributions from water stored in the vadose zone [5,6]. Therefore, in order to describe ET under non-steady-state conditions, the assumption of negligible changes in subsurface water storage must be relaxed and Equation (1) must be accordingly generalized [1,3,5]: ET = f(P,ET 0 ,∆S)…”

“…The Budyko framework [1] is the name given to an implicit mathematical relationship between catchment evapotranspiration (ET) and two macro-climatic variables, precipitation (P) and potential evapotranspiration (ET 0 ), originally put forth by Budyko [2] (pp. 322-323).…”