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into account the change in terrestrial water storage S. The variation in storage amount S is taken as negative when withdrawn from the area at stake and used for evaporation and positive otherwise, when removed from the precipitation and stored in the area.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.

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