The Budyko functions relate the evaporation ratio

The Budyko framework has become a simple tool that is widely used within the
hydrological community to estimate the evaporation ratio

The Budyko framework was initially limited to steady-state conditions on long
timescales, under the assumption of negligible change in soil water storage
and groundwater. Hydrological processes leading to changes in water storage
are not represented and the catchment is considered closed without any
anthropogenic intervention: precipitation is the only input and evaporation
and runoff

Different expressions for the Budyko curves under steady-state conditions.

Different expressions for the Budyko curves under non-steady-state conditions.

Representation of the change in soil water storage

Upper and lower limits of the feasible domain (in grey) of
evaporation in the Turc space (

With the extension of the Budyko framework to non-steady-state conditions being a
real challenge, this paper aims to propose a new formulation inferred from a
clear physical rationale and compared to other non-steady formulations
previously derived. The paper is organized as follows. First, we present the
new formulation under non-steady-state conditions: its upper and lower
limits, its generic equations under restricted evaporation in the Budyko
space (

In the Budyko framework, each catchment is characterized by the three
hydrologic variables (

First, we present the upper and lower limits in the Turc space under steady-state
conditions, when all the water consumed by evaporation comes from the
precipitation, the change in water storage

Under non-steady-state conditions, either a given amount of water

In the Turc space, the case where evaporation is at its maximum value is
visualized as the upper limit in Fig. 2c and e (all the available water is
used for evaporation). For both cases,

Translating the above equations into the Budyko space (Fig. 2d, f) yields the
following for the upper limits:

We now examine the case where the evaporation rate is lower than its maximum
possible rate. In the Turc space, under non-steady-state conditions (

In the Turc space, the lower limit

The ML formulation in the Budyko space with the Fu–Zhang
relationship Eq. (14a, b) for

Following the same reasoning as above, the lower limit, the energy limit and
the water limit of

In the following, these new generic formulae (Eqs. 8 and 9 for

Any Budyko equation

The ML formulation with the Fu–Zhang Eq. (21a, b) in the space
[

As mentioned in the introduction, some authors (Chen et al., 2013; Du et al.,
2016) have dealt with the non-steady conditions by modifying the Budyko
reference space and replacing the precipitation

Any Budyko formulation

A mathematical development, similar to the one of Sect. 2.1, 2.2 and 2.3, is
conducted in Appendix A using the dimensionless parameter

When evapotranspiration exceeds precipitation (corresponding herein to the
case

Relationship (Eq. 23) between the parameter

Example showing the similarity of the ML formulation Eq. (14a) and
the equation of Greve et al. (2016) Eq. (22) (with

The derivative of Eq. (22) gives

Figure 6 compares the ML formulation Eq. (14a) with Greve et
al.'s (2016) analytical solution Eq. (22) for

The formulations proposed by Chen et al. (2013) and Du et al. (2016) in the
space

For

In the space [

All four formulations (ML; Greve et al., 2016; Chen et al., 2013; and Du et
al., 2016) have two parameters each: one for the shape of the curve and
another for its shift due to non-steady conditions:

It is worth noting that for

The ML formulations constitute a general mathematical framework which allows
any standard Budyko function developed at catchment scale under steady-state
conditions (Table 1) to be extended to non-steady conditions (Table S1 in the
Supplement). They take into account the change in catchment water storage

The new formulations are inferred from an evaluation of the feasible domain
of evaporation in the Turc space, adjusted for the case where additional
(

Appendix A presents the set of equations when scaling the change in soil
water storage

In the Turc space, the upper limits of evapotranspiration

The translation in the Budyko space yields the following for the upper limits:

In the Supplement, Fig. S1 shows the upper and lower limits of the feasible
domain of evaporation in the Turc and Budyko spaces, drawn with the parameter

We distinguish two cases:

Equations (15), (16), (17a) and (17b) yield the following for the upper limits:

Replacing

The authors are very grateful to the reviewers, L. Gudmundsson and F. Jaramillo, and to the editor, M. Coenders-Gerrits, for their constructive comments of the manuscript. They also gratefully acknowledge the scientific and financial support of the UMR LISAH, as well as the always inspiring advice of A. Gaby. Edited by: M. Coenders-Gerrits Reviewed by: L. Gudmundsson and one anonymous referee