This Technical Note documents and analyzes the puzzling similarity of two widely used water balance formulas: Turc–Mezentsev and Tixeront–Fu. It details their history and their hydrological and mathematical properties, and discusses the mathematical reasoning behind their slight differences. Apart from the difference in their partial differential expressions, both formulas share the same hydrological properties, and it seems impossible to recommend one over the other as more “hydrologically founded”: hydrologists should feel free to choose the one they feel more comfortable with.

The Turc–Mezentsev (Mezentsev, 1955; Turc, 1954) and Tixeront–Fu (Fu, 1981; Tixeront, 1964) formulas were introduced to model long-term water balance at the catchment scale. Both formulas are almost equivalent numerically (but differ nonetheless). Surprisingly, comparisons are rare: Tixeront knew the work of Turc (1954), which he cites, but it seems that he did not realize that Turc's formulation was numerically equivalent to the one he proposed. Similarly, Fu knew the work of Mezentsev (1955) because he precisely starts his 1981 paper by discussing it, but it seems that he did not realize that the formulation he obtained was so close numerically.

As far as we know, Yang et al. (2008) were the first to compare the Turc–Mezentsev and Tixeront–Fu formulas and to conclude that both formulas were “approximately equivalent”. In this note we further elaborate on the similarity between the two formulas and contribute complementary explanations of their underlying hypotheses.

The TM and TF formulas use as inputs long-term average precipitation

Turc–Mezentsev (TM) and Tixeront–Fu (TF) water–energy balance
formulations (

Partial derivatives of the Turc–Mezentsev formula (

Partial derivatives of the Tixeront–Fu formula (

We need to clarify here that the TM and TF formulas can be found in the hydrologic literature under different names. The naming convention we adopted is explained as follows: Eqs. (1) and (2) are named “Turc–Mezentsev” (TM) because Turc (1954) and Mezentsev (1955) worked independently and published the same equation almost simultaneously. Equations (3) and (4) are named “Tixeront–Fu” (TF) because although Tixeront's original publication predates Fu's by almost 20 years, both publications were independent, and the name of Fu has already gained wide international recognition. Both formulas are sometimes referred to as “Budyko-type,” although none of them were actually used by Budyko (1963/1948), who instead used a parameter-free formula derived from the work of Oldekop (1911) (for a synthesis of Oldekop's work and how it was used by Budyko, see Andréassian et al., 2016). Other authors have published papers containing the TM formula (see, e.g., Hsuen-Chun, 1988, and Choudhury, 1999), and their names are sometimes used to designate it.

In our interpretation of the TM and TF formulas, we will use their partial derivatives, which we present in Tables 2 and 3.

We mentioned in the introduction that the first paper comparing the TM and
TF formulas was published by Yang et al. (2008), who note that
the TM and TF formulas are “approximately equivalent” and that their
parameters have a “perfectly significant linear correlation relationship”,
which they identify as in Eq. (13):

Illustration of the similarity between the values of the
Turc–Mezentsev (TM) and Tixeront–Fu (TF) formulas for a range of values of

Note that Eq. (13) is an experimental relationship
obtained by regression. It gives slightly more satisfying results than the
“theoretical” relationship (found by equating both equations for

Figure 1, which illustrates the numerical proximity of the two formulas,
speaks for itself: while we tested a wide range of (

Illustration of the similarity between the Turc–Mezentsev (TM) and
Tixeront–Fu (TF) formulas for a range of values of

Figures 2 and 3 also present the differences between the partial derivatives of the TM and TF formulas. The reason for this is that both formulas are sometimes used to predict the hydrological impact of climatic change, i.e., to evaluate the evolution or differences between future and current conditions. Again, both formulas appear numerically equivalent.

Illustration of the similarity between the Turc–Mezentsev (TM) and
Tixeront–Fu (TF) formulas for a range of values of

The TM and TF formulas share a set of hydrological properties that we summarize in Tables 4 and 5, following the presentation proposed by Lebecherel et al. (2013).

Hydrological interpretation of the Turc–Mezentsev and Tixeront–Fu
formulas, applied to streamflow (

Hydrological interpretation of the Turc–Mezentsev and Tixeront–Fu
formulas, applied to actual evaporation (

The Appendix summarizes the underlying mathematical reasoning presented by
the authors of the TM and TF formulas and by Zhang et al. (2004) and Yang et
al. (2008). What can be concluded from the analysis presented in the Appendix
is that both formulations are based on very similar but nonetheless slightly
different hypotheses: Table 6 illustrates them after rewriting the partial
differentials to make

For the Turc–Mezentsev formula, Table 6 shows that

For the Tixeront–Fu formula, Table 6 shows that

Comparison of the partial differentials of the Turc–Mezentsev and
Tixeront–Fu formulas (

What can we conclude from this? Does this make the TF formula (slightly) more general and the TM formula (slightly) more restrictive? Perhaps, but from the user's point of view, both formulas are so close numerically (see Fig. 1 and also compare the maps presented by de Lavenne and Andréassian, 2018) that any data-based distinction is impossible.

The Turc–Mezentsev and Tixeront–Fu formulas are two sound and numerically
equivalent representations of the long-term water balance at the catchment
scale. This note investigated the underlying assumptions of the two formulas
and showed that the Tixeront–Fu formula is slightly more general than the
Turc–Mezentsev formula, because its partial differences can be written both
as a function of the

No data sets were used in this article.

Lucien Turc was a French soil scientist. He produced his formula while
working on his PhD thesis, defended in April 1953 (and published in 1954 in
the

Varfolomeï Mezentsev was a Soviet geographer, working at the University
of Omsk in Siberia. He published his formula in 1955, and continued working
on it throughout his life (Mezentsev, 1955, 1982, 1993). Mezentsev (1955, 1982, 1993)
started his analysis from a formula proposed by Bagrov (1953) (Eq. A2):

Mezentsev (1955) reasoned that in order to find a generic solution,
Bagrov's formula could be rewritten as follows:

Equation (A3) can be integrated analytically and yields Eq. (A4):

Jean Tixeront (1901–1984), a graduate of Ecole Nationale des Ponts et
Chaussées, was a French hydrologist who spent most of his professional
career in Tunisia. The most accessible reference for his formula is a paper
published in the proceedings of the General Assembly of the IAHS in 1964
(Tixeront, 1964). The formula had been first published in 1958, in the note
accompanying a map of mean annual runoff in Tunisia (Berkaloff and Tixeront,
1958). There, the authors give more explanation
of their reasoning, stating that two desirable properties of such a formula
would be that (i) “when precipitation increases, runoff tends to equal
precipitation minus potential evapotranspiration” and (ii) “when
precipitation tends towards zero, the runoff to the precipitation ratio tends
towards zero.” They proposed Eq. (A5) as the “simplest formula satisfying
these conditions”:

Bao-Pu Fu was a Chinese hydrologist working at the University of Nanjing. He
published his formula in 1981, and an English abstract of his computation is
given in the appendix of the paper by Zhang et al. (2004). It is interesting
to note that Fu's (1981) paper starts with a well-informed review of the
formulas in the literature, where he cites the works of Bagrov (1953) and
Mezentsev (1955). Then he makes assumptions about a system of differential
equations that should be respected by an actual evaporation formula (Eq. A1
in Zhang's paper):

Yang et al. (2008) were not only the first to compare the Turc–Mezentsev and
Tixeront–Fu formulas, but they also made a mathematical analysis of the
Turc–Mezentsev formula that we reflect on now. They start to write down a
system of differential equations that should be respected by an actual
evaporation formula (Eq. 14 in their 2008 paper):

Straightforward computations show that the functions given by Eq. (A18) do
not satisfy Eq. (A21), although they satisfy Eq. (A20). This is the reason
why Yang et al. (2008) missed the solution given by Tixeront and Fu's formula
in their demonstration. For the functions

We present here another interpretation of both equations, which is partly
mathematical and partly hydrological. For this, we define two simple
functions, which we tentatively call “

We can now hydrologically interpret the TM formula by saying that it states
that catchment-scale actual evaporation

TS did the maths; VA did the history and the hydrological interpretation.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge the review provided by Laurène Bouaziz, Maik Renner, Charles Perrin and an anonymous reviewer, which all contributed to clarifying the manuscript.

This paper was edited by Martijn Westhoff and reviewed by Maik Renner, Laurène Bouaziz, and one anonymous referee.