In this study, we start from Eq. (1) in (which is equivalent to Eq. 3 in and was traced back to by ):

1Ea‾=Ep‾P‾P‾n+Ep‾n1/n, with Ep‾ the mean annual potential evaporation, Ea‾ the mean annual evaporation, P‾ the mean annual precipitation, and n a shape factor, assumed to represent catchment characteristics (e.g., vegetation, soils). To express the ratio Ea‾/P‾ as a function of Ep‾/P‾ (see also Supplement S5), both sides of Eq. () were divided by P‾ and the resulting nominator and denominator on the right hand side were again divided by P‾. Following , Ep‾ was expressed as a function of mean annual net radiation (λEp‾=Rn‾), and all fluxes were transformed in terms of energy by multiplying by the latent heat of vaporization λ:

2λEa‾λP‾=Rn‾λP‾Rn‾λP‾n+1-1/n.

In a similar way, Eq. () can be transformed into an equation with the fraction of λP‾/Rn‾ as the independent variable (see also Supplement S5), by first dividing both sides of Eq. () by Ep‾, followed by a division of the nominator and denominator of the right hand side by Ep‾ and a conversion to energetic units:

3λEa‾Rn‾=λP‾Rn‾λP‾Rn‾n+1-1/n.

For fits to single data points (i.e., one catchment/study site), the equations were solved analytically using open-source software (sympy, https://www.sympy.org/, last access: 10 February 2022, essm https://pypi.org/project/essm/, last access: 10 February 2022). The exponent n in Eqs. () and () was fitted to data of multiple catchments with a non-linear least squares fit based on the Levenberg–Marquardt algorithm python scipy.optimize.curve_fit, https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html, last access: 10 February 2022,. The mean square error (MSE) was used in order to assess the goodness of fit:

4MSE=1m∑i=1mYi-Y‾i2, with m the number of observations, Yi an individual observation, and Y‾i the predicted value.

In order to capture a variation of climates, the study focuses on several sites along a precipitation gradient in Australia. The study sites include flux tower sites where the VOM has been tested previously as well as several larger catchments. An additional analysis based on a selection of 357 catchments of the CAMELS dataset is presented in Supplement S3, Fig. S3.8–S3.11.

The Vegetation Optimality Model VOM, is a combined water and vegetation model that optimizes vegetation properties, such as rooting depths and foliage cover, in order to maximize the net carbon profit (NCP), defined here as the difference between carbon taken up by photosynthesis and the carbon invested into maintenance of leaves, roots, and water transport tissues. The model code and documentation can be found online (https://github.com/schymans/VOM, last access: 10 February 2022, https://vom.readthedocs.io, last access: 10 February 2022) and a more detailed description can be found in , as well as Supplement S6. VOM version v0.6 (https://github.com/schymans/VOM/tree/v0.6, last access: 4 March 2022) was used in this study.

Three relatively simple conceptual hydrological models were selected in order to compare with the more complex VOM. These were selected in order to represent a commonly used model (the TUWmodel, a version of the HBV model), a slightly more complex model with more calibration parameters (FLEX), and a model with a very parsimonious number of parameters (the GR4J model). Details about the models can be found in Supplement S6.

The general approach was to first run the models using site-specific meteorology, and then conduct several numerical experiments where only precipitation is increased or decreased by a constant factor, while all other meteorological variables and model parameters remain unaltered. Precipitation was chosen as a representation of climate change, as it affects all models in a similar way, whereas potential evaporation is not used as input for the VOM. Since the VOM requires extended meteorological input over the entire modeling period (solar irradiance, vapor pressure), whereas the conceptual hydrological models require streamflow data for calibration, they were run for different sets of sites, with only the six catchments in Australia providing adequate input data for all models. For the Australian catchments and flux tower sites, the positions in the Budyko framework for the VOM and the hydrological models were all determined for the period 1 January 1985 until 31 December 2005 for consistency.

The steps of the experiment are summarized in Fig. . First, the base steps of the experiment were as follows.

Optimization VOM. The VOM was fully optimized for the flux tower sites, the 36 additional locations along the NATT, and the six Australian catchments for the meteorological data as observed.

In the next step, the meteorological forcing was altered by multiplying the precipitation (P) with factors ranging from 0.2 to 2.0 in steps of 0.2. The models were run with the optimized vegetation properties (VOM) or calibrated model parameters (hydrological models) from the first step, specifically as follows.

Re-run the VOM for each P scenario. The VOM was run for each P scenario with the vegetation parameters that were obtained in the unmodified situation based on observed P, representing perturbations in climate where long-term vegetation properties are not (yet) affected by changes in P.

In the third and last step, it was assumed that vegetation has fully adapted to each perturbation of precipitation. This was only possible for the VOM.

Re-optimize the VOM for all P scenarios. The VOM was re-optimized for all P scenarios at each of the Australian study sites (i.e., catchments, flux tower sites, and additional locations), representing perturbations in rainfall where vegetation has fully adapted to each perturbation.

The hydrological models were not run in this last step, as there is no way to represent the adaptation of vegetation to a hypothetical situation in these models.

VOM simulations with optimized vegetation properties led to a closer convergence along the Budyko curve and higher n values than VOM simulations without vegetation (Fig. ). For the flux tower sites where observations of λE were available (Fig. a), the optimized VOM plotted much closer to the flux tower observations than the VOM runs without vegetation. At the same time, the curve fit for Eq. () was much better for the optimized VOM, as indicated by the lower mean square error (3.86×10-3 vs. 2.004×10-2). The 36 additional locations along the NATT (Fig. b) and the six Australian catchments (Fig. c) also resulted in higher n values for the optimized vegetation compared to no vegetation, as well as a higher convergence to the curve (lower mean squared errors, with 1.788×10-3 vs. 2.339×10-2 and 1.249×10-3 vs. 5.504×10-3, for the curves with and without optimized vegetation, respectively).

Simulated responses to systematic shifts in precipitation (P) only tracked along the Budyko curve at individual flux tower sites if vegetation was re-optimized (i.e., green and gray symbols did not plot along the lines of the same color in Fig. , but the black symbols plotted much closer to their curve). This is also evident in the substantially lower mean squared errors for the black lines (see figure legends). The deviations from the Budyko curves were systematic, in that simulations with reduced P (higher Rn/λP) fell above the curve and simulations with higher P (lower Rn/λP) fell below the curve.

The n value fitted for 36 additional locations along the NATT similarly reduced from 1.183 to 1.095 after increasing the total rainfall by 20 % if the long-term vegetation properties were not re-optimized (Fig. ). After re-optimizing the vegetation properties in the VOM for the increased precipitation, n returned close to its original value for unperturbed precipitation (black triangles in Fig. , with an n value of 1.176). This was associated with increased values of the water-use parameters (cλf,p and cλe,p), vegetation cover, and rooting depth of the perennial vegetation (Fig. a), whereas the seasonal vegetation parameters changed less. Re-optimization for increased P also resulted in greater water and carbon fluxes in the VOM, which was more pronounced for the perennial than the seasonal vegetation (Fig. b).

The simulated responses to changes in precipitation (P) at six Australian catchments revealed that the re-optimized VOM followed a Budyko curve most closely (i.e.,lowest values of the mean square error, Fig. ) compared with the conceptual hydrological models FLEX, TUWmodel, and GR4J (see Supplement S3, Fig. S3.1–S3.7 for time series of meteorological data, model performances, and resulting discharges). The hydrological models, with the same model parameters for each P scenario, deviate from the Budyko curve in a similar way as the VOM without re-optimization (too flat around the unperturbed P). When the hydrological models were run for a selection of CAMELS catchments (see Supplement S3, Fig. S3.8–S3.11), their n values also reduced for a 20 % increase in precipitation and constant model parameters, but these changes remained relatively small.

In the next step, the n values were treated as site-specific properties, and the sites and catchments were each fitted to Eq. () individually and per precipitation (P) scenario (i.e.,a separate n value for each data point in Figs.  and ). The n values resulting from re-optimized long-term vegetation properties for each P scenario were considerably less variable at each site than those obtained for constant long-term vegetation properties, both at each NATT site (Fig. ) and across the 36 additional sites along the NATT (Fig. a). For these sites, increasing precipitation by 20 % without re-optimizing the vegetation resulted in a reduction in n values by around 0.10 (blue box, Fig. a), whereas the re-optimized VOM simulations (red box, Fig. a) did not result in a systematic change in n. Surprisingly, for re-optimized vegetation, the variation in n between sites was greater than at each individual site, even though P was varied by an order of magnitude (Figs.  and a). The opposite was the case if the long-term vegetation properties were not re-optimized for each perturbed P (Figs.  and b), or if the conceptual hydrological models with constant parameters were used (Fig. c–e). In all these latter model simulations, the n values systematically declined with positive perturbations in P and increased with negative P perturbations. In an additional analysis, the three hydrological models were applied to a selection of catchments of the CAMELS data to prove the generality of our findings. Interestingly, in this analysis we found that the sensitivity of n to changes in P was relatively similar (median decrease in n by 0.1 in response to a 20 % increase in P) compared with the non-optimized VOM (Fig. a), although the models were run over very different sets of conditions, i.e.,the CAMELS catchments in the US for the hydrological models and sites along the NATT in Australia for the VOM (see also Supplement S3, Fig. S3.8–S3.11 for all results).

The first hypothesis formulated in the introduction (H1) is that convergence of catchments along the Budyko curve, i.e., close to the maximum possible long-term evapotranspiration, results from vegetation optimality.

H1: model simulations based on vegetation optimality lead to a better reproduction of the empirical Budyko curve than model simulations without self-optimized vegetation.

This hypothesis is clearly supported by the following findings: (a) the VOM with fully optimized vegetation follows more closely Budyko curves with realistic n values (1.1–1.5) than the VOM without vegetation (Fig. ). In comparison, literature n values often range between 1.5 and 2.6 e.g.,. Note that the values found here are lower than the recommended value of 1.8 by or the value of 2.0 used in the Turc–Pike relationship , but compare well with the n value of 1.49 found by for 176 flux tower sites. In contrast, the n values of 0.62–0.78 produced by the VOM simulations without vegetation (bare soil evaporation only) are well below any values reported in the literature. Many other authors have shown previously that the evaporative fraction decreases due to land clearing. In fact, found a positive relation between the curve shape parameter and vegetation cover, implying an increasing evaporative fraction with increasing vegetation cover. As discussed below, our results suggest on top of this that for a changing climate, this shape parameter would only be conserved if vegetation is free to co-evolve with the climate, which would not be the case in a cleared catchment. (b) Simulations with perturbed precipitation (P) revealed that the VOM with re-optimized vegetation for each P setting followed the same Budyko curve very closely, whereas none of the other models did (Figs. , and ). This happened consistently for different study sites and will be discussed in more detail in the next section.

Altogether, these results suggest that vegetation optimality has a strong tendency to push the catchment water balance closer towards the envelope compared to simulations without vegetation, but also to keep it on a catchment-specific Budyko curve as climate changes (n varied more between catchments than within each catchment as P was varied by an order of magnitude in Fig. ). So far, the Budyko framework was related to optimality theory only a few times, mostly by applying thermodynamic optimality principles and constraints in numerical experiments . To our knowledge, the results presented here illustrate for the first time that an ecologically motivated optimality principle (maximum net carbon profit) leads to a close reproduction of the Budyko curve. Previously, suggested that convergence towards the Budyko curve may be a result of plants optimizing their rooting depths to maximize transpiration in a given environment. However, here we found that convergence on the Budyko curve is likely the result of rooting depths, vegetation cover, and water-use strategies playing together in a way to satisfy a biological optimality principle.

Our second hypothesis (H2) reflects a common belief expressed in the literature whenever the Budyko framework is used to study the effects of climate change.

H2: the empirical parameter n stays constant as climate changes, as long as vegetation cover and rooting depths stay constant.

Based on our results for varying precipitation (P), this hypothesis has to be clearly rejected. Constant long-term vegetation properties in the VOM and constant parameters in the conceptual hydrological models led to a large spread in n values, with a systematic decline in n for increasing P and increase in n for reduced P (Figs. , ). In our study, only full optimization of vegetation properties to changing precipitation led to a constant n as P was changed step-wise, by up to an order of magnitude (Figs.  and a). But this optimization, in fact, represents a change in vegetation properties as P changes, disproving the above hypothesis in the context of the four models applied in the present study. Interestingly, the VOM with constant vegetation properties and the conceptual models showed similar responses to perturbations of precipitation, even though these models are extremely different in nature. The VOM specifically models photosynthesis by representing the vegetation as two big leaves, resulting in carbon and water exchange with the atmosphere, whereas the conceptual hydrological models simulate vegetation in a more simple manner. In these models, the root zone storage capacity is often seen as the key variable determining the vegetation-related fluxes e.g.,. Note that the rooting depths of the VOM along with the soil water retention curves could be used to calculate root zone storage capacities, which would have a similar meaning to those in the conceptual models, but such a comparison and its interpretation is beyond the scope of this study. The conceptual hydrological models do not differ strongly between each other in their formulations of transpiration, but they do differ in their representations of hydrological processes, such as overland flow and interception evaporation (included in the FLEX model), or their number of free calibration parameters (e.g., just six for the GR4J model).

However, our findings bring forward one of the main deficiencies that many conceptual hydrological models currently have, namely the missing ability to adjust system properties (i.e.,parameters) in response to environmental change, an ability that would be needed for predictions under change . Calibrating conceptual model parameters to past observations and then using the calibrated models for prediction under environmental change implicitly assumes that the model parameters represent static catchment properties, which should not change as, e.g., the climate changes. The flaw in this assumption becomes immediately clear if one considers that some of these parameters encode vegetation functioning, which cannot be assumed static as the environment changes. The VOM and other process-based models offer a clearer separation between parameters related to physical catchment properties and those encoding vegetation functioning. Especially the optimality part enabled the VOM to track the Budyko curve under changing P, which is very encouraging for the implementation of vegetation optimality for predictions under change, as already shown for predicting vegetation response to elevated CO2 .

The last hypothesis relates to the underlying reasons for a change in n values.

H3: changes in n values are a result of slowly varying, long-term vegetation properties.

As already hinted at in the previous section, this hypothesis has to be rejected too as our results suggest the opposite. Only if the slowly varying, long-term vegetation properties were re-optimized in response to changing precipitation (P) did n stay constant, otherwise n changed in the opposite direction to P (Fig. ). This seems in line with the long-term character of the Budyko curve itself, but actually conflicts with findings that variations around the curve correlate with variations of the long-term average annual vegetation coverage , forest cover , or fPAR values . Interestingly, the simulated changes in rooting depths in response to increased P (Fig. ) agree with the general observation that rooting depths increase with increasing precipitation amounts . In a previous study, we found that the VOM reproduces this pattern along a precipitation gradient , whereas here we demonstrate that the VOM also maintains this pattern over time for constant physical catchment properties.

Our finding that n varied less in response to changes in P than between catchments (Fig. ) suggests that the n values are indeed site-specific. However, our analysis did not reveal whether n depends on properties of climate (e.g., temperature, seasonality of net radiation and precipitation, rainfall intensity), physical catchment properties (e.g., soil water-holding capacity), or both. Previous studies also suggested that n is a catchment-specific variable depending on climate properties (other than mean annual P and Rn), physical catchment properties, and/or vegetation .

However, our findings challenge the common belief formulated in H3 that changes in n are a result of changes in the physical catchment properties or vegetation, whereas n should stay constant if only mean annual P or Rn change . Our results suggest quite the opposite, as changes in the long-term, slowly varying vegetation properties were needed to bring a certain catchment back to the original n value after a change in P. Since vegetation reacts to the climate, the parameter n can be considered a combined result of climate, vegetation, and landscape properties . In some studies, n values were found to correlate stronger with catchment properties e.g.,, but one could argue that catchment properties influence vegetation and vice versa. Interestingly, vegetation optimality suggests that the vegetation response to changes in climate stabilizes n. Our results also emphasize that the dynamic evolution of the system is important, as argued before by, e.g., and . If a climatic shift happens quickly, and the state of the system did not yet adapt to the new situation, we may expect different n values than for a system in a (dynamic) steady-state. Note that other authors were not able to find that catchments followed a single Budyko trajectory over time .

Different ways to plot the Budyko framework have been used in the literature, either using a dryness index (Rn/λP) or a wetness index (λP/Rn) as the independent variable. For this reason, we repeated our analyses in Supplement S4 for different projections of the Budyko framework. The previous results in Sect. – remained valid after using a Budyko space based on a dryness index (Fig. S4.1–S4.5 in Supplement S4). Nevertheless, it must be noted that the resulting n values differ for a Budyko space with a wetness index instead of a dryness index. More specifically, Fig. S4.5 shows, for example, that the resulting n values for the CAMELS data slightly differ. However, the n values still similarly change for a Budyko plot with the dryness index λP/Rn as the independent variable as for a Budyko plot with a wetness index (Supplement S3, Fig. S3.8).

Note also that the results in Sect.  would not change for a different Budyko projection, as these n values could be solved analytically. However, when using a dryness index (i.e., plotting E/P as a function of Rn/λP), the interpretation of the results becomes more difficult, as e.g., an increased P would decrease E/P at constant E. Therefore, changes in E do not directly translate to changes in E/P. For this reason, we used a wetness index in the main paper (i.e., plotting λE/Rn as a function of λP/Rn), in which case a change in P only results in a change on the vertical axis if E changes.

In the present study, we perturbed precipitation in isolation, i.e., without perturbing any other climate variables, which does not reflect any realistic climate change scenario. Moreover, changes in precipitation are in reality linked to changes in vapor pressure, net radiation (e.g., through cloud formation), air temperatures, stronger sensible heat fluxes, warmer and drier soils, and likely also different rainfall intensities. We also attempted to vary atmospheric water demand but were limited by the differences of the VOM and conceptual hydrological models. The hydrological models use potential evaporation as input, whereas the VOM uses vapor pressure, temperature, and incoming shortwave radiation to drive evaporation. VOM simulations with altered vapor pressure and incoming shortwave radiation are presented in Supplement S2. Especially when altering the vapor pressure, results became difficult to interpret, as net longwave radiation and hence the position along the x axis of the Budyko plots is only indirectly affected by vapor pressure Eq. 39,. Here, the results strongly depended on the projection in Budyko space, i.e., whether a dryness or wetness index was used (see Supplement S2, Fig. S2.3). In contrast, changes in shortwave radiation confirmed that vegetation optimality reduces departures from the Budyko curve compared to static vegetation (Supplement S2, Fig. S2.1 and S2.2), but in a less systematic way than precipitation. This related to shortwave radiation being a strong driver for photosynthesis in the VOM, which profoundly affects all components of vegetation.

As the study used 37 years of data, changes at the study sites during this time window may have occurred. The flux tower sites are relatively undisturbed, but suffer occasionally from fires . At the same time, at Howard Springs it has been noted that trends exist in gross primary productivity and water-use efficiency . Also at the catchments and additional locations along the transect, changes and trends may have occurred during this period. However, only the full time series of the climate data were used as input for the numerical experiments, and hence any trends would not make a difference to our final results. Note also that discrepancies between flux tower observations (i.e., blue triangles in Fig. ) and model results relate to the long-term character of the Budyko framework, with flux tower time series being much shorter than the simulated time series.

The VOM and the hydrological models used in this study are spatially lumped so they should not be expected to adequately represent the water and energy partitioning of large heterogeneous catchments. Here we just used these models as simplified representations of the key processes determining this partitioning and to test our general hypotheses about the Budyko curve.

The range of hydrological models used in this study is extremely limited. Many hydrological models use more complex approaches to model evapotranspiration, which are likely better suited for climate change studies. Here, three relatively simple models were chosen to test the robustness of our finding that changing precipitation leads to changing n if vegetation does not adapt. The changing n in these simple models without explicit vegetation component illustrates most convincingly that the change is not brought about by vegetation processes, as suggested in the literature previously.

This study did not aim to predict responses of vegetation and the water balance to any realistic climate change scenario, and the finding of a stabilizing effect of vegetation optimality on the value of n as precipitation changes may not apply to human-induced climate change at all. This is most obvious when considering that rising atmospheric CO2 concentrations are the main driver of human-induced climate change, as they have a profound, direct effect on vegetation water use (which can be buffered, enhanced, or even reversed by vegetation optimality, depending on the climate), which would modify a site-specific n value permanently, even in the absence of any other climatic changes .