Letter | Published:

# Global patterns and dynamics of climate–groundwater interactions

## Abstract

Groundwater, the largest available store of global freshwater1, is relied upon by more than two billion people2. It is therefore important to quantify the spatiotemporal interactions between groundwater and climate. However, current understanding of the global-scale sensitivity of groundwater systems to climate change3,4—as well as the resulting variation in feedbacks from groundwater to the climate system5,6—is limited. Here, using groundwater model results in combination with hydrologic data sets, we examine the dynamic timescales of groundwater system responses to climate change. We show that nearly half of global groundwater fluxes could equilibrate with recharge variations due to climate change on human (~100 year) timescales, and that areas where water tables are most sensitive to changes in recharge are also those that have the longest groundwater response times. In particular, groundwater fluxes in arid regions are shown to be less responsive to climate variability than in humid regions. Adaptation strategies must therefore account for the hydraulic memory of groundwater systems, which can buffer climate change impacts on water resources in many regions, but may also lead to a long, but initially hidden, legacy of anthropogenic and climatic impacts on river flows and groundwater-dependent ecosystems.

## Main

Groundwater flow systems exist in dynamic balance with the climate, connecting interacting zones of recharge (the replenishment of water in the subsurface) and discharge (the loss of groundwater from the subsurface), with multiple feedbacks. As climate varies, changes in the quantity and location of natural groundwater recharge lead to changes in groundwater storage, water table elevations and groundwater discharge1. These changes in time and space play a central role in controlling the exchange of moisture and energy across the Earth’s land surface5,6 and connect processes critical to hydro-ecology, for example, as well as carbon and nutrient cycling7. Climate–groundwater interactions may also have played a key role in the evolution of our own and other species8 and continue to be critical in setting the availability of water for abstraction by humans in coupled food–water–energy systems1. Recent global mapping of water table depths9 and the critical zone10 suggest where interactions of climate and groundwater may be most tightly coupled. However, they do not resolve where groundwater systems are most sensitive to changes in climate and vice versa, or the timescales over which such changes may occur.

Here, we derive and combine global-scale analytical groundwater model results and other hydrologic data sets to provide the first global assessment of the sensitivity of groundwater systems to changes in recharge in both space and time (Fig. 1), and discuss their utility as an emergent constraint in understanding and modelling groundwater interactions with climate and other Earth systems at the global scale.

We have characterized the mode of groundwater–climate interactions as being either principally bidirectional or unidirectional using an improved formulation of the water table ratio (WTR)11,12 mapped globally at high resolution (Fig. 1a and Supplementary Figs. 1 and 2). The WTR is a measure of the relative fullness of the subsurface and thus the extent of the water table’s interactions with topography. Values of WTR > 1 indicate a topographic control on water table conditions broadly correlating to shallow (<10 m below ground level, m b.g.l.) water table depths (WTDs) globally (see Methods and Supplementary Fig. 3). This is indicative of a prevalently bidirectional mode of groundwater–climate interaction (Fig. 1c) where the climate system can both give to the groundwater system in the form of recharge, and receive moisture back via evapotranspiration if WTDs are shallow enough.

The land surface in such regions rejects a proportion of the potential recharge, and groundwater can have a limiting control on land–atmosphere energy exchanges5; a tight two-way coupling between groundwater and surface water is also common. In contrast, in ‘recharge controlled’ areas where WTR < 1, water tables are more disconnected from the topography and, while groundwater may still receive recharge from the land surface, the extent of two-way interaction between climate and groundwater is limited and the mode of interaction is predominantly unidirectional (Fig. 1c).

We find that regions where WTR > 1 cover around 46% of the Earth’s land area (see Methods and Fig. 1a,b) and contribute to the large, but until recently underestimated, extent of groundwater–vegetation interactions globally10,13,14. Consistent with previous regional analyses and the form of the governing equation (see Methods), our results indicate that bidirectional interactions are more likely to occur in areas with high humidity, subdued topography and/or low permeability. In contrast, regions with WTR < 1 are more common in drier climates or more mountainous topography11.

To assess the large-scale temporal sensitivity of climate–groundwater interactions we have used an analytical groundwater solution to quantify groundwater response times (GRTs) globally and at high resolution. GRT is a measure of the time it takes a groundwater system to re-equilibrate to a change in hydraulic boundary conditions15. For example, the GRT estimates the time to reach an equilibrium in baseflow to streams (or other boundaries) after a change in recharge rate, potentially from climate or land-use change. Our results indicate that groundwater often has a very long hydraulic memory with a global median GRT of nearly 6,000 yr, or approximately 1,200 yr when hyper-arid regions, where recharge is <5 mm yr−1, are excluded (Fig. 1d,e). Only 25% of Earth’s land surface area has response times of less than 100 yr (herein called ‘human timescale’). However, this is equivalent to nearly 44% of global groundwater recharge flux, calculated by aggregating contemporary recharge over the land area where GRT < 100 yr, expressed as a proportion of the total global recharge. Around 21% by area has unidirectional climate–groundwater interactions and response times on human timescales, mostly associated with high permeability geology suggesting a strong lithological control (Fig. 2a).

The remainder (4%) in areas with bidirectional climate–groundwater interactions is mostly located in the humid, lowland, tropical regions with unconsolidated sediments (for example, Amazon and Congo basins, Indonesia), low-lying coastal areas (for example, Florida Everglades, Asian mega-deltas) or in high-latitude, low-topography humid settings (for example, northeastern Canada, parts of northern Europe).

A powerful advantage of using analytical groundwater equations such as the WTR is that they allow us to directly assess the spatial sensitivity of the mode of climate-groundwater interactions. By taking the derivative of WTR with respect to recharge (Supplementary Fig. 4) we have a measure of the sensitivity of the relative fullness of the subsurface to changes in recharge (see Methods). Our results indicate that the mode of climate–groundwater interaction is very insensitive to relative changes in recharge (Fig. 2b and Supplementary Fig. 5), with only 5% of the Earth’s land surface switching mode for a 50% relative change in recharge rate. This represents a large change in natural groundwater recharge in the context of projections for the coming century16. However, when absolute recharge rate changes are considered, more sensitivity is apparent and a pattern emerges (Supplementary Figs. 6 and 7) that indicates the strong inverse relationship between the spatial and temporal sensitivity of groundwater systems to changes in recharge that we observe (Fig. 3b). At small, local scales, our calculations may have relatively large uncertainties, stemming from the uncertainties in the global data sets used for the analysis, particularly for hydraulic conductivity (see Methods). However, at the larger scales considered here, Monte Carlo experiments (MCEs) indicate that, once the variance in each parameter is combined, the global estimates have relatively small standard deviations (Figs. 1 and 2 and Supplementary Fig. 2).

The global pattern of GRT (Fig. 1d) indicates a propensity for longer hydraulic memory in more arid areas. Despite the expected scatter due to geomorphological and lithological heterogeneity, there is a power law relationship between median GRT and groundwater recharge R such that GRT  1/Ry with y ~ 2 (Fig. 3a). This discovery is not directly expected from the form of the governing equations, but is rather an emergent property of groundwater system interactions with the Earth’s land surface and climate system. The principal control on the observed power law is the distribution of perennial streams (Supplementary Fig. 8) to which the GRT is most sensitive, and which itself is strongly controlled by climate (Supplementary Figs. 911). How to characterize, quantitatively, this climatic control on the perennial stream distributions is a pertinent question for further hydro-geomorphological research.

We should not therefore expect GRTs to be static nor consider them as ‘time constants’, despite being mathematically equivalent to other diffusion processes. Rather, GRTs will evolve in time as both climate and geology vary the geometry and hydraulic properties of groundwater flow systems. This will occur over long but diverse timescales associated with changing river geometries.

Despite its importance, most global climate, Earth system, land surface and global hydrology models exclude groundwater or do not allow groundwater to flow between model grid cells17,18,19. While our results suggest that the spatial distribution of the mode of climate–groundwater interactions may be rather static over century-long timescales, we have shown that nearly a half of the world’s groundwater flux is responsive on 100 yr timescales. Hence, to capture the important mass and energy transfers correctly, which may affect regional precipitation and temperature dynamics5,6, lateral flow circulation of groundwater must be incorporated into the next generation of global models rather than assuming within-grid-cell hydrological closure of the water budget as is currently often assumed20,21,22. Our GRT calculations provide direct estimates of spin-up times to improve groundwater-enabled global models, without having to use the currently employed methods of extrapolation21. Given the long GRTs present over much of the Earth’s land surface, defining initial conditions with an equilibrium water table calculated for present-day climate conveniently, but wrongly, assumes stationarity in groundwater levels and fluxes. Since groundwater is known to be the part of the hydrological system that takes longest to achieve equilibrium23, new approaches that incorporate the existence of long-term transience should continue to be developed24.

The global distribution of GRTs suggests that widespread, long-term transience in groundwater systems persists in the present day due to climate variability since at least the late Pleistocene in many semi-arid to arid regions (Fig. 3a). This is consistent with observations of larger than expected groundwater gradients, given the current low recharge, that have been observed in present-day arid zones24. While groundwater residence time and groundwater response time are fundamentally different concepts, we also note the correspondence between high GRT and significant volumes of fossil-aged groundwater storage in arid regions2,25. The outcome of this result is that groundwater discharge to oases, rivers or wetlands in otherwise dry landscapes will be particularly intransient in comparison to climate change, in as much as climate controls the variations in groundwater recharge. However, our results also indicate that groundwater response times tend to be greater in regions where water tables are most sensitive to changes in recharge (Fig. 3b). This follows from the fact that both the groundwater response time and the derivative of the water table ratio share a strong dependence on the square of the distance between perennial streams (L, compare equations (10) and 14)).

Away from these more arid contexts, the responsiveness of groundwater systems has recently been demonstrated to be as important as climate controls for the development of hydrological drought26. For example, low GRT systems tend to enhance the speed of propagation of meteorological drought through to hydrological drought, whereas higher GRT systems attenuate climate signals to a greater extent but also show greater lags in recovery from drought. Thus, even within relatively small geographic areas, geological variations can lead to very different drought responses even under similar climate variability. By way of a specific example, increasing lags between meteorological and hydrological drought indicators have been observed between the two most significant aquifers in the UK27 in a manner consistent with what would be expected from our estimates of GRT (Cretaceous Chalk limestone—GRTs of months to years; Permo-Triassic sandstone—GRTs of years to hundreds of years; Fig. 1d).

Our analysis therefore provides a new framework for understanding global water availability changes under climate change. First, the discovery of a power law relating groundwater recharge and GRT suggests that important areas of groundwater discharge in naturally water-scarce parts of the world are likely to be more resilient to climate fluctuations than humid areas. However, where groundwater response times are higher, water tables also tend to be most sensitive to changes in recharge in the long term. Hence, accounting appropriately for groundwater–climate interactions within analyses of global water scarcity in the context of climate change is thus of great importance when explicitly considering the contribution of groundwater storage changes28. Second, the long memory of groundwater systems in drylands also means that abrupt (in geological terms) changes in recharge or widely distributed groundwater abstraction will leave longer legacies. There may also be initially ‘hidden’ impacts on the future of environmental flows required to sustain streams and wetlands in these regions. It is critical therefore that climate change adaptation strategies that shift reliance to groundwater1 in preference to surface water should also take account of lags in groundwater hydrology29 and include appropriately long timescale planning horizons for water resource decision making. Third, robust assessments of the impact of climate change on hydrological drought require estimates of ‘groundwater responsiveness’26. The timescale of such responses can be directly informed by our results and improve the decision-making process with regard to adaptation strategies to changing drought frequencies under climate change.

## Methods

### Derivation of equations

#### Governing groundwater flow equations

The governing equations were formulated by considering an ideal homogeneous, horizontal unconfined aquifer bounded at one end (x = L/2) by a stream assumed to be a constant head boundary and at the other (x = 0) by a no-flow boundary representing a flow divide (Supplementary Fig. 13). The one-dimensional (1D) (Boussinesq) equation of groundwater flow for such an aquifer receiving homogeneous recharge can be given as

$$\frac{\partial }{{\partial x}}\left( {Kh\frac{{\partial h}}{{\partial x}}} \right) = S\frac{{\partial h}}{{\partial t}} - R(t)$$
(1)

where K is hydraulic conductivity [LT−1], S is storativity [–], h(x,t) is hydraulic head [L], t is time [T], x is distance [L] and R(t) is groundwater recharge [LT−1].

If changes in transmissivity due to fluctuations in groundwater heads are assumed to be negligible, equation (1) may be linearized as follows:

$$T\frac{{\partial ^2h}}{{\partial x^2}} = S\frac{{\partial h}}{{\partial t}} - R(t)$$
(2)

where T is transmissivity [L2T−1], and T = KH, with H the average saturated thickness [L].

The lateral boundary conditions are as follows:

$$\frac{{\partial h(0,t)}}{{\partial x}} = 0,h\left( {\frac{L}{2},t} \right) = b$$
(3)

Parameter L is thus a characteristic length equivalent to the distance between perennial streams which act as fixed head groundwater discharge boundaries.

#### WTR derivations

For steady-state flow, where h(x,t) becomes h(x), the solution to equation (1) for the stated boundary conditions is given by

$$h(x) = \left( {b^2 + \frac{R}{K}\left( {\frac{{L^2}}{4} - x^2} \right)} \right)^{0.5}$$
(4)

At the flow divide, x = 0, therefore

$$h(0) = \sqrt {b^2 + \frac{{RL^2}}{{4K}}}$$
(5)

For steady-state flow, the solution to the linearized form, equation (2), for the stated boundary conditions is

$$h(x) = \frac{R}{{2T}}\left( {\frac{{L^2}}{4} - x^2} \right) + b$$
(6)

At the flow divide, x = 0, therefore

$$h(0) = \frac{{RL^2}}{{8T}} + b$$
(7)

The WTR is defined12 as the ratio of the head at the flow divide above the fixed head boundary (that is, $$h_0 - b$$) to the maximum terrain rise above the fixed head boundary, d [L]. This yields a new, nonlinearized, form of the WTR, from equation (5) as follows:

$${\mathrm{WTR}}_{\mathrm{NL}} = \frac{{\sqrt {b^2 + \frac{{RL^2}}{{4K}}} - b}}{d}$$
(8)

For the linearized form, from equation (7), and as originally given by ref. 12, the WTR is

$${\mathrm{WTR}}_{\mathrm{L}} = \frac{{RL^2}}{{8Td}} = \frac{{RL^2}}{{8KHd}}$$
(9)

Equations (8) and (9) become equivalent for combinations of small L or R, or large K.

All maps and analyses presented in this paper use the nonlinear form of the WTR (equation (8)) with the exception of Supplementary Fig. 1 where the two versions are compared, and are calculated using the L parameters derived using a minimum river discharge threshold of 0.1 m3 s−1. Comparisons of global maps and frequency distributions for the linear and nonlinear forms are shown in Supplementary Figs. 1 and 2. The frequency distribution comparison (Supplementary Fig. 2) shows that the new nonlinear formulation has a narrower and more symmetric distribution with a median closer to zero than the linearized form. This is indicative of its better physical representation such that the extent of higher WTRs is limited by the feedback between higher water table elevation and concomitant increases in transmissivity inherent in the nonlinear Boussinesq equation (equation (1)).

The WTR is a measure of the relative fullness of the subsurface and thus the extent of the water table’s interactions with topography. We have therefore used the WTR to characterize the dominant mode of groundwater–climate interactions as being either principally bidirectional or unidirectional based on whether they are ‘topographically controlled’ (WTR > 1) or ‘recharge controlled’ (WTR < 1), respectively. This is a reasonable approximation since a global comparison with WTDs (Supplementary Fig. 3) indicates that WTR > 1 broadly correlates to shallow (<10 m below ground level) water table conditions. This condition is indicative of a prevalently bidirectional mode of groundwater–climate interaction where the climate system can both give to the groundwater system in the form of recharge, and receive moisture back where local variations in WTDs enable evapotranspiration to occur from groundwater directly. In contrast, areas with WTR < 1 show increasingly large WTDs well beyond plant rooting depths, leading to predominantly unidirectional climate–groundwater interactions where the groundwater system receives recharge from the climate system but there is more limited potential for feedback in the other direction.

The sensitivity of the WTR to changing recharge is given by differentiating equation (8) with respect to R:

$$\frac{{{\mathrm{dWTR}}_{\mathrm{NL}}}}{{{\mathrm{d}}R}} = \frac{{L^2}}{{8Kd}}\left( {b^2 + \frac{{RL^2}}{{4K}}} \right)^{ - 0.5}$$
(10)

This equation represents the sensitivity of the maximum head to recharge relative to the topography, which can be understood as the sensitivity of the ‘fullness’ of the subsurface to changes in recharge.

Following from equation (8), we calculate the recharge required for the WTR to equal 1 for every grid cell as

$$R_{{\mathrm{WTR}} = 1} = \frac{{4K}}{{L^2}}(d^2 + 2db)$$
(11)

The difference between R and the values given in equation (11) then gives an expression for the change in recharge (ΔR) needed to effect a change in the WTR across the transition between topography control (bidirectional climate–groundwater interactions) and recharge control (unidirectional climate–groundwater interactions) modes. In absolute terms this is

$$\Delta R_{\mathrm{abs}} = R - R_{{\mathrm{WTR}} = 1}$$
(12)

and in relative terms it becomes

$$\Delta R_{\mathrm{rel}} = \frac{{R - R_{{\mathrm{WTR}} = 1}}}{R}$$
(13)

#### GRT definition

The GRT is, in general terms, a measure of the time it takes a groundwater system to respond significantly (as defined below) to a change in boundary conditions15,31,32,33,34,35 and is defined here as follows:

$${\mathrm{GRT}} = \frac{{L^2S}}{{\beta T}}$$
(14)

where β is a dimensionless constant, T is transmissivity [L2T−1], S is storativity [–] and L is the distance between perennial streams [L]. To illustrate why this equation defines a time of response, consider a groundwater mound such as that shown in Supplementary Fig. 13. Let the initial shape of the mound (of maximum height A), due to some steady recharge, be given by

$$h\left( {x,0} \right) = A{\mathrm{cos}}\left( {\frac{{\pi x}}{L}} \right)$$
(15)

If recharge suddenly ceases (that is, a step change) then it can be shown, in the manner of ref. 33, that the solution to the linearized equation (2) without recharge (that is, R(t) = 0) is

$$h\left( {x,t} \right) = h\left( {x,0} \right){\mathrm{exp}}\left( { - \frac{t}{{\mathrm{GRT}}}} \right)$$
(16)

for β equal to $$\pi ^2$$.

Hence, for this case, the GRT controls the timescale for the groundwater levels to decay exponentially to reach 63% re-equilibrium after a change in boundary (recharge) conditions (that is, an ‘e-folding’ timescale). This value for β was chosen to be consistent with mathematically equivalent uses of ‘time constants’ (often denoted τ), in other branches of science.

As outlined by ref. 34, comparing the timescale of a particular forcing to the GRT can be a useful measure of the degree of transience a groundwater system will manifest in terms of variations in lateral groundwater flow. However, there is an important difference to note in the case of a step change in conditions, as used to define GRT in equation (14), in comparison with a periodic variation in the forcing recharge (of period P). For the step change case outlined above, both heads and fluxes decay exponentially after the change in recharge. However, in the periodic case, where GRT » P, variations in recharge lead to very stable groundwater fluxes (including at the downstream lateral boundary) but large temporal changes in groundwater head across much of the aquifer35. Thus, it is important to distinguish between the control of GRT on the degree of transience in either heads or fluxes, depending on the nature of the boundary conditions.

### Spatial input data and manipulation

#### Global mapping of the distance between perennial streams (L)

The distance between perennial streams (L) was calculated using a globally consistent river network provided by the HydroSHEDS database36, which was derived from the 90 m digital elevation model of the Shuttle Radar Topography Mission (SRTM). For this study, we extracted the global river network from the HydroSHEDS drainage direction grid at 500 m pixel resolution by defining streams as all pixels that exceed a long-term average natural discharge threshold of 0.1 cubic metres per second, resulting in a total global river length of 29.4 million km. Smaller rivers with flows below this threshold were excluded as they are impaired by increasing uncertainties in the underpinning data. However, the sensitivities of the most important results of this paper to the chosen threshold are considered in our uncertainty analysis below. Estimates of long-term (1971–2000) discharge averages have been derived through a geospatial downscaling procedure37 from the 0.5° resolution runoff and discharge layers of the global WaterGAP model (version 2.2, 2014), a well-documented and validated integrated water balance model16,38. Only perennial rivers were included in the assessment; intermittent and ephemeral rivers were identified through statistical discharge analysis (lowest month of long-term climatology is 0) and extensive manual corrections against paper maps, atlases and auxiliary data, including the digital map repository of National Geographic39. L was calculated for every pixel of the landscape (Supplementary Fig. 8) by identifying the shortest combined Euclidean (straight-line) distance between two river locations at opposing sides of the pixel. Neighbourhood low-pass filters (5 × 5 kernel size) were applied to remove outlier pixels and speckling. All calculations were performed in the ESRI ArcGIS environment using custom-made scripts.

#### Global mapping of the WTR, GRTs and other expressions

Global WTR maps were created from the above equations using the recharge rate (R in m yr−1), based on ref. 30, a minimum saturated thickness of the aquifer (b) set to 100 m (refs. 40,41), the distance between two perennial streams (L, in m, as described above), intrinsic permeability values (m2) reported in ref. 40, which were converted to hydraulic conductivity (m s−1) by assuming standard temperature and pressure (1 × 107 multiplication factor) and then converted to units of m yr1. The maximum terrain rise between rivers (d, in m) was based on the range of elevations in the 250 m GMTED2010 data set42.

The GRT was mapped using the same L data and hydraulic conductivity values as for the WTR calculations. Transmissivity (T, m2 yr−1) was calculated by multiplying the hydraulic conductivity with a fixed saturated thickness of 100 m (refs. 40,41). It was assumed that storativity (S) for unconfined aquifers is dominated by the specific yield and that this can be approximated by mapped porosity values43. Owing to the significant uncertainties in these assumptions for calculating T and S values, the parameters were subjected to a Monte Carlo analysis as described in the following.

Each of the data sets was prepared to match a global equal-area projection with a grid size of 1 km × 1 km, and the calculations of the data sets were performed in ArcGIS. To avoid mathematical problems, for zero values of d and R, 1 and 0.00001 were added, respectively. For WTR estimates, regions where contemporary groundwater recharge was estimated as <5 mm yr−1 (ref. 30) were excluded from the analysis due to the increasingly large relative uncertainties in recharge below this range, and the resulting unrealistic sensitivity of the resulting WTR estimates. For deriving the frequency distributions and comparisons of parameters from the range of derived geospatial data sets, point values were taken from each raster of interest for 10,000 randomly distributed locations across the Earth’s land surface. Global distributions of the parameters d, K and S are provided in Supplementary Fig. 10 and relationships between R and L, d and WTR, and R and WTR are explored in Supplementary Figs. 9 and 11. All areal calculations ignore the Antarctic landmass.

Although we have made best use of coherent available global data sets at high (1 km) resolution for the calculations, our results are intended for appropriate large-scale interpretation, not detailed local analysis.

### Justification of the model assumptions

Our calculations are based on mapped surface lithology only and, as such, they represent a first estimate of the response of unconfined groundwater across the global land surface. The more complex responses of regional or local confined aquifers, which may be locally important to discerning groundwater–climate interactions, are not considered. However, such confined aquifers only cover around 6–20% of the Earth’s surface44, are often located in more arid parts of the world and are, by definition, inherently less connected to the land surface and climate-related processes.

Using 1D analytical solutions to the groundwater flow equations gives a powerful advantage over the use of more complex models in enabling the sensitivity of the key parameters controlling patterns and timescales of climate–groundwater interactions to be analysed analytically. This, for example, allows us to sample the entire parameter space directly rather than a restricted subset via a limited ensemble of more computationally expensive numerical model runs. Equation (1) assumes the validity of the Dupuit–Forchheimer approximation whereby the water table is assumed to be a true free surface governed by effective hydraulic parameters and that water pressure in the direction normal to the flow is approximately hydrostatic. This is a good approximation when the ratio of the lateral extent of the average saturated depth is more than approximately five times its depth12, that is, H/(L/2) < 0.2 (Supplementary Fig. 13). Calculating the maximum saturated depth hmax as the smaller of d + b or h0, and approximating the average saturated depth as (hmax + b)/2, we find that the criterion H/(L/2) < 0.2 is met in 96% of our global grid calculations. Locations that fail this test are all in mountainous regions where equation (1) cannot account accurately for steep hillslope groundwater hydraulics and hence our results may be less reliable in such areas.

The GRT is a parameter that consistently appears in solutions to the groundwater flow equations and has been used for decades32 as a robust estimate for the timescale of re-equilibration of a groundwater system following a change in boundary conditions8,15,29,31,32,33,34,35,43,45,46,47,48. Thus it is an appropriate metric for long-term transience, which is currently impossible to model in state-of-the-art coupled groundwater–surface water models, which are limited to short run times even for regional-scale analyses due to their massive computational demands. More realistic aquifer geometries and initial water table configurations lead to behaviours that are more complex than the case of a simple exponential decay46, and non-uniform flow fields (strong convergence or divergence) can also lead to variations in GRT (refs. 45,47,48). We have therefore included these factors in an uncertainty analysis as outlined below.

While the models used here cannot represent the detailed process interactions in the way that a distributed fully coupled 3D model would, they have a strong theoretical basis and show consistency with other large-scale studies based on very different model assumptions and data sets. Justification for the approach of using WTR as a proxy for the mode of climate–groundwater interaction is given in at least four ways. First, at a global scale, similarities of WTR to shallow WTD globally9 are strong (Supplementary Fig. 3), given the very different model assumptions and data sets employed in the two studies. Second, at a continental scale for the contiguous USA, a recent study compared the results of a physically based, 3D, fully coupled surface water–groundwater model validated against water table depth data, against the WTR metric41. The results show scatter as expected due to variations in the derivation of the comparative characteristic length scales used in the comparison. However, general trends and geographic patterns at a regional scale compare well for the WTD computed by the fully coupled model and the calculated WTRs. Third, also at a continental scale for the contiguous USA, a systematic relationship has been shown between WTR and mean stream junction angles, indicative of a strong coupling between the surface and subsurface49. Finally, comparisons of WTR calculations against a more complex 3D regional groundwater flow model have indicated that the WTR is a robust indicator of groundwater’s connection to the land surface as it is a strong predictor of the propensity for local versus regional flow conditions50. Our analyses thus allow us to make a robust first global scale estimate of the sensitivity of climate–groundwater interactions, while enabling the range of uncertainty to be fully and directly appreciated.

### Uncertainties and Monte Carlo Experiments

We ran 10,000 Monte Carlo Experiments (MCEs) at 10,000 randomly distributed locations across the Earth’s land surface to investigate the range of uncertainty due to parameter uncertainties as well as model structural simplifications.

Hydraulic conductivity (K) was allowed to vary log-normally within the uncertainty ranges defined in refs. 40,51, this parameter having by far the highest uncertainty of any used in our calculations51. Groundwater recharge (R) values were taken from ref. 30 but allowed to vary through a normal distribution with a standard deviation of 22% of this baseline, chosen according to the difference with a contrasting global recharge distribution52,53 commonly used in other global hydrological calculations. Storativity (S) was sampled from a normal distribution with standard deviations of 25% of the mapped value53. Although the absolute error in the digital elevation model (DEM) used is only 1–2 m, we allowed the maximum terrain rise (d) to vary normally with a standard deviation of 10% to allow for uncertainties due to gridding. The minimum saturated thickness of the aquifer (b) was allowed to vary log-normally around 100 m with a standard deviation of 0.3 orders of magnitude. Sampled distributions were cut off at zero to stop meaningless negatives being included in the calculations.

Parameter uncertainty in the distance between perennial streams (L), calculated from the variation in L for an order of magnitude change in discharge threshold used to define the stream network (from 0.1 to 1 m3 s−1), gives a median uncertainty of a factor of 1.9. However, there is additional uncertainty in L due to the choice of the 1D groundwater flow solutions applied, which ignore non-uniform (that is, convergent or divergent) flow fields, which are common in real catchments. To account for the maximum likely range of possible uncertainty, we compared the 1D analytical solutions used here to cases of radial flow, which represent an extreme 2D non-uniform flow end-member for natural groundwater flow systems. By considering the distance between perennial streams (L) to be equal to the radius of the flow domain for the equivalent radial solutions, we can estimate the impact of this choice on both WTR and GRT. For WTR, by replacing equation (6) with equation (30.11) from ref. 54, the average error is approximately a factor of 2. For the GRT, comparison of recession timescales for 1D and radial flow cases (for example, appendix A of ref. 46) indicates a similar level of uncertainty due to non-uniform flow as for the WTR. We therefore added a log-normal variation in L with a standard deviation of 0.3 orders of magnitude to accommodate the likely range of combined parameter and structural uncertainty.

## Data availability

Digital data sets of the main geomatic results for the water table ratio and groundwater response times maps are freely available for download as geotiffs from https://doi.org/10.6084/m9.figshare.7393304.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

The authors acknowledge funding for an Independent Research Fellowship from the UK Natural Environment Research Council (NE/P017819/1) (to M.O.C.); the German Science Foundation DFG (Cluster of Excellence ‘CliSAP’, EXC177, Universität Hamburg) and Bundesministerium für Bildung und Forschung Project PALMOD (ref. 01LP1506C) (to J.H.); the German Federal Ministry of Education and Research (BMBF) (grant no. 01LN1307A) (to N.M.); the Agence Nationale de la Recherche (ANR grant ANR-14-CE01-00181-01) and the French national programme LEFE/INSU (to A.S.); and the Natural Sciences and Engineering Research Council of Canada (NSERC) (Discovery grant RGPIN/341992) (to B.L.).

## Author information

### Affiliations

1. #### School of Earth and Ocean Sciences, Cardiff University, Cardiff, UK

• M. O. Cuthbert
2. #### Water Research Institute, Cardiff University, Cardiff, UK

• M. O. Cuthbert
3. #### Connected Waters Initiative Research Centre, UNSW Sydney, Sydney, New South Wales, Australia

• M. O. Cuthbert

• T. Gleeson

• T. Gleeson
6. #### Leibniz Centre for Tropical Marine Research (ZMT), Bremen, Germany

• N. Moosdorf
7. #### Department of Civil and Architectural Engineering, University of Wyoming, Laramie, WY, USA

• K. M. Befus
8. #### Sorbonne Université, Milieux environnementaux, transferts et interactions dans les hydrosystèmes et les sols, METIS, Paris, France

• A. Schneider
9. #### Institute for Geology, Center for Earth System Research and Sustainability (CEN), Universität Hamburg, Hamburg, Germany

• J. Hartmann

• B. Lehner

### Contributions

The idea for the paper was conceived by M.O.C. and T.G. Analyses were carried out by all authors. The manuscript was written by M.O.C. with input from all authors.

### Corresponding author

Correspondence to M. O. Cuthbert.

## Supplementary information

1. ### Supplementary Information

Supplementary Figures 1–13, Supplementary References.