James W. Kirchner

The science of hydrology is on the threshold of major advances, driven by new hydrologic measurements, new methods for analyzing hydrologic data, and new approaches to modeling hydrologic systems. Here I suggest several promising directions forward, including (1) designing new data networks, field observations, and field experiments, with explicit recognition of the spatial and temporal heterogeneity of hydrologic processes, (2) replacing linear, additive “black box” models with “gray box” approaches that better capture the nonlinear and non‐additive character of hydrologic systems, (3) developing physically based governing equations for hydrologic behavior at the catchment or hillslope scale, recognizing that they may look different from the equations that describe the small‐scale physics, (4) developing models that are minimally parameterized and therefore stand some chance of failing the tests that they are subjected to, and (5) developing ways to test models more comprehensively and incisively. I argue that scientific progress will mostly be achieved through the collision of theory and data, rather than through increasingly elaborate and parameter‐rich models that may succeed as mathematical marionettes, dancing to match the calibration data even if their underlying premises are unrealistic. Thus advancing the science of hydrology will require not only developing theories that get the right answers but also testing whether they get the right answers for the right reasons.

This paper is the outcome of a community initiative to identify major unsolved scientific problems in hydrology motivated by a need for stronger harmonisation of research efforts. The procedure involved a public consultation through online media, followed by two workshops through which a large number of potential science questions were collated, prioritised, and synthesised. In spite of the diversity of the participants (230 scientists in total), the process revealed much about community priorities and the state of our science: a preference for continuity in research questions rather than radical departures or redirections from past and current work. Questions remain focused on the process-based understanding of hydrological variability and causality at all space and time scales. Increased attention to environmental change drives a new emphasis on understanding how change propagates across interfaces within the hydrological system and across disciplinary boundaries. In particular, the expansion of the human footprint raises a new set of questions related to human interactions with nature and water cycle feedbacks in the context of complex water management problems. We hope that this reflection and synthesis of the 23 unsolved problems in hydrology will help guide research efforts for some years to come.

Water fluxes in catchments are controlled by physical processes and material properties that are complex, heterogeneous, and poorly characterized by direct measurement. As a result, parsimonious theories of catchment hydrology remain elusive. Here I describe how one class of catchments (those in which discharge is determined by the volume of water in storage) can be characterized as simple first‐order nonlinear dynamical systems, and I show that the form of their governing equations can be inferred directly from measurements of streamflow fluctuations. I illustrate this approach using data from the headwaters of the Severn and Wye rivers at Plynlimon in mid‐Wales. This approach leads to quantitative estimates of catchment dynamic storage, recession time scales, and sensitivity to antecedent moisture, suggesting that it is useful for catchment characterization. It also yields a first‐order nonlinear differential equation that can be used to directly simulate the streamflow hydrograph from precipitation and evapotranspiration time series. This single‐equation rainfall‐runoff model predicts streamflow at Plynlimon as accurately as other models that are much more highly parameterized. It can also be analytically inverted; thus, it can be used to “do hydrology backward,” that is, to infer time series of whole‐catchment precipitation directly from fluctuations in streamflow. At Plynlimon, precipitation rates inferred from streamflow fluctuations agree with rain gauge measurements as closely as two rain gauges in each catchment agree with each other. These inferred precipitation rates are not calibrated to precipitation measurements in any way, making them a strong test of the underlying theory. The same approach can be used to estimate whole‐catchment evapotranspiration rates during rainless periods. At Plynlimon, evapotranspiration rates inferred from streamflow fluctuations exhibit seasonal and diurnal cycles that agree semiquantitatively with Penman‐Monteith estimates. Thus, streamflow hydrographs may be useful for reconstructing precipitation and evapotranspiration records where direct measurements are unavailable, unreliable, or unrepresentative at the scale of the landscape.