By rigorously accounting for dimensional homogeneity in physical laws, the

Galileo is credited as the first scientist to have used dimensional analysis and scaling. In his 1638 “Dialogues Concerning Two New Sciences”

Looking at some of the most striking applications of the

The presence of emerging scaling laws in geophysics has been widely recognized for a long time, and the related power laws have been observed in rainfall and streamflow statistics, in landscape and river-network geometry, as well as in the aggregation properties of soils and aquifers (among many others, see, e.g.,

In this paper, we revisit a series of fundamental problems in hydrology and geomorphology and scrutinize them under the common lens of dimensional analysis with the goal of sharpening our intuition of the underlying physical processes. After a brief review of the concepts of dimensional analysis and scaling in Sect. 2, in Sect. 3 we consider three different instances of the partitioning of water and soils in natural landscapes. Because of their complexity, availability of global data, and the lack of governing equations from first principles, these phenomena provide a particularly fertile test bed for dimensional techniques.

Dimensional analysis is a chapter of the more general group theory

On the theoretical front, the

The word scaling, in the sense used here, refers to the existence of a property that allows one to go from one scale to another (upscaling or downscaling) using the same mathematical law and, therefore, it is related to the absence of a preferred scale or unit of measure. From a mathematical viewpoint, this property is linked to the fact that the solutions of dimensional physical problems are generalized homogeneous functions

The defining property of a homogeneous function of degree

For the purposes of this article, it may be useful to note how in mathematics one almost always assumes to deal with dimensionless quantities. For example, in Eqs. (1) and (2), the argument of the function

As we will see, scaling laws appear naturally in the application of the

The starting point of a dimensional analysis is the formulation of a “physical law”,

With Eq. (

The number

Enforcing dimensional homogeneity in Eq. (

A comparison of Eqs. (

Depending on the governing variables involved in the physical law, there may be freedom in choosing the

In summary, starting from a physically meaningful law (1), the

When one or more of the

Thus, assuming for example self-similarity with respect to the group

Incomplete self-similarity seems to be more rare and is often difficult to distinguish from the case of complete self-similarity, especially for experimental or numerical problems where there are transitions to different regimes

It is not infrequent in the literature to come across a point of view, made explicit by

Compared to the aforementioned (Sect.

As the reader may imagine, this line of reasoning has attracted both stern skepticism and enthusiastic support, leading to interesting debates and controversies, with defenders of rigor and objectivity on one side and advocates of a flexible approach on the other. In the writer's experience, the subject is always a source of interesting discussion, if not else because, when one writes in favor of it, the editors somehow always manage to select a reviewer who belongs to the skeptical camp.

The most emblematic case of controversy is perhaps the well-known Rayleigh–Riabouchinsky controversy

The resolution of this controversy (see Appendix A for more details) shows that it is not a matter of the arbitrariness of which dimensions are considered but of the level of description intended for a problem. Whether to include the specifics of the molecular motion depends on the size of the considered object. If one, following Rayleigh, accepts a thermodynamic approach, then the details of the disorderly molecular energy are irrelevant and temperature can be treated as an independent quantity compared to the kinetic energy of the mean motion. Formally, choosing a greater number of fundamental units (i.e., the number of primary dimensions) is made possible by the addition of corresponding dimensional unifiers

Of a similar nature, and perhaps even more subtle and controversial, is directional dimensional analysis, which is based on the fact that in some cases distinguishing between vertical and horizontal dimensions provides more informative results (i.e., fewer dimensionless groups).

Directional dimensional analysis and its generalizations have been used successfully in a variety of problems, including applications in mechanics and atmospheric sciences

Some of the most important questions of terrestrial geophysics are related to the partitioning of water and soil minerals at the land surface. Figure

The three partitionings analyzed in this paper using dimensional analysis: the rainfall partitioning taking place at the land surface, the soil mineral partitioning by chemical dissolution (weathering) and transport processes, and the related partitioning of soil sediments responsible for landscape evolution and the formation of drainage networks.

The scaling laws obtained from dimensional analysis shed light on the dominant soil, vegetation and climate controls for these hydrologic partitionings. Specifically, in the first application it will be seen how the dimensionless dryness and storage indices determine the long-term rainfall partitioning. The second application will reveal the key nonlinear control of the dryness index on weathering rates, while the third application will analyze new macroscopic relations among average variables' landscape evolution, uncovering an intriguing analogy between self-similar topographies and turbulent flow fields.

The rainfall reaching the soil surface is either lost by runoff or infiltrates into the soil, where in turn it is lost either by evapotranspiration or percolation. The fate of this partitioning is essentially controlled by the properties of the soil–plant system, which – in a sense – acts as a geophysical valve, not only for the entire hydrologic cycle, but also for the energy and carbon cycles. This fundamental hydrological problem presents robust behaviors for its macroscopic (i.e., averaged) patterns as well as rich and complex controls at the detail level, where the role of temporal fluctuations and spatial heterogeneities becomes important.

We will indicate the mean rainfall rate as

While they did not use dimensional analysis, Turc and Budyko started their work by making what is perhaps the simplest hypothesis of a physical law for the rainfall partitioning

If one chooses

Budyko's hypothesis is tested in Fig.

Rainfall partitioning.

While the dryness index captures the main variability of the data, the scatter in Fig.

To account for this effect, we may add a quantity

Thus, to achieve a further refinement in the Pi-theorem formulation, in combination with the storage depth

These five hydrologic spaces provide different, if related, scaling laws that allow us to focus on the role of specific combinations of parameters and emphasize different hydrologic conditions. It is useful to note that the dimensionless group

Transient trajectories of the time-varying ratio

With the goal of analyzing the suitability of different hydrologic spaces for capturing the information available in global datasets on hydrologic partitioning,

The analysis by

It is logical to wonder about the effects of adding other potentially important variables to the physical law of rainfall partitioning. These could include variables describing seasonality, soil and vegetation properties, and other details of the climatic forcing. With the goal of investigating the role of seasonality in rainfall and evapotranspiration,

It is clear that more detailed analyses to disentangle the role of different ecohydrological variables in the rainfall partitioning should focus on specific aspects of the space–time variability of evapotranspiration, which instead are lost in the spatially lumped, long-term evapotranspiration rates of Budyko's type analyses. The combination with simple physically based models may be a valuable way to sharpen the hypotheses of the physical laws, especially when trying to unravel the effects of the covariation of some of the variables. These regard questions of how the rooting depth, and thus the storage capacity of the active soil layer, may depend on the dryness index and whether such covariations may imply some adaptation of vegetation and soil properties to the hydroclimatic characteristics. Along these lines, the minimalist stochastic model developed by

Shedding light on the role of higher-order controls in rainfall partitioning requires charting of more detailed model–data investigations, branching off the beaten path of Budyko-type analyses. Long-term averages may only contain a weak signal of those interactions, which may be easily overwhelmed by noise and other data limitations. This suggests a need for adapting dimensional analysis to these other aspects of the rainfall partitioning and formulating more sophisticated physical laws that capture the more subtle controls by the soil–plant–atmosphere system (see

A partitioning of the soil mineral component also takes place, on much larger timescales, on the land surface

The input of solid material to the soil from breaking up parent rock is called rock denudation. While the dissolved minerals and the very fine particles resulting from denudation can be transported away by water, the remaining particles stay in the ground, where they continue to be weathered until they too can be transported away. As the soil ages and transforms chemically, the soil may also be deformed by the action of internal stresses and move as creeping flow. As a result, if one assumes, for simplicity, that the input and output balance

To analyze this partitioning and help resolve the complex interaction between chemical weathering, climate, and the hydrologic cycle,

To verify this functional relationship,

Both these expressions are suggestive of self-similarity. At a first glance, looking at Eq. (

Equation (

The scatter shown by the data in Fig.

Besides the macroscopic effects of wetness on weathering discussed in the previous section, the loss of minerals driven by the coupled water and sediment fluxes on the topographic surface is also responsible for the formation of complex topographic patterns, which in turn impact ecohydrologic processes

Steady-state simulation of landscape elevation given by Eqs. (

Towards the goal of performing dimensional and self-similarity analyses, here we will consider only (spatially) averaged quantities in idealized geometries for which the solutions may be expected to depend only on a limited number of dimensionless quantities. The effort of formulating meaningful physical laws for average landscape quantities is facilitated by the existence of simplified, semi-empirical landscape evolution models (LEMs)

The coupled Eqs. (

In steady state, spatial averaging leads to the partitioning of soil material into soil creep (SC) and the soil loss by stream erosion (SL),

Averaging spatially Eqs. (

These results suggest a tantalizing analogy with the analysis of wall-bounded turbulent shear flows. In fact, the regular behavior of

It is also useful to note that in

Average sediment partitioning according to Eq. (

A second interesting fact is the plateauing of the curves at large values of

In a system with relatively small diffusive transport and dominated by erosion and uplift,

Mean elevation profiles for increasing values of

As observed by

Our final example follows

Given three dimensions

In the asymptotic limit of relatively high

Figure

From a geomorphological point of view, the connection between the exponent

Finally, from a practical point of view, the spectral scaling of landscape elevation could be profitably utilized in developing efficient numerical simulations of landscape evolution

We have presented several examples of applications of the

If these observations confirm the utility of dimensional analysis, it should also be clear that these methods are not a foolproof set of rules to achieve miraculous solutions. Rather, they are an iterative procedure to sharpen our hypotheses on physical processes. Thinking of dimensional arguments as a form of modeling allows an “explication of the role abstraction and multiple realisability; not as compatibility with other possible worlds but as compatibility with different fictional descriptions of our own world”

In this Appendix we discuss in more detail the Rayleigh–Riabouchinsky controversy related to augmented dimensional analysis. In particular, after a brief historic background, we show how the thermodynamic approach of Rayleigh corresponds to a self-similar solution of the first kind with respect to the dimensionless group obtained from the Boltzmann constant, which serves as the dimensional unifier when augmenting the dimensional analysis from mechanics to thermodynamics.

As mentioned in the main text,

This

In a very short comment published soon after Rayleigh's paper,

It would indeed be a paradox if the further knowledge of the nature of heat afforded by molecular theory put us in a worse position than before in dealing with a particular problem. The solution would seem to be that the Fourier equations embody something as to the nature of heat and temperature which is ignored in the alternative argument

Rayleigh's reply was authoritative and sensible but not completely satisfactory. As pointed out by

Thus, following Rayleigh, we specifically choose length (

Choosing the heat flux

We follow here

The surface water flux

Here we report the equation for the streamwise velocity

No data sets were used in this article.

The contact author has declared that there are no competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Much of this work has been the result of invaluable collaborations with Edoardo Daly, Jun Yin, Xue Feng, Salvatore Calabrese, Sara Bonetti, Milad Hooshyar, and Shashank Anand. The author is also grateful to Paolo D'Odorico, Gaby Katul, Luca Ridolfi, and Ignacio Rodriguez-Iturbe for continued friendship, support, and advice. We thank Stefan Hergarten, Demetris Koutsoyiannis, Lamberto Rondoni, and two anonymous reviewers for very valuable suggestions.

This research has been supported by the US National Science Foundation (NSF) grant nos. EAR-1331846 and EAR-1338694, the BP through the Carbon Mitigation Initiative (CMI) at Princeton University, and the Moore Foundation.

This paper was edited by Nunzio Romano and reviewed by Demetris Koutsoyiannis, Stefan Hergarten, and two anonymous referees.