Comments on: Thinking on Optimal Theories in Hydrology

I think the comments made by Christian Massari on yesterday post require some deepening.

He wrote:

C - .. I have some comments from your text….

C - I went trough it and I enjoyed a lot the reading. The patterns like perspective is something that is there, closer to reality than the classical Eulerian approach where objects are divided in elements and physical quantities are attempted to be described for a number of discretized points (with point-based physical laws ) of what you called “shapes”. Is this approach really valid?
R - The question is well posed. To really answer to this, we should be able to build a “statistical mechanics” out of the finer scales to obtain the behavior at the larger scales.
This is, for instance, the case of Richards equation which is, in its essence, mass conservation plus a hypothesis about pore filling and emptying, plus an intrinsic assumption about the randomness of the medium (pore dimensions are connected randomly). These hypotheses leaves behind macropores and preferential flow (I know how to include them, however) but produces a working equation at the Darcy scale. It is less clear for other compartments of the hydrological cycle.
If we have a catchment, the traditional lumped approach is to consider it composed by hydrologic response units (HRUs) that then interacts to get the whole catchment rules (the big picture, often called semi-distributed). To my knowledge, the model Topkapi (Liu and Todini, 2002) are obtained by integration of the smaller scale hydrology, Topmodel (Beven and Kirkby, 1979, Beven and Freer, 2001) is another type of aggregation (related to the production of the runoff by saturation excess), the Geomorphic Instantaneous Unit Hydrograph (GIUH, Rodriguez Iturbe and Valdez, 1979; Rigon et al, 2016) a way to aggregate HRUs using travel time concepts. (I wrote about this here). All of them are types of aggregation of fluxes driven by a scope (getting the saturated area, getting the discharge, a.k.a the hydrologic response) and it is not clear if they can be aggregated when the goal is wider (for instance getting all the fluxes and the energy budget).
Gray (1982) and coworkers, (references later) envisioned a method of integration over space and time to make emerge laws bottom up that were subsequently popularised by the work by Reggiani et al. (1999, 2003). However, their work is based on the naive assumption that the topology of the interactions is irrelevant. Topology of interactions is instead fundamental to get the right fluxes at the large scale and it is explicit (and simplified) in both Richards and GIUH (but, as we said, at the price to let something out). So we should envision a way to built HRUs and make them to interact properly. This is also when known in physics, where the process of aggregation implies the “renormalization” of interactions. Kenneth Wilson got the Nobel prize for getting a clue of it. For renormalization working, however, the system must have certain properties of scale invariance, in which the form of the equation remain invariant but the coefficients change of magnitude. Notably Richards equation is almost scale invariant (e.g Sposito, 1997) and we are now able to verify it numerically. Most of the systems we deal with are, however, not self-similar and equations must change when changing scale.
In this context a guiding principle could be to search for emerging conservation laws (e.g. Baez et al., 2018), from which extract budgets' equations.
At present we can only make hypotheses, and, classically, think HRUs as reservoirs connected by empirical laws of fluxes, whose behavior can be tested in various ways, including the use of tracers. It is the practical trick that many of us use with some satisfaction (maybe the more clear statements about this approach are in the work by Fabrizio Fenicia: see Fenicia et al. (2016) for an example). On this type of models we wrote a paper, yesterday accepted in WRR. In this paper we deal with the representation of the models, but in reality representation methods reveal the assembly of compartmental models and a give clear suggestions on how to obtain travel times equations (the topic of an incoming paper). However the answer to your question is: at present we do not really know.
C - Is the whole system the simple sum of the the parts?
R - I said before, talking about Reggiani et al. that no, this is not the case. The topology of interactions counts.
C - Maybe this is true as we move to the microscopic scale but even at this scale things are organized in shapes (i.e., molecules). So it seems a scale dependent problem. So we have the chance for every system or compartment to use the “right" scale to describe processes and properties as patterns.
R -True
C -At that scale there is an undoubtedly existing organizing/optimality principle that is able to make that shape recognizable and distinguished from the rest, a complex system behaving as whole and having certain relation with the rest.
R -Right

C - You talked about three main processes/systems like turbulence, water flow in vegetation and river network organization but of course we could identify others which are not only related to water movement but also for instance to soil properties, meteo forcing organization (e.g. rainfall and temperature and humidity patterns induced by landscape) and so on…
R -Right

C - This could a be direction we could take for example by looking at pattern like modelling (Grimm et al. 2005).
R - Pattern based dynamics is a successful story that works with individuals interacting by a rule. This is not very much different to our lumped model, except for the fact that these interactions can be at discrete times. However, they are not essentially different. The issue remains to get the law of interaction right (or approximately right). These systems, as we did with systems of ordinary differential equations are representable by Petri nets, on graphs, and topological methods are available to get some clue of their collective behavior.
C - I am honest I am not in the topic so I need to study it but it seems something interesting. How we could join our forces?
Because we do not have the machinery (yet, but probably forever) to perform renormalisations, we need to deduce empirically both the patterns and the fluxes among patterns at the aggregated scale. To this scope smart use of remote sensing is essential. Eventually we could be able to do an operation of reverse engineering and understand how to deduce them from basic (finer scale laws).
We, observers, have the task of identifying these patterns or shapes based on observations (ground and remote sensing) also by unconventional ways (i.e. Clemens), you modelers, will have the task of translating in mathematical ways the the existence of these shapes and patterns as well as the relation between them (which are likely again the results of optimality principles).
R - Right.

References

• Baez, J. C., Lorand, J., Pollard, B. S., & Sarazola, M. (2018). Biochemical Coupling Through Emergent Conservation Laws. arXiv.org, 1–13.
• Beven,K., and M. J. Kirkby (1979), A physically based, variable contributing area model of basin hydrology, Hydrol. Sci. Bull.,24, 43-69
• Beven, K., & Freer, J. (2001). A dynamic TOPMODEL. Hydrological Processes, 15(10), 1993–2011. http://doi.org/10.1002/hyp.252
• Fenicia, F., Kavetski, D., Savenije, H. H. G., & Pfister, L. (2016). From spatially variable streamflow to distributed hydrological models: Analysis of key modeling decisions. Water Resources Research, 52(2), 954–989. http://doi.org/10.1002/2015WR017398
• Gray WG. Constitutive theory for vertically averaged equations describing steam-water ̄ow in porous media. Water Resour Res 1982;18(6):1501±1510.
• Grimm, V., Revilla, E., Berger, U., Jeltsch, F., Mooij, W. M., Railsback, S. F., et al. (2005). Pattern-Oriented Modeling of Agent-Based. Science, 310, 987–992.
• Z. Liu, E. Todini. Towards a comprehensive physically-based rainfall-runoff model. Hydrology and Earth System Sciences Discussions, European Geosciences Union, 2002, 6 (5), pp.859-881.
• Reggiani, P., Hassanizadeh, S. M., Sivapalan, M., & Gray, W. G. (1999). A unifying framework for watershed thermodynamics: constitutive relationships. Advances in Water Resources, 23(1), 15–39. http://doi.org/10.1016/S0309-1708(99)00005-6
• Reggiani, P., & Schellekens, J. (2003). Modelling of hydrological responses: the representative elementary watershed approach as an alternative blueprint for watershed modelling. Hydrological Processes, 17(18), 3785–3789. http://doi.org/10.1002/hyp.5167
• Rigon, R., Bancheri, M., Formetta, G., & de Lavenne, A. (2015). The geomorphological unit hydrograph from a historical-critical perspective. Earth Surface Processes and Landforms, 41(1), 27–37. http://doi.org/10.1002/esp.3855
• Rodríguez-Iturbe I, Valdés JB. 1979. The geomorphologic structure of hydrologic response. Water Resources Research 15(6): 1409–1420.
• Sposito, G. (1997). Scaling Invariance and the Richards Equation (pp. 1–23), in G. Sposito (Ed. Scale dependence and scale invariance in hydrology, Cambridge University Press

Thinking on Optimal Theories in Hydrology

In Nature we have to deal with forms that we somewhat recognize and distinguish from the rest (Thom, R., 1975). These forms (shapes), as we know since D'Arcy Thomson (1917, see also Ball, 2013), have functionalities that are shaped by some dynamics that we struggle since then (and maybe before) to find and understand. Because forms and functional forms are so ubiquitous we are brought to think that there is some design to produce them (Zanella,  G. Sopra una conchiglia fossile nel mio studio, About a Fossil shell in my room) but this is from the science point of view an error of perspective (Monod, J., 1970) and an undecidable question. K. Lorenz, on the other side warns often that evolution does not produce always functional forms (or behaviours) but let grow also unnecessary “neutral” stuff which is not useful nor a handicap. The discussion can grow very general and philosophical, and pursuing it could be the topic of another post.

We are interested here to grasp the grow of forms and patterns by means of methods that are proper of mathematics, physics and chemistry (on the illuminating example of E. Schroedinger, What is life ?, 1944). It is to be remarked here that to appeal to hard sciences does not mean a priori a reductionist approach, in which the systems are pruned apart and loose their quality. This is especially true for living systems but also for complex Earth science cycles as the hydrological one, and we aim to keep the systems and their dynamics together, emphasizing the interactions that makes forms to emerge at various scales. 

The general but qualitative understanding of the physics and mathematics related to these problems, is that, deprived of its teleology to be investigated elsewhere and eventually, behind forms and patterns there is some “optimality principle” or, stated in other words, that Newtonian mechanics, physics and chemistry of complex systems evolve solutions that require spatio-temporal structures and patterns. The creation of “forms” is extended not only to the immediately appearance that we perceive with our (highly biased by evolution, though) senses, but are minimal or maximal solutions when observed from the point of view of a certain variable. Unfortunately we are not really able to move, based on the present knowledge, from the basic principles to the appropriate laws of structures interactions, just through derivation of statistical law or integration over degrees of freedom.

The theories of optimality assume among the driving forces moving the dynamics of a system in Nature there are some extensive quantities like, for instance, entropy that are pushed to increase. For instance entropy, because,  entropy of a closed thermodynamic system is shown to grow. In fact this is a principle of Equilibrium Thermodynamics. However,  in Non Equilibrium Thermodynamics  it can be derived, at least  for some simple systems, i.e. is not anymore a principle,  (e.g. see the Thermodynamics derivation of evaporation in Monsoon and Baldocchi, 2014).

Moreover, in open systems, entropy of a subsystem can decrease, so in such context it cannot be used to understand the asymptotic state of the system. Fortunately,  the rate of production of entropy could. As a matter of fact, in non-equilibrium, non-autonomous, open systems, asymptotic states could not be so relevant, and instead what matters are “intermediate asymptotics” as pointed out for fluid dynamics by Barenblatt, 1996 and, before Baranblatt, by Ilya Prigogine work on dissipative structures, i.e. structures that are identified because, they represents steady (or at least somewhat persisting) states of a (thermo)dynamic system out of equilibrium.

It can happen, for instance, that intermediate asymptotics are obtained by minimizing energy dissipation, i.e. the quantity of energy that is transformed into heat (or non-usable energy). This is the case of river networks (Rodriguez-Iturbe et al., 1992) and implies that the maximum entropy of the comprehensive system is obtained as slowly as possible, meaning that the available energy is used at its best for producing work.

This behavior seems logically correct for living systems, and seems justified by evolution and selection: the most efficient survives and reproduces (Virgo, 2001) but is intriguing the fact that river networks are not living systems and anyway they obey such a type law.  Therefore it seems implied that some general dynamic law is the origin of all (e.g. Prigogine, 1945). The mathematical problem is all but trivial, and such minimization (or maximization) problems has received recently the attention of the Field prize (e.g. see the work of Alessio Figalli), but, forgive me if I dare to say that is simpler than the physical problem to understand why certain equations that bring to optimisation problems are valid. Anyway, optimisation problems are mathematically obtained by expressing the large scale dynamics of the (river) networks as a functional to be optimised (actually expressed in Rodriguez-Iturbe et al work in discrete form). In optimal channel network (by Rodriguez-Iturbe et al, 1992), this functional is, obtained by using the basic Newton law (in appropriate form) and some additional hypothesis, derived from observations or educated guesses. One would like to obtain it without heuristics but this seems out of our present possibilities notwithstanding the large literature on optimal transportation networks (e.g Barabasi, Network science, 2018). It should be noted, however, that the case of networks is produced by some twofold optimality: a tradeoff between maintaining an optimal transport and optimally maintaining the the structure that conveys optimally the transported.

An epitome of structure analysis is also the Navier-Stokes equation and turbulence. Here we have the formation of structures that dissipate kinetic energy, the vortexes, actually of all dimension from the scale of observation to the dissipation scale.
We recognize these patterns as preferential flow of energy, or, in the case of turbulence as a form of quasi-random disorganization whose structure is particularly evident in the 4/3 Kolmogorov law. In this case, the equation is directly the Newton law (plus Newton’s hypothesis on stresses), so, despite the complexity of the outcomes, the physics is very directly reduced to mathematics which, BTW, in this case is unable to completely solve the problem.

A third piece of the elephant is the water flow in vegetation. It happens to maintain the temperature of the plant system in a range acceptable for plants to comfortably survive subject to weather and climate forcings and at the same time, not secondarily or maybe primarily, fix $CO_2$ to build their structure and carbohydrates trough photosynthesis. So plants need to optimise their fit to varying weathers for maximizing their production. A plant can be decomposed in its main structural parts: root, steams, leaves and their functional counterparts, xylem and phloem (e.g. Stroock et al., 2014), each part can be disassembled back to the singles cells and chloroplasts. But after the reductionist operation, plants overall functioning remain partially elusive and resistant to quantitative treatment if we do not treat plant’s part as a system (with a lot of osmoregulatory subsystems, e.g. Perri et al., 2019) and ecosystems where plants of various species interacts in competition, cooperation and coopetition for light, water, nitrogen, or phosphorus. The structure of plant and ecosystems and their interactions, evidently does not violate basic physical laws, their functioning respect mass and energy conservation, and momentum (of water, for instance) is peculiarly dissipated to obtain the scope of water supply to very high height and very negative pressures (up to -30 MPa). Optimization here involves various aspects, including the scaling of xylem dimensions (e.g. Olson et al., 2014), to obtain optimal sapping performances. Besides, recent work by Hildebrandt et al. (2016) shows evidence of optimal use of water when the energy budget is properly accounted for.

Soils are not a passive medium, first because themselves contain a lot of aggregated microbic biota (so far mainly neglected in hydrological analyses) and secondly because it is the environment where soil-roots interact. Also in soil, even according to more traditional views, there are optimisation processes when, during evaporation, the rate of water uptake is maintained constant up to critical soil water content (stage I evaporation) after which, evaporation strongly decreases (stage II evaporation). This is provided by a series of feedbacks among small and large soil pores, viscous forces and cohesion processes still not well understood (but kind of well described in Lehman et al., 2008).

Hydrology in the critical zone (CZ, the elephant) is therefore overwhelming difficult because it is the compendium of optimisation processes regulated by networks, vegetation, NS equations and water flowing in soil. According to what we focus on, we can isolate various non trivial dynamics. However the challenge is to model their comprehensive interplay for which still we do not have appropriate mathematics, observations and tools.

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References

• Ball, Philip (7 February 2013). "In retrospect: On Growth and Form". Nature. 494 (7435): 32–33. doi:10.1038/494032a.
• Barabási, Albert-László (2018). Network science. Cambridge University Press. ISBN 978-1107076266.
• Barenblatt, G.I. (1996), Scaling, self-similarity, and intermediate asymptotics, Cambridge University Press, 1996
• D’Arcy W. Thomson (2017), On growth and forms, Cambridge university Press
• Hildebrandt, A., Kleidon, A., & Bechmann, M. (2016). A thermodynamic formulation of root water uptake. Hydrology and Earth System Sciences, 20(8), 3441–3454. http://doi.org/10.5194/hess-20-3441-2016
• Lehmann, P., Assouline, S., & Or, D. (2008). Characteristic lengths affecting evaporative drying of porous media. Physical Review E, 77(5), 354–16. http://doi.org/10.1103/PhysRevE.77.056309
• Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology by Jacques Monod, New York, Alfred A. Knopf, 1971, ISBN 0-394-46615-2
• Olson M.E., AnfodilloT., Rosell J.A., Petit G., Crivellaro A., Isnard S., León-Gómez C., Aalvarado CardenasL.O., Castorena M. (2014). Universal hydraulics of the flowering plants: Vessel diameter scales with stem length across angiosperm lineages, habits and climates. Ecology Letters 17 (8), 988–997.
• Perri, S., Katul, G. G., & Molini, A. (2019). Xylem‐ phloem hydraulic coupling explains multiple osmoregulatory responses to salt‐stress. New Phytologist, nph.16072–51. http://doi.org/10.1111/nph.16072
• Prigogine, Ilya (1945). "Modération et transformations irréversibles des systèmes ouverts". Bulletin de la Classe des Sciences, Académie Royale de Belgique. 31: 600–606
• Rodriguez-Iturbe I. , Rinaldo R., R. Rigon, Bras R.L., Marani A. and Ijjasz- Vasquez E.J. (1992), Energy dissipation, runoff production, and the 3-dimensional structure of river basin, Water Resources Research, (28)4, 1095-1103.
• Schroedinger, E. (1944), What is Life ?, Cambridge University Press
• Stroock, A. D., Pagay, V. V., Zwieniecki, M. A., & Michele Holbrook, N. (2014). The Physicochemical Hydrodynamics of Vascular Plants. Annu. Rev. Fluid Mech., 46(1), 615–642. http://doi.org/10.1146/annurev-fluid-010313-141411
• Thom, R. (1975), Structural stability and morphogenesis, An Outline of a General Theory of Models, Addison-Wesley
• Virgo, N. (2011, March 23). Thermodynamics and structure of Living Systems, Ph.D. dissertation, University of Sussex

Growth and Form - Morphogenesis

In my studies about the growth and form of river networks, I actually also touch several other works on the same subject and in other disciplines. I would not say that they are unrelated (see, for instance the Banavar et al, PNAS paper, where the very same principles we apply to river networks are used to explain the Kleiber's law).

Anyway some of these readings are summarised in web page of the Imperial College dedicated to morphogenesis of plants, that I reproduce here verbatin:

"The word morphogenesis is a composition of the Greek words morphê (shape) and genesis (creation) to indicate the process of generation of a particular shape. Morphology, the science of forms, was first made esplicit by Johann Wolfgang von Goethe (1749-1832) in his botanical writings “Zur Morphologie“ (1817). The classic work by D'Arcy Thompson (1860-1948), "On Growth and Form" (1917), completely embraced the idea of a quantitative study of the way forms are created (morphogenesis) in living organisms. The third major contribution towards a refinment in the analysis of morphogenesis is due to Alan Turing in his seminal paper "The Chemical Basis of Morphogenesis" (1952), where a family of mathematical models (reaction-diffusion models) is proposed to describe examples of pattern formation during morphogenesis in living organisms.

  When studying morphogenesis, it is important to define a method to measure forms (morphometrics) and the transformation of forms in time (morphodynamics). "