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?

C - Is the whole system the simple sum of the the parts?

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.

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.

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…

C - This could a be direction we could take for example by looking at pattern like modelling (Grimm et al. 2005).

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 - 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.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

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