Linking landscape structure and rainfall runoff behaviour in a thermodynamic optimality context

Abstract:

The fact that persistent spatial organization in catchments exists has inspired many scientists to speculate whether this is the manifestation of an underlying organizing principle (e.g. Dietrich and Perron, 2006). Catchments (open thermodynamic systems in general) may reveal different fingerprints of spatial organization as for instance a spatial correlation in patterns of states and properties (Kondepudi and Prigogine, 1998). In line with this a spatial covariance of soil hydraulic properties reflects spatially organized storage of soil water within a given soil, while a soil catena reflects organized formation of different soil types along the gradient driving lateral water flow. Also the omnipresence of spatially connected preferential flow paths across many scales and compartments alongside with their similarity is often regarded as evidence for spatial organisation (Bejan et al., 2008; Howard, 1990). In this study we investigate whether a thermodynamic perspective might better explain why and better predict how catchment heterogeneity and its spatial organization controls their rainfall runoff behaviour.

To make this connection, a necessary first step is to express hydrologic processes and fluxes in thermodynamic terms. At the very basic level, the second law of thermodynamics tells us that gradients are depleted by the fluxes that are caused by these gradients, no matter if we deal with energy or water (mass) fluxes (e.g., Kleidon et al., 2013). This direction of the second law is the foundation for expressing hydrologic fluxes in the common way as a product of a conductance (or an inverse resistance) and a potential gradient. Gradient fields in hydrologically relevant potentials (soil and plant water potentials, piezometric heads and surface water levels) determine the (thermodynamic) forces driving fluxes of either capillary soil water sustaining evapo-transpiration or free water sustaining different runoff components. As these gradients are associated with spatial differences in associated forms of free energy (defined as a thermodynamic quantity equivalent to the capacity of a system to perform work: i.e. to accelerate a (water) mass (as overland flow), to lift a (water) mass against gravity (as capillary rise) or to enlarge a potential gradient), rainfall runoff processes are associated with conversions of capillary binding energy (in fact chemical energy), potential energy and kinetic energy. These conversions reflect energy conservation and irreversibility as they imply small amounts of dissipation of free energy into heat and thus production of entropy; which is exported from the critical zone to maintain a spatially organized configuration far from equilibrium (Kleidon et al., 2012).

Energy conversions and dissipation during rainfall runoff transformation are, though being very small when compared to energy conversions of the surface energy balance, nevertheless of key importance, because they are related to the partitioning of incoming rainfall mass into runoff components and storage dynamics (Zehe et al., 2013). This splitting and the subsequent subsurface dynamics are strongly controlled by preferential flow paths, which in turn largely influence hydrologically relevant resistance fields in larger control volumes. The field of subsurface flow resistances depends for instance on soil hydraulic conductivity, its spatial covariance and soil moisture. Apparent preferential pathways reduce, depending on their density, topology and spatial extent, subsurface flow resistances along their main extent, resulting in accelerated fluxes against the driving gradient. This implies an enlarged power in the subsurface flux (Kleidon et al., 2013), thereby either an enlarged free energy export from the control volume or an increased depletion of internal driving gradients, and thus a faster relaxation back towards local thermodynamic equilibrium (Zehe et al., 2010; 2013).

Thermodynamic optimality principles allow for a priory optimization of the resistance field at a given gradient (Porada et al. 2011), not in the sense how they exactly look like but in the sense how they function with respect to export and dissipation of free energy associated with rainfall runoff processes. Several authors suggest that water flow in catchments and catchment structure (especially the river net) is in accordance with candidate optimality principles that characterise the associated energy conversion and related thermodynamic limitations (Paik and Kumar, 2010; Phillips, 2010). Woldenberg (1969) showed that basic scaling relationships of river basins can be derived from optimality assumptions regarding stream power. Similarly, Howard (1990) described optimal drainage networks from the perspectives that these minimize the total stream power. Rodriguez-Iturbe et al. (1992) and explain river networks as "least energy structures" minimizing local energy dissipation and based on this they reproduced observed fractal characteristics of river networks. Related to these energetic minimization principles, the community debates several principles that seem to state exactly the opposite (Paik and Kumar, 2010): that systems organize themselves to maximize steady state power (MAXP proposed by Lotka (1922), steady state net reduction of free energy (MRE, Zehe et al., 2010, 2013) or steady state maximized entropy production (MEP Paltridge, 1979) associated with environmental flows. The MEP hypothesis has been corroborated within studies that allowed a) successful predictions of states of planetary atmospheres (Lorenz et al., 2001) b) identification of parameters of general circulation models (Kleidon et al., 2006) or c) identification of hydrological model parameters to estimate the annual water balances of the 35 largest catchments on Earth (Porada et al 2011). Kleidon et al. (2013) recently explored whether the formation of connected river networks is in accordance with MAXP and thus whether free energy transfer to sediment flows is maximized. This implies fastest depletion of morphological gradients, which remain however due to the external feedback of isostatic rebound in steady state (at least at the human time scale). These outlined maximization and minimisation principles are largely two sides of the same medal, because local minimization of frictional dissipation of kinetic energy increases the flows ability to transport matter against the driving macroscale gradient and thus to deplete it. MAXP, MEP and maximum net reduction of free energy (MRE) are furthermore equivalent during steady state conditions in closed but not isolated systems (Kleidon, 2012).

The validity and the practical value of MEP/MAXP/MRE is, however, still under a vital debate. We will thus further explore this alternative thermodynamic perspective, particularly a) whether thermodynamically optimal model structures allow acceptable uncalibrated predictions of rainfall-runoff in different soil landscapes and b) to define alternative descriptors for hydrological behavior. To this end we recently proposed thermodynamic framework to quantify conversion and dissipation of free energy associated with rainfall runoff proposed by Zehe et al. (2013). Key idea is that hydraulic equilibrium in soil corresponds to local thermodynamic equilibrium (LTE) as a state of minimum Helmholtz free energy. Deviations from LTE occur either due to evaporative losses, which increase absolute values of negative capillary binding energy of soil water and reduce its potential energy, or due to infiltration of rainfall, which increases potential energy of soil water and reduces the strength of capillary binding energy. This framework allows analysing the free energy balance of hillslopes and catchments either within numerical experiments or based on available data, with respect to its dependency on climate, vegetation, soil hydraulic functions, topography and density of macropores. Application of this theory within numerical experiments corroborated, as outlined, that simulations based on the thermodynamically optimal model structures were in good accordance with observed discharge in two distinctly different hydro-climatic settings without tuning.

However, the nature of these optima suggests there might be two distinctly different thermodynamically optimal regimes of rainfall runoff behaviour. In the capillary- or c--regime, free energy dynamics of soil water is dominated by changes in its capillary binding energy, which is the case for cohesive soils. Soil wetting during rainfall in the c-regime implies pushing the system back towards LTE, especially after long dry spells. Dead ended macropores (roots, worm burrows which end in the soil matrix) act as dissipative wetting structures by enlarging water flows against steep gradients in soil water potential after long dry spells. This implies accelerated depletion of these gradients and faster relaxation back towards LTE during rainfall runoff. In the c-regime several optimum macropore densities with respect to maximization of net reduction of free energy exist (Figure 1). This is because the governing equation is a second order polynomial of the wetting rate, which depends on macropore density, the slope of the soil water retention curve, topography and depth to groundwater. An uncalibrated long term simulation of the water balance of the 3.5 km² Weiherbach catchment based on the first optimum macroporosity performed almost as well as the best fit when macroporosity was calibrated to match rainfall runoff.

In the other regime called potential- or p-regime, free energy dynamics of soil water is dominated by changes in its potential energy, which applies to non-cohesive soils and a pronounced topography. Soil wetting during rainfall in the p-regime implies to push the system away from LTE. This can be compensated by preferential pathways which connect directly to the riparian zone or the groundwater body, because these drainage structures enhance export of potential energy from the critical zone. However, in the p-regime no local optimum exists because potential energy reduction rates scale linearly with the drainage rate (there is at best an optimum at the margin of the parameter space). Nevertheless, in this case one can define a "distinguished" density of vertical and lateral preferential flow paths that assures steady state conditions of the potential energy balance of the soil. This applies when average storage of potential energy is compensated by average potential export (Figure 1 b). When applying this idea to the Mallalcahuello catchment in Chile model, which is characterized by non-cohesive soils, high annual rainfall and steep terrain, simulations performed close to the value that yielded the best fit of rainfall runoff behaviour obtained during a calibration exercise. Secondly this idea allowed a robust a priory estimate of the annual runoff coefficient in accordance with long term observations (Zehe et al., 2013).

We will further extend the outlined analysis within numerical experiments for a different hydro-pedological setting in an alpine environment (Wienhöfer and Zehe, 2014). Particularly we further explore the feasibility of classifying rainfall runoff and related soil water dynamics happens in two different dynamic regimes (c- or p-regime) and whether these regimes can be separated based on a dimensionless number defined as the ratio of temporally averaged changes in capillary binding energy and potential energy associated with soil wetting and soil water flow. This dimensionless number is expected to depend on climate and topography, the soil pattern and related soil water retention properties, vegetation, density of preferential pathways and ground and surface water levels as suggested in Zehe et al. (2013).

References

Bejan, A., Lorente, S., and Lee, J.: Unifying constructal theory of tree roots, canopies and forests, Journal of theoretical biology, 254, 529-540, 10.1016/j.jtbi.2008.06.026, 2008.
Dietrich, W. E., and Perron, J. T.: The search for a topographic signature of life, Nature, 439, 411 - 418, 2006.
Howard, A. D.: Theoretical model of optimal drainage networks, Water Resour. Res., 26, 2107-2117, 1990.
Kleidon, A., Fraedrich, K., Kirk, E., and Lunkeit, F.: Maximum entropy production and the strength of boundary layer exchange in an atmospheric general circulation model, Geophysical Research Letters, 33, L06706
10.1029/2005gl025373, 2006.
Kleidon, A.: How does the earth system generate and maintain thermodynamic disequilibrium and what does it imply for the future of the planet?, Philosophical Transactions of the Royal Society a-Mathematical Physical and Engineering Sciences, 370, 1012-1040, 10.1098/rsta.2011.0316, 2012.
Kleidon, A., Zehe, E., Ehret, U., and Scherer, U.: Thermodynamics, maximum power, and the dynamics of preferential river flow structures at the continental scale, Hydrology And Earth System Sciences, 17, 225-251, 10.5194/hess-17-225-2013, 2013.
Kondepudi, D., and Prigogine, I.: Modern thermodynamics: From heat engines to dissipative structures, John Wiley
Chichester, U. K., 1998.
Lorenz, R. D., Lunine, J. I., Withers, P. G., and McKay, C. P.: Titan, mars and earth: Entropy production by latitudinal heat transport, Geophysical Research Letters, 28, 415-418, 2001.
Lotka, A. J.: Natural selection as a physical principle, Proc Natl Acad Sci USA, 8, 151-154, 1922.
Paik, K., and Kumar, P.: Optimality approaches to describe characteristic fluvial patterns on landscapes, Philos. Trans. R. Soc. B-Biol. Sci., 365, 1387-1395, 10.1098/rstb.2009.0303, 2010.
Paltridge, G. W.: Climate and thermodynamic systems of maximum dissipation, Nature, 279, 630-631, 10.1038/279630a0, 1979.
Phillips, J. D.: The job of the river, Earth Surface Processes And Landforms, 35, 305-313, 10.1002/esp.1915, 2010.
Porada, P., Kleidon, A., and Schymanski, S. J.: Entropy production of soil hydrological processes and its maximization, Earth Syst. Dynam., 2, 179-190, 10.5194/esd-2-179-2011, 2011.
Rodriguez-Iturbe, I., Rigon, R., Bras, R. L., Ijjasz-Vsaquez, E., and Marani, A.: Fractal structures as least energy patterns: The case of river networks, Geophysica Research Letters, 889-892, 1992.
Wienhöfer, J., and Zehe, E.: Predicting subsurface stormflow response of a forested hillslope - the role of connected flow paths, Hydrology And Earth System Sciences, 18, 121-138, 10.5194/hess-18-121-2014, 2014.Woldenbe.Mj: Spatial order in fluvial systems - hortons laws derived from mixed hexagonal hierarchies of drainage basin areas, Geological Society of America Bulletin, 80, 97-&, 10.1130/0016-7606(1969)80[97:soifsh]2.0.co;2, 1969.
Zehe, E., Blume, T., and Bloschl, G.: The principle of 'maximum energy dissipation': A novel thermodynamic perspective on rapid water flow in connected soil structures, Philos. Trans. R. Soc. B-Biol. Sci., 365, 1377-1386, 10.1098/rstb.2009.0308, 2010.
Zehe, E., Ehret, U., Blume, T., Kleidon, A., Scherer, U., and Westhoff, M.: A thermodynamic approach to link self-organization, preferential flow and rainfall-runoff behaviour, Hydrology And Earth System Sciences, 17, 4297-4322, 10.5194/hess-17-4297-2013, 2013.

Figures

Figure 1: Panel a) presents the average reduction rates of free energy of soil water plotted against the normalized conductivity of the macropores system, fmac (normalized with hydraulic conductivity of the soil matrix) in the Weiherbach catchment. Panel b) shows the sum of averaged changes in potential energy in soil water and averaged export by means of groundwater recharge and subsurface storm flow in the Mallalcahuello catchment.