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Some Critical and Emerging Areas OVERVIEW Earth processes are driven by two engines. The sun maintains the external engine that is responsible for the weather, surface erosion, and most oceanic processes. Radioactivity and primordial heat drive the internal engine that maintains the dynamic plate system and cre- ates global topography. Hydrologic science plays a fundamental role in key mechanisms by which the external and internal engines make the earth such a singular planet. This chapter presents some critical and emerging areas in hydrologic science. It is not exhaustive; the intention is to convey the flavor of the challenges and frontiers that make hydrology so critical a field of study in understanding the earth system. Toward this goal, the connection between hydrology and the earth's internal engine is explored. It is precisely through hydrologic processes that some of the most important interactions between the internal and the external engines occur. The tectonic system and the hydrologic system come together in the earth's rigid outer skin, mainly in the upper 10 km of the continental crust. Hydrologic processes play an important role in the tectonic system; for example, subsurface waters, in responding to changing thermal and stress conditions, can have a significant impact on the mechanics of earthquakes. The evolution of sedimentary basins and the genesis of ore deposits are fundamentally influenced by ground water flows operating on time scales of 102 to 106 years and spatial scales 62

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SOME CRITICAL AND EMERGING AREAS 63 of tens to hundreds of kilometers. The vastness of the scales in- volved brings enormous variability in the properties of the physical system, and new models to understand transport processes and their media of occurrence are being explored. The subsurface is also where one of the major environmental impacts of human activities takes place. This is the deposition of different types of waste and their water-borne migration from original deposit sites. The greatest concern lies in predicting the temporal evolution of a contaminant plume under highly heterogeneous soil and rock conditions, and where it is subject to a wide range of geochemical and biochemical transformations. Fractured rocks and karst terrain present particularly difficult challenges in understanding solute transport processes. Within the upper part of the earth's crust, rocks undergo an important sequence of chemical and physical changes, collectively called weathering, which gradually convert the rocks to soil. Soil lies at the intersection of the two major systems of the external engine, the physical climate system and the biogeochemical cycles. These two systems exchange energy and matter through their interactions, many of which are hy- drologically controlled. Whether adequate soils survive in which to grow crops; whether rivers are navigable; whether there is magnificent scenery: each depends on geomorphic processes driven by water. Much remains to be learned about the processes of erosion and sediment transport, including the effects of varying climate and land use. Rivers are the conduits for the transport of the water, sediment, and nutrients that control the fertility of floodplains. A quantitative understanding of the mechanisms that will allow the prediction of long-term landscape evolution and the effects of major human inter- ventions is missing. The mechanisms of transport in a river basin are organized around the channel networka tree-like structure with remarkable properties. How topography differentiates into channels and hillslopes is one of the key questions in its development. What are the unifying principles behind the three-dimensional network geometry? These principles are central to the runoff-generating pro- cess, which is intimately linked to the growth and development of the drainage network. River runoff itself is a key flux in the physical climate system. It is an input to ocean dynamics and an output from the convergence of atmospheric water vapor. This flux highlights the relationship of hydrology and climate. One challenge we still face is to improve our understanding of the interaction between the hydrologic cycle and the general circulation of the ocean-atmosphere system. There is a

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64 OPPORTUNITIES IN THE HYDROLOGIC SCIENCES constant exchange of water between the reservoirs of these two sys- tems, mainly through precipitation, runoff, and evaporation, but the time and space scales of the exchanges vary greatly among the com- ponents and with location. Unanswered questions relate to the at- mospheric pathways of evaporated moisture and to the sensitivities of atmospheric dynamics to the exchanges of heat and moisture between land and atmosphere. The operational tools in these studies are the atmospheric general circulation models (GCMs) that are being developed to reproduce the basic patterns and processes of atmospheric systems. Only recently have these models been used to study the spatial and temporal patterns in the atmospheric and surface branches of the hydrologic cycle. It is critical to intensify these efforts to quantify the role of land surface-atmosphere feedbacks in the maintenance of climatic systems. For a GCM to successfully simulate climate and be useful in regional hydrologic studies, realistic modeling of land surface processes is essential. Given that GCM grids are typically 104 to 105 km2, the sig- nificant effects of spatial heterogeneities in surface hydrologic processes must be defined. Identifying those effects, what controls them, their magnitudes, and their appropriate parameters is among the challenges that lie ahead. Of all the processes in the hydrologic cycle, precipitation in its various forms has perhaps the greatest impact on everyday life. At- mospheric processes that produce precipitation operate over a variety of space and time scales. They exhibit control and feedback mechanisms, and they interact with surface topography, soil moisture, and vegetation. A characteristic feature of rainfall is its extreme variability over time intervals of minutes to years and in space ranges of a few to thousands of square kilometers. One of the major challenges for hydrologists, meteorologists, and climatologists is to measure, model, and predict the nature of this variability. In hydrology, a primary interest lies in the dynamics governing the time and space distributions of rainfall, especially heavy rainfall that can produce floods, and in understanding the dynamic interaction of the drainage basin with these storms. This requires a link between deterministic models of rainfall dynamics and stochastic models of rainfall fields. The interaction between land surface processes and regional weather is another exciting frontier in hydrologic science. For instance, under what conditions will the spatial distribution of evaporation generate regional circulations that could influence mesoscale rainfall and regional climate? In this and other questions, it is becoming clear that spatial distribution of the phenomena plays important roles in controlling the strength of the feedback mechanisms between the surface and Me atmosphere.

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SO1\dE CRITICAL AND EMERGING AREAS 65 Surficial processes are those involving the transport of mass and energy through the interface between the lower atmosphere and the earth's surface. Once again it is necessary to understand the relevant processes on different temporal and spatial scales. How can local observations of infiltration and soil moisture be translated to larger regions? At the laboratory scale, many important issues remain theoretically unresolved, an example being the effect of the chemical constituents of the soil on its hydraulic properties. At the hillslope scale, debate exists over the role of different factors on the effective- ness of the various flow paths. At the mesoscale, much progress is needed in the formulation of appropriate parameters for regional evaporation, and we need to learn more about which phenomena in the atmospheric boundary layer control evaporation from the land surface and from large water bodies. The frozen environment presents its own challenges: the behavior of surface and subsurface waters at all scales of description is complicated by phase changes and by the peculiar properties of ice. In alpine terrain, there is a need for methods that integrate the radiation balance over large areas to provide estimates of times and rates of snowmelt. Surficial processes not only provide key interactions between ter- restrial surface moisture and energy and atmospheric dynamics, but also constitute a vital link between the physical climate system and biogeochemical cycles. The hydrologic cycle provides a useful framework for interpret- ing key biological processes. From an ecosystem perspective, water is important as a carrier, a cooler, a substrate, and a mechanical force. The dependence of life on water is fundamental since water is the major constituent in essentially all functioning organisms. Their intricate life cycles are organized in most cases around their access to water. Thus the hydrologic cycle represents a fundamental physical template for biological processes. Hydrology and biology interact over a wide range of spatial scales, from the microscale of small habitats, through the mesoscale of drainage basins, to the macroscale of continents. Similarly, their temporal interactions range over minutes to centuries. It is precisely in the interplay between the different scales that the hydrologic cycle of- fers unique scientific challenges in the search for general principles. In plant dynamics it is known that the abundance of species and their spatial distribution are related to environmental conditions called ecological optima. A natural speculation is that these conditions are the preferred operating domain of the climate-soil-vegetation system. The key question relates to what the optimality criteria are that direct the functioning of this system. Where water is the driving force of

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66 OPPORTUNITIES IN THE HYDROLOGIC SCIENCES the system, a logical hypothesis is that the optimality criteria are related to the key hydrologic variables. Among the exciting scientific challenges are the search for the optimality criteria under different types of constraints and the mathematical representation of these cri- teria. The fluxes of the key biological variables (e.g., biomass) are also intertwined with the operating chemical processes (e.g., carbon assimilation), which in turn are linked to hydrologic processes (e.g., evapotranspiration). From such relationships it is clear that hydrology is a fundamental structural component of the biogeochemical cycles. Knowing where a parcel of water has been, and for how long, often is the key to understanding its chemical evolution. Hydrologic residence times are basic unknowns related to many contemporary environmental concerns. Acid rain is a clear example of the importance of understanding and predicting the effects of the chemical composition of precipitation. The use of rainfall composition data as tracers of the hvdrolo~ic cycle offers singular opportunities to better understand J ~ J V 1 1 the relationship between the chemistry of rainfall and the chemistry of soils, ground water, and surface waters. Hydrology and a number of chemical processes also are tightly connected in understanding the effects of soil and vegetation systems on the biogeochemical cycles of nutrients and toxic elements that affect water quality. Sediment transport, a process long of interest to hydrologists, is receiving much renewed attention as an important vehicle for the storage and movement of chemical species. The nature of organic coatings on stream sediments and the transport rate of polluted aggregates play important roles in the quality of stream waters. Understanding the interaction of processes at widely different scales is again a pivotal challenge. For example, soil history at a point is largely controlled by microscale chemical kinetics that can be studied in the laboratory, but how is this related to weathering rates at the regional scale? The biogeochemical cycles of elements like carbon and nitrogen are intimately linked with hydrologic processes. The importance of this linkage cannot be overstated because it affects the very nature of life on the planet and is a major component of the earth's external engine. A recurrent theme throughout this exploration of the frontiers of hydrologic science is the dynamic nature of the processes involved. These processes are highly nonlinear and have a wealth of feedback mechanisms operating over a wide range of temporal and spatial scales. These features are common to many scientific fields, and their study draws on the same mathematical ideas.

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SOME CRITICAL AND EMERGING AREAS 67 How does one contend with the scale issues that are so pervasive in hydrologic phenomena? Is there hope for unifying relationships across scales? A challenging task is to uncover the organizing struc- ture hidden in the highly irregular patterns that hydrologic phenomena show at different scales. A striking feature of many natural processes is that changes in their scale of description lead to fluctuations that look statistically similar except for a factor of scale. A characteristic irregularity seems to exist that reflects an underlying structure. This characteristic irregularity can be described in terms of what is called the fractal dimension. Recent analyses of spatial rainfall and channel gradients in drainage networks suggest a more subtle type of structure where more than one factor of scale is involved. This is called multiscaling invariance, and it offers an exciting perspective for bringing unifying principles across scales to highly erratic hydrologic phenomena. The issues of nonlinear dynamics and the limits of predictability are other frontiers of contemporary science intimately related to hydrology. Recent developments in the theory of dynamical systems show that many nonlinear deterministic phenomena are sources of intrinsically generated complex behavior and unpredictability. In fact, they look as if they were a stochastic process, and thus the phenomenon is called deterministic chaos. Is this phenomenon detectable in hydrologic processes? If so, then, there is hope that through its characterization many important features can be understood regarding the complex nonlinear dynamics underlying the processes. What follows is a more detailed examination of selected frontiers in hydrologic science. In choosing these topics, the committee has subjectively sought the interesting and exciting, seeking to transmit the flavor of the science rather than to provide either an exhaustive or a rank-ordered list of the most important opportunities. HYDROLOGY AND THE EARTH'S CRUST Introduction Geoscientists describe the earth in terms of its three major struc- tural zones: the core, the mantle, and the crust. The crust is the rigid outer skin of the earth; in continental areas, it varies in thickness from approximately 15 to 70 km. Two aspects of the upper 10 km of the continental crust that make it unique are that (1) it is the only region of the earth's subsurface to which humans have direct access, and (2) it forms the interface between the earth's two major dynamic

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68 OPPORTUNITIES IN THE HYDROLOGIC SCIENCES systems the hydrologic system and the tectonic system. The tec- tonic system and the concept of plate tectonics involves a grouping of processes that lead to the formation and deformation of crustal rocks. Whereas the hydrologic system is set in motion primarily by solar energy, the tectonic system is driven by the earth's own internal thermal energy. When the role of hydrology in tectonic processes is considered, the depth scale of interest is kilometers, with horizontal distances usually on the order of tens to hundreds of kilometers. The processes of interest are the movement of fluids and the transport of mass and energy in the earth's crust. Hot springs provide an example to illustrate the nature of the interaction between the tectonic and the hydrologic systems. Hot springs develop when meteoric water originating from rainfall or snowmelt circulates to depths of several kilometers, adsorbs heat from the surrounding rock matrix, and then is able to move relatively quickly to the ground surface along fault zones. Figure 3.1 shows how heat flow from deeper levels of the earth's crust can be captured by the ground wa- ter flow system and diverted to a major fault zone. In areas where subsurface temperatures are increased by local intrusions of molten - O 4 -\ ~ ~ 6 ' e- cO~ ^ J -2 ~ ",,1 - .; . . . . . ... ... . . .. _ JO ::::::::::::::::.:; L-.-.. = Vat . - . ................... ...................... ........ : ::::::::::: ::::::: ::! _ be;. ................ : :::: :,: l: ::::::: :::::::: :: :: :~;; .; ............ .~ O ~ ~ ~ ~ 6 ~, ~ ' 2] :-:-:-:-:-: :~:-:-:-:-:-:-:-:-:-:-:;:;:-:;:;:-: :;:-:~. ,; ... ;.;.;.;.;.;.;.;.;.;.;... L:: :, ,:,:,:.:,:.:, .:.:, I:,:.: , .:.:.:.:., , . ; _ = = = = by: =: 8 10 12 DISTANCE (km) FIGURE 3.1 Patterns of fluid flow (dotted lines) and heat transfer (dashed lines) in an asymmetric mountain valley. This diagram shows how ground water flow can modify the subsurface thermal regime, by adsorbing heat and transferring it to a fault zone (thick line). A warm spring discharges in the valley. SOURCE: Reprinted, by permis- sion, from Forster and Smith (1989). Copyright O 1989 by the American Geophysical Union.

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SOME CRITICAL AND EMERGING AREAS 69 rock, hydrothermal systems may develop. The hot springs and gey- sers of Yellowstone National Park are dramatic examples. Active hydrothermal systems are potential sources of geothermal energy. Relic geothermal systems are targets for mineral exploration because metals may have been transported by the hot fluids, and geochemical conditions may have promoted precipitation of those metals to form ore deposits. Different issues arise when we focus on processes occurring within the upper several hundred meters of the earth's surface. This vadose zone normally comprises the weathered, unconsolidated soil material that is present at the land surface. In the vadose zone, both air and water are present within the open pore spaces between the solid grains. The medium is the site of innumerable chemical transforma- tions mediated by solar radiation, wet and dry atmospheric deposition, and biologic activity. The vadose zone is a storage component of the hydrologic cycle, a reservoir of water, air, and reactive inorganic and organic solid matter. It influences the runoff cycle and ground water recharge by affecting both the flow patterns and the quality of surface and percolating subsurface waters. The water table marks the transition from the vadose zone to the deeper saturated ground water zone, where all the pore spaces are filled by water. Like the vadose zone, the saturated zone is a reservoir of water and supports a range of chemical reactions. Issues at this scale center on the characterization of the physical, chemical, and biologic processes occurring within the subsurface hydrologic envi- ronment, their link to hydrologic processes occurring on the earth's surface, and the development of techniques for quantifying these processes and monitoring their effects. Many research questions here have direct relevance to the serious environmental problems facing our society. A number of processes must be addressed at the microscale, that is, the scale of the individual pore spaces within a soil or rock. The transport of chemical species dissolved in the water is a complex and dynamic process. Solutes entering the subsurface can interact with other dissolved solutes, with the solid matrix, and with the native ground water and can take part in the life cycle of microbes present within the subsurface. There is feedback between these biochemical processes and the patterns and rates of fluid flow. Greater under- standing at the microscale is necessary to build a framework for developing predictive models that apply at the mesoscale, the scale at which it is feasible to tackle most applied problems in subsurface hydrology.

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70 OPPORTUNITIES IN THE HYDROLOGIC SCIENCES Some Frontier Topics The Role of Ground Water in Tectonic Processes To what extent can the app! ication of quantitative hydrogeologic concepts provide new insights into geo- fogic processes that occur in the earth's upper crust? l- Water originating as precipitation is thought to be able to pen- etrate to depths of at least 10 km. Water present within the void spaces of sediment or rock is a central feature in a number of geologic processes because (1) fluid pressures influence the strength of sediments and rocks to resist shearing and thus influence processes such as landslides, faulting, and earthquakes, and (2) fluid flow is the key process for large-scale redistribution of mass and heat within crustal rocks. Although rates of ground water flow are much lower than those in the upper few hundreds of meters of the earth's surface, and time scales may approach 106 years or longer, from a geologic viewpoint ground water circulation within the upper crust is no less important than the near-surface component of the hydrologic cycle. Permeability is the parameter that quantifies the ability of a fluid to flow through the interconnected pore spaces of a rock or soil. Figure 3.2 identifies typical values of permeability for a variety of geologic deposits and rock types. Hydraulic conductivity, the permeability when the fluid is water, often is used to characterize the flow of water through near-surface soils or rocks. More permeable sediments or rocks, capable of transmitting significant quantities of water, are referred to as aquifers. The wide range of variation in permeability implies that subsurface fluid fluxes (flow per unit area) can vary by orders of magnitude, depending on the nature of the geologic setting. Recognition of the significant role of circulating fluids in tectonic environments is not a recent development. The geologic literature contains a vast array of models that have been proposed to explain innumerable sets of data. However, for many years the science pro- ceeded no further than well-reasoned qualitative analysis. To quan- tify this link between the hydrologic and tectonic systems, the nonlinear interaction of the hydraulic, geochemical, stress, and thermal regimes must be tackled. Stresses in the crust originate from movements of the earth's tectonic plates, and more locally, from the weight of over- lying rock units. A benchmark paper by M. King Hubbert and William

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SOME CRITICAL AND EMERGING AREAS unfractured metamorphic and igneous rocks - shale unweathered marine clay ~ sandstone limestone and dolomite fractured igneous and metamorphic rocks permeable basalt- - karst limestone - ~11~, c,allu - clean sand 1 1 1 ~ I 1 1 1 1 10-12 1011 10-10 10-9 10-8 10-7 71 gravel 1 1 1 1 1o 2' 10-20 10-19 10-18 10-17 1o~16 10-15 1014 10-13 PERMEABILITY (m2) FIGURE 3.2 Permeability of common geologic media. SOURCE: Adapted, by permis- sion, from Freeze and Cherry (1979). Copyright (31979 by Prentice Hall, Inc. Rubey demonstrated the importance of pore fluid pressures . . ha, . . - . .. ~ . . . . . . ~ in the mecnamcs ot faulting, and ultimately, in mountam-oullamg processes (Hubbert and Rubey, 1959~. This work, probably more than any other, set the stage for interactions between ground water hydrologists, ge- ologists, and solid-earth geophysicists. In the early 1970s, Barry Ra- leigh, Jack Healy, and John Bredehoeft, in what have become known as the Rangely experiments, provided field confirmation that fluid pressure can be a key parameter in triggering earthquakes associated with faulting (Raleigh et al., 1972~. Others documented the fact that filling a large reservoir behind a new dam can induce local earthquake activity. Dennis Norton and his colleagues at the University of Arizona were among the first to promote a quantitative framework to link geochemical processes, fluid circulation patterns, and heat transfer (Norton, 1984~. Recently, attempts have been made to quantify the role of ground water flow in regional metamorphism, where, for ex- ample, a rock such as limestone is transformed to marble. The concepts and tools of hydrologic analysis are being adopted to solve a number of fundamental geoscience problems, and opportunities abound for collaborative research. Earthquake Cycle Earthquakes occur when slip is initiated along a fault and stored energy arising from long-term tectonic movement is abruptly released. Subsurface waters, in responding to changing pressure, thermal, and stress conditions, can have a significant impact on the

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72 OPPORTUNITIES IN THE HYDROLOGIC SCIENCES EM. 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SOME CRITICAL AND EMERGING AREAS 203 (the time rate of change of X equals the nonlinear dynamics plus the stochastic forcing). Here X denotes the variable to be predicted, and f is the nonlinear dynamical part of the evolution, which includes the effects of feedback, of radiation, and so on, and also depends on a parameter \. The term F(t) represents a stochastic forcing. In the simplest form, F(t) is gen- erally assumed to have no correlations and to have a normal prob- ability law. It is called the Gaussian white noise. The principal features of the evolution predicted by the above equation may be summarized as follows. Suppose that the system starts in one of its two stable attractors. If the strength of the stochastic forc- ing F(t) is small, then during a long period of time the system will perform a small-scale jittery motion around a level corresponding to this attractor. But sooner or later, there is bound to be a fluctuation capable of overcoming the "barrier" separating this state from the second available state. When this happens, the system finds itself in another attractor in a very short time interval. Subsequently, it will again undergo a small-scale random motion around this new state until a new fluctuation drives it back to the previous state, or to a third one if such is available. This intermittent evolution looks very much like the record of Figure 3.33. More generally, it provides us with an archetype for understanding other hydrologic processes beyond our specific example, for instance, river flows that seem to exhibit abrupt transitions. Of course, a more quantitative view requires that the function f(X, \) and the noise strength F(t) be known. A minimal model of f should involve nonlinearities giving rise to stable states whose number and characteristics are identical to those of the plateaus found from the statistical analysis of the record. Having chosen the dominant nonlinearity, one can actually determine most of the model parameters from the data. An interesting question pertains to the residence times, that is, the time the system spends in a given attractor state or, alternatively, to the transition times between attractors. It would obviously be quite interesting to predict such times, since this would be equivalent to predicting the duration of an ongoing drought or to forecasting a forthcoming one. The theory of stochastic processes allows one to make statistical predictions of these times for the systems described by the above equation. Applied to the Sahel record shown in Figure 3.33, this type of an analysis predicts that the mean transition time from the dry state is much larger than the time from the more humid state. The theory also predicts an appreciable dispersion around these mean values.

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204 OPPORTUNITIES IN THE HYDROLOGIC SCIENCES In conclusion, dynamical systems theory suggests new techniques of data analysis. It also allows us to formulate the key questions pertaining to the dynamical behavior of complex hydrologic phe- nomena from a novel point of view. New physical insights and pre- dictive capabilities will emerge from such analyses in the future. Nonlinear Dynamics and Predictability of Hydrologic Phenomena Weather and climate processes of hydrologic interest, such as rainfall, exhibit a complex and highly variable structure in time and space. The general approach for making predictions of these processes, as, for example, for real-time flash flood forecasting, has been to use nonlinear deterministic equations governing atmospheric dynamics and to solve these equations numerically using high-speed computers. This is called numerical weather forecasting. Within the last two decades the complexity of numerical models has grown commensurately with the capacity and speed of computers. However, despite substantial progress in short-term weather forecasting, the reliability of forecasts has not increased much. Recent developments in the theory of dynamical systems show that many nonlinear deterministic phenomena are sources of intrinsically generated complex behavior and unpredictability. As explained earlier in this section, solutions of many nonlinear dynamical systems can take any one of many possible states called attractors. Which of these alternate states is chosen by the system depends on the initial conditions. This high degree of sensitivity to initial conditions confers a markedly random-looking character to the evolutions governed by purely deterministic dynamical equations. Ordinarily, in mathematical modeling or in laboratory experiments, the state (physical) variables are known in advance, and one deals with a well-defined set of evolution laws for these variables. However, this full information is seldom available for a natural system. Rather, only an observed time series of a climatic variable, say, rainfall rate, is available at one or several locations in space. Recent advances in dynamical systems theory have been instrumental in the development of new techniques to provide important qualitative information about process dynamics from the observed time series at one or several locations in space. They do not depend on specific model assumptions and details of the nonlinear dynamics. Therefore an important problem is to learn more about the underlying dynamics of weather and climate processes, independent of any modeling, from the observed time se- ries, and to find to what extent they are predictable.

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SOME CRITICAL AND EMERGING AREAS Are there strange attractors in hydrologic time series? What are the limits of preclictability of hydrologic phenomena? 205 Suppose that but), k = 0, 1, . . ., r- 1, are the state variables actually taking part in the dynamics. The mathematical space in which these variables take values is called the phase space. Xk's are assumed to sat- isfy a set of first-order nonlinear equations whose form is unknown, but which, given a set of initial data Xk(0), produce the full details of the evolution. By successive differentiation in time, this set of r equations can be reduced to a single, highly nonlinear, rth order equation for any one of these variables. For example, instead of but), k = 0,1, . . ., r- 1, one can take Apt) and its (r- 1) successive derivatives to be the r state variables spanning the full phase space. Now, the most impor- tant point to notice is that both Xo~t) and its (r -1) derivatives can be deduced from a single observed time series, Xo~t~), Xo~t2), . . ., Xottn), where to is the initial time, /` = t2 - to = t3 - t2 = . . . = In - to_ is the sampling time, and n is the total number of observations. So, in principle, an observed time series contains sufficient information about the multi- dimensional character of the system's dynamics. Important scientific issues, such as the extent of the predictability of a natural system, depend on the nature of the trajectories of the dynamical system in phase space, i.e., the "geometry" of the phase space. In order to identify this geometry from observed time series data, one typically wants an estimate of the minimum number r of variables that captures the essential features of the long-term evolu- tion of the climatic or weather system. This number also denotes the dimension of the phase space. In addition, one wants to test for the possible existence of an attractor in the phase space that represents this evolution. In a dissipative dynamical system like rainfall, the attractor occu- pies only a reduced portion of the phase space and therefore has a lower dimension than that of the phase space. One might visualize this scenario with the example of a simple pendulum. Its trajectories lie in a two-dimensional phase space, defined by an angle ~ with the vertical direction and the angular velocity d8/dt. If the pendulum loses energy to friction, then the trajectories gradually spiral inward toward a point that represents the state of no motion. In this case, the attractor is a zero-dimensional point. If the energy supplied to the pendulum exactly balances the energy dissipated by friction, then a steady state is reached in the form of a repeating loop in the phase space. In this

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206 OPPORT~ES ~ ME HYDRO[OGIC SCONCES so :~ I? is/> ~ ~SS~ S?~S~S: ~ S~S~>s~e~n~ce I ~~;s ~~ ~t h iA~-~ ~~s~<~ss~ i::: fF~<~l1 s-~ stewed ~ _ S S S .S ~ S SO ~ ~ S #:: S AS ~ S SS SO S S ~ :> ~ :: S ::> ~ :::: i. ~ S ::S:' ~ :; S :SS: US ~S:S S ~ i. ~s> S ~S? Sib. S >: so is. AS ~ >> S:S:: . ~ :S:S S #> sit . SS ~ ~~ ~~ci~31 ~~l~r~ fr^~s~i~i#~:~

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SOME CRITICAL AND EMERGING AREAS case, the attractor is a one-dimens space and is called a limit cycle. 207 tonal closed curve in the phase Trajectories of many natural systems like rainfall do not converge with time either to a point or to a limit cycle. Even though the attractor has a dimension smaller than that of the phase space, the trajectories do not cross themselves, do not repeat themselves, and contain every possible frequency in a broadband spectrum. To fulfill these conditions, the attractor has to have some strange geometrical attributes. For example, its dimension turns out to be a fraction rather than a positive integer and therefore is known as a fractal. It is called a strange attractor. The existence of a strange attractor means that trajectories, which are initially close, ultimately diverge into completely different paths. Therefore, beyond this time, predictability is no longer possible. The limits of predictability are set by the rate of divergence of the trajec- tories from the initial conditions close to one another. This rate of divergence is measured by the so-called Lyapunov exponents. The inverse of the largest positive Lyapunov exponent gives the time limit of predictability. The calculation of these exponents is an area of active research. Applications of these techniques of phase space reconstruction from time series are beginning to appear in the literature. Some recent examples include the identification of chaotic attractors governing the weather over Western Europe, the climate dynamics of Quaternary glaciations, and the mesoscale dynamics of certain extratropical storms in the United States. These techniques hold the potential to enhance understanding of different dynamic scenarios in diverse hydrologic processes, e.g., river flows, sediment flows, and rainfall, which is necessary both for developing physical descriptions of these processes and for making predictions. SOURCES AND SUGGESTED READING Hydrology and the Earth's Crust Back, W., and R. A. Freeze. 1983. Chemical Hydrogeology. Benchmark Papers in Geol- ogy. Vol. 73. Hutchinson Ross, Stroudsburg, Pa., 413 pp. Back, W., J. S. Rosensheir, and P. R. Seaber. 1988. Hydrogeology. Geology of North America. Geological Society of America, 524 pp. Bethke, C. M., W. J. Harrison, C. Upson, and S. Altaner. 1988. Supercomputer analysis of sedimentary basins. Science 239:261-267. Biggar, J. W., and D. R. Nielsen. 1976. Spatial variability of the leaching characteristics of a field soil. Water Resour. Res. 12(1):78-84. Blot, M. A. 1941. General theory of three-dimensional consolidation. J. Appl. Phys. 12:155-164.

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208 OPPORTUNITIES IN THE HYDROLOGIC SCIENCES Dagan, G. 1986. Statistical theory of ground water flow and transport: pore to labora- tory, laboratory to formation, and formation to regional scale. Water Resour. Res. 22:1305-1345. de Jong, de Josselin, and S. C. Way. 1972. Dispersion in Fissured Rock. Unpublished report. New Mexico Institute of Mining and Technology, Socorro, N. Mex. Delhomme, J. P. 1979. Spatial variability and uncertainty in groundwater flow patterns. Water Resour. Res. 15(2):269-280. Forster, C., and L. Smith. 1989. The influence of groundwater flow on thermal regimes in mountainous terrain: A model study. J. Geophys. Res. 94(B7):9439-9451. Freeze, R. A. 1975. A stochastic conceptual analysis of one-dimensional ground water flow in nonuniform homogeneous media. Water Resour. Res. 11(5):725-741. Freeze, R. A., and W. Back. 1983. Physical Hydrogeology. Benchmark Papers in Geology. Vol. 72. Hutchinson Ross, Stroudsburg, Pa., 431 pp. Freeze, R. A., and J. Cherry. 1979. Groundwater. Prentice-Hall, Englewood Cliffs, N.J., 604 pp. Gelhar, L. W. 1976. Effects of hydraulic conductivity variations in groundwater flows. In Proceedings Second International IAHR Symposium on Stochastic Hydraulics. International Association of Hydraulic Research, Lund, Sweden. Hubbert, M. K., and W. W. Rubey. 1959. Role of fluid pressure in the mechanics of overthrust faulting, I, Mechanics of fluid-filled porous solids and its application to overthrust faulting. Geol. Soc. Am. Bull. 70:115-166. Jury, W. A. 1985. Spatial Variability of Soil Physical Parameters in Solute Migration: A Critical Literature Review. EPRI Report EA-4228. Electric Power Research Insti- tute, Palo Alto, Calif., September. Jury, W. A., D. Russo, G. Sposito, and H. Elabd. 1987. The spatial variability of water and solute transport properties in unsaturated soil. Hilgardia 55(4):1-32. Lachenbruch, A. H., and J. H. Sass. 1980. Heat flow and energetics of the San Andreas fault zone. J. Geophys. Res. 85:6185-6207. Long, C. S. J., P. Gilmour, and P. A. Witherspoon. 1985. A model for steady fluid flow in random three-dimensional networks of disc-shaped fractures. Water Resour. Res. 21(8):1105-1115. National Research Council. 1984. Groundwater Contamination. Studies in Geophysics. National Academy Press, Washington, D.C., 179 pp. Norton, D. 1984. Theory of hydrothermal systems. Annul Rev. Earth Planet. Sci. 12:155-177. Raleigh, C. B., J. H. Healy, and J. O. Bredehoeft. 1972. Faulting and crustal stress at Rangely, Colorado. American Geophysical Union Monograph Series, Vol. 16, pp. 275-284. Sibson, R. H. 1973. Interactions between temperature and fluid pressure during earth- quake faulting A mechanism for partial or total stress relief. Nature 243:66-68. Sun, Ren Jen, ed. 1986. Regional aquifer system analysis program of the U.S. Geologi- cal Survey. Summary of Projects, 1978, 1984. USGS Circular 1002, 26 pp. Theis, C. V. 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground water storage. Trans. AGU 16:519-524. Hydrology and Landforms Ahnert, F. 1976. Brief description of a comprehensive three-dimensional process-re- sponse model of landform development. Z. Geomorphol., Suppl. 25:29-49. Cooke, R. U., and R. W. Reeves. 1976. Arroyos and Environmental Change in the American Southwest. Oxford University Press, London, 213 pp.

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