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OCR for page 62
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
OCR for page 63
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 network—a 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
OCR for page 64
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.
OCR for page 65
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.
OCR for page 67
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
OCR for page 68
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
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................ : :::: :,: l:
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:-:-:-:-:-: :~:-:-:-:-:-:-:-:-:-:-:;:;:-:;:;:-: :;:-:~.
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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.
OCR for page 69
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
OCR for page 72
72
OPPORTUNITIES IN THE HYDROLOGIC SCIENCES
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OCR for page 203
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.
OCR for page 204
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.
OCR for page 205
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
OCR for page 206
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#~:~
OCR for page 207
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.
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Representative terms from entire chapter:
ground water
