Sediment Yield in Intermountain West Headwater Basins as Indicators of Environmental Change and Landscape Denudation

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A Spectral View of Watershed Processes
James W. Kirchner U. C. Berkeley
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Other stuff we do

• Earth surface processes

Especiallycosmogenic isotope geochemistry (to
measure long-term rates of landscape evolution
and chemical weathering)
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• Other stuff we do

How (much) does riparian vegetation stabilize
streambanks against erosion?

• Field sites
• So. Fork Kern R.
• Sacramento R.
• Lower Truckee R.

4

• Some other stuff we do

Dynamics of extinction and diversification inferre
d from the fossil record
with Anne Weil (Duke University)
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Some other stuff we do
Evolutionary ecology of pathogens and their hosts

• with B.A. Roy (ETH Zurich)

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Snow hydrology and snowmelt chemistry
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Central Sierra Snow Laboratory, Donner Pass,
CA Annual avg. precip. 1.3m, snowfall 10.4m
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Central Sierra Field Research Stations Sagehen
Creek Field StationCentral Sierra Snow
LaboratoryOnion Creek Experimental ForestNorth
Fork ReserveChickering Reserve
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Watershed hydrology and geochemistry

• how (and how fast) does rainfall travel to the
  stream?
• what happens to it chemically on its way there?

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A Spectral View of Watershed Processes
James W. Kirchner U. C. Berkeley
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• spectral (spek'trel), adj.
• 1. of or relating to a specter ghostly phantom
• 2. of or relating to a spectrum or spectra

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With thanks to Xiahong Feng (Dartmouth) and
Colin Neal (Inst. of Hydrol) Outstanding
in their field!
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And thanks to Brit Lisa Skjelkvåle Norwegian
Institute for Water ResearchTom
Clair Environment CanadaSimon Langan Macaulay
Land Use Research Institute, ScotlandChris
Soulsby University of AberdeenSteve Kahl and
Steve Norton University of MaineGene
Likens Hubbard Brook Ecosystem Study
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Watersheds are gatekeepers of the hydrologic
cycle...
...controlling stream flow, soil moisture,
evapo-transpiration and groundwater supply.
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Watersheds are gatekeepers of landform
evolution...
...controlling rates and patterns of erosion and
sediment delivery.
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Watersheds are gatekeepers of biogeochemical
cycles...

• ...thus controlling
• solute fluxes,
• evolution of atmos. CO2,
• nutrient dynamics,
• and pollutant delivery to downstream waters.

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Watersheds are important test cases for
environmental analysis...

• complex
• heterogeneous(at all scales)
• material properties are hard to measure

• what can we learn about how they work, by
  observing how they behave?

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Key questions

• what controls water and solute fluxes?
• how does rainfall travel to the stream?
• how fast does it get there?
• what happens to it chemically on its way to the
  stream?

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Stream flow and chemistry reflect the integrated
behavior of catchments, so...
analyses of inputs and outputs should clarify
catchment-scale processes.
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Primary field sitesSmall (1 km2) headwater
catchments at Plynlimon, Wales
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Plynlimon study Water fluxesevery 15
minutesfor 30 years Complete chemistry(40
elements)weekly for 22 years Chloridedaily for
3 years(Cl- from seasalt conc's highly variable)
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Storm runoff -- the standard model, 1950's-1980's
(...2003?)
...traveling rapidly through watersheds via
overland flow or macropores...
Peak stormflow assumed to be mostly "new water"
(recent rainfall)...
...superimposed on baseflow of "old water"...
...traveling slowly through watersheds via
groundwater flow.
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The "old water - new water" paradox
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Observation Stream discharge responds strongly,
and promptly, to rainfall inputs
Inference Stormflow must be "new" (storm) water
that reaches the stream promptly
WRONG!
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Observation Inert tracers (here, Cl-) in
rainfall are very strongly damped in the stream
Implication Storm runoff is predominantly "old"
(pre-storm) water!
Puzzle How do catchments retain water for long
periods, then discharge it promptly during storm
events?
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Question

• How old is this "old" water?

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In daily time series,streamflow Cl is strongly
damped
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On weekly time scales,streamflow Cl is strongly
damped
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Seasonal Cl fluctuationsare also strongly damped
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Stream Cl even exhibits "memory effects" over
timescales gt1 year
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Refined Question

• How old is this "old" water?might mean...
• What is the residence time of water stored in the
  catchment?

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The 'box model' view of a watershed
Mean residence timeto volume/flow rate
Fluctuations on timescales shorter than to are
damped by mixing.
Fluctuations on timescales much longer than to
are transmitted without attenuation.
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Random noise series Longer residence time
implies more long-term persistence, less
short-term transience.
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Cl data show both short-term transience and
long-term persistence
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• Implication catchments do not behave like a
  well-mixed "box", or like any simple collection
  of "boxes".

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Nonetheless, the tracer behavior should tell us
something about storage and mixing in the
watershed.
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Spectral analysis 101
White light is a mixture of many wavelengths
In white light, each wavelength has (roughly)
equal intensity
A prism separates light into its component
wavelengths
Intensity
Hence this is called a "white noise" spectrum
Red long ?, low f
Blue short ?, high f
Wavelength
A plot of intensity versus wavelength or
frequency is called a power spectrum
Thus creating a spectrum
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Spectral analysis 102
Reddish light is more intense at long wavelengths
than at short ones
Intensity increases as a function of wavelength
So the red end of the spectrum is brighter than
the blue end
Intensity
So this is called a "reddened" spectrum
Red long ?, low f
Blue short ?, high f
Wavelength
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Spectral analysis in nature the eye
(source Raven Johnson, 1999)
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Spectral analysis in nature the eye
Color vision comes from light-absorbing pigments
in outer segments of cone cells
(source Raven Johnson, 1999)
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Spectral analysis in nature the eye
Different amino acid sequences on cone cell
pigments cause band-pass filtering of three
wavelength bands.
(source Raven Johnson, 1999)
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The Fourier transform measures the amplitude of
the best-fit sine wave at each wavelength. The
square of this amplitude is the spectral power.
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Note! Analyzing unevenly spaced data (i.e.,
environmental data!) requires special spectral
and wavelet methods Scargle, J. D., The
Astrophysical Journal, 343, 874-887,
1989. Foster, G., Astronomical Journal, 112(4),
1709-1729, 1996.
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White noise (residence time zero)
Time series
Power spectrum
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The 'box model' view of a watershed
Mean residence timeto volume/flow rate
Fluctuations much longer than to are transmitted
with little damping
Mixing damps fluctuations on timescales at and
below to -- faster fluctuations are damped more.
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Mixing implies greater damping at shorter
wavelengths
Time series
Power spectrum
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How about the spectra of chloride concentrations
in rainfall and streamflow?
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Chloride spectra show strong damping on all
timescales lt a few years
Kirchner et al., Nature, 2000
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Note that power-law slope is 1, not 2 (time
series is self-affine, a.k.a. fractal)
Kirchner et al., Nature, 2000
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A plea

• Don't be too alarmed by the mention of fractals!
• (It will turn out that there's nothing too weird
  going on here...)

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• Fractal filtering behavior tells us something
  about the distribution of travel times for water
  traveling through the catchment.

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'Box model' watershed
Travel time distribution
t travel timeto mean residence time
volume/flow rate
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Alternative travel time distributions
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Stream concentration is...
a convolution of...
the travel time distribution...
and past rainfall inputs
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Convolution of two time functions...
...is the same as multiplying their Fourier
transforms. (also used elsewhere, e.g. Duffy and
Gelhar, Water Resources Research, 22(7),
1115-1128, 1986.)
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Different travel time distributions imply
different degrees of wavelength-dependent damping
in the stream
And thus different power spectra (via the
convolution theorem)
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Conventional travel-time models (exponential and
A-D) don't fit observed spectrum, but gamma does
Kirchner et al., Nature, 2000
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Gamma travel-time distribution has long tail
Kirchner et al., Nature, 2000
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Implication

• Long-tailed travel time distribution means that
  contaminant cleanup will be much slower than one
  would otherwise expect.

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And now for the bad news For chemically
reactive contaminants, adsorption/desorption
(i.e. "retardation") will delay recovery even
further.
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Question
How general is this fractal scaling phenomenon?
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Mid-sized Nova Scotia stream (17 km2)
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Large Nova Scotia stream (279 km2)
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Birkenes, Norway stream Cl spectrumis 1/f
steeper than rain Cl spectrum
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Kaarvatn, Norway stream Cl spectrumis 1/f
steeper than rain Cl spectrum
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Question
What's the physical mechanism behind the observed
travel time distribution (and thus fractal
scaling)?
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Approximation
Hillslope length is ?100x soil thickness, so flow
system is approximately one-dimensional
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But also
Fracture flow does occur!
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Assumptions
1. Advection happens 2. Dispersion happens 3.
Rainfall inputs are spatially distributed across
the catchment
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Working Hypothesis Advection and dispersion of
spatially distributed rainfall inputs
(Kirchner et al., J. Hydrol., 2001)
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But wait! Didn't I just say that
advection-dispersion models don't work for these
data?
(Kirchner et al., J. Hydrol., 2001)
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Governing equation advection-dispersion equation
(Kirchner et al., J. Hydrol., 2001)
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Two key parameters mean travel time ?o
L/2? Peclet number Pe ?L/2D
(Kirchner et al., J. Hydrol., 2001)
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Need to integrate this twice once for advection
and dispersion along any path, and again for all
paths simultaneously
(Kirchner et al., J. Hydrol., 2001)
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Analytic solution is complicated...
But travel-time distribution depends on only two
parameters -mean residence time (?0) -Peclet
number (Pe) ratio of dispersive to advective
timescales
(Kirchner et al., J. Hydrol., 2001)
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Model agrees with observed spectrum...
(Kirchner et al., J. Hydrol., 2001)
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...if Peclet number is ?1 (highly dispersive)
(Kirchner et al., J. Hydrol., 2001)
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Similar fits to a second stream
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Observation
Fractal behavior can arise from non-fractal
(indeed, downright prosaic) mechanisms.
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Comments
Other models probably work too(e.g., fracture
flow models, transmissivity feedback models,
fractal velocity distributions). But no simple
collection of "boxes" will work.
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Question
How does flow regime affect catchment-scale
advection and dispersion?
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Cl is more variable during higher flows
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Can analyze each flow regime separately
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Can analyze each flow regime separately
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Can analyze each flow regime separately
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Similar scaling in each flow regime, with higher
spectral power at higher flows
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But note! At high flows, advection and
dispersion are both faster (?proportionally)
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Clock time Time as we see it
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Flow-equivalent time Time as dispersion sees it
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Similar scaling in each flow regime, with higher
spectral power at higher flows
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In flow-equivalent time, each flow regime has
same spectrum (same process, just faster at
higher flows)
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Question
Can we measure catchment-scale reaction (as
distinct from transport) processes, by comparing
solutes with different degrees of reactivity?
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Streamflow Na varies less than Cl on all
timescales
Cl damped by dispersion Na damped by dispersion
and ion exchange Concentration scales adjusted
by seasalt Na/Cl ratio (Neal and Kirchner,
Hydrology and Earth System Sciences 4, 295-310,
2001)
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Na more damped than Cl--reflects cation buffering
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Kaarvatn, Norway Na and Cl fluctuations in
rainfall show same scaling (reflecting seasalt
source)
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Kaarvatn, Norway In streamflow, Na
systematically more damped than Cl (reflecting
cation buffering)
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Kaarvatn, Norway Na and Cl in streamflow are 1/f
steeper than in rainfall (reflecting advection
dispersion)
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Adsorption/desorption of Na adds a retardation
factor Rd to our governing equation
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Extensive soil sampling (340 cores at
Kaarvatn) Analyzed concentrations of exchangeable
cations (Ca, Mg, Na, H, Al)
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Exchangeable cation inventories yield
whole-catchment retardation factors
Rd ? Kd / ?
For Kaarvatn, Rd ? 10-20
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Using observed Cl spectrum and retardation factor
calculated from soil analyses...
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We can predict the spectrum for Na... which is
more strongly damped than the actual Na spectrum
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The observed Na spectrum implies 5-10x less
retardation than predicted from the soils data
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Na is damped much less than expected from the
soil cation inventory. Why?
If flow through macropores bypasses most of the
soil... the effective watershed inventory of Na
could be much smaller than the whole-soil value.
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Na more damped than Cl--reflects cation buffering
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Question
What can we learn from applying spectral analysis
to flow data? Probably a lot! Here's just a
nibble....
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Water fluxes track the propagation of hydraulic
potentials ("pressure waves") in the subsurface
Rainfall inputs are transmitted promptly to the
stream, with little damping.
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Water flux spectra show little damping except on
timescales lt a few days
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Power spectra of water fluxes reflect propagation
of hydraulic potentials (rather than propagation
of the water itself).
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Power spectra of low and intermediate flows are
similar high flows are different
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Power spectra of low and intermediate flows are
similar high flows are different
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In flow-equivalent time, spectra of low and
intermediate flows coincide (same process, just
faster) high flows are different
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In flow-equivalent time, spectra of low and
intermediate flows coincide (same process, just
faster) high flows are different
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Power spectrum
Transferfunction (output spectrum normalized to
input)
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Transfer function
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Water fluxes damped on much shorter timescales
than Cl
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Where do we go from here?

• Analyses of data from diverse catchments
• Intensive field measurements
• Laboratory simulation models

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Benchtop watersheds
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Analyses of data from diverse catchments
Wales Norway Scotland
Hafren Birkenes Allt a' MharcaidhHore Storgam
a Dargall LaneTanllwyth Langtjern White
Laggan Kårvatn Green Burn
Nova ScotiaMersey RiverMoose Pit BrookPine
Marten Brook Maine East BearWest Bear New
HampshireHubbard Brook VermontSleeper's
River GeorgiaPanola Mountain
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Typical chemical sampling is far too infrequent
to capture flow dynamics
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Even daily sampling misses the fine structure of
stream chemistry
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Revealing hydrochemical dynamics requires
continuous monitoring
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Intensive field measurements

• Sample rainfall and streamflow daily for ? 2
  years...
• with higher-frequency sampling during selected
  storm events.
• Complete chemicalanalysis, including 18O
• Monitor major elements,pH, etc. continuouslyvia
  online autoanalyzers

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Watershed monitoring data are not just a history
book (a record of what has happened). Th
ey are also a Rosetta Stone for decoding the
"language" of watershed systems, helping us to
understand why they behave the way they do -- and
how they will respond in the future.
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For further information, see...
Kirchner, Feng, and Neal, Nature, 403, 524-527,
2000. Neal and Kirchner, Hydrology and Earth
System Sciences, 4, 295-310, 2000. Kirchner,
Feng, and Neal, J. of Hydrology, 254, 81-100,
2001. Feng, Kirchner and Neal, J. of Hydrology,
292, 296-307, 2004. Kirchner, Feng, Neal, and
Robson, Hydrological Processes, 18, 1353-1359,
2004. And lots of stuff yet to come...
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Is Cl a sufficiently nonreactive tracer?Compare
Cl and 18O -- same site, same dates...
Data from C. Soulsby, U. of Aberdeen
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Cl and 18O spectra exhibit similar damping of
precipitation in streamflow
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Spectral analysis in nature the ear
Auditory nerves originate in the cochlea
(source Raven Johnson, 1999)
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The cochlea unwound Our auditory nerves begin at
hair cells that sense vibrations in the basilar
membrane. Different tones resonate at different
points along the membrane, stimulating different
hair cells.
(source Raven Johnson, 1999)
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Spectral analysis in nature the ear
The cochlea unwound Hair cells sense the
vibration of the basilar membrane. At the near
end, short, stiff fibers in the basilar membrane
make it resonant at high frequencies (short
wavelengths).
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Spectral analysis in nature the ear
At the far end, the fibers in the basilar
membrane are 5 times longer and 100 times more
flexible. Thus it resonates at a frequency 500
times lower at one end than the other. Different
tones resonate at different points along the
membrane, stimulating different hair cells.
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Mixing implies greater damping at shorter
wavelengths
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A plea

• Don't blame me for all the silly things that have
  been done with fractals over the years!
• (It will turn out that there's nothing too weird
  going on here...)

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The other Plynlimon streams show similar fractal
"1/f" scaling
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Random walk (residence time infinite)
Time series
Power spectrum
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Small Nova Scotia stream (1.3 km2)
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Birkenes, Norway (note rain is 1/f0.45 and stream
is 1/f1.50, so damping is roughly 1/f)
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Storgama, Norway
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Langtjern, Norway
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Kaarvatn, Norway
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Longer records for Cl show clearer scaling
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Observation
Catchments act as fractal filters, converting
white noise inputs into fractal 1/f noise
outputs.
Question

• So what...?
• ...does this tell us about how water moves
  through catchments?

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Observation

• Fractal spectra gt long-tailed travel times gt
  slow contaminant cleanup.

Comment
And now for the bad news This is for a
chemically inert tracer. Chemical reactions (i.e.
"retardation") will make this problem even worse.
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Research Topics Watersheds

• Long-term effects of acid rain on forests and
  streams
• Karl-Joseph Meiwes, Martin Novak and Gene Likens
  
• Trace element mobilization by clearcutting acid
  rain
• Colin Neal
• Isotopic and chemical fractionation during
  snowmelt
• Xiahong Feng, Carl Renshaw and Susan Taylor

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William of Ockham (ca. 1286 - 1347)

• It is futile to do with more what can be done
  with fewer. (Frustra fit per plura quod potest
  fieri per pauciora.)
• When a proposition comes out true for things, if
  two things suffice for its truth, it is
  superfluous to assume a third. (Quando
  propositio verificatur pro rebus, si duae res
  sufficiunt ad eius veritatem, superfluum est
  ponere tertiam.)
• Plurality should not be assumed without
  necessity. (Pluralitas non est ponenda sine
  necessitate.)
• No plurality should be assumed unless it can be
  proved by reason, or by experience, or by some
  infallible authority. (Nulla pluralitas est
  ponenda nisi per rationem vel experientiam vel
  auctoritatem illius, qui non potest falli nec
  errare, potest convinci.)

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Question
What's the physical mechanism behind this travel
time distribution?
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Simple or complex...

• ...in behavior?
• ...in external forcing?
• ...in internal configuration?
• ...in internal processes?

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3-way puzzle

• Stormflow responds very rapidly to rainfall...
• BUT is almost all isotopically old water, just
  like low flow...
• BUT often bears no chemical resemblance to low
  flow or rainfall...

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• Chaos theory
• simple (but nonlinear) complex mechanisms
  behavior
• Complexity theory
• interactions between elaborate many simple
  parts patterns
• Watersheds
• complex systems simple complex forcing
  behavior?

157

• Confronting unfamiliar ideas is like being hit in
  the face with a custard pie...
• the few bits that stick may not resemble the
  original very much.

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Other Research Topics

• Transport and fate of mercury in mine waste
• Dyan Whyte
• Correlations between extinction and orgination in
  the fossil record
• Anne Weil
• Evolution, ecology genetics of pathogen
  resistance and tolerance
• Barbara "Bitty" Roy

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Two major advances 1 better (longer, higher
frequency) data sets 2 spectral and wavelet
methods for unevenly spaced data Scargle, J. D.,
The Astrophysical Journal, 343, 874-887,
1989. Foster, G., Astronomical Journal, 112(4),
1709-1729, 1996.
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Everything should be madeas simple as
possible,but not simpler. Albert Einstein
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Paradox

• If residence time is Long Short
• persistence is high low
• and transients are weak strong
• But real catchment data combine
  high persistence and strong transients.