Mod2/3-E Lake Ecology - Major Ions & Nutrients

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Transcript Mod2/3-E Lake Ecology - Major Ions & Nutrients

LAKE ECOLOGY
Unit 1: Module 2/3 Part 5 - Major Ions and Nutrients
January 2004
Modules 2/3 overview
 Goal – Provide a practical introduction to
limnology
 Time required – Two weeks of lecture (6
lectures) and 2 laboratories
 Extensions – Additional material could be used
to expand to 3 weeks. We realize that there are
far more slides than can possibly be used in
two weeks and some topics are covered in
more depth than others. Teachers are expected
to view them all and use what best suits their
purposes.
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s2
Modules 2/3 outline
Introduction
Major groups of organisms; metabolism
Basins and morphometry
Spatial and temporal variability – basic
physical and chemical patchiness (habitats)
5. Major ions and nutrients
6. Management – eutrophication and water
quality
1.
2.
3.
4.
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s3
5. Water chemistry:
Gases, major ions & nutrients
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s4
5. Water chemistry:
Gases, major ions & nutrients
 Gases
 Oxygen (O2)
 Carbon dioxide (CO2)
 Nitrogen (N2)
 Hydrogen sulfide (H2S)
 Major ions (anions and cations)
 Nutrients (phosphorus and nitrogen)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s5
Water chemistry: gases
 What are the ecologically most important gases ?
 O2
 CO2
 N2
 H2S
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s6
Gas solubility
 The maximum amount of gas that can be
dissolved in water (100% saturation) is
determined by temperature, dissolved ion
concentration, and elevation
 solubility decreases with temperature
“warm beer goes flat”
 solubility decreases with higher dissolved ion
content (TDS, EC25, salinity)
“DO saturation is lower in saltwater than freshwater
(for the same temperature, solids “drive out” gases)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s7
Water chemistry: O2
 ~ 21% of air
 Very soluble (DO)
 Highly reactive and concentration is dynamic
 Involved in metabolic energy transfers (PPr & Rn)
 Major regulator of metabolism (oxic-anoxic)
 Aerobes (fish) vs anaerobes (no-fish, no zoops)
 Types of fish
 Salmonids = high DO (also coldwater because of
DO)
 Sunfish, carp, catfish = low DO (also warmwater)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s8
O2 variability
 Diel (24 hr) variation due to ____________?
 Seasonal variation due to _____________ ?
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s9
Major sources of O2
 Sources
 Photosynthesis (phytoplankton, periphyton,
macrophytes)
 Air from wind mixing
 Inflows
 tributaries may have higher or lower DO
 groundwater may have higher or lower DO
 Diffusion (epilimnion to hypolimnion and vice
versa)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s10
Major sinks of O2
 Sinks
 Respiration
 bacteria, plants, animals; water and sediments
 Diffusion to sediment respiration
 Outflow (tributary or groundwater)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s11
Gases: wind mixing from storms
 Oxygen from a storm – How many mixing
“events” can you find for Halsteds Bay in Lake
Minnetonka, MN in this 1 year record?
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s12
Gases: seasonal wind mixing
 Oxygen varies seasonally and the entire water
column lake may be fully saturated at certain
times. How often did this happen in Ice Lake, MN
in this 5+ year record?
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s13
O2: Human significance
 Not a direct threat to humans
 Directly affects fish physiology and habitat
 Indirectly affects fish and other organisms via toxicants
associated with anoxia:
 H2S
 NH4+ (converts to NH4OH and NH3 above ~pH 9)
 Indirectly affects domestic water supply
 H2S (taste and odor)
 Solubilizes Fe (staining)
 Indirectly affects reservoir turbines
 Via H2S corrosion and pitting (even stainless steel)
 Via regulation of P-release from sediments (mediated via
Fe(OH)3 adsorption)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s14
Gases: N2
 ~ 78% of air
 Concentrations in water usually saturated
because it is nearly inert
 Supersaturation (>100 %) can occur in reservoir
tailwaters from high turbulence
 May be toxic to fish (they get “the bends)
 N2 -fixing bacteria and cyanobacteria (bluegreen “algae”) convert it to bio-available NH4+
 Denitrifying heterotrophic bacteria convert NO3to N2 and/or N2O under anoxic conditions
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s15
Gases: CO2
 Only about 0.035% of air (~ 350 ppm)
 Concentration in H2O higher than expected based on low
atmospheric partial pressure because of its high solubility
Gas
Concentration @
Concentration @
(at 10oC)
1 atm (mg/L)
normal pressure (mg/L)
N2
23.3
18.2
O2
55.0
11.3
CO2
2319
0.81
How long does your soda pop fizz after shaking it?
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s16
CO2 reactions in water
 <1% is hydrated to form carbonic acid:
CO2 + H2O
H2CO3
 Some of the carbonic acid dissociates into bicarbonate and
hydrogen ions which lowers the pH:
H2CO3
HCO-3 + H +
 As the pH rises, bicarbonate increases to 100% at a pH of
8.3. Above this, it declines by dissociating into carbonate:
HCO-3
CO3-2 + H+
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s17
Inorganic - C equilibria
H2CO3
HCO3
CO3
pH
Note – 100% CO2 for pH< ~ 4.5; 100% bicarbonate for pH ~ 8
and 100% carbonate for pH > ~12
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s18
Inorganic - C: Major sources and sinks
Sources:
 Atmospheric CO2 (invasion)
 Respiration and other aerobic and anaerobic
decomposition pathways in the water and sediments
 Groundwater from soil decomposition products
 Groundwater from volcanic seeps
Sinks:
 pH dependent conversions to bicarbonate and
carbonate
 Precipitation of CaCO3 and MgCO3 at high pH
 Photosynthesis
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s19
CO2 supersaturation – killer Lake Nyos
 In 1986, a tremendous explosion of
CO2 from Lake Nyos, in Cameroon,
West Africa, killed >1700 people and
livestock up to 25 km away.
 Dissolved CO2 seeps from volcanic
springs beneath the lake and is
trapped in deep water by hydrostatic
pressure. Nearby Lake Manoun is
similar in nature
 Although unconfirmed, a landslide
probably triggered the gas release
Visit http://www.biology.lsa.umich.edu/~gwk/research/nyos.html and
http://perso.wanadoo.fr/mhalb/nyos/index.htm for detailed information
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s20
Soda pop chemistry
www.saddleback.cc.ca.us/faculty/thuntley/ms20/seawaterprops2/sld013.htm
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s21
CO2 and the inorganic carbon system
• Carbon dioxide diffuses from the atmosphere
into water bodies and can then be incorporated
into plant and animal tissue
• It is also recycled within the water with some
being tied up in sediments and some ultimately
diffusing back into the atmosphere
• Fixed carbon also enter the water as
“allocthonous” particulate and dissolved
material
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s22
CO2 and the inorganic carbon system - 2
• Alkalinity, acid neutralizing capacity (ANC),
acidity, carbon dioxide (CO2), pH, total
inorganic carbon, and hardness are all related
and are part of the inorganic carbon complex
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s23
CO2 chemistry: Alkalinity
 Alkalinity – the ability of water to neutralize acid; a measure
of buffering capacity or acid neutralizing capacity (ANC)
 Total Alkalinity (AlkT) = [HCO3-] + 2[CO32-] +[OH-] - [H+]
 Typically measured by titration with a strong acid. The
units are in mg CaCO3/L for reasons relevant to drinking
water treatment (details in Module 9)
 Can be used to estimate the DIC (dissolved inorganic
carbon) concentration if the [OH-]
 Conversely, direct measurements of DIC by infrared
analysis or gas chromatography, together with pH and the
carbon fractionation schematic can be used to estimate
alkalinity (* see slide notes)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s24
Alkalinity and water treatment
 Advanced wastewater treatment (domestic sewage)
 Phosphorus nutrient removal by adding lime (Ca(OH) 2)
or calcium carbonate (CaCO3)
 As pH increases >9, marl precipitates adsorbed PO4-3
 Settle and filter the effluent to obtain 90-95% removal
 Used for particle (TSS) removal also
 Drinking water treatment
 For TSS removal prior to disinfection
 Acid-rain mitigation to whole lakes
 Lime or limestone added as powdered slurry to increase
impacted lake pH
 Also broadcast aerially to alkalize entire watersheds
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s25
CO2 chemistry: Hardness
 Hardness - the total concentration of multi-valent (i.e. >2)
cations
 Ca+2 + Mg+2 + Fe
+3
(when oxic) + Mn+2 (when oxic); all other
multivalent cations are typically considered to be negligible
 Sources Minerals such as limestone (Ca and Mg) and gypsum (Ca)
 Water softeners and other water treatment processes such as
reverse osmosis and ion exchange
 Evaporation can increase hardness concentration
 Drinking water effects (no real health effects)
 Soap scums and water spots on glasses and tableware
 Deposits (scaling) can cause clogging problems in pipes,
boilers and cooling towers
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s26
Water chemistry – Major ions
Ion balance for typical fresh water
Anions
Percent
Cations
Percent
HCO32-
74
Ca2+
63
SO42-
16
Mg2+
17
C1-
10
Na+
15
SiO2
<1
K+
4
Note: plant nutrients such as nitrate, ammonium and phosphate that can
cause algae and weed overgrowth usually occur at 10’s or 100’s of partsper-billion and along with other essential micronutrients usually represent
<1% of the actual amount of cations or anions present in the water
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s27
Major ion concentrations - freshwater
Anions
mg/L
Cations
mg/L
HCO3-
58.4
Ca+2
15.0
SO4-2
11.2
Mg+2
4.1
Cl-
7.8
Na+
6.3
SiO2
13
K+
2.3
NO3-
~1.0
Fe+3
~0.7
Total = ~91.4 anions + ~28.4 cations = ~ 120 mg/L (TDS)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s28
Nutrients – phosphorus
 Essential for plant growth
 Usually the most limiting nutrient in lakes
 Derives from phosphatic rock – abiotic, unlike
nitrogen
 No gas phase, but can come from atmosphere
as fugitive dust
 Adsorbs to soils
 Naturally immobile unless soil is eroded or excess
fertilizer is applied
 Phosphorus moves with sediments
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s29
Nutrients – phosphorus
 Not toxic
 Algae have physical adaptations to acquire
phosphorus
 High affinity (low k)
 APA
 Storage
 Luxury uptake
 Single redox state
 Phosphorus cycle is closely linked to the iron
(Fe) cycle
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s30
Phosphorus – basic properties
 No redox or respiration reactions directly
involved (organisms are not generating energy
from P chemistry)
 PO4–3 highly adsorptive to cationic sites (Al+3,
Fe+3, Ca+2)
 Concentration strongly affected by iron redox
reactions
 Ferric (+3) – insoluble floc
 Ferrous (+2) – soluble, unless it reacts with
sulfide, causing FeS to precipitate
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s31
Phosphorus levels in the environment
 Major factors affecting phosphorus levels,
cycling, and impacts on water quality include:
 Soil properties
 Land use and disturbance
 Transport associated with runoff
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s32
Where does phosphorus come from?
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s33
Phosphorus – external sources
 Nonpoint sources
 Watershed discharge from tributaries
 Atmospheric deposition
 Point sources
 Wastewater
 Industrial discharges
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s34
Phosphorus – nonpoint sources
 Watershed discharges from tributaries
 Strongly tied to erosion (land use management)
 Stormwater runoff (urban and rural)
 Agricultural and feedlot runoff
 On-site domestic sewage (failing septic systems)
 Sanitary sewer ex-filtration (leaky sewer lines)
 Atmospheric deposition
 Often an issue in more pristine areas
 Arises from dust, soil particles, waterfowl
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s35
Phosphorus – point sources
 Wastewater
 Municipal treated wastewater
 Combined sewer overflows (CSOs)
 Sanitary sewer overflows (SSOs)
 Industrial discharges
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s36
Phosphorus – internal sources
 Mixing from anoxic bottom waters with high
phosphate levels is closely tied to iron redox
reactions
 O2 > 1 mg/L – Insoluble ferric (+3) salts form that
precipitate and settle out, adsorbing PO4-3
 O2 < 1 mg/L (anoxic) – ferric ion reduced to
soluble ferrous ion (Fe+2) – allowing sediment
phosphate to diffuse up into the water
 Wind mixing (storms and fall de-stratification)
can re-inject high P water to the surface,
causing algal blooms
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s37
Phosphorus – Lake budget
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s38
Nutrients – phosphorus cycle
 Major pools and
sources of P in lakes
 “Natural” inputs are
mostly associated
with particles
 Wastewater is mostly
dissolved phosphate
 P is rapidly removed
from solution by
algal-bacterial uptake
or by adsorption to
sediments
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s39
Phosphorus cycling – major sources
 Sewage
 Dissolved
 Tributaries and
deposition
 Particulate
 Erosion
 Particulate
 Sediments
 Particulate
and dissolved
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s40
Phosphorus cycling – internal recycling
 Rapid PO4-3
recycling
 Bacterial uptake
 Algal uptake
 Adsorption to
particles
 Detritus
mineralization
 Zooplankton
excretion
 Fish excretion
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s41
Phosphorus cycle – major transformations
 The whole
phosphorus
cycle
Modified from Horne and
Goldman, 1994.
Limnology. McGraw Hill.
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s42
Nitrogen – basic properties
 Nitrogen is relatively scarce in some
watersheds and therefore can be a limiting
nutrient in aquatic systems
 Essential nutrient (e.g., amino acids, nucleic
acids, proteins, chlorophyll)
 Differences from phosphorus
 Not geological in origin
 Unlike phosphorus, there are many oxidation
states
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s43
Nitrogen – biologically available forms
 N2 – major source, but usable by only a few
species
 Blue green algae (cyanobacteria) and anaerobic
bacteria
 Nitrate (NO3-) and ammonium (NH4+) – major
forms of “combined” nitrogen for plant uptake
 Also called dissolved inorganic nitrogen (DIN)
 Total nitrogen (TN) – includes:
 DIN + dissolved organic nitrogen (DON) +
particulate nitrogen
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s44
Nitrogen – general properties
 Essential for plant growth
 Not typically limiting but can be in:
 Highly enriched lakes
 Pristine, unproductive lakes located in
watersheds with nitrogen-poor soils
 Estuaries, open ocean
 Lots of input from the atmosphere
 Combustion NO2, fertilizer dust
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s45
Nitrogen – general properties
 Mobile – in the form of nitrate (soluble), it goes
wherever water flows
 Ammonium (NH4+) adsorbs to soil particles
 Blue green algae can fix nitrogen (N2) from the
atmosphere
 Nitrogen has many redox states and is involved
in many bacterial transformations
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s46
Nitrogen – sources
 Atmospheric deposition
 Wet and dry deposition (NO3- and NH4+)
 Combustion gases (power plants, vehicle
exhaust, acid rain), dust, fertilizers
 Streams and groundwater (mostly NO3-)
 Sewage and feedlots (NO3- and NH4+)
 Agricultural runoff (NO3- and NH4+)
 Regeneration from aquatic sediments and the
hypoliminion (NH4+)
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s47
Nitrogen - toxicity
 Methemoglobinemia – “blue baby” syndrome
 > 10 mg/L NO3--N or > 1 mg/L NO2--N in well
water
 Usually related to agricultural contamination of
groundwater
 NO3- – possible cause of stomach/colon cancer
 Un-ionized NH4+ can be toxic to coldwater fish
 NH4OH and NH3 at high pH
 N2O and NOx – contribute to smog, haze, ozone
layer depletion, acid rain
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s48
Nitrogen – many oxidation states
 Unlike P there are many oxidation states
 Organisms have evolved to make use of these
oxidation-reduction states for energy
metabolism and biosynthesis
-3
0
+1
+2
+3
+5
NH4+
N2
N2O
NO2
NO2-
NO3-
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s49
Nitrogen – bacterial transformations
NH4+-N
Organic N
•Decomposition
NH4+-N
NO3-
•Nitrification
NO3-
N2 (gas)
•Denitrificatio
n
N2 (gas)
Organic N
•Nitrogen fixation
Heterotrophic ammonification or
mineralization. Associated with oxic
or anoxic respiration.
Involves oxygen (oxic). Autotrophic
and chemosynthetic ("burn” NH4+-N
to fix CO2).
Anoxic process. Heterotrophic.
("burn" organic matter and respire
NO3-, not O2).
Some blue green algae are able to
do this.
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s50
Nutrients
– nitrogen
NutrientsThecycle
Nitrogen Cycle
•modified from Horne and Goldman. 1994. Limnology. McGraw Hill.
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s51
Chemical forms of nitrogen in aquatic systems
organism-N + detrital-N
+ dissolved organic-N
Org–N
Ammonium:
basic unit for
biosynthesis
NO3- -
NO3
Nitrate: major
runoff fraction
Dissolved
Fixed or
inorganic-N
available-N
(DIN)
NH4 + +
NH4
NO2- -
NO2
N2 = largest reservoir
but cannot be used by
most organisms
N2
N2O
NO2
0
+1
+2
Nitrite:
usually
transient
•gases
Oxidation state
-3
-2
-1
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
+3
+4
+5
U1-m2/3Part 5-s52
Functionally in the lab using filters…
Total-N = particulate organic-N + dissolved organic-N
+ particulate inorganic-N + dissolved inorganic-N
TN = PN + DON + DIN
Dissolved inorganic-N = [Nitrate + Nitrite]-N + ammonium-N
DIN = NO3-N + NO2-N + NH4-N
Notes:
• Nitrate+nitrite are usually measured together.
• Nitrite is usually negligible.
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s53
Main N-cycle transformations
Assimilation
Assimilation
Denitrification
Mineralization
(algae + bacteria)
Org-N
NO2-
Assimilation
NH4+
NO3-
Nitrification 2
Nitrification 1
(oxic bacteria)
Ammonification
Anammox
N2 - Fixation
Denitrification
(anoxic bacteria)
- Soil bacteria
- Cyanobacteria
- Industrial activity
- Sulfur bacteria
(anoxic bacteria)
N2
N2O
NO2
0
+1
+2
•gases
Oxidation state
-3
-2
-1
Developed by: R.Axler and C. Hagley
+3
Draft Updated: January 13, 2004
+4
+5
U1-m2/3Part 5-s54
Whole lake N-budget
N2
Tribs, GW, Precip
DON, PON, NO3-, NH4+
N2-fixation
Assimilation
NH4
Nitrification
DIN
PON
DON
Mineralization
Mixing
oxic
anoxic
NO3Nitrification
Mineralization
Outflow
Algae
+
NH4+
Sedimentation
Sedimentation
NO3
-
Ammonia
volatilization
NO2-, N2O
NO
diffusion
Burial
Surficial Sediments
Burial
Deep Sediments
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s55
Nutrients – summer vertical profiles
•Oligotrophic
•0
•0
•Eutrophic
T
T
NO3
NH4
O2
PO4
•
NO3
O2
anoxia
PO4
anoxia
Developed by: R.Axler and C. Hagley
NH4
Draft Updated: January 13, 2004
U1-m2/3Part 5-s56
Sulfide and iron – summer vertical profiles
•0
•Oligotrophic
•0
•Eutrophic
T
T
O2
Soluble Fe
O2
•
Soluble Fe anoxia
H2S
H2S
anoxia
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s57
Developed by: R.Axler and C. Hagley
Draft Updated: January 13, 2004
U1-m2/3Part 5-s58