lake water balance and mass balance - university of...
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Lake Water Balance and Mass Balance
Objectives
Surface-ground water interaction may have substantial influence on water levels and solute concentrations in lakes and wetland ponds.
Water and mass balance of a lake can be seen as an “integrated” measurement of surface-ground water exchange fluxes over the entire lake.
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Objectives
1. Understand the effects of surface-ground water exchange on lake water and mass balance.
2. Estimate the lake-scale average exchange rates from water balance simulation.
Textbook chapterRosenberry and Hayashi (2013. Assessing and measuring wetland
hydrology, In: Wetland techniques, Springer, vol. 1, pp. 87-225).
Lake Water Balance Equation
V: lake water volume (m3)
dV/dt: rate of volume change (m3 d-1)
A: lake water area (m2)
h: water depth (m)dt
dhA
dt
dVQQ outin
=
=−
input output
Pcp ETIS
OSR
SD
2
OS
IG OG
RSD
Pcp: precipitation
IS: stream inflow
IG: groundwater inflow
R: diffuse runoff
ET: evaporation & transpiration
OS: stream outflow
OG: groundwater outflow
Input flux (m3 d-1) Output flux (m3 d-1)
SD: snow drift (in or out)
Qin
Q
h
Simple ‘tank’ model of water balance
Case 1: Qin = const.h = 0 at t = 0
h
t
h
t
Case 2: Qin = 0h = 1 at t = 0
3
Qoutt t
Qout ∝∝∝∝ h →→→→ negative feedback
wa
ter
lev
el
time
Natural systems are usually in the steady-state, averaged over a long term.
Float-pulley system
Staff gauge
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Pressure transducer
Stilling well
Rosenberry and Hayashi (2013. Assessing and measuring wetland hydrology. In: Wetland
techniques)
Precipitation Gauges
Non-recording rain gauge
Tipping bucket rain gauge
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• Inexpensive• Frequent attendance required• No data for rainfall intensity
1 tip = 0.2 mm1 tip = 0.2 mm
Can we measure snow with these?
Hendricks (2010. Intro. to Physical Hydrol.)
Weighing Precipitation Gauge
WindshieldWindshield
Antifreeze Antifreeze solutionsolution
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Weight Weight sensorsensor
Wind-Induced “Undercatch” of Snow
no shield
Catch deficiency =Uncorrected precip.
Actual precipitation
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no shield
with Alter shield
Dingman (2002, Physical hydrology)Singh and Singh (2001, Snow and glacier hydrology)
• Installation in sheltered area• Measure wind speed
Even shielded snow gauges require wind correction.
Archived climate data are often ‘uncorrected’, resulting in inaccuracy and inconsistency.
Example: A recent study by Meteorological Service of Canada (Mekis & Vincent. 2011. Atmosphere-Ocean 49: 163-177) showed substantial differences between corrected and uncorrected precipitation data.
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Climate data interpretation for long-term trend analysis requires a special attention.- What type gauge was used?- Was correction made? How?
uncorrected precipitation data.e.g. Calgary annual pecipitation (1981-2010)
420 mm → 475 mm (13 % increase by correction)
Snow Drift or Blowing Snow
Wind driven transport can move a large amount of snow, particularly on smooth, non-vegetated surfaces.
Snow drift also enhances the sublimation loss of snow.
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sublimation loss of snow.
Do lakes lose or gain snow during blowing snow events?
What is the effect of tall stubble in farm fields?
Rain or Snow?
Automated meteorological stations record total amounts of precipitation, but not rain and snow individually.
Climate models calculate total precipitation and temperature. How can we separate rain and snow??
93.7%
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Cu
mu
lati
ve
fre
qu
en
cy (
%)
10
3.5%
2 oC
0
20
40
60
-30 -25 -20 -15 -10 -5 0 5 10 15 20Daily mean air temperature (°C)
Cu
mu
lati
ve
fre
qu
en
cy (
%)
Data by Alberta Agriculture, Food and Rural Development
Background Information for Evaporation
What is relative humidity?
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Vap
or
pre
ssu
re (
hP
a)
Amount of water vapour in the atmosphere is expressed as partial pressure. Maximum possible amount at a given temperature is called saturation vapour pressure.
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0
20
40
60
0 10 20 30 40
Vap
or
pre
ssu
re (
hP
a)
Temperature (oC)
slope = ∆∆∆∆
The slope of the temperature-
vapour pressure curve (∆∆∆∆) has a special significance in the estimation of evaporation.
Solar radiation has relatively short wavelengths, while the radiation from the earth has long wavelengths.
Radiation: Energy Source for Evaporation
(Christopherson, 2000. Geosystems, Prentice-Hall)
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Net radiation = incoming – outgoing radiation= (incoming SW + LW) – (outgoing SW + LW)
Radiation Balance
Ratio of outgoing SW / incoming SW is called albedo.
Christopherson (2000, Fig. 4-1)
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Lake Energy Balance
Qn + Qa – Qh – Qe ≅≅≅≅ Qw (all terms in W m-2)
Qn: net radiationQa: net advection of energy by
streams (and groundwater)Qh: sensible heat fluxQe: latent heat flux
wind
Qa
Qh Qe
Qn
Qe: latent heat fluxQw: rate of energy storage in
lake water
Qa
Qw ∝∝∝∝ volume ×××× temp. change
Evaporation rate, E (m s-1) is proportional to latent heat flux.Qe = E ×××× density of water ×××× latent heat of vaporization
How is E affected by meteorological conditions?
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Imagine a box over a lake. Each air ‘parcel’ within the box contains numerous molecules. Parcels near the water
Turbulent Flux of Latent Heat and Sensible Heat
surface contain more water vapor than the ones far from the surface.
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As the wind causes turbulent mixing within the box, random motion of the parcels lead to the net upward transfer of water vapor →→→→ Latent heat flux
Does the same principle apply to sensible heat flux?
What controls the magnitude of flux?
Bowen Ratio
Wind speed and temperature (or humidity) controls the flux.Qe ∝∝∝∝ f(u) ×××× (es – ea)Qh ∝∝∝∝ f(u) ×××× (Ts – Ta)
u: horizontal wind speed (m s-1)f(u): wind function (m s-1); e.g. f(u) = a + bu.es: vapour pressure at the lake surface (hPa)ea: vapour pressure in the air above the lake (hPa)T : temperature of the lake surface (°°°°C)
a
Ts: temperature of the lake surface (°°°°C)Ta: air temperature above the lake (°°°°C)
Same wind function for Qe and Qh. Why?
The wind function is complex, dependent on many factors (what are they?), but the ratio of Qh to Qe is relatively simple.
ββββ = Qh / Qe = γγγγ (Ts – Ta) / (es – ea) Bowen ratioγγγγ: psychrometric constant (≅≅≅≅ 0.66 hPa °°°°C-1 at sea level)
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Estimation of Lake Evaporation
From the lake energy balance,
Qh + Qe ≅≅≅≅ Qn + Qa – Qw
Using the Bowen ratio, Qh = ββββQe
∴∴∴∴ Qe = (Qn + Qa – Qw) / (1 + ββββ)
←←←← Available energy
Written in a different form, the Priestley-Taylor equation is:
Qe = (Qn + Qa – Qw) ×××× αααα ×××× ∆∆∆∆ / (∆∆∆∆ + γγγγ)∆∆∆∆: slope of vapour pressure-temperature curveαααα: dimensionless constant
The equation with αααα = 1.26 has been shown to give reasonably accurate estimates of evaporation from shallow lakes and wetlands (Rosenberry et al., 2004, Wetlands, 24:483).
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Measurement of meteorological variables
Rosenberry and Hayashi (2013. In: Wetland techniques)18
Radiation sensor
Evaporation panLangston et al. (2013. Water Resour. Res.,49:5411-5426) 19
Groundwater Exchange with Lakes
Lakes are almost always connected to groundwater.
The amount and direction of groundwater exchange depends on topographic setting, geology, climate, and many other factors.
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factors.
Water balance of some lakes are dominated by groundwater exchange, while other lakes are dominated by surface water inputs and outputs.
Winter et al. (1998. USGS Circular 1139)
Stream Inflow and Outflow
V-notch weirFor lakes with inflow and outflow streams, accurate flow measurement is critical for lake water balance.
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Outflow is controlled by lake water level →→→→ negative feedback
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Flow metering
0
1
2
0.5 0.6 0.7 0.8lake water level (m above bench mark)
str
eam
ou
tflo
w (
m3 s
-1)
Lake O’Hara
Runoff and Stream Inflow: Watershed Hydrology
A
Surface water input (m3) = runoff (m) ×××× Ac (m2)
What controls runoff?
- Climate
- Topography
- Soil thickness
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A: lake area AC: catchment area
AC
- Soil thickness
- Geology
- Vegetation and landuse
-
Basin MorphologyVolume (V) - Area (A) – Depth (h) Relation
A
Vh
23Hayashi and van der Kamp (2000, J. Hydrol., 237: 74)
A ∝∝∝∝ h2
V ∝∝∝∝ h3
A ∝∝∝∝ h0
V ∝∝∝∝ h1
A ∝∝∝∝ h2/p
V ∝∝∝∝ h(1+2/p)
p : parameter representing the slope ‘profile’.
p = 1 for straight slope
p > 1 for concave slope
0
2000
4000
6000
A (
m2)
calculated from
bathymetry
Volume (V) - Area (A) - Depth (h) Model
pp
h
h
p
hAV
h
hAA
/
max
maxmax
/
max
max
/
212
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+
+=
=
Amax: maximum area
hmax: maximum depth
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0 0.4 0.8 1.2 1.6h (m)
0
1000
2000
3000
4000
0 0.4 0.8 1.2 1.6h (m)
V(m
3)
40 m survey points
Simple Lake Water Balance Simulation
dt
dVQQ outin =−
For lakes with negligible snow drift and diffuse runoff,
Pcp ETIS
O
∆∆∆∆V: volume change∆∆∆∆t: time interval
(e.g. 1 day)
Pcp + IS + IG – ET – OS – OG = ∆∆∆∆V/∆∆∆∆t
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OS
IG OG
Pcp, ET, and surface flows can be measured, but groundwater components are very difficult to measure. We will use the water balance equation to estimate net groundwater flow.
IG – OG = ∆∆∆∆V/∆∆∆∆t – Pcp + ET – IS + OS
Simple Spreadsheet Exercise of Water Balance Simulation
We will use Microsoft Excel to demonstrate a simple simulation of lake water balance.
Field data from a study site in Canada will be used as examples.
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Location of Calgary and Banff
Lake O’HaraLake O’Hara
L. LouiseL. Louise
Lake O’Hara Watershed (14 km2)Elevation: 2000-3500 m
BanffBanff
Lake O’HaraLake O’Hara
2 km
N
L. O’HaraL. O’Hara
Image from Google Earth
Lake O’Hara Hydrological Study
Issue: Climate change impacts on glaciers and water resources
Is groundwater a significant part of the hydrologic cycle?
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Lake O’Hara at 2000 m altitude Opabin Glacier at 2500 m
Supplementary reading: Hood et al. (2006, Geophys. Res. Lett., 33, L13405)
Area = 0.26 km2
Max depth = 42 m0
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0 5 10
temperature (oC)
Lake O’Hara Characteristics
Frozen from November to May.
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200 m
Depth contour = 5 m
10
20
30
40
dep
th (
m)
June 26
Jul-28
Aug-24
Evaporation Estimate by Priestley-Taylor Eqn.
Qe = (Qn + Qa – Qw) ×××× αααα ×××× ∆∆∆∆ / (∆∆∆∆ + γγγγ)
Qn: net radiation, measured (photo)Qa: advection by streams, ignored (expected to be minor)Qw: energy storage in lake, from temperature profiles
For June 3-15, 2005, Qn = 72 W m-2 Qw = 23 W m-2
Avg. temp = 4.1 oC →→→→ ∆∆∆∆ = 0.57 hPa K-1
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Avg. temp = 4.1 C →→→→ ∆∆∆∆ = 0.57 hPa K
At 2000 m elev., γγγγ = 0.52 hPa K-1
Assume αααα = 1.26
Qe = 32.3 W m-2
Latent heat (Lv) = 2.49 ×××× 106 J kg-1
Density (ρρρρw) = 1000 kg m-3
E = Qe / (Lv ρρρρw) =
Precipitation and Stream Flow Measurements
Estimated uncertainty in flow measurements ≅≅≅≅ 10 %
31Jaime Hood gauging a stream. Tipping bucket rain gauge.
See the handout for step-by-step instructions.
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Simple Spreadsheet Exercise of Water B
In this computer exercise we will estimate net groIn this computer exercise we will estimate net grobalance equation. It is based on Hood et al. (200
The water balance equation of a lake is;
IG – OG = ΔV/Δt – Pcp + ET – IS + OS
Lake area (A) is a function of water level (h) in gthat A is constant at 0.26 km2.
Your data set contains lake water level h (m) withprecipitation P (mm), estimated daily evaporations-1) and outflow OS (m3 s-1). Note that there are fostreams.
(a) For 03/06/2005 (Row 3), calculate the volumthe lake by multiplying P by the lake area:the lake by multiplying P by the lake area:
Pcp = P mm × 0.001 m mm-1 × (0.26 ×In terms of cell formula, Eq. (2) can be writte
H3 = D3*0.001*C3*1e6/86400
(b) Similarly, calculate the volumetric rate of evmultiplying E by the lake area:
ET = E mm × 0.001 m mm-1 × (0.26 × 1In terms of cell formula, Eq. (3) can be writte
I3 = E3*0.001*C3*1e6/86400
(c) Calculate the rate of storage change:ΔV/Δt = (Change in water level between Ju
Or in terms of cell formula,J3 = (B4 – B3)*C3*1e6/86400
(d) Calculate the net groundwater flow rate IG –K3 = J3 – H3 + I3 – F3 + G3
(e) Repeat the same calculation up to October 17
(f) Plot the time series of h on a chart.
(g) Plot the time series of IS and IG – OG on a sinf (f) i h l d f hfrom (f). Discuss the seasonal trends of these
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Balance Simulation
oundwater input to an alpine lake using the wateroundwater input to an alpine lake using the water 6, Geophysical Research Letters, 33, L13405).
(1)
eneral, but in this simple example we assume
h respect to a local bench mark, daily n E (mm), and daily average stream inflow IS (m3
our inflow streams, and IS is the total of all four
metric rate of precipitation Pcp (m3 s-1) falling on
106 m2) / 86400 s (2)en as:
aporation ET (m3 s-1) leaving the lake surface by
06 m2) / 86400 s (3)en as:
une 3 and June 4) × (0.26 × 106 m2) / 86400 s
OG from Eq. (1) using the cell formula:
7 (Row 139) by copying the cell formula.
ngle chart, compare it with the water level chart i ble variables.
0.5
0.6
0.7
0.8
0.9
6/2 6/17 7/2 7/17 8/1 8/16 8/31 9/15 9/30 10/15
wa
ter
lev
el (m
)
snowmelt period
Lake Water Level (w.r.t. Bench Mark) and Flux
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0
0.5
1
1.5
6/2 6/17 7/2 7/17 8/1 8/16 8/31 9/15 9/30 10/15
flo
w (
m3 s
-1) stream inflow
groundwater
6/2 6/17 7/2 7/17 8/1 8/16 8/31 9/15 9/30 10/15
Lake Solute Mass Balance Equation
Solute mass balance is similar to water balance. Each term is multiplied by the concentration. For example;
CP (kg m-3) ×××× Pcp (m3 d-1) = mass flux (kg d-1) in precip.
C (kg m-3) ×××× V (m3) = total mass (kg) in the lake.
Mass balance equation is:
[CPPcp + CISIS + CIGIG - C(OS + OG) + RXN] ∆∆∆∆t = ∆∆∆∆(CV)
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[CPPcp + CISIS + CIGIG - C(OS + OG) + RXN] ∆∆∆∆t = ∆∆∆∆(CV)
C: Concentration in lake (kg m-3)RXN: Reaction rate (kg d-1)
CPPcpCISIS
COS
CIGIG COG
Solute Mass Balance
[CPPcp + CISIS + CIGIG - C(OS + OG) + RXN] ∆∆∆∆t = ∆∆∆∆(CV)
The concentration of outflow terms is equal to C. What is the underlying assumption?
Why is ET not in the equation?
Reaction term (RXN) represents all other processes. What are those?
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XN
What are those?
- Dissolution/precipitation of minerals
- Biological production (e.g. CO2) and uptake (e.g. N and P)
- Atmospheric exchange
- Diffusive exchange with the sediment
Combining water (WB) and mass balance (MB)
WB: Pcp + IS + IG – ET – OS – OG = ∆∆∆∆V/∆∆∆∆t
MB: CPPcp + CISIS + CIGIG - C(OS + OG) + RXN = ∆∆∆∆(CV)/ ∆∆∆∆t
Two equations can be solved simultaneously for IG and OG.
Example: GW flow through a pond with no surface flow.Conservative tracer, e.g. chloride.
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WB: Pcp + IG – ET – OG = ∆∆∆∆V/∆∆∆∆t
MB: CIGIG – COG = ∆∆∆∆(CV)/ ∆∆∆∆t
Pcp, ET, V, C can be easily measured or estimated. If we have a good estimate of CIG, we can determine IG and OG
on a daily time step.
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0 1 2 3 4 5
Ch
lori
de (
mg
L-1
)
Wate
r d
ep
th (
m) water depth
chloride
Chloride Tracer Experiment in a Proglacial Pond
0 1 2 3 4 5Time (day)
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•Tracer was released.
•Average concentration was determined daily from spatially distributed measurements. 0
200
400
600
800
1 2 3 4 5
GW
flo
w (
m3
d-1
)
Inflow
Outflow
Langston et al. (2013. Water Resour. Res.,49:5411-5426)