4 Estuarine Mixing
Initial concepts: tides and salinity
Tide-resolving models
Tidal-average models
Tracers for model calibration
Mixing diagrams
Residence time
Dual tracers
What is an estuary?
A semi-enclosed coastal body of water
which has a free connection with the
open sea and within which sea water is
measurably diluted with fresh water
derived from land drainage (Pritchard,
1952)
Where the river meets the ocean
Like a river but with tides and salinity
gradients
Tidal motion
Tidal Channel
Ocean
η(t)
2ao
Tt
Head
2ξo
ao = tidal amplitude
2ao = tidal range
Mouth
Tt = tidal period
Gravitational and centrifugal
acceleration (E with M & S)
2ξo = tidal excursion
Ocean range ~ 0.5 m
Coastal waters may have
much larger ranges
Equilibrium tide; moon only
Low
Water surface
M
High
E
High
Low
At any time: 2 high and 2 low tides;
At any location: ~ 2 high and 2 low tides per day
Combined sun and moon
T=6.8d
Lunar
29.5d month
S
27.3d
13.6d
20.5d
Sun and moon aligned (full and new moon) => spring tide;
Sun and moon opposed (1st and 3rd quarters) => neap tide
Because the earth revolves, period of spring-neap cycle =
365d/[(365/27.3)-1] = 29.5 days
Number of full moon’s per year
And because the moon revolves
24.8
E
Lunar day
24 h
M
Lunar day = 29.5 d /(29.5 – 1) = 24.8 hours
Dominant (lunar semi-diurnal tidal) period is 12.4 h
Also a diurnal period
“Side” View
In general a number of tidal
constituents are required to
compose an accurate tidal
signal
L
H
L
Because of the earth’s
declination higher latitudes tend
to experience a single (diurnal)
cycle per rotation
H
“Top” View
Full Moon
Last Quarter
New Moon
Tidal displacement above reference
Moon Phase
First Quarter
Los Angles, California
(outer harbor)
o
o
33 43' N, 118 16' W
12.4 hr
Mixed tide (with
strong semi-diurnal
component; lower
latitude)
5
0
14.7d
St. Michael, Alaska
(on the being sea)
o
o
63 29' N, 162 02' W
24.8 hr
Diurnal tide (higher
latitude)
Spring-neap cycle
5
0
5
10
15
20
25
30
Date in July 1963
Ippen, 1969
Figure by MIT OCW.
Idealized (linear) tidal motion
η(t)
Qf
η (t ) = a cos(ωt )
u (t ) =
Qf
A
2ξo
+ u max cos(ωt + φ )
ξ (t ) = u t +
2ξ o ( x) =
ω = 2π/Tt
u max Tt
sin(ωt + φ )
2π
u max Tt
Tidal excursion
π
Pu ( x) = Vu ,high ( x) − Vu ,lowh
Upstream tidal prism
Pu(x)
x
L
High Tide
Low Tide
2ξο(x)
High Tide
Vu(x)
Low Tide
2ξ(x)
Pu(x)
0
x
L
Now introduce salinity
River
Estuary
Tidal,
Freshwater
Qf
Ocean
Salinity
Intrusion
+
+
+
Mouth
Head of Tide
10
S=0
20
30
S=35 psu
PSU = practical salinity unit,
an operational definition of salinity (mass fraction: ppt, o/oo or g/kg)
Equation of State
ρ = ρ (T ) + ∆ρ ( S ) + ∆ρ (TSS )
⎡
ρ (T ) = 1000⎢1 −
(Gill, 1982; ch 6)
(Also pressure at deep depths)
⎤
T + 288.9414
(T − 3.9863) 2 ⎥
508929.2(T + 68.12963)
⎦
⎣
∆ρ ( S ) = AS + BS 3 / 2 + CS 2
A = 0.824493 − 4.0899 x10 −3 T + 7.6438 x10 −5 T 2 − 8.2467 x10 −7 T 3 + 5.3875 x10 −9 T
B = −5.72466 x10 −3 + 1.0227 x10 − 4 T − 1.6546 x10 −6 T 2
C = 4.8314 x10 − 4
1 ⎤
⎡
−3
∆ρ (TSS ) = TSS ⎢1 −
10
x
⎣ SG ⎥⎦
ρ = kg/m3, T in oC, S in PSU (g/kg), TSS in mg/L
Seawater Density ( σ1 Units )
44
-8
40
-6
σt = 1000*(ρ-1)
-4
36
Temperature (oC)
Fischer, et al. (1979)
-10
48
-2
(ρ in g/cm3)
0
32
2
4
28
6
8
10
24
12
14
20
16
18
16
20
22
12
24
26
8
28
30
4
0
0
4
8
12
16
20
24
SALINITY (% )
28
32
36
40
Figure by MIT OCW.
Seawater Density ( σ1 Units )
44
-8
40
-6
Salt water – Freshwater
density difference ∆ρ o ρ
-4
36
Temperature (oC)
Example:
-10
48
-2
0
32
Ocean salinity
~ 35 psu
Freshwater salinity
0 psu
2
4
28
6
8
10
24
12
14
20
16
18
16
Temperatures 0 to 30C
20
22
12
∆ρ o ρ = [28-0]/1000=0.028
24
26
8
28
(0C)
30
4
0
0
4
8
12
16
20
24
28
SALINITY (% )
Figure by MIT OCW.
32
36
40
= [22-(-4)]/1000=0.026
(30C)
Estuary classification
10
S=0
20
30
S=35 psu
Well mixed: isohaline lines approach vertical (Delaware R)
Partially mixed: isohaline lines slant
Vertically stratified (salt wedge): isohaline lines approach horizontal
(Mississippi R.)
Desire to classify to know what type of model/analysis to use;
several options available; none is perfect
Estuary classification, cont’d
Densimetric Estuary number (Harleman & Abraham, 1966;
Thatcher & Harleman, 1972)
2
Pt = tidal prism; Q f = freshwater flow rate;
Pt Fd
Ed =
Q f Tt
Fd =
Tt = tidal period
uo
g (∆ρ o / ρ )h
Fd is a densimetric Froude number
u o = maximum tidal velocity; h = estuary depth;
∆ρ o /ρ = salt water − fresh water density difference
Estuary classification, cont’d
Estuary Richardson number (Fischer, 1972; 1979)
R=
∆ρ o gQ f / W
~ Ed
ρu t
3
−1
W = estuary width;
u t = RMS tidal velocity ≅ 0.71u o
R ~ potential energy rate/kinetic energy rate
R < 0.08
well-mixed
0.08 < R < 0.8
partially stratified
0.8 < R
vertically stratified (salt wedge)
Example later
Estuary classification, cont’d
The definitions are related
E d ~ R −1 ~
ut
3
u f ud
2
Each involves 3 velocities:
u t = RMS tidal velocity
Tends to mix estuary
u f = fresh water velocity = Q f /A
Tends to stratify estuary
u d = density velocity = g(∆ρ o /ρ )h Tends to stratify estuary
Hanson-Rattray (1966)
10
10
-1
3.3
x
10
-2
P
=
10
P=
1
x
„
3.
3
x
10
-4
Increases w/ P, decreases w/ Fm
=
„
P
Fm = 10
-3
10
=
P
-4
-3
10
x
3.
3
=
P
10-2
10-3
1
10
us
uf
salinity stratification
δS S o = ( S b − S s ) S
Fm = 10
10
=
P
Fm = 10
-2
-2
-3
-1
Fm = 10
P
=
3.
3
δS -1
S0 10
Fm = 1
< - Salinity stratification
Semi-empirical
Predicts
-1
2
2
10
Velocity stratification ->
Figure by MIT OCW.
10
3
Velocity stratification
average surf vel /
u s /u f =tidal
tidal and depth aver vel
Decreases w/ Fm
uf
uf
; Fm =
P=
ud
ut
Tide resolving models
Well-mixed (1-D) estuary
∂c
∂c 1 ∂ ⎛
∂c ⎞ q L (c L − c )
′
+ ∑ ri + ∑ re
+ u (t )
=
⎜ AE L (t ) ⎟ +
A
∂t
∂x A ∂x ⎝
∂x ⎠
Major difference between river and well-mixed estuary are
1) u is time-varying, 2) EL is constrained by reversing tide.
Look at 2) first
Characteristic dispersion time scales
EL ~ Uc2Tc ~ u*2Tc
(Fischer et al., 1979)
For rivers, two possible time scales, Tc:
„
Ttm ~ B2/ET and Tvm ~ h2/Ez
„
Ttm >> Tvm => EL ~ u*2 Ttm
(after transverse mixing)
For estuaries, additional possibility: Tc = Tt/2
„
Ttm >> Tt/2 ~ Tvm => EL ~ u*2 Ttm or u*2 Tt/2
Previous example, B = 100 m, H = 5 m, u = 1 m/s
Tvm = 750 s, Ttm = 34000 s, Tt/2= 22000 s (6.2 h)
Dispersion in reversing flow
“narrow” channel
B
t=0
Tc=Ttm
Dispersion governed by Ttm,
0.5Tt
2
u* B 2
EL ~
ET
Dispersion in reversing flow, cont’d
“narrow” channel
B
t=0
Ttm
Dispersion governed by Ttm, E L ~
2
u* B
ET
0.5Tt
2
“wide” channel
B
l ~ ET Tc
t=0
Tc = 0.5Tt
2
2
u* ET Tt
2⎛ l ⎞
E
~
u
T
~
⎟ t
Dispersion governed by Tt,
* ⎜
L
B
B2
⎝ ⎠
2
Effects of reversing u(t)
Mass continuously injected at x = 0
C
land
ocean
High
tide
Low
tide
x
0
2ξo
An actual simulation
Harleman, 1971
1.0
0.5
N = 400.5
N = 400.0
L.W.S
H.W.S
N = 30.5
N = 30.0
0.0
-2.0
-1.5
-1.0
-0.5
0.0
0.5
Figure by MIT OCW.
1.0
1.5
2.0
x / 2ξ o
Continuous injection at x = 0; output after 30, 400 tidal
periods (high slack) and 30.5 and 400.5 tidal cycles (low slack)
Tidal-average models
Perhaps we don’t care to resolve intratidal time-dependence
Strong non-uniformities prevent
resolution of intra-tidal variability
Long term calculations more efficient
with tidal-average time step
However, averaging obscures physics
Tidal-average models, cont’d
Analogous, in principle, to time and cross-sectional averaging
u = u + u ′′′
Triple bars imply tidal average
c = c + c ′′′
Insert into GE and tidal-average
∂c
∂ c 1 ∂ ⎛⎜
∂ c ⎞⎟
AE L ⎟ + ∑ ri + ∑ re
+u
=
⎜
∂t
∂x A ∂x ⎜
∂x ⎟
⎠
⎝
Tidal average Tidal average
velocity
disp coef
Structurally similar to
equation for river
transport => similar
solutions
Tidal average dispersion
Tidal pumping (shown)
„
„
Ebb
„
Asymmetric ebb (a) & flood (b)
Tidal averaging => mean
velocity (c)
Trans mixing + trans velocity
gradients => dispersion!
Similar drivers
Flood
„
„
„
Net
Tidal trapping
Coriolis + density
Depth-dependent tidal reversal
EL ~ (2ξo)2/Tt
General result
A
B
C
c
Conservative Tracer;
3 injection locations
x
c
Non-conservative
tracer; middle location
x
c
Non-conservative
tracer; 3 locations
x
Comments
For conservative tracer, c(x)
„
„
Is independent of xd for x > xd
Decrease with xd for x < xd
If you must pollute, do it downstream
(more discussion later)
Several specific solutions in notes
Conclusion applies loosely even if not 1-D
Signell, MWRA
(1999)
One example
m&
q" =
A
Rectangular channel; no through flow
0=
d ⎛ dc ⎞
⎜ E L ⎟ − kc
dx ⎝ dc ⎠
E L ~ ( 2ξ o ) 2 / Tt = αx 2
2
dc
2d c
− kc
0 = 2αx + αx
2
dx
dx
0
xd
L
Solution
q ′′ ⎡ x −1 2−κ 2
x −1 2+κ 2 ⎤
c + ( x, x d ) − c L =
⎢ 1 2−κ 2 − κ 1 2−κ 2 ⎥
ακ ⎣ x d
L xd
⎦
x > xd
q " ⎡ x −1 2+κ 2
x −1 2+κ 2 ⎤
c _( x, x d ) − c L =
⎢ 1 2+κ 2 − κ 1 2−κ 2 ⎥
ακ ⎣ x d
L xd
⎦
x < xd
κ = 1 + 4k / α
x
WE4-1 Proposed relocation of
Gillette’s Intake
Proposal to shorten Fort Point Channel
as part of the Big Dig threatened to limit
Gillette’s cooling water source
Details
Boston Harbor
1700m
1700m
2ao=2.9 m;
h=6 m;
k=0.1 day-1
700
600
Discharge (xd)
Intake (xi)
400
Qo = 1.4 m3/s;
∆To = 6C
0
“Existing” Channel
“Modified” Channel
Proposed remedies: move discharge and/or intake downstream.
How far?
Results of analysis
3
Temperature (C)
2.5
Existing
Mod Chan
2
MC + Disch
1.5
1
0.5
xd
0
0
400 m
500
1000
1500
2000
Distance (m)
Existing: Ti (x=600) ~ 0.8C; Modified: Ti ~ 2.4C
Moving intake 400 m downstream (x=600) yields Ti ~ 0.8C
Moving discharge 300 m downstream (x=900) also yields Ti ~ 0.8C
Tidal Prism Method
Treats whole channel as single
well-mixed box
High tide
Low tide
Qf
Mass that leaves on ebb does
not returns
Except for harbors/short
channels, this overestimates
flushing; underestimates c.
Hence common to “discount” P
by defining the effective volume
P’ of “clean” water. E.g., P’ =
0.5 P
Formal ways to compute return
factor using phase of circulation
outside harbor
Pf
= Qf
Tt
Pctp
Tt
= m&
m& Tt
ctp =
P
P = total tidal prism
f = “freshness” =(So-Sn)/So
Modified Tidal Prism Method
Divides channel into segments of length 2ξo
Assumes EL = (2ξo)2/Tf <=> net ds transport during Tt is Pn
2ξo,n
Pn
Qf
High tide
Low tide
Vn
Pn f n
= Qf
Tt
Pn cn
= m&
Tt
cn =
Vn+1=Vn+Pn
m& Tt
Pn
fn = “freshness” =(So-Sn)/So
mass injected continuously upstream of
section n (behaves like freshwater)
Comments
Modified Tidal Prism Method has been
modified and re-modified many times
Ad-hoc assumption => not always
agreement with data
Non-conservative contaminates reduced in
concentration by χ
r
1 − (1 − r )e − kTt
r = 2a / h
χ=
Salinity as tracer to measure EL
Steady, tidal average flow
d
(u f AS ) = d ( AE L dS )
dx
dx
ds
Integrate with
S = dS/dx at head, x=0
EL =
uf S
dS / dx
Example: Delaware R (WE 4-2)
Measured salinity profiles
5.0
10.0
8.0
14.0
12.0
18.0
16.0
22.0
20.0
7.0
26.0
24.0
28.0
8.0
9.0
20.0
40.0
60.0
DISTANCE FROM BAY MOUTH (km)
100.0
120.0
12
4
5.0
6.0
6
10
18
7.0
3.0
22
4.0
26
14
8.0
9.0
60.0
40.0
0.0
20.0
DISTANCE FROM BAY MOUTH (km)
November 1987
0.0
Should it be?
What is EL?
22
18
14
6
24
10
20
9.0
12.0
140.0
0.0
18.8
Salinity profiles show
river to be well-mixed.
26
4
8.0
11.0
80.0
20.0
12
7.0
11.0
100.0
2
6.0
10.0
120.0
40.0
16
5.0
10.0
12.0
140.0
24
DRBC RIVER MILES
55.9
37.3
74.6
2.0
24
20
12
60.0
80.0
C&D
16
0.0
18.8
Cheater
C&D
8
2
26
April 1987
PRESSURE (db)
PRESSURE (db)
4.0
Cheater
2.0
3.0
DRBC RIVER MILES
55.9
37.3
16
DISTANCE FROM BAY MOUTH (km)
October 1986
74.6
8
9.0
12.0
140.0
0.0
22
8.0
11.0
80.0
18
7.0
11.0
100.0
14
10
0.0
18.8
20
4
6.0
10.0
120.0
6
2
5.0
10.0
12.0
140.0
Cheater
3.0
C&D
2.0
C&D
6.0
4.0
DRBC RIVER MILES
55.9
37.3
74.6
4.0
2.0
6.0
0.0
18.8
PRESSURE (db)
PRESSURE (db)
4.0
Cheater
2.0
3.0
DRBC RIVER MILES
55.9
37.3
74.6
28.0
8
120.0
100.0
80.0
60.0
40.0
DISTANCE FROM BAY MOUTH (km)
April 1988
20.0
0.0
Figure by MIT OCW.
Kawabe et al. (1990)
Å Head
3.0
PRESSURE (db)
4.0
Cheater
2.0
C&D
74.6
DRBC RIVER MILES
55.9
37.3
8
12
2
4
5.0
6.0
6
16
10
18
7.0
Mouth (ocean)
0.0
18.8
24
20
22
26
14
November 1987
8.0
9.0
10.0
11.0
~ h (m)
EL =
uf S
12.0
140.0
120.0
100.0
80.0
60.0
40.0
20.0
0.0
Figure by MIT OCW.
DISTANCE FROM BAY MOUTH (km)
≅
(Q f / A) S
dS / dx
∆S / ∆x
(260)(8)
≅
(1.5 x10 4 )(8) / 20000)
≅ 350 m 2 / s
Qf = 260 m3/s;
A = 1.5x104 m2;
S = 8 psu (80 km);
∆S/∆x = (12-4) psu/20 km
(70-90 km)
Should river be well-mixed?
∆ρ
R=
≅
ρ
g
ut
Qf
W
3
(0.025)(10)(260) / 4000
≅ 0.02 < 0.08
3
1
Yes!
Box models
Qf
Qf
Qf
Q12
c1
Qf
Q23
c2
Q34
c3
c4
Q f + f 2 Q1, 2 = f 1 (Q1, 2 + Q f )
f1 (Q1, 2 + Q f ) + f 3 Q2,3 = f 2 (Q1, 2 + Q2,3 + Q f )
f 2 (Q2,3 + Q f ) + f 4 Q2,3 = f 3 (Q2,3 + Q3, 4 + Q f )
n equations in n unknowns; boxes dictated by geometry
Salinity as direct measure of c
Qf
x
0
xd
Use measured salinity distribution S(x) resulting from
river discharge Qf entering at head (x=0) to infer
concentration distribution c(x) of mass entering
continuously at downstream location xd.
L
Qf
x
0
xd
L
S/So
1
S/So
0
0
xd
L
Qf
x
0
xd
S/So, f
f = (So-Sx)/So
f
1
L
Freshness: ND concentration
of fresh water
S/So
0
0
xd
L
Qf
x
0
xd
L
Effective downestuary transport rate, Qeff
Qeff =
Qeff
Qf
fx
Hypothetical flow rate necessary to transport
freshness downstream by advection only (no
tidal dispersion)
df
f = Q f = Q f f − EL A
dx
Qeff = Q f −
E L A df
f
dx
Qeff really accounts for both
advection and dispersion
Qf
x
0
xd
S/So, f
Qeff = Qf/f
L
Qeff/Qf
Qeff/Qf
4
f
1
S/So
2
0
0
0
xd
L
Qf
x
0
xd
L
Downstream from xd, mass is transported like freshness
Qeff c = m&
Qeff f = Q f
cx
m&
=
fx Qf
⎛ So − S x
c x = ⎜⎜
⎝ So
Concentration at xd
⎞ m&
⎟⎟
⎠ Qf
⎛ So − Sd
c d = ⎜⎜
⎝ So
⎞ m&
⎟⎟
⎠ Qf
Qf
x
0
xd
S/So, f
⎛ S − Sx
c x = ⎜⎜ o
⎝ So
L
⎞ m&
⎟⎟
⎠ Qf
Qeff/Qf
Qeff/Qf
4
c
f
1
S/So
2
0
0
0
xd
L
Qf
x
0
xd
Upstream from xd, mass is transported like salinity
cx S x
=
cd S d
S o − S d m& S x
cx =
So Q f Sd
cd
L
Qf
x
0
xd
L
S/So, f c = S o − S d m& S x
x
So
Qeff/Qf
Qeff/Qf
Q f Sd
4
c
f
1
S/So
2
0
0
0
xd
L
(Conservative) Mixing Diagrams
Concentration of
conservative contaminant
discharged at head (using
freshness as tracer)
c
cmax
⎛ So − S x
c x = ⎜⎜
⎝ So
x
c x = a − bS x
0
0
⎞ m&
⎟⎟
⎠ Qf
So
m&
= c max
S a=
Qf
aka C-S (or T-S, etc.) diagram, or property-salinity diagram
Uses for Property-S diagrams
Determine end-member concentration
& )
and loading (So, Qf known, but not m
Identify extraneous sources (we think
we know m& = Q f co but cmax > co)
Distinguish different water masses
Predict quality of mixed water masses
Detect non-conservative behavior
Determining end member c
1) Extrapolate to get cmax
c
2)
m& = Q f c max
3) If cmax > measured co,
difference is extraneous
source(s)
0
0
So
Distinguishing water masses
N-S diagram for Massachusetts Bay, Kelly (1993)
Used to identify coastal water vs offshore waters
Non-conservative
behavior
c
cmax
x
0
S
0
So
Non-conservative
behavior
c
cmax
+
x
-
0
S
0
So
Non-conservative
behavior
c
cmax
+
x
-
0
S
0
So
Note that conservative mixing curve is only linear if
conditions are steady and there is a single source
Two conservative sources look
like one NC source
c
1
2
0
0
So
Two conservative sources look
like one NC source
c
1
1+2
2
0
0
So
Transient Conditions
WE4-2 Nitrate-Salinity
diagrams in Delaware R
200
X
Fall
100
Ciufuentes, et al. (1990)
200
Summer
Solid lines are predictions for
conservative tracer & salinity at 4
times (not linear because river flow
varies in space and time)
Nitrate (µM)
100
200
Spring
100
Symbols are data for nitrate &
salinity
200
Water
100
Why the discrepancy in fall, spring?
0
16
Salinity (% )
32
Figure by MIT OCW.
Residence times
Why? Compare with k-1
„
„
tres >> k-1 => reactions are important
tres << k-1 => reaction not important
Also to determine if model has reached
steady state
Approaches
„
„
Continuous tracer
Instantaneous tracer
Related time scales
Continuous tracer release; c(x,y,z)
monitored after steady state
V
t res =
∫ cdV
0
m&
=
M
m&
SS inventory over renewal rate;
heuristic interpretation
m&
Types of Tracers
Advantages and Disadvantages of each
m&
Deliberate tracer (e.g., dye)
Tracer of opportunity (e.g. trace metals
from WWTP)
Freshwater inflow (freshwater fraction
approach; residence time sometimes
called flushing time)
V
t res =
∫ fdV
0
Qf
WE 4-4 Trace metals to calculate
residences times for Boston Harbor
Residence Time in Boston Harbor
2500
3.4 Days
10 Days
Zn
2000
Cu
1500
Cr
2 Days
1000
Ni
500
PCB
0
Pb
Cd
0 Hg 20
cV
t res =
m&
(2.5 x10 −6 kg / m 3 )(6.3 x10 2 m 3 )
=
= 3.4d
5
(1.7 x10 kg / yr ) /(365d / yr )
PAH
40
60
80
100
120
140
160
180
Total Load to Harbor (kg/yr)
(Thousands)
Figure by MIT OCW.
Shea and Kelly, 1992
Comments
Ignore re-entries (by convention)
If multiple sources, tres is average time
weighted by mass inflow rate
Assumes steady-state, but “fix-ups”
applicable to transient loading
Residence time reflects injection
location; not property of water body…
unless well mixed, in which case:
c( x, y, z ) = c = const
t res
cV
=
m&
Tres depends on discharge location
A
B
C
V
t res =
c
x
c
x
c
x
tres
A
∫ cdV
0
m&
> tres
=
M
m&
B
> tres
C
Instantaneous Release; c(x,y,z,t)
monitored over time
Unit mass
m&
m*
1
1
0
t
Rate of injection
f * (t ) = −
f*
0
t
Mass remaining in system
dm *
dt
∞
∫ f * (t )dt = 1
0
0
Mass leaving rate
t
Instantaneous release, cont’d
f* is also distribution of residence times (mass leaving no longer
resides). By definition, tres is mean (first temporal moment) of f*
∞
t res = ∫
0
∞
∞
dm&
∞
f *tdt − ∫ tdt = −m * t 0 + ∫ m * (t )dt
dt
0
0
1st moment of f*
0th moment of m*
For mass of arbitrary loading Mo (not necessarily one)
∞
t res =
∞
∫ f (t )tdt ∫ M (t )dt
0
Mo
=
0
Mo
M(t) = mass remaining
f(t) is mass leaving rate
Thus two more operational definitions of residence time: 1st
temporal moment of f(t) and 0th temporal moment of M(t)
WE 4-5 Residence time of CSO
effluent in Fort Point Channel
Rhodamine WT injected instantaneously at channel head on three dates;
results for one survey:
N
Boston
Inner Harbor
18
18
ft
ft
Northern Ave.
Congress St.
Summer St.
18
ft
Gillette
Dorchester Ave.
Broadway
Adams, et al. (1998)
BOS 070
Figure by MIT OCW.
Dye 100
Meters
0
500
Fort Point Channel dye release, cont’d
Total Mass of Dye (Kg)
12
10
∞
8
6
t res =
4
2
0
0
20
40
60
80
100
∫ M (t )dt
0
Mo
≅ 2.7 day
120
TIME (Hours after injection)
Figure by MIT OCW.
tres
Adams, et al. (1998)
Comments
f(t) can be obtained from time rate of
change of M(t); or from measurements
of mass leaving (at mouth)
Residence times for continuous and
instantaneous releases are equivalent
f(t) of f*(t) conveniently used to assess
first order mass loss.
∞
F = ∫ f * (t )e − kt dt
0
F = total fraction of mass that leaves
Mass loss from FPC (%/10hrs)
WE 4-6 Residence time of bacteria in CSO
effluent in Fort Point Channel
(Adams et al., 1995)
40
1.6
35
1.4
30
1.2
f*(t) 1990
25
1
20
0.8
15
0.6
1
10
5
0.4
0.25
0.5
0.2
2
0
50
100
150
0
200
Time (h)
∞
F = ∫ f * (t )e − kt dt
0
1− F
Residence time distributions f(t)
determined from distributions of
m(t).
Indicator bacteria “disappear”
(die or settle) at rates of 0.25
to 2 d-1
What fraction of bacteria would
disappear for 1990 conditions?
Figure by MIT OCW.
Fraction (of viable bacteria) that leave
Fraction that are removed within channel
k=2.0 d-1 => F=0.15 (85% removed); k=0.25 d-1=> F=0.55 (45% removed)
Relative advantages of 3 approaches?
M (t)
C(V)
f (t)
Mo
t
t
V
Instantaneous Instantaneous Continuous
t1 =
V
∞
∞
∫ M (t )dt
0
Mo
t2 =
∫ f (t )tdt
0
Mo
t3 =
∫ cdV
0
m&
Amount of tracer (e.g., dye) required?
Effort to dispense?
Number of surveys and their spatial extent?
Total duration of study?
Other related time scales
Flushing time use to describe decay of initial
concentration distribution (convenient for
numerical models); used by EPA for WQ in
marinas (see example)
Age of water (oceanography): time since
tracer entered ocean or was last at surface
(complement of tres)
Concepts often used interchangeably, but in
general different; be careful
Dual Tracers
Used to empirically distinguish fate from transport: introduce two
tracers (one conservative; one reactive) instantaneously. Applies
to any time of water body, but consider well mixed tidal channel
dM c
= −k f M c
dt
dM nc
= −k f M nc − kM nc
dt
Mass of conservative tracer
declines due to tidal flushing
Mass of NC tracer declines due
to tidal flushing and decay
d ⎛ M nc
⎞ = −k ⎛ M nc
⎞ Ratio of masses declines due to
⎜
⎟
⎜
⎟ decay
M
M
c⎠
c⎠
⎝
dt ⎝
M nc
=⎡
M c ⎢⎣
M nc
⎤ e − kt
M c ⎥⎦ o
WE 4-7 Fort Point Channel again
R
2
1.8
1.6
1.4
1.2
Best-fitted line
1
0.8
R
k3 = 0.25 d-1
0.6
Fluorescent pigment
particles (yellow DayGlo
paint) were injected with
dye. Pigment particles
settle as well as flush.
R = (Mp/Mpo)/(Md/Mdo)
k = ksettle = 0.25 d-1
0.4
k = ws/h
0.2
0
20
40
Time (hr)
Figure by MIT OCW.
Adams, et al. (1998)
60
80
ws = kh = (0.25d-1)(6m)
=1.5 m d-1
More in Chapter 9