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2 Turbulent Diffusion
TurbulenceTurbulent (eddy) diffusivitiesSimple solutions for instantaneous and continuous sources in 1-, 2-, 3-D.Boundary ConditionsFluid ShearField Data on Horizontal & Vertical DiffusionAtmospheric, Surface water & GW plumes
Turbulence
Turbulent flow (unstable, chaotic) vslaminar flow (stable)Turbulent sources: internal (grid, wake), boundary shear, wind shear, convectionTurbulent mixing caused by water movement, not molecular diffusionTwo-way exchange--contrast with initial mixing (one-way process)
Initial mixing
H
Qo
Q
cb
co
cUa
xs
Kinetic Energy Spectrum
ηππ /2/2 L
Sk = kinetic energy density
k = 2π/λ = wave number
Sk~ε2/3k-5/3
mean flow turbulence
could also use frequency
implied gap
Sk = kinetic energy/mass-wave number [U2/L-1 = L3/T2]
L = size of largest eddy
ε = energy dissipation rate [U2/T = U2/(L/U) = U3/L = L2/T3]
η = Kolmogorov (inner) scale = (ν3/ε)1/4 [L]
Turbulent Averaging
qtqtq
ctctcqtqtqctctc
−=
−=
−=−=
)()('
)()(')()(')()('
cc
Conservative mass transport eq.
Both q and c fluctuate on scales smaller than environmental interest
Therefore average. Two choices: time average, ensemble average; equivalent if ergotic.
( ) cDcqtc 2∇=⋅∇+
∂∂ r
c(t)c Time average
Ensemble average
t
Turbulent Averaging, cont’d( ) cDcq
tc 2∇=⋅∇+
∂∂ r
Expand q and c; time average
( ) ( )( )[ ][ ]
( )( ) cDcqcq
tc
cqcq
cqcqcqcq
ccqqcq
2''
''
''''
''
∇+⋅−∇=∇⋅+∂∂
⋅∇+∇⋅=
+++⋅∇=
++⋅∇=⋅∇
rr
rr
rrrr
rr
0=⋅∇ qr
( )''
''''''''
cw
cwz
cvy
cux
cq∂∂
+∂∂
+∂∂
=⋅∇r
Eddy correlation.
Continuity
Closure problem: we only want but we must deal with c’
c
Turbulent DiffusionInst. flux (M/L2-T)
)'('' 1
ccwcwJ z
+==z
zcLwJ
Lzccc
ccwJ
z
z
∂∂
−=
∂∂
+=
−=
'
)('
12
21
Mean flux (M/L2-T)
Ezz
(2)
(1)w’
z L
cc 'cc +
Eddy (or turbulent) diffusivity
Eddy Diffusivity LuE '~
Structurally similar to molecular diffusivity D, but much larger (due to fluid motion, not molecular motion) => often drop DE is a tensor (9 components, Exx, Exy, etc.) but often treated as a vector (Ex, Ey, Ez)Depends on nature of turbulence; in general neither isotropic nor uniformEddy diffusivity ~ conductivity ~ viscosityIndividual plumes not always Gaussian; but ensemble averages -> Gaussian
Turbulent transport eqn
( )
∑
∑
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=
∂∂
+∂∂
+∂∂
+∂∂
+∇∇=∇⋅+∂∂
rzcE
zycE
yxcE
x
zcw
ycv
xcu
tc
rcEcqtc
zyx
r
Cartesian coordinates;diagnolized diffusivity
How to measure eddy diffusivity
Measure u’, c’, etc. and correlateMeasure something else (e.g. dissipation) that correlates with EMeasure concentration distribution and calibrate E (more later)Model it
Less direct
Models of turbulent diffusion
2'
2'
2'~'
222 wvuTKEkku ++==
turbulent kinetic energy (don’t confuse with wave number)
1) k-L model (two eqn model)
Lofsinks&sources
kofsinks&sources
~E
±⋅⋅⋅=∂∂
±⋅⋅⋅=∂∂
tLtk
Lk
2) k model (one eqn; solve only for k; L is hardwired)
Models of turbulent diffusion, cont’d
2'
2'
2'~'
222 wvuTKEkku ++==
turbulent kinetic energy
3) k-ε model (two eqn model)
ε ~ k/τ ; τ = time scale of turb. ~ L/k 1/2
ε ~ k 3/2/L or L ~ k 3/2/ε
ε−⋅⋅⋅=∂∂
tkε = turbulent dissipation rate:
ε/~~ 2kLkE
εε ofsinks&sources±⋅⋅⋅=∂∂
t
A gazillion analytical solutions
Instantaneous Point Source (Sec 2.2)Instantaneous Line Source (Sect 2.3)Instantaneous Plane Source (Sect 2.4)Continuous Point Source (Sect 2.5)Continuous Line Source (Sect 2.6)Continuous Plane Source (Sect 2.7)
Simple ones, e.g., u = const, given in following
Instantaneous (point) source in 3Dy
M
ut=0 x
( )
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+++−
−=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+++−
−=
−∂∂
+∂∂
+∂∂
=∂∂
+∂∂
ktzyutxMc
kttE
ztE
ytE
utxEEEt
Mc
kczcE
ycE
xcE
xcu
tc
zyxzyx
zyxzyx
zyx
2
2
2
2
2
2
2/3
222
2/12/3
2
2
2
2
2
2
222)(exp
)2(
444)(exp
)(8
σσσσσσπ
π
Instantaneous (line) source in 2D(e.g. extending over epilimnion)
yM/h (or m’)
ut=0 x
( )
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++−
−=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++−
−=
−∂∂
+∂∂
=∂∂
+∂∂
ktyutxhMc
kttE
ytE
utxEEthMc
kcycE
xcE
xcu
tc
yxyx
yxyx
yx
2
2
2
2
22
2/1
2
2
2
2
22)(exp
)2(/
44)(exp
)(4/
σσσσπ
π
Instantaneous (plane) source in 1Dc
u
x(or m’’)M/A at t=0
( )
⎭⎬⎫
⎩⎨⎧
+−
−=
⎭⎬⎫
⎩⎨⎧
+−
−=
−∂∂
=∂∂
+∂∂
ktutxMc
kttE
utxEt
Mc
kcxcE
xcu
tc
xx
xx
x
2
2
2/1
2
2/12/1
2
2
2)(exp
)2(
4)(exp
2
σσπ
π
Continuous (point) source in 3Dy
m&
c(y,z)ux
(or q)
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++−=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
++−=
−∂∂
+∂∂
+∂∂
=∂∂
uxkuzuyumc
uxk
xEuz
xEuy
EExmc
kczcE
ycE
xcE
xcu
zyzy
zyzx
zyx
2
2
2
2
22
2/1
2
2
2
2
2
2
22exp
)2(/
44exp
)(4
σσσσπ
π
&
&
Continuous (line) source in 2D(e.g. extending over epilimnion)
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−=
−∂∂
=∂∂
uxk
xuyuhmc
uxk
xEuy
xuEhmc
kcycE
xcu
yy
yy
y
σσπ
π
2exp
)2(/
4exp
)(2/
2
2/1
2
2/1
2
2
&
&
y
hm /&
c(y)
(or q’)
ux
Continuous (plane) source in 1D
0exp
0exp
2
2
<⎭⎬⎫
⎩⎨⎧
≅
>⎭⎬⎫
⎩⎨⎧−≅
−∂∂
=∂∂
xExu
Aumc
xuxk
Aumc
kcxcE
xcu
x
x
&
&
c
uxAm /& (or q’’)
A few comments re solutions
Spatial integration of point source => line source => plane sourceTemporal integration of inst source => Continuous sourceRelationship between σ’s and E’s found from spatial moments (as before)
Comments, cont’d
c ~ t-1/2, t-1, t-3/2 for instantaneous 1, 2, 3-D sourcesc ~ x-0, x-1/2, x--1 for continuous 1, 2, 3-D sources (difference: negligible Ex)Assumes E’s are constant. If not, E’s are ‘apparent’ (more later)Most common method to determine E is to fit to measured concentration distribution (tracer, drogues)
Boundary Conditions
bndryopenoncgeconstc 0.,., ==Type I:
II
III bndrysolidonncgeconst
nc 0.., =
∂∂
==∂∂u I Type II:
inletatxcEucucge
constncbac
x+
− ⎟⎠⎞
⎜⎝⎛
∂∂
−=
=∂∂
+
00
..
,Type III(mixed):
ImagesA B C
No flux
c = 0
-+
D E∑+= imagereal ccc
Inst. Pt. Source in Linear Shearu(z)
z
M at t=0
( )[ ] ( )[ ]
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
++++
⎭⎬⎫
⎩⎨⎧
+−−
−+
=
∂∂=∂∂=++=
∫
x
zz
x
yy
zyx
z
t
yo
zyx
zyzyo
EE
EE
kttE
ztE
yttE
tzydttux
tEEEt
Mtzyxc
zuyuzytuu
222
22
2
2
02/122/12/3
121
4414
)(21')'(
exp1)()4(
),,,(
//)(
λλφ
φ
λλ
φπ
λλλλ
Inst. Pt. Source in Linear Shear, cont’d
( )
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
=
→
→+=
zy
xx
xx
xx
EzuE
yut
tEEt
EEttEE
222
22
22
12
',large
',small1'
φ
φ
(1) (2) (1) (2)
C ~ t-3/2
C ~ t-5/2
Longitudinal (Shear) Dispersion
(1) differential longitudinal advection
(2) transverse mixing
Ex’ is really a dispersion coefficient
Okubo (1970)
110-9-1
10-8-1
10-7-1
T-2.47
T-1.41
Variation of Peak Concentration with Time
10-6-1
10-5-1
10-4-1
2
2
5
5 10 20
Time (hrs)
CM
AX
/M(m
-3)
50 100 200 500 1000
2
5
2
5
2
2
5
5
Release #3Release #4
August
April -
{ Release #5Release #6
Figure by MIT OCW.
Fluorescent Tracer
Rhodamine WT (red dye; fluoresces orange)Injected as neutrally buoyant liquidFlow thru or in situfluorometer (I ~ c)Detection ~ 10-10
Lamp
Many wavelengths of light
Specific wavelengths of light
Cuvette or sample cell
Emission filter
Light detector
Wavelengths specific to the compound
Wavelengths created by thecompound, plus stray light
Digital readout
Excitation filter
555 nm
580 nm
Figure by MIT OCW.
ExampleD.L.
D.L.
Start
D.L.
10
50
1000
A
A'
C
500250
100100
107.7 ppb100
End
B
B'
C'
Figure by MIT OCW.
Injected as gas dissolved in waterSampled with Niskin bottle or equiv (profiles collected w/ Rosette sampler)Analyzed w/ shipboard GC w/ electron captureDetection ~10-17
SF6
Vent
Carrier Gas Column
Injection Port
Oven
Detector
Reference
Sensing
Recorder
Figure by MIT OCW.
North Atlantic Tracer Release Experiment (NATRE)
Mass of SF6: 139 kgLocation: 1200 km W of Canary Is.Depth = 310 mTime: 5-13 May, 1992References:
Ledwell et al., Nature, 1993Ledwell et al., JGR, 1998
Images: Kim Van ScoySix Months After Release
26 N100 km
25 N
24 N31 W 30 W 29 W
Two Weeks After Release
Release Pattern
20 km
Figures by MIT OCW.
NATRE, cont’d
Images: Kim Van Scoy
Six Months After Release
26 N100 km
25 N
24 N31 W 30 W 29 W
One Year After Release
25
300 kmLa
titud
e
30N
20
Longitude3540 30 25 W
Figure by MIT OCW. Figure by MIT OCW.
Drogues (drifters)
Floats w/ large drag at constant depthHave flag or periodically rise to surfacePosition viewed from above or recorded using GPS
Horizontal DiffusionHistorically analyzed using vertical line source in cylindrical coordinates (rather than x, y)
y Actual patch
x
Equiv. circular patch
rx, y are relative (to center of mass) coordinates
injection
Cylindrical Coordinates, cont’d
tEr
r
ktr
r
kt
ro
r
ryx
ryx
rr etEehMeehMc
Edrrc
drrcr
MM
hMm
vu
EEE
ryx
42
0
0
2
22
222222
2
2
2
4)/()/(
42
2
/'
0
−−−−
∞
∞
==
===
=
==
==
=+=>=+
∫
∫
ππσ
π
πσ
σσσ
σ
Diffusivity assumed horizontally isotropic, independent of coordinate system
vertical line source
If Er is const. (or treated as such)
4 (vs 2) because σr2 = 2σx
2
Gaussian; if Er = const cmax ~ t-1 but obs show cmax ~ t-2 or t-3
NATRE
HorizontalDiffusionDiagram(Okubo 1971)
107
108 100 m
1 km
10 km
100 kmt2.3
107106105
t (sec)104103
Hour Day Week Month
109
1010
1011
1012
σ r (c
m2 )
2
1013
1014
Rheno
North Sea
Off California
Off Cape Kennedy
1964 V1962 III1962 II1961 I
New York Bight
# 1# 2# 3# 4# 5# 6
# a# b# c# d# e# f
Banana river
Manokin river
Figure by MIT OCW.
Horizontal Diffusion Summary
15.1
15.1
34.12
34.22
017.0
085.0
006.04
011.0
l=
=
==
=
r
rr
rr
r
E
E
tdt
dE
t
σ
σ
σcgs units; some data from pt source, some from line source (not quite proper but…)
rσ4=l arbitrary length scale of patch
Example
100 kg of paint spilled in Mass Bay over a depth of 10m; how widely will it have spread in one week?
σr2 = 0.011 t2.34 = 3.7x1011 cm2 = 3.7 x 107 m2
σr = 6000 m
Peak concentration?
22 /2
/rrkt
r
eehMc σ
πσ−−= k = r = 0; h = 10m; M = 100 kg; t
=86400x7=600,000 s
c = 8.6x 10-8 kg/m3 = 8.6 x 10-5 mg/L
Gaussian fit; actual peak may be higher
σr2=3.7x1011 cm2
σr = 6000 m
Figure by MIT OCW.
107
108 100 m
1 km
10 km
100 kmt2.3
107106105
t (sec)104103
Hour Day Week Month
109
1010
1011
1012
σ r (c
m2 )
2
1013
1014
Rheno
North Sea
Off California
Off Cape Kennedy
1964 V1962 III1962 II1961 I
New York Bight
# 1# 2# 3# 4# 5# 6
# a# b# c# d# e# f
Banana river
Manokin river
A few more comments
Three ways to relate tracer spreading: σ(t), E(t), E(σ)E(σ) => scale dependent diffusion.Not truly stationary => ensemble average not same as individual realization (absolute vs relative diffusion; more later)
is arbitrary; others choose rσ4=l
σσ 12,5.3 == ll
A few more comments, cont’d
tE
dtdE
rra
rr
4
42
2
σ
σ
=
= Diffusivity
σr2
Ave slope ~ Ea
Local slope ~ E
σr2 ~ t2.34
Apparent
diffusivity
tRichardson’s 4/3 law
4/3 rather than 1.15; theoretical (but not empirical) basis
3/43/1~ lεE
Interpretation of Scale-dependent horizontal diffusivity & 4/3 law
Eddy Soup: As patch increases in size it encounters eddies of increasing size (eddies smaller than patch spread patch while larger eddies merely advect it)4/3 Law interpreted as shear dispersion:
( )
3/4
3/2322
2
2
~
~~~
~1
σ
σσσ
φ
E
ttdt
dE
tEt
=>=>
>>
Interpretation of 4/3 law, cont’d4/3 law in inertial sub-range
ηππ /2/2 L
Sk = kinetic energy density
k = wave number
Sk~ε2/3k-5/3 Inertial sub-range
E = diffusivity ~ u’L
ε= dissipation rate ~ dk/dt ~ u’2/t ~ u’2/(L/u’) ~ u’3/L = const
u’ ~ L1/3
E ~ u’L ~ L4/3
Summary
Fickian Okubo 4/3 Law Gen’l
σ2(t) σ2 ~ t σ2 ~ t2.34 σ2 ~ t3 σ2 ~ tq
E ~ tq-1
E ~σ(2q-2)/q
E(t) E~const E ~ t1.34 E ~ t2
E(σ) E~const E ~ σ1.15 E ~ σ4/3
Absolute vs Relative Diffusion
y
σu
Absolute vs Relative Diffusion
σ
y
u
Absolute vs Relative Diffusion
y
Σσu
Absolute diffusion (Σ2) > Relative diffusion (σ2); ratio decreases with time
Do the values of Er differ (and if so, which is right)?
T(z)h
A B Csmall point source line source
drogue of dye of dyecluster
z
Do the values of Er differ (and if so, which is right)?
h T(z)
A B Csmall point source line source
drogue of dye of dyecluster
z
Er < Er < Er
(drogue) (point) (line)
Okubo et al. (1983)
103
Time (seconds)
107
108
σ r (c
m2 )
109
104 105
DyeDrogue
March 10, 1981
Figure by MIT OCW.
OK, the values of Er differ, but which is right?
h T(z)
All can be right
Key: use the same equation for modeling as calibration
z
⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂∂
+∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂∂
+∂∂
xcE
xxcu
tc
cEzx
cExx
czutc
xave
zzx)(
Shear & diffusion excluded from g.e. include effects in Ex calibration => use line source of dye
Vertical shear & diffusion included explicitly in g.e. => don’t want them influencing Ex => use drogues
Diffusivities in numerical models (with finite grid sizes)
2/1222
5.0⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∆∆=yv
xu
yv
xuyxEh α
Smagorinski and Lilly (1963)
α = Smagorinski coefficient (0.1-2.0)
α = 0.16 theoretically; higher empirical values account for vertical shear
Vertical Diffusion
Fit to large scale property distributions
Flux gradient method (lakes & reservoirs)Upwelling diffusion (ocean)
Measured rate of spread of tracer second momentRates of measured dissipationOthers
Decreasing time scale
Flux-gradient methodz
Below depth of other sources/sinks, thermal energy increases only by turbulent diffusionApplicable to relatively long time steps (e.g. weeks or more)
T
z
z
zTzA
dztzTzAt
tzE
∂∂
∂∂
=∫
)(
),()(),( 0
z
A(z) t
North Anna Power Station (WE2-1)
Pond 1
Elk Creek
0 5,000 10,000 FT.
Millpond Creek
Waste HeatTreatment Facility
Dike 1
Dike 2
ACL
N
NMeteorologicalTower
Lake Anna
Dike 3
Pond 2
Dam
Pond 3
North Anna Power Station
Figure by MIT OCW.
Temperature and DO profiles
June-August average Ez = 0.11 m2/d (0.013 cm2/s)
0.14 m2/d (0.016 cm2/s)
0.46 m2/d (0.053 cm2/s)
Pre-operational
< one unit
~ two units
Temperature
10
20
0
10
1413 15
98 87
01
8
10
778910 65
3
35
28 2622
12
14
18 25 10 8 789
41
0 2 35 2
27
19
April May June July Aug Sept Oct April May June July Aug Sept Oct
Dissolved Oxygen
Temperature Dissolved Oxygen
Temperature Dissolved Oxygen
10
20
0April May June July Aug Sept Oct April May June July Aug Sept Oct
10
20
0April May June July Aug Sept Oct April May June July Aug Sept Oct
31
29
2725
16
12
8
1976
1982
1983
Depth (m)
Figure by MIT OCW.
Dominion Power Co.
Vertical Diffusion from NATRE
Ledwell, et al., 1993 yr)(2/scm17.0
mo)(6/scm11.02
2
2
≅
≅=t
E zz
0.0-60
-40
-20
0.2
Concentration (scaled)
Hei
ght (
m)
0.4 0.6 0.8 1.0
0
20
40
60
00
100
50 100 150 200Time (days)
Seco
nd m
omen
t (m
2 )
200
300
400
t
σz2
σz
Nov
Oct
May
Figure by MIT OCW.
Measured dissipationFrom previous discussion
ηππ /2/2 L
Sk = kinetic energy density
k = wave number
Sk~ε2/3k-5/3Inertial sub-range
Turbulent velocities
generated by mean flow
Dissipated by molecular diffusion
Turbulent temperature variations similar to turbulent velocity variations
Temperature Micro-profile
<T>
T = <T> + T’
T(z)
Measured with temperature microstructure probe; resolution < 1 mm
zGeneration (of temp variance)
~ Ez (d<T>/dz)2
Dissipation (of temp variance)
~ κ (dT’/dz)2
Ez = turbulent eddy diffusivity
κ = molecular thermal diffusivity
Formulae based on measured dissipation
Osborn-Cox (1972), Sherman-Davis (1995)
( )
( )2
2
2
2
/
)/'(
)/'(2
/2
zT
zTIE
zTI
zTE
zz
z
z
∂∂
∂∂=
∂∂=
∂∂=
κ
κχ
χ
Osborn (1980)
χ = temp variance dissipation rate [K2s-1]
T(z) = <T> + T’
κ = molecular thermal diffusivity [m2s-1]
I ~ 3 (accounts for gradient in T’ in 3 directions)
N2 = (g/ρ)(dρ/dz) [s-2]
ε = TKE dissipation rate [m2s-3]
γmix = const <= 0.22N
E mixz
εγ=
Examples
Stevens, et al. 2000
20
Dep
th (m
)
4024 25.5
σt (kg/m3)
Profiles of (a) density and temperature gradient, (b) KT, (c) ε and (d) centered-displacementlengthscale LC. The estimates of KT and ε include low-pass filtered versions.
27
0-2 0 2
dT0/dz
-7log10KT (m2s-1)
0 -12 -6-9
log10(ε, m2s-3) LC (m)0.1 101
Figure by MIT OCW.
Langmuir Circulation
Oil Streaks
Wind
Figure by MIT OCW.
Formulae for EzOpen waters, near surface
Ichiye (1967); z = depth; Hw, Tw, Lw = significant wave height, period and length
wLz
w
wz e
THE /4
2028.0 π−=
In presence of stratification and shear
( )2
2/3
//)/(
3101
dzdudzdg
Ri
RiEE zoz
ρρ=
⎥⎦⎤
⎢⎣⎡ +=
−
Munk & Anderson (1948);
Ri = gradient Richardson no
Ezo = value at neutral stratification
Formulae for EzStratification only (near surface)
Koh and Fan (1970)
[Ez in cm2/s; dρ/dz in g/cm4]zEz ∂∂
=−
/10 6
ρ
Stratification only (deep waters)
Broecker and Peng (1982)
[Ez in cm2/s; dρ/dz in g/cm4]zxEz ∂∂
=−
/104 9
ρ
Typical ocean
11.0 ≅≅ zz EE (local) (basin average) Ez in cm2/s
Formulae, cont’dRivers
u* = friction velocity, h = water depth, z = height above bottomhuE
hzzuE
z
z
*
*
07.0
)/1(
=
−= κ
Estuaries
2/2
23
22
)1()(
)1()(
−−
−
+−
+
+−
=
RieTH
hzhz
Rih
zhzuE
wLz
w
w
z
βζ
βη
π
Pritchard (1971); η = 8.59 x 10-3, ζ = 9.57 x 10-3, β = 0.276
u = mean tidal speed
Koh and Fan (1970) 100(dρ/dz) (g/cm4)
10-70.01
0.1
1
10
10-6
102
Verti
cal D
iffus
ion
Coe
ffic
ient
, Ez
(cm
2 /se
c)
Density gradient, l (m-1)ρ
-
103
10-5 10-4
Natre
10-3 10-2 10-1
zEz ∂∂
=−
/10 6
�
zxEz ∂∂
=−
/104 9
�
dρdz
Kolesnikov 1961
Jacobsen (Defant 1961)Foxworthy 1968 (patch)
Harremos 1967
Foxworthy 1968 (plume)Foxworthy 1968 (point source)
Figure by MIT OCW.
Application: coastal sewage discharge from multi-port diffuser
b = 300 mH = 30 mh = 10 mu = 0.1 m/sNF dilution SN = 100How far ds until SF =10?
(ST=SNSF=1000)Formal solution by Brooks in
Section 2.8; approximate solution follows
bL
x
y
z
H
u
h
Assume
Sewage discharge, cont’d
kmmttux
ststtt
cmcmbbxat
of
for
r
rorfro
14000,14)000,22000,162)(1.0()(
000,162011.0000,120;000,22
011.0000,12;
011.0011.0
000,12010;000,124.06
,0
34.2/1234.2/1234.2/1234.22
==−=−=
=⎥⎦
⎤⎢⎣
⎡==⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡==>=
==≅≅≅=
σσ
σσσ
Contrast with 30 m!
b L
x-xo 0 xf
u
to = 22000s t = 162000sx = u(t-to) = (0.1)(162000-22000) = 14 km
107
108 100 m
1 km
10 km
100 kmt2.3
107106105
t (sec)104103
Hour Day Week Month
109
1010
1011
1012
σ r (c
m2 )
2
1013
1014
Rheno
North Sea
Off California
Off Cape Kennedy
1964 V1962 III1962 II1961 I
New York Bight
# 1# 2# 3# 4# 5# 6
# a# b# c# d# e# f
Banana river
Manokin river
(120m)2 = σro2
SF = 10(σro
2 = 100σro2)
Figure by MIT OCW.
Neglect of vertical diffusion
Reasonable?
scmx
z
E
cmgxz
S
cmg
z
N
o
o
os
/3103
1010
/1031000/0003.0
0003.0
03.0
/00.103.1
27
66
47
3
≅≅
∂∂
=
≅≅∂∂
≅∆
≅∆
≅≅
−
−−
−
ρρ
ρ
ρρρρ
ρρ
σT =
(ρ-1)x1000
ρ
h
3029.7
ρo = 1.000 1.0297 1.030
conservatively small
Vertical diffusion, cont’d
9.1580
10808.10
117000)140000)(3)(2(580
2
8.5)2(121
2
2
22
==
==
+=
+=
≅=
Fv
z
zzoz
zo
S
mcm
tE
mh
σ
σσ
σ
hσo
concentration reduction = 9% of that due to horizontal mixing; even smaller if stronger density gradient chosen
Koh and Fan (1970)
10-70.01
0.1
3
0.3
1
10
10-6
102
Verti
cal D
iffus
ion
Coe
ffic
ient
, Ez
(cm
2 /se
c)
Density gradient, l (m-1)ρ
-
103
10-5 10-4 10-3 10-2 10-1
dρdz
Kolesnikov 1961
Jacobsen (Defant 1961)Foxworthy 1968 (patch)
Harremos 1967
Foxworthy 1968 (plume)Foxworthy 1968 (point source)
Figure by MIT OCW.
Atmospheric, surface water and ground water plumesSimilarities
Same transport equation (porosity included in some GW terms)Scale-dependent dispersion. Similar mechanisms: non-uniform flow (differential longitudinal advection plus transverse mixing)Ex > Ey >> Ez
Differences, too
Atmospheric PlumesModest NF mixing (wind quickly dominates)Often large “point”sourcesTime scales: minutes to daysNon-uniform wind caused by shear and density stratification
Image courtesy of usgs.gov.
Stratification
For examples of plume types, please see:http://www.environmenthamilton.org/projects/stackwatch/plume_types.htm
Typical analysis
Image source for ground level exposureNF mixing handled by virtual elevationCooper and Alley (1994)
H
A
B
h
Region of mathematicaldispersion underground
H h
Real source
Image source
Region of reflection
-H -h
Figure by MIT OCW.
Diffusion diagrams
Turner (1970); Cooper and Alley (1994)
0.1
10
100
σ y, m
eter
s
σ z, m
eter
s
1000
10,000
1Distance Downwind, km
lateralX
10 100 0.11.0
10
100
1000
5000
1Distance Downwind, km
verticalX
10 100
D
F
E
A
CB
BA
C
D E
F
stablestable
Figure by MIT OCW.
STABILITY CLASSIFICATIONS*
*
Surface WindSpeeda (m/s)
< 22-33-55-6> 6
Notes:-a) Surface wind speed is measured at 10 m above the ground.b) Corresponds to clear summer day with sun higher than 60o above the horizon.c) Corresponds to a summer day with a few broken clouds, or a clear day with the sun 35-60o above the horizon.d) Corresponds to a fall afternoon, or a cloudy summer day, or clear summer day with the sun 15-35o.e) Cloudiness is defined as the fraction of sky covered by clouds.f) For A-B, B-C, or C-D conditions, average the values obtained for each.
A = Very unstable, B = Moderately unstable, C = Slightly unstable, D = Neutral, E = Slightly stable,and F = Stable.Regardless of wind speed, Class D should be assumed for overcast conditions, day or night.
unst
able
AA-BBCC
A-Bf
BB-CC-DD
BCCDD
EEDDD
FFEDD
Strongb
Day Incoming Solar Radiation Night Cloudinesse
Moderatec Slightd Cloudy ( 4/8)>_ Clear ( 3/8)>_stable
Figure by MIT OCW.
Groundwater PlumesNo (dynamic) NFDistributed, poorly characterized sourcesMultiple phases (contaminant and medium)Laminar (turbulent fluctuations replaced by heterogeneity)Time scales: months to decades
HeteorogeneityCauses non-uniform flow => macro-dispersionOften poorly resolved: handled stochasticallyPlumes often (very) non-Gaussian
MADE experiments at CAFB
Please see:
http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1408&context=lbnl
Dispersivity, α [L]
λα
λλτ
α
~
~~~ 22 uu
uu
uE =Length scale of hydraulic conductivity correlation
100 102
Scale (m)104
10-2
100
102
HighIntermediateLow
Long
itudi
nal D
ispe
rsiv
ity (m
)
"one tenth"
Reliability
Xu and Eckstein
104
Gelhar, Welty and Rehfeldt (1992) Dispersivity DataFigure by MIT OCW.
Superposition: Puff modelsMIT Transient Plume Model
[ ] [ ]⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧ −
+−
−
=
−+=+=
∑
),(),(
),(),(
exp
),(),,(
),,,(
5.0)2/(cos18.0)2/(cos34.0
2
2
2
2
2
kr
kc
kr
kc
k kr
ko
ttttyy
ttttxx
ttttzm
tzyxc
tvtu
σσ
πσ
πωπω
x
yA B
x
Instantaneous location of end of NF
y
σx= σy
21
1
0.5
0.5
1.5
Figure by MIT OCW. Adams (1995)