2 Turbulent Diffusion
Turbulence
Turbulent (eddy) diffusivities
Simple solutions for instantaneous and
continuous sources in 1-, 2-, 3-D.
Boundary Conditions
Fluid Shear
Field Data on Horizontal & Vertical Diffusion
Atmospheric, Surface water & GW plumes
Turbulence
Turbulent flow (unstable, chaotic) vs
laminar flow (stable)
Turbulent sources: internal (grid, wake),
boundary shear, wind shear, convection
Turbulent mixing caused by water
movement, not molecular diffusion
Two-way exchange--contrast with initial
mixing (one-way process)
Initial mixing
xs
Q
Ua
c
H
cb
Qo
co
Kinetic Energy Spectrum
Sk = kinetic energy density
implied gap
Sk~ε2/3k-5/3
could also use frequency
2π / L
mean flow
2π / η
k = 2π/λ = wave number
turbulence
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
r
∂c
+ ∇ ⋅ (qc ) = D∇ 2 c
∂t
c
c(t)
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.
c
Time average
c
Ensemble average
c' (t ) = c(t ) − c
q ' (t ) = q (t ) − q
t
c' (t ) = c(t ) − c
q ' (t ) = q (t ) − q
Turbulent Averaging, cont’d
r
∂c
+ ∇ ⋅ (qc ) = D∇ 2 c
Expand q and c; time average
∂t
r
r
∇ ⋅ (qc ) = ∇ ⋅ q + q' (c + c')
r
r r
r
= ∇ ⋅ q c + qc'+ q ' c + q ' c'
r
r
r
Continuity
∇⋅q = 0
= q ⋅ ∇c + ∇ ⋅ q ' c '
r
∂c r
+ q ⋅ ∇c = −∇ ⋅ q ' c' + D∇ 2 c
Closure problem: we
∂t
only want c but we
r
∂
∂
∂
∇ ⋅ q ' c' = u ' c' + v' c' + w' c'
must deal with c’
∂z
∂x
∂y
[(
[
)
( )
( )
]
]
( )
w' c'
Eddy correlation.
Turbulent Diffusion
Inst. flux (M/L2-T)
J z = w' c1
z
= w' (c + c' )
Mean flux (M/L2-T)
J z = w' (c1 − c2 )
(2)
z
L
w’
(1)
c
c
c + c'
∂c
c2 = c1 +
L
∂z
∂c
J z = − w' L
∂z
Ezz
Eddy (or turbulent) diffusivity
Eddy Diffusivity
E ~ u' L
Structurally similar to molecular diffusivity D, but
much larger (due to fluid motion, not molecular
motion) => often drop D
E 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 uniform
Eddy diffusivity ~ conductivity ~ viscosity
Individual plumes not always Gaussian; but ensemble
averages -> Gaussian
Turbulent transport eqn
∂c r
+ q ⋅ ∇c = ∇(E∇c ) + ∑ r
∂t
∂c
∂c
∂c
∂c
+u
+v +w
∂t
∂x
∂y
∂z
Cartesian coordinates;
diagnolized diffusivity
∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c ⎞
⎟⎟ + ⎜ E z
= ⎜ E x ⎟ + ⎜⎜ E y
⎟+ ∑r
∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠
How to measure eddy diffusivity
Less direct
Measure u’, c’, etc. and correlate
Measure something else (e.g. dissipation)
that correlates with E
Measure concentration distribution and
calibrate E (more later)
Model it
Models of turbulent diffusion
u' ~ k
u '2 v'2 w'2
k = TKE =
+
+
2
2
2
turbulent kinetic energy (don’t confuse
with wave number)
1) k-L model (two eqn model)
E ~ kL
∂k
= ⋅ ⋅ ⋅ ± sources & sinks of k
∂t
∂L
= ⋅ ⋅ ⋅ ± sources & sinks of L
∂t
2) k model (one eqn; solve only for k; L is hardwired)
Models of turbulent diffusion,
cont’d
u' ~ k
u '2 v'2 w'2
k = TKE =
+
+
2
2
2
turbulent kinetic energy
3) k-ε model (two eqn model)
ε = turbulent dissipation rate:
∂k
= ⋅⋅⋅ − ε
∂t
ε ~ k/τ ; τ = time scale of turb. ~ L/k
ε ~ k 3/2/L or L ~ k 3/2/ε
E ~ kL ~ k2 /ε
∂ε
= ⋅ ⋅ ⋅ ± sources & sinks of ε
∂t
1/2
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 3D
y
M
u
t=0
x
∂c
∂c
∂ 2c
∂ 2c
∂ 2c
+ u = E x 2 + E y 2 + E z 2 − kc
∂t
∂x
∂x
∂y
∂z
⎧⎪ ( x − ut ) 2
⎫⎪
M
y2
z2
exp− ⎨
+
+
+ kt ⎬
c=
3/ 2
1/ 2
4E yt 4Ezt
8(πt ) ( E x E y E z )
⎪⎩ 4 E x t
⎪⎭
⎧⎪ ( x − ut ) 2
⎫⎪
y2
z2
exp− ⎨
c=
+
+
+ kt ⎬
2
2
2
3/ 2
(2π ) σ xσ yσ z
2σ y
2σ z
⎪⎩ 2σ x
⎪⎭
M
Instantaneous (line) source in 2D
(e.g. extending over epilimnion)
y
M/h
(or m’)
u
t=0
x
∂c
∂c
∂ 2c
∂ 2c
+ u = E x 2 + E y 2 − kc
∂t
∂x
∂x
∂y
⎧⎪ ( x − ut ) 2
⎫⎪
M /h
y2
+
+ kt ⎬
c=
exp− ⎨
1/ 2
4E yt
4(πt )( E x E y )
⎪⎩ 4 E x t
⎪⎭
⎧⎪ ( x − ut ) 2
⎫⎪
M /h
y2
c=
+
+ kt ⎬
exp− ⎨
2
2
(2π )σ xσ y
2σ y
⎪⎩ 2σ x
⎪⎭
Instantaneous (plane) source in 1D
c
u
M/A at t=0
(or m’’)
x
∂ 2c
∂c
∂c
+ u = E x 2 − kc
∂x
∂x
∂t
⎧ ( x − ut ) 2
⎫
M
+ kt ⎬
exp− ⎨
c=
1/ 2
1/ 2
2(πt ) E x
⎩ 4 Ext
⎭
⎧ ( x − ut ) 2
⎫
M
c=
+ kt ⎬
exp− ⎨
2
1/ 2
(2π ) σ x
⎩ 2σ x
⎭
Continuous (point) source in 3D
y
u
c(y,z)
m&
(or q)
∂c
∂ 2c
∂ 2c
∂ 2c
u = E x 2 + E y 2 + E z 2 − kc
∂x
∂x
∂y
∂z
⎧⎪ y 2u
m&
z 2u
x ⎫⎪
exp− ⎨
+
+k ⎬
c=
1/ 2
4πx( E x E z )
u ⎪⎭
⎪⎩ 4 E y x 4 E z x
⎧⎪ y 2u
m& / u
z 2u
x ⎫⎪
exp− ⎨
+
+k ⎬
c=
2
2
(2π )σ yσ z
u ⎪⎭
2σ z
⎪⎩ 2σ y
x
Continuous (line) source in 2D
(e.g. extending over epilimnion)
y
u
c(y)
m& / h
(or q’)
∂c
∂ 2c
u = E y 2 − kc
∂x
∂y
⎧⎪ y 2u
m& / h
x ⎫⎪
exp− ⎨
+k ⎬
c=
1/ 2
2(πuE y x)
u ⎪⎭
⎪⎩ 4 E y x
⎧⎪ y 2u
m& / uh
x ⎫⎪
exp− ⎨
+k ⎬
c=
1/ 2
(2π ) σ y
u ⎪⎭
⎪⎩ 2σ y x
x
Continuous (plane) source in 1D
c
u
m& / A
x
(or q’’)
∂c
∂ 2c
u = E x 2 − kc
∂x
∂x
m&
⎧ xk ⎫
exp− ⎨ ⎬
c≅
Au
⎩u⎭
⎧ xu ⎫
m&
exp⎨ ⎬
c≅
Au
⎩ Ex ⎭
x>0
x<0
A few comments re solutions
Spatial integration of point source =>
line source => plane source
Temporal integration of inst source =>
Continuous source
Relationship 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
sources
c ~ 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
Type I:
c = const , e.g ., c = 0
on open bndry
Type II:
∂c
∂c
=0
= const , e.g . =
∂n
∂n
on solid bndry
II
u
I
III
Type III
(mixed):
∂c
= const ,
∂n
∂c ⎞
⎛
e.g . uc 0− = ⎜ uc − E x ⎟
∂x ⎠ 0+
⎝
at inlet
ac + b
Images
A
B
C
No flux
D
E
+
c=0
c = creal + ∑ cimage
Inst. Pt. Source in Linear Shear
z
u(z)
M at t=0
u = uo (t ) + λ y y + λz z
c ( x, y , z , t ) =
λ y = ∂u / ∂y λz = ∂u / ∂z
M
1 ⎛ 2 Ey
2 E ⎞
+ λz z ⎟⎟
φ = ⎜⎜ λ y
12 ⎝
Ex
Ex ⎠
2
[
(4πt ) 3 / 2 ( E x E y E z )1/ 2 1 + (φt )
]
2 1/ 2
2
t
⎡⎧
⎤
⎫
1
⎢ ⎨ x − ∫ uo (t ' )dt '− (λ y y +λz z )t ⎬
⎥
2
2
2
⎢
⎭ + y + z + kt ⎥
exp− ⎢ ⎩ 0
⎥
2
4
E
t
4
E
t
(
)
+
4
1
φ
E
t
t
y
z
x
⎢
⎥
⎢
⎥
⎣
⎦
[
]
Inst. Pt. Source in Linear Shear, cont’d
(
E x ' = E x 1 + φ 2t 2
)
small t , E x ' → E x
C ~ t-3/2
large t , E x ' → E xφ 2t 2
2
2
⎤
⎡
t ⎛ ∂u ⎞
u
∂
⎛ ⎞
= ⎢⎜⎜ ⎟⎟ E y + ⎜ ⎟ E z ⎥
12 ⎢⎝ ∂y ⎠
⎝ ∂z ⎠
⎥⎦
⎣
(1) (2) (1) (2)
2
C ~ t-5/2
Longitudinal (Shear) Dispersion
(1) differential longitudinal advection
(2) transverse mixing
Ex’ is really a dispersion coefficient
Okubo (1970)
10-4-1
Variation of Peak
Concentration with Time
5
2
Release #3
Release #4
Release #5
Release #6
April -
10-5-1
T-1.41
5
August
{
CMAX/M(m-3)
2
10-6-1
5
2
10-7-1
5
T-2.47
2
10-8-1
5
2
10-9-1
1
2
5
10
20
50
Time (hrs)
Figure by MIT OCW.
100
200
500
1000
Fluorescent Tracer
Digital readout
Light detector
Wavelengths specific
to the compound
Emission filter
Lamp
580 nm
Excitation
filter
Wavelengths created by the
compound, plus stray light
Cuvette or
sample cell
Many wavelengths
of light
Specific wavelengths
of light
555 nm
Figure by MIT OCW.
Rhodamine WT
(red dye;
fluoresces orange)
Injected as
neutrally buoyant
liquid
Flow thru or in situ
fluorometer (I ~ c)
Detection ~ 10-10
Example
D.L.
D.L.
Start
A
10
50
1000
A'
D.L.
B
500
250
100
100
C
B'
107.7 ppb
100
C'
End
Figure by MIT OCW.
SF6
Injected as gas dissolved in water
Sampled with Niskin bottle or equiv
(profiles collected w/ Rosette sampler)
Analyzed w/ shipboard GC w/ electron
capture
Detection ~10-17
Vent
Reference
Oven
Detector
Injection
Port
Carrier Gas
Figure by MIT OCW.
Recorder
Sensing
Column
North Atlantic Tracer Release Experiment (NATRE)
20 km
Release Pattern
Two Weeks After Release
26 N
100 km
Mass of SF6: 139 kg
Location: 1200 km W of
Canary Is.
Depth = 310 m
Time: 5-13 May, 1992
References:
„
25 N
„
24 N
31 W
30 W
29 W
Six Months After Release
Ledwell et al., Nature,
1993
Ledwell et al., JGR, 1998
Images: Kim Van Scoy
Figures by MIT OCW.
NATRE, cont’d
N
30
100 km
Latitude
26 N
25
25 N
300 km
20
24 N
31 W
30 W
29 W
Six Months After Release
Figure by MIT OCW.
40
35
Longitude
30
One Year After Release
Figure by MIT OCW.
Images: Kim Van Scoy
25 W
Drogues (drifters)
Floats w/ large drag
at constant depth
Have flag or
periodically rise to
surface
Position viewed from
above or recorded
using GPS
Horizontal Diffusion
Historically analyzed using vertical line source in
cylindrical coordinates (rather than x, y)
y Actual patch
r
x
x, y are relative (to
center of mass)
coordinates
Equiv. circular patch
injection
Cylindrical Coordinates, cont’d
=> σ x + σ y = σ r
x2 + y2 = r 2
2
E x = E y = Er
2
2
Diffusivity assumed horizontally isotropic,
independent of coordinate system
u=v=0
vertical line source
m' = M / h
∞
σr
2
M2
=
=
Mo
2
π
2
rcr
dr
∫
0
∞
∫ 2πrcdr
0
c=
( M / h )e
πσ r 2
− kt
−
e
r2
σr2
4 (vs 2) because σr2 = 2σx2
= 4 Er
If Er is const. (or treated as such)
( M / h )e
=
4πEr t
− kt
−
e
r2
4 Er t
Gaussian; if Er = const
cmax ~ t-1 but obs show
cmax ~ t-2 or t-3
NATRE
1013
σr2 (cm2)
Horizontal
Diffusion
Diagram
(Okubo 1971)
1014
1012
t2.3
Rheno
1964 V
1962 III
1962 II
1961 I
#1
#2
#3
#4
#5
#6
100 km
North Sea
Off Cape
Kennedy
10 km
New York Bight
1011
1010
#a
#b
#c
#d
#e
#f
Off
California
1 km
Banana river
Manokin river
109
108
100 m
Hour
107 3
10
Figure by MIT OCW.
Day
104
105
t (sec)
Week Month
106
107
Horizontal Diffusion Summary
σ r = 0.011t
2
2.34
cgs units; some data from pt
source, some from line source
(not quite proper but…)
dσ r
1.34
Er =
= 0.006t
4dt
1.15
Er = 0.085σ r
2
Er = 0.017l
1.15
l = 4σ r
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?
c=
M /h
πσ r
2
2
− kt − r 2 / σ r
e e
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
1014
σr2=3.7x1011 cm2
σr = 6000 m
σr2 (cm2)
1013
1012
t2.3
Rheno
1964 V
1962 III
1962 II
1961 I
#1
#2
#3
#4
#5
#6
100 km
North Sea
Off Cape
Kennedy
10 km
New York Bight
1011
1010
#a
#b
#c
#d
#e
#f
Off
California
1 km
Banana river
Manokin river
109
108
100 m
Hour
107 3
10
Day
104
105
t (sec)
Figure by MIT OCW.
Week Month
106
107
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)
l = 4σ r is arbitrary; others choose
l = 3.5σ , l = 12σ
A few more comments, cont’d
dσ r
Er =
4dt
σr2
2
Era =
σr
2
4t
Diffusivity
Ave slope ~ Ea
Local slope ~ E
Apparent
diffusivity
σr2 ~ t2.34
t
Richardson’s 4/3 law
E ~ε l
1 / 3 4 / 3 4/3 rather than 1.15; theoretical (but not
empirical) basis
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:
(φt )2 >> 1
E ~ t2
dσ 2
=> σ 2 ~ t 3 => t ~ σ 2 / 3
E~
dt
E ~ σ 4/3
Interpretation of 4/3 law, cont’d
4/3 law in inertial sub-range
Sk = kinetic energy density
Sk~ε2/3k-5/3 Inertial sub-range
2π / L
2π / η
k = wave number
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
σ2(t)
σ2 ~ t
σ2 ~ t2.34 σ2 ~ t3
E(t)
E~const E ~ t1.34
E(σ)
E~const E ~ σ1.15 E ~ σ4/3
E ~ t2
Gen’l
σ2 ~ tq
E ~ tq-1
E~
σ(2q-2)/q
Absolute vs Relative Diffusion
y
u
σ
Absolute vs Relative Diffusion
y
u
σ
Absolute vs Relative Diffusion
y
u
σ
Σ
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
z
A
B
small
drogue
cluster
point source
of dye
C
line source
of dye
Do the values of Er differ (and if
so, which is right)?
h
T(z)
z
A
B
small
drogue
cluster
point source
of dye
Er
(drogue)
<
Er
(point)
C
line source
of dye
<
Er
(line)
Okubo et al. (1983)
109
March 10, 1981
σr (cm2)
Dye
Drogue
108
107
103
104
Time (seconds)
105
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
∂c
∂c ∂ ⎛ ∂c ⎞ ∂ ⎛ ∂c ⎞ Vertical shear & diffusion included
+ u ( z ) = ⎜ E x ⎟ + ⎜⎜ E z ⎟⎟ explicitly in g.e. => don’t want them
∂t
∂x ∂x ⎝ ∂x ⎠ ∂z ⎝ ∂ z ⎠
influencing Ex => use drogues
∂c
∂c ∂ ⎛ ∂c ⎞
+ u ave
= ⎜ Ex ⎟
Shear & diffusion excluded from g.e.
∂t
∂x ∂x ⎝ ∂x ⎠
include effects in Ex calibration => use
line source of dye
Diffusivities in numerical models
(with finite grid sizes)
⎡⎛ ∂u ⎞ ⎛ ∂v ⎞
⎛ ∂u ∂v ⎞
Eh = α∆x∆y ⎢⎜ ⎟ + ⎜⎜ ⎟⎟ + 0.5⎜⎜ + ⎟⎟
⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠
⎝ ∂x ∂y ⎠
2
2
Smagorinski and Lilly (1963)
α = Smagorinski coefficient (0.1-2.0)
α = 0.16 theoretically; higher empirical
values account for vertical shear
2 1/ 2
⎤
⎥
⎥⎦
Vertical Diffusion
Fit to large scale property
distributions
„
„
Flux gradient method (lakes &
reservoirs)
Upwelling diffusion (ocean)
Measured rate of spread of
tracer second moment
Rates of measured dissipation
Others
Decreasing
time scale
Flux-gradient method
z
T
z
t
A(z)
∂
A( z )T ( z , t )dz
∫
∂t 0
E ( z, t ) =
∂T
A( z )
∂z z
z
Below depth of
other sources/sinks,
thermal energy
increases only by
turbulent diffusion
Applicable to
relatively long time
steps (e.g. weeks or
more)
North Anna Power Station (WE2-1)
N
Meteorological
Tower
N
L
Dike 1
Lake Anna
C
Dike 2
North Anna
Power Station
0
5,000
Pond 1
Pond 2
A
Dam
Dike 3
Pond
3
Waste Heat
Treatment
Facility
10,000 FT.
Elk Creek
Millpond Creek
Figure by MIT OCW.
Temperature and DO profiles
1976
0
April May June July Aug Sept Oct
27
25
18
19
10
20
10
4
1
14
5
12
Temperature
Depth (m)
0
April May June July Aug Sept Oct
9
8
8
10
7
2
2
0
3
Pre-operational
Dissolved Oxygen
1982
April May June July Aug Sept Oct
22
28
26
April May June July Aug Sept Oct
8
9
7
8
10
13
20
June-August average Ez =
0.11 m2/d (0.013 cm2/s)
15
14
Temperature
3
5
< one unit
0
1
0.14 m2/d (0.016 cm2/s)
Dissolved Oxygen
1983
0
10
April May June July Aug Sept Oct
31
10 9 8 7
29
12
8
20
April May June July Aug Sept Oct
27
25
16
Temperature
7
6
5
8
0.46 m2/d (0.053 cm2/s)
~ two units
3
1
0
Dissolved Oxygen
Dominion Power Co.
Figure by MIT OCW.
Vertical Diffusion from NATRE
60
sz
400
sz2
40
300
0
Second moment (m2)
Height (m)
20
May
-20
Oct
-40
-60
200
100
Nov
0.0
0.2
0.4
0.6
0.8
1.0
0
0
50
Concentration (scaled)
100
150
Time (days)
t
Figure by MIT OCW.
Ledwell, et al., 1993
Ez =
z
2t
≅ 0.11 cm 2 /s (6 mo)
≅ 0.17 cm 2 /s (2 yr)
200
Measured dissipation
From previous discussion
Sk = kinetic energy density
Inertial sub-range
Sk~ε2/3k-5/3
Turbulent
velocities
generated by
mean flow
Dissipated by
molecular diffusion
2π / L
2π / η
k = wave number
Turbulent temperature variations similar to turbulent velocity variations
Temperature Micro-profile
T(z)
<T>
T = <T> + T’
Measured with temperature
microstructure probe;
resolution < 1 mm
z
Generation (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)
Ez =
χ
χ = temp variance dissipation rate [K2s-1]
2(∂ T / ∂z )
2
T(z) = <T> + T’
χ = 2κI (∂T ' z / ∂z )
Ez =
Iκ (∂T ' z / ∂z )
/ ∂z )
Osborn (1980)
Ez =
(∂ T
γ mixε
N
2
2
2
κ = molecular thermal diffusivity [m2s-1]
I ~ 3 (accounts for gradient in T’ in 3
directions)
2
N2 = (g/ρ)(dρ/dz) [s-2]
ε = TKE dissipation rate [m2s-3]
γmix = const <= 0.2
Examples
dT0/dz
Depth (m)
0
-2
0
2
24
25.5
27
20
40
σt (kg/m3)
-7
0
log10KT (m2s-1)
-12
-9
-6
log10(ε, m2s-3)
0.1
1
10
LC (m)
Profiles of (a) density and temperature gradient, (b) KT, (c) ε and (d) centered-displacement
lengthscale LC. The estimates of KT and ε include low-pass filtered versions.
Figure by MIT OCW.
Stevens, et al. 2000
Langmuir Circulation
Oil Streaks
Wind
Figure by MIT OCW.
Formulae for Ez
Open waters, near surface
2
0.028 H w − 4πz / Lw
e
Ez =
Tw
Ichiye (1967); z = depth;
Hw, Tw, Lw = significant wave
height, period and length
In presence of stratification and shear
−3 / 2
⎡ 10 ⎤
E z = E zo ⎢1 + Ri ⎥
3 ⎦
⎣
( g / ρ ) dρ / dz
Ri =
(du / dz )2
Munk & Anderson (1948);
Ri = gradient Richardson no
Ezo = value at neutral
stratification
Formulae for Ez
Stratification only (near surface)
10 −6
Ez =
∂ρ / ∂z
Koh and Fan (1970)
[Ez in cm2/s; dρ/dz in g/cm4]
Stratification only (deep waters)
4 x10 −9
Ez =
∂ρ / ∂z
Broecker and Peng (1982)
[Ez in cm2/s; dρ/dz in g/cm4]
Typical ocean
E z ≅ 0.1 (local)
E z ≅ 1 (basin average) Ez in cm2/s
Formulae, cont’d
Rivers
E z = κu* z (1 − z / h)
E z = 0.07u*h
u* = friction velocity, h =
water depth, z = height above
bottom
Estuaries
z 2 (h − z ) 2
−2
Ez = η u
Ri
(
1
+
β
)
h3
z (h − z ) H w − 2πz / Lw
+ζ
e
(1 + βRi ) − 2
h
Tw
Pritchard (1971);
η = 8.59 x 10-3,
ζ = 9.57 x 10-3,
β = 0.276
u = mean tidal speed
103
Kolesnikov 1961
Harremos 1967
Vertical Diffusion Coefficient, Ez (cm2/sec)
Jacobsen (Defant 1961)
Foxworthy 1968 (patch)
Foxworthy 1968 (plume)
Foxworthy 1968 (point source)
102
10 -6
Ez =
¶ / ¶z
10
4 x10 -9
Ez =
¶ / ¶z
1
0.1
0.01
10-7
Natre
10-6
10-5
10-4
10-3
10-2
10-1
Density gradient, - l dρ (m-1)
ρ dz
Koh and Fan (1970)
Figure by MIT OCW.
100(dρ/dz) (g/cm4)
Application: coastal sewage
discharge from multi-port diffuser
Assume
b = 300 m
H = 30 m
h = 10 m
u = 0.1 m/s
NF dilution SN = 100
How far ds until SF =10?
H
(ST=SNSF=1000)
Formal solution by Brooks in
Section 2.8; approximate
solution follows
L
u
b
x
y
h
z
Sewage discharge, cont’d
u
L
b
-xo
at x = 0, σ ro ≅
0
b
6
xf
x
≅ 0.4b ≅ 12,000 cm; σ rf = 10σ ro = 120,000 cm
1 / 2.34
⎡ σ r2 ⎤
⎡12,000 2 ⎤
⎡120,000 2 ⎤
2
2.34
; to = ⎢
= 22,000s; t f = ⎢
= 162,000 s
σ r = 0.011t => t = ⎢
⎥
⎥
⎥
⎣ 0.011 ⎦
⎣ 0.011 ⎦
⎣ 0.011⎦
x = u (t f − t o ) = (0.1)(162,000 − 22,000) = 14,000m = 14km
Contrast with 30 m!
1 / 2.34
1 / 2.34
1014
σr2 (cm2)
1013
1012
#1
#2
#3
#4
#5
#6
100 km
t2.3
Rheno
1964 V
1962 III
1962 II
1961 I
North Sea
Off Cape
Kennedy
10 km
New York Bight
1011
SF = 10
(sro2 = 100sro2)
1010
#a
#b
#c
#d
#e
#f
Off
California
1 km
Banana river
Manokin river
109
(120m)2 = sro2
108
100 m
Hour
107 3
10
Day
104
105
Week Month
106
107
t (sec)
to = 22000s
t = 162000s
x = u(t-to) = (0.1)(162000-22000) = 14 km
Figure by MIT OCW.
Neglect of vertical diffusion
Reasonable?
ρ s ≅ 1.03 ρ o ≅ 1.00 g / cm
∆ρ o
≅ 0.03
ρ
∆ρ o
≅ 0.0003
SN ρ
29.7 30
∂ρ
≅ 0.0003 / 1000 ≅ 3 x10 −7 g / cm 4
∂z
10 −6
10 −6
2
Ez =
≅
≅
3
cm
/s
−7
∂ρ
3 x10
ρ∂z
(ρ-1)x1000
h
3
ρo = 1.000
σT =
1.0297 1.030
conservatively small
ρ
Vertical diffusion, cont’d
σ zo =
1
12
(2h) ≅ 5.8 m
σ z 2 = σ zo 2 + 2 E z t
= 580 2 + (2)(3)(140000)
σo
= 117000 cm 2
σ z = 10.8 m
S Fv
1080
=
= 1.9
580
concentration reduction = 9% of that
due to horizontal mixing; even smaller if
stronger density gradient chosen
h
103
Kolesnikov 1961
Harremos 1967
Vertical Diffusion Coefficient, Ez (cm2/sec)
Jacobsen (Defant 1961)
Foxworthy 1968 (patch)
Foxworthy 1968 (plume)
Foxworthy 1968 (point source)
102
10
3
1
0.3
0.1
0.01
10-7
10-6
10-5
10-4
10-3
Density gradient, - l dρ (m-1)
ρ dz
Koh and Fan (1970)
Figure by MIT OCW.
10-2
10-1
Atmospheric, surface water and
ground water plumes
Similarities
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 Plumes
Modest NF mixing
(wind quickly
dominates)
Often large “point”
sources
Time scales: minutes
to days
Non-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
A
H
h
Region of mathematical
dispersion underground
B
Real source
H
-H
h
-h
Region of reflection
Image source
for ground
level
exposure
NF mixing
handled by
virtual
elevation
Cooper and
Alley (1994)
Image
source
Figure by MIT OCW.
Diffusion diagrams
5000
10,000
C
D
A
C
B
B
A
F
100
1000
σz, meters
σy, meters
1000
E
stable
D
100
E
F
stable
10
10
0.1
1
10
Distance Downwind, km
100
1.0
0.1
X
lateral
1
10
Distance Downwind, km
100
X
vertical
Figure by MIT OCW.
Turner (1970); Cooper and Alley (1994)
Moderatec
<2
2-3
3-5
5-6
>6
A
A-B
B
C
C
A-Bf
B
B-C
C-D
D
Slightd
_ 4/8)
Cloudy ( >
B
C
C
D
D
E
E
D
D
D
_ 3/8)
Clear ( >
F
F
E
D
D
stable
Strongb
unstable
Night Cloudinesse
Day Incoming Solar Radiation
Surface Wind
Speeda (m/s)
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.
STABILITY CLASSIFICATIONS*
Figure by MIT OCW.
Groundwater Plumes
No (dynamic) NF
Distributed, poorly
characterized sources
Multiple phases
(contaminant and
medium)
Laminar (turbulent
fluctuations replaced by
heterogeneity)
Time scales: months to
decades
Heteorogeneity
Causes non-uniform flow => macro-dispersion
Often poorly resolved: handled stochastically
Plumes often (very) non-Gaussian
MADE experiments at CAFB
Please see:
http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1408&conte
xt=lbnl
Dispersivity, α [L]
Length scale of hydraulic conductivity correlation
E = αu
~u τ ~u
2
2
λ
u
Reliability
~ uλ
104
Longitudinal Dispersivity (m)
α ~λ
"one tenth"
Xu and Eckstein
High
Intermediate
Low
102
100
10-2
100
102
Scale (m)
104
Gelhar, Welty and Rehfeldt (1992) Dispersivity Data
Figure by MIT OCW.
Superposition: Puff models
MIT Transient Plume Model
y
y
A
B
x
x
2
1
1.5
1
σx= σy
0.5
0.5
Instantaneous
location of
end of NF
u = 0.34 cos (ωt + π / 2)
v = 0.18 cos (ωt + π / 2) − 0.5
mo ( z , t , t k )
c ( x, y , z , t ) = ∑
2
k πσ r (t , t k )
⎧⎪ [x − xc (t , t k )]2 [ y − y c (t , t k )]2 ⎫⎪
+
exp− ⎨
⎬
2
2
⎪⎩ σ r (t , t k )
σ r (t , t k ) ⎪⎭
Figure by MIT OCW.
Adams (1995)