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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

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Page 1: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 2: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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)

Page 3: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

Initial mixing

H

Qo

Q

cb

co

cUa

xs

Page 4: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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]

Page 5: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 6: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 7: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 8: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 9: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

Turbulent transport eqn

( )

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

+∇∇=∇⋅+∂∂

rzcE

zycE

yxcE

x

zcw

ycv

xcu

tc

rcEcqtc

zyx

r

Cartesian coordinates;diagnolized diffusivity

Page 10: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 11: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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)

Page 12: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 13: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 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

Page 14: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 15: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 16: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 17: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

&

&

Page 18: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 19: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

Continuous (plane) source in 1D

0exp

0exp

2

2

<⎭⎬⎫

⎩⎨⎧

>⎭⎬⎫

⎩⎨⎧−≅

−∂∂

=∂∂

xExu

Aumc

xuxk

Aumc

kcxcE

xcu

x

x

&

&

c

uxAm /& (or q’’)

Page 20: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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)

Page 21: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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)

Page 22: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

Boundary Conditions

bndryopenoncgeconstc 0.,., ==Type I:

II

III bndrysolidonncgeconst

nc 0.., =

∂∂

==∂∂u I Type II:

inletatxcEucucge

constncbac

x+

− ⎟⎠⎞

⎜⎝⎛

∂∂

−=

=∂∂

+

00

..

,Type III(mixed):

Page 23: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

ImagesA B C

No flux

c = 0

-+

D E∑+= imagereal ccc

Page 24: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 25: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 26: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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.

Page 27: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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.

Page 28: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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.

Page 29: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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.

Page 30: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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.

Page 31: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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.

Page 32: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

Drogues (drifters)

Floats w/ large drag at constant depthHave flag or periodically rise to surfacePosition viewed from above or recorded using GPS

Page 33: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 34: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 35: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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.

Page 36: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 37: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 38: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 39: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 40: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

Page 41: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

=>=>

>>

Page 42: 2 Turbulent Diffusion - MIT OpenCourseWare · 2 Turbulent Diffusion Turbulence Turbulent (eddy) diffusivities Simple solutions for instantaneous and continuous sources in 1-, 2-,

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

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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

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Absolute vs Relative Diffusion

y

σu

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Absolute vs Relative Diffusion

σ

y

u

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Absolute vs Relative Diffusion

y

Σσu

Absolute diffusion (Σ2) > Relative diffusion (σ2); ratio decreases with time

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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

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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)

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Okubo et al. (1983)

103

Time (seconds)

107

108

σ r (c

m2 )

109

104 105

DyeDrogue

March 10, 1981

Figure by MIT OCW.

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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

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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

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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

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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

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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.

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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.

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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.

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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

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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

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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=

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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.

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Langmuir Circulation

Oil Streaks

Wind

Figure by MIT OCW.

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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

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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

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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

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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.

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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

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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

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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.

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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

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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

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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.

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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

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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.

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Stratification

For examples of plume types, please see:http://www.environmenthamilton.org/projects/stackwatch/plume_types.htm

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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.

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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.

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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.

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Groundwater PlumesNo (dynamic) NFDistributed, poorly characterized sourcesMultiple phases (contaminant and medium)Laminar (turbulent fluctuations replaced by heterogeneity)Time scales: months to decades

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HeteorogeneityCauses non-uniform flow => macro-dispersionOften poorly resolved: handled stochasticallyPlumes often (very) non-Gaussian

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MADE experiments at CAFB

Please see:

http://repositories.cdlib.org/cgi/viewcontent.cgi?article=1408&context=lbnl

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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.

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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)