Dr. Martin T. AuerMTU Department of Civil & Environmental Engineering

CE5504Surface Water Quality

ModelingLab 5. One-Dimensional Models

Thermal Stratification, Vertical Mass Transportand Interactions with Trophic State

Temperature and the Density of Water

1.0000

0.9995

0.9990

0.9985

0.9980

0.9975

0.9970

0.9965

0.9960

0.99550.9950

Den

sity

(g∙

cm3 )

0 5 10 15 20 25 30

Temperature (°C)

MaximumDensity3.94 °C

Thermal Stratification

spring summer

Temperature (°C)

Thermal Stratification

Seasonality in Stratification and Mixing

= °C

Dollar Bay - Temperature

0

1

2

3

4

5

6

7 8

Dep

th (

m)

0 5 10 15 20

Temperature (°C)

0

1

2

3

4

5

6

7

8

Dep

th (

m)

M A M J J A S O N D

Productivity and Oxygen

Trophic State and Oxygen Profiles

Orthograde

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25

De

pth

(m

)

Clinograde

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25

De

pth

(m

)

Oligotrophic Lake Eutrophic Lake

Temperature (°C) Temperature (°C)

Dollar Bay - Temperature

0

1

2

3

4

5

6

7 8

0 5 10 15 20

Temperature (°C)

0

1

2

3

4

5

6

7

8

Dep

th (

m)

M A M J J A S O N D

Dollar Bay – Dissolved Oxygen

Dep

th (

m)

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12

Dissolved Oxygen (mg∙L-1)

M A M J J A S O N D

Onondaga Lake – Dissolved Oxygen

Oneida Lake – Dissolved Oxygen

Lake Erie – Dissolved Oxygen

Gulf of Mexico – Dissolved Oxygen

8 Lake Superior – 4/28/00

Completely-Mixed Lake

8

8

Lake Superior – 8/25/00

Vertical Resolution - Stratification

Advection and Diffusion

Diffusion

Scales of Diffusion

11 2 1

33

3 3

'dC

V E C Cdt

g m gm

m d d m

g g

d d

11

3 23 2

c

dCV J A

dt

g gm m

m d m d

g g

d d

Mathematics of Diffusion I

2

2 3

2 2

dCJ E

dx

g m g

m d d m mg g

m d m d

Fick’s 1st Law

X-direction

E

C

Mathematics of Diffusion II

dCJ E

dx

2 1

3 3

C CdC

dx l

g g

m m m m

l

Mathematics of Diffusion III

1 2 11

23 2

3 3

c

dC C CV E A

dt l

g m gm m

m d d m m

g g

d d

11 2 1

2 23

3 3

( )cE AdCV C C

dt l

g m m gm

m d d m m

g g

d d

Mathematics of Diffusion IV

3 2 2

3 3

' cE AE

l

m m m

d d m

m m

d d

11 2 1

33

3 3

' ( )dC

V E C Cdt

g m gm

m d d m

g g

d d

E, turbulent diffusion coefficient, m2/dE’, bulk diffusion coefficient, m3/d

Mathematics of Diffusion V

Vertical Mass Transport

11 2 1' ( )dC

V E C Cdt

' cE AE

l

Application to Vertical Mass Transport

11 2 1

33

3 3

' ( )dC

V E C Cdt

g m gm

m d d m

g g

d d

11 2 1

3 23 3

( )t t

dCV v A C C

dt

g m gm m

m d d m

g g

d d

turbulent diffusion coefficient thermocline heat transfer coefficient

Calculation of the Thermocline Heat Transfer Coefficient

Epilimnion Heat Balance

3 33 2 2

3 3 3 2 3

( ) ( )ee p p in p e s t p t h e

dTV C Q C T t Q C T J A v C A T T

dt

g cal C m g cal m g cal cal m g calm C C m m C

m g C d d m g C d m g C m d d m g C

cal cal cal cal cal

d d d d d

changein heat

tributaryheat in

dischargeheat out

surfaceheat flux

thermoclineheat exchange

Calculation of the Thermocline Heat Transfer Coefficient

Hypolimnion Heat Balance

3 23 3

( )hh p t p t h e

dTV C v C A T T

dt

g cal C m g calm m C

m g C d d m g C

cal cal

d d

changein heat

thermoclineheat exchange

assumes no direct heating ofhypolimnion or sediments

upper mixed layer(~epilimnion)

lower mixed layer(~hypolimnion)

thermocline vt∙As

Calculation of the Thermocline Heat Transfer Coefficient

( )hh p t p t e h

dTV C v C A T T

dt

h t t t te e

h h

dT v A v AT T

dt V V

t th

h

v A

V

Canceling the and Cp terms, dividing by Vh and expanding …

And defining …

Yields …

hh e h h

dTT T

dt

uml

lml

Calculation of the Thermocline Heat Transfer Coefficient

, (1 )h ht th h i eT T e T e

,

,

ln e h iht

t s e h s

T TVv

A t T T

and re-arranging, yields …

Which, for Th at the beginning of the calculation = Thi, yields …

,

,

1ln e h ih

st t e h s

T TVt

v A T T

further re-arranging, yields …

where a plot of ts versus termyields 1/vt as the slope

term

Application to Dollar Bay

Dollar Bay - Temperature

0

1

2

3

4

5

6

7 8

0 5 10 15 20

Temperature (°C)

0

1

2

3

4

5

6

7

8

Dep

th (

m)

M A M J J A S O N D

Dollar Bay – Dissolved Oxygen

Dep

th (

m)

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10 12

Dissolved Oxygen (mg∙L-1)

M A M J J A S O N D

0

2

4

6

8

10

12

14

16

18

20

0 50 100 150 200 250 300 350

For diffusion calculation, need to choose an interval where Tepi is stable and Thyp is warming.

Dollar Bay – Temperature Calculations

,

,

1ln e h ih

st t e h s

T TVt

v A T T

( )h t te h

h

dT v AT T

dt V

Hypolimnion temperature mass balance.

Sediment oxygen demand reflects the diffusion of oxygen into the sediment to meet the metabolic needs of microorganisms participating in the aerobic decomposition of organic matter. SOD is also exerted through the aerobic oxidation of reduced species byproducts of anaerobic respiration.

The mass balance, for SOD alone is …

Sediment Oxygen Demand

uml

lml

2 12,SOD gO m d

2sed

dOV SOD A

dt

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

For the SOD calculation, we need to choose the interval of oxygen depletion.

Dollar Bay – Oxygen Calculations

22, 2,

t t sede h

h h

v A SOD AdOO O

dt V V

(the question of the cones)

Time-Variable vt

Values for Onondaga Lake (Doerr et al. 1996)