PAP_038

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Regional Distribution Of Thermal Stratification Properties

In Japanese Inland Waters Kohji MICHIOKU

(Department of Civil Engineering)

Abstract

Thermal régimes in Japanese limnological waters such as lakes and reservoirs are

investigated from a hydrodynamic point of view. Focus here is placed on seasonal

variation patterns of thermal properties such as stratification behaviors, heat exchange at

an air-water interface, etc.. Dependency of lakes’ thermal regimes on meteorology and

water depth dimension is analyzed by making a mixed-layer modeling. Surface heat

exchange is formulated as a function of an equilibrium temperature. Applying field data ofthe equilibrium temperature to the model, maps of lakes’ thermal properties are produced

and thermal régimes in Japanese inland water areas is discussed.

keywords: lakes thermal régimes, thermal stratification, mixed-layer model, equilibrium temperature

1 Introduction

Inland waters like lakes and reservoirs are used for many purposes of our life such as water

resources, flood protection, resort activities. They are deeply involved in our life throughmany environmental processes. We learned from our history that human activities should be

harmonized with nature, otherwise catastrophic changes in environment bring serious troubles

to us very immediately.

Among many environmental aspects, water temperature is most fundamental as well as

important because many of hydrological, meteorological and biological processes aresignificantly affected by temperature. When we deal with thermal environment in lakes and

reservoirs, one of the most important factors we have to consider is thermal stratification,

since heat and mass transfer is governed by buoyancy effects of density stratification.

In an enclosed water area, where river through-flow scarcely affects heat balance, a major

factor governing thermal process is heat exchange across an air-water interface. Watertemperature is directly governed by the surface heat flux. At the same time, surface heat

exchange is strongly affected by water temperature through radiation, conductive and latent

heat transfer. In such a feed back system, a water body’s temperature structure governs the

whole thermal process through internal hydrodynamics in a stratified water body.

The author developed an integral mixed-layer model for prediction of a lake’s temperaturestructure. In the model the surface heat flux is assumed to be proportional to a difference

between an equilibrium temperature and a water surface temperature, [1], [2]. In this

formulation, all of the meteorological conditions concerning heat budget are described in

1 Department of Civil Engineering, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan

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terms of only one variable, say the equilibrium temperature.

A variety of thermal régimes are found in lakes and reservoirs. For instance, a shallow lake

is uniformly mixed throughout a year; some lakes have so stable density stratification that little

water mass is vertically exchanged. These thermal characteristics are considered to bedependent on a lake’s geographical dimensions, local meteorological conditions, etc..

In this paper, thermal structure is analyzed by the model and it is examined how a lake’s

thermal properties depend on the relating factors. From field data of equilibrium temperature,

maps of a lake’s thermal properties are produced and thermal environment in Japanese lakes

is discussed.

2 Analytical procedures

2.1 Summary of an integral mixed-layer model [1], [2]

In order to describe seasonal variation of a lake’s thermal structure, a mixed-layer model wasdevised as shown in Fig.1 [1]. Following to a concept of a mixed-layer model proposed by

Kraus and Turner [3], a system is assumed to be vertically one-dimensional and consisting of

two layers, say a uniformly mixed layer and a non-turbulent lower layer with a continuous

density stratification. The layers are bounded by a thermocline interface with a temperature

jump of ∆T . Under a time-dependent surface heat flux F (t ), a heat conservation equation for

a water column and an entrainment law regarding vertical mixing rate are applied and

numerically integrated; solutions for temperature and thickness of a mixed-layer, i.e. T m and

hm, are obtained. The model can describe not only summer stratification but also winter

stratification with an inverse temperature stratification. In the model steady heat conduction

in an ice plate is assumed to analyze an ice-cover régime as well. Details of the modelshould be referred to our previous papers [1] and [2]. Despite of its simplicity, model’s

performance is fairly satisfactory.

2.2 Surface heat flux

A surface heat flux per unit area F (t ) is described in terms

of an equilibrium temperature T e as follows.

F (t ) = -k (T e-T m) (1)

where, T m is a mixed-layer temperature and k is a heat

exchange coefficient which was identified to bek =0.45m/sec from the author’s previous research [2].

In Eq.(1) variables relating surface heat budget, for

instance, atmospheric humidity, air temperature, radiation

heat flux are integrated in one variable, say the equilibrium

temperature T e.

In this study a seasonal variation of T e is approximated to

be

T e = ∆T e sin (ωt -φ) +T ea

(2)where ∆T e: an amplitude in a seasonal variation of

equilibrium temperature, T ea: an annual average of

equilibrium temperature, ω=1/365 (1/day): an angular frequency of annual cycle, t : a time

Fig.1 A schematic ofmixed-layer model.

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coordinate in day and φ: angular phase lag.

Giving (∆T e, T ea) in Eqs.(1) and (2), solutions for T m, hm and F (t ) can be obtained from the

mixied layer model. Seasonal variations of lake’s thermal régime are completely dependent

on (∆T e, T ea) and a water depth H .

3 Equilibrium temperature in Japan

Equilibrium temperature were given by Uchijima [4], from water temperature measurements inpaddy fields. Rearrangement of his data gives regional distributions of (∆T e, T ea) as in

Figs.2 and 3. A variation of ∆T e is much smaller compared to that of T ea. T ea strongly

depends on altitude and latitude.

4 Comparison between the analysis and field data

In order to verify the model, analytical solutions are compared with field data from Japanese

lakes. Applying values of (∆T e, T ea) from Figs.2 and 3 to each lake, seasonal development

for hm and T m is. A temperature profile in a continuously stratified lower layer is described by

assuming that a mixed-layer temperature at heating stage is reserved in the course of amixed-layer shallowing.

Figs.4 through 6 show field data compared with the computed results, where temperature

isopleth and mixed-layer temperature T m are represented. Depending on its meteorological

conditions and depth scale, each lake undergoes a totally different seasonal variation of

thermal régimes. As shown in Fig.4 only summer stratification develops in Lake Sai and

water temperature is never below 4℃. This is very contrary to the case of Lake Kawaguchi

shown in Fig.5, where weak winter stratification with an inverse temperature gradient is

observed in winter. The two lakes show such different thermal properties, despite they are

locating very close each other. This well indicates that lake’s thermal régimes are dependentnot only on meteorological conditions but also on lake’s geographical conditions, especially

on a depth dimension. The field data in Fig.6 show that Lake Kuttara are stratified both in

summer and winter. The water surface is frozen in midwinter, which is also reproduced in themodel.

Lon itude

20

128 130 132 134 136 138 140 142 144 146

31

33

35

37

39

41

43

45℃

T

31

33

35

37

39

41

43

45

128 130 132 134 136 138 140 142 144 146Longitude

5

10

15

20

25

Fig.2 Distribution of annual average Fig.3 Distribution of seasonal amplitude

of equilibrium temperature T ea. of equilibrium temperature∆T e.

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5 Thermal régimes in Japan

5.1 Definition of lake’s thermal régimes

Lake’s thermal régimes possibly occurred in a lake are listed in Table 1 [2]. Ten thermal

régimes are considered. A code name is given to each of them in the first column of the

table. In the second column, it is noted what kind of phenomena occur. Regarding surface

heat flux, all of the thermal régimes are classified into the two categories, i.e. a heating and a

cooling periods as shown in the third column. The fourth column relates water surface

temperature T m. Summer and winter stratification develop in cases of T m>4℃ and T m>4℃,

respectively, and a water surface is frozen under conditions of surface temperature below 0℃.

Mixed-layer behaviors are described in the fifth column; destratification or vertical mixing

makes a mixed-layer thicker, i.e. dhm/dt >0, while stratification makes the layer shallower, i.e.

dhm/dt <0. “hm=H ” means that a water body is uniformly mixed. When a water body is

ice-covered, no mixing occurs and hm is kept constant. The final column in the table

documents ice behaviors, say no ice, ice thickening and ice melting take place, respectively.

7060

50

40

30

20

10

0

0 100 200 300Day

55 6

8

1012

141618202224

(a) Observed temperature isopleth

5

5

101520

70

56

42

28

14

0

0 100 200 300 Day

(b) Computed temperature isopleth

Lake Sai

Lake Sai

J

J

J

J

J

J

J

J

J

J J

J

J

J

J

J

J

J

J

JJ J

0

10

20

30

0 100 200 300 Day

J Observed

Computed

Lake Sai

(c) Seasonal variation of observed surfacetemperature compared with the analysis

22 4 66 88 10

10

10

12

12

14

14

16

16

18

18

20 20

22 22

4

10

8

6

4

2

0

0

100

200

300

Day

4



Lake Kawaguchi

10

8

6

4

2

0

0 100 200 300 Day

3

3

3

4

26

24

22

18

16

18

14

16

12

12

6

8

10

6

8

10

14

4

Lake Kawaguchi

J

J

J

J

J

J

J

J J

J

J

J

J

J

J

J

J

J

JJ J JJ

J

J

J

0

10

20

30

0 100 200 300 Day

J Observed

Computed

(c) Seasonal variation of the surface

Lake Kawaguchi

D e p t h [ m ]

D e p t h [ m ]

Fig.4 Observed temperature structures Fig.5 Observed temperature structures

compared with computed ones, compared with computed ones,

Lake Sai. Lake Kawaguchi.

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5.2 Seasonal cycle patterns of thermal régimes

Under conditions of equilibrium temperature ina range of Figs.2 and 3, a seasonal variation of

lake’s temperature structure is computed for

water depth ranging between H =2-100m.Through the analysis it is found that only sevenpatterns of seasonal thermal cycle can actually

take place. They are listed in Table 2,

although more patterns with different

combination of thermal régimes are

theoretically possible [2].

Both of (A) and (J) are patterns found in shallow

water areas in which the system is uniformly

mixed in all the seasons; difference between

the two is that the latter has an ice régime. Inthe patterns, (B), (C) and (L), summer

stratification develops but not winterstratification. The pattern (L) undergoes an

ice régime in winter. In the cases of (I) and (O),

a stratified density structure develops both in

summer and winter, in other words, full-scale

overturning occurs two times in autumn and

spring. A lake is ice-covered in the pattern(O).

Seasonal cycle patterns of lake’s thermal

régimes are shown as a function of (T ea, H ) in

Table 1 Stages possibly occurring in a seasonal variation of a lake’s thermal régimes.

Regime

code

Phenomena Surface heat

flux, F(t)

Mixed-layer

temp., Tm(℃)

Mixed-layer

thickness, hm Ice thickness, δ

SPT Spring turnover and full

scale mixing

Tm <4℃ &

Tm>4℃ hm=H

S1 Development of summer

stratification

(dhm/dt)<0

stratifying

S2 Erosion of summerstratification with

warming

S3 Erosion of summer

stratification with cooling

Summerstrati-

fication

(dhm/dt)>0

destratifying

FLT Fall turnover and full scale

mixing

Tm<4℃ &

Tm>4℃ hm=H

W1 Development of winter

stratification

(dhm/dt)<0

stratifying

W2 Erosion of winter

stratification with cooling

Winter

strati-

fication

(dhm/dt)>0

destratifying

I1 Freezing Tm=0℃ &

TI<0℃

(dδ/dt)>0

Freezing

I2 Melting Tm=0℃ &

TI=0℃

hm=const.

unchanged (dδ/dt)<0

Melting

W3 Erosion of winter

stratification with

warming

Winter

stratification

0<Tm<4℃

(dhm/dt)>0

destratifying

No ice cover

δ=0

0

55

10 15 20

0

50

40

30

20

10

0

0 100 200 300 Day


Lake Kuttara

50

40

30

20

10

0

0 100 200 300 Day

5

4 3


J

J

J

J

JJ

JJ

J

J

J

JJ

JJ

JJ

J

J

-5

0

5

10

15

20

25

0 100 200 300 Day

J Observed

Com uted

(c) Seasonal variation of the surface

temperature compared with the analysis

Lake Kuttara

Fig.6 Observed temperature structures

compared with computed ones,Lake Kuttara.

H e a t i n g

F ( t ) < 0

C o o l i n g

F ( t ) > 0

T m

> 4

℃

0 < T m

< 4 ℃

N o i c e c o v e r

δ = 0

H e a t i n g

F ( t ) < 0

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Fig.7, where computations are carried out

for ∆T e=12℃ which is an average of ∆T e

from Fig.3. Three curves drawn in this

diagram classify lakes into the seven

categories. The curve (1) is a criterion

classifying lakes into a “uniformly mixedone” and a “stratified one”. Fig.7 welldocuments that a system is more

uniformly mixed in a shallower lake and

more stratified in a deeper lake,

respectively. The curve (2) is a criterion

if a lake is frozen or not. A lake is more

possibly ice-covered in colder regions, where T ea is lower. The curve also verifies our

experience that a deep lake is scarcely ice-covered. The curve (3) classifies lakes into the

two categories, i.e. stratified only in summer or stratified both in summer and winter. The

former type is so called a “tropical lake” and the latter a “temperate lake”, respectively. In

colder regions the lake locates and deeper the lake is, more possibly the winter stratificationdevelops. A temperate lake is overturned two times every year, in spring and autumn. The

seven cycle patterns of thermal régimes correspond to the sub-areas divided by the three

curves as noted in the figure.

5.3 Maps of lake’s thermal régimes

Applying data from Figs.2 and 3, seasonal variation of a lake’s thermal structure can be

computed by the model. Maps for lake’s thermal properties are shown as follows.

(1) Seasonal cycle patterns of thermal régimesFig.8 shows which pattern of seasonal cycle happens in lakes with different water depth

scales H . Fig.8(a) documents only two types, i.e. (A) and (J), are possible in a shallow lake

of H =2m. Both of them are uniformly mixed-type. In lakes of H >10m, summer stratification

develops. In lakes locating in high altitude or in high latitude, an ice-cover and winterstratification are developing. The ice régime is more possibly observed in shallower lakes.

Table 2 Seasonal cycle patterns of lake’s

thermal régimes which actually exist.

Pattern Thermal cycle history

(A) SPT→FLT→SPT

(B) SPT→S1→S2→FLT→SPT

(C) SPT→S1→S2→S3→FLT→SPT(I) SPT→S1→S2→S3→FLT→W1→W2→W3→SPT

(J) SPT→FLT→ I1→ I2→SPT

(L) SPT→S1→S2→S3→FLT→ I1→ I2→SPT

(O) SPT→S1→S2→S3→FLT→W1→ I1→ I2→W3→SPT

Fig.7 Seasonal cycle patterns of thermal régimes as a function of (T ea, H ).

The computation is carried out for ∆T e=12℃ which is an average of ∆T e=12℃ from Figs.2

and 3.

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More winter stratification develops in deeper lakes. No winter stratification is found in south

areas of Japan.

Fig.8 Seasonal cycle patterns of thermal régimes for different water depths H .

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Fig.9 Life span of summer stratification (H =2m, 10m, 50m and 100m).

(2) Life span of temperature stratification

Life span of the two types of temperature stratification, say summer stratification and winter

stratification, is distributed as shown in Figs.9 and 10. Fig.9 shows that summer

stratification is kept longer in warmer and deeper lakes. In the case of winter stratification in

Fig.10, a longer life span is observed in colder and deeper water areas.

(3) Ice life span

From Fig.11 it is confirmed that an ice-cover is kept longer in shallower and colder region,

which is expected from Fig.7.

6 Concluding remarks

Applying field data of equilibrium temperature to a mixed-layer model, seasonal variation

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patterns of lake’s thermal régimes are analyzed and discussed. In the model a thermal

structure is described as a function of the three parameters; an annual variation amplitude of

equilibrium temperature ∆T e, an annual average of equilibrium temperature T ea and a lake’s

depth dimension of H .

The present analysis leads to the following findings.In a rage of equilibrium temperature in Japan, seven patterns of seasonal variation could

occur, which are classified by the three criteria that (1) a lake is homogeneous or stratified, (2)

a water surface is frozen in winter or not and (3) water temperature is kept below or above

4℃.

In warmer and deeper lakes, the summer stratification is kept longer.

More ice régime and winter stratification are observed in regions of higher altitude and

latitude.

A surface ice is kept longer in shallow lakes. A winter stratification is stronger in deeper

lakes.In colder regions more heat is exchanged across the water surface not only in summer but

also in winter.

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Fig.10 Life span of a winter stratification (H =2m, 10m, 50m and 100m).

Fig.11 Life span of ice (H =2m, 10m, 50m and 100m).

References

[1] Michioku,K. and Kadoyu,K.: Parametric analysis on annual variation of surface heat

exchange and thermal structure in lakes, J.Hydroscience and Hydraulic Engineering, Vol.10,

No.1, pp.77-94, (1992).

[2] Michioku,K.: Thermal régimes of impounded water body experiencing summer andwinter stratification, Proc. 25th IAHR Congress, P15, pp.457-464, (1993).

[3] Kraus,E.B. and Turner,J.S.; A one-dimensional model of the seasonal thermocline II.

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The general theory and its consequences, Tellus, 19, pp.98-105, 1967.

[4] Uchijima,Z.: A physico-climatological study of the water temperature in the paddy field,

Bulletin of the National Institute of Agricultural Sciences, A-7, pp.131-181, 1959 (in Japanese).

[5] Arai,T.: Theory of water temperature (Suionron), Kyoritsu Press 1974 (in Japanese).

[6] Muraoka,K. and Hirata,T.: Thermal stratification and internal wave in Lake Chuzenji,

Research Report from the National Institute for Environmental Studies, Vol.69, pp.5-35, 1984(in Japanese).

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