Calculation of temperature variations of small mountain streams

TEMPEF, ATURE VARIATIONS OF SMALL MOUN-

H . F : VUGTS

Departmen$ of Meteorology, Free University, Amsterdam (The Netherlands) (Accepted for publication February 4, 1974)

ABSTRACT Vugts, H.F., 197 ~. Calculation of temperature variations of small mountain streams. J, Hydro]., 23: 267--278, Temperatures. of small mountain streams can be accurately predicted by using an energy balance method. The technique was tested by performing micro-meteorological measurement~ along the Vigilbach (Rio di San VigfiiolDolomites, northern Italy). The water temperature was measured at 8 points from the source down to a distance of 9 kin. From the experimental results it can be concluded that, starting from meteorological observations, stream temperature changes up to 7.0°C were predicted within 10% accuracy. It appeared that during the day on unshaded stretches the net radiation and the transfer of heat by conduction and turbulence are the predominant energy som'~es, a,~d that during the night the energy gain due to friction of the water with the b o t t o m plays an ~mportant role in the energy budget. In this paper the various en2rg-y balance terms are discussed and some results are shown.

INTRODUCTION

Clearcutting practices in forestry and heated industrial effluents (thermal pol]iufion) cause a temperature increase in rivers and lakes. The knowledge of the dependence of the water temperature of rivers and lakes on meteorological parameters becomes therefore important. The water quality depends strongly on the water temperature which affects a ~ d e range of biological, che~nical and physical processes: A biological process affected by water temperature was reported by Cairns (1956). The activity of lower plants and animal~ such as algae and pathogenic bacteria, generally follows the "Van 't Hoff" ru!e. The rule expresses in this case that a 10°C temperature increase causes a doubling of the reaction rate, which tends to increase oxygen consumption° But the physical capacity of water to hold oxygen decreases. In such a situao tion the competition for oxygen in :an aquatic community becomes a critica! effects of temperature on fish life are reported by Bret$ (1956L A $heoreticai study on a larger scale was carried out by Nobel (1971), who

268

calculated the self-qooling capacity of a river as a lunch;ion.of the distance between two successive power stations situated along t;he ~ver. Before developil the C pan: his

1 _

of the Nile are mentioned in his meteorological tables; More sys~matic observations along the were c iea out by (1826--1888) between 1826 and 1833. From this period up to 1885 many investiga~rs measured more or less regularly the water temperatures. Based on these results Guppy (1892-1897} was able to analyze the t h e r m ~ regime of the Nile in a manner sufficiently accurate for the purpose of a preliminary investigatiom Guppy draws the following conclusions. (1) It has been well established that rivem of shallow depths, say between 0.5 m and 1.5 m, have either much the same temperature on top and bottom, or display only minor temperature differences. Obviously the more rapid currents of the shallow rivers, result in a more complete mixing process. (2) The hour after sunrise constitutes the most suitable time for observing the lowest daily temperature of rivers. During summer the maximum temper° ature should be observed between 15h and 16h. ]In spring and autumn the maximum takes place between 14h and 15h, and in win~.er~ime at 13h. (3) The mean temperature is probably at'~ained between 10h and 11h in summer~ during ~he spring and the autumn at 11h30 and in wintertime at 12h. Guppy's conclusions do not appear to apply to very small rivers, bu~ ~o rivers which display a daily ~empera~ure range of about 2°C. Although claiy ~emperature measurements constituted the basic da~a, i~ is unfortunate tha~ we are left with monthly mean values of the water ~emperature. This body of measurements was used by Guppy only for comparisons with ai~ ~emperature. No quantitative relation was derived, however; at best a qualitative rela~ionship is some$imes g~ven. Similarly ~he paper of Wund~ (1940) and ~he ~hesis of Petsche (1947) on rNer ~emperatures con~:[n only incidental remarks on meteorological parameters. Pe~che even sta~;es that t:he air ~empera~ure can be assumed equal for all the river regions in half of Austria. The f i ~ reasonable quantitative treatment is given by Eckel and Reuter (1950)° Their paper reports on much theoretical work and their formulae are checked w~th me~eorolol~cal observations ~ad wa~er temperature measurement. Final]y~ ~wo other important experiments are known in l~era~ure: one by Edinger e~ al. (19~8) and one by Brown (1969). The recen~ inves~iga~ion~ were s~imulated by the in~eres~ in good. managemen~ of wa~er quali~y, for which wa~er temperature i~ an essential environmental index°

269

THEORY

A general energy budget equation m a y be written as: (1) Qn = net radiation; Qe = evaporative re flux; Qb = b o t t o m conduction fl~ux; are assumed positive if energy is added es occur. With the help of eq.1 it is possible to predict the variations o f the water temperature from measurements ~ f t h e meteorological parameters. The results of the t h e o r y can be checked by measuring the water temperature at several points along a stream. For the purpose of this study the Vigllbach, Dolomites, northern Italy (46¢N 11~) was aelected. The terms o f the energy budget equation will now be discussed in de,taft. The net radiation (Qn) was measured with a Thornthwaite portable net radiometer (hr.603). The radiometer sensor was placed at about 30 cm above the water surface of the Vigilbach, to measure the total incomhng and outgoing all-wave radiation. Because the stream runs through partially w e o d e d areas the percentage of shaded area along the stream was esthnated for three representative m o m e n t s of the summer day. These e s t i m a ~ s are listed in Table I; Fig.1 indicates the location of the Vigilbach. In the calculations the net radiation has been corrected for the percentage of shadow. TABLE I Some physical parameters of the four sections of the Vigflbach Traject

I II IH IV

% shadow : 09h 13h 15 0 12 15

15 0 7 9

Diff. in height (m)

Length (m)

Mean depth ( cm )

Time (rain)

17h

Mean effecfive width L (m )

21 0 12 24

10 35 15 15

140 15 60 150

3,300 1,150 1,750 2,750

33 27 $1 37

63 27 25 41

Heat exchange at the surface of the stream (Qe) is caused by condensation or evaporation. For this study the formula of Brutsaert and Yu (1968) was used: Qe = .15 " 10 -3 U2 (A~) -°'1~ (ew - - e : )

(2)

where Qe = evaporative flux (J cm -2 rain-l); U2 = wind velocity at an elevatk~n of two meters (cm sec -~ ); e w = saturation vapour pressure correspondh~g wRh t h e temperature of ~he water surface (mbar); e I = vapour pressure of the a~' (mbar); and A = evaporating area of the water surface (cm2).

270

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Fig. 1. Location of the instruments along the Vigfibach.

T e path length L of the wind across the stream is m the htera~ure a,,~sumed equal L = A . In this paper the path length L is called effective width of the stream. Table I ]ists some estimated values of L along the Vigfibach. As the measured humidity e i i~ supposed not to be influenced by t h e evaporating water the humidity was measured at the windward side of the stream. Eq.2 has an advantage above other ones in that it takes into account the dimensions of the stream. Convection of heat (Ql), also occurs in the air above the stream surface and to represent this process in the caIculation t]he Bowen ratio has been ua~d° h

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271

1"ne Bowen zafio describes arelati0nship between heat convection and evaporao tion. In this case the relation is: QI = 6.1 - 10-* p ~ Qe ew : e t

(3)

where p = : ~ pressure (mbar); Tw = wa~ertemperature (°C); and T1 = air ternperature (°C). The :use of the Bowen ratio is restricted to special cond.~tions, which were not always present during thefield work. Energy m a y b e added to the stream by the addition of water from side branches o f a different temperature (Qa). This is important when the difference in temperature between the main stream and the side branches is considerable and also when the discharge from the tributaries m sufficiently large with respect to the main stream. At four junctions of the main stream and the side branches the temperature was measured continuously. Discharge measurements were carried out occasionally> so that the advective energy term Qa could be adjusted by a simple mixing ratic. The conduction heat transfer process between bottom and water could not be measured (Qb). As the contribution of this process maid be important during the night an estimation of the term Qb had to be made. For this purpose a formula suggested by Eckei and Reuter was used. This fonxmla is based on the assumption that the variation of the water temperature is a periodic function, as follows: T w = constant + T o cos

(2,rt

~

"--~--o/

(4)

where T O = amplitude of the water temperature variation; t = time; and t o = period = :one day = 86~400 sec, The conductive flux derived by Eckel and Reuter (1950} can be expressed as: 1 Qb = t 2 _ t ~

(<) 2~

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s,

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('

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)

(5)

where ~b = thermal conductivity of the bottom; and a = thermal diffusivity. For the material constants we used values from the literature, namely a = 1() -3 crn =sec -J and X= 5.10 -2 J cm -2 rain -1. Fig.2 shows values of Qb vs. time for the four different trajects. Although the contribution of the frictional heat Qf is usually n e # e c ~ d in experiments on energy budget analyses of streams, the effect of friction of ~be . water . . . . . . . . .rna . Y be not ..... :unim P :ortant . The impol~ance of the term Q f h ~ been reported in the l i ~ r a t u r e a long :time ago (Mayer, 1842). After a fall of about 5(} m the wa~er temperature has increased by 0.t C~ because of the conservao ~;ion of e n e r ~ a~suxning that the stream velocity has:a constant value° The

272 0'~0

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t~ng.2. The daffy pattern i;n the b o t t o m conductive flux for the four sections of the Vigflbacho

height differences al[ong the stream (Table I) give thus an indication on the magnitude of Qf. As shown by Eckel and Reuter the water temperature change in a stream can be predicted as a function o f the total applied heat and the volume of" water which is warmed up, Their result is given by:

dT~,= 1 dt phc Qs

(6)

or, if the time intervals are shorter than 2 h: A Tw

1 ~hc Qs ~ '

(7)

where p := densiW of the water; c = specific heat. capacity of the water; and h = depth of the stceamo From eq+7 it is clear that A Tw is indirectly a func~i.on of ~he foliow~g met,eorologic~l parame~;e~s, which de~;ermine the ma~iSude o f Qs: water ~em= perai;ta°e {Tw,, ew), air Sempera~ta-e (TI), vapour pressure (el).~ wind veloci~ (U~) ~ d radiation (Qn).

273

OBSERVATIONS

The above prediction theory was tested on a smaU stream, the Vigilbach, tream was spl~t up rested land and in considered in the arge o f about 1.2 ff 5.6°C (in 1971 ). In section II the braided stream flows through unforested areas. In the reach of section II most of the radiation measurements and additional micro-meteorological observations were carried out. In the third section the stream runs through the village of San~Vigfiio, and in the fourth section two important side branches flow into the main stream. Over a total d~stemce of about 9 km the travel time of the water is about 2.5 h. The stream temperature was continuously measured at 8 points along the stream, as wel! as along the four side branches. The meteorological station was located near the village of San V ~ i o , halfway the stretch of the river. Additional measurements were performed in the other sections, but no significant differences were observed. The water temperature was measured with an accuracy of + 0.15°C; the temperature recorders were checked twice daily with calibrated mercury thermo° meters. RESULTS

Fig.3 shows a diagram of the energy budget components measured at section II along the Vigfibach on July 9, 1971, a bright sunny day. The maxin~um temperature reached 28°C and the mean wind velocity hovered about 2 m sec-~° Table H lists the results of calculations for all sections for 6 representative periods during that day. For each of the four sections the e n e r ~ / b u d g e t was computed for water which had left the Sankt Veitsquellen at 00h00, 04h00, 08h00, 12h00, 16h00 and 20h00, respectively. The units used for Qn~ Ql, Qe, Qb and ~'total" are cal. cm "2 rain-l; Qa and Qf and the temperatures are given in ~Co In the last two columns the differences t)etween calculated and measured values are given per section and for all sections combined° From the last column it is clear that the predicted values of T w agree well with the measured values at night. During the daytime, however, a cons~tent difference between measured and calculated values was observed. Especially for the water which leaves the sources at 08h00, the largest dev~.gon was observed in the first section° A plat~ib!e explanation for the difference is ~hat the percentage of shadow in section i has been underestimated, which ~ l l result in temperature v~ues that are too high° For water which leaves the sources at 12h00 and 16h00 the l ~ e s t de~ations w e r e observed m section HL !n this region the mean increase m temperature is a~ways tess than in the other regions. The discrepancy

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can be explained by infiltration of cold ground yeast from the alluvial fan at San V i ~ i o into the river. Fig,4 is a graphical representation of the last columns of Table Ii° The pic~ ture shows how the te~peratt~e increases with increasing distance from +~he sources. An observer who would t.~ve! with the water would experience this strong dependence of the ware. temperature on the distance° Per ~fferent poin~ ~ong the s~eam the ~empera~re-~at~.on has been shown in Figo&:This would be measured by an observer sitting alongside ~he s¢~e~an ~a~ch.mg h~s the~nometero

276

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Fig,4. Observed and predicted teraperatures for the four sections of the Vigilbaeh o ~ 9 July 1 9 7 i , The water had left the sources at 00h00, 04h00, 08h00, i 2 h 0 0 , 16he0 aiad 20he0, respectively.

CONCLUSIONS

{1) A s ~ p l e ene~'gy budzet eq~.m%ion(eqot) a~;d measmoemen% of five meo ~o~ologmal p~ameCe~ :rm~d~it pessib!e $o I:,r.d~.;t ~he waCe~~,em~ Cm.e (2) The minb:mm waSer Sempemtuze ~¢as reached atotmd

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277

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A F T E R I-3RAtDiNG.

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Fig. 5. Observed and predicted four hourly water temperatures ag four ~ocafSons along the Vigfibach on 9 July 1971.

morning' which is in agreement with Guppy (1892--1897) and Eckel and Reu1~er (1950). (gee also Fig.&) (3) The maximum temperaNre is observed around 14h00, an hour after ~ e maximum air t e m p e ~ t ~ e and the maximum net radia'i,ion. This differs from ~he observations of most of the other authors.. The fast response of the ~ate_ ~empera~ure to ~he meteorological element~ is the result of the shallow° hess of the stream. (4) The ~r~ean deJbz water terape~°a~ure, obser~ed for instance near She vil= lage of Z~schenwasser, occurs a~ abouf~ l o g o 0 in the morning. This is in

agreement with ~;hefiadings of Guppy°

278

(5) The daily amplitude of ¢he water temperature increases i n a downstream direc~,;ion. ACKNOWLED GEMENTS

The a u t h o r wishes to thank Mr. G:J:W. Krajenbrinkand Mr. W.J. Willems for their assistance in the measurement program. I: am also indebted tO Prof' P. Groen and Drs. J.T.F. Zimmerman for many helpful discussions.

REFERENCES ~rett, J.R., 1956. Some principles in the thermal requirements of fishes. Q. Rev. Biol., 31: 75--81. Brown, G.W., 1969. Predicting temperatures of small streams. Water Resour. Res., 5: 68w75. Brutsaert, W. a~id Yu, ~.I,.,1968. Mass transfer aspects of pan evaporation. J. Appl. Meteorol., 7: 563--566. Cairns, J., 1956. Effects of increased temperature on aquatic organisms. Industrial Wastes, 1: 150--153. Coutelle, M., 1799--1801. On the Nile temperature. Description de l'Egypte. Hist. Nat., tome 2, p.334° Ecke], O. and Reuter, H., 1950. Zur Berechnung des sommerlichen W~rmeumsatzes in Flussl~ufen. Geogr. Ann., 7: 188--209. Edinger, J.E., Duttweiler, D.W. and Geyer, J.C., 1968. The response of water temperatures to meteorological conditions. Water Resour. Res., 4: 1137--1143. Guppy, H.B., 1892--1897. River temperature. Proc. R. Phys. Soc. Edinb., part I, vol.12, pp.286--312; part If, vol.13, pp.33--61; part IIL vol.13, pp.204--214. Hay, R., 1826--1833. On the Nile temperature. Manuscript in British Museum Add. London, M.S. 29858, Mayer, R.J., 1842. Mechanik der W~rme. Liebigs Anna]. 1867o Collected papers. Nobem, L., 1971. The Analytical Determination of the Heat Transfer Coefficient between the Free Surface of a River and the Atmosphere. EUR commission of the European Communities. EUR 4631 e, Luxembourg, 38 pp. Petsche, W., 1947. Flusstemperaturen in OberSsterreich. Dissertation, Wien, 34 pp. Wundt, W., 1940. Beitr~ge zur Temperatur der fliessenden Gew~sser. Peterm. Geogr. Mitt., 86: 399--406.