The Solar to Stream Power Ratio

RIVER RESEARCH AND APPLICATIONS

River Res. Applic. (2012)

Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/rra.2579

THE SOLAR-TO-STREAM POWER RATIO: A DIMENSIONLESS NUMBER EXPLAININGDIEL FLUCTUATIONS OF TEMPERATURE IN MESOSCALE RIVERS

O. LINKa*, A. HUERTAa, A. STEHRb, A. MONSALVEa, C. MEIERa and M. AGUAYOb

a Civil Engineering Department, Universidad de Concepción, Concepción, Chileb Center of Environmental Sciences EULA-Chile, Universidad de Concepción, Concepción, Chile

ABSTRACT

The diel variation of temperature in mesoscale river reaches (catchment area> 1000 km2) is analysed using concurrent measurements ofwater temperature and of those meteorological (incident short-wave radiation, air temperature, relative humidity and wind speed variables)and hydraulic variables (streamflow, top width, channel slope and flow depth) controlling the thermal regime. Measurements were takenalong two river reaches located in central Chile, on the Itata (11 290 km2, Strahler’s order 6, reach length 30 km, Qbankfull = 400m

3 s�1)and Vergara (4340 km2, Strahler’s order 5, reach length 20 km, Qbankfull = 85m

3 s�1) rivers. The measuring frequency was 15 min. The rele-vant energy fluxes at the air–water interface, that is, atmospheric long-wave radiation, net short-wave radiation, radiation emitted by the waterbody, evaporation (latent heat) and conduction heat are computed and analysed for four scenarios of 12 days duration each, representingtypical conditions for the austral winter, spring, summer and autumn. We find large differences in the diel river temperature range betweenthe two sites and across seasons (and thus, flows and meteorological conditions), as reported in previous studies, but no clear relationship withthe controlling variables is overtly observed. Following a dimensional analysis, we obtain a dimensionless parameter corresponding to theratio of solar-to-stream power, which adequately explains the diel variation of water temperature in mesoscale rivers. A number of ourown measurements as well as literature data are used for preliminary testing of the proposed parameter. This easy-to-compute number isshown to predict quite well all of the cases, constituting a simple and useful criterion to estimate a priori the magnitude of temperature dielvariations in a river reach, given prevailing meteorological (daily maximum solar radiation) and hydrologic–hydraulic (streamflow, mean topwidth) conditions. Copyright © 2012 John Wiley & Sons, Ltd.

key words: thermal regime; diel variability; river temperature; dimensional analysis; fluvial hydraulics

Received 12 October 2011; Revised 3 April 2012; Accepted 21 April 2012

INTRODUCTION

The temperature of a river determines fundamental fluidproperties such as water density and viscosity, as well asthe solubility of dissolved gases. It alters the flow turbulentintensities, thus affecting river transport and mixing capacity,and controls the velocity of chemical reactions, with strongeffects on important processes such as stream metabolism,productivity and the decomposition of organic matter. Watertemperature governs the growth rate of aquatic organismsand influences their distribution and abundance through theirspecific tolerance ranges. Because the thermal regime of ariver reach is such a fundamental control of water qualityand biota in the fluvial ecosystem, it must be adequatelyassessed for any effective environmental management(Caissie, 2006; Webb et al., 2008).River temperature depends on many different, interrelated

factors (Johnson, 2003, 2004), which can be classified in fourgroups, following Caissie (2006), with slight modifications:(i) meteorological (atmospheric) conditions (i.e. incident

*Correspondence to: O. Link, Civil Engineering Department, Universidadde Concepción, Barrio Universitario s/n, casilla 160-C, Concepción, Chile.E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

short-wave radiation, air temperature, relative humidity andwind speed variables), (ii) shading effects, (iii) hydrologicaland hydraulic variables (i.e. streamflow, top width, channelslope and flow depth) and (iv) streambed conditions.Meteoro-logical conditions, reviewed by Edinger et al. (1968), arefound to be the most important factor in many differentsettings (e.g. Brown, 1969; Mosley, 1983; Sinokrot andStefan, 1994; Webb and Zhang, 1997, 1999; Evans et al.,1998; Gu et al., 1998; Younus et al., 2000; Meier et al.,2003). Specifically, net short-wave radiation, net long-waveradiation, air moisture and wind speed are the atmosphericvariables that mainly explain the heat fluxes at the air–waterinterface (Mosley, 1983; Evans et al., 1998). Shading effectsare due to a combination of the riparian vegetation andsurrounding landscape (Beschta, 1997; Chen et al., 1998;Johnson, 2004) but also depend on the stream’s aspect(orientation with respect to the sun’s trajectory and the nearbyhills; Rutherford et al., 1993). Hydrological (i.e. streamflow)and dependent, hydraulic variables such as top width andflow depth all affect thermal inertia —the capacity for heatstorage— along a reach (Edinger et al., 1968; Smith, 1972;Rutherford et al., 1993; Gu et al., 1998; Sinokrot and Gulliver,2000). According to Gu and Li (2002), river temperatures can

O. LINK ET AL.

be as sensitive to streamflow as to weather conditions,specifically when dealing with maximum instead of meanvalues. Finally, streambed conditions include heat conduc-tion into the river bed material (Brown, 1969; Hondzo andStefan, 1994; Webb and Zhang, 1999; Meier et al., 2003;Neilson et al., 2009), as well as the effects of hyporheicexchanges (Mosley, 1983; Johnson and Jones, 2000;Neilson et al., 2009) and groundwater contributions(Webb and Zhang, 1999).In the case of larger, wider streams and rivers (i.e. width

50m), local shading effects such as those caused by riparianvegetation and surrounding hills tend to become less impor-tant (Brown, 1969; Evans et al., 1998); in which case, solarradiation will be the main heating mechanism during cleardays (Meier et al., 2003). Such larger rivers will run deeperand have larger streamflows so that the influence ofsubstrate and groundwater contributions will also decrease,in relative terms (Evans et al., 1998), allowing for theseterms to be neglected (Gu et al., 1998). Mosley (1983) andArscott et al. (2001) show that there can be ample spatialvariability of water temperature across the differentchannels, in the case of braided rivers. On the other hand,very large, deep and slow-moving rivers can stratify(Bormans and Webster, 1998), requiring a two-dimensionalvertical approach. In this paper, we refer to mesoscale riversas those cases where the shading and streambed effects onwater temperature can be neglected, but where there isadequate vertical and transversal mixing, so that its variabil-ity can still be described in a one-dimensional framework. Insuch cases, the behaviour of river temperature will becontrolled by an interaction between atmospheric variablesand hydrologic–hydraulic conditions of the flow.Even though Smith (1981) argued that ‘the diurnal

variation of river water temperature is probably the mostsensitive index of any thermal modification’, the biologicaleffects of diel fluctuations in river temperature have beenscarcely studied (e.g. see Cox and Rutherford, 2000a,2000b), and there is little literature on estimating rivertemperature fluctuations at the daily scale. Many authors(e.g. Stefan and Preud´Homme, 1993; Caissie et al., 2001;also, see review in Caissie, 2006) have used stochasticmodels to calculate daily water temperatures, but these needto be calibrated for each river reach under the local bound-ary conditions. Deterministic solutions, on the other hand,establish energy budgets to estimate river temperatures.Because of the rather complex processes involved, and toaccount for variability in time and space, most such modelsare solved numerically (Sinokrot and Stefan, 1993, 1994;Hondzo and Stefan, 1994; Kim and Chapra, 1997; Younuset al., 2000; Tung et al., 2006, 2007). This type of approachdoes not allow for a priori estimation of the diel fluctuationsthat a river reach would experiment, as the models need cali-bration, or else, they require hard to measure, not commonly


available meteorological variables, such as hourly airtemperature, relative humidity and wind speed.Only Edinger et al. (1968; Equation 27) and Gu et al.

(1998; Equations 8 and 9) have proposed direct equationsfor estimating diel fluctuation of water temperature in riverson the basis of the concept of equilibrium temperature Te,the water temperature for which the net rate of heatexchange between air and water is zero. As the equilibriumtemperature Te continuously adjusts to the changing weatherconditions (radiation, air moisture, wind speed, etc.), thewater temperature tends towards it but with a lag. In bothpapers, it is assumed that the amplitude of the diel variationfollows a sinusoidal fluctuation, which is derived by alsoimposing a sinusoidal behaviour for Te. This approachrequires evaluation of the thermal exchange coefficientK between the air and the water, as the net rate of heatexchange is given by multiplying the difference (Te� Tw)by K. In both cases, detailed measurement or estimationof a series of atmospheric variables, such as dew-pointtemperature or air moisture content, solar radiation, airtemperature, wind speed, and so on, is needed.The aim of this paper is to derive a simple equation that

allows for a priori estimation of the magnitude of the dieltemperature fluctuations in water temperature in the caseof mesoscale rivers on the basis of commonly available data.Measurements of river temperature and of the variablescontrolling the heat fluxes at the air–water interface, andthus the thermal regime of mesoscale rivers, are presentedand analysed. Dimensional analysis is applied to obtain acontrolling dimensionless parameter of the temperature dielvariation. We use our own measurements as well as litera-ture data for preliminary testing of the proposed equation.The paper is organized as follows. First, the processes,governing equations and parameters involved in the thermalregime of a mesoscale river are briefly reviewed. Second,the study sites and measuring techniques are presented.Third, field measurements of the involved variables andrelevant heat fluxes for two different rivers are analysed.Finally, a new dimensionless number explaining the dielvariation of river temperature is obtained and tested againstfield measurements and literature data. The paper concludeswith final remarks on the obtained results.

PROCESSES, GOVERNING EQUATIONS ANDPARAMETERS

The processes involved in the thermal regime of a river aremainly the advective transport, mixing, and heat and radi-ation exchanges between the water body and the surround-ing environment, that is, the air at the water surface andthe streambed. The effects of shade by riparian vegetationand surrounding topography, as well as the upwelling of


DOI: 10.1002/rra

THE SOLAR-TO-STREAM POWER RATIO

cold groundwater and heat exchange with the bed, areneglected in this analysis because the study sites correspondto mesoscale rivers where local shading effects such as thosecaused by riparian vegetation and surrounding hills tend tobecome less important (Brown, 1969; Evans et al., 1998),and the influence of substrate and groundwater contributionswill also decrease, in relative terms (Evans et al., 1998),allowing for these terms to be neglected (Gu et al., 1998).Mathematically, the governing equation of the river thermalregime is the advection–diffusion–reaction equation forwater temperature:

A@T

@tþ @ QTð Þ

@x� @

@xDLA

@T

@x

� �¼ HTW

Cwrw(1)

where A (m2) is the transverse flow area, T(�C) the section-averaged water temperature, t (s) the time, Q (m3 s�1) theriver discharge, x (m) the longitudinal distance along thechannel, DL (m2s�1) the longitudinal dispersion coefficient,HT (W m�2) the thermal energy flow through the free sur-face,W (m) the top channel width, Cw (J kg�1 �C�1) the spe-cific heat of water and rw (kgm�3) the water density.The longitudinal dispersion coefficient is a mixing param-

eter that can be computed according to several availableequations. Because of its validity range, we preferred thatproposed by Vargas and Ayala (2001):

DL ¼ 7:39W

RH

� ��1:86 Q2

U�R3H

(2)

where RH (m) is the hydraulic radius and U* (m s�1) is theshear velocity. However, both study reaches have a very highPéclet number (defined as the ratio of the rate of advection of aphysical quantity by the flow to the rate of diffusion of thesame quantity driven by an appropriate gradient), indicativeof flows that are clearly advection dominated. As a result,diffusion effects on temperature changes in time are negli-gible, and thus, the choice of a specific formula for estimationof the longitudinal dispersion coefficient is irrelevant.The sources/sinks of heat where calculated from the

balance of energy flows following

HT ¼ HSW þ HLW � HB � HE � HC (3)

where HSW is the short-wave net radiation, HLW the netlong-wave incoming radiation, HB the long-wave radiationemitted by the water surface, HE the evaporation heat andHC the conduction heat.


Short-wave net radiation, HSW

The short-wave net radiation is computed as

HSW ¼ HSWin � HSWref (4)

where the incident solar radiationHSWin is directly measuredwith a radiometer whereas the reflected solar radiation,HSWref , is given by

HSWref ¼ RSHSWin (5)

with RS being the reflectivity, that is, the fraction of the solarradiation that is reflected at the water surface given by

Rs ¼ a180p

a� �b

(6)

where a is the solar altitude in radians and a and b arecoefficients that depend on the cloud cover, Cb. Accordingto Martin and McCutcheon (1999),

If Cb≥0:9 ⇒a ¼ 0:33 b ¼ �0:45If 0:5≤Cb < 0:9 ⇒a ¼ 0:95 b ¼ �0:75If 0:1≤Cb < 0:5 ⇒a ¼ 2:20 b ¼ �0:97If Cb < 0:1 ⇒a ¼ 1:18 b ¼ �0:77

(7)

The cloud cover Cb is determined as

Cb ¼ 1� Hrs sunactualHrs suntheo

(8)

The actual sunny hours, Hrs sunactual, were determinedfrom the measured incident solar radiation, considering thata sunny hour occurs if the solar radiation during that hourexceeds 120W m�2. Cb was determined on a daily basis.

Net long-wave incoming radiation, HLW

The net long-wave radiation was computed assuming that3% of the incident long-wave radiation is reflected:

HLW ¼ 0:97s Tair þ 273:16ð Þ4eair (9)

where s= 5.67� 10� 8 Wm�2 K�4 is the Stefan-Boltzmannconstant and Tair (�C) the air temperature. The air emissivity,eair, was estimated after Swimbank (1963), modified byWunderlich (1972) as

eair ¼ a0 1þ 0:17Cbð Þ Tair þ 273:16ð Þ2 (10)

where a0 is a proportionality constant equal to 0.937 � 10� 5.


DOI: 10.1002/rra

O. LINK ET AL.

Long-wave radiation emitted by the water, HB

The long-wave radiation emitted by the water surface followsStefan–Boltzmann’s law:

HB ¼ es T þ 273:16ð Þ4 (11)

where e =0.97 is the emissivity of water and T (�C) the watertemperature.

Evaporation heat, HE

The latent heat flux to vapourize water from the liquid to thegas phase is given by

HE ¼ rwLwE (12)

where rw (kgm�3) is the water density, Lw the latent heat ofvapourization of water equal to 1000(2499� 2.36 T) (J kg�1)and E (m s�1) is the evaporation rate, computed as follows:

E ¼ F wSð Þ es � eað Þ (13)

whereF wSð Þ is a linear function of the wind speed, wS (m s�1);

F wSð Þ ¼ a1 þ b1ws (14)

where a1 (mb�1ms�1) and b1 (mb�1) are site-specific coeffi-cients. Following Magnus–Tetens equation, the saturatedvapour pressure at the water temperature, es (mb), was calcu-lated as

es ¼ e7:5T

Tþ237:3þ0:7858ð Þ ln 10ð Þ (15)

The vapour pressure at the air temperature, ea (mb), wascomputed as

ea ¼ es � HR

100(16)

where HR is the relative humidity.

Conduction heat, HC

The conduction heat is computed as

HC ¼ rwLwCBowenF wSð ÞPa

PT � Tairð Þ (17)

where CBowen=0.61mb �C�1 is the Bowen coefficient, Pa (mb)is the atmospheric pressure and P (mb) is a referencepressure at sea level. In this study, we assume P = Pa

because both of our study reaches are at low elevationsabove sea level.Note that all of the previously described heat flows can be

computed by measuring only four hydrometeorologic


variables, namely (i) incident short-wave radiation, (ii) airtemperature, (iii) air relative humidity and (iv) wind speed.

STUDY SITE AND FIELD INSTRUMENTATION

The study sites correspond to the lower Itata (36�12′–37�16′and 71�00′–73�10′) (Link et al., 2009) and the Vergara(37�29′–38�14′ and 71�36′–73�20′) (Stehr et al., 2010)rivers, located in Central Chile. These reaches have lengthsof 32 and 20 km, average altitudes of 68 and 27masl, catch-ment areas equal to 11 290 and 4340 km2 and Strahler ordersat the study reach outlet of 6 and 5, respectively. The aver-age altitude, annual rainfall and minimum and maximumannual ambient temperature for the catchments are 582masl,1042mm, 6.8 and 19.9 �C for the Itata and 431masl,1650mm, 8.0 and 18.0 �C for the Vergara, respectively.Discharges are measured by the Dirección General de Aguas(National Water Agency) at the gauge stations ‘Coelemu’,located just at the downstream end of the Itata reach and‘Tijeral’ located in the middle of the Vergara reach. Add-itionally, two meteorological stations were installed for thisstudy at the river reaches, equipped with a data logger(Campbell CR1000), pyranometer (CS300-L Apogee),temperature and relative humidity probe (Vaisala HMP60-L),wind monitor (RMYoung 05103-5) and a barometric pressuresensor (Campbell CS106) for recording solar radiation, airtemperature, relative humidity, wind speed and direction andatmospheric pressure. For measurements of water temperature,loggers (HOBO Pendant UA-002-64) were used. The measur-ing frequency for all variables was 15min. Figure 1 shows thestudy reaches and the measuring points.Bathymetric surveys were conducted using a differential

GPS system (Leica SR530). The average longitudinal slopesfor the Itata and Vergara reaches are 0.3% and 0.4%,whereas their bankfull discharges are 400 and 85m3 s�1,respectively. Under bankfull conditions, average river topwidth, maximum flow depth, transverse section area andsection-averaged velocity take values of 412m, 3.0m,600m2 and 0.7m s�1 at Itata and 50m, 3.3m, 108m2 and0.8m s�1 at Vergara.

DIMENSIONAL ANALYSIS

To explore the dimensional relation between the diel vari-ability in water temperature and the variables controllingthe thermal regime of a river, we first need a dimensionlessparametrization for the daily temperature range. Herein, weuse Tdiel variation, defined as the difference between themaximum and minimum daily temperature divided by areference temperature, for example, the annual averageambient temperature in the basin.


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Figure 1. Study reaches and measuring points


Tdiel variation ¼ Tdaily max � Tdaily min

Tbasin average(18)

The functional relation is then expressed as

Tdiel variation ¼ f1 Q;W ; S0; g; n; cw;HSWmax; Tair;HR;wsð Þ(19)

where Q is the daily mean discharge in the river (m3 s�1), W(m) the mean wet channel width corresponding to Q, S0 (�)the average longitudinal slope of the river reach, HR (%) thedaily average relative air humidity, g the specific weight ofthe water equal to 9810 kgm�2 s�2, n the water viscosityequal to 1� 10�6m2 s�1 at 20 �C, Cw the specific heat ofwater equal to 4186 J kg�1 �C�1, Tair (�C) the mean dailyair temperature, HSWmax (Wm�2) the daily maximum solarradiation and ws (m s�1) the maximum daily wind speed.Applying the p-Buckingham theorem, the following dimen-sionless parameters can be formulated:

Tdiel variation ¼ f2 p1; p2; p3; p4; p5; p6ð Þ (20)

where p5 and p6 can be combined by multiplication into onesingle dimensionless parameter:

p1; p2; p3; p4; p5ð Þ ¼ HR;gv

HSWmax;g4Q2cwTairH4

SWmax

;g2Qws

H2SWmax

;HSWmaxW

gQS0

� �

(21)

Note that p5 represents the solar-to-stream power ratio:


HSWmaxHW

gQHS0¼ HSWmaxW

gQS0(22)

where H is the section-averaged flow depth. As a first ap-proximation, we assume that the longitudinal slope presentsa small range of variation across mesoscale rivers; as it isdimensionless, it can be dropped without violation of thep-Buckingham theorem, obtaining

p1; p2; p3; p4; p5ð Þ ¼ HR;gv

HSWmax;g4Q2cwTairH4

SWmax

;g2Qws

H2SWmax

;HSWmaxW

gQ

� �

(23)

At both study reaches, water temperature measurementswere collected during 48 h with thermistor chains, both atthe cross-section centre and at a distance of 2m from eachborder. Preliminary results showed that the water tempera-tures did not present significant differences (<0.5 �C) acrossthe sections so that the reaches could be considered wellmixed. Consequently, it seemed reasonable to apply a one-dimensional approach at both study sites. Next, we comparethe measured, controlling hydrometeorological variables andthe resulting heat fluxes for the winter and summer scenariosat the Itata and Vergara rivers. We apply dimensional analysisto derive a parameter explaining diel temperature variations,which we then test with data corresponding to typicalscenarios for austral (i.e. Southern Hemisphere) winter,spring, summer and autumn.


DOI: 10.1002/rra

O. LINK ET AL.

RESULTS

Hydrometeorological variables

Figure 2 shows the diel variation of the six hydrometeorologi-cal variables that were measured, namely water temperature atthe reach end, discharge, solar radiation, air temperature,relative air humidity and wind speed, for the winter (left)and summer scenario (right) at the Itata reach. The averagedwater temperature remained nearly constant around 9.6 and24.0 �C, during the winter and summer scenarios, respect-ively. The daily range for T was negligible during winter, at0.7 �C, but significant during summer, at about 8–10 �C. Peakwater temperatures were observed around 16:00 h, whereasminimum daily water temperatures occurred at about 04:00 h.The discharge was constant and very low (about 8m3 s�1)

for the summer scenario and unsteady during the winterscenario, with a flood peak discharge equal to 900m3 s�1.Note that a high temperature diel variation is detected duringthe summer (with constant discharge), whereas during the

Figure 2. Water temperature at the reach end, discharge, solar radiation, a(left) and summer scenarios


winter, we do not observe any diel fluctuation nor a director inverse correlation with the varying discharge.Solar radiation during sunny days reached peaks of up to

450 and 950W m�2 in winter and summer, respectively,whereas it was about half these values during overcast days.Air temperature varied from 4 to 18 �C and from 15 to

35 �C during winter and summer, respectively. In bothseasons, the daily behaviour of air temperature closely followsthat of solar radiation, with the temperature peak lagging by afew hours. On the other hand, the air and water temperaturecurves behave similarly during summer, but such similitudeis lost during the winter scenario, when the diel variabilityof water temperature goes close to zero.Relative humidity during winter varies from 90% to

35–50% during the warmer days. Minima are observedaround midday. During summer days, relative humidityvaries from 80% to 40–50% and even down to 15–25%during dry days. A marked diel variation of the relativehumidity is observed during the summer scenario only.

ir temperature, relative air humidity and wind speed, for the winter(right) at the Itata reach


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Wind speed goes to a base level during winter of ~0.5m s�1

with peak events of up to 6.5m s�1. During the summerscenario, wind speed is generally higher, with marked dielvariations from ~1m s�1 at night to peaks of 4–7m s�1,occurring during the afternoon.Figure 3 shows the diel variation of the six measured

hydrometeorological variables for the winter (left) andsummer scenario (right) at the Vergara reach.The daily average water temperature remained nearly

constant during the winter and summer scenarios, at 9.6and 24.0 �C, respectively. As in the Itata, peak temperaturesduring the day were observed around 16:00 h, whereasminima occurred at about 04:00 h. In the Vergara, however,the daily range in water temperatures was negligible bothfor summer and winter, with mean diel fluctuations of about1.0 and 0.5 �C, respectively.The discharge was constant and very low during the

summer scenario (12–15m3 s�1) and variable during the win-ter scenario going from a baseflow of 55m3 s�1 to a floodpeak discharge of 235m3 s�1. Note that during the summer

Figure 3. Water temperature at the reach end, discharge, solar radiation, a(left) and summer scenarios (ri


(with a nearly steady streamflow), we detected a high dielvariation in river temperature, whereas no temperature dielvariation nor a direct or inverse correlation with dischargemagnitude was observed in winter.Solar radiation during sunny days reached peaks of up to

500 and 1000W m�2 in winter and summer, respectively.Note that this is slightly higher than at the Itata site. Airtemperature varied from�1 to 15 �C and from 6 to 35 �C dur-ing winter and summer, respectively. Air and watertemperature curves follow similar trends during summer butnot in winter. Relative humidity in winter varies from 100down to 60% during some warmer days.Minima are observedat noon. During the summer, relative humidity varies from 90to ~20%. In winter, wind speed goes to a base level of about0.3m s�1 at nights, with peak events of up to 8.0m s�1. Forthe summer scenario, average wind speed is higher than inwinter, with minima around 1–2m s�1 during the early morn-ing and maxima of up to 7m s�1 in the afternoon.Note that there is a major difference in behaviour between

both rivers for the summer scenario. Even though incident

ir temperature, relative air humidity and wind speed, for the winterght) at the Vergara reach


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O. LINK ET AL.

short-wave radiation, discharge and other variables weresimilar at both rivers, the Itata exhibits a high temperaturediel variation, whereas the Vergara does not.

Heat fluxes

Figure 4 shows the relevant heat fluxes for the Itata reach,namely net short-wave radiation, net long-wave incomingradiation, radiation emitted by the water body, latent evapor-ation heat, sensible conduction heat and the general balancefor the winter (left) and summer (right) scenarios, respectively.The marks at the horizontal axis indicate the beginning of theday, that is, midnight.Diel variability of the net short-wave radiation is pro-

nounced during both measured periods, with maxima of about200–400W m�2 during winter and 600–900W m�2 duringthe summer, depending on cloud cover.Net long-wave incoming radiation is very similar during

winter and summer, equal to ~300W m�2, exhibiting a smalldiel variation in the order of 50–100W m�2. Radiation emit-ted by the water surface shows a nearly constant value during

Figure 4. Net short-wave radiation, net long-wave incoming radiation, radconduction heat and the general heat balance for the winte


winter, equal to �340W m�2 and a small diel variabilityaround �450W m�2 during the summer, practically cancel-ling the long-wave radiation balance.Evaporation heat is always negative. It is very small, less

than 50W m�2 during winter, when low ambient tempera-tures and high relative humidity are observed. During thesummer, sensible heat becomes important in the overall bal-ance, reaching peaks of up to �700W m�2 at noon.Conduction heat is very small compared with the other

fluxes, fluctuating around 0W m�2 year round. Neverthe-less, during the summer, higher conduction heat fluxes canbe observed, with maximums of ~100W m�2.The general heat balance exhibits a diel variation in the

range of �100 to 450W m�2 in winter and between �350and 700W m�2 during summer. During nights, especiallyin summer, the energy balance is negative because ofevaporation heat.Figure 5 shows the relevant heat fluxes at the Vergara river

for the winter (left) and summer (right) scenarios, respectively.Diel variability of the net short-wave radiation is

pronounced during both measured periods, with maxima

iation emitted by the water body, latent evaporation heat, sensibler (left) and summer (right) scenarios at the Itata reach


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Figure 5. Net short-wave radiation, net long-wave incoming radiation, radiation emitted by the water body, latent evaporation heat, sensibleconduction heat and the general heat balance for the winter (left) and summer (right) scenarios at the Vergara reach


of about 400W m�2 during winter and ~900W m�2 duringthe summer, depending on cloud cover. Net long-wave in-coming radiation is instead very similar during winter andsummer with values around 300W m�2, exhibiting a smalldiel variation in the order of 50–100W m�2. The radiationemitted by the water shows a nearly constant valueof �320W m�2 during winter and about �400W m�2 insummer.Evaporation heat was negligible during the whole winter

scenario. During the summer, sensible heat peaked at valuesof up to �120W m�2 at noon. Conduction heat is againvery small compared with the other fluxes, fluctuating closeto 0W m�2 during winter and summer.The general heat balance exhibits a diel variation between

�150 and 420W m�2 during winter and between �180and 870W m�2 in summer. Again, the heat balance isnegative at nights, which, especially in summer, is accen-tuated by the evaporation.Overall, diel variability of the heat fluxes is smaller in

winter than summer. Considerably higher short-wave radi-ation during summer does not necessarily cause a higher diel


variability in the water temperature as shown, for example,in the summer scenario for the Vergara river. Evaporationheat can reach sizeable values during summer nights, result-ing in lower minimum water temperatures (see the summerscenario for the Itata river). The diel variability of the differ-ent heat fluxes and in the general energy balance is not ableto explain the diel variability of the water temperature on itsown. On clear days, the behaviour of the general energybalance is driven by that of the incident short-wave radiation,during the sunny hours and also, on a lesser scale, by evapor-ation. At nights, it depends basically on the evaporationheat flux.Figure 6 shows Tdiel variation as a function of the dimen-

sionless numbers p1, p2, p3,p4 and p5 for all of the days ofthe previously presented scenarios on the Itata and Vergarastudy reaches.Out of the five parameters in (23) and in Figure 6, the

solar-to-stream power ratio, p5, clearly is the only dimen-sionless number consistently explaining diel temperaturevariation over the range of scenarios, at both locations. Notethat all variables in p5 are daily averages, except the


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Figure 6. Diel temperature variation over Π parameters for Itata and Vergara study reaches

O. LINK ET AL.

maximum solar radiation HSWmax. At this stage, the prelim-inary data suggest the following relationship between thetwo variables:

Tdiel variation ¼ 1:0HSWmaxW

gQ

� �0:6

(24)

with an R2 = 0.98.To verify the predictive value of the solar-to-stream

power ratio, Figure 7 shows Tdiel variation over p5 for datacorresponding to the Bere and Piddle rivers published byWebb and Zhang (1999) and to GaoShan creek by Tunget al. (2006, 2007).Again, p5 correlates quite well with the observed diel

fluctuations, as indexed by the dimensionless parameter Tdielvariation, with the data suggesting a linear relationship.


DISCUSSION

When tested against a number of actual measurements, aswell as literature data, the ratio of solar-to-stream powerwas shown to adequately explain the diel variation of watertemperature in mesoscale rivers. This index basically weighsin the two most important factors explaining watertemperature in cases where both shading and bed effectscan be neglected: on one hand, solar radiation is the mainvariable affecting available energy, whereas on the other,streamflow determines thermal inertia.The physics behind the solar-to-stream power ratio is

quite simple: On clear days (as well as overcast conditionsif cloud cover can be assumed constant), the maximumincident solar radiation, HSWmax, occurs at astronomicalnoon and is proportional to the daily total energy comingas short-wave radiation. In the case of mesoscale rivers, as


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Figure 7. Diel temperature variation over Π5 for Bere and Piddle riverspublished byWebb and Zhang (1999) and for GaoShan creek by Tung

et al. (2006, 2007)



defined in the Introduction section, diel fluctuations in watertemperature will be mostly driven by the variability in solarradiation. For a given reach length, the product of dailymaximum solar radiation times river wet width will beroughly proportional to the total energy available for heatingwater. Dividing this by g, Q, the weight flow of water, yieldsa measure of the energy available to heat a unit weight ofwater, which should be proportional to the temperaturefluctuation.Even though the proposed parameter performed well for

the available data, important deviations from the assumedconditions in the controlling factors, that is, in (i) meteoro-logical (atmospheric) conditions, (ii) shading effects, (iii)hydrological and hydraulic variables or (iv) streambed con-ditions, could generate a significantly different behaviour ofthe diel variation of river temperature. For example, if aclear window was to occur around astronomic noon in anotherwise overcast day, the maximum solar radiation wouldstrongly overestimate the total short-wave radiation energyavailable, and the computed diel fluctuation would be largerthan the actual value. In the case of smaller rivers, shadingeffects could decrease incident short-wave radiation, orgroundwater contributions could become relevant, resultingin smaller than expected diel variability. Thus, the proposeddimensionless number needs more testing across differingsituations to restrict or generalize its applicability.In the case of reaches located close to a source of warm

(springs in wintertime) or cold (glaciers, hypolimneticreleases from a dam, springs in summertime) water, thebehaviour of temperature diel variability might differ fromthat of the proposed criteria.Both the Itata and the Vergara rivers are sand-bed

channels, where hyporheic flows are usually restricted;mesoscale gravel-bed rivers could have important ground–surface water exchanges, affecting the applicability of theproposed parameter.

Anthropogenic changes altering controlling factors ofthe thermal regime are expected to change the behaviourof river temperature. As an example, Gu et al. (1998),Sinokrot and Gulliver (2000), Meier et al. (2003) andVal et al. (2006) show the effect of diversion dams onwater temperature.It could be argued that both streamflow and wet width

change along the longitudinal dimension of a river, whichcould invalidate our approach. Nevertheless, well-knownconcepts of downstream hydraulic geometry (e.g. Dingman,2009) indicate that such changes are very progressiveand predictable in the case of alluvial rivers. If the solar-to-stream power ratio were applied to a mesoscale river withnon-alluvial reaches immediately above the site of interest,one should expect some deviations from the predicteddiel fluctuations.

CONCLUSION

The diel variation of temperature in mesoscale river reacheswas analysed using concurrent measurements of watertemperature and of those meteorological and hydraulicvariables controlling the thermal regime.Large differences in the behaviour of diel temperature

variation were found between the two study sites and acrossseasons. By using dimensional analysis, we derived adimensionless parameter corresponding to the ratio ofsolar-to-stream power, which adequately explains the dielfluctuations of water temperature in mesoscale rivers. Anumber of our own measurements as well as literature datawere used for preliminary testing of the proposed parameter.The ratio of the solar-to-stream power is an easy-to-

compute number, constituting a simple and useful criterionto estimate a priori the magnitude of temperature dielvariations in a river reach, given prevailing meteorological(daily maximum solar radiation) and hydrologic–hydraulic(streamflow, mean top width) conditions.

ACKNOWLEDGEMENTS

The financial support provided by Grant 1090428 of theChilean Research Council CONICYT is greatly acknowledged.The chilean Dirección General de Aguas (National WaterAgency) provided discharges measured at gauge stations‘Coelemu’ and ‘Tijeral’. Professor Peter Goodwin from theUniversity of Idaho facilitated the GPS for bathymetric survey.Special thanks are given to our students Christian Barahonaand Miguel Barahona and to our technicians Hector Alonsoand René Iribarren for their engangement and collaborationduring the field work. Professor Ching-PinTung and


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O. LINK ET AL.

Dr. Tsung-Yu Lee from the National Taiwan University gentlyprovided the data for GaoShan and ChiChiaWan creeks tovalidate the proposed parameter.

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The Solar to Stream Power Ratio

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