HYDROLOGICAL PROCESSES Hydrol. Process. 17, 3069–3084 (2003) Published online 21 August 2003 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.1280
Water–air temperature relationships in a Devon river system and the role of flow
B. W. Webb,* P. D. Clack and D. E. Walling School of Geography and Archaeology, University of Exeter, Amory Building, Rennes Drive, Exeter EX4 4RJ, UK
Abstract: The nature of the water–air temperature relationship, and its moderation by discharge, were investigated for catchments ranging in size from 2Ð1 to 601 km2 in the Exe basin, Devon, UK and for data relating to hourly, daily and weekly time bases. The sensitivity and explanatory power of simple water–air temperature regression models based on hourly data were improved by incorporation of a lag, which increased with catchment size, although relationships became more sensitive and less scattered as the time base of data increased from hourly to weekly mean values. Significant departures from linearity in water–air temperature relationships were evident for hourly, but not for daily mean or weekly mean, data. A clear tendency for relationships between water and air temperatures to be stronger and more sensitive for flows below median levels was apparent, and multiple regression analysis also revealed water temperature to be inversely related to discharge for all catchments and time-scales. However, discharge had a greater impact in accounting for water temperature variation at shorter time-scales and in larger catchments. Copyright 2003 John Wiley & Sons, Ltd.
KEY WORDS water–air temperature relationships; discharge effects; catchment scale
INTRODUCTION In investigating the thermal regimes of streams and rivers, which are of considerable significance to aquatic ecology, water quality and the utilization of water resources, attention often has been given to the nature of statistical relationships between water and air temperatures (e.g. Johnson, 1971; Kothandaraman, 1972; Smith, 1981; Crisp and Howson, 1982; Webb, 1987; Stefan and Preud’homme, 1993; Caissie et al., 1998; Mohseni et al., 1998). Air temperature is commonly used as the independent variable in regression analysis of stream and river temperatures because it can be viewed as a surrogate for the net changes in heat flux that affect the water surface, and also because it approximates the equilibrium temperature of a water course. The latter is the temperature of the water surface at which no net exchange of energy occurs with the atmosphere (Edinger et al., 1968; Dingman, 1972). Linear regression models between air and water temperature have been developed successfully for data relating to different time periods, including 2-h values (Stefan and Preud’homme, 1993), daily, weekly and monthly and annual means (e.g. Webb, 1987; Crisp, 1988; Webb and Walling, 1993; Erickson and Stefan, 1996, 2000; Webb and Nobilis, 1997), and daily maxima and minima (Smith, 1979). Water–air temperature regression models offer a simple means of predicting water temperatures, not least because data on air temperature are usually much more readily available than information on the wide range of hydrometeorological parameters that are required to apply more sophisticated physically based models based on river energy budgets and mass transport of heat calculations (e.g. Bartholow, 1989; Sinokrot and Stefan, 1993; Kim and Chapra, 1997; Younus et al., 2000). Furthermore, such regression models are
* Correspondence to: B. W. Webb, School of Geography and Archaeology, University of Exeter, Amory Building, Rennes Drive, Exeter EX4 4RJ, UK. E-mail: [email protected] Received 16 July 2002 Copyright 2003 John Wiley & Sons, Ltd. Accepted 8 January 2003 3070 B. W. WEBB, P. D. CLACK AND D. E. WALLING frequently characterized by high levels of explained variance, even when the meteorological station recording air temperature is located at a distance of several tens of kilometres from the river temperature monitoring site (Crisp and Howson, 1982; Mohseni et al., 1998; Pilgrim et al., 1998). Some studies (Sullivan et al., 1990) have cautioned that estimating water temperature from air temperature data collected at a remote location may lead to inaccuracies, although such problems have been found to be most significant in regions where air temperature gradients are steep and display abrupt changes (Lewis et al., 2000). Water–air temperature regression analysis has also proved a useful tool in studies seeking to predict future stream and river temperatures that may result from global warming (Mackey and Berrie, 1991; Webb, 1996), especially where information is required for ecological purposes and for large geographical areas (e.g. Eaton and Scheller, 1996; Mohseni et al., 1999). Although simple linear regression models linking water to air temperatures have often been developed and utilised, the subtleties and complexities of the water–air temperature relationship are being increasingly recognized. For example, a study of the Straight River, Minnesota, USA has demonstrated that the water–air temperature regression relationship becomes steeper and less scattered as the time interval of the data increases from 2-h, through daily averages to weekly means (Stefan and Preud’homme, 1993). Strong relationships are also usually encountered when monthly mean values are used in regression analysis, but weaker and less sensitive relationships are evident when annual mean values have been analysed for a 90-year period in the River Krems, Austria (Webb and Nobilis, 1997). The lack of well-defined water–air temperature regression relationships associated with annual data probably reflects the limited variability of annual means of water and air temperature (Pilgrim et al., 1998; Erickson and Stefan, 2000). Regression relationships between daily minimum water and air temperatures also tend to be more scattered than those for daily mean or maximum values. This has been attributed to the fact that the higher thermal capacity of water prevents development of the low nocturnal minima characteristic of air temperature (Smith, 1979). Water temperature variations also tend to lag behind fluctuations in air temperature (Jeppesen and Iversen, 1987). For larger rivers, with higher flow volumes and greater thermal capacities, this effect becomes more pronounced and should be taken into account in developing water–air temperature relationships. Grant (1977) suggested that the maximum water temperature on a given day in the Ngaruroro River, New Zealand should be predicted not only from the maximum air temperature on that day but also from the maximum air temperature on the preceding day. Stefan and Preud’homme (1993) have shown more generally that regression relationships based on daily mean values can be significantly improved by introducing a lag into the data. This lag was 0 days for catchments 300 km2 or less in size, but more than 8 days for basins in excess of 4 ð 105 km2. The assumption that the water–air temperature relationship is linear has also been questioned. Several studies have demonstrated departures from linearity as air temperature falls below 0 °C (e.g. Crisp and Howson, 1982; Webb and Nobilis, 1997) and this can be ascribed to the release of latent heat with ice formation, which prevents water temperatures falling much below 0 °C. More recently, it has been suggested that the water–air temperature relationship derived from weekly mean values also departs from linearity at high air temperatures (>ca.25°C). Increases in the moisture-holding capacity of the atmosphere, which promote greater evaporation from the water surface and, in turn, increase evaporative cooling of the water course, together with enhanced back radiation as water temperatures rise, are considered responsible for this effect (Mohseni et al., 1998, 1999, 2002). Testing of data from 584 rivers in the USA has demonstrated that a continuous S-shaped curve, based on a non-linear logistic regression function, successfully represented departures of the water–air temperature relationship at both high and low air temperatures. Mohseni and Stefan (1999) have offered a general physical interpretation of this S-shaped relationship in terms of equilibrium temperatures and natural upstream temperatures (groundwater, snowmelt and dew point) in rivers. The influence of other factors on the water–air temperature relationship in water courses also has been recognized. For example, in a study of Minnesota rivers, Stefan and Sinokrot (1993) showed that different regression models existed for river sites open to high solar radiation receipts and those heavily shaded from incoming solar radiation by riparian vegetation. More recently, Erickson and Stefan (2000) reviewed the impacts that impoundments and reservoirs, groundwater inflows, wastewater inputs and stream shading and
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) RIVER-WATER–AIR TEMPERATURE RELATIONSHIPS 3071 wind sheltering have on linear water–air temperature regression parameters. Changing sources and volumes of runoff are also known to affect water temperature behaviour (e.g. Shanley and Peters, 1988; Gu, 1998a,b; Kobayashi et al., 1999, Langan et al., 2001), but there have been relatively few studies that have investigated the impact of stream and river discharge on the water–air temperature relationship. For rivers affected by seasonal snow- and ice-melt runoff, hysteresis has been identified and represented by looped relationships (Webb and Nobilis, 1994; Mohseni et al., 1998, 1999). Smith and Lavis (1975) showed that water–air temperature correlations for daily maxima and minima were weaker in a small upland catchment of northern England when flows exceeded 70 L s 1. They suggested that the water course was less sensitive to air temperature fluctuations at higher discharges because of the increased thermal capacity and reduced travel time of flow through the channel system. Multiple regression analysis also has been used to relate spot measurements of water temperature in the Hurunui River, New Zealand to maximum daily air temperatures and to discharge (Hockey et al., 1982). However, Crisp and Howson (1982) investigating streams in northern England found that introduction of discharge information produced little or no improvement in explanation of water temperature variations beyond that associated with simple water–air temperature regression models. Furthermore, Mohseni et al. (1999) demonstrated that stream flow had only a small effect on the parameters of logistic functions fitted to water and air temperature data for three catchments of different size, located in cold, warm and dry, and wet regions of the USA. The purpose of the present study was to examine the nature of water–air temperature relationships in a principal river system of the UK, with particular attention to the role of river discharge in moderating that relationship. The investigation provided the opportunity to examine how water–air temperature relationships vary with catchment scale and with information collected over different time bases. The latter included data collected at hourly intervals, which have rarely been studied in investigations of water–air temperature relationships.
STUDY CATCHMENTS AND METHODS The present study investigated four catchments of contrasting scale within the Exe basin in Devon and Somerset, UK (Figure 1), which encompasses a wide variety of terrain types, ranging from upland moorlands on rocks of Devonian age in the north and north-west to intensively farmed agricultural lowlands underlain by Permo-Triassic strata in the south and east of the basin. The Black Ball Stream (2Ð1km2), the smallest water course studied, drains an area of massive siliceous sandstones and interbedded shales overlain by surface water gley soils on the interfluves and stagnopodzols on the valley slopes. The mean annual precipitation of this small catchment exceeds 1600 mm, and flow during the 5-year study period, which ran from 1 January 1994 to 31 December 1998, averaged c.84Ls 1. The channel bed at the monitoring site comprised a mixture of gravel, silt and sand. Shallow groundwater in the peat feeds springs, which sustain flows in the Black Ball Stream. The small catchment of the Jackmoor Brook (10Ð2km2) is underlain by breccias and conglomerates and by a fine-sand unit. The soils developed over the bedrock include coarse loams, pelosols and brown earths. Mean annual precipitation is of the order of 1000 mm, and for the study period the discharge averaged c. 165 L s 1. Discharge in the Jackmoor Brook is influenced by significant inflows of groundwater from Permian aquifers. The channel has a sand bed at the monitoring site. The River Barle (128 km2) is the major tributary of the Exe, which drains a succession of grits, slates, sandstones and minor limestones underlying the Exmoor region (Figure 1). The catchment is dominated by podzolic and surface-water gley soils, and mean annual precipitation ranges from in excess of 2000 mm in the north-western headwaters to less than 1300 mm in the vicinity of the monitoring station. The average discharge for the study period was 6 m3 s 1. The channel floor at the monitoring site is bedrock. The main stream of the Exe (601 km2) was the largest water course investigated in the present study. In addition to the tributaries draining the upland area of Exmoor, this catchment receives runoff from a large area of Carboniferous sandstones and shales on which brown earths and surface-water gleys are the dominant soil
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) 3072 B. W. WEBB, P. D. CLACK AND D. E. WALLING Figure 1. The study catchments
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) RIVER-WATER–AIR TEMPERATURE RELATIONSHIPS 3073 types. Mean annual precipitation ranges from >2000 mm in the headwaters of the Exe to c. 1000 mm in the vicinity of the monitoring site, where the Exe is a typical gravel-bedded river. Discharge during the study period averaged c.18m3 s 1. The Exe basin is free from significant discharges of heated effluent, but flows in the main stream of the Exe are regulated by releases from Wimbleball Lake, a sizeable reservoir situated in the eastern part of Exmoor (Figure 1). However, the reservoir has no impact on water temperatures in the Black Ball Stream, Jackmoor Brook and River Barle, and has only a small influence in moderating peak summer temperatures of the main stream of the Exe at the monitoring site, under exceptional drought conditions. Comparison of temperature records at a network of sites throughout the Exe basin for extreme low flow conditions before and after the construction of Wimbleball Lake suggests regulation has suppressed daily maximum temperatures in July by c.0Ð5 °C, although this effect is absent in most years (Webb, 1995). Water temperature in the study catchment was recorded using metallic resistance sensors linked to data loggers, from which values were extracted for hourly intervals. In the case of the Black Ball Stream and Jackmoor Brook, hourly discharge records for the study period were obtained from small gauging structures installed by the Department of Geography, University of Exeter and rated through a series of flow gaugings. Hourly discharge data for the Rivers Barle and Exe at the temperature monitoring sites were available from the Environment Agency. The former river is gauged by means of a rated section and the latter by a flat- vee control structure. Information on air temperature for the present study was largely obtained from the Meteorological Office British Atmospheric Data Centre. Hourly data from two stations (Figure 1) were used primarily in the analysis of the water temperature records. Information from the station at Liscombe was used to characterize the macro-air temperature environment (Lewis et al., 2000) of the water temperature monitoring sites in the Black Ball Stream and River Barle, which are located a distance of 5 and 8Ð5km, respectively, from this station. The macro-air temperature environment of the Jackmoor Brook and River Exe monitoring sites was represented by information from Exeter Airport, which is situated c.11kmaway from the former and 10 km distant from the latter. Information on air temperature from the Meteorological Office station at Dunkeswell Aerodrome and the Devon County Council roadside ice monitoring equipment at Burn near Bickleigh (Figure 1) was used, via the establishment of strong linear regression relationships, to infill some small gaps in the air temperature records for Liscombe and Exeter Airport, respectively. During the 5-year study period, occasional failure of water temperature, discharge and air temperature monitoring equipment led to loss of data. Incompleteness of data sets ranged from less than 5% of hourly sampling observations for the Jackmoor Brook to more than 15% in the case of the River Barle. Linear regression analyses of water temperature were undertaken using the statistical functions in Microsoft Excel. Testing of the significance of differences in explained variance associated with linear regression equations derived from different data sets was based on the Fisher’s Z normalizing procedure. In comparing the slopes of linear regression relationships fitted to different data sets, a significant difference was considered to exist if there was no overlap in the range defined by two standard errors around each exponent value. Following Mohseni et al. (1998), non-linear logistic functions of the form ˛ µ Tw D µ C 1 1 C e ˇ Ta where Tw is the estimated water temperature, Ta is measured air temperature, µ is estimated minimum water temperature, ˛ is estimated maximum stream temperature, ˇ is air temperature at the inflection point of the function and is a measure of the steepest slope of the function—were fitted using Microsoft Excel Solver, which uses a non-linear optimization code. Multiple regression analyses of daily and weekly water temperatures were undertaken using the statistical functions in Microsoft Excel. In the case of hourly values, the multiple regression models included the non-linear logistic function (Equation 1) and were also fitted using Microsoft Excel Solver. Beta coefficients for the discharge term in the multiple regression equations were calculated as q ˇq D bq 2 t
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) 3074 B. W. WEBB, P. D. CLACK AND D. E. WALLING
where ˇq is the beta coefficient, bq is the regression coefficient for the discharge term, q is standard deviation of discharge data and t is standard deviation of water temperature data.
THERMAL CHARACTERISTICS OF THE STUDY CATCHMENTS Water temperatures in the study catchments reflect the temperate maritime climate of South-west England and the relatively modest relief of the region. Summary statistics derived from the 5-year study period (Table I) reveal that the average temperature varied from 9Ð3 °C in the headwater tributary on Exmoor to 11Ð1 °Catthe site furthest downstream. There was no seasonal freezing of rivers, and temperatures 0 °C occurred for only a small percentage of the time. Maximum temperatures did not exceed 20 °C in the two smaller tributaries and did not rise above 24 °C in the main stream. Air temperature measurements for the upland site on Exmoor revealed the occurrence of freezing temperatures for 5% of the study period with minimum values of 7Ð8 °C. Equivalent values for the lowland site in the lower Exe Valley were 3% of the time and 7Ð5 °C. Although air temperature maxima of 28Ð8 and 30Ð6 °C were recorded at the upland and lowland stations, respectively, values ½25 °C were rare and occurred for c.0Ð1 of the time in the uplands and for c.0Ð5% of the time in the lowlands.
WATER–AIR TEMPERATURE RELATIONSHIPS Time-scale and daily extremes The strength of the water–air temperature relationship, indexed by the level of explained variance, and its sensitivity, indicated by the exponent of the regression line, increased at all sites as the time base was extended from hourly to weekly mean values (Table II). Weekly mean air temperature accounted for more than 90% of the variation in weekly mean water temperature in all the study rivers, but the r2 value fell below 70% for the relationship based on hourly data for the River Exe. At an hourly time-scale, the extent to which water temperature variation could be accounted for by air temperature fluctuation declined as catchment size increased, but this scale effect was not clearly apparent for daily or weekly mean values (Table II). For all time-scales studied, the slope of the water–air temperature relationship was greater for the main stream sites (Rivers Barle and Exe), than for the headwater stations close to shallow or deeper groundwater sources (Black Ball Stream, Jackmoor Brook). The exponent of the regression model approached 1Ð0 for weekly mean values in the River Exe, but was close to 0Ð5 for hourly values in the Black Ball Stream and the Jackmoor Brook. Parameters of regression models fitted to daily extremes of water and air temperature (Table II) showed, with the exception of daily minima for the River Exe, that the r2 value associated with these models exceeded 80%. However, in all cases except for daily maxima in the River Exe, levels of explained variance were lower for the regression relationships fitted to daily extreme than to daily mean data. Relationships were stronger for daily maximum than daily minimum values with the exception of River Barle (Table II). As with the daily
Table I. Summary statistics of water temperatures ( °C) measured during the study period
Catchment Minimum Mean Maximum Frozena (%)
Black Ball Stream 2Ð19Ð318Ð20Ð0 Jackmoor Brook 0Ð610Ð418Ð20Ð0 River Barle 0Ð410Ð421Ð10Ð3 River Exe 0Ð111Ð124Ð00Ð3
a Percentage of study period for which water temperature was 0 °C.
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Table II. Parameters of water–air temperature relationships fitted by linear regression analysis to data based on different time periods for the study catchments: r2 per cent explained variance; n, number of samples; Lag (in hours), established by cross-correlation analysis
Catchment Hourly Hourly values Daily Daily Daily Weekly values with lag mean values maximum values minimum values mean values
Black Ball Stream r2 (%) 83Ð887Ð485Ð484Ð080Ð591Ð0 Exponent 0Ð512 0Ð523 0Ð547 0Ð523 0Ð533 0Ð590 Intercept 4Ð88 4Ð78 4Ð63 4Ð30 5Ð31 4Ð24 n 39 551 39 544 1571 1571 1571 194 Lag 2 Jackmoor Brook r2 (%) 76Ð384Ð891Ð589Ð380Ð194Ð9 Exponent 0Ð492 0Ð519 0Ð593 0Ð525 0Ð554 0Ð636 Intercept 5Ð10 0Ð671 4Ð03 3Ð90 5Ð51 3Ð57 n 41 822 41 818 1727 1727 1727 238 Lag 3 River Barle r2 (%) 74Ð279Ð485Ð580Ð881Ð194Ð2 Exponent 0Ð649 0Ð671 0Ð749 0Ð666 0Ð735 0Ð816 Intercept 4Ð64 4Ð43 3Ð73 3Ð42 5Ð01 3Ð10 n 37 050 36 966 1482 1482 1482 192 Lag 4 River Exe r2 (%) 67Ð473Ð284Ð287Ð162Ð992Ð2 Exponent 0Ð667 0Ð699 0Ð844 0Ð793 0Ð699 0Ð946 Intercept 4Ð11 3Ð80 2Ð30 0Ð80 5Ð71 1Ð27 n 40 685 40 629 1635 1635 1635 218 Lag 6 mean values, the slopes of the fitted water–air temperature relationship for daily extreme values were higher for main stream than headwater study sites.
Lags and hysteresis Previous studies in UK rivers, and investigations in small catchments more generally, have not taken into account the lagged response of water in relation to air temperature when examining water–air temperature relationships. Cross-correlation analysis of hourly water and air temperature time-series was used to investigate the extent to which water temperature variations lagged behind air temperature fluctuations in the study rivers. The magnitude of the lag was found to increase with catchment scale from 2 h for the smallest to 6 h for the largest water course investigated (Table II). Incorporation of the lag effect into the regression analysis led to a statistically significant improvement in the strength of the water–air temperature relationships for all the study rivers. The r2 value was increased by between 3Ð6 for the Black Ball Stream and 8Ð5% for the Jackmoor Brook (Table II). The slope of the water–air temperature relationship fitted by linear regression was also increased at all sites by incorporation of a lag effect. The increase was smallest for the Black Ball Stream and greatest for the River Exe. Water–air temperature relationships based on weekly mean data were examined for the presence of seasonal hysteresis. Plotting of the data classified into four 3-month periods of January–March, April–June, July–September and October–December (Figure 2) did not reveal any significant hysteresis in the water–air temperature relationships at the study sites.
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) 3076 B. W. WEBB, P. D. CLACK AND D. E. WALLING January to D r(1 C) C) ° ° Weekly Mean Air Temperature ( Temperature Mean Air Weekly Weekly Mean Air Temperature ( Temperature Mean Air Weekly
Pynes Cottage
Jackmoor Brook at Jackmoor
Thorverton River Exe at Exe River -5 0 5 10 15 20 25 -5 0 5 10 15 20 25
5 0 5 0
-5 -5 25 20 15 10 25 20 15 10
C) ( Temperature Water Mean Weekly C) ( Temperature Water Mean Weekly ° ° he study catchments. Data have been classified according to month of the yea 1 - 3 4 - 6 7 - 9 10 - 12 C) C) ° ° December) and the dashed line indicates the fitted linear regression model D 12 Weekly Mean Air Temperature ( Temperature Mean Air Weekly Weekly Mean Air Temperature ( Temperature Mean Air Weekly
Black Ball Stream Black Farm at Lyshwell
Brushford
River Barle at River -5 0 5 10 15 20 25 -5 0 5 10 15 20 25
5 0 5 0
-5 -5 25 20 15 10 25 20 15 10
C) ( Temperature Water Mean Weekly C) ( Temperature Water Mean Weekly ° ° Figure 2. Relationships between weekly mean water and air temperatures in t
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) RIVER-WATER–AIR TEMPERATURE RELATIONSHIPS 3077
Departures from linearity To examine whether water–air temperature relationships in the study rivers exhibited any significant departure from linearity at low and high air temperatures, non-linear logistic regression functions were fitted to hourly, daily and weekly data. For hourly data sets, which also incorporated the lag effect identified for each river, logistic functions (Figure 3) were associated with small but statistically significant improvements in the level of explained variance. These ranged from 1Ð3% for the Jackmoor Brook to 0Ð8% for the River Barle. The logistic functions indicated that the rate of increase in water temperature with rising air temperature tends to decline as air temperature approaches and exceeds 20 °C. The curves also showed that the relationship based on hourly values departs from linearity as air temperature approaches and falls below 0 °C. Fitting of logistic functions to water–air temperature relationships based on daily data resulted in a statistically significant increase of explained variance, compared with a simple linear regression model, only in the case of daily maximum values for the Black Ball Stream and Jackmoor Brook. In these water courses, a tailing off in the increase in the daily water temperature maximum was clearly apparent as daily maximum air temperature approaches and exceeds 25 °C (Figure 4). Inspection of the water–air temperature plots for weekly mean values (Figure 2) provides no evidence of departure from linearity at low or high temperatures. Fitting of logistic curves to these data did not result in improved levels of explained variance that were statistically significant.
THE EFFECTS OF RIVER DISCHARGE Flow classes Previous studies have suggested that water–air temperature relationships vary in their strength according to the range of flow conditions for which they are developed (Smith and Lavis, 1975). To test whether this finding can be applied to the study catchments, separate linear regression models were established for hourly, daily and weekly data subdivided according to whether corresponding discharge values were above or below the study period median (Table III). The hourly data incorporated the water–air temperature lag established by cross-correlation analysis (Table II), and mean, maximum and minimum values were examined for daily data. Comparison of relationships based on hourly data for flows above and below the median discharge revealed differences that are not entirely consistent between the study catchments. In the smaller water courses (Black Ball Stream and Jackmoor Brook), water–air temperature relationships derived from data when flows were less than the median level are less scattered with a significantly higher r2 value (Table III). However, exponent values indicate there are not large contrasts in the sensitivity of water temperature to air temperature variations in these catchments at flows below, compared with above, median discharge. However, for the Jackmoor Brook, the higher b value derived from analysis of flows below median discharge is statistically significant. In the larger catchments (Rivers Barle and Exe), the exponent values are markedly higher for water–air temperature relationships based on flows below median discharges, but these relationships are not associated with higher levels of explained variance than those based on flows above median discharge. Regression models for daily mean data indicate that water–air temperature relationships relating to daily mean flows below the study period median are less scattered and more sensitive than those relating to daily mean flows above the study period median (Table III). Levels of explained variance associated with the former models are significantly higher for the River Exe, Jackmoor Brook and Black Ball Stream, and almost 10% greater for the latter catchment. The b values for relationships based on data collected at lower flows are also higher for all sites, although differences are more pronounced for the larger catchments, and are not statistically significant for the Black Ball Stream. Similar differences in the scatter and slope of regression models developed for the greater than, and less than, median daily mean flow classes are apparent for daily minimum data (Table III). For the Black Ball Stream, explained variance is more than 12% greater for
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) 3078 B. W. WEBB, P. D. CLACK AND D. E. WALLING 74.3% 23.0 2.6 12.8 86.1% 0.17 16.8 3.0 9.8 = 0.18 = = =
= =
= =
=
=
2
2 r µ γ β α r µ γ β α ble II, and the field of nd the parameters of this curve C) C) ° ° Air Temperature Air Temperature ( Air Temperature Air Temperature ( e depicts the linear regression model fitted to the data
Pynes Cottage
Jackmoor Brook at Jackmoor Thorverton River Exe at Exe River -10 -5 0 5 10 15 20 25 30 35 -10 -5 0 5 10 15 20 25 30 35
6 2 6 2
-2 -2 26 22 18 14 10 26 22 18 14 10
C) ( Temperature Water C) ( Temperature Water ° ° udy rivers. Water temperature data have been lagged by the value given in Ta 80.2% 19.5 -0.4 7.6 0.15 = e depicts the non-linear logistic regression function fitted to the data, a = = 88.6% =
16.2 = 2.7 8.9
0.19
2 = = = =
r γ µ α β =
2 r γ µ α β C) C) ° ° ) associated with it are given in the inset. The dashed lin 2 r Air Temperature Air Temperature ( Air Temperature Air Temperature ( ) and the explained variance (
Lyshwell Farm Lyshwell Brushford
River Barle at River and Black Ball Stream at Black ˇ, , -10 -5 0 5 10 15 20 25 30 35 -10 -5 0 5 10 15 20 25 30 35 ˛
( 6 2 6 2
-2 -2 26 22 18 14 10 26 22 18 14 10
C) ( Temperature Water C) ( Temperature Water ° ° Figure 3. Relationships between hourly water and air temperatures in the st data points, rather than individual observations, are shown. The solid lin
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) RIVER-WATER–AIR TEMPERATURE RELATIONSHIPS 3079
30
C) Black Ball Stream ° 25 at Lyshwell Farm
20
15
10
α = 16.4 5 β = 11.6 γ = 0.19 µ = 2.7 0 r2 = 86.0% Daily Maximum Water Temperature ( Temperature Water Maximum Daily
-5 -5 0 5 10 15 20 25 30 35 Daily Maximum Air Temperature (°C)
30 C) ° 25 Jackmoor Brook at Pynes Cottage
20
15
10
5 α = 17.5 β = 10.7 γ = 0.16 0 µ = 1.5
Daily Maximum Water Temperature ( Temperature Water Maximum Daily r2 = 91.0% -5 -5 0 5 10 15 20 25 30 35 Daily Maximum Air Temperature (°C)
Figure 4. Relationships between daily maximum water and air temperatures in the Black Ball Stream and Jackmoor Brook. The solid line depicts the non-linear logistic regression function fitted to the data, and the parameters of this curve (˛, ˇ, and ) and the explained variance (r2) associated with it are given in the inset. The dashed line depicts the linear regression model fitted to the data the regression model developed for the lower compared with the higher flow class. However, there is no significant difference between the r2 levels for daily minimum water–air temperature relationships for the two flow classes in the River Exe. The exponent of the water–air temperature relationship for daily minimum values is significantly different between the two flow classes for all study catchments (Table III). Water–air temperature relationships based on daily maxima exhibit the least consistent differences according to flow classes based on the median daily mean discharge at the study sites (Table III). Levels of explained variance are significantly higher for data collected during flows below the median value only for the Black Ball Stream and River Exe, whereas b values are greater for the lower flow class only in the Rivers Exe and Barle. For all catchments, the water–air temperature relationship fitted to weekly mean and air temperature data associated with weekly mean discharge values below the study period median exhibit less scatter, and the
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) 3080 B. W. WEBB, P. D. CLACK AND D. E. WALLING
Table III. Parameters of water–air temperature relationships for the study catchments fitted by linear regression analysis to data divided into different flow classes: r2, explained variance (%); b, exponent of relationship; n, number of samples. All regression relationships are statistically significant. Italic values indicate the differences in regression parameters for flow classes below and above median discharge in the study period are statistically significant
Flow Black Ball Stream Jackmoor Brook River Barle River Exe class r2 bnr2 bnr2 bnr2 bn
Hourly with lag
Multiple regression The combined influence of air temperature and flow on hourly, daily and weekly water temperature variations was examined through the use of multiple regression analysis (Table IV), which allows the effect of discharge to be isolated. For hourly variations, the non-linear logistic functions between lagged water and air temperatures were incorporated into the regression models. The results show that for all the study rivers hourly water temperature is inversely and significantly related to flow, but the amount of explained variance associated with the addition of discharge as an independent variable is modest (Table IV). The rise in R2 gained by adding flow to the regression equation is greater as catchment size increases. The sensitivity to flow, as indexed by the beta coefficient for the discharge term, also increases from the smallest to the largest study catchment (Table IV). Daily mean flow data were used in multiple regression analysis of both daily mean and daily extreme values of water temperature. For all regression equations based on daily data, discharge is present as a significant variable and exhibits an inverse relationship with water temperature (Table IV). The amount of explained variance associated with the discharge term is generally lower for daily data than for hourly values with the exception of daily minimum water temperatures in the River Exe. In the latter case, inclusion of daily mean flow increases the coefficient of determination by more than 11%. As with the analysis based on hourly data, the extent to which discharge accounts for variation in daily water temperatures and the sensitivity of river
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) RIVER-WATER–AIR TEMPERATURE RELATIONSHIPS 3081
Table IV. Parameters of multiple regression equations linking water temperature ( °C) values in the study catchments ° 3 1 to air temperature ( C) and river flow (m s ). Discharge values subjected to a log10 transformation: b0, constant; 2 ba, exponent of air temperature term; bq, exponent of discharge term; ˇq, beta coefficient of discharge term; R total, overall explained variance (%); R2 increase, explained variance (%) associated with inclusion of discharge term; n, number of observations Parameter Black Ball Stream Jackmoor Brook River Barle River Exe
Hourlya b0 0Ð607 1Ð048 0Ð192 3Ð676 ba 0Ð963 0Ð975 0Ð977 0Ð929 bq 0Ð689 0Ð916 1Ð230 2Ð634 ˇq 0Ð1006 0Ð1163 0Ð1642 0Ð2756 R2 total 89Ð387Ð182Ð380Ð1 R2 increase 0Ð71Ð02Ð15Ð8 n 39 544 41 818 36 966 40 629 Daily mean b0 4Ð186 3Ð703 4Ð451 5Ð157 ba 0Ð532 0Ð572 0Ð711 0Ð756 bq 0Ð438 0Ð544 0Ð756 1Ð927 ˇq 0Ð0621 0Ð0708 0Ð1018 0Ð1927 R2 total 85Ð791Ð986Ð387Ð4 R2 increase 0Ð30Ð40Ð83Ð2 n 1571 1727 1482 1635 Daily maximum b0 3Ð855 3Ð650 4Ð415 3Ð033 ba 0Ð512 0Ð507 0Ð624 0Ð730 bq 0Ð473 0Ð495 0Ð990 1Ð358 ˇq 0Ð0594 0Ð0629 0Ð1313 0Ð1358 R2 total 84Ð389Ð682Ð288Ð4 R2 increase 0Ð30Ð31Ð41Ð3 n 1571 1727 1482 1635 Daily minimum b0 4Ð691 4Ð606 5Ð835 9Ð637 ba 0Ð516 0Ð517 0Ð687 0Ð597 bq 0Ð552 1Ð149 1Ð033 3Ð215 ˇq 0Ð0819 0Ð1497 0Ð1408 0Ð3551 R2 total 81Ð182Ð082Ð874Ð2 R2 increase 0Ð61Ð91Ð711Ð3 n 1571 1727 1482 1635 Weekly mean b0 4Ð004 3Ð393 3Ð637 3Ð839 ba 0Ð582 0Ð619 0Ð785 0Ð854 bq 0Ð233 0Ð360 0Ð538 1Ð593 ˇq 0Ð0311 0Ð0461 0Ð0725 0Ð1665 R2 total 91Ð195Ð094Ð694Ð1 R2 increase 0Ð10Ð10Ð41Ð9 n 194 238 192 218
a Hourly values of air temperature incorporate non-linear logistic relationships and lags. temperature to flow, indexed by the beta coefficient, is generally greater for the larger than for the smaller study catchments. Multiple regression analysis reveals that discharge has the smallest impact on variations in weekly mean temperatures (Table IV), and discharge is not a statistically significant variable in the analysis of weekly mean
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) 3082 B. W. WEBB, P. D. CLACK AND D. E. WALLING water temperatures in the Black Ball Stream. As with the analysis of hourly and daily data, beta coefficients for the discharge term increase with catchment size.
DISCUSSION Water and air temperatures are strongly related in the study catchments, and the results presented indicate that river temperatures can be successfully correlated with the macro-air temperature environment (Lewis et al., 2000) measured at some distance from the river channel. Even fluctuations in water temperature at an hourly time-scale are well explained by variations in air temperature, with a minimum r2 of almost 70%, which reflects the relatively small size and therefore low thermal capacity of the study rivers. Incorporation of a lag into the water–air temperature relationship improved the sensitivity and explanatory power of simple regression relationships derived for hourly data. Although this approach has been advocated in the past for larger rivers and for daily mean water temperatures, the present study suggests that it may also be worth adopting for small catchments and shorter term fluctuations, in order to take into account the time that even small bodies of water require to respond to changing air temperature. In common with a range of rivers studied in the USA (Stefan and Preud’homme, 1993; Pilgrim et al., 1998), simple linear relationships between water and air temperature for the study catchments become less scattered and steeper as the time base of the data increases from hourly to weekly mean values. Regression relationships between weekly mean water and air temperatures in the study area are very strong and show no evidence of seasonal hysteresis. The absence of the latter phenomenon is unsurprising given the broad climatic setting of South-west England, which dictates that water courses are not subjected to prolonged periods of freezing. Fitting of non-linear logistic functions suggests that the relationship in the study area between water and air temperature, defined by hourly values, do depart significantly from linearity as air temperatures exceeds 20 °C or falls below freezing, which is consistent with a recent physically based interpretation of the stream- temperature–air-temperature relationship (Mohseni and Stefan, 1999). However, when data in the present study are aggregated to the daily or weekly time-scale, there is much less evidence that relationships between water and air temperatures depart significantly from linearity, although the rate of increase in daily maximum water temperature clearly slows as daily maximum air temperature rises above 25 °C in the two smallest study basins. The implication of these results is that in using regression relationships based on temperatures averaged over daily or weekly time periods to predict future water temperatures and their ecological effects, it may not be necessary for South-west England, and probably many parts of the UK, to allow for the effects of enhanced evaporational cooling and back radiation in suppressing the impact of higher air temperatures under scenarios of global warming. The overall influence of flow in moderating the water–air temperature relationship in the study catchments is generally modest, which reflects the relatively strong relationships that exist between water and air temperatures at all time-scales investigated in the present study. Nevertheless, there is a clear tendency for relationships between water and air temperatures to be stronger and more sensitive for flows in the range below median discharge than those above this threshold. This finding echoes those of Smith and Lavis (1975) with respect to the effects of increased thermal capacity and reduced travel time in decreasing the response of stream temperatures to air temperature fluctuations. Multiple regression analysis for all sites and time-scales show water temperature to be inversely related to discharge and suggest that once the influence of varying air temperature had been accounted for, increasing flow volume and thermal capacity give rise to reduced water temperatures. This is consistent with previous multivariate analysis of river temperatures (e.g. Hockey et al., 1982), but is in contrast to findings from a study of a small upland catchment in north-east Scotland where the inverse relationship between river temperature and flow was viewed as non-causal, and as reflecting the greater incidence of high temperatures in winter months when air temperatures and water temperatures are also low (Langan, 2001). In general,
Copyright 2003 John Wiley & Sons, Ltd. Hydrol. Process. 17, 3069–3084 (2003) RIVER-WATER–AIR TEMPERATURE RELATIONSHIPS 3083 the multivariate analyses indicate that discharge has a greater influence in accounting for water temperature variation as catchment size increases and the time interval of the data decreases. However, the results of the present study also demonstrate that statistical relationships linking water temperature to both air temperature and flow are very strong for all the study rivers, and offer a means of simply and successfully predicting water temperature time-series ranging from hourly instantaneous to weekly mean values.
ACKNOWLEDGEMENTS The support of a Natural Environment Research Council research studentship awarded to Paul Clack and the assistance of the Meteorological Office in providing access to air temperature data held in the British Atmospheric Data Centre are gratefully acknowledged.
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