Spectral scaling of heat fluxes in streambed sediments

A. Wörman,1 J. Riml,1 N. Schmadel,2 B. T. Neilson,2 A. Bottacin-Busolin,1

and J. E. Heavilin2

Received 3 October 2012; revised 24 October 2012; accepted 1 November 2012; published 7 December 2012.

[1] Advancing our predictive capabilities of heat fluxes instreams and rivers is important because of the effects onecology and the general use of heat fluxes as analogues forsolute transport. Along these lines, we derived a closed-formsolution that relates the in-stream temperature spectra tothe responding temperature spectra at various depths in thesediment through a physical scaling factor including theeffective thermal diffusivity and the Darcy flow velocity.This analysis considers the range of frequencies in tempera-ture fluctuations that occur due to diurnal and meteorologicalvariation both in the long and short term. This approachprovides insight regarding the key frequencies for analysingtemperature responses at different depths within the sedimentand also provides a simple and accurate method that offersquantitative insight into heat transport and surface waterinteractions with groundwater. We demonstrate for SävaBrook, Sweden, how the values of effective thermal diffu-sivities can be estimated based on the observed in-stream andsediment temperature time series and explain the temporalscaling of the heat transport as a function of a dimensionlessfrequency number. We find that the lower limit of periods ofsignificance for the analysis increases with depth, and werecommend further research regarding appropriate frequencywindows. Citation: Wörman, A., J. Riml, N. Schmadel, B. T.Neilson, A. Bottacin-Busolin, and J. E. Heavilin (2012), Spectralscaling of heat fluxes in streambed sediments, Geophys. Res. Lett.,39, L23402, doi:10.1029/2012GL053922.

1. Introduction

[2] Because biogeochemical processes depend on temper-ature, understanding heat transfer in streams is key to betterunderstanding stream ecology. Significant heat exchangewithin streams occurs at two primary interfaces with theatmosphere and the subsurface [Neilson et al., 2010; Webbet al., 2008]. Heat exchange at the stream-subsurface inter-face has been specifically used to investigate groundwater-surface water interactions [Conant, 2004; Evans and Petts,1997]. Various techniques have been employed to quantifythese exchanges, including the use of fibre optic cables toprovide distributed temperature data along stream reaches[Selker et al., 2006] and temperature sensors placed verticallywithin the stream sediments [Constantz et al., 2002; Hatch

et al., 2006; Keery et al., 2007]. The latter methods com-monly assume one-dimensional (1D) convective and con-ductive heat transport through the sediments [Lautz, 2010;Rau et al., 2010]. These approaches provide estimates ofseepage rates and effective thermal diffusivities based on thediurnal period of the instream temperature [e.g., Hatch et al.,2006]. However, the temperature time series for streamsoften exhibits a spectrum of different frequencies that reflectthe variations in air mass temperature, solar radiation andprecipitation. Vogt et al. [2010] and Gordon et al. [2012]used real harmonic series to extract information from fre-quencies, amplitude and phase, but they rely on the diurnalperiod to evaluate thermal diffusivity and vertical fluxes. Tofurther extract information from the coupled spectrum oftemperature frequencies in the stream water and bed-sediments, we propose a physically based spectral transformthat relates the in-stream temperature fluctuations to those atspecific sediment depths. Such frequency spectra in thetemperature fluctuations persist due to diurnal cycles, as wellas both short and long-term variations in atmospheric con-ditions. Here we develop the appropriate theory and illustratehow the effective thermal diffusivity and Darcy flux can beevaluated from in-stream temperature data at various sedi-ment depths and locations. These results are compared withresults from laboratory tests as well as an optimized finite-element model.

2. Physical Spectral Transform of TemperatureTime Series

[3] The temperature T(z,t) (�C) resulting from the thermalprocesses in the sediment bed can be expressed as a con-volution of the boundary condition, i.e., the in-stream tem-perature T0, and a unit or impulse response function yd ofthe form

T z; tð Þ ¼ T0*y� ð1Þ

where the unit response function yd = Td/A , Td is the solu-tion to the process based governing heat transport equationvalid for a unit pulse at the boundary T0,d = T(z = 0, t) = Ad(t),d(t) is the Dirac delta function (s–1), A is the temperatureintegrated over a given period [s � �C], z is the depth coor-dinate (m) and t is the time (s). The Fourier transform ofequation (1) becomes F [T] = F [T0] ⋅ F [yd], with time treplaced by the fundamental frequency w = 2 p/l, where l isthe time period of basic harmonics inherent to the time series(s). Because the power spectrum can be expressed asPT z;wð Þ ¼ F T½ ��F T½ � ¼ F T½ �j j2, where the over-bar denotesthe complex conjugate, equation (1) yields the power spec-trum as:

PT z;wð Þ ¼ PT0 z;wð Þ⋅Py�z;wð Þ ð2Þ

1Department of Land and Water Resources Engineering, Royal Instituteof Technology, Stockholm, Sweden.

2Utah Water Research Laboratory, Utah State University, Logan, Utah,USA.

Corresponding author: A. Wörman, Department of Land and WaterResources Engineering, Royal Institute of Technology, Teknikringen 76,SE-100 44 Stockholm, Sweden. (worman@kth.se)

Published in 2012 by the American Geophysical Union.

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L23402 1 of 6

Consequently, the main implication of the suggestedapproach is that two temperature power spectra – at an upperboundary level (generally in-stream) (PT0) and at a lowersediment depth (PT) – are related through a physical scalingfactor Py�

wð Þ that represents the thermal processes in thesediment. This factor is key to the transformation of the timeseries spectra and can be referred to as the square of themodulus of the Fourier transform of the impulse responsefunction [Miller, 1974, p. 133].[4] This basic principle given by equation (2) can be

applied to different situations for which the physical scalingfactor is derived. The fundamental transport equation forsuch derivations is the heat conduction-convection model forsaturated soil [Constantz et al., 2002; Keery et al., 2007]:

∂T∂t

¼ ke∂2T∂z2

� q

g∂T∂z

ð3Þ

where ke is the effective thermal diffusivity (which includesthe thermal dispersion and diffusion), q is the Darcy flowvelocity (porosity multiplied by pore velocity), g = (r c)/(rf cf), (r c) is the product of the density and the specific heatcapacity of the saturated sediment-fluid system [J/(K m3)]and (rf cf) is the product of the density and the specific heatcapacity of the fluid [J/(K m3)]. In the auxiliary material(section S1 of Text S1) we propose three possible cases forwhich the physical scaling factor Py�

wð Þ is derived and whichare physically motivated: 1) effective conduction-convectionwith semi-infinite depth (T(z = ∞, t) = 0), 2) effective con-duction with semi-infinite depth and 3) thermal dispersionwith limited depth, L (m).1 For the first case, one can derivethe relationship between the temperature power spectra onthe form of (section S1 of Text S1)

PT ¼ PT0⋅Py�¼ PT0⋅ F y�½ �j j2

¼ PT0⋅ exp Pe 1�ffiffiffi2

p

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 16

zPe

� �4svuut

0B@

1CA

264

375 ð4Þ

where the Peclet number Pe = (q/g) z /ke is the ratio of theconvective flux to the conductive flux and the conductive-frequency number z = z (w/ke)

0.5 is the ratio of the frequencyto the inherent conduction frequency. Note that the expo-nential factor represents the solution to the physical scalingfactor Py�

wð Þ. The second case with pure effective conduc-tion is obtained as a special case of equation (4) when Pe≪ z,which yields PT ¼ PT0⋅ exp � ffiffiffiffiffiffiffiffiffiffiffiffiffi

2w=ke

pz

� �. Rearrangement

yields a direct estimate of ke as function of two observedpower spectra in the form of

ke wð Þ ¼ 2w z2 ln PT wð ÞP�1T0

wð Þ� �h i�2

ð5Þ

The third case with depth limited thermal dispersion isdefined by the solution to equation (2) for a no-flux surface ata certain depth, which introduces the stratification-frequencynumber Y = L (w/ke)

0.5 (see section S1 of Text S1). This caseapplies for conditions with decreased mixing with depth inthe hyporheic zone [Elliott and Brooks, 1997;Wörman et al.,

2006; Bottacin-Busolin andMarion, 2010] or stratification ofthe subsurface soil with decreased thermal diffusivity belowsome depth. In the case of a no-flux surface vertical advec-tion may be neglected, which leads to an additional transformequation for the temperature power spectra described insection S1 of Text S1.[5] Prior to spectral analyses the time series were

“de-trended” and Hamming windowing, generally accept-able for linear problems [Brillinger, 1981], was applied toimprove the quality of the results. The numerical proceduresare described in sections S2 and S3 of Text S1 and the opti-mization of the model parameters are described in section S5of Text S1.

3. Interpretation of Heat Fluxes in Säva Brook

[6] Temperatures were measured at seven sites along16 km of the Säva Brook in Uppland County, Sweden.Temperature sensors were used to record temperature at5-minute intervals in the main-stream channel and in thesediment at depths ranging from 3 to 100 cm. The datacollection period was 18 to 30 days, depending on the sta-tion location during May 2011, while the flow was mainlydecreasing. Sediment samples were also obtained for labo-ratory measurements of thermal diffusivity. See section S4of Text S1 for details on the data collection.[7] The temperature time series obtained at Site 2 at dif-

ferent depths in the sediment show strong fluctuations(Figure 1). We also simulated the temperature time series byusing COMSOL Multiphysics to indicate from this conven-tional approach to solve equation (3) that the temperaturefluctuations at different depths can be accurately related tothe physical description of the heat transport. This finiteelement solution considers a statistical optimization of anexponential distribution of effective thermal diffusivity(ke(z) = 8�10�8 + 3�10�7�(1-exp(�z/0.1)) as described inmore detail in section S4 of Text S1.[8] Figure 2 (top) compares the observed power spectrum

at 3 cm depth at Site 1 and equation (4) with the optimizedvalues of the Darcy velocity and the effective thermaldiffusivity, i.e., Pe and z (see section S5 of Text S1).When the optimization is limited to periods longer than1 day (w0.5 < 0.02 in Figure 2, bottom) the Darcy velocityis found to be close to zero (<10�10 m/s) and the effectivethermal diffusivity is essentially the same as that obtainedunder the assumption of pure conduction (i.e., equation (5)).Hence, for w0.5 < 0.02 the power spectrum in Figure 2(bottom) follows the proportionality ln(PT) ∝ w�0.5

appearing in equation (5), which indicates that the heattransport is controlled primarily by effective conduction (i.e.,Pe ≪ z). For higher frequencies, both the observed powerspectra and equation (4) indicates a non-linear behaviour.However, we find that equation (4) cannot be well fitted toobservations for plausible combinations of ke and the q/g(see section S5 of Text S1) and, hence, the more constantbehaviour of the power spectrum at high frequencies isprobably not because of convection dominance. An expla-nation can be conceived from Figure 2 (top) in which,according to equation (5), the thermal diffusivity in thestream sediments is independent of the frequency of the heatinput for all periods longer than 5–7 hours. Although, forshorter periods, the observed time series include increasingnoise and a drift of kewith frequency. High-frequency events

1Auxiliary materials are available in the HTML. doi:10.1029/2012GL053922.

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of energy flux can be caused by short-term variations inmeteorological patterns (e.g., air mass temperature, wind andradiation), but their statistical significance seems to be lowrelative to the noise and the damping of temperature fluc-tuations with sediment depth. The noise can be reducedslightly by limiting the interval of the periods considered inthe analysis and, especially, the lower limit of the significantperiods tends to increase with depth (see section 4.1). Somevariability in this lower limit between sampling sites isattributed to site-specific factors (e.g., sediment properties,landscape and vegetation).[9] Thus, for pure conduction (Pe ≪ z), the slope of the

power spectrum of temperature scales exponentially with theconductive-frequency number exp � ffiffiffi

2p

z� �

. If Pe ≪ z,convection dominates the heat transport and the powerfunction is independent of frequency. An optimization of thethree derived power functions (conductive and conductive-convective with semi-infinite depth and conductive withlimited depth) provides the same values of effective thermaldiffusivity at all sampling sites. The stratification-frequencynumber Y = L (w/ke)

0.5 is also shown to be of minorimportance for heat transport in Säva Brook sediments.[10] With this understanding, the temperature signal at

certain sediment depths can be used as a boundary conditionfor the response at a lower depth and one can evaluate thedepth variation of the effective thermal diffusivity layer bylayer for the Säva Brook data. Figure 3 (top) shows a com-parison of the results of the spectral approach for differentperiods with the laboratory values of ke from site-specificsediment core samples and the optimized results usingCOMSOL Multiphysics. The power spectrum results indi-cate that the thermal diffusivity at the depths of 50 cm and100 cm decreases with an increase in the lower limit of the

window of periods considered. We found that at thesedepths, periods up to at least 2 days had to be excluded toobtain a convergent result. The spatial variability of theestimated effective thermal diffusivity between sites is small(Figure 3, bottom), which is expected due to the relativelyhomogenous geological setting. The effective thermal dif-fusivity increases and asymptotically approaches a constantvalue with depth. A possible explanation for the relativelylow thermal diffusivity at the sediment surface is the pres-ence of gas [Cuthbert et al., 2010] that was qualitativelyobserved at the site (methane or nitrogen) with higherporosity and organic content in the shallow sediments. Theresults from sampling Site 3 diverge from this pattern,exhibiting a constant or even decreasing value of effectivethermal diffusivity with depth (Figure 3, bottom).[11] The Fourier spectral approach presented here can also

be used to analyze how thermal properties change over timein a window moving along the time series as previouslyshown for analyses of runoff processes in watersheds[Wörman et al., 2010]. The analysis presented in section S6of Text S1 illustrates that the effective thermal diffusivity fora period of a couple of days shows an increase withincreasing flood stage. This increase may be due to theincreased pumping occurring with increasing stream veloc-ities [Elliott and Brooks, 1997], which would increase thethermal dispersion, or to the scour and deposition of thesediments that influences the location of the boundarycondition.

4. Discussion of Implications

4.1. Feasibility of the Method

[12] The transform method developed here offers a simpleway to relate one observed temperature power spectrum to

Figure 1. The temperature time series measured at sampling Site 2 in Säva Brook at different levels in the bed sediment,down to 1 m depth (solid curves), and the optimized model results from a numerical solution to the partial differential equa-tions governing heat transport (dashed curves). The blue curve indicates the instream temperature.

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another by multiplication with a physical scaling factor(cf. Matlab code in section S3 of Text S1). The basic scalingequation (2) applies to different heat transport models,whereas equation (4) provides a general spectral solution forthe physical scaling factor Py�

wð Þ based on 1D conduction-convection. The convection term can often be neglected(see section 4.2 and section S1 of Text S1) and for conduction-dominated heat transport, effective thermal diffusivities canbe obtained from the exact solution either separately for eachfrequency used in the transform (equation (5)) or fitted to therange of frequencies. A scale transform that accounts fordepth-limited conduction is also included in section S1 ofText S1 (equation (S12)). The applicability of these differenttransforms must be assessed using site-specific temperaturedata as subsequently demonstrated for Säva Brook.[13] Thermal diffusivities obtained from Säva Brook using

the spectral method, numerical solutions and laboratory

measurements show excellent agreement. There is a lowerlimiting period that avoids noise and drift in the thermaldiffusivity, which for Säva Brook was found to be 5–7 hoursat sediment depths above about 10–20 cm. However, therequired limit increases with depth and from 50–100 cm indepth we found drift in the thermal diffusivity for periods upto at least 2 days. Hence, a key finding in the evaluation ofthe thermal diffusivity using the spectral method is that atsediment depths deeper than 50 cm one may have to excludethe diurnal signal commonly used in heat transport studies.

4.2. Heat Mixing in Säva Brook Bed Sediments

[14] Temperature measurements have previously beenused to identify up- or downward water fluxes in streamsediments [Silliman et al., 1995; Gordon et al., 2012].However, in Säva Brook the conduction was found to be thedominant heat transport mechanism in the sediments.

Figure 2. Spectral representation of the temperature time series. (top) The smoothed power spectra from 3 cm depth at Site1 evaluated using equation (4) for a window length up to 18 days. The model fitting is based on a window of periods eitherfrom 10 minutes to 18 days or from 1 to 18 days. (bottom) Thermal diffusivity evaluated for each time period l at Site 2using equation (5) applicable for Pe≪ z (i.e., pure conduction). Relatively constant thermal diffusivities are found for periodslonger than 0.2–0.3 days and agree well with laboratory data (grey-shaded band). The noise that appears at shorter periodsindicates greater uncertainties in the result, which is also responsible for the drift in the constant thermal diffusivity.

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Previous studies by Hatch et al. [2006] and Jensen andEngesgaard [2011] stress the need to consider both ampli-tude and phase when estimating conduction and convectionsimultaneously. However, while the optimization based onthe power spectrum gave a Darcy velocity close to zero(<10–10 m/s), optimization based on the phase spectrumresulted in finite, but still small velocities (<10–7 m/s). Evenby varying the range of different frequencies considered andexpressing the optimization based on a combination of powerand phase spectra, the vertical advection was found to beinsignificant. Use of the power transform, i.e., equation (4),or the phase transform, equation (S7) in Text S1, results insimilar value of the effective thermal diffusivity.[15] Previous studies on Säva Brook sediments, have

indicated that hyporheic mixing of solutes is limited to about

10 cm [Wörman et al., 2002; Jonsson et al., 2003]. Conse-quently, we hypothesize that the physical properties of thestream sediment, which mainly consist of clay, restrict con-vective heat transport at depths greater than about 10 cm.Even if hyporheic mixing is likely caused by shallow flowsin the bed sediment due to pumping [Thibodeaux and Boyle,1987], the large-scale subsurface flows are sufficiently smalland are not identified in this study as vertical 1D advection.The temporary increase in the effective thermal diffusivityidentified for a couple of days in connection with increasingflood stage is likely caused by either small-scale convective/pumping processes confined to shallow bed-sediments and/or remobilisation of an upper layer of the stream-bed sedi-ment. The latter would move the in-stream temperaturefluctuations closer to the corresponding points in the

Figure 3. Variability of the effective thermal diffusivity over depth based on the optimization of the power spectrum for therange of significant frequencies, as performed for Figure 2b. (top) Comparison of the optimal effective thermal diffusivitybased on equation (5) at Site 2 (red curve) with similar optimized results from a numerical solution to the equations govern-ing heat transport (blue curve), as well as laboratory data (circles, red from Site 2). (bottom) Small differences in thermaldiffusivity between sites using equation (5) on the temperature data from each depth that was related individually to instreamtemperature (dashed lines) and when the temperatures at nearby sediment depths are related (solid lines) assuming homo-geneous conditions in both cases. The shaded grey band denotes the range of laboratory results.

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sediment (i.e., the depth of the sensor would change) andresult in an increase in the thermal dispersion. In theseshallow sediments, there are also factors that tend todecrease the thermal diffusivity, such as the existence of gasand organic material (see section 3.2) or the variation ofeffective thermal diffusivity, which appears to be highestclose to the bed surface (Figure 3, bottom).

4.3. Scaling of Heat Fluxes in Sediments

[16] The exact spectral solution enables the evaluation of theconsistency in the physical model (e.g., the 1D convective-conductive equation) over the significant range of streamtemperature fluctuations. Some of the shortest periods are notwarranted because of a lack of periodicity or incompleterepresentation, but the general vertical heat responses in bed-sediments scales in equation (4) with the Peclet number,Pe = (q/g) z/ke and the conductive-frequency number z ¼z w=keð Þ0:5 . Therefore, the physical scaling factor resultingfrom the spectral approach clearly separates scaling due to thePeclet number, which is frequency independent, from theconductive-frequency numbers, which are frequency depen-dent. In this paper we demonstrated how the different scalingnumbers are related through the physical-mathematical for-mulation of the heat transport problem in bed-sediments.These relationships should help develop the understanding ofheat transport in streams and assist in developing futureobservation and analysis methods.

[17] Acknowledgments. Part of the project was funded by the stra-tegic research project STandUp for Energy (KTH), the Swedish Hydro-power Centre (SVC) and The Utah Water Research Laboratory at UtahState University. Thanks also go to anonymous reviewers who helpedimprove this paper.[18] The Editor thanks the anonymous reviewers for their assistance in

evaluating this paper.

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