EARTH SURFACE PROCESSES AND LANDFORMS Earth Surf. Process. Landforms 36, 1577–1589 (2011) Copyright © 2011 John Wiley & Sons, Ltd. Published online 25 May 2011 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/esp.2167

Near‐surface temperatures and heat balance of bare outcrops exposed to solar radiation Yann Gunzburger* and Véronique Merrien‐Soukatchoff LAEGO, Nancy‐Université, Parc de Saurupt, CS 14234, F‐54042 Nancy, France Received 22 November 2010; Revised 8 April 2011; Accepted 12 April 2011 *Correspondence to: Y. Gunzburger, LAEGO, Nancy‐Université, Parc de Saurupt, CS 14234, F‐54042 Nancy, France. E‐mail: [email protected]‐nancy.fr

ABSTRACT: Subsurface temperatures in rocks naturally fluctuate under the influence of local meteorological conditions. These fluctuations play a role in mechanical weathering, thus creating the environmental conditions conducive to natural hazards such as rockfalls and providing important sediment source terms for landscape evolution. However, the physics that control heat penetration into rocks are not fully understood, which makes the underground thermal state difficult to interpret when temperature measurements are available and even more difficult to estimate for unmonitored sites. This is an important lacuna given possible impacts of future climate change on mechanical weathering processes. The natural daily variations of subsurface temperatures were investigated on a bare gneiss outcrop exposed to solar radiation, where temperatures at various depths (up to 50 cm), as well as the solar radiation reaching a pyranometer, were monitored hourly for several months. This detailed times series of thermal data was used to gain insight into the heat balance at the inclined free surface of the rock mass. Attention was focused on two major contributors to the heat balance; the heat flux entering the rock mass through conduction and the incoming shortwave (solar) radiation. A Fourier decomposition of the temperature measurements provided an estimate of the in situ thermal conductivity of the rock and was used to calculate the conductive term. The shortwave radiation term was determined on the basis of the pyranometer measurements adjusted to account for the angle of incidence of the sun. It is shown that, throughout clear‐sky periods, heat exchanges at the surface are mainly controlled by direct solar radiation during the day, and by a roughly constant outgoing heat flux during the night. Subsurface temperatures can be reliably estimated with a semi‐infinite medium model whose boundary condition is derived from an analytical insolation model that takes atmospheric attenuation into account. Copyright © 2011 John Wiley & Sons, Ltd. KEYWORDS: temperatures; conduction; solar radiation; heat balance; pyranometer

Introduction Along bare outcrops, underground rock masses are directly subjected to the influence of local meteorological conditions (e.g. sunlight, clouds, wind) and exchange heat with the atmosphere. The thermal energy flux through the surface dictates the temperature field at shallow depths. Detailed knowledge of the thermal state of the ground represents essential input data for extremely varied research topics, including agronomy, ecology, hydrology, climatology, and architecture. The study of ground temperatures is also essential for explaining mechanical weathering, as first postulated by Blackwelder (1933) and Griggs (1936), and for understanding the occurrence of some natural hazards, such as rockfalls, in mountainous areas. Nevertheless, because external thermal forcing fluctuates continuously on both a daily and annual basis, steady‐state thermal equilibrium is never achieved in rocks. Consequently, the subsurface temperature field is complex, heterogeneous, and variable in time. It is not easy to interpret when dense or high‐frequency temperature measurements are available and even more difficult to estimate at unmonitored sites.

The objective of this study was to take advantage of a detailed, continuous time series of data acquired hourly at a field site in order to explore the natural daily variations of temperatures in the shallowest part of a rock mass and to gain insight into the heat balance at its free surface. This will help in predicting the temperature field outside of monitoring periods and/or in virgin areas, which can, in turn, lead to a better understanding, or even prevention, of some rock slope instabilities induced by temperature variations. In the presence of water, such instabilities may be generated either by the volume increase of interstitial water when it freezes (Coutard and Francou, 1989; Matsuoka, 1994, 2001, 2008; Matsuoka et al., 1997; Matsuoka and Sakai, 1999) of by the growth of segregation ice lenses (Walder and Hallet, 1985; Hallet et al., 1991; Anderson, 1998; Hales and Roering, 2005, 2007). The complete thaw of permafrost resulting from atmospheric warming has also been suggested as a possible explanation for some rockfalls (Gruber et al., 2004). Even in the absence of water, repeated temperature variations may lead to ‘thermal fatigue’, a rock weathering process resulting from the accumulation of micro‐cracks, due to differential thermal expansion of the rock grains (Simmons and

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Cooper, 1978; Yong and Wang, 1980; Fredrich and Wong, 1986; Hall et al., 2008). A so‐called ‘thermal shock’ may occur if the rates of temperature change exceed 2°C min–1 (Jenkins and Smith, 1990; Hall, 1997, 1999; Hall and André, 2001; McKay et al., 2009; Molaro and McKay, 2010). Recent studies have also pointed out the existence of thermally‐induced, non‐reversible movements of rock blocks along pre‐existing macro‐fractures (Matsuoka, 2001; Gunzburger et al., 2005; Ishikawa et al., 2004; Watson et al., 2004; Vlcko et al., 2005; Vargas et al., 2009). Recordings of temperature variations and the establishment of temperature–depth logs close to the Earth’s surface have previously been produced at a number of sites around the world (most of the above‐cited references are based on such recordings), including up to depths of several hundreds of metres for the reconstruction of past climates (Lachenbruch and Marshall, 1986; Guillou‐Frottier et al., 1998; Huang et al., 2000; Demezhko and Shchapov, 2001). However, to date, attention has essentially been focused on temperatures in the ground, while few investigations have paid attention to the heat balance at rock surfaces, despite the fact that it is known that ground surface temperatures only crudely reflect air temperatures (Jenkins and Smith, 1990). Additionally, the physics that establish the ground temperature are not fully understood, especially for mountainous areas, where the local orientation of free surfaces with respect to the sun plays a key role. In the present study, the surface heat balance schematically depicted in Figure 1 is presented in the following form: qcond ¼ ð1−αÞðqbeam þ qind Þ−qlw −qconv

(1)

where qcond designates the heat flux (W m‐2) transmitted toward depth by conduction; qbeam and qind represent the direct (beam) and indirect shortwave solar radiation reaching the surface, respectively; qlw is the net longwave (infrared) radiation lost by the heated rock; qconv corresponds to the convective heat flux associated with air movements along the outcrop; and α represents the albedo (reflectivity) of the outcrop. To quantify the relative importance of these different contributions to the heat balance, we analysed a time series of temperatures at various depths (up to 50 cm) and of the total (direct and indirect) solar radiation recorded for several months on an inclined rock slab located in the southern French Alps. We focused our attention on the terms qcond and qbeam. The first of these was deduced from the temperature–depth logs using Fourier’s law, and the in situ thermal conductivity of the rock was calculated from the decay of the amplitude and the phase shift of temperature variations with depth. The second term was deduced from insolation measurements carried out by a horizontally‐lying pyranometer, by taking into account the

Figure 1.

angle of incidence of the sun. Most who have approached the problem of the surface heat balance of rock slopes before have lacked these radiative data that we have collected and thus could not take the solar radiation into account properly. We show that, during cloudless days, the heat flux penetrating the rock mass can be reliably reproduced by a simple analytical insolation model during periods when sunbeams reach the free surface and, otherwise, by a roughly constant term corresponding to the emitted longwave radiation and the convective heat flux. As a result, the diurnal fluctuations of the thermal state underground can be determined by means of the conduction law, provided the local meteorological conditions are calm and stable. This result was verified for a selected day, and because the input data for the model are restricted to basic parameters (such as thermal conductivity and albedo), this method for acquiring a first estimate of these temperatures is assumed to be applicable to other periods of time and to other sites. The case of periods with overcast skies was also briefly considered, though only from a semi‐quantitative perspective.

Heat Flux Penetrating the Rock by Conduction Measurements of subsurface temperatures The study site is a steep gneiss slope, several hundreds of metres high, known as ‘Rochers de Valabres’, which is located at a longitude of E7°05’41’’ and a latitude of N44°07’40’’ (Figure 2a). After two rockfalls in 2000 and 2004, it is feared that additional instabilities might occur, which has motivated research concentrated on field observations, monitoring and numerical modelling. Among the many potential causes for these falls, temperature variations have previously been pointed out as one possible priming factor (Gunzburger et al., 2005). Since 2002, a section of the slope, located at an altitude of approximately 760 m a.s.l. and consisting of a large, valley‐ dipping, planar surface (herein referred to as a ‘facet’ or a ‘slab’), without any soil or vegetation, and that is almost never covered by snow (Figure 2b), has been studied as an experimental site (Dünner et al., 2007; Clément et al., 2009). Thermal and meteorological measurements have been collected there at 1 h intervals since April 2006, using the following sensors: two surface temperature thermocouples (installed at less than 1 cm depth), a thermometric multipoint probe that records the temperatures at five different depths (10, 20, 30, 40 and 50 cm) in a borehole perpendicular to the free surface, and a pyranometer for radiation measurements. In analysing the long‐term time series, we faced the problem of acquisition gaps corresponding to temporary breakdowns of

Heat balance at the free surface of the rock.

Copyright © 2011 John Wiley & Sons, Ltd.

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the monitoring system. Therefore, the seasonal evolution of the data cannot be fully investigated at present. Instead, we focused on the daily variations over periods of continuous measurement without gaps (lasting for several days, weeks or

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months) that correspond to different times of the year and offered different sky conditions. The time periods explored in this study are summarized in Figure 3 and in Table I, along with their main features. Due to the relatively low sampling

Figure 2.

Location maps (a) and northward view of the ‘Rochers de Valabres’ study site showing the instrumented area (b).

Figure 3.

Time periods of the years 2006 and 2007 that were analysed in this study.

Table I. Time periods of temperature and radiation recordings analysed in this study. n corresponds to the Julian day of the year (n = 1 for 1 January, for example) Designation A A′ B C C′ C′1 C′2

Dates 14 June – 3 July 2006 (n = 165 – 184) 22 June – 3 July 2006 (n = 173 – 184) 1 December – 31 December 2007 (n = 335 – 365) 21 June – 20 September 2007 (n = 172 – 263) 30 August – 8 September 2007 (n = 242 – 251) 30 August 2007 (n = 242) 8 September 2007 (n = 251)

Copyright © 2011 John Wiley & Sons, Ltd.

Duration 20 days 11 days 31 days 3 months 10 days 24 hours 24 hours

Interesting features Typical set of days near the summer solstice,with one temperature measurement per hour Portion of period A with very stable meteorological conditions Typical set of days near the winter solstice,with one temperature measurement per hour Entire summer season,with one temperature measurement per hour Portion of period C during which the sampling frequency was increased to four measurements per hour One selected day of period C′ with clear sky and some passing clouds One selected day of period C′ with very clear sky and no clouds

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frequency of the temperature data (1 measurement per hour), very rapid temperature fluctuations, such as those reputed to induce thermal shocks (dT/dt > 2°C min‐1), could not be confidently detected and were not considered. Figures 4a and 5a provide an example of the temperatures measured during periods A and B, respectively, which were selected because they correspond to times of the year with strongly contrasting meteorological conditions, near the summer and winter solstice. More or less periodic daily oscillations were observable at all depths, with the half‐ amplitude at the surface varying from 3–10°C during period A and 1–6°C during period B. A decay in the amplitude and an increasing phase shift in the signal with depth are clearly visible in both cases. It is also obvious that the time series departs from the purely sinusoidal model, which is frequently

used in the literature as a first approach. Here, the surface signal is strongly asymmetrical, with a rapid increase from sunrise to a peak in the early afternoon, followed by a monotonic decline of temperature, first quite rapid during the evening and then slower through the night. Interruptions of the regular pattern are expressed as damped temperature swings, for example, on 17 June 2006 (Julian day n = 168 on Figure 4a). They are linked to greater cloudiness, as indicated by higher values of the Sky Overcast Index (SOI) (Figures 4d and 5d). The SOI is a unitless qualitative estimate of the daily average cloud cover provided by the French Meteorological Office, which varies from 0 (completely clear sky) to 8 (completely overcast sky). For practical reasons, the SOI used in this case was not determined at the test site but was taken from the meteorological station of the city of Nice,

Figure 4. Thermal state of the subsurface during period A (near the summer solstice). Temperatures measured hourly at the surface and at five depths down to 50 cm (a) were used to calculate the heat flux entering the rock by conduction (b) as well as the corresponding cumulative thermal energy (c). The measurements of two surface temperature sensors were averaged into a single temperature for simplicity. The sky overcast index (SOI) measured at the city of Nice, 50 km away, is also shown (d). Copyright © 2011 John Wiley & Sons, Ltd.

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Figure 5.

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Same data as in Figure 4, but for period B (near the winter solstice).

approximately 50 km away (Figure 2a), where the weather may be significantly different from that at the study site on some days.

temperatures varied almost periodically with a 1‐day period. The temperature at the free surface could then be expressed as the Fourier series: ∞

In situ thermal diffusivity of the rock Owing to the low porosity and permeability of the instrumented rock, we assumed an absence of heat advection and latent heat sources and sinks within it. We also ignored the possible role of light‐transmissive minerals, because they have an influence only to depths of several millimetres (Gomez‐Heras et al., 2006; Hall, 2011). Consequently, because the free surface is locally planar, the thermal problem was approximated as a conduction problem in a semi‐infinite solid with thermal diffusivity a (m2 s‐1). Over a period of several days with stable meteorological conditions, seasonal fluctuations were not perceptible, and Copyright © 2011 John Wiley & Sons, Ltd.

T ðz ¼ 0; t Þ ¼ T0 þ ∑ An cos ðnωt−φn Þ

(2)

n¼1

with a mean temperature of T0 and an angular frequency of 2π ω ¼ 243600 rad s‐1. The temperature at depth z is given by ∞

T ðz; t Þ ¼ T0 þ ∑ An ðz Þ cos ðnωt−φn ðz ÞÞ

(3)

n¼1

pffiffi (Carslaw and Jaeger, 1986; Holman, 2002). An ðz Þ ¼ An e −z n=δ represents the half‐amplitude of the nth harmonic (nth partial pffiffi z n ‘temperature wave’ ) at depth z,pand ffiffiffiffiffiffiffiffiffiffiffi φn ðz Þ ¼ φn þ δ represents its phase constant. δ ¼ 2a=ω is the damping depth of the first harmonic.

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The exponential amplitude decay of the harmonics with depth was used to acquire an estimate of the in situ thermal diffusivity of the rock. For this, a Fourier decomposition of the temperature time series at the six different measurement depths was performed for period A′, which presents regular temperature swings (Figure 4a). Linear fits were calculated for the logarithm of the amplitudes (Table II and Figure 6a), and one value of a was determined for each of the three first harmonics (n = 1,2,3) using the following relationship: rffiffiffiffiffi ln An ðz Þ ln A ω pffiffiffi ¼ pffiffiffi n − z (4) 2a n n Their mean value was a = 2.0 × 10‐6 m2 s‐1. Because of the limited measurement frequency (one measurement per hour), the higher order harmonics were not considered because the accuracy of the calculated diffusivity decreases rapidly as n increases. Another diffusivity estimate was obtained from the linear increase of the phase shift with increasing depth: φn ðz Þ φn pffiffiffi ¼ pffiffiffi þ z n n

rffiffiffiffiffi ω 2a

(5)

The results are presented in Figure 6b and Table III. On average, a = 1.8 × 10‐6 m2 s‐1. Finally, a mean value of a = 1.9 × 10‐6 m2 s‐1 was taken as an estimate of the in situ thermal diffusivity of the rock.

Calculation of the heat flux associated with conduction Figure 7 displays the temperature–depth profiles at different times of day C′2 (8 September 2007), on which the sky was very clear. Between 10:45 and 16:45, the temperatures were greater at the surface than at 10 cm depth (continuous lines), and thus, the heat transfer proceeded inwards in the rock mass. From 00:00 to 10:45 and from 16:45 to 24:00, the temperatures were lower at the surface (dashed lines), and the heat proceeded outwards to be transmitted to the atmosphere through the free surface. The temperature at 50 cm depth varied only slightly around 22°C, which illustrates the damping of temperature oscillations. The heat flux qcond entering the rock mass through its free surface and transmitted downward by conduction was calculated using Fourier’s law and approximated by the following difference quotient:

Table II. Amplitudes An(z) (in °C) of the three first harmonics (n = 1, 2, 3) of the Fourier series of the temperatures at six different depths z and the calculated values of the in situ thermal diffusivity a

0 cm 10 cm 20 cm 30 cm 40 cm 50 cm a (m2 s‐1) deduced from Equation 4 z

A1(z)

A2(z)

A3(z)

6.18 °C 3.99 2.68 1.76 1.07 0.68 1.88 × 10‐6

2.41 1.28 0.73 0.41 0.21 0.13 2.10 × 10‐6

0.79 0.32 0.16 0.08 0.04 0.04 2.01 × 10‐6

Figure 6. Exponential decay of the amplitude (a) and increasing phase shift (b) with depth of the three first partial waves of temperature during period A′. The calculation was performed by means of a Fourier decomposition of the measured temperatures. Copyright © 2011 John Wiley & Sons, Ltd.

qcond ¼ λ

∂T Tsurf − T10 cm ð z ¼ 0Þ ≈ λ ∂z 0:1

(6)

Table III. Phase lag φn(z) (in radians) of the three first harmonics (n = 1, 2, 3) of the Fourier series of the temperatures at six different depths z and the calculated values of the in situ thermal diffusivity a

0 cm 10 cm 20 cm 30 cm 40 cm 50 cm a (m2 s‐1) deduced from Equation 5 z

φ1(z)

φ2(z)

φ3(z)

4.034 rad 4.383 4.758 5.164 5.683 6.204 1.94 × 10‐6

0.562 0.876 1.253 1.679 2.247 2.785 1.82 × 10‐6

2.322 2.512 2.898 3.388 4.097 4.531 1.68 × 10‐6

Figure 7. Temperature–depth profiles at different times of the day on 8 September 2007 (day C′2). The value shown on each curve indicates the hour of recording. Dashed lines correspond to surface temperatures that are lower than the temperature at 10 cm depth (heat leaving the rock mass through its free surface), and continuous lines indicate the opposite phenomenon (heat entering the rock mass). Earth Surf. Process. Landforms, Vol. 36, 1577–1589 (2011)

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Tsurf and T10 cm represent the temperatures at the surface and at 10 cm depth, respectively. The thermal conductivity, λ = aρcp= 4.21 W m‐1 K‐1, was determined by considering a rock density of ρ = 2.7 × 103 kg m‐3 (measured on rock samples) and a specific heat capacity of cp = 830 ± 30 J kg‐1 K‐1 (deduced from values found in the literature for similar rock types, e.g., Berest and Weber, 1988). The results are shown in Figure 8a for days C′1 (30 August 2007, continuous line) and C′2 (8 September 2007, dashed line). The conductive flux on 8 September was positive from 10:45 to 16:45 (the highest value was approximately 350 W m–2) and negative for the rest of the day (roughly stable at a value of approximately −80 W m–2). The pattern was very similar on 30 August (dashed line), but at least two troughs were visible in the curve (around 12:00 and 14:00), corresponding to two periods during which clouds obscured the sunlight. Figures 4b and 5b display the calculated heat flux for periods A and B, respectively, for which temperature fluctuations are displayed in Figures 4a and 5a. The differences between the winter and summer were much more obvious for the temperatures than for the heat fluxes, which had amplitudes that only slightly varied from one period to the other. The main difference was that the peaks were broader near the summer solstice (period A) than near the winter solstice (period B), which is explained by the fact that the daily duration of sunshine is longer

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in the first case (15.3 h on a flat unobstructed surface) than in the second (8.7 h). The mismatch between the positive and negative parts of the flux curves resulted in an increase in the cumulative thermal energy that penetrated the rock during period A (Figure 4c) and in a decrease during period B (Figure 5c). In the first case, the rock mass gains thermal energy from day to day (with a mean rate of approximately +1–2 × 106 J m‐2 day–1), while it loses energy (−1×106 J m‐2 day–1) in the second case.

Solar Radiation Input Incoming shortwave radiation measured by the pyranometer A pyranometer was installed horizontally near the temperature sensors to register the total incoming shortwave radiation in the wavelength range 0.3–1.1 µm: 0 0 0 qsw ¼ qbeam þ qind

(7)

where qbeam represents the direct incoming shortwave solar radiation (also known as ‘beam radiation’) and qind is the indirect shortwave input, which includes the diffuse radiation (received from the sun after its direction has been changed by atmospheric scattering) and the radiation reflected by adjacent slopes (Figure 1). The subscript ‘0’ is used to indicate that the target surface is horizontal. An example of pyranometer measurements with a 10 min sampling frequency is presented in Figure 8b for days C′1 (dashed line) and C′2 (continuous line). The radiation was null for a large portion of both days: before 7:05 (dawn) and after 19:55 (dusk). Between these two limits, it was low (typically less than 100 W m‐2) before 10:45 (local sunrise on the facet) and after 15:15 (local sunset on the site), when only indirect radiation reached the sensor. At 10:45, it increased abruptly up to values between 650 and 820 W m‐2 and then rapidly decreased at 15:15. The same overall pattern was observed for the two selected days, but the curve for day C′1 was more irregular, showing at least two major periods of lower radiation (around 12:00 and around 14:00) corresponding to cloudy spells, and some minor fluctuations resulting from partial obscuring of the sun by isolated clouds. Before sunrise and after sunset, the diffuse radiation appeared to be higher on this more overcast day, which is interpreted as a consequence of increased scattering by the atmosphere due to higher relative air humidity.

Direct shortwave radiation estimated by an analytical attenuation model The direct shortwave radiation reaching a horizontal surface 0 ) depends on (qbeam

Figure 8. Heat flux entering the rock by conduction (a) and total solar radiation measured by the pyranometer (b) on days C′1 (slightly cloudy, dashed lines) and C′2 (very clear sky, continuous lines). Because the two days are close in the calendar, the times of dawn, dusk, local sunrise and sunset show only little variation from one day to the other. Copyright © 2011 John Wiley & Sons, Ltd.

• the solar irradiance reaching the outer surface of the Earth’s atmosphere, known as the solar constant (q* = 1370 W m–2); • the solar altitude angle (h), which corresponds to the local height of the sun in the sky above the horizon (Figure 9) and determines the incidence coefficient (sin h) of the sunbeams on the horizontal target surface; • atmospheric attenuation caused by gases, aerosols, and cloud particles, which depends on the thickness of the air section crossed by the sunbeams (proportional to 1/sin h to a first approximation) and on the optical density of the atmosphere. To characterize the atmospheric attenuation, we used the Linke turbidity factor, TL (Linke, 1922), which varies between 2 Earth Surf. Process. Landforms, Vol. 36, 1577–1589 (2011)

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The declination of the sun (δ) and its hour angle (ω) indicate the position of the sun with respect to the Earth:

˚

δ ¼ 23:45  sin



n −81  360 365



˚

(10)

and ω¼

Figure 9. The solar altitude angle (h) and the solar azimuth angle (A) determine the local position of the sun in the sky above a horizontal plane. They can be calculated with Equations 9 and 14, respectively.

and 3 for a very clear sky and 7 and 8 for an extremely polluted atmosphere. The direct shortwave radiation can then be estimated using the exponential attenuation model proposed by Kasten (1980): 0 qbeam

 −TL ¼ q sin h ⋅ exp 0:9 þ 9:4 sin h 



(8)

The solar altitude angle is given by Gauss’ formula: sin h ¼ cos δ cos ω cos φ þ sin δ sin φ

(9)

clock time− 1hour  360 −ðL −Lst Þ−180 24hours

˚

˚

(11)

φ and L are the latitude and longitude of the site, respectively. Lst is the longitude of the standard meridian for the local time zone (Lst = −15° for the Central European Time Zone). A one‐ hour subtraction from clock time is required when the clock time is Daylight Savings Time; otherwise, the clock time can be used without correction. Figure 10a presents a comparison between the pyranometer 0 (thick continuous line) and the direct solar input recordings qsw 0 qbeam derived from Equation 8 (dashed lines) for different values of TL in the range corresponding to clear‐sky conditions. The calculated clock times for sunrise (7:15) and sunset (19:55) on the horizontal surface (defined as the times of the first and last positive value of h on the considered day, respectively) are in excellent agreement with the times of dawn and dusk at the site deduced from Figure 8. They differ from the effective times of local sunrise (10:45) and sunset (15:15) because the neighbouring slopes obscure sunlight in the early morning and in the late afternoon, when only indirect radiation reaches the sensor.

Figure 10. Contribution of solar radiation (qsw) and conductive heat flux (qcond) to the heat balance at the surface on days C′2 and C′1, respectively: 0 (thick continuous lines) were first compared to the direct solar 8 September (a and c) and 30 August 2007 (b and d). Pyranometer measurements qsw 0 (dashed line) derived from the exponential attenuation model with TL = 2.25, 2.75 and 3.25 (a and b). After incidence correction (c and d), radiation qbeam the total solar radiation reaching the inclined facet qsw (thin continuous line) can be compared to the heat flux qcond penetrating the rock (thick continuous line). Copyright © 2011 John Wiley & Sons, Ltd.

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Between the local sunrise and sunset, when sunbeams reach the pyranometer, the model predictions follow the measurements closely and are only slightly lower than these data. The difference between the two values corresponds to the indirect 0 (shown in grey in Figure 10a). We shortwave radiation qind considered that the latter term would evolve continuously throughout the day and would not present any peaks at sunrise or sunset, in contrast to the direct radiation. Consequently, the value TL = 2.75 was selected as the value that most reliably describes the very stable atmospheric conditions prevailing on day C′2 (a completely cloudless day). Then, the total amount of direct and indirect solar energy that actually reached the sensor on this day (1.3 × 107 J m‐2) presents a ratio of R = 66% of the theoretical amount of direct solar energy that would have reached a flat unobstructed surface, as estimated with the formula (2.0 × 107 J m‐2). The difference between the two values is explained by the local configuration of the surrounding obstacles to sunbeams before local sunrise and after local sunset. Such a detailed analysis is impossible for overcast days, such as C′1 (Figure 10b), because the value of TL that should be used for comparison probably varied greatly on this day and cannot be estimated easily. However, the local maxima of the pyranometer measurements still roughly coincided with the model prediction obtained using TL = 2.75, which indicates that the highest values of solar radiation are essentially of direct origin. Due to the troughs in the curve, the ratio between the measured (0.89 × 107 J m‐2) and theoretically received solar energy (2.2 × 107 J m‐2) decreased to R = 41% on this day.

The total shortwave radiation that reaches the inclined facet (qsw = qbeam + qind) is different from that measured by the pyranometer because the orientations of the two target surfaces with respect to the sunbeam are different. Thus, an incidence correction had to be introduced. For the direct radiation, this takes the form: cos θ 0 q sin h beam

(12)

θ, which represents the angle between the sunbeam and the normal to the inclined facet, is given by cos θ ¼ sin i cos h cos ðA−γ Þ þ cos i sin h

cos δsin ω cos h

(14)

The indirect radiation has no well‐defined preferential orientation. Most of the corrections proposed in the literature assume that it is proportional to the fraction of the sky dome visible from the surface considered (Hay and McKay, 1985; Duguay, 1993). In practice, these corrections are not fit for use in mountainous environments due to the complex geometry of the surrounding slopes, so we followed another approach, as described below. Copyright © 2011 John Wiley & Sons, Ltd.

0 qsw ≈ qsw

cos θ sin h

(15)

At other periods of the day (before local sunrise, after local sunset, or during cloudy periods), no direct radiation reached the sensor, so the pyranometer measured the indirect portion of the solar radiation at these times. Because the latter form of radiation is of low intensity compared with the values corresponding to when sunbeams are received, the influence of the sky‐view correction was neglected in this case: 0 qsw ≈ qsw

(16)

Figures 10c and 10d display the shortwave radiation qsw arriving at the inclined facet (thin continuous line) calculated with this method for days C′2 and C′1. The facet received more solar radiation than the pyranometer because it is more perpendicular to the sunbeam due to its south‐ eastward dip.

Discussion

For days C′1 and C′2, the curves corresponding to qcond and qsw have similar shapes, but their amplitudes differ by a factor of 3 to 4 (Figure 10c and 10d). The same pattern was observed for most of the year, as illustrated by Figure 11a for the entire period C′. Our objective was to determine which part of the heat flux entering the rock mass could be attributed to insolation. Once the incoming solar radiation has fallen to zero at dusk (19:55 for dayC′), the rock mass loses part of its previously accumulated thermal energy through convective exchanges with the air (qconv) and longwave radiation (qlw). The corresponding heat balance is qcond ¼ −ðqconv þ qlw Þ≤0

(17)

(13)

where i corresponds to the dip angle of the target surface (i = 0° for a horizontal surface and i = 40° for the facet in our case) and γ represents its strike counted westwards from the south (γ = 0° for a south‐facing slope and γ = −27° for the facet). The solar azimuth angle A (Figure 9) is given by sin A ¼

At times when the pyranometer measurement was close to the value derived from the exponential attenuation model using TL = 2.75 (i.e. between the local sunrise and sunset, on days with a clear sky), the incident solar radiation was almost entirely of direct origin, and the correction of its indirect component was considered to be of secondary importance. Therefore, Equation 12 could be applied directly to the pyranometer measurement:

Contribution of solar radiation to the surface heat balance

Total shortwave radiation reaching the inclined facet

qbeam ¼

1585

According to Figure 8a, the heat flux leaving the rock mass first slowly increases (qcond becomes more and more negative) and then stabilizes at around 50–100 W m‐2 after midnight, until dawn (7:05). The convective heat transfer term (qconv) depends on, among other factors, the wind velocity (forced convection) and the difference between the temperature of the rock surface and that of the air in its vicinity, which was measured by a weather station (free convection). This difference is shown, for example, in Figure 11b. The two temperatures exhibited similar trends but differed from each other by values up to 15°C, especially during the day. This confirms that the temperature of the rock is not simply determined by its contact with the surrounding air and that convective exchanges between the rock and the atmosphere do exist. The longwave radiation term (qlw) includes the grey‐ body type radiation emitted by the heated rock surface Earth Surf. Process. Landforms, Vol. 36, 1577–1589 (2011)

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Figure 11. Contribution of solar radiation qsw (thin line) and conductive heat flux qcond (thick line) to the heat balance at the surface during period C′ (a). The difference between the temperature of the rock surface and that of the air in its close vicinity, as measured by a weather station (b), shows that there were necessarily some convective heat exchanges between the rock and the atmosphere.

and the infrared heat flux back‐radiated by the atmosphere toward it: 4 4 qlw ¼ εrock σTsurf −εatm σTatm

(18)

where σ is the Stefan–Boltzmann constant (5.67 × 10‐8 W m‐2 K‐4). The thermal emissivity of the rock (εrock) may be estimated at 0.95 for gneiss (Rubio et al., 1997), while that of the atmosphere (εatm) and its absolute temperature (Tatm) to be used in Equation 18 are unknown because, under clear‐sky conditions, the rock surface ‘sees’ the temperature of the atmosphere well above it (Anderson, 1998). Thus, Tatm could not simply be measured, for example, by a weather station located on the facet. During the periods of the day between dawn (7:05) and local sunrise (10:45) and those between local sunset (15:15) and dusk (19:55), only indirect solar radiation reaches the surface, and the insolation term has a low positive value. The corresponding heat balance is qcond ¼ ð1− αÞqind − ðqconv þ qlw Þ

(19)

with α representing the albedo of the rock surface. Between local sunrise (10:45) and sunset (15:15), sunbeams do reach the rock, and insolation increases rapidly to large positive values. Thus, an incoming direct solar radiation term (qbeam) must be added to the previous terms: qcond ¼ ð1−αÞðqbeam þ qind Þ−ðqconv þ qlw Þ ≥0

(20)

During this period of the day, any change in cloudiness is easily observable both in the pyranometer measurements and in the conductive heat flux term (Figure 10d). The heat flux entering the rock mass closely follows the incoming shortwave Copyright © 2011 John Wiley & Sons, Ltd.

radiation (after deducting the albedo‐proportional reflected part), which may be estimated on the basis of the exponential attenuation model of Equation 8.

Modelling of the subsurface thermal state under clear‐sky conditions Temperature changes within the rock are dictated by the mismatch between the incoming and outgoing heat fluxes, so we wanted to assess how well the measured temperature can be explained or reproduced, at least during cloudless days, using a simplified history of heat flux as a boundary condition at the free surface. We focused attention on the diurnal warm‐up. Because the heat flux leaving the rock during the night, which is responsible for the nocturnal cool‐down, fluctuates much less than the solar heat input (Figure 11a), we considered as a first approximation, that the term qconv + qlw is roughly constant over the whole day during the studied period, with a value of approximately 80 W m–2. Neglecting for the moment the fact that the albedo may change with the angle of incidence of the sun to the facet (Oke, 1999) and thus using a single value, the albedo could be estimated as approximately α = 55%. The temperature field inside the rock, considered as a semi‐ infinite medium, was calculated using the heat conduction law and a Fourier transformation. The surface heat flux used, denoted as q, is shown in Figure 12 (dashed line) and can be compared with the conductive flux measured on day C′2 (thick continuous line). q has a 24 h periodicity and was generated by distinguishing two periods of the day: • before local sunrise and after local sunset: Earth Surf. Process. Landforms, Vol. 36, 1577–1589 (2011)

NEAR‐SURFACE TEMPERATURES AND HEAT BALANCE OF BARE OUTCROPS

Figure 12. Heat flux used as a boundary condition in the model (dashed line). It may be compared with the conductive heat flux deduced from a difference quotient using the measured temperatures (thick continuous line) or the calculated temperatures (thin continuous line) on day C′2.

q ¼ −ðqconv þ qlw Þ ¼ −80W m−2

(21)

• between sunrise and sunset, it was calculated by the exponential attenuation model, considering an albedo of α = 55% and a Linke attenuation factor of TL = 2.75, and taking into account the incidence correction through the angle θ:  −TL q ¼ ð1−αÞ⋅q exp ⋅ cos θ − ðqconv þ qlw Þ (22) 0:9 þ 9:4 sin h 



A constant value of qconv + qlw was maintained during this period and the term qind was neglected. The mean temperature could not be calculated this way because the flux boundary condition only determines temperature variations and not the temperatures themselves. Nevertheless, this would be sufficient for the investigation of thermally induced stresses or strains and their effect on slope stability when the possible existence of subzero temperatures is not taken into account. When we assigned a mean temperature of 22°C (the mean temperature measured at the site during day C′2), the calculated temperature histories displayed a high degree of similarity with the measurements (Figure 13). The sharper edges of the calculated curves are explained by the pronounced jump

discontinuities of the imposed boundary heat flux due to the fact that we did not consider the periods before local sunrise or after local sunset, during which only indirect radiation reaches the surface. The small, high‐frequency oscillations observable near the edges are numerical artefacts due to convergence of the Fourier series in the vicinity of the jumps (Gibbs phenomenon). These oscillations die out with increasing depth. This method can be used to predict temperatures outside of the monitoring periods and at other sites with a planar free surface subjected to solar radiation. For this purpose, measurements of temperatures at various depths are necessary as a first step to calibrate the attenuation of temperature variations with depth (in situ thermal diffusivity). Once this is done, they are no longer necessary. Similarly, the calibration of the heat balance at the surface (albedo coefficient, times of effective sunrise and sunset, amount of heat lost by longwave radiation and convection) initially requires measurements of the solar radiation and temperature at two different depths as close as possible to the surface. Once the model has been fully calibrated, the temperature variations around their mean value inside the rock mass can be calculated on cloudless days by considering a semi‐infinite medium model while using a simplified heat flux deduced from the exponential attenuation model as a boundary condition. It is worth noting that this calculation does not require the measurement of the air temperature.

Uncertainties regarding the terms involved in the heat balance The value of albedo we found (of α = 55%) is not unrealistic for the light‐grey gneiss of the studied site, but it is quite high compared with other values for similar rock types (α = 30 to 40%) that may be found in the literature (Sellers, 1965; Anderson, 1998). This discrepancy may be explained either by an underestimation of the conductive heat flux qcond, or by an underestimation of qconv + qlw. A possible underestimation of qcond may result from the use of a difference quotient in Equation 6 instead of a true temperature derivative with depth. The corresponding bias may be quantified by comparing the heat flux (q) used as a boundary condition in our model (Figure 12, dashed line) with its estimate resulting from a temperature difference quotient based on the temperatures calculated with the model (Figure 12, thin continuous line). The constant rate of heat loss imposed between 15:15 and 10:45 is quite well reproduced by the difference quotient, especially after midnight, but the difference quotient underestimates the daily maximum heat input by about 20%. Correcting this bias would decrease the calculated value of the albedo and thus make it closer to the expected values. However, this is not easy to do systematically, as the underestimation ratio of the flux varies during the day. The difference quotient is also responsible for the delayed response of the conductive heat transfer term to the jump in the insolation input at local sunrise and sunset (Figures , 10c, 10d, and 12). The smooth shape of the conduction curve is explained by the fact that very rapid increases and decreases in the solar radiation take some time to be transmitted from the surface to 10 cm depth. The characteristic transmission velocity is v¼

Figure 13. Temperature histories at the different depths on day C′2, as determined numerically, using the simplified heat flux imposed at the free surface (dashed lines), and measured in situ (continuous lines). Copyright © 2011 John Wiley & Sons, Ltd.

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pffiffiffiffiffiffiffiffiffiffi 2aω:

(23)

This leads to a time lapse of approximately 1.6 h for 10 cm, which is in accord with the observations. A possible underestimation of the term qconv + qlw may result from the fact that it was supposed to be constant throughout the Earth Surf. Process. Landforms, Vol. 36, 1577–1589 (2011)

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day, at the same value as during the night. Actually, since the difference between the temperatures of the air and of the rock surface is maximum around midday (Figure 11b), we may expect a larger value of the convection term during the day than during the night. In the same way, because the surface temperature is maximum in the early afternoon, the radiation term may be greater at this time of the day. Given that our simple model quite nicely captures the temperature field within the rock, at least in a 1‐day signal (Figure 13), it is doubtful whether introducing a variable value for the term qconv + qlw would significantly improve the fit. Yet, it would make the problem more cumbersome to solve analytically (due in particular to the 4th power of the temperature, even if some linearization is possible) and require some numerical values (such as the temperature of the high atmosphere and its emissivity, and the Newtonian coefficient of convection) that are unknown at this stage of our work. Future studies should be aimed at obtaining a better estimation of the longwave and convective terms and discussing to what extent their sum may be considered as constant on a daily and annual basis. Another improvement would be to perform temperature measurements with a shorter depth spacing for a better calculation of the heat flux with difference quotient. An alternative approach would be to deduce the temperature derivative with depth from a second‐ order fit of the temperature–depth curve, rather than from a linear approximation of it.

Influence of cloud cover on the boundary condition Figures 4 and 5 show that days with overcast skies are characterized by lower temperatures and smaller daily temperature amplitudes. Calculation of the heat flux boundary condition for these days is impossible without further information about additional factors such as changes in the atmospheric transmissivity. In this study, we limited ourselves to a semi‐ quantitative approach aimed at determining how strongly the local meteorological conditions (characterized by the SOI) influence the heat flux reaching the rock surface. Figure 14 presents the ratio R between the total incoming solar energy measured by the pyranometer and the direct solar energy estimated by means of Equation 8 for a flat unobstructed surface, using TL = 2.75 for all of the days of period C. It shows that the cloud cover can decrease the value of R by a

Figure 14. Ratio R between the total incoming solar energy measured by the pyranometer and the direct solar energy estimated by means of the exponential attenuation model with TL = 2.75 for a flat unobstructed surface. Copyright © 2011 John Wiley & Sons, Ltd.

factor of up to 2.5. A threshold seems to exist at SOI ≈ 3, below which the influence of meteorological conditions is quite limited. The next step in this work will be to investigate whether the subsurface temperature field inside the rock can be reliably predicted on days with overcast skies. The results of these investigations are of great practical importance because rapid changes in local meteorological conditions (e.g. clouds obscuring the sun) have been shown to be responsible for high rates of temperature fluctuations within rock, which can significantly contribute to mechanical weathering (Jenkins and Smith, 1990; Hall, 1997, 1999; Hall and André, 2001; McKay et al., 2009).

Conclusion Hourly measurements of solar radiation and subsurface temperatures in an inclined rock slab were used to develop a theoretical model of the heat exchanges through the rock surface, especially for cloudless days. The heat flux penetrating the rock mass could be reliably reproduced by a simple insolation model during the day and by a roughly constant rate of heat loss during the night. Once the key parameters of this model (in situ diffusivity, albedo, atmospheric attenuation, constant heat flux during the night) have been calibrated, it can be used as a boundary condition to explore the expected temperature variations inside the rock out of the monitoring periods and at other sites where no measurements are available. The case of periods with overcast skies was also briefly considered, though only from a semi‐quantitative perspective. Given that cloudiness cannot be forecast precisely, the model is not able, at its present stage, to predict the temperatures during overcast periods. It may however help to predict the temporal and spatial variability of subsurface temperature amplitude by considering, for example, extreme weather conditions. This study offers a quantitative contribution to the characterization of temperatures at shallow depth and their evolution in time, which is profoundly important for the quantification of mechanical weathering. The analysis developed could be a useful addition to landscape evolution models as well as provide for interesting studies of rock weathering processes and associated hazards under past and future climate change. However, the relevant characteristics of the thermal state will depend upon the weathering process of concern: if the dominant mechanism is frost wedging and the subsequent thawing of frozen rock, or of cracks that are holding shattered rock together, maximum weathering will occur at sites that experience the greatest number of oscillations across 0°C (Hales and Roering, 2007). The penetration depth of the freeze–thaw front in the bedrock then controls the size of the detachable blocks (Matsuoka, 1994; Matsuoka and Sakai, 1999). If the segregation ice growth mechanism dominates, temperature oscillations about 0°C are not required. Most intense weathering will take place at sites that spend the greatest amount of time within the frost‐cracking window (temperature range of approximately −3 to −8°C), provided liquid water is available at the same time (Walder and Hallet, 1985; Hallet et al., 1991; Anderson, 1998; Hales and Roering, 2005, 2007). Finally, if thermal shock is considered, McKay et al. (2009) suggested that the percentage time spent above a given threshold value of dT/dt multiplied by the damping depth of a given frequency may be a measure of weathering efficacy for thermal oscillations of that frequency. Earth Surf. Process. Landforms, Vol. 36, 1577–1589 (2011)

NEAR‐SURFACE TEMPERATURES AND HEAT BALANCE OF BARE OUTCROPS Acknowledgements —This study was performed within the framework of the research project ‘STABROCK’, which seeks to assess the influence of climate change on rock slope stability and is sponsored by the French National Research Agency (ANR). The authors are indebted to those of their students at the Ecole des Mines de Nancy, France, who contributed to this study.

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