Abstract

The surface radiation budget of a groomed ski run is important to ski racing. Variables such as snow-surface temperature and liquid water content depend upon the surface radiation budget and are crucial to preparing fast skis. This case study focuses on downwelling longwave radiation, measurements of which were made at a point on a ski run on Whistler Mountain, British Columbia, Canada, throughout a 5-day clear-sky intensive observation period. Tall trees often dominate the horizon of a point on a ski run, and so contributions to total downwelling longwave radiation from trees and sky were treated separately. The “LWRAD” longwave radiative flux model estimated the total downwelling longwave radiation by first calculating thermal contributions from the trees, incorporating regressions for tree temperature that use routine meteorological measurements. Contributions from each azimuth direction were determined with horizon-elevation angles from a theodolite survey. Thermal emissions were weighted accordingly and summed. Sky contributions were estimated using the “libRadtran” radiative transfer model with input of local atmospheric profiles of temperature and humidity and were added to tree emissions. Two clear-sky emissivity parameterizations using screen-height measurements were tested for comparison. LWRAD total downwelling longwave radiation varies between 235 and 265 W m−2 and compares well to measurements, with correlation coefficient squared (r2) of 0.96. These results can be used to improve estimates of downwelling longwave radiation for a groomed ski run.

1. Introduction

The surface radiation budget in snow-covered mountainous terrain is important in hydrology (Fierz et al. 2003), snowmelt modeling (Plüss and Ohmura 1997; Sicart et al. 2006), and glacier energy balance investigations (Brock et al. 2010). Avalanche forecasting and safety studies (McClung 2002a,b; McClung and Schaerer 2006) show the significance of the radiation budget for a natural alpine snowpack. Ski racing takes place on manually prepared ski pistes. It has been shown that snow surface temperature and snow liquid water content can affect ski–snow friction (Colbeck 1988, 1992) and hence the outcome of ski races (Fauve et al. 2005).

The snow-surface temperature of a ski run is affected by the four components of the radiation budget: upwelling and downwelling shortwave and longwave radiation. Colbeck (1994) observed that a 4°C temperature variation is possible between shady and sunny conditions, showing the importance of downwelling shortwave radiation. The upwelling shortwave radiation is dependent upon the snow albedo, defined as the ratio of reflected radiation to total incoming radiation integrated over the shortwave spectrum (Bakermans and Jamieson 2006). Snow albedo varies with snow type and wetness (Gray and Male 1981; Hicks and Martin 1972; Warren 1982), but in general its estimation depends upon empirical formulas. The upwelling longwave radiation is directly proportional to the fourth power of the absolute snow-surface temperature, by the Stefan–Boltzmann law.

The three components of net radiation just described are important but relatively simple to calculate in comparison with downwelling longwave radiation, especially under clear skies. The latter is dependent upon all sources of infrared radiation in the upper hemisphere above a plane parallel to the surface of interest. We present a case study of the downwelling longwave radiation for a groomed ski run under clear skies.

a. Radiative transfer in a plane-parallel atmosphere

According to Petty (2006), Schwarzchild's equation is the most fundamental description of radiative transfer:

 
formula

where dI is the change in radiant intensity I over pathlength ds and B is the blackbody radiance by air molecules along the path (I and B are both dependent on wavelength). Intensity I will increase or decrease depending upon whether B or I is greater at a particular point along the path. Parameter βa is an absorption coefficient that depends on the wavelength of the radiation and on the physical medium. Since scattering is negligible for most of the infrared (IR) band (Petty 2006), we need only consider the absorption coefficient βa.

Integrating Eq. (1) down through the whole atmosphere to the surface gives the downward-emitted radiation in terms of atmospheric transmittance (Petty 2006):

 
formula

where I(0) is the downward radiation as seen by an upward-looking sensor at the surface z = 0, I(∞) is the intensity of radiation that enters the top of the atmosphere (i.e., from the sun), z = ∞ represents an arbitrary point beyond the top of the atmosphere, tr* = exp(−τ*/μ) is the transmittance from the surface to the top of the atmosphere (τ* is the total atmospheric optical depth, which is a function of the absorption coefficient βa, and μ ≡ cosθ is the weighting function for radiant intensities with angle θ relative to the zenith; dz = μds), B(z) is the blackbody radiance for an atmospheric temperature profile, and the weighting function W(z) is given by −d[tr(0, z)]/dz, since the transmittance between the surface and altitude z decreases with increasing height while absorption increases.

Equation (2) can be rewritten as

 
formula

where is the transmission-weighted Planck function for the entire atmosphere. Equation (3) from Petty (2006) states that the total radiant intensity received by an upward-looking sensor viewing downwelling radiation in an isothermal atmosphere is the sum of 1) the transmitted intensity from any source above the top of the atmosphere (e.g., the sun) plus 2) the product of Planck's function and the emissivity of the entire atmosphere. Integrated over all wavelengths and directions, the second term in Eq. (3) becomes the Stefan–Boltzmann law. In reality, the atmosphere is never isothermal and modeling the radiative transfer is complex, and therefore for this study an existing numerical radiative transfer model is used (see section 4a and  appendix A).

b. Longwave radiation from trees

Groomed ski runs often have a reduced sky-view factor as a result of tall evergreen trees on either side (Fig. 1). Howard and Stull (2013, hereinafter HS13) showed that longwave radiation from the trees can enhance the total downwelling longwave radiation at the snow surface of a groomed ski run and that it should not be neglected. For greater accuracy, the tree radiation must be divided into separate thermal emissions from needles and trunks (HS13; Pomeroy et al. 2009).

Fig. 1.

Schematic showing ski piste with reduced sky view due to trees. Terms IRsky, IRneedles, and IRtrunks refer to infrared radiation emitted by sky, tree needles, and tree trunks, respectively, all of which contribute to the total downwelling longwave flux at the snow surface. “Path effects” refer to the absorption and emission effects of the intervening air between the trees and the snow surface on the amount of longwave radiation received relative to that emitted. The approximate width of the groomed ski run in this case study is 50 m.

Fig. 1.

Schematic showing ski piste with reduced sky view due to trees. Terms IRsky, IRneedles, and IRtrunks refer to infrared radiation emitted by sky, tree needles, and tree trunks, respectively, all of which contribute to the total downwelling longwave flux at the snow surface. “Path effects” refer to the absorption and emission effects of the intervening air between the trees and the snow surface on the amount of longwave radiation received relative to that emitted. The approximate width of the groomed ski run in this case study is 50 m.

The air between the trees and the snow surface can absorb and emit longwave radiation along its transmission path, affecting the amount that reaches the snow surface from the trees. Equation (3) describes this effect where the “source” of radiation is the trees. HS13 showed that the assumption of constant temperature is a reasonable approximation for the short pathlengths between the trees and the center of a groomed ski run (~20 m). They found that, although the effect of the air is small (see section 4d), it is desirable to include a correction for it.

2. Method

Total downwelling longwave radiation is estimated for a site on Whistler Mountain (known as RC Whistler; 50°5′16″N, 122°57′41″W) in the southwest coastal mountain range of British Columbia, Canada. RC Whistler is in the subalpine of a recreational ski area, situated on a ski piste as seen from downhill in Fig. 2. Section 3 has a detailed site description. Since we are concerned with integrated contributions from the entire upper hemisphere above the snow surface, we separately model longwave radiative fluxes for sky and trees and then add them together, weighted by their view factors.

Fig. 2.

Photograph showing snow-covered ski piste with tall evergreen trees that reduce the sky view. (The photograph was taken by the authors at Whistler Mountain in 2008.)

Fig. 2.

Photograph showing snow-covered ski piste with tall evergreen trees that reduce the sky view. (The photograph was taken by the authors at Whistler Mountain in 2008.)

For downwelling longwave radiative flux from the sky, as discussed in section 1a, the Stefan–Boltzmann equation represents the downwelling longwave flux :

 
formula

where Vs is the sky-view factor, ɛs is the effective sky emissivity, σ = 5.67 × 10−8 W m−2 K−4 is the Stefan–Boltzmann constant, and Ts is the effective absolute brightness temperature of the sky. Since T varies with height in the atmosphere, the sky downwelling longwave radiation is estimated with an existing radiative transfer model code called libRadtran (described in  appendix A) that uses temperature and humidity profiles from radiosonde ascents (see section 3) as input and solves equations similar to Eq. (2).

For downwelling longwave radiative flux from the trees (see section 4c), Eq. (3) can be rewritten as

 
formula

where Vt is the tree-view factor (Vs + Vt = 1), ɛt is the tree emissivity, Tt is the effective brightness temperature of the trees, trF is the flux transmittance between the trees and the snow surface, ɛa is the flux emissivity of the air between the trees and snow surface (equal to 1 − trF), and Ta is the absolute air temperature. Flux is estimated by using Eq. (5), and regression equations for needle and trunk temperatures from HS13, in which routine meteorological measurements from RC Whistler were used, give Tt.

Sky emissivity is discussed in section 4b. Tree emissivity varies with species (Leuzinger et al. 2010) mostly as a result of the differing components of the tree such as leaves, branches, and trunk. It is frequently taken as unity (Leuzinger et al. 2010; Jarvis et al. 1976; Spronken-Smith and Oke 1998) since data are often unavailable for a particular species. Dirmhirn (1964) uses a value of ɛt = 0.97 for conifers. Rutter (1968) claims that this small difference from unity can lead to substantial differences in the emission of thermal radiation. A simple calculation shows that, for tree temperatures ranging from −10° to 10°C, the difference in longwave radiation emitted by applying an emissivity of 0.97 versus using unity ranges from 8.2 to 10.9 W m−2. These values could be important to the radiation balance. We will use ɛt = 0.97 for tree emissivity throughout this study, since a perfect blackbody is rare in nature (Petty 2006).

The sky-view factor can be calculated from Müller and Scherer (2005):

 
formula

where is the horizon-elevation angle computed or measured for N discrete azimuth angles φi, i = 1, N. If one looks in a particular azimuth direction, the horizon-elevation angle is the angle above horizontal at which the top of any local or nonlocal object (tree, mountain, etc.) is observed, above which is only sky. Larger N gives a more accurate sky-view factor. A theodolite survey (Howard and Stull 2011) at RC Whistler with N = 72 gave a sky-view factor of Vs = 0.65.

Adding together the results from the sky and tree components of the downwelling longwave radiative flux gives an estimate of the total downwelling longwave radiation at the surface of a groomed ski run. According to Plüss and Ohmura (1997), the error from calculating the downwelling longwave radiation in this way is similar to the typical error of radiation instruments, assuming isotropic radiance from the sky and terrain. Estimates at RC Whistler are validated by data described in section 3.

3. Site description and data

a. Site description

Data have been collected at RC Whistler, an active piste during winter with skiers and snowboarders during day and Snowcat-type grooming machines at night. The elevation at this point is 1335 m MSL. This measurement station was in place during the 2010 Vancouver Winter Olympic Games (hereinafter referred to as the Olympics), situated about one-third of the way down the men's downhill skiing racecourse. A summary of instruments and their locations is given in  appendix B. Figure 3 shows the location of RC Whistler and two locations at which Environment Canada collected ceilometer and radiosonde/manual observations, respectively [TFT (50°05′31″N, 122°58′55″W; elevation 782 m MSL) and VOC (50°07′44″N, 122°57′17″W; elevation 658 m MSL)].

Fig. 3.

Map showing the locations of RC Whistler (RCW), the Environment Canada ceilometer site (TFT), and the Environment Canada manual observation and radiosonde release site (VOC). Also shown is Whistler Village, nearby lakes, and Whistler Mountain's peak. Elevation contours are plotted every 200 m MSL.

Fig. 3.

Map showing the locations of RC Whistler (RCW), the Environment Canada ceilometer site (TFT), and the Environment Canada manual observation and radiosonde release site (VOC). Also shown is Whistler Village, nearby lakes, and Whistler Mountain's peak. Elevation contours are plotted every 200 m MSL.

Instruments at RC Whistler were suspended over the racecourse (for skiers' safety and to directly monitor the snow surface beneath) on a platform 9 m above bare ground level, as shown in Fig. 4. All data from RC Whistler were sampled at a frequency of 2 s and were averaged and recorded by a CR3000 datalogger (Campbell Scientific, Inc.) every 10 s. An intensive observation period (IOP) during the Olympics, 18–22 February 2010, included measuring of upwelling and downwelling shortwave and longwave radiation by a CNR1 net radiometer (Kipp & Zonen, Inc.) located on the suspended instrument platform. Measurements of downwelling longwave radiation from the CNR1 lie within the broad band of 4–50 μm and are used to verify total downwelling longwave estimations.

Fig. 4.

Photograph at the RC Whistler site showing instrument platform (circled) and local trees that affect the radiation budget of a point on the snow surface below the instrument platform. The photographer is facing south-southeast looking across the piste. (The photograph was taken by the authors at Whistler Mountain in 2009.)

Fig. 4.

Photograph at the RC Whistler site showing instrument platform (circled) and local trees that affect the radiation budget of a point on the snow surface below the instrument platform. The photographer is facing south-southeast looking across the piste. (The photograph was taken by the authors at Whistler Mountain in 2009.)

Downwelling shortwave radiation measurements from the CNR1 indicated mainly clear skies during the IOP. A nearby CT25K laser ceilometer (Vaisala, Inc.) located at TFT (see Fig. 3) confirmed this condition. Hourly manual data (EnvironmentCanada 2012) from VOC (see also Fig. 3) reported “clear” or “mainly clear” for nearly all of the IOP, with only one report of “mostly cloudy” at 0800 Pacific standard time (PST: local time) on 19 February 2010. Manual observations were not possible at RC Whistler during the IOP because of Olympics security.

A Campbell Scientific SR50 sonic ranger (accuracy of ±3.2 cm) located on the suspended instrument platform measured the distance to the snow surface. Radiosondes (Vaisala RS92-SGP) were released 4 times daily by Environment Canada from VOC during the Olympics. The daytime soundings (1715 and 2315 UTC) during the RC Whistler IOP were used as input to the libRadtran single-column radiative transfer model to determine the sky downwelling longwave flux at the snow surface. Since the application here is alpine ski racing, nighttime is ignored. The radiosondes were launched from the valley floor at an elevation of 658 m MSL, whereas RC Whistler is on the slope of Whistler Mountain at 1335 m MSL; therefore, the temperature and humidity profiles were adjusted, as described next.

b. Constructing atmospheric profiles for RC Whistler

New profiles have been constructed as if the radiosondes were released from RC Whistler. During the IOP, snow-surface temperature was measured at RC Whistler by an infrared thermometer (Apogee Instruments, Inc., IRR-P). Two Campbell Scientific HC-S3-XT temperature and relative humidity probes contained within R. M. Young Co. 41003-X radiation shields measured temperature and relative humidity at approximately 2- and 10-m heights above ground level. Data from these instruments were smoothed using a running-average window of 30 min, which is the approximate average time that it took for the VOC radiosondes to reach the top of the troposphere (~10 km). These three measurement heights form the first three points of the new RC Whistler profiles from the ground upward. Surface relative humidity was assumed to be saturated with respect to water, because the sunshine during the IOP caused snowmelt.

Figure 5 shows an example of how the RC Whistler atmospheric soundings were produced from VOC soundings and from RC Whistler surface and near-surface measurements, for 18 February 2012. Potential temperature and specific humidity are shown for the morning (1715 UTC) and afternoon (2315 UTC). For morning profiles, there is a residual mixed layer above the surface/near-surface measured values up to or just above tree canopy height (~25 m AGL). A neutral or weakly stable residual layer exists above this layer (Erven 2012), and the RC Whistler profiles merge well with the VOC profiles for both temperature and moisture.

Fig. 5.

Soundings constructed for RC Whistler (dashed lines merging into solid lines above) on the basis of VOC radiosonde profiles (solid lines), RC Whistler surface and near-surface measurements (black dots), and the literature, for 18 Feb 2010 for (top) the potential temperature and (bottom) the specific humidity for (left) the morning sounding at 0915 PST and (right) the afternoon sounding at 1515 PST. Above the height where the dashed line merges with the solid line, the RC Whistler sounding is taken to be the same as the VOC sounding. The vertical coordinate is height above mean sea level.

Fig. 5.

Soundings constructed for RC Whistler (dashed lines merging into solid lines above) on the basis of VOC radiosonde profiles (solid lines), RC Whistler surface and near-surface measurements (black dots), and the literature, for 18 Feb 2010 for (top) the potential temperature and (bottom) the specific humidity for (left) the morning sounding at 0915 PST and (right) the afternoon sounding at 1515 PST. Above the height where the dashed line merges with the solid line, the RC Whistler sounding is taken to be the same as the VOC sounding. The vertical coordinate is height above mean sea level.

By the afternoon, the VOC profiles all had an inversion layer at some height above RC Whistler (~1700 m MSL in Fig. 5) capping the mixed layer. This mixed-layer height was used as guidance for constructing the RC Whistler mixed-layer height. There was substantial insolation on these clear sky days, with the average daily maximum for the IOP at ~520 W m−2. The near-surface air temperatures rose above freezing each day because of warming by sunlight on the dark trees, while the piste snow-surface temperature remained below freezing. These observations suggest a profile with a well-mixed layer above a thin stable layer and a smoother transition with height than in the morning because of increased turbulent mixing. Potential temperature and specific humidity are assumed to be relatively constant with height within the turbulent mixed layer. Above the mixed layer, potential temperature increases with height and specific humidity decreases with height, as measured by the VOC radiosondes.

The boundary layer structure and daily evolution described here are supported by several studies in mountainous terrain. A conceptual model described in de Wekker (2002) shows the evolution of the potential temperature profile throughout the day, with a mixed layer developing throughout the afternoon that is capped with a stable layer. The conceptual model of Whiteman (1982) is verified by tethersonde observations over a snow surface that indicate a “top down” inversion erosion throughout the daytime. Even by late afternoon the inversion was not completely destroyed. These observations were for the center of a valley, like the VOC profiles.

A numerical study in an idealized valley (Colette 2003) models the boundary layer throughout the day for a location at midmountain elevation on a west-facing slope, a location similar to RC Whistler. The structural evolution is very similar to that of Whiteman (1982) and the RC Whistler profiles, with large convective cells forming that warm the valley atmosphere considerably from sunrise into late afternoon. Colette (2003) also modeled valley-floor profiles and found the pattern of evolution to be similar to that of the mountainside profiles but with greater range in potential temperature. This result is the justification for using the same mixed-layer heights for the RC Whistler profiles as were observed in the VOC profiles.

Radiosonde and tethersonde observations [Kalthoff et al. (1998) and Kossmann et al. (1998), respectively] from two midmountain sites during the month of September agree with the proposed structure of the RC Whistler profiles, aside from the surface layer where they observe considerable warming throughout the day. Aside from the study of Whiteman (1982), who made observations over snow, the main difference between the cited literature and this case study is the surface. The surface at RC Whistler is snow covered, and observations indicate a persistent stable surface layer throughout the day, whereas the literature generally shows the development of a superadiabatic surface layer for non-snow-covered surfaces.

Anderson and Neff (2008) observed the boundary layer for a snow surface over flat terrain in Antarctica with radiosondes and mast-mounted thermometers. Measurements show a stable surface layer with a shallow mixed layer above. The stable boundary layer is almost ubiquitous in polar regions (Anderson and Neff 2008) because of large longwave cooling from the snow surface, as well as high snow albedo reflecting a large amount of what little insolation is received. The RC Whistler site is in subalpine terrain surrounded by many dark-colored trees that can be heated by solar radiation to temperatures greater than freezing, allowing development of a mixed layer. The combination of a cold snow-covered surface surrounded by warm trees on a mountainside leads to the construction of RC Whistler profiles as described here.

Constructing the atmospheric profiles in this way is subjective, and therefore the sensitivity of the downwelling longwave radiation at the surface to the atmospheric profile near the surface (up to ~2 km MSL above the measured stable layer) was tested. Hock (2005) states that the largest portion of the longwave radiation reaching the surface originates from the lowest layers of the atmosphere. Profiles were varied as follows, above which the original VOC sounding data were used: 1) absolutely unstable, in which environmental lapse rate (ELR) = 11 K km−1; 2) neutral, in which ELR = the dry-adiabatic lapse rate of 9.8 K km−1; and 3) absolutely stable, in which ELR = 5 K km−1.

These profiles were input to libRadtran to obtain estimates of the downwelling longwave radiative flux at the surface for each stability criterion. Figure 6 shows the downwelling longwave radiative flux from libRadtran for the morning (1715 UTC) and afternoon (2315 UTC) profiles on each of the five RC Whistler IOP days for an absolutely stable, neutral, and absolutely unstable boundary layer. Also plotted is the downwelling longwave radiation from libRadtran for the new RC Whistler profiles shown in Fig. 5.

Fig. 6.

Downwelling longwave radiation output by libRadtran at the surface for RC Whistler for absolutely stable (open circles), neutral (times signs), and absolutely unstable (triangles) boundary layers. Also shown is the downwelling longwave radiation from libRadtran (filled dots) for the new RC Whistler profiles shown in Fig. 5. The abscissa shows morning (0915 PST) and afternoon (1515 PST) radiosonde launch times for five consecutive days, 18–22 Feb 2010.

Fig. 6.

Downwelling longwave radiation output by libRadtran at the surface for RC Whistler for absolutely stable (open circles), neutral (times signs), and absolutely unstable (triangles) boundary layers. Also shown is the downwelling longwave radiation from libRadtran (filled dots) for the new RC Whistler profiles shown in Fig. 5. The abscissa shows morning (0915 PST) and afternoon (1515 PST) radiosonde launch times for five consecutive days, 18–22 Feb 2010.

As expected, the unstable, convective boundary layer causes the least downwelling longwave at the surface since it is colder above the surface. The stable boundary layer causes the most downwelling longwave radiation. The mean absolute difference between these two extremes over the five case-study days is 6.5 W m−2. The maximum absolute difference is 8.9 W m−2, giving an estimate of the potential error in using the new RC Whistler profiles. The output from libRadtran lies mostly within the bounds of absolutely stable and absolutely unstable. Of the three occasions on which RC Whistler output lies outside the extremes, the maximum absolute difference is 1.3 W m−2 for the afternoon sounding on 20 February 2010.

4. Modeling total downwelling longwave radiation at the snow surface

The “LWRAD” model has been written (using Matlab, v7.8, proprietary software) to estimate the total downwelling longwave radiative flux as described in section 1. The output from libRadtran is input to LWRAD, which then determines the downwelling longwave radiation contribution from the trees and adds them together. The model estimates the total downwelling longwave radiation at user-specified height(s) vertically above the surface on the basis of calculated sky-view and tree-view factors. For this case study, heights ranged from 0 to 30 m in 1-cm increments. This range allows the observations at instrument platform level to be corrected to the snow surface. This height difference is determined by input of “distance to snow surface” measurements from the sonic ranger. There is also a correction for longwave transmittance and emissivity of the air along the vertical “height correction” path, the method of which is described in section 4d.

a. Modeling downwelling longwave radiation from the sky

The radiative transfer model LibRadtran (described in  appendix A) is used to compute the sky downwelling longwave radiation for this case study. A spectral range of 4.9–70 μm is chosen, the closest available option to the CNR1 radiometer spectral range of 4–50 μm, for comparison.

The standard midlatitude winter background atmosphere was truncated at 1335 m MSL, the elevation of RC Whistler, so that the concentration of constituents is representative. Each newly constructed RC Whistler profile (based upon VOC profiles as described in section 3) replaced the temperature and humidity profile in the background atmosphere for a realistic estimate of the downwelling longwave radiative flux at that time. Temperature and humidity above the radiosonde data are taken from the background atmosphere (RC Whistler profiles have a mean maximum height of ~23 km MSL whereas the background profile has a maximum height of 120 km MSL).

b. Comparison with sky downwelling longwave parameterizations in the literature

Hock (2005) states that longwave radiation incident at the earth's surface is mostly emitted by water vapor, carbon dioxide, and ozone. Variations in downwelling longwave radiation are mostly due to variations in cloudiness and in the amount and temperature of water vapor because carbon dioxide and ozone are relatively constant in comparison (Hock 2005). According to Kondratyev (1969), longwave irradiance correlates well with air temperature and vapor pressure measurements at screen level, 2 m above the surface.

Sedlar and Hock (2009) tested seven clear-sky downwelling longwave radiation parameterizations at Storglaciären, Sweden. Clear-sky emissivity is approximated using screen-level vapor pressure, absolute temperature, or both. They found that the three parameterizations that include atmospheric moisture performed comparably to each other and better than the four without moisture. This result supports the assertion that water vapor is the most effective absorber in the longwave part of the electromagnetic spectrum (Petty 2006) that is most relevant to ski pistes.

Two emissivity parameterizations that use screen-height measurements rather than a full sounding are compared with the sky downwelling longwave radiation from libRadtran for RC Whistler. The first is that of Brutsaert (1975), which can be written as

 
formula

where a1 = 1.24 and b1 = 7 are empirically calculated, e is vapor pressure, and T is absolute temperature. The second parameterization is that of Ohmura (1982), which can be written as

 
formula

where a2 = 8.733 × 10−3 and b2 = 0.788 are empirically calculated. Equations (7) and (8) both gave clear-sky emissivities that are close to 0.7, with mean values of ɛs,Brutsaert = 0.67 and ɛs,Ohmura = 0.73, both with standard deviations of 0.01. A clear-sky emissivity of 0.7 is commonly cited in the literature (Konzelmann et al. 1994; Kuhn 1987; Marty and Philipona 2000).

These emissivities were used with screen-level absolute temperature from RC Whistler to estimate the sky downwelling longwave radiation as in Eq. (4), with Vs = 0.65. The downwelling longwave radiation from the trees (section 4c) was added to compare with LWRAD output and measurements.

c. Modeling downwelling longwave radiation from trees

HS13 showed that the longwave radiation emitted by tall evergreen trees at RC Whistler can greatly enhance the downwelling longwave radiative flux at the surface of the ski run under clear skies by 75.6 W m−2 for their case study in February 2012. HS13 derived linear regressions between tree (trunk and needle) temperature, measured with a handheld FLIR Systems, Inc., E40 infrared camera, and measurements of air temperature, relative humidity, and wind speed data from the RC Whistler site. Note that for different weather conditions (e.g., cloud/precipitation/snow-covered trees) alternative regressions or methods should be used to calculate tree temperature.

Using meteorological data from the RC Whistler IOP, the tree (trunk and needle) temperatures and resulting longwave radiation emissions were calculated with the regressions and the Stefan–Boltzmann law. Horizon-elevation angles from a theodolite survey at RC Whistler (Howard and Stull 2011) were input to LWRAD, and the weighted longwave radiation contribution from each azimuth direction was summed to calculate the total amount of longwave radiation from the trees, according to the tree view (=1 − sky view).

The theodolite survey at RC Whistler (Howard and Stull 2011) included notes on whether the terrestrial “object” changing the horizon elevation was close or far, and tree or mountain, as follows:

  1. a close tree is a tree whose individual branches create defined ground shadows,

  2. a far tree is a tree whose individual branches do not create defined ground shadows,

  3. a close mountain is a mountain horizon that is only a few kilometers away (e.g., parts of Whistler Mountain), and

  4. a far mountain is a mountain horizon that is far away (e.g., mountains across the valley from Whistler Mountain).

For close and far mountains and far trees, the absolute air temperature at RC Whistler was used with the Stefan–Boltzmann law to calculate thermal emissions from those azimuth directions. This is a reasonable assumption since these categories contribute such a small amount to the entire hemispherical view. Also, the air along the path of transmission between the far mountains and the snow surface will alter the longwave radiation received at the snow surface.

d. Corrections to modeled downwelling longwave radiation from trees and to measured downwelling longwave radiation

As mentioned in section 1, the air between the trees emitting longwave radiation and the snow surface receiving longwave radiation can itself absorb and emit longwave radiation. This can decrease or increase, respectively, the total amount reaching the snow surface. Recall that water vapor is considered to be the single most important absorber in the IR band (Petty 2006). The change in radiation due to water vapor content in the air can be corrected for by finding the flux emissivity of the air along the path of radiation:

 
formula

where k is the mass extinction coefficient, ρυ is the absolute humidity of the air along the path of radiation, and Δz is the pathlength.

Measurements of air temperature and relative humidity by the 2-m-height HC-S3-XT probe enabled calculation of the absolute humidity at RC Whistler. The pathlength is finite and is taken to be Δz = 20 m, which is approximately the distance from treetop height to the ski piste at RC Whistler. HS13 demonstrated that for cold weather a value of k = 0.3 m2 kg−1 is reasonable over the pathlength Δz = 20 m for this case-study site. Calculation of emissivity in this way allows Eq. (5) to be applied and the longwave radiation emitted by the trees to be corrected for the discussed air effects. An air emissivity mean value of ɛa = 0.03 was found for the RC Whistler IOP, resulting in a small adjustment.

A further correction is needed to adjust the observed longwave radiation from instrument platform height (order of 8 m) to the snow surface. LWRAD determined the difference in longwave radiation between these two heights. “Distance to target” measurements from the SR50 sonic ranger (see section 3) gave the height of the instrument platform above the surface.

A second necessary correction to longwave radiation measurements arises from solar radiation heating the CNR1 pyrgeometer window, causing temperature differences between the window and the instrument body (Philipona et al. 1995). Michel et al. (2008) claim that this error can be corrected by reducing the longwave value by 1.5% of the concurrent downwelling shortwave measurement. This correction was made only to the afternoon measurements since the sun was not shining on the CNR1 at 0915 PST, the morning measurement time. Calculations determining sun versus shade (Howard and Stull 2011) verify this condition, as do digital camera pictures at RC Whistler. All of the above corrections to the observations allow comparison with modeled downwelling longwave radiation at the surface from LWRAD.

5. Results and discussion

a. Model demonstration and validation

Figure 7 shows output from LWRAD for a simplified, idealized case with uniform sky temperature = −20°C, uniform tree temperature = 0°C, air temperature = −5°C, relative humidity = 100%, and emissivities and other parameters as described above. In this idealized case, the measured horizon-elevation angles for RC Whistler (Howard and Stull 2011) are used to calculate sky-view factors for sensor heights from 0 to 30 m above the snow surface, with 0.1-m resolution. The total downwelling longwave radiation is the sum of the sky and tree radiation corrected at each height by the sky-view factor. The single solid straight line indicates what the downwelling longwave radiation from the sky would be if there were no trees and the horizon were to be flat around a full azimuth circle.

Fig. 7.

Output from LWRAD showing how downwelling longwave radiation from different sources varies with height, for an idealized clear-sky case with an isothermal atmosphere, plotted every 1 m above the snow surface (LWRAD height resolution is 0.1 m). Sky only (solid line) is constant with height, sky corrected for sky view (open circles) increases with height, trees corrected for tree view (times signs) decrease with height, and total downwelling (sky plus trees; filled dots) decreases with height. Note the enhancement by trees of the total downwelling longwave radiation at the surface for these conditions.

Fig. 7.

Output from LWRAD showing how downwelling longwave radiation from different sources varies with height, for an idealized clear-sky case with an isothermal atmosphere, plotted every 1 m above the snow surface (LWRAD height resolution is 0.1 m). Sky only (solid line) is constant with height, sky corrected for sky view (open circles) increases with height, trees corrected for tree view (times signs) decrease with height, and total downwelling (sky plus trees; filled dots) decreases with height. Note the enhancement by trees of the total downwelling longwave radiation at the surface for these conditions.

Note the enhancement by trees of total downwelling longwave radiation at the surface, adding nearly 50 W m−2 to the 100% sky-view factor value, for this idealized case. Also, the trees contribute nearly as much longwave radiation at the surface as the sky, with a difference of only 2 W m−2. The height of the tallest tree at RC Whistler, derived from the theodolite survey, is 24.5 m. Above this exact height, Fig. 7 shows that there is no more contribution of downwelling longwave radiation from the trees, providing some model validation.

b. Comparison of LWRAD with measurements

The modeled and measured total downwelling longwave radiation are compared in Fig. 8 for each radiosonde release time. The original measurements at instrument platform height are plotted, as are the corrected-to-surface measurements. The height difference between the snow surface and the instrument platform as measured by the SR50 sonic ranger varied from 7.9 to 8.0 m. A large improvement in accuracy is seen as a result of the LWRAD height correction. The sky downwelling longwave radiation modeled by libRadtran and the tree longwave radiation modeled by LWRAD are also shown, and “total modeled” is the sum of these two components.

Fig. 8.

Comparison between total LWRAD modeled (times signs), platform-measured (filled dots), and corrected-measured (open triangles) downwelling longwave radiation during the RC Whistler IOP, 18–22 Feb 2010. Also shown are individual contributions from sky (open circles) and trees (plus signs). Times are when radiosonde releases were made during the day, allowing estimation of sky longwave radiation with resulting temperature and humidity profiles. There were two radiosonde releases per day, 0915 and 1515 PST (1715 and 2315 UTC), for each of the five consecutive days.

Fig. 8.

Comparison between total LWRAD modeled (times signs), platform-measured (filled dots), and corrected-measured (open triangles) downwelling longwave radiation during the RC Whistler IOP, 18–22 Feb 2010. Also shown are individual contributions from sky (open circles) and trees (plus signs). Times are when radiosonde releases were made during the day, allowing estimation of sky longwave radiation with resulting temperature and humidity profiles. There were two radiosonde releases per day, 0915 and 1515 PST (1715 and 2315 UTC), for each of the five consecutive days.

LWRAD (times signs in Fig. 8) gives a good estimation of the total downwelling longwave radiation measured at the surface (open triangles in Fig. 8), where the mean absolute error (MAE) = 1.7 W m−2 and root-mean-square error (RMSE) = 2.1 W m−2. Figure 9 shows the linear correlation between modeled total downwelling longwave radiation at the surface from LWRAD versus corrected measured downwelling longwave radiation, with correlation coefficient squared r2 = 0.96. The linear correlation is reasonable for downwelling longwave radiation between 235 and 265 W m−2.

Fig. 9.

Modeled total downwelling longwave radiation at the surface from LWRAD vs measured downwelling longwave radiation corrected to the surface for the RC Whistler IOP, 18–22 Feb 2010, at radiosonde release times (0915 and 1515 PST) each day.

Fig. 9.

Modeled total downwelling longwave radiation at the surface from LWRAD vs measured downwelling longwave radiation corrected to the surface for the RC Whistler IOP, 18–22 Feb 2010, at radiosonde release times (0915 and 1515 PST) each day.

c. Comparison of LWRAD and measurements with published parameterizations

Parameterizations from Brutsaert (1975) and Ohmura (1982) with added tree radiation from LWRAD are compared with corrected measurements and LWRAD total downwelling longwave radiation output (Fig. 10). Parameterization longwave radiation output was adjusted for sky-view factor before adding the tree contributions from LWRAD, as with the output from libRadtran in LWRAD. Table 1 shows morning, afternoon, and daily MAE and RMSE for total downwelling longwave radiation from LWRAD and using the Brutsaert (1975) and Ohmura (1982) parameterizations. The parameterizations are based only on surface shelter-height meteorological conditions.

Fig. 10.

Comparison between modeled total downwelling longwave radiation using the parameterizations from Brutsaert (1975) (asterisks) and Ohmura (1982) (downward-pointing open triangles) for sky longwave radiation with added longwave contributions from the trees, and corrected-measured downwelling longwave radiation (upward-pointing open triangles), for the RC Whistler IOP, 18–22 Feb 2010, at radiosonde release times (0915 and 1515 PST) each day. Modeled total downwelling longwave radiation from LWRAD (times signs) is also shown.

Fig. 10.

Comparison between modeled total downwelling longwave radiation using the parameterizations from Brutsaert (1975) (asterisks) and Ohmura (1982) (downward-pointing open triangles) for sky longwave radiation with added longwave contributions from the trees, and corrected-measured downwelling longwave radiation (upward-pointing open triangles), for the RC Whistler IOP, 18–22 Feb 2010, at radiosonde release times (0915 and 1515 PST) each day. Modeled total downwelling longwave radiation from LWRAD (times signs) is also shown.

Table 1.

Morning (AM), afternoon (PM), and overall MAE and RMSE for total downwelling longwave radiation from LWRAD and using the Brutsaert (1975) and Ohmura (1982) parameterizations calculated for the RC Whistler IOP. Model output is compared with measurements from the CNR1 after corrections, as discussed in section 4d.

Morning (AM), afternoon (PM), and overall MAE and RMSE for total downwelling longwave radiation from LWRAD and using the Brutsaert (1975) and Ohmura (1982) parameterizations calculated for the RC Whistler IOP. Model output is compared with measurements from the CNR1 after corrections, as discussed in section 4d.
Morning (AM), afternoon (PM), and overall MAE and RMSE for total downwelling longwave radiation from LWRAD and using the Brutsaert (1975) and Ohmura (1982) parameterizations calculated for the RC Whistler IOP. Model output is compared with measurements from the CNR1 after corrections, as discussed in section 4d.

LWRAD, which is based on the whole sounding, performs best for morning, afternoon, and overall. The Brutsaert (1975) method does well in the mornings, but the error increases in the afternoon, greatly affecting the overall performance and tripling the MAE and RMSE when compared with LWRAD. This result indicates that there are processes occurring by the afternoon that are not being captured by the screen-height measurements. Ohmura (1982) overestimates in all cases and in general does a poor job. This helps to confirm the importance of including vapor pressure in downwelling longwave radiation parameterizations even under clear skies.

The measurements corrected to the surface include output from LWRAD. Nonetheless, the comparison here is relevant since it is assumed that the parameterizations are for longwave radiation at the surface. Also, when correcting from platform height to snow surface, LWRAD showed that the trees add on average 70.5 W m−2, and the radiation from the sky at the surface (including sky-view factor) is 46.9 W m−2 less than that at platform height, which is probably due to the stable surface layer, with a net gain of 23.7 W m−2; most of the correction is due to the trees.

For this case study there were no measurements of sky downwelling longwave radiation as unaffected by the trees to verify the parameterizations prior to adding the tree contributions. Instead, we compare the differences between the platform-height measurements and the modeled sky downwelling longwave radiation from libRadtran, the Brutsaert (1975) parameterization, and the Ohmura (1982) parameterization. The differences are not expected to be zero, and the standard deviation of the differences for libRadtran is 2.2 W m−2, for the Brutsaert (1975) method it is 8.4 W m−2, and for the Ohmura (1982) method it is 8.9 W m−2. The small variance in the differences for libRadtran indicates that it is likely capturing processes that affect the sky downwelling longwave radiation over the course of the 5 days, unlike the parameterizations. The addition of tree downwelling longwave radiation will not remove the variance in the two emissivity parameterizations, implying that libRadtran does better at modeling the sky downwelling longwave radiation, as expected.

6. Conclusions

a. Summary

The total downwelling longwave radiative flux at the surface of a groomed ski run has been successfully estimated by determining the weighted contributions of thermal emissions from a cloudless sky plus the surrounding trees. A radiative transfer model using vertical profiles of temperature and humidity as input satisfactorily estimated the downwelling longwave radiation from the sky at RC Whistler, whereas parameterizations from the literature using only screen-height measurements overestimated this quantity.

The longwave radiation from the trees appears to offset the effect of the stable surface layer on sky downwelling longwave radiation. The screen-height measurements at RC Whistler used in the sky parameterizations do not incorporate the complex radiative processes happening below and above treetop height sufficiently. In this case, using atmospheric profiles of temperature and water vapor is better than using screen-height measurements in parameterizations, as expected. Also, the time taken to run libRadtran here is trivial (less than 1 s on a quad-core Intel Corporation i5 processor) and the resulting improvement in accuracy is substantial.

For this case study, the trees contributed on average 82.6% as much downwelling radiation as the sky did. This confirms that the contribution of longwave radiation from the trees at RC Whistler must be considered in order to correctly model the total downwelling longwave radiation at the surface.

Measurements at RC Whistler were made from a platform suspended over the center of the ski run, meaning that corrections to the surface are necessary for certain variables (e.g., downwelling longwave radiation). The fact that LWRAD models the total downwelling longwave radiation at multiple heights above the snow surface allows the appropriate correction to be made so that the model can be verified. Once this correction was made, the modeled total downwelling longwave radiation at the surface compared well to observations, with overall MAE = 1.7 W m−2 and RMSE = 2.1 W m−2. Corrections were also made for the effect of the intervening air between the trees and the snow surface. That effect is small.

During the IOP there were no radiosonde measurements available for RC Whistler or other sites on a mountainside over a snow surface having a similar climate. The constructed RC Whistler profiles utilized boundary layer profiles to merge in situ surface and mast data with nearby radiosonde data. The sky downwelling longwave radiation was tested for sensitivity to variations in atmospheric stability up to about 700 m AGL. Between extreme cases of absolute stability and instability, the sky downwelling longwave radiation varied by a maximum of only 8.9 W m−2 (higher for the warmer stable profile).

b. Recommendations

Using empirical parameterizations with only screen-height measurements to calculate the sky downwelling longwave radiation, as suggested by the literature, does not give accurate results for this case study, since the screen-height measurements are likely affected by the tall trees and the sky downwelling longwave is affected by processes above treetop height. Hence, the emissivity models of Brutsaert (1975) and Ohmura (1982) using screen-height measurements are not the best choice in this situation.

Radiosondes should ideally be released from the groomed ski run of interest so that approximations are not necessary, despite agreement across the literature regarding the boundary layer structure in mountainous terrain. If in situ profiles are not available but nearby soundings of temperature and humidity are, the latter can be used to construct radiosonde profiles, as in this case study. This approach is preferable to using screen-height measurements of temperature and humidity in a parameterization like the ones tested for RC Whistler. The Brutsaert (1975) model performed reasonably well, however, and therefore could serve as an alternative if no atmospheric profiles are available.

When modeling the total downwelling longwave radiation for a groomed ski run, it is recommended to estimate the contributions from trees (Howard and Stull 2013) as well as the sky and to do these estimations separately. It is also necessary to correct longwave measurements from the instrument height down to the snow surface. If measurements of temperature and humidity are available, then it is advisable to correct for the effects of the air between the trees and the snow surface.

c. Future work

If measurements of temperature and vapor pressure were taken just above treetop height (as if over a flat surface), then it may be appropriate to use parameterizations from the literature that employ these measurements to calculate the sky downwelling longwave radiation, rather than using a radiative transfer model. Atmospheric profiles of temperature and humidity are not always available and there is a lack of literature on profiles for midmountain snow-covered locations. Empirically determined equations with above-treetop measurements could be better than screen-height measurements in this case. Further work could also include comparison of libRadtran with other radiative transfer models, such as the Moderate Resolution Atmospheric Transmission (MODTRAN) model, for calculating the sky downwelling longwave radiation.

Variables such as snow-surface temperature and liquid water content are highly relevant to ski technicians when preparing race skis. These variables in turn depend upon the surface energy budget. Improved estimations of downwelling longwave radiation as modeled here can be used in surface heat budget calculations for a groomed ski run. Modeling these variables is very useful, in addition to real-time in situ measurements, since this allows forecasting and better preparation ahead of time, for example, when waxing skis the night before a race.

The clear-sky downwelling longwave radiation has been estimated here; however, mountainous coastal terrain frequently has cloud cover. LibRadtran allows user input of clouds at different heights above the surface, so further work should include different cloud-cover scenarios.

Acknowledgments

Funding was provided by Own the Podium 2010, the Vancouver Olympic Committee, the Natural Sciences and Engineering Research Council of Canada, and the Geophysical Disaster Computational Fluid Dynamics Center of The University of British Columbia. Thanks are given to Dr. Thomas Nipen, George Hicks, and Bruce Thomson of the Weather Forecast Research team of The University of British Columbia for their valuable input. We also thank Dominic Lessard, Zoran Nesic, and Dr. Andy Black of the Biometeorology Department of The University of British Columbia for providing expert advice and assistance regarding instrumentation methods and deployment, and we thank Dr. Phil Austin of the Department of Earth and Ocean Sciences of The University of British Columbia for help and support using libRadtran. Daniel Casanova is commended for his extensive theodolite surveys at many points along the Olympics site's pistes. In regard to the use of EC Climate Data Online, the website is official work that is published by the government of Canada, and the reproduction has not been produced in affiliation with or with the endorsement of the government of Canada.

APPENDIX A

LibRadtran Model Description

This appendix briefly describes the libRadtran radiative transfer model, which is used throughout this case study to compute the sky downwelling longwave radiation. LibRadtran is a numerical model that computes downwelling longwave radiative transfer in the earth's atmosphere using an appropriate radiative transfer solver. Discrete ordinate code 2.0 (DISORT2; Stamnes et al. 1988, 2000) solves the radiative transfer equations in a one-dimensional plane-parallel atmosphere and was chosen for this study. The spectral resolution is determined with a correlated k-distribution method, which, according to Hock (2005), provides a compromise between speed and accuracy.

Petty (2006) argues that accuracy is not sacrificed since the method is based upon the idea that the integration with respect to frequency of a complex line spectrum can be replaced with an equivalent integration over a much smoother function. This allows large discretization steps and reduced computational effort. Fluxes and heating rates can be calculated with errors of less than 1% and are done so with three orders of magnitude fewer computational resources than would be required by a line-by-line calculation (Petty 2006). The Fu and Liou (1992) correlated k-distribution method is used since it is fast and covers the terrestrial longwave part of the electromagnetic spectrum.

The standard midlatitude winter profile (Anderson et al. 1986) provided the background atmosphere with properties and constituents of absolute temperature, air density, water vapor, ozone, and nitrogen dioxide. See Mayer and Kylling (2005) for model technical details.

APPENDIX B

Instrumentation and Locations

Table B1 outlines the instrumentation used for this study and their locations.

Table B1.

Instrumentation used in this study, and their locations; CS = Campbell Scientific, RCW = RC Whistler, and TFT and VOC are Environment Canada measurement stations. During the RC Whistler IOP, the suspended platform height varied between 7.9 and 8.0 m above the snow surface. See Fig. 3 for a map.

Instrumentation used in this study, and their locations; CS = Campbell Scientific, RCW = RC Whistler, and TFT and VOC are Environment Canada measurement stations. During the RC Whistler IOP, the suspended platform height varied between 7.9 and 8.0 m above the snow surface. See Fig. 3 for a map.
Instrumentation used in this study, and their locations; CS = Campbell Scientific, RCW = RC Whistler, and TFT and VOC are Environment Canada measurement stations. During the RC Whistler IOP, the suspended platform height varied between 7.9 and 8.0 m above the snow surface. See Fig. 3 for a map.

REFERENCES

REFERENCES
Anderson
,
G.
,
S.
Clough
,
F.
Kneizys
,
J.
Chetwynd
, and
E.
Shettle
,
1986
: AFGL atmospheric constituent profiles (0-120 km). Air Force Geophysics Laboratory Tech. Rep. AFGL-TR-86-0110, 43 pp.
Anderson
,
P. S.
, and
W. D.
Neff
,
2008
:
Boundary layer physics over snow and ice
.
Atmos. Chem. Phys.
,
8
,
3563
3582
.
Bakermans
,
L.
, and
B.
Jamieson
,
2006
: Measuring near-surface snow temperature change over terrain. Proc. 2006 Int. Snow Science Workshop, Telluride, CO, ISSW,
377
386
.
Brock
,
B. W.
,
C.
Mihalcea
,
M. P.
Kirkbride
,
G.
Diolaiuti
,
M. E. J.
Cutler
, and
C.
Smiraglia
,
2010
:
Meteorology and surface energy fluxes in the 2005–2007 ablation seasons at the Miage debris-covered glacier, Mont Blanc Massif, Italian Alps
.
J. Geophys. Res.
,
115
,
D09106
,
doi:10.1029/2009JD013224
.
Brutsaert
,
W.
,
1975
:
On a derivable formula for long-wave radiation from clear skies
.
Water Resour. Res.
,
11
,
742
744
.
Colbeck
,
S. C.
,
1988
:
The kinetic friction of snow
.
J. Glaciol.
,
34
,
78
86
.
Colbeck
,
S. C.
,
1992
: A Review of the Processes that Control Snow Friction. Cold Regions Research and Engineering Laboratory Monogr. 92-2, U.S. Army Corps of Engineers, 40 pp.
Colbeck
,
S. C.
,
1994
:
Bottom temperatures of skating skis on snow
.
Med. Sci. Sports Exercise
,
26
,
258
262
.
Colette
,
A.
,
2003
:
A numerical study of inversion-layer breakup and the effects of topographic shading in idealized valleys
.
J. Appl. Meteor.
,
42
,
1255
1272
.
de Wekker
,
S. F. J.
,
2002
: Structure and morphology of the convective boundary layer in mountainous terrain. Ph.D. thesis, The University of British Columbia, BC, Canada, 138 pp.
Dirmhirn
,
I.
,
1964
: Das Strahlungsfeld im Lebensraum (The Radiation Field in the Environment). Akademische Verlagsgesellschaft, 426 pp.
EnvironmentCanada
, cited
2012
: National climate data and information archive website. [Available online at http://www.climate.weatheroffice.gc.ca/climateData/canada_e.html.]
Erven
,
L.
,
2012
: An observational study of slope air and free air wintertime temperatures in Whistler Valley, British Columbia, Canada. M.S. thesis, Dept. of Atmospheric Science, The University of British Columbia, 112 pp.
Fauve
,
M.
,
D.
Buhl
,
H. U.
Rhyner
,
M.
Schneebeli
, and
W.
Ammann
,
2005
: Influence of snow and weather characteristics on the gliding properties of skis. Science and Skiing III, E. Müller et al., Eds., Meyer and Meyer Sport, 401–410.
Fierz
,
C.
,
P.
Riber
,
E. E.
Adams
,
A. R.
Curran
,
P. M. B.
Föhn
,
M.
Lehning
, and
C.
Plüss
,
2003
:
Evaluation of snow-surface energy balance models in alpine terrain
.
J. Hydrol.
,
282
,
76
94
.
Fu
,
Q.
, and
K. N.
Liou
,
1992
:
On the correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres
.
J. Atmos. Sci.
,
49
,
2139
2156
.
Gray
,
D. M.
, and
D. H.
Male
,
1981
: Handbook of Snow: Principles, Processes, Management and Use. Pergamon Press, 776 pp.
Hicks
,
B. B.
, and
H. C.
Martin
,
1972
:
Atmospheric turbulent fluxes over snow
.
Bound.-Layer Meteor.
,
2
,
496
502
.
Hock
,
R.
,
2005
:
Glacier melt: A review of processes and their modelling
.
Prog. Phys. Geogr.
,
29
,
362
391
.
Howard
,
R.
, and
R.
Stull
,
2011
:
Forecasting sun versus shade in complex terrain for the 2010 Winter Olympic and Paralympic Games
.
Bull. Amer. Meteor. Soc.
,
92
,
1303
1309
.
Howard
,
R.
, and
R.
Stull
,
2013
:
IR radiation from trees to a ski run: A case study
.
J. Appl. Meteor. Climatol.
,
52
,
1525
1539
.
Jarvis
,
P. G.
,
G. B.
James
, and
J. J.
Landsberg
,
1976
: Coniferous forest. Case Studies, J. L. Monteith, Ed., Vegetation and the Atmosphere, Vol. 2, Academic Press, 171–240.
Kalthoff
,
N.
,
H. J.
Binder
,
M.
Kossmann
,
R. V.
Ögtlin
,
U.
Corsmeier
,
F.
Fiedler
, and
H.
Schlager
,
1998
:
Temporal evolution and spatial variation of the boundary layer over complex terrain
.
Atmos. Environ.
,
32
,
1179
1194
.
Kondratyev
,
K. Y.
,
1969
: Radiation in the Atmosphere. Academic Press, 912 pp.
Konzelmann
,
T.
,
R. S. W.
van de Wal
,
W.
Greuell
,
R.
Bintanja
,
E. A. C.
Henneken
, and
A.
Abe-Ouchi
,
1994
:
Parameterization of global and longwave incoming radiation for the Greenland ice sheet
.
Global Planet. Change
,
9
,
143
164
.
Kossmann
,
M.
,
R.
Vögtlin
,
U.
Corsmeier
,
B.
Vogel
,
F.
Fiedler
,
H. J.
Binder
,
N.
Kalthoff
, and
F.
Beyrich
,
1998
:
Aspects of the convective boundary layer structure over complex terrain
.
Atmos. Environ.
,
32
,
1323
1348
.
Kuhn
,
M.
,
1987
:
Micro-meteorological conditions for snowmelt
.
J. Glaciol.
,
33
,
24
26
.
Leuzinger
,
S.
,
R.
Vogt
, and
C.
Körner
,
2010
:
Tree surface temperature in an urban environment
.
Agric. For. Meteor.
,
150
,
56
62
.
Marty
,
C.
, and
R.
Philipona
,
2000
:
The clear-sky index to separate clear-sky from cloudy-sky situations in climate research
.
Geophys. Res. Lett.
,
27
,
2649
2652
.
Mayer
,
B.
, and
A.
Kylling
,
2005
:
The libRadtran software package for radiative transfer calculations—Description and examples of use
.
Atmos. Chem. Phys.
,
5
,
1855
1877
.
McClung
,
D. M.
,
2002a
:
The elements of applied avalanche forecasting, Part I: The human issues
.
Nat. Hazards
,
26
,
111
129
.
McClung
,
D. M.
,
2002b
:
The elements of applied avalanche forecasting, Part II: The physical issues and the rules of applied avalanche forecasting
.
Nat. Hazards
,
26
,
131
146
.
McClung
,
D. M.
, and
P.
Schaerer
,
2006
: The Avalanche Handbook. 3rd ed. The Mountaineers Books, 342 pp.
Michel
,
D.
,
R.
Philipona
,
C.
Ruckstuhl
,
R.
Vogt
, and
L.
Vuilleumier
,
2008
:
Performance and uncertainty of CNR1 net radiometers during a one-year field comparison
.
J. Atmos. Oceanic Technol.
,
25
,
442
451
.
Müller
,
M. D.
, and
D.
Scherer
,
2005
:
A grid- and subgrid-scale radiation parameterization of topographic effects for mesoscale weather forecast models
.
Mon. Wea. Rev.
,
133
,
1431
1442
.
Ohmura
,
A.
,
1982
:
Climate and energy balance on the Arctic tundra
.
Int. J. Climatol.
,
2
,
65
84
.
Petty
,
G. W.
,
2006
: A First Course in Atmospheric Radiation. 2nd ed., Sundog, 459 pp.
Philipona
,
R.
,
C.
Fröhlich
, and
C.
Betz
,
1995
:
Characterization of pyrgeometers and the accuracy of atmospheric long-wave radiation measurements
.
Appl. Opt.
,
34
,
1598
1605
.
Plüss
,
C.
, and
A.
Ohmura
,
1997
:
Longwave radiation on snow-covered mountainous surfaces
.
J. Appl. Meteor.
,
36
,
818
824
.
Pomeroy
,
J. W.
,
D.
Marks
,
T.
Link
,
C.
Ellis
,
J.
Hardy
,
A.
Rowlands
, and
R.
Granger
,
2009
:
The impact of coniferous forest temperature on incoming longwave radiation to melting snow
.
Hydrol. Processes
,
23
,
2513
2525
.
Rutter
,
A. J.
,
1968
: Water consumption by forests. Water Deficits and Plant Growth, T. T. Kozlowski, Ed., Vol. 2, Academic Press, 23–84.
Sedlar
,
J.
, and
R.
Hock
,
2009
:
Testing longwave radiation parameterizations under clear and overcast skies at Storglaciären, Sweden
.
Cryosphere
,
3
,
75
84
.
Sicart
,
J. E.
,
J. W.
Pomeroy
,
R. L. H.
Essery
, and
D.
Bewley
,
2006
:
Incoming longwave radiation to melting snow: Observations, sensitivity and estimation in northern environments
.
Hydrol. Processes
,
20
,
3697
3708
.
Spronken-Smith
,
R.
, and
T. R.
Oke
,
1998
:
The thermal regime of urban parks in two cities with different summer climates
.
Int. J. Remote Sens.
,
19
,
2085
2104
.
Stamnes
,
K.
,
S.-C.
Tsay
,
W.
Wiscombe
, and
K.
Jayaweera
,
1988
:
Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media
.
Appl. Opt.
,
27
,
2502
2509
.
Stamnes
,
K.
,
S.-C.
Tsay
,
W.
Wiscombe
, and
I.
Laszlo
,
2000
: DISORT, a general-purpose Fortran program for discrete-ordinate-method radiative transfer in scattering and emitting layered media: Documentation of methodology. Stevens Institute of Technology Dept. of Physics and Engineering Physics Tech. Rep., 112 pp. [Available online at http://www.met.reading.ac.uk/~qb717363/adient/rfm-disort/DISORT_documentation.pdf.]
Warren
,
S. G.
,
1982
:
Optical properties of snow
.
Rev. Geophys.
,
20
,
67
89
.
Whiteman
,
C. D.
,
1982
:
Breakup of temperature inversions in deep mountain valleys: Part I. Observations
.
J. Appl. Meteor.
,
21
,
270
289
.