Topographic controls on the surface energy balance of a high Arctic valley glacier - Arnold, Hodson and Kohler (2006)

Topographic Controls on Surface Energy Balance

Introduction

• The surface energy balance, along with winter snow accumulation, primarily determines the mass balance of glaciers and ice sheets at middle and high latitudes, especially during summer.
• A positive surface energy balance warms the surface, leading to melting if it reaches 0°C. Models for understanding glacier melt rates vary in complexity:
  • Simple empirical relationships between meteorological variables and melt (e.g., Willis et al., 1993).
  • Degree day approaches (e.g., Laumann and Reeh, 1993; Braithwaite and Zhang, 2000).
  • Zero- or one-dimensional energy balance studies (e.g., Braithwaite and Olesen, 1990; Munro, 1990; Oerlemans, 1993).
• Two-dimensional approaches are needed to accurately capture the spatial and temporal complexities of the surface energy balance.
• Distributed energy balance models commonly find that net shortwave radiation dominates the summer surface energy balance, accounting for up to 99% of the energy [Arendt, 1999], but more typically around 75% [e.g., Greuell and Smeets, 2001; Oerlemans and Klok, 2002].
• Incoming solar radiation is affected by:
  • Glacier surface topography (slope and aspect), which influences the incident angle of direct solar radiation.
  • Surrounding terrain, which controls shading patterns and affects the proportion of visible sky, influencing diffuse radiation.
• These topographic effects are more pronounced in the high Arctic because of high solar zenith angles.

Study and Model

• Distributed energy balance models can assess the impact of topography on solar energy receipt using digital elevation models (DEMs).
• Traditional DEMs are often smoothed due to interpolation of data from maps, ground surveys, or air photography.
• This study uses a high-resolution DEM derived from airborne lidar data, preserving small-scale topographic features and their influence on local variations in slope and aspect.
• Investigates whether the long duration of sunshine compensates for higher solar zenith angles in the surface energy balance and the role of shading by surrounding topography, slope, and aspect on solar radiation.

Study Site and Data Collection

• Midre Love´nbreen is a 6 km2 valley glacier in northwest Spitsbergen, Svalbard, flowing northward from the southern side of Kongsfjord.
• The glacier tongue is north-facing with a bend at 300-350 m asl into an east-facing upper basin.
• Elevation ranges from ~50 m asl at the snout to ~500 m at the headwall.
• Annual mass balance has been measured since 1968.
• The glacier snout has retreated over 1000 m during the 20th century, losing approximately 10,000,000 m3 of ice [Hagen et al., 2003].
• Data requirements:
  • DEM of the catchment.
  • Start-of-summer water equivalent snow depth.
  • Hourly meteorological data.
  • Surface roughness and albedo parameterizations.
• Primary model runs used data from May 8, 2000, to September 5, 2000.
• DEM from airborne lidar data collected in summer 2003 at 0.7-1.5 m resolution, converted to a 20 m spatial resolution DEM by averaging elevations within each grid cell.
• RMS error varies from $±0.05$ m on the upper glacier to $±0.1$ to 0.15 m on the lower glacier, with horizontal position errors >0.025 m.
• Snow depth was measured at 29 points using an avalanche probe and GPS on April 14, 2000.
• Snow depth (Ds) in meters was empirically related to elevation (E) in meters and surface slope (Z0) in degrees:
  $Ds = 0.9436 + 0.00166E - 0.016Z0$ $(1)$
• Snow depth data were converted to water equivalent (Dswe) using a mean snow density of 331 kg m-3.
• Meteorological data from an automatic weather station (GWS) on the glacier at ~125 m elevation, including air temperature, relative humidity, wind speed, global radiation, and net all-wave radiation.
• Data supplemented with synoptic weather station (SWS) data from Ny-A˚ lesund, approximately 5 km from the glacier, including downward longwave radiation and diffuse component of solar radiation for validation.
• 2000 was an unusually cool summer
  • Maximum recorded temperature: $10.34°C$
  • Mean temperature: $1.18°C$
  • Compared to the mean (1997-2003): $2.14°C$
• Albedo was measured using a Kipp and Zonen CM7B albedometer at snow depth measurement locations.

Energy Balance Model

• The model, based on Arnold et al. [1996] and developed by Brock et al. [2000a, 2000b] and Arnold [2005], calculates surface energy balance (G) components:
  • Net shortwave (solar) radiation (Q*).
  • Net longwave (terrestrial) radiation (L*).
  • Sensible turbulent heat (S).
  • Latent turbulent heat (T).
  • Conductive flux from the surface into the glacier.

Solar Radiation

• Shortwave radiation is considered the mot important. The downward instantaneous flux of direct and diffuse radiation in each grid cell ($Q0_i$) is
• For each DEM grid cell, value is modified depending on local circumstances.
• Determine if the cell is shaded by surrounding topography, if so, both direct and diffuse radiation reach the cell.
• Incoming direct component ($Q0<em>{idir}$) is calculated from measured direct radiation at GWS ($Q0</em>{dir}$), from incoming global radiation at the GWS ($Q0$) minus the diffuse component at the SWS ($Q0_{dif}$). Solar zenith angles and azimuth are used:

$Q0<em>{idir} = Q0</em>n sin Z cos Z0 + cos Z sin Z0 cos(A − A0)$ $(2)$

• Where:
  • $Q0_n$ is direct solar radiation received by a surface normal to the solar beam.
  • $Z$ is the angle of the sun above the horizon.
  • $A$ is the solar azimuth.
  • $A0$ is the azimuth of the surface slope.
  • $Q0_{ndir}$ is set to zere for shaded cells.
• Measured diffuse radiation at the SWS ($Q0<em>{dif}$) is asumed to be representative of the whole catchment. For each cell recieving diffuse radiation, the measured value is multiplied by a sky view factor ($f</em>s$), to account for the variation in the proportion of the sky visible from any given grid cell.

$f<em>s = \sum</em>{j=0}^{360} cos^2 q \frac {\Delta j} {360}$ $(4)$
where $q$ is the local horizon angle at a given azimuth, $j$. Experimentation with $\Delta j$ showed that very little change in the calculated values of fs occurred when $\Delta j$ <12°

• The value is supplemented by the reflected radiation from surrounding topography:

$Q0<em>t = a</em>t (1 - f_s)Q0$ $(5)$

Where $a<em>t$ is the albedo of thge surrounding visible areas (including both glaciated and ice free areas). For this study, a value of 0.25 was used. This gives a final valud for the diffuse radiation received by a given cell ($Q0</em>{idif}$):
$Q0<em>{idif} = f</em>s Q0<em>{dif} + Q0</em>t$ $(6)$

• Net solar radiation flux (Q*) is calculated from:

$Q* = (1 - a) (Q0<em>{idir} + Q0</em>{idif})$ $(7)$

*   $a$ is DEM cell albedo.

• Snow depth was the only statistically significant control on albedo of snow covered surfaces ($a_s$), giving the realtionship:

$a<em>s = 0.743 - (a</em>i - 0.372) exp \frac {-D_{swe}}{0.4501}$ $(8)$

• The albedo of the underlying ice surface is modeled with an empirical relationship with elevation

$a_i = 0.4474 - 0.5878 exp(\frac {-E}{65.0057})$ $(9)$

• If a DEM cell contains snow, equation 8 is used, if all the snow has melted, equate 9 is used to calculate albedo.
• An albedo correction factor ($a<em>c$) factor is added to $a</em>s$ or $a_i$ to allow for the nonisotropic reflectance properties of snow and ice surfaces:

$a_c = 0.16 (3 + 4 cos z)^3 - 1$ $(10)$

• where $z$ is the solar zenith angle at the DEM cell.

Longwave Radiation, Ice Temperature, and Heat Flux

• Net longwave radiation is the sum of radiation emitted by the glacier surface and received from the sky and surrounding terrain.
• Surface temperature of the glacier varies, using a two-layer subsurface scheme (based on Klok and Oerlemans [2002]).
• Energy input at the surface warms the surface layer:

$\frac {DT<em>s}{Dt} = k \frac {T</em>2 - T<em>s}{Dz} \frac {\rho c}{d</em>s} + \frac {G}{\rho c d_s}$ $(11)$

*   Where:
    *   $T_s$ is the temperature of the surface layer.
    *   $T_2$ is the temperature of the second layer.
    *   $d_s$ is the depth of the surface layer.
    *   $k$ is the thermal diffusivity.
    *   $c$ is the specific heat capacity.
    *   $\rho$ is the density.
    *   $G$ is the sum of Q*, L* (from the previous time step), S, and T.

• Heat loss from the surface layer forms input to the subsurface layer:

$\frac {DT<em>2}{Dt} = k \frac {T</em>3 - T<em>2}{Dz d</em>2} - \frac {T<em>2 - T</em>s}{Dz d_2}$ $(12)$

*   Where:
    *   $d_2$ is the depth of the second layer.
    *   $T_3$ is the temperature of the main body of the glacier.

• $T_3$ is set to the average annual air temperature from the synoptic weather station at Ny-A˚ lesund, corrected for local surface elevation using a lapse rate of $6.5°C km^{-1}$.
• Outgoing longwave radiation flux ($L_{out}$) is calculated by:

$L<em>{out} = \sigma T^4</em>s$ $(13)$

*   Where $\sigma$ is the Stefan-Boltzmann constant.

• Incoming longwave radiation from the sky ($L_{insky}$) uses measured radiation at the synoptic weather station, corrected for glacier surface elevation and sky view factor:

$L<em>{insky} = L</em>{0dc} f_s$ $(14)$

$L_{0dc}$ is the elevation-corrected incoming longwave radiation.

• Stefan-Boltzmann equation and $L<em>{0d}$ are used to calculate the “effective emissive temperature” of the sky; this is corrected with the standard elevation lapse rate and then $L</em>{0d}$ is recalculated. Over elevation range of teh glacier, this leads to a maximum variation in $L_{0d}$ of 10–12 $W m^2$.
• Longwave radiation from the sky is supplemented by radiation emitted by surrounding terrain ($L_{interr}$):

$L<em>{interr} = (1 - f</em>s) \sigma T^4_{terr}$ $(15)$

*   Where $T_{terr}$ is the average surface temperature of the visible terrain.
*   $T_{terr}$ is set to the elevation-corrected air temperature recorded by the GWS for each time step.

• The total net longwave flux (L*) is then

$L* = L<em>{insky} + L</em>{interr} - L_{out}$ $(16)$

Turbulent Heat Fluxes

• Air temperature is calculated using an assumed lapse rate of $6.5°C km^{-1}$.
• Relative humidity and wind speed are assumed constant, and a lapse rate of 10 kPa km-1 is used to calculate air pressure.
• Iterative scheme calculates the Obukhov length scale [Munro, 1990] for sensible and latent heat fluxes.
• Surface roughness length (z0) scale is required, and mean values for snow and ice are used [Arnold and Rees, 2003].

Precipitation

• Measured precipitation at GWS is assumed for the whole catchment
• Precipitation falls as snow if the lapse rate corrected air temperature is below a threshold temperature (1°C). Snow fall its added to depth of snow, liquid is asumed to run off wothout affecting balance.
• Energy fluxes are added for each DEM cell each hour to give the total surface energy flux.
• Positive energy leads to melt, negative to cooling.

Results

Model Validation

• Validated using summer mass balance measurement at centerline stakes, surface lowering measurements at GWS with an ultrasonic ranger, and net all-wave radiation measurement at the GWS.
• Modeled net all-wave fluxes are calculated as topographically corrected measured incoming shortwave radiation, minus reflected shortwave radiation derived from this value and surface albedo, plus measured incoming longwave radiation, corrected for DEM cell sky view factor, minus calculated outgoing longwave flux based on surface temperature in the DEM cell.
• Correlation coefficient between modeled and measured mass balance values: r = 0.982 (P < 0.01).
  *For surface lowering: r = 0.847. The Nash-Sutcliffe measure of model efficiency (N) is 0.6901 for this period.
• Good statistical agreement between modeled and measured net all-wave radiation; r = 0.8717, N = 0.7018.

Impact of Topography on the Surface Energy Balance

• Melt generally decreases at higher elevations
• Spatial distribution of four main surface energy balance components shows variability.
  • Turbulent fluxes decrease with elevation due to decreasing air temperature and lower roughness of snow.
  • Decrease is not uniform; a wide band of similar shortwave fluxes occurs between 200 and 350 m, with a local maximum around 300m.
  • Longwave flux increases at higher elevations due to surface temperatures and longer periods below 0°C.
  • Radiative fluxes dominate turbulent fluxes.
  • Turbulent fluxes are due in large part to the low average summer temperatures durning 2000, and the smooth surfaces found on Midre Love´nbreen

Derived Topographic Parameters

• Slope and aspect of the glacier surface, shading by surrounding topography and the proportion of the sky hemisphere visible from any location impact radiative fluxes.
• Bulk of glacier shows relatively shallow slopes between 4 and 7°.
• Most of the glacier is oriented just to the east of due north.
• Shading generally increases with elevation, but is also affected by the valley walls.
• The proportion of sky visible is similar to shading; the tongue has the hightest proportion, decreasing up glacier and toward the valley sides.
• Total incoming solar radiation (ISR) generally decreases eith elevation, but shows a very high degree of spatial complexity.

Sensitivity to Topographic Controls

• Slope and aspect, shading, have the largest impacts on incoming shortwave radiation 200 $MJ m^{-2}$, sky view has a lower impact of around 60 $MJ m^{-2}$.
• The effect of shading leads to a high degree of spatial complexity in the model, leading to significant variations in melt profiles across the glacier surface.
• Overall, shading and the sky view factor reduce incoming radiation, while slope and aspect affect the cross-glacier variations.
  • Those factors which increase Q* typically lead to a decrease in S and an increase in T

Discussion

• Presents a large complex interactions between glacier topography, the spatial distribution of the main energy balance components, and the resulting spatial distribution of melt, and summer mass balance.
• Increase in incoming shortwave radiatio, in the centralpart seem to be the main cause for flattening of the summer mass balance/elevation gradient.
• There may be feedback mechanisms within the model which amplifies the change in slope and aspect. This change between snow and ice surfaces that amplifies the incoming solar radiation, leads to marked reduction in the mass balance/elevation gradient that overcomes the generally greater snow depths at higher altitutdes on the glacier, and the general decrease in melt with altitutde.
• The high latitude of teh glacier seems to overide some of the issues related to incoming shortwave radiatio, as averaged over the whole glacier, it initially seems less sensitive to topographic controls vs. other glaciers.
  • The solution fo this apparent paradox, is because the glacier has a high latitutde, there is about 4 months of 24 hr daylight.
• The rapid retreat of the snout of the glacier seems to be more affected on the eastern side vs. western side. This may be because the higher radiation receipts on the eastern side contribute to more rapid retreat.
• The complexity in the spatial patterns of the turbulat fluxes, coupled with the fact that they change to changes in the incoming short wave radiative flux, must imply feedbacks exist between the various energy fluxes. The key to these feed backs seem to depend on the changing surface temperatutre of the glacier. This is beause the suface temperature has a big impact on changing the outgoing flix of longwave radiation and the turbulant fluxes. The surface temperature acts as a negative feedback mechanism that damps down changes in energy flux.
• Feedbacks opperate both over time and spatially, drivem by complex patterns of incoming solar radiatio.

Conclusions

• Paper presents results from two-dimentional energy balance model for glacier.
• Model performace is found to be very high compared with measurements at stakes drilled ober glacier, radiation treatmen within model also produces totals that agree very well with measured values from weather station.
• Preserves small-scale topographic variation compared with DEM's. Topography plays vital role in explaining spatial complexity in the modeled fluxes.
• This is due to the very complex pattersn of incoming solar radiation. Shading and slope and aspect has a complex role in determining the pattersn of this radiatio.
• Observed melt rates and overall new radiation flux can nly be simulated accurately by including an albedo correction factor to account for anisotropic reflectance of snow/ice at high sun zenith angles.
• Changes can be affected by feedbacks withing model throught changes in the suface temperature do to changes with solar radiation
• Highlughts the difficulty in using stake lines to infer gradient
  *Complex feedbacks for the need to intergrate mor detailed snow hydrology models with surface energy balance models.