Modeling Energy Balance, Surface Temperatures, Active Layer Depth and Permafrost Thickness around Longyeardalen, Svalbard
A project funded by the University Courses on Svalbard (UNIS) 2000-2005
Ole Humlum, UNIS, Department of Geology, Svalbard, Norway
Objectives
A dynamic numerical 3D terrain model entitled VirtualWorldForWindows (VW4W) is currently being developed and tested. This model, which runs on normal PC's, takes topographic data, terrain surface characteristics (geomorphology and vegetation) and meteorology (air temperature at sea level, wind speed, wind direction and cloud cover) as input and yields as output information on various phenomena such as terrain surface net radiation balance, snow cover thickness, snow cover duration, glacier mass balance, ground surface temperatures, growing and freezing degree days, maximum active layer thickness, stable permafrost thickness and the amount of melt water discharge during the summer.
Introduction
By tradition, most terrain elements within a landscape are seen as representing semi-permanent features, with only few elements, such as glaciers, representing dynamic exceptions from this main rule. Modern research has, however, clearly demonstrated that the organization of the landscape usually is much more complex, and that there is a close link between climate, vegetation and terrain surface processes. In reality, a given landscape should be considered as a highly dynamic system consisting of a number of elements currently adjusting according to present climate, and a number of relict features, produced under past climatic conditions.
Whether a climatic change is of geomorphic significance depends on its magnitude and duration, and on the properties of the landform considered. The larger or more resistant a landform, the longer, in general, it takes to adjust to a change in climate. The rate of adjustment can be studied by measuring the sediment flux through and within river basins or smaller catchments (by establishing sediment budgets), as well as by mapping of recent landform changes and dating of older forms. In this way, analytic studies of landscapes may yield information on character and duration of both past and present environments.
Due to albedo-induced feedback effects, the variability of climate is usually expected to be greater in polar areas than elsewhere. Climatically controlled changes in a landscape are therefore also expected to be pronounced within high-latitude areas, especially as the rate of individual surface processes - besides from being directly controlled by local climate - often is partly controlled by vegetation and snow cover, features themselves controlled by climate.
As a means to investigate the climatic control on the evolution of different landscape elements, a analytic numeric model is presently being developed. This model (VW4W) aims at yielding information on the different energy fluxes between the atmosphere and the surface of a terrain element considered. The model output (the virtual reality) is currently being tested by observations of local meteorological conditions around Longyearbyen, where year-round observations is possible.
A few of the pertinent considerations made during programming are shortly outlined below, while others, such as the calculation of the wind field, surface temperatures and snow drift, are not. Please find also below a few examples of the output themes from the model. VW4W is coded in VisualBasic6.0®.
Topographic model for Longyearbyen and surroundings used in the numerical experiment outlined below. Longyearbyen is located in the lower part of the valley trending towards the lowermost corner. Horizontal dimension of model is 8700 x 6700 m. Contour equidistance 50 m. Viewed towards SW.
Outline of model approach used for snow- and ice covered terrain surfaces
Melting (ablation) of ice or snow covered terrain surface represents a complicated problem which needs to be addressed for dynamic terrain modeling in polar regions. The associated processes are controlled by the energy balance at the surface, which can be described as:
c(dTe/dt)+M = G(1-α)+εLi-εσTs4+S+L (1)
Energy fluxes from the atmosphere to the glacier surface are on the right side of the equation. They may change the mean absolute temperature (Te) of the snow or ice layer (glaciers) or cause loss or gain of water in all phases (M), usually dominated by the melting of snow/ice or the refreezing of melt water. The symbol c represents the heat capacity of the upper part of the glacier considered. G is the short-wave (global) radiation emitted by the sun and reaching the surface as direct or indirect (diffuse) sky radiation. Part of this is reflected, controlled by the surface albedo (α). Li represents the incoming long-wave radiation emitted in the atmosphere and ε is the emission (and absorption) coefficient of snow or ice. For practically purposes, it can be set to 1 (Müller 1984), that is, ice and snow absorbs and emits long-wave radiation much like a perfect black body. The next term represents the long-wave radiation emitted from the glacier surface, where σ is the Stefan Boltzmann's constant (6.670x10-8 mW/cm2(deg K)4) and Ts the surface temperature (degrees Kelvin). S and L are the turbulent fluxes of sensible- and latent-heat, respectively.
The short-wave radiation G decreases with the zenith angle of the sun and with cloudiness. The albedo α depends on surface characteristics: typical values are 0.7 to 0.9 for fresh snow, 0.4 to 0.6 for old snow and 0.2 to 0.4 for exposed glacier ice.
The incoming long-wave radiation Li increases with cloudiness, air temperature and humidity. The incoming long-wave radiation is given by
Li = ε*σTa4 (2)
where ε* is the effective emissivity of the sky and Ta air/cloud temperature (Kelvin). The effective emissivity ε* can be expressed in terms of cloud cover n, and the emissivity of the clear sky εo by (Ohmura 1981)
ε* = (1 + kn)εo (3)
where k is a constant depending on cloud type. Ohmura (1981) listed k values for eight different cloud types, but a constant value of 0.26 may be assumed for most practical purposes (Braithwaite and Olesen 1990).
As the temperature of an ice/snow surface cannot exceed the melting point for ice (0oC), the maximum value of the outgoing long-wave term εσTs4 is 316 Wm-2.
Estimated mean annual ground surface temperature (MAGST), based upon meteorological observations 20.August 1999-19.August 2000. Horizontal dimension 8700 x 6700 m. Viewed towards SW.
The sensible-heat flux S involves the transfer of heat directly from the surface to layers of air immediately adjacent to it by the processes of conduction and convection. The latent-heat flux L involves the transfer of heat from the surface via the evaporation of water; as water evaporates from the surface, latent-heat is extracted, only to be released to the atmosphere later when the water condenses. The turbulent-heat fluxes S and L will therefore increase with the vertical gradients of potential temperature and humidity, respectively, in the layer just above the glacier surface (Kuhn 1979, Escher-Vetter 1985, Hay and Fritzharris 1988). Furthermore, they will also increase with mean wind speed and intensity of turbulence in this layer. The relative importance of sensible- and latent-heat mechanisms in the transfer of heat is sometimes characterized by the Bowden ratio (S/L).
The englacial temperature in glaciers Te or in thick snow layers may vary with depth below the surface. If, at a particular depth, temperature is at the melting point and further energy is supplied, melting occurs. Most melt water are, however, produced at the surface. If the glacier surface is covered by snow, melt water will percolate downwards, where it refreezes and heats up the surrounding snow if the local temperature is below the melting point. By this, the englacial temperature changes due to conduction and refreezing of melt water. If all snow layers are at the melting point and saturated, the melt water runs off.
The basic processes of heat exchange at the glacier or snow surface are the same whether the surface is at its melting point or not. Many of the details are different, however, because in one case the temperature of the surface can change while in the other case it is fixed at the melting point. These two conditions have different effects on the temperature gradients in the air immediately above the glacier surface. In most cases where ablation is considered during the summer season, ice/snow temperatures can be assumed to be at the melting point. Below freezing conditions are well documented in the study by Liljequist (1957) in the Antarctica.
The relative importance of the individual terms in the energy equation can vary considerably from year to year and between different geographical sites. In general, however, net radiation is considered the dominant term in the energy equation. The net radiation constitutes more than half of the total energy supply in most published glacier surface energy balance studies (see e.g. Paterson 1994). From these, it appears that the relative importance of radiation as a heat source increases with altitude above sea level and with continentality (dry summers) of climate.
Estimated growing degree days (GDD; >5oC), based upon meteorological observations 20.August 1999-19.August 2000. Horizontal dimension 8700 x 6700 m. Viewed towards SW.
Braithwaite and Olesen (1990) investigated the relative importance of the various energy sources for ablation on two glaciers in SW Greenland. They found that the largest single source of energy is the short-wave radiation followed by sensible-heat flux and long-wave radiation. The net long-wave radiation was always negative during the ablation period June-August, and more or less tended to cancel out most of the sensible-heat flux, which always was positive. The latent-heat flux was found to be very small on average but this was due to substantial fluctuations between negative and positive daily fluxes, i.e. evaporation and condensation, respectively, which nearly cancel out over longer periods. These findings agree well with estimates made by Braithwaite and Olesen (1984) and with the results of measurements by Knudsen et al. (1987), also in Greenland.
Summing up, the net short-wave radiation apparently accounts for most of energy available for ablation on glaciers and snow covered terrain surfaces in the Arctic, and, as a first approximation, it may approximate the energy amount which is available for ablation during the summer season, as the remaining sources of energy more or less tend to cancel each other out. This assumption, however, needs further field investigations to be verified or falsified.
Sun-Earth astronomical relationships adopted in the model
The earth revolves around the sun in an elliptical orbit with the sun at one of the two foci. The amount of solar radiation reaching the earth is inversely proportional to the square of its distance from the sun; an accurate value of the sun-earth distance is, therefore, important. The mean distance from the earth to the sun ro is called one astronomical unit:
ro = 1.496 x 108 km (4)
The minimum sun-earth distance is about 0.983ro (3.January, perihelion), and the maximum approximately 1.017ro (4.July, aphelion). During a year, earth-sun distance variation is thus ro ±2% and the actual amount of solar energy reaching the earth corresponds to the average incoming energy ±4%.
In order to determine the sun-earth distance for a given day, a eccentricity correction factor Eo of the earth's orbit can be defined (Spencer 1971):
Eo = (ro/r)2 (5)
also called the reciprocal of the square of the radius vector of the earth, which can be expressed as (Spencer 1971):
Eo = 1.000110 + 0.034221cosΓ + 0.001280sinΓ
The term Γ is called the day angle and should be measured in radians. It is calculated as
Γ = 2π(dn - 1)/365 (7)
where dn is the day number (DOY) of the year, ranging from 1 on 1.January to 365 on 31.December. February is always assumed to have 28 days, and because of the leap year cycle the accuracy of (7) will vary slightly. From equations (4) to (7), the sun-earth distance r may be calculated for any given day.
The plane of revolution of the earth around the sun is called the ecliptic plane. The earth itself rotates around the polar axis, which is inclined at approximately 23.5o from the normal to the ecliptic plane. The earth's rotation around the polar axis causes diurnal changes in solar radiation and the position of the polar axis relative to the sun causes seasonal changes. Both the angle between the polar axis and the normal to the ecliptic plane, and the angle between the earth's equatorial plane and the ecliptic plane, remains constant.
Estimated annual net radiation, based upon meteorological observations 20.August 1999-19.August 2000. Note the net radiation maximum associated with the exposed mountain plateaus, while especially the upper part of the deeply incised Longyearvalley is in radiation shade. The difference in net radiation between Longyearbreen (rather low values) and the somewhat more exposed Larsbreen (rather high values) is presumably partly explaining why Longyearbreen terminates at lower attitudes than Larsbreen. Horizontal dimension 8700 x 6700 m. Viewed towards SW.
The angle between the earth's equatorial plane and a vector from the center of the earth to the center of the sun, however, changes at all times. This angle is called the solar declination δ. It is zero at the vernal and autumnal equinoxes and has a value of about +23.5o at the summer solstice and about -23.5o at the winter solstice. The declination changes every instant, but within a 24h period, the maximum rate of change (which occurs at the equinoxes) is less than 0.5o/day. Therefore, for most practical purposes, the solar declination δ (in radians) can be considered constant for a given day and may be calculated as (Spencer 1971):
δ = (0.006918 - 0.399912cosΓ + 0.070257sinΓ - 0.006758cos2Γ + 0.000907sin2Γ - 0.002697cos3Γ + 0.00148sin3Γ) * (180/π) (8)
By way of the above equations (4) to (8) the sun's position in the sky, relative to that of the earth, may be calculated. However, in order to calculate the incoming solar radiation on any arbitrarily inclined terrain surface, it is necessary to use the trigonometric relationships between the solar position in the sky and local surface coordinates on the earth.
For a given geographical position, the sun's position in the sky in relation to a horizontal surface can be determined by the two following equations:
cosθz = sinΔ = sinδsinφ + cosδcosφcosω (9)
and
cosψ = (sinΔsinφ - sinδ)/cosΔcosφ (10)
where θz is the zenith angle (degrees), Δ is the solar altitude or the solar elevation above the horizon (degrees; Δ = 90 - θz), ω is the hour angle (noon zero, morning positive), φ is the geographic latitude (degrees; north positive), ψ is the solar azimuth (degrees; south zero, east positive) and δ is the declination (8), the angular position of the sun at solar noon with respect to the plane of the equator (degrees; north positive).
The solar azimuth ψ is the angle at the local zenith between the plane of the observer's meridian and the plane of a great circle passing through the zenith and the sun. By convention it is measured east positive, west negative (south zero) and varies between 0o and ±180o. The hour angle ω is the angle measured at the celestial pole between the observer's meridian and the solar meridian. Counting from midday, it changes 15o per hour. By way of (9) and (10), from any position on earth, it is possible to calculate the sun's position in the sky in relation to a local horizontal plane.
Also of great interest is the angle of incidence θ, i.e. the angle between the normal to any arbitrarily orientated surface, at any geographical position on earth, and the earth-sun vector.
The angle of incidence θ can be obtained from the following equation (Benrod and Bock 1934, Kondratyev 1969, Coffari 1977):
cosθ = (sinφcosß - cosφsinßcosγ)sinδ + (cosφcosß + sinφsinßcosγ)cosδcosω + cosδsinßsinγsinω (11)
where β is the slope of the surface measured from horizontal position (degrees) and γ is the surface azimuth angle, i.e. the deviation of the normal to the surface with respect to the local meridian (degrees; east positive). All other terms have the same meaning as in (9) and (10).
Using equations (1) to (11), it is now possible to answer the day-or-night question for a given terrain surface segment, arbitrarily orientated, at any geographical position on earth, for any time. If the sun is found to be visible from that surface at that time, the angle of incidence can be calculated by means of equation (11), and the net short-wave radiation on the surface estimated. The above equations were all coded in the numerical model.
Estimated stable permafrost thickness, based upon meteorological observations 20.August 1999-19.August 2000. This is, however, only a crude estimate which does not take into account climate during the previous centuries. Horizontal dimension of diagram is 8700 x 6700 m. Viewed towards SW.
Estimated stable permafrost thickness (m) in Nordenskiöldland, based upon meteorological observations 20.August 1999-19.August 2000. As was the case for the figure above, this is only a crude estimate which does not take into account climate during the previous centuries.
Click here for an example of snow avalanche modeling approach, using VW4W.
Outlook
The model is now developed further and now also handles the remaining energy sources in (1), i.e. the turbulent heat fluxes.
One immediate use of the model is estimation of the mass balance of glaciers for a given set of meteorological data. Another issue may be a forecasting of discharge from areas with glaciers or a seasonally snow cover, e.g. in connection with hydroelectric projects. A third area of interest may be represented by snow cover studies, using photographic or satellite documentation for the gradual disappearance of the seasonal snow cover within an area considered.
From the automatic camera research scheme outlined elsewhere on this website it will be uncomplicated to determine the day when the snow cover disappear during summer at various points in the landscape. From this, provided access to local meteorological observations on cloud cover, wind and air temperature from the melting period, it will then be possible to use the model in reverse to obtain an estimate of the amount of snow (mm w.e.) present at these points shortly before the onset of spring melting. This represents a new method to estimate the effective (real) distribution of precipitation (snow) at the end of the winter season.
The above approach may also have various research potentials within arctic and alpine biology.
The initial version of the software were, however, made for pure geomorphological purposes, in connection with current research on the meteorological control on past and present geomorphic processes in Greenland, on the Faroe Islands and on Svalbard. The present study thus represents a spin-off from these research initiatives.
A word of caution
Is computer-based modeling in general the best way to handle complex associations of processes controlling landscape and climate ?
Climate and landscape processes are in a continuous dynamic state of flux, representing an analogue system, where everything is happening simultaneously. In contrast to this, computer models are digital, trying to solve a problem by repetitive calculations (iterations), before moving on to solving the next problem, etc. This is clearly a significant drawback for computer-based modeling of nature.
While the laws of physics may be immutable, it is not always predictable which law or process will predominate over which when a maelstrom of competing laws are acting simultaneously as is the case of climate and landscapes. The description of the individual laws in a model are hopefully correct defined in the equations used, but the dominance or subservience of one law to dozens of others is defined by the modeler, not by the model itself. The modeler decides that issue in the way the code is written.
In the end, the computer model therefore simply mirrors the intellectual choices of the modeler and only puts numbers to them. If those choices are based on flawed reasoning or insufficient observational evidence, it is naive to believe that the model will somehow iron out the problem through sheer number crunching power. That would be to attribute qualities of judgment to models which they simply do not have. In essence, a model does not relieve the intellectual burden of determining which variable is dominant over which. The modeler has to choose when writing the code, and this choice then becomes integral to the model. Anybody who plan to make use of a model not coded by him- or herself should consider this scientific aspect very carefully.
For those simple reasons, computer models can never be superior to the knowledge based understanding derived from experiments and classic field observations. Models may prove powerful instruments in developing our understanding of complicated laws and process associations. However, until the empirical knowledge coded into them is perfect they still have to be considered as predictive tools with many limitations. Never put the chart before the horse !
A word about science
Philosopher-scientist Herbert Spencer once defined science as "organized knowledge." But clearly science is much more than just that. One aspect of science may be defined as the design or conduct of reproducible experiments to test how nature works, or the creation of theories that can themselves be tested by such experiments. Another aspect of science is the orderly mapping and observation of events - past as well as present - that cannot yet be manipulated or understood, and, ultimately, the testing of many different such observations as the basis for theories to explain the events.
This makes science the one human activity that seeks knowledge in an organized way. It's not only the knowledge that's organized, it is also the seeking process itself.
Science doesn't guess, doesn't hope, doesn't wish, doesn't trust, doesn't believe. Science seeks.
PS: If you want to see what the meteorological conditions are like in the real world around Longyearbyen, please click here.
References
Benrod, F. and Bock, J.E. 1934: A time analysis of sunshine. Transactions American Illumination Engineering Society, 34:200-218.
Braithwaite, R.J. 1981: On glacier energy balance, ablation, and air temperature. Journal of Glaciology, Vol.27(97):381-391.
Braithwaite, R.J. and Olesen, O.B. 1984: Ice ablation in West Greenland in relation to air temperature and global radiation. Zeitschrift für Gletscherkunde und Glazialgeologie, Bd.20:155-168.
Braithwaite, R.J. and Olesen, O.B. 1990: A simple energy-balance model to calculate ice ablation at the margin of the Greenland ice Sheet. Journal of Glaciology, 36(123):222-228.
Coffari, E. 1977: The sun and the celestial vault. In "Solar Energy Engineering" (ed. Sayigh, A.A.M.), Chapter 2. Academic Press, New York.
Escher-Vetter, H. 1985: Energy balance calculations for the ablation period 1982 at Vernagtferner, Öetztal Alps. Annals of Glaciology, 6:158-160.
Hay, J.E. and Fritzharris, B.B. 1988: A comparison of the energy-balance and bulk-aerodynamic approaches for estimating glacier melt. Journal of Glaciology, 34(117):145-153.
Knudsen, N,T., Ottosen, O. and Svendsen, L.M. 1987: Energy balance on outlet glaciers from the Indland ice, West Greenland. Grønlands Geologiske Undersøgelse. Rapport, 135:99-105.
Kondratyev, K.Y. 1969: Radiation in the Atmosphere. Academic Press, New York.
Kuhn, M. 1979: On the computation of heat transfer coefficients from energy-balance gradients on a glacier. Journal of Glaciology, 22(87):263-272.
Liljequist, G.H. 1957: Norwegian-British-Swedish Antarctic Expedition 1949-52. Scientific Results. Norsk Polarinstitut, Oslo, Vol2, Pt.1.
Müller, H. 1984: Zum Strahlungshaushalt im Alpenraum. Mitteilungen der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, Nr.71.
Ohmura, A. 1981: Climate and energy balance on Arctic tundra. Axel Heiberg Island, Canadian Arctic Archipelago, spring and summer 1969, 1970 and 1972. Zürcher Geographische Schriften, 3.
Paterson, W.S.B. 1994: The Physics of Glaciers. 3rd Edition. Oxford, Pergamon Press. 480 pp.
Spencer, J.W. 1971: Fourier series representation of the position of the Sun. Search, 2(5), 172.
Latest update: 3. January 2006.