The radiative heat flux

The International Association of Scientific Hydrology has recommended that comparison of a glacier’s heat and ice budgets be made over unit area of its horizontally projected surface ([International Hydrological Decade], 1970, p. 13). The radiative heat flux to a level glacier is then simply the measured net radiation, whose components need not be resolved. This simplicity does not extend to sloping glaciers where the contributions of long-wave, direct short-wave, and diffuse short-wave radiation must each be resolved and adjusted to the horizontally-projected surface. Reference KondratyevKondratyev (1969) has shown that the flux of isotropic sky radiation on a shallow slope (< 30°) is approximately equal to that on a horizontal surface reduced by the cosine of the slope angle. The horizontal projection of the slope has an area equal to the slope’s area reduced by the same factor. Thus the flux of isotropic radiation on a horizontal radiometer at the Hodges Glacier site was effectively that on a horizontal projection of its shallow (15°) slope. This approximate result was used to adjust for slope the reflected short-wave, diffuse short-wave, and net long-wave radiations, all of which were assumed isotropic and measured by horizontal sensors. The reflected short-wave radiation was measured directly at the glacier; the diffuse short-wave radiation was not measured in situ but instead was obtained from a horizontal radiometer at the Grytviken meteorological station. The Grytviken measurements were applied directly to the glacier because skyline plots showed similar topographic screening at both sites. In addition, they usually experienced the same cloudiness and the effect of their altitude difference was negligible in comparison with the accuracy of the radiometer measurements. The net long-wave radiation was obtained as a residual in the radiation balance. The approximations employed for isotropic radiations (i.e. Kondratyev’s method and the transposition of diffuse short-wave radiation from Grytviken) were justified because the radiations were required only for slope adjustment of net radiation measured in situ over a shallow gradient.

The treatment of the direct solar beam is more complicated. To simplify the problem we first consider the instantaneous radiation geometry where the radiation components and angles are as defined in Table II.

Table II.Symbols Used for Radiation Components and Angles

On unit area of a horizontal sensor the instantaneous flux is

The flux required for comparison with vertical changes in ice storage is the flux per unit of horizontally projected area given by

Because L’, S’_d, and S’_o are assumed isotropic, the instantaneous short-wave radiation flux per unit of horizontally projected area is:

and the instantaneous net radiation flux on the projection is:

From this treatment we obtain the components of radiative heat flux to be

and

Reference Garnier and OhmuraGarnier and Ohmura (1968) developed a method for calculating the magnitude of I(θ) cos θ’ for any latitude, date, time, slope angle, and aspect, using accepted values of the solar constant and assuming a power-law absorption of radiation in the atmosphere. Their method allows the calculation of a factor α by which the direct insolation on a horizontal surface at the meterological site is reduced by inclining it parallel to the glacier surface. Mean daily values of α were determined throughout the summer and were used to convert the direct insolation on a horizontal sensor to that on unit area of sloping glacier. This conversion, calculated for cloudless conditions, also applied for days during which sunshine was evenly distributed, during the summer of 1973–74 one of these criteria was generally satisfied at Hodges Glacier. However, during January and February, there was a 2:1 bias of sunshine towards the afternoons; such asymmetry was accounted for by a factor n in the radiation formulae.

The daily short-wave radiation flux S _g, and the daily net all-wave radiation flux on a horizontal projection of the glacier (F _r) were calculated with daily measurements from horizontal sensors by

and

where n = 1 for days of uniform cloudiness.

The global short-wave component S’_g was measured with a possible error of 5%. The uncertainty in diffuse short-wave radiation S’_d (transposed from Grytviken) was estimated to be 10%. The specified precision of the Funk net radiometer was 5% but Reference LatimerLatimer (1963) has shown that a carefully maintained Funk radiometer is accurate to between 5% and 10%. The reliability of radiation measurements from Hodges Glacier was tested by comparing net long-wave radiation deduced from the algebraic sum of F’_r, S’_o, and S’_g with values calculated from an empirical formula compiled by Reference WilliamsWilliams (1974). This incorporates the Swinbank counter-radiation formula with the local temperature gradient, the amount and type of cloud cover, and the surface slope. The agreement between totals summed over a four-month period was within 6% indicating good long-term reliability of the radiation measurements. This supports the stated instrumental accuracies which led to likely errors of 5% and 15% in the summer totals of global short-wave radiation and net radiation, respectively.

The sensible- and latent-heat fluxes

Fluxes of sensible and latent heat were calculated from aerodynamic estimates of the heat and water vapour transferred by vertical eddy diffusion in turbulent air over the glacier. The fluxes were assumed proportional to the vertical gradients of temperature and water-vapour pressure, and to the rates at which eddies of heat and vapour were transferred vertically (Reference PatersonPaterson, 1969, p. 50–51):

where A _h is the eddy diffusivity for heat, A ₁ the eddy diffusivity for water vapour, T the temperature of air, e the water-vapour pressure, C _p the specific heat of air at constant pressure, ρ the density of air, M the molecular weight of water, R the universal gas constant, L the latent heat of vaporization of water, and z the height. Equation (3) applies strictly to potential temperatures but we used ambient values since, during summer, the temperature lapse between the thermometer and the glacier surface was generally much greater than the adiabatic lapse.

The fluxes of sensible and latent heat were paralleled by the turbulent transfer of momentum represented by τ, the shearing stress of wind on the glacier surface. This was assumed to be proportional to the vertical gradient of wind-speed and the rate at which eddies of momentum were transferred vertically (Reference PatersonPaterson, 1969, p. 51):

where u is the wind-speed and A _m is the eddy diffusivity for momentum.

The fluxes of sensible heat, latent heat, and momentum are conserved with height in air moving steadily without advection across a homogeneous surface (Reference SellersSellers, [^c1965], p. 124–25, 142). In this situation, the eddy diffusivities for heat, water vapour, and momentum can be compared through observed profiles of T, e. and u. Published profiles suggest that the diffusivities are approximately equal at a given level in neutrally buoyant air over ice (Reference PatersonPaterson. 1969, p. 49–54; Reference HolmgrenHolmgren, 1971). Such air is characterized by logarithmic profiles of temperature, water-vapour pressure and wind-speed (see also Reference Grainger and ListerGrainger and Lister, 1966). Glaciologists commonly assume equal diffusivities in near-neutral and mildly stable stratifications where approximately logarithmic profiles prevail (Reference Gilbert, Gilbert, Jamieson, Lister and PendlingtonGilbert and others, 1969; Reference WilliamsWilliams, 1974; Reference MartinMartin, 1975).

Wind profiles at the meteorological site were studied on 17 d during December 1973 and January 1974. During this period the upward movement of the temporary snow-line exposed firn surfaces above the site and ice surfaces below it. The mean daily values of wind-speed recorded at 2.8 m, 1.0 m, and 0.2 m were plotted against the logarithms of the heights. Deviations from logarithmic variation were less than 10% as long as the wind-speed at 2.8 m was below 6.1 m s⁻¹. Since the mean summer wind-speed was only 3.8 m s⁻¹ and mean daily wind speed exceeded 6.1 m s⁻¹ on only 15% of the days, the general assumption of logarithmic wind profiles seemed justified. The variation of wind-speed with height can be approximated (Reference HolmgrenHolmgren, 1971, p. 5) by:

where u _*is the friction velocity , k is von Keánnán’s constant ( = 0.40). and z ₀ is a logarithmic scaling length or surface roughness parameter for wind.

The thermal influence of Hodges Glacier on the overlying air created mildly stable stratifications with a mean lapse of 3 deg through the lowest metre during the summer. The predominance of logarithmic wind profiles suggested such stability did not significantly suppress turbulence (Reference SellersSellers, [^c1965], p. 152) and the assumption of equal diffusivities for heat, water vapour, and momentum therefore seemed plausible. Equations (5) and (6) then give

If logarithmic profiles are also assumed for temperature and water-vapour pressure. Equations (3) and (4) become (Reference HolmgrenHolmgren, 1971, p. 12)

and

where T _o is the temperature of the ice surface, T _m the average temperature of the air layer. e _o the water-vapour pressure at the ice surface, and r _o and s _o the logarithmic scaling lengths, or surface roughness parameters, for temperature and water-vapour pressure, respectively.

Reference Munro and DaviesMunro and Davies (1977) have shown that the lowest metre of air over a melting glacier reliably forms a boundary layer where near-logarithmic profiles prevail. Wind speeds from 1.0 m and 0.2 m were therefore used to compute daily values of In z _o from Equation (6) during the 17 d study in December and January. The logarithmic mean value of z _o was (1.33 ± 0.25)× 10⁻³ m where the uncertainty is estimated from the standard error of the logarithmic mean. This result compares well with those tabulated by Reference KuhnKuhn (1979) for six melting glaciers five of which had values of z _o in the range (1.1−2.0) × 10⁻³ m.

For most of the summer, the meteorological site was not surrounded by a uniform surface of snow, firn, or ice. The temporary snow-line and/or the firn–ice boundary generally lay within the equilibration distance for airflows reaching the site (within c. 100 m, Reference SellersSellers, [^c1965], p. 147). Down-glacier winds arrived at the site over surfaces where snow predominated, whereas up-glacier winds arrived over surfaces where ice predominated. Thus, the mean surface roughness during the 17 d study reflects the distribution and development of a variety of surface conditions around the site. The variety was believed to be representative of the glacier during summer, and the mean surface roughness was therefore used to estimate aerodynamic heat fluxes throughout the melt season. On small cirque glaciers, it seems inappropriate to associate summer roughness parameters with a single surface type.

Profiles of temperature and water-vapour pressure were not available from Hodges Glacier to determine the corresponding roughness parameters r _o and s _o. The identity of r _o, s _o, and z _o is assumed in some glacier models (Reference Streten and WendlerStreten and Wendler, 1968; Reference KuhnKuhn, 1979) but Reference HolmgrenHolmgren (1971) has plausibly argued the identity of r _o and s _o alone. He reviewed values of r _o, s _o, and z _o from profiles measured over melting glaciers in Devon Island, Spitsbergen, and Greenland. The values of z_o lay in the narrow range (1–3) × 10 ⁻³ m, whereas r _o and s _o ranged broadly from (0.3–10) × 10⁻⁵ m. Since the value of z _o from Hodges Glacier (1.33 × 10⁻³ m) appeared typical, it was anticipated that values of r _o and s _o would also fall within the corresponding range reported by Holmgren. From his review, it was estimated that, to the nearest order of magnitude, r _o = s _o = 10⁻⁵ m. (This was also Holmgren’s own best value from his extensive measurements in Devon Island.) The fluxes of sensible and latent heat for melting conditions at the pressure altitude of the meteorological site (950 mbar) then become, from Equations (7) and (8)

and

where T _m was 274.7 K and u ₁, t ₁ , and e ₁ are the wind-speed (m s⁻¹), air temperature (° C), and water-vapour pressure (mbar) at one metre above the surface, respectively.

Equations (9) and (10) are semi-empirical formulae derived for short-period averages of temperature, wind-speed, and humidity. These variables should exhibit mainly random fluctuations during each averaging period which should therefore be short enough to exclude coupling effects such as the parallel diurnal variation of wind with temperature and water-vapour pressure. In practice, the maximum averaging period which satisfies this criterion is about 60 min (Reference SellersSellers, [^c1965], p. 144). The influence of coupling over periods of one day was examined through an “index of coupling”, P . This was the ratio of the sum of hourly sensible heat fluxes to the sum of fluxes computed from daily means of the meteorological variables. Values of P were calculated thoughout the first half of February 1974, when ablation was at its height. The index of coupling reached 163% on a day with a föhn wind and on a day with active thermals. On such days, most sensible heat was transferred when wind and temperature peaked simultaneously; their coupling usually lasted only a few hours and was unrepresented in fluxes calculated with daily means. On days without discernible weather trends the index fell as low as 102%. The values of P showed that computations from daily means always underestimated the flux because wind and temperature are, in general, positively correlated. Owing to the very changeable nature of weather on South Georgia, the average values of P consistently approached 114% for periods of a week or more. This result was used to simplify the evaluation of the sensible heat flux during each budget interval. The flux was calculated by Equation (9) with daily mean values of wind and temperature, and then increased by 14% to account for the consistent effect of coupling.

The assumption of neutral or mildly stable stratifications up to one metre above the glacier is believed to be valid even during föhn winds because the stabilizing effect of high temperatures is then balanced by vigorous mixing. The meteorological site was over 150 m from the up-wind glacier edge and so was beyond the likely equilibration distance for the rock-ice boundary (Reference SellersSellers, [^C1965], p. 147). However, the deep cirque containing Hodges Glacier is subject to violent rotor winds which may occasionally have advected air to the site by routes other than along the glacier surface. Errors in the estimates of sensible heat flux arose mainly from the assumption of simple turbulent transfer with constant surface roughness, the restriction of temperature measurements to one level, and the simplified treatment of coupling effects. Instrumental precision was not a significant constraint on accuracy. From these considerations, the error in the sensible heat flux was estimated to be ± 20%.

The latent-heat fluxes were computed from Equation (10) with daily means of wind and water-vapour pressure. No correction was applied for coupling because error sources were dominated by the uncertainty in calibration of the dew-cell. This caused possible systematic errors of ±0.4 deg in the measured dew points and ±0.3 MJ m⁻² in the daily totals of latent-heat flux.

Summer heat and ice balances

Daily values of the radiative, sensible, and latent-heat fluxes were computed for the summer period of 155 d. Table III lists the components of the heat balance. Radiation supplied (54 ± 6)% of the heat income while the remaining (46 ± 6)% came as sensible heat. The net loss of latent heat was insignificant. Almost all the heat income was available to melt ice. The mean summer albedo (= 0.54) was probably lower than usual because ice would be exposed relatively early in such a warm melt season.

Table III.

Summer Heat Balance, Hodges Glacier 1 November 1973–4 April 1974 (155 d)

The reduction of insolation by slope and screening at the meteorological site was studied by the method of Gamier and Ohmura (1968). The daily direct insolation on the glacier surface under clear skies was compared with that on an unscreened plateau at the same latitude (Fig. 4). During summer, the deficit ranges from 40% at the equinox to 10% at the solstice; for the melt season (November–March) it averages 25%.

Fig. 4.

Daily direct insolation under clear skies at (a) the meteorological site on Hodges Glacier and at (b) an unscreened plateau in the same latitude.

The principal components of the ice balance are shown in Table IV together with the melt determined from the heat balance. The ablation of summer snowfalls accounted for (7 ± 3)% of the observed melt; wastage of winter snow, firn, and glacier ice accounted for the remaining (93 ± 3)%. The melt computed from the heat balance was only 2% greater than the observed melt, the small difference being well within the estimated errors.

Table IV.

Summer Ice Balance. Hodges Glacier Meteorological Site. 1 November 1973–4 April 1974

Heat and ice balances of the budget intervals

The effect of albedo changes and föhn winds on the radiative and sensible heat fluxes was studied by dividing the melt season according to periods of high albedo (intervals I–VI), intermediate albedo (VII–VIII), and low albedo (IX–XV). For each period Table V shows the surface albedo, the incidence of föhn winds, the heat fluxes, and their ratio. Corresponding data are also shown for the entire summer.

Table V.

Factors Affecting the Radiative and Sensible Heat Fluxes

The surface albedo of the glacier decreased consistently as firn and then ice were exposed by summer melting. Radiative heating was greatest during the period of intermediate albedos occurring 2–4 weeks after the summer solstice. The heating was about 50% less during the initial high-albedo period, and about 20% less during the period of low albedos towards the end of summer. The latter reduction arose from enhanced mountain screening as the sun’s declination moved northwards (Fig. 4) and not from an increase in cloud cover (Fig. 3).

Föhn winds were most frequent and intense during the final period of low surface albedos. Sensible heating was then more than twice that during the earlier periods. Although there were no föhn winds during the period of intermediate albedos, sensible heating was greater than during the initial period (when there were three föhn winds) because air temperatures were generally higher (Fig. 3).

The relative importance of radiative and sensible heating was calculated for each period by the ratio F _r : F _h. During the first period, radiative heating was limited by the high albedo but still predominated over sensible heating, which was heavily constrained by the low temperatures and the infrequency of föhn winds. During the second period radiation was the dominant heat source because albedos were lower and there.were no föhn winds to augment sensible heating. During the final period the high incidence of föhn winds caused sensible heating to predominate, even though radiative heating was sustained by the low surface albedo. The variations in F _r : F _h (Table V) reflect changes in the incidence of föhn winds rather than the development of lower surface albedos during summer. This is true in spite of the fact that the three periods were defined by albedo regime.

The effect of föhn winds in controlling the summer heat partition on Hodges Glacier can be shown for budget intervals as well as for periods of differing albedo. During the intervals in which föhn winds occurred, the ratio F _r : F _h had an average value of 47 : 53, whereas, in intervals without them, the average value was 63 : 37.

The greatest rate of melting occurred during interval IX when ice surfaces of low albedo enhanced radiative heating and two föhn winds brought rapid transfers of sensible heat (Fig. 5a and b). Melting was slowest during interval II when air temperatures were generally below 0° C (Fig. 3) and most of the available heat was used for evaporation. Interval II was the only period in which convection failed to contribute heat for melting.

Fig. 5.

Components of heat balance (a) and comparison of heal and ice balances (b) at the meteorological site on Hodges Glacier. Components and balances are both shown for 15 budget intervals (I–XV) and for the entire summer 1 November 1973 to 4 April 1974. Intervals V and VI are combined in (b)—see text. Arrows (a) denote days with vigorous föhn winds as: ↓ föhn with 15° C> maximum air temperature >10^ºC and relative humidity <60%. ⬇ föhn with 20° C > maximum air temperature >15° C and relative humidity <60%.

Although latent-heat fluxes were generally insignificant, the results suggested (Fig. 5a) that there was net evaporation in early and late summer when water-vapour pressure was relatively low (Fig. 3). In mid-season, however, as water-vapour pressures rose, condensation prevailed and became pronounced during intervals IX, X and XII, when föhn winds were particularly active.

For most intervals there was broad agreement between the heat and ice balances (Fig. 5b). For ease of analysis, heat balance intervals began and ended at midnight but ice balance intervals were defined by stake measurements made during daytime. The differences in scheduling (c. 12 h) were short in relation to the length of each interval (6–21 d) and only once caused a significant error. This was on 28 December when a vigorous föhn wind occurred between the end of heat balance interval V (midnight on 27 December) and the end of the corresponding ice balance interval (afternoon of 28 December). In this case, scheduling errors were eliminated by combining intervals V and VI for comparison of heat and ice balances.

The differences between the heat and ice balances in each interval gave no indication that any component of the heat-balance estimates was a predominant source of error. The overall agreement was 11% for intervals III–X when melting temperatures prevailed and the meteorological site was fully instrumented. Thereafter, the site was gradually dismantled and the deterioration in agreement was attributed to an increasing reliance on extrapolated meteorological data. Although the agreement between summer balances was within 2%, this was probably fortuitous and the 11% agreement found over intervals III–X is considered more indicative of the general accuracy attained.