Estimating glacier melt from bulk-exchange coefficients

Sir,

There are two ways to study ablation-climate relations. The first is to measure energy exchanges at the glacier surface and the second is to find simple correlations between ablation and selected climate elements. Supporters of the first approach claim that theirs is more physical, while supporters of the second are sure that their approach is more useful. Kuhn (1979) made an admirable step towards reconciling these approaches with his heat-transfer coefficient α, which is physically based and is useful for calculating sensible-heat flux from air temperature. The paper by Hay and Fitzharris (1988) can be similarly welcomed as describing physical models to estimate turbulent-heat fluxes from simple meteorological data, but Ambach (1986) has already done this in greater detail.

The calculation of sensible- and latent-heat fluxes by Ambach (1986) is based upon energy-balance data from the ablation area of the Greenland ice sheet (Ambach, 1963) and from the accumulation area (Ambach, 1977). Although the paper’s title refers to nomographs, all assumptions are clearly presented so that the reader can choose between graphical and numerical methods. The sensible-heat flux Q_s for a Prandtl-type boundary layer over a melting glacier surface is given by:

(1)

where b is the average atmospheric pressure at the site in question, v is wind speed, and T is air temperature. The subscript “2” denotes a measuring height of 2 m above the glacier surface. The coefficient K_s depends upon surface-roughness parameters z _0w and z _0T, referring respectively to wind and temperature profiles. Ambach (1986) gave different K _s values for snow and ice surfaces because of their differing surface roughness.

For a similar assumption of a melting glacier surface, Hay and Fitzharris (1988) expressed their sensible-heat flux Q _H by:

(2)

where p _a is the density of air, c _p is the specific heat of air at constant pressure (1010 J kg⁻¹ deg⁻¹), and K is a dimensionless bulk-exchange coefficient for the surface layer. Equation (2) can be criticized as the site elevation is only implicit in the density p _a. This can be overcome by setting p _a = p ₀(b/b ₀) where p ₀ is the density of air (1.29 kg m⁻³) at standard temperature and pressure, b ₀ is standard air pressure (101 300 Pa), and b is the actual air pressure at the site which depends on elevation. Substituting into Equation (2) gives:

(3)

where γ = (P ₀ c _p/b ₀. Tne resemblance between Equations (1) and (3) is clear with:

(4)

For sensible-heat flux in J m⁻²s⁻¹ units, K _s = 2.46 × 10⁻⁵ for the assumptions of Ambach (1986), and γ = 1.29 10⁻². Substitution of these into Equation (4) gives K = 1.91 × 10⁻³, which is lower than 3.9 × 10⁻³ found by Hay and Fitzharris for Ivory Glacier, New Zealand. However, in the latter case, the same bulk-exchange coefficients have been assumed for wind, temperature, and humidity profiles, which implies that the same roughness parameters are also assumed for the three quantities. The surface roughness on Ivory Glacier is 0.014 m, which is much greater than the 0.002 m assumed by Ambach (1986) for the wind profile over an ice surface. Substituting this greater roughness for both wind and temperature profiles into the Ambach model gives K = 6.83 × 10⁻³, which is higher than the value found by Hay and Fitzharris.

The above discussion indicates a factor-of-two agreement between the approaches of Hay and Fitzharris (1988) and Ambach (1986), which is very encouraging and supports the basic concept of a simple relation between sensible-heat flux and the product of temperature and wind speed as expressed by Equations (1) and (3). However, the aerodynamic assumptions by Ambach (1986) seem less restrictive than those of Hay and Fitzharris, and may therefore be more correct. More important, Ambach (1986) took account of the aerodynamic differences of snow and ice surfaces, e.g. the sensible-heat flux to a snow surface is only 70% of the flux to an ice surface with the same temperature and wind conditions. This is definitely needed for any serious attempt to explain the past behaviour of glaciers with high accumulation rates like Ivory Glacier.

My colleague and I have previously stressed the simple relation between ablation and temperature in analysis of our ablation data from Greenland (Braithwaite and Olesen, 1985, in press). However, we have recently tested a simple energy-balance model using turbulent-flux equations from Ambach (1986) and long-wave radiation equations from Ohmura (1981). Ablation calculations by the model are surprisingly accurate and will be described in a future paper.

Roger J. Braithwaite

Grønlands Geologiske Undersøgelse, Øster Voldgade 10, DK-1350 København K, Denmark

18 July 1988