Notation

• A Dimensionless bulk-transfer coefficient

• A_N As above, neutral conditions

• A_s As above, stable conditions

• b Atmospheric pressure (Pa)

• b ₀ Standard atmospheric pressure (1.013 × 10₅Pa)

• C_p Specific heat of air, constant pressure (1005 J kg⁻¹ deg⁻¹)

• g Gravitational acceleration (9.81 m s⁻²)

• H Turbulent sensible-heat flux (W m⁻²)

• H _N As above, neutral conditions (W m⁻²)

• H _S As above, stable conditions (W m⁻²)

• k von Karman’s constant (0.41)

• K _H Coefficient of eddy diffisivity (m²s⁻¹

• K _M Coefficient of eddy viscosity (m²s⁻¹)

• L obukhov scale length (m)

• N Sample size (d)

• R Product-moment correlation coefficient

• Ri Bulk Richardson number

• T Air temperature at 2 m above glacier (deg)

• T _K Absolute air temperature (K)

• T ₀ Surface temperature of glacier (0deg)

• u Wind speed 2 m ahove glacier (ms⁻¹)

• u _* Friction velocity (ms⁻¹)

• z instrument height (2 m)

• z ₀ Surface roughness for sensible-heat flux (1.7 × 10⁻⁴m)

• z _0u Surface roughness for wind speed (2 × 10⁻³m)

• z _0T Surface roughness for temperature (6 × 10⁻⁶m)

• α Empirical parameter for sensible-heat flux (5)

• α _U As above, log-linear wind profile

• α _T As above, log-linear temperature profile

• β Empirical beat-transfer coefficient(Wm⁻² deg⁻¹)

• Γ Adiabatic lapse rate (9.8 × 10⁻³ deg m⁻¹)

• ρ Density of air (k gm⁻³)

• ρ ₀ Standard density of air (1.29 k gm ³)

• τ Turbulent shear stress at glacier surface (kg m⁻¹ s⁻²)

Introduction

The energy supply for melting at glacier surfaces comes from several sources, net radiation and turbulent sensible-heat flux being the main ones. The relative magnitudes of the energy sources vary with situation but net radiation is generally the largest energy source on most glaciers (Reference PatersonPaterson, 1969, p. 58–61), and results from the margin of the Greenland ice sheet (Reference Braithwaite and OlesenBraithwaite and Olesen, 1990a) agree with this pattern. However, if air temperatures rise, e g. due to the enhanced greenhouse effect, more than half of the increased ablation will be caused by an increase in sensible-heat flux (Reference Braithwaite and OlesenBraithwaite and Olesen, 1990c). This statement refers to sensible-heat flux calculated for a neutral boundary layer, although rather stable aerodynamic conditions can be expected (Reference Grainger and ListerGrainger and Lister, 1966), and a temperature increase will involve increased stability. The present paper therefore re-examines the calculation of sensible-heat flux to assess the effect of aerodynamic stability and to see if previous conclusions about increased melting from Greenland need revision.

Background

The vertical turbulent sensible-heat flux H is expressed in flux-gradient form as:

where ρ is the density of air, c _p is the specific heat of air, K _H is the coefficient of turbulent diffusivity, dΤ/dz is the vertical temperature gradient and Γ is the adiabatic lapse rate. As the present paper only deals with the air layer close to the glacier surface (instrument height 2 m) with large air-temperature gradients, Γ is neglected compared with dΤ/dz. In the present paper, sensible-heat flux towards the glacier surface is taken as positive.

Despite the analogy of Equation (1) with the classic heat-conduction equation, the eddy diffusivity K _H is not a simple property of air. There is a substantial literature on how K _H varies with height over the surface and with the conditions of turbulence (Reference Panofsky and DuttonPanofsky and Dutton, 1984; Reference GarrattGarratt, 1992).

The turbulent shear stress τ at the glacier surface is the vertical flux of horizontal momentum:

where K _M is the coefficient of eddy viscosity and du/dz is the vertical gradient of the horizontal wind speed. In laboratory studies of turbulence, the quantity [τ/ρ)^0.5 is called the friction velocity u_*.

The essential point of the flux-gradient approach is to assume some kind of similarity, i.e. the Reynolds analogy, between K _H, and K _M so that K _H can be estimated from the vertical wind profile immediately above the surface. Reference Grainger and ListerGrainger and Lister (1966) discussed wind profiles that have been used in glaciological studies.

Logarithmic Wind Profile

The simplest, and possibly most elegant, treatment of sensible-heat flux assumes that the wind speed increases as the natural logarithm of the height above the surface, i.e. the so-called logarithmic profile (Reference GarrattGarratt, 1992, p. 45; Reference PatersonPaterson, 1994, p. 60–66):

where u _* is the friction velocity, k is von Karman’s constant that appears in laboratory studies of turbulence, and z _0U is the surface roughness, representing the very small height above the glacier surface where wind speed u is zero. The logarithmic profile is based on the assumption that fluxes of momentum and heat are constant with height within the surface boundary layer (inertial sub-layer) immediately over the glacier and that the eddy viscosity K _M is proportional to height z:

This is only strictly valid for a neutral atmosphere but Reference Grainger and ListerGrainger and Lister (1966) suggested that the logarithmic profile is applicable over a wide range of stability conditions and it has been used without modification by Reference FöhnFöhn (1973), Reference MartinMartin (1975), Reference PoggiPoggi (1977) and Reference Hogg, Paren and TimmisHogg and others (1982). From the above, turbulent flux in a neutral atmosphere H _N is:

where z _0T is the surface roughness for the temperature, i.e. the height where Τ = T ₀ the surface temperature of the glacier. Throughout this paper, the glacier surface is assumed to be melting so that T ₀ = 0 deg. Elimination of the friction velocity between Equation (4) and (5) gives:

where A is a dimensionless bulk-transfer coefficient. For neutral conditions, this is equal to A _N:

The wind speed u and temperature Τ are measured at the height z (2 m for present purposes) and A _N is about 0.002–0.004 for melting snow and ice surfaces (Reference PatersonPaterson, 1994, p. 65). The density of air ρ in Equation (6) is estimated from the atmospheric pressure b:

where ρ0 is the standard density of air and b ₀ is the standard pressure, and b is the mean atmospheric pressure for the particular site that can be estimated from its elevation above sea level using a standard atmosphere equation.

Equations (6)–(8) mean that sensible-heat flux can be calculated from wind and temperature data at one height z if the bulk-transfer coefficient, which depends on the surface roughness lengths z _0T and z _0U, is known. The principle of calculating snow or ice ablation from simple meteorological data has been known for a long time (Reference ÅngströmÅngström, 1933; Reference SverdrupSverdrup, 1995, Reference WilsonWilson, 1941) and the specific form in Equation (6) and (7) has been used in Greenland by Reference AmbachAmbach (1986), Reference Braithwaite and OlesenBraithwaite and Olesen (1990a,Reference Braithwaite and Olesenc) and Reference Wal, van de and Russellvan de Wal and Russell (1994).

A wide range has been reported in the hterature (Table 1) for surface roughness z _0U over ice, mainly reflecting the effect of micro- and meso-scale topography on surface roughness (Reference MunroMunro, 1989), although Reference MorrisMorris (1989) suggested that some of the larger roughness values may be caused by slope errors. This may account for the values in Table 1 for the GIMEX profile, in West Greenland, which are especially problematic because the roughness apparently increases from the ice-sheet margin (site 4) towards the interior (site 9), contrary to intuition.

Table 1.

Surface roughness for wind speed over ice in units 10⁻³m. Updated from Reference KuhnKuhn (1979) and Reference MorrisMorris (1989)

Reference AmbachAmbach (1986)suggested different roughness for wind speed, air temperature and humidity for ice and snow surfaces (Table 2) on the basis of energy-balance studies in both the ablation area (Reference AmbachAmbach, 1963) and accumulation area (Reference AmbachAmbach, 1977). Reference HolmgrenHolmgren (1971) also found that the surface raughness is lower for snow than ice, but Reference MunroMunro (1989) and Reference Bintanja and van den BroekeBintanja and van den Broeke (1994) found the opposite, and some textbooks give snow a greater roughness than ice (Reference OkeOke, 1978; Reference Panofsky and DuttonPanofsky and Dutton, 1984). There is general agreement that the surface roughness of snow increases with wind speed due to the effects of drifting (Reference GarrattGarratt, 1992, p. 97–103), thus accounting for some large snow-roughness values, but drift is largely excluded if one is talking about melting snow and is absent for melting ice. Day-to-day variations of several orders of magnitude in surface roughness have also been reported for melting snow (Reference Plüss and Mazzoni.Plüss and Mazzoni 1994), presumably reflecting real variations as well a: measurement errors, which must be considerable.

Table 2.

Surface roughness for wind speed, air temperature and humidity profiles according to Reference AmbachAmbach (1986). Units are metres

Less is known about the surface roughness z _0T for temperature but Reference SverdrupSverdrup (1995), Reference HolmgrenHolmgren (1971) and Reference AmbachAmbach (1986) have agreed that it is about two orders of magnitude smaller than z _0U. Although some author assume the roughness is the same for both profiles, there i no reason, in principle, why they should be the sam because the transfer processes are different close to the surface, e.g. there is no flux of momentum through the interficial sub-layer (of thickness z _0U) while heat transfer is by molecular diffusion. This prompted Reference AndreasAndreas (1987) to express the ratio z _0T/z _0U as a function of the roughness Reynolds number, and with the appropriate values Andreas’s model predictsz _0T that sen should be 10–100 times smaller than z _0U in rough agreement with Reference AmbachAmbach (1986) but Reference MunroMunro (1989) and Reference King and AndersonKing and Anderson (1994) found to the contrary.

In view of the above disagreements, to simplify the treatment, and to anticipate the result that Arnbachs surface roughness lengths need not apply to the ice-area, even if they apply to the upper ablation area, it is convenient to follow Reference Morris and HardingMorris and Harding (1991) in assuming that z _0U = z _0T = z ₀ the effective roughness for sensible-heat flux such that:

For Equation (9) to give the same numerical value for A _N as Equation (7) with the ice roughness from Reference AmbachAmbach (1986), i.e. z_0U = 2 × 10⁻³m and z_0T = 6 × 10⁻⁶ m for ice, an effective value of z0 = 1.7 × 10⁻⁴ m is assumed.

The sensible heat flux (Fig. 1) is calculated according to Equation (6) and (9) for different values of temperature and wind speed. The atmospheric pressure is 90 kPa, corresponding to an elevation of about 1000 m a.s.l., which is typical of the middle of the ablation area in West Greenland. The range of temperature and wind is chosen to be typical of those prevailing at Nordbogletscher and Qamanârssûp sermia (West Greenland) during the summer months June–August. The sensible-heat flux in the model is linear with respect to increasing temperature. This is true as long as the glacier surface is but surface temperatures need not remain at the point even with air temperatures above 0 deg (Reference KuhnKuhn, 1987), so there might be greater sensible-heat flux to the (non-melting) glacier surface at the lower air temperatures.

Fig. 1.

Sensible-heat flux for the logarithmic wind profile with wind speeds 1—9 m s⁻¹. Temperature and wind at 2 m above melting glacier surface.

It is now necessary to consider the effect of stability on the sensible-heat flux that has been neglected up to this point.

Aerodynamic Stability Over Melting Ice

When an air parcel is displaced upwards by turbulence from its usual height, it cools at the adiabatic lapse rate Γ. The air parcel experiences negative or positive buoyancy forces according to whether it is now warmer (less dense) or cooler (more dense) than its new surroundings and the buoyancy forces tend to amplify or inhibit the further movement of the air parcel. In the case of a melting glacier in the summer, the air temperature generally increases in the first few metres over the surface and buoyancy forces inhibit turbulence (Reference Grainger and ListerGrainger and Lister, 1966). The surface layer is said to be aerodynamically stable. The derivation of the sensible-heat flux for a logarithmic wind profile neglects the effect of stability and the flux in Figure 1 may therefore be overestimated.

The stability, or otherwise, of the surface Layer is described by the bulk Richardson number (Ri) which relates the relative effects of buoyancy to mechanical forces (Reference OkeOke, 1978). The bulk Richardson number at height z over a melting surface (T ₀ = 0 deg) is:

where g is the acceleration of gravity and T _k is the air temperature on the absolute scale (K). Ri is positive in a stable atmosphere.

The variations of Richardson number with wind speed u and temperature T (both at 2 m) are shown in Figure 2 where the glacier surface is once agatn assumed to be melting. The boundary between “fully forced convection” (neutral)and “damped forced convection” (stable) is Ri = 0.01 (Reference OkeOke, 1978) and the Critical value of Ri for the transition to “no convection” is about + 0.2 (Reference WebbWebb, 1970) The variations of Ri (Fig. 2) show that turbulent conditions may be close to neutral for higher wind speeds, i.e. 7 ms⁻¹ or more, up to quite high temperatures. Stable conditions may be expected for medium wind speeds, 3–5 m s⁻¹, over most of the temperature range while turbulence essentially disappears for low wind speed, 1 m s ⁻¹ or less.

Fig. 2.

Bulk Richardson number at 2 m above a melting ice surface with wind speeds of 1—9 m s⁻¹.

Some authors correct the sensible-heat flux over melting snow or ice for effects of stability, either by functions of the bulk Richdrdson number or by using the Monin–Obukhov stability function ϕ (Reference Panofsky and DuttonPanofsky and Dutton, 1984; Reference GarrattGarratt, 1992) that is related to Ri.

Snowmelt studies by Reference Price and DunnePrice and Dunne (1976) and Reference BraunBraun (1985) used:

where H _S is the sensible-heat flux in stable conditions and H _N is for neutral conditions. The correction factor (Fig. 3) shows only small variations up to quite high temperatures for higher wind speeds but is small for low wind speeds where the Richardson number is high. Another correction factor has been used for snowmelt by Reference WeismanWeisman (1977) and Reference MooreMoore (1983), and on a glacier by Reference Hay and FitzharrisHay and Fitzharris (1988):

This correction factor (Fig. 4) is essentially the same as the previous one (Fig. 3) for higher wind speeds (5–9 m s⁻¹) but drops to zero much more rapidly for lower wind speeds ((1 m s⁻¹), i.e. it shows a stronger cut-off of sensible-heat flux due to stability.

Fig. 3.

Stability correction factor 1 /1 + 10Ri)² at 2 m above a melting glacier.

Fig. 4.

Stability correction factor (1–5Ri)² at 2 m above a melting glacier.

The sensible-heat flux using the logarithmic wind may be overestimated under the very stable conditions encountered at low wind speeds over a melting glacier. On the other hand, stability effects are fairly small at the high wind speeds that are common on the Greenland ice sheet. As an alternative to using the correction factors in Figures 3 and 4, the use of a different wind profile is now considered.

Log–Linear Wind Profile

For the log–linear profile, based on the work of Monin and Obukhov (Reference GarrattGarratt, 1992, p, 52—54), the wind at height z is given by:

where α is an experimental parameter and L is the scale height of Reference ObukhovObukhov (1971):

Some authors, e.g. Reference KrausKraus (1973), have included an extra term in the denominator of L to incorporate the effects of vapour stratification on buoyancy but Obukhov’s original formulation is used here. By analogy with Equation (9), and assuming again the same parameters for wind and temperature profiles, the bulk-transfer coefficient A _s for stable conditions is:

There is an extensive literature on α, summarized by Reference GarrattGarratt (1992, p.289) but this study follows Reference MunroMunro (1989) in assuming that α = 5 for both wind and temperature profiles in stable conditions as proposed by Reference DyerDyer (1974).

The log–linear profile has been used over glaciers by Reference Grainger and ListerGrainger and Lister (1966), (Reference KrausKraus 1973,Reference Kraus1975), Reference Casinière and delaCasiniére (1974), Reference DerikxDerikx (1975), Reference Munro and DaviesMunro and Davies (1977), and (Reference MunroMunro 1989, Reference Munro1990), and in Greenland by Reference Duynkerke and van den BroekeDuynkerke and van den Broeke (1994). One apparent difficulty is that the Obukhov length L is needed to calculate the sensible-heat flux while one needs to know the sensible-heat flux to calculate L. Reference MunroMunro (1989) overcame this vicious circle by an iterative procedure whereby Η is first calculated for z/L = 0 (neutral case), then a new L is calculated from H and an updated H is calculated from the new L, the whole procedure being repeated for a number of iterations. The present study agrees with Reference MunroMunro (1989) that the iterations quickly converge and the calculation of sensible-heat flux for the log–linear profile is therefore hardly more time-consuming than for the logarithmic profile.

Sensible-heat fluxes for the log–linear profile (Fig. 5) are generally similar to those for the logarithmic profile (Fig. 1) except that (1) sensible-heat flux is no longer exactly linear with temperature and (2) fluxes are lower for the same wind and temperature conditions as before, reflecting the inhibition of turbulence by stability. In particular, sensible-heat flux becomes zero at very low winds speeds (1 ms⁻¹), although there is a non-zero heat flux (too small to he seen in Figure 5) at both low wind speeds and low temperatures (1—2 deg).

Fig. 5.

Sensible-heath flux for the log—linear wind profile with wind speeds 1–9 ms⁻¹. Temperature and wind at 2 m above melting glacier surface.

Relative Effects of Stability and Roughness

Reference Braithwaite and OlesenBraithwaite and Olesen (1990a) calculated the energy balance at two sites in West Greenland with a model that uses the logarithmic wind profile for the sensible-heat flux. This seemed a reasonable thing to do at the time because (1) Reference Grainger and ListerGrainger and Lister (1966) implied that the logarithmic profile is valid over a wide stability range, and (2) ablation calculated for neutral conditions was found to fit observed ablation rather well on the average. However, the calculation of the sensible-heat flux is now repeated using the log—linear profile, with all other data and assumptions as before. For convenience, only cases with Τ ≥ 0 deg are considered (386 d at Nordbogletscher and 480 d at Qamanârssûp sermia) so a frozen glacier is largely excluded.

The mean sensible-heat flux with the log–linear profile is respectively 74 and 84% of the corresponding values for the logarithmic profile (Table 3). In round figures, stability reduces sensible-beat flux by about one-fifth compared with neutral conditions, with a smaller reduction at Qamanârssûp sermia (mean air temperature 5.4 deg and mean wind speed 5.0 ms ⁻¹ both at 2 m above the glacier surface) than at Nordbogletscher (4.1 deg and 3.3 m s ⁻¹, because stability is weaker due to the higher mean wind speed.

Table 3.

Mean sensible-heat fluxes for two sites in West Greenland. Cases with air temperature above 0 deg

The reduction in calculated sensible-heat flux by stability appears to imply that the energy-balance model of Reference Braithwaite and OlesenBraithwaite and Olesen (1990a) overestimates ablation by the same amount that sensible-heat flux is overestimated, i.e. equivalent to overestimating mean ablation by 6 and 8% at the two sites. However, such an overestimation is excluded because there was on average good agreement between measured and calculated ablation at the two sites (Reference Braithwaite and OlesenBraithwaite and Olesen, 1990a). There must therefore be another source of error that more-or-less compensates for the effect of stability. The most likely candidate is the choice of roughness. The preceding calculations all use z ₀= 0.00017 m, which actually refers to the relatively smooth ice studied by Reference AmbachAmbach (1963) in the upper ablation area. The roughness of ice in the outer ablation area, as at Nordbogletscher and Qamanârssûp sermia, could well be an order-of-magnitude greater.

The curves in Figure 6 show that a ten-fold increase in surface roughness z ₀ has a larger effect on sensible-heat flux than the effect of stability as expressed by the difference between logarithmic and log–linear profiles (wind speed 5 ms⁻¹ at 2 m for all cases).

Fig. 6.

Sensible-heat fluxes for different surface roughness and different wind profiles.

Neglect of Wind Variations

The logical extension to calculating sensible-heat flux from temperature and wind-speed data at one height is to use temperature data alone. This is the basis of a method whereby wind speed is “lumped” into a bulk heat-transfer coefficient (Reference KuhnKuhn, 1979; Reference MooreMoore, 1983; Reference Moore and OwensMoore and Owens, 1984: Reference Escher-VetterEscher-Vetter, 1985; Reference Hay and FitzharrisHay and Fitzharris, 1988). Such a simplification is attractive because it is often difficult to get data for wind speed in glacier areas, and especially over the whole Greenland ice sheet, while temperature data can be easily extrapolated from a distant station. The definition of the heat-transfer coefficient β is:

where H is the mean sensible-heat flux over many days and T is the mean temperature over the same period. By comparison with Equation (6), (8)and (15), it is obvious that β must depend implicitly upon mean wind speed, measurement height and the density of air, and thereby elevation above sea level. The transfer coefficient is at most a parameter (quantity constant in the case considered but varying in different cases; The Concise Oxford Dictionary, sixth edition) rather than a strict constant. For the logarithmic wind profile (Fig. 1), sensible-heat flux is certainly proportional to temperature for a constant wind speed and is nearly proportional for most of the log–hear profile (Fig. 5).

For varying wind speed, β also depends upon the correlation between wind speed and temperature, which is positive at both sites due to an association of high wind speeds with high temperatures under Föhn-type events (correlation coefficients of + 0.41 and + 0 42, respectively, for Nordbogletscher and Qamanârssûp sermia). As mean wind speed is implicit in β, a correlation between transfer coefficients and mean wind speeds in different situations may he expected (Reference KuhnKuhn, 1979), although Reference FunkFunk (1985, p. 165–75) did not find notably high correlations (the expected correlations were probably obscured by other effects).

One way of isolating the temperature sensitivity of sensible-heat flux is by simple least-squares regression of the calculated sensible-heat flux on air temperature, i.e. a kind of re-sampling procedure. The resulting regression equation represents the flux-temperature relation for the average wind conditions prevailing in the data set. The regression equation for the log—linear profile at Nordbogletscher (days with Τ ≥ 0 deg) is:

where R is the correlation coefficient and N a the sample size in days. The corresponding equation for the logarithmic profile at Nordhogletscher is:

The equations for Qamanarssûp sermia (days with Τ ≥ 0 deg) ar:

and

The negative intercepts in all four equations have no deep physical significance and probably arise because temperature and wind speed are positively correlated, increasing the slope of the regression line and forcing the intercept below zero. If there were no correlation between wind speed and temperature, the intercept would be essentially zero and the slopes would be less than here.

The slopes in the regression equations have the nature of heat-transfer coefficients. The effect of stability is to reduce these coefficients, by 32 and 16%, respectively, for the present cases, or by about one-quarter in round figures, although, as said previously, this could be offset again by using larger roughness to recalculate the sensible-heat fluxes.

The heat-transfer coefficients for the present cases are compared with others from the literature (Table 4), showing a fairly wide range, which presumably reflects variations in wind speed and roughness. The heat-transfer coefficient of Reference KuhnKuhn (1979) refers to an alpine glacier and is rather larger than the Greenland values in Table 4 despite its greater altitude. Presumably, the surface roughness is large and more than offsets the effect of lower atmospheric pressure (as discussed in the next section).

Table 4.

Heat-transfer coefficients for turbulent heat flux according to various authors

The three transfer coefficients in the middle of Table 4, which have been used for modelling (Reference Oerlemans and HoogendoornOerlemans and Hoogendoorn, 1989; Reference OerlemansOerlemans, 1991,Reference Oerlemans1992), refer to assumed values at the glacier snouts that decrease with altitude but no reasons for choosing different values for the different models are given. The correct choice of transfer coefficient is important for assessing the sea-level rise due to increased melting of mountain glaciers, which Reference Oerlemans and FortuinOerlemans and Fortuin (1992) said is less than previously estimated.

The most serious objection to the widespread use of heat-transfer coefficients is the problem of choosing a suitable value to take account of wind conditions in Greenland, i.e. mean wind speed and surface roughness. Recent research on the dynamics of the boundary layer over the Greenland ice sheet (Reference Oerlemans and VugtsOerlemans and Vugts, 1993) may help us to guess wind-speed distributions where observations are lacking. For example, Reference Meesters, Henneken, Bink, Vugts and CannemeijerMeesters and others (1994) suggested that wind speeds are higher over the upper ablation area than over the tundra and glacier margin.

Geographical Variations of Sensible-Heat Flux

The dependence of sensible-heat flux on the density of air ρ also implies a geographical variation in the size of sensible-heat fluxes, because glaciers in different regions are located at different altitudes. This point is illustrated (Fig. 7) by calculations for different values of atmospheric pressure. 90kPa (typical of the mid-ablation area of the Greenland ice sheet), 70 kPa (Alps and Rockies) and 50 kPa (Andes and Himalaya).

Fig. 7.

Effect cf atmospheric pressure variations on turbulent sensible-heat flux.

All things being equal, the heat flux in the Andes and Himalaya should only be about one-half of that in Greenland for the same temperature and wind speed, although variations within Greenland are relatively small, because of the restricted range of ablation-area altitudes. There are probably also differences in surface roughness for different glacier regions, e.g. due to different frequencies of surface debris, weathering crust and surface features like hummocks, suncups and penitents. For example, in Greenland, the ice surface is more bumpy close to the margin of the ice sheet than close to the equilibrium line. Different temperature sensitivity can therefore be expected for sensible-heat flux in different glacier regions and may contribute to inter-regional variations in positive degree-day factors (Reference BraithwaiteBraithwaite, 1995), although these have not yet been clearly identified.

Discussion

The bulk-exchange coefficient A _s given in Equation (15) is only approximately correct. A more exact formula is:

where different roughness and α parameters are used for wind and temperature profiles. The roughness lengths determine the magnitude of sensible heat for neutral conditions and the α parameters determine the shape of the stability function, i.e. the way in which sensible-heat flux is reduced with increasing stability. More research is needed to find the most appropriate values of these parameters to use in any particular situation. A minimum requirement would be to take account of differences between ice and snow surfaces, as well as accounting for broad differences in surface types.

It is easy to criticize the use of the effective surface roughness z ₀ in the present paper, because there is evidence, both empirical and theoretical, that surface roughness for wind and temperature profiles is generally different. However, it is not immediately clear what values should be used. For example, most evidence suggests that z _0T is much less than z _0U but some authors have suggested the contrary. The effective roughness z ₀ must lie between z _0U and z _0T and, as realistic values of sensible-heat flux can be calculated with z ₀ values that are lower than z _0U values from the literature, the present study implies that z _0T is indeed much less than z _0U.

The important issue is the precise form of stability function, or of α values, to be used in large-scale hydrological and climatological models for climate-change experiments. The assumption of the same α for wind and temperature profiles under stable conditions (α = 5) is taken from Reference DyerDyer (1974) but other values have been proposed (Reference GarrattGarratt, 1992, p. 289). A particularly respected recent study outside Greenland by Reference HögströmHögström (1988) suggests higher α values of 6.0 and 7.8, respectively, for wind and temperature profiles.

For a whole generation, the best information on boundary-layer conditions in Greenland has been from Reference AmbachAmbach (1963,Reference Ambach1977), Reference Lister and TaylorLister and Taylor (1961) and Reference Grainger and ListerGrainger and Lister (1966) hut no discussion on α has been possible. However, in the early 1990s, extensive new data were collected in West Greenland by Dutch and Swiss expeditions (Reference Oerlemans and VugtsOerlemans and Vugts, 1993; Reference OhmuraOhmura and others, 1994), including the first direct measurements in Greenland of turbulent fluxes using eddy-correlation instruments. Preliminary results from there studies (Reference Forrer and RotachForrer and Rotach, 1994; Reference HennekenHenneken, 1994; Reference OhmuraOhmura and others, 1994) indicate a valueseven larger than those proposed by Reference HögströmHögström (1988). The stability factor (1–5Ri)² (Fig. 4) agrees closely with the log–linear profile (α = 5) and a larger α value implies an even sharper cut-off of sensible-heat flux with increasing stability.

The effect of stability on sensible-heat flux over the Greenland ice sheet is well illustrated by Figure 7 in Reference OhmuraOhmura and others (1994), which compares field measurements with curves from Reference WebbWebb (1970) (essentially the same as the Reference DyerDyer (1974) model used here) and Reference HögströmHögström (1988) as well as from the ECHAM 3 global climate model (GCM). The field measurements show a stronger stability effect than either the Webb or Högström models but it is particularly disturbing that the GCM shows a much weaker stability effect, implying serious overesthnation of sensible-heat flux by this GCM. This be remedied before the GCM is used to calculate the impact of climate changes on the Greenland ice sheet.

Conclusions

Aerodynamic stability reduces sensible-heat flux over a melting ice surface compared to that predicted for a stable boundary layer. The stability effect is fairly small for the high wind speeds that are common over the Greenland ice sheet but it is very large at low wind speeds.

The uncertainty in surface roughness probably causes greater error in sensible-heat-flux calculations than the neglect of stability. However, the log–linear wind profile is only slightly more difficult to use than the logarithmic profile and should be used for future calculations of sensible-heat flux, because it is more realistic.

Lack of wind-speed data over the Greenland ice sheet is a problem for calculating sensible-heat flux. The use of a bulk heat-transfer coefficient avoids the need for wind data but there is still an uncertainty about the choice of the coefficient itself.

Sensible-heat flux depends on air pressure and this implies a geographical variation of heat flux because glaciers in different regions are located at different altitudes.

It cannot he concluded definitely that an earlier estimate of increased melting from the Greenland ice sheet (Reference Braithwaite and OlesenBraithwaite and Olesen, 1990a) is too high, because calculated ablation in that study agreed fairly with observed ablation. The effect of neglecting stability was probably offset by underestimation of the surface roughness for sensible-heat flux.