Introduction
The bulk aerodynamic method provides the most practical means of estimating the sensible heat component of ice and snow melt because the only data requirements are wind speed and temperature measurements at one height above the surface. Extreme precision is not required, so thermograph data can be useful. Robust instrumentation can thus be used to allow continued application of the method over long time periods under difficult field conditions.
There remains, however, much uncertainty about bulk-transfer theory, particularly the calculation of exchange coefficients (Reference KuhnKuhn, 1979; Reference MooreMoore, 1983; Reference BraithwaiteBraithwaite, 1988; Reference Hay and FitzharrisHay and Fitzharris, 1988). The problem stems from the variety of conclusions about how to treat stability effects and, in particular, the large range of reported roughness lengths (Reference InoueInoue, 1989a). Seldom does one find a situation in which the bulk-transfer procedure can be compared with a good independent method, particularly if one is dealing with a glacier site.
Such a comparison was available for this study, in the form of eddy-correlation measurements. Attempts to calculate suitable bulk-transfer coefficients, to obtain agreement between the two methods, showed that accurate roughness-length estimates are by far the most important requirement. Therefore, they are the focus of this paper.
Theoretical Background
Eddy correlation (Оке, 1987) is the most direct procedure for measuring the sensible heat-flux density, Q H:
where ρ is the air density, c p is the specific heat of air at constant pressure, w’ and T are the respective deviations of vertical velocity and temperature from their mean values. The method is quite accurate if proper care is taken with sensor design, and field deployment. It is not a convenient method because it uses delicate sensors which require constant care in the field. There is always a substantial risk of damage during bad weather, which can occur unexpectedly in the mountainous surroundings of glacier sites.
The bulk-transfer procedure is preferred over melting ice and snow because the required instrumentation is simple to deploy and to maintain, thus allowing long periods of unattended use in most weather conditions. The theory can be developed from an expression analogous to Equation (1):
in which u* is the friction velocity, or wind-speed scaler, and T* is a temperature scaler. They are estimated from wind-speed and temperature profiles by taking logarithms of the heights, and adding a suitable stability correction.
If the surface is melting ice, or snow, the scalers can also be estimated from the wind speed, uz and temper ature, Tz, measured at height z above the ground:
where к is von Kàrmàn–s constant (0.41), and z0 are, z t respectively, the roughness lengths for wind speed and temperature. The empirical constant, α is part of the stability correction, which varies according to the value of the
Monin-Obukhov length scale, L:
where g is the gravitational acceleration and T is the absolute temperature of the air between the surface and the measurement height. The bulk-transfer formula for sensible heat exchange is obtained by substituting Equation (3) into Equation (2):
The inherent assumption that surface temperature is 0°C for melting ice and snow eliminates a difficult measurement problem. There remains, however, a procedural problem in that the accurate calculation of L requires a priori knowledge of u* and QH. Also, it has been assumed that the stability corrections for wind speed and temperature are similar, a view which is not universally shared, but against which there is not very compelling evidence (Reference MooreMoore, 1983).
Many investigators prefer to avoid the use of L altogether (Reference MooreMoore, 1983; Reference Marcus, Moore and OwensMarcus, 1985; Reference Hay and FitzharrisHay and Fitzharris, 1988), expressing stability in terms of the bulk Richardson number, because it can be evaluated directly from windspeed and temperature data. The Monin-Obukhov length is retained here because much of the good stability-analysis work is expressed in terms of L (Reference DyerDyer, 1974), and the formulation is preferred by many others, particularly those who work with eddy correlation (Reference Banke, Smith and AndersonBanke and others, 1976; Reference InoueInoue, 1989a). The problem of requiring a priori knowledge is circumvented by iteration, an approach which has been successfully applied elsewhere (Reference Berkowicz and PrahmBerkowicz and Prahm, 1982; Reference LeeLee, 1986). In the first iteration, Equations (3a) and (5) are evaluated for the neutral case, where z/L = 0. The results are used to calculate the first estimate of L, which is applied to the subsequent evaluation of u* and QW The cycle is repeated until no significant change occurs, a condition which is satisfied in five iterations (Table I). Most of the correction occurs in the second iteration.
table
The effect of the stability correction, taking α = 5 (Reference DyerDyer, 1974), is to reduce Q H to approximately 80% of the neutral flux estimate in both of the iteration sequences shown (Table 1). Thus, such variation as there is in the α literature signifies a relatively small uncertainty in Q H The two sets of flux densities differ by a factor of two, however, reflecting the order of magnitude difference in the values of z 0 and Z t Roughness lengths reported from various ice and snow studies vary by at least an order of magnitude (Reference MooreMoore, 1983; Reference Marcus, Moore and OwensMarcus, 1985) and, as noted by lnoue (1989a), can do so even for a single study. This inspires very little confidence in using borrowed values, so it seems best to obtain new values, either from profile data, or from microtopographic information.
Profile estimates of the roughness lengths begin with the log-linear equation, in the form expressed by Reference WebbWebb (1970):
where Yu = ∆u /∆(ln z), Yt = ∆T/∆(ln z) and X = ∆ z /∆)(ln z). Plots of Yu and Yt against X can be fitted by linear regression (Fig.1), so Y u = u */k and Y t = u */k at X = 0. It follows that Equation (2) can be evaluated from profile data without knowledge of either α or L.
Fig.1. Example of (a) wind speed, and (b) lemperature-profile analysis for 13.00h. 25 August 1971.
Given the scalers, roughness lengths are obtained by re-arranging Equation (3):
Correction for stability is obtained by re-arranging Equation (6) to solve for α/L and substituting the solution into Equation (7):
Each measurement height, z, is associated with one or more sets of X and Y, depending upon the number of measurements taken. The plot in Figure 1 is based upon four measurements, so it yields six roughness lengths per profile from which to compute an average.
It is rare in ice and snow studies to have access to the precise instrumentation required for the evaluation of Equation (8). Therefore, some investigators have used Reference LettauLettau’s (1969) method of roughness-element description to estimate z0:
where h* is an effective height for the roughness elements, S is their density, or frequency per unit area, and s is the silhouette area of roughness elements measured in a vertical plane perpendicular to the wind direction. An equivalent expression does not exist for zt because it is a function of the flow (Reference Owen and ThompsonOwen and Thompson, 1963), which is characterized by the roughness Reynolds number, Re* = u * z 0/v where v is the kinematic viscosity of the air. Various attempts to model ln(zt/z0 ) as a function of Re* have been described (Reference Owen and ThompsonOwen and Thompson, 1963; Reference Garratt and HicksGarratt and Hicks, 1973; Reference AndreasAndreas, 1987).
The idea that z 0 can be computed from microtopographic information seems suspicious to those who prefer to trust instrumentation, leaving Equation (9) as a method of last resort.
It will be shown here that such trust may be misplaced when measurements are taken close to the surface, as they are here, and that Equation (9), combined with a suitable model for ln(zt/z0 ), is likely to provide the simplest, most accurate approach to dealing with the roughness-length problem.
Experimental Details
Micrometeorological data were collected at two sites on Peyto Glacier during three summers. One site was the exposed ice of the glacier tongue, which exhibits a distinctive grain, aligned parallel to the ice flow. The grain is imparted by a continuous cover of elongated ice hummocks, of various sizes, the long axes of which point down the glacier tongue. The other site was a ripe snow cover in the upper basin of the glacier, described by Reference FöhnFöhn (1973) as comprising sun cups, where there appeared to be no preferred orientation to the pattern. Katabatic forces are important, even dominant at times, in establishing the surface-wind regime (Reference Munro and DaviesMunro and Davies, 1977). Therefore, down-slope winds which blow approximately parallel to the direction of the ice grain are common over the glacier tongue, particularly during anticyclonic weather conditions (Reference Stenning, Banfield and YoungStenning and others, 1981).
Data were first collected in 1971, when detailed temperature and wind-profile measurements were made at the ice site to determine boundary-layer thickness and stability relationships. The instruments were mounted on independent masts located 10 m apart across the ice grain. The sensors were a sensitive anemometer system (Model 106, Thornthwaite Associates, U.S.A.), with a precision of ±1% of indicated wind speed, and a ventilated thermometer system (modified design of Reference Lourence and PruittLourence and Pruitt (1969), with a precision of ±0.01°C The data were collected on strip charts and subsequently digitized.
Among more than 600 profiles collected, only 122 were found to have sufficient resolution for boundary-layer analysis, so they are the basis of the roughness-length estimates reported here. Reference Munro and DaviesMunro and Davis (1977) found that in such cases only the first four measurement levels, at 0.25, 0.5, 0.75, and I m above the ice, were positioned within the boundary layer, so the application of Equation (8) is restricted to these levels. Later, Reference Munro and DaviesMunro and Davies (1978) found equality between the stability corrections for temperature and wind speed, with α = 4.5, a value which is close to the number 5 quoted by Reference DyerDyer (1974) as being typical of findings in stable conditions.
Simultaneous bulk-transfer and eddy-correlation measurements were made at the ice site in 1987, and at the snow site in 1988. Instruments were mounted at I m above the surface, a height which corresponds with expected boundary-layer thickness. Wind speed was measured with a contact closure anemometer (Model 014A, Campbell Scientific. U.S.A.) to ±1.5% of the measured speed, while temperature was measured with a resistance thermometer (Model 207, Campbell Scientific, U.S.A.) to a precision of ±0.1°C. Temperature and wind-speed measurements were repeated at 5 m above the ice and 4 m above the snow. The data were processed by a compact data logger (Model CR 21X, Campbell Scientific, U.S.A.) and recorded on tape at hourly intervals.
The same type of data logger was used to record eddy correlation data taken at 1 m above the surface, using a commercially available sensor (Model CA 27, Campbell Scientific, U.S.A.), which follows the design of Campbell and Unsworth (1979). The 1 m height was chosen as a compromise between being far enough away from the surface to sense all significant eddy fluctuation, and close enough to be within the boundary layer. A 5 Hz sampling rate was chosen, the fastest which could be accommodated by the data logger, and 5 min averages were stored on tape. They were averaged further in the subsequent data analysis to produce hourly sensible heat-flux values against which to compare the bulk-transfer estimates.
Although eddy correlation can be precise to within ±10%, it is subject to systematic error (Reference TannerTanner, 1988). The sampling rate may be too slow to sense all the turbulence, thus reducing QH. This is mitigated by locating the sensor as far away from the surface as possible, though the maximum distance possible here was only I m. Both sites were located on gently sloping surfaces, where the mean vertical wind speed can deviate from zero, thus changing QH from its true value. The decision to record 5 min averages may have filtered out significant error from this source. Weather presented a problem, as only 71 h of eddycorrelation data were collected over the ice before a summer storm stopped operation of the equipment. The same situation occurred at the snow site after just 6 h of data collection.
Microtopographic data were also collected on the ice in 1971 by conducting a micro-relief survey on a 20 m by 20 m grid, aligned with the ice grain. Elevations were leveled at 0.5 m intervals across the grain, and I m intervals along the grain, to establish a zero reference level for the instrument masts. The grid also serves as an excellent data source for roughness-length work because it comprised transects which can be analysed either across the grain, or along the grain of the ice. Detailed snow-surface measurements were made near the snow site in 1970 by Reference FöhnFöhn (1973), who represented the sun cups as sine waves and calculated z0 accordingly.
Other methods of collecting microtopographic data were tried in 1987 and 1988. One consisted of laying down a 5 m long aluminium pole across the ice grain, the direction most likely to be perpendicular to the wind, and allowing it to melt into position for a few days. Then 30 measurements of the height of the ice surface, relative to the centre line of the pole, were taken at equal intervals along the length of the pole. The pole was subsequently moved to five other sites, where the procedure was repeated. This was not attempted on the snow in 1988, but the pattern of ripe snow stood out clearly in photographs of the instrumentation. Because instrument dimensions are known quantities, it was found that a typical size and density could be assigned to the sun cups by comparing them to the instruments.
Results And Discussion
Mean roughness lengths obtained from wind-speed and temperature profile analysis were found to be extremely sensitive to measurement height ( Fig.2). The effect of adding 0.1 m to each z in Equations (7) and (8) is to increase z0 and zt by an order of magnitude. The corresponding effect upon the sensible heat flux is to double the value of QH . Ice hummocks can easily have a local relief of 0.3 m, or more, thus making it impossible to define a zero reference by eye with sufficient accuracy for profile work. Even if one can set a zero reference from a micro-relief survey, as was done in this case, there is no assurance that it represents the zero level for the flow, which responds to surface conditions far up-wind of the limits of the micro-relief survey.
Fig.2. Effeci of height correction on z0 (dashed line). Zt (dotted line), and QH (solid line).
The zero-referencing problem has also been noted in Antarctica by Reference InoueInoue (1989b), who consequently did not use measurements closer than 1 m to the surface. Such a strategy cannot be employed here because the katabatic wind layer on Peyto Glacier is thin by comparison with that of the Antarctic Plateau, so all the useful measurements are in the first meter above the ice. Zero referencing is not so critical for bulk-transfer work, where an error of 0.1 m in a 1 m measurement height causes only a 3% change in QH . It is crucial to have accurate roughness lengths, however, and therefore important to see what can be done with microtopographic data.
The rationale for microtopographic estimates of z0 is outlined schematically in Figure 3, where a cross-grain surface profile from the 1971 work is plotted. The value of h* is taken to be twice the standard deviation of the elevations, where the mean elevation is set equal to zero. The number of positive extensions of the plot above the mean elevation determines the frequency of roughness elements per unit length, and the square of the frequency defines S. Thus, the roughness elements are characterized as equally spaced rectangles of area s.
Fig.3. Example of a cross-grain height profile (solid line), and its representation by the standard-deviation approach (dashed line). The area measured by planimetry is shaded.
The characterization differs from that which has been used by some others (Reference WeilerWeiler, 1968; Reference FöhnFöhn, 1973; Reference PricePrice, 1977) in that no attempt is made to replicate the typical roughness-element form. However, it has the advantage of computational simplicity, and it avoids the need to make judgements about form. Such judgements might be relatively easy to make for ripe snow, but very difficult, if not impossible, to make for melting ice. Also, the silhouette area of the roughness elements portrayed in Figure 3 was checked by planimetry and found to be only 12% greater than that which was yielded by the standard-deviation approach. This translates to a 2% error in bulk-transfer estimates, which is likely to decrease as the number of z0 samples increases.
The z0 results for ice and snow are remarkably consistent, despite differences in timing and measurement procedures (Table II). The 1971 ice estimate is the average of 21 z0 values based upon cross-grain traverses which are each 20 m in length. The 1987 value is virtually the same, yet it is based upon only six cross-grain traverses, each of which was only 5 m long. Also, Reference FöhnF±hn–s (1973) value agrees quite well with the photographic estimate of 1988. Although photographic information should probably not be preferred over micro-relief data, it is interesting to see that it can be useful.
The main disagreement is between z0 values obtained from traverses taken across the grain and along the grain of the ice. The latter are smaller by a factor of 4. Such directional variation has recently been documented from wind-profile data as well (Reference InoueInoue, 1989a). Thus, wind direction could be significant in heat transfer over glacier ice. The same is not likely to apply to melting snow because sun cups are arranged in isotropic patterns. Therefore, the movement of the snow line is significant not only because it marks a change in surface roughness, as noted by Arnbach (1985), but also because the orientation of the surface with respect to the wind affects z„. The wind direction on the glacier tongue was approximately along the grain, so the concern was to get the cross-grain z0 values to agree with those from the wind profiles.
Agreement between profile and micro-relief estimates of z0 was forced by adding 0.17 m to the original z values. This implies that the roughness length is controlled primarily by the form of the ice hummocks rather than the fine structure of the ice itself, as expressed in crystal size and cavities created through differential melting. The implication seems reasonable in view of the work of Reference Banke, Smith and AndersonBanke and others (1976), who were able to relate the drag coefficient of sea ice to the root mean square of the surface elevation. It is different from the interpretation of Weller (1968), that profile-derived z0 values which are smaller than those obtained from roughness-element description signify control by the fine structure of the surface. However, none of these considerations applies to the analysis of zt because it is a property of the flow.
Although there is probably no good physical reason to do so, the behaviour of zt was examined by applying the 0.17 m height adjustment to the temperature profiles, in order to make the analysis consistent with that of z0 . Values of ln(zt/z0 ) were computed for each pair of profiles, and plotted against Re* ( Fig.4). It appears that ln(zt/z0 ) decreases with increasing Re* in a manner which is parallel to the model proposed by Reference AndreasAndreas (1987). It also supports Andreas– contention that flow over ice tends to be aerodynamically rough. Although it is possible that the tendency for Andreas– model to underpredict the experimental results suggests a need to change the model, it is more likely that a zero-reference error is the cause. The experimental points could be forced to agree with the model by arbitrarily removing the 0.17 m height correction from the temperature profile, while retaining it for the wind-speed profile. This is quite possibly the correct action to take, given the IOm distance between the two masts.
Fig.4. Fig. 4. Experimental results of the III(Z1z/zq) analysis (dots) compared with the Reference AndreasAndreas (1987) prediction (line plot).
Given that experimental data confirm the shape, if not the absolute values predicted by Andreas– model, it is important to see what can be accomplished by using the model in conjunction with the results listed in Table II. In rough flow, the model takes the form
which is easily embedded within the iteration procedure used to calculate QH from Equation (5). Also, since the roughness-length estimates for each type of surface are so consistent, the average of the earlier and more recent estimates is used, giving 2.46 mm for ice and 5.5 mm for snow. The commonly accepted value, α = 5, is used for stability corrections.
On the whole, excellent agreement between bulk transfer and eddy correlation is achieved, though it is unfortunate to have so few data points for snow ( Fig.5). A few of the eddy-correlation values are low, even to the point of a reversal in sign. Although such small values could be the result of measurement error, say a lack of response to high-frequency turbulence in very stable flow, it may also be the consequence of a collapse, or thinning of the boundary layer due to low-level wind maxima (Reference Munro and DaviesMunro and Davies, 1977). The smallest values, relative to those predicted by bulk transfer, occurred during a very warm 4 h period, when stability may have been particularly strong. Similar behaviour has been documented in Antarctica by lnoue (1989a), who has suggested both the presence of low-level wind maxima and lack of response to high-frequency turbulence as likely causes.
Fig.5. Comparison of eddy correlation and bulk-transfer estimates of QH over ice (solid symbols) and snow (open symbols). Square symbols denote suspected cases of anomalous flow conditions. Siraight-Iine plot shows 1:1 relationship.
If the values which seem to be the consequence of anomalous flow conditions are dropped from the comparison, there is only a small bias in the bulk-transfer results over ice, though the root-mean-square error shows that there are substantial differences for individual comparisons (Table III). The effect of boundary-layer thinning cannot easily be seen in bulk-transfer values because Equation (5) is not very sensitive to anomalies in the flow structure. It may even continue to provide reasonable QH estimates provided thinning does not extend all the way to the surface. Nevertheless, there is good reason for concern about measurement height in relation to boundary-layer thickness. The use of 5 m measurements over ice results in a substantial overestimate of QH > (Table III). The comparison was not made over snow because only six samples were available.
The effect of changing the roughness length is also shown in Table III, taking z0 = 0.67 along the grain of the ice surface. The bias, though substantial, is not as large as one might expect in view of Figure 2 because as z0 decreases so does Re*.
The result is larger zt values which compensate to some extent for the change in z0 Therefore, one should not be too concerned about the small errors in z0 which might arise from variance between the standard-deviation approach to roughness-element description and the true profile of the surface, because the modelling scheme for QH is inherently conservative. This point is illustrated in Table Ш, where z0 values of 3.08 and 1.84 correspond to a ±25% change in the cross-grain roughness length, twice the difference noted with respect to the analysis portrayed in Figure 3. Yet the mean bias error is within 2 W m−2 in each case.
The mean bias error, 
where the prime refers to the bulk-transfer value. The root-mean-square error,
Conclusions
The microtopographic approach to roughness-length estimation makes the tacit assumption that z0 depends principally upon the surface form, an assumption which is probably reasonable for any rigid surface. This view is preferred by Reference AndreasAndreas (1987), and supported by the results of Reference Banke, Smith and AndersonBanke and others (1976). Since the method is simple to apply, it would be interesting to see roughness-element description employed at more glacier sites, to see how consistent the results might be. The extremes in the Z0 values for snow reported by Reference MooreMoore (1983) were all derived from profile data which, in addition to the uncertainty of sampling, suffer the problems of dealing with stability and height definition. Although one may argue about how to best represent surface form, roughness-element description is essentially a problem in sampling.
Even though Reference AndreasAndreas (1987) derived his model for use over sea ice, its usefulness for glacier snow and ice is not surprising, given that it depends upon the properties of the air and the flow itself. Also, because it is easy to incorporate the model into any bulk-transfer computation scheme, there is no good reason to use a fixed value for zt . Thus, questions about the thermal-roughness length should not concern the size of Zt relative to z0, but rather the validity of the Andreas model. The model seems valid according to the results presented here, but the sample is small, so further validation studies would be useful. It is especially important in this regard for micrometeorologists to include microtopographic measurements in their work, as Reference WeilerWeiler (1968) did, so that the two approaches may be reconciled.
Because the instrumentation is simple, bulk transfer is likely to remain the method of choice over melting ice and snow. It is probably a good choice provided it is used in conjunction with microtopographic data, obtained from a suitable sampling scheme. Sampling has not been a strong feature of this study because the importance of good microtopographic data came to be appreciated only as other aspects of the work were completed. It is a weakness which is common to studies of this kind, a problem which must be addressed in the future.
Acknowledgements
The work is supported by the Natural Sciences and Engineering Research Council of Canada. Institutional support from McMaster University, the University of Toronto, the University of Colorado and especially National Hydrology Research Institute, Environment Canada, has been vital to the work. Gratitude is also expressed to Parks Canada for granting permission to work at the site. Research assistance was provided by D. Chisholm, K. Munro, B. Alberton, A. Munro, and T. Bain.
