## 1. Introduction

Over glaciers, sea ice, and snow-covered ground, the atmospheric surface layer is often stably stratified. Estimating the contributions from the surface sensible and latent heat fluxes to the surface energy budget for such surfaces usually relies on Monin–Obukhov similarity theory to deal with these stratification effects. This method, in turn, requires knowing how to parameterize the roughness lengths for wind speed (*z*_{0}), the so-called scalar roughness length for temperature (*z*_{T}) and humidity (*z*_{Q}), and the stratification corrections to the usual semilogarithmic profiles for wind speed, temperature, and humidity.

*U*), potential temperature (

*T*), and specific humidity (

*Q*) as functions of height (

*z*) in the atmospheric surface layer obey Here,

*k*(=0.40) is the von Kármán constant;

*T*

_{s}and

*Q*

_{s}are the temperature and specific humidity at the surface;

*L*is the Obukhov length, a stratification parameter; and

*ψ*

_{m}and

*ψ*

_{h}are the stratification corrections to the semilogarithmic profiles. In (1.1c), I make the usual assumption that

*ψ*

_{h}is the same for both the temperature and humidity profiles. Over surfaces of snow and ice, we commonly take

*Q*

_{s}to be the saturation specific humidity at temperature

*T*

_{s}.

*u*∗ is the friction velocity, and

*t*∗ and

*q*∗ are analogous temperature and humidity flux scales such that the sensible (

*H*

_{s}) and latent (

*H*

_{L}) heat fluxes are Here,

*ρ*is the air density;

*c*

_{p}, the specific heat of air at constant pressure; and

*L*

_{υ}, the latent heat of vaporization or sublimation. Since this review concentrates on stable stratification, I follow Nieuwstadt (1984) and treat

*u*∗,

*t*∗,

*q*∗, and

*L*as local scales.

*H*

_{s}and

*H*

_{L}: Here,

*C*

_{Hz}and

*C*

_{Ez}are called the scalar transfer coefficients: the transfer coefficients for sensible and latent heat at reference height

*z.*Combining (1.1), (1.2), and (1.3), we evaluate these coefficients to be where

*ζ*=

*z*/

*L.*

Thus, to reiterate, estimating *H*_{s} and *H*_{L}, in general, requires knowing how to parameterize the roughness lengths *z*_{0}, *z*_{T}, and *z*_{Q} and the stratification corrections *ψ*_{m} and *ψ*_{h}. My focus here is on how to parameterize *z*_{T} and *z*_{Q} over surfaces of ice or snow. The subject of how to parameterize *z*_{0} over such surfaces is much more complex and must await a dedicated review, although Kind (1976), Jackson and Carroll (1978), Banke et al. (1980), Chamberlain (1983), Inoue (1989), Raupach (1992), Andreas and Claffey (1995), and Andreas (1995), among many others, offer insights into this parameterization.

Finding *z*_{T} and *z*_{Q} involves either calculating *C*_{Hz} and *C*_{Ez} from measurements of *H*_{s} and *H*_{L} using (1.3) and then solving (1.4) for *z*_{T} and *z*_{Q} or fitting profile measurements with (1.1) to obtain the flux scales and the roughness lengths. Either way, knowing the functional forms for *ψ*_{m} and *ψ*_{h} is crucial. I therefore first review and assess several published expressions for *ψ*_{m} and *ψ*_{h}. The *ψ*_{m} and *ψ*_{h} functions from Paulson (1970), for example, are suitable for treating unstable stratification, and I discuss these no further. On the other hand, because the atmospheric surface layer over surfaces of ice or snow is often stably stratified, I focus on the forms of *ψ*_{m} and *ψ*_{h} in stable stratification, where there is little consensus on how to represent these.

## 2. Profile metrics

*ϕ*

_{m}(

*ζ*) and

*ϕ*

_{h}(

*ζ*) is easier than using the profile functions

*ψ*

_{m}(

*ζ*) and

*ψ*

_{h}(

*ζ*). These gradient functions are related to the surface-layer profiles of wind speed, potential temperature, and specific humidity as (e.g., Dyer 1974) Comparing (1.1) and (2.1), we get the expression that links the

*ψ*and

*ϕ*functions (e.g., Panofsky 1963):

A host of *ϕ*_{m} and *ϕ*_{h} functions for stable stratification (i.e., *ζ* > 0) have been suggested. Of course, not all of these have proper theoretical behavior—some are simply empirical fits. Here I introduce four profile metrics to help us decide which *ϕ*_{m} and *ϕ*_{h} functions have proper behavior, especially in the limit of very stable stratification. These metrics are the gradient Richardson number Ri, the Deacon numbers for wind speed *D*_{m} and potential temperature *D*_{h}, and the turbulent Prandtl number Pr_{t}.

### a. Gradient Richardson number

*g*is the acceleration of gravity and

*T*

_{υ}is the virtual temperature. From (2.1a) and (2.1b), we see that (2.3) can be written The group of variables in the front of (2.4) is just

*ζ,*where Consequently,

*ζ,*the gradient Richardson number is a stratification parameter. In stable conditions, turbulence is presumed to cease and the flow becomes laminar when the Richardson number exceeds a critical value Ri

_{cr}. Thus, we should expect accurate

*ϕ*

_{m}and

*ϕ*

_{h}functions to predict this critical value through (2.6). That is,

Traditionally, Ri_{cr} is assumed to be 0.20–0.25 (Okamoto and Webb 1970; Busch 1973; Businger 1973; Nieuwstadt 1984). But Mahrt (1981) and Heinemann and Rose (1990) report that a larger value is sometimes indicated. Lyons et al. (1964) report nighttime data from Brookhaven, New York, that show “no clear ‘critical’ Richardson number” for Ri values up to at least 0.99 but also point out that these data do suggest the decreasing potential for turbulence for Ri greater than 0.25–0.50. Kondo et al. (1978) likewise report that turbulence can persist up to Ri values of 1 but conclude that the turbulence is only intermittent for Ri values between 0.2–0.3 and 1. Woods (1969) explained this apparent range in critical Richardson numbers somewhat differently by demonstrating how hysteresis can affect Ri_{cr}. He concluded that a turbulent flow becomes laminar when Ri exceeds 1, but a laminar flow does not become turbulent until Ri falls below 0.25 (see also Plate 1971, p. 76). Canuto et al. (2001) partially corroborate this scenario by reporting that, in large-eddy and discrete numerical simulations, turbulence persists for Richardson numbers up to 1. In his observations at the South Pole, however, Lettau (1979) frequently found turbulence to exist even when Ri exceeded 1 and, thus, concluded that there is no critical Richardson number. Monin and Yaglom (1971, p. 440 f.) and Yamamoto (1975) also argue that no critical Richardson number seems to exist.

In light of this controversy, I conclude that the critical Richardson number for an existing turbulent flow is probably larger than the traditional value of 0.20 or 0.25; perhaps it is of order 1. Still, some have speculated that Ri does not reach a critical value at all: that turbulence does not cease as *ζ* increases.

### b. Deacon numbers

Lettau (1957, 1979; see also Viswanadham 1979, 1982) uses two quantities to characterize profile curvature in the atmospheric surface layer, the Deacon numbers for wind speed (*D*_{m}) and potential temperature (*D*_{h}). Andreas (1998) also discusses these.

In neutral stratification, where *ζ* = 0, *D*_{m} and *D*_{h} are both 1. As the stratification increases, however, reliable *ϕ*_{m} and *ϕ*_{h} functions should predict limiting *D*_{m} and *D*_{h} values that agree with theory and experiment. I will discuss theoretical limits for *D*_{m} and *D*_{h} as *ζ* gets large shortly, but the experimentally determined limits are inconclusive. For example, from profile observations at the South Pole, Lettau (1979) concludes that *D*_{m} = 1/4 and *D*_{h} = −1/2 in the limit of *ζ* → ∞. On the other hand, Viswanadham (1982) suggests that *D*_{m} is small but slightly positive; he gives 0.04 as the limit for *D*_{m} with increasing stratification. He did not evaluate *D*_{h}.

One reason for this absence of definitive results must surely be the difficulty in measuring second derivatives of *U* and *T* in the atmospheric surface layer. As a minimum, such measurements require four levels of well-calibrated sensors.

### c. Turbulent Prandtl number

*K*

_{m}) and sensible heat (

*K*

_{h}). These fluxes are related to the respective wind speed and potential temperature gradients through these diffusivities (e.g., Dyer 1974): From (2.1) and (2.12), we thus see that As a result, the turbulent Prandtl number is

Much of the discussion regarding Pr_{t} has concentrated on its value at neutral stability, *ζ* = 0 (e.g., Businger et al. 1971; Kader and Yaglom 1990; Högström 1996). Again, since I am interested more in the behavior of the *ϕ*_{m} and *ϕ*_{h} functions in the limit of very stable stratification, I focus on the limit of Pr_{t}(*ζ*) as *ζ* gets large.

Because Pr_{t} contains the ratio *ϕ*_{h}/*ϕ*_{m}, as does Ri, if the Richardson number is unbounded as *ζ* increases, Pr_{t} could be too. Monin and Yaglom (1971, p. 440 f.) therefore believe that Pr_{t} increases without bound as *ζ* increases. Mahrt (1998) reaches essentially the same conclusion, explaining that pressure fluctuations in the atmosphere caused by gravity waves can transfer momentum but not sensible heat (cf. Beljaars and Holtslag 1991). Observations by Kim and Mahrt (1992) seem to substantiate this conclusion. Using aircraft data collected over Kansas and Oklahoma, they show Pr_{t} increasing without bound for Ri values up to almost 1. In contrast, Howell and Sun (1999) show surface-layer data from the Microfronts experiment in Kansas in 1995 for which the turbulent Prandtl number remains near 1 for *ζ* up to 10.

### d. In laminar flow

*ζ*approaches infinity—a turbulent flow eventually becomes laminar, at least in a laboratory setting. Here molecular processes alone must support the fluxes of momentum and sensible heat. Thus, for the momentum flux, rather than (2.12a), we have (e.g., Tennekes and Lumley 1972, p. 160) where

*ν*is the kinematic viscosity of air. As a result,

*D*is the thermal diffusivity of air. Consequently,

_{t}is a bound quantity that is of order 1. Of course, this extension to laminar flow does not consider intermittent processes in the stable atmosphere, such as wave breaking, that can periodically destroy the laminar flow and foster momentum exchange with little heat transfer, as Monin and Yaglom (1971, p. 440 f.) and Mahrt (1998) explain.

In summary, these laminar limits are targets for *D*_{m}(*ζ*), *D*_{h}(*ζ*), and Pr_{t}(*ζ*) in the limit of very stable stratification. We realize though that contrary experimental and theoretical results call into question the idea of relying strictly on these laminar limits for flows in the atmospheric boundary layer. Nevertheless, real flows should not stray far from these laminar limits as the stratification increases.

## 3. A sampling of gradient functions for stable stratification

Dozens of expressions for the *ϕ*_{m} and *ϕ*_{h} functions have been published over the last 40 years. Dyer (1974), Yamamoto (1975), Yaglom (1977), and Sorbjan (1989, p. 74 ff.), for example, summarize some of these functions. Since I cannot hope here to provide an encyclopedic review of all of these functions, I select for discussion a small sample of representative functions. Readers can evaluate other functions themselves with the techniques I will describe.

### a. Log–linear

*ϕ*

_{m}

*ζ*

*ϕ*

_{h}

*ζ*

*αζ,*

*ϕ*

_{m}and

*ϕ*

_{h}in stable stratification. The constant

*α*is generally reported to be in the range from 5 (Webb 1970; Dyer 1974; Large and Pond 1981) to 7 (Wieringa 1980; Large and Pond 1982; Högström 1988).

Figure 1 shows plots of *ϕ*_{m} and *ϕ*_{h} for *α* values of 5 and 7, while Table 1 lists the Deacon, gradient Richardson, and Prandtl numbers for these functions in the limit of large *ζ.* The Deacon and Prandtl numbers for these log–linear functions reach the proper laminar theoretical limits of 0 and order 1, respectively. The critical Richardson numbers implied by the *α* = 5 and *α* = 7 versions of the log–linear gradient functions, 1/5 and 1/7, respectively, are approximately in the classical range, 0.20–0.25. As I explained, however, these critical Richardson numbers are too small for atmospheric flows in light of more recent theory and observations.

### b. Lettau

*ϕ*

_{m}and

*ϕ*

_{h}: Figure 1 depicts these functions. Lettau's

*ϕ*

_{h}function predicts by far the steepest temperature gradient among any of the functions that I survey.

Table 1 lists the Deacon, Richardson, and Prandtl numbers that Lettau's functions imply in the limit of large *ζ.* Lettau specifically formulated his *ϕ*_{m} and *ϕ*_{h} functions to produce the rather unusual *D*_{m} and *D*_{h} values of 1/4 and −1/2 because these values mirror the results of his profile observations at South Pole. His predicted Richardson number does not approach a limit at large *ζ* but, rather, continues increasing in proportion to *ζ.* Likewise, his turbulent Prandtl number does not reach a laminar limit but increases as *ζ*^{3/4}. As I mentioned above, some have speculated that such unbounded Richardson and Prandtl numbers could exist in an intermittently turbulent, stable boundary layer. None, however, have corroborated the steepness of Lettau's *ϕ*_{h} function.

### c. Brutsaert

*ϕ*

_{m}and

*ϕ*

_{h}but constrains these to values of 6 or less. That is, Kondo et al. (1978) also suggest that

*ϕ*

_{m}is limited by 6. Figure 1 shows plots of the functions in (3.3). At large

*ζ,*they yield the smallest

*ϕ*

_{m}and

*ϕ*

_{h}values for any of the functions I am surveying.

Table 1 lists the Deacon, Richardson, and Prandtl numbers predicted by Brutsaert's functions in the limit of large *ζ.* His functions predict rather unusual limiting Deacon numbers of *D*_{m} = *D*_{h} = 1. His functions also predict a Richardson number that is unbounded; it increases as *ζ*/6. In other words, he predicts that no critical Richardson number exists. In a break with the other functions in Table 1 that predict an unbounded Richardson number [except those from King et al. (1996)], however, his functions predict that the turbulent Prandtl number reaches a limit of 1.

### d. Holtslag and de Bruin

*ϕ*

_{m}and

*ϕ*

_{h}that are specially adapted for very stable stratification (cf. Launiainen and Vihma 1990): Figure 1 shows plots of these functions. They are fairly close to Brutsaert's (1982) in the plotted range but continue increasing slowly with

*ζ,*while his functions are constant at 6 for

*ζ*larger than 1.

Table 1 lists the limiting values of the Deacon, Richardson, and Prandtl numbers implied by the Holtslag and de Bruin functions. All these values reach reasonable limits as *ζ* increases. The Deacon numbers *D*_{m} and *D*_{h} both go to zero, the limit for laminar flow. The turbulent Prandtl number is always 1, which is the approximate order of the molecular Prandtl number. Finally, Holtslag and de Bruin's functions predict that the critical Richardson number is 1.43, in line with my earlier discussion that modern ideas suggest a critical Richardson number of order 1.

### e. Beljaars and Holtslag

*ϕ*

_{m}and

*ϕ*

_{h}functions given by Holtslag and de Bruin (1988) because, presumably, these new functions “are more consistent with critical Ri considerations.” The Beljaars and Holtslag functions are Figure 1 shows plots of these functions.

Table 1 lists the values of the Deacon, Richardson, and Prandtl numbers implied by Beljaars and Holtslag's functions in the limit of large *ζ.* The limiting Deacon number for the wind speed profile, 0, agrees with the laminar limit. The Deacon number for the temperature profile, −1/2, on the other hand, does not; though it is the same as Lettau's (1979) value. Beljaars and Holtslag's functions also predict that no critical Richardson number exists; for their functions, Ri increases as [(2/3)*ζ*]^{1/2}. Likewise, their predicted turbulent Prandtl number does not have a limiting value but also increases as [(2/3)*ζ*]^{1/2}.

### f. King et al

*ϕ*

_{m}and

*ϕ*

_{h}on log–linear relations but limiting these functions to a maximum of 12. That is, their formulation is Here the additive constants and the coefficients of the

*ζ*terms come from King and Anderson's (1994) profile measurements at Halley Station on the Antarctic continent. Figure 1 shows plots of these

*ϕ*

_{m}and

*ϕ*

_{h}functions.

Table 1 lists values for the profile metrics implied by the functions suggested by King et al. in the limit of large *ζ.* Both *D*_{m} and *D*_{h} equal 1 in very stable conditions, contrary to the predictions from laminar flow theory and at odds with all results except Brutsaert's (1982). The King et al. functions also do not produce a critical Richardson number; in their formulation, Ri increases monotonically as *ζ*/12. Notice that the King et al. functions are the only ones that I consider for which the turbulent Prandtl number at *ζ* = 0 is not 1; their functions imply Pr_{t} = 0.95 at neutral stability, in line with Högström's (1996) recent review. In very stable stratification, their predicted turbulent Prandtl number is 1, as predicted by several other sets of functions in Table 1.

### g. Summary

We see from Fig. 1 that, for 0 ≤ *ζ* ≤ 0.5, where most profile data for evaluating the *ϕ*_{m} and *ϕ*_{h} functions have been collected, the seven candidate sets of functions show only minor differences. As the stratification increases, however, and the turbulence likely becomes intermittent, the *ϕ*_{m} and *ϕ*_{h} functions show diverging opinions. The two log–linear expressions for *ϕ*_{m}—which admittedly were never intended for extrapolating into very stable stratification—suggest very large values. The other five functions, which are intended explicitly for treating very stable stratification, imply *ϕ*_{m} values typically between 6 and 12.

For *ϕ*_{h}, again the log–linear functions imply very large values in very stable stratification but really should not be extrapolated into this region. Lettau's (1979) function, in contrast, suggests even larger values and was formulated specially to treat very stable stratification. Nevertheless, it has no corroboration that I know of; I must thus assume it is unrealistically large. The other four *ϕ*_{h} functions in Fig. 1 typically predict values between 6 and 14 in very stable stratification.

I use the values of the four profile metrics, *D*_{m}, *D*_{h}, Ri, and Pr_{t}, in the limit of large *ζ* to judge which of these functions have realistic behavior. I eliminate the two log–linear sets immediately because they were never intended to treat very stable stratification. I eliminate Lettau's (1979) functions because these imply the most unusual profile metrics of all the functions in Table 1 and because *ϕ*_{h} seems too large. Though mathematically simple, I also eliminate the Brutsaert (1982) and King et al. (1996) functions because of their implied Deacon numbers in the limit of large *ζ.*

Finally, I eliminate Beljaars and Holtslag's (1991) functions because I have seen no hard evidence that turbulence persists without limit as the stratification increases. In particular, Howell and Sun's (1999) Microfronts data do not show Ri increasing as rapidly with *ζ* as Beljaars and Holtslag's functions predict and do not give Ri values larger than 0.6 for *ζ* up to 10. In other words, most analyses that have addressed the question suggest a critical Richardson number of order 1 exists. Likewise, Howell and Sun suggest the turbulent Prandtl number is also of order 1 in very stable stratification contrary to Beljaars and Holtslag's prediction of no limiting Prandtl number.

The functions developed by Holtslag and de Bruin (1988) imply profile metrics that agree with these assessments and are thus the functions I recommend for representing stratification effects in surface-layer wind speed, temperature, and humidity profiles in stable stratification. Launiainen (1995), Vihma (1995), and Jordan et al. (1999, 2001), among others, have also settled on these stability functions for treating stable stratification.

The radiative flux divergence in the atmospheric surface layer is a process that may affect the conclusions above. Although a full evaluation of this term in the heat budget of the atmospheric surface layer is beyond my scope, I want to raise the issue as an area that needs research. Briefly, near-surface water vapor absorbs and emits longwave radiation. Depending on the near-surface temperature and humidity profiles, this exchange may ruin the assumption that a surface layer exists in which the vertical flux of sensible heat is constant with height. In turn, the *ϕ*_{h} function especially would then not obey Monin–Obukhov similarity.

Normally, the radiative flux divergence is negligible. But in very stable stratification, when the winds are light and the magnitude of the sensible heat flux is small, the radiative flux divergence can lead to a significant variation in the sensible heat flux with height (e.g., Coantic and Seguin 1971). Coantic and Seguin (1971), Garratt and Brost (1981), and Narasimha and Vasudeva Murthy (1995), among others, investigated the effects of radiative flux divergence on atmospheric surface layer profiles but did not treat the specific cases of snow-covered surfaces or temperatures well below freezing, when the water vapor density in the atmospheric surface layer will be small. Hence, I cannot reliably infer what their results might say about the effects of the radiative flux divergence over surfaces of ice or snow. We need a thorough study of the possible interactions between turbulence and radiation in the atmospheric surface layer at temperatures well below freezing to evaluate whether these interactions can explain the variety of *ϕ*_{m} and *ϕ*_{h} functions reported in the literature.

## 4. Scalar roughness over snow and ice

Andreas (1987) built on the surface-renewal models of Brutsaert (1975) and Liu et al. (1979) to produce the only theoretically based model that specifically predicts *z*_{T} and *z*_{Q} over ice and snow-covered surfaces. Although I had scant data with which to test that model when I published it, sporadic tests have been published since (e.g., Munro 1989; Bintanja and Van den Broeke 1995). More importantly, since that model is the only one specifically for snow and ice surfaces, many have been using it for numerical modeling (e.g., Morris 1989; Launiainen and Cheng 1998; Jordan et al. 1999), though, to my mind, it has not been adequately validated.

*z*

_{s}from the roughness Reynolds number

*R*∗ (=

*u*∗

*z*

_{0}/

*ν*),

*z*

_{s}

*z*

_{0}

*b*

_{0}

*b*

_{1}

*R*

*b*

_{2}

*R*

^{2}

*z*

_{s}is either

*z*

_{T}or

*z*

_{Q}. Table 2 lists the polynomial coefficients,

*b*

_{0},

*b*

_{1}, and

*b*

_{2}.

Comparing (4.1) and the coefficients summarized in Table 2, we see that (4.1) is a piecewise-continuous function with three pieces. For aerodynamically smooth flow, *z*_{s}/*z*_{0} is independent of the roughness Reynolds number because here molecular effects control the exchange of both momentum and scalars. That is, both scale similarly with *u*∗. For aerodynamically rough flow, in contrast, the viscous boundary layer continually thins with increasing *u*∗, the surface roughness elements protrude farther above this layer, and pressure forces become more important in transferring momentum. The effect is that *z*_{0} increases with *u*∗; while *z*_{s}, which is still dictated by molecular processes, does not change as rapidly. Hence, *z*_{s}/*z*_{0} is a monotonically decreasing function of *R*∗ for aerodynamically rough flow. Andreas (1987) simply made a log–log interpolation between the limits of smooth and rough flow to fill in the transition region in Table 2.

I want to test this model here with several datasets that I have located or that have become available since Andreas (1987) appeared. Hicks and Martin (1972), Thorpe et al. (1973), Joffre (1982), King and Anderson (1994), and Calanca (2001) have all reported measurements of quantities related to *z*_{T} or *z*_{Q}; but these are small datasets, or the reported data were not in a form that I could use.

The primary reason for the scanty data is the difficulty in making the required measurements. Analyses for *z*_{T} and *z*_{Q} rely on (1.3), but the fluxes *H*_{s} and *H*_{L} are generally smaller in magnitude in stable stratification than in unstable stratification and can be especially small over snow and ice surfaces. That is, often the magnitudes of *H*_{s} and *H*_{L} are comparable to the experimental uncertainty in the measurements of these values. Likewise, the required gradients in (1.3), *T*_{s} − *T*(*z*), and *Q*_{s} − *Q*(*z*), are often small with large experimental uncertainties. In total, then, over surfaces of snow and ice, experimental uncertainties can often swamp values of *C*_{Hz} and *C*_{Ez} calculated from (1.3) and, thus, values of *z*_{T} and *z*_{Q} calculated from (1.4). Still, I have located five datasets that are modestly sized and seem to have enough signal-to-noise ratio to provide estimates of *z*_{T} and *z*_{Q}. For these sets, I was also usually able to obtain and analyze the raw data.

### a. Munro's data

Munro (1989) measured profiles of wind speed and temperature at four levels between 0.25 and 1.00 m over the melting Peyto Glacier (Alberta, Canada; e.g., Goodison 1970). The value of this dataset is that the surface temperature is well known, 0°C, and the surface–air temperature difference is unusually large—sometimes as large as 12°C.

Munro's original paper contains his analysis of *z*_{T}/*z*_{0} as a function of *R*∗, but almost all of his plotted values are above the line set by (4.1). To obtain his results, however, Munro used the log–linear form for *ϕ*_{m} and *ϕ*_{h} with *α* = 5; assumed *k* = 0.41; and most importantly, modified each of his wind speed *and* temperature profile heights by adding 0.17 m to each original height. He evidently based this 0.17-m correction on the typical trough-to-peak height of the microtopography of the glacier and what he viewed as ambiguity in his zero-reference height.

I have three concerns with these manipulations. That 0.17 m seems akin to a displacement height, but displacement heights are always *subtracted* from the measured height. I am not sure what adding 0.17 m to the height means physically. Second, because momentum can be transferred by pressure forces acting on the roughness elements, Thom (1971) interpreted the displacement height as the height at which the roughness elements absorb momentum. Because pressure forces do not transfer heat, however, it seems unlikely that the temperature profile should exhibit the same displacement height as the wind speed profile, as Munro acknowledges himself. Finally, Andreas (1995) adapted a form-drag model developed by Raupach (1992) to investigate momentum transfer over a surface covered with sastrugi—that is, over a fairly rough surface like that of the Peyto Glacier. I infer from this modeling that the ratio of displacement height to the height of the roughness elements is much less than one. Consequently, Munro's choice of 0.17 m seems too large by a factor of, at least, three. The upshot is that, since Munro's measurement heights were 0.25, 0.50, 0.75, and 1.00 m, adding 0.17 m to each height causes significant changes in the apparent curvature of the profiles and could easily explain the difference between his results and the Andreas (1987) model.

I therefore obtained Munro's raw profile data (S. Munro 1995, personal communication) and analyzed these myself to estimate *z*_{T} and *z*_{0}. This analysis simply involved fitting his wind speed and potential temperature profile data iteratively with (1.1a) and (1.1b). This fitting yields *u*∗, *t*∗, and *L.* Since *U*(*z*_{0}) = 0, (1.1a) then gives *z*_{0}; and since *T*(*z*_{T}) = *T*_{s} = 0°C, (1.1b) gives *z*_{T}. Unlike Munro, I assumed no displacement height; set *k* = 0.40; and on the basis of the last section, used the Holtslag and de Bruin (1988) functions for *ψ*_{m} and *ψ*_{h}.

Figure 2 demonstrates the success of this fitting with five representative pairs of wind speed and potential temperature profiles. In each panel, the left profile is the second most stable run in the dataset; the right profile is the one nearest neutral stratification. The three middle profiles get more nearly neutral from left to right. The correlation coefficients of the fits depicted in Fig. 2 are all at least 0.997. Of the 122 pairs of Munro's profiles that I analyzed, the smallest correlation coefficient for my fitting was 0.996. Figure 2 and these correlation coefficients are testimony to my choice of the Holtslag and de Bruin (1988) stability corrections and to my decision to ignore Munro's suggestion that a displacement height is necessary.

*z*

_{T}/

*z*

_{0}. Plots of

*z*

_{s}/

*z*

_{0}are common in this business because this quantity lets us estimate the neutral-stability scalar transfer coefficient at reference height

*z*(i.e.,

*C*

_{sNz}) without the added steps of calculating

*z*

_{s}and

*z*

_{0}individually. That is, from (1.4a) with

*ζ*= 0 (cf. Garratt and Hicks 1973), Here,

*C*

_{DNz}is the neutral-stability drag coefficient at

*z*(e.g., Andreas 1998),

The data and the model in Fig. 3 seem to agree startlingly well, both with respect to magnitude and to *R*∗ dependence. In particular, the data and the model agree much better than in Munro's (1989) original analysis.

Careful readers will realize, however, that *z*_{0} appears in both the dependent and independent variables in Fig. 3. This shared variable could thus lead to an artificially good (or bad) correlation. And, as I explain in the appendix, Fig. 3 apparently does suffer from this effect: The artificial correlation seems to explain the tendency for *z*_{T}/*z*_{0} to decrease with increasing *R*∗ at approximately the same rate as the model predicts. Consequently, I hesitate to conclude that Fig. 3 confirms the predicted *R*∗ dependence in *z*_{T}/*z*_{0}. The artificial correlation, on the other hand, has no influence on the typical magnitude of the *z*_{T}/*z*_{0} values, which the model predicts well for *R*∗ between 0.1 and 10.

### b. Kondo and Yamazawa's data

Kondo and Yamazawa (1986) report measurements of the wind speed and temperature profiles at six levels over snow-covered ground in Japan. Using a profile analysis like the one I just described, they obtained from these profiles *u*∗, *L,* *C*_{DN1}, and *C*_{HN1}—the latter two being neutral-stability values of the drag coefficient and the sensible heat transfer coefficient at a reference height of 1 m. Although their paper shows plots of some of these quantities, the details were not sufficient for my reanalysis. But J. Kondo (1986, personal communication) kindly provided me a table of their entire *u*∗, *L,* *C*_{DN1}, and *C*_{HN1} dataset. I earlier converted these *C*_{DN1} and *C*_{HN1} values to *C*_{HN10} (i.e., a 10-m reference height) and compared these with the model predictions in Andreas (1987). Here I further convert these tabulated values to *z*_{T}, *z*_{0}, and *R*∗.

*z*= 1 m. Thus, from the values tabulated in the Kondo and Yamazawa dataset, I could easily compute

*z*

_{0},

*z*

_{T}, and

*R*∗.

Figure 4 compares my calculations of *z*_{T}/*z*_{0} and *R*∗ from the Kondo and Yamazawa dataset with the Andreas (1987) model. Only three of the markers here reflect unstable stratification; all the other runs were made in stable stratification, though the stratification was never very strong. The data are generally within an order of magnitude of the model but, contrary to Fig. 3, tend to suggest values lower than the model. Although I have not confirmed this as I did with Fig. 3, the trend in *z*_{T}/*z*_{0} with *R*∗ probably reflects some self-correlation because of the shared *z*_{0}.

### c. Ishikawa and Kodama's data

Ishikawa and Kodama (1994) report other profile and flux measurements over snow-covered ground in Japan. N. Ishikawa (1995, personal communication) kindly provided me all their raw data so I could reanalyze them for *z*_{0}, *z*_{T}, *u*∗, and *L.*

Ishikawa and Kodama measured *H*_{s} with a sonic anemometer/thermometer. They also measured the snow-surface temperature, which was usually very near 0°C, and the temperature at a height of 1 m. From (1.3a) we see that these measurements provide *C*_{H1}, the sensible heat transfer coefficient appropriate at a reference height of 1 m.

*U*at heights

*z*

_{1}and

*z*

_{2}—either 1 and 5 m or 0.1 and 1 m. From (1.1a), I could relate these measurements to

*u*∗; that is, Once this equation yields

*u*∗, I could use (1.1a) to compute

*z*

_{0}. Substituting this

*z*

_{0}value and

*C*

_{H1}into (1.4a) let me also calculate

*z*

_{T}. Of course, these computations are iterative because

*L*depends on

*u*∗ and the measured sensible heat flux. I again used the Holtslag and de Bruin (1988) functions for

*ψ*

_{m}and

*ψ*

_{h}in (1.4a) and (4.5). All of Ishikawa and Kodama's data were collected in stable stratification.

Figure 5 shows the results of my reanalysis of the Ishikawa and Kodama data. Although the centroid of the data cloud agrees fairly well with the model's predictions, the trend in the data with *R*∗ is contrary to the model. I attribute this difference to the need to use (4.5) to evaluate *u*∗. Winds in this dataset were very light: The highest extrapolated 10-m wind speed for the data points plotted in Fig. 5 was 4.5 m s^{−1}, and most values were between 1 and 3 m s^{−1}. As a result, the difference in wind speed, *U*(*z*_{2})–*U*(*z*_{1}), in (4.5) was often small; the consequent large relative uncertainty in that difference therefore made the *u*∗ evaluation imprecise. In fact, in creating Fig. 5, I eliminated 62 runs that had implied 10-m wind speeds less than 1 m s^{−1}. These runs with light winds led to unrealistically large *z*_{0} and *R*∗ values.

### d. Bintanja and Van den Broeke's results

*z*

_{0},

*z*

_{T},

*u*∗, and thus

*R*∗ from these data. Their paper contains a plot of

*B*

^{−1}

_{H}

*R*∗ bins, where (e.g., Garratt and Hicks 1973) I digitized this plot and converted the averaged

*B*

^{−1}

_{H}

*z*

_{T}/

*z*

_{0}averages.

Figure 6 shows my replotting of the Bintanja and Van den Broeke results. This plot represents many hours of data over a wide *R*∗ range and is thus a valuable test of the Andreas (1987) model. The data in Fig. 6 do seem to corroborate the model—both with respect to the magnitude of *z*_{T}/*z*_{0} and the trend in this ratio with *R*∗. All but three of the data markers in Fig. 6 are within half an order of magnitude of the model's predictions.

### e. Bintanja and Reijmer's results

Bintanja and Reijmer (2001) report recent measurements of *z*_{0}, *z*_{T}, and *z*_{Q} in the vicinity of Svea. They based their analysis on five-level profiles of wind speed, temperature, and specific humidity measured between 0.4 and 10 m above the surface. They restricted their analysis to cases of blowing and drifting snow, when *u*∗ is above a threshold of 0.30 m s^{−1}, and deduced *z*_{0}, *z*_{T}, and *z*_{Q} by fitting the profiles with semilogarithmic functions. That is, they made no corrections for the curvature in the semilogarithmic profiles caused by stratification and modeled by the *ψ*_{m} and *ψ*_{h} functions in (1.1).

*u*∗: for

*u*∗ ≤ 0.30 m s

^{−1},

*z*

_{0}

*z*

_{T}

*z*

_{Q}

^{−4}

*u*∗ > 0.30 m s

^{−1}, In all of these, the roughness lengths are in meters for

*u*∗ in m s

^{−1}.

The approximate quadratic dependence of *z*_{0} on *u*∗ in (4.7b) is in line with earlier assessments of how *z*_{0} should behave in drifting and blowing snow (e.g., Owen 1964; Chamberlain 1983; Andreas and Claffey 1995). But the large exponents of the *u*∗ terms in (4.7c) and (4.7d) imply that both *z*_{T} and *z*_{Q} get much larger than *z*_{0} at large *R*∗, a result not supported by any theory or by any other data.

As an example of (4.7), I include in Fig. 6 the relation for *z*_{T}/*z*_{0} as a function of *R*∗ that (4.7) implies. Although this new model is not unreasonable for *R*∗ less than 5, for larger *R*∗, Bintanja and Reijmer's model does not fit Bintanja and Van den Broeke's (1995) data.

### f. Barry and Munn's data

The only dataset of even modest size that I have found suitable for my reanalysis and that has anything to say about *z*_{Q} over snow or ice is Barry and Munn's (1967). They released tritiated water vapor at the surface or at a height of 10 m over snow-covered ground in Chalk River, Ontario, Canada. Using radioactivity detection techniques, they measured the downwind water vapor concentration at a height of 0.3 m 45–90 m from the release point and could estimate the surface water vapor flux from a time series of snow-surface samples. They also measured the mean wind speed at five heights between 0.25 and 2 m and could therefore estimate *u*∗ and *z*_{0}. Their tabulated data include *R*∗ and enough other information for me to estimate *z*_{Q}/*z*_{0}. Figure 7 shows the results for their highest-quality runs.

Figure 7 also shows Andreas's (1987) model for *z*_{Q}/*z*_{0} and Bintanja and Reijmer's (2001) prediction of the same quantity based on (4.7). The data in Fig. 7 tend to be significantly below both model predictions. The *z*_{Q}/*z*_{0} data do, however, decrease with increasing *R*∗ as the Andreas model predicts—a distinct contrast between it and Bintanja and Reijmer's results. Still, Fig. 7 is inconclusive and thus implies that we have more work to do in evaluating *z*_{Q}.

Though Barry and Munn's is the only dataset I could find for this review, it is far from ideal. Because of their experimental design, I suspect their data suffer from nonstationarity and the horizontal inhomogeneity of their site. They also made no corrections for stratification in their analysis of the wind speed profiles (e.g., see Fig. 2), an omission that could have had a large effect on their *u*∗ and *z*_{0} values. In summary, though Barry and Munn's work was impressive for its time, I have enough doubts about their data that I present Fig. 7 primarily for its historical value and to motivate further work.

## 5. Discussion

Bin-averaging often clarifies relationship in plots of wildly scattered data. Therefore, following the example of Bintanja and Van den Broeke (1995) (Fig. 6), I bin-averaged the individual *z*_{T}/*z*_{0} and *R*∗ values in the Munro (Fig. 3), Kondo and Yamazawa (Fig. 4), and Ishikawa and Kodama (Fig. 5) datasets to see if such averaging gives a better picture of the *R*∗ dependence in *z*_{T}/*z*_{0}. To be faithful to the depictions in these plots, my averaging was geometric rather than arithmetic. That is, for both *z*_{T}/*z*_{0} and *R*∗, I calculated the average of the logarithms of the values.

I identified just two *R*∗ bins in the Munro data in Fig. 3: *R*∗ values less than 0.01, and *R*∗ values greater than 0.01. Likewise, I used just two bins in averaging the Kondo and Yamazawa data in Fig. 4. The cluster of three points with *R*∗ values between 1 and 2 are in one bin; all other points are in the second bin. Because the Ishikawa and Kodama data in Fig. 5 do not show much variation in *z*_{T}/*z*_{0} with *R*∗, I lumped all these data in a single bin.

Figure 8 shows the results from this bin-averaging. The four points with *R*∗ values between 0.7 and 20 that represent most of the data agree quite well with the model—both with respect to magnitude and to *R*∗ dependence. The point in the aerodynamically smooth regime (below *R*∗ = 0.135) suggests the model's predictions may be low here. But measuring in such light winds is often difficult, so eight points obtained in difficult experimental conditions may be suggestive but are not sufficient to refute the model. Besides, the more extensive dataset from Bintanja and Van den Broeke (Fig. 6) does not corroborate this discrepancy between the model and the measurements at small *R*∗. In fact, Figs. 6 and 8 now provide fairly strong support for the Andreas (1987) model for *z*_{T}/*z*_{0} over the *R*∗ range from 10^{−4} to 100.

One caveat in this conclusion, though, is that most of the *z*_{T} data that I have reviewed were collected in fairly warm conditions where snow would not have been blowing or drifting. The Bintanja and Van den Broeke (1995) set is the one exception. The issue is that Bintanja and Reijmer (2001) suggest that both *z*_{T}/*z*_{0} and *z*_{Q}/*z*_{0} increase with *R*∗ when snow is drifting. My analyses here have not shown this effect, however. In particular, the Bintanja and Van den Broeke set, Fig. 6, which likely included drifting snow during the measurements with the large *R*∗ values, does not substantiate Bintanja and Reijmer's empirical expressions for *z*_{T} and *z*_{0} in drifting snow.

## 6. Conclusions

This review of current procedures for estimating the fluxes of sensible and latent heat over surfaces of snow and ice leads to some recommendations. First, because the atmospheric surface layer over snow and ice is often stably stratified, we need to reach some consensus on which functions, *ψ*_{m} and *ψ*_{h}, to use to model this stratification. On defining four profile metrics—the critical Richardson number, the Deacon numbers for wind speed and temperature, and the turbulent Prandtl number—I reviewed probable values for these in the limit of very stable stratification. Observations and theory suggest that Ri_{cr} and Pr_{t} are both bounded and of order 1, while *D*_{m} and *D*_{h} are approximately 0. Of the *ψ*_{m} and *ψ*_{h} functions that I reviewed here, the set that Holtslag and de Bruin (1988) developed satisfies these limits best. I recommend these functions for handling stable stratification in general and for treating stable stratification over snow and ice in particular.

The Andreas (1987) model is the only one specifically adapted to predict the scalar roughness lengths *z*_{T} and *z*_{Q} over snow and ice; though to date, it has had only sporadic and incomplete testing. Here I have reanalyzed five datasets collected over snow and ice for the explicit purpose of testing this model. My comparison of the *z*_{Q} results with the model in Fig. 7 is inconclusive because of presumed shortcomings in the data and, therefore, highlights the difficulty in measuring *z*_{Q} at low temperatures. For intellectual reasons and because of the technical challenges, we need to concentrate on making the measurements required for evaluating *z*_{Q}.

The other four datasets (Figs. 3–6), in contrast, agree fairly well with the model's predictions for *z*_{T}/*z*_{0}. Since the model's independent variable, *R*∗ = *u*∗*z*_{0}/*ν,* also contains *z*_{0}, however, plots of *z*_{T}/*z*_{0} versus *R*∗ may suffer from fictitious correlation. In the appendix, I demonstrate that this self-correlation indeed affected the shape of the plot for Munro's (1989) data (Fig. 3). The implied correlation between *z*_{T}/*z*_{0} and *R*∗, even if *z*_{T} were not related to any other variable, coincides closely with the model's predicted dependence for *z*_{T}/*z*_{0} on *R*∗. This means that making scatterplots from individual datasets may not be a reliable way to judge a model's veracity.

Comparing data from various sources, however, at least mitigates the effects of bias errors in individual datasets. My bin-averaging of the Munro (1989), Kondo and Yamazawa (1986), and Ishikawa and Kodama (1994) datasets yielded data points that generally agree both in magnitude and in *R*∗ dependence with the Andreas (1987) model (Fig. 8). This summary plot and the Bintanja and Van den Broeke (1995) results (Fig. 6) therefore finally provide fairly strong support for the Andreas model's predictions for *z*_{T}/*z*_{0} for *R*∗ values between 10^{−4} and 100.

Although I have been unable to similarly test the *z*_{Q} model directly, the success of (4.1) in representing the *z*_{T}/*z*_{0} data supports it indirectly. Equation (4.1) is theoretically based; consequently, Figs. 6 and 8 essentially support that theory for heat transfer. Because most evidence from the atmospheric surface layer suggests that heat and moisture are transferred by similar processes, we can presume that the theory applies equally well for moisture transfer. Consequently, until we learn differently, (4.1) with the appropriate coefficients for humidity from Table 2 is also a reasonable model to use for predicting *z*_{Q}/*z*_{0}.

The Andreas (1987) model for *z*_{T}/*z*_{0} and *z*_{Q}/*z*_{0} contains nothing that makes it specific for snow and ice surfaces. Only the parameterization for *z*_{0} from Banke et al. (1980) that enabled me to predict *C*_{H} and *C*_{E} [i.e., see (1.4), (4.2), or (4.4)] limited it to use over snow and ice. In other words, the model's predictions for *z*_{s}/*z*_{0} should be just as valid for other solid surfaces with small roughness elements—such as sand, bare soil, or mud flats—as they are for snow and ice.

## Acknowledgments

This work would not have been possible if Junsei Kondo, Scott Munro, and Nobuyoshi Ishikawa had not generously sent me their datasets. I would also like to thank the three reviewers for helpful comments. The U.S. Department of the Army supported this work through Project 4A1611AT24; the U.S. National Science Foundation supported it with Awards OPP-98-14024 and OPP-00-84190.

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,*J. Meteor. Soc. Japan***53****,**189–195.

## APPENDIX

### Self-Correlation in *z*_{s}/*z*_{0}-versus-*R*∗ Plots

Figures 3–7 show plots of log(*z*_{s}/*z*_{0}) versus log(*R*∗). The trends in the data in these plots could result from fictitious correlation because of the shared variable *z*_{0} (e.g., Hicks 1978; Kenney 1982). Using ideas from Hicks (1978), I look at this question.

*z*

_{s}

*z*

_{0}

*a*

*R*

*b.*

*X*and

*Y*values is where cov(

*X,*

*Y*) is the

*X*–

*Y*covariance and

*s*

_{X}and

*s*

_{Y}are the sample standard deviations of

*X*and

*Y.*Likewise, the slope

*a*and intercept

*b*in (A.1) are where

*X*

*Y*

*X*and

*Y*values. That is, where an overbar denotes the sample average.

*ρ,*

*a,*and

*b*analytically by assuming that

*z*

_{s},

*z*

_{0},

*u*∗, and

*ν*are all uncorrelated. For example, from the definition of the sample standard deviation, where

*N*is the sample size, Here,

*i*is the index for the sample.

*u*∗,

*z*

_{0}, and

*ν*are all assumed to be uncorrelated for this analysis, from (A.10) we get

*s*

^{2}

_{X}

*s*

^{2}

_{lnu∗}

*s*

^{2}

_{lnz0}

*s*

^{2}

_{lnν}

*u*∗, ln

*z*

_{0}, and ln

*ν.*Similarly,

*s*

^{2}

_{Y}

*s*

^{2}

_{lnzs}

*s*

^{2}

_{lnz0}

*s*

^{2}

_{lnzs}

*z*

_{s}.

*X*–

*Y*covariance can be evaluated similarly: Thus, under the assumption that all the variables are uncorrelated, the only covariance between

*z*

_{s}/

*z*

_{0}and

*R*∗ results because of the shared variable

*z*

_{0}:

*X,*

*Y*

*s*

^{2}

_{lnz0}

*z*

_{s}/

*z*

_{0}) and ln

*R*∗ is always negative if it results strictly because of the shared

*z*

_{0}. From (A.5), (A.11), and (A.15), we likewise get for the slope of the artificially correlated variables which is again always negative. Finally, we can calculate

*b*from (A.6)–(A.8) and (A.17).

Least squares linear regression, on which the above analysis is based, implicitly assumes that the *X* values are known perfectly and that the only uncertainty is in the *Y* values. This assumption is rarely true in geophysical data series and is certainly not true of our *R*∗ values. Consequently, I like to also fit *X* versus *Y* with a least squares line and take the bisector of the two fitting lines as the “best” fit.

*X*

*a*

*Y*

*b*

*X*-versus-

*Y*data is the same as for the

*Y*-versus-

*X*data. From (A.5), we can also immediately write Hence, from (A.12) and (A.15),

*Y*

*âX*

*b̂.*

*b̂*

*Y*

*â*

*X*

I have tested the effects of this self-correlation by evaluating *ρ,* *â,* and *b̂* for Munro's (1989) dataset (see Fig. 3). Figure A1 shows the discouraging result.

*u*∗ values in the Munro (1989) dataset is small and because

*s*

_{lnz0}

*s*

_{lnzT}

*â,*is between −1 and −2. That is, even if

*z*

_{T}is not correlated with anything, because of the shared

*z*

_{0}, the correlation between ln(

*z*

_{T}/

*z*

_{0}) and ln

*R*∗ is still high and the implied slope is similar to the slope that both the data and the model display. The dashed line in Fig. A1 shows the regression line (A.22) that results solely from this fictitious correlation.

In summary, at least for the Munro (1989) dataset, the fictitious correlation between the *z*_{T}/*z*_{0} and *R*∗ values means that we cannot reliably evaluate the *R*∗ dependence predicted by the Andreas (1987) model, except perhaps at small *R*∗, where the data tend more toward the model than toward the artificial regression line. The artificial correlation, however, does not influence the mean value of *z*_{T}/*z*_{0}. The model and the data in Fig. A1 tend to agree that, for *R*∗ between 0.1 and 10, the typical value for *z*_{T}/*z*_{0} is 1.

Although I have not made similar calculations for the other four datasets, because of (A.16), (A.17), and (A.20), ln(*z*_{s}/*z*_{0}) in all of these should decrease with increasing ln*R*∗ if the data are strongly influenced by artificial correlation caused by the shared *z*_{0}. Clearly, the data in Figs. 4, 6, and 7 display this trend. My reanalysis of Ishikawa and Kodama's (1994) data, shown in Fig. 5, shows a positive trend between *z*_{T}/*z*_{0} and *R*∗, however. To produce this trend, *z*_{T} or *z*_{0} in this dataset must be correlated with some of the other variables [see (A.14)], either physically or as a consequence of my analysis procedure. Thus, again, the data tend to corroborate the mean *z*_{T}/*z*_{0} level predicted by the model but are ambiguous when it comes to testing the model's predicted *R*∗ dependence.

From (A.16), we can infer some ways to minimize the fictitious correlation in plots of *z*_{s}/*z*_{0} versus *R*∗. For example, if the standard deviation of ln*z*_{0}, *s*_{lnz0}*s*_{lnzs}*ρ* in (A.16)—is small. Likewise, if the *u*∗ range is large so that *s*_{lnu∗} is much larger than *s*_{lnz0}*ρ* is small.

A comparison of the predictions of several sets of ϕ_{m} and ϕ_{h} functions in the limit of very stable stratification. The laminar case is a possible nonturbulent limit

Values of the coefficients to use in (4.1) for estimating the scalar roughness lengths in the three aerodynamic regimes