Turbulent heat fluxes

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sensible and latent heat may be carried to or from a snow surface by the action of turbulent eddies in the surface boundary layer (Morris, 1989). Formally, these fluxes may be written as the covariance of fluctuations in vertical velocity, w, with those either in temperature, Ta, or specific humidity, Q, i.e.:

Hl = Pa L vi w' Q', where the overbar denotes a mean over time, primes denote deviations from time averaged values, pa is the air density and cp,a is the specific heat of air at constant pressure (1.01 x 103 J kg-1 K-1). Given suitable fast-response instrumentation, it is possible to measure these covariances and hence obtain direct estimates of the fluxes. However, such measurements are rarely available and in many experimental applications or modeling studies it is necessary to parameterize these fluxes. Most commonly this is done using a bulk transfer formulation, in which fluxes are expressed in terms of differences between surface variables and the values of those variables at some reference height, zref, in the surface boundary layer, i.e.:

Here, Ta(zref), Q(zref) and M(zref) are the air temperature, specific humidity, and wind speed respectively at the reference height zref, while To and Qo are the temperature and specific humidity at the snow surface. Qo may be taken as the specific humidity of air saturated with respect to ice at temperature To. to is the surface stress and CD, CH, and CQ are the bulk transfer coefficients for momentum, heat, and water vapor, respectively. In order to calculate fluxes using (3.11), it is necessary to determine these coefficients that, in general, will depend on both surface roughness and atmospheric stability. This is most satisfactorily accomplished through the framework of the Monin-Obukhov surface layer similarity theory (see, e.g. Garratt, 1992, pp. 4958). This approach also takes into account that the reference height Zref = zref — HS above a snowpack of depth HS will change with time as HS increases or decreases.

The bulk transfer coefficients are related to the integrated forms of the surface layer similarity functions by:

Ch = K2[ln(Zref/zo) " (Zref/Lo)]" 1 [ln(Zref/zh) " (Zref/Lo)]"1,

Cq = K2[ln(Zref/zo) " (Zref/Lo)]" 1 [ln(Zref/zq) - (Zref/Lo)]-1,

where k is von Karman's constant (generally taken to be around 0.4), zo, zH, and zq are the roughness lengths for momentum, heat, and water vapor respectively and , , are the corresponding integrated forms of the surface layer similarity functions ^^:


= j (1 " $$(Z'/Lo)) d(ln Z'); § = M, H, Q; Zo = zo, zh, zq .

The surface layer similarity functions express how profiles of wind speed, temperature and humidity deviate from the logarithmic forms that are observed under neutral conditions (Zref/LO = 0) as a result of stability effects. These functions depend solely on the dimensionless height Zref/LO, where LO is the Obukhov length defined as:

where u * is the friction velocity, Tmean is the mean air temperature in the layer of depth Zref, and g is the acceleration due to gravity. Note the negative sign on the right-hand side of Equation (3.14), which results from the chosen sign convention (see Stull, 1988). Since LO is a function of the fluxes to be calculated, the equation set (3.11, 3.12 and 3.14) must generally be solved iteratively. However, as shown below, if some simplifying assumptions are made concerning the form of the ^ -functions, a direct solution is possible under some circumstances.

Experimental studies have determined the forms of the similarity functions in (3.12) even though little has been based on measurements over snow covers (Morris, 1989). Under stable conditions, where the sensible heat flux is directed towards the surface (HS < 0), it is found that, for 0 < Zref/LO < 1:

^M = j1M Zref/ L O, ^H = ^Q = j H Zref/L o, while for unstable conditions (HS > 0), for —5 < Zref /LO < 0:

^m = 2ln[(1 + xm )/2] + ln[(1 + xw)/2] — 2 tan-1 xm + n/2, ^h = ^q = 2ln[(1 + yH )/2],

Measurements indicate j 1M ~ j 1H ~ 5 and y 1 ~ Y2 ~ 16, butthere is a considerable range in experimentally determined values (see Garratt 1992, appendix 4).

In what follows, we shall concentrate on stable conditions, since this regime tends to prevail over snow covers. In winter, particularly in high latitudes, RN is positive for much of the time, leading to a downward heat flux, while advection of warm air over a melting snow cover also leads to the establishment of stable stratification since To cannot rise above 0 °C. If we simplify (3.16) by assuming j 1M = j 1H = j 1, manipulation of (3.12) and (3.14) yields an explicit expression for the transfer coefficients in terms of the bulk Richardson number (Garratt, 1992):


Xm = (1 + Yl Zref/Lo)1/4, yH = (1 + Y2 Zref/L o)1/2.

where RiB > 0 for stable conditions and Tmean is the mean temperature in the layer of depth Zref = Zref — (HS + Zo).

Ch = K2[ln(Zref/Zo)ln(Zref/Zh)]- (1 - P1 R?b)2 = Chn f (RiB), (3.20)

Cq = K2[ln(Wzo)ln(Wze)]-1 (1 - PRb)2 = Cgnf (RiB), and for RiB > P1-1:

In practice, there may be problems with using (3.20) and (3.21) as presented above. In runs of coupled surface boundary layers - snow-cover models that parameterize turbulent fluxes using (3.20) and (3.21), it has been found that, when strong radiative cooling is imposed, snow surface temperatures can drop to the point where RiB = P-1, at which point the fluxes are "switched off" and the snow surface becomes effectively decoupled from the atmosphere. This leads to further rapid (and unrealistic) cooling as the surface temperature evolves towards a pure radiative equilibrium (e.g. Morris et al., 1994). Measurements (e.g. King, 1990) show that small but non-zero heat fluxes persist even when RiB > P1-1 and this effect should be incorporated into any heat flux parameterization used in snow models. Theoretical understanding of turbulent transport in this high-stability regime is still developing but a number of practical alternative schemes to (3.20) and (3.21) have been proposed to avoid the problem of decoupling (Beljaars and Holtslag, 1991; King and Connolley, 1997).

The ratio of HS/HL is known as the Bowen ratio, B. Over snow surfaces, the air is often close to saturation, so Q(z) may be related to Ta(z) through the Clausius-Clapeyron equation. It can be demonstrated (Andreas, 1989) that, if supersaturation is forbidden and if both HS and HL are directed downwards, B is limited by:

T=To where pv sat is the saturation water vapor density. Andreas and Cash (1996) have extended this result to other combinations of HS and HL and have deduced general formulations for B over saturated surfaces. Such relationships can be of value if estimates of HL are required but no humidity measurements are available.

In order to calculate the turbulent fluxes, it is necessary to know the appropriate roughness lengths for momentum, heat and water vapor. The momentum (or aerodynamic) roughness length, zo, is related to the geometric roughness characteristics of the snow surface. Snow cover is one of the smoothest land surface types encountered in nature and, consequently, measurements of zo over snow (Table 3.1) indicate small values, of the order 10-4 to 10-3 m. Usually, zq and zH are assumed to be one

Table 3.1 Aerodynamic roughness lengths measured over various snow and ice surfaces.


Roughness length z0(m)



Seasonal snow cover

2.0 x 10~4 to 4.0 x 10~3 2.0 x 10~4 to 2.0 x 10~2 2.5 x 10~3



Kondo and Yamazawa (1986) Harding (1986) Konstantinov (1966) Sverdrup (1936)

Antarctic ice shelves

5.6 x 10-5 1.0 x 10~4 1.0 x 10~4

King and Anderson (1994) Heinemann (1989) König (1985)

Antarctic blue ice

2.8 x 10~6

Bintanja and van den Broeke (1995)

Sea ice


3 to 5 x 10~4

Joffre (1982)

Subantarctic glacier

Fresh snow

2.0 x 10~4

Poggi (1976)

Alpine glacier

Accumulation zone Ablation zone Sastrugi

Undulating wet snow

(1.10 ±0.25) x 10~2 (6.8 ± 1.4) x 10"3

van den Broeke (1997) van den Broeke (1997) Grainger and Lister (1966) Grainger and Lister (1966)

Icelandic glacier

Ablation zone

2 x 10~3 to 1 x IQ"1

Smeets etal, (1998)

order of magnitude smaller, as suggested by Garrat (1992) and Morris (1989). With values this small, it is clear that roughness elements of the scale of individual snow grains must be making the greatest contribution to the surface drag, with larger micro-topographic features, such as sastrugi, making a lesser contribution (Kondo and Yamazawa, 1986; Inoue, 1989). Ablating glaciers can develop large roughness elements on their surfaces, leading to aerodynamic roughness lengths of up to 0.1 m. The roughness of such surfaces can change rapidly as surface features develop during the ablation season (Smeets et al., 1998). Bare ice surfaces have particularly small roughness lengths (Bintanja and van den Broeke, 1995; see Table 3.1). Flow over such surfaces is "aerodynamically smooth," i.e. the surface Reynolds number:

la where |a is the kinematic viscosity of air and z*, the scale of the roughness elements, is less than about 5. In this low Reynolds number regime, flow around individual roughness elements is laminar and the roughness length is given by:

If the surface stress is great enough to generate blowing snow (see Section 3.4), suspended snow grains may contribute to the momentum transfer from atmosphere to surface and may thus cause an apparent change in zo. Owen (1964), Tabler (1980), Chamberlain (1983), and others have suggested that, under these conditions, the apparent roughness length is increased by drag exerted on saltating snow and will therefore be proportional to the surface stress, i.e.

Experimental evidence for such a relationship is variable. Tabler (1980), Tabler and Schmidt (1986) as well as Pomeroy and Gray (1990) present extensive quantitative measurements that support such behavior with values of c1 of 0.1203 over continuous snowfields (Pomeroy and Gray, 1990) and c1 of 0.02648 over a mixture of snow and lake ice (Tabler, 1980).

Bintanja and van den Broeke (1995), however, suggest that, in some cases, the alteration of surface characteristics by the transport of fresh snow onto a smoother underlying snow or ice surface may be the dominant process leading to an apparent increase in zo with wind speed. More observations are needed to resolve this issue.

Fewer measurements exist for the scalar roughness lengths, zH and zq, largely because of the difficulty of defining To and Qo other than over a melting snow surface, when To may be taken as 0 °C. Heat and water vapor transfer at the snow surface must ultimately be accomplished purely by molecular diffusion, since there is no equivalent to the form drag of roughness elements that is responsible for the majority of the momentum transport. Andreas (1989) developed a theory that predicts the ratio of zo to the scalar roughness lengths as a function of the surface Reynolds number (see Equation 3.23). In the aerodynamically smooth regime, zH/zo = 3.49 and zq/zo = 5.00. As Re increases, the ratio of scalar roughness length to momentum roughness length decreases rapidly and, for moderate wind speeds over a typical snow cover, the scalar roughness lengths will be one or two orders of magnitude smaller than zo. Measurements (Kondo and Yamazawa, 1986; Bintanja and van den Broeke, 1995) generally support this functional dependence. However, in many modeling applications, the scalar roughness lengths are set equal to zo for simplicity, on the grounds that the ensuing errors in surface fluxes will be no greater than the uncertainties resulting from other parts of the flux computation procedure.

The results of Section 3.3.4 are strictly applicable only over an extensive and uniform snow cover, where the atmospheric conditions at the reference height used for the flux computation are in equilibrium with the underlying surface. This will not, in general, be the case over a patchy snow cover. Areas of bare ground will have different roughness and albedo characteristics from snow-covered areas. Advection of air warmed over bare ground onto a snow-covered area will lead to an enhanced downward heat flux at the upwind edge of the snow patch, with heat fluxes decreasing with increased fetch over the snow as the air comes into a new equilibrium with the snow surface. For instance, Weisman (1977) showed that the dimensionless sensible heat flux H's at any point downwind of the leading edge of a snow patch varies with snow-covered fetch distance as where X is a dimensionless distance downwind of the leading edge of the snow patch (influenced by roughness length) and a4 is controlled by a temperature difference stability parameter (Weisman, 1977).

Marsh and Pomeroy (1996) proposed that the additional sensible heat advected to a snow patch is a function of the snow-free fraction of the upwind domain,

3.3.5 Heat fluxes over a non-uniform snow cover

where HS s is the sensible heat flux to snow over a completely snow-covered fetch, HS b is sensible heat over bare ground, SCA is the proportion of snow-covered area and hS,b expresses the portion of bare ground sensible heat that is advected to the snow patch. Neumann and Marsh (1998) showed that hS b declines from 0.3 to 0.001 as SCA decreases from 1 to 0.01 and that for SCA of 0.5, hS b ranges from 0.02 to 0.2 depending on wind speed and snow patch size. Even more extreme effects may occur over a sea-ice cover interspersed with leads of open water (Claussen, 1991). Interactions with the vegetation will also affect the heat fluxes. They will be discussed in more detail in Sections 3.5.4 and 3.5.5.

The computation of surface fluxes over such non-uniform surfaces is very important for climate simulation (see Section 4.3) and it is a developing subject that requires the application of numerical or analytic models of varying degrees of complexity. A detailed description of such techniques is beyond the scope of the present work and the reader is referred to Weisman (1977), Liston (1995), Essery (1997), and Marsh et al. (1997) for further information.

To conclude, few authors have investigated the variation of the turbulent fluxes in heterogeneous terrain although it was termed a priority research topic by Garratt (1992). Based on such an investigation, a simplified approach is presented in Pluss (1997).

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