I. Introduction

The fact that snow melts at a temperature of T _s = 0^°C has often lead to the simplified assumption that it also melts at air temperatures of T _a = 0^°C. This statement disregards the intricacies of the energy budget of the snow surface in which radiation fluxes and turbulent exchange of latent heat operate independently of air temperature. In the following, a formulation is derived that permits the assessment of the meteorological conditions for snow melt in terms of absorbed radiation, humidity, and temperature of the atmospheric boundary layer above the snow surface.

The treatment of the problem is essentially an exercise in micro-meteorology. The partition of the energy budget obeys the conservation of energy law and is formally straightforward. Its application to real snow, however, is complicated by three facts:

• (i) The separate variables chosen to describe a particular situation are not entirely independent, for instance, the atmospheric vapor density ρ _va is limited by air temperature T _a, and the long-wave downward radiation flux L↓ is to a certain degree coupled to T _a and ρ _va. Such interconnections need to be considered when specifying a particular set of variables and will be explicitly stated in two examples in the text.

• (ii) Because of the diurnal variation of the energy fluxes, stationarity is only approximated.

• (iii) The transfer of heat through a turbulent boundary layer depends on external parameters such as wind speed and air temperature as well as internal ones like surface temperature and surface roughness.

In view of these complications, an a priori determination of the transfer coefficient is not attempted here. A value of 18.5W m⁻² K⁻¹, as found from long-term observations on Alpine firn and ice (Reference KuhnKuhn, 1979), was used in the quantitative examples in section IV and in Figure 1. Various ways of expressing turbulent-heat transfer and its dependence on stability are briefly summarized in the Appendix.

Fig. 1.

Air temperature T_a at the onset of melting for various values of absorbed radiation R = S↓ — S↑ + L↓ and water-vapor density ρ_va. Curves show saturation and 20% relative humidity. The dotted line is explained in the text. This diagram is valid for a thermal resistance r_H = 65 s m^-1 (equivalent to a transfer coefficient α_H = 18.5 W m^-2 K^-1 at p_a = l.2kg m^-3).

II. Parameterization of the Energy Budget

Energy fluxes at the surface balance such that the sum of short-wave (S↓, S↑) and long-wave (L↓, L↑) radiation fluxes, sensible (H), and latent (V) turbulent-heat fluxes are available for either changing the temperature or the phase of the snow at a rate Q.

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Surface temperature T _s, air temperature T _a, and vapor density of air p _va and surface p _vs are obvious choices of meteorological parameters in specifying this budget. As downward long-wave fluxes L↓ are only weakly related to

T _a (while L↑ is unambiguously tied to T _s), the first three of the four radiation terms are taken as one independent variable

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The turbulent fluxes are taken to be proportional to the differences (T _S – T _a) and (ρ _va – ρ _va), where the factor of proportionality can be expressed as a transfer coefficient, or resistance, which depends on stability as explained in the Appendix. The budget can then be written as

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where ϵ_s is the surface emissivity and is approximately equal to unity, σ = 5.67 x 10⁻⁸ W m⁻² K⁻⁴ is the Stefan–Boltzmann constant, ρ _a is the air density, c _ρ is the specific heat of air, r _H and r _v are the resistance to heat and vapor transfer, respectively, L _v = 2.5 MJ kg⁻¹ the latent heat of evaporation, and the other terms have been explained before.

Considering that surface vapor density is determined by surface temperature

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there are four independent variables (R, T _s, T _a, and ρ _va) and one dependent variable Q for given values of ρ _a, r _H, and r _v in Equation (3).

III. Melting Conditions

At the onset (or cessation) of melting, however, Q = 0, T _s = 0°C, ϵ_sσT_s ⁴ = 315 W m⁻², and ρ*_vs = 4.8 x 10⁻³ kg m⁻³. This situation facilitates the analysis of Equation (3) by reducing the number of variables to three (R, T_a, and ρ_va), each one of which can be studied as it reacts to the other two. That means, for given values of R and ρ_va, there is only one value of T_a that fulfils the condition of incipient melting.

By taking partial derivatives in Equation (3), one can determine the values of (∂T_a/∂ρ_va)_R, (∂Ta/∂R)_ρva, (∂R/∂ρ_va)T_a , which may be considered as sensitivity coefficients that indicate how much one variable has to change in order to compensate for an imbalance caused by one of the others.

VI. Numerical Examples

Choosing, for example, values for ρ_ac_p = 1200 J kg⁻¹K⁻¹ and r_H = r_v = 65 s m⁻¹, Equation (3) becomes

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or T_a = 27.0 – R/18.5 – 2.08ρ_va.

From this equation, the sensitivity coefficients can be derived

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which means that a decrease in atmospheric vapor density by 1 g m⁻³ must be counterbalanced by an increase in air temperature of 2.1 K or, in other words, less condensation (stronger evaporation) is compensated by an increased flux of sensible heat towards the surface.

Similarly,

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A decrease in absorbed radiation by 1 W m⁻² needs to be compensated by an increase of T _a by 0.054 K. Inspecting the reciprocal of this value, one finds that it is equal to ρa^cp^r _H ⁻¹ and thus to the transfer coefficient for sensible heat.

Finally,

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A decrease in atmospheric vapor density by 1 g m⁻³ requires an increase in absorbed radiation by 38.5 W m⁻² for compensation.

It is clear from Equations (3) and (5) that the co-efficients just described are independent of the value of the respective third variable (R, ρ _va, T _a) but that they change with the assumptions made for ρ _a and r_H or r_v.

Let us now use the example in Equation (5) in order to estimate a reasonable range of air temperatures under which melting might be expected.

For a given value of absorbed radiation R, the maximum conceivable air temperature will occur in an absolutely dry atmosphere. A somewhat less stringent lower limit for T _a is set at ρ _va = ρ^* _va(T _a).

In Figure 1, which shows a graphical solution of Equation (5), the two limits ρ ^* _va and ρ _va = 0 are entered as well as the more likely lower limit of 20% relative humidity.

The extreme values of T _a for melting conditions are further determined by extremes of R. Although these cannot be calculated exactly, some likely situations will be discussed in the following examples.

2. Intermediate values, R = L↑

A particular situation arises when the absorbed radiation equals that emitted (L↑ = 315 W m⁻² for T _s = 0° C). Equation (5) then describes T _a as the dry-bulb temperature of a psychrometer for a fixed wet-bulb temperature of 0°C and

can be recognized as the psychrometric constant.

3. Maxima of R

From Equation (2), maxima of R are to be expected for snow exposed to a normally incident solar beam under conditions of high atmospheric transparency (maximum S↓), low albedo (minimum S↑) with warm, humid air (maximum L↓). Extremes of S↓ and L↓ are unlikely to occur simultaneously; as a matter of fact, the vertical changes

and

are nearly equal and opposite under Alpine conditions. In Figure 1, high values of R are associated with low values of T _a which is found in extra-polar mountains.

In order to avoid excessive speculation, let us use midsummer, Alpine measurements (Wagner, Reference Wagner1979, Reference Wagner1980) over an almost horizontal snow surface (S↓ = 1020 W m⁻², L↓ = 280 W m⁻²) and an extreme albedo of 0.2 as applicable to dirty glacier ice or a thin, translucent ice cover on a dark rock. This means S↑ = 204 and R = 1096 W m⁻², yielding a debatable T _a = –32°C. This result is questionable, as it implies a strong lapse-rate above the snow, and turbulent exchange should be significantly higher than initially assumed. If we take only half the resistance as before in Equation (5), (now r _H = 32 s m⁻¹) the resulting t _a becomes –16°C.

V. Conclusions

Since a change in resistance by a factor of 2 is not at all unreasonable and may occur in small space and time intervals, it is impossible to make a quantitative but generally true statement about the upper and lower limits of T _a, except, maybe, that the range –10°< T _a < +10°C is likely to be encountered above snow at the beginning or end of melting (Q = 0).

Figure 1 treats both stable (T _a > 0°C) and unstable situations with the same heat-transfer resistance but it is not intended to suggest that r _H might be a constant for snow. To predict a value of r _H, a number of micrometeorological hypotheses need to be resolved and values of T _a, u, u_*, and z ₀ need to be known, as described in the Appendix. However, for climatological applications of the present problem, the range of r _H is small and likely to be centered around the value of 65 s m⁻¹ found in the ablation period of an Alpine glacier.

Considering that the determination of r_H is a crucial problem in calculating the energy budget, we may even venture to use a form of Equation (3) to determine r _H = r_v from measurements of R, T _a, and ρ _va at a time when Q = 0

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taking care that R is sufficiently different from 315 W m⁻²