Microscale Weather and Climate - Radiation and heat transfer in water, snow and ice

CA-DBB station after a significant snowfall event

Today’s learning objectives

Explain what makes snow and water different compared to most other surface-atmosphere interfaces.

Describe the transmission, absorption and reflection of radiation in water and snow.
Describe how we can define the energy budget of a snow-pack or water body.

Properties of snow and water surfaces?

Thermal Properties of Water

Figure 1: For a given amount of energy input (J), higher heat capacity means a volume of a substance will warm more slowly; higher thermal admittance means a surface will warm less

Treating Snow and Water as Volume Interfaces

In contrast to opaque land surfaces, land-atmosphere interfaces of liquid and solid water transmit radiation into lower layers.

This means we must know the transmission (and absorption, reflection and emission) of radiation in those media in order to predict energy exchange.
As a consequence, we must treat water, snow and ice ‘surfaces’ as 3D-volume interfaces.

Transmission of short-wave radiation through a glacier (Photo: USGS Webpage)

Beer’s Law

Describes the reduction in flux density of a parallel beam of monochromatic (single wavelength) radiation through a homogeneous medium.

\[ R_z = R_0 e^{-z\mu} \qquad(1)\]

Assuming that absorptivity (and hence transmissivity) are constant:
- Radiative flux density decays exponentially with depth
  - \(\mu\) is the attenuation coefficient, not thermal admittance

Attenuation Coefficient

The constant of proportionality \(\mu\) is called attenuation coefficient (m_-1). It depends on:

Wavelength
Nature of the medium
Impurities (i.e. algae, plankton, chemicals)

Attenuation Coefficient (iClicker)

Which line (A or B) has a higher attenuation coefficient?

Figure 2: Beam attenuation as a function of z and \(\mu\)

Why is water blue?

Not only an overall reduction of the radiative flux density but also a shift in the maximum wavelength towards blue.

\(\mu\) of water depends strongly on wavelength \(\lambda\). Absorption is very high in near infrared (NIR, 0.7 to 3 \(\mu m\)), and lowers in the visible range (0.4 to 0.7 \(\mu m\)).

Attenuation in Water

Figure 3: Beam attenuation as a function of depth in liquid water for selected wavelenghts of visible light

Liquid vs. Solid

The liquid and solid state of water have similar attenuation coefficients (with some exceptions).

We can use Beer’s law to describe the decay of radiation with depth z in water bodies, snow or ice for different wavelengths.

Exhaust plume from Amundsen-Scott South Pole Station stratifies into the very stable layer (Photo: J. Dana Hrube)

Spectral Transmittance of Ice

Effect of Impurities: Warren et al (2006)

Ice has high transmissivity in the UV-A and blue visible wavelengths and decreasing transmissivity in the red part of the visible spectrum.

Measuring transmission (Photo: Website U of Washington / Atmos Sci)

How Does this Effect the Surface Energy Balance?

Photo: A. Christen

Radiation Balance of Snow and Ice

One of the most important characteristics of snow and ice is their high albedo

One of the most important characteristics of snow and ice is their high albedo  overall low energy status. The introduction of even a thin snow cover over the landscape has dramatic effects. In a matter of a few hours a natural landscape can experience a change in albedo from approximately 0.25 to perhaps 0.80. Thereafter α declines as the snow pack ages (becomes compacted, and soiled), but with a fresh snow-fall it rapidly increases again. The wavelength dependence of the albedo of snow helps to explain the ease with which human skin becomes sunburnt, especially on sunny days on snow covered mountains.

Human skin is very sensitive to ultra-violet (UV) radiation, with peak sensitivity at about 0.3 \(\mu m\). The potential for sunburn stretches from 0.295 to 0.330 \(\mu m\). The lower limit is governed by the almost total absorption of these wavelengths by ozone in the high atmosphere (Reading Package Lectures 3-4, Figure 5, Panel b). With a fresh snow cover the receipt of potentially burning ultra-violet radiation by exposed skin is almost doubled, because in addition to that received from the incoming beam there is a very significant proportion received after surface reflection (due to the high value of αλ at these short wavelengths). The upward flux is responsible for sunburn on earlobes, throat and within nostrils, areas which are sensitive and normally in shade.

Radiation Balance of Snow and Ice: Antarctica

Surface: The internal transmission of radiation through snow and ice gives problems in formulating the surface balance and in observation. For example, measurements of reflected shortwave from an instrument mounted above the surface include both surface and sub-surface reflection. The albedo calculated from such measurements is therefore a volume not a surface value. LW: However, although their emissivity ε is high the absolute magnitude of L↑ is usually relatively small because the surface temperature T0 is low. Since clouds are also close to being full radiators the net longwave exchange (L∗) between a fresh snowpack and a complete overcast is simply a function of their respective temperatures. Should the cloudbase be warmer than the snow surface there will actually be a positive L∗ budget at the snow surface. With clear skies L∗ is almost always negative, as is true of most other surfaces. Net allwave radiation. In general it may be said that the daytime net radiation surplus (Q∗) of snow and ice is small by comparison with most other natural surfaces. This is directly attributable to the high surface albedo. In the case of the Antarctic early summer budget shown in Figure 1 the surface albedo is 0.80. This results in so little shortwave radiation absorption that when it is combined with a net longwave loss, that is quite similar to other environments, the midday net all-wave radiation is less than 10% of the incoming solar radiation K↓. In- deed even though the daylength at this high latitude is about 21 h, the total daily net radiation is negative (approximately −1.0 MJ m−2 d−1).

Variation of the radiation budget components over snow at Mizuho Station, Antarctica (70◦S), on 13 November 1979 (after Yamanouchi et al., 1981).

Net-radiation in snow and ice

We can calculate net radiation \(R_n(z)\) for each depth layer.
In snow, long-wave radiation is relatively quickly absorbed, but short-wave radiation is less reduced with depth. Also long-wave emission to the atmosphere is limited to a shallow layer.
During daytime, net radiation \(R_n(z)\) in a certain layer is the sum of short-wave \(SW^*_z\) and long wave \(LW^*_z\) in this layer.

Snow Temperature Profiles

Due to the difference in radiative absorption of the long-wave and the short-wave radiation, daytime net radiation \(R_n\) is greatest just below the surface, creating a subsurface temperature maximum.
If \(R_n\) dominates the melting process, this subsurface layer shows first snow-melting → ‘loose’ or ‘hollow’ character of a melting snow pack.

T.R. Oke (1987): ‘Boundary Layer Climates’ 2nd Edition.

Photo: Joe Shea (UBC Geography)

The energy balance of snow and ice

Beside the 3D framework, also the phase changes of water play an important role in the energy balance of a snow and ice volumes.

The energy balance of snow and ice

Beside the 3-d framework, also the phase changes of water play an important role in the energy balance of a snow and ice volumes.

Phase changes of water are accounted by a special term in the energy balance, \(\Delta M\); the energy flux density associated with latent heat of fusion (freezing/melting)
- Lf = 0.334 MJ kg^-1 at 0ºC

\[ \color{orange}{R_n} = \color{red}{H} + \color{blue}{LE} + \color{green}{\Delta S} + \color{grey}{\Delta M} \qquad(2)\]

The State of a Snow Pack

‘Cold’ snow pack

TS well below 0ºC. Only solid state of water. No water available for LE or \(\Delta M\).

‘Wet’ snow pack

TS at 0ºC and often isothermal. Both solid and liquid state of water makes LE or \(\Delta M\) important. \(H_r\) is the heat input by rain (i.e. TR > TS)

‘Wet’ snowpacks. A wet snowpack during the melt period the surface temperature will be held very close to 0◦C but the air temperature may be above freezing. Precipitation may then be as rain and the energy balance (Figure 2b) becomes: Here\(H_r\) heat supplied by rain if its temperature is greater than that of the snow. In some mid-latitude locations\(H_r\) can be a significant energy source for melt, especially where the area is open to storms originating over warm oceans. During active melting both radiation (Q∗) and convection (QH + LE) act as energy sources (Figure 2b) to support the change of phase (ice to water). The temperature of the snow changes very little in this process; therefore the large change of energy storage is due to latent rather than sensible heat uptake, i.e. ∆QM (latent heat storage change) rather than ∆QS.

The role of LE in the case of ‘wet’ snow is interesting. The surface vapour pressure of a melting ice or snow surface is of course the saturation value (e∗) at 0◦C which equals 611 Pa (See Oke, 1987, Table A3.2, p. 394). In absolute terms this is a low value and it is very common to find the warmer air above the surface has a greater vapour pressure. In that case an air-to- surface vapour pressure gradient exists, so that any turbulence contributes a downwards flux of moisture, and condensation on the surface. Since at 0◦C the latent heat of vaporization (Lv) released upon condensation is 7.5 times larger than the latent heat of fusion (Lf ) required for melting water, for every 1 g of water condensed sufficient energy is supplied to melt a further 7.5 g. Under these conditions LE is an important energy source. Of course should the air be drier than 611 Pa the vapour gradient would be the reverse, evaporation would occur

Energy Balance Melting Snow

An example of the energy balance of a melting snow cover from the Canadian Prairies in Spring is given in Figure 3. The primary source of energy for the melt in this case is the net radiation. There is a small input of sensible heat by turbulent transfer throughout the day. This is because the air-mass was slightly warmer than the snow surface. Prior to noon the heat input to the pack is almost entirely put into storage, ∆QS. This is used to raise the snow temperature from its overnight value to the freezing point and then to change the proportions of ice and water in the pack. The melt peaks in the afternoon period and then declines as the pack cools again. The decline is retarded somewhat by the release of latent heat of fusion (Lf ) as the upper layers re-freeze.

In Figure 3 heat conduction into the soil is not large enough to warrant inclusion.

Albedo of a water surface

Albedo of liquid water strongly depends on the angle at which the direct solar beam hits the water surface.

Specular and Diffuse Reflection

Specular: beam reflected at same angle (like mirror)

Diffuse: beam diffused isotropically (Lambertian)

Albedo of a Water (iClicker)

A smooth water surface will under clear (cloud-free) skies will have the highest albedo at:

Noon
Sunset
Midnight
All of the above

Influence of waves on albedo

Radiation Balance of Open Water

Open water surfaces (rivers, lakes, oceans) have the unique feature compared to land surfaces, that turbulent exchange is important on both sides, the atmosphere and the hydrosphere.

Similar to air, turbulent exchange is much more efficient than molecular heat conduction in water.
Further advective energy flux densities (ΔQA) are almost all the time significant.

This means that heat transfer within water is possible not only by conduction and radiation, but also by convection and advection. Where ∆QS change of heat storage in the layer, ∆QA net horizontal heat transfer due to water currents. Radiation absorption by a water body is strong and it should be noted that for large portions of the daytime more of this energy is being used to heat the water (∆QS ). At night this energy store becomes the source of energy which sustains an upward flow of heat to the atmosphere throughout the period. Any water body therefore acts as a major heat sink by day, and a major heat source at night. On an annual basis for large water bodies ∆QS can be assumed negligible (i.e. zero net heat storage). The energy balances of the major oceans are then as given in Table 3.1 which strikingly illustrates the dominant role played by evaporation (LE ) as an energy sink for a water body. On an annual basis approximately 90% of Q∗ is used to evaporate water, and this leads to characteristically small Bowen ratio (β) values of approximately 0.10.

Take home points

Beer’s law describes the transmission of radiation through a medium (snow, water) at a given wavelength. attenuation Coefficient \(\mu\) for water is changing from low (VIS, blue) to high (NIR).
Radiation balance of a snow pack volume can cause \(R_n\) maximum below surface, and hence subsurface temperature maximum that causes melting and ‘hollow’ snow pack.
The energy balance for a snow-pack or ice volume needs to consider the latent heat of fusion. Depending on snow/ice temperature we distinguish between dry ‘cold’ and melting ‘warm’ snowpacks.