Snowpack Properties, Evolution and Ablation
The discussion in the preceding sections has
emphasized the various processes of metamorphism that control the snow bulk properties.
Thermal properties that depend only on density
(specific heat, latent heat) are well defined.
However, those that depend on conductivity or
permeability of the snowpack are affected by
sintering, particle size, ice layers and depth hoar.
The specific and latent heats of snow are the
simplest thermal properties to determine since the
contributions from air and water vapour can be
discounted; each property is simply the product
of the snow density and the corresponding
property for ice.
The temperature dependence of the specific heat
of ice given by Dorsey (1940) is:
C = 2.115 + 0.00779T
1
where C is the specific heat (kJ kg-1 K-1), and T
(oC) is temperature.
The latent heat of melting of ice at 0oC and
standard atmospheric pressure is 333.66 kJ kg-1.
For one-dimensional, steady-state heat flow by
conduction in a solid the thermal conductivity is
the proportionality constant of the Fourier
equation:
F = -K dT/dz
where F is the heat flux (W m-2) and dT/dz is the
temperature gradient.
The thermal conductivity of snow is a more
complex property than specific heat because its
magnitude depends on such factors as the density,
temperature and the microstructure of the snow.
2
The thermal conductivity of ice varies inversely
with temperature by about 0.17% oC-1; the same
may be expected for snow.
A temperature gradient could induce a transfer of
vapour and the subsequent release of the latent
heat of vapourization, thereby changing the
thermal conductivity value.
In non-aspirated dry snow the heat transfer
process involves: conduction of heat in the
network of ice grains and bonds, conduction
across air spaces or pores, convection and
radiation across pores (probably negligible) and
vapour diffusion through the pores.
Because of the complexity of the heat transfer
processes, the thermal conductivity of snow is
generally taken to be an “apparent” or “effective”
conductivity Ke to embrace all the heat transfer
processes.
The degree of surface packing (for example,
hardness) also affects the flow of heat through
3
snow, probably because a surface crust of low air
permeability inhibits ventilation in the upper
snow layer.
The thermal conductivity of snow, even when
dense, is very low compared to that of ice or
liquid water; therefore snow is a good insulator.
This is an important factor affecting heat loss
from buildings and the rate of freezing of lake
and river ice.
Typical numerical models of snow use three
prognostic variables to define a snowpack: snow
depth, snow water equivalent, and temperature.
From snow depth and snow water equivalent, one
can infer the snow density from:
ρs = ρw(w/s)
4
where w (m) is the snow water equivalent, s (m)
is the snow depth, and ρs and ρw are the snow and
water densities, respectively.
Apart from snow depth and snow water
equivalent, the heat content or temperature of the
snowpack is required to describe the system
completely.
The snow temperature is directly related to its
heat content H (J) by:
T = H/(ρw w C).
The energy balance of a snowpack is complicated
not only by the fact that shortwave radiation
penetrates the snow but also by water movement
and phase changes.
The energy balance of a snow volume depends
upon whether it is a “cold” (< 0oC) or a “wet”
(0oC, often isothermal) snowpack.
Recall the energy balance of the snowpack:
5
Q* + QP = QH + QE + QG + ΔQS + QM.
A term is added here to the energy balance to
consider the heat transported by precipitation
(QP), either snowfall or rainfall.
In the case of a cold snowpack, such as is
commonly found in mid-latitudes during winter
with little or no solar input, QE and QM are likely
to be negligible.
Similarly, heat conduction within the snow will
be small because of the low thermal conductivity
of snow and the lack of solar heating, so that ΔQS
and QG are also negligible.
The energy balance therefore reduces to that
between a net radiative sink Q* and a convective
sensible QH heat source.
Although snowcover reduces the available energy
at the surface because of its high albedo to solar
radiation and high emissivity of longwave
6
radiation, its insulative properties exert the
greatest influence on soil temperature regime.
Snow acts as an insulating layer that reduces the
upward flux of heat, resulting in higher ground
temperatures than would occur if the ground was
bare.
In Canada, average ground temperatures are
about 3oC warmer than average air temperatures.
In the case of a “wet” snowpack during the melt
period, the surface temperature will remain close
to 0oC, but the air temperature may be above
freezing.
Since snow is porous, liquid water infiltration is
also important in transporting energy within the
snowpack and into soils.
If meltwater freezes within the snowpack, there is
latent release, warming snowpack layers to the
freezing point.
7
Most of the energy exchanges between snow and
its environment occur at the atmosphere or
ground interfaces; however, because snow is
porous, some radiation and convective fluxes that
occur within the top few centimetres of the
snowpack.
The important fluxes that can directly penetrate
the snowpack are radiation, conduction,
convection, and meltwater or rainwater
percolation.
Temperature regimes in dry snowpacks are
exceedingly complex and are controlled by a
balance of the energy regimes at the top and
bottom of the snowpack, radiation penetration,
effective thermal conductivity of the snow layers,
water vapour transfer, and latent heat exchange
during metamorphism.
Temperature stratification within dry snowpacks
is usually unstable (warm temperatures below
cold temperatures) from formation until late
8
winter and spring, as energy inputs from the soil
boundary exceed those from the atmosphere and
upper layers.
As a result, temperatures become warmer with
depth, with gradients as high as 50oC m-1 in
shallow subarctic and arctic snowpacks during
early midwinter.
In cold climates with frozen soils, an inversion
can develop in late winter where the upper
snowpack warms to higher temperatures than the
lower layers; this reflects higher energy inputs
from the atmosphere (often due to long sunlit
periods in the northern spring) than from the
frozen soil.
For a given climate, the thermal regime in the
snowpack strongly depends on the
amount of snowfall early in the winter season.
Heavy snowfall early in the winter will tend to
maintain the snowpack relatively warm, whereas
9
shallow snowcovers will adjust more rapidly to
the air temperatures.
For a deep snowpack a midwinter rainfall would
increase density and decrease depth.
Subarctic and arctic snowpacks can undergo melt
in upper layers whilst maintaining snow
temperatures significantly below the freezing
point in the lower layers.
Internal heat fluxes in wet snow, or in partially
wet snow, are principally driven by conduction
and by latent heat release due to refreezing of
liquid water.
10
Ref:
Sun et al. (2004)
11
Ref: Lynch-Stieglitz (1994)
12
Ref: Bartelt and Lehning (2002)
13
Ref: Stieglitz et al. (2003)
14
Ref: Stieglitz et al. (2003)
15
Snowpack Ablation
In many countries snow constitutes a major water
resource; its release in the form of melt water can
significantly affect agriculture, hydro-electric
energy production, urban water supply and flood
control.
The ablation of a snowcover or the net volumetric
decrease in its snow water equivalent is governed
by the processes of snowmelt, evaporation and
condensation, the vertical and lateral transmission
of water within the snowcover and the infiltration
of water to the underlying ground.
In turn, water yield and streamflow runoff
originating from snow are governed by these
same processes as well as the storage and the
hydraulics of movement of water in channels.
The rate of snowmelt is primarily controlled by
the energy balance near the upper surface, where
melt normally occurs.
16
Shallow snowpacks may be considered as a
“box” to which energy is transferred by radiation,
convection, and conduction.
Early in the melt sequence vertical drainage
channels develop in the snow contributing further
to its heterogeneity.
The internal structure significantly influences the
retention and movement of melt water through
the snow, making a detailed analysis of the
transmission process extremely difficult.
When the pack is primed to produce melt it is at a
temperature of 0oC throughout and its individual
snow crystals are coated with a thin film of
water; also, small pockets of water may be found
in the angles between contacting grains, usually
amounting to 3 to 5% of the snow by weight.
Any additional energy input produces melt water
which subsequently drains to the ground.
17
When melt rates are at their highest, 20% (by
weight) of the pack or more may be liquid water,
most of which is in transit through the snow
under the influence of gravity.
The amount of energy available for melting snow
is determined from the energy budget equation.
Shortwave radiation
There are two main types of radiation affecting
snowmelt: shortwave and longwave radiation.
The amount of solar radiation penetrating the
earth's atmosphere to be received at the surface
varies widely depending on latitude, season, time
of day, topography (slope and orientation),
vegetation, cloud cover and atmospheric
turbidity.
While passing through the atmosphere radiation
is reflected by clouds, scattered diffusely by air
molecules, dust and other particles and absorbed
18
by ozone, water vapour, carbon dioxide and
nitrogen compounds.
The absorbed energy increases the temperature of
the air, which in turn, increases the amount of
longwave radiation emitted to the earth's surface
and to outer space.
Short-wave radiation reaching the surface of the
earth has two components: a direct beam
component along the sun's rays and a diffuse
component scattered by the atmosphere but with
the greatest flux coming from the direction of the
sun.
Figure 9.1 shows the annual variation in daily
values of solar radiation received by a horizontal
surface at several latitudes assuming a mean
transmissivity of unity, implying that all the
energy reaches the surface.
The influence of transmissivity is illustrated in
Figure 9.2.
19
The time of year obviously is an important factor
governing the solar radiation flux incident on the
earth's surface, and hence on the melt rate.
As a rule, the longer the spring melt is delayed
the greater the danger of flooding.
This is due partly to increases in the radiative
flux and partly to the increased probability of
rain.
The transmissivity is highest in winter and lowest
in summer because the atmosphere contains more
water vapour during summer. It also varies
somewhat with latitude, increasing northwards.
Snow on a south-facing slope melts faster than
snow on a north-facing slope, the reason being
that the orientation of the slope affects the
amount of direct beam solar radiation the area
receives.
The results are symmetric about a north-south
line; as might be expected the influence of
20
orientation diminishes towards the summer
solstice.
Even on a 10o slope the effect of orientation can
be significant; e.g., at 50oN on April 1, a southfacing slope receives approximately 40% more
direct beam radiation than a north-facing slope.
Longwave Radiation
The net longwave radiation at the snow surface
L* is composed of the downward radiation L↓
and the upward flux L↑ emitted by the snow
surface.
Over snow L↑ is normally greater than L↓ so that
L* represents a loss from the snowpack.
The longwave radiation emitted by the snow
surface is calculated with the Stefan-Boltzmann
law on the assumption that snow is a near perfect
black body in the longwave portion of the
spectrum.
21
In alpine areas topographical variations have a
significant influence on the longwave radiation
received at a point, e.g., in a valley the
atmospheric radiation is reduced because a part
of the sky is obscured by its walls.
However, the valley floor will gain longwave
radiation from the adjacent slopes in amounts
governed by their emissivities and temperatures;
the reflected longwave radiation from snow and
most natural surfaces is almost negligible.
Thus in areas of high relief the radiation incident
at a site includes longwave emission from the
atmosphere and the adjacent terrain.
To a first approximation the radiation emitted by
cloud can be obtained by assuming black-body
emission at the temperature of the cloud base.
Hence, the net longwave radiation exchange
between the overcast sky and the snow can be
approximated as an exchange between two black
22
bodies having temperatures Ts (snow surface) and
Tc (cloud base), i.e., L* = σ(Tc4 - Ts4).
Sensible, Latent, and Ground Heat Fluxes
The convective and latent energy exchanges, Qh
and Qe, respectively, are of secondary importance
in most snowmelt situations when compared to
the radiation exchange, but still need to be
considered to assess melt rates.
Both Qh and Qe are governed by the complex
turbulent exchange processes occurring in the
first few metres of the atmosphere immediately
above the snow surface.
Heat is transferred to the snow by convection if
the air temperature increases with height
(commonly occurring when the snow is melting);
and water vapour is condensed on the snow
(accompanied by release of the latent heat of
vapourization) if the vapour pressure increases
with height.
23
The ground heat flux QG is a negligible
component in daily energy balances of a
snowpack when compared to radiation,
convection or latent heat components, so that the
total snowmelt produced by QG over short
periods of time can be ignored.
However, QG does not normally change direction
throughout the winter months and consequently
its cumulative effects can be significant over a
season.
In areas where snow temperatures remain near
the freezing point and ground temperatures are
relatively warm, melt can be produced as a result
of QG.
Although the amount of water produced may be
small, its resultant effect on the thermal
properties and infiltration characteristics of the
underlying soil may be important.
24
Heat exchanges between soils and snow follow
the simple Fourier equation for heat transfer used
in heat transfer in snow alone.
Rain on Snow
The heat transferred to the snow by rain water is
the difference between its energy content before
falling on the snow and its energy content on
reaching thermal equilibrium within the pack.
Two cases must be distinguished in this energy
exchange:
1) Rainfall on a melting snowpack where the rain
does not freeze;
2) Rainfall on a pack with temperature < 0oC
where the water freezes and releases its latent
heat of fusion.
The first case can be described by the expression:
QP = ρw Cp(Tr - Ts)Pr/1000
25
where QP is the energy supplied by rain to the
snowpack, ρw is the density of water, Cp is the
heat capacity of water, Tr the temperature of the
rain, Ts is the snow temperature, Pr is the depth of
rain or precipitation rate.
When rain falls on a snowpack which has a
temperature <0oC, the situation is more
complicated since the pack freezes some of the
rain thereby releasing heat by the fusion process.
Snowmelt
The amount of meltwater can be calculated from:
wm = QM /(ρw Lf B)
where wm is the meltwater (m), Lf is the latent
heat of fusion, and B is the fraction of ice in a
unit mass of wet snow. B usually has a value of
0.95 to 0.97.
26
Net radiation and sensible heat largely govern the
melt of shallow snowpacks in open
environments.
At the beginning of the melt, radiation is the
dominant flux with sensible heat growing in
contribution through the melt.
If a complete set of meteorological measurements
is not available, then temperature index models
may be used to predict snowmelt. Index models
relate melt to air temperatures such that:
wm = Mf (TA - TB)
where TA is the mean air temperature over a given
time period and TB is a base temperature below
which melt does not occur (usually 0oC).
The melt factor Mf varies from 6 to 28 mm oC-1
day-1 for snowmelt on the Canadian Prairies.
27
Although index models are simple, they should
be used with caution as the melt factors tend to
vary from year to year and with location.
Streamflow Generation
Streamflow generated by snowmelt water that
directly runs off rather than infiltrating or from
water that infiltrates and then moves downslope
through a shallow subsurface soil of high
permeability.
During snowmelt, frozen or saturated soils
restrict infiltration and evaporation is relatively
low; this promotes a water excess over a basin
and permits relatively large runoff generation for
the amount of water applied to the ground.
As a result, peak annual streamflows often occur
directly after snowmelt.
28
The constituent water of this freshet comprise
both snowmelt water and water expelled from
soils by infiltrating snowmelt water, with
important implications for stream chemistry.
For point scales, the influence of snow water
equivalent on infiltration and runoff generation
varies for different soil types.
The effect of a deep forest environment
snowpacks in promoting warm soils causes forest
runoff to drop with increasing snow water
equivalent for deep snow and dry soils.
In northern forests, from 40 to 60% of annual
streamflow is derived from snowmelt, with
increases in snowmelt runoff of from 24 to 75%
when the forest is removed by harvesting or fire.
In cold, semiarid environments (arctic, northern
prairies, steppes), greater than 80% of annual
streamflow is derived from snowmelt, even
29
though snowfall accounts for less than 50% of the
annual precipitation.
Snowmelt in the western cordillera of North
America and mountain systems of central Asia is
the major source of water when carried as
streamflow to semiarid regions downstream.
Snowmelt water sustains arctic, alpine, prairie,
and boreal forest lakes and wetlands, which are
primary aquatic habitats in their respective
ecosystems.
30
Ref: Barnett et al. (2005)
31
Ref: Déry et al. (2005)
32