Temperature distribution in snow, firn and ice

firnBy Bianca

The temperature plays an important role in the behaviour of snow.

If it’s too warm the snow will just melt, but if it’s cold enough the snow turns into firn and eventually, into glacier ice. How long the transition from snow to ice takes strongly depends on the temperature.

The temperature that has occurred at the surface can take up to millennia to propagate through the glacier ice. That means temperature with depth can provide information on past variations in surface temperatures.

For my study of a firn compaction model, the temperature plays an important role in the speed of snow densification, and so we need to find a way to calculate the temperature distribution with depth.

For a large ice-covered area like Antarctica we don’t have access to much information and are only able to use surface temperatures from available climate models (e.g. ECMWF, NCEP, RACMO). These models provide surface temperatures for each location, calculated from measured temperatures. However, due to the climatic conditions in Antarctica not many observations exist and data is interpolated from the few available observations, together with the knowledge about climatic conditions.

So, how do we tackle this problem?

Firstly, we need some thermal parameters of firn and ice. The thermal properties of pure ice are well known from measurements. The thermal properties of snow and firn, however, have to be calculated using equations. For this, you need to consider thermal conductivity, which depends on snow texture and thus is density dependent, and the specific heat capacity, which varies with temperature, not density. Another needed parameter is the thermal diffusivity, which can be calculated for any density and temperature using the thermal conductivity and specific heat capacity.

To analyse changes in the temperature distribution of surface layers, Fourier’s law of heat conduction can be used. It states that the heat flux is proportional to the temperature gradient (temperature change with increasing depth) per unit length.

Simplified, temperature changes at the surface can be considered as a cyclic variation with fluctuations, with the temperature perturbation at depth approximating a wave of fundamental frequency. This allows us to express near surface temperatures as harmonic series. In reality, surface temperature variations are a complicated function of time that can solely be represented as model estimations.

In regions where the temperature rises above the freezing point during summer months, meting processes need to be considered as well. Melt-water percolates downwards and refreezes once it reaches a cold enough environment, which warms the deeper layers rapidly due to the warmer temperatures of the water.

In summary, temperature variations with depth in a firn layer originate with changes in surface temperature, water infiltration, and conduction.

depthFor glacier ice the vertical temperature profile is not just influenced by the previously mentioned causes that affect the firn layer, but additionally by vertical and horizontal ice flow (advection) and geothermal flux.

In general, a glacier on its own is an effective thermal insulator with the warmest temperatures near the bed and a strong surface temperature influence on top.

Different heat sources and processes can be found within a glacier and vary in importance.

Estimated temperature distribution for the first 25m depths for two locations in Antarctica.

Estimated temperature distribution for the first 25m depths for two locations in Antarctica.

Often a simplified steady-state solution is assumed to obtain a vertical temperature profile. Although the temperature distribution in a glacier is never in a steady state, a large part of measured temperature profiles can actually be understood by examining steady-state distributions. While steady state models help to understand prominent features of temperature profiles, they can never deliver an accurate solution for the temperature distribution in firn and ice, as no ice sheet is ever in a steady state.

Therefore, more of interest are time-varying temperature profiles, where the temperature distribution depends on time, depth and the previously mentioned parameters.

For further information on temperature distribution I refer to the book  “The Physics of Glaciers” by Cuffey & Paterson, 2012 once again.

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