Lake Ice and 
Snow Science 

SNOW: Conductive Heat Flux through Snow on Lake Ice

What is conductive heat flow?

1.  When water freezes, latent heat is produced.

2.  In the case of water freezing on the bottom of a lake ice (or sea ice) cover, the latent heat is conducted through the ice and snow to the atmosphere along the negative temperature gradients (red).

3. The magnitude of the conductive heat flow is determined by the thermal conductivity of and the temperature gradients in the snow and ice. The latter are a function of the depth of the snow and ice, and the temperature at the top and bottom of the snow surface and at the bottom of the ice.

4. To calculate the conductive heat flow from the bottom of the ice to the top of the snow, we only need to know the snow variables (depth, top and bottom temperature, density [from which we derive thermal conductivity]).

Snow temperature gradient (T_grad)

The change in the temperature of the snow with depth expressed by the equation:

T_grad = (T_s - T_b) / Z_s

where    T_s: snow surface temperature (°C),
              T_b: snow bottom temperature (°C), and
              Z_s: snow depth (m).

The snow temperature gradient increases as air temperatures (snow surface temperatures) decrease.

Snow Density (ρ)

Snow density is the mass per volume and is expressed by the equation:

ρ = Mass (g)/Volume (cm³) with units g cm^-3 (g/cm³)
ρ = Mass (kg)/Volume (m³) with units kg m^-3 (kg/m³).

Snow densities vary between as little as 50 kg/m³ (new snow) and 500 kg/m³ (wet snow). Higher density (heavier) snow typically results from higher temperatures and/or winds while lower density (lighter snow) usually results from colder air with less wind. The density will increase over time due to snow settlement.

Snow Thermal Conductivity
(k_eff, i.e. effective thermal conductivity)

The thermal conductivity of the snow cover can be calculated as a function of its density.

If snow density is ρ< 0.156 g cm^-3,      then k_eff = 0.023 + 0.234ρ
If 0.156 > ρ < 0.6 g cm^-3,                     then k_eff = 0.138 - 1.01ρ + 3.2332ρ²

k_eff is expressed as W m^-1 K^-1 (W/m/K, Watts per metre per Kelvin)

Conductive Heat Flow (F_a)

F_a = (T_grad)( k_eff )

F_a is expressed as W m^-2 (W/m²).

QUESTIONS:

1. What is the conductive heat flux through the top pair of snow covers?


2. What difference does the snow density make?



3. What is the conductive heat flux though the bottom pair of snow covers?


4. What difference does the temperature gradient make?