# Sea ice thickness

Sea ice thickness is an important climate-related variable whose determination from satellite measurements is still an unsolved problem. While ice concentration is often used as a marker for climate change, the more important variable is sea ice volume which can be determined by multiplying concentration with thickness and integrating over the ocean surface. Sea ice thickness determines a number of important fluxes such as heat flux between the air and ocean surface—see below—as well as salt and fresh water fluxes between the ocean since saline water ejects much of its salt content when frozen—see sea ice growth processes. It is also important for navigators on icebreakers since there is an upper limit to the thickness of ice any ship can sail through.

## Determining surface heat flux

The rate of heat transfer between ocean and air through an ice sheet can be determined from the ice thickness and the prevailing weather conditions. The net heat flux is related to the ice thickness and surface temperature as follows:

$hQ^{*}=k(T_{s}-T_{w})$ where h is the ice thickness, Q* net heat flux, k is the thermal conductivity of the ice, Ts is the ice surface temperature and Tw is the water temperature which is assumed to be freezing—note that this varies with the salinity of the water, see freezing-point depression. Thermal equilibrium is assumed, true if the weather conditions are changing slowly or the ice is not too thick. If we do not assume thermal equilibrium, numerical solution of the heat equation based on past weather conditions is required.

The net heat flux can be decomposed into four components: latent heat, sensible heat, longwave and shortwave fluxes:

$Q^{*}=Q_{E}+Q_{H}+Q_{LW}+Q_{SW}$ The latent heat flux is due to the latent heat stored and released from the sublimation of water between the ice and air. It is normally calculated from a parameterization based on the wind speed and the difference in water vapor partial pressure above the ice (2 or 10 m) and at the ice surface:

$Q_{E}=k_{E}L\rho v\left[RHe(T_{a})-e(T_{s})\right]$ where kE is a constant, $\rho$ is the air density, L is the latent heat of sublimation, v is windspeed, RH is relative humidity, Ta is the air temperature at the 2 or 10 m level and e is a function for determining saturation water vapor pressure--see also dewpoint. The sensible heat flux refers to actual physical heat transfer and its major component is due to surface convection. It too is calculated from a paramerization, this time based on the difference in air and surface temperature:

$Q_{H}=k_{H}C_{p}\rho v(T_{a}-T_{s})$ where kH is a constant and Cp is the heat capacity of air.

The longwave flux can be approximated from the Stefan-Boltzmann law. A typical formulation looks something like this:

$Q_{LW}=\epsilon _{i}\sigma \left[0.39(1-k_{cc}(\phi )cc^{2})T_{s}^{4}+T_{s}^{3}(T_{s}-T_{a})\right]$ where $\epsilon _{i}$ is the emissivity of the ice and $\sigma$ is the Stefan-Boltzmann constant. Note the corrections for cloud-cover, cc, using a latitude-dependent cloud-cover coefficient, kcc. Meanwhile, the shortwave flux is determined primarily from geometric consideration, with the cloud-cover and albedo acting as multiplication coefficients:

$Q_{SW}=S(1-0.62cc)(1-a)\cos \theta (\phi ,t)$ where S is the solar constant, a is the albedo of the ice surface and $\theta$ is the angle of the sun's rays relative to the Earth's surface as a function of latitude, $\phi$ , and time of day and year—see insolation.  

## Measurement

Ice thickness can be measured directly by taking an ice core and measuring it or drilling through the ice. Since this is laborious and time-consuming, more automated methods are normally sought. Measurements of ice depth below the waterline (or draft) by submarine sonar or RADAR systems can give good estimates of ice thickness provided there isn't too much snow (which is less dense than ice) on top. The E-M Bird ice thickness meter, designed by the Alfred Wegener Institute for Polar and Marine Research, is carried aloft by helicopter and measures ice thickness with a combination of a pair of inductance coils that measure the ice-water interface based inductance variations—similar to a metal detector--and a laser altimeter which measures the ice surface.  It was used on a small scale in 2007 to supplement microwave radiometer measurements during the Pol-Ice campaign and on a much larger scale during the GreenICE (Greenland Arctic Shelf Ice and Climate Experiment) campaign conducted in 2004 and 2005.

### Satellite instruments

There are a number of satellite-mounted altimeters capable of measuring ice thickness from space. ICESat, for instance, measured ice surface (of primarily glacial ice pack) using laser altimetry. Unfortunately, variations in surface elevation caused by differing ice thickness are so small that only averages over a relatively long time scale (one month, for instance) are significant.

The microwave emissivity of sea ice is found to vary quite significantly with thickness. This is caused mainly by changes in the salinity, particularly in the surface salinity which is a result of growth processes.  Martin et al.  use the following, empirical, equation to determine the thickness of new ice in the Chukchi Sea from satellite microwave radiometer measurements:

$h=\exp \left[{\frac {T_{h}(37GHz)}{\alpha T_{v}(37GHz)}}\right]-\gamma$ where $T_{p}(\nu )$ is the brightness temperature at polarization, p (horizontal or vertical) and frequency, $\nu$ , and $\alpha$ and $\gamma$ are constants. The relationship, being highly dependent on growth conditions, was not found to hold everywhere in the polar oceans.

Currently, the most accurate method for determining sea ice thickness from satellite data is based on infrared radiometry using satellites such as the Advanced Very High Resolution Radiometer (AVHRR). Since the emissivity of sea ice (indeed of most classes of objects) is fairly constant at infrared frequencies, measurements are used to estimate the physical surface temperature of the ice. This, in turn, is used along with the prevailing weather conditions, which can be derived from circulation models or buoys to calculate the heat flux from which follows the ice thickness—see first section.   The main problem with this method is that it can only be used during cloud-free conditions.

A promising new satellite instrument for detecting ice thickness is the Soil Moisture and Ocean Salinity (SMOS) instrument. SMOS is a polarimetric radiometer operating at 1.4 GHz (or L band). Generally, the lower the frequency of radiation, the more weakly it interacts with matter (the photons don't have enough energy to induce many energy transitions) thus L band radiation penetrates quite deeply into sea ice. This implies that the instrument will supply at least some information on ice thickness.