This post is acting as an appendix for a blog post I'm writing at the moment. In this post I describe a very simple physical model of sea ice growth and the conversion to a Freezing Degree Days format.
Last year I had been re-reading Semtner 1975 "A Model for the Thermodynamic Growth of Sea Ice in Numerical Investigations of Climate", PDF. Semtner includes a very simple sea ice growth model at the end of that paper. Thanks are due to Steven over at the sea ice forum for getting me on the right track back in September last year, when a blunder in a spreadsheet made me think I'd not got the following correct.
The simplest thermodynamic model of sea ice growth is:
k(To-Ts) / h = p l dh/dt [eg. 1]
In plain english...
The heat flux though a sheet of ice floating on water is determined by the temperature difference through the sheet of ice and the ice thickness (h). The temperature difference across the ice (To - Ts), which is the difference between the surface temperature (Ts) and the temperature at the ice/ocean boundary (To). To, the temperature where ice and ocean meet at the underside of the ice is set to -1.8degC, that's roughly the freezing point of sea water. So that's the k(To-Ts) / h bit, where k is the thermal conductivity of ice ~2.2W/m degC.
I'll assume that all this heat flux goes into making ice at the ice/ocean boundary. This means I'm neglecting heat flux from the ocean, if needed the ocean heat flux can simply be deducted from the heat flux through ice to reduce the amount of heat flux that makes new ice at the ice/ocean boundary (underside of the ice). This will change equation 1 into equation 2:
k(To-Ts) / h = p l dh/dt +Oflux. [eq. 2]
Where Oflux is the ocean heat flux. In other words the Oflux term diverts off some of the energy that would otherwise have gone into making new ice. Note that I will also keep it simple by neglecting snow, snow would insulate the surface of the ice reducing heat flux through it. Neglecting both ocean heat flux and snow means this simplest of models will produce thicker ice than is seen in reality. But that's OK. If we want volume or thickness of sea ice we can use the sparse observations or a numeric model such as PIOMAS or HYCOM/CICE that includes such complex factors and addresses mechanical processes like transport and ridging. As will be seen, that is not my aim, I want to know the sort of thickening that will occur due to winter cold over the Arctic Ocean, not those other factors.
Back to this simple model. I have dealt with the heat flux through ice resulting from the temperature difference between the upper and lower surfaces of the ice. This heat flux grows ice at the ice/ocean boundary by removing heat and forcing the water to freeze, that is how ice thickens thermodynamically when floating on the sea. So I need to convert the heat flux through the ice into a measure of joules (energy) per cubic metre of ice. The latent heat of fusion is the energy that water gives out as it turns into ice, that's l ~ 333.4kj/kg. The density of sea ice (p) is about 917kg/cubic metre (m^3).
kj/kg * kg/m^3 = kj / m^3. That's our conversion from energy into volume of ice.
So that's the p l dh/dt bit. In plain English that is; for each incremental interval of time multiply the latent heat of fusion by the density of ice to get the energy during that interval that goes into making ice, as the units are in seconds the time interval is in seconds.
So if you want to have a change in ice thickness equation you just rearrange the first equation to give:
k(To-Ts) / h p l = dh/dt
This physical equation is the basis of the growth/thickness feedback, because the denominator contains 'h' (while p and l are unchanging), as the ice gets thinner so the growth rate goes up. Of course through a single melt season the ice is thickening, but consider when ice starts the melt season at only a fraction of a metre thick versus starting at a metre thick. The thinner ice will have a far faster rate of growth than the thicker ice, so as the ice thins it rebounds better with thicker growth.
With some playing around the equation can be converted to a form that uses Freezing Degree Days (FDDs) to calculate how much ice would thicken over that number of FDDs.
OK, it's not quite there yet. But the integral (tall wiggle) from time = 0 to time = t of the temperature difference (To - Ts in the above notation for equation 1) can easily be worked into Freezing Degree Days (FDDs). Putting the above stated values for k, p, and l (noting here that my 'p' s really the Greek symbol 'rho'), and a 'wiggle factor' to convert from seconds to days, and we get to equation 4.
Ice thickness = SQRT(InitialThickness^2+(FDD/804.2082)) [Eq 4].
Where I've changed the notation into something most people can probably get right away. So the maximum ice thickness that can be grown for a given FDD, allowing for initial thickness, can easily be calculated.
What are Freezing Degree Days? It's easiest to grasp for a single point, like a weather station. Say you have a winter's worth of data from the weather station, you calculate Freezing Degree Days by summing the temperatures for all the days when the temperature is below zero. It's that simple.
I have taken NCEP/NCAR Reanalysis 2 data, because Reanalysis 1 is making my conversion Excel Add On crash, and have worked out area weighted average FDDs for the regions north of 70degN and north of 80degN for all years from 1979 to 2013. Note for example that the year 2013 is for days 244 to 365 of 2013 and days 1 to 120 of 2014. Note also that the definition of freezing used is 'less than -1.8degC', as this is the freezing point of sea water and it is sea ice that interests me.
Considering the decline of FDDs shown above, it is worth noting that by using equation 4 above an FDD of 4000 implies ice growth from open water of 2.23m.
To virtually guarantee no sea ice by the end of the following summer we could halve that peak winter thickness to 1.12m. Such a thickness would need the FDD to drop to 1000.