The topic of this post is “cold content”. This is an important quantity in the world of snow science, but sometimes it’s not explained or understood very well. It’s always helpful to have an order of magnitude understanding of these types of physical quantities. Cold content (CC) is the amount of energy required to heat all of the ice in the snowpack to a temperature of 0°C. It’s not completely obvious that this is the case, but all observations of real world snowpacks show that significant ablation only happens when the entire snowpack is isothermal at 0°C (in other words, the snow has a cold content of zero) ¹.

The equation for cold content, for a uniform single layer snowpack, is given by

\[CC = \rho \cdot c_p \cdot h \cdot (0 - T_{snow})\]

Where \(\rho\) is the density of the snow, \(c_p\) is the specific heat capacity of ice, T\(_{snow}\) is the temperature of the snow layer, and h is the height of the snow. The specific heat capacity is the only constant that we probably don’t have a good familiarity with. For ice, it’s 2.1E3 J/kg/°C ².

So, it’s a pretty straightforward quantity. But there are a few caveats. It doesn’t just depend on how cold the snow is. A deep snowpack with warm snow can have the same CC as a cold, shallow snowpack. The density of the snow also matters. The above equation is just for one layer of snow, but in reality both the snow density and the snow temperature vary throughout the pack. So it might be better to write CC as something like this:

\[CC = \sum_i \rho_i \cdot c_p \cdot h_i \cdot (0 - T_{snow,i})\]

Where the \(i\) subscript indicates the layer of the snowpack. Fresh low density snow near the top of a snowpack may very well be much colder than deeper, higher density snow near the bottom of the pack — but its contribution to the total CC is less, since there is simply less mass to heat up in the low density snow layer. We might get excited when we stick a thermometer on the top of fresh snow on top of a snowpack and see cold temperatures, but from the perspective of the entire energy balance of the snow and CC, that might be a relatively small contribution given how low-density it is.

How much energy does it take to erode cold content?

Let’s put some numbers together. Let’s say we have a 1 m snowpack with a 300 kg/m\(^3\) density that has an average temperature of -1 °C. That is equivalent to 300 mm of SWE (snow water equivalent). This is probably not that far off from a middle or low elevation snowpack somewhere in the Sierra Nevada in the late winter. The math becomes:

\[\begin{aligned} CC &= 3E2 \times 2E3 \times 1 \times 1 \\ &= 6E5 \, \text{J/m}^2 = 0.6 \, \text{MJ/m}^2 \end{aligned}\]

So such a snowpack has 6E5 J/m\(^2\) of CC to overcome before melt can begin. Is that a lot? If you’re like me then it’s hard to have a good idea of how much energy that really is.

Let’s consider a heating rate of 10 W/m\(^2\) averaged across an entire day. Daily solar insolation is probably something like ~200 W/m\(^2\) (averaged across an entire day) at your mid-latitude study site in late winter³, for some context. Recalling that one Watt is one Joule per second, we can then do a nice conversion keeping in mind that there are 8.64E4 seconds in a day. So that means that, with a heating rate of 10 W/m\(^2\), we have transferred 8.64E5 J/m\(^2\) (0.864 MJ/m\(^2\)) to the snowpack! This is larger than the cold content of the snowpack, and we did that in a single day!

How much energy does it take to melt snow versus deplete cold content?

Once the snow has warmed up to an average temperature of 0°C, we still have to melt the snow for it to totally disappear. The latent heat of fusion of ice is 3.34E5 J/kg. So for our example snowpack from earlier, we simply multiply this number by the snow density (\(\rho\)) and height (\(h\)). Let’s make up a new quantity — the melt-energy (ME).

So,

\[\begin{aligned} ME &= L_v \cdot \rho \cdot h = 3.34E5 \cdot 3E2 \cdot 1 \\ &= 1E8 \, \text{J/m}^2 = 100 \, \text{MJ/m}^2 \end{aligned}\]

We can see that this is a large number. It only took 0.6 MJ to warm up the snow, but it takes 100 MJ to completely melt it.

A general expression for cold content (CC) vs melt energy (ME)

So we can think about this even more clearly by just doing some more algebra. If we look at the ratio of ME to CC, the density and the depth fall out of the equation, and only the snow temperature term remains. Since the specific heat capacity of ice (c\(_p\)), is about two orders of magnitude less than the latent heat of fusion (L\(_v\)), we are in a situation where the snow has to be really really cold relative to 0°C in order for the CC to be on-par with the energy required for melt.

\[\frac{CC}{ME} = \frac{c_p}{L_v} (0 - T_{snow}) = \frac{2.1E3}{3.34E5} (0 - T_{snow})\]

The snow would have to be about -150 °C in fact for \(\frac{CC}{ME} \approx 1\).

ME vs. CC at the daily timescale

Let’s assume that one-inch per day of SWE completely melts. This is a somewhat rapid rate. That’s 2.5E-2 m of melt over the course of the day. In units of energy, that is 8 MJ/m\(^2\). If we pretend that the CC of our -1°C snowpack from our previous example was overcome prior to melting at that rate, then the total energy required is ~8.6 MJ/m\(^2\). This is equivalent to an energy imbalance of approximately 100 W/m\(^2\) over the course of the day. And it’s apparent that only a small amount of that energy is assosciated with overcoming CC; over 90% is used for melting, rather than heating up, the ice.

Conclusions

Nothing in this post is new content per-se, and I think I recall doing a homework during the first year of my M.S. thesis that was something close to this exercise. In my M.S. paper I also came to appreciate these facts in greater detail when I did some experiments seeing how the atmosphere responded to different land surface snow anomalies. These are all things that are encoded in most (all? I hope so) snow models that researchers use. To develop intuition I think it’s important to actually pull apart the equations every now and again to get a sense of the numbers and orders of magnitude.

The numbers used in this post are fairly conservative and may undersell the importance of CC; in mid-latitude snowpacks we can easily have > 1000 mm of SWE. If we had 3x the SWE as my example, and the same bulk snow temperature of -1°C, then there would be ~1.8 MJ/m\(^2\) of cold content. You can easily plug in other numbers to get a handle on the amount of energy required.