Guest Post by Willis Eschenbach
One of the best parts of new tools is new discoveries. So the tools to calculate the heat constants of the ocean and land as described in my last post, Lags and Leads, reveal unknown things to me.
A while back I wrote a post called “Wrong Again”, about a crucial mistake I’d made that was pointed out by a commenter, might have been Mosher. In any case, what I’m wrong about this time is that I always thought that over the course of a year, much more energy per square metre of surface was stored in and subsequently released from the ocean than was stored in the land. I figured that in part this was a result of the difference in heat capacities (specific heats) of water and soil, which are in a proportion of about three to one per cubic metre. That is to say, it takes about three times the energy to warm a cubic metre of ocean by one degree C that it takes to warm a cubic metre of soil. I also figured that because the heat can penetrate the ocean more deeply, it would store more of the heat than the land would. Finally, I figured that the differences in albedo would favor the ocean over the land.
One of the joys of writing for the web is that commenters set off whole trains of thought about what I’m investigating. It’s like having a set of global colleagues. Sometimes rambunctious colleagues, to be sure, but well worth it. In this case a comment from Bruce Ploetz led me to look into the question of the implied heat storage in the thermally lagged system.
In my last post I used the example of putting an aluminum pan on the fire versus putting a cast-iron pan on the fire. The aluminum pan has a small heat constant “tau”, while the cast-iron pan has a large heat constant tau.
An oddity of this is that we can calculate the relationship between tau and the actual size of the thermal mass. For the earth system, it works out to a thermal mass per square metre of 7.9 metres depth of water. This is the amount of thermal mass that is involved in the annual temperature swings. I usually call that 8 metres for quick first-cut calculations. (I’m not sure where I got that number, 7.9 metres thermal mass per month of tau in the earth system, although it was from a trusted source of some kind. Any assistance in backing up that number would be appreciated.)
As a result of my newly-gained ability to calculate the time constants tau for the ocean and the land (on the order of 3+ months and 1 month respectively), we get about 24+ metres and 8 metres of thermal mass for the ocean and land respectively.
So … given that we have that much thermal mass for the land and the ocean, how much energy goes into and out of the thermal mass per year?
Now, I’m a great fan of rules of thumb, which I keep in my head for back-of-the-envelope calculations. One such rule of thumb is that a watt per square metre of incoming energy applied over a year (1 W-yr/m2) will raise a cubic metre of seawater by 8°C.
This means that for the land, with an involved thermal mass of 8 metres depth, 1 W-yr/m2 changes the temperature of the thermal mass by 1°C. And similarly, for the ocean’s involved thermal mass of 24+ metres depth, it takes about 3+ times that or 3+ W/m2 to change the temperature by 1°C.
Here’s the key graph, from my last post:
The land is easy, because the involved thermal mass warms and cools at one degree C per watt-year/m2. It swings 28°C, but remember that is in half a year. So the rate is 56°C per year. And conveniently, this is also 56 watts/m2. So every year, in back-of-the-envelope terms, there is a flux of about half a hundred watts first into and then out of the land.
Next, the ocean. The swing of the ocean temperatures is 7.7°C per half-year, or a rate of about 15°C per year. The involved thermal mass of the ocean requires 3+ watts per square metre per degree. How nice, the thermal flux in W-yr/m2/°C is equal to tau. That works out to 45+ watts/m2. Now, the “+” I carried through the calculations was plus 10%, as the true tau for the ocean is 3.3. So we need to add an additional 10% to the 45+ watts, giving us 50 watts first into and then out of the ocean … versus 56 watts for the land. Half a hundred either way.
Of course, now that I think about it, it makes perfect sense. The smaller involved thermal mass will heat up more, the larger involved thermal mass will heat up less, and they both actually store about the same amount of energy per square metre. Of course, in global terms, the 70:30 ratio of the sizes of ocean and land comes into play … but per square metre, they each have a flux of about fifty watts per square metre, first into and then out of the thermal mass each year. Good to know.
Always more to learn,
My Usual Request: If you disagree, do us all the favor of telling us exactly what you disagree with. There’s only one way to do this, which is to quote the exact words you disagree with. Anything else is your interpretation. We can only be clear what you think is incorrect if you quote the precise words they actually used.
Here’s the actual calculations in comma separated format.
Type, time constant tau, depth heat mass, W-yr/m2/°C, Swing °C, Equiv W/m2
Land, 1, 7.9, 1.0, 28.3, 55.8
Ocean, 3.3, 26.1, 3.3, 7.7, 50.3
Note that this is only valid with tau expressed in months.