The first figure in the article (right) reminded me of something I've been wondering -- how sensitive are atmospheric trends to changes in the 93.4% number for the percentage of heat that goes into the ocean?

I think I can estimate it with some simplications. The answer is, if I'm right: surface and tropospheric temperature trends are

*very*sensitive to changes in ocean heating. Answer below (in red), using some very basic physics that is a bit pedantic.

Let Q

_{in}be the amount of heat coming into the Earth over some period of time. (So the units of Q

_{in}are energy per unit time, whereas the symbol Q is usually used for just energy.) Let's assume in all goes either into the ocean or into the troposphere, so Q

_{in}= Q

_{O}+ Q

_{t}, where Q

_{O}is the amount of the heat that goes into the ocean, and Q

_{t}the amount into the troposphere.

Now consider two scenarios. In the first, the one we're living in, a fraction f1 (= 0.934) of the heat goes into the ocean, and in the second, a fraction f2 goes into the ocean.

So

Q

_{O,1}= f1Q_{in}and Q_{O,2}= f2Q_{in}.By assumption, the total amount of heat going into the Earth is the same, so

Q

_{in}= Q_{O,1}+ Q_{t,1}= Q_{O,2 }+ Q_{t,2}_{}where "t" stands for troposphere. A dash of rearranging and a smidgen of algebra gives

(f1-f2)Q

_{in}= Q_{t,2}- Q_{t,1}What is Q

_{in}? It's Q

_{O,1}/f1. We know what f1 is: 0.934, from the figure. If we assume all the heat coming into the ocean goes into the 0-2000 m region (that's the top half of the ocean -- and not a bad assumption), then Q

_{O,1 }can be estimated from 10-years worth of Argo data for that region. When I do that I get 0.93e22 J/yr.

That finishes the ocean part. For the troposphere

Q

_{1}= m_{t}c_{t}ΔT_{1}and Q_{2}= m_{t}c_{t}ΔT_{2},where ΔT is the amount of temperature change from whatever the baseline is: ΔT

_{1}= T

_{1}- T

_{base}and likewise for scenario 2. (Strictly speaking it's the change of temperature with time, viz. a trend, but as with Q above I'll just call it T.) Then

_{t,2}- Q

_{t,1}= m

_{t}c

_{t}(T

_{2}- T

_{1})

where we want to calculate the temperature difference as a function of f2.

Putting this all together then gives

_{2}-T

_{1}= (1-f

_{2}/f

_{1})Q

_{O,1}/m

_{t}c

_{t}

All this would surely look better if (1) I knew more HTML, and (2) I was using Wordpress, which has an equation-maker plug-in, but I don't and I'm not.

Now, what's the mass of the troposphere? About 80% of the total atmospheric mass, or 4.1e18 kg.

(Interestingly, you can calculate the atmosphere's mass without leaving your chair: m = P

_{s}A/g, where P

_{s}is the surface pressure, A is the Earth's area, and g the acceleration due to gravity at the surface. Of course, g varies with altitude, but that's a small effect.)

Here c

_{t}is the troposphere's specific heat. I'll assume the troposphere is homogeneous throughout, and approximate its specific heat by the specific heat of air at typical room conditions and 40% humidity, which is 1,012 J/kgK.

Now we just calculate. If f2 = 0.935 -- that is, it's just 0.1 percentage points above f1 -- I find the difference in trends is

T

_{2}-T_{1}= -0.024°C/decade
which seems like it could be in the ball park. At least, it doesn't seem ridiculous. And the sign is right: more heat into the ocean means heat taken out of the troposphere, which lowers its temperature.

So a

*very small change*in how much heat goes into the ocean can lead to noticeable changes in the temperature trends. If f2 is 0.5 percentage points above f1, then
T

_{2}-T_{1}= -0.12°C/decade
which could well put a damper on the greenhouse warming of a decade or two. So just a little more heat going into the ocean (relatively little -- it's still a huge number of Joules) can noticeably cool the troposphere. Or even more noticeably for the surface, which holds even less heat then the troposphere (for, say, the top 2 meters of air above the surface).

That's what I suspected, and the rough numbers seem to support that.

That's what I suspected, and the rough numbers seem to support that.

## 2 comments:

OK, can't argue with the algebra or even the very simple model. But what causes "a little more heat to go into the ocean"?

Rob: What causes the US to be colder than the rest of the globe in November?

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