Earth's Energy Balance as Seen in Ocean Heat Uptake

 ... or, "is it plausible that the 'missing heat' has gone deeper than 2 km?"

Background

What follows is a post from WUWT in response to Chic Bowdrie about a number of different things which I reproduce here in full mainly because of the number-crunching I do to justify at least a portion of various energy imbalance estimates seen in the likes of papers such as:


Trenberth, Fasullo and Kiehl (2009), Earth's Global Energy Budget
Stephens et al. (2012), An update on Earth’s energy balance in light of the latest global observations



Chic's comments in indented paragraphs, everything else is mine.

Begin Post
----------------

Brandon Gates

February 13, 2016 at 4:14 am

Chic Bowdrie,

I get the notices by email and will keep my eye out. I think our discussions here and on the Briffa post are converging on the reliability of the surface and satellite measurements. Perhaps we should continue here exclusively?

I don’t think that post is coming out of limbo any time soon, but I saved some of it which I use below.

I was interested in your response to the following exchange earlier:
The switch from ERSST v3b to v4 was … (drumroll) … net cooling:
From 1850 something that may be the case. However, Christy showed that since 1995, that change biased NOAA trends upwards relative to Hadcrut4 and UAH sea surface measurements.
The changes to ERSST, whether justified or not, do constitute a warming trend since 1995.

And my counterpoint remains: it’s the trend over the entire interval which figure more into climate sensitivity estimates. So if the IPCC are trying to pump up CO2 potency for political leverage, and NOAA are complicit in helping them do it, the last thing they’d be expected to do is reduce the overall trend.

Now concerning the Monckton issue,
I did, and attempted to respond to his assertion masquerading as a question: “Why is the ocean warming not from above but below?”
But it wasn’t an assertion.

“The ocean is warming not from above but below” is implicit. It’s an assertion which has been loaded into the question.

It was a question that deserves an answer.

It’s an assertion that wants substantiation before it can be answered. I’ll demonstrate:
1) Why does CO2 warm the planet?
2) Why will AGW be catastrophic?
3) Why are you still beating your wife?
It’s one of the oldest rhetorical tricks in the book. People who habitually do it are not to be trusted in my opinion.

I would answer that there is insufficient evidence to know where the warming comes from.

Welllll … maybe you should give an earful to Monckton in addition to me.

The errors in measurement are too large to say with confidence that warming comes from increasing CO2, or excess solar insolation, or from ocean warming originating from who knows how long ago. Solar insolation is greatly affected by clouds and I’m not aware of any definitive data indicating how much SW radiation is actually absorbed by the surface.

My standard response: all I can do is present evidence I find compelling. If it doesn’t meet your standards of “definitive” or “confidence” there isn’t much else I can do other than appeal to theory.

Had the intent been an assertion, I surmise that it would have been something like “global warming can’t be caused by fossil fuel emissions, because the troposphere hasn’t warmed as it would have had to according to AGW theory. His substantiation was that you would have to deny satellite data to claim that the troposphere is warming the surface.

The troposphere’s absolute temperature is cooler than the surface. Surface warms the troposphere, not the other way around. 2nd law of thermodynamics.

Vertical convection in the ocean is not favored, because the temperatures are lower and density greater with depth.

Bingo.

Turbulence at the surface and thermohaline circulation are other phenomena.

Yes.

If the circulation is causing the warming, what does that have to do with fossil fuel emissions?

I don’t assert circulation is causing the warming. Here’s the basic argument: Sun warms first few tens of meters. GHGs modulate the rate at which oceans radiatively dissipate energy. Turbulent mixing carries some of that heat to further depths, thermohaline circulation also moves some of it, the balance is diffusive.
CO2 is a GHG, and burning fossil fuels is what’s causing the rise.

If that were obvious there wouldn’t be such a debate about it, as you and I are having.

Yes that’s true. I was arguing that point as if you were Monckton, whom I don’t consider to be acting in good faith. It’s bloody well obvious to me; however, I should not impose that on you just because of my animus toward him.

So what mechanisms and magnitudes do you attribute to the sources that are heating the surface(s) of the Earth?

I had a bunch of calcs and scribblings for a previous post that ended up on the cutting room floor, let me see … ah. Some of this is has been covered already:

sea water (290 K) 4.006 kJ/kg K 
air (300 K)       1.005 kJ/kg K

mass of oceans     1.35E21 kg
mass of atmosphere 5.15E18 kg

energy to change ocean temps by 1 K = 1 K * 4.006 kJ/kg K * 1.35E21 kg = 5.41E+21 kJ
energy to change atmos temps by 1 K = 1 K * 1.005 kJ/kg K * 5.15E18 kg = 5.18E+18 kJ

From Bintanja (2008) I estimate that the ratio of temperature change at the surface to deep ocean is ~5:1 so a 1 K change to surface would be expected to change ocean temps by 0.2 K, thus:

5.41E+21 kJ/K * 0.2 K = 1.08E+21 kJ to oceans for every 1 K change in surface temps. We can check this assumption against instrumental observation since 1955:



Regressing that against HADCRUT4 surface temps over the same interval, I get a slope of 7.60, R^2 = 0.76, so the deep ocean model from Bintanja (2008) is in the ballpark with modern instrumental observation.

The oceans are deeper than 2,000 m; average depth is 3,682.2 m according to a 2010 estimate: http://www.livescience.com/6470-ocean-depth-volume-revealed.html (satellites again), so the 2,000 m layer works out to about 54% of the ocean’s total volume, and therefore mass. We can check this another way by doing some math against ocean heat content estimates for the 2000 m layer:



Mean ΔOHC works out to 5.47E21 J/yr, and ΔT works out to 0.0019 K/yr on average, so:

4.006 kJ/kg K * 1,000 J/kJ * 0.0019 K/yr / 5.47E21 J/yr = 1.39E-21 1/kg = 7.19E+20 kg
7.19E+20 kg / 1.35E21 kg = 0.53

That checks out within 1% of the expected result, which is rather better than I thought I’d get. From the same uppper 2,000 m OHC data, more calcs:

5.47E21 J/yr / 3.16E07 s/yr = 1.73E14 J/s (W)
1.73E14 W / 5.10E14 m^2 = 0.34 W/m^2

Using HADCRUT4 over 1957-2015 as a proxy for net atmospheric temperature change:

0.0130 K/yr * 5.18E+18 kJ/K * 1,000 J/kJ = 6.72E19 J/yr
6.72E19 J/yr / 3.16E07 s/yr = 2.13E12 J/s (W)
2.13E12 W / 5.10E14 m^2 = 0.004 W/m^2

Given that the uncertainty of Stephens (2013) is +/- 0.4 W/m^2, 0.004 W/m^2 is looking like nothing more than rounding error no matter whether surface, radiosonde, or satellites are used to estimate the temperature change. However, noting that 0.004 / 0.6 * 100 = 0.7%, it’s in line with the 1% of net energy absorbed by the system the IPCC say we should expect.

My 0.34 W/m^2 calculation from 2,000 m OHC estimates is 0.26 W/m^2 lower than the 0.6 central estimate, and as such falls within the +/- 0.4 W/m^2 uncertainty interval. However it still wants an accounting. What “should” OHC’s contribution be? According to the IPCC (going by memory here), the estimated relative fluxes are as follows:

0.93 oceans
0.05 land
0.01 latent heat (surface evaporation/ice melt)
0.01 atmosphere
----
1.00 total

0.6 W/m^2 * 0.93 = 0.56 W/m^2 - 0.34 W/m^2 = 0.22 W/m^2

Given that the upper 2,000 m accounts for just over half of the oceans’ mass, it seems plausible to me that my missing heat is diffusing/circulating to lower layers. OTOH, the imbalance could be on the lower end of the Stephens (2013) central estimate. Some of both.

Mechanisms.

First the Sun. Here’s Lean’s monthly TSI reconstruction from 1882 through October 2015:



The average annual change over the entire interval is 9.06E-03 W/m^2/yr. However, that does not account for the fact that only half the planet is illuminated at any given time, nor that the angle of incidence is < 90 on the sunlit side everywhere except where the sun is directly overhead, so we need to divide by 4, giving 2.27E-03 W/m^2/yr. We've then got to knock off 30% to account for albedo, which gives 1.59E-03 W/m^2/yr. Or, to be fair, the ocean's mean albedo is 0.06, not 0.30, so call the maximum plausible upper bound 2.13E-03 W/m^2/yr.

0.34 W/m^2 / 59 yr = 5.76E-03 W/m^2 yr (upper 2km oceans)
2.13E-03 W/m^2 yr / 5.76E-03 W/m^2 yr = 2.7

… so AT BEST the increase in TSI since the late 19th century could only account for between a quarter and a third of the observed OHC increase in the upper 2 km layer over the past ~60 years.
CO2:

ΔCO2 = 1.5 ppmv/yr
Δln(C/C0) = 4.21E-03 1/yr
4.21E-03 1/yr * 5.35 W/m^2 = 2.25E-02 W/m^2/yr

The net change over 1958-2015 is:

ln(400 ppmv/315 ppmv) * 5.35 W/m^2 = 1.28 W/m^2

And for parity with the solar calcs since 1882:

ln(400 ppmv/290 ppmv) * 5.35 W/m^2 = 1.71 W/m^2 :CO2
9.06E-03 W/m^2/yr * 135/yr = 1.21 W/m^2 :TSI
1.21 W/m^2 * (1-0.06) / 4 = 0.29 W/m^2 :TSI accounting for albedo and geometry

Ratio comparison to solar forcing:

1.28 W/m^2 / 0.29 W/m^2 = 4.4 (CO2:1958-2015 / Solar:1882-2015)
1.71 W/m^2 / 0.29 W/m^2  = 5.9 (CO2:1882-2015 / Solar:1882-2015)

… which gives a lot of wiggle room for CO2 vis a vis the energy imbalance estimate uncertainty and magnitude IF the 5.35 W/m^2 forcing coefficient is correct. I haven’t accounted for negative anthropogenic forcings (mainly aerosols) which have the effect of reducing solar influence. There’s also methane, albedo change (ice/snow loss, black carbon), water vapor feedback, etc., to consider. This is wayyy to long already and I’m too tired to get to it in this post anyway.

We could attempt to juice solar by going all the way back to the Maunder Minimum:



… problem is, the shape of the OHC curve is all wrong.

TSI peaked ~1960 and has been in a slight decline. If the runup from 1700-1960 caused most of the observed imbalance, we might still expect to see OHC rising, but at a decelerating rate. The data above show the exact opposite. The shape of the OHC curve best matches the shape of ln(CO2). I find it very difficult to call that sheer coincidence.

I already referred to evidence from Stephens et al. (2013) reporting that incoming SW radiation (gross minus reflected) is 240 +/- 2 W/m2. They also report outgoing as 240 +/- 3.3 W/m2. Trenberth et al. (2009) report incoming and outgoing from various sources ranging from 225 to 245 W/m2 for incoming SW and 236 to 254 W/m2 for outgoing LW. By your own calculations, the increase in ocean heat content is approximately 0.34 W/m2 which constitutes to over 90% of the increase in global heat content. Therefore we can account for less than 0.5 W/m2 of the energy imbalance whatever it may actually be. Since the imbalance from solar is possibly several times greater than that, I conclude that all of the increase in heat content could be due to the amount of solar SW insolation absorbed by the planet.

Emphasis added. A lot of things are possible. I know of no TSI reconstructions that match OHC from 1957 and surface temp change from 1850. It doesn’t work. If it’s not CO2 and not the Sun, it’s something else we haven’t found, and that seems extremely unlikely to me. Logically I cannot rule it out, OTOH, I don’t think it irrational to believe what the consilience of multiple lines of evidence and radiative physics suggest: it’s CO2.

To change my mind requires that someone identifies the … Force X … applies a plausible physical model for it, and demonstrates it with empirical evidence such that it is a better “predictor” of the past than the CO2 model. I wasn’t kidding about supporting a Red Team to do such a thing. I found it curious that I met with such resistance in this forum when I expressed that sentiment.
I cannot write another word. I hope some of the above makes sense.

---------------- 

End Post

Could stand some for some cleanup.  Mostly I just wanted to preserve these calcs here as a ready reference when I wish to refer to them.