Guest Post by Willis Eschenbach
In my last post I investigated the mathematical relationship between the amount of total precipitable water vapor (TPW) in the atmosphere, and the clear-sky greenhouse effect. Here is the main figure from that post showing the relationship:
Figure 1. Scatterplot, TPW (horizontal scale) versus Atmospheric Absorption (vertical scale). Dashed yellow line shows theoretical value based on TPW. Dashed vertical line shows area-weighted global average value. Dotted vertical lines show the range of the global average value over the period. The slope of the curve at any point is 62.8/TPW (W/m2 per degree)
In this post I’m looking at the other half of the relationship. The other half is the relationship between the ocean surface temperature and the total precipitable water. The good news is that unlike the CERES data which is only about 15 years, we have TPW records since 1988 and sea surface temperature for the period as well. Figure 2 shows the relationship between the two:
As you can see, the relationship is regular but not simple. I first thought that the relationship was logarithmic, but it turns out not to be so. It is also very poorly represented by a power function. After unsuccessfully investigating a variety of curves, I found it could be approximated by an inverse sigmoid function (shown in yellow above). Now, given the number of very smart folks here, I suspect someone will be able to give a physical reason complete with the right equation, but this one suffices for my purposes.
Now, the relationship between water vapor and atmospheric absorption is clearly logarithmic, as is predicted by theory. On the other hand, I don’t know of any simple theory relating SST to total precipitable water. For example, the curve doesn’t match the Clausius-Clapeyron increase in water vapor. And clearly, my method is purely heuristic and brute-force … but that’s OK because I’m not claiming that it is explanatory. My purpose in doing it is quite different—I want to figure out how much change there is in the precipitable water per degree of change in the sea surface temperature (SST). And for that, the main quality is that the function needs to be differentiable.
So let me recap where we stand. In the last post I derived a mathematical relationship between the two variables shown in Figure 1. Those are clear-sky atmospheric absorption of upwelling longwave radiation from the surface, and the total precipitable water content (TPW) of the atmosphere.
And above in this post, I’ve derived a mathematical relationship between the two variables shown in Figure 2. Those are the total precipitable water content (TPW) of the atmosphere, and the sea surface temperature (SST).
That means that by substituting the latter into the former, I can derive a mathematical relationship between the SST and the atmospheric absorption.
Of course I wanted to ground-test my formula that converts from sea surface temperature to atmospheric absorption. I only have the CERES data for the absorption, so this covers a shorter period than that shown in Figure 2. Since the overall relationship was established using the Reynolds sea surface temperature data, I used that for the comparison.
Dang, I’m pretty satisfied with that as a comparison of theoretical and observed atmospheric absorption. A few comments. First, the difference below 0°C is because CERES and Reynolds are measuring slightly different things below freezing, when there is ice in the picture. CERES is measuring the average temperature of the ice and the water, and Reynolds is measuring water temperature alone.
Next, the slight bend in the black line from 0°C to 25°C is not completely captured by the red line. This is because I’ve included the data below freezing, which has slightly distorted the results. Probably should have left it out, but I figured for completeness …
Next, the slight bend in the black line from 0°C to 25°C is due to the fact that surface radiation is proportional to the fourth power of the temperature. If absorption were calculated against surface upwelling radiation rather than temperature, it would plot as a straight line … go figure. I’ve done it this way because there is much discussion about the value of the “water vapor radiative feedback” which is measured per degree C. I could get a slightly closer fit by including the T^4 relationship, but my conclusion was that the gain wasn’t worth the pain … if I need greater accuracy I can redo the figure, but it is more than adequate for the present purposes.
The amount of the feedback is calculated as the slope of the red line in Figure 3. The slope is the change in the absorption for a 1°C change in the sea surface temperature. Figure 4 shows the amplitude of the water vapor radiative feedback across the range of ocean temperatures:
That is a very interesting shape. Now, given the general shapes of Figure 1 and Figure 2, I might have expected the shape … but it came as a surprise anyhow. Over much of the world, the two tendencies cancel each other out and the clear-sky water vapor radiative feedback is about 3-4 W/m2 per degree C. But in the tropics, where the water is warm, the water vapor feedback goes through the roof.
So … with such a large radiative feedback from water vapor, three to four watts per square metre per degree and much higher in the tropics, why is there not runaway feedback? I mean, the so-called “climate sensitivity” claimed by the IPCC says that 2-3 W/m2 of additional radiation will cause one degree of warming. And according to observations above, when it warms one degree, we get additional downwelling radiation from water vapor of 3-4 W/m2. And that amount is claimed to be sufficient to warm it more than one additional degree … a recipe for runaway positive feedback if I ever saw one. So … with that large a radiative feedback, why isn’t there runaway feedback?
Well, you might start by perusing Dr. Roy Spencer’s discussion of the subject, yclept Five Reasons Why Water Vapor Feedback Might Not Be Positive. The TL;DR version is that as the amount of water vapor in the air increases, downwelling radiation does indeed increase … but there are plenty of other things that change as well.
To expand a bit on one of the things Dr. Roy mentioned, in his discussion of evaporation versus precipitation he said:
While we know that evaporation increases with temperature, we don’t know very much about how the efficiency of precipitation systems changes with temperature.
The latter process is much more complex than surface evaporation (see Renno et al., 1994), and it is not at all clear that climate models behave realistically in this regard.
Let me add a bit to that. Rainfall goes up with increasing atmospheric water as shown in Figure 5:
Note the size of the cooling involved … not watts per square metre, but hundreds of watts per square metre. As precipitable water goes from about forty to fifty-five kg per square metre, evaporative cooling goes from fifty to two hundred fifty watts per square metre or more … that’s a serious amount of cooling, about ten watts of additional cooling per additional kg of precipitable water.
We can compare that to the slope of increasing water vapor radiative feedback in Figure 1. The slope in Figure 1 is 62.8 W/m2 divided by TPW, so at a TPW of 50 kg/m2 that would be about 1.2 W/m2 of additional radiative warming per additional kg/m2 of water … versus 10 W/m2 of rainfall evaporative cooling per additional kg/m2 of water.
But wait … there’s more. Figure 6 shows the rainfall evaporative cooling versus sea surface temperature (SST). Since SST and precipitable water are closely related, Figure 6 is quite similar to Figure 5.
As in Figure 5, at the hot (right hand) end of the scale, the rainfall evaporative cooling goes from about 50 to about 200 W/m2 very quickly. However, in this case it makes that change as the SST goes from about 27° to 30°. And that gives us a net cooling of about 50 W/m2 per degree … kinda dwarfs the 3-4 W/m2 per degree of water vapor based warming …
There is another interesting aspect of Figure 6 … the empty area at the lower right. I have long stated that the thermoregulatory phenomena like thunderstorms are based on temperature thresholds. The blank area in the lower right corner of Figure 6 shows that above a certain sea surface temperature … it’s gonna rain and cool it down. And not only will it rain, but the hotter it gets, the greater the rainfall evaporative cooling overall, and the greater the minimum evaporative cooling as well.
Nor do the cooling effects of water vapor end there. Increasing water vapor also increases the amount of solar energy absorbed as it comes through the atmosphere. As with the absorption of the upwelling longwave, the relationship is logarithmic. Figure 7 shows that relationship.
Logarithmic relationships of the form “m log(x) + b” have a simple slope, which is m / x. The slope of the equation shown in Figure 7 is 31.6/TPW (W/m2 per degree). Now, earlier we saw that the slope of the warming from increasing water was 62.8/TPW (W/m2 per degree). This means that at any point, half of the warming due to water vapor radiative feed back is cancelled out by the loss in downwelling sunlight due to increased water vapor.
Nor is this the end of the related phenomena … Figure 8 shows the correlation between total precipitable water and cloud albedo:
As you can see, over much of the tropics, as precipitable water increases so does the cloud albedo (red-yellow). Makes sense, more water in the air means more clouds. Again, this has a cooling effect.
Nor is this an exhaustive list, I haven’t discussed changes in downwelling longwave radiation due to clouds … the relationships go on.
The center of climate action is the tropics. Half of the available sunlight strikes the earth between 23° north and south. The main phenomena regulating the amount of incoming solar energy occur in the tropics. And as the graphs above show, the amount of water in the atmosphere is at the heart of those phenomena.
So … is the feedback of water vapor positive or negative? Overall, I’d have to say it is well negative, for two reasons. The first is the long-term stability of the global climate system (e.g. global surface temperature only changed ± 0.3° over the entire 20th century). This implies negative rather than positive feedback.
The second reason I’d say it’s negative is the relative sizes of the various feedbacks above. These are dominated by the evaporative cooling due to rainfall and by the changes in reflected sunlight due to albedo, both of which are much larger than the 3-4 W/m2 in increased water vapor radiative warming.
However, there is a very large difficulty in isolating the so-called “water vapor feedback” from the myriad of other phenomena. This difficulty is embodied in what I refer to as my “First Rule Of Climate”, which states:
In the climate system, everything is connected to everything else … which is turn is connected to everything else … except when it isn’t.
For example, the temperature affects the water vapor – when the temperature goes up, the water vapor goes up. When the water vapor goes up, clouds and rain go up. When clouds and rain go up, temperatures go down. When temperatures go down, water vapor goes down … you can see the problem. Rather than having things which are clearly cause and clearly effect, the whole system is what I describe as a “circular chain of effects”, wherein there is no clear cause and no clear boundaries.
Anyhow, those are the insights that I got from examining the total precipitable water dataset … like I said, no telling where a new dataset will take me.
And speaking of precipitable water, it is sunset here on our hillside. As I look out the kitchen window towards the ocean I see the fog washing in from the Pacific. It is pouring in waves over the far hills, swallowing redwood trees as it rolls on toward our house … it came and visited last night as well.
I love that sea fog. It reeks of my beloved ocean, with the smell of fishing boats and slumbering clams, of hidden coves and youthful dreams. And when the fog comes in, it brings with it the sound of the foghorn at the mouth of Bodega Bay. It’s about seven miles (ten kilometres) from my house to the bay, but the sound seems to get trapped in the fog layer, and when the fog comes I hear that foghorn calling to me in the far distance, a mournful midnight wail. I took frequent breaks from my scientific research and writing last night to sit outside on a bench, where I let the fog wreathe around my head and bear me away. I breathe in the precipitated water, and I emerged refreshed …
My best to everyone, and for each of you, I wish for whatever fog it is that carries you away in reverie and washes off the mask of socialization …
My Usual Request: Misunderstandings suck, but we can avoid them by being specific about our disagreements. If you disagree with me or anyone, please quote the exact words you disagree with, so we can all understand the exact nature of your objections. I can defend my own words. I cannot defend someone else’s interpretation of some unidentified words of mine.
My Other Request: If you believe that e.g. I’m using the wrong method or the wrong dataset, please educate me and others by demonstrating the proper use of the right method or identifying the right dataset. Simply claiming I’m wrong about methods or data doesn’t advance the discussion unless you can point us to the right way to do it.
The math … I start with the equation for relationship between absorption (A) and total precipitable water (TPW) shown in Figure 1:
A = 62.8 Log(TPW) – 60
To this I add the inverse sinusoidal relationship between TPW and sea surface temperature, as shown in Figure 2:
TPW = – 13.5 Log[-1 + 1/(0.00368 SST + .887)] -19.1
Combining the two gives us:
A = 62.8 Log[-19.1 – 13.5 Log[-1 + 1/(0.887 + 0.00368 SST)]] – 60
Differentiating with respect to sea surface temperature gives the result as shown in Figure 3:
dA/dT = 3.13/((-1 + 1/(0.887 + 0.00368 SST)) (0.887 + 0.00368 SST)^2 (-19.1 – 13.5 Log[-1 + 1/(0.887 + 0.00368 SST)]))
Further Reading: NASA says water vapor feedback is only 1.1 W/m2 per degree C …
Reynolds SST data, NetCDF file at the bottom of the page
TRMM data, NetCDF file at the bottom of the page