Archive for September, 2016

Matthew to Arrive 4,000 days after Last Major Hurricane

Thursday, September 29th, 2016

Updated 7:30 a.m. EDT Saturday, Oct. 1.

Major Hurricane Matthew was briefly a Category 5 hurricane overnight, the first Cat 5 in the Atlantic in nine years. It now has 155 mph sustained winds, making it a strong Category 4 storm on the Saffir-Simpson scale.

Matthew is over the south-central Caribbean, traveling slowly westward, but a turn to the north is expected on Sunday. Matthew is expected to cross eastern Cuba Tuesday morning and possibly make U.S. landfall somewhere on the East Coast around next Friday or Saturday.

Thursday will mark exactly 4,000 days after Major Hurricane Wilma’s landfall.

Hurricane Wilma, the last major hurricane (Cat 3 or stronger) to hit the U.S., struck Florida on October 24, 2005. Will Matthew arrive as the first major hurricane to strike the U.S. in almost 11 years? Only time will tell. (Sandy was Cat 1 at landfall, and technically not a hurricane at that time. Hurricane Ike, 2008, was a Cat 2.)

Here is the latest GFS model forecast for Matthew on midnight Sunday, Oct. 9 (graphics courtesy of


That particular forecast, which remains very uncertain this far in advance, has Matthew making landfall at Cape Hatteras, Cape Cod, and then going inland in Maine. Here is the spread of model forecasts from NOAA’s GEFS ensemble forecast system:


The Faster a Planet Rotates, the Warmer its Average Temperature

Wednesday, September 28th, 2016

This is a followup to my post from yesterday where I provided time-dependent model results of the day-night cycle in lunar temperatures.

One of the fascinating things about the model result (which I would not have expected) is that all other things being equal, the faster a solar-illuminated planet rotates, the warmer its average temperature will be. The calculations I provided are for planets without an atmosphere (e.g. the Moon).

Before examining the issue, I would have guessed that the rotation rate would not matter. Or, maybe I would have guessed that a more-slowly rotating planet would get warmer, since the period of sunlight is longer and higher daytime temperatures would be achieved.

But I would have been wrong.

Simple Thought Experiment

The reason is very simple, and is related to the non-linearity of the Stefan-Boltzmann equation, which can be used to estimate how warm a body gets based upon the rate at which it absorbs solar energy when its only mechanism to cool is through thermal emission of radiation:

Fig. 1. The non-linearity of the Stefan-Boltzmann equation can lead to very different average planetary temperatures given the same long-term average absorbed solar flux.

Fig. 1. The non-linearity of the Stefan-Boltzmann equation can lead to very different average planetary temperatures given the same long-term average absorbed solar flux.

Imagine a body with a realistic heat capacity that uniformly absorbs a solar intensity of 1,000 Watts per sq. meter for 1 second, then 0 W/m2 for one second, over and over. Think of it as a 2 sec long diurnal cycle. That rapidly flickering energy source would be too fast for the temperature to come into equilibrium with the absorbed sunlight (or lack of sunlight). It would, in effect, be like a continuous energy source of 500 W/m2 in intensity, and the resulting S-B temperature (assuming a thermal radiative emissivity of 1.0) would be about 307 Kelvin, taken from the curve in Fig. 1.

Now imagine the energy source stays on for a very long time, say 10,000 days, then stays off for 10,000 days (a 20,000 day diurnal cycle…the Moon has a 29.5 day diurnal cycle). From Fig. 1 we see that during the daytime the temperature would approach 365 Kelvin, and at night it would approach 0 Kelvin. In this case the average temperature would be about 182 Kelvin…which is 125 deg. colder than 307 K!

The only difference in the two imaginary cases is the length of the day/night cycle. The long-term average rate of absorbed sunlight is the same.

Yesterday I showed that the difference in rotation rate between the Earth and the Moon caused the more-slowly rotating Moon to be about 55 deg. colder than the Earth, all other things being equal (no atmosphere, the same albedo and IR emissivity, and a surface bulk heat capacity which gives model temperatures than match actual lunar observations). The effect is muted the greater the surface bulk heat capacity, since that also reduces the diurnal temperature range.

Basically, any process which increases the day-night temperature range (such as a longer diurnal cycle) will decrease the average temperature of a planet, simply because of the non-linearity of the S-B equation. I suspect the effect does not exist if the surface being heated has zero heat capacity, since the temperature of the surface will instantly come into equilibrium with the absorbed sunlight; in that case the length of day would not matter. But since that is physically impossible, it does not apply to real planets.

Errors in Estimating Earth’s No-Atmosphere Average Temperature

Tuesday, September 27th, 2016

While the non-linearity of the Stefan-Boltzmann equation leads to at least a 60 deg. C overestimate of the Moon’s average surface temperature if a global-average solar flux is used in place of computing temperatures over a sphere with a diurnal cycle, the error is only about 5 deg. C for the Earth. The difference is due the the very long lunar day (29.5 Earth days), which causes a very large diurnal cycle in temperature, which enhances the errors arising from the nonlinearity of the S-B equation.

PrintThe greenhouse effect is often claimed to cause an average warming of the Earth’s surface of about 33 deg. C, from an atmosphere-free value of about 255 K to the observed value of around 288 K. In the no-atmosphere case, the absorbed solar flux heats the surface up until the thermal emission of longwave radiation matches the intensity of absorbed sunlight.

Typically this theoretical average surface temperature is computed using a global average of the absorbed solar flux, and then using the Stefan-Boltzmann equation to find the emitting temperature that matches it.

But the strong nonlinearity of how the S-B flux depends upon temperature can lead to a warm bias in the no-atmosphere temperature estimate if a wide range of solar fluxes are used in a single average:

Fig. 1. The non-linearity of the Stefan-Boltzmann equation leads to a warm bias if a global average solar flux is used to estimate a global average equivalent emitting temperature.

Fig. 1. The non-linearity of the Stefan-Boltzmann equation leads to a warm bias if a global average solar flux is used to estimate a global average equivalent emitting temperature.

If the absorbed solar flux does not vary much over the spherical shape of a planet without an atmosphere, then using a global-average solar flux will give a pretty good estimate of the global average surface temperature.

But the absorbed solar flux actually varies a lot over a spherical planet.

So, just how large of an error is introduced by the use of a global average flux to calculate an average temperature? (My recent discussions with David South, an Auburn forestry professor, led me to reexamine this issue.)

In the case of the Moon, the error is very large. As has been pointed out elsewhere (e.g. by Willis Eschenbach here, and Nikolov & Zeller here), extreme day-night temperature swings on the Moon can cause a single-solar flux estimate of surface temperature to be biased very high, due to the nonlinearity of the S-B equation. The error can be many tens of degrees C.

Clearly, the 33 deg. C estimate for the Earth’s atmospheric greenhouse effect depends upon how accurate our estimate is of the average surface temperature of the Earth without an atmosphere. (I won’t go into the reasons why we really don’t know what the Earth would look like without an atmosphere, which affects it’s albedo and thus how much sunlight it would absorb, which in turn will impact the temperature calculation).

Since the non-linearity induced error depends upon just how hot surface temperature gets during the daytime, you need to do the calculations using a diurnal cycle, including how deep the solar heating (and nighttime cooling) penetrates into the surface. Also, obviously, the calculations need to be done on a sphere.

So, I put together this model spreadsheet that allows you to change planets (through the assumed solar irradiance), the assumed solar albedo of the atmosphere-free planet, surface longwave emissivity, and how deep of a water/soil layer is assumed to change in its temperature in response to imbalances between absorbed sunlight and thermally-emitted longwave radiation.

The time-dependent calculations are done in 96 increments of a day, which is 15 minutes for the Earth, at latitudes of 5, 15, 25, 35, 45, 55, 65, 75, and 85 degrees separately at assumed equinox conditions. Cosine latitude weighting then gives a pretty good estimate of the area averaged temperature over the sphere. It can take up to a couple weeks for the polar regions to finally equilibrate when the model is initialized at absolute zero temperature. The plots that follow are after 40 day-night cycles of the model run.

When I run the model for the Moon, which has a 29.5 Earth-day diurnal cycle, I found that I needed a soil layer of about 0.05 meters depth (about 2 inches) to match actual temperature measurements on the Moon (see Willis’s post here for some actual lunar temperature measurements). This is the thickness of soil assumed to be uniform in temperature that responds to solar heating and IR cooling. Of course, in reality the very top of the soil surface will get the hottest/coldest, with the temperature swings dampening strongly with depth; the model just uses a thin, uniform-temperature layer that approximates the average behavior of the real, thicker layer.

Fig. 2. Diurnal cycle in lunar surface temperatures at different latitudes calculated from a simple time-dependent model during equinox conditions.

Fig. 2. Diurnal cycle in lunar surface temperatures at different latitudes calculated from a simple time-dependent model during equinox conditions.

Significantly, the resulting global area average lunar temperature of 212 K is 61 K colder than the 273 K one gets by just putting the global average absorbed solar flux through the S-B equation to get a single temperature. As discussed by Willis, this shows the large bias that can result from S-B equation calculations when one doesn’t bother to average over a wide range of temperatures.

So, How Large is the S-B Bias in Earth Temperature Calculations?

Just how big is this warm bias effect when computing what the Earth’s global average surface temperature would be in the absence of an atmosphere?

If I repeat the model calculations in Fig. 2 and only change the length of the diurnal cycle, from 29.5 Earth days (for the Moon) to 1 day, we get (obviously) a greatly reduced diurnal range in temperature (22 deg. C diurnal range, global average, versus 209 deg. C diurnal range for the Moon), and a global average surface temperature of 267 K. This is only 6 deg. below the 273 K value using a single solar flux in the S-B equation:

Fig. 3. As in Fig. 2, but with a 24 hr (Earth) diurnal cycle rather than 29.5 days (lunar diurnal cycle).

Fig. 3. As in Fig. 2, but with a 24 hr (Earth) diurnal cycle rather than 29.5 days (lunar diurnal cycle).

If I use the more traditionally-used Earth albedo value of 0.3, I get a global average surface temperature of 251 K, which is only 5 deg. C below the single solar flux calculation of 256 K. Thus, the error caused by using a single global average solar flux to estimate a global average terrestrial temperature in the S-B equation is much less for the Earth than it is for the Moon.

Fig. 4. As in Fig. 3, but using a solar albedo of 0.3 rather than 0.1.

Fig. 4. As in Fig. 3, but using a solar albedo of 0.3 rather than 0.1.


Using the S-B equation with a global average absorbed solar flux to compute the global average emitting temperature of the Moon leads to a very large warm bias, as reported by others.

But that lunar bias (about 60 deg. C) is mostly due to the very long period of daylight on the moon, which is 29.5 times longer than on Earth. When the Earth’s diurnal cycle length is used, the warm bias is only about 5 deg. C.

One might then wonder if this means that the 33 deg. C greenhouse effect on Earth should really be 38 deg. C?

Maybe…but I would say that the 33 deg. C number is suspect anyway. First, because it depends upon an albedo of 0.3, which is probably too high. If I use a lunar albedo for the Earth, then the GHE becomes only 21 deg. C with the new calculations. One might wonder if the no-atmosphere Earth would be ice covered, with a very high albedo and very low surface temperatures, but the existence of water would lead to evaporation/sublimation, and a water vapor atmosphere. So an ice Earth is, I believe, incompatible with the assumption of no atmosphere. But I’m open to different arguments on this point.

Secondly, the 33 deg. C number isnt really the greenhouse effect, anyway. It’s more of a total “atmosphere effect”, the final result after atmospheric convection has cooled the surface substantially below the very high temperatures the greenhouse effect would cause in the case of pure radiative equilibrium (see Manabe and Strickler, 1964).

So, you can get a wide variety of numbers for the estimated surface warming effect of the atmosphere (combined effect of greenhouse warming and convective cooling). They depend on what assumptions you make in your calculations related to what an atmosphere-free Earth would look like, which are at the very least uncertain, and at most, physically impossible.

The bottom line, though, is that neglect of the nonlinearity of the S-B equation leads to about a 5 deg. C overestimate of the average surface temperature of the Earth in the absence of an atmosphere.

NOTE: Most of the comments on this post will likely be off-topic, centering around the extreme minority view of a few people that there is no atmospheric “greenhouse effect” involving the atmosphere emitting infrared radiation toward the surface.

UAH Global Temperature Update for August, 2016: +0.44 deg. C

Thursday, September 1st, 2016

August Temperature Up a Little from July

NOTE: This is the seventeenth monthly update with our new Version 6.0 dataset. Differences versus the old Version 5.6 dataset are discussed here. Note we are now at “beta5” for Version 6, and the paper describing the methodology is back to the journal editors from peer review.

The Version 6.0 global average lower tropospheric temperature (LT) anomaly for August 2016 is +0.44 deg. C, up a little from the July, 2016 value +0.39 deg. C (click for full size version):


The global, hemispheric, and tropical LT anomalies from the 30-year (1981-2010) average for the last 20 months are:

2015 01 +0.30 +0.44 +0.15 +0.13
2015 02 +0.19 +0.34 +0.04 -0.07
2015 03 +0.18 +0.28 +0.07 +0.04
2015 04 +0.09 +0.19 -0.01 +0.08
2015 05 +0.27 +0.34 +0.20 +0.27
2015 06 +0.31 +0.38 +0.25 +0.46
2015 07 +0.16 +0.29 +0.03 +0.48
2015 08 +0.25 +0.20 +0.30 +0.53
2015 09 +0.23 +0.30 +0.16 +0.55
2015 10 +0.41 +0.63 +0.20 +0.53
2015 11 +0.33 +0.44 +0.22 +0.52
2015 12 +0.45 +0.53 +0.37 +0.61
2016 01 +0.54 +0.69 +0.39 +0.84
2016 02 +0.83 +1.17 +0.50 +0.99
2016 03 +0.73 +0.94 +0.52 +1.09
2016 04 +0.71 +0.85 +0.58 +0.94
2016 05 +0.55 +0.65 +0.44 +0.72
2016 06 +0.34 +0.51 +0.17 +0.38
2016 07 +0.39 +0.48 +0.30 +0.48
2016 08 +0.44 +0.55 +0.32 +0.50

The July-August pause in cooling as La Nina approaches is unusual compared to the few other dissipating El Nino events in the satellite period of record; recent weeks’ ENSO predictions from CPC have suggested the coming La Nina won’t be as stong as previously forecast. Also, warmth elsewhere is offsetting cooling in the tropical Pacific, keeping global average temperatures up; the CFSv2 model average surface temperature for August as compiled at was +0.42 deg. C.

To see how we are now progressing toward a record warm year in the satellite data, the following chart shows the average rate of cooling for the rest of 2016 that would be required to tie 1998 as warmest year in the 38-year satellite record:

Based upon this chart, as we enter the home stretch, it now looks like a horse race to see whether 2016 will or won’t exceed 1998 as a new record-warm year (since the satellite record began in 1979).

The “official” UAH global image for August, 2016 should be available in the next several days here.

The new Version 6 files (use the ones labeled “beta5”) should be updated soon, and are located here:

Lower Troposphere:
Lower Stratosphere: