## A Tmin bias?

I ended up in a discussion with Roger Pielke Sr about a claim that there is a warm bias in Tmin. It ended rarely sourly when I pointed out that accusing an entire scientific discipline of being dysfunctional because they appear to disagree with you, is not very constructive. Anyway, Roger’s basic point appears to be that Tmin warms faster than Tmax. It seems that this is what Sou is discussing in this post. The planetary boundary layer is the lowest region of the atmosphere, and is a region in which vertical mixing is strong. At night it is much thinner than during the day and so the extra energy is compressed into a thinner layer and hence Tmin increases faster than Tmax; at least I think that is about right.

Roger seemed to be claiming that this produces a warm bias in Tmin. Well, this seems a bit odd if it is real; if Tmin really does increases faster than Tmax, then it’s not really a bias, it’s actually happening. Roger’s next claim was that Tmin should then not be used when assessing global warming. What he’s referring to, I think, is what he and I discussed in a joint post about assessing anthropogenic global warming. You can write down a basic 1D climate model in the following way

$N(t) = N(0) + \Delta F(t) - \lambda \Delta T(t),$

where $N$, is the planetary energy imbalance, $\Delta F$ is the change in forcing, $\lambda$ is the feedback factor, $\Delta T$ is the change in global mean temperature, and $t$ is time. If, for example, you have a forcing time series, a temperature time series, and some estimate for $\lambda$ you can evolve the above equation in time to see how, for example, the ocean heat content should change. That way you can assess anthropogenic global warming.

Roger’s point seems to be that the global mean temperature is determined by averaging Tmin and Tmax, is therefore warm biased, and shouldn’t be used to assess global warming. Well, if Tmin really is warming faster than Tmax, then surely that doesn’t mean that the mean temperature is warm biased; it has to be some combination of minimum and maximum temperatures.

However, it is kind of true that, in the above equation, we want a mean temperature that allows us to produce a reasonable representation of the energy fluxes. However, it’s firstly not clear that the manner in which the mean global temperature is determined is not suitable. Secondly, everything in the above equation is globally averaged, and so as long as the feedback factors (such as the Planck response) are determined in a way that is consistent with the way in which the global mean temperature is computed, I can’t really see the issue.

Additionally, as Victor pointed out, if a warm bias in Tmin produces a warm bias in the mean temperature, then that would imply that the climate sensitivity estimates from energy balance models would be slightly too high. Given that these are already lower than many other estimates suggests, would seem to make this rather unlikely. So, I can’t really see what Roger is getting at. I could, of course, just be confused.

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### 41 Responses to A Tmin bias?

1. Michael Hauber says:

On the face of it Tmin increasing faster than Tmax seems to be good news. But then I’ve heard that heatwaves are at their most deadly when temps are high over multiple days and don’t drop much at night for the human body to have a chance to recover, so maybe its not.

2. kap55 says:

Tmin increases faster because the greenhouse effect is *relatively* stronger at night than during the day. During the day, greehouse is swamped by incoming solar. During the night, ALL the incoming is greenhouse. So when greenhouse is enhanced, it’s felt at night most.

This is, by the way, one of the ways we know that the warming is greenhouse and not solar.

And it also explains why the diurnal temperature range on Venus is flat zero.

3. Michael,
Yes, that was mentioned during the Twitter discussion

4. Chris Colose says:

Much of Roger’s erroneous reasoning through the years comes from his obsession that there is one right way to characterize the climate system, e.g., ocean heat content.

Any measurement, including Tmin, Tmax, water vapor concentration, etc, emerges from many complex interactions. People are affected by Tmin, among other things. Tmin is real.

GMST anomaly is defined by GMST anomaly, not an energy balance equation. So if you have an equation that you think links the two, and it doesn’t work, then the equation is wrong (even if it’s useful) and may be modified by higher-order terms, better consideration of the spatial structure of temperature or radiative forcing, or whatever. It does not mean your measurement is a poor reflection of what you are measuring.

Really, as the self-declared arbiter on how science is done, he should understand this.

5. Pielke’s problem is not that Tmin warms faster than Tmax. That is basic physics, which I think all parties agree on.

The planetary boundary layer is not always where “vertical mixing is strong”. Sometimes it is also stable. This happens when the soil is cold relative to the air (typically due to radiative cooling). This cold surface cools the air close to the surface. This cold air is heavy. If you would lift it, it would be surrounded by warmer lighter air and thus want to sink again. Thus in this situation vertical movements and thus turbulent fluxes are suppressed. This also means that strong vertical changes are possible, while the temperature only changes slowly when the atmosphere is well-mixed. A stable is common in the Arctic in winter, and can happen in our latitudes at night.

Pielke’s problem is that due to global warming, the radiative cooling at night becomes smaller (the additional CO2 will radiate back to the surface) and the average temperature profile will thus change in cases where the atmosphere is stable. There is a paper that shows this for an idealized boundary layer model which assumes perfectly flat and uniform terrain (as these models typically do). As far as I know there is no paper yet which quantifies the effect for the global average.

Then Pielke starts calling every temperature, which is not at the height of the measurement, biased. That is technically true, but outside of a laboratory the temperature of a system A will always be biased relative to system B. It does not say much.

In a climate model, the stability of the atmosphere is also taken into account. And if there is thus a clear change in the temperature profile the climate model would thus model this. If there are any differences between the idealized computations and those of climate models, I would at the moment prefer the ones of climate models because they attempt to take into account that the surface is not uniform and flat. If Pielke wants to make the claim that comparisons of model and observations are biased he will actually have to show that his idealized model is better because everyone would expect the reverse.

If Pielke wants to make the case that the change of the temperature profile very close to the surface is a problem with your type of “basic 1D climate model”, then I would currently expect that these models have larger problems. The climate sensitivities that Nic Lewis is able to obtain with such models are not physical (or at least in need of a physical mechanism that explains the strong stability and why such values are not found in other estimates). That seems a much larger and more urgent problem than a change in the temperature profile near the surface.

6. Ethan Allen says:

ATTP,

You might want to read this paper …
An alternative explanation for differential temperature trends at the surface and in the lower troposphere
http://onlinelibrary.wiley.com/doi/10.1029/2009JD011841/full

Correction to “An alternative explanation for differential temperature trends at the surface and in the lower troposphere”
http://onlinelibrary.wiley.com/doi/10.1029/2009JD011841/full

Peter Thorne might be your best ‘objective’ source in puzzling out this situation. TLT is front and center, but has the baggage of the largest structural uncertainties.

Senior seem obsesses with this issue though, he published something recently (oh say a year or so ago) on temperature measurements at different heights.

That paper does not appear to have, how do I say this, had a reasonable following in the wider climate science community.

7. Joshua says:

Granted, I’m not able to understand any of the technical aspects of the science, but it seems to me that if you’re looking at trends, it wouldn’t make much difference whether you’re looking at Tmax or Tmin*. Is there a difference in the rate of change between Tmax and Tmin?

* But then I thought that about the UHI as well…and clearly from reading “skeptic” arguments, I was wrong about that. 🙂

8. Chris Colose says:

Joshua,

There is a little bit of a difference. Night temperatures have been going up faster than daytime temps. But, that is real, nighttime is real, and people are affected by nighttime. You can’t just wish it away because boundary layer physics behaves differently.

As far as mechanisms, it’s often said that this reduction in the diurnal temperature range is a GHG signature (I never found that very compelling or with a solid ground in the literature, aerosols and cloudiness I expect have a stronger role). With respect to feedbacks, I am not convinced it matters at all. I don’t even understand what Roger is trying to argue on that front…

9. Joshua says:

Thanks Chris –

I did go back and read Anders’ OP again (“hence Tmin increases faster than Tmax; “), and looked at Sou’s post…

10. lorcanbonda says:

It’s hard to be certain what Pielke’s point is, but it sounds like he is saying that we are really using the temperature data as an analog for energy data. In other words, we are always talking about Global Warming because of its temperature impact, but what we really care about is the change in Energy Balance between the radiate energy reaching the planet and the radiant energy leaving the planet — usually discussed in Watts/meter2. Temperature is our measure of energy, but it is only a measure for energy when you include a mass term.10 kg of air at 50C contains ten times the energy of 1 kg of the same air at 50C.

If that is the case, then Pielke is right. The boundary layer at Tmin is lower than at Tmax which means the energy content of the atmosphere is retained in a smaller mass of air. In Global Warming terms, we are mostly concerned about the change in energy content from ~1950 to today. Assuming that the boundary layer height has not changed since the 1950s (which is probably not a good assumption), then you would need to include a factor for the lower boundary layer which corresponds to the lower height.

For a very simplified comparative example, lets say the boundary layer is 2000 meters during the day,but only 1,000 meters at night. The temperature during the night time has risen 1 degree C since the baseline was established, whereas the temperature during the day has risen 0.5 degree C since the baseline.

In this case, the averaged of Tmin and Tmax is 0.75 degrees C. However, Tmin has twice the temperature change, but the same energy change as the daytime period. The net calculation would be complicated, but for sake of argument, it would be the equivalent of 0.67 degrees C and the Tmin has biased the total data by 0.08 degrees C for that day.

If that’s Pielke’s argument, then he is right.

Of course, we would also need to consider the amount of time spent at or close to Tmin vs. Tmax. I assume there would be a seasonal difference which would hopefully average out — but we could consider something like that also. If Tmin is higher during the winter and the boundary layer is much lower, then these estimations can get very complicated.

11. Chris,

GMST anomaly is defined by GMST anomaly, not an energy balance equation. So if you have an equation that you think links the two, and it doesn’t work, then the equation is wrong (even if it’s useful) and may be modified by higher-order terms, better consideration of the spatial structure of temperature or radiative forcing, or whatever. It does not mean your measurement is a poor reflection of what you are measuring.

Thanks, this is a good point. This is kind of what I was trying to get at with my comment about how we determine the globally averaged feedback responses, but you’ve said it more clearly than I did.

Victor,

If Pielke wants to make the case that the change of the temperature profile very close to the surface is a problem with your type of “basic 1D climate model”, then I would currently expect that these models have larger problems.

Seems that you and Chris are saying the same kind of thing.

Ethan,
Thanks, I’ll have a look.

12. lorcanda,
I think that might be related to Pielke’s argument, but as Victor and Chris point out, a 1D models needs an input and the mean temperature we’re determining is essentially the mean temperature. If putting that into a 1D model produces some kind of error, it’s more the model that needs to be modified, not the measurement.

13. David says:

The trend I found in Tmin is 0.08 C/dec. In Tmax it’s 0.10 C/dec. Since 1878 Tmax has warmed faster than Tmin in CET. I also checked it from 1951 onwards, when greenhouse gas concentrations would have had a greater impact than over the course of the whole series. Since 1951 I found that the CET Tmin trend is 0.14 C/dec and in Tmax it’s 0.20 C/dec.

Is this to be expected, given the comments above re Tmin should be warming faster than Tmax? Perhaps CET is unrepresentative of the global picture. Perhaps using annually averaged Tmin/max data isn’t appropriate.

14. David,
I suspect that CET is simply not representative. You can look at the global data here.

Perhaps using annually averaged Tmin/max data isn’t appropriate.

The point, I think, people are making is that if you define this as your mean, then it’s the mean. As long as you’re being consistent, it’s fine.

15. David,
I suspect one issue (and maybe Victor can clarify) is that you have to be careful when considering land-only data.

16. David says:

ATTP,

Thanks. I checked the BEST anomalies from your link for both Tmin and Tmax and found that over the whole series (since 1850) Tmin does indeed have a faster warming trend than Tmax (0.12 C/dec for min versus 0.07 for max).

However, since 1951 the trends are much closer. Tmin is still warming slightly faster than Tmax globally, but since 1951 the respective rates are 0.20 and 0.19 C/dec. Should we not expect the warming rates between Tmin and Tmax to diverge further, rather than close up, as greenhouse gas concentrations increase?

17. Eli Rabett says:

The base problem with RPS’s argument about energy content is that we don’t have useful data (the only thing that counts really is the heat content of the oceans) going back more than your average millenial. It is an argument that if cows were horses pigs would fly. We use 2m surface measurements because that is what we have the longest record of.

Years ago when he started this, he triumphantly waved the early Argo data about. Now, with Argo more mature, he keeps gish galloping the depth that is the “real indicator”

18. David,
Are you sure you did the trends properly. Using the Skeptical Science trend calculator, I get a trend in the mean Berkeley Earth temperature since 1951 as 0.12C/deg.

19. David says:

ATTP,
I just used the ‘linest’ function on Excel for the BEST annual Tmin and Tmax anomaly data. Obviously if that disagrees with the SkS trend calculator you should defer to it!

20. David,
They only have the trends for the mean, so I was just assuming that if the mean is calculated using the max and the min, that the trend shouldn’t be less than these two trends. I’ll have a check myself.

21. David says:

Update: I used the BEST ‘land only’ data. Perhaps including the ocean data is what creates the biggest difference in the global Tmin/Tmax trends? That might also explain the CET results.

22. Yes, that probably will explain it. The land warms about 1.5 times faster than the global mean.

23. Chris Colose says: “As far as mechanisms, it’s often said that this reduction in the diurnal temperature range is a GHG signature (I never found that very compelling or with a solid ground in the literature, aerosols and cloudiness I expect have a stronger role).

On average the trend in Tmin should be about 1.5 times the trend in Tmax due to global warming. Thus the difference between Tmax and Tmin should decrease. This is also called the Diurnal Temperature Range (DTR). Its change has been studied a lot because, as Chris said, it is seen as a signature of global warming.

It has turned out to be enormously difficult because the DTR also change due to the circulation, clouds, aerosols, land-use changes and instrumental changes, especially radiation protection improvements caused a long-term trend bias, but also relocations typically produce large changes in DTR. (And I probably missed many more factors of influence.)

Thus that the DTR range in the Central England Temperature (CET) does not behave like the global average is no surprise. Any small circulation change can already produce such deviations. A lot of care was put into the creation of the CET, but it was not statistically homogenized, which makes me even more careful in interpreting the change in DTR.

For the same reason, using only Tmin or only Tmax is a much worse quality indicator of climate change than Tmean. (Because of how easy it is perturbed it is also harder to make global maps.)

And using only Tmax (or Tmin) would remove a large part of the oldest measurements we have where the mean temperature is computed from fixed hour measurements and not from Tmin and Tmax. A long term perspective is important if you want to understand climate and see real climatic changes.

24. paulski0 says:

ATTP,

I suspect you’re calculating Berkeley L+O and David Land-only(?)

One question is whether Tmin has warmed faster than Tmax or Tmax has warmed slower than Tmin. Ok, that may seem a silly question, but anthropogenic aerosols strongly reduce surface forcing which will tend to reduce Tmax temperatures. Anthropogenic GHG increases may also have small negative shortwave influences (the canonical 3.7W/m-2 CO2 forcing is the net of something like +3.8W/m-2 longwave and -0.1W/m-2 shortwave), affecting Tmax. Likewise shortwave effects of increased water vapour can apparently slightly reduce Tmax relative to Tmin. Shortwave cloud feedbacks can also be a factor.

25. paulski0 says:

Victor,

A lot of care was put into the creation of the CET, but it was not statistically homogenized, which makes me even more careful in interpreting the change in DTR.

I was going to ask if you would be able to comment on that. As far as I could tell they essentially use expert judgement to reduce the Tmin trend in order to account for urbanisation. I think their Tmax/Tmin is quite different from Berkeley for the same region.

On average the trend in Tmin should be about 1.5 times the trend in Tmax due to global warming.

Where does that number come from? Looking at CMIP5 models the difference generally seems to be much smaller than that.

26. paulski0, no idea any more what the source was. I hope it is not wrong. It would be the number for the trend difference due to the increase in the greenhouse effect. The CMIP5 models may have another difference because of the increase in aerosols, which mainly cools Tmax.

27. In raw data the trend in Tmean from Tmin & Tmax is the same as trend in Tmean from hourly data. ht Steve Mosher.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3982162/

The Tmean from hourly data would be affected much less from changes in the temperature profile because most of the time the boundary layer is well mixed.

(Could still be different in homogenized data.)

28. Victor,
Very interesting, thanks. Have you mentioned that to Roger?

29. Robert says:

” Secondly, everything in the above equation is globally averaged, and so as long as the feedback factors (such as the Planck response) are determined in a way that is consistent with the way in which the global mean temperature is computed, I can’t really see the issue.”

Speaking of the Planck response, I have been thinking about “heat balance based empirical estimate of climate sensitivity” such as the one proposed by SteveF at the Blackboard. That particular exercise has a lot of problems, but one issue in particular pertains to this Tmin issue: since AGW is in general warming cooler places faster (nights more than days, poles more than the tropics, Tmin more than Tmax) wouldn’t that imply that heat loss to space is getting less “efficient” over time, as the warmth of the earth gets less “lumpy” (over both space and time) with fewer “hot spots” that lose heat more readily than the background?

It seems to me that heat balance estimates are going to underestimate climate sensitivity since they fail to take into account that as less “lumpy” planet must be warmer overall to lose the same heat via the Planck response.

30. Chubbs says:
31. Physics, have just tweeted Pielke. He should know it, the article cites his.

Robert, also the spatial variance is reduced, which would need a small correction in an 1D model, if you want to use it quantitatively and not just for educational purposes.

32. Robert,
Something to bear in mind is that the 288K temperature that we regard as the mean surface temperature is already based on energy balance. It’s not really an average of the temperature; it is the temperature of a blackbody that would radiate as much energy per square metre per second as is radiated by our surface. The mean temperature anomaly is then the average of variations away from local means; so it is telling us how much – on average – we have warmed.

You can write the Planck response as

$dF = 4 \epsilon \sigma T^3 dT,$

which comes from Taylor expanding the blackbody flux equation and keeping the first term

$F = 4 \epsilon \sigma T^4.$

If you keep an extra term, you would then have a Planck function of

$dF = 4 \epsilon \sigma T^3 dT + 6 \epsilon \sigma T^2 dT^2.$

If you use 288K for $T$, 0.61 for $\epsilon$, then the first term (without the dT) has a value of 3.3 W/m^2/K, and the second (without the dT^2) has a value of 0.017 W/m^2/K. So, it does seem rather insensitive to higher-order terms as long a dT is small enough.

Chubbs,
Thanks, very interesting.

33. Robert says:

“Something to bear in mind is that the 288K temperature that we regard as the mean surface temperature is already based on energy balance. It’s not really an average of the temperature; it is the temperature of a blackbody that would radiate as much energy per square metre per second as is radiated by our surface.”

I did not know that. #todayIlearned

So if I understand you correctly, lumpy temperatures will lose more energy than a smooth blackbody warmth, but the effect is quite small. Do I have that right?

Pardon me as I try to explain that equation to myself like I’m five. So if you warm an area, like the Arctic (or a time, like nighttime) from an average of 0C (273K) by 5C, you will lose about 85% of the energy via the Planck response as you would if you started at 288K (273^3/288^3). That’s based on the first term; the extra term is small enough to neglect for smallish values of dT.

So what I was suggesting (not arguing, just clarifying for my own edification) is that a climate model, which “knows” that a disproportional amount of the change in the mean temperature is occurring in the Arctic, or in the winter, or at night, will come to a higher and potentially more accurate estimate of climate sensitivity than a heat balance model, which cannot take the difference between heating a cool thing (greater average change in the global mean temperature to balance a given forcing) and heating a hot thing (less required, and hence a lower estimate of climate sensitivity.)

You obviously have incomparably greater knowledge of this subject compared with me, and if you tell me this effect is too tiny to ever make a difference, I will believe you.

34. Robert,
I think that if the lumpiness is simply small perturbations relative to the mean, then the effect should be small. Of course, I mean small relative to 288K, not necessarily small relative to what might be significant for us.

Pardon me as I try to explain that equation to myself like I’m five. So if you warm an area, like the Arctic (or a time, like nighttime) from an average of 0C (273K) by 5C, you will lose about 85% of the energy via the Planck response as you would if you started at 288K (273^3/288^3). That’s based on the first term; the extra term is small enough to neglect for smallish values of dT.

Yes, this sounds right, and probably answers the basic question. Imagine we have 5C of warming in the polar regions, but that averaged over the globe was an increase of 1C (i.e., it accounts for 20% of the area), then you would over-estimate the Planck response by about 17%. However, this is so unlikely that it probably represents the highest possible error. The poles do certainly warm faster than the tropics, but this is probably a small enough difference that the error this introduces is probably relatively small; especially given all the other uncertainties.

So what I was suggesting (not arguing, just clarifying for my own edification) is that a climate model, which “knows” that a disproportional amount of the change in the mean temperature is occurring in the Arctic, or in the winter, or at night, will come to a higher and potentially more accurate estimate of climate sensitivity than a heat balance model,

I agree. If you have a three-dimensional, time-dependent climate model, then it will be able to take all these variations into account. A 1D model is always going to be an approximation.

35. Robert says:

Thanks, this improved my understanding a great deal (although, like the Arctic, I am starting from a low baseline.)

36. Chris Colose says:

By the way, on the real Earth, and in GCM’s, the change in outgoing radiation is not T**4. You have to consider the whole vertical structure of the column, including changes in water vapor opacity. The water vapor feedback makes Earth’s climate more sensitive by reducing curvature in the slope of OLR vs. T. GCM’s do not do radiative transfer by considering just one level at the ground, at night or otherwise.

37. Chris,

You have to consider the whole vertical structure of the column, including changes in water vapor opacity.

Indeed, I was just approximating.

GCM’s do not do radiative transfer by considering just one level at the ground, at night or otherwise.

Yes, which is think what Robert was really getting at. All this discussion about 1D models is rather irrelevant if we consider full GCMs that don’t use these simple approximations.

38. John Hartz says:

Although much of the OP and comments are above my pay-grade, this discussion is an excellent example of how open-minded and serious scientists reach common understanding of complex issues. Kudos to ATTP and his colleagues.

39. Robert says:

“Yes, which is think what Robert was really getting at. All this discussion about 1D models is rather irrelevant if we consider full GCMs that don’t use these simple approximations.”

That’s my perspective exactly. Seems like a sufficiently advanced heat-balance equilibrium estimate is indistinguishable from a climate model.

40. izen says:

Perhaps another way to frame it is that the Tmax is cold biased. Primarily by the thermal inertia of the oceans. 70% of the global surface has a quite small DTR, so using Tmean or Tmax may not make a significant difference.

That the land warms ~1.5 times as fast as the oceans may be derived from observation as much as theoretical modelling. If full global Tmax is closer to SST Tmean then it may be a better metric from which to derive the heat content of the climate system. The subsequent variation of DTR on land is of interest because it is where we live.

41. anoilman says:

This image will update every day;

May 7, 2016, we’re pushing 5-10C hotter than normal through out my province.