Probability of Setting versus holding records.

There’s been a lot of talk of records lately, and while I’m thinking about doing a full blown analysis myself, I thought I’d just comment on some information about the probability that a record is set at a given time in a record of something. Note that this is a somewhat different question from the probability a particular time is the record holder. Let’s suppose that the change in some variable relative to whatever it’s previous value was is a continuous random variable (so that ties have 0 probability) with a symmetric distribution with  mean value of zero: in other words, let’s assume that our variable changes randomly, and over a sufficiently long time should show no change. For simplicity let’s assume this also means that it has an equal probability that any point will be higher or lower than any given point in the time series. A given point has an equal probability of being higher or lower than the point preceding it or any point for that matter.  So, when the record starts, every single possible record has a probability of 1 of being “set” since there is no previous value to compare to. The next point has a probability of .5 of being lower than the previous point and .5 of being higher, which will be true of the next point relative to it. In order for a point to set a record, however, it must be lower or higher than every preceding point, not merely the previous point. So say it’s the mean temperature of a particular day of the year. The expected number of records set is N warm “records” and N cold “records” the first year, where N is the number of locations (since we could also go for high and low records for min and max separately, and do every day of the year, we could multiply the number by two and also by the number of days in a year (for simplicity, let’s not concern ourselves with the fact that since we are looking at records on calendar days of the year, another day is periodically added, or the fact that the temperature of a particular calendar day should drift back and forth in sync with the realignment of the calendar like that) and then the numbers could get larger). The next year the expected number of new warm records is N/2 and the expected number of new cold records is N/2 because half of locations should have that day colder and half warmer than that day the previous year-note that spatial correlation may lead to the stations doing those things being clustered together, which means that if the “clusters” are of the size of the area containing our N locations, these values won’t tend to be right. As long as the geographic region can easily contain within it a large number of “clusters” at a given time, the “cool” clusters should be as common and large as the “warm” clusters, however if clusters can encompass large portions of the Earth’s surface, differences from the expected behavior may occur. But regardless, given our previous assumptions about the behavior of our hypothetical temperature variable, the expected number of records continues to decay, in year three to N/4 each, then N/8 each, and so on (N/(2^year-1). If you have 10000 locations with records starting at the same time, the expected number of new records both warm and cold being set would drop to less than one in just 16 years. The main point here is that the probability that a record will get set decays with time rapidly until it gets close to zero at which point in decreases quite slowly. So it would not be surprising if one were to find the rate at which new records are set going down over time. However, if one is looking at a time series showing number of current record holders, that is an entirely different story. The first year, for instance, sets a record for both the warmest and coldest for that date automatically, but, since the next year will be warmer or colder (we have stipulated a tie is impossible), the maximum number of records that the first year might retain is merely half those it set-the maximum, not the expected number held. In point of fact, the number of records that may be held is of course 2N in total, spread out over all the years. The probability that after a long time, the first year still holds one of it’s set records will be exceedingly small.Year two has half of the records held by the first year, half by the second. The third year breaks a quarter of the standing records, and half of those records are from the first year and half the second, so the distribution of records would be .375, .375, and .125. And as the record gets longer, the probability of record retention by early years gets lower and lower, and the distribution gets flatter. For a very long dataset, the probability a record is held by a particular year is generally low, and not very different from the year before it. While there is still a higher probability of records in the earliest years, the decrease with time is not nearly so dramatic. Now, what all this means is that graphs that show the number of records decreasing with time, and the rate of that decrease becoming smaller with time, is entirely consistent even with a static climate, and would not require a climate to be changing in any particular way. Another point is that it is not obvious, since the climate is of course, definitely not completely static, what impact a change in climate would have on records. Presumably a climate that gets warmer would see an increase in high temperature records (given the character of twentieth century changes and also changes in the last thirty or so years, those would be generally exceptionally warm winter nights more so than exceptionally warm summer days) but also a decrease in the number of cold temperature records, relative to the naive statistical expectation. Since those two effects are in opposite directions it is conceivable that they would cancel one another and thus records wouldn’t necessarily be a metric indicative of a change in climate or lack there of because they could conceivably not change relative to the expectation even if there was a change in the different kinds of records due to a change in climate. Some would suggest that the ratio of one set of records to the other would be more appropriate. This however would be potentially highly misleading. Even the warm records decreased relative to expectations, in the cold records decreased relative to expectations faster, the ratio of warm to cold would go up-in other words, a ratio doesn’t tell you whether the change was in the numerator or the denominator, and moreover as the denominator approaches zero, even if the numerator is very small, as long as it is larger, the ratio becomes huge, blowing up at a rapid rate exaggerating what is going on and obscuring important details. Moreover, a change in the number of records is not necessarily and indication of a change in how extreme the climate is: if high temperature records are being set for nights in the dead of winter, but not so much in summer, that represents a decrease in the variance of climate and one becoming on the whole less extreme. As far as I can tell, that’s exactly what has been happening, and is not a particularly alarming way for climate to change at all. As a last point, has anyone really considered how much of a difference between new records and old records (which may have been standing, against difficult odds, for upwards of a hundred years) there generally is? In the real world the variable is generally rounded to whole degrees and not purely continuous (which in turn allows for the possibility of ties) so a record that gets broken may have only a single degree difference from the new record, which, even if the entire difference between the records could be attributed to anthropogenic changes (highly unlikely given we are talking about pairs of data points) these would represent small trends.

So, basically, I see people getting all worked up over records getting set or broken, and how scary that is, I have to admit to being mildly amused. To paraphrase Inigo Montoya, “You keeping using that statistic. I do not think it means what you think it means.”

Keeping The World Unfrozen

In previous work, we discussed the mathematical nature of the climate response to “forcing.” We determined that there were several difficulties in determining the “climate sensitivity”-one needs to know the forcing at work, one needs to know the response time of the system, the forcing needs to be spatially uniform, and so on. I have figured out a possible way around the response time problem. On a timescale of billions of years, the difference between the actual temperature and the equilibrium value is negligible, so we may neglect it and use a zero dimensional model for radiation balance. While I do tend to believe that we are mostly concerned with the climate response over centuries a most, the long term evolution of the Earth system might still offer some constraints on the kind of response we might expect. Of course, over thousands of years, feedback processes emerge in the carbon cycle and ice sheets that would alter the value over hundreds of years: to the extent that those are positive feedbacks, the long term sensitivity is higher than the sensitivity we care about. To the extent that there is an even longer term carbon cycle feedback involving silicate weathering (Walker et al 1981, Volk 1987) which is probably negative, we may underestimate the sensitivity. However, I have made s0me improvements over the model in the earlier post: I am treating radiation balance more explicitly (using the Stefan-Boltzmann law and estimates of the current solar “constant” from Kopp and Lean 2011, the Albedo from Goode et al 2001 and a tuned value of the emissivity that would yield the current climate of ~288 K mean temperature as “baseline” values.) and allowing for the possibility of nonlinear feedback (by treating feedback not as a linear dependence of radiation fluxes on temperature, but as whatever dependence of the ratio of 1-albedo/emissivity on temperature will balance the system). The climate change we are assessing, is the “Faint Young Sun Paradox” first identified by Sagan and Mullen (1972). Essentially, the problem is that our understanding of the evolution of the sun (based upon physics and assessment of sun like stars) calls for the sun to have been substantially dimmer than it presently is billions of years ago, gradually brightening over time (Gough 1981): a constant reflectance of the Earth and a constant tendency of the atmosphere to impede radiative emission would imply that about 2.7 billion years ago, the Earth’s mean surface temperature should have just risen above the freezing point of water (I have produced this result from my own model). But liquid water and a stable ocean was present on Earth by 3.9 billion years ago (Pinti 2005), and possibly as early as 4.4 billion years ago (Wilde et al 2001).  To be sure, the presence of liquid water does not require the mean temperature be above zero, since the water may be present in the Tropics, as long as they remain significantly above the mean temperature. Nevertheless, there is an absence of evidence of widespread glaciations through much of the period when the mean temperature should have been below freezing (Young 1991) although a mid-latitude glaciation probably occurred 2.9 billion years ago (Young et al 1998). Some might take the position that an absence of evidence for glaciation requires a mean climate warmer than the present, but this is frankly implausible on physical grounds-simply put, the greenhouse effect from higher CO2 at the time would not have produced sufficient forcing to lead to warmer temperatures (there is an extensive overview of this discrepancy here starting at page 99), and cooling temperatures over time would lead to, by the aforementioned weathering feedback, long term increases, not decreases, in CO2 levels-which, combined with a brightening sun, surely should have lead to warming temperatures, even though we just stipulated that they were cooling. If there really was an absence of ice, it might mean that there was a change in the meridional heat transport, which would possibly render assessment of the climate sensitivity more difficult. I will instead assume here that the warming over time was slow, such that the Earth was on average above freezing for 4.5 billion years, and we might plausibly claim that the temperatures for much of Earth’s history were sufficiently similar to the present so as to have little ice present (the glacial cycles in recent climate history were associated with mean temperatures about 5 degrees colder than the present, which happens, with our gradual warming, about 1.5 billion years ago) and keep in mind, slower warming would demand a lower climate sensitivity. (Note that if the Earth was still generating a significant amount of it’s own heat for several hundred million years after it’s formation, the very early Earth should have been very warm and I’d have to chop out a significant time period of Earth’s history from my model as invalid. However, the result is not particularly dependent on the climate of the Hadean, it is mostly dependent on the Earth having warmed at a rate since the Archean to the present such that if extrapolated back to the formation of the Earth, would imply freezing temperatures at the Earth’s formation. Although that period is technically included in the calculations of feedback, the feedback is explicitly not linear and is different in the assumed much colder climate of that era that is perhaps incorrect.) Anyway, I’ve kept you all in suspense as to the results of this work: I get a sensitivity to doubling CO2 from roughly present conditions of about .65 K. This is very low compared to “official” estimates from models, although it is compatible with careful assessment of the radiation flux data from CERES (but not careless analysis). It is possible this low sensitivity is due to a long term negative CO2 feedback, which might mean that the sensitivity we’d get over centuries rather than billions of years would be higher to some extent. Still, a long term sensitivity which is much higher than that throws into question how the Earth could possibly have stayed unfrozen through so much of it’s history. I note that I am not the first person to suggest a low sensitivity could resolve the Faint Young Sun Paradox, and for much of the inspiration for this I am indebted to Rondanelli and Lindzen (2010). A key difference between their work and mine is that they identify a specific mechanism which might lead to low sensitivity, I have shown that resolution of the paradox can be achieve by low climate sensitivity regardless of the origin of the negative feedback-other than that, their study is more comprehensive in it’s treatment of climate physics. While this does not constitute absolute “proof” it is does make, in my opinion, a good argument for sensitivity being pretty low. Anyway, you can download a spreadsheet that goes through the determination of this value for sensitivity here.

Goode, P. R., J. Qiu, V. Yurchyshyn, J. Hickey, M.‐C. Chu, E. Kolbe, C. T. Brown, and S. E. Koonin (2001), Earthshine observations of the Earth’s reflectance, Geophys. Res. Lett., 28(9), 1671–1674, doi:10.1029/2000GL012580.

Gough, D. (1981), Solar interior structure and luminosity variations, Sol. Phys., 74(1), 21– 34.

Kopp, G., and J. L. Lean (2011), A new, lower value of total solar irradiance: Evidence and climate significance, Geophys. Res. Lett., 38, L01706, doi:10.1029/2010GL045777.

Pinti, D. (2005), The origin and evolution of the oceans, in Lectures in Astrobiology, vol. 1, edited by M. Gargaud et al., pp. 83–111, Springer, New York.

Sagan, C., and G. Mullen (1972), Earth and Mars: Evolution of atmospheres and surface temperatures, Science, 177(4043), 52–56.

Rondanelli, R., and R. S. Lindzen (2010), Can thin cirrus clouds in the tropics provide a solution to the faint young Sun paradox?, J. Geophys. Res., 115, D02108, doi:10.1029/2009JD012050.

Volk, T. (1987), Feedbacks between weathering and atmospheric CO2 over the last 100 million years, Am. J. Sci., 287(8), 763– 779.

Walker, J., P. Hays, and J. Kasting (1981), A negative feedback mechanism for the long-term stabilization of the Earth’s surface temperature, J. Geophys. Res., 86, 9776–9782.

Wilde, S., J. Valley, W. Peck, and C. Graham (2001), Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago, Nature, 409, 175– 178.

Young, G. (1991). The Geologic Record of Glaciation: Relevance to the Climatic History of Earth. Geoscience Canada, 18(3).

Young, G.M., V. von Brunn, D.J.C. Gold, W.E.L. Minter, Earth’s oldest reported glaciation; physical and chemical evidence from the Archean Mozaan Group (∼2.9 Ga) of South Africa, J. Geol. 106 (1998) 523–538.

The Equator to Pole Temperature Difference

How much colder, on average, is the North Pole (geographically) than the near equator area in the Northern Hemisphere? What about in the Southern Hemisphere? And how would people expect these to be changing over time? What is the seasonality of these things?

Well, one must take the data I will present with a grain of salt: they come from the NCAR-NCEP reanalysis daily data (t2m, downloaded from KNMI), which uses a weather prediction model to  get complete fields based on regular re-initializations with observed data from around the world. I have to wonder how much of any trends is simply a consequence of changing data or biases in model behavior…or alternatively if those factors are hiding trends rather than creating them. But anyway, I found the results a little surprising. Well, some of the results.

On average, the annual average difference in the Northern Hemisphere between the lowest ~1 degree latitude band and the highest is almost 44 degrees over the whole reanalysis period. In the Southern Hemisphere, it’s almost 73 degrees (note: this is about a degree further from the Pole, I think, since . That’s not terribly surprising, I think: high elevation of the South Pole probably makes it much colder than the North Pole-well, that’s my first guess why, anyway. I am also not terribly surprised that, over the period, there has been a reduction in the difference in both Hemispheres on an annually averaged basis:

Equator to Pole Temperature difference in the Northern Hemisphere, 366 day running average and 3651 day centered average.

Same as the Above, but for the Southern Hemisphere.

What is a bit surprising is that the Northern Hemisphere difference appears stable until the late eighties, whereas the Southern Hemisphere difference is a bit more continuous, but mostly . I expected a larger decline earlier in the record for the Southern Hemisphere, based on Antarctic station based data I’ve seen, with not much if any in the last twenty years, and a continuous decline since the seventies in the Northern Hemisphere (keep in mind that the Reanalysis begins in the late forties, after the warmer temperatures of the thirties in the Arctic). At any rate, such a reduction in Equator to Pole Temperature differences is suggestive of enhanced heat transport and, at least in the Northern Hemisphere, the ice-albedo effect. This next finding is also not surprising, at least given the above: annual maximums in the Equator to Pole temperature differences has declined too:

366 day running maximum of Northern Hemisphere Equator to Pole Temperature difference.

Same as above, but for the Southern Hemisphere.

Please note that neither of these trends can plausibly be related to the ice-albedo feedback effect, since they are associated with the winter season, during the six months of basically continuous darkness at the Poles. This next result may be surprising for some readers:

366 running minimum of Northern Hemisphere Equator to Pole temperature differences.

Same as above, but for the Southern Hemisphere.

Now, since I looked at the breakdown of annual maximum and minimum of daily temperatures in the Arctic before, this wasn’t terribly surprising to me, but are readers surprised by the fact that, according to this data, the annual minimum of the Equator to Pole temperature differences are increasing? In the Northern Hemisphere, this seems to be occurring because, in the Arctic, air temperatures in the reanalysis seem to hit a ceiling every year that’s near the triple point of water (coincidence?) which they basically can’t seem to exceed: I looked at that and there is basically no trend in annual maximum daily temperatures in the deep Arctic. While there isn’t an obvious summer “ceiling” nor is there any connection to being close to the triple point of water the Southern Hemisphere, there isn’t any tendency of the annual maximum to warming there at all it, appears. These changes seem to indicate that there are seasonally dependent trends in the meridional transport of heat. Such a thing could be brought about by significantly spatially and seasonally heterogeneous forcing. This information cannot confirm that definitively, nor can it identify the source of such a forcing. But spatial variations in cloud coverage would be my first guess for a natural factor. Also note that, as I previously stated, one cannot really test “global mean” sensitivity on the basis of spatially heterogeneous forcing. If recent climate change is due in significant part to very spatially (and seasonally!) heterogeneous forcing, studies of “global mean” sensitivity will almost inevitably give misleading results.

Climate as A Differential Equations Problem

Cross posted at The Air Vent.

I find that when discussing problems of what are ostensibly physical systems, little progress can be made until people formulate their ideas in coherent mathematical terms. At that point there can be no arguing, a statement is either mathematically correct or it isn’t, and it will be completely unambiguous. The only ambiguous questions should be the values of various coefficients and constants, as long as we are sure of equations that characterize the problem. However, much confusion in discussions of climate comes about because the equations that are thought to characterize the problem are rarely stated, and even when they are, they are difficult for most people to understand, partly due to a common lack of numeracy at the necessary level (in this case, understanding of at least some calculus would be helpful, perhaps necessary) and also the fact that no attempt is made to explain the underlying equations to the laymen (I honestly know of zero counter examples). I am going to attempt to explain what I presently understand to be the equations which characterize the climate problem as it is commonly thought about, and try my best explain what the equations mean.

Let’s start off with the equation that relates a function that causes the system to change to the parameter that it changes. First, let’s start off with the common simplifications: we characterize the climate by a single function, the global mean temperature (the natural asymptote, the “equilibrium” is made the zero point here), T(t), the forcing function was the global mean top of the atmosphere radiation flux change caused by some factor, f(t). Well, what is the math that relates these? Let’s start by eliminating obviously wrong relations. To begin with, the equation that is appropriate is not:

Equation (1)

Where lambda is the “sensitivity” that converts forcing to the temperature change. It is not hard to see the problem with this simple equation: it assumes that when there is some force that acts to bring the Earth out of thermal equilibrium (more so than usual!) the Earth would instantaneously get into equilibrium. We know from our daily experience that this can’t be correct: the peak daily temperatures occur an hour or two after local solar noon. Moreover, the seasons lag behind the solstices significantly, depending on which season, the latitude, and whether the location is continental or on/near the ocean: the lag for mid-latitude continental climates averages about a month. To be fair, these examples are not perfectly analogous to our simplified picture of a single global temperature parameter determined by a single global mean forcing, and also, the individual locations experience energy and even mass flow of the atmosphere from other places, not just a flux to space. Even so, it is obvious from other examples that bodies do not warm or cool instantaneously in reaction increased energy input. Consider a cool room, now place a space heater in it, and turn it on. Now, the device itself does not reach its full power instantly, even so, it takes much longer for the temperature of the room to heat up most of the way. So it is obvious that equation (1) cannot be correct. Now, the kind of system we are describing can be modeled as something called a Linear Time Invariant system. Now, for our purposes, getting into what that term means, instead, let’s look at what a first order equation of this type typically looks like:

Equation (2)

Equation (2) is a generic, first order LTI system. What it means is that if you add the rate of change to a constant multiple (inverse tau) of the value of something relative to it equilibrium value, the result is the input function to the system that is causing the change. In the case of a system which begins at a state F₀ different from its natural equilibrium, and receives zero input, the solution to Equation (2) is:

Equation (3)

Now what that equation means is that the system, absent an input, will tend to approach equilibrium with exponential decay. Also, this means that unless the initial value happened to be the equilibrium value, climate would be an initial value problem. That initial values are expected to decay to a negligible amount after thirty years perhaps gives us a clue as to the commonly assumed response time of the system. Tau is the “response time” of the system, or the time constant, and represents the time t at which the system has reached e^-1 of the initial value, or about 37%. Many electrical engineers must by now (and probably earlier) recognize the kind of equation we are talking about, it is of the same form as equations that describe simple RL and RC circuits. Specifically, tau is the ratio of the inductance to the resistance in the former, and the product of the resistance and the capacitance in the latter. This makes for a good jumping off point for talking about the “doubling CO₂” input function to the climate equations, which is how I will derive the specific function I understand to characterize current understanding of climate. The step response of and RC circuit looks like this. Basically, the value that will eventually be taken, after infinite time following a step input, is a tau times the constant A by which the Heaviside function u(t) was multiplied in the input. In other words, the solution to equation (2) when g is an arbitrary instantaneous step input with initial value set to zero is:

Equation (4)

So the new equilibrium value is A times tau. Now, in the case of climate, the common hypothetical is the step function from a doubling of CO₂, which is said to have a value of about 3.7 Watts per square meter (the value isn’t really important to our discussion just yet) and the equilibrium response is that value multiplied by lambda, the “sensitivity.” Now, in this particular case we get these equations:

Equation (5)

Equation (6)

Equation (6) must be generalized for situations where the forcing function is arbitrary, so we must isolate the constants that characterize the system (lambda and tau) from the CO₂ doubling function (3.7 times the Heaviside function). The result:

Equation (7)

Equation (7) is at last the form of the climate equation, or what seems to be about the form usually used. But what does this equation mean? Well, for smaller response times for a given sensitivity, the Temperature more closely resembles the time evolution of the forcing, for longer response times the forcing function more closely resembles the time derivative of the temperature.

What might we say about the response time of the climate system? Well, climate usually is defined as a thirty year average, presumably because scientists expect there to be very little remaining of decaying initial values per equation (3). If negligible contribution from those decaying values is 10%, then that happens in about 2.3 time constants, so the thirty year average implies an assumed time constant of about 13 years. If a negligible contribution is 25% it happens after about 1.4 time constants and the implied time constant assumed is a little less than 22 years. So we see that the thirty year average implies climate scientists think the time constant is on the order of several years. This doesn’t mean they are right, and what level of averaging is necessary depends on the amount of attenuation desired as well as the response time (also note that, contrary to what is commonly asserted, the unforced solution of (7) does not directly depend on sensitivity, since it can be canceled out when f(t) is zero, thought it does depend the response time, in terms of its time scale, but not it’s magnitude).

Can we estimate the response time of the system? Only if we know the climate forcing, the climate evolution, and either the sensitivity or some relationship between the sensitivity and the response time. But we can look at some lines of evidence that might imply certain things about the response time. Consider the ice core data: in them, the temperature appears to vary in proportion to the greenhouse gas forcing, which would imply that, relative to the time scale of the glaciation cycles, the response time is short. Only one problem with this line of reasoning: We know that the earliest temperature changes preceded the CO₂ changes, implying a positive CO₂ feedback (and we shall discuss the issue of feedback in a bit) and it is the CO₂ which has a short response time to temperature relative to length of a glaciation cycle. What about the forcing that actually causes the change to begin with? It turns out that his implies the exact opposite! According to Roe (2006), the changes in insolation track the rate of change of climate. This result would seem to require a very long response time. However, as with the seasons example, Milankovitch climate forcing is something which cannot fit into our simplified model of the climate system, as it is not a globally averaged top of the atmosphere flux change: in fact, the global insolation change is essentially tiny or nonexistent compared to the change in July insolation near the Arctic Circle that Roe correlates with the rate of change in ice volume. So this example cannot tell us what kind of response time to expect from a global forcing like CO₂ (although later we will discuss some reasons why this response time may be longer than the one with which we are concerned anyway). Likewise, the seasons cannot establish a very short response time. But there is an example of a global mean flux change forcing which the nature of the response to which may imply something useful about the response time. Lindzen (1995) found that the response time has an interesting effect on the response of the climate system to relatively close in time volcanic eruptions: if the response time is long, the cooling impact of closely spaced together in time volcanic eruptions will build on one another, leading to long term cooling absent a significant warming forcing to offset this (a finding built on later by Lindzen and Gianitsis (1998) in a context of the climate’s sensitivity) in particular the response time which lead to accumulating cooling was sixteen years or longer by Lindzen’s calculations. Note that, if climate scientists really do believe the time constants I think are implied by thirty year average representing forced climate, then values close to this threshold but slightly above or below are considered reasonable. But at least according to Lindzen’s calculations, the assumed response times are generally significantly longer than 16 years (which would implied much less attenuation of unforced variability at thirty years) and thus would imply volcanic cooling building on volcanic cooling that is incompatible with the temperature record. Of course, it could be that the temperature record back then is not good enough to capture the real variation in climate associated with those volcanoes, or it could be that Lindzen’s calculation of the threshold at which volcanic cooling build up is an underestimate. It could even be the case that a large warming forcing canceled the cooling, although both solar variability and greenhouse gases seem inadequate. Still, very long response times seem unreasonable based on the available evidence-middle of the road estimates might be about right, not as low as say, Willis Eschenbach recently estimated at WUWT (months or less) but not as high as models assume (decades or even centuries, evidently).

Now to the question of including feedback in this consideration. At my own blog, I have discussed the equations used to estimate lambda from considering changes in the radiation flux with temperature (and also estimated the relationship between changes in flux and temperature). Let’s consider those equations for a moment:

Equation (8)

Equation (9)

Now, in my original posts, I did not really discuss the nature of these equations. For instance, why does equation (8) look like it does? Well, the reason is because it is describing a process which essentially acts behaves like a Geometric series. To see why this is the case, let’s consider a feedback process at work: Let’s say I add a unit value to the system, and the system responds to that change by adding half of that to the value over again, to which it responds again with half of that. This particular geometric sum is convergent, meaning it approaches value less than infinity as the number of times the process repeats approaches infinity. That particular sum is equal to two. It turns out there is a general formula for feedbacks less than adding the full value added again, and even negative feedbacks. That formula is:

Equation (10)

Now, equation (10) is interesting, since it carries out an infinite sum. The like appears in many areas, for example, in economics (please try to contain yourselves) it is the form of the infamous Keynesian “multiplier”, which also involves an infinitely repeated process (of partial spending and partial saving of marginal income). Now, I believe the use of such formulas in economics acts more to obscure than it does to elucidate, and the Keynesian models are a load of nonsense. But back to climate, where this formula is acceptable because we are dealing with a physical system. It is crucial, for this formula to be useful, that the timescale associated with the feedback processes be very short compared to the system response time. The climate feedbacks we are generally interested in involve cloud and water vapor processes that are very fast and probably are a lot shorter than reasonable estimates of the response time we are interested in. But recall when I said of the Milankovitch response time “we will discuss some reasons why this response time may be longer than the one with which we are concerned anyway”—well, now I intend to. It is obvious that the feedbacks at the glaciation cycle timescale are much longer than the reasonable estimates of the response time: the CO₂ feedback and the formation and melting of ice age continental ice sheets are slow process, taking hundreds or thousands of years. For this reason both the sensitivity and response time to Milankovitch forcing appear to be very different from what we might reasonably expect from a doubling of CO₂ and so that timescale being looked at for clues to the sensitivity and response time of present interest, will be extremely misleading. Finally, note the nature of the functional form implies that feedback factors close to one imply very large sensitivities, and a feedback factor of one corresponds to infinity sensitivity (greater than one, bizarrely, leads to a system with negative sensitivity, but such values are obviously unphysical). This means that as long as f is positive, slight variations in its value lead to large variations in sensitivity, which is why the range of sensitivities typically given is usually quite large, but if the feedback is negative, even large relative errors in the size of the value of f lead to a small uncertainty band about a low sensitivity.

Finally, can we relate the climate sensitivity to the response time? That is, might it be possible to reduce this problem from one of identifying the value of two parameters to the value of a single parameter? Actually, it turns out there may be a relationship between the two. According to Lindzen (1995) the response time is in fact basically proportional to the gain (or one over one minus f) of the system (following Hansen et al (1985) so this rough relationship for values of tau that are plausible is not controversial, although the exact proportion may not be so). His proportionality implies that the typically assumed sensitivity range implies response times of 80 years or more. This is much longer than I estimated from the attenuation of natural variability by the thirty year average climate. Can climate scientists really believe in such long response times? But the feedback necessary to double the no feedback sensitivity apparently requires such a long response time, and that merely constitutes the “low end” of the IPCC’s sensitivities. Yet perhaps this proportionality is not right. Schwartz (2007) originally claimed a response time of about five years and a sensitivity approximately equal to the no feedback value (gain of one) which seems to suggest a gain of two for response times of only about ten years (note here that it is not relevant that Schwartz revised his estimates of both, only that his proportionality between the two appears to be very different from Lindzen’s). This is the issue in this post about which I know the least and welcome input as to what, precisely, determines the relationship between the sensitivity and the response time.

I certainly hope that I have provided a useful framework for moving discussion of the climate problem forward in a mathematical framework. Commenters, if you find an error in my calculations, I welcome it, for that is how mathematics (as opposed to the usual arm waving climate arguments) works. If you have suggestions for improvements on the assumptions and simplifications necessary to develop the mathematical climate theory, I welcome those too, provided they are mathematically rigorous and logically and scientifically justifiable.

References:

Hansen, J., G. Russell, A. Lacis, I. Fung, and D. Rind (1985) Climate response times: dependence on climate sensitivity and ocean mixing. Science, 229, 857-859

Lindzen R.S. (1995) Constraining possibilities versus signal detection. pp 182-186 in Natural Climate Variability on Decade-to-Century Time Scales, Ed. D.G. Martinson, National Academy Press, Washington, DC

Lindzen, R.S. and C. Giannitsis (1998) On the climatic implications of volcanic cooling. J. Geophys. Res., 103, 5929-5941.

Roe, G. (2006) In defense of Milankovitch. Geophys. Res. Ltrs., 33, L24703, doi:10.1029/2006GL027817

Schwartz, S. E. (2007), Heat capacity, time constant, and sensitivity of Earth’s climate system, J. Geophys. Res., 112, D24S05, doi:10.1029/2007JD008746.

Daily Rainfall At WPBIA

Lately things have been quite wet around here in my part of Florida. This is about what one would expect if we are heading into an El Nino year, but El Nino is not yet official, I think. Anyway, I was curious about what the typical rainfall in our area tends to be from day to day. The nearest climate station is apparently up at the West Palm Beach  International Airport. It has (somewhat incomplete) data from July of 1938 until roughly the present (data available at KNMI run up until the 16th as of now) and the daily rainfall history looks like this:

Figure 1, Daily Precipitation Totals in inches, WPBIA

You probably notice that point near the beginning that is higher than any other point: specifically, according to the data the rainfall total for April 17 1942 was more than fifteen inches! Well, I’m not sure about that particular data point since it is so exceptional, but I’m going to assume it, and the rest of the data, are accurate. It does not appear there is any obvious trend within the daily data, though of course this mostly because it is just so noisy. What does the annual variation look like?

Figure 2, Daily Precipitation Totals plotted against “Fraction of the Year”

The above plot shows rainfall totals plotted against the “Fraction of the Year” which here is taken as the sum of the month minus one divided by twelve and the day of the month minus one divided by thirty one divided by twelve. That plot has 26420 data points. So I decided to see what the 1321 point center averages of that as a series is, each point of which representing the average daily precipitation in approximately one twentieth of the year. But first, I put a second version of the above series at the end of it, so that I could then get a full annual cycle in a smooth. So here is a full cycle of one twentieth of the year centered averages, plotted against ay of the year (well, almost, it’s actual “Fraction of the Year” times 365.2425, rounded down to the nearest integer, plus one):

Figure 3, one twentieth year centered averages of the Daily Precipitation Totals

Interestingly, the annual cycle in precipitation at WPBIA has not one wet season peak, but two. The wet “half” of the year (the above curve’s first and last points above it’s overall average) lasts from roughly mid to late May to the end of October. The two wet peaks occur during the above average parts of the wet “half” of the year (In the same manner as as the dates for the above average half year were determined, only this time with the average as the average of points above the average for all points)  from the end of May to till just after Independence Day (ie early July) and from mid-August to mid-October, with the sharp peak of the latter period occurring in in mid to late September. Now, obviously this is just on average, and the edges of these periods are fuzzy even in the smoothed data by a few days. But the double peak is an interesting feature of our local seasonal weather variation. Anyone have any idea why this may occur?

Revisting Category 4&5 Tropical Cyclones…Again

A little less than a year ago, I updated my work examining whether there has been any increase in the number of category four and five tropical cyclones, first in the Northern Hemisphere and then, more cautiously, the entire world since 1987 (that is, when aircraft reconnaissance ceased in the Northern West Pacific Basin) (in the case of the Southern Hemisphere, this means the 1987-88 season at the start). Well I figure now is as good a time as any to update to include the year 2011 in the analysis (the 2010-2011 season for the Southern Hemisphere.

First, there is the Northern Hemisphere data:

As you can see, 2011 was a slightly above average year in terms of the number of category four and five storms in the Northern Hemisphere (in terms of overall activity it was still below average). Nevertheless, the trend remains ever so slightly negative which is contrary to the hypothesis that the number of such storms would increase with higher sea surface temperatures. But what about the world? Well, as mentioned before, because the Southern Hemisphere seasons occur during the end of one year and the beginning of the next, it is difficult to combine seasons. So I did it two ways: in one time series I added the Southern Hemisphere seasons to the Northern Hemisphere totals in the year associated with the beginning of their season, while in the other added them to end year time series.

Here they are, in green and red, respectively:

The conclusion with respect to the global data is even stronger now than last year: there are no trends! The slight positive trend in the green time series has vanished, in fact it appears it is now very, very slightly negative; the red trend is still very slightly negative. So I remain confident in my initial assessment of this data: there is no evidence that higher sea surface temperatures are increasing the number of strong tropical cyclones.

The Story of Where I’ve Been

So I haven’t blogged in quite some time, and I haven’t been commenting on various sites for a while either. Well there are a couple of reasons for this:

To begin with, I am typing this on my (relatively) new laptop. That’s because my old laptop experience a malware meltdown, and after the hard drive was wipe, decided to cease to function anyway. That situation took me months to sort out, and I prefer not to blog from other computers. But even if I would have, I was also quite busy with school work. I don’t have the Summer off, but I only have one class and it should be any easy one. The Fall semester ends for me with my last Final Exam on Wednesday.

Also, I’m pretty damn upset about the state of things right now. Let’s leave it at that.

Anyway, I will probably resume commenting semi-regularly soon, and if I think of anything to blog about, I will.

Wiki Climate Lies Infect Non-Climate Articles

The bias of Wikipedia in dealing with climate issues has been well documented. I tend to find that the more controversy their is over any topic, actually, the less reliable wiki gets. But who would expect that should be mostly factual articles would be infected with climate bullshit? It would be one thing to expect unreliable, biased information on local politics to infect an article about a major city. It is quite surprising to look at an article about a major city and find a falsehood about climate being promoted, and referenced to “scientists” who ought to know better, given that they work for an organization that gathers data that contradicts their own claims.

But when, with the intention of merely identifying the coordinates of the City of Austin, Texas, to examine temperature records nearby (prompted to do so by this post), the above described situation is in fact what actually happened to me. I will use screen captures in case someone comes to their senses and realizes this is a really stupid, factually inaccurate thing to put in an article: here.

What is wrong with this statement? Well, let’s think carefully about it. What “these kinds of droughts will have effects that are even more extreme in the future, given a warming and drying regional climate” can be interpreted as meaning is pretty unambiguous. Droughts will get worse in the future, given regional warming and drying as facts. Now, weasels will say, that he is saying that, “if” we take such future trends as a “given” then “of course” that would be true. If that is what is being said, someone needs to go back to grade school, because the actual statement is saying that given the present tense trends, that is what will happen. So no weaseling for you fools. The facts are that this statement, that Texas is warming and drying is false both as two separate statements, and as a combined statement. The long term records for Texas annual temperature trend since 1895, according to the National Climate Data Center is 0.00 degF / Decade, precipitation 0.08 Inches / Decade, in other words, the numbers given by the NOAA organization that specifically monitors climate directly contradict the claims of an NOAA “scientist” who clearly should know better: The trends are pretty clearly not different from zero, if you look at the data themselves, although the temperature trend is technically slightly negative. So the claim that Texas is drying and is warming, is FALSE. If anything the long term trends are toward wetter and cooler. You can see the data for yourself here.

Anyway, I am glad I got that off my chest. I really can’t stand that “scientists” make claims that can be shown to be wrong by a mere amateur with a couple of mouse clicks. I am even more frustrated that such claims get parroted unquestioningly in “factual” articles about major cities. Thankfully it would appear that no such stupid statements are present on the West Palm Beach, Florida article (the nearest major weather station (the airport) to where I live, although that’s not particularly close), even though we have also had a bad drought (well, bad by our standards, and since we are subtropical, and very wet normally, “bad drought” here is nowhere near as bad as in Texas), although ours is safely over for now, with quite a lot of rain recently (it may return next year if La Niña persists). How does it come to be that climate idiocy finds its way into non-climate articles? Is nothing sacred?

Critical Points

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0.

The above is from Wikipedia. As you all should know, I know Calculus. Now, technically, no climate data is a truly differentiable  function, since they are not provided in continuous form. Nevertheless, one can estimate the instantaneous rate of change of a function from slopes of lines between discrete points. So with timeseries that are not continuous in a mathematical sense, one can nevertheless identify points at which the rate of change abruptly switches sign. I am looking into this as a way to identify intervals of change for better estimating variance adjustments for LT versus surface temperatures. So my interest is in finding something analogous to “critical points” in the surface or LT data, specifically ones associated with changes in the sign of temperature change. So I will want to identify when, say, three month slopes change sign, and then identify those intervals between them that are useful for assessing short term amplification. I am just getting started, but will start with a plot of identified intervals in the surface data (here I will be using an average of GISS, HadCRUT, and NCDC over 1979-almost the present (deviations from the 1981-2010 mean), and three month averages to make the points of change stand out more. Note that later when finding the magnitude of temperature changes I will probably just use the unsmoothed data over identified intervals). The plot shows those intervals I chose to be sufficiently long, taking breaks in the intervals that were short as not seperate intervals put part of the longer intervals, and making them overlap on the maxima/minima that characterize the change in rate.

Now, having identified intervals, I examine the ratio of changes in the troposphere over said intervals: two problems occur: Near zero the ratios vary a little too much, and also a couple of times the rates of change are opposite in sign. This can be seen in a plot of the ratios with the the amount of atmospheric change as the independent variable:

See, now, a deceptive thing I could have done was not tell you that my new approach was failing to overcome the noise. But I am showing you this plot so you can see why I am going to say: I am disappointed, because this is not the kind of clean result I was hoping for, showing the short term amplification without a lot of noise. So anyway, a couple of things that might help: Eliminating ratios associated with changes too close to zero, and unphysical ratios less than zero. The result (eliminating ratios where the absolute value of the atmospheric change less than .1)  is that the mean ratio goes from about .64 to 1.04 and the standard deviation goes from .81 to .46, so while it does converge closer to be within the realm of my previous estimates (which were roughly from 1 to 1.5 with a best estimate of about 1.3, very close to the theoretical/model value of 1.2) it clearly is very uncertain. So I would consider this not a great method to get the right value for the amplification ratio for short term changes. I am still most satisfied with taking various different approaches and averaging them. So remember our previous estimates: 1.46, 1.07, 1.49, 1.32, 1.49, .954, 1.56, and 1.38, and add in the new estimate, the average is ~1.3 so a conservative estimate of the short term amplification remains about 1.2, with some uncertainty still to be figured out. (Excluding the estimates based on the smooth trend removal method, since it is a little difficult to explain and I am not sure it is a detrending method that is reliable anyway, the new mean is 1.28)

So I feel I have not made much progress in the variance adjustment estimate. Since I don’t yet have a good method to get it, I am going to be faced with the issue of continuing to search for a good method. Well, for now, I have the best I can do.

ENSO and US Climate/Weather Revisited

I was curious what would happen if I used my “Invariant ENSO Index” to look at ENSO impacts in the US. So here is El Nino versus La Nina temperature composites:

 

And Now El Nino Versus La Nina Precipitation:

 

So now let me go out on a limb here and say that, what with us  just having come out one La Nina and going into another one, having had a warm summer across most of the Central and Eastern US and dry conditions in most of the South Central US as a result, it looks like we will be in for more of the same for this next La Nina. This is bad news for Texas and Florida, as it means that we will have a repeat of the drought conditions just when we thought we were out of the woods.

Of course, one can be sure that typical, predictable weather patterns will be played for all they are worth in the media as somehow evidence of our evil human ways, even though years just like the last one and what will likely happen in the coming year, have happened throughout the recorded history of climate in the US, and undoubtedly longer than that. The observational record offers us many opportunities to identify past patterns of weather and the conditions that gave rise to them. What the record tells us is often that the weather we can expect looks a lot like weather we have had in the past. Sometimes that past weather was quite nasty. We better listen and be ready.