Saturday, October 24, 2020

How can climate be predictable if weather is chaotic?

[This is a transcript of the video embedded below. Some parts of the text may not make sense without the graphics in the video.]

Today I want to take on a question that I have not been asked, but that I have seen people asking – and not getting a good answer. It’s how can scientists predict the climate in one hundred years if they cannot make weather forecasts beyond two weeks – because of chaos. The answer they usually get is “climate is not weather”, which is correct, but doesn’t really explain it. And I think it’s actually a good question. How is it possible that one can make reliable long-term predictions when short-term predictions are impossible. That’s what we will talk about today.


Now, weather forecast is hideously difficult, and I am not a meteorologist, so I will instead just use the best-known example of a chaotic system, that’s the one studied by Edward Lorenz in 1963.

Edward Lorenz was a meteorologist who discovered by accident that weather is chaotic. In the 1960s, he repeated a calculation to predict a weather trend, but rounded an initial value from six digits after the point to only three digits. Despite the tiny difference in the initial value, he got wildly different results. That’s chaos, and it gave rise to the idea of the “butterfly effect”, that the flap of a butterfly in China might cause a tornado in Texas two weeks later.

To understand better what was happening, Lorenz took his rather complicated set of equations and simplified it to a set of only three equations that nevertheless captures the strange behavior he had noticed. These three equations are now commonly known as the “Lorenz Model”. In the Lorenz model, we have three variables, X, Y, and Z and they are functions of time, that’s t. This model can be interpreted as a simplified description of convection in gases or fluids, but just what it describes does not really matter for our purposes.

The nice thing about the Lorenz model is that you can integrate the equations on a laptop. Let me show you one of the solutions. Each of the axes in this graph is one of the directions X, Y, Z, so the solution to the Lorenz model will be a curve in these three dimensions. As you can see, it circles around two different locations, back and forth.

It's not only this one solution which does that, actually all the solutions will end up doing circles close by these two places in the middle, which is called the “attractor”. The attractor has an interesting shape, and coincidentally happens to look somewhat like a butterfly with two parts you could call “wings”. But more relevant for us is that the model is chaotic. If we take two initial values that are very similar, but not exactly identical, as I have done here, then the curves at first look very similar, but then they run apart, and after some while they are entirely uncorrelated.

These three dimensional plots are pretty, but it’s somewhat hard to see just what is going on, so in the following I will merely look at one of these coordinates, that is the X-direction. From the three dimensional plot, you expect that the value in X-direction will go back and forth between two numbers, and indeed that’s what happens.

Here you see again the curves I previously showed for two initial values that differ by a tiny amount. At first the two curves look pretty much identical, but then they diverge and after some time they become entirely uncorrelated. As you see, the curves flip back and forth between positive and negative values, which correspond to the two wings of the attractor. In this early range, maybe up to t equals five, you would be able to make a decent weather forecast. But after that, the outcome depends very sensitively on exactly what initial value you used, and then measurement error makes a good prediction impossible. That’s chaos.

Now, I want to pretend that these curves say something about the weather, maybe they describes the weather on a strange planet where it either doesn’t rain at all or it pours and the weather just flips back and forth between these two extremes. Besides making the short-term weather forecast you could then also ask what’s the average rainfall in a certain period, say, a year.

To calculate this average, you would integrate the curve over some period of time, and then divide by the duration of that period. So let us plot these curves again, but for a longer period. Just by eyeballing these curves you’d expect the average to be approximately zero. Indeed, I calculated the average from t equals zero to t equals one hundred, and it comes out to be approximately zero. What this means is that the system spends about equal amounts of time on each wing of the attractor.

To stick with our story of rainfall on the weird planet, you can imagine that the curve shows deviations from a reference value that you set to zero. The average value depends on the initial value and will fluctuates around zero because I am only integrating over a finite period of time, so I arbitrarily cut off the curve somewhere. If you’d average over longer periods of time, the average would inch closer and closer to zero.

What I will do now is add a constant to the equations of the Lorenz model. I will call this constant “f” and mimics what climate scientists call “radiative forcing”. The radiative forcing is the excess power per area that Earth captures due to increasing carbon dioxide levels. Again that’s relative to a reference value.

I want to emphasize again that I am using this model only as an analogy. It does not actually describe the real climate. But it does make a good example for how to make predictions in chaotic systems.

Having said that, let us look again at how the curves look like with the added forcing. These are the curves for f equals one. Looks pretty much the same as previously if you ask me. f=2. I dunno. You wouldn’t believe how much time I have spent staring at these curves for this video. f=3. Looks like the system is spending a little more time in this upper range, doesn’t it? f=4. Yes, it clearly does. And just for fun, If you turn f up beyond seven or so, the system will get stuck on one side of the attractor immediately.

The relevant point is now that this happens for all initial values. Even though the system is chaotic, one clearly sees that the response of the system does have a predictable dependence on the input parameter.

To see this better, I have calculated the average of these curves as a function of the “radiative forcing”, for a sample of initial values. And this is what you get. You clearly see that the average value is strongly correlated with the radiative forcing. Again, the scatter you see here is because I am averaging over a rather arbitrarily chosen finite period.

What this means is that in a chaotic system, the trends of average values can be predictable, even though you cannot predict the exact state of the system beyond a short period of time. And this is exactly what is happening in climate models. Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels.

This video was sponsored by Brilliant, which is a website that offers interactive courses on a large variety of topics in science and mathematics. In this video I showed you the results of some simple calculations, but if you really want to understand what is going on, then Brilliant is a great starting point. Their courses on Differential Equations I and II, probabilities and statistics cover much of the basics that I used here.

To support this channel and learn more about Brilliant, go to Brilliant.org/Sabine and sign up for free. The first 200 subscribers using this link will get 20 percent off the annual premium subscription.



You can join the chat about this week’s video, tomorrow (Sunday, Oct 25) at 5pm CET, here.

42 comments:

  1. Because climate is predictable to a certain extent, that's why we use scenarios.

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  2. As a graduate student I did something similar with the Lorentz equations. I added not a constant term, but an oscillating F = f sin(ωt) driving term. I did though find similar behavior. The system would exhibit a measure of resonance that depended on f. The system would enter an orbit that was less chaotic, and for f very large the path was an elliptical orbit. I did not numerically compute the average spent in the two loops, but I computed the Hausdorff dimension and it approached 1 as f became large. I never published anything on this, and I was just playing around with this.

    I think it was Neil de Grasse Tyson in his Cosmos series who showed a man walking with a dog. The man would follow a fairly straight path while the dog would wander all around near that path. For weather watch the dog, but for climate watch the man.

    On a more social dynamics note, it will not matter at this point whether one uses reason or data to make any point on global warming to a near majority. The social condition with politics has reached a sort of critical mass with Qanon and insane conspiracies to the point that 1/3 of people in the US see the world entirely through different mental lenses. I can only hope this does not cross a threshold into a majority, for once that happens, we, the meaning of the word we means not just Americans but the entire world, are in a whole lot of big trouble. Because of this a substantial percentage of people think this is a hoax. Look at sepp.com and wattsupwiththat,com to look as some of this. The last one touts itself as the most widely clicked on website devoted to climate. No matter how much you can debunk this stuff some people refuse to listen.

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  3. “Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels.”
    I am not sure what you mean. If you delete “in 2079” or replace it with “in the future” I agree.
    But to predict the average rainfall in 2079 is in my view still weather forecast.
    In Germany we have dry years (600 mm/a) and wet years (1000 mm/a) and everything in between. I have some doubts that someone can predict that 2079 is a dry or wet or average year.

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    Replies
    1. Yes, I am sorry, of course this refers to the average rainfall within a range of uncertainty.

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  4. But is it possible to predict the climate if not all causes for climate change are known and if the known causes are not known to even Lorenz's 3 decimal accuracy?

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    1. The answer is evidently yes, because it has been done.

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    2. Hehe. Ok, it has been done, but the future has not arrived yet so we don't know whether the predictions are correct!

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    3. They started making predictions in the 1960s. That was 60 years ago. These predictions were remarkably accurate, see eg here. The claim that climate models have not made predictions is a typical climate change denier argument. It is bluntly wrong and demonstrates that you probably didn't do as much as Google the question.

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    4. In 1983, I was a student and had to give a presentation about a paper describing an early one-dimensional climate model using extrapolated CO2 concentrations. All it's predictions about rising temperatures have come true.

      But see this Science article for an evaluation:
      www.sciencemag.org/news/2019/12/even-50-year-old-climate-models-correctly-predicted-global-warming

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  5. There are some underappreciated and potentially serious consequences of climate change. First, wind energy will be greatly diminished in the northern hemisphere as the Polar Regions warm. Next, increased cloud cover will reduce the efficiency of solar power. The location of cloud cover increase may not match up with solar power siting plant locations.

    Global warming will weaken wind power, study predicts
    https://www.theguardian.com/environment/2017/dec/11/global-warming-will-weaken-wind-power-study-predicts

    Climate change itself may make the green power revolution ineffective as a solution to climate change.

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    1. I am not a panegyric for geo-engineering, but I suspect this will be forced on us. Methods to increase the atmospheric albedo, such as small particles in the upper atmosphere, will be a big part of this. We will also have to reduce carbonic acid in the oceans, which will involve using iron or other substances to increase phytoplankton activity, There may be other methods, even clouds of micron scale particles at the L1 point to Mie scatter light.

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  6. Le'usband: We have models, just as Dr. Hossenfelder was using. We can try, systematically, to see what the climate model does with a specific level of CO2 in the atmosphere, and then repeat that with a slightly varied level of CO2: Not wildly varied, but within the measurement error of CO2. Just as she was doing with her "+f" forcing constant. Doing that we can predict hundreds of models using values in a tiny "cloud" around a given CO2 level, and get the average result.

    And as she demonstrated, the average can be quite stable and predictable, even if the specific values cannot be predicted, other than to say they are going to be in a small cloud centered on the average result.

    If we then do the same for all the possible CO2 values, then we can predict, if CO2 continue to increase on its current trend, where the climate is going to be in 10 years, or 50 years, or 100 years, assuming the trend continues.

    Of course, nothing can go on forever; at some point, if the trend continues, the disruptions to food chains, arability, storm severity, droughts, water shortages, forest fires, desert growth, and other phenomenon will disrupt society, cause mass extinctions, and end the technology age.

    In other words, Global Warming may be a problem that sorts itself out, by killing us in sufficient numbers that we stop causing it. Of course we might also end up in a runaway heat trap like Venus, where no life-as-we-know-it can survive. But at least no one will be claiming it's all a hoax; we will have eliminated ignorance.

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    1. OK. But. The climate models have a large number of variables (not only CO2) few of which are exactly known (e.g. cloud cover) and not even all factors affecting climate are included. So how much confidence can you have in those models? And does the "attractor" idea apply to climate models given that they all lead to wildly different outcomes (the only common factor being that they all show warming)?

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    2. As you say correctly, they all show a common trend, which means the outcomes are not "wildly different." To quantify the confidence you can have in these predictions, the IPCC currently mostly relies on using the predictions from different models to generate a possible range. One can debate whether that's the best way to do it, but that's a debate which is better left to people who work in the field.

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    3. Le 'usband: We can have reasonable confidence; we aren't starting from scratch. We develop confidence by applying the new models we develop to past data that has known outcomes.

      The same way we developed weather prediction in the first place.

      The "science" is in developing the models, and seeing if, using data gathered in years past, how well they agree with the outcomes already known. That is one way to develop confidence levels.

      If they work predictively, we believe the models are accurate (within the limits we found empirically, also informed by the theoretical mathematical limits of the statistical distributions we are using, such as the extreme value distribution).

      To the extent that our models are NOT accurate on past data, we may investigate other variables we might use to improve the fit (subject to technical considerations about over-fitting the data).

      But to the extent our models ARE accurate, we presume additional variables must not have been significant enough to collectively sway the result any more than the error we observe.

      YES, the models can still have attractors; but as Dr. Hossenfelder notes, that doesn't mean the attractors are wildly different; particularly with regard to global temperature. There is no scenario where we triple the greenhouse gases in the atmosphere and stumble into some permanent springtime attractor; all the possible attractors will be unlivable.

      Finally, it is important to note that chaotic systems do not have to be permanently chaotic, as Dr. Hossenfelder's experiments with the forcing factor demonstrate, an externality can kick a chaotic system with attractors into a single stable state. So that is another possibility; we may kick our climate system hard enough, by dumping methane and CO2 into our atmosphere, to knock it into a stable unlivable state, like Venus.

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    4. The models since the 1970s have indeed been checked against global. And 15 out of 17 models predicted temperature changes right.

      www.scientificamerican.com/article/climate-models-got-it-right-on-global-warming/

      Climate Models Got It Right on Global Warming

      Even models in the 1970s accurately predicted the relationship between greenhouse gas emissions and temperature rise

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  7. Has the distribution of X been calculated rather than just the average? I could try to do this myself, but if it has already been done....

    I have an idea that the average of X could be analagous to the polarisation vector [= "climate"] for an electron while that distribution (say, cos or sin curve) could be equivalent to the distribution of individual measured spin vectors [= "weather"]. Cosine or sine distribution here would, according to my calculations, conform to overall results of measurements complying with Malus's Law.

    Austin Fearnley

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  8. You raise an interesting question "does the
    state of Earth's atmosphere
    display chaotic behaviour on climate time scales?"
    But how can an analysis of an equation
    that, according to you, "does not actually describe
    the real climate", provide any explanation for
    the claim that climate does not display chaotic
    behaviour?!

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    Replies
    1. We know that climate trends are predictable, so at least in the range that predictions have been made they are not chaotic. I merely provided an example for how this predictability is compatible with weather being chaotic.

      Delete
    2. On a time scale of years it still
      seems chaotic, why is not also on
      time scale of decades? You promised
      an explanantion at the beginning
      of your piece...

      Delete
    3. The question I asked is "how can they make predictions", and that's what I explained. Maybe read it again.

      Delete
  9. As a comment about the group chat, the discussion got mired in whether there is even global warming. One person in particular became very testy that I did not have data at hand showing CO_2 increase was human caused. This was not to be the focus of it.

    Global warming tends to get people's dander up. Websites devoted to astronomy have largely dropped AGW, because anti-AGW types come out of the woodworks and pounce. The Universetoday site has not had an AGW topic in a long time, because the denier crowd would bomb it with posts.

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    Replies
    1. Sorry, I couldn't make it yesterday. In any case, I am actually planning a video about exactly this question (though I'll probably not get there this year).

      Delete
    2. The skeptical science website answers the question of the origin of the CO2:
      www.skepticalscience.com/co2-increase-is-natural-not-human-caused.htm

      It will take some work to assemble the primary literature as this matter was settled decades ago. Also, human activities have altered the greenhouse gases during the whole Holocene by deforestation and land use changes.

      Delete
  10. "What this means is that in a chaotic system, the trends of average values can be predictable, even though you cannot predict the exact state of the system beyond a short period of time. "

    The question that remains is if this is also true for dynamical systems other than the example you discussed here. In particular for the actual climate of Earth.

    "And this is exactly what is happening in climate models. Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels. "

    Why is that true? What are the references?

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    Replies
    1. Search the IPCC report for precipitation.

      Delete
    2. I meant it more as a theoretical result. Are there any theorems that say that for some systems of PDE this is true. And specifically for the ones used in climate science.

      IPCC stands for intergovernmental panel on climate change. I am not interested in climate change, just in climate predictability.

      Delete
    3. If there were any theorems they wouldn't help you because, needless to say, they can't actually integrate the equations exactly. In any case, I don't understand your problem because if they can make predictions, predictions that fit on the data, then that's better proof that they can make predictions than any theorem.

      Delete
    4. I don't know what the predictions are based on, that's why I ask the question. Is it based on analysis of the actual equations, or is it based on a model of best fitting? It would be a lot more interesting if it was analysis of PDE. And you don't need to integrate the equations in order the prove theorems like that. Take for example the stability of Minkowski space-time.

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    5. "Is it based on analysis of the actual equations, or is it based on a model of best fitting?"

      Climate models are based on general circulation (air) models (equations) that include surface albedo and coupling with the oceans. They simulate the effects of changes in atmospheric composition (greenhouse gases).

      Difficult parts where information is lacking are ocean mixing (removes heat and CO2) and cloud formation (changes the albedo).

      Predictions in time are difficult as the release of greenhouse gasses depends on the economy, technology, and politics.

      Delete
  11. Of course anything including climate is predictable. And of course the average is less chaotic than the variable. And of course one can manipulate the outcome by adding constants to the model. And of course some of the predictions from the 60s are more accurate than others (likely discarded). And of course if a model is curve-fitted to the past, as they always are and should be, it will continue to be accurate as long as the past trend continues.

    Here is what can still go wrong:
    1. Trend may change in the future, thus diverging from the model.
    2. Accurate models may be misinterpreted. IOW, the best way to go about changing the desired outcome may be different than thought.
    3. What exactly are we trying to optimize? Total human population? Quality of life? Number of plants/animals?
    4. How bad is the problem? So what if [say] Norway's climate turns into that of Sicily? If so, will Brazil be less livable than Greenland today? So what?
    5. Who and how much gets affected? Living x%, children y%, grandchildren z%? Concept of climate "emergency" is not productive.
    6. Should we trust older generations with solving our problems? Will grandchildren have better solutions than us?
    7. What are the known unknowns and more importantly the unknown unknowns? Let's look at history: Malthus, peak coal, peak oil, peak humans, etc.

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    1. "If so, will Brazil be less livable than Greenland today? So what?"

      If it is that bad, several billion people will have to emigrate. And there is no guarantee that agricultural production will stay at its current level

      But the fact that you do not seem to care about even the population of Brazil (200M) tells us enough, I think.

      Delete
    2. @ Nonlin.org

      Thank you.
      One of the things I most despise about the climate debate, from either side,

      Is the resistance to,
      The defensiveness against,
      And the aggression towards,
      - anyone asking questions.
      ( Remember Socrates and Plato ? )

      Best wishes



      Delete
    3. "Is the resistance to, The defensiveness against,"

      Currently, there is now some 50 years of scrutinized research on Climate Change. However, the questions posed are still about the very earliest of problems. Even the most general searches on your favorite search engine will give you all the answers you will ever need. Which begs the question: Why are these people coming here for their questions? And are they even interested in the answers?


      In many respects, most questions "against" climate change sound like "How do you know the earth is round? Where is the evidence?" or "Is COVID-19 just like a cold or the flu?".

      And you are surprised scientists lose their patience after 50 years?

      Delete
    4. "Currently, there is now some 50 years of scrutinized research on Climate Change."

      You might want to reflect on the events of 2008, when financial risk models, devised, in many cases by economic refugees from the physics community, proved catastrophically wrong. Why? The models were built on the prior 50 years of data and did not incorporate the most recent market disruption which occurred in 1929.

      The point being that 50 years of data was insufficient in the context of modern financial markets which are at best a few hundred years old. In the context of the earth's climate 50 years isn't even a blip.

      Certainty is a poison pill to science; doubt is the only trustworthy coin of the realm in scientific research.

      Delete
    5. Financial risk models were not "built on data". What the heck are you even talking about? Financial risk models are not models of the financial system, they are exactly what the name says, they're models to evaluate a supposed financial risk to help rich people become even richer.

      Delete
    6. bud rap: 50 years of scrutinized research on Climate Change, it has not been limited to the last 50 years of data. The research includes literally hundreds of thousands of years of data, using ice cores from both poles, soil samples and tree rings going back thousands of years, etc. We can measure or infer temperature, CO2 levels, oxygen levels and much more. The longest continuous historic record of temperature goes back two million years; but plant fossils and other ancient data gives us spot values going back to the dinosaurs.

      You are right, 50 years by itself could be a blip, but 50 years in the context of a few million years is not a blip; that is enough to identify a rare-beyond-reason anomaly.

      Delete
  12. In March 2019, the German magazine SPIEGEL published an interview with the climate scientist Bjorn Stevens (Max-Planck-Institut für Meteorologie, Hamburg, Professor at University Hamburg and IPCC author)
    (https://www.spiegel.de/wissenschaft/warum-die-vorhersagen-zur-erderwaermung-so-schwierig-sind-a-00000000-0002-0001-0000-000163037012).
    He works in climate modelling for 20 years and claims that there is no real progress in the last 40 years (sounds like HEP). The uncertainty in the predictions was not improved. It is still 3K +-1.5K for double CO2 content. He mentioned that he cannot even predict whether glaciers in the Alpes will disappear or grow again. The problem is the modelling of the clouds.
    That is not what you read and hear every day. Pretty confusing.

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  13. The Climatereanalyzer site, run by the University of Maine, provides a preview of what’s in store for us as the season progresses. What’s most revealing is the “Snow depth-MSLP” (Mean Sea Level Pressure, I think). I’ve been tracking it almost daily since mid-summer. Having gone a few days without checking it, I was startled by the enormous increase in snow cover both in Canada and the US. This could very well presage a brutal winter here. Come Friday the forecast is for a mixed rain and snow event with up to 4 inches of snow in southern New Hampshire. I’m hoping it doesn’t materialize as I need another 190 miles to reach this year's 2650 mile cycling season goal.

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    Replies
    1. You talk about weather not climate.
      Weather like this is not unusual for a continent where all the mountains run from north to south and not west to east like in Europe.

      Delete
    2. True, but the annual expansion/contraction of the snow cover in North America and Eurasia must surely be factored into long term climate models. That winter mantle of snow would increase the average albedo of our planet, functioning, I imagine, like a feedback mechanism in helping to cool our planet. Another feedback mechanism serving as a planetary air conditioner is the formation of giant storms like cyclones and hurricanes. These quite literally operate as giant heat engines transferring heat from the ocean surface to the upper atmosphere where it radiates out into space in the infrared band. In the process they perform work on the environment, imparting rotation to physical mass, just like any manmade heat engine – internal combustion, steam engine, etc.

      I had firsthand experience of such ‘work’ performed by one of these atmospheric heat engines in 1991 when hurricane Bob plastered the south coast of New England. It’s a day I will never forget. I was living in an old estate in Falmouth, Massachusetts that was converted into multiple apartment dwellings. As the winds increased in intensity suddenly the top half of a huge pine tree toppled over just yards from my building. That was enough for me. I called my neighbors in the rear most building who had a basement. With winds topping out at 90 MPH I raced to their building as trees slammed down in front and behind me, leaping over and crawling under others to get to the safety of their basement. We all huddled fearfully in the dark basement for some time until the storm passed. When we emerged the transformation of our little complex was startling. What was once a thickly wooded old estate was now a shoulder level solid thicket of downed trees and brush covering the parking lots and grassy areas. Took us a week to clean up the mess and power was out for 11 days. But luckily only superficial damage was done to the buildings.

      On the bright side, that hurricane cooled our planet by a little bit. But this year’s unusually large number of Atlantic hurricanes and tropical storms is likely our planet’s response to the ever rising accumulation of greenhouse gases with their heat trapping effect. That, of course, is a rather ominous trend for coastal regions around the world that are most impacted by tropical storms.

      Delete
  14. Talk about chaotic weather, what we are experiencing in the US for the last 6 or 7 days is truly remarkable. Temperature records dating back 100 years in 46 cities from the Midwest to the East coast have been either tied or outright shattered. Meanwhile a huge surge of arctic air has invaded the western part of the United States, bringing far below normal temperatures. This has been a godsend for us on the eastern half of the nation for outdoor activities and especially leaf cleanup in yards. I ended up with a huge yard, unintentionally, that I’ve been raking for 3 days now, about 2 to 3 hours each day before muscle exhaustion sets in. Today is the last day of this incredible record warmth between the high 60’s and low to mid 70’s Fahrenheit (20-23 C.). Hopefully, I’ll get the last leaves carted away today before the howling winds accompanied by whipping rain, snow squalls, and plunging temperatures arrives over the next few days.

    ReplyDelete

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