 So, good morning. So, we will continue with our study of climate sensitivity and climate dynamics correct. What is that? So, what is the problem we considered in the last class? Problem number? 50 50 was it is solar flux reaching the earth f s equal to sigma t e to the power of 4 where t is equilibrium temperature considering the earth to be a black body that turns out to be 255 Kelvin. Therefore, what do you get? 1 by t e d t e by t e this is correct, but we know that this is also equal to, so if you look at this, this is a very fine distinction, t e is the equilibrium temperature of the earth, t s is the global mean surface air temperature okay, but the rate of change in the global surface air temperature with respect to f that the partial d of t s with respect to f will be the same as d t d t by d f s in the case of I mean there is no feedback from other things. So, basically the temperature as such is a consequence of the solar radiation coming in and the emission that is a basic that is a basic assumption from then like a pizza you add on toppings are there those toppings are water vapor feedback carbon dioxide feedback base is something is coming otherwise earth will not get heated at all something is coming and something is emitted back that equilibrium causes some basic temperature. Now, we are trying to see what is the sensitivity with respect to change in some of this forcing if that f s itself changes what happens that is are you getting the point instead of the 239 watts per meter square it becomes 240 watts per meter square what will happen okay. So, everything is benchmarked with respect to the change in that f s, so that is that is how we define the radiative forcing okay. So, how much was this 255 so the sensitivity has got the units of Kelvin divided by watts per meter square it looks like the units of heat transfer coefficient watts per meter square per Kelvin correct mechanical engineers will it will ring a bell these are units of H okay do not get confused it is not it is far away from the H it is nowhere related okay it is it had some point I made some CRC complex kind of definition in the last class. So, the earth equivalent blackbody radiation sorry the earth equivalent blackbody temperature rises by 1 Kelvin for every 3.76 watts per meter square per Kelvin of downward radiative forcing incident of the earth from the top of the atmosphere okay now all these climate all climate change experts are trying to calculate what is the watts per meter square okay or I will just we can okay so we can you can leave it like this also one Kelvin requires 3.76 watts per meter square. So, all these climate change climate change and climate science experts are trying to figure out if the Albedo reflectivity changes how many watts per meter square if carbon dioxide changes how many watts per meter square what are vapor changes how many watts per meter square. So, you add all this and these forcing are generally are additive you calculate the forcing separately for calculating the carbon dioxide forcing you have to solve the radiative transfer equation what is the are you getting the point if the carbon dioxide changes how will the forcing change that is not so easy to obtain. So, what you do is you will you will solve the equation of radiative transfer in the atmosphere with 390 ppm and get one value then with 395 ppm or 400 ppm you will get a value and then find out how this changes with respect to. So, I will tell you later on that there is a relationship called ln of C by C0 into something that is how it changes with respect to carbon dioxide. So, you will have to do or you have to take controlled air you have to do I score measurements but you go deep into Vladivostok you go to Vladivostok where somewhere north I have right you go to Vladivostok and then drill a hole kilometers into the snow then get the I score in that I score air will be trapped then there is a dating formula will find out for that depth what will be the time period that time period that air will contain carbon dioxide find out what is the carbon dioxide concentration in the atmosphere at that point in time then the the funda that carbon dioxide concentration is universal then you apply that whatever is in Vladivostok or whatever is the same in Velachery then you reconstruct the carbon dioxide concentration though you do not have direct measurements before 1945 when some keeling set up that biggest laboratory to measure carbon dioxide in Hawaii is it okay that is how they these are all hardcore the whole scientifically established it is not like alchemy or some somebody's astrology or something it is not like that. So, this carbon dioxide concentration people are because normally when you say how are you confident how are you confident people who people who do not understand they will say how are you confident how did people measure carbon dioxide 100 years back where was the instrument 200 years back where is instrument all that air is trapped now now I have instrument then archaeology or paleo climatology can never be studied people have man has found out ways of studying the past alright so this is a very important this thing so something like this is called radiative forcing so two things for one Kelvin it is 3.76 watts per meter square and we are talking about radiative forcing not forcing by wind forcing by convection radiation is very important for climate convection winds and all these pressure distribution and all this orography topography all these things are land breeze sea breeze are important for weather what will happen in the evening what will happen in the afternoon all that is weather but compared to 20th century what happens to 21st century 22nd century is a bigger picture is a bigger problem so here radiation has to be taken into account okay now this is fine let us get back though we solve the problem in the last class I think now you understood it better right okay where is are you done I think you went all over the place you are not to be seen Bharat Darshan you figure out you ask him you get you do Delhi and Bombay Darshan if you are on a flight because normally the very difficult to get the ATC clearance you will keep on circling Bombay right free time is lost but alright okay I promise to you I will teach at least two hours of atmospheric dynamics but I forgot to realize that there is a very important lecture on climate change which I will have to which is a fast paced power point presentation which I will have to deliver tomorrow so we are just left with only one lecture for the atmospheric dynamics I will quickly give the governing equations and some simplification and tell you two approximation the cyclone approximation and the geostrophic approximation will solve a simple problem and then close the course that is a plan on Thursday tomorrow's plan is to bombard you with a PPD okay I will convert it into PDF and send it to you by tomorrow I will also send you the PDF which will which gives the schedule schedule whatever from 9 o'clock to 1 o'clock 1 o'clock Syrahul will come I think 1245 or this cloud seating two people are coming two groups so 1015 and 1030 so I will everybody has to be present okay you just cannot afford to come only for your presentation unless you have some this exam that exam and so you and you are at your creative best so if it's a genuine reason it's okay why are you late to class brilliant expression of creativity why are you late to class one two three four five seconds just come like a missiles okay for also just thrown off God how creative students can get alright now this we are looking at so DTS by this is the lambda right yeah what was the equation number for this one was it one we wrote it in one of the earlier classes two okay okay now see why is a TS is the global mean surface air temperature okay F is the forcing okay flux so this is this DTS by DF is how much the temperature will change if for a given change in the forcing that is a lambda that is a climate sensitivity right so we are qualifying the whole of climate with just one quantity what is it TS and climate sensitivity by one more quantity which is lambda okay climate means global mean surface air temperature sensitivity means with respect to forcing this has to be understood so this is a partial D is partial to a TS to partial F is basically when there are no feedbacks plus you add the feedback but the feedbacks were including auxiliary variables these auxiliary variables are carbon dioxide water vapor clouds all those things okay now this D y I by DF can be taken to be this is okay no this is some playing with calculus partial differentiation okay therefore plus sigma it is too much okay all right this is called the feedback factor it is qualified by subscript I where I can be 1 2 3 4 5 1 could be water vapor 2 could be snow cover 3 could be cloud cover and so on okay now F equal to this is that okay so we are able to relate we are able to calculate all the feedback factors get the sum of the feedback factors and call that as F and find out 1 minus F and then do the simple do TS by do F divide by 1 minus F you can find out the climate sensitivity of course this equation will cup with F equal to 1 that is all right that is okay but now some equation number for this so this is lambda naught this is lambda correct lambda is the sensitivity of the climate with feedback lambda naught is the sensitivity of the climate without feedback therefore lambda by lambda naught should be what conceptually it is a gain it is G right what is the climate sensitivity with feedback divided by what is a climate sensitivity without feedback so now from equation 10 it is clear that G is G is 1 by 1 minus F now provided F is less than 1 so F is greater than equal to 1 infinite sensitivity unstable so the challenge is to be able to calculate this F so you can use your zero order model first order model second order model and whatever you can use your full CFDT transfer model and all this if you do all this calculate this feedbacks and give projections and you are a climate science or a climate change expert okay all right so so much about climate sensitivity okay but we have not studied one thing so far whether it is the atmosphere whether it is atmosphere or the ocean it has got a finite inertia whether it is atmosphere or the ocean it has a finite inertia or you just change the forcing by another 2 watts per meter square the next second will the temperature change so there is something called the transient response and the equilibrium response even if we change that instantaneously solar load is changed instantaneously next second the temperature will not become 255 because there is a mcp of the of the atmosphere there is a are you getting the point there is a mcp of the ocean correct there is a thermal inertia so we will have to look at this response so then we will have to solve we have to solve a initial value problem now so we have to so the most simplified approach will be a considering it to be a first order system okay let us consider a simplified first order system and then proceed okay so this will be transient versus equilibrium response what do you mean by this F much much greater than 1 even small changes in the forcing can cause large changes in the temperature as a highly unstable equilibrium okay like I do not think we are in that situation now as of now yes as of now we are not in that situation okay it will not happen tomorrow or Wednesday or before the end sem I mean so okay so transient heat capacity of earth is very large hence PS shows a delayed response to change in F correct is the first point you have to understand so this is called the adjustment time for the atmosphere it is a few months for the ocean mix layer take a guess 73% 71% of the it is a few years it takes a few years it does not mean that we can be irresponsible what I am what we are saying is it would not happen overnight continental ice sheet it will take several centuries okay now we are going to consider an example we will consider ocean mix layer okay so I am not going to consider layer wise and then make it a big complicated problem there is a M there is a CP there is a temperature your baseline temperature of T0 and the temperature increase or fluctuation is the T dash the change now we are going to write a equation for this so okay this is okay okay so T dash the transient response to forcing which is the forcing which is the source term correct okay so if you are at a loss to understand what is going on okay so you if you're not able to conceptualize what is going on you take a bucket of water you're heating it in the hostel for example you want to take hot water bath there is a deeper heater okay it gives you a heating input of Q okay so this has got a mass of M and CP this mass can be heater plus wall plus water and all that let everything will be at one temperature now it is also losing heat by convection mass plus radiation also let's neglect that so the equilibrium response of this will be MCP DT by D tau equal to Q-HA into T-T as it starts getting heated this temperature increase therefore it will lose heat by convection but heat is continuously input so the curve it will go like this when it reaches a steady state this term will vanish that will be the equilibrium temperature of the bucket if it is more than 100 degrees you are finished when will it become more than 100 degrees you sit down without the water okay otherwise you can do your math and it will come to 80 or 85 degrees if you're using a 1 kilowatt I give this problem in one of the courses right which one was that you forgot heat transfer okay one of these courses now we are able to instead of the instead of the heater we have some change in carbon dioxide or something instead of this water in the bucket we have water in the ocean okay alright so you want to be able to solve this lambda is whatever your gain okay so we can bring the equation we can lambda is the climate sensitivity okay yeah please solve this what what what is that why is it negative sign the same equation minus T by lambda yeah because the forcing term will be positive this will be negative no you solve it we will see now please solve this equation let's do I'll answer your question that's okay now let tau be let tau be mcp into lambda therefore q dash by what will be right hand side q dash by tau into lambda by tau no problem so please solve equation 3 it consists of two parts the complementary function and the particular integral so then I land the solution will answer your question so you can also have it as q dash into lambda by tau keep it that way also particular integral is what is the particular integral for this q dash into lambda go figure out what is this particular integral that particular integral if you take the derivative the first time will vanish and we substitute then left hand side will be equal to right hand side then it is a particular integral all right is it correct Sneha I'm believing what you're saying okay therefore the can we proceed okay so T dash how do you get a now a is the constant at time T equal to 0 what is T dash at time T equal to 0 T dash is 0 because it's equal to the initial temperature T naught what is T dash that is why that minus T dash by this thing is coming now you got it if there is no forcing left to itself this will decay understood okay fine so at T equal to 0 0 a plus therefore q dash lambda okay so there will be a rapid response initially after some time depending on the time constant there will be a slow response when T tends to infinity we thought that if you apply a forcing immediately it will reach q dash lambda that q lambda q dash lambda will take a long time that T equal to infinity is few months for atmosphere few years for the ocean few centuries for the continental ice okay it is a characteristic time are getting the point so any first order system will behave like this so if you put the thermometer in the mouth doctor will not get the temperature immediately for him he has to wait his time constant is 2 minutes if you put a thermocouple you will wait for no change in reading for 10 minutes you will say it should not change by more than 0.1 degrees 10 minutes you will wait some experiment will wait for hours so there is a time constant for everything doctor will not get the exact temperature at the end of 2 minutes you'll get a reasonable temperature anyway it is Dolos 650 111 isn't it whether it is 101 or 102 okay all right so this nice right so you got how would it response also we have seen we have we saw the sensitivity and all this now what about the feedbacks I will give you some stats climate feedbacks and then we'll solve a problem and close what a vapor f equal to find point 5 people have calculated if f equal to 0.5 g equal to 2 very good so that you understood okay next cloud clouds contribute to greenhouse effect if the cloud cover is more than whatever radiation is going watch watch what I am saying very carefully if the cloud cover is more whatever radiation is coming back from the earth this cloud will intercept it and it will not allow it to go out okay so the cloud will continuously build up but correct that's what you're thinking but there is a the cloud will also reflect the sense radiation from the top if there is no cloud everything will come isn't it so the net effect is 0 clouds are not causing climate change otherwise big anti-cloud movement would have taken place. So is that okay the cloud also created by nature now why unnecessarily we had to trouble so net cloud forcing is I am not going to write it down net cloud forcing is there is some cloud computing also right so g is 0 okay so clouds are harmless ice albedo feedback it is very small it is not a cause for concern there's something called snowball earth where the earth when it's a tipping point is ice albedo feedback and make the whole earth freeze but we are not at that tipping point now but somebody who is taking the snowball earth if you reach that tipping point they will present that it will everything will work in such a way that the feedback will it's an infinite do loop it will repeat and this thing and then the earth will keep on it will like crazy it will build up ice and snow and all that but now we are not there so we are fine okay so no worry so the national logo of Australia you know is you know T-shirt what they were no worries Australia no worries Australia right so no worries ice albedo feedback the carbon dioxide feedback is dangerous it positively contributes to GHG greenhouse gases and people have figured out if CO is your initial concentration and C is your act is a new concentration 5.35 that is arrived by radiative transfer calculation if you do that then you have to add that F and then calculate the lambda and then you will calculate the G and so on okay now problem number 51 estimate the apparent climate sensitivity estimate the apparent climate sensitivity the apparent climate sensitivity is given by what do you mean by this T s at the temperature T s at one point in time minus another point in time divided by F at one point this is like the Desi way of otherwise you have to get get the curve and then calculate d y by dx and then so it is an apparent climate okay estimate the apparent climate sensitivity based on differences in T s and F based on differences between TS and F between the current climate based on differences in TS and F between the current climate and the climate 20,000 years ago and the climate 20,000 years ago called LGM the 20,000 last glacial maximum okay with the following data TS current minus TS LGM is 5 degree centigrade CO2 LGM 180 ppm CO2 LGM 180 ppm CO2 now 360 ppm okay change in albedo 0.01 that's all only 2 FX so can you proceed you can calculate this del F okay so numerator is how much 5 so the challenge is to calculate the denominator okay there are only 2 feedbacks carbon dioxide only calculate the carbon dioxide feedback 5.35 ln 2 how much is it okay 3 point correct 342 isn't it I already get this 342 okay 239 what is that FS run 239.4 what is the albedo of the earth 0.7 so incoming is 342 but what is contributing to sigma T to the power of 4 is 342 into 0.7 we have done this again and again in the problem that results in 239 okay so out of this 342 if again changes by 0.01 that is 3.42 so please add this so you're all becoming climate change experts now given the data you know how to calculate but this is coming from some model and all that you can improve your models you can consider interaction FX okay and then you can look at all these changes of 30 years 40 years 50 years you can put your current consumption of oils and this thing and CO2 emission all that then it will become a stud model now we are using a very simple model okay 5 by 7 point okay this is 0.7 what that's all what do you infer from this G is we solve the problem number 15 know it was 0.266 without feedback right that is 1 by 0.266 gives 3.76 for 1 Kelvin so this is the gain or the amplification of this amplification of the sensitivity because of feedback how much is this how much is it 2 point okay in simple English due to the 2 feedbacks the apparent sensitivity increased 2.6 times what was an original value of 0.266 which is coming because of solar radiation now 2 things have changed the reflectivity or the albedo has changed carbon dioxide has changed so if they are going at this level then we are in trouble but this is 20,000 years the time we are talking about is 20,000 years 5 degree centigrade but now next 5 degree centigrade many things will melt we lose surfaces we lose continents we lose countries and so on but it may happen much more before 20,000 years because if it has taken 20,000 years for carbon dioxide double from 180 to 360 now it is doubling at a mad rate 360 now it is 390 already in calculations you are doing 390 400 so I tomorrow I will show you a plot where carbon dioxide last 100 years just goes like this so all this so this 5 degrees is not a great thing but the time is coming down okay that is the problem I will stop here tomorrow we will run through a presentation I will give you a power point on climate change and then we will wind up the course on Thursday with just a very brief one lecture course on dynamics