WEBVTT 00:01.500 --> 00:02.570 RONALD SMITH: Well, today we're going to talk 00:02.567 --> 00:05.997 about mixing and dilution. 00:06.000 --> 00:09.270 When you add something to the atmosphere, gases or 00:09.267 --> 00:12.567 particles, what happens to it? 00:12.567 --> 00:13.767 How does it mix in? 00:13.767 --> 00:15.967 To what degree does it get diluted? 00:15.967 --> 00:20.797 This is a subject that has applications to human-induced 00:20.800 --> 00:23.730 air pollution, but also to a large number of natural 00:23.733 --> 00:26.673 processes in the atmosphere. 00:26.667 --> 00:32.567 So it's really a set of quite fundamental ideas. 00:32.567 --> 00:36.127 And I could be thinking about either small particles or 00:36.133 --> 00:38.473 gases added to the atmosphere. 00:38.467 --> 00:42.467 Basically, something that's going to move with the air and 00:42.467 --> 00:44.027 mix in, in a way. 00:44.033 --> 00:46.833 If the particles are large enough, of course, they might 00:46.833 --> 00:50.173 gravitationally fall out after awhile. 00:50.167 --> 00:53.497 And that effect would not be included in the calculations 00:53.500 --> 00:54.030 I'm doing here. 00:54.033 --> 00:58.103 So I'm assuming the particles are small or it's a gas so 00:58.100 --> 01:02.500 that it will just move along with the air itself. 01:02.500 --> 01:05.570 So the easiest case-- 01:05.567 --> 01:10.427 maybe it's a little bit, unnatural, but not so much 01:10.433 --> 01:15.673 --is that you're in a confined valley and you know the 01:15.667 --> 01:16.997 dimensions of the valley-- 01:17.000 --> 01:19.700 horizontal dimensions of the valley, and the valley is 01:19.700 --> 01:21.770 capped, in some way. 01:21.767 --> 01:25.727 I've written here that it's capped by an inversion. 01:25.733 --> 01:30.703 Now, I'll give a more careful and physical interpretation of 01:30.700 --> 01:35.670 inversion later on, but an inversion is defined as a 01:35.667 --> 01:38.427 layer of air in which the temperature 01:38.433 --> 01:42.133 increases with height. 01:42.133 --> 01:44.403 And for reasons that are not at all clear, from what I've 01:44.400 --> 01:48.970 just said, but will become clear, it's very difficult for 01:48.967 --> 01:52.697 mixing to occur across an inversion. 01:52.700 --> 01:56.100 An inversion tends to act almost like a rigid lid. 01:56.100 --> 02:03.030 It holds air and pollutants below it or in any way, 02:03.033 --> 02:04.633 prevents it from crossing. 02:04.633 --> 02:08.333 In this case, it traps these 02:08.333 --> 02:09.703 pollutants below the inversion. 02:09.700 --> 02:14.800 So leaving that as only partially defined concept for 02:14.800 --> 02:20.030 a moment, we'll know then the volume of the air into which 02:20.033 --> 02:24.503 the pollution from my little power plant here has reached. 02:24.500 --> 02:28.600 The volume into which the substance has mixed is then 02:28.600 --> 02:32.600 the product of the two horizontal dimensions, L1 and 02:32.600 --> 02:36.570 L2, and the depth of the layer, that's the depth up to 02:36.567 --> 02:38.367 the inversion. 02:38.367 --> 02:42.027 The mass of the air into which the pollutant has mixed is 02:42.033 --> 02:46.633 just the product of the air density and that volume. 02:46.633 --> 02:50.903 And that's going to have units of kilograms. So here comes a 02:50.900 --> 02:52.230 very simple example. 02:52.233 --> 03:00.073 Let's that over some period of time we put in 40 metric tons 03:00.067 --> 03:02.167 of a pollutant. 03:02.167 --> 03:07.297 Maybe it's small particles, for example, or a gas. 03:07.300 --> 03:11.070 A metric ton, you recall, is a thousand kilograms. So I've 03:11.067 --> 03:16.297 written that in SI units as 40 times 10 to the 3 kilograms. 03:16.300 --> 03:18.300 That's the amount of pollutant we've put in. 03:18.300 --> 03:21.270 That's quite a bit, but it depends on the problem we're 03:21.267 --> 03:22.327 working on. 03:22.333 --> 03:25.433 Let's say that the horizontal dimensions of the valley are 03:25.433 --> 03:29.633 10 kilometers in one dimension and 10 kilometers in the other 03:29.633 --> 03:32.233 dimension and that the depth to the 03:32.233 --> 03:35.073 inversion is one kilometer. 03:35.067 --> 03:43.327 So then the concentration of the pollutant is going to be 03:43.333 --> 03:48.173 the mass of the pollutant divided by the mass of the air 03:48.167 --> 03:50.467 into which it has mixed. 03:50.467 --> 03:53.097 I didn't put that definition up here, but be sure you get 03:53.100 --> 03:54.030 that in your notes. 03:54.033 --> 03:59.733 We're defining concentration in this case as a mass ratio. 03:59.733 --> 04:02.833 The mass of pollutant that you've added to the atmosphere 04:02.833 --> 04:06.703 divided by the mass of the air into which it has mixed. 04:09.867 --> 04:12.967 So here's the formula then. 04:12.967 --> 04:16.327 There's the mass of pollutant that we added, 40 times 10 to 04:16.333 --> 04:20.003 the three kilograms, and here's the mass of the air 04:20.000 --> 04:24.600 into which it mixed, it's the air density multiplied times 04:24.600 --> 04:26.270 the volume of the air-- 04:26.267 --> 04:30.567 L1 L2 times D. So it's 10 kilometers expressed in 04:30.567 --> 04:34.427 meters, 10 kilometers again, and one kilometer 04:34.433 --> 04:35.803 expressed in meters. 04:35.800 --> 04:40.000 You work all that out, you get 0.4 times 10 to the 6 04:40.000 --> 04:41.700 kilograms per kilogram. 04:41.700 --> 04:44.200 You could cancel it out and say it has no units. 04:44.200 --> 04:45.170 That's fine. 04:45.167 --> 04:48.067 Or you could leave it as kilograms per kilogram, just 04:48.067 --> 04:49.967 to remind you what it is. 04:49.967 --> 04:53.867 It's a kilogram of pollutant mixed into a certain number of 04:53.867 --> 04:59.167 kilograms of the air into which you mixed it. 04:59.167 --> 05:04.897 So since this is 10 to the minus 6 and it's a mass ratio, 05:04.900 --> 05:09.870 I could also write this as 0.4 ppmm, parts 05:09.867 --> 05:12.497 per million by mass. 05:12.500 --> 05:15.800 The per million, the pm there, refers to the 10 05:15.800 --> 05:17.500 to the minus 6. 05:17.500 --> 05:20.170 And the last m there refers to the fact 05:20.167 --> 05:23.527 that it's a mass ratio. 05:23.533 --> 05:23.803 Question? 05:23.800 --> 05:24.100 Yes. 05:24.100 --> 05:26.370 STUDENT: Why is it kilograms of pollutant? 05:26.367 --> 05:29.197 PROFESSOR: Well, because I've got the kilograms 05:29.200 --> 05:32.430 here, but it's kilograms of pollutant divided by the 05:32.433 --> 05:35.333 kilograms of air into which it's been mixed. 05:35.333 --> 05:36.303 STUDENT: So they don't cancel out? 05:36.300 --> 05:37.230 PROFESSOR: They could. 05:37.233 --> 05:38.033 That's what I mentioned. 05:38.033 --> 05:39.103 You could cancel that out. 05:39.100 --> 05:41.070 I wouldn't mind it at all if you did. 05:41.067 --> 05:44.727 But I sometimes like to keep it there just to remind myself 05:44.733 --> 05:46.433 of what that definition is. 05:46.433 --> 05:48.133 Either way would be fine. 05:48.133 --> 05:50.303 It'd be perfectly fine to cancel that out. 05:50.300 --> 05:52.900 Normally, concentration is considered to be a quantity 05:52.900 --> 05:54.930 that has no units. 05:54.933 --> 05:57.573 But sometimes I like to keep it there just as a reminder. 05:57.567 --> 05:58.067 Yeah? 05:58.067 --> 05:58.897 STUDENT: Why are the tens raised to the fourth power? 05:58.900 --> 05:59.030 Oh, it's meters. 05:59.033 --> 06:00.333 OK. 06:03.600 --> 06:05.000 PROFESSOR: Where are you looking? 06:05.000 --> 06:06.970 STUDENT: The concentration calculation. 06:06.967 --> 06:08.267 PROFESSOR: Yeah. 06:08.267 --> 06:09.067 Well, that's right. 06:09.067 --> 06:11.467 So this was 10 kilometers. 06:11.467 --> 06:13.827 And a kilometer is a 1,000 meters. 06:13.833 --> 06:19.603 So the 10 times the 3 gives 4, 10 to the fourth meters. 06:19.600 --> 06:21.030 Yeah. 06:21.033 --> 06:22.773 STUDENT: What is L1 and L2 again? 06:22.767 --> 06:23.827 PROFESSOR: The horizontal 06:23.833 --> 06:25.273 dimensions of the valley. 06:25.267 --> 06:27.227 STUDENT: Oh, so the area. 06:27.233 --> 06:30.533 PROFESSOR: Well the product is the area of the 06:30.533 --> 06:35.173 valley, but the L1 is the dimension in one direction, L2 06:35.167 --> 06:36.697 is the dimension in the other direction. 06:40.600 --> 06:43.100 If you want to know the base area of the valley, 06:43.100 --> 06:44.330 that's L1 times L2. 06:47.567 --> 06:49.827 Other questions? 06:49.833 --> 06:50.303 Yes? 06:50.300 --> 06:52.600 STUDENT: Where did you get 10 to the third? 06:52.600 --> 06:54.030 PROFESSOR: 10 to the third? 06:54.033 --> 06:55.273 STUDENT: It's the mass of the air below the inversion? 06:59.767 --> 07:01.667 PROFESSOR: Well the mass of the air below the 07:01.667 --> 07:03.427 inversion is what's in the denominator here. 07:03.433 --> 07:07.773 It's the density of the air times the volume of the air 07:07.767 --> 07:10.067 and the volume was given by that product, which you see 07:10.067 --> 07:11.967 written out here. 07:11.967 --> 07:14.397 Yeah. 07:14.400 --> 07:16.230 It's good to get this one really clear, because the next 07:16.233 --> 07:17.203 one is even tougher. 07:17.200 --> 07:21.830 So any other questions on this one? 07:21.833 --> 07:22.233 OK. 07:22.233 --> 07:26.703 So now we move to case two, which is unconfined mixing. 07:26.700 --> 07:31.330 We don't have this predetermined valley in a nice 07:31.333 --> 07:33.273 inversion in which we know things 07:33.267 --> 07:36.167 are going to be trapped. 07:36.167 --> 07:38.467 We know how much pollutant we're putting in-- 07:38.467 --> 07:41.297 I'm going to use the same value as I did here --but we 07:41.300 --> 07:46.070 don't know into what volume will it be mixed. 07:46.067 --> 07:48.197 But we're going to assume that there's a certain amount of 07:48.200 --> 07:50.570 turbulence in this atmosphere. 07:50.567 --> 07:56.067 The air is kind of mixing and spinning and moving around a 07:56.067 --> 07:56.897 little bit. 07:56.900 --> 07:59.000 And we're going to define something called the 07:59.000 --> 08:04.330 dispersion coefficient which will help us decide how fast 08:04.333 --> 08:09.233 that added material will diffuse or will disperse into 08:09.233 --> 08:10.603 the atmosphere. 08:10.600 --> 08:14.630 I'll call it capital K and it has units of meters squared 08:14.633 --> 08:15.873 per second. 08:17.933 --> 08:22.503 And you can estimate a magnitude for that quantity in 08:22.500 --> 08:23.270 the following way. 08:23.267 --> 08:26.397 You can take a typical turbulent velocity-- 08:26.400 --> 08:29.370 let's say there are eddies in this room that are turning 08:29.367 --> 08:32.627 over and that air speed has a certain value, let's say one 08:32.633 --> 08:34.373 meter per second. 08:34.367 --> 08:40.397 And a typical eddy size, maybe the eddies are filling the 08:40.400 --> 08:43.030 space between the floor and the ceiling, 08:43.033 --> 08:45.833 that's about four meters. 08:45.833 --> 08:48.233 So if that were the case, if the typical speed were one 08:48.233 --> 08:52.273 meter per second, and the eddy size were four meters, that 08:52.267 --> 08:56.967 would give a dispersion coefficient of four meters 08:56.967 --> 08:57.997 squared per second. 08:58.000 --> 08:59.100 It's simply that product. 08:59.100 --> 09:00.800 See, that'll give you the right units. 09:00.800 --> 09:03.030 That has meters per second. 09:03.033 --> 09:04.633 This has meters. 09:04.633 --> 09:06.633 And so that's going to give meters squared per second, 09:06.633 --> 09:09.003 when you take that product. 09:09.000 --> 09:12.430 So this is sometimes very difficult to determine, what 09:12.433 --> 09:15.373 that dispersion coefficient is going to be for a particular 09:15.367 --> 09:17.397 atmospheric application. 09:17.400 --> 09:21.170 But you can get a rough estimate, if you can estimate 09:21.167 --> 09:24.767 the size of the eddies and the speed at which those eddies 09:24.767 --> 09:28.967 are turning over in this turbulent atmosphere. 09:28.967 --> 09:31.797 Let's say somehow we know what that is. 09:31.800 --> 09:35.800 Then how fast will that mix things into the atmosphere? 09:35.800 --> 09:40.500 If I have a point source sitting on the ground, one 09:40.500 --> 09:42.330 direction is as good as any other direction. 09:42.333 --> 09:47.333 It's going to mix to the left, to the right, and up equally 09:47.333 --> 09:52.873 fast. So it's going to produce a kind of hemisphere of 09:52.867 --> 09:58.327 pollutant, a half sphere that's growing in time as that 09:58.333 --> 10:03.003 material diffuses outward and upward. 10:03.000 --> 10:06.070 And the radius of that hemisphere is given by this 10:06.067 --> 10:10.397 formula, the square root of the dispersion coefficient 10:10.400 --> 10:12.470 times time. 10:12.467 --> 10:18.297 So the longer you wait, the larger that hemisphere will 10:18.300 --> 10:24.130 be, and of course, the more dilute will be the pollutant 10:24.133 --> 10:26.903 because you've put a fixed amount of pollutant in, unless 10:26.900 --> 10:28.030 you're continuing to add it. 10:28.033 --> 10:30.803 I'm assuming, the problem I'm going to work out here, you 10:30.800 --> 10:34.000 put in just a fixed amount of pollutant and then gradually 10:34.000 --> 10:39.000 you watch it dilute itself as it mixes into a larger and 10:39.000 --> 10:44.200 larger part of the atmosphere. 10:44.200 --> 10:46.430 So what does that say? 10:46.433 --> 10:50.273 It goes as a square root of time. 10:50.267 --> 10:54.667 That function, the square root function, increases rapidly at 10:54.667 --> 10:58.927 first and then more slowly later on. 10:58.933 --> 11:02.173 And that is the nature of turbulent dispersion. 11:02.167 --> 11:06.897 Rapidly at first and then more slowly later in time. 11:06.900 --> 11:09.500 And you may have had that experience. 11:09.500 --> 11:11.700 If you burn something in the kitchen and you're right 11:11.700 --> 11:14.730 there, you smell it very quickly. 11:14.733 --> 11:18.333 A couple of meters away, it takes a lot longer to get 11:18.333 --> 11:23.173 there and even further away even longer, but not linearly. 11:23.167 --> 11:26.497 It actually gets slower and slower and slower. 11:26.500 --> 11:32.800 The advance of that burned smell moves slower and slower 11:32.800 --> 11:34.000 as time progresses. 11:34.000 --> 11:38.900 So rapidly at first and then more slowly later on describes 11:38.900 --> 11:44.930 how things disperse away from their source. 11:44.933 --> 11:46.803 So let's carry out this example then. 11:46.800 --> 11:50.430 Let's say the dispersion coefficient on this day is a 11:50.433 --> 11:56.533 100 meters squared per second and that the time we're going 11:56.533 --> 11:59.373 to wait after the pollutant was suddenly 11:59.367 --> 12:01.997 put in is 10 minutes. 12:02.000 --> 12:05.000 Well, that's 600 seconds. 12:05.000 --> 12:07.330 So the first thing is to compute the radius of that 12:07.333 --> 12:10.503 hemisphere that contains the pollutant. 12:10.500 --> 12:14.870 It's the square root of 100 times 600. 12:14.867 --> 12:18.997 And that's about 245 meters. 12:19.000 --> 12:21.100 By the way, check the units on this. 12:21.100 --> 12:23.330 Is this going to give you a length? 12:23.333 --> 12:25.333 How can that be? 12:25.333 --> 12:29.673 Well, if I multiply this times the time, the seconds are 12:29.667 --> 12:31.427 going to cancel out. 12:31.433 --> 12:33.533 I'm going to have meters squared 12:33.533 --> 12:36.403 inside the square root. 12:36.400 --> 12:39.230 Taking the square root, that gives me units of meters. 12:39.233 --> 12:42.233 And that's just what I need to be physically consistent with 12:42.233 --> 12:45.833 this quantity, R, which is the radius of the hemisphere. 12:45.833 --> 12:49.133 So these units work out nicely. 12:49.133 --> 12:52.173 So the radius of that's going to be 245 meters. 12:52.167 --> 12:57.027 We want to know the volume or the mass of the air into which 12:57.033 --> 12:59.503 the pollutant has spread. 12:59.500 --> 13:03.530 So I've written here the formula for the air mass. 13:03.533 --> 13:09.133 It's the density of the air, rho, times the volume of half 13:09.133 --> 13:11.403 of a sphere. 13:11.400 --> 13:15.370 You may recall the formula for the volume of a sphere is 4/3 13:15.367 --> 13:16.597 pi r cubed. 13:18.967 --> 13:23.267 Well, I've taken half of that, because this is a hemisphere. 13:23.267 --> 13:24.297 It's half of a sphere. 13:24.300 --> 13:25.570 So it's 2/3 pi r cubed. 13:28.100 --> 13:30.370 And I put the density in front, so I get the mass of 13:30.367 --> 13:34.027 the air and it's 3.7 times 10 to the 7 kilograms. So what 13:34.033 --> 13:35.273 have I done? 13:37.400 --> 13:39.500 I'm going to use the same source here. 13:39.500 --> 13:42.900 I've mixed in 40 times 10 to the third kilograms of 13:42.900 --> 13:48.700 pollutant into 3.7 times 10 to the seventh kilograms of air. 13:48.700 --> 13:52.970 So the concentration is given by that ratio and it's 1.1 13:52.967 --> 13:55.027 times 10 to the minus 3. 13:55.033 --> 13:58.733 Again, you could cancel that out or you could say kilograms 13:58.733 --> 13:59.533 per kilogram. 13:59.533 --> 14:05.473 Kilogram of pollutant per kilogram of air. 14:05.467 --> 14:05.867 Question? 14:05.867 --> 14:06.327 Yes. 14:06.333 --> 14:09.633 STUDENT: So you don't know the radius of the hemisphere 14:09.633 --> 14:10.333 unless you know the dispersion coefficient 14:10.333 --> 14:12.973 and the time which--? 14:12.967 --> 14:14.197 PROFESSOR: That's correct. 14:14.200 --> 14:14.900 It's right. 14:14.900 --> 14:18.370 So unlike the valley case, where we kind of knew from the 14:18.367 --> 14:22.827 initial dimensions of the system, how big or how much 14:22.833 --> 14:27.233 the dilution will be, here we have to estimate it based on 14:27.233 --> 14:30.103 the dispersion coefficient and the time that's left. 14:30.100 --> 14:32.670 Obviously, for longer times, that's going to continue to 14:32.667 --> 14:35.197 grow and this concentration is going to get less 14:35.200 --> 14:36.330 and less and less. 14:36.333 --> 14:41.773 We haven't removed any pollutant, but we've diluted 14:41.767 --> 14:45.597 it into a larger and larger amount of air. 14:45.600 --> 14:46.000 Yes? 14:46.000 --> 14:50.170 STUDENT: And so the air density is not 14:50.167 --> 14:52.867 affected by the pollutant? 14:52.867 --> 14:54.127 PROFESSOR: No. 14:54.133 --> 14:57.433 I'm assuming that this is a small amount of pollutant 14:57.433 --> 14:58.673 relative to the air. 14:58.667 --> 14:59.497 And so no. 14:59.500 --> 15:01.270 It doesn't change the air density. 15:01.267 --> 15:04.197 It just mixes in with the air. 15:04.200 --> 15:05.500 Yeah. 15:05.500 --> 15:06.870 Was there another question? 15:06.867 --> 15:07.127 Yes. 15:07.133 --> 15:09.073 STUDENT: So this only tells you overall concentration 15:09.067 --> 15:13.367 rather than concentration within the different smaller 15:13.367 --> 15:14.797 hemispheres it could go to? 15:14.800 --> 15:15.930 PROFESSOR: Yeah, so 15:15.933 --> 15:16.873 that's a very good question. 15:16.867 --> 15:22.567 The question is this only tells you what's going on 15:22.567 --> 15:23.927 generally within that sphere. 15:23.933 --> 15:27.703 It doesn't give you details of what's inside the sphere. 15:27.700 --> 15:31.300 And of course, the model that I've described here is not 15:31.300 --> 15:32.600 precisely true. 15:32.600 --> 15:37.270 If I were to plot concentration across this 15:37.267 --> 15:39.867 distance, it would be very small outside. 15:39.867 --> 15:43.427 It would begin to rise smoothly, probably would be at 15:43.433 --> 15:45.503 maximum somewhere in the middle, and 15:45.500 --> 15:46.770 then decrease again. 15:46.767 --> 15:50.267 Whereas what I've indicated to you, which is a little bit 15:50.267 --> 15:54.727 off, is if there would be zero outside and uniform 15:54.733 --> 15:58.003 concentration inside and then zero again outside. 15:58.000 --> 15:59.370 And that's not exactly true. 15:59.367 --> 16:02.567 So your vision of it is actually more 16:02.567 --> 16:03.597 accurate than mine. 16:03.600 --> 16:08.200 It'd be more of a continuous distribution, large closer to 16:08.200 --> 16:10.670 the source, smaller further away. 16:10.667 --> 16:16.327 But this is an easy-to-compute estimate of what those 16:16.333 --> 16:17.733 concentrations would be. 16:17.733 --> 16:21.533 You would not have a sharp boundary here with this 16:21.533 --> 16:24.303 concentration inside and zero outside. 16:24.300 --> 16:27.130 It wouldn't be like that, there'd be more of a gradual 16:27.133 --> 16:27.633 transition. 16:27.633 --> 16:28.433 So this is kind of an 16:28.433 --> 16:30.733 approximation to the real condition. 16:30.733 --> 16:32.633 That's an excellent question. 16:32.633 --> 16:33.873 Other questions on this? 16:36.500 --> 16:39.900 So there would be our concentration 16:39.900 --> 16:42.770 after time, 600 seconds. 16:42.767 --> 16:46.597 You could redo the computation for a later time and you'd get 16:46.600 --> 16:49.830 a larger value in the denominator and therefore, a 16:49.833 --> 16:52.303 smaller value of concentration. 16:52.300 --> 16:56.530 That's the way dilution works. 16:56.533 --> 16:59.433 Anything else on this? 16:59.433 --> 16:59.773 OK. 16:59.767 --> 17:03.267 Now we're going to add another element of complexity here. 17:03.267 --> 17:06.067 For case three, we're going to add a wind. 17:06.067 --> 17:08.627 So there's a wind blowing in this system. 17:08.633 --> 17:13.003 So it's unconfined, no valley walls, but now 17:13.000 --> 17:15.170 there's a wind blowing. 17:15.167 --> 17:18.327 So I've drawn some little cartoons here. 17:18.333 --> 17:21.433 There is the source. 17:21.433 --> 17:23.173 Maybe it's a-- what I'm envisioning here is not just 17:23.167 --> 17:26.867 an instantaneous release of pollutant, like I was in case 17:26.867 --> 17:31.767 one and case two, but here I'm envisioning a steady state 17:31.767 --> 17:34.897 source of pollution, like a power plant that's putting out 17:34.900 --> 17:38.770 a certain amount of smoke from the stack per unit time 17:38.767 --> 17:39.997 continuously. 17:42.467 --> 17:46.697 If I had a side view of that, and the wind was blowing from 17:46.700 --> 17:49.330 left to right, I would find that pollutant would be 17:49.333 --> 17:51.703 confined in a kind of-- 17:51.700 --> 17:55.770 we call this a plume, a plume of pollutant, moving 17:55.767 --> 18:00.797 downstream, getting deeper the further it goes downstream due 18:00.800 --> 18:04.200 to the turbulent mixing. 18:04.200 --> 18:07.630 If I had a top-view of that same situation-- 18:07.633 --> 18:10.473 here's the top-view -there's the power plant, there's the 18:10.467 --> 18:12.967 source, the wind is from left to right. 18:12.967 --> 18:15.967 Now we see that plume is spreading in the horizontal 18:15.967 --> 18:20.597 dimensions as well, again by turbulent mixing and 18:20.600 --> 18:21.430 dispersion. 18:21.433 --> 18:27.203 And if I was to slice this here or here and look at it 18:27.200 --> 18:31.030 along the direction of the wind, at any particular 18:31.033 --> 18:36.933 distance downstream, x, it would look like half a circle. 18:39.533 --> 18:43.273 And the radius of that circle will be larger the further 18:43.267 --> 18:47.327 downwind that I go, because dispersion is enlarging that 18:47.333 --> 18:50.133 plume the further we get from the source. 18:56.033 --> 18:58.033 So how are we going to work this one out? 18:58.033 --> 19:02.603 Again, it's just a matter of computing dilution, but now 19:02.600 --> 19:06.970 this is a steady-state pollution and we're going to 19:06.967 --> 19:10.167 have to take into account the effect of wind moving that 19:10.167 --> 19:11.797 material downwind. 19:11.800 --> 19:15.370 So I wanted to define a transit time, the time it 19:15.367 --> 19:21.867 takes the air to go from the pollutant source to whatever x 19:21.867 --> 19:23.967 location we're interested in. 19:23.967 --> 19:27.767 Maybe your house is five kilometers downwind of the 19:27.767 --> 19:28.527 power plant. 19:28.533 --> 19:32.733 Well then, x is going to be five kilometers. 19:32.733 --> 19:36.473 And the time it takes the air to get from the power plant to 19:36.467 --> 19:40.227 your house is going to be x divided by the wind speed. 19:40.233 --> 19:42.373 Wind speed is capital U in this case. 19:42.367 --> 19:45.267 Notice that'll have the right units, distance over a 19:45.267 --> 19:47.327 distance per time. 19:47.333 --> 19:49.273 That's going to work out to have units of time. 19:51.900 --> 19:55.370 The radius of this plume is given by the same formula we 19:55.367 --> 20:02.667 had earlier, except now it's this transit time we put in to 20:02.667 --> 20:05.097 that formula. 20:05.100 --> 20:08.400 It's a steady-state situation, so it's not changing in time, 20:08.400 --> 20:12.330 but yet there is this transit time that plays the role of 20:12.333 --> 20:15.073 time in the original problem. 20:15.067 --> 20:18.997 So I'll put in x over t into this formula and I get kx over 20:19.000 --> 20:24.400 u square root for this quantity, r, that you see in 20:24.400 --> 20:28.200 each of these three diagrams. Question, yes. 20:28.200 --> 20:28.870 STUDENT: What is U? 20:28.867 --> 20:32.567 PROFESSOR: U is the wind speed, typically in 20:32.567 --> 20:33.797 meters per second. 20:38.300 --> 20:40.900 Now, by the way, the cross-sectional area of this 20:40.900 --> 20:43.770 plume, if I were to slice across like this, 20:43.767 --> 20:46.327 it is half a circle-- 20:46.333 --> 20:49.073 and you know the area of the circle is pi r squared --and 20:49.067 --> 20:53.297 so the plume cross-sectional area is pi r squared divided 20:53.300 --> 20:55.600 by two, because it's half a circle. 21:00.333 --> 21:03.733 Now here comes probably the trickiest part of the 21:03.733 --> 21:04.673 derivation. 21:04.667 --> 21:09.297 I'm going to assume that I know the source strength and 21:09.300 --> 21:13.030 that's how much pollutant is being added to the air per 21:13.033 --> 21:15.173 unit time by the power plant. 21:15.167 --> 21:18.467 That might have units, for example, of kilograms per 21:18.467 --> 21:23.167 second, kilograms of smoke, let's say, being added to the 21:23.167 --> 21:26.197 atmosphere every second. 21:26.200 --> 21:30.030 Well, in a steady state, the amount of pollutant being 21:30.033 --> 21:33.473 added at the source must be the same amount that's being 21:33.467 --> 21:36.127 carried away by the wind. 21:36.133 --> 21:40.733 And so by equating those two, I can derive a formula for the 21:40.733 --> 21:42.903 concentration. 21:42.900 --> 21:45.870 So this is the key step, I'm going to equate the rate at 21:45.867 --> 21:48.467 which pollutant's being added to the rate at which it's 21:48.467 --> 21:49.897 being carried away by the wind. 21:49.900 --> 21:52.970 So on the left-hand side is a rate which it's being added, 21:52.967 --> 21:54.127 this is the rate at which it's being 21:54.133 --> 21:55.773 carried away by the wind. 21:55.767 --> 21:57.597 Let me go through this right-hand side. 21:57.600 --> 22:00.400 That's the air density, that's wind speed, that's the 22:00.400 --> 22:04.870 cross-sectional area of the plume. 22:04.867 --> 22:08.227 And so that product by itself, those three things together, 22:08.233 --> 22:14.603 is the amount of air per unit time passing a way within the 22:14.600 --> 22:18.270 plume-- being carried downwind by the wind. 22:18.267 --> 22:21.327 And I just have to multiply that times the concentration 22:21.333 --> 22:23.433 to convert that from an air mass flux to a 22:23.433 --> 22:27.403 pollutant air mass flux. 22:27.400 --> 22:32.730 So then solving this formula for concentration, I get 2 22:32.733 --> 22:38.133 times the source, I've plugged this in there as well, 2 times 22:38.133 --> 22:43.033 the source, divided by the density of the air, divided by 22:43.033 --> 22:47.403 pi times k times x. 22:47.400 --> 22:50.670 And you're going to want to leave an inch or two in your 22:50.667 --> 22:54.497 notes to check the units on this as well. 22:54.500 --> 22:59.830 For example, the units on the concentration, we expect if 22:59.833 --> 23:03.233 previous calculations are right, it'll be either 23:03.233 --> 23:06.303 kilograms per kilogram, if you prefer, or it'll 23:06.300 --> 23:09.130 have no units at all. 23:09.133 --> 23:12.533 And so all the units over here should cancel out. 23:12.533 --> 23:18.103 And leave an inch or two of your notes to deal with that. 23:18.100 --> 23:22.300 One very interesting thing is that the velocity of the wind 23:22.300 --> 23:24.530 canceled out. 23:24.533 --> 23:25.803 How did that happen? 23:29.367 --> 23:33.367 I had it here because the rate at which material is being 23:33.367 --> 23:34.767 blown downwind is going to be 23:34.767 --> 23:37.027 proportional to the wind speed. 23:37.033 --> 23:45.373 But you see the radius here, well the area, depends on r, 23:45.367 --> 23:48.027 and r depends on U underneath. 23:48.033 --> 23:50.603 So when you square the radius, you get a U underneath, which 23:50.600 --> 23:51.230 cancels that out. 23:51.233 --> 23:56.873 So the width of the plume will change with the wind speed, 23:56.867 --> 23:59.797 but not the concentration within the plume. 23:59.800 --> 24:01.270 And when you do-- 24:01.267 --> 24:03.227 you're going to do a couple of examples like this in your 24:03.233 --> 24:04.773 next problem set. 24:04.767 --> 24:08.067 When you're doing that, you might spend a few minutes 24:08.067 --> 24:11.997 puzzling over what was the role of the wind speed. 24:12.000 --> 24:14.530 The wind speed is very important in this problem, but 24:14.533 --> 24:17.803 in that particular formula, it is canceled out. 24:17.800 --> 24:20.800 And so you should try to resolve that little dilemma-- 24:20.800 --> 24:25.070 how, if the wind speed is important, how did it cancel 24:25.067 --> 24:27.367 out in that particular formula. 24:27.367 --> 24:32.997 So think a bit about how that would work. 24:33.000 --> 24:34.800 Any questions on this? 24:34.800 --> 24:35.100 Yes. 24:35.100 --> 24:36.000 STUDENT: What is A? 24:36.000 --> 24:38.470 PROFESSOR: A is the-- 24:38.467 --> 24:41.197 is this thing, the cross-sectional area of the 24:41.200 --> 24:43.970 plume, pi r squared over 2. 24:43.967 --> 24:44.427 Yes. 24:44.433 --> 24:46.673 STUDENT: What is the x in the equation over there? 24:46.667 --> 24:50.367 PROFESSOR: X is the distance between the 24:50.367 --> 24:55.497 source of the pollutant and the location of interest: your 24:55.500 --> 24:59.770 house or the school yard, where the kids are playing, or 24:59.767 --> 25:02.427 whatever it is that you're interested in finding out what 25:02.433 --> 25:03.873 the concentration is. 25:03.867 --> 25:08.467 It's the distance away from the pollutant source. 25:08.467 --> 25:13.767 Now obviously, if the wind is from the west and you're 25:13.767 --> 25:17.227 living down here somewhere, that plume is not going to hit 25:17.233 --> 25:18.773 you at all. 25:18.767 --> 25:22.897 So this calculation is irrelevant for you. 25:22.900 --> 25:25.470 So the wind direction comes into this in a very important 25:25.467 --> 25:28.327 way also, I haven't mentioned that up until now, but it's 25:28.333 --> 25:31.233 probably quite obvious that the wind direction is a key 25:31.233 --> 25:32.773 player in this. 25:32.767 --> 25:36.097 It's only if the wind is blowing generally in the 25:36.100 --> 25:40.400 direction from the source to your location that this 25:40.400 --> 25:44.200 calculation becomes relevant. 25:44.200 --> 25:44.700 Yeah. 25:44.700 --> 25:47.170 STUDENT: Does the plume get wider as you go 25:47.167 --> 25:48.127 farther away from it? 25:48.133 --> 25:50.833 So if you live south of it, far enough away-- 25:50.833 --> 25:51.803 PROFESSOR: That's true. 25:51.800 --> 25:52.770 That's what I've sketched. 25:52.767 --> 25:54.767 This is a top view. 25:54.767 --> 25:57.767 It is getting wider, fast at first, and then slower. 25:57.767 --> 26:00.467 But it never stops widening, it keeps getting wider and 26:00.467 --> 26:03.067 wider and wider the further you are downwind. 26:03.067 --> 26:03.667 So that's true. 26:03.667 --> 26:07.927 If you lived a great distance downwind, but not in the 26:07.933 --> 26:11.233 exactly in the wind direction, you might still get the 26:11.233 --> 26:12.773 pollutant, because the pollution 26:12.767 --> 26:15.067 will widen the plume. 26:15.067 --> 26:18.597 So keep track of that when you do these calculations. 26:18.600 --> 26:19.000 Yes. 26:19.000 --> 26:20.330 STUDENT: What does source represent 26:20.333 --> 26:21.733 in that final equation? 26:21.733 --> 26:24.203 PROFESSOR: Source would be the rate at which 26:24.200 --> 26:27.000 pollutant is being added to the atmosphere at the power 26:27.000 --> 26:28.100 plant, let's say. 26:28.100 --> 26:32.770 It could be in units of kilograms per second. 26:32.767 --> 26:36.467 But if it's particles, it could be the number of 26:36.467 --> 26:39.697 particles per second that's being added. 26:39.700 --> 26:42.100 And then your concentration would have to be in the 26:42.100 --> 26:43.100 appropriate-- 26:43.100 --> 26:45.630 by the way, that's the curve ball I've given you the 26:45.633 --> 26:47.003 problem set. 26:47.000 --> 26:51.570 I think I gave you a problem in which the source was quoted 26:51.567 --> 26:54.497 as a number of particles per second. 26:54.500 --> 26:59.130 And so the concentration then was in units of particles per 26:59.133 --> 27:03.733 cubic meter, rather than kilograms per cubic meter. 27:03.733 --> 27:05.573 So just be wary, you have to have consistent 27:05.567 --> 27:10.097 units when you do this. 27:10.100 --> 27:12.830 OK, now we're going to leave that there and move on to 27:12.833 --> 27:18.333 another aspect of mixing which is very important also. 27:18.333 --> 27:21.833 And that's the role of the temperature lapse rate. 27:25.033 --> 27:27.903 So I called this section lapse rate and buoyancy effects. 27:30.433 --> 27:33.973 First of all, I want to define what I mean by the lapse rate. 27:33.967 --> 27:35.827 Lapse rate is the rate of change of 27:35.833 --> 27:38.103 temperature with height. 27:38.100 --> 27:42.330 If you launch a balloon and get that data back, you can 27:42.333 --> 27:48.173 measure the values of temperature with height and 27:48.167 --> 27:49.767 you can define the lapse rate. 27:49.767 --> 27:55.267 It has units typically of degrees Celsius per meter or 27:55.267 --> 27:58.527 perhaps you would want to use degrees Celsius per kilometer. 27:58.533 --> 28:02.133 It's basically how fast is the temperature change in the 28:02.133 --> 28:04.573 atmosphere as you go up and down. 28:04.567 --> 28:08.797 Typically, in the troposphere, you remember, it gets colder 28:08.800 --> 28:10.570 as you go up. 28:10.567 --> 28:13.227 And an average value-- 28:13.233 --> 28:18.133 I'm using lower gamma for this --typical lapse rate for the 28:18.133 --> 28:25.133 troposphere is about 6.5 degrees Celsius per kilometer. 28:25.133 --> 28:28.103 On average, every kilometer you go up, it 28:28.100 --> 28:30.770 cools about 6.5 degrees. 28:30.767 --> 28:32.067 But don't take that too literally. 28:32.067 --> 28:33.067 That's just an average. 28:33.067 --> 28:36.697 Sometimes it's very different than that, 28:36.700 --> 28:39.070 sometimes it's even positive. 28:39.067 --> 28:40.627 For example, in an inversion-- 28:40.633 --> 28:42.273 remember how I defined an inversion? 28:42.267 --> 28:45.627 An inversion was a layer where the temperature 28:45.633 --> 28:46.673 increases with height. 28:46.667 --> 28:50.327 Well, that would mean a positive lapse rate, instead 28:50.333 --> 28:53.203 of a negative one. 28:53.200 --> 28:55.200 So for some layers-- and then in the stratosphere, remember 28:55.200 --> 28:57.530 you have a positive lapse rate, the temperature gets 28:57.533 --> 29:01.003 warmer as you go up. 29:01.000 --> 29:03.630 The other really important quantity is a more of a 29:03.633 --> 29:05.873 theoretical one, but it's equally important. 29:05.867 --> 29:07.127 It's the adiabatic lapse rate. 29:09.733 --> 29:12.403 This is the lapse rate that a parcel of air 29:12.400 --> 29:14.700 experiences when it rises. 29:18.267 --> 29:21.027 Why would the temperature of an air parcel 29:21.033 --> 29:23.133 change when it rises? 29:23.133 --> 29:24.103 Well, it's pretty-- 29:24.100 --> 29:26.100 it's fairly straightforward. 29:26.100 --> 29:30.630 When you take an air parcel of a certain volume and lift it-- 29:30.633 --> 29:32.833 you know the pressure decreases as you go up, so 29:32.833 --> 29:34.833 when you lift a parcel up, it's going to 29:34.833 --> 29:36.803 expand a little bit. 29:36.800 --> 29:40.270 You lift it further, it's going to expand still more. 29:40.267 --> 29:44.867 And when air expands, it does work on its environment by 29:44.867 --> 29:47.767 pressing out and expanding. 29:47.767 --> 29:51.597 It decreases the amount of energy stored in the parcel 29:51.600 --> 29:54.770 because it's done work on the environment. 29:54.767 --> 29:57.297 And its temperature drops. 29:57.300 --> 29:59.100 It's called adiabatic expansion 29:59.100 --> 30:00.570 or adiabatic cooling. 30:00.567 --> 30:06.467 The word adiabatic here means without adding 30:06.467 --> 30:09.267 or subtracting heat. 30:09.267 --> 30:12.027 And you may wonder, how can I change the temperature without 30:12.033 --> 30:13.103 adding heat? 30:13.100 --> 30:16.200 The point is I'm changing the temperature by having that 30:16.200 --> 30:20.330 parcel do work on its environment, not by adding 30:20.333 --> 30:23.733 heat from some energy source. 30:23.733 --> 30:28.403 It's purely a mechanical process of expanding the air 30:28.400 --> 30:33.430 and watching it cool because of that adiabatic expansion. 30:33.433 --> 30:35.533 That's quite a powerful effect, as you'll 30:35.533 --> 30:37.703 see in just a moment. 30:37.700 --> 30:40.470 So the adiabatic lapse rate is defined as a rate of cooling 30:40.467 --> 30:41.897 as an air parcel rises. 30:41.900 --> 30:46.830 It is a reversible phenomenon, so if a parcel sinks back down 30:46.833 --> 30:51.003 in the atmosphere, it will compress and its temperature 30:51.000 --> 30:53.070 will increase at the same rate. 30:57.567 --> 31:00.797 I'll use capital gamma for this quantity, 31:00.800 --> 31:03.700 adiabatic lapse rate. 31:03.700 --> 31:07.000 It can be computed as a ratio of the acceleration of gravity 31:07.000 --> 31:09.570 to the heat capacity of the air. 31:09.567 --> 31:12.227 If this were a course on thermodynamics, I would derive 31:12.233 --> 31:14.803 that for you. 31:14.800 --> 31:19.870 For the earth, and our atmosphere made of air, the 31:19.867 --> 31:23.497 value is about minus 9.3 times [correction: 9.8] 31:23.500 --> 31:27.230 10 to the minus 3 degrees Celsius per meter or 31:27.233 --> 31:35.673 expressing that in kilometers, it's about minus 9.8 degrees 31:35.667 --> 31:37.027 Celsius per kilometer. 31:37.033 --> 31:37.473 Question. 31:37.467 --> 31:38.867 STUDENT: What's the denominator again? 31:38.867 --> 31:40.727 PROFESSOR: The heat capacity of constant 31:40.733 --> 31:44.073 pressure for air. 31:44.067 --> 31:48.867 You could look it up in a table of physical constants. 31:48.867 --> 31:50.327 You can Google it. 31:50.333 --> 31:54.303 Its value is about a 1,004, I believe, in SI units. 31:57.733 --> 32:01.703 So I have to confess, I'm a little bit sloppy, that number 32:01.700 --> 32:05.800 is so close to 10 that I very often round it off to 10 when 32:05.800 --> 32:08.600 I'm doing quick calculations. 32:08.600 --> 32:11.270 And I've done that here in my little example. 32:11.267 --> 32:14.197 Let's say I've got air at sea level that's at 10 degrees 32:14.200 --> 32:18.000 Celsius and I move it up in the atmosphere three 32:18.000 --> 32:21.330 kilometers, 3,000 meters. 32:21.333 --> 32:23.903 What temperature will it be at when it 32:23.900 --> 32:25.500 reaches that higher elevation? 32:25.500 --> 32:29.730 Well, if I round that off to 10, it's cooling 10 degrees 32:29.733 --> 32:33.903 for every kilometer that I lift it, so it's going to cool 32:33.900 --> 32:38.600 by 30 degrees Celsius, approximately. 32:38.600 --> 32:41.400 Please, I would ask you not to do that rounding. 32:41.400 --> 32:46.030 In your problem sets, let it be 9.8, not 10. 32:46.033 --> 32:48.273 But for the purposes of illustration, it's going to 32:48.267 --> 32:54.067 cool by approximately 30 degrees Celsius. 32:54.067 --> 32:58.467 It starts at 10, so it's going to be minus 20 Celsius when it 32:58.467 --> 33:02.627 gets to 3 kilometers. 33:02.633 --> 33:04.473 You can do that in Kelvins as well, it will work 33:04.467 --> 33:04.967 out the same way. 33:04.967 --> 33:06.327 If it starts at-- 33:06.333 --> 33:10.173 let's see if it's 10 degrees Celsius, that's going to be 33:10.167 --> 33:17.227 283 Kelvins, and subtract 30 from that, it's going to be 33:17.233 --> 33:20.903 253 Kelvins when it gets up there. 33:20.900 --> 33:23.870 So you might want to put both the Celsius and the Kelvins in 33:23.867 --> 33:26.927 your notes to be sure you're clear on that question. 33:26.933 --> 33:28.533 STUDENT: So that first one there is it's shown as 33:28.533 --> 33:30.273 expanded in that diagram? 33:30.267 --> 33:31.227 PROFESSOR: That's right. 33:31.233 --> 33:34.033 I've drawn it that way to remind you that the reason 33:34.033 --> 33:36.603 it's cooled is because it has expanded. 33:36.600 --> 33:40.400 It moved into a region where the pressure was less and 33:40.400 --> 33:43.070 therefore, it naturally expanded because of that 33:43.067 --> 33:44.527 lowered pressure. 33:44.533 --> 33:45.003 Question. 33:45.000 --> 33:47.230 STUDENT: So it doesn't matter the size of the air parcel? 33:47.233 --> 33:47.973 PROFESSOR: No. 33:47.967 --> 33:49.297 So it's so simple. 33:49.300 --> 33:52.970 It doesn't matter the size of the parcel, it's just that 33:52.967 --> 33:54.697 elevation difference is all that matters. 33:54.700 --> 33:57.500 That tells you how much, on a relative basis, it has 33:57.500 --> 34:00.530 expanded, and therefore, how many degrees 34:00.533 --> 34:03.403 Celsius it has cooled. 34:03.400 --> 34:05.400 Now remember I said this is reversible. 34:05.400 --> 34:08.730 If I take that parcel and move it back down, it'll compress 34:08.733 --> 34:12.203 back down to its original volume, and it'll return to 34:12.200 --> 34:14.630 its original temperature. 34:14.633 --> 34:16.603 So it's a reversible process. 34:16.600 --> 34:17.830 STUDENT: Are you assuming the parcel was on the ground? 34:21.400 --> 34:22.700 PROFESSOR: The question is whether we're 34:22.700 --> 34:24.030 assuming it was on the ground. 34:24.033 --> 34:25.673 No, not necessarily. 34:25.667 --> 34:28.797 This would happen, I could start at any elevation and 34:28.800 --> 34:31.370 move it to any other elevation. 34:31.367 --> 34:33.767 So this is independent of where the ground was. 34:33.767 --> 34:36.927 I just happened to give you an example with the ground. 34:36.933 --> 34:39.673 Just a three kilometer lifting, no matter where you 34:39.667 --> 34:43.867 started, would give you an approximate 330 degrees 34:43.867 --> 34:45.127 Celsius cooling. 34:48.400 --> 34:52.170 Questions on that? 34:52.167 --> 34:52.567 OK. 34:52.567 --> 34:57.127 Now, we can begin to do some calculations then of what 34:57.133 --> 35:00.203 happens to air as you move it up and down in the atmosphere. 35:00.200 --> 35:05.000 Using these two concepts, the measured lapse rate and the 35:05.000 --> 35:06.230 adiabatic lapse rate. 35:06.233 --> 35:09.903 So normally in the textbooks, you'll see diagrams like this, 35:09.900 --> 35:12.670 where the temperatures have been put on the x-axis, 35:12.667 --> 35:17.097 altitude's been put on the y-axis, and some reference 35:17.100 --> 35:19.800 lines are drawn on here, where the slope, 35:19.800 --> 35:21.270 given by capital gamma. 35:24.800 --> 35:27.700 And then I've drawn in a black curve, which represents the 35:27.700 --> 35:32.600 actual lapse rate on the day, under consideration. 35:32.600 --> 35:35.500 Let's say we've launched a balloon, measured temperature 35:35.500 --> 35:39.330 as a function of height, and I've plotted that up as that 35:39.333 --> 35:40.733 black line. 35:40.733 --> 35:43.033 Now, the question is what's going to happen to an air 35:43.033 --> 35:47.033 parcel as it's moved upwards or downwards in 35:47.033 --> 35:48.303 this kind of a situation. 35:51.167 --> 36:02.467 If I take an air parcel from this elevation and lift it, 36:02.467 --> 36:04.827 it's going to cool. 36:04.833 --> 36:09.403 It's going to cool moving along the dashed blue curves, 36:09.400 --> 36:10.870 because that's the adiabatic lapse rate. 36:10.867 --> 36:13.967 So this parcel, when I take it out of its environment and 36:13.967 --> 36:17.467 lift it a little ways, it's going to move along like this 36:17.467 --> 36:18.727 to its new elevation. 36:21.000 --> 36:24.070 Its new temperature is going to be there and the 36:24.067 --> 36:26.497 temperature of its new environment 36:26.500 --> 36:29.270 is going to be there. 36:29.267 --> 36:31.367 Here's the tricky part of this argument. 36:31.367 --> 36:34.927 When you take an air parcel and lift it, it's cooling at 36:34.933 --> 36:42.233 one rate, its environment is changing at a different rate. 36:42.233 --> 36:44.603 And we're interested in knowing, after we lift it a 36:44.600 --> 36:47.670 certain amount, what is its temperature relative to the 36:47.667 --> 36:48.697 environment? 36:48.700 --> 36:52.730 It's new, the air that it's now next to. 36:52.733 --> 36:58.933 In this case, the parcel went to that temperature, it moved 36:58.933 --> 37:01.773 into cooler air above, but that wasn't 37:01.767 --> 37:02.597 such a great effect. 37:02.600 --> 37:07.030 And actually, the parcel is now colder than its new 37:07.033 --> 37:09.133 environment. 37:09.133 --> 37:12.533 Environment cooled, the parcel cooled, the parcel cooled 37:12.533 --> 37:16.773 more, so the parcel is now cold, relative to its new 37:16.767 --> 37:17.367 environment. 37:17.367 --> 37:20.627 What's going to happen to that parcel? 37:20.633 --> 37:23.773 It's going to sink because it's more dense than its 37:23.767 --> 37:24.727 environment. 37:24.733 --> 37:28.303 So I lifted it up there, it looks around and says, oh my 37:28.300 --> 37:31.230 lord, I am dense than everything up 37:31.233 --> 37:32.973 here, back down I go. 37:32.967 --> 37:34.697 I don't belong here in this crowd. 37:34.700 --> 37:38.470 I've got to go back, I'm negatively buoyant, I'm heavy, 37:38.467 --> 37:40.727 I'm going to sink back. 37:40.733 --> 37:41.433 You can repeat it. 37:41.433 --> 37:44.533 You could push that parcel down in the atmosphere, it'll 37:44.533 --> 37:46.203 move to here. 37:46.200 --> 37:47.970 There's a new temperature. 37:47.967 --> 37:50.167 There is a new temperature of its environment. 37:53.600 --> 37:56.330 It got warmer, as it went down. 37:56.333 --> 38:00.633 The environment got warmer too, but it got warmer faster. 38:00.633 --> 38:03.233 Therefore, it's warmer than its new environment. 38:03.233 --> 38:05.003 It's now buoyant. 38:05.000 --> 38:09.030 It's going to want to bob back up to its original condition. 38:09.033 --> 38:12.833 This is called a stable atmosphere. 38:12.833 --> 38:15.303 This term is very important, this is called a stable 38:15.300 --> 38:20.230 atmosphere because a parcel that is lifted upwards wants 38:20.233 --> 38:22.933 to sink back down, a parcel that's pushed down wants to 38:22.933 --> 38:25.033 bob back up. 38:25.033 --> 38:29.233 And the atmosphere is usually in this state, but not always. 38:31.967 --> 38:37.867 I want to look at this second example, here, which is again, 38:37.867 --> 38:40.327 temperature plotted versus altitude. 38:40.333 --> 38:42.873 The same reference lines have been drawn on 38:42.867 --> 38:44.727 here for capital gamma. 38:44.733 --> 38:49.703 But I put a different actual lapse rate on that curve. 38:49.700 --> 38:52.470 It's a lapse rate where the temperature 38:52.467 --> 38:55.697 increases with height. 38:55.700 --> 38:58.070 This is an example of an inversion. 39:04.067 --> 39:11.197 If I take a parcel from here and lift it, it gets colder. 39:11.200 --> 39:12.200 I'm going to take it from that 39:12.200 --> 39:15.270 elevation up to that elevation. 39:15.267 --> 39:17.627 There's the parcel temperature. 39:17.633 --> 39:20.233 It's actually been moving into warmer air, however. 39:22.767 --> 39:26.167 So the new environment is there. 39:26.167 --> 39:30.427 So this is an extreme example of stability. 39:30.433 --> 39:34.503 This is a very stable environment where the parcel 39:34.500 --> 39:37.500 gets colder as you go up, the environment gets warmer. 39:37.500 --> 39:41.470 And so you develop these restoring forces, this 39:41.467 --> 39:44.767 tendency for the parcel to want to quickly remove, go 39:44.767 --> 39:50.597 back to where it was, are amplified, in this case, 39:50.600 --> 39:52.300 because of the inversion. 39:52.300 --> 39:57.170 That's the reason why air cannot mix effectively through 39:57.167 --> 39:58.697 an inversion. 39:58.700 --> 40:01.670 Because the air parcels want to cool as they rise, and yet 40:01.667 --> 40:07.397 there's warmer air aloft, the two things add to each other, 40:07.400 --> 40:11.100 making it virtually impossible for air parcels to mix upwards 40:11.100 --> 40:12.830 or downwards, for that matter. 40:12.833 --> 40:17.933 The air tends to lay in stable layers, stratified stable 40:17.933 --> 40:23.673 layers with very little turbulent mixing in a 40:23.667 --> 40:24.967 situation like this. 40:24.967 --> 40:28.997 Now the situation that I didn't do, and you can fill in 40:29.000 --> 40:32.330 your notes, do one last case where now-- 40:32.333 --> 40:37.973 tilt this curve over so it's flatter than 40:37.967 --> 40:40.727 the reference curves. 40:40.733 --> 40:42.973 Leave some space in your notes for this. 40:42.967 --> 40:46.167 Take that observed temperature, the black line, 40:46.167 --> 40:51.497 rotate it over until it's more horizontal than the blue 40:51.500 --> 40:52.530 reference lines. 40:52.533 --> 40:54.903 And then repeat the argument. 40:54.900 --> 40:57.070 What you're going to find there is it if you look to 40:57.067 --> 41:01.567 parcel, it suddenly becomes buoyant relative to its 41:01.567 --> 41:04.627 environment and it'll continue to rise. 41:04.633 --> 41:06.373 That's called an unstable atmosphere. 41:08.900 --> 41:11.270 Same thing would happen if you pushed it down. 41:11.267 --> 41:15.397 It would become negatively buoyant and it would then drop 41:15.400 --> 41:19.400 further, another illustration that it's an unstable 41:19.400 --> 41:20.900 atmosphere. 41:20.900 --> 41:24.030 So this sort of thing will have a big impact on how 41:24.033 --> 41:26.933 materials mix around in the atmosphere. 41:26.933 --> 41:30.533 And as we will see next time, it also has a big impact on 41:30.533 --> 41:31.973 how clouds develop. 41:31.967 --> 41:36.027 Next week, we're going to get into the issue of water vapor 41:36.033 --> 41:39.233 in the atmosphere, how clouds form. 41:39.233 --> 41:43.573 And then by Wednesday or so, how precipitation forms in the 41:43.567 --> 41:44.697 atmosphere. 41:44.700 --> 41:47.000 And I think we'll call it quits today. 41:47.000 --> 41:48.700 But that will be the subject for next week.