WEBVTT 03:29.400 --> 03:35.370 RONALD SMITH: So last time, we derived a formula for the 03:35.367 --> 03:39.127 equilibrium temperature of a planet. 03:39.133 --> 03:41.373 And I'm not going to go back over that much. 03:41.367 --> 03:45.727 The first column of information here is a 03:45.733 --> 03:46.833 continuation of that. 03:46.833 --> 03:54.373 But the way we did it was to consider the Earth as a kind 03:54.367 --> 03:59.197 of reservoir of heat that can store heat. 03:59.200 --> 04:01.030 It can receive heat from the Sun, and it can 04:01.033 --> 04:03.773 radiate heat to space. 04:03.767 --> 04:06.967 And then we tried to consider how that system might come 04:06.967 --> 04:08.527 into a steady-state balance. 04:08.533 --> 04:14.303 So we were thinking of this very much like the water tank. 04:14.300 --> 04:21.270 If I plotted temperature versus time-- 04:21.267 --> 04:24.097 oh, I have to excuse myself for the slow writing today. 04:24.100 --> 04:27.170 I think I've dislocated my shoulder, so I'm going to be 04:27.167 --> 04:29.927 doing it two-handed. 04:29.933 --> 04:31.773 The lesson of the day is wear your 04:31.767 --> 04:33.897 helmet when you're biking. 04:33.900 --> 04:39.470 I did, so my head is OK, but my shoulder's not. 04:39.467 --> 04:43.427 So if I started the Earth at some temperature but I was-- 04:43.433 --> 04:47.403 let's say I started it at zero temperature, zero Kelvin. 04:47.400 --> 04:48.400 This is unrealistic. 04:48.400 --> 04:50.030 It probably had some temperature when 04:50.033 --> 04:50.873 it was first formed. 04:50.867 --> 04:53.397 But let's imagine I started at zero Kelvin. 04:53.400 --> 04:59.830 Now, the Stefan-Boltzmann Law says that an object at zero 04:59.833 --> 05:03.203 temperature does not radiate. 05:03.200 --> 05:08.070 So the Earth would be receiving heat from the Sun 05:08.067 --> 05:09.297 but not radiating it. 05:09.300 --> 05:12.370 So the amount of heat stored in the Earth would be 05:12.367 --> 05:12.967 increasing. 05:12.967 --> 05:14.967 That is to say, its temperature would be 05:14.967 --> 05:20.527 increasing with time, getting hotter and hotter each day 05:20.533 --> 05:24.533 that went by, receiving more heat from the Sun, not 05:24.533 --> 05:27.003 radiating it to space. 05:27.000 --> 05:29.530 Well, of course, as soon as it develops some temperature, 05:29.533 --> 05:31.273 it's going to start to radiate. 05:31.267 --> 05:33.827 Remember, the Stefan-Boltzmann Law says the power per unit 05:33.833 --> 05:37.073 area is the sigma times T to the fourth. 05:37.067 --> 05:40.467 So as soon as T begins to rise, you get a little bit of 05:40.467 --> 05:41.667 radiation to space. 05:41.667 --> 05:46.127 So this curve will begin to arc over a little bit, but 05:46.133 --> 05:51.803 eventually, it'll reach a constant temperature. 05:51.800 --> 05:55.730 And that'll be the case when the rate of heat received from 05:55.733 --> 05:59.873 the Sun equals the rate at which energy is 05:59.867 --> 06:01.667 radiating to space. 06:01.667 --> 06:02.827 That's the steady state. 06:02.833 --> 06:05.673 So we made the assumption that we were in this state. 06:05.667 --> 06:09.727 We equated the incoming radiation and the outgoing 06:09.733 --> 06:13.033 radiation and found the temperature that would allow 06:13.033 --> 06:14.633 that balance to be reached. 06:14.633 --> 06:18.673 Remember, if the temperature's too cold, it won't radiate 06:18.667 --> 06:23.867 enough to keep up with what the Sun is delivering. 06:23.867 --> 06:27.527 If the temperature is too hot, it's radiating more than the 06:27.533 --> 06:29.973 Sun is delivering, and the temperature of the Earth would 06:29.967 --> 06:31.497 cool with time. 06:31.500 --> 06:33.900 But in steady state, that balance is reached. 06:33.900 --> 06:37.070 And what we did was solve for that temperature, that single 06:37.067 --> 06:40.967 temperature that would allow the balance to be reached. 06:40.967 --> 06:44.797 And we derived this nice little formula, and we 06:44.800 --> 06:46.000 computed some values from it. 06:46.000 --> 06:49.700 So we can summarize that calculation by looking at this 06:49.700 --> 06:54.400 little list of the things that control Earth's temperature. 06:54.400 --> 06:58.700 The Sun diameter and the Sun's temperature, because that 06:58.700 --> 07:03.300 controls how much radiation is emitted from the Sun. 07:03.300 --> 07:06.030 You can use the Stefan-Boltzmann Law to figure 07:06.033 --> 07:10.333 out how much each square meter of the Sun is emitting and 07:10.333 --> 07:12.533 then multiply times the surface area of the Sun to get 07:12.533 --> 07:17.533 the total amount of Sun's radiant emission. 07:17.533 --> 07:20.403 It also depends on the Earth-Sun distance, because as 07:20.400 --> 07:23.870 that radiation moves out away from the Sun, it diverges, it 07:23.867 --> 07:25.467 spreads out. 07:25.467 --> 07:27.327 And so the local intensity-- 07:27.333 --> 07:30.333 that is the watts per square meter-- 07:30.333 --> 07:31.573 decreases. 07:31.567 --> 07:34.267 So if you take a square meter and put it right at the Sun's 07:34.267 --> 07:37.267 surface, you'll get one value. 07:37.267 --> 07:40.727 You take that same square meter and move it further way, 07:40.733 --> 07:43.273 it'll have a lesser value, because some of that radiation 07:43.267 --> 07:47.967 has spread out and is missing your little one square meter. 07:47.967 --> 07:50.567 So the Earth-Sun distance is important. 07:50.567 --> 07:55.397 These two things together gave us the solar constant, the 07:55.400 --> 07:58.670 thing we called capital S, which you'll recall is about 07:58.667 --> 08:02.597 1,380 watts per square meter. 08:02.600 --> 08:04.970 It depends on the Sun diameter, the Sun temperature, 08:04.967 --> 08:07.027 and the Earth-Sun distance. 08:09.867 --> 08:12.197 The albedo also came in. 08:12.200 --> 08:16.300 The albedo was the fraction of the radiation that hits the 08:16.300 --> 08:20.770 Earth that is reflected back to space. 08:20.767 --> 08:23.027 Now, you know what are the bright objects 08:23.033 --> 08:24.273 and the dark objects. 08:26.600 --> 08:28.730 Clouds are very bright. 08:28.733 --> 08:30.933 They reflect light back to space. 08:33.600 --> 08:37.030 Snow fields are very bright. 08:37.033 --> 08:39.733 Bright desert sands and even bright sand 08:39.733 --> 08:42.403 beaches are very bright. 08:42.400 --> 08:44.100 They have a high albedo. 08:44.100 --> 08:47.070 Forests have a very low albedo. 08:47.067 --> 08:49.197 Ocean has a very low albedo. 08:49.200 --> 08:52.500 The value we used here-- we used 33% for the Earth-- 08:52.500 --> 08:54.770 that's an average of all of those. 08:54.767 --> 08:58.127 The bright objects, the dark objects, sum them all up on 08:58.133 --> 09:01.103 average, for Earth, it's about 33%. 09:01.100 --> 09:02.500 For other planets, it's different. 09:02.500 --> 09:04.500 That was important. 09:04.500 --> 09:09.400 And from these considerations, we came up with that formula 09:09.400 --> 09:12.100 predicting the planet's temperature, and we found that 09:12.100 --> 09:16.270 it was a little bit low. 09:16.267 --> 09:19.027 And we argued that the reason it was low is because we had 09:19.033 --> 09:21.073 neglected the greenhouse effect. 09:21.067 --> 09:24.027 We had assumed that the long-wave radiation emitted 09:24.033 --> 09:27.233 from the Earth's surface could just escape to space. 09:27.233 --> 09:31.503 Whereas, in fact, the greenhouse gases in our 09:31.500 --> 09:36.900 atmosphere, especially water vapor, CO2, N2O, NO, some 09:36.900 --> 09:41.630 ozone as well, absorbed some of that outgoing long-wave 09:41.633 --> 09:46.333 radiation in the atmosphere, heats up the atmosphere, and 09:46.333 --> 09:48.873 some of it gets re-radiated back to the 09:48.867 --> 09:50.167 surface of the Earth. 09:50.167 --> 09:53.167 That's an extra warming, if you like. 09:53.167 --> 09:58.027 So our estimates for planetary temperatures were universally 09:58.033 --> 10:01.373 too cold, because we had neglected that. 10:01.367 --> 10:06.767 Now, for Earth, it was a pretty serious discrepancy. 10:06.767 --> 10:09.797 You've got the values in your notes, but I think it was 10:09.800 --> 10:12.530 something like 30 degrees-- 10:12.533 --> 10:17.603 we were 30 degrees too cold in our simple estimates using 10:17.600 --> 10:21.670 these items. And that makes a difference between a habitable 10:21.667 --> 10:26.267 planet and an uninhabitable planet, at least for those 10:26.267 --> 10:28.597 living creatures that depend on having water 10:28.600 --> 10:29.870 in the liquid form. 10:32.767 --> 10:36.267 So we've learned a lot of lessons in last time's 10:36.267 --> 10:39.697 lecture, but one of them is that, well, the greenhouse 10:39.700 --> 10:41.330 effect is our friend. 10:41.333 --> 10:45.133 In other words, it makes this planet habitable. 10:45.133 --> 10:47.503 Were it not for that, the temperature would be 10:47.500 --> 10:50.400 universally below the freezing point for water, and we 10:50.400 --> 10:53.170 wouldn't have the kind of life that we have. Now, later in 10:53.167 --> 10:55.667 the course when we're talking about global warming, we'll 10:55.667 --> 10:59.127 come back to this subject and see maybe the greenhouse 10:59.133 --> 11:02.503 effect is increasing, which may not be a good thing. 11:02.500 --> 11:08.400 But on a broad-brush analysis, looking at the habitability of 11:08.400 --> 11:11.270 the planet, the greenhouse effect is great. 11:11.267 --> 11:15.397 It warms us up to a place where we can live. 11:15.400 --> 11:16.630 Any questions on that? 11:19.700 --> 11:20.770 So that's in the way of review. 11:20.767 --> 11:25.027 Now, I want to go on and do a different subject today. 11:25.033 --> 11:28.073 You may have noticed, or maybe you haven't, but I'm marching 11:28.067 --> 11:31.867 through these first couple of weeks of the course some of 11:31.867 --> 11:36.467 the most fundamental ideas in atmospheric science. 11:36.467 --> 11:38.027 They're not in any particular order. 11:38.033 --> 11:41.303 I've tried to organize them a bit so they make sense, but 11:41.300 --> 11:47.000 they're kind of my picks for the half-dozen or so most 11:47.000 --> 11:50.030 important concepts in atmospheric science. 11:50.033 --> 11:53.403 And this basic heat budget thing was one of them. 11:53.400 --> 11:54.700 We did that last time. 11:54.700 --> 11:59.000 Today I want to do another one called hydrostatic balance. 11:59.000 --> 12:02.300 The basic idea behind hydrostatic balance is that 12:02.300 --> 12:05.430 air has weight. 12:05.433 --> 12:07.933 Air is heavy. 12:07.933 --> 12:14.033 And the man who first made this so obvious was the 12:14.033 --> 12:19.873 Frenchman Pascal, who has the pressure unit named after him. 12:19.867 --> 12:25.397 He took a mercury barometer like the one that you had in 12:25.400 --> 12:30.670 your lab the other day and carried that from the base up 12:30.667 --> 12:35.167 to the top of a mountain in southern France. 12:35.167 --> 12:37.427 Puy-de-Dome is the name of the mountain. 12:37.433 --> 12:38.103 I've been up there. 12:38.100 --> 12:41.770 I drove up there as a historical gesture to Pascal. 12:41.767 --> 12:43.927 I wanted to see the place. 12:43.933 --> 12:48.203 And he noted that the pressure decreases as you go up. 12:48.200 --> 12:49.770 That was a great thing to notice. 12:49.767 --> 12:51.367 But he also explained it. 12:51.367 --> 12:56.897 He said, listen, the pressure at each level pushing up has 12:56.900 --> 13:00.970 to support the weight of the air above it. 13:00.967 --> 13:02.767 And so as you climb up a mountain, you're 13:02.767 --> 13:04.827 putting some air-- 13:04.833 --> 13:07.103 every time you step up, you're putting some air below you. 13:07.100 --> 13:11.970 So the amount of air above you decreases the higher you go, 13:11.967 --> 13:15.027 and therefore the pressure will decrease as well. 13:15.033 --> 13:18.073 Because the less air you have above you, the less weight you 13:18.067 --> 13:21.067 have to support with that pressure pushing up. 13:21.067 --> 13:23.067 This was remarkable. 13:23.067 --> 13:25.927 It seems so simple today, but yet it really did 13:25.933 --> 13:27.633 revolutionize the way we thought about the 13:27.633 --> 13:29.103 atmosphere and so on. 13:29.100 --> 13:30.130 So here's the idea. 13:30.133 --> 13:30.903 We're going to repeat his 13:30.900 --> 13:35.300 calculation, hydrostatic balance. 13:35.300 --> 13:38.670 "Static" here meaning there's no acceleration-- 13:38.667 --> 13:40.427 it's just a static balance-- 13:40.433 --> 13:41.773 "hydro" referring to fluid. 13:41.767 --> 13:43.867 Now, we're talking about air, but this applies 13:43.867 --> 13:46.367 to liquids as well. 13:46.367 --> 13:52.297 So here is a column of air extending from sea level up 13:52.300 --> 13:55.270 until we're outside the atmosphere. 13:55.267 --> 13:57.427 So of course, the atmosphere doesn't have a definite top, 13:57.433 --> 14:02.073 but we'll go up 10, 20 scale heights, and so we're above 14:02.067 --> 14:04.397 virtually all the atmosphere. 14:04.400 --> 14:07.770 So there's no pressure pushing down at the top of this, 14:07.767 --> 14:09.667 because there's no air up there. 14:09.667 --> 14:13.227 But there is a pressure pushing up. 14:13.233 --> 14:16.933 It's the ground pushing up on the air or the air pushing 14:16.933 --> 14:17.903 down on the ground, either one. 14:17.900 --> 14:20.100 They have to be in balance. 14:20.100 --> 14:25.400 The pressure force pushing up on this column of fluid is the 14:25.400 --> 14:27.230 value of the pressure times the base 14:27.233 --> 14:30.033 area of that cylinder. 14:30.033 --> 14:33.573 Think of it as a cylinder going up to space. 14:33.567 --> 14:36.467 So P times A has units of force. 14:36.467 --> 14:40.097 So remember, pressure is a force per unit area. 14:40.100 --> 14:42.330 So when I multiply it times an area, I'll get something with 14:42.333 --> 14:48.703 units of force-- that is to say, Newtons in the SI system. 14:48.700 --> 14:55.300 Now, this air column has a certain amount of mass, and 14:55.300 --> 14:58.600 I'll call that capital M. And any time you've got mass and 14:58.600 --> 15:00.470 you put it in the gravitational field of a 15:00.467 --> 15:04.067 planet, that object will have weight. 15:04.067 --> 15:06.497 And you compute the weight by multiplying the mass times the 15:06.500 --> 15:09.230 acceleration of gravity. 15:09.233 --> 15:12.973 So now, here's where the word "balance" comes in. 15:12.967 --> 15:16.267 I'm going to equate those two things. 15:16.267 --> 15:19.967 I'm going to say that this is a static system, so the weight 15:19.967 --> 15:24.067 of that air is held up by the pressure force pushing up at 15:24.067 --> 15:25.067 the bottom of the column. 15:25.067 --> 15:26.467 So there I've done it. 15:26.467 --> 15:28.527 The pressure times the area-- 15:28.533 --> 15:30.433 that's the vector up-- 15:30.433 --> 15:34.133 is equal to M times g-- that's the weight 15:34.133 --> 15:36.873 vector pulling down. 15:36.867 --> 15:38.167 I've just rearranged that. 15:38.167 --> 15:42.027 I've divided through by A and divided through by g to get M 15:42.033 --> 15:46.503 over A. That's the mass per unit area is equal to the 15:46.500 --> 15:49.870 surface pressure divided by the acceleration of gravity. 15:49.867 --> 15:53.267 This is the fundamental form in which we use it. 15:53.267 --> 15:58.297 This has units of kilograms per square meter. 15:58.300 --> 16:00.230 And now leave an inch in your notes and 16:00.233 --> 16:01.433 work out these units. 16:01.433 --> 16:04.133 This should have the same units as that. 16:04.133 --> 16:07.073 This one is obvious, kilograms per square meter. 16:07.067 --> 16:08.527 This one's not so obvious. 16:08.533 --> 16:10.603 You're going to have to remember what the units for 16:10.600 --> 16:13.600 pressure are and for g and work that out to see if the 16:13.600 --> 16:16.570 units check on that. 16:16.567 --> 16:20.767 So that's the basic assumption, and that's the 16:20.767 --> 16:22.627 basic equation that results from it. 16:22.633 --> 16:26.533 The mass per unit area in a column reaching from sea level 16:26.533 --> 16:30.633 to outside the atmosphere is just given by the pressure at 16:30.633 --> 16:33.103 sea level divided by the acceleration of gravity. 16:37.600 --> 16:40.630 Now, we could do a quick calculation of this. 16:40.633 --> 16:44.073 For surface pressure, I would put in about 1,000-- 16:44.067 --> 16:45.227 let's see-- 16:45.233 --> 16:55.833 101,300 Pascals, and I would divide that by 9.81. 16:55.833 --> 16:58.003 Let's round that off to 10. 16:58.000 --> 17:01.100 So I'll just knock a decimal place off here, and that'll be 17:01.100 --> 17:03.270 one 3 with one 0. 17:03.267 --> 17:06.327 That's going to be kilograms per square meter. 17:09.500 --> 17:14.130 And that's about 10 metric tons, 10 17:14.133 --> 17:18.433 tons per square meter. 17:18.433 --> 17:21.033 In other words, if you walk outside and imagine this 17:21.033 --> 17:25.473 one-square-meter base area and that column going up, there's 17:25.467 --> 17:30.427 about 10 tons of air in that column. 17:30.433 --> 17:33.173 Any questions on that? 17:33.167 --> 17:36.467 A pretty simple calculation. 17:36.467 --> 17:36.997 And it works. 17:37.000 --> 17:40.570 It's very accurate, actually. 17:40.567 --> 17:45.897 Now, that's a big step, and it was pretty easy. 17:45.900 --> 17:49.770 I've got another big step that's awful easy too. 17:49.767 --> 17:53.527 Now that I know the amount of mass per unit area, I can 17:53.533 --> 17:56.173 quickly compute the total amount of mass in the Earth's 17:56.167 --> 17:58.427 atmosphere. 17:58.433 --> 18:02.073 I just have to know the area of a sphere. 18:02.067 --> 18:04.967 We used that the other day for another purpose. 18:04.967 --> 18:07.427 That's 4 pi r squared. 18:07.433 --> 18:14.403 So if each square meter of Earth's surface has this much 18:14.400 --> 18:18.870 mass standing above it, then I just multiply that by 4 pi r 18:18.867 --> 18:22.597 squared, which is the total number of square meters on the 18:22.600 --> 18:25.370 surface of the Earth, and I get the atmospheric mass. 18:25.367 --> 18:29.427 So the atmospheric mass is given by P over g, which is 18:29.433 --> 18:33.833 the same as M over A, times 4 pi r squared. 18:33.833 --> 18:37.203 Now, this r is the radius of the planet. 18:37.200 --> 18:38.900 So I've done the calculation here again. 18:38.900 --> 18:44.630 I've put in 1,013, 9.81, like I did there. 18:44.633 --> 18:46.533 4 pi. 18:46.533 --> 18:49.003 There is the radius of the planet. 18:49.000 --> 18:51.600 Now remember, I had to get that in meters. 18:51.600 --> 18:55.200 Everything has to be in consistent units. 18:55.200 --> 18:58.200 You square that out, and you get 54 times 10 to the 17 18:58.200 --> 19:03.200 kilograms. So just like that, we got the mass of the Earth's 19:03.200 --> 19:05.970 atmosphere. 19:05.967 --> 19:07.197 Questions on that? 19:15.133 --> 19:16.733 I'm going to extend that calculation a little bit 19:16.733 --> 19:18.603 further now. 19:18.600 --> 19:27.230 If we've measured with some device the mixing ratio of CO2 19:27.233 --> 19:28.003 in the atmosphere-- 19:28.000 --> 19:30.630 how much CO2 is mixed in-- 19:30.633 --> 19:35.033 we measure it locally and assume that the ratio of CO2 19:35.033 --> 19:37.603 to air is the same everywhere. 19:37.600 --> 19:39.200 It's not a bad approximation. 19:39.200 --> 19:42.430 CO2 is pretty well mixed in the atmosphere. 19:42.433 --> 19:45.703 It varies by a few percent, but not much more than that, 19:45.700 --> 19:48.030 around the whole Earth's atmosphere. 19:48.033 --> 19:49.433 Then I could compute the amount of 19:49.433 --> 19:54.873 mass of CO2, basically. 19:54.867 --> 19:57.767 Remember, though, there's a couple of ways to represent 19:57.767 --> 20:01.067 mixing ratio. 20:01.067 --> 20:04.267 For example, you might see the mixing ratio of CO2 in square 20:04.267 --> 20:08.667 brackets sometimes written at about 390 parts per million by 20:08.667 --> 20:10.027 volume (390ppmv). 20:10.033 --> 20:11.603 That's a molecular count. 20:11.600 --> 20:15.900 That's how many molecules of CO2 per molecule of air. 20:18.667 --> 20:22.027 Or you can forget the pp-- that's parts per million-- 20:22.033 --> 20:25.503 and write it 390 times 10 to the minus six by volume. 20:25.500 --> 20:29.800 But you might want it by mass instead, in which case you'd 20:29.800 --> 20:33.030 have to multiply up front by the ratio of 20:33.033 --> 20:34.633 the molecular weights. 20:34.633 --> 20:35.703 CO2 is 44. 20:35.700 --> 20:37.170 Air is 29. 20:37.167 --> 20:40.097 You put the same number in there, you get 550. 20:40.100 --> 20:44.470 That's the concentration of carbon dioxide by mass, parts 20:44.467 --> 20:46.467 per million by mass (ppmm). 20:46.467 --> 20:50.227 It ends with an m instead of a v. 20:50.233 --> 20:55.103 Well then, if you want to know the amount of carbon dioxide 20:55.100 --> 20:59.200 in the atmosphere, you've got the total amount here, just 20:59.200 --> 21:01.800 multiply that number times that number. 21:01.800 --> 21:04.770 Don't forget, there's a 10 the minus six in this parts per 21:04.767 --> 21:07.297 million thing here. 21:07.300 --> 21:09.230 That's 10 to the minus six. 21:09.233 --> 21:13.103 And bingo, you've got the amount of CO2 in the Earth's 21:13.100 --> 21:13.200 atmosphere. 21:13.200 --> 21:18.570 So from a few simple ideas, our quantitative understanding 21:18.567 --> 21:23.897 of the atmosphere is advancing by leaps and bounds with these 21:23.900 --> 21:25.130 simple calculations. 21:27.633 --> 21:28.033 Yes. 21:28.033 --> 21:28.833 STUDENT: I'm sorry. 21:28.833 --> 21:29.233 What was the 29, again? 21:29.233 --> 21:30.933 PROFESSOR: The 29 is the 21:30.933 --> 21:33.773 molecular weight of air. 21:33.767 --> 21:36.627 You'll recall that the molecular weight of nitrogen 21:36.633 --> 21:43.703 is 28, and the molecular weight of oxygen is 32. 21:43.700 --> 21:48.470 There's more nitrogen than oxygen, so the average is a 21:48.467 --> 21:50.967 little closer to 28 than to 32. 21:50.967 --> 21:54.967 It comes out to be almost exactly 29 for the average 21:54.967 --> 22:00.767 molecular weight of air, being a mixture of N2 and O2. 22:00.767 --> 22:02.597 That's how that number comes from. 22:02.600 --> 22:03.870 Good question. 22:03.867 --> 22:05.127 Other questions or comments? 22:08.133 --> 22:11.003 Now, there's another form of the hydrostatic relation. 22:11.000 --> 22:14.870 This is for the total column. 22:14.867 --> 22:18.267 There's another form which involves just looking at a 22:18.267 --> 22:20.167 little segment of the atmosphere. 22:20.167 --> 22:22.627 So imagine a little cylinder. 22:22.633 --> 22:24.373 You might want to still consider it a cylinder, 22:24.367 --> 22:27.597 perhaps, but you slice it here, and you slice it there, 22:27.600 --> 22:31.130 and you just consider this little stub of an atmosphere. 22:31.133 --> 22:33.103 And you want to know how the pressure is different at the 22:33.100 --> 22:36.430 bottom than the top. 22:36.433 --> 22:41.033 And while I do this derivation, I have a volunteer 22:41.033 --> 22:43.803 who is going to take the Kestrel that you're familiar 22:43.800 --> 22:51.030 with and run up to the top of KBT and measure the pressure 22:51.033 --> 22:54.803 at the ground floor and the top floor. 22:54.800 --> 22:58.370 And when she comes back in class, we will have finished a 22:58.367 --> 23:02.727 prediction of what she will measure for that pressure 23:02.733 --> 23:05.003 difference, based on the derivation 23:05.000 --> 23:06.630 I'm about to do here. 23:06.633 --> 23:07.873 Thanks very much. 23:11.067 --> 23:15.427 Now, in many ways-- in some ways, the calculation is 23:15.433 --> 23:16.273 similar to this one. 23:16.267 --> 23:19.067 But instead of looking at the whole atmospheric column, 23:19.067 --> 23:22.797 we're just going to consider this little stub of a column. 23:22.800 --> 23:26.400 So there's a pressure at the bottom, which I'll call P sub 23:26.400 --> 23:29.170 B. That's the pressure at the bottom 23:29.167 --> 23:30.827 of that little cylinder. 23:30.833 --> 23:36.003 The pressure at the top is P sub T. The horizontal area of 23:36.000 --> 23:43.200 that little chunk is capital A. Its height is delta z. 23:43.200 --> 23:44.400 In other words, that's the difference between the 23:44.400 --> 23:47.330 altitude at the bottom and the altitude at the top. 23:47.333 --> 23:50.133 I usually use z in this course for altitude. 23:50.133 --> 23:50.903 STUDENT: What is A? 23:50.900 --> 23:51.570 PROFESSOR: Sorry? 23:51.567 --> 23:52.197 STUDENT: What is A? 23:52.200 --> 23:57.570 PROFESSOR: A is the top area of the cylinder. 23:57.567 --> 24:01.197 So now there are three forces acting on this thing. 24:01.200 --> 24:06.070 There's a pressure at the bottom acting on area A 24:06.067 --> 24:07.967 pushing up, and that's the little vector 24:07.967 --> 24:09.227 I've drawn in there. 24:12.367 --> 24:17.597 There's a pressure at the top acting over the same area A, 24:17.600 --> 24:23.100 so it's P Top times A. That's the vector pointing down. 24:23.100 --> 24:26.000 And there's another down vector, and that's the weight 24:26.000 --> 24:31.100 of the mass in that chunk. 24:31.100 --> 24:36.000 And if you want to know the mass of a fluid, you multiply 24:36.000 --> 24:40.400 the density, if you have it, times the volume. 24:40.400 --> 24:45.500 So I've written it here as Greek letter rho for density, 24:45.500 --> 24:48.400 and then the volume is the product of the area and the 24:48.400 --> 24:49.230 height of the cylinder. 24:49.233 --> 24:50.973 Do you remember that from trigonometry? 24:50.967 --> 24:54.097 The volume of the cylinder is just the area times 24:54.100 --> 24:56.370 the height of it. 24:56.367 --> 24:57.267 So that's the volume. 24:57.267 --> 25:01.167 So those three things together-- rho, A, delta z-- 25:01.167 --> 25:04.267 that's the mass of the air. 25:04.267 --> 25:09.427 It has units of kilograms, like the M over here. 25:09.433 --> 25:12.733 And then I multiply it times g to get the weight. 25:15.567 --> 25:19.197 Now, I want to simplify that formula a little bit. 25:19.200 --> 25:24.030 Notice A appears in each of the three terms. So I cancel 25:24.033 --> 25:29.933 that out, I bring the PT over, and I keep the rho g delta z 25:29.933 --> 25:30.773 on the right-hand side. 25:30.767 --> 25:37.497 And this is the formula that I've then derived for the 25:37.500 --> 25:43.530 hydrostatic law extending over some small range of altitudes. 25:43.533 --> 25:47.233 It says that the pressure at the bottom is greater than the 25:47.233 --> 25:51.233 pressure at the top by an amount given by the product of 25:51.233 --> 25:55.003 the air density, the acceleration of gravity, and 25:55.000 --> 25:57.870 the height of the object, the height of the air chunk, the 25:57.867 --> 25:59.097 air parcel. 26:01.967 --> 26:03.197 Questions on that? 26:05.967 --> 26:08.767 Let's do a calculation then. 26:08.767 --> 26:10.527 She's gone up to Kline Biology Tower. 26:10.533 --> 26:14.003 She'll take a measurement here, take the elevator up to 26:14.000 --> 26:15.630 the 12th floor, take a measurement there, and then 26:15.633 --> 26:16.373 come back to class. 26:16.367 --> 26:19.567 So we've got to hurry up and get a prediction 26:19.567 --> 26:20.297 of what that is. 26:20.300 --> 26:22.730 So we're interested in the difference between P at the 26:22.733 --> 26:28.133 bottom and P at the top. 26:28.133 --> 26:31.233 So here's what we need to do. 26:31.233 --> 26:35.773 We're going to use the standard value for air density 26:35.767 --> 26:37.067 at sea level. 26:37.067 --> 26:40.067 It varies a little bit, but for most purposes we can 26:40.067 --> 26:45.497 assume that it's 1.2 kilograms per cubic meter. 26:50.233 --> 26:53.873 g is 9.81. 26:53.867 --> 26:57.967 Units on that are meters per second squared. 27:00.667 --> 27:03.427 And we have to get the height, the delta z. 27:03.433 --> 27:07.873 Now, this is 12 floors, and I'm assuming that there's 27:07.867 --> 27:12.767 about four meters from floor to floor. 27:12.767 --> 27:17.067 For example, here, I think that's about nine or 10 feet, 27:17.067 --> 27:18.327 which is about three meters. 27:18.333 --> 27:21.073 But then there's a certain thickness before you get up to 27:21.067 --> 27:23.097 the next floor. 27:23.100 --> 27:24.570 So this is a pretty crude estimate. 27:24.567 --> 27:26.867 I've never measured it, but I'm going to assume that 27:26.867 --> 27:33.767 there's four meters per floor, so that's 48 meters for the 27:33.767 --> 27:34.997 height of Kline Biology Tower. 27:35.000 --> 27:37.670 We could measure this, and I probably would find that I 27:37.667 --> 27:38.567 have some error here. 27:38.567 --> 27:43.727 But to a rough approximation, that's going to be 48 meters. 27:43.733 --> 27:46.933 Now, check the units on this, because when I multiply these 27:46.933 --> 27:48.273 things together, I've got to get something 27:48.267 --> 27:51.467 with units of Pascals. 27:51.467 --> 27:55.067 And let me see what I come up with when I do that 27:55.067 --> 27:56.297 calculation. 28:00.267 --> 28:07.997 Yes, this gives me 565. 28:08.000 --> 28:10.170 If anyone has a calculator, please check me on this, 28:10.167 --> 28:13.127 because I'm not sure I can read my own handwriting here. 28:13.133 --> 28:18.773 565 Pascals. 28:18.767 --> 28:20.097 Would someone check me on that, please, with a 28:20.100 --> 28:21.600 calculator? 28:21.600 --> 28:28.770 1.2 times 9.81 times 48. 28:28.767 --> 28:29.267 Is that about right? 28:29.267 --> 28:29.997 It's 565? 28:30.000 --> 28:30.600 OK. 28:30.600 --> 28:39.270 Now, the instrument she's reading, the Kestrel reads out 28:39.267 --> 28:44.267 in hectoPascals, which is a hundredth of a Pascal. 28:44.267 --> 28:47.067 So we are predicting that she will find a pressure 28:47.067 --> 28:54.797 difference of 5.65 hectoPascals, which is also 28:54.800 --> 28:57.970 millibars, by the way. 28:57.967 --> 29:00.467 And so we'll see when she comes back. 29:00.467 --> 29:07.767 Now, this is a pretty useful idea, and you've used it 29:07.767 --> 29:08.967 already in your lab. 29:08.967 --> 29:12.227 For example, when you were using the-- a question. 29:12.233 --> 29:12.703 Yes. 29:12.700 --> 29:13.670 STUDENT: Sorry. 29:13.667 --> 29:17.997 Is a hectoPascal 100 Pascals or one hundredth of a Pascal? 29:18.000 --> 29:19.730 PROFESSOR: A hectoPascal is 29:19.733 --> 29:22.373 100 Pascals, yes. 29:22.367 --> 29:27.897 So 565 Pascal is 5.65 hectoPascals. 29:27.900 --> 29:30.900 I get that switched all the time. 29:30.900 --> 29:33.730 We've used the hydrostatic law in your lab a couple of times. 29:33.733 --> 29:38.103 For example, when you're using the mercury barometer, you are 29:38.100 --> 29:41.170 assuming that hydrostatic balance held within the 29:41.167 --> 29:43.197 mercury fluid. 29:43.200 --> 29:44.900 The atmospheric pressure was pushing down on 29:44.900 --> 29:46.330 the top of the reservoir. 29:46.333 --> 29:49.673 There's a vacuum above, so there's nothing pushing down 29:49.667 --> 29:50.767 on the mercury. 29:50.767 --> 29:55.127 But the weight of that mercury column is balanced by 29:55.133 --> 29:57.873 atmospheric pressure pushing on the reservoir. 29:57.867 --> 29:58.797 Let me just sketch that out. 29:58.800 --> 30:06.370 I know you did this in class also, but the mercury lies in 30:06.367 --> 30:09.727 here and up to there. 30:09.733 --> 30:14.973 So atmospheric pressure acting down here supports that column 30:14.967 --> 30:15.467 of mercury. 30:15.467 --> 30:19.727 And this calculation would work perfectly for that. 30:19.733 --> 30:22.133 Just remember, there's a vacuum up here, so there's no 30:22.133 --> 30:25.873 pressure pushing down, but the pressure pushing up is exactly 30:25.867 --> 30:30.527 balancing the weight of the column of mercury standing 30:30.533 --> 30:33.573 above the base level. 30:33.567 --> 30:37.297 I believe you also used it when you were correcting your 30:37.300 --> 30:40.100 pressure measurement to go down to sea level. 30:40.100 --> 30:45.830 You measured the pressure here in the laboratory, but that's 30:45.833 --> 30:49.533 a few meters above sea level, and so you needed to compute a 30:49.533 --> 30:54.873 correction to add to that to get the pressure at sea level. 30:54.867 --> 30:59.597 Are there any questions about the hydrostatic law before I 30:59.600 --> 31:00.430 change gears? 31:00.433 --> 31:02.203 I'm waiting for her to come back, but I've got a few more 31:02.200 --> 31:03.900 things I want to say about this. 31:03.900 --> 31:05.970 But I'd prefer to take your questions now. 31:08.900 --> 31:10.170 Here we go. 31:13.067 --> 31:16.927 Could you just call out the pressures for this? 31:16.933 --> 31:21.703 STUDENT: For the ground level, it was 1,007.9. 31:21.700 --> 31:25.470 PROFESSOR: 1,007.9. 31:25.467 --> 31:30.667 STUDENT: And then the 12th floor was 1,002.2. 31:30.667 --> 31:37.297 PROFESSOR: 1,002.2. 31:37.300 --> 31:39.970 So that difference is 5.7. 31:43.467 --> 31:44.897 Wow. 31:44.900 --> 31:46.100 Did we do good? 31:46.100 --> 31:47.230 Look at that. 31:47.233 --> 31:49.233 5.65, 5.7. 31:49.233 --> 31:50.003 We did good on that. 31:50.000 --> 31:52.070 It doesn't usually come out that well, actually. 31:52.067 --> 31:52.897 Congratulations. 31:52.900 --> 31:54.230 You're good. 31:54.233 --> 31:55.333 You're good. 31:55.333 --> 31:58.133 You jiggied those numbers just right, so we 31:58.133 --> 31:59.173 got the right answer. 31:59.167 --> 31:59.897 You get the point. 31:59.900 --> 32:04.070 Now, you didn't scale up the outside of the building. 32:04.067 --> 32:05.927 You went in and took the elevator up, I presume, right? 32:05.933 --> 32:06.473 STUDENT: Yes. 32:06.467 --> 32:10.327 PROFESSOR: So the question is how was she able 32:10.333 --> 32:14.573 to get good numbers like this inside the building? 32:14.567 --> 32:16.467 Isn't the inside of the building different than 32:16.467 --> 32:21.367 outside, where things are free to the rest of the atmosphere? 32:21.367 --> 32:26.067 Not really, because buildings leak around the windows and 32:26.067 --> 32:26.427 everywhere. 32:26.433 --> 32:32.373 So when you walk into a building, the pressure changes 32:32.367 --> 32:33.427 very, very little. 32:33.433 --> 32:37.573 The outside pressure, inside pressure is about the same, 32:37.567 --> 32:40.267 with a couple of exceptions. 32:40.267 --> 32:44.127 If you play tennis at one of these inflatable tennis 32:44.133 --> 32:45.833 courts-- you know what I'm talking about, they blow these 32:45.833 --> 32:47.003 things up?-- 32:47.000 --> 32:49.370 you walk into that building, you have to 32:49.367 --> 32:50.827 kind of pull the door-- 32:50.833 --> 32:53.533 high pressure inside, so you kind of have to 32:53.533 --> 32:54.873 push to get in there. 32:54.867 --> 32:59.697 If you feel when you're entering that building a 32:59.700 --> 33:02.500 pressure resistance when you go through the doors-- 33:02.500 --> 33:04.570 and most buildings don't have that-- 33:04.567 --> 33:06.967 but if you feel it, there's some kind of active air 33:06.967 --> 33:09.397 conditioning that's keeping the pressure inside higher or 33:09.400 --> 33:11.530 lower than outside. 33:11.533 --> 33:14.433 And you might want to worry then about whether your 33:14.433 --> 33:16.303 measurements are the same inside and outside. 33:16.300 --> 33:19.800 But for the most part, you can measure the pressure inside a 33:19.800 --> 33:21.530 building, and it's the same as it is outside. 33:21.533 --> 33:24.633 So in the laboratory, when I had you measure the pressure 33:24.633 --> 33:28.503 using the mercury barometer in Room 120, that's the same as 33:28.500 --> 33:30.000 the atmospheric pressure outside. 33:30.000 --> 33:33.430 There's no pressure differential in that building. 33:33.433 --> 33:34.933 You couldn't do that with temperature. 33:34.933 --> 33:36.873 Temperature's different inside, humidity's different 33:36.867 --> 33:40.167 inside, but the pressure likes to equalize. 33:40.167 --> 33:45.727 And so this is a very robust calculation, as the agreement 33:45.733 --> 33:48.173 shows there. 33:48.167 --> 33:51.267 Other questions about hydrostatic? 33:51.267 --> 33:54.427 Let's spend a few minutes on this. 33:54.433 --> 33:55.303 Questions on hydrostatic law? 33:55.300 --> 33:55.730 Yeah. 33:55.733 --> 33:57.933 STUDENT: Where do you get the 1.2? 33:57.933 --> 33:59.703 PROFESSOR: That is a standard 33:59.700 --> 34:04.970 value, sea-level density. 34:04.967 --> 34:06.997 You can compute it from the perfect gas law if you know 34:07.000 --> 34:08.870 the pressure and the temperature. 34:08.867 --> 34:10.127 I think we did that. 34:10.133 --> 34:13.833 If you measure the temperature in this room and the pressure 34:13.833 --> 34:15.973 in this room, you can use the perfect gas law 34:15.967 --> 34:16.927 to solve for that. 34:16.933 --> 34:19.333 And I think we did that earlier. 34:19.333 --> 34:21.073 Now, as you go up in the atmosphere, of course, that 34:21.067 --> 34:22.797 value changes. 34:22.800 --> 34:29.330 By the way, this simple formula has a value for 34:29.333 --> 34:32.403 density in it. 34:32.400 --> 34:35.970 That means you couldn't use it to predict pressure difference 34:35.967 --> 34:38.467 over a very great height difference, because the 34:38.467 --> 34:41.897 density changes over a great height difference. 34:41.900 --> 34:45.730 And so you wouldn't know what value to put in for rho. 34:45.733 --> 34:51.803 So this formula is best used over distances of a couple of 34:51.800 --> 34:55.700 hundred meters or less. 34:55.700 --> 34:59.230 If you're going to compute pressure difference over a 34:59.233 --> 35:03.273 much greater distance, I would use that exponential formula 35:03.267 --> 35:05.597 that we had in class a week ago. 35:05.600 --> 35:11.370 So this is for short distances delta z, because it's assumed 35:11.367 --> 35:14.867 a constant value for density. 35:14.867 --> 35:20.267 Now, by the way, this same law operates in the oceans. 35:20.267 --> 35:24.197 As you go down in the oceans, the pressure increases at a 35:24.200 --> 35:28.130 rate given quite precisely by the hydrostatic law. 35:28.133 --> 35:31.703 It's accurate to within at least one part in 1,000, 35:31.700 --> 35:34.070 probably one part in 10,000, so a very accurate 35:34.067 --> 35:35.567 calculation. 35:35.567 --> 35:38.027 And for that, it's a bit easier, because the density of 35:38.033 --> 35:40.733 water doesn't change very much as you go down in the ocean. 35:40.733 --> 35:44.333 Water is nearly an incompressible liquid. 35:44.333 --> 35:46.933 It doesn't change its volume when you put it 35:46.933 --> 35:49.503 under a high pressure. 35:49.500 --> 35:53.000 But remember, the density for water is much, much greater 35:53.000 --> 35:54.130 than that for air. 35:54.133 --> 35:55.273 What is the density of water? 35:55.267 --> 35:56.327 Anybody know? 35:56.333 --> 35:57.773 I bet you know from your problem set. 36:00.733 --> 36:01.073 One-- 36:01.067 --> 36:03.067 STUDENT: 1,000 kilograms-- 36:03.067 --> 36:07.627 PROFESSOR: 1,000 kilograms per cubic meter, 36:07.633 --> 36:09.673 about 1,000 times that value. 36:09.667 --> 36:14.667 So water is about 1,000 times denser than air. 36:14.667 --> 36:17.327 So I've been trying to impress on you today that air has 36:17.333 --> 36:20.503 mass, but of course, water has much more mass. 36:20.500 --> 36:25.730 Water is much denser liquid than air is. 36:25.733 --> 36:28.773 Other comments or questions about the hydrostatic law? 36:31.667 --> 36:32.097 Yes. 36:32.100 --> 36:38.800 STUDENT: I understand the practical applications of this 36:38.800 --> 36:39.700 for the ocean, but what are some ways this is used in 36:39.700 --> 36:40.970 meteorology? 36:47.533 --> 36:51.573 PROFESSOR: Let me give you an example of that. 36:51.567 --> 36:54.167 Notice that the rate at which the pressure increases as you 36:54.167 --> 36:58.227 go down depends on the density of the air. 36:58.233 --> 37:03.673 So let's say in the center of a hurricane, you had 37:03.667 --> 37:04.897 a lot of warm air. 37:07.467 --> 37:12.767 If you look at the perfect gas law, you can find that warm 37:12.767 --> 37:16.167 air at the same pressure is less dense. 37:16.167 --> 37:22.597 So you have a value of rho here that is relatively small 37:22.600 --> 37:24.900 compared to the rho out here, which is larger. 37:27.467 --> 37:32.597 Small rho in the center of the hurricane, large rho outside 37:32.600 --> 37:33.100 the hurricane. 37:33.100 --> 37:36.500 Now, let's say you've got the same pressure everywhere here, 37:36.500 --> 37:39.570 but as you come down in the atmosphere here, through the 37:39.567 --> 37:42.197 low-density region, pressure increases. 37:42.200 --> 37:46.600 But it increases slowly, because rho is smaller. 37:46.600 --> 37:49.900 Here, pressure increases, but it increases more rapidly, 37:49.900 --> 37:51.030 because rho is larger. 37:51.033 --> 37:53.573 Remember, the story here in this box is that the pressure 37:53.567 --> 37:59.227 increases as you go down at a rate given by the air density. 37:59.233 --> 38:03.303 So if the pressure is the same at all locations up here, by 38:03.300 --> 38:05.530 the time I work my way down under here, I'm going to have 38:05.533 --> 38:10.103 low pressure and relatively higher 38:10.100 --> 38:13.000 pressure outside the hurricane. 38:13.000 --> 38:18.070 So this hydrostatic law is the reason why there is low 38:18.067 --> 38:19.897 pressure in the center of a hurricane. 38:19.900 --> 38:21.530 We'll talk about this later on in the course. 38:21.533 --> 38:22.503 We'll talk about hurricanes. 38:22.500 --> 38:26.770 But we see it now, this basic law is responsible for this 38:26.767 --> 38:30.527 very important property of hurricanes having low pressure 38:30.533 --> 38:32.533 in the center. 38:32.533 --> 38:36.333 Because there's warm air aloft, and the warm air is 38:36.333 --> 38:38.773 less dense, has a smaller value of density. 38:38.767 --> 38:41.727 Then just use that formula, and you'll get that very nice 38:41.733 --> 38:44.603 and very practical, very important result. 38:44.600 --> 38:48.230 In fact, we typically-- very often, we measure the strength 38:48.233 --> 38:52.973 of a hurricane by how low that pressure reaches in the center 38:52.967 --> 38:53.897 of the hurricane. 38:53.900 --> 38:56.600 What you're really measuring when you do that is the amount 38:56.600 --> 38:59.870 of warm air aloft. 38:59.867 --> 39:02.567 And that's a good thing, because that's an important 39:02.567 --> 39:05.697 aspect of the structure of a hurricane. 39:05.700 --> 39:08.770 So there's a good example for that question. 39:08.767 --> 39:10.067 Other issues, other questions? 39:14.433 --> 39:15.733 Let's quit a bit early today. 39:15.733 --> 39:18.173 I'll see you on Friday.