GG 140: The Atmosphere, the Ocean, and Environmental Change
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The Atmosphere, the Ocean, and Environmental Change
GG 140 - Lecture 13 - Global Climate and the Coriolis Force
Chapter 1: Three-Cell Circulation Model of the Earth’s Atmosphere [00:00:00]
Professor Ron Smith: Well, so we’re talking about the general circulation of the atmosphere. Very important, very broad subject. And I’m going to re-hash a couple of things here, just to get us started on today’s lecture. But I did bring us to the point last time of looking at a kind of a textbook cartoon like this, that outlines in very simple form the three cell circulation model for the Earth’s atmosphere. You know, in its most basic form, this is a very simple problem. You’re just putting more heat in near the equator, and taking more heat out near the pole, and the differential heating causes a circulation.
But the nature of that circulation becomes rather complicated because, well, of the spherical geometry of the Earth, the fact that the Earth is spinning, which distributes the solar heat pretty well zonally around the equator, but then the other big thing is the Coriolis force. We’re going to be talking about Coriolis force today in some detail, but for right now, I’ll just mention it as one of the reasons it gives rise to this three cell, rather complex structure.
Now of these three cells, the one that is the easiest to see, in day to day weather, in satellite images, is the Hadley cell. So if you’re not going to memorize everything on this, be sure you know the Hadley cell. That represents the rising air near the equator, poleward air moving aloft, and then sinking and then returning towards the equator at the surface of the earth in a kind of a symmetric way around the equator. We’ll see it’s a little more complicated than that, but this is a good starting point. It’s a double kind of a circulation pattern driven by the excess heat put in from the sun along the equator. That’s the Hadley cell.
The Ferrel cell is less easy to see in the data, and so is the Polar cell, but what’s easy to see are these westerlies. Here you have easterly flow near the surface, here westerlies, and we’ll talk about that, as well as this being the belt of storms, which I’ll define for you in just a moment.
Chapter 2: Geostationary Satellite Images of Clouds [00:02:28]
Now, I showed these last time but they’re so loaded with information and detail, that I wanted to say a few more words about it. What this is, is a composite of several–Let me get my globe up here for a minute. So, the globe is spinning, of course, like this.
But there are several satellites in orbit of a type called geostationary satellites. If you put a satellite in orbit, approximately this far from Earth, about 42,000 kilometers, the time it takes to go once around in its orbit just happens to be 24 hours. The time it takes to go around depends on the radius of that orbit, and that particular radius takes 24 hours to go around. Well, if in addition, you’ve put that satellite into an equatorial orbit, and put it in the pro-grade direction, so it’s moving from west to east, I think you can see where I’m going with this, then as the Earth spins around, that satellite stays over the same point. So, Earth is spinning once every 24 hours, it takes 24 hours for the satellite to go around, that’s called the geostationary satellite. The beauty of that is, that then it looks like the Earth is just fixed, and you can see all the cloud motions beautifully. OK, now you’re only seeing one side of the Earth, but in fact there are several of these geostationary satellites in orbit at the present time. The United States has two, the Europeans have one, the Indians have one, the Japanese have one, all generally in their part of the world. And so, you’ve got multiple geostationary satellites. So with a little bit of computer wizardry, you can stitch those movies together, and make something like this.
Now, as you’ll see, there are an occasional little gap in the data where something went wrong. The other thing is that we’re not looking at reflected sunlight. You would think, well, the clouds are white. No, we’re not looking at reflected sunlight. Notice that the full globe seems to be illuminated. How could that be? There’s always a day side and a night side, so you could never have a situation like that. Well, we’re not looking at reflected sunlight. We’re looking at emitted radiation in the thermal infrared, the TIR. The wavelengths being used here are roughly in the range of eight to twelve microns. There happens to be in atmospheric window there which allows those photons to move through the atmosphere without being strongly absorbed. And therefore, what we’re seeing is the intensity of that radiation emitted by the earth reaching the satellite. In the areas where there are no clouds, we’re seeing radiation emitted from the land, and the land is hot. So the radiation is strong. In other areas where there are high clouds, those cloud tops are cold. They emit less intensely, and we show that with kind of a false–Notice we’ve kind of inverted the color scheme. It’s actually the radiation coming from here is large, the radiation coming from there is small, but we’re–we’ve coded this as white and this as black, in order to make it look a little bit like reflected sunlight, so the clouds look white.
So we’re trying to play a trick on you, or all of us, by coding that emitted radiation to make it look a little bit like clouds are white and land is dark and ocean is dark. So don’t be fooled by that, this is really emitted radiation. So it’s a temperature map, but we can easily find these high clouds because they are higher up in the troposphere. Remember, it gets colder as you go up, so those high clouds have a colder temperature and they emit less. There’s no doubt that these are clouds all right, but we’re seeing them because they emit less radiation. Any questions on that?
So then when I put this into motion, I want you to watch for various things here. I think I’ll start it and then stop it. So you’re seeing clouds generally along in the equatorial region. These are high, deep, convective clouds with heavy rain under them, and they’re generally moving from east to west. We’ll see in just a minute that they’re moving along with the easterly trade winds in those latitudes. Whereas in–down here and up here, you’re seeing these comma shaped clouds, notice they’re upside down in the south, so the commas look like this in the north, but they’re kind of reversed in the south. We’ll see later today that has to do with the flip in the sign of the Coriolis force. And they’re generally moving from west to east because this is a belt of westerlies. Westerly winds, or winds moving from west to east, and they’re carrying those storms eastward. In between, there are belts where there are fewer clouds, and that–fewer clouds, of course, means less rain as well, so this happens to be the belt of deserts, for example, the Sahara Desert lies in that range, and so does some deserts in Argentina, and Africa, and Australia, and so on.
So we’re going to be spending a lot of time on understanding the distribution of climate around the globe, but here’s our first hint at how this whole system works, when we’re studying the general circulation. So let me finish up this little film loop. And then I’ll move on to the next, which is very similar, except it’s taken in a different wavelength. Instead of near 10 microns, it’s about 6.7 microns, which is not a window. It’s not a wavelength for which the atmosphere is transparent. It’s actually a wavelength for which the atmosphere is rather opaque. It’s opaque because water vapor absorbs at that particular wavelength. So we call this a water vapor image, because the radiation you’re seeing there is coming from water vapor molecules in the atmosphere.
I think they look really cool, because you start to see, it looks like you’re making some kind of a cake, and you’re stirring the chocolate in with the vanilla. You see the dry and the moist air kind of being mixed in together in these mid latitude frontal cyclones. You still see the clouds all right, the clouds show up just as they did before, but in addition you’re seeing the water vapor that’s in and around those clouds. And there’s a dry zone in the north and south in these belts of deserts, and then there’s moisture, again, down here, being stirred in both high–northern and southern mid-latitudes by these frontal cyclones. Let me put this into motion. So again, you see the general sense of motion, east to west through here, but west to east here and here. I’ll finish up that loop. Any questions on this? Yes?
Student: The tropical rainforest, like in the Amazon, falls in that area that you said there aren’t like that many clouds but how does it rain so often there?
Professor Ron Smith: Well, this is one season of the year. So, this one happens to be September. And this belt of cloudiness, that you see over the oceans and then down over the continents, we call it the Intertropical Convergence Zone, and I’ll define it in just a moment, but it does move a little bit with season. But it is those clouds that is responsible for the rainforests. But you want to take a look at the whole year to see when it’s going to be raining there, and when it might not be raining. So this is only a few days in September, and it doesn’t give you the full picture. But we’ll talk about the seasonality of this in a couple days, and I think that’ll answer your question. But basically, yes. The rainforests are connected with this belt of rain and cloudiness, but perhaps not at the exact September moment that I’m showing here. OK, I think I can move on to the next.
Chapter 3: Climate Terminology [00:11:40]
So we’re going to be speaking on this subject together for the next week or two, and we’ll be failing to communicate unless we have a common terminology. And so, I just want to be sure that you understand these terms before I go any further.
First of all, I’ve broken this into kind of zones at various latitudes. And the first one I want to talk about is what’s called the equatorial zone. Of course, it’s the zone going around the Earth east to west. Some authors differ on these definitions, but for the present purposes I’m going to say from five degrees north to five degrees south around the globe will be what we call the equatorial zone. Some of the things that goes on in that zone is that at least part of the year, that’s where the Intertropical Convergence Zone is. That’s where the northeast and the southeast trade winds converge at low levels, and cause rising air, which goes back to the question was just asked. And in there somewhere will be the belt of rainforest, because it’s that rain that produces that kind of ecology, that forest that’s adapted to receiving a whole lot of rain every year.
This word doldrums, you’ll often find in the–sometimes in the scientific literature, sometimes in the broader literature, it refers to a region over the oceans where the winds are usually very weak. And that’s because you’re right in between these two trade wind zones. So in the early days, when the sailors were trying to travel across the equator, they would often become becalmed in this zone, and then the name doldrums came from that experience.
Then as we go north and south a little bit more, north or south, we enter what’s called the tropics. And again, authors may differ a little bit on their definitions. I’ve defined it as being five degrees to 23 degrees. Now I mean by that five degrees north, to 23 north, and five south to 23 south. So I’m referring to both the northern and southern hemisphere tropics.
Now I’m sure you know where that 23 comes from, actually should be 23 1/2 degrees. The tilt of the Earth is 23 1/2 degrees from normal to the plane of the ecliptic, and based on that we define the so-called Tropics of Cancer and Capricorn, that at least in the traditional literature, define what we mean by the tropics. It’s not so precise when we’re applying it to meteorology and climate. It’s more of an astronomical definition of the tropics than it is a climatological definition, but that’s OK, it’s close enough for our purposes. So in most cases I’m happy to define the tropics in that way as being these two belts from about five degrees up to the Tropics of Capricorn and Cancer.
In that region is where you have these steady easterly trade winds, and the Hadley cell is operating in both hemispheres, generally in that range of latitudes. Any questions so far on this?
Well, the sub tropics then, we’ll take to be from 23–and here authors again differ quite a bit, I’m going to take it 23 to 35, so it’s somewhere in here. Florida would be in the sub tropics. Generally you’ve got high pressure, slightly higher average pressure around the globe in that belt. So it’s called sometimes the STHP, the Subtropical High Pressure, it’s also the belt of deserts, it’s where the air in the Hadley cell is descending. And you know what that means, when air descends, it compresses adiabatically, it warms, and the relative humidity decreases. That air becomes drier and drier in the sense of relative humidity, so you’re unlikely to get clouds and precipitation in any area where the air is descending, therefore that’s the belt of deserts.
Over the oceans, it has a special name. In the open literature, sometimes it’s called the horse latitudes. That too is a reversal point, because you’ve got easterly trade winds equatorward of that, and westerlies poleward of that, so there’s a zone, again, where the average winds are very weak. The tradition of this term is a bit odd, but in the early days, the Spanish were carrying horses to the New World, and when they became becalmed in the middle of the Atlantic ocean, and their water and their food ran out, of course, the horses would die or they’d be eaten by the sailors, and then the carcasses would be thrown overboard. So a later sailor or ship sailing through that region, would find the carcasses of dead horses floating in the ocean, it became known as the horse latitudes. But what that really is referring to is the fact that there’s generally weak winds there. So if you’re depending on the wind for your propulsion, then that’s a difficult part of the ocean to cross. Questions on that?
OK, then we get into mid latitudes. That’s where we live. New Haven’s latitude is about 41 degrees north, so we’re clearly in that. Generally it’s a zone of westerlies, it’s the belt of frontal storms, remember all those little comma clouds we saw zipping from west to east? That really characterizes this part of the world. The jet stream is here, and the so-called polar front, which is a boundary between colder air to the north and warmer air to the south in the northern hemisphere is found in the mid-latitude zone. So, these are all really important terms. Are there any questions on them? OK.
Chapter 4: Dynamics that Drive Atmospheric Motion [00:18:11]
Now, the general circulation is pretty complicated. And other aspects of atmospheric motion such as storm development are pretty complicated. And in a course like this, we can’t go into all the details, but I did want to give you a broad outline of how the atmosphere moves. What drives it, and kind of what effects what effects what, cause and effect around the loop. I constructed this diagram, which I’m simply calling atmospheric dynamics, it’s kind of a box diagram that tries to point out some of the causality. The reason I think it’s appropriate to bring in at this point in the course, is that you’ve already dabbled with a number of these concepts. So I think you can appreciate the way some of these linkages work. Let me run through it very briefly. And this applies, by the way, not only to the general circulation, but to really every type of atmospheric motion, including sea breezes and thunderstorms, everything kind of works off this in one way or another.
So, differential heating is the ultimate cause of all the atmospheric motion on our planet. And of course, it would cause temperature differences. It would heat up some parts of the Earth more than others, the air in that region would become warmer or cooler relative to other regions. Now, the temperature differences generally cause density differences. We can understand that partially through the perfect gas law. Although remember, there are three variables in the perfect gas law, not just two. But generally, temperature differences give rise to density differences, as we have argued before in this course. Density differences give rise to pressure differences, not so much through the perfect gas law, but through the hydrostatic law. In other words, if I have cold, dense air here, hydrostatically, it’s going to have higher pressure beneath it. If I have warm, less dense air over here, hydrostatically, it’s going to have lower pressure beneath it. So it’s largely through the hydrostatic law that we develop pressure differences on a horizontal plane, so called sea level pressure differences, or pressure differences at other horizontal levels.
The pressure differences want to make the air move. Like when I pucker up and blow through my lips, high pressure in my cheeks, low pressure in the room, the air wants to move under the influence of that pressure difference. Now, the atmosphere doesn’t work exactly like that, as I’ll talk about later today. But generally, pressure differences want to get the air moving. In this case, however, the law that controls that to a first approximation, is the geostrophic law, which I’ll be talking about later today. And then it loops back on itself, because the wind will have an influence back on the temperature of the air. It does that in two ways. If I’ve got cold air here, and the wind is blowing, it’s going to carry that cold air to a different location. Last time we talked about if air moves, heat moves too, and that’s what I’m talking about here. So, air will advect from one place to another, carrying its temperature field with it, so that’s going to cause a change in some region’s temperature.
And the other thing, if air converges horizontally, and air rises, then you get adiabatic cooling. Or if air descends, you get adiabatic warming. So, there’s a couple of ways in which the wind can influence the temperature, and then the loop just continues. So, in order to understand atmospheric dynamics, one has to understand how all of these things work together in a system.
And if you took a course in fluid mechanics or in atmospheric dynamics, you’d be learning how to deal with that simultaneous and consistent action of all of those variables. But you already, with your knowledge, can understand the pieces, and we’ll try to describe how these different systems work in terms of how the pieces play a role in that. Any questions there?
Chapter 5: Coriolis Force [00:22:38]
OK, so we can’t go any further in atmospheric dynamics without knowing about the Coriolis force. And here’s just a few bullet points. The name comes from a French mathematician who discovered it. He was hired by the French Army to try to figure out why, when they were developing longer range cannons, that the cannon was hitting to the right–the shells were hitting to the right of where they aimed it. They didn’t notice it when the cannons were firing only a short distance, but as they got longer and longer trajectories, they noticed a systematic bend to the right, and he figured out it had to do, of all things, with the rotation of the Earth. Pretty fundamental discovery.
Now what is the Earth rotate with respect to? There’s always a question of whether there is some reference frame, some ultimate Newtonian reference frame, from which one can measure such things as the rotation of the Earth. Well, it seems like there is. If you look at the most distant stars and use that as a reference frame, and measure the rotation of the Earth relative to that, I think we’ve convinced ourselves, I think it’s true, that that is the absolute rotation of the Earth. We can take those distant stars as zero rotating inertial reference frame. So when I say rotation of the Earth, that’s what I’m referring to.
As we’ll see when we do some examples today, it’s in some sense you might consider to be a rather weak force. You’ve probably never noticed it as you walked around are throwing baseballs or footballs, you’ve probably never noticed the Coriolis force. Nevertheless, it is acting, but it’s only when you get up to larger scale systems, such as the atmosphere and the ocean that it turns out to be important. And not just important, but actually, in some ways dominant. In some ways it it’s the most important force that acts on the atmosphere and the ocean. It has the characteristic, the odd characteristic, of always being a deflecting force. In other words, whatever motion you already have with your object, it doesn’t try to speed you up or slow you down, it just tries to turn you. So it’s always just a deflecting force. It acts at right angles to the motion vector. I’ll come back to that.
And who’s ever seen a Foucault pendulum? Show of hands. Only a few of you. Well, it’s a remarkable way to see the action of the Coriolis force, it is evident in a Foucault pendulum. So, I’m going to take a minute just to describe a Foucault pendulum. So, imagine that I’ve taken a string and brought it down from the ceiling, and put a massive weight on it, like your coffee urn here, and I get it swinging back and forth, just as a simple pendulum. Turns out that if you can design a pendulum that will–of course, that would probably damp out in half an hour and we wouldn’t see any motion. But if it’s a massive enough bob, that weight on the end of the pendulum’s called a bob, it might keep going for several hours, or even a day or so. And if you did that, you’d begin to notice that the plane of that oscillation would begin to rotate.
And this is why, because we’re looking down now at the pendulum, so it’s swinging back and forth like this, and when it’s moving in this direction, this is a Coriolis force, it’s acting to deflect it to the right of the motion. Well that’s going to put it probably over onto this trajectory, but then when it’s swinging back in the other direction, again the Coriolis force acts to the right of the motion, the motion has reversed, and so has the Coriolis force. So that’s going to rotate the plane of that oscillation even more. So you can see, slowly through time the plane of that oscillation is going to rotate, in this case I’ve drawn the Coriolis force acting to the right of the motion vector. And so, the Foucault pendulum’s going to precess in the clockwise direction, the clockwise direction, through the action of the Coriolis force. Here’s a picture of one. And I’m not sure which one this is, but they’ve done the same trick that I’ve seen in the one in the Smithsonian. And that is, they put little markers along here, so that you can go look at the pendulum and go visit some other exhibits in the museum, and then come back a couple hours later and see whether new markers have been knocked down. And if they have, that will give you an indication of the procession of the plane of that oscillation. And notice here, these have been knocked down, and these have not. So we can see that thing is processing in this direction here, and you see these have been knocked down here, and not these. So indeed, this one must be in the Northern hemisphere, because it is processing in the clockwise direction. Is that clear? Questions on this?
This is a Foucault pendulum. Wonderful way for seeing this marvelous, mysterious force that is so important in the atmosphere and the ocean. Anything? Yeah?
Student: Does that mean in the Southern Hemisphere it goes counterclockwise?
Professor Ron Smith: That’s correct. It would go opposite in the Southern Hemisphere, it would go in a counterclockwise direction.
Student: Does that have anything to do with flushing a toilet?
Professor Ron Smith: I’ll get to that. I’ve got a subject today called toilet bowl mythology, which I’ll get to. Thanks for the question. So, this thing called a Coriolis force, there’s a nice, simple formula for it, and here it is. The magnitude of the Coriolis force is given by the mass of the object, times 2 times the speed of the object. I’m using capital Greek letter Omega to represent the rotation rate of the Earth and the sine of the latitude. Rotation rate of the Earth once around is 2 pi radians. It takes one day to do that, so if I express that in SI units, that’ll be units of radians per second, or just inverse seconds, and the value is this. Now being in this field, I carry that number around in my head, but there’s no reason for you to. Also, it’s so easy to re-derive. You just put 2 pi up front, and then figure out how many seconds there are in a day, and divide by that, and you’ll get that famous number, capital Omega, the rotation rate of the Earth, expressed in radians per second.
Student: What is U?
Professor Ron Smith: Use the speed of the object. Omega is the rotation right of the Earth. Phi is the latitude that you’re at. So that tells you about the magnitude of it. The direction, the formula doesn’t tell you, but you can remember. In the Northern hemisphere, it acts to the right of the motion vector. In the Southern hemisphere, it acts to the left of the motion vector. Remember, there always has to be motion, if there’s no motion, there’s no Coriolis force. So there always has to be–the Coriolis force only acts on moving objects. If an object is stationary in the Earth reference frame, there’s no Coriolis force acting on it. So there’s never any ambiguity about the direction, because it already has a motion, and we use that to figure out what direction of the Coriolis force is. Julia?
Student: So even with–so you were saying when you throw a football you don’t see the Coriolis force, is that just ‘cause the distance you throw it isn’t long enough?
Professor Ron Smith: Yeah, long enough to see it, exactly. If you were some Superman that could fire it 4-5 kilometers, it would land a meter or so to the right of where you threw it. But, you know, a normal football quarterback doesn’t throw it that far.
Student: So does that act on anything that moves?
Professor Ron Smith: Anything that moves.
Student: Like even a car?
Professor Ron Smith: Anything, anything. As I’m walking across this floor, there’s a Coriolis force acting on me at the moment. I can’t feel it because it’s too small, but it’s there. Anything that moves on the earth feels the Coriolis force, whether it’s a–well, just anything at all, anything that moves. Absolutely. And there’s no Coriolis force at the equator. And you’ll see that right here, because if you put in zero degrees for latitude at the equator, the sine of zero is zero. And so, there’s no Coriolis force at the equator, and of course it’s strongest at the poles. At the poles, latitude is 90, the sine of 90 is one, and you get everything else here for the strength of the Coriolis force. Questions there?
Let me do a little example, I put one up on the board here. I’ve written that same formula out, did I get it right? Yes. Let’s say we’ve got a mass of one kilogram, at a latitude of 30 degrees north, moving at 10 meters per second, on Earth, so there’s the rotation rate of Earth. I plug it into the formula, one kilogram times two, times the speed, times the rotation rate. The sine of 30 degrees, if you don’t remember, is 0.5. That comes out to be 7.27x10-4 Newtons. That’s a force unit in the SI system. And what direction does it act? Well, it depends on what direction the motion is. I haven’t told you that. If the object is moving towards the northeast, the Coriolis force will be towards the southeast. If the object happens to be moving towards the southeast, the Coriolis force will be towards the southwest. So always to the right of the motion vector, whatever the motion vector is. Coriolis force.
Chapter 6: Geostrophic Balance [00:33:07]
Now, here’s where it comes in to the atmospheric application. Very often, we find that air is moving along, around the atmosphere, in the atmosphere, in a state of geostrophic balance. Geostrophic balance is a particular kind of force balance. Remember hydrostatic balance was a kind of force balance, it’s a balance between weight and the vertical pressure gradient force. Well, this is more of a force balance in the horizontal. So it’s horizontal forces that we’re talking about. Pressure gradient force and the Coriolis force, when they come into balance, we call that geostrophic, geo meaning Earth, stroph coming from a root meaning turning. So you know it has to do something with the turning of the Earth, well, in this case it has to do with the Coriolis force.
It results in some very odd things that I’ll show you, as we move through this section. For example, the wind blows parallel to the isobars instead of across them. And the speed of the wind is related to the isobar spacing. The geostrophic force balance applies actually most of the time in the free atmosphere. I hesitate to give you a number, but I would say maybe 90 or 95% of the time, when you’re above the boundary layer, the air is moving along at a pretty good state of geostrophic force balance. This is why it’s so important for us to understand this particular characteristic of the atmosphere. It is invalid, however, you do not have geostrophic force balance down in the frictional boundary layer, where there’s a lot of turbulence, or in strong storms or other disturbances. So it’s not universally applicable, but it’s widely applicable.
Here’s a little cartoon then, of describing what I mean by geostrophic balance. So if I have–this a plan form view. So I’m looking down at the surface of the Earth, north, south, east, west. And if there happens to be high pressure to the south on this particular day, and low pressure to the north, then there’s going to be a pressure gradient force acting on a parcel of air sitting here from high to low. If I’ve got an object, no matter how small it is, if there’s slightly higher pressure on one side and lower pressure on the other side, there’s going to be a net force on that. That net force is what we call the pressure gradient force. We call it a pressure gradient force because it arises because there is a gradient. Gradient means a change in pressure with position.
So, high pressure here, low pressure there, means that object is going have a slightly higher pressure on the southward side, a little lower pressure on the northern side, and the net force is going to be to the north, called the pressure gradient force. Now, if it’s in geostrophic balance, it has to have a Coriolis force that is equal and opposite to that. This is a vector balance, so the speed and–the magnitude and the direction have to be exactly opposite to that. And here’s how the reasoning goes. If the force must be like that, then what must the air be doing, how must it be moving. It must be moving, then, from west to east, so that the Coriolis force which is at right angles to it, has the orientation given by the green vector. So this is the only consistent–once I draw in that pressure gradient force, than that vector and that vector are locked in, that’s the only way I can draw them if that’s air parcel is to be in geostrophic balance.
These lines of constant pressure are called isobars, and you could label them with a pressure, 1,020, 1,010, 1,000, 990; those are just lines of constant pressure drawn on a map called isobars. And so, you’ll notice that in this circumstance, instead of the air blowing from high pressure to low, like you would have expected, because of the Coriolis force, it moves along the isobars, not across them. Quite a surprise, because in our common world, you know blowing up a tire to a bicycle, or whatever, air tends to move from high pressure to low. But on a larger scale, where the Coriolis force plays a role, it’s more like this. Air moves along the isobars rather than across them. Questions here?
Now, so mathematically, we want to write down an expression for this geostrophic balance, so let me write that down. First of all, I need a formula for the pressure gradient force, and that’s given by the pressure gradient times the volume of the object. The pressure gradient is defined as the rate at which pressure changes with position, as units of pascals per meter. And the volume is just the volume of the object in cubic meters. The mass of any object would be given by its mass density times its volume. So, going back a couple of slides where I have the formula for the Coriolis force, and adding these two formulas to it, I can come up with this very important, slightly lengthy equation. But on the left, is the pressure gradient force given by this, and on the right, is the Coriolis force, given here on the board still. But for mass, I’ve put in the product of the density and the volume of the air parcel that’s being considered. So now, we’re not considering any old object–it’s not a, car it’s not a freight train, it’s not a howitzer shell, it’s a little chunk of air.
We’re trying to come up with a geostrophic balance formula for a chunk of air and rho is the density of that air, volume, 2UΩ sin(φ). Now, the volumes are going to cancel out from the left and right, and I can solve this formula for u, I’ll put a subsequent geostrophic on it, it’s given by the pressure gradient divided by 2ρΩ sine of the latitude. This is a rather remarkable formula. It doesn’t solve every problem in atmospheric dynamics, but it’s still rather remarkable. It says if you know something about the pressure field, I can tell you something about the wind. It’s a relationship between spatial variations in pressure and the wind speed and direction, called the geostrophic formula.
Let’s see. Let me do the problem on the board first, and then we’ll go on here. Hope you can see this. So again, we’re looking down on a map, north, south, east, west. And on this particular day, the isobars look like this. This one’s labeled 1,012 millibars, 1,011, 1,010, 1,009, so these are lines of constant pressure, isobars, constant pressure. And let’s say for the sake of argument that the spacing between those isobars is about 100 kilometers. About 50 miles between these different things. So the pressure gradient is derived as the rate at which pressure changes with distance, so it’s going to be one millibar divided by that distance. So it’s ΔP/L, one millibar divided by 100 kilometers, that’s 100 Pascals, divided by 10 to the fifth meters, so the answer is 10-3 Pascals per meter. That’s the pressure gradient, the magnitude of the pressure gradient in that region. It’s a reasonable value.
Now I’m ready to plug it into the formula. Let’s see if I’ve transcribed it right, pressure gradient over 2ρΩ sine of the latitude, OK. The pressure gradient, we decide, is 10-3. I’m going to use sea level density, 1.2 kilograms per cubic meter, here comes the rotation rate of the Earth, capital Omega (Ω). And for this problem I’m going to say we’re in New Haven, the latitude’s around 42 north, sine of 42 is about 0.67. Plug all that in I get 0.08 times 102, and just moving the decimal place over, that’s eight meters per second. So under this circumstance, the speed would be eight meters per second. And I know what the direction is going to be too, because it’s going to be parallel to the isobars, with the high pressure on the right. High pressure on the right, so that’s going to be the wind vector, eight meters per second, under that kind of a pressure situation. Any questions on that? An easy calculation to do, once you know something about how the how the isobars look. Question there?
So, if I were to imagine another isobar pattern that was in the form of a circle, or a bullseye, whenever you have an isolated low pressure region with high pressure all the way around, it you normally refer to that in meteorology as a cyclone. Sometimes the word cyclone has other connotations. It may imply some kind of severe storm, or a tornado, or something like that. In the Wizard of Oz, when Dorothy was sucked up into the tornado, she was referring to that as a cyclone. Right? Cyclone. But in meteorology, it has somewhat broader and less dangerous meaning. It’s just a low pressure center with a high pressure around it. And so, I’ve drawn the isobars in this way. And let’s imagine a parcel sitting there, kind of halfway out from the center.
So, there’s high pressure here, and low pressure there, so the pressure gradient force is going to be from high to low, so the red arrow’s going to be the pressure gradient force, PGF. If it’s to be in geostrophic balance, the Coriolis force must be equal and opposite to that. And the only way, given the properties of the Coriolis force, the only way it can have a Coriolis force in that direction is if the wind velocity was to the north. Because then, the Coriolis force is to the right of the motion vector. So I conclude, by that reasoning, that the air must be moving northwards, or I could say it’s a southerly wind at that location. If I repeated the calculation there, and there, and there, here I would find that the wind must be blowing in that direction, here I would find it must be blowing in that direction, here, in that direction. So in fact, the air must be going around the cyclone in a counterclockwise direction, if we’re in the Northern hemisphere. Is that clear? So, low pressure, cyclone, Northern hemisphere, air goes around counterclockwise. We say that’s the cyclonic direction, the air is moving cyclonically around that low pressure zone.
But now, we want to take a somewhat broader view of this, so I’m going to introduce–OK, this is a big jump, but I think we’re ready to do this. So, here’s the equator. So Northern hemisphere, Southern hemisphere. There’s the problem I just worked out, there’s the low pressure, and that’s called a cyclone. I’m going to introduce an anticyclone in the Northern hemisphere. And then in the Southern hemisphere, I’m going to have a cyclone and an anticyclone. And we’re going to understand what way the wind blows around each of these systems. Well, let’s stay in the Northern hemisphere, with high pressure in the center, the pressure gradient forces it outwards, so the Coriolis force must be inwards, so the air must go around this way. It must go around in the clockwise direction. Around an anticyclone in the Northern hemisphere.
In the Southern hemisphere, we’ve got the low pressure in the center again, because it’s a cyclone, but now, the pressure gradient force is again in towards the center. But because the Coriolis force acts to the left of the motion vector, we end up with just the opposite circulation, direction to the circulation. It’s clockwise here, and counterclockwise around the anticyclone.
So, this is a weather map taken from yesterday. At 500 millibars, so about five kilometers up in the atmosphere, you can consider these black lines to be the isobars. And they’re from individual balloon launches. Every place there’s a balloon launch, they’ve picked off the wind speed and direction, and put a wind barb at the location of that balloon launch. And notice first of all, that the wind vectors kind of parallel the isobars. There’s high pressure down here, low pressure up there, you might expect the wind to be blowing from high to low, but no, it blows along the isobars instead. And that’s what we expect from geostrophic balance.
Notice also that the wind is stronger here, it’s getting up to be 70 or 80 knots, where the isobars are packed closely together. But where the isobars are far apart, the wind is very weak. We’ll take a look at this formula, this example we did for a minute. If those isobars were further apart, then the L would be greater. If L is greater, the pressure gradient is less, and putting the pressure gradient into the geostrophic formula, I get a weaker wind. So directly, whenever you have closely spaced isobars, you have strong winds. Whenever the isobars are far apart, you have weak winds. That follows directly from the geostrophic formula. For example, here’s a map, and no winds are given. This is a sea level pressure map. No winds are given. What can you conclude from this? Well, a lot, now that you know the geostrophic law. Because you know that around the anticyclone, the winds are generally going in this direction. And because those isobars are spaced out quite a bit, there’s a rather weak wind blowing around here. Here near this low pressure, near this cyclone, with closely spaced isobars, there are very strong winds going around in a counterclockwise direction. We’re out of time, but we’ll continue this on Monday. Have a good weekend.
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