GG 140: The Atmosphere, the Ocean, and Environmental Change
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The Atmosphere, the Ocean, and Environmental Change
GG 140 - Lecture 20 - Ocean Water Density and Atmospheric Forcing
Chapter 1: Ocean Depth Profiles [00:00:00]
Professor Ron Smith: So now we are into this new section of the course, oceanography. And last time I gave an overview of the nature of the ocean basins, basically the geometry of the basins in which the water sits. We connected that to plate tectonics, both to make the point that those ocean basins are changing through geologic time, but also to get at this curious issue of how the oceans are not just kind of a random roughness on the Earth, but they really represent two basic levels of Earth crust.
The continental crust which sits a bit higher, and the ocean crust which sits a bit lower. And that’s why there are these vast areas of the ocean that are at about the same depth below sea level, these abyssal planes. It’s an oversimplification to say, of course, that the ocean basins have flat bottoms like a bathtub or something.
But there is a little bit of a tendency that way because they do have—they do have ocean crust beneath them which floats at a certain level and gives you the depth of about 5 kilometers in many, many places around the world ocean. Some places are deeper, some places are shallower, but there is that reference level from which we work.
Any questions on that? Then we moved into a discussion of how to measure salinity and temperature in the oceans, and I talked about some of the methods to do that. And I wanted to pursue that just a little bit further, and then get into some quantitative methods for estimating how the atmosphere forces various things that go on in the ocean. So let me just start here, and I showed you this last time. This is a typical ocean sounding. Zero refers to sea level, and then depth is in meters below that, and this one going down to 4,000 meters.
The temperature is shown here with a distinct cooling as you go down that starts pretty quickly below the surface. Maybe in this case, just a couple hundred meters below the surface you go from a mixed layer to a strong gradient region called the thermocline, “cline” referring to change, and “thermo,” of course, referring to temperature. Then you get down to temperatures below that that are four, five, six degrees Celsius and colder, and that fills most of the interior of the ocean basin.
So this point I was emphasizing last time about how the surface temperature does not represent the deep ocean temperature is shown nicely here. Salinity is somewhat similar in that you can have strong gradients near the air surface, but more uniform conditions below. In this case, it’s saltier, then gets a bit fresher, than gets very slowly saltier below. But always staying in this remarkably narrow range between 34 and 35 and 1/2 parts per thousand.
Then the dynamical quantity we’re interested in is the seawater density with units that you’re familiar with, kilograms per cubic meter, for example. And that is a generally a function of temperature and salinity. The warmer the water is, the more it expands a little bit and its density is less. The colder it is, the more it contracts, the density is greater. And salt, when you add salt to water, the salt goes a little bit into the pores between the water molecules, which increases the density.
So the greater the salt content the greater is the density, and the fresher the water is the less is the water density. And since density is what gravity acts on, we’re particularly interested in this. For example, if you wanted to apply the hydrostatic relation to find out how fast pressure increases with depth, you’d want to use the density derived from the salinity and the temperature.
The word pycnocline is used when you’re referring to this gradient region as it applies to density. So here it’s the halocline referring to the salt, there the thermocline referring to the temperature, and the pycnocline refers to that combined quantity which is the density of the water.
These profiles vary from place to place in the world ocean. For example, the high latitudes where you might have a lot of precipitation and because it’s cold, not very much evaporation. The surface waters, notice the salinity scale is reversed on this diagram. The water might be a little less saline at the surface than it is down deep. Whereas in the tropics, for example, in the belt of deserts, the descending branch of the Hadley cell, you’d have a lot of evaporation, but very little precipitation. So the water near the surface would be salty there, but not at the bottom. The bottom is more homogeneous.
Chapter 2: Salinity [00:06:05]
Let’s take a look at the salt itself, and this will connect a little bit with the lab that you’re currently writing up. So if you take a kilogram of seawater, about 965 grams of that is water, H2O, and about 34.4 of that is in weight, in mass, grams, is the salt. Then if you break up that little salt wedge into its chemical compositions you get this little pie chart here. It shows the most abundant ions. So assuming that this quantity, these chemicals break apart when they dissolve in the water into their positive and negative ions. By far the largest contribution is the chlorine, 18.96 grams of the 34, a little more than half of that salt mass is due to the chlorine.
And let’s look at the other negative ions here. There’s a sulfate radical, SO4, and there’s a bicarbonate, HCO3 with minus signs for the charge. Then we come to the cations, which is what you’re studying in the lab this week. The dominant ones are sodium, which is by far and away the most abundant. But also we have magnesium, calcium, and potassium. Those are the four things that you have data for in the lab as you go from fresh water river into the ocean.
So you might want to compare these numbers that I have here in lecture, the 10.56 grams per kilogram of seawater. The 1.27 for the magnesium, 0. 4 for calcium, and 0.38 for potassium with the values you had. But just remember, when we got to Long Island Sound we only had a salinity of about half that of ocean water. So you should divide these numbers approximately in half, or double your numbers before you do the comparisons. But I think that’d be useful to try to understand how our measurements agree with the textbook values for what makes up the salts.
The reason they can make a diagram like this is these relative proportions are the same everywhere you go in the world ocean. Even when the salinity varies a little bit, maybe down to 34 or up to 36. These all increase in proportion. So that these relative numbers are quite stable when you go from place to place in the world ocean. And of course, our simple theory of this is that this represents an accumulation over geologic time of small amounts of these chemicals, these salts, that have come into the ocean from rivers. And you’re testing that idea in using your lab data.
Any questions there?
Chapter 3: Stability in the Ocean [00:09:36]
Now, you remember in the atmosphere we spent a good deal of time talking about static stability, that is, we looked at the role of the atmospheric lapse rate–how the temperature changes in the vertical–to whether parcels can rise and fall easily or not. Or whether that atmosphere might even be unstable and break down to convection. So we defined an unstable lapse rate, a stable lapse rate, the inversion, which was an example of a very stable lapse rate. We’re interested in the same kind of analysis in the ocean. But there’s a couple of essential differences.
One is that the ocean density depends not just on temperature, but on salinity too. So we’ll have to take into account both of those quantities. We’ll do that simply by computing the density. We’ll base our stability analysis on water density, rather than on temperature or salinity alone. The other big difference is that while air is compressible, so as a parcel lifts in the atmosphere and moves to lower pressure, it expands–does work on its environment and its temperature changes. That affect is very, very small in the oceans.
It’s not exactly zero, but it is so small that for many quick calculations we ignore that volume change and that so-called adiabatic cooling or adiabatic warming as air parcels rise up and down in the atmosphere. We ignore that in the ocean. So because of these two differences, the idea of what determines stability in the atmosphere and the ocean are significantly different.
The basic principle is that while temperature and salinity can either increase or decrease with depth, the density derived from them must always increase with depth. We saw that in the earlier diagram here that the salinity, for example, decreased with depth, then increased with depth. But when I computed the density it increased smoothly. If there was ever a layer where the density got less with depth, that would be an unstable layer, and it would immediately cause convection and would mix.
Years ago when I first starting out, I was looking at some CTD data from a ship and thought I found an unstable layer. Went running up to the chief scientist to show him this remarkable discovery, and he quickly pointed out where I’d made a mistake in my calculations. So once again, the ocean was found to be stable. It’s been proven many, many times. The point is, you could have an unstable layer, but it would exist only minutes because immediately convection would begin, that layer would mix, and you might produce a layer of constant density from that mixing, but you would be unlikely to sustain any layer that is unstable.
So you should always expect to find this kind of relationship, and if you don’t, you’ve either made a wonderful discovery, or somewhat more likely you’ve screwed up your calculations when you compute a density from temperature and salinity. So it’s a very important principle. Probably almost universally true in the atmosphere—in the ocean, sorry. So yeah, so there it is.
Chapter 4: Density [00:13:24]
Now, this diagram is nice because it combines together hundreds of CTD and Nansen bottle data sets from three of the major world ocean basins–the Pacific Ocean, the Indian Ocean, and the Atlantic Ocean. What I like even better about this diagram is that it’s superimposed. These balloons of data are superimposed on a diagram that has lines of constant density on it. Let me walk you through those lines first. So on this axis is the salinity in parts per thousand that we’re used to working with, PPT. On this axis is temperature in degrees Celsius. And then these lines are lines of constant seawater density.
And by the way, so what is this unit here? This unit is the density of seawater with the 1 and the 0 dropped off. So you would read this as 1,028.5 kilograms per cubic meter. And since it only changes in those last couple of digits, we don’t want to keep writing that 10 in front of all of those values. But you can see that the density is greater down here, 29, 28.5, and 28. And that corresponds to salty water and low temperatures. And the seawater density is less up here, 25, 24.4–24.5, and 24 for high temperatures and low salinity. So that’s as we thought it would be.
Are there any questions on that?
There’s a couple of other things I’d like you to notice about this diagram. Up in the warmer temperature region, up near the top of the diagram, because of the way these lines are tilted, the density is most sensitive to temperature changes. Whereas down in the colder regions where the lines are tilted more like this, the density is more sensitive to salinity changes.
So when you’re talking to a tropical oceanographer, and we have one in our department, Professor Fedorov. He is usually most interested in talking to you about the temperature profiles in the ocean. Of course, he hasn’t forgotten that salinity plays a role in this. But in the tropical—in the warm parts of the ocean, the temperature is the primary control on the density.
If you’re talking to an Arctic oceanographer, and we have one of those in our department, Mary-Louise Timmermans. She’s usually most interested in understanding the salinity field because the salinity field hacking down here, as you can see, is more in control of the density field in the colder regions of the ocean.
So I don’t want to exaggerate that too far. Both are always playing a role, but in warm areas it’s mostly the temperature, in cold areas it’s mostly the salinity that’s controlling the density.
Any questions on this diagram?
So let’s look at these big kind of balloons. The Pacific Ocean is generally a little fresher, and maybe even a little bit warmer than the Atlantic Ocean. There is a little part down here which is common to both oceans, they come together down there, called the Antarctic Bottom Water. We’re going to be talking about that probably next time. It’s a mass of water that’s produced near Antarctica. It flows Northward along the bottom of the ocean, and we find it in both the Atlantic and the Pacific.
So there’s a common element. But other than that, there is a general systematic difference between the Pacific and the Atlantic Ocean. And that, of course, has to do with atmospheric controls, which I’m just about to get to. If you do a cross-section, this happens to be a North-South cross-section through the Atlantic Ocean from high latitudes to low. Perhaps you can’t read in the back, but the Equator is there. 10 degrees North, 20, on up to 60 degrees North, and on down to minus 70 degrees latitude, 70 South.
Here’s a depth scale going from 0 down to 6 kilometers. And then the horizontal scale is what I indicated. And contoured here is temperature. Temperature actually gets a little bit below 0 here. Minus 0.2. But by the time you’re up in here it’s 1.2 degrees, 2.4. And then you’ve got some strong gradients near the top, well that’s the thermocline, and then near the surface of the ocean in the mid-latitudes and low latitudes, you have this little lens of warm water floating on the surface.
Many of you I expect have been swimming in a fresh water lake in early summer. You may have noticed that as you swim out, the water’s pretty comfortable in terms of this temperature, but when you stop swimming and let your legs dangle down a bit, suddenly it’s a lot colder down there. What’s happening is that that fresh water that’s warm is a little bit less dense than the cold water. So it floats in a little layer on top. Well that’s exactly what’s going on here.
There’s salinity involved, but mostly it’s a temperature control, and you’ve got a warm layer of water kind of floating in a broad, deep, cold layer of ocean. And you’ll find this as you go North to South in all of the world’s oceans, you’ll find this thermocline that separates this warm sphere from the deep cold sphere. That’s kind of a common element in both the Atlantic, the Pacific, and the Indian ocean. So we’ll talk more about that later on.
Any questions on what this diagram shows? By the way, this blue water here, that’s that Antarctic Bottom Water that I was talking about. I’ll come back to that. Are there any other simple analogies to this? Well, I would point you to the Great Salt Lake as kind of an analogy to the world ocean. The Great Salt Lake unlike most lakes doesn’t have an outflow. When water flows into the Great Salt Lake in Utah, it doesn’t spend a few weeks there and then flow in some river into the ocean. No. It basically stays there until it evaporates.
This is a no outflow lake. And for that reason, you get the same kind of concentration of salts, the same process, as you do in the ocean. So it’s a little mini ocean, if you like, in the sense that it has to evaporate its water, get rid of it, and therefore, it concentrates the salts.
Now, in this case, it concentrates the salts even much more than the ocean does. The salinity is three to five times saltier than in the ocean. Has anybody swum in the Great Salt Lake? Dip your body in it? But you’ve heard about it, right? The remarkable thing is how high you float. You’re sitting there, and maybe from here on up you’re out of the water. Why? Because the water is salty, and therefore more dense. Following Archimedes rule, the denser the water is, the more mass you’re displacing, the more the buoyancy force, the higher you float. So this is a good thing to think about when you’re trying to understand basically how oceans work. It’s kind of a little mini ocean.
Chapter 5: Atmospheric Forcing of the Ocean [00:22:08]
Now, the subject I was going to get into today, and I will get into it, is atmospheric forcing of the ocean. For the most part, any motions you have in the ocean, in fact, almost everything that goes on in the ocean, is driven from above by either sunlight hitting the ocean, or some other interaction with the Earth’s atmosphere. Now is there any forcing from below at all? Well there is a little bit of geothermal heat coming out of the bottom of the ocean basins, but that’s quite small. So except for that, it’s pretty much the oceans are driven by the atmosphere from above. These three things are how that happens.
You can add or remove heat at the top of the ocean, either by radiation, the Sun’s radiation, or long-wave radiation emitted from the ocean surface. Or by contact with the atmosphere. if you have cold air, that’ll suck heat out of the ocean. If you have warm air that’ll conduct heat into the ocean. Or you can add fresh water by precipitation, or remove fresh water by evaporation, leaving the salt behind. These two things will change the salinity of the surface, and therefore, the density. And this one will change the temperature and, therefore, the density. Or you’ve got the wind stress. You’ve got the wind blowing over the ocean surface transferring some momentum to the ocean, and getting the ocean moving in that way.
So we’re going to need to understand each of these three mechanisms. For example, here is our best estimate of the heat flux in and out of the ocean. Basically, where you get, for example, cold water and warm air, the heat is going into the ocean, and that’s shown as positive on this diagram. Where heat is coming out you’ve got the blue or the red color shown as negative. But it varies from place to place. It depends on the air temperature. It depends on how much sunlight is being received. It depends on the water temperature, whether heat is going into the ocean or whether heat is coming out. We’ll talk about this quantitatively in just a moment.
Then the other one is E minus P or P minus E. This one is evaporation minus precipitation. The sign is derived in that way. So whenever water is evaporating, you get a yellow signature here, and the units on this are in millimeters per day. How much water is being peeled off and evaporated in terms of a layer depth per day.
Whereas in the blue, and you notice it coincides with the ITCZ, for example, where you have heavy precipitation, precipitation exceeds evaporation. So the sign of this quantity of negative shows up as blue. Then you get up into mid-latitudes again where it’s colder, there’s less evaporation, but you have the frontal storms, and once again, you’re getting an excess of precipitation over evaporation. So this is driving motions in the ocean as well.
Then the final one is the winds over the ocean, which change a bit with the season. There’s January, there’s July. But generally you’ve got Westerlies in the mid-latitudes and Easterlies in the low latitudes in both seasons. And that is generating motions in the ocean itself. So that is where we’ll end the pictures today. And I want to get into quantifying these things.
So here I’ve repeated those three ways in which the atmosphere forces the ocean. Adding or removing heat, precipitation and/or evaporation, and the wind stress. Now, I want to make a further distinction from this. I’ve got a little—I’ve drawn a little cartoon down here. Here’s the cloud that’s precipitating, adding fresh water to the surface of the ocean. There’s some evaporation, that’s this one. Here’s some heat being added or removed, either by radiation or by the effect of the atmosphere itself. And here’s the wind stress–I’m going to use Greek letter Tau (τ) to represent that.
But I’ve indicated that there’s a layer which feels these inputs directly, and that’s called the depth of that capital D. That will vary from time to time and place to place, but typically that’s a very small fraction of the ocean depth. When I do examples, I often set D equal to 100 meters. Roughly only the first few tens or couple of hundred meters feels these inputs directly. The rest of the world ocean will feel them, but it’ll be an indirect influence. Something coming not directly from the atmospheric input, but something then coming out of this other layer that feels a direct influence.
Chapter 6: Atmospheric Forcing of the Ocean: Adding and Removing Heat [00:27:50]
So be aware that I’m making this distinction for all three of these categories of forcing. My first job is to figure out what does it do to this layer that feels the influence directly. The first one will be adding heat. And you remember that if you add a certain amount of heat to a mass, M, with a heat capacity Cp , it’s temperature will change by delta T. We’ve had that formula before. If I write that in a per unit area basis, I’ll divide the Q by A, and the mass by A, and I’ll leave everything else the same. I’ve just divided both sides through by the area that I’m considering, which in oceanography we don’t usually even consider that area. We just do these calculations on a per unit area basis.
The mass per unit area, if you think of a piece of the ocean, a little cylinder of area A, its total mass is going to be its volume times the density of the water. And the volume is going to be the product of A and D where D is the depth of that cylinder. So the mass per unit area here, which divides out the A is going to be the product of ρ D, the water density times the depth of the layer that I’m considering.
So then let me just take that and plug it into this formula. I get Q over A–that’s the heat being added per unit area. Plugging this and I get ρ D Cp delta T, and solving that for delta T, I get Q over A over ρ D Cp. And I think we’ve seen that formula before, but there we have it again. This will tell you how much the temperature of the surface layer of the ocean changes when you put in a certain amount of heat per unit area. And it depends on the density of the seawater, the depth of the layer over which you’re distributing that heat, and the heat capacity of water.
So a quick example of that–I’m going to take a solar flux of 1,000 watts per square meter. Let’s say we have sunlight hitting the surface of the ocean with that kind of intensity. That persists for one day, which is 86,400 seconds. So let’s say that persists for one day. And that heat is distributed over a depth of 1,000 meters. How much will the temperature of the surface ocean change in that case? Let’s just put the numbers into the formula that I’m just extending this formula over.
So to get the heat, I need to multiply–remember this is watts. That’s a rate of putting in heat. A watt is a Joule per second. So I need to multiply the time by the watts to get Joules per square meter, which is what Q over A means, that’s Joules per square meter. So that’s going to be 8.64 times 10 to the seventh Joules per square meter up top–that’s just the product of those two numbers.
And then down below I’m going to put in a estimate for seawater density, a depth of 100 meters, and the heat capacity for water which is about 4,200 units of Joules per kilogram per degree. That’s Cp. you work that out, and you might want to check me, but I got 0.2 degrees Celsius.
So the warming turned out to be pretty small because that heat got distributed over a pretty great depth, and the mass of that water is large, and its heat capacity is large. So although I put in almost 10 to the eighth joules for every square meter, the rise in temperature was pretty small. Yeah.
Student: Is that supposed to be 1000 meters?
Professor Ron Smith: That’s supposed to be–I made the mistake here, 100 meters, sorry. Thanks for pointing that out. 100 meters, yeah. That’d be too deep. Very rarely does the ocean mix down to 1,000 meters, as you saw in the diagram. By the time you get down to 1,000 meters you’ve got a lot of gradients and things like that, so it doesn’t usually mix that deep.
So you can apply this to a wide variety of circumstances, depending how you alter the input numbers. The heat could come from the Sun like I’ve done, it could be a loss of heat by radiating to space. It could be a heat transport from the atmosphere in or out of the ocean, depending where–remember the diagram I showed you–some parts of the world is positive, some parts it’s negative. So this number’s going to depend on where you are, the season of the year, and so on.
Chapter 7: Atmospheric Forcing of the Ocean: Precipitation and Evaporation [00:33:47]
Now, the next one will be number two. This is a little trickier. Let me define the salinity as the mass of the salt over the mass of the water in any particular sample that you have. I’m going to be looking at change, so I’m going to define salinity at time 1 as being the mass of salt at time 1 over the mass of water at time 1. And salinity at time 2 as being MS2 over MW2.
But I’m not going to consider cases where I’m adding and removing salt. That doesn’t happen very often in the ocean. Instead I’m going to consider cases where I add and remove fresh water. Either it rains, so I’m adding fresh water, or it evaporates. When it evaporates it leaves all the salt behind, so you can say that I’m subtracting fresh water. So I’m going to assume that in the changes I’m about to describe that MS1 is equal to MS2. So I’m not changing the mass of salt in the sample, but I’m going to change the mass of salt water.
I’m going to define the change in salinity as being S2 minus S1, and just using these formulas that’ll be MS over M water 2 minus MS over M water 1. And notice I’ve dropped the subscript now on the MS because I’m going to assume that the mass of salt doesn’t change.
With just a little bit of manipulation, I can rewrite this as MS over M salt water 1, M water 1 over M water 2 minus M Salt over M water 1. What I’ve done is just multiply and divide the first term by M water 1. You see it there and there. I’ve just multiplied and divided by the same number. But now this can be rewritten as the salinity at time 1 times the bracket M water 1, M water 2 minus 1 (S1(MW1/MW2 – 1).
So the change in salinity is related to the salinity you started with, plus the ratio of the fresh water you started with to the fresh water you ended with. That’s an MW1, and an MW2, the fresh water that you have.
Now I’m going to try to motivate this a little bit geometrically. Imagine I’ve got a column of seawater, perhaps some little section of the surface layer of the ocean I just sliced out with a cookie cutter. And then I add a layer on top of it. It’ll be capital D in its depth, and then I’m going to put a little thin layer on top of it with depth little d. Or alternatively, I could strip off, I could evaporate, a layer of depth little d. That would make the seawater saltier.
With that geometric interpretation, I can write the ratio in my formula MW1 over MW2 as being capital D over capital D plus little d. In other words, I’m just adding water, and that’s going to change the mass ratio before and after. That allows me to rewrite this formula finally to–can you see it if I do it here?
So I’m going to take this formula, divide through by the S1 and use little added formula. So delta S over S1 can be written finally as big D over D plus little d minus 1 (D/(D+d)-1), which can be simplified to little d over big D plus little d (d/(D+d)). That’s the formula we’re going to use.
If that seems confusing in any way, all I’ve done is conserved salt, but change the amount of fresh water, and therefore the salinity changes. So if I add a layer of fresh water of depth D with a minus sign there, that’s going to make the water less salty. The minus sign reminds us of that. If D is positive, if I’ve added fresh water, the ocean gets less salty. If I subtract by evaporating, then D would be negative. Negative times a negative is positive, and the salinity would increase. So that’s the way you’d use this formula.
Let’s do an extreme example to be sure this formula’s correct. Let’s say that I doubled the depth. I put a whole thick layer of fresh water on there equal to what I had originally of seawater. Well then it would be minus capital D over capital D plus capital D. That quantity is going to be minus 1/2. That means the salinity’s going to drop in half of what it was before because I’ve added an equal amount of fresh water. So that salt now had to mix into twice the amount of fresh water. That’s going to drop the salinity exactly in half.
Now this would be an extreme example. Usually in the ocean we’re just talking about adding some small fraction. So let me do a little more realistic example of this. My example would be adding 1 meter of fresh water to 100 meters of ocean water. How much will the salinity change?
So ΔS over S1 is going to be minus 1 over 100 plus 1. So that is approximately minus 0.01. This is approximately 35 parts per thousand let’s say, the original salinity. So ΔS is going to be 1/100th of that. It’s going to be 0.35 parts per thousand, which means the new S, the new salinity, I’ll call it here S2, is going to be 34.65 parts per thousand.
I started with 35 parts per thousand, 100 meter column, added 1 meter of fresh water on top, mixed it all in, the salinity drops by 0.35 parts per thousand, which takes me from 35 to 36.65. That would be a typical situation. And that’s a big difference, considering that the full range of ocean salinity is only from about 34 to 35 and 1/2. That’s a big difference in ocean salinity. And that’ll cause circulations to begin, because the seawater density will have changed.
Any questions on that?
Student: Is that only for the surface?
Professor Ron Smith: Pardon me?
Student: Is that only for the surface?
Professor Ron Smith: Yeah I don’t imagine–well that’s right. So that would affect this top 100 meters. Then what happens later on to the rest of the ocean, that would remain to be seen. If, for example, in this case, I made it less dense, because I made it less saline, that water would probably remain floating there. If I had made it more dense, it might find itself so dense that it would then fall to the bottom of the ocean. But that’s a separate calculation. Here I’m just trying to understand what happens to the surface water when you change its salinity.
Chapter 8: Atmospheric Forcing of the Ocean: Wind Stress [00:43:23]
Now we turn to the third type of forcing, which is wind stress. This is perhaps not quite as obvious as the others. I don’t think I’ve greatly surprised you by either of these calculations here. It’s pretty much common sense.
But what happens when the wind blows over the ocean? When the wind blows over the ocean it produces a wind stress. Frictionally, just like when you move your hand across the table pushing down, there’s a stress being applied on that table. That table starts to move sometimes in the direction that you’re pushing. And that’s my question. If I put the wind in that direction, from left to right in this diagram, does the ocean water begin to move in the same direction?
Well, you’re probably on your guard because you know that the Coriolis force exists so that some things are not quite as obvious cause and effect-wise when you have the Coriolis force. In fact, here’s what happens. So I’m going to draw a plan view here now, North, South, East and West. I’m going to imagine I’ve got a Westerly wind, that is a wind from West to East. So the wind is blowing in that direction. The wind stress will also be in that same direction.
I’d like to prove to you, in the remaining three or four minutes, that when I push the water in that direction it goes to the right. It doesn’t go in the direction I’m pushing. This will involve a new kind of force balance called the Ekman force balance. It has something in common with geostrophic force balance, but it’s also quite different.
So let’s say that I start to put this wind stress, it’s going to–in the first few minutes it is going to do the obvious thing. It’s going to start the water moving towards East. But then as soon as it gets moving, if feels the Coriolis force and it will begin to bend. After a few hours, I will argue now, that the flow will be towards the South–that’ll be the Ekman flow. To the right angles of that will be the Coriolis force acting on that Ekman flow. And that’s going to be equal and opposite to the wind stress itself.
And so when you push on the water on a rotating Earth with the frictional stress from the wind, instead of moving in the direction you push, it moves at right angles. To the right in the Northern Hemisphere, to the south—I’m sorry, to the left in the Southern Hemisphere. We could do a calculation. I probably don’t have time to finish this today, but I can do a calculation of how fast that water will move.
If I have, again, a column of depth D, it has a mass given by ADρ across the area, the height of the column, and the density of seawater. If it begins to move you’ll have a Coriolis force given by 2, the mass, the speed, the rotation rate of the Earth, and the sign of the latitude (2MUΩsinϕ). That’s going to have to balance the wind stress.
There’s a nice empirical formula for wind stress that I’ll give you. I won’t derive it, it’s just an observational quantity. I’ve written there empirical, which means it’s an experimentally-derived formula. It’s given by constant 0.003 times the density of the air blowing over the water’s surface times the wind speed squared. And this has units of Newtons per square meter.
So now, let me just equate those two things following the prescription for the Ekman layer force balance. I’m going to say when you’ve got a well established balance of this sort, then the Coriolis force which is 2, let me substitute this in for M, Dρ then comes U omega sin(latitude)ϕ, must be equal to τ acting on that same area A. Since that’s a force per unit area, I need to multiply it by a surface area to get a Newton. So I got Newtons and Newtons on both sides.
And then solving this for the speed of the flow in the Ekman layer, I can cancel the A’s, and I have τ over 2DρΩsinϕ, sign of the latitude. That is the box formula for this part of how the atmosphere drives the ocean. If I know the wind speed I can compute τ. If I know τ, the depth of the layer that’s being influenced directly, the density of seawater, the rotation rate of the earth, and the latitude, I can compute how fast this water moves off to the right under the influence of the wind stress.
So we’re out of time today. This needs a lot more discussion. But you’ve got a problem you’re working on for the problem sets where you need to use that formula, I believe. And I will talk more about the physics of this on Friday.
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