Session 11: Gas Exchange Across the Air-sea Interface

The following content is provided under a Creative Common License. Your support will help MIT Open Courseware continue to offer high-quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT Open Courseware at ocw. mmit. edu. — Okay, so gas exchange. So you have um this is sort of the last lecture on how does stuff get into and out of the ocean. Um and then after the midterm we'll deal a lot more with uh what goes on within the ocean in terms of internal cycling and dynamics. And so what we're interested in is for soluble gases there's going to be fluxes in and out of across the RC interface. And we're interested in these fluxes which we're going to call F. [clears throat] Um typically the way this is formulated or discussed is in terms of two components. It's going to be the flux is going to equal a kinetic component. times a thermodynamic component. Um, and if any of you have had transport theory like you know diffusion um, uh, thermal conductivity uh, this should look very familiar. It's it's basically almost everything in this formulation has been stolen from physical chemistry and chemical engineering adapted uh and there's a lot of crossover between the chemical engineering and the environmental uh communities in terms of gas exchange. The kinetic component is going to depends on things like properties of the water and turbulence. So this kinetic component we're going to try to set up so that lots of the kinetic component are independent of the particular gas. And we'll also show you how to um reference you know gas one to say gas 2. The thermodynamic component is going to depend specifically on individual gases. and is essentially a measure of things like you know the solubility of the gas the atmospheric and ocean concentrations. Oh, do you want to move I can I can't — I was just going to move the guys there. So um and this is pretty standard this separation uh into a kinetic component and the thermodynamic component. So we're going to first we're going to talk about solubilities and the thermodynamic component. So you need to know a little bit about the composition of the air and there's some notes uh or a table in the notes. You know most of the atmosphere is nitrogen. It's about 78%. You know next comes oxygen which is 21%. And then everything else uh argon is 1%. if you look at this well it actually looks like it already adds up to 100%. Everything else in the atmosphere is small. It's either in parts per million or parts per trillion — by volume — by well yeah I'll get there in a second. Um the way these are usually reported are as partial pressures. So you guys all remember you know PV= NRT. um the partial pressure is the gas is the pressure that a gas would exert

um independent of the other gases all by itself at that particular concentration or molar ratio and temperature. um you'll see it traded back and forth whether it's reported as parts per million or mole fraction because the pressure that a gas exerts depends upon how many moles are there not on its molecular weight. So ppm is either the um is the mole fraction or it's the partial pressure relative to the total pressure. For most gases you can assume ideal behavior that is that u the u you can actually use PV equals NRT without any corrections for some gases example CO2 you actually need to look at the fugacity um it's a fairly minor correction but it's basically deviations from the ideal gas law associated with the fact that CO2 is interacting with the other gases. Um, also almost all the airc gas exchange measurements are reported for dry air. So they've actually dried the air and remove the water vapor. So it would be ppm, you know, parts per million relative to dry air, not the total pressure. So when you go back and apply total pressure, you just have to make sure you know um what's you just need to keep track of what's been done with the water vapor because it can be several percent particularly in the tropics. Okay. Um, usually we're going to formulate our equation as uh in terms of not the partial pressure but the um but the concentration that would be in trying to see what it would be the concentration in the air se in the air sea surface that would be in equilibrium with the atmosphere. So if you have say a partial pressure of some gas C, we want to find out right at the sea surface what would be the concentration that would be in solubility equilibrium with that partial pressure. So rather than working in partial pressure in one side and concentration in the other, we're going to convert partial pressure into a uh corresponding solubility equilibrium concentration. Now there's a couple ways of doing that. Um and I will warn you that the literature is not consistent. Um if you remember from physical one form of Henry's law it's basically relating the partial pressure of a gas to a solubility concentration as a function of an equilibrium constant temperature and the ideal gas uh the ideal gas constant. You also see things often reported like this which is technically the Bunson solubility law where this K equilibrium is now actually a bunch solubility coefficient. Um but in the literature people are very sloppy and sometimes you see it sometimes you see the Bunson solubility law written as um Henry's law. So they go back and forth and you just have to be very aware. Uh sometimes if this is also written as a bet theta um you just need to be aware of where the data is coming from and what they're talking about — and units will help that. — Yes. Well units — well use them. — Yeah if they use them. — Okay. — Yeah. Give me pressure of your gas C in that those last equations, not pressure gas A. Oh, this is supposed to be. Yeah, it's sorry. It's pressure of it's the atmospheric pressure of C. So, I guess I

could put a little C here if you wanted. It is the pressure of C, but it's the Yeah, sorry, I didn't match those very well. It's the atmospheric pressure of those of compound C. It's not Yes, that's a good point. It's not total pressure, right? It's the partial pressure of that particular gas. Thanks. Um units. the modern standard for gas exchange literature is gases are reported in micro moles per kilogram. Um unfortunately you will see all sorts of things. Uh the most confusing particularly if you go back in the older literature is that oxygen um which is ubiquitous not just for gas exchange but for you know hydrography and for biology was often reported as milliliters per liter. Um and you say milliliters of what? Um, it's actually milliliters at STP or standard temperature and pressure. And there's a conversion in the notes, but is essentially 1 milll per liter equals about 43. 6 microles per kilogram. And since we're assuming ideal gas behavior and it's at standard temperature, pressure, this But so you'll often see the older literature oxygen values that are about a factor of 50 lower than if you had them in micro moles per kilogram. Okay. So, what determines the this Bunson coefficient or the Henry Block coefficient? Well, there's essentially three things. You've got a gas that you're trying to stick into water. So, we've got all these water molecules that are happily bound to each other. Remember, they're all sitting there undergoing their lovely little um hydrogen um hyd have all their little hydrogen bonds. They're quite happy. You got to put a hole in the water in order to put a gas in. And so, the first thing is going to be uh the energy to create a cavity. And that's going to just depend upon the solvent, right? It's just basically you got to break the water, break those hydrogen bonds and make room to stick your gas in. The second thing is the um the energy to keep the solute in the cavity [snorts] and that's going to depend upon the solute. Um, you know, for example, a big solute might bounce around more slowly, so it stays in the cavity more easily. A smaller solute um has a much harder time and has a higher diffusion rate and it's going to bounce out uh quite quickly. And then the third component is the solute solvent interaction which of course depends upon both the solute and the solvent. Um, and this is things like, you know, we know that water is polar unless it's 400° and 5,000 bars. [snorts] Um, water is going to be polar and so polar gas molecules are going to be more soluble than non-polar gas molecules, right? Because it's easier you get you gain energy back from the solute solvent interaction and it helps if things are forward. Now, there's uh some examples in the notes. Um we're going to do a fair amount of stuff looking at noble gases. Um so the noble gases are helium up to xenon x zeon xenon. Yeah. Um and a lot of the gas exchange work has focused on those not because those are the geochemical things that you want to know. um but they don't have other physical and biological sources and syncs and so if you can isolate using the noble gases what's going on with gas exchange and then you

can apply them to your molecule of interest whether it's dimethyl sulfide or CO2 or oxygen or something like that. So a couple of rules about solubility. Um if we were to plot now the Bunson solubility against temperature, what we would see is that helium would be down here, xenon would be up here. So there's a couple things. One is um solubility, sorry, increases as you decrease temperature. And that makes sense. Colder water is going to hold more gas. If anybody's ever played with a soda can or a soda bottle, you know, if it's warm, you open it up, it spills all over the place. Well, that's because a lot of the CO2 and the carbonated water has degassed and is sitting over the lid where if you cool it down, put it in the refrigerator, more of that gas goes into the water and you have less pressure, a smaller pressure head built up underneath the cap. The other thing is that um there tends to be a relationship in this case for the noble gases where the larger gases are more soluble. Um, and that's, um, if you want, uh, details on that, um, there's more, there's nice figure in your notes and tables and the references to go look up coefficients. Um I should warn you that often these betas uh just like what we saw for say temperature and salinity where um you're given an equation um a very complicated polomial that's also the way that these are calculated for seawater because of salinity effects and temperature effects is it will usually be some long polomial as a function of temperature and salinity that will give you an exact value. Um but there are you know tables of curves up. Uh the other thing is that salenity works to disrupt some of these solute solvent interactions. Um because of all the the large ion charges bouncing around and all of these all of the uh well basically that yeah because of all the high ionic strength it disrupts some of the solvent interactions which actually help make the gas more soluble. And so as you increase salinity, you decrease solubility. And this is often ris written as I'm not going to test spelling on this one. It's the set and shower relation which basically says if you had um your Bunson coefficient for fresh water, whatever salinity uh those are related and you can you could actually calculate them as a function of salinity. This is an approximation. In reality we're going to use the full polomials but it basically says that as salinity goes up solubility goes down and it's a fairly substantial effect. I mean it's order of you know 20 30%. — Does that also hold for nonpolar gases? — Well even like the noble even the noble gases have can have a small inducible polarity — something like methane. Oh yeah. Methane as well. But even like xenon has a salting out effect because it can be inducibly polar which is why you can actually make some you know you can actually do some stuff with noble gases. Okay. So that's all we're going to cover on solubility. Any questions on that? If anybody wants to, you know, we got one of the world's experts on this sort of stuff here, Bill Jenkins, and he can you can go talk to him and have coffee and spend the afternoon. Okay, great. So, so we've done we'll go back to our sort of simple flux model. We're not going to we're going to start

to delve into what we're going to call the kinetic component. It's often approximated by this coefficient little K and sorry for the that this is just notational um gas exchange is notational health that's come from so many different fields um the sunshaw relation uses a little K with an S K is just what is typically used for the kinetic component and gas exchange and I think I've kept straight in the notes um but usually it's going to be formulated as something like uh a kinetic component times the concentration in the water. So I'm going to use C is the concentration in the bulk water and C 0 is the atmospheric equilibrium concentration. So C 0 is a fictional quantity. It would be what the water right at the very surface that's just touching the gas would have as a concentration. You don't actually measure C 0. What you do is you measure the atmospheric partial pressure and then you compute C 0. So C 0 would be that Budsen coefficient times the pressure in the atmosphere. So this is kind of going back to remember the kinetic component and the thermodynamic component. This is going to be the kinetic thermodynamic component. So the thermodynamic component you go out and you make measurements of the partial pressure in the atmosphere. The confusing thing is often the way people make concentration measurements in the water is they take a bob of a blob of blob a blob of water, a parcel of water and then the way they instead of making concentration measurements, they may take this and actually compute its own seawater partial pressure. So sometimes you'll see this instead of being written as K C minus C 0 you might see it as K P minus P 0 where or P SW minus P A just for some gases it's instead of it's easier to make partial pressure measurements for the seawater rather than to make concentration measurements. Just depends upon what the easiest chemical effect is. Everybody set with that? Okay. So, there are several models of gas exchange and these are really simple conceptual models to give you uh some uh framework in which to interpret observations. The first one is called the stagnant film model. And the stagnant film model is going to try to simplify everything by saying that if we were to plot depth and we're going to plot concentration. So we're going to Start off with that C0. We're going to say that the atmosphere is well mixed. And for most of the gases we're going to be talking about in this class, it's a pretty good assumption that most of the resistance to gas exchange is on the water side, not on the air side. Uh because they're what are called uh sparingly soluble gases. So things like CO2, O2, N2, the noble gases, um they're soluble in seawater, but they're not significant, you know, it's not significantly soluble in seawater. Um so um you can basically make the assumption that the atmosphere the transport is so quick that um the atmosphere is a well-mixed So this C 0 you make an atmospheric measurement and then you can say okay I'm going to know what the concentration

in equilibrium is right at the surface for the stagnant film model it's just like what it said stagnant film where the only processes going on are diffusion molecular So we have that relationship. Remember we have flux, right? The flux across this stagnant film is going to be constant. We're going to assume there's no chemical reactions and the biological reactions. It's a very small film. So if you have molecular diffusion going on um and you have no sources and syncs and you have a constant flux, can anybody tell me what the profile is going to look like through that stagnant film layer? Anybody have molecular diffusion? — Straight line. Good guess. Well, or good knowledge. um across that stagnant film there's actually going to be a straight line and then there's going to be we're going to assume that there's some bulk concentration C and that the ocean side is actually well mixed as well. So all the turbulence is going on down here keeps this infinitely well mixed. There's this stagnant film layer and then an infinitely well mixed atmosphere. The reason why this is a straight line, if you remember from fickian diffusion, um thickian diffusion for molecular diffusion says that the flux is simply equal to a diffusion coefficient times a gradient. Well, if the flux is fixed and the diffus diffusivity is fixed, right, you have a constant flux. Otherwise, you'd be draining or adding gas in and out of the stag film layer. Only way for that to occur is if you have a constant gradient and a constant. Okay, [snorts] where the stagnant film comes in is we're going to say that this stagnant film has the thickness delta Z. So we're going to approximate the thickian diffusion as flux equals minus D delta C over delta Z. Right? So we're just turning the little D DC and DZ into delta C and delta Z. Then we're going to say, well, we have minus D delta Z and then the delta C is just the C minus C 0, right? So these things we can measure. We know D. We can look that up in a table. It's a diffusivity. So we could rearrange this equation. Um if we could estimate F from some measurements, we could then turn around and calculate delta Z. It's like well you know I'm not really interested in delta C for or delta Z per se but if I had a gas if I had some set of gases for which I could measure the fluxes and then compute delta Z I could then use that delta Z for other gases for which I couldn't make flux measurements that make sense and often Let's um oh yeah let's do one thing the um diffusivities the units on those are length squared over time. So meters squar per second for example is a typical diffusivity. It depends upon temperature and the solute and the solvent. So you actually need to know what gas it is and what whether it's in water. You know water has different diffusivities than magma or methanol. [snorts] Um and you also know the temperature. Um D

tends to increase with increasing temperature. So if warmer waters diffusion tends to occur faster and also um as molecular weight goes up, diffusivity goes down. So bigger molecules tend to um tend to diffuse slower. Now if you're you know for the noble gases there's nice sets of relationships. As soon as you get a molecule that is not spherical. Um then it's not just molecular weight, it's also you know its polarity and its actual shape. Uh but these can be these are are recorded have been measured in the lab. They've lab for fresh water. as a function of salinity and temperature. Okay. So let's show an example of how you might do this. So, we're going to talk um about radon. Everybody remembers from Bill Martin's lecture on radioactive isotopes, radioucleides. Everybody remembers that radon 222 is a daughter of what? Radium 22, — right? — Radon 222 is a daughter of radium 226 or yeah, radium 226. Um, and so this is a gas. This is not a gas. And so because the radium is always decaying to radon 222, we can use look at the imbalance between how much radon you would expect to see based on how much radium is there versus how much is actually there. Remember um we would expect the activity of radon 222 to equal the activity of radium 226 unless there's some process that's removing the rad. Right? This is the secular equilibrium that Bill talked about quite a bit. So if you were to look in the field, what you'll often see is that in the deep water um the so this would be depth and this would be say 100 meters. Um what you might see is that the activity of RA of radon 222 is low in the surface and then tends to increase at depth. And this dash line is the activity of 226. And so deep in the water column they are in secular equilibrium with each other. And right up near the surface in the mix layer, there tends to be a deficit of of radar. And that's just simply because radon has been out the top. Now [snorts] everybody have that. Can I switch? You guys sat at MIT? Okay. So, what we want to do is set up a mass balance [cough and clears throat] for radon 222. And so, and we're going to do it over the mix layer. So, remember the little diagram I had, right? where that's radon 222 and this is the radon 226. We're going to approximate and say that there's going to be some mixed layer depth H and we're just going to look at the mix layer and we're going to assume that everything is well mixed within that mixed layer. If you had to do the calculation for real, you know, you make you could integrate over that and find out averages and all that, but typically things are pretty well mixed in the mix layer and there for first order that's a good assumption. So, we're going to say we want to know the um [snorts]

first look at the production terms. The production term of radon 222 is simply going to be the mix layer depth times the activity of 226, right? So radioactive decay is producing radon and we can if we know the rate the rad activity that gives us the production rate of radon. There's going to be two loss terms. One is simply going to be the mixed layer depth times the activity of 222. The second one is going to be a flux term which is going to depend upon a piston velocity um which we'll I'll get to in a minute times a delta concentration. Remember that was the flux form that we showed previously. So from the stagnant film model, this is going to be this term stays the same. We're going to assume that the atmosphere has no radon 222. And that's not a bad assumption. The halflife for radon is about four days. And so over the ocean the amount of radon in the atmosphere is really small. That's not true over land. Um because you have outgassing of you know you have uranium rich soils and rocks has a lot of radon out fills up your basement fills up your shower. Um so we're going to assume that the C atmosphere for radon goes to zero. And we're also going to assume a stagnant film. So then what we would need to know is simply the diffusion rate of radon over the stagnant film depth. Remember that's the piston velocity. time the concentration of radon 222. Now, usually this is formulated in — Did I drop out? — Okay, — we lost them. We can't hear you either. So, we'll give it a second. — The answers. — Do we have you back? — Okay, — cool. Um, how far back did you miss — below the word loss? — Oh, okay. So what I was saying was that there's two components to the loss term, right? There's going to be radioactive decay for radon and then there's going to be gas exchange, right? And the gas exchange is going to depend upon the concentration difference across the interface and a piston velocity. We're going to assume that the concentration of radon, so this would be for radon, is going to be zero. And so all you need to know is the concentration of rad of radon in the water because that's going to drive gas exchange across the interface. So all I did was then expand this loss term. So this would still be loss is going to equal the radioactive loss term. And we're going to replace this K by the stagnant film model. Right? So we have the diffusivity of radonog. We have the thickness of the stagnant film. So this is that And then we need a concentration. And the way we're going to get that concentration is we're going to go back and remember

what the definition of activity is because usually these are measured and reported in activities. So we want n and n is just going to be the activity over the decay constant. So then this would be the activity of radon over a lambda for radar. So if we assume steady state which for when they first went out and started to do these experiments in the 60s you know they weren't able to make the experiments long enough to see what actually was changing over time state is that production equals loss. And I won't go through but what you can find if your notes well is you can rearrange the equation for delta Z, right? Because everything else in here is known, right? H is known from your measurements. The activity of 226 is known. The activity of 222, diffusion coefficients, lambdas, etc. And so you can rearrange and find delta Z as some function of the activities of 222 and the activities of 226. And what they found was that um the delta Z's tended to be in the tens of microns for stagnant films. And you can go through the whole calculation in the notes and I would recommend it because it's a pretty useful both in terms of remembering secular equilibrium and radioactive decay and also gas exchange. — Can I ask a quick question? — Sure. — What's this velocity? — Um, good question. I should have covered that. Okay, I'm going to slide this up a little bit. Do you guys have everything? Um remember I wrote F equals K delta C the very beginning. Um and this K can also be it's the D over delta Z for the stagnant film model. I think I wrote it as K C minus C 0 but that's the same thing. Um if you look at the units D has units of me squar length squar time because it's a molecular diffusivity. Z has units of 1 over m or one over length. So this term K or capital D over delta Z has units of meters per second and it's often called a transfer velocity. I actually like transfer velocity because it sort of hearkens back to transport uh transport theory, transport phenomenon, but for reasons that escape me, it's also called a piston velocity. And I think the idea was that you can imagine that as sort of you know this piston sitting at the bottom of the mix layer pushing up pushing the gas from the mix layer into the atmosphere or similarly pushing gas from the atmosphere down into the ocean. It's really terrible. It's really terrible nomenclature but it has it has leaked into the notation. Um but it's essentially this is the kinetic it's the kinetic. Okay. The stagnant film model is not the best model. Oh and I should say these are order of tens. K is typically tens of centimeters per hour. Um, and that's just also a historical um a historical thing that ks are often reported at centimeters per hour rather than meters/s. And that that's just so the units look more uh are closer to one. Okay. So the stagnant film model is a really good pedagogical tool. um and useful for the initial interpretation. But what was found was it's actually not a very good model for open ocean conditions. You think about it, you've got this little

um you're supposed to have this little stagnant film that just sits there and nothing happens to it. Well, if anybody's gone out to the open ocean, particularly when the wind picks up, it's hard to imagine a little constant film that stays there indefinitely. Um, so a a new model or a competing model if you like is the film replacement model. And the argument for the film replacement is that Let's say you have some stagnant film sitting up at the surface. So it would be if you were to do this in cartoon, you know, you might have some stagnant film that then stays there for some length of time, but then some Eddie comes along and pulls that film off um and you have no stagnant for a while [clears throat] and then a new stagnant film would develop. So that film basically gets wholesale replaced every so often. And so then the problem becomes you're going to have uh say a diffusional flux into that film over time. And then whatever gas has built up, you know, it's so like one and then you know maybe you've got lots of gas that's built up. All of that gas gets transported down into the interior when that film gets replaced or overcharged and then you start off with no gas and then it would build up with time. This would be this direction. So then we no longer have a steady state problem and we need to look at the um the flux into the film as a function of time. Okay. So f time into a infinite slab. So we'll assume the film is thick enough that we can treat it as an infinite slab. Um is going to equal again the concentration difference where this is the concentration difference from the atmosphere to the bulk fluid. Right? Same thing that we've been approximating before. uh and it's going to depend upon diffusivity in the water. Um except now it's going to depend instead of being depending upon linearly on diffusivity, it's going to depend upon the square root of diffusivity. Um it's going to depend on pi which is just a constant and time. And that says that the when you bring this when you bring um this parcel up, what you'll notice is that this equation is actually um unstable at zero time. It says you have this infinite flux. And that says that right when you bring it up, you're exposing water that has had that essentially has zero pertabbation to the gas concentration right at the surface. So you're exposing it across just that molecular layer of the atmosphere to the water. Um but uh and so you have this very rapid rate for a very short period of time. But as it as time progresses, scourge all the way down the way. Um, as time progresses, this term gets smaller. And it basically says as this film, as time progresses, this film fills up with gas. And so the flux gets less and less because this film starts to become equilibrated with the atmosphere. That make sense? And the only way to increase the flux is then to rip that film off, put it down, and then bring up a new film from you know, a new uncontaminated film. Um, what we actually want to know is the mean flux over some time theta. That's going to be the mean life, the mean life of the film. And that's just going to be um the mean flux is going to be 1 / theta. I'm just going to integrate from 0 to theta. Integrate the flux. And that's just has a very similar form except now instead of having t I have

theta in the denominator. And so what we've done is instead of having remember in stagnant film we had the transfer velocity piston velocity was equal to d over delta z. In the film replacement model it's now 2 over the square roo of diffusivity * theta. Now there's a very useful non-dimensional coefficient that we use a lot which is the Schmidt number and the Schmidt number SH equals the viscosity of water over the diffusivity of a gas. So this is the viscosity water diffusivity of the gas. Now the nice thing about this is that um this gives you some information about relative transport, right? If the viscosity of the fluid were to go up, you'd expect turbulence to be damped. So it gives some measure of turbulence. And if the diffusivity of the gas goes up, you'd expect the gas to be more effectively transported. The Schmidt number looks a lot like, you know, you've got a D term. We can then instead of just computing D, we'll actually report these K's in terms of Schmidt numbers. And Remember at the beginning I was saying one of the things we want to do is how to go from you know compute K for one gas and then go to a second gas. That was one of the goals. We're going to try to compute gas exchange with things we can measure and then we're going to apply those other gases. The way we're going to relate them is through the Schmidt numbers. So we have for the stagnant film model, we're going to say we have some we're going to have K is proportional to D to the 12. Right? That's what this says up here. Well, and this is proportional. Well, that's also proportional to Schmidt number to the minus 12, right? Because D is in the denominator of the number. So, one of the things you will see a lot and um I think did we put it on switch numbers? — I haven't done the last question. Okay. Um is that the generalized forms of the flux equation for the replacement model is K0. So this might be some piston velocity you measure say with radon or some other uh no a noble gas would then be you'd have a Schmidt number correction. So you'd have the Schmidt number that you originally made the reference measurement with for K0 and then the new Schmidt number where this is the Schmidt number for whatever gas that you're interested in. So this would be the the flux you're trying to compute for new gas times the concentration gradient — that would be a subscript in parentheses — uh that's a these are both schmid numbers and that's just supposed to be a zero so it's the same k0 number zero so if I had made a measurement with radon I would plug in the piston velocity for radon, the Schmidt number for radon. Oh, excuse me. And then minus [snorts] 0. 5 for the replacement fil model. And that's simply because the piston velocity is [snorts] proportional to numbers.

Have I lost you guys? How you guys doing up in MIT? Yeah, riveting stuff, huh? sometimes you'll see this written as um people will say well I don't know whether it's whether I believe the stagnant film model or some other model and they will write it as um same general form Schmidt number over Schmidt number reference to the minus n where n might be the exponent that they're actually trying to find from their experiment. — Because I'm sorry because for the stag model it's — yeah if we were to go back remember uh K if we go K was proportional to diffusivity for the stagnant film model. So if we come down here that would say that K for the stagnant film model would be proportional to one over the Schmidt number. So Schmid number to the minus one. So if the stagnant film model were dominant then n would be one where for the film replacement model it's 0. 5. Quick clarification for the equations up there. The um — not right. — Um — you were pointing out. Yeah. Oh, that's a two, right? — That's a two. Yeah, — yeah, I do. I try — I thought it was — Yeah, I try to cross my Z's. So if I Yeah, I usually pretty good about that. Makes up for some of my for — this is the atmosphere that the atmosphere. — Yeah, it's it just depends upon the concentration gradient whether it's going to be from the atmosphere water the atmosphere. Okay. So now I've told you how if I have computed a K0, we went through an example with radon. Um we've now gone through how to ex transfer from gas to gas. And there's actually a table of Schmidt numbers uh for seawater in your notes that you can look at. Um it it's basically one over you can think of it as one over diffusivity. So just invert everything we said about diffusivities in terms of sensitivity to temperature and molecular weight. But what does K what else does K depend upon? It depends upon the solute but it also depends upon other environmental conditions. And um there's a fairly longunning debate in the literature about the values of K and how to measure them. Um, and I don't know whether you know I don't know whether it's actually getting resolved or not with time or whether we're just headed into to more confusion. Right? So we had flux equals K Schmid number zero to some exponent delta C. Um, one thing is there's a couple of things that we think increase K or K0 would be wind speed. So higher wind leads to more gas exchange. Well, that sort of makes sense. You know, higher wind leads to a lot more turbulence. it leads to wave breaking uh bubbles all sorts of things like that. So that that kinetically is makes some intuitive sense. Um the second thing that people have tried to do so people have spent a lot of time trying to build models or empirical models relating K0 to wind speed. Uh the theory on this is somewhat weak. It's how does wind speed then affect ical measurements. You could go out and make radon measurements, for example, um in high wind speed conditions and then go out a few weeks later or a season later and make them under low wind

conditions and then look at the radon deficit and look at how K differed under those two conditions and try to, you know, basically map that out. You could look at bubbles and bubbles are usually approximated by white cap uh coverage [snorts] and it makes sense that bubbles should increase uh gas exchange, right? If you think about it, you know, you're taking um you're taking gases out of the air and you're down in the water. uh some of that gas is dissolving before that bubble reaches back to the surface and that would be a net flux into the water. Similarly, you can also think of bubbles as increasing the surface area between the water and the air. And by increasing the surface area, if you had something that was super saturated in the water, you could diffuse gas into those bubbles. And some of those bubbles are going to make it back up to the surface. when they make it up back up to the surface, they're going to lose gas to the atmosphere because they're going to pop. Um, and they're going um so those both would increase the um uh transfer velocity. It should increase with fetch. Does everybody know what fetch is? Okay, fetch is basically the distance the wind is blowing over. The fetch is basically from one side of the lake to the other. The fetch is the distance across the lake. And if you looked at the lake, if you were on the say the wind was blowing this side, uh if you looked at the waves, the waves would be really small on one side of the lake and then they might be really big on the other side of the lake. And that's just because it takes a while for the wind for the momentum in the wind to be transferred into the water and to build up a wave state. And since turbulence, you know, you think the same thing might be true for turbulence. It might depend upon the wave state uh and how long the wind has been acting on the water. Most of the time in the open ocean, fetch isn't a problem. But sometimes you'll also see it as wave state. So if the wind suddenly just kicked up in speed, um the turbulence might not have caught up to it yet. And one way to do that would be to look at the wave state. Um, if you go out and look at the gas exchange measurements, and there's an example in the book, um, and this is where things get rather contentious in the literature. If you plot piston velocity or transfer velocity as a function of wind speed, the data kind of looks like this. And, you know, depending on how you interpret the data, whose data you decide to keep, throw out. Um, you can pass a number of different lines through that curve. Um, some of the curves there there's a classic curve by listener. List. And you can tell that this field isn't as mature as other fields because the relationships are still sort of we still talk about individual people's curves rather than saying, you know, this is so and so's law. This is just somebody's estimate. So listen, Merlovat said, well, you know, we view it kind of goes like this. There's actually three broken lines. So it' be three um the right for that um three discrete lines that are joined together. It's a better word for that. — Is you wind speed? — Oh yes, you is. Sorry. You was wind speed. Thanks. Um and it's usually referenced you'll often see it as U10, which is the wind speed at 10 meters because wind varies height above the surface. So you try to reference it to a common height. Um is basically three discrete lines. So that it's it's um they're basically linear in each region and they peacewise it's peacewise linear peace wise linear. Um another famous one that goes through the same cloud is the wine and cough relationship.

which is K is proportional to the square [snorts] of the wind speed. So it's a quadratic relationship. Now from there it gets really ugly. Um wine cop also now has a cubic. Um you have other people who are arguing it's actually linear with wind speed. And in some ways the data isn't strong enough yet to partition between those. The problem is you know obviously the quadratic and the cubic when you get to very high wind speed you know the piston velocity keeps on shooting up. We don't have a lot of data at very high wind speed. It's very hard to go out and make gas exchange measurements in hurricane. Um both technically and also because you just don't know where the hurricane's going to be. you basically have to position yourself ahead of a hurricane um and know where it was in advance and all that. Um there are attempts to um resolve that with some other measurements which I'll talk about in a moment. But I do want to touch on one other thing that comes up a lot in exams and general exams and things like that which is the yeah hint hint um which is the gas exchange resonance time. So remember back we had that little model where we had you know some thickness h which was the mix layer. Well, you could sort of think that um gas exchange is going to try to push super saturated waters towards equilibrium. Similarly, it's going to try to push undersaturated waters to equilibrium. And you can approximate that as a time scale. This would be a time scale for equilibration that's going to be proportional to H, right? The deeper the mix layer, the bigger volume you're looking at, the longer you would expect it to take to come into equilibrium, right? Because your flux doesn't depend upon mix layer depth. So your flux doesn't know how big the mix layer is. So if you have a 10 meter mix layer, the equilibriation is going to go very fast. If you have a thousand meter mix layer, it's going to go very slow. and your piston velocity or transfer velocity. If you think about that, that has units of meters. This has units of meters per second. Um, and so you're left with something that is time. And typically typical values are tens of days for most gases. So, you know, it's going to take, you know, a few weeks to a month or so for um things to start equilibrating. And this is technically it's the one over E. E folding time. So if you started with a concentration difference after one uh efolding time you'd be down to one over e the original perturbation and the two efolding times you' be one over e^2 etc. I don't understand the term better — if you're so ddt let's say you had some pertabation — um is equal to say minus k c right if [snorts] you solve that remember dc over c= minus k dt and so it actually looks like radioactive decay it's going to C at any time is going to equal C 0 E to the minus KT. And so when time um when when times when you get to one over E holding depth, this is going to be this value would be one. And so then it would be C= C 0. That's just called folding. That's just it's just conventional notation. It's called one time. — Um there are some caveats to the resonance time. I went slow. Oh no, that clock's fast. That's — I went slow today. Um there's a couple

of caveats to the resonance time. Um for CO2, um remember Most of the CO2 in water is in what? Most of the inorganic carbon in seawater is in what form? — Bicarbonate. So most of the dissolved organic carbon is in bicarbonate. And so if you want to equilibrate the CO2 system, let's say you had 14 CO2. So you had radioisotope CO2 diffusing into the water. It's got to fill not only the dissolved [snorts] CO2 gas, it's also got to fill, you know, H2 CO3 star bicarbonate ions, right? So, there's actually a correction that T is going to equal H over K, just like what we wrote, times total amount of CO2 in the water, the total DIC dissolved carbon over CO2 aquous. And this latter factor is somewhere between 100 to 200, right? So the gas exchange time scale for C14 is going to be about 100 to 200 times longer than a normal gas because CO2 has this huge reservoir of other compounds that aren't actually gases. Remember um hydrated the hydrated form the bicarbonate and the carbonate ion they are not gases. So they cannot exchange across the air interface. But in terms of a resonance time or equilibration time, you have to fill up um you'd have to put C14 and plug it into all of these other reservoirs. Fill up all those other reservoirs before you can equilibrium. So that means that T for C14 is about 10 years. Now, the last little tidbit I want to get to and the there's a section in the notes on um there's a long methods and you can read that um and your problem set actually goes through one of the examples of one of the methods. But the last thing I want to touch on is what happens for CO2 gas that's not C14. Right? So you have regular gases approximately you know weeks to months you have 14 CO2 which is t is approximately 10 years. It turns out for CO2 gas, T is intermediate between those two. It's about one year. And it has to do with the fact that as you add CO2 to the water, what are you doing to in you're increasing total CO2 or DIC like Um, but what else are you doing as you add CO2 to water, you change it, you change its pH, so you are actually decreasing the pH, right? And one way of thinking about that is you're actually shifting the acid base equilibrium from bicarbonate to CO2 aquous, right? H2 CO3 star. And so that actually speeds up the equilibration because you're actually you're actually filling up more CO2 aquous. So you're coming into equilibrium with the atmosphere faster. And so you get an intermediate time scale between the 10 years you would expect. If the pH didn't change in seawater, CO2 would have the same equilibration time as 14 CO2. This is because um pH changes. Okay, we're out of time. Um there's some discussion in the notes that are on the web page about all sorts of fun different ways that people have measured CO2 or excuse me gas change. Um read those through. Um oh yeah what I said was that um the

notes have discussion of the different methods that have been used to measure gas exchange and you should read those. But for problem sets and stuff, I think some of these equations, the stagnant film model, the film replacement model, and the resonance time are are more useful and might be more amenable to being an exam question. Yeah, I was going to ask does that mean in areas where surface water pH is lower than that? — Well, the um the pH isn't going to affect it's the change in pH as you equilibrate because this is the time scale to reach equilibrium. But if you've already — Well, it does change. There's something called the It's called the Rebel factor, — and it actually changes. It actually changes slower in low pH environments. — Yeah, we will get to the revel factor later in the semester. [clears throat] — Okay, great. see some of you and — question. — Yeah. — Oh, hang on. Can you one more time? — What does it have to do with it being carbon 14? What do you Why is it — I don't understand what that has. — The difference is if you're just adding an isotope. Okay, this would be the equilibration time if the pH in seawater were fixed or buffered by something else. Okay, but that's not true. But it is true if you're adding like an isotope of CO2 because you're not actually changing you're adding so little of this that you're not actually changing the seawater pH or the seawater chemistry where if you add CO2 gas you actually are changing the seawater chemistry and then it speeds up because you're basically you're getting if you add CO2 you're making it more acidic and you're actually increasing the aquous CO CO2 concentration faster and so you're equilibrating with the atmosphere faster. Does that help? — It's a It is a tough concept. We can probably we should probably go over it. We'll go over this again in recitation on Monday.