nm0885: going on here er this gentleman is from i think the Linguistics department om0886: nm0885: right and he's he's doing study of th-, of the language that academics use and things like that er so he's he's taping a random sampling of lecturers around the university so if you're wondering why i'm replacing all technical terms with the names of frudg-, fruits and vegetables this time it's basically i'm having a go at him so [laugh] er well what we are at is that formula and we spent a lot of time yesterday hacking our way up to this thing hacking that's a good word er [laugh] and and we got it and we're really happy about it er what is it [laugh] what is D-of-L-nu sf0887: decrease of the radiance nm0885: right and some r-, if some radiation is passing through a slab of atmosphere a thin layer of atmosphere er some of it's going to get lost and the amount that gets lost is the change in radiance D-L-nu okay er what happens to the radiance that gets lost where does it go there are a couple of different things that could happen to it sf0888: could get scattered or absorbed nm0885: right it can get scattered or absorbed okay and so now the amount we worked out depends on a couple of different things okay some of them are pretty obvious y-, er you lose more if you have more going in so it depends on the amount of radiance that's incident on our little slab of atmosphere it depends on something called K what does K tell us about sf0889: nm0885: er scattering is one of the things that go that g-, that goes into it sf0890: er is it the extinction coefficient nm0885: yeah it it's it's the extinction coefficient which is the sum of the scattering coefficient plus the absorption coefficient okay and so what it's telling us about is it's telling us about the characteristics of the material is this material an effective scatterer or an effective absorber and we'll be getting into later on in the course what it is actually that makes the material a good scatterer or absorber and then it depends on rho-secant-theta-D-Z what's that that's the easy one sf0891: pathlength nm0885: pardon sf0891: pathlength nm0885: yep this one's the pathlength why do we have a rho in there sf0892: nm0885: yeah it's the density of the material that's doing the interaction so if your total this thing up pathlength times the density that's the amount of material per unit area okay so that's for a very thin slab and that means and because it's a very thin slab we can say well things like density the amount of material extinction coefficents all all of these things don't vary much as you go across the slab i mean in principle they can vary so if we have a finite slab of material if you have a finite slab of material so instead of just a thin layer D-Z here we've actually got a thick layer which ranges between er you know a finite slab of material from Z-one to Z-two so we've got two heights involved and so if we say Z-one is the top where it's entering and Z-two is the bottom where it's going out er we can integrate this expression to add up the changes in the er add up the amount of depletion in all the different layers and give us the amount coming out the bottom so if we've got an initial radiance L-nought-sub-nu in our direction here passes through the slab then what comes out is at the bottom is something less and i'll call that L with a superscript- D so that's er the depleted beam so it's lost some stuff and that's L-nu and it's of course coming out in the same direction theta-phi okay so if we have a finite slab of material from Z-one to Z-two we can integrate this expression er to give D direct compo-, of the emerging beam 'cause remember the emerging beam is also going to have contributions from emission and contributions from scattered stuff which we're going to have to work on a little bit later but er but the direct part of it L- D and actually yeah i m-, i made a mistake that D stands for direct we can get that by doing the integral so what we do is we take this expression and er you notice we've got L-nu on both sides so we'll take this one over to the other side so we'll write this as D-L-nu over L-nu and that's excuse me going to be equal to minus- K- nu-E- rho-secant- theta-D-Z okay now we integrate this across the layer er so integral signs on both sides we're now going to sum up those layers so the integral on the right hand side we're integrating with respect to Z so we're integrating in height what are the limits of integration what's the bottom one sf0894: nm0885: pardon sf0894: nm0885: Z-one that's right it's where the stuff's going in and it goes up to Z- two where the stuff's coming out now on the left hand side we're integrating over radiance D-L-nu so what's the starting radiance where we start the integration sm0895: L-nought-sub-nu nm0885: yeah L-nought- sub-nu and finishing up at L-D- sub-nu so that's our integral it's now a matter of doing it now left hand side is pretty simple what's the integral of D-L-over-L sf0896: nm0885: right log-L so on this side we get log of L-nu and you're ranging from L-nought to L-D- sub-nu and on this side er what do we get there this is a trick question so have a guess at it and y-, and y-, and you'll be wrong but but you won't be have to be embarrassed because it's it's a trick question sf0896: Z nm0885: pardon sf0896: er Z times all that nm0885: Z times that stuff okay and that is indeed the wrong answer and [laugh] the r-, r-, but it's exactly what i was hoping you'd say because that would be true if all this stuff was constant then you could just pull it out of the integral sign but these properties may well the secant-theta shouldn't vary but the other stuff might potentially vary as you go through the layer and certainly you know if you imagine a beam of sunlight going down through the atmosphere er the density of just about everything is going to be increasing as as you go down to the surface and probably the only significant thing where that's not really true is going to be something like ozone where it's really varying wildly or clouds or something so we can't do much with that side so what i'm going to do is i'm just going to leave that side as an integral from Z-one to Z-two K-nu-E-rho- secant- theta- D-Z and er i'm going to give it a name and the name is going to be optical thickness or optical depth so i'm going to write that as minus- delta lower case delta and the er and we'd better put a subscript-nu on that because this is going to be a function of frequency so where delta-nu is equal by definition to this integral Z-one to Z-two extinction coefficient times density times pathlength secant-theta-D-Z is the optical thickness or optical depth these two terms mean the same thing and people will tend to use them in-, interchangeably so what that is is that's a m-, is that's going to be some measure of how much stuff is getting pulled out of the beamer or what or what fraction of it and if we define the optical depth this way then we can say that our direct beam L-D-of-nu is going to be equal to the initial beam times E-to-the-minus- delta-nu just rearranging that equation and this is a very important relationship i mean in some sense in in the theory of radiative transfer this was the first really quantitative law that was discovered and er and so it's named after its discoverer not what he was drinking at the time Beer's law er you will also find this is sometimes called Lambert's law and it is also sometimes called Bouguer's law and sometimes it's called the Beer-Lambert law and sometimes it's called the Lambert-Bouguer law and sometimes it's called the Beer-Lambert-Bouguer law or whatever er i think it probably has to do with which nation you are and which nationality this person these people are er 'cause this 'cause this relationship was probably discovered independently several times we're going to call it Beer's law in this course just for simplicity okay and so what it says is that the amount of radiation that's coming through directly is er decreasing exponentially in the optical thickness so the fraction of energy the fraction of well radiance that emerges is given by the monochromatic transmittance and it's monochromatic because we're always doing this as a function of wave number or or sorry as a function of er frequency but we can always add up over frequency and that is defined as tau lower case Greek letter tau and that by definition is going to be the amount that's coming out the bottom L-D divided by the amount that's going in the top L-nu and you can see immediately that that's directly related to the optical depth E-to-the-minus-delta-nu so there's a couple of different ways that we can think about how much is going through we can think about a transmittance wh-, so you may have a transmittance of point- five which says half the stuff gets through er you may have an optical depth here and the thing about the optical depth is that's going to be directly proportional to things like pathlength and density and so on so if you double the density you double the optical depth and er yeah so that that's that's the amount that gets through okay so that's transmittance but of course remember there are two other things two other ways that energy is coming out the bottom of the slab what were they again sf0897: scattering nm0885: scattering and sf0898: emission nm0885: emission thank you it's one of these cases where everyone's whispering the answer but no one's saying it loud enough all right so so the slab can be emitting radiation and also radiation that's entering the slab from other directions can be getting bounced into this direction so let's let's write down some expressions for those ones as well and we'll start by taking emission so energy emitted by the slab now once again this is going to depend on a couple of different factors so this depends on what's going to determine how much energy the slab is emitting at some given wavelength temperature is going to be a big one so this depends on temperature and what's the relationship that tells us how much emission we're going to get at a given temperature sm0899: Wein's law nm0885: er Wein's law's going to tell us the wavelength where we get the most emission but what's going to give us the whole function at a as a function of any wavelength sm0900: Stefan sf0901: nm0885: er the Stefan-Boltzmann one is isn't the one i was looking for because that's giving you the total i'm saying if you just have some arbitrary wavelength that's not at the peak what's the function that's going to tell you how much you get at that wavelength we've done it in this course we've written it out on the board i've shown plots of it on the projector sf0902: Planck's law nm0885: Planck's law that's right so the Planck function so the energy emitted depends on the temperature and that's given by the er Planck function and er well what else is it going to depend on so if if it's at a given temperature what else is going to affect the amount of emission that we get sf0903: nm0885: sorry sf0903: is it the amount of material nm0885: the amount of material absolutely the amount of material does it depend on anything else sf0904: density of material nm0885: er that's another w-, i would i'd say it's another way of saying amount okay sm0905: the heat capacity of the material nm0885: er probably i'm i w-, i wouldn't write it in terms of heat capacity but it does depend on the type of material different materials emit with with er wi- , with with different efficiency so it's it's also going to er depend on the er type of material so some materials will be better emitters than others and especially better emitters at given wavelengths and this'll depend on the molecular structure and things like that which w-, which we'll see later on er something that you well i'll come back to that in a second let's write it down okay so let's write the emitted radiance and so i'm going to denote that as L with a superscript-capital-E for the emitted part of the radiance subscript-nu 'cause we're at one given frequency again and oops and let me not forget to write down the direction theta-phi so this is going to depend on these various properties so it's going to depend on the Planck function which is a function of temperature now you'll notice i've written a B here whereas before i wrote an E when i wrote down the expression before i wrote down the formula for the irradiance emitted so the total of what's coming out in all directions but in this case we're actually thinking about just one direction from the slab so we need the radiance but it's easy to go between radiance and irradiance for a black body why why is that sf0906: isotropically nm0885: yeah that's right a black body emits isotropically so it emits the same in any direction and that means that if we know the total then we know what the amount going in any given direction is and remember and so let's actually write this down where B-nu-of-T is the Planck function in radiance units you use the Planck Planck function in radiance units and that means that er if you'll recall and i don't necessarily expect you to remember this just at this moment but you might want to remember it for the exam is that er the difference between the radiance and the irradiance for an isotropic emitter is just a factor of pi so in one direction it's going to be E-nu- of-T divided by pi so we've got the er the temperature dependence in there because now we've got the Planck function for the radiance okay the next thing we want to do is we want to put in something about the type of material and we'll write this in terms of the mass absorption coefficient so K-A-of-nu is the mass absorption coefficient which we've already def-, which we've already mentioned why am i writing the mass absorption coefficient when i'm talking about emission here this is a question for somebody who's done the right physics course this is not something you're going to be able to figure out on the spot unless your name is Kirchoff and you're really clever or you might be really clever but if your name isn't Kirchoff they still won't name the law after you [laugh] sf0907: is it because good emitters are good abs-, good absorbers nm0885: that's exactly right and that that's Kirchoff's law and what Kirchoff's law says and this is what what you have to wor-, work your way through in a physics course is Kirchoff's law says that if you're in thermodynamic if you're in a local thermodynamic equilibrium then a property must be equally good at absorbing and emitting and the reason for that is if you're in thermodynamic equilibrium you're er you've got to be at the same temperature in in different parts that are that are close together and if you're at and if you're at the same temperature but one part of th-, th-, b-, but your substance is a good emitter er but a very bad absorber or something like that then what that means is that you'll end up with a flux of energy going from one part to the other and one part of your substance will try and heat up relative to the other and that's a contradiction with the thermodynamic equilibrium so if you're going to have an equilibrium and have nearby stuff at the same temperature you've got to be equally good at absorbing and emitting so that there's no net flow of energy from one place to another by radiation so anyway i mean th-, that that kind of detail doesn't matter but what matters for our purposes is is that for any part of the atmosphere that we're interested in and this means not the the far outer thermosphere or whatever er we're in thermodynamic equilibrium at least locally and the absorption coefficient and the emission coefficient are going to be the same thing so er K-A-of-nu er well gives the emission properties of the material for substances in thermodynamic equilibrium and i'll just put in brackets here that this is er Kirchoff's law so it's the same for absorption so K is the same for absorption and emission okay final bit we need is the amount of material how are we going to write the amount of material this is an easy one 'cause we've done it before sf0908: density nm0885: sorry sf0908: nm0885: density times sm0909: volume nm0885: volume well we're going to do it per unit area so density times pathlength and that gives you the am-, the amou-, the amount of mass per unit area and so that's going to be rho-secant-theta- D-Z we've already worked that one out okay so that's the emission the emitted part of it and as long as we know the temperature we know the properties of the material that we're dealing with and and i've sort of glossed over how this one comes about so we're going to have to go back to this later in the course and if you know the pathlength then you can work out how much radiance is going to be emitted from a given slab of atmosphere and going off in a given direction but of course the emission's isotropic so here it's going to be the same in all the different directions anyway okay final point that we need to get in this is we need the scattered bit so our next thing to talk about here is radiation er scattered in from other directions so if we're sitting below our slab here and we're observing a scattered radiance L-S-of-nu which is coming out in that direction er that radiance could be getting into that direction from anywhere else and it could be coming from above the slab it could be coming from below the slab or whatever so there could be a contribution to this scattered irradiance coming from any direction so by the way is this pen fading out too much can you still read it okay give m-, give me a sh-, give me a shout if it's if it's getting bad so radiance can be scattered can be scattered into direction theta-phi from any other direction and so we'll write an arbitrary other direction as a er as a theta-primed- phi-primed so from any direction theta-prime-phi-prime the energy could potentially be scattered in into the direction that we're interested in so what's the scattered energy going to depend on sf0910: single scattering albedo nm0885: right it's going to depend on the single scattering albedo can you remember what that defines what does the single scattering albedo measure sf0911: er nm0885: er sounds right yeah it's it's going to measure if the radiation interacts with the material what fraction of it is going to be scattered rather than absorbed so big albedo omega-equals-one everything that interacts at all is going to be scattered small albedo omega-equals-zero anything that interacts at all is going to be absorbed so so the er scattered radiation depends on a single scattering albedo omega-nu now of course it de-, if it depends on the sing-, single scattering albedo that's what happens to it if it interacts in the first place but er what's going to determine the probability that it interacts in the first place what are we going to need to worry about there sf0912: pathlength nm0885: yeah it's going to depend on on the pathlengths or it's going to depend on the amount of material which of course you realize is going to be given by the density times the pathlength er what else is it going to depend on sf0913: size of particle nm0885: er it is actually but i wasn't going to tell you that until next week er [laugh] let's say for the moment it depends on again the properties of the material and you're right it's going to turn out that size of the particles or size of the molecules is the p-, is the property that matters most but it's going to depend on the er type of material or properties of the material and so that's going to be given by the er well we we can write it in terms of the extinction coefficient K-E-of-nu which is the probability that it's going to interact with the radiation or equivalently we can write it as the mass scattering coefficient K-S-nu and the reason for that is because if we know the single scattering albedo that's the ratio between the two of these so if all we if we have any two of these three we can work out what the third one is so it's going to depend on the type of material it's going to depend on how much material er there's another pretty obvious one here suppose that er the sun goes very high in the sky and you now have more radiation incident what's going to happen to the scattering then sm0914: nm0885: yeah it's going to increase if you have more incident radiation you're going to get more scattered into any direction so it's going to depend on the incident radiance and so that's going to be L-nu in what direction is the radiation coming in that we want to worry about someone's doing this which i think means any direction yeah so any direction so if we just pick an arbitrary direction we can say in a direction theta-prime-phi-prime but this brings us on to the last factor which is really crucial here is er the contributions from different directions theta-prime-phi-prime because if radiation gets scattered off some particle a molecule or a wa-, water drop an ice crystal whatever it's not going to you're not going to get the same amount going off in every direction the scattering is not isotropic the radiation will interact with the particle and it'll be sent off in different amounts in different directions so the other thing that we're going to have to know here is we're going to have to know that property of of of of the material we're going to have to know which direction the scattered radiation is going to go in in any given interaction so and the direction directional dependence of the scattering and so we're going to need to define an expression for this so let's define the phase function so the phase function which we'll write with a er capital-P from direction omega-prime into direction omega and i've used solid angle here so omega-prime would be theta-prime-phi- prime and omega would be theta-phi so what this is going to be is this is going to be the probability that a photon incident in direction omega-prime is scattered into the direction omega so some radiance comes from or or is initially travelling in in direction omega- prime then in this certain pathlength and given this certain nature of material it turns out that it interacts with one of the particles there now it could in principle get sent off in any direction but the proportion that gets sent off in the direction that we're interested in omega omega is given by the phase function so the phase function says whichever angle you come from what are the odds that you're going to get scattered into the direction that we want and er again when we get on to talking about scattering in more detail we're going to have to think about what this phase function looks like so what direction does the radiance get scattered okay so er what this means then now is that if we want to get the radiance which is scattered into our direction omega what we're going to need to do is we're going to need to integrate over this phase function 'cause we're going to need to add up the contributions from all the different possible directions that get bounced into our direction and i'm fed up with this pen so i'm going to throw it away lovely okay so er so we must integrate over all incident directions omega-prime to to get the total emerging radiance in direction omega so what does this mean well what this means is if we want to write our scattered radiance so L-superscript-capital-S- sub-nu and er forgot here let's in the direction omega we can now write this as er what are we going to have well we've got to get in all our little bits of stuff here it's going to be in terms of the er of the well let's ca-, let's write it as the mass scattering coefficient so i could also write this as omega-nu times the extinction coefficient let's write it as as the scattering coefficient then an integral over all possible directions of er of the incident radiance in that direction so L-nu- omega-prime and er the multiply by the phase function which is the probability that er you're going to go from omega-prime into direction omega and let me clear a little bit more space here then we're going to need to write the er pathlength which is part of the probability that you're going to get scattered at all so we've got a rho-secant-theta- D-Z and er and then we're going to need to integrate over all directions omega-prime and actually i mean it's a f-, a few things like like the D-Z we can take out of that integral but the pa-, but the the s-, the secant-theta part you know the pathlength is going to depend on which direction the radiance comes in because if something's coming in at a different angle it's going to have a different probability of getting scattered if it's coming in at a low angle it's going to get scattered much more so what we'll do is we'll integrate this over the whole sphere and that means we'll need to divide out four-pi we're integrating over the whole sphere all possible directions because radiation can can get into the slab from above and below why are we dividing out four-pi in the context of solid angle and sphere what does four-pi represent sf0915: whole nm0885: yeah four-pi is is the ra-, is is the solid angle of the whole sphere so we're adding up contributions from all of this and then we're dividing out the total solid angle so 'cause remember radiance is in units of per steradian so we want to keep it as per steradian okay so with that hideous mess we now have all the contributions to the er to the radiance so the radiance emerging from a slab is then going to be what is it going to be it's going to be L-nu in some direction theta-phi and it's going to be the the sum of all these bits that we've put together so it's going to be a sum of the direct beam L-D-nu- theta-phi plus the er emitted that's right L-E-nu again what's emitted in our direction theta-phi although emission of course is going to be isotropic and it's going to be added to the scattering L-S of theta- phi so our emission at the bottom of of the slab is er is given by these three terms but remember as as we were mentioning in the last lecture these terms are not all always important at certain wavelengths the radiation is going to be dominated by er by different parts of this for example with solar radiation coming into a fra-, solar wavelengths the emission by the atmosphere is going to be negligible the atmosphere's just too cold to emit those high energy photons but on the other hand scattering is going to be pretty important and er and of course there will be some depletion of the direct beam as well okay so that's kind of cool that basically covers the er definition section of the course we now know what we're talking about now for the rest of the course we can talk about it so what we're going to do during the rest of the course is er it's going to be divided into two big chunks okay first big chunk is we're going to do scattering and we're going to look at scattering in quite a bit of detail we're going to look at what it is in the atmosphere that does the scattering how those properties determine what the scattering does and then if you actually look up in the sky how does that knowledge of scattering actually explain the stuff that we see so we're going to do fairly obvious or reasonably obvious things like why is the sky blue why is the er why is the er well why why do rainbows form we're going to get into slightly more subtle things like why do photographers always use polarizing filters when they want to look want to photograph clouds and a few other things like that and then after that we're going to do emission and absorption and remember emission and absorption are actually sort of two sides of the same coin so we're going to do that part together and that means the direct beam that depletion bit and the emission bit we're going to handle them together and er and there we're going to get into a little bit of the properties of the particles so we're going to get into a little bit of quantum mechanics we're not going to do it in an-, in any sort of great detail but just enough that we can see why some particles are better at certain things than some other particles and er and then we're going to get into at the end how you actually do these calculations if you happen to be interested in weather or climate or something and you want to know how to put those effects into a model or into some kind of explanation of how climate's going to change or how er or how you know is there going to be ice on the roads tomorrow morning this kind of thing we're going to get into er how you'd actually do those calculations but we can only do that once we've actually gotten into detail of all the processes so we've just got five minutes left today so i want to er point out one slight subtlety of what we've done earlier on today now you remember optical depth is a measure of how much stuff you lose from the beam but optical depth gets used a lot you'll hear that term a lot but you want to be slightly careful with it because as we've seen optical depth is composed of two bits it's composed of absorption and scattering and those can have very different effects you know if if say you've got er a water cloud in the atmosphere with an optical depth of point-five or something like that er on a day like today which actually actually the o-, what would you guess the optical depth was going to be on a day like today that's that's that's a good question sli-, slightly subtle let me give it to you as a multiple choice ten-million one or one-over-ten-million ten-to-the-minus-seven well think of it this way remember the transmittance the fraction of the radiation that's getting through okay suppose the optical depth were ten-to-the-minus-seven what's E-to-the-minus ten-to-the-minus-seven is that going to be a big number little number let's draw a graph so E-to-the- minus-X as a function of X what's it going to be at X-equals-zero sf0916: one nm0885: sorry sf0916: one nm0885: one okay going to be one what's it going to be as X gets big sm0917: nm0885: sorry sm0917: it goes small nm0885: yeah it's going to fade away er where is it going to get to be ten-and- one-halfish sort of thing how fast is it fading away so if there's point-one- five is that point there is that going to be closer to ten-to-the-minus-seven or ten-million or to one or which of those three sf0918: one nm0885: pardon sf0918: nm0885: yeah it's go-, it's going to be kind of oneish 'cause if if if if X is one then that's going to be E-to-the-minus-one which is going to be one-over-E which is one-over-two-point-seven whatever so so basically if you have an optical depth which is really small sort of ten-to- the-minus-seven or something like that then basically what you're saying is the transmission is pretty much one all the light's getting through so on a day like today a lot of the light is not getting through so the optical depth is not going to be that small on the other hand if you've got an E that's something like ten-million then E is going to be very small and you're saying only a very very very small fraction of the radiation is getting through it's going to be very dark and it's not that dark er so it's going to be something of order one but that's because the radiation is scattered suppose we had an abs-, an absorbing case suppose we had a single scattering albedo in in our cloud today of zero then what would the transmission look like i'm asking that question in a co-, kind kind of confusing way the transmission what's directly getting through depe-, depends on depends on the optical depth and that's not a problem er the problem is relating what you see when you look up at the sky to the trans-, to the transmission because what we see when we look up at the sky is actually very little direct beam and it's almost all scattered radiation so actually the optical depth today is pretty big but because of the fact that the single scattering albedo is pretty close to one most of that light that's getting taken out of the direct beam is getting bounced around and some fraction of it maybe half of it or something is getting down to us so we can still see where we're going if we had a smoke cloud like say from a forest fire or something like that and th-, did you guys see the Australian bush fires a couple of years ago on T-V there's showing pictures of that and basically it was black under tho-, if you got close to those things and that was because you had a cloud which in terms of optical depth was probably pretty similar to what we have today but because it had a very low single scattering albedo it was very dark so you got to watch out with optical depth optical depth tells you how much is taken out of the direct beam but it doesn't tell you how much light you're actually going to see on any given day that was actually the point that i wanted to make in a slightly roundabout way okay that's it for today see you next week