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pslct027

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<title>Dynamical equations</title></titleStmt>

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<availability><p>The British Academic Spoken English (BASE) corpus was developed at the

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The recordings and transcriptions used in this study come from the British

Academic Spoken English (BASE) corpus, which was developed at the

Universities of Warwick and Reading under the directorship of Hilary Nesi

(Warwick) and Paul Thompson (Reading). Corpus development was assisted by

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<u who="nm0877"> so today <pause dur="0.7"/> we carry on yeah last week we was a bit waffly went into the <pause dur="1.5"/> i went into the introduction <pause dur="0.2"/> and some <pause dur="0.2"/> early history of numerical modelling of weather and <pause dur="0.7"/> # atmospheric modelling <pause dur="1.5"/> today <pause dur="0.4"/> # we get into more nitty gritty of this and more of the numerics <pause dur="0.2"/> so we're going to get into dynamical equations in this lecture <pause dur="0.9"/> and then in the after lunch there in the old Meteorology building <pause dur="0.5"/> we're going to go across and start doing some time discretization <pause dur="1.3"/> now <pause dur="0.8"/> the next week as well we do horizontal discretization vertical discretization <pause dur="0.3"/> a lot of this some of this stuff you've covered already in the last term in <gap reason="name" extent="2 words"/>'s course <pause dur="0.6"/> but you go into a bit of a deeper level with this in this one so <pause dur="0.4"/> this'll be a bit like a review for some of the things you'll be reviewing a bit but we will do a bit more <pause dur="0.5"/> some new things as well <pause dur="1.0"/> and that'll get a bit more <pause dur="0.4"/> that'll be further than <gap reason="name" extent="1 word"/> took it and this'll go a bit further as well <pause dur="0.9"/> so before we do that the first thing is well what

are the equations what you know we're trying to solve numerically <pause dur="0.7"/> trying to simulate the atmosphere numerically <pause dur="0.5"/> # <pause dur="0.4"/> which <pause dur="0.2"/> equations do we use yeah <pause dur="1.3"/> and # it's not obvious <pause dur="3.3"/> so <pause dur="0.9"/><kinesic desc="puts on transparency" iterated="n"/> the <pause dur="0.5"/> layout the format for this morning's going to be <pause dur="2.7"/> dynamical equations <pause dur="0.7"/> i'm going to go through the <pause dur="1.0"/> we don't have just one meteorological model we have many different models <pause dur="0.5"/> # <pause dur="0.2"/> so i'm going to go through some of the different hierarchy of models the the whole range of models that we can use in meteorology <pause dur="1.2"/> and explain why sometimes it's useful to use a hierarchy of models it's good to use not <pause dur="0.2"/> if you're going to use a model it's better to use <pause dur="0.3"/> a couple of models rather than just one all the time <pause dur="1.0"/> # it's a bit like a map <pause dur="0.2"/> if you had a <pause dur="0.5"/> trying to find your way around you'd have a map of the world but you'd also have a map of <gap reason="name" extent="1 word"/> as well you'd use a couple of things <pause dur="0.2"/> to <pause dur="0.4"/> more than one model is always a good idea <pause dur="1.7"/> # <pause dur="0.4"/> so i'll talk about models and their complexity <pause dur="0.9"/> then i'm going to talk about some of the

waves in the atmosphere this is a bit of a review of some of the fluid dynamics that you probably did last term <pause dur="0.2"/> in the fluid dynamics <pause dur="0.4"/> course <pause dur="0.9"/> # <pause dur="0.2"/> the speed of these waves is very important for how you solve the equations so if you have very fast waves <pause dur="0.2"/> it makes it very hard to solve the equation so <pause dur="0.7"/> knowing which waves you're trying to simulate is very important <pause dur="1.7"/> do some basic equations just as a review of that <pause dur="0.6"/> and then i'll <pause dur="0.4"/> i'll try and present some ideas from sort of dynamical systems theory <pause dur="0.2"/> there's a branch of mathematics called dynamical systems theory <pause dur="0.7"/> # that was started about the beginning of the twentieth century <pause dur="0.7"/> some of the ideas from dynamical systems theory are very relevant for <pause dur="0.2"/> understanding the atmosphere and climate so <pause dur="0.3"/> i'll give you a little <pause dur="0.6"/> a very <pause dur="0.4"/> <shift feature="voice" new="laugh"/> very <shift feature="voice" new="normal"/>brief introduction into that <pause dur="1.0"/> # so that's the plan for today and if there's any questions please interrupt or <pause dur="0.7"/> # <pause dur="0.5"/> feel free to ask any questions you like yeah <pause dur="2.0"/> so <pause dur="2.3"/><kinesic desc="changes transparency" iterated="y" dur="10"/> this <pause dur="3.2"/><vocal desc="clears throat" iterated="n"/><pause dur="0.8"/> hierarchies and models <pause dur="0.2"/> # <pause dur="1.5"/> some

people this isn't in your notes you might want to copy this down this is a <pause dur="0.2"/> this in an extra <pause dur="1.6"/> this is from an article by Hoskins <pause dur="0.8"/> Brian Hoskins <pause dur="0.3"/> in nineteen-eighty-three <pause dur="1.0"/> Q-J-R-M-S # volume one-o-nine <pause dur="0.8"/> pages one to twenty-one <pause dur="1.4"/> Brian Hoskins said well this is the wrong way to look at it some people look at it this way so they say oh <pause dur="0.6"/> there's some observations <pause dur="0.5"/> they get fed into our big complex numerical models <pause dur="1.2"/> # <pause dur="0.5"/> we have some theoretical models which are completely detached from this these are just sort of mathematicians amusing themselves <pause dur="0.4"/> # no connection to <pause dur="0.2"/> # <pause dur="0.2"/> these other models <pause dur="0.8"/> and then there's some basic ideas you know some <pause dur="0.4"/> theories of what <pause dur="0.2"/> simple ideas and concepts that we use <pause dur="0.4"/> and these sort of feed in <pause dur="0.2"/> now this is completely the wrong way to <pause dur="0.7"/> this is not a helpful way to look at models <pause dur="2.3"/> a better way <pause dur="0.2"/><kinesic desc="changes transparency" iterated="y" dur="4"/> # <pause dur="0.4"/> tell me if i'm <pause dur="0.2"/> if i take this off too quickly let me know yeah <pause dur="1.5"/> the better way to look at it is <pause dur="0.7"/> well there's <pause dur="0.3"/> observations <pause dur="1.0"/> they're getting used in a range of dynamical models from

complex ones <pause dur="0.2"/> medium complexity ones to very simple ones so you have a nice range of models so you can understand things <pause dur="1.0"/> and you have evolving conceptual models <pause dur="0.3"/> that sort of feed into understanding how you should observe which things you should observe <pause dur="0.2"/> and how you develop the models so <pause dur="0.4"/> it's more <pause dur="0.4"/> there's a whole range of hierarchy model and there's more <pause dur="0.6"/> # <pause dur="0.4"/> connection between these things there's no separate theoretical model as a separate <pause dur="0.2"/> thing yeah <pause dur="0.2"/> everything is <pause dur="0.8"/> you think of it as a continuum of different types of models <pause dur="1.1"/> it's dangerous people see a very realistic <pause dur="0.2"/> climate model a big one and then they think oh that's the only model <pause dur="0.3"/> all the others are wrong <pause dur="0.3"/> and that's a wrong view of it it's more <pause dur="0.5"/> all models are sort of wrong <pause dur="0.6"/> # <pause dur="0.8"/> what you need is a range of them you need a good range of models <pause dur="0.7"/> so that's a better way to look at the whole modelling exercise yeah <pause dur="2.4"/> did you get that down is that <pause dur="0.3"/> you're still <pause dur="0.3"/><vocal desc="laughter" iterated="y" dur="1"/><pause dur="0.4"/> go fast <vocal desc="laughter" iterated="y" dur="1"/><pause dur="1.9"/> okay i'll give you a

few seconds to get that <pause dur="1.0"/> <trunc>st</trunc> if i write that on the board it takes me longer so these overheads are dangerous 'cause you <pause dur="0.2"/> put things up too quickly yeah <pause dur="5.6"/> well the two differences on these is that the <pause dur="0.4"/> these are very static and don't <pause dur="0.3"/> change <pause dur="0.2"/> the ideas stay the same <pause dur="0.3"/> these are actually <pause dur="0.3"/> by looking at these models your concepts can change things are evolving <pause dur="0.6"/> it's also there's a hierarchy <pause dur="0.5"/> all you don't see it as two separate boxes here you see it as a whole range of models yeah <pause dur="1.0"/> that's the two differences <pause dur="2.3"/> so okay <pause dur="0.3"/> you nearly there <pause dur="0.9"/> # <pause dur="0.5"/> so <pause dur="0.5"/> you might say # <pause dur="0.4"/> well why don't we <pause dur="1.0"/> <trunc>le</trunc> well we're going start i'm going make a hierarchy of models <pause dur="0.4"/> and we'll <pause dur="0.2"/> think of <kinesic desc="writes on board" iterated="y" dur="30"/> the different hierarchy <pause dur="2.2"/> whoops <pause dur="3.4"/> my hands are cold <pause dur="2.0"/> so at the top of the <pause dur="0.2"/> we're going to start with the most complex <pause dur="1.9"/> and we go down to the most <trunc>sim</trunc> <pause dur="0.3"/> simple <pause dur="4.1"/> now just because it's complex or simple doesn't mean it's better or worse sometimes complex models <pause dur="0.5"/> sometimes simple models are you believe them more <pause dur="0.3"/> # <pause dur="0.7"/> very simple models if you get a

result from that you can have more confidence in the result if it's a very complicated model <pause dur="0.7"/> # you have less confidence sometimes <pause dur="1.1"/> now <pause dur="0.4"/> the first way you might say well why don't you just simulate if you could <pause dur="0.2"/> the atmosphere is made up of molecules why don't you just simulate the motion of all the molecules <pause dur="0.7"/> you know we could use Navier <pause dur="0.2"/> we could use Newton's laws of motion <pause dur="0.8"/> and we could <pause dur="0.2"/> simulate all the molecule <pause dur="0.8"/> molecular dynamics <pause dur="2.6"/> so we know <pause dur="0.5"/> we know Newton's laws controls the motion of each molecule <pause dur="0.7"/> # we work out in the future where each molecule's going to go <pause dur="0.8"/> well <pause dur="0.4"/> that's obviously stupid <pause dur="0.6"/> # because there are ten-to-the-forty-five <pause dur="1.1"/> molecules in the atmosphere <pause dur="2.6"/> # now that's <pause dur="0.9"/><kinesic desc="changes transparency" iterated="y" dur="6"/> these these are in your notes these this this is on a <pause dur="0.3"/> a <pause dur="0.2"/> that sheet's been photocopied there <pause dur="0.8"/> but there are ten-to ten-to-the-forty-five molecules and you don't really need to know if you want to know what the weather's going to do tomorrow you don't need to know where each each molecule's going to be <pause dur="0.2"/> you're

more interested in big things you're not really interested in each molecule <pause dur="0.8"/> so it's a bit <pause dur="0.6"/> you'd be <pause dur="0.5"/> it'd be you'd have to do a lot of calculation there it would be impossible to do that calculation actually <pause dur="0.5"/> computers <pause dur="0.6"/> # <pause dur="0.3"/> there's too many molecules <pause dur="0.4"/> and it's also would give you <pause dur="0.2"/> too much information you don't need all that information you're not really interested in each molecule so <pause dur="2.2"/><kinesic desc="changes transparency" iterated="y" dur="7"/> next thing you do is you could say well i don't want to consider the atmosphere as individual molecules i can assume <pause dur="0.4"/> it's a continuum a fluid <pause dur="0.9"/> we know it's not a really a fluid it's full of little molecules but <pause dur="0.6"/> you make an assumption called the <kinesic desc="writes on board" iterated="y" dur="12"/> continuum approximation <pause dur="1.5"/> so this called the continuum <pause dur="1.6"/> this is used in all the fluid mechanics <pause dur="0.9"/> 'cause all fluids are made of molecules in the end <pause dur="1.3"/> so what you do is you say well it's not individual molecules it's a sort of fluid a continuous fluid <pause dur="1.0"/> it's a big <trunc>ajum</trunc> big assumption <pause dur="0.3"/> # <pause dur="0.6"/> doesn't work in outer space when you get into the upper air <pause dur="0.2"/> upper

atmosphere up near where the space stations are <pause dur="0.4"/><kinesic desc="changes transparency" iterated="y" dur="6"/> this continuum approximation is is not right because there's so few molecules <pause dur="0.5"/> you can't treat it as a continuous fluid <pause dur="1.5"/> # <pause dur="0.6"/> so the next thing you could do then is use Navier-Stokes <pause dur="1.7"/><kinesic desc="writes on board" iterated="y" dur="8"/> Navier-Stokes equations for fluids <pause dur="1.2"/> and use those to solve the atmosphere <pause dur="2.3"/> so now you're treating in as a fluid so <pause dur="0.3"/> a lot of people who don't know anything about meteorology or oceanography <pause dur="0.4"/> say well why you don't you just <trunc>u</trunc> they just use Navier-Stokes equations you know people who do aircraft simulation think we use Navier-Stokes <pause dur="1.1"/> and they're always a bit surprised when we say no we don't <pause dur="0.7"/> # <pause dur="1.4"/> now <pause dur="1.0"/> the problem is with Navier-Stokes equations it it involves it includes lots of waves or possible waves it includes <pause dur="0.7"/><kinesic desc="writes on board" iterated="y" dur="25"/> # sound waves <pause dur="1.8"/> # <pause dur="1.0"/> gravity waves <pause dur="0.3"/> # we're going to come back to waves again <pause dur="1.2"/> # <pause dur="0.4"/> it kind of includes inertial waves <pause dur="0.8"/> these are the sort of <pause dur="0.4"/> different types of waves you tend to get in the atmosphere <pause dur="0.2"/> so <pause dur="1.1"/> whoops <pause dur="0.5"/> Rossby waves <pause dur="0.9"/><vocal desc="cough" iterated="n"/><pause dur="0.8"/> oh there's also

Kelvin waves along edges <trunc>th</trunc> <pause dur="0.2"/> Kelvin waves will go along edges mainly so they're a bit strange <pause dur="0.9"/><kinesic desc="changes transparency" iterated="y" dur="3"/> so in the atmosphere <pause dur="0.5"/> or in a in a fluid in general <pause dur="0.4"/> in in the in the compressible fluid in the atmosphere <pause dur="0.3"/> there are sound waves there are gravity waves <pause dur="0.3"/> there are inertial waves <pause dur="0.3"/> there are Rossby waves and there are Kelvin waves <pause dur="0.5"/> those are all the waves you need to know about to understand the atmosphere <pause dur="1.7"/> now you've probably noticed that <pause dur="0.7"/> well of all these waves these ones are very fast <pause dur="0.6"/> sound waves move at about three-hundred metres a second <pause dur="1.6"/><kinesic desc="writes on board" iterated="y" dur="2"/> so if i scream i can't get to the <trunc>li</trunc> i can't run out of the room and <pause dur="0.6"/> you know # move very quickly # sound <pause dur="0.8"/> # now <pause dur="0.3"/> the point is sound waves don't really interact very much with the <pause dur="0.5"/> atmosphere <pause dur="0.9"/> if you go outside and scream you can't affect the weather you know you can't <pause dur="0.2"/> by producing a <trunc>lo</trunc> even a big rock concert <pause dur="0.2"/> doesn't produce cyclones or anything you know there's not much interaction <pause dur="0.6"/> these waves the sound waves <pause dur="0.7"/> don't really interact with the

other waves <pause dur="1.4"/> so to do a weather forecast in the early days in the beginning of the twentieth century people said well <pause dur="0.4"/> we should include sound waves because that's all part of the fluid <pause dur="0.8"/> then they realized oh really the sound waves don't do anything to the atmosphere <pause dur="0.7"/> so they're not really relevant <pause dur="1.5"/> the <trunc>pad</trunc> problem with including sound waves <pause dur="0.2"/> is <pause dur="0.7"/> you have to use a very small timestep you have you you've come across this # Courant-<pause dur="0.2"/>Friedrich-Levy <pause dur="0.3"/> condition yeah <pause dur="0.4"/> have you seen have you heard that from <gap reason="name" extent="1 word"/>'s <pause dur="1.2"/> there's a <pause dur="1.5"/><kinesic desc="writes on board" iterated="y" dur="9"/> a condition for numerical stability called Courant-<pause dur="1.5"/>Friedrich-<pause dur="1.6"/>Levy <pause dur="1.8"/> so we just call it C-F-L <pause dur="0.8"/> C-F-L says that <pause dur="0.2"/> # <pause dur="0.4"/> your timestep of your model <pause dur="0.5"/> has got to be less than the <pause dur="0.4"/> grid spacing divided by the speed <pause dur="0.3"/> the maximum speed of propogation <pause dur="1.6"/> so if i have a <pause dur="0.3"/> hundred kilometre <pause dur="0.5"/> hundred kilometre grid <pause dur="0.5"/> we could work this out if you work this out <pause dur="0.6"/> suppose delta-X i'm trying to simulate a hundred kilometre resolution <pause dur="1.3"/><kinesic desc="writes on board" iterated="y" dur="24"/> suppose i've got a sound wave in there going at three-hundred

metres a second <pause dur="1.3"/> yeah <pause dur="0.8"/> now i don't know what that comes out as if you work that out let's work that out <pause dur="0.3"/> hundred <pause dur="1.8"/> hundred-thousand divided by three-hundred <pause dur="0.9"/> yeah <pause dur="0.6"/> so that's # <pause dur="0.4"/> cross those off <pause dur="1.1"/> that gives you a thousand if i <trunc>ri</trunc> that gives you about three-hundred <pause dur="2.5"/> three-hundred second <pause dur="0.5"/>

that's not right <pause dur="3.9"/> that's right <pause dur="1.1"/> # <pause dur="3.0"/> yeah i've got that right haven't i think <pause dur="0.6"/> yeah got three-hundred seconds <pause dur="0.5"/> so three-hundred seconds is like five <pause dur="0.2"/> five minutes <pause dur="0.6"/> so if i wanted to simulate this sound wave on a thousand kilometre <pause dur="0.2"/> grid horizontal <trunc>se</trunc> <pause dur="0.2"/> this is a <trunc>hos</trunc> horizontal sound wave <pause dur="0.5"/> i would need a <pause dur="0.2"/> a timestep of five minutes in the model <pause dur="0.7"/> but it's worse than that <pause dur="1.0"/> because the sound waves can go vertical as well when i <pause dur="0.2"/> when i scream sound goes upwards as well <pause dur="0.7"/> # the spacing in the models can be a hundred metres in the vertical <pause dur="0.3"/> you want to resolve the troposphere nicely <pause dur="0.4"/> so really that distance shouldn't be a hundred kilometres should be a hundred metres really <pause dur="0.9"/><kinesic desc="writes on board" iterated="y" dur="6"/> divided by three-hundred metres a second <pause dur="0.7"/>

you get a third of a second <pause dur="0.7"/> so if you wanted to include <pause dur="0.7"/><kinesic desc="changes transparency" iterated="y" dur="7"/> # <pause dur="0.4"/> <trunc>ti</trunc> sound waves in your model you'd have to timestep the model every third of a second that would be the maximum timestep you could use <pause dur="0.6"/> so you have to really <pause dur="0.2"/> do lots of simulation you know <pause dur="0.4"/> and you're not really interested in them anyway you know you say well i don't really need them anyway for meteorology so <pause dur="0.6"/> you know <trunc>ho</trunc> why how stupid you going to have to timestep all the time <pause dur="0.6"/> # very quickly <pause dur="0.2"/> just so i can simulate these stupid sound waves <pause dur="0.6"/> and then they don't not useful anyway so <pause dur="0.8"/> # <pause dur="0.4"/> so what you can do then is filter them out you know <pause dur="1.4"/> so <pause dur="1.2"/><vocal desc="cough" iterated="n"/><pause dur="0.2"/> so the next set of equations have got no sound waves <kinesic desc="writes on board" iterated="y" dur="6"/> in them <pause dur="1.2"/> # the Euler equations <pause dur="2.4"/> now the Euler equations <pause dur="0.9"/> # <pause dur="0.9"/> you make an assumption of <pause dur="0.3"/> # anelastic <pause dur="1.4"/><kinesic desc="writes on board" iterated="y" dur="23"/> yeah <pause dur="0.4"/> so you make the approximation to go here is anelastic <pause dur="2.2"/> so you assume <pause dur="1.0"/> # <pause dur="0.8"/> before in the density equation <trunc>y</trunc> <pause dur="0.2"/> in the full one you'd have D-rho-by-D-T <pause dur="0.4"/> plus divergence of rho-<pause dur="0.2"/>U <pause dur="0.5"/> equals zero <pause dur="1.3"/> now if you have that that can

produce <trunc>sou</trunc> that that's responsible that equation's necessary to get the sound waves <pause dur="0.4"/> you need the compressibility to get sound waves it it's compression in density <pause dur="1.0"/> so if you get rid of this equation and sort of say well <pause dur="0.2"/> okay <pause dur="0.2"/> approximate that one by <pause dur="0.3"/><kinesic desc="writes on board" iterated="y" dur="14"/> div-<pause dur="0.4"/>rho-<pause dur="0.4"/>U <pause dur="0.2"/> equals zero <pause dur="0.3"/> yes <pause dur="1.1"/> then you get rid of sound waves <pause dur="0.7"/> and that approximation is called an anelastic approximation <pause dur="2.6"/> now nearly every <pause dur="1.0"/> well every meteorological model i've seen even the very <pause dur="0.2"/> cloud resolving ones or small models <pause dur="0.4"/> all make this approximation <pause dur="0.7"/> 'cause they don't want sound waves in there <pause dur="0.6"/> nobody wants sound waves in their meteorological model <pause dur="2.0"/> so <trunc>i</trunc> it's important to know which approximation so if i i do them in <pause dur="2.1"/><vocal desc="cough" iterated="n"/><pause dur="0.9"/> so the first approximation we use was a <trunc>continu</trunc> ooh <pause dur="2.0"/> was a <pause dur="3.4"/> continuum approximation <pause dur="0.5"/> and the next approximation we use is an anelastic one <pause dur="2.9"/> now <pause dur="3.6"/> now the problem with that is if you still have that they're still not very good because <pause dur="0.5"/> the next wave that moves very quickly are these gravity waves <pause dur="0.5"/> so if you've been

on a near a mountain <pause dur="0.7"/> you can see this on mountains i saw this this last week <pause dur="0.2"/> # the weekend i was near the Alps <pause dur="0.7"/> and you could see <pause dur="0.5"/> a beautiful mountain wave if you're ever near a big mountain you <pause dur="0.3"/> you have a big mountain like this say <pause dur="2.1"/> if there's air coming across the mountain like this what the air does it goes up over the mountain and then it does little oscillations at the back of the mountain yeah <pause dur="1.6"/> and sometimes you can see clouds develop in the <pause dur="0.5"/> in this area these <pause dur="0.4"/> these little stripes so if you see <pause dur="0.4"/> you see the top of the mountain you see these little bands of white cloud <pause dur="0.5"/> have you ever seen <trunc>tho</trunc> has anybody <pause dur="0.6"/> seen these <pause dur="0.4"/> sometimes you see them <pause dur="0.6"/> roll clouds even without mountains but quite often near mountains you see these roll clouds <pause dur="4.2"/> now <pause dur="0.2"/> this thing behind me is a gravity wave <pause dur="0.8"/> yeah so it's created because air had to bob up here and then it's it's trying to relax again it's the # conservation of mass that's creating this <pause dur="0.3"/> gravity wave <pause dur="1.0"/> now some of the gravity waves

can actually they don't just go horizontal some of them can shoot off <trunc>u</trunc> <pause dur="0.2"/> upwards as well <pause dur="0.2"/> they go vertically <pause dur="1.0"/> so <pause dur="0.3"/> mountains produce <trunc>ver</trunc> vertically propagating gravity waves just they're big mountains <pause dur="1.8"/> now the vertically propagating gravity waves also are pretty fast and the spacing's pretty small here so you'd have <pause dur="0.4"/> something like a hundred metres a <trunc>sec</trunc> <pause dur="0.2"/> hundred metres <pause dur="0.5"/> divided by say <pause dur="0.4"/> fifty metres a second yeah <pause dur="1.0"/> so you'd still need a timestep of two seconds if you had the vertically propagating ground <trunc>w</trunc> <pause dur="0.3"/> gravity waves <pause dur="1.2"/> and they're not important for the big weather systems they're not important for cyclones and <pause dur="0.2"/> those sort of things you can live without vertically propagating gravity waves <pause dur="1.5"/> so <pause dur="1.2"/> the next step <pause dur="0.2"/> to go down is to filter out the <pause dur="0.4"/> vertically progagating gravity waves and the way you do that <pause dur="0.9"/><kinesic desc="writes on board" iterated="y" dur="6"/> is the hydrostatic approximation <pause dur="2.7"/> you probably wondered why you keep <trunc>se</trunc> did you see in the fluid course you saw all these approximations like hydrostatic

approximation geostrophic and all these <pause dur="0.6"/> well there's a reason for why you want to approximate them <pause dur="0.9"/> the hydrostatic approximation <pause dur="0.8"/> if you're interested in things longer than say ten seconds or something a bit <pause dur="0.2"/> longer timescale than that <pause dur="0.5"/> you don't really care about vertical <pause dur="0.5"/> gravity <trunc>wa</trunc> <trunc>vertica</trunc> <trunc>bob</trunc> <pause dur="0.3"/> you don't care about little air bobbing behind the back of mountains like this very quickly <pause dur="0.6"/> # <pause dur="0.5"/> you don't need to have <pause dur="0.7"/> you don't need to have vertical gravity you can make this approximation <pause dur="0.7"/> # <pause dur="0.5"/> D <pause dur="0.9"/> D-P-by-D-Z equals minus-rho-G <pause dur="1.8"/> if you make that approximation vertically propagating gravity waves disappear from the equations <pause dur="1.1"/> and you end up with these set of equations which everybody <pause dur="1.2"/> likes using in meteorology it's called the primitive equations <pause dur="3.2"/> so <pause dur="0.8"/> okay <pause dur="0.2"/> so it's a series of sort of approximations go down and this is very <trunc>pop</trunc> these are very popular yeah <pause dur="0.6"/> these these equations are the ones that are used in <pause dur="0.3"/> most of the atmosphere models the weather

forecasting models the <pause dur="0.2"/> ocean models as well use the primitive equations <pause dur="0.5"/> these are a good set of equations for <pause dur="0.4"/> fluids on the sphere <pause dur="1.8"/> # <pause dur="1.5"/> now <pause dur="1.7"/> yeah and in these equations you still have you've got rid of sound waves you've got rid of vertically propagating gravity waves but you still have <pause dur="0.4"/> horizontal gravity waves you still have inertial waves you still have Rossby waves and you still have Kelvin waves so <pause dur="0.5"/> and the storm systems tend to be Rossby waves <pause dur="0.4"/> so # things like storms are <pause dur="0.4"/> mid-latitude storms are basically instable Rossby waves <pause dur="0.3"/> so you can still simulate all those storm systems you <trunc>do</trunc> you didn't need all these <pause dur="0.4"/> sound waves and vertical <pause dur="0.3"/> propagation <pause dur="1.8"/> well those equations are still pretty foul to to solve them <pause dur="0.2"/> # <pause dur="0.3"/> i'm going to show you <pause dur="0.6"/> # <pause dur="0.7"/> i'll show well i'll show you the equation <trunc>act</trunc> yeah <pause dur="0.7"/> you see them <pause dur="2.3"/> these are good ones to understand because they're ones we use all the time in meteorology so <pause dur="2.2"/><vocal desc="sneeze" iterated="n"/><pause dur="0.9"/> so these are some basic equations here <pause dur="2.7"/> # <pause dur="0.8"/> so <pause dur="3.2"/><vocal desc="cough" iterated="n"/><pause dur="0.3"/> check i'm <gap reason="inaudible" extent="1 sec"/> to switch

that thing off <pause dur="4.0"/> so <pause dur="1.0"/> i've just written down a few of these equations here but the <pause dur="0.3"/> these are the primitive equations there are five equations <pause dur="1.3"/> so there's equation <pause dur="0.3"/> these are written in vector form but there's the two components of the wind <pause dur="0.5"/> # U and V <pause dur="1.0"/> # there's a D-U-by-D-T <pause dur="0.2"/> that includes the advection term as well it's the material <pause dur="0.2"/> derivative <pause dur="0.9"/> Coriolis force and a pressure gradient <pause dur="0.6"/> yeah <pause dur="0.2"/> so there's basically a pressure gradient some Coriolis force and a bit of advection yeah <pause dur="0.3"/> that's all there is in the that bit <pause dur="1.2"/> # <pause dur="0.6"/> this D-phi-by-D-P plus R-T-over-P is a different way of writing <pause dur="0.2"/> hydrostatic balance hydrostatic approximation <pause dur="0.6"/> this is a bit like if you like is a <pause dur="0.3"/> is a <pause dur="0.8"/> a modified version of the W equation normally Navier-Stokes should have three equations should have one for U <pause dur="0.3"/> one for V and one for W <pause dur="0.9"/> but because we made this hydrostatic approximation we really <pause dur="0.3"/> made a mess of the W equation <pause dur="0.7"/> the hydrostatic balance is a bit like a <pause dur="0.3"/> # <pause dur="0.8"/> you can look at that as a sort of modified <pause dur="0.7"/>

double equation for W yeah <pause dur="1.7"/> # <pause dur="0.4"/> mass is conserved so we made this anelastic approximation here <pause dur="0.2"/> this is this conservation <pause dur="0.2"/> what flows in must flow up <pause dur="0.2"/> basically so if <trunc>a</trunc> if air comes into a region it has to go up <pause dur="0.6"/> you notice we got rid of all the compression all the <pause dur="0.2"/> D-rho-by-D-T disappeared from this equation so <pause dur="0.7"/> so that's sort of an anelastic <pause dur="0.9"/> version of the conservation of mass <pause dur="1.0"/> and then this is the heat equation <pause dur="0.4"/> # the <pause dur="0.2"/> this is the conservation of potential temperature or temperature <pause dur="0.7"/> # this telling how the atmosphere is heated which is very important for the <pause dur="0.4"/> # <pause dur="0.5"/> that heating is extremely important for the atmosphere without heating there'd be no <pause dur="0.2"/> no winds <pause dur="0.4"/> it's all to do with heating that caused the wind <pause dur="0.4"/> so <pause dur="0.5"/> those are the five equations so the basic conservation this is conservation of <pause dur="0.4"/> horizontal momentum <pause dur="0.6"/> this is sort of conservation of vertical momentum <pause dur="0.5"/> this conservation of mass and this is conservation of energy <pause dur="0.9"/> so there are five <pause dur="0.8"/> those are the basic laws <pause dur="1.2"/>

# that we use <pause dur="1.5"/> is that okay <pause dur="0.2"/> so far <pause dur="0.6"/> are you happy with that <pause dur="0.7"/> # </u><pause dur="0.4"/> <u who="sm0878" trans="pause"> <gap reason="inaudible" extent="1 sec"/> what you said about <unclear>it</unclear></u><pause dur="0.6"/> <u who="nm0877" trans="pause"> pardon </u><pause dur="0.6"/> <u who="sm0878" trans="pause"> <gap reason="inaudible" extent="1 sec"/> what you call the F-K <gap reason="inaudible" extent="1 sec"/></u><u who="nm0877" trans="latching"> oh F-K that's the Coriolis term <pause dur="1.1"/> that's the thing if you # <pause dur="0.7"/> few years ago there was a discussion in # <pause dur="0.4"/> in the Guardian about # <pause dur="0.4"/> if you were near the North Pole and you tried to shoot a polar bear <pause dur="0.9"/> # which eye would you shoot if you want to hit it between the eyes yeah <pause dur="0.5"/> 'cause that's the best way to kill a polar bear <pause dur="0.6"/> # if you do that which eye do you aim at <pause dur="0.6"/> because there's a Coriolis force <pause dur="0.4"/> when you fire something on the sphere then it bends <pause dur="0.7"/> so basically you aim for the left eye i think it is on the North Pole so if you see a polar bear you <trunc>shou</trunc> shoot at the left eye and then the bullet sort of turns a bit and <pause dur="0.6"/> so <pause dur="0.5"/> this swerving of the air mass air is the <pause dur="0.6"/> is the Coriolis <pause dur="1.1"/> another example of that a nice one is the # <pause dur="1.0"/> if you look at the whole planet <pause dur="1.3"/><vocal desc="cough" iterated="n"/><pause dur="0.2"/> normally air is hot it's hot here in the <pause dur="0.2"/> equator and it's cold at the pole <pause dur="1.2"/> so air would normally if there was no Coriolis <gap reason="inaudible" extent="1 sec"/><pause dur="0.2"/> this

air here would <pause dur="0.7"/> you'd be <pause dur="0.3"/> air that came out of the Hadley cell at the top would <pause dur="0.2"/> would flow towards the pole <pause dur="0.9"/> but because of the Coriolis force it it it swerves that way and produces the westerly jet <pause dur="1.0"/>

so the westerly jets that we get in mid-latitudes are because of the Coriolis force <pause dur="0.5"/> causing this air to swerve to the right <pause dur="1.1"/> so this is the same reason if you were to shoot a polar bear its have its two eyes would be there <pause dur="1.1"/> you'd fire at that eye and then the thing would swerve a bit and <pause dur="0.2"/><vocal desc="laughter" iterated="y" dur="1"/><pause dur="0.7"/> always remember this if you're shooting polar bears <vocal desc="laughter" iterated="y" dur="1"/><pause dur="0.9"/> and it's all <trunc>t</trunc> <pause dur="0.2"/> well you'd have to be a long way away from the polar you can work it out actually i think in the the newspaper article people starting working out how far away the polar bear would be and <pause dur="0.5"/> you know how many <trunc>mi</trunc> millimetres it would <shift feature="voice" new="laugh"/> move <shift feature="voice" new="normal"/><pause dur="0.4"/> needs to be <shift feature="voice" new="laugh"/>a long <shift feature="voice" new="normal"/>way away to see the effect of the <pause dur="0.4"/> thousand kilometres or something <shift feature="voice" new="laugh"/>so <shift feature="voice" new="normal"/><pause dur="1.2"/> # <pause dur="1.1"/> okay so that's the those are pressure <pause dur="0.2"/> pressure grade normally

most fluids people who do just regular fluid dynamics don't have Coriolis forces <pause dur="0.6"/> so we have that in geophysical fluids <pause dur="0.2"/> but # <pause dur="0.5"/> on on the rotating earth so <pause dur="0.3"/> this is normally it's just pressure gradient <pause dur="0.4"/> causes the movement of air or # the <pause dur="0.2"/> the velocity <pause dur="0.7"/> we're a bit different 'cause we have this rotation term <pause dur="2.0"/> so that's the primitive equations <pause dur="0.2"/> # <pause dur="0.6"/> yep <pause dur="0.4"/> # now <pause dur="1.4"/> now i know it's a bit tricky to understand sometimes <pause dur="0.3"/> 'cause the it's still the solution to these equations they don't look too nasty mathematically but the solution is very non-linear <pause dur="0.5"/> and it's very complicated to try and solve those things on a sphere and things <unclear>yeah</unclear> <pause dur="0.8"/> so <pause dur="0.3"/> people have made a lot of <pause dur="0.7"/> progress using things like shallow water <pause dur="2.4"/><vocal desc="cough" iterated="n"/><pause dur="2.2"/> now <pause dur="1.1"/> shallow water equations <pause dur="1.9"/> what you do there is you say <pause dur="0.4"/> if you look at these equations here there's a lot of vertical structure <pause dur="1.0"/> so this works for a three-dimensional atmosphere this is <pause dur="0.3"/> this has got all these <pause dur="0.4"/> # <trunc>D-do</trunc> D-omega-by-D-Ps and things so things vary in the vertical complicated <pause dur="1.0"/> #

sometimes it's not as bad as that some some waves you see in the atmosphere have got pretty much the same structure all the way up <pause dur="0.6"/> or they have the same <pause dur="0.4"/> they'll have a certain vertical structure which stays the same it's just the horizontal bit that changes <pause dur="0.8"/> # these waves that propagate on the tropical Pacific for El Niño <pause dur="0.3"/> the Kelvin waves and the Rossby waves the equatorial ones <pause dur="0.5"/> # <pause dur="0.6"/> they <trunc>do</trunc> you don't really need the primitive equation to understand why they're there <pause dur="0.2"/> you you don't need the vertical structure they're not very <pause dur="0.2"/> they're not varying a lot vertically <pause dur="1.1"/> in time <pause dur="1.0"/> so the the <pause dur="1.1"/> shallow water equations is a <trunc>ve</trunc> they're a very nice set of equations actually <pause dur="1.8"/> # <pause dur="0.7"/> you can't use them for everything but the <pause dur="1.7"/> they're good for testing numerical schemes <pause dur="1.2"/> and they work at the equator <pause dur="0.2"/> which is quite nice <pause dur="0.8"/> so what you have is basically a <pause dur="0.3"/> # <pause dur="1.0"/> you have the <pause dur="0.4"/> <unclear>something</unclear> <pause dur="0.6"/> think of the ocean if you like you can think of this as the atmosphere or the ocean <pause dur="0.4"/> imagine there's two types of fluid <pause dur="1.5"/> so

# <pause dur="0.4"/> they originally started for <trunc>a</trunc> <pause dur="0.2"/> oceans so you'd say if this was air <pause dur="1.3"/> and this was # water <pause dur="0.2"/> yeah <pause dur="1.2"/> and then there's a certain height <pause dur="1.5"/> a certain depth of the fluid yeah <pause dur="1.1"/> now you want to understand how it's a bit like your water in the bath you know how does it move around basically <pause dur="0.2"/> there's no vertical structure as such there's only two types of fluid really here <pause dur="1.0"/> # you can use these equations even in just the ocean or the atmosphere you just say the <pause dur="0.2"/> the things <pause dur="0.2"/> in the <pause dur="0.2"/> in the tropical Pacific what they do is they say <pause dur="0.3"/> oh well this is like <pause dur="0.6"/> # <pause dur="0.3"/> this is warm ocean <pause dur="0.3"/> so <pause dur="0.2"/> the ocean above the thermocline <pause dur="0.5"/> for the ocean normally this is the thermocline <pause dur="3.3"/> below this certain gradient in the ocean there's a very cold ocean water at four degrees celsius <pause dur="0.2"/> so that's deep ocean <pause dur="1.5"/> so if you go down deep in the ocean you find water at four degrees celsius <pause dur="0.3"/> it's not at zero <pause dur="1.3"/> so you can say there's cold deep dense water underneath and then there's this nice warm stuff <pause dur="0.3"/> and this thermocline in the tropical

Pacific is about <pause dur="0.4"/> from about a hundred to three-hundred metres <pause dur="0.4"/> in depth <pause dur="1.3"/> so <pause dur="0.3"/> you only have one interface you only have this sort of line in between <pause dur="0.3"/> you you don't care about all the vertical detail <pause dur="1.2"/> and so you have one one <trunc>h</trunc> <pause dur="0.6"/> instead of having Ws and things you <trunc>ha</trunc> just have a H <pause dur="0.3"/> in this shallow water equations you have <pause dur="0.7"/> conservation of <pause dur="0.7"/> # <pause dur="1.1"/> conservation of U <pause dur="0.5"/> # of the zonal wind <pause dur="0.2"/> conservation of the meridional wind <pause dur="0.5"/> those two equations are very similar to are almost <pause dur="0.2"/> identical actually to these equations up there <pause dur="1.1"/> and then you have # <pause dur="0.4"/> conservation of mass really here <pause dur="0.2"/> this was sort of a mixture of this one <pause dur="0.7"/> # <pause dur="1.2"/> yep <pause dur="0.2"/> it's a bit like a mixture of this and this together produce this third equation here <pause dur="0.7"/> this is basically just saying the convergence of fluid <trunc>i</trunc> if a lot of <pause dur="0.2"/> fluid converges on a particular point <pause dur="0.4"/> the <trunc>w</trunc> the the height would have to go up at that point <pause dur="0.7"/> so in this thing here <pause dur="0.5"/> the reason that's a big blob up there is because fluid converged in <pause dur="0.8"/> at that

point <pause dur="1.7"/> so there's a very simple set of equations <pause dur="1.1"/> # <pause dur="0.7"/> i've got an exercise for you if you'd like to try this <pause dur="0.3"/> which is quite <pause dur="0.2"/> a good one to do <pause dur="0.6"/> is here the equations for U and here's an equation for V <pause dur="1.1"/> now what i would like you to do is try and find out <pause dur="0.7"/> # <pause dur="1.0"/> # <pause dur="0.4"/> try and find out an equation for the vorticity and for the <pause dur="0.5"/> # <pause dur="0.4"/> divergence <pause dur="0.3"/> so <pause dur="0.6"/> the exercise <pause dur="3.3"/> oh <pause dur="3.1"/> exercise a bit like this <pause dur="1.3"/> just remind you <pause dur="0.9"/> this is useful to understand <pause dur="0.6"/> where things come from <pause dur="0.4"/> so the <pause dur="0.8"/> the <pause dur="0.2"/> # relative <pause dur="2.2"/> relative <pause dur="0.4"/> vorticity <pause dur="2.3"/> is defined as <pause dur="0.2"/> # D <pause dur="0.5"/> this horrible Greek symbol i can never pronounce right D <pause dur="0.4"/> this will be bad for your linguistics one that's your V <pause dur="0.4"/> the D <pause dur="0.6"/> no it's <pause dur="0.2"/> <trunc>x</trunc> <pause dur="0.3"/> xi <pause dur="0.3"/> in Greek it's X-I <pause dur="1.1"/> actually <pause dur="0.4"/> you might be able to help me <pause dur="0.6"/> how do you pronounce it </u><pause dur="0.6"/> <u who="om0880" trans="pause"> <distinct type="sampa">[xi]</distinct></u><pause dur="0.6"/> <u who="nm0877" trans="pause"> <distinct type="sampa">[xi]</distinct> </u><pause dur="0.4"/> <u who="om0880" trans="pause"> <gap reason="inaudible" extent="1 sec"/></u><u who="nm0877" trans="latching"> is that right <distinct type="sampa">[xi]</distinct> </u><pause dur="0.4"/> <u who="om0880" trans="pause"> <gap reason="inaudible" extent="1 sec"/></u><pause dur="0.3"/> <u who="nm0877" trans="pause"> okay <pause dur="0.5"/> xi <pause dur="0.4"/> # this is relative vorcity <pause dur="0.2"/> xi yeah <pause dur="0.4"/> # it's always a bit tricky to pronounce so <pause dur="0.7"/> D <pause dur="0.2"/> D-xi-<pause dur="0.2"/>by-D-X <pause dur="0.2"/> minus <pause dur="0.2"/> D-<pause dur="0.2"/>xi-<pause dur="0.6"/>by-<pause dur="0.4"/>D-<pause dur="0.3"/>Y <pause dur="0.5"/> that's the definition of relative vorticity yeah <pause dur="2.1"/> the <pause dur="0.7"/> the and <pause dur="1.0"/> oh sorry what am i doing no it's not the definition <pause dur="0.3"/> i'm

<trunc>f</trunc> getting <trunc>con</trunc> confused <pause dur="0.7"/> no relative vorticity is <pause dur="0.2"/> is is <trunc>r</trunc> <pause dur="0.3"/> is xi <pause dur="0.6"/> and it's equal to D-<pause dur="0.2"/>V-by-D-X <pause dur="0.7"/> minus D-U-by-D-Y <pause dur="1.3"/> so if you like is the spin of the fluid <pause dur="0.2"/> mm <pause dur="1.8"/> it's how much the <trunc>s</trunc> <trunc>flu</trunc> fluid is <trunc>s</trunc> spinning around or swirling <pause dur="1.7"/> there's also the divergence <pause dur="0.2"/> which is a lot easier to say in Greek <pause dur="0.7"/> # divergence usually use the symbol delta <pause dur="1.7"/> and delta is defined as D-U-by-D-X <pause dur="0.9"/> plus D-V-<pause dur="0.4"/>by-D-Y <pause dur="1.1"/> yeah <pause dur="0.9"/> so the difference there both of them involve Us and Vs and X and Ys but they're mixed up <pause dur="0.5"/> yeah <pause dur="0.2"/> so one's like that and one's got a minus sign in <pause dur="1.4"/> now the point is i <pause dur="0.2"/> i've given you an equation <pause dur="0.2"/> # <pause dur="0.5"/> D-U-by-D-T equals nananana <pause dur="0.2"/> and i've given you an equation D-V-by-D-T equals <pause dur="0.3"/> dadadada <pause dur="1.3"/> now what i would like <pause dur="0.2"/> which is what you solve normally you don't solve these equations with vectors in in the <pause dur="0.2"/> in the big models <pause dur="0.3"/> you solve equations for the vorticity and the divergence <pause dur="1.2"/> so <pause dur="0.7"/> if you work out now i'll just set you off on the right track if you write down <pause dur="0.4"/> D-<pause dur="0.5"/>xi-<pause dur="0.3"/>by-D-T <pause dur="1.0"/> that's going to be <pause dur="1.3"/> D-by-D-T <pause dur="0.6"/> # <pause dur="0.4"/> you plug in that

definition D-V-by-D-X <pause dur="1.0"/> minus D-U-by-D-Y <pause dur="2.1"/> and then <pause dur="0.2"/> you can switch the <pause dur="0.2"/> derivatives so the D-by-D-T and the D-by-D-X you can change the order of them <pause dur="0.6"/> yeah you know <pause dur="0.3"/> can you see that okay from there the <pause dur="1.7"/> so this thing you can then write as D-<pause dur="0.2"/>by-D-X <pause dur="1.3"/> D-V-by-D-T <pause dur="1.3"/> minus D-<pause dur="0.4"/>by-D-Y <pause dur="1.2"/> D-U-by-D-T <pause dur="1.5"/> okay <pause dur="1.9"/> now you've got the equation i gave you the equations for D-U-by-D-T and D-V-by-D-T you've got them written down so you just plug those equations in now <pause dur="0.7"/> substitute in there and then sort out all the D-Xs and D-Ys and you get <pause dur="0.4"/> you'll end up with two nice little equations you'll have one for <pause dur="0.6"/> D-xi-by-D-T <pause dur="0.9"/> and you'll have another one that's for D-delta-by-D-T <pause dur="3.2"/> okay <pause dur="0.3"/> so if <pause dur="0.2"/> try that exercise you know go away <pause dur="0.2"/> it doesn't take you that long to do it probably <pause dur="0.2"/> takes about fifteen minutes or so <pause dur="0.6"/> but it's quite nice 'cause you've then derived a vorticity equation by yourself because <pause dur="0.3"/> sometimes you don't really care about the divergence <pause dur="0.2"/> sometimes you're more interested just in vorticity yeah <pause dur="0.7"/> Rossby

waves are more interested in vorticity than they are in divergence so <pause dur="0.5"/> Rossby wave dynamics is all <pause dur="0.2"/> vorticity stuff <pause dur="0.7"/> so this sort of separating of the two fingers is nice and <pause dur="0.5"/> and you can recast those equations in terms of vorticity and divergence so <pause dur="0.2"/> it's a good little exercise to try <pause dur="1.3"/> shouldn't tax you too much hopefully <pause dur="1.1"/> # <pause dur="1.2"/> and you can also in this one <pause dur="0.2"/> # <pause dur="2.4"/> # <pause dur="0.6"/> so this one you can <pause dur="0.7"/> # <pause dur="1.4"/> now if you set <pause dur="0.3"/> once you've solved the you'll have three equations then one for D-xi-by-D-T one for D-delta-by-D-T and one for D-phi-by-D-T <pause dur="0.8"/> you can see here this is already delta isn't it D-U-by-D-X plus D-V-by-D-Y is delta <pause dur="0.6"/> so the only coupling between <pause dur="0.9"/> is a vorticity equation a diverged one and a <pause dur="0.2"/> and a height <pause dur="0.4"/> phi is a sort of height <pause dur="0.8"/> # <pause dur="0.2"/> the height is only coupled to the divergence equation it's not coupled to the vorticity one <pause dur="1.1"/> and and if you set the # <pause dur="0.4"/> if you set the phi to <trunc>z</trunc> steady phi so D-phi-by-D-T is zero the <trunc>dele</trunc> divergence is then zero <pause dur="0.5"/> and then you'll get a just a vorticity equation yeah <pause dur="1.2"/> which comes to my

next set of simple things <pause dur="0.8"/> which is <pause dur="0.3"/> vorticity equations <pause dur="2.5"/> so if you say i don't care about divergence <pause dur="0.9"/> so basically you said delta is equal to zero <pause dur="0.2"/> then you get sort of divergence <trunc>eq</trunc> <pause dur="0.2"/> i mean vorticity equations <pause dur="0.5"/> and you can get a vorticity equation from this quite nicely if you do this thing over there <pause dur="0.3"/> and then just set delta to zero <pause dur="0.3"/> you'll end up with a nice little <pause dur="0.4"/> vorticity equation <pause dur="2.1"/> and your vorticity equation that you'll get <pause dur="0.2"/> is called <pause dur="0.2"/> if you do that you'll get the <pause dur="0.7"/> # barotropic <pause dur="0.2"/> vorticity equation <pause dur="1.2"/> which is the one you're going to be using in the exercise in the <trunc>n</trunc> practical assignment when we do the <trunc>numeri</trunc> <pause dur="0.2"/> the computer <pause dur="0.4"/> practical <pause dur="1.7"/> there are quite a few different vorticity equations there's a barotropic one <pause dur="0.8"/> which is what you're going to have <pause dur="1.8"/><vocal desc="cough" iterated="n"/><pause dur="0.4"/> # <pause dur="0.4"/> there's also you you've seen ones probably <pause dur="0.6"/> # <pause dur="1.4"/> yeah <pause dur="0.2"/> so that's sometimes called the barotropic <pause dur="0.2"/> so we call that B-V-E <pause dur="0.2"/> the baratropic vorticity equation <pause dur="0.9"/> # <pause dur="0.4"/> you can also have ones which have <pause dur="1.2"/> bit more complicated stuff <pause dur="0.2"/>

quasi-<pause dur="0.4"/>geostrophic <pause dur="2.9"/> vorticity equation <pause dur="2.7"/> but you have different equations for the spin basically of the fluid yeah <pause dur="0.9"/> and those are quite useful for understanding things like cyclones and instabilities things <pause dur="0.2"/> we consider cyclones to be things in vorticity really rather than <pause dur="0.5"/> we're not really that interested in divergence usually <pause dur="2.1"/> # <pause dur="0.4"/> right the next set of models # again <pause dur="0.5"/> kind of takes a while this <pause dur="0.4"/> these are all three-dimensional sort of things this well this was these were if you look at this this was three-D <pause dur="1.4"/> this was a three-D model <pause dur="0.2"/> this was a three-D model <pause dur="0.5"/> three-D <pause dur="0.7"/> # these are now two dimension <trunc>be</trunc> really because we've only got <pause dur="0.7"/> we only had a height equation we got rid of a lot of the vertical we only had one height everything was a function of either <pause dur="0.5"/> in in these equations here everything was a function of X <pause dur="0.3"/> Y <pause dur="0.2"/> and T <pause dur="0.2"/> there was no Z in there there was no height <pause dur="1.4"/> even height the variable H is just a function of <pause dur="0.4"/> where you are <pause dur="1.9"/> so the <pause dur="0.2"/> the the these we've made a big step from

going from here <pause dur="0.3"/> to here we went from three-dimensional set of equations down to two-dimensional flow <pause dur="1.4"/> these are two-D sort of things <pause dur="1.5"/> # <pause dur="0.7"/> now what you can do is you can average equations so sometimes people # <pause dur="0.6"/> they do zonal mean models <pause dur="1.0"/> or zonal mean <pause dur="0.3"/> another type of <pause dur="0.2"/> two-D equation is zonal mean <pause dur="1.3"/> so they're interested in say the Hadley cell or the <pause dur="0.4"/> the general <pause dur="0.3"/> they want to look at just the cross section they don't want to get into details about longtitude they just want latitude and height <pause dur="1.2"/> # now they make sometimes they make these zonally average models where they've been averaged everything in the longtitude direction <pause dur="1.4"/> and # <pause dur="0.2"/> to get those models they use those <pause dur="1.1"/> another type of model that's used quite a lot <pause dur="0.2"/> is a very useful type of model actually is a one-D model <pause dur="1.4"/> sometimes these are sort of like functions the <trunc>th</trunc> the thing will be a <pause dur="0.5"/> for instance could be U <pause dur="0.2"/> as a function of <pause dur="0.6"/> # <pause dur="0.8"/> X and Z <pause dur="0.6"/> and T <pause dur="1.9"/> oh sorry Y Z T they're not interested in X they're not interested in longtitude <pause dur="1.8"/>

so it's just a cross section in <pause dur="0.6"/> # <pause dur="0.5"/> goes from the pole to the other <pause dur="0.2"/> pole to the equator or something like that and up and down it <trunc>d</trunc> they don't care about where you are <pause dur="0.4"/> in longtitude <pause dur="0.9"/> one-D models sometimes people are interested in just models that vary <pause dur="0.3"/> # use <pause dur="0.4"/> the function of height and the function of time <pause dur="1.3"/> # for instance when you <trunc>s</trunc> fire off a radiosonde in the atmosphere <pause dur="0.6"/> # when you do the radiosonde you just get this sort of measurement with height and time <pause dur="0.6"/> you don't know about <pause dur="0.3"/> you don't really know about X and Y much <pause dur="0.9"/> and <pause dur="0.2"/> the models people use here <pause dur="0.2"/> quite often the nice ones are these radiative <pause dur="1.7"/> convective models <pause dur="4.5"/>

if you like those are the models people use to # <pause dur="0.8"/> you can look at a particular point on the planet so you say well i'm interested in <gap reason="name" extent="1 word"/> <pause dur="0.3"/> i'd like to know what <pause dur="0.5"/> how much radiation comes down to hits the ground <pause dur="0.3"/> and how much convection goes on at this point over <gap reason="name" extent="1 word"/> <pause dur="0.3"/> i just assume there's no horizontal mixing at all i just ignore all horizontal flow <pause dur="0.4"/> and i'm just looking at each point individually <pause dur="0.9"/> and oceanographers do that sometimes they have one-dimensional models of the ocean at particular points yeah <pause dur="0.9"/> so they're quite handy things to have <pause dur="1.4"/> and the final sort of model is a <pause dur="0.2"/> zero-dimensional model so it doesn't have any <pause dur="1.1"/> any space or time variation in it <pause dur="0.2"/> any space version <pause dur="0.7"/> # <pause dur="0.2"/> the nice example of that <pause dur="0.2"/> is these energy balance models <pause dur="2.0"/> so when

people <pause dur="0.7"/> talk about climate change <pause dur="2.4"/> they sometimes use an energy balance model <pause dur="1.0"/> and they'll get a <pause dur="0.3"/> they'll get an equation <pause dur="1.0"/> for instance you can look at the whole Earth <pause dur="1.2"/> you can say here's <pause dur="0.3"/> here's planet Earth <pause dur="0.6"/> it receives so much solar radiation comes in <pause dur="0.8"/> we only get a quarter of that because it hits just one side of the planet <pause dur="0.9"/> # <pause dur="0.2"/> but then we lose infra-red radiation out of the <pause dur="0.2"/> all sides <pause dur="1.1"/> so the amount of stuff we lose goes as <pause dur="0.3"/> sigma-T-to-the-fourth as a black body <pause dur="0.9"/> so you can say sigma-T-to-the-fourth is equal to the incoming solar <pause dur="0.6"/> that's energy balance so you said <pause dur="0.4"/> what came in as far as energy onto Earth <pause dur="0.2"/> must go out yeah <pause dur="0.3"/> so it's <pause dur="0.3"/> coming in as solar and it's leaving as <trunc>r</trunc> infra-red radiation <pause dur="1.1"/> that's the simplest energy balance model <pause dur="1.7"/> if you work that out the solar constant is thirteen-seventy <pause dur="0.3"/> watts per metre squared <pause dur="1.1"/> # this is the Stefan-<pause dur="0.2"/>Boltzmann Stefan-Boltzmann constant <pause dur="0.2"/> you find in a thermodynamics book for black body radiation <pause dur="0.6"/> and you'll find a temperature for the

Earth of two-hundred-and-fifty-five degrees kelvin if you do that <pause dur="0.7"/> which is too cold <pause dur="0.5"/> the average on Earth the average temperature on Earth the observed one <pause dur="0.7"/> if you average over the whole planet is two-hundred-and-eighty-eight kelvin <pause dur="1.7"/> so that's an energy balance model the reason it didn't work very well it was too cold was thirty degrees too cold is because we didn't include any <trunc>abs</trunc> we didn't include a nice little atmosphere round here that would absorb infra-red radiation <pause dur="0.6"/> so there's no <pause dur="0.2"/> this is <pause dur="0.3"/> that that little balance model is just <pause dur="0.5"/> # <pause dur="0.2"/> for a planet <pause dur="0.3"/> there's no atmosphere there <pause dur="0.7"/> if you put the little blanket round it then you get another thirty degrees and you get <pause dur="0.7"/> you get what they call the greenhouse effect <pause dur="0.2"/> not the enhanced one the <pause dur="0.3"/> it's just natural <pause dur="0.3"/> yeah <pause dur="0.4"/> yeah </u><pause dur="0.2"/> <u who="sm0879" trans="pause"> <gap reason="inaudible" extent="3 secs"/></u> <pause dur="0.5"/> <u who="nm0877" trans="pause"> no they don't no <pause dur="0.3"/> no <pause dur="0.4"/> no you can have you can develop more complicated ones but <pause dur="0.4"/> there've been an awful lot of <pause dur="0.2"/> nice little <pause dur="0.2"/> things like that one </u><u who="sm0879" trans="latching"> yeah </u><pause dur="0.5"/> <u who="nm0877" trans="pause"> yeah they don't they can get you can make one-D energy balance

models <pause dur="0.4"/> the thing with those is they start getting more <pause dur="0.9"/> the more complicated ones like that <pause dur="0.3"/> # <pause dur="0.7"/> don't <pause dur="0.7"/> once you go go to higher higher number of when you want the more space variation then all the flow comes into it you know <pause dur="0.3"/> so you really need the fluid dynamics you can't there's a limit to how far you can get on energy balances so <pause dur="0.5"/> they sort of work nice when you average over the whole planet but they don't <pause dur="0.6"/> they <trunc>w</trunc> <pause dur="0.2"/> they're not so sweet when you look at them in certain regions yeah <pause dur="1.9"/> but yeah you're good point yeah it's # <pause dur="1.1"/> anyway <pause dur="0.4"/> that's the range of different models that you come across <pause dur="0.4"/> # this is sort of the most complex and this is the simplest <pause dur="0.8"/> as far as complexity it doesn't mean it's <pause dur="0.2"/> doesn't mean it's not got interesting results <pause dur="0.2"/> actually this little model here <pause dur="0.6"/> <trunc>t</trunc> can tell you by playing around with this you can find out that <pause dur="0.7"/> # Arrhenius in the <pause dur="0.4"/> this person called Arrhenius <pause dur="1.1"/> Swedish scientist in the <pause dur="2.1"/> in the <pause dur="0.2"/> eighteen-ninety-six <pause dur="1.2"/> he used a little model like this <pause dur="0.6"/> and he found

the greenhouse effect he found the enhanced one he got a he said well if carbon dioxide changed we'd get so many degrees kelvin using a little model like that <pause dur="0.6"/> he was way before his time yeah <pause dur="1.2"/> # <pause dur="0.7"/> and so even without a big climate model you can you can get confidence that by changing a bit of carbon dioxide that will cause temperatures to warm up and <pause dur="0.3"/> it's based on a very simple little model <pause dur="0.2"/> so that's a nice model you know you <pause dur="0.5"/> you don't want to believe a big complicated thing you know if if a little thing like that on a back of an envelope <trunc>c</trunc> <pause dur="0.3"/> can convince you of climate change then <pause dur="0.7"/> that's pretty good <shift feature="voice" new="laugh"/>yeah <shift feature="voice" new="normal"/><pause dur="1.0"/> well it makes you feel better about if somebody just said well i've got a big <trunc>complica</trunc> i i've solved <pause dur="0.4"/> if somebody said i've solved every molecular dynamics and i've found out it was going to get warmer you know <pause dur="0.4"/> you'd just say well i don't believe you you know so <pause dur="0.3"/> there's that you can do on a bit of paper yourself and <pause dur="0.5"/> convince yourself quite quickly <pause dur="1.1"/> so # <pause dur="0.2"/>

simple models can be pretty good <pause dur="1.4"/> okay is that <pause dur="0.2"/> that's a sort of quick run-through the important thing <pause dur="0.9"/> like a lot of these things in this numerical course you'll find out <pause dur="0.4"/> the important thing is to know which approximations are being made <pause dur="0.4"/> so if somebody <pause dur="0.3"/> comes along and gives you a model <pause dur="0.2"/> or tell say your supervisor or whatever comes along eventually and says <pause dur="0.5"/> # <pause dur="0.4"/> here's a nice model to use you know <pause dur="0.6"/> it's a good one we use it round here <pause dur="0.2"/> everybody likes it round here <pause dur="0.6"/> the things to know are what are the approximations that went into your model <pause dur="0.9"/> and a lot of people are not very good at that not knowing that what they <pause dur="0.5"/> their models are good at some things for instance <pause dur="0.7"/> this model because primitive equations because it's had the hydrostatic approximation <pause dur="0.6"/> and the anelastic approximation <pause dur="0.3"/> is absolutely useless at doing anything to do with sound waves <pause dur="0.4"/> or anything to do with vertically propagating mountain waves <pause dur="1.0"/> so if somebody came along with the big climate model say a primitive

equation model and said <pause dur="0.4"/> oh we're looking at vertically propagating waves round the back of a mountain you're using this model <pause dur="0.6"/> complete rubbish you know it's not made for that it's the approximations <pause dur="0.2"/> got rid of those sort of things <pause dur="0.7"/> so you should be very careful about which <pause dur="0.6"/> each model involves certain approximations so <pause dur="0.3"/> if you're clever you know which approximations have been used for your model <pause dur="0.9"/> that tells you which things you can use your model for you know so <trunc>y</trunc> <pause dur="0.2"/> it'd be <pause dur="0.2"/> plain <pause dur="0.3"/> stupidity you couldn't look at sound waves using primitive equation models so <pause dur="0.8"/> it's designed for looking at bigger things <pause dur="2.2"/> and people even clever people these days <pause dur="0.5"/> get confused with that so they'll say oh we've reduced the resolution of our very complicated climate model <pause dur="0.2"/> and we're going to go down like the Japanese now have a project frontier project <pause dur="0.5"/> to make a one <pause dur="0.2"/> five kilometre grid on their model so they're going to get a really small resolution you know and look at <pause dur="0.4"/> very local

things <pause dur="0.9"/> well <pause dur="0.3"/> they're still using the primitive equations <pause dur="0.9"/> so their model's still not going to get gravity waves properly so even if they got five kilometre resolution <pause dur="0.5"/> round the back of a mountain or something they're not going to get it right <vocal desc="laugh" iterated="n"/><pause dur="0.8"/> so <pause dur="0.2"/> you get a very funny impression they say <pause dur="0.5"/> ah but it's got a five kilometre resolution it's resolving everything great you know <pause dur="0.3"/> well it's not because it's got the wrong equations to resolve everything correct yeah <pause dur="0.5"/> so <pause dur="0.5"/> you've got to be aware of the weaknesses of your equations <pause dur="2.1"/> okay <pause dur="0.2"/> # <pause dur="1.0"/> now i'll move on to this <pause dur="0.8"/> the final bit was this # <pause dur="1.4"/> where we got to <pause dur="1.2"/> yeah <pause dur="0.3"/> so i've gone through the equations are you sort of happy with that does it help review a bit <pause dur="0.5"/> well hopefully this course reviews some of the other things you've seen in the last term # <pause dur="1.2"/> # <pause dur="0.4"/> there's nothing like making a model to <pause dur="0.2"/> make you know what you're doing in the subject <pause dur="0.9"/> i've talked about hierarchy models we talked about waves in the <trunc>atmosphe</trunc> <pause dur="0.8"/> oh i didn't show you the

waves <pause dur="0.2"/> oh hang on <pause dur="0.5"/> what i am doing <pause dur="1.4"/> oh the waves <pause dur="2.6"/> yes <pause dur="1.9"/> sorry <pause dur="0.5"/> missed a crucial slide here <pause dur="2.3"/> well it's better <trunc>ne</trunc> you now know the equations so if we look at this <pause dur="1.8"/> this is a quite a nice little diagram this was a meteorologist Green <pause dur="0.6"/> who <pause dur="0.2"/> who <pause dur="0.4"/> drew up this sort of <trunc>d</trunc> schematic diagram of different waves in the atmosphere <pause dur="0.6"/> so the waves are <pause dur="0.2"/> the atmosphere's a big compressible atmosphere <pause dur="1.6"/> at this axis <pause dur="0.2"/> is phase phase velocity <pause dur="0.6"/> so <pause dur="0.2"/> call it C <pause dur="0.3"/> yeah <pause dur="1.0"/> this sort of yellowy line going across here is three-hundred metres a second <pause dur="0.3"/> that's the speed of sound <pause dur="0.4"/> yeah <pause dur="0.2"/> anything above that line is supersonic <pause dur="1.0"/> not very important because you <trunc>d</trunc> haven't seen any supersonic weather systems i don't know if anybody's ever seen a supersonic weather system but <pause dur="0.3"/> i certainly haven't seen a supersonic weather system <pause dur="0.9"/> in principle you could have if you look at these lines on here <pause dur="0.5"/> the atmosphere could support supersonic waves <pause dur="1.2"/> but it'd never do that because they produce a shock wave and it would <pause dur="0.2"/> you know

when you've got Concorde going supersonic then <pause dur="0.4"/> you have to put so much energy in to get it <pause dur="0.2"/> when it produce the sonic boom it <pause dur="0.2"/> creates so much dissipation that <pause dur="0.6"/> you need a lot of energy to keep it moving so <pause dur="0.2"/> if <pause dur="0.2"/> if suddenly one of those storms became supersonic <pause dur="0.4"/> it would slow down very quickly it would it would lose all its energy so <pause dur="0.8"/> chance of seeing a supersonic weather system is low <pause dur="0.6"/> very low <pause dur="1.5"/> sound waves are obviously in the atmosphere 'cause you can hear what i'm saying <pause dur="0.2"/> so <pause dur="0.2"/> and that line's there <pause dur="2.1"/> # <pause dur="0.4"/> then there's some other waves so <pause dur="0.5"/> this bunch of waves this is sorry # spatial wavelength the horizontal wavelength yeah <pause dur="0.3"/> in kilometres <pause dur="0.4"/> these are logarithmic scales as well so <pause dur="0.5"/> this is phase speed this is <trunc>logarith</trunc> <pause dur="0.2"/> # horizontal wavelength so <pause dur="0.6"/> ten-thousand kilometres is called planetary scale <pause dur="0.2"/> in meteorology usually <pause dur="0.8"/> one-thousand kilometres is synoptic scale that's the size of a storm <pause dur="0.3"/> in mid-latitudes <pause dur="1.0"/> # <pause dur="0.3"/> anything between a thousand kilometres and ten kilometres is called

mesoscale <pause dur="0.6"/> so most of your favourite weather systems are in mesoscale <pause dur="0.8"/> but the big things are <pause dur="0.3"/> the big systems are the synoptic systems are a thousand kilometres <pause dur="0.9"/> # then there's things less than ten kilometres which are called microscale <pause dur="0.4"/> so the three sort of <pause dur="0.5"/> the four scales on the planet are <pause dur="0.4"/> you know really big planetary scale things <pause dur="0.2"/> synoptic scale so El Niño the big El Niño response is a planetary scale phenometa <pause dur="0.2"/> phenomena <pause dur="0.8"/> then there's synoptic then there's mesoscale then there's micro <pause dur="0.8"/> and then there's these different speeds <pause dur="1.0"/> and so <pause dur="0.5"/> # <pause dur="0.2"/> the free surface one doesn't matter 'cause the atmosphere doesn't have a free surface it doesn't have a top on it really it just <pause dur="0.2"/> the atmosphere just gets less dense as it goes out to space <pause dur="0.3"/> if it did have a top surface on it would go at this speed but it it <pause dur="0.2"/> doesn't really have a <pause dur="0.8"/> top surface <pause dur="1.1"/> # then there's gravity waves <pause dur="0.2"/> # these are the <pause dur="0.4"/> gravity waves here have quite a range of speeds <pause dur="0.2"/> so they can go very fast so very <pause dur="0.5"/> very # <pause dur="0.3"/> long

wavelength gravity waves <pause dur="0.6"/> # move very quickly <pause dur="1.1"/> very short wavelength ones move slower yeah <pause dur="1.4"/> # <pause dur="0.2"/> but then there's an area in the mean between where the phase velocity doesn't change much so the gravity waves are non-dispersive <pause dur="0.2"/> so over quite a long spatial scale from synoptic right down to <pause dur="0.2"/> about ten kilometres <pause dur="0.4"/> gravity waves move at about the same speed they don't <pause dur="0.9"/> they're not that dispersive as waves <pause dur="0.7"/> # once you get below ten kilometres gravity waves become dispersive again <pause dur="1.3"/> # <pause dur="1.0"/> there's also some of these gravity waves get mixed up with these inertial waves that you can have <pause dur="0.8"/> # the easiest way to see why you have an inertial wave <pause dur="1.1"/> is in your equations here <pause dur="0.9"/> if you had D-U-<pause dur="0.4"/>by-D-T equals minus-F-V and D-V-by-D-T equals F-U <pause dur="0.2"/> yeah <pause dur="0.8"/> so that's just Coriolis that's <pause dur="0.7"/> the Coriolis force acting on both of those <pause dur="0.2"/> no pressure gradients imagine no pressure gradients <pause dur="0.7"/> if you solve those equations there you get D-squared-U-by-D-T-squared <pause dur="0.8"/> # equals F-squared-<pause dur="0.2"/>U <pause dur="0.5"/> and the same sort of thing for V <pause dur="0.5"/> so that's an

oscillation <pause dur="0.4"/> if you look at the thing and the period of the oscillation is one-over <pause dur="1.2"/> period is one-over-F <pause dur="0.7"/> oh sorry two-pi-over-F <pause dur="1.8"/> so <pause dur="0.4"/> it depends on where you are on the planet <pause dur="0.2"/> the the winds <pause dur="0.3"/> the <pause dur="0.2"/> U and V components oscillate <pause dur="0.2"/> they go round <pause dur="0.3"/> so if <pause dur="0.2"/> when people do measurements of the ocean <pause dur="0.7"/> # the when they put a probe down they'll see that the velocities are are turning round at a certain speed <pause dur="0.6"/> and that speed is determined by the Coriolis parameter at that point <pause dur="0.3"/> and that's an inertial wave <pause dur="1.5"/>

you don't tend to see them as much in the atmosphere <pause dur="1.7"/> as an inertial well they call it inertial oscillations <pause dur="2.0"/> you don't see those things much in the atmosphere <pause dur="0.3"/> # <pause dur="0.2"/> 'cause it's not stable enough so you don't you see it more in the ocean <pause dur="0.8"/> # <pause dur="0.2"/> but those things can mix up with gravity waves so you get this sort of funny little mixture wave here <pause dur="0.4"/> these shorter ones the <pause dur="0.3"/> you get these things called inertia gravity waves <pause dur="0.5"/> so the two things have combined a bit you know <pause dur="1.6"/> # <pause dur="0.3"/> right so those are the gravity waves and

then up here on the long large scale stuff <pause dur="0.3"/> <trunc>r</trunc> from planetary down to synoptic are the Rossby <pause dur="0.4"/> Rossby waves <pause dur="1.1"/> now <pause dur="0.5"/> if you're interested in forecasting weather and looking at storm synoptic systems coming through which is what the original <pause dur="0.6"/> point of doing weather forecasting was was to <trunc>s</trunc> get <pause dur="0.2"/> try and get the storm systems <pause dur="0.8"/> you're interested in getting the Rossby waves right yeah <pause dur="0.2"/> these especially the synoptic ones down here <pause dur="0.5"/> typical speeds less than ten metres a second yeah <pause dur="1.3"/> you don't really <pause dur="0.3"/> care much about these very big thing these gravity waves that <pause dur="0.3"/> that go at very high speeds yeah <pause dur="0.8"/> you're not really interested in those very <trunc>lon</trunc> very large scale gravity waves that move quickly that that <pause dur="0.5"/> those aren't relevant so <pause dur="0.5"/> the trick is to try and filter some of those things out a little bit in your equations <pause dur="0.2"/> you're more interested in getting these waves <pause dur="0.8"/> and you don't really want sound waves so <pause dur="0.2"/> by doing the <pause dur="0.4"/> anelastic approximation you get rid of all the sound waves and all the

supersonic stuff <pause dur="0.7"/> by doing <pause dur="0.6"/> # the <pause dur="0.3"/> # <pause dur="1.2"/> by doing the hydrostatic approximation you get rid of quite a lot of these fast gravity waves <pause dur="0.4"/> a lot of the fast ones are going vertically <pause dur="0.6"/> these are vertically propagating ones <pause dur="0.6"/> and you end up with waves that then all move at a slower speed yeah <pause dur="0.2"/> and that means you can then <pause dur="0.3"/> simulate the thing much nicer use a bigger timestep <pause dur="0.5"/> and you've <trunc>k</trunc> still kept the essential part of <pause dur="0.5"/> you've still kept the Rossby waves and the the synoptic systems you've got rid of the fast stuff <pause dur="1.7"/> does that make sense are you are you happy with that <pause dur="0.8"/> all the ways we cheat in writing our equations down yeah <pause dur="0.2"/><vocal desc="laugh" iterated="n"/><pause dur="0.6"/> so anybody who says why did you why did you use primitive equations <pause dur="0.6"/> well the reason is to get the timestep down in the model really you know <pause dur="1.5"/> # <trunc>y</trunc> so so you don't have to use a large timestep <pause dur="2.5"/> okay <pause dur="0.2"/> # <pause dur="1.1"/> now the final bit <pause dur="1.3"/> i was going to go is <pause dur="1.9"/> # <pause dur="2.7"/> yeah <pause dur="1.1"/> the ideas <pause dur="3.2"/> oh we're running out of time very quickly <pause dur="0.4"/> we've got <pause dur="0.4"/> a hierarchy of models we've got waves in

the atmosphere some basic equations i was just going to show you how you can write these things in a general <pause dur="0.4"/> as a dynamical system <pause dur="0.8"/> so it's all a dynamical system and <pause dur="0.4"/> so the <pause dur="1.5"/> in the last few minutes just show you this <pause dur="3.6"/> you can write all of those sort of equations that i've just gone through there in a quite a simple way <pause dur="0.9"/> you can't solve them easily but you can write the equations simply <pause dur="3.4"/> so <pause dur="1.1"/> if <pause dur="0.2"/> we <pause dur="0.7"/> write the equation this is a general equation for the ocean and the atmosphere <pause dur="0.4"/> and you can write down <pause dur="1.1"/> D-xi-by-D-T <pause dur="0.4"/> so it's a first order equation <pause dur="0.2"/> you don't usually get second order D-squared-xi-by-D-Ts <pause dur="1.2"/> there's some horrible <pause dur="1.1"/> non-linear operator called Q <pause dur="0.4"/> which is some <pause dur="0.2"/> depends on the state of the system so this is like the advection terms <pause dur="2.8"/> it's actually not that non-linear <pause dur="0.3"/> # compared to what you <trunc>c</trunc> it's only quadratic usually that's why i call it Q actually <pause dur="0.6"/> so when you look at your equations you have things like U <pause dur="0.3"/> D-U-by-D-X et cetera <pause dur="1.3"/> so you have some sort of quadratic operator here <pause dur="1.9"/> and you

have <pause dur="0.3"/> some forcing on this side forcing and dissipation <pause dur="0.5"/> that depends on the state of the system xi <pause dur="0.9"/> the time <pause dur="0.3"/> and some parameters of your model <pause dur="0.4"/> <trunc>d</trunc> parameters of <pause dur="0.4"/> albedo of the Earth et cetera et cetera <pause dur="0.6"/> so <pause dur="0.2"/> xi here is a general xi is the state of the system <pause dur="1.7"/> the state of either the atmosphere or the ocean <pause dur="2.4"/> and <pause dur="0.4"/> that for instance if you're doing the primitive equations would be a little vector of <pause dur="0.3"/> U <pause dur="0.2"/> it would consist of V <pause dur="0.2"/> W <pause dur="0.5"/> # <pause dur="0.7"/> # <pause dur="0.2"/><vocal desc="sigh" iterated="n"/><pause dur="0.4"/> T <pause dur="0.4"/> what else have we got <pause dur="0.7"/> about that <pause dur="0.9"/> yeah <pause dur="0.4"/> # <pause dur="0.2"/> and here you'd have this'd be a function of latitude <pause dur="0.2"/> <trunc>long</trunc> <pause dur="0.6"/> longitude latitude <pause dur="1.0"/> and time <pause dur="0.4"/> and that'd be a function of longitude latitude time <pause dur="0.7"/> and height sorry <pause dur="0.3"/> or this Z <pause dur="1.5"/> latitude longitude <pause dur="0.2"/> <trunc>ti</trunc> height and <pause dur="0.2"/> time <pause dur="0.9"/> longitude latitude <pause dur="0.8"/> height and time yeah <pause dur="0.5"/> so you have four fields you have <pause dur="0.2"/> the U field the V field the W field and the <trunc>ti</trunc> the temperature field <pause dur="0.5"/> would all be <pause dur="0.4"/> would all be define the state of the atmosphere <pause dur="0.5"/> so if i know all those things i know what the state the atmosphere's in <pause dur="0.6"/> so that's what

i mean by xi it's sort of a <pause dur="0.2"/> big <pause dur="0.5"/> <trunc>beg</trunc> <pause dur="0.4"/> big field summarizing that lot <pause dur="0.6"/> this this xi the state of the system evolves like that basically <pause dur="0.7"/> that's the general equation for most atmosphere ocean systems <pause dur="1.2"/> this bit on this side gets called <pause dur="0.2"/> dynamics quite often <pause dur="1.4"/> by <pause dur="0.2"/> quite a lot of people <pause dur="0.3"/> and this bit is called the physics <pause dur="2.0"/> so this is the force in <pause dur="0.4"/> the force this includes force in <pause dur="0.4"/> so radiation and things <pause dur="0.4"/> it also <trunc>invo</trunc> includes dissipation <pause dur="1.1"/> so the atmosphere's got <pause dur="0.2"/> diffusion and drag it's got it's losing energy as well so it's <pause dur="0.2"/> it's gaining energy from the sun but it's also losing <pause dur="0.3"/> energy when it drags along the ground <pause dur="1.0"/> so these <pause dur="0.6"/> # <pause dur="0.6"/> well we people do <trunc>pr</trunc> <trunc>m</trunc> numerical models they like to split it into two so they call this part the <trunc>dymani</trunc> dynamical part of the model <pause dur="0.3"/> and this part is the physics the parameterization of all the radiation all the messy bits of the clouds and things <pause dur="1.1"/> # <pause dur="0.6"/> sometimes this is called the dynamical core <pause dur="0.6"/> so this is the fluid part of this is basic just basic

fluids and waves <pause dur="0.3"/> this is the physics how it's forced yeah <pause dur="0.3"/> so that's a useful way to look at climate models <pause dur="0.7"/> # how they usually break up people look at the dynamical schemes and then they look at the physical schemes in the model <pause dur="1.1"/> # the current models tend to spend fifty per cent of the computer time doing this and fifty per cent of the computer time doing this so <pause dur="0.7"/> when i do all this fluid all those fluid equations i wrote before i ignored that completely actually <pause dur="0.7"/> i only showed you the dynamical bit <pause dur="0.9"/> so in reality <pause dur="0.5"/> # the actual real climate models and weather models are spending fifty per cent of the time doing physical calculations like what's the radiation and what's the <pause dur="0.6"/> what's the latent heat release so <pause dur="0.2"/> physics physical things in models are very important it's not just a pure fluid <pause dur="1.9"/> # <pause dur="0.9"/> i'll try and finish off quickly <pause dur="1.0"/> # <pause dur="0.4"/> what we <pause dur="0.8"/> you can do when <pause dur="0.6"/> these sort of things you can't really <pause dur="1.3"/> this state of the system here <pause dur="0.8"/> depends on <pause dur="0.4"/> # <pause dur="0.2"/> longitude <pause dur="0.2"/> lambda <pause dur="0.6"/> latitude phi <pause dur="0.6"/> the height <pause dur="0.5"/> Z <pause dur="0.4"/>

and T yeah <pause dur="1.1"/> now <pause dur="0.4"/> it's a continuum this <trunc>d</trunc> this is everywhere you know so even at this point that point it's very <pause dur="0.2"/> this is a <pause dur="0.2"/> an infinite number of places this is defined at yeah <pause dur="0.4"/> this is a continuum field <pause dur="0.7"/> now obviously you can't put that onto a little computer 'cause the computer's only got a finite memory it's only <pause dur="0.2"/> you've got to break it up <pause dur="1.2"/> so what you do <pause dur="0.2"/> is you break it up <pause dur="0.2"/> like this so you say <pause dur="0.6"/> well i won't look at the field everywhere <pause dur="0.6"/> i'll discretize this <pause dur="0.5"/> so i'll have a bunch of <trunc>lati</trunc> longitudes a bunch of latitudes <pause dur="1.0"/> a bunch of heights of levels <pause dur="0.5"/> and a bunch of timesteps <pause dur="0.5"/> Ns <pause dur="1.3"/> right <pause dur="0.4"/> so <pause dur="0.3"/> these are sort of like the grid points in the model <pause dur="0.5"/> and you label them with an index here this is labelled with I as an index <pause dur="0.8"/> so <pause dur="0.7"/> I equals one to <pause dur="0.8"/> # and it can go up to a very large number P <pause dur="0.3"/>

usually about ten-to-the-five up to about ten-to-the-seven in current models <pause dur="0.3"/> of grid points yeah <pause dur="2.0"/> so you've broken up space into a little <pause dur="0.7"/> you've broken it into a big <pause dur="0.2"/> a grid in the horizontal and a grid in the vertical as well so you've <pause dur="0.3"/> you've got a lattice if you like a like a climbing frame <pause dur="0.6"/> and at each point you've only defining the variable at those points so you've <pause dur="0.3"/> replaced <pause dur="0.5"/> # this wonderful continuous field <pause dur="0.7"/> by a horrible <pause dur="0.2"/> discrete bunch of things yeah <pause dur="0.6"/> about ten-to-the-five variables yeah <pause dur="1.2"/> and these you can label these ones <pause dur="0.8"/> time as well you break up into steps <pause dur="0.3"/> so you discretize time so you discretized all the space and the time dimensions into <pause dur="0.6"/> broken <unclear>them up</unclear> as a grid <pause dur="1.3"/> this you can then label as <pause dur="0.3"/> one label here <pause dur="0.4"/> I for the space index <pause dur="0.5"/> and a little label at the top N <pause dur="0.2"/> meaning the time

index <pause dur="0.6"/> so that's the <trunc>times</trunc> that tells you what timestep it is <pause dur="2.2"/> and that tells you which grid point variable that's the grid point <pause dur="1.8"/> yeah <pause dur="1.6"/> so now you've replaced this continuous equation here by a bunch of very finite <pause dur="0.2"/> just that you've <trunc>g</trunc> only a certain number of variables phi Is <pause dur="0.5"/> and then you do them in different times yeah <pause dur="1.3"/> so # <pause dur="0.4"/> i think i'm going to stop there because it's going to # <pause dur="0.2"/> we're going to run into lunch otherwise we can pick it up again in the old <pause dur="0.7"/> old Meteorology building afterwards so <pause dur="0.6"/> have have lunch and have a think about this and then we'll <pause dur="0.2"/><vocal desc="laugh" iterated="n"/><pause dur="0.9"/> we're going to discretize time in after lunch yeah we're going to <pause dur="0.3"/> do that a bit more

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