nm0877: so today [0.7] we carry on yeah last week we was a bit waffly went into 
the [1.5] i went into the introduction [0.2] and some [0.2] early history of 
numerical modelling of weather and [0.7] er atmospheric modelling [1.5] today 
[0.4] er we get into more nitty gritty of this and more of the numerics [0.2] 
so we're going to get into dynamical equations in this lecture [0.9] and then 
in the after lunch there in the old Meteorology building [0.5] we're going to 
go across and start doing some time discretization [1.3] now [0.8] the next 
week as well we do horizontal discretization vertical discretization [0.3] a 
lot of this some of this stuff you've covered already in the last term in 
namex's course [0.6] but you go into a bit of a deeper level with this in this 
one so [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 [0.5] some new things as well [1.0] 
and that'll get a bit more [0.4] that'll be further than namex took it and 
this'll go a bit further as well [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 [0.7] trying 
to simulate the atmosphere numerically [0.5] er [0.4] which [0.2] equations do 
we use yeah [1.3] and er it's not obvious [3.3] so [0.9] the [0.5] layout the 
format for this morning's going to be [2.7] dynamical equations [0.7] i'm going 
to go through the [1.0] we don't have just one meteorological model we have 
many different models [0.5] er [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 [1.2] and explain why sometimes it's useful to use a hierarchy of 
models it's good to use not [0.2] if you're going to use a model it's better to 
use [0.3] a couple of models rather than just one all the time [1.0] er it's a 
bit like a map [0.2] if you had a [0.5] trying to find your way around you'd 
have a map of the world but you'd also have a map of namex as well you'd use a 
couple of things [0.2] to [0.4] more than one model is always a good idea [1.7] 
er [0.4] so i'll talk about models and their complexity [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 [0.2] in the fluid dynamics [0.4] course [0.9] 
er [0.2] the speed of these waves is very important for how you solve the 
equations so if you have very fast waves [0.2] it makes it very hard to solve 
the equation so [0.7] knowing which waves you're trying to simulate is very 
important [1.7] do some basic equations just as a review of that [0.6] and then 
i'll [0.4] i'll try and present some ideas from sort of dynamical systems 
theory [0.2] there's a branch of mathematics called dynamical systems theory [0.
7] er that was started about the beginning of the twentieth century [0.7] some 
of the ideas from dynamical systems theory are very relevant for [0.2] 
understanding the atmosphere and climate so [0.3] i'll give you a little [0.6] 
a very [0.4] very brief introduction into that [1.0] er so that's the plan for 
today and if there's any questions please interrupt or [0.7] er [0.5] feel free 
to ask any questions you like yeah [2.0] so [2.3] this [3.2] [0.8] hierarchies 
and models [0.2] er [1.5] some 
people this isn't in your notes you might want to copy this down this is a [0.
2] this in an extra [1.6] this is from an article by Hoskins [0.8] Brian 
Hoskins [0.3] in nineteen-eighty-three [1.0] Q-J-R-M-S er volume one-o-nine [0.
8] pages one to twenty-one [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 [0.6] there's some 
observations [0.5] they get fed into our big complex numerical models [1.2] er 
[0.5] we have some theoretical models which are completely detached from this 
these are just sort of mathematicians amusing themselves [0.4] er no connection 
to [0.2] er [0.2] these other models [0.8] and then there's some basic ideas 
you know some [0.4] theories of what [0.2] simple ideas and concepts that we 
use [0.4] and these sort of feed in [0.2] now this is completely the wrong way 
to [0.7] this is not a helpful way to look at models [2.3] a better way [0.2] 
er [0.4] tell me if i'm [0.2] if i take this off too quickly let me know yeah 
[1.5] the better way to look at it is [0.7] well there's [0.3] observations [1.
0] they're getting used in a range of dynamical models from 
complex ones [0.2] medium complexity ones to very simple ones so you have a 
nice range of models so you can understand things [1.0] and you have evolving 
conceptual models [0.3] that sort of feed into understanding how you should 
observe which things you should observe [0.2] and how you develop the models so 
[0.4] it's more [0.4] there's a whole range of hierarchy model and there's more 
[0.6] er [0.4] connection between these things there's no separate theoretical 
model as a separate [0.2] thing yeah [0.2] everything is [0.8] you think of it 
as a continuum of different types of models [1.1] it's dangerous people see a 
very realistic [0.2] climate model a big one and then they think oh that's the 
only model [0.3] all the others are wrong [0.3] and that's a wrong view of it 
it's more [0.5] all models are sort of wrong [0.6] er [0.8] what you need is a 
range of them you need a good range of models [0.7] so that's a better way to 
look at the whole modelling exercise yeah [2.4] did you get that down is that 
[0.3] you're still [0.3] [laughter] [0.4] go fast [laughter] [1.9] okay i'll 
give you a 
few seconds to get that [1.0] st-, if i write that on the board it takes me 
longer so these overheads are dangerous 'cause you [0.2] put things up too 
quickly yeah [5.6] well the two differences on these is that the [0.4] these 
are very static and don't [0.3] change [0.2] the ideas stay the same [0.3] 
these are actually [0.3] by looking at these models your concepts can change 
things are evolving [0.6] it's also there's a hierarchy [0.5] all you don't see 
it as two separate boxes here you see it as a whole range of models yeah [1.0] 
that's the two differences [2.3] so okay [0.3] you nearly there [0.9] er [0.5] 
so [0.5] you might say er [0.4] well why don't we [1.0] le-, well we're going 
start i'm going make a hierarchy of models [0.4] and we'll [0.2] think of the 
different hierarchy [2.2] whoops [3.4] my hands are cold [2.0] so at the top of 
the [0.2] we're going to start with the most complex [1.9] and we go down to 
the most sim-, [0.3] simple [4.1] now just because it's complex or simple 
doesn't mean it's better or worse sometimes complex models [0.5] sometimes 
simple models are you believe them more [0.3] er [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 [0.7] er you have less confidence sometimes [1.1] now [0.4] 
the first way you might say well why don't you just simulate if you could [0.2] 
the atmosphere is made up of molecules why don't you just simulate the motion 
of all the molecules [0.7] you know we could use Navier [0.2] we could use 
Newton's laws of motion [0.8] and we could [0.2] simulate all the molecule [0.
8] molecular dynamics [2.6] so we know [0.5] we know Newton's laws controls the 
motion of each molecule [0.7] er we work out in the future where each 
molecule's going to go [0.8] well [0.4] that's obviously stupid [0.6] er 
because there are ten-to-the-forty-five [1.1] molecules in the atmosphere [2.6] 
er now that's [0.9] these these are in your notes these this this is on a [0.3] 
a [0.2] that sheet's been photocopied there [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 [0.2] you're 
more interested in big things you're not really interested in each molecule [0.
8] so it's a bit [0.6] you'd be [0.5] it'd be you'd have to do a lot of 
calculation there it would be impossible to do that calculation actually [0.5] 
computers [0.6] er [0.3] there's too many molecules [0.4] and it's also would 
give you [0.2] too much information you don't need all that information you're 
not really interested in each molecule so [2.2] next thing you do is you could 
say well i don't want to consider the atmosphere as individual molecules i can 
assume [0.4] it's a continuum a fluid [0.9] we know it's not a really a fluid 
it's full of little molecules but [0.6] you make an assumption called the 
continuum approximation [1.5] so this called the continuum [1.6] this is used 
in all the fluid mechanics [0.9] 'cause all fluids are made of molecules in the 
end [1.3] so what you do is you say well it's not individual molecules it's a 
sort of fluid a continuous fluid [1.0] it's a big ajum-, big assumption [0.3] 
er [0.6] doesn't work in outer space when you get into the upper air [0.2] 
upper 
atmosphere up near where the space stations are [0.4] this continuum 
approximation is is not right because there's so few molecules [0.5] you can't 
treat it as a continuous fluid [1.5] er [0.6] so the next thing you could do 
then is use Navier-Stokes [1.7] Navier-Stokes equations for fluids [1.2] and 
use those to solve the atmosphere [2.3] so now you're treating in as a fluid so 
[0.3] a lot of people who don't know anything about meteorology or oceanography 
[0.4] say well why you don't you just u-, they just use Navier-Stokes equations 
you know people who do aircraft simulation think we use Navier-Stokes [1.1] and 
they're always a bit surprised when we say no we don't [0.7] er [1.4] now [1.0] 
the problem is with Navier-Stokes equations it it involves it includes lots of 
waves or possible waves it includes [0.7] er sound waves [1.8] er [1.0] gravity 
waves [0.3] er we're going to come back to waves again [1.2] er [0.4] it kind 
of includes inertial waves [0.8] these are the sort of [0.4] different types of 
waves you tend to get in the atmosphere [0.2] so [1.1] whoops [0.5] Rossby 
waves [0.9] [cough] [0.8] oh there's also 
Kelvin waves along edges th-, [0.2] Kelvin waves will go along edges mainly so 
they're a bit strange [0.9] so in the atmosphere [0.5] or in a in a fluid in 
general [0.4] in in the in the compressible fluid in the atmosphere [0.3] there 
are sound waves there are gravity waves [0.3] there are inertial waves [0.3] 
there are Rossby waves and there are Kelvin waves [0.5] those are all the waves 
you need to know about to understand the atmosphere [1.7] now you've probably 
noticed that [0.7] well of all these waves these ones are very fast [0.6] sound 
waves move at about three-hundred metres a second [1.6] so if i scream i can't 
get to the li-, i can't run out of the room and [0.6] you know er move very 
quickly er sound [0.8] er now [0.3] the point is sound waves don't really 
interact very much with the [0.5] atmosphere [0.9] if you go outside and scream 
you can't affect the weather you know you can't [0.2] by producing a lo-, even 
a big rock concert [0.2] doesn't produce cyclones or anything you know there's 
not much interaction [0.6] these waves the sound waves [0.7] don't really 
interact with the 
other waves [1.4] so to do a weather forecast in the early days in the 
beginning of the twentieth century people said well [0.4] we should include 
sound waves because that's all part of the fluid [0.8] then they realized oh 
really the sound waves don't do anything to the atmosphere [0.7] so they're not 
really relevant [1.5] the pad-, problem with including sound waves [0.2] is [0.
7] you have to use a very small timestep you have you you've come across this 
er Courant- [0.2] Friedrich-Levy [0.3] condition yeah [0.4] have you seen have 
you heard that from namex's [1.2] there's a [1.5] a condition for numerical 
stability called Courant- [1.5] Friedrich- [1.6] Levy [1.8] so we just call it 
C-F-L [0.8] C-F-L says that [0.2] er [0.4] your timestep of your model [0.5] 
has got to be less than the [0.4] grid spacing divided by the speed [0.3] the 
maximum speed of propogation [1.6] so if i have a [0.3] hundred kilometre [0.5] 
hundred kilometre grid [0.5] we could work this out if you work this out [0.6] 
suppose delta-X i'm trying to simulate a hundred kilometre resolution [1.3] 
suppose i've got a sound wave in there going at three-hundred 
metres a second [1.3] yeah [0.8] now i don't know what that comes out as if you 
work that out let's work that out [0.3] hundred [1.8] hundred-thousand divided 
by three-hundred [0.9] yeah [0.6] so that's er [0.4] cross those off [1.1] that 
gives you a thousand if i ri-, that gives you about three-hundred [2.5] three-
hundred second [0.5] 
that's not right [3.9] that's right [1.1] er [3.0] yeah i've got that right 
haven't i think [0.6] yeah got three-hundred seconds [0.5] so three-hundred 
seconds is like five [0.2] five minutes [0.6] so if i wanted to simulate this 
sound wave on a thousand kilometre [0.2] grid horizontal se-, [0.2] this is a 
hos-, horizontal sound wave [0.5] i would need a [0.2] a timestep of five 
minutes in the model [0.7] but it's worse than that [1.0] because the sound 
waves can go vertical as well when i [0.2] when i scream sound goes upwards as 
well [0.7] er the spacing in the models can be a hundred metres in the vertical 
[0.3] you want to resolve the troposphere nicely [0.4] so really that distance 
shouldn't be a hundred kilometres should be a hundred metres really [0.9] 
divided by three-hundred metres a second [0.7] 
you get a third of a second [0.7] so if you wanted to include [0.7] er [0.4] ti-
, 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 [0.6] so you have to 
really [0.2] do lots of simulation you know [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 [0.6] you know ho-, why how stupid you going to have to 
timestep all the time [0.6] er very quickly [0.2] just so i can simulate these 
stupid sound waves [0.6] and then they don't not useful anyway so [0.8] er [0.
4] so what you can do then is filter them out you know [1.4] so [1.2] [cough] 
[0.2] so the next set of equations have got no sound waves in them [1.2] er the 
Euler equations [2.4] now the Euler equations [0.9] er [0.9] you make an 
assumption of [0.3] er anelastic [1.4] yeah [0.4] so you make the approximation 
to go here is anelastic [2.2] so you assume [1.0] er [0.8] before in the 
density equation y-, [0.2] in the full one you'd have D-rho-by-D-T [0.4] plus 
divergence of rho- [0.2] U [0.5] equals zero [1.3] now if you have that that 
can 
produce sou-, that that's responsible that equation's necessary to get the 
sound waves [0.4] you need the compressibility to get sound waves it it's 
compression in density [1.0] so if you get rid of this equation and sort of say 
well [0.2] okay [0.2] approximate that one by [0.3] div- [0.4] rho- [0.4] U [0.
2] equals zero [0.3] yes [1.1] then you get rid of sound waves [0.7] and that 
approximation is called an anelastic approximation [2.6] now nearly every [1.0] 
well every meteorological model i've seen even the very [0.2] cloud resolving 
ones or small models [0.4] all make this approximation [0.7] 'cause they don't 
want sound waves in there [0.6] nobody wants sound waves in their 
meteorological model [2.0] so i-, it's important to know which approximation so 
if i i do them in [2.1] [cough] [0.9] so the first approximation we use was a 
continu-, ooh [2.0] was a [3.4] continuum approximation [0.5] and the next 
approximation we use is an anelastic one [2.9] now [3.6] now the problem with 
that is if you still have that they're still not very good because [0.5] the 
next wave that moves very quickly are these gravity waves [0.5] so if you've 
been 
on a near a mountain [0.7] you can see this on mountains i saw this this last 
week [0.2] er the weekend i was near the Alps [0.7] and you could see [0.5] a 
beautiful mountain wave if you're ever near a big mountain you [0.3] you have a 
big mountain like this say [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 [1.6] and sometimes you can see 
clouds develop in the [0.5] in this area these [0.4] these little stripes so if 
you see [0.4] you see the top of the mountain you see these little bands of 
white cloud [0.5] have you ever seen tho-, has anybody [0.6] seen these [0.4] 
sometimes you see them [0.6] roll clouds even without mountains but quite often 
near mountains you see these roll clouds [4.2] now [0.2] this thing behind me 
is a gravity wave [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 er conservation of mass that's 
creating this [0.3] gravity wave [1.0] now some of the gravity waves 
can actually they don't just go horizontal some of them can shoot off u-, [0.2] 
upwards as well [0.2] they go vertically [1.0] so [0.3] mountains produce ver-, 
vertically propagating gravity waves just they're big mountains [1.8] now the 
vertically propagating gravity waves also are pretty fast and the spacing's 
pretty small here so you'd have [0.4] something like a hundred metres a sec-, 
[0.2] hundred metres [0.5] divided by say [0.4] fifty metres a second yeah [1.
0] so you'd still need a timestep of two seconds if you had the vertically 
propagating ground w-, [0.3] gravity waves [1.2] and they're not important for 
the big weather systems they're not important for cyclones and [0.2] those sort 
of things you can live without vertically propagating gravity waves [1.5] so [1.
2] the next step [0.2] to go down is to filter out the [0.4] vertically 
progagating gravity waves and the way you do that [0.9] is the hydrostatic 
approximation [2.7] you probably wondered why you keep se-, did you see in the 
fluid course you saw all these approximations like hydrostatic 
approximation geostrophic and all these [0.6] well there's a reason for why you 
want to approximate them [0.9] the hydrostatic approximation [0.8] if you're 
interested in things longer than say ten seconds or something a bit [0.2] 
longer timescale than that [0.5] you don't really care about vertical [0.5] 
gravity wa-, vertica-, bob-, [0.3] you don't care about little air bobbing 
behind the back of mountains like this very quickly [0.6] er [0.5] you don't 
need to have [0.7] you don't need to have vertical gravity you can make this 
approximation [0.7] er [0.5] D [0.9] D-P-by-D-Z equals minus-rho-G [1.8] if you 
make that approximation vertically propagating gravity waves disappear from the 
equations [1.1] and you end up with these set of equations which everybody [1.
2] likes using in meteorology it's called the primitive equations [3.2] so [0.
8] okay [0.2] so it's a series of sort of approximations go down and this is 
very pop-, these are very popular yeah [0.6] these these equations are the ones 
that are used in [0.3] most of the atmosphere models the weather 
forecasting models the [0.2] ocean models as well use the primitive equations 
[0.5] these are a good set of equations for [0.4] fluids on the sphere [1.8] er 
[1.5] now [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 [0.4] horizontal gravity waves you still have inertial waves you 
still have Rossby waves and you still have Kelvin waves so [0.5] and the storm 
systems tend to be Rossby waves [0.4] so er things like storms are [0.4] mid-
latitude storms are basically instable Rossby waves [0.3] so you can still 
simulate all those storm systems you do-, you didn't need all these [0.4] sound 
waves and vertical [0.3] propagation [1.8] well those equations are still 
pretty foul to to solve them [0.2] er [0.3] i'm going to show you [0.6] er [0.
7] i'll show well i'll show you the equation act-, yeah [0.7] you see them [2.
3] these are good ones to understand because they're ones we use all the time 
in meteorology so [2.2] [sneeze] [0.9] so these are some basic equations here 
[2.7] er [0.8] so [3.2] [cough] [0.3] check i'm to switch 
that thing off [4.0] so [1.0] i've just written down a few of these equations 
here but the [0.3] these are the primitive equations there are five equations 
[1.3] so there's equation [0.3] these are written in vector form but there's 
the two components of the wind [0.5] er U and V [1.0] er there's a D-U-by-D-T 
[0.2] that includes the advection term as well it's the material [0.2] 
derivative [0.9] Coriolis force and a pressure gradient [0.6] yeah [0.2] so 
there's basically a pressure gradient some Coriolis force and a bit of 
advection yeah [0.3] that's all there is in the that bit [1.2] er [0.6] this D-
phi-by-D-P plus R-T-over-P is a different way of writing [0.2] hydrostatic 
balance hydrostatic approximation [0.6] this is a bit like if you like is a [0.
3] is a [0.8] a modified version of the W equation normally Navier-Stokes 
should have three equations should have one for U [0.3] one for V and one for W 
[0.9] but because we made this hydrostatic approximation we really [0.3] made a 
mess of the W equation [0.7] the hydrostatic balance is a bit like a [0.3] er 
[0.8] you can look at that as a sort of modified [0.7] 
double equation for W yeah [1.7] er [0.4] mass is conserved so we made this 
anelastic approximation here [0.2] this is this conservation [0.2] what flows 
in must flow up [0.2] basically so if a-, if air comes into a region it has to 
go up [0.6] you notice we got rid of all the compression all the [0.2] D-rho-by-
D-T disappeared from this equation so [0.7] so that's sort of an anelastic [0.
9] version of the conservation of mass [1.0] and then this is the heat equation 
[0.4] er the [0.2] this is the conservation of potential temperature or 
temperature [0.7] er this telling how the atmosphere is heated which is very 
important for the [0.4] er [0.5] that heating is extremely important for the 
atmosphere without heating there'd be no [0.2] no winds [0.4] it's all to do 
with heating that caused the wind [0.4] so [0.5] those are the five equations 
so the basic conservation this is conservation of [0.4] horizontal momentum [0.
6] this is sort of conservation of vertical momentum [0.5] this conservation of 
mass and this is conservation of energy [0.9] so there are five [0.8] those are 
the basic laws [1.2] 
er that we use [1.5] is that okay [0.2] so far [0.6] are you happy with that [0.
7] er [0.4] 
sm0878: what you said about it [0.6] 
nm0877: pardon [0.6] 
sm0878: what you call the F-K 
nm0877: oh F-K that's the Coriolis term [1.1] that's the thing if you er [0.7] 
few years ago there was a discussion in er [0.4] in the Guardian about er [0.4] 
if you were near the North Pole and you tried to shoot a polar bear [0.9] er 
which eye would you shoot if you want to hit it between the eyes yeah [0.5] 
'cause that's the best way to kill a polar bear [0.6] er if you do that which 
eye do you aim at [0.6] because there's a Coriolis force [0.4] when you fire 
something on the sphere then it bends [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 shou-, shoot 
at the left eye and then the bullet sort of turns a bit and [0.6] so [0.5] this 
swerving of the air mass air is the [0.6] is the Coriolis [1.1] another example 
of that a nice one is the er [1.0] if you look at the whole planet [1.3] 
[cough] [0.2] normally air is hot it's hot here in the [0.2] equator and it's 
cold at the pole [1.2] so air would normally if there was no Coriolis [0.2] 
this 
air here would [0.7] you'd be [0.3] air that came out of the Hadley cell at the 
top would [0.2] would flow towards the pole [0.9] but because of the Coriolis 
force it it it swerves that way and produces the westerly jet [1.0] 
so the westerly jets that we get in mid-latitudes are because of the Coriolis 
force [0.5] causing this air to swerve to the right [1.1] so this is the same 
reason if you were to shoot a polar bear its have its two eyes would be there 
[1.1] you'd fire at that eye and then the thing would swerve a bit and [0.2] 
[laughter] [0.7] always remember this if you're shooting polar bears [laughter] 
[0.9] and it's all t-, [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 [0.5] you know 
how many mi-, millimetres it would move [0.4] needs to be a long way away to 
see the effect of the [0.4] thousand kilometres or something so [1.2] er [1.1] 
okay so that's the those are pressure [0.2] pressure grade normally 
most fluids people who do just regular fluid dynamics don't have Coriolis 
forces [0.6] so we have that in geophysical fluids [0.2] but er [0.5] on on the 
rotating earth so [0.3] this is normally it's just pressure gradient [0.4] 
causes the movement of air or er the [0.2] the velocity [0.7] we're a bit 
different 'cause we have this rotation term [2.0] so that's the primitive 
equations [0.2] er [0.6] yep [0.4] er now [1.4] now i know it's a bit tricky to 
understand sometimes [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 [0.5] and it's very complicated to try and solve those things on a 
sphere and things yeah [0.8] so [0.3] people have made a lot of [0.7] progress 
using things like shallow water [2.4] [cough] [2.2] now [1.1] shallow water 
equations [1.9] what you do there is you say [0.4] if you look at these 
equations here there's a lot of vertical structure [1.0] so this works for a 
three-dimensional atmosphere this is [0.3] this has got all these [0.4] er D-do 
D-omega-by-D-Ps and things so things vary in the vertical complicated [1.0] er 
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 [0.6] or they have the 
same [0.4] they'll have a certain vertical structure which stays the same it's 
just the horizontal bit that changes [0.8] er these waves that propagate on the 
tropical Pacific for El Niño [0.3] the Kelvin waves and the Rossby waves the 
equatorial ones [0.5] er [0.6] they do-, you don't really need the primitive 
equation to understand why they're there [0.2] you you don't need the vertical 
structure they're not very [0.2] they're not varying a lot vertically [1.1] in 
time [1.0] so the the [1.1] shallow water equations is a ve-, they're a very 
nice set of equations actually [1.8] er [0.7] you can't use them for everything 
but the [1.7] they're good for testing numerical schemes [1.2] and they work at 
the equator [0.2] which is quite nice [0.8] so what you have is basically a [0.
3] er [1.0] you have the [0.4] something [0.6] think of the ocean if you like 
you can think of this as the atmosphere or the ocean [0.4] imagine there's two 
types of fluid [1.5] so 
er [0.4] they originally started for a-, [0.2] oceans so you'd say if this was 
air [1.3] and this was er water [0.2] yeah [1.2] and then there's a certain 
height [1.5] a certain depth of the fluid yeah [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 [0.2] there's no vertical structure as such there's only two types of 
fluid really here [1.0] er you can use these equations even in just the ocean 
or the atmosphere you just say the [0.2] the things [0.2] in the [0.2] in the 
tropical Pacific what they do is they say [0.3] oh well this is like [0.6] er 
[0.3] this is warm ocean [0.3] so [0.2] the ocean above the thermocline [0.5] 
for the ocean normally this is the thermocline [3.3] below this certain 
gradient in the ocean there's a very cold ocean water at four degrees celsius 
[0.2] so that's deep ocean [1.5] so if you go down deep in the ocean you find 
water at four degrees celsius [0.3] it's not at zero [1.3] so you can say 
there's cold deep dense water underneath and then there's this nice warm stuff 
[0.3] and this thermocline in the tropical 
Pacific is about [0.4] from about a hundred to three-hundred metres [0.4] in 
depth [1.3] so [0.3] you only have one interface you only have this sort of 
line in between [0.3] you you don't care about all the vertical detail [1.2] 
and so you have one one h-, [0.6] instead of having Ws and things you ha-, just 
have a H [0.3] in this shallow water equations you have [0.7] conservation of 
[0.7] er [1.1] conservation of U [0.5] er of the zonal wind [0.2] conservation 
of the meridional wind [0.5] those two equations are very similar to are almost 
[0.2] identical actually to these equations up there [1.1] and then you have er 
[0.4] conservation of mass really here [0.2] this was sort of a mixture of this 
one [0.7] er [1.2] yep [0.2] it's a bit like a mixture of this and this 
together produce this third equation here [0.7] this is basically just saying 
the convergence of fluid i-, if a lot of [0.2] fluid converges on a particular 
point [0.4] the w-, the the height would have to go up at that point [0.7] so 
in this thing here [0.5] the reason that's a big blob up there is because fluid 
converged in [0.8] at that 
point [1.7] so there's a very simple set of equations [1.1] er [0.7] i've got 
an exercise for you if you'd like to try this [0.3] which is quite [0.2] a good 
one to do [0.6] is here the equations for U and here's an equation for V [1.1] 
now what i would like you to do is try and find out [0.7] er [1.0] er [0.4] try 
and find out an equation for the vorticity and for the [0.5] er [0.4] 
divergence [0.3] so [0.6] the exercise [3.3] oh [3.1] exercise a bit like this 
[1.3] just remind you [0.9] this is useful to understand [0.6] where things 
come from [0.4] so the [0.8] the [0.2] er relative [2.2] relative [0.4] 
vorticity [2.3] is defined as [0.2] er D [0.5] this horrible Greek symbol i can 
never pronounce right D [0.4] this will be bad for your linguistics one that's 
your V [0.4] the D [0.6] no it's [0.2] x-, [0.3] xi [0.3] in Greek it's X-I [1.
1] actually [0.4] you might be able to help me [0.6] how do you pronounce it [0.
6] 
om0880: [xi] [0.6] 
nm0877: [xi] [0.4] 
om0880: 
nm0877: is that right [xi] [0.4] 
om0880: [0.3] 
nm0877: okay [0.5] xi [0.4] er this is relative vorcity [0.2] xi yeah [0.4] er 
it's always a bit tricky to pronounce so [0.7] D [0.2] D-xi- [0.2] by-D-X [0.2] 
minus [0.2] D- [0.2] xi- [0.6] by- [0.4] D- [0.3] Y [0.5] that's the definition 
of relative vorticity yeah [2.1] the [0.7] the and [1.0] oh sorry what am i 
doing no it's not the definition [0.3] i'm 
f-, getting con-, confused [0.7] no relative vorticity is [0.2] is is r-, [0.3] 
is xi [0.6] and it's equal to D- [0.2] V-by-D-X [0.7] minus D-U-by-D-Y [1.3] so 
if you like is the spin of the fluid [0.2] mm [1.8] it's how much the s-, flu-, 
fluid is s-, spinning around or swirling [1.7] there's also the divergence [0.
2] which is a lot easier to say in Greek [0.7] er divergence usually use the 
symbol delta [1.7] and delta is defined as D-U-by-D-X [0.9] plus D-V- [0.4] by-
D-Y [1.1] yeah [0.9] so the difference there both of them involve Us and Vs and 
X and Ys but they're mixed up [0.5] yeah [0.2] so one's like that and one's got 
a minus sign in [1.4] now the point is i [0.2] i've given you an equation [0.2] 
er [0.5] D-U-by-D-T equals nananana [0.2] and i've given you an equation D-V-by-
D-T equals [0.3] dadadada [1.3] now what i would like [0.2] which is what you 
solve normally you don't solve these equations with vectors in in the [0.2] in 
the big models [0.3] you solve equations for the vorticity and the divergence 
[1.2] so [0.7] if you work out now i'll just set you off on the right track if 
you write down [0.4] D- [0.5] xi- [0.3] by-D-T [1.0] that's going to be [1.3] D-
by-D-T [0.6] er [0.4] you plug in that 
definition D-V-by-D-X [1.0] minus D-U-by-D-Y [2.1] and then [0.2] you can 
switch the [0.2] derivatives so the D-by-D-T and the D-by-D-X you can change 
the order of them [0.6] yeah you know [0.3] can you see that okay from there 
the [1.7] so this thing you can then write as D- [0.2] by-D-X [1.3] D-V-by-D-T 
[1.3] minus D- [0.4] by-D-Y [1.2] D-U-by-D-T [1.5] okay [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 [0.7] substitute in 
there and then sort out all the D-Xs and D-Ys and you get [0.4] you'll end up 
with two nice little equations you'll have one for [0.6] D-xi-by-D-T [0.9] and 
you'll have another one that's for D-delta-by-D-T [3.2] okay [0.3] so if [0.2] 
try that exercise you know go away [0.2] it doesn't take you that long to do it 
probably [0.2] takes about fifteen minutes or so [0.6] but it's quite nice 
'cause you've then derived a vorticity equation by yourself because [0.3] 
sometimes you don't really care about the divergence [0.2] sometimes you're 
more interested just in vorticity yeah [0.7] Rossby 
waves are more interested in vorticity than they are in divergence so [0.5] 
Rossby wave dynamics is all [0.2] vorticity stuff [0.7] so this sort of 
separating of the two fingers is nice and [0.5] and you can recast those 
equations in terms of vorticity and divergence so [0.2] it's a good little 
exercise to try [1.3] shouldn't tax you too much hopefully [1.1] er [1.2] and 
you can also in this one [0.2] er [2.4] er [0.6] so this one you can [0.7] er 
[1.4] now if you set [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 [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 [0.6] so the only coupling between [0.9] is a vorticity equation a 
diverged one and a [0.2] and a height [0.4] phi is a sort of height [0.8] er [0.
2] the height is only coupled to the divergence equation it's not coupled to 
the vorticity one [1.1] and and if you set the er [0.4] if you set the phi to z-
, steady phi so D-phi-by-D-T is zero the dele-, divergence is then zero [0.5] 
and then you'll get a just a vorticity equation yeah [1.2] which comes to my 
next set of simple things [0.8] which is [0.3] vorticity equations [2.5] so if 
you say i don't care about divergence [0.9] so basically you said delta is 
equal to zero [0.2] then you get sort of divergence eq-, [0.2] i mean vorticity 
equations [0.5] and you can get a vorticity equation from this quite nicely if 
you do this thing over there [0.3] and then just set delta to zero [0.3] you'll 
end up with a nice little [0.4] vorticity equation [2.1] and your vorticity 
equation that you'll get [0.2] is called [0.2] if you do that you'll get the [0.
7] er barotropic [0.2] vorticity equation [1.2] which is the one you're going 
to be using in the exercise in the n-, practical assignment when we do the 
numeri-, [0.2] the computer [0.4] practical [1.7] there are quite a few 
different vorticity equations there's a barotropic one [0.8] which is what 
you're going to have [1.8] [cough] [0.4] er [0.4] there's also you you've seen 
ones probably [0.6] er [1.4] yeah [0.2] so that's sometimes called the 
barotropic [0.2] so we call that B-V-E [0.2] the baratropic vorticity equation 
[0.9] er [0.4] you can also have ones which have [1.2] bit more complicated 
stuff [0.2] 
quasi- [0.4] geostrophic [2.9] vorticity equation [2.7] but you have different 
equations for the spin basically of the fluid yeah [0.9] and those are quite 
useful for understanding things like cyclones and instabilities things [0.2] we 
consider cyclones to be things in vorticity really rather than [0.5] we're not 
really that interested in divergence usually [2.1] er [0.4] right the next set 
of models er again [0.5] kind of takes a while this [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 [1.4] this was a three-D model [0.2] this was a three-D model 
[0.5] three-D [0.7] er these are now two dimension be-, really because we've 
only got [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 [0.5] in in 
these equations here everything was a function of X [0.3] Y [0.2] and T [0.2] 
there was no Z in there there was no height [1.4] even height the variable H is 
just a function of [0.4] where you are [1.9] so the [0.2] the the these we've 
made a big step from 
going from here [0.3] to here we went from three-dimensional set of equations 
down to two-dimensional flow [1.4] these are two-D sort of things [1.5] er [0.
7] now what you can do is you can average equations so sometimes people er [0.
6] they do zonal mean models [1.0] or zonal mean [0.3] another type of [0.2] 
two-D equation is zonal mean [1.3] so they're interested in say the Hadley cell 
or the [0.4] the general [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 [1.2] er now they make sometimes they make these zonally average models 
where they've been averaged everything in the longtitude direction [1.4] and er 
[0.2] to get those models they use those [1.1] another type of model that's 
used quite a lot [0.2] is a very useful type of model actually is a one-D model 
[1.4] sometimes these are sort of like functions the th-, the thing will be a 
[0.5] for instance could be U [0.2] as a function of [0.6] er [0.8] X and Z [0.
6] and T [1.9] oh sorry Y Z T they're not interested in X they're not 
interested in longtitude [1.8] 
so it's just a cross section in [0.6] er [0.5] goes from the pole to the other 
[0.2] pole to the equator or something like that and up and down it d-, they 
don't care about where you are [0.4] in longtitude [0.9] one-D models sometimes 
people are interested in just models that vary [0.3] er use [0.4] the function 
of height and the function of time [1.3] er for instance when you s-, fire off 
a radiosonde in the atmosphere [0.6] er when you do the radiosonde you just get 
this sort of measurement with height and time [0.6] you don't know about [0.3] 
you don't really know about X and Y much [0.9] and [0.2] the models people use 
here [0.2] quite often the nice ones are these radiative [1.7] convective 
models [4.5] 
if you like those are the models people use to er [0.8] you can look at a 
particular point on the planet so you say well i'm interested in namex [0.3] 
i'd like to know what [0.5] how much radiation comes down to hits the ground [0.
3] and how much convection goes on at this point over namex [0.3] i just assume 
there's no horizontal mixing at all i just ignore all horizontal flow [0.4] and 
i'm just looking at each point individually [0.9] and oceanographers do that 
sometimes they have one-dimensional models of the ocean at particular points 
yeah [0.9] so they're quite handy things to have [1.4] and the final sort of 
model is a [0.2] zero-dimensional model so it doesn't have any [1.1] any space 
or time variation in it [0.2] any space version [0.7] er [0.2] the nice example 
of that [0.2] is these energy balance models [2.0] so when 
people [0.7] talk about climate change [2.4] they sometimes use an energy 
balance model [1.0] and they'll get a [0.3] they'll get an equation [1.0] for 
instance you can look at the whole Earth [1.2] you can say here's [0.3] here's 
planet Earth [0.6] it receives so much solar radiation comes in [0.8] we only 
get a quarter of that because it hits just one side of the planet [0.9] er [0.
2] but then we lose infra-red radiation out of the [0.2] all sides [1.1] so the 
amount of stuff we lose goes as [0.3] sigma-T-to-the-fourth as a black body [0.
9] so you can say sigma-T-to-the-fourth is equal to the incoming solar [0.6] 
that's energy balance so you said [0.4] what came in as far as energy onto 
Earth [0.2] must go out yeah [0.3] so it's [0.3] coming in as solar and it's 
leaving as r-, infra-red radiation [1.1] that's the simplest energy balance 
model [1.7] if you work that out the solar constant is thirteen-seventy [0.3] 
watts per metre squared [1.1] er this is the Stefan- [0.2] Boltzmann Stefan-
Boltzmann constant [0.2] you find in a thermodynamics book for black body 
radiation [0.6] and you'll find a temperature for the 
Earth of two-hundred-and-fifty-five degrees kelvin if you do that [0.7] which 
is too cold [0.5] the average on Earth the average temperature on Earth the 
observed one [0.7] if you average over the whole planet is two-hundred-and-
eighty-eight kelvin [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 abs-, we didn't include a nice little atmosphere round here 
that would absorb infra-red radiation [0.6] so there's no [0.2] this is [0.3] 
that that little balance model is just [0.5] er [0.2] for a planet [0.3] 
there's no atmosphere there [0.7] if you put the little blanket round it then 
you get another thirty degrees and you get [0.7] you get what they call the 
greenhouse effect [0.2] not the enhanced one the [0.3] it's just natural [0.3] 
yeah [0.4] yeah [0.2] 
sm0879: [0.5] 
nm0877: no they don't no [0.3] no [0.4] no you can have you can develop more 
complicated ones but [0.4] there've been an awful lot of [0.2] nice little [0.
2] things like that one 
sm0879: yeah [0.5] 
nm0877: yeah they don't they can get you can make one-D energy balance 
models [0.4] the thing with those is they start getting more [0.9] the more 
complicated ones like that [0.3] er [0.7] don't [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 [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 [0.5] they sort of work 
nice when you average over the whole planet but they don't [0.6] they w-, [0.2] 
they're not so sweet when you look at them in certain regions yeah [1.9] but 
yeah you're good point yeah it's er [1.1] anyway [0.4] that's the range of 
different models that you come across [0.4] er this is sort of the most complex 
and this is the simplest [0.8] as far as complexity it doesn't mean it's [0.2] 
doesn't mean it's not got interesting results [0.2] actually this little model 
here [0.6] t-, can tell you by playing around with this you can find out that 
[0.7] er Arrhenius in the [0.4] this person called Arrhenius [1.1] Swedish 
scientist in the [2.1] in the [0.2] eighteen-ninety-six [1.2] he used a little 
model like this [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 
[0.6] he was way before his time yeah [1.2] er [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 [0.3] it's based on a very 
simple little model [0.2] so that's a nice model you know you [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 c-, [0.3] can convince you of climate change then [0.
7] that's pretty good yeah [1.0] well it makes you feel better about if 
somebody just said well i've got a big complica-, i i've solved [0.4] if 
somebody said i've solved every molecular dynamics and i've found out it was 
going to get warmer you know [0.4] you'd just say well i don't believe you you 
know so [0.3] there's that you can do on a bit of paper yourself and [0.5] 
convince yourself quite quickly [1.1] so er [0.2] 
simple models can be pretty good [1.4] okay is that [0.2] that's a sort of 
quick run-through the important thing [0.9] like a lot of these things in this 
numerical course you'll find out [0.4] the important thing is to know which 
approximations are being made [0.4] so if somebody [0.3] comes along and gives 
you a model [0.2] or tell say your supervisor or whatever comes along 
eventually and says [0.5] er [0.4] here's a nice model to use you know [0.6] 
it's a good one we use it round here [0.2] everybody likes it round here [0.6] 
the things to know are what are the approximations that went into your model [0.
9] and a lot of people are not very good at that not knowing that what they [0.
5] their models are good at some things for instance [0.7] this model because 
primitive equations because it's had the hydrostatic approximation [0.6] and 
the anelastic approximation [0.3] is absolutely useless at doing anything to do 
with sound waves [0.4] or anything to do with vertically propagating mountain 
waves [1.0] so if somebody came along with the big climate model say a 
primitive 
equation model and said [0.4] oh we're looking at vertically propagating waves 
round the back of a mountain you're using this model [0.6] complete rubbish you 
know it's not made for that it's the approximations [0.2] got rid of those sort 
of things [0.7] so you should be very careful about which [0.6] each model 
involves certain approximations so [0.3] if you're clever you know which 
approximations have been used for your model [0.9] that tells you which things 
you can use your model for you know so y-, [0.2] it'd be [0.2] plain [0.3] 
stupidity you couldn't look at sound waves using primitive equation models so 
[0.8] it's designed for looking at bigger things [2.2] and people even clever 
people these days [0.5] get confused with that so they'll say oh we've reduced 
the resolution of our very complicated climate model [0.2] and we're going to 
go down like the Japanese now have a project frontier project [0.5] to make a 
one [0.2] five kilometre grid on their model so they're going to get a really 
small resolution you know and look at [0.4] very local 
things [0.9] well [0.3] they're still using the primitive equations [0.9] so 
their model's still not going to get gravity waves properly so even if they got 
five kilometre resolution [0.5] round the back of a mountain or something 
they're not going to get it right [laugh] [0.8] so [0.2] you get a very funny 
impression they say [0.5] ah but it's got a five kilometre resolution it's 
resolving everything great you know [0.3] well it's not because it's got the 
wrong equations to resolve everything correct yeah [0.5] so [0.5] you've got to 
be aware of the weaknesses of your equations [2.1] okay [0.2] er [1.0] now i'll 
move on to this [0.8] the final bit was this er [1.4] where we got to [1.2] 
yeah [0.3] so i've gone through the equations are you sort of happy with that 
does it help review a bit [0.5] well hopefully this course reviews some of the 
other things you've seen in the last term er [1.2] er [0.4] there's nothing 
like making a model to [0.2] make you know what you're doing in the subject [0.
9] i've talked about hierarchy models we talked about waves in the atmosphe-, 
[0.8] oh i didn't show you the 
waves [0.2] oh hang on [0.5] what i am doing [1.4] oh the waves [2.6] yes [1.9] 
sorry [0.5] missed a crucial slide here [2.3] well it's better ne-, you now 
know the equations so if we look at this [1.8] this is a quite a nice little 
diagram this was a meteorologist Green [0.6] who [0.2] who [0.4] drew up this 
sort of d-, schematic diagram of different waves in the atmosphere [0.6] so the 
waves are [0.2] the atmosphere's a big compressible atmosphere [1.6] at this 
axis [0.2] is phase phase velocity [0.6] so [0.2] call it C [0.3] yeah [1.0] 
this sort of yellowy line going across here is three-hundred metres a second [0.
3] that's the speed of sound [0.4] yeah [0.2] anything above that line is 
supersonic [1.0] not very important because you d-, haven't seen any supersonic 
weather systems i don't know if anybody's ever seen a supersonic weather system 
but [0.3] i certainly haven't seen a supersonic weather system [0.9] in 
principle you could have if you look at these lines on here [0.5] the 
atmosphere could support supersonic waves [1.2] but it'd never do that because 
they produce a shock wave and it would [0.2] you know 
when you've got Concorde going supersonic then [0.4] you have to put so much 
energy in to get it [0.2] when it produce the sonic boom it [0.2] creates so 
much dissipation that [0.6] you need a lot of energy to keep it moving so [0.2] 
if [0.2] if suddenly one of those storms became supersonic [0.4] it would slow 
down very quickly it would it would lose all its energy so [0.8] chance of 
seeing a supersonic weather system is low [0.6] very low [1.5] sound waves are 
obviously in the atmosphere 'cause you can hear what i'm saying [0.2] so [0.2] 
and that line's there [2.1] er [0.4] then there's some other waves so [0.5] 
this bunch of waves this is sorry er spatial wavelength the horizontal 
wavelength yeah [0.3] in kilometres [0.4] these are logarithmic scales as well 
so [0.5] this is phase speed this is logarith-, [0.2] er horizontal wavelength 
so [0.6] ten-thousand kilometres is called planetary scale [0.2] in meteorology 
usually [0.8] one-thousand kilometres is synoptic scale that's the size of a 
storm [0.3] in mid-latitudes [1.0] er [0.3] anything between a thousand 
kilometres and ten kilometres is called 
mesoscale [0.6] so most of your favourite weather systems are in mesoscale [0.
8] but the big things are [0.3] the big systems are the synoptic systems are a 
thousand kilometres [0.9] er then there's things less than ten kilometres which 
are called microscale [0.4] so the three sort of [0.5] the four scales on the 
planet are [0.4] you know really big planetary scale things [0.2] synoptic 
scale so El Niño the big El Niño response is a planetary scale phenometa [0.
2] phenomena [0.8] then there's synoptic then there's mesoscale then there's 
micro [0.8] and then there's these different speeds [1.0] and so [0.5] er [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 [0.2] the atmosphere just 
gets less dense as it goes out to space [0.3] if it did have a top surface on 
it would go at this speed but it it [0.2] doesn't really have a [0.8] top 
surface [1.1] er then there's gravity waves [0.2] er these are the [0.4] 
gravity waves here have quite a range of speeds [0.2] so they can go very fast 
so very [0.5] very er [0.3] long 
wavelength gravity waves [0.6] er move very quickly [1.1] very short wavelength 
ones move slower yeah [1.4] er [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 [0.2] so over quite a long spatial scale from synoptic right 
down to [0.2] about ten kilometres [0.4] gravity waves move at about the same 
speed they don't [0.9] they're not that dispersive as waves [0.7] er once you 
get below ten kilometres gravity waves become dispersive again [1.3] er [1.0] 
there's also some of these gravity waves get mixed up with these inertial waves 
that you can have [0.8] er the easiest way to see why you have an inertial wave 
[1.1] is in your equations here [0.9] if you had D-U- [0.4] by-D-T equals minus-
F-V and D-V-by-D-T equals F-U [0.2] yeah [0.8] so that's just Coriolis that's 
[0.7] the Coriolis force acting on both of those [0.2] no pressure gradients 
imagine no pressure gradients [0.7] if you solve those equations there you get 
D-squared-U-by-D-T-squared [0.8] er equals F-squared- [0.2] U [0.5] and the 
same sort of thing for V [0.5] so that's an 
oscillation [0.4] if you look at the thing and the period of the oscillation is 
one-over [1.2] period is one-over-F [0.7] oh sorry two-pi-over-F [1.8] so [0.4] 
it depends on where you are on the planet [0.2] the the winds [0.3] the [0.2] U 
and V components oscillate [0.2] they go round [0.3] so if [0.2] when people do 
measurements of the ocean [0.7] er the when they put a probe down they'll see 
that the velocities are are turning round at a certain speed [0.6] and that 
speed is determined by the Coriolis parameter at that point [0.3] and that's an 
inertial wave [1.5] 
you don't tend to see them as much in the atmosphere [1.7] as an inertial well 
they call it inertial oscillations [2.0] you don't see those things much in the 
atmosphere [0.3] er [0.2] 'cause it's not stable enough so you don't you see it 
more in the ocean [0.8] er [0.2] but those things can mix up with gravity waves 
so you get this sort of funny little mixture wave here [0.4] these shorter ones 
the [0.3] you get these things called inertia gravity waves [0.5] so the two 
things have combined a bit you know [1.6] er [0.3] right so those are the 
gravity waves and 
then up here on the long large scale stuff [0.3] r-, from planetary down to 
synoptic are the Rossby [0.4] Rossby waves [1.1] now [0.5] if you're interested 
in forecasting weather and looking at storm synoptic systems coming through 
which is what the original [0.6] point of doing weather forecasting was was to 
s-, get [0.2] try and get the storm systems [0.8] you're interested in getting 
the Rossby waves right yeah [0.2] these especially the synoptic ones down here 
[0.5] typical speeds less than ten metres a second yeah [1.3] you don't really 
[0.3] care much about these very big thing these gravity waves that [0.3] that 
go at very high speeds yeah [0.8] you're not really interested in those very 
lon-, very large scale gravity waves that move quickly that that [0.5] those 
aren't relevant so [0.5] the trick is to try and filter some of those things 
out a little bit in your equations [0.2] you're more interested in getting 
these waves [0.8] and you don't really want sound waves so [0.2] by doing the 
[0.4] anelastic approximation you get rid of all the sound waves and all the 
supersonic stuff [0.7] by doing [0.6] er the [0.3] er [1.2] by doing the 
hydrostatic approximation you get rid of quite a lot of these fast gravity 
waves [0.4] a lot of the fast ones are going vertically [0.6] these are 
vertically propagating ones [0.6] and you end up with waves that then all move 
at a slower speed yeah [0.2] and that means you can then [0.3] simulate the 
thing much nicer use a bigger timestep [0.5] and you've k-, still kept the 
essential part of [0.5] you've still kept the Rossby waves and the the synoptic 
systems you've got rid of the fast stuff [1.7] does that make sense are you are 
you happy with that [0.8] all the ways we cheat in writing our equations down 
yeah [0.2] [laugh] [0.6] so anybody who says why did you why did you use 
primitive equations [0.6] well the reason is to get the timestep down in the 
model really you know [1.5] er y-, so so you don't have to use a large timestep 
[2.5] okay [0.2] er [1.1] now the final bit [1.3] i was going to go is [1.9] er 
[2.7] yeah [1.1] the ideas [3.2] oh we're running out of time very quickly [0.
4] we've got [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 [0.4] as a dynamical system [0.8] so it's all a 
dynamical system and [0.4] so the [1.5] in the last few minutes just show you 
this [3.6] you can write all of those sort of equations that i've just gone 
through there in a quite a simple way [0.9] you can't solve them easily but you 
can write the equations simply [3.4] so [1.1] if [0.2] we [0.7] write the 
equation this is a general equation for the ocean and the atmosphere [0.4] and 
you can write down [1.1] D-xi-by-D-T [0.4] so it's a first order equation [0.2] 
you don't usually get second order D-squared-xi-by-D-Ts [1.2] there's some 
horrible [1.1] non-linear operator called Q [0.4] which is some [0.2] depends 
on the state of the system so this is like the advection terms [2.8] it's 
actually not that non-linear [0.3] er compared to what you c-, it's only 
quadratic usually that's why i call it Q actually [0.6] so when you look at 
your equations you have things like U [0.3] D-U-by-D-X et cetera [1.3] so you 
have some sort of quadratic operator here [1.9] and you 
have [0.3] some forcing on this side forcing and dissipation [0.5] that depends 
on the state of the system xi [0.9] the time [0.3] and some parameters of your 
model [0.4] d-, parameters of [0.4] albedo of the Earth et cetera et cetera [0.
6] so [0.2] xi here is a general xi is the state of the system [1.7] the state 
of either the atmosphere or the ocean [2.4] and [0.4] that for instance if 
you're doing the primitive equations would be a little vector of [0.3] U [0.2] 
it would consist of V [0.2] W [0.5] er [0.7] er [0.2] [sigh] [0.4] T [0.4] what 
else have we got [0.7] about that [0.9] yeah [0.4] er [0.2] and here you'd have 
this'd be a function of latitude [0.2] long-, [0.6] longitude latitude [1.0] 
and time [0.4] and that'd be a function of longitude latitude time [0.7] and 
height sorry [0.3] or this Z [1.5] latitude longitude [0.2] ti-, height and [0.
2] time [0.9] longitude latitude [0.8] height and time yeah [0.5] so you have 
four fields you have [0.2] the U field the V field the W field and the ti-, the 
temperature field [0.5] would all be [0.4] would all be define the state of the 
atmosphere [0.5] so if i know all those things i know what the state the 
atmosphere's in [0.6] so that's what 
i mean by xi it's sort of a [0.2] big [0.5] beg-, [0.4] big field summarizing 
that lot [0.6] this this xi the state of the system evolves like that basically 
[0.7] that's the general equation for most atmosphere ocean systems [1.2] this 
bit on this side gets called [0.2] dynamics quite often [1.4] by [0.2] quite a 
lot of people [0.3] and this bit is called the physics [2.0] so this is the 
force in [0.4] the force this includes force in [0.4] so radiation and things 
[0.4] it also invo-, includes dissipation [1.1] so the atmosphere's got [0.2] 
diffusion and drag it's got it's losing energy as well so it's [0.2] it's 
gaining energy from the sun but it's also losing [0.3] energy when it drags 
along the ground [1.0] so these [0.6] er [0.6] well we people do pr-, m-, 
numerical models they like to split it into two so they call this part the 
dymani-, dynamical part of the model [0.3] and this part is the physics the 
parameterization of all the radiation all the messy bits of the clouds and 
things [1.1] er [0.6] sometimes this is called the dynamical core [0.6] so this 
is the fluid part of this is basic just basic 
fluids and waves [0.3] this is the physics how it's forced yeah [0.3] so that's 
a useful way to look at climate models [0.7] er how they usually break up 
people look at the dynamical schemes and then they look at the physical schemes 
in the model [1.1] er 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 
[0.7] when i do all this fluid all those fluid equations i wrote before i 
ignored that completely actually [0.7] i only showed you the dynamical bit [0.
9] so in reality [0.5] er 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 [0.6] what's the latent heat release so [0.2] physics 
physical things in models are very important it's not just a pure fluid [1.9] 
er [0.9] i'll try and finish off quickly [1.0] er [0.4] what we [0.8] you can 
do when [0.6] these sort of things you can't really [1.3] this state of the 
system here [0.8] depends on [0.4] er [0.2] longitude [0.2] lambda [0.6] 
latitude phi [0.6] the height [0.5] Z [0.4] 
and T yeah [1.1] now [0.4] it's a continuum this d-, this is everywhere you 
know so even at this point that point it's very [0.2] this is a [0.2] an 
infinite number of places this is defined at yeah [0.4] this is a continuum 
field [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 [0.2] you've got to break it up 
[1.2] so what you do [0.2] is you break it up [0.2] like this so you say [0.6] 
well i won't look at the field everywhere [0.6] i'll discretize this [0.5] so 
i'll have a bunch of lati-, longitudes a bunch of latitudes [1.0] a bunch of 
heights of levels [0.5] and a bunch of timesteps [0.5] Ns [1.3] right [0.4] so 
[0.3] these are sort of like the grid points in the model [0.5] and you label 
them with an index here this is labelled with I as an index [0.8] so [0.7] I 
equals one to [0.8] er and it can go up to a very large number P [0.3] 
usually about ten-to-the-five up to about ten-to-the-seven in current models [0.
3] of grid points yeah [2.0] so you've broken up space into a little [0.7] 
you've broken it into a big [0.2] a grid in the horizontal and a grid in the 
vertical as well so you've [0.3] you've got a lattice if you like a like a 
climbing frame [0.6] and at each point you've only defining the variable at 
those points so you've [0.3] replaced [0.5] er this wonderful continuous field 
[0.7] by a horrible [0.2] discrete bunch of things yeah [0.6] about ten-to-the-
five variables yeah [1.2] and these you can label these ones [0.8] time as well 
you break up into steps [0.3] so you discretize time so you discretized all the 
space and the time dimensions into [0.6] broken them up as a grid [1.3] this 
you can then label as [0.3] one label here [0.4] I for the space index [0.5] 
and a little label at the top N [0.2] meaning the time 
index [0.6] so that's the times-, that tells you what timestep it is [2.2] and 
that tells you which grid point variable that's the grid point [1.8] yeah [1.6] 
so now you've replaced this continuous equation here by a bunch of very finite 
[0.2] just that you've g-, only a certain number of variables phi Is [0.5] and 
then you do them in different times yeah [1.3] so er [0.4] i think i'm going to 
stop there because it's going to er [0.2] we're going to run into lunch 
otherwise we can pick it up again in the old [0.7] old Meteorology building 
afterwards so [0.6] have have lunch and have a think about this and then we'll 
[0.2] [laugh] [0.9] we're going to discretize time in after lunch yeah we're 
going to [0.3] do that a bit more