nm0925: okay [1.1] er [1.9] problem sheet four i wanted to say a word or two 
about problem sheet four [0.2] which er [0.3] er i guess [0.2] you had back the 
other week [0.5] er [0.4] i had [1.2] vibrations from people that er [0.5] er 
people were a bit shocked at the marks [0.5] er [0.5] and there was er [0.2] a 
very straightforward reason why the marks on that sheet [0.3] were somewhat 
lower than the marks on the earlier ones [0.6] and the reason was [0.4] that 
nearly not quite but very nearly all of you [0.4] got in a mess with signs [0.
7] and the result was that er [0.3] er one way or another you nearly all [0.3] 
not quite all but nearly all got [0.3] incorrect answers because somewhere 
along the line [0.3] you'd got into a mess with signs [0.8] and i'm afraid i 
did mark it harshly because i think it's quite important to make the point [0.
5] that we're now at a stage where signs matter [0.4] they're not just a sort 
of frill that we add on at the end of a calculation [0.3] you've actually got 
to keep track of the signs as you go through it [0.3] otherwise [0.2] you'll 
get a silly answer [0.7] so i i want to just remind you of a couple of points 
which 
will come in again and again in [0.4] er later problems [0.3] er to do with 
getting signs right i suspect these calculations [0.3] involving thermal wind 
shear [0.5] are amongst the most awkward [0.6] or have the most potential [0.4] 
for getting signs wrong [0.3] but the same goes for other calculations you'll 
do [0.7] first thing to remind you is [0.4] er [0.2] that er in nearly all 
these problems [0.3] you do best if you break the wind down into [0.3] 
components [0.6] so let me just remind you of the usual convention that we use 
[0.4] we imagine an X axis which points from the west to the east [0.7] a Y 
axis that points from the south to the north [0.8] and if you are told [0.7] 
that there is a wind of particular magnitude [1.8] shall we call it U [1.7] 
with a particular direction and there's various ways of measuring that 
direction [0.3] the meteorologists [0.4] talk about the direction from which 
the wind blows and they measure the angle from true north [0.4] the [0.2] er 
mathematicians and this is probably the convention i will stick to [0.3] will 
measure the angle of the wind vector [0.2] from the X axis in the anticlockwise 
sense so if we have [0.5] theta there [0.3] which describes the [0.3] er 
direction of the wind vector [0.4] then 
we have two components [0.6] we have a component along the X axis which i'll 
call [0.3] little-U [0.6] and little-U is big-U [0.4] times [0.4] the cosine [0.
2] of that angle theta [1.1] and then we have [0.3] a component [0.3] parallel 
to the [0.3] Y axis i call that little-V [0.4] and little-V is capital-U [0.2] 
times [0.3] the [0.2] sine [0.2] of theta [1.2] now [1.0] the way i've drawn it 
there it's easy because [0.3] er [0.3] both U and V are positive [0.9] if the 
wind vector points into these various other quadrants then different signs 
apply [0.5] so here [0.2] we have both U-positive [0.8] and V-positive [1.9] if 
the wind vector was down in this quadrant [1.2] then of course U [0.3] would be 
still positive but in this case V would be directed down the negative Y axis so 
[0.4] V would be negative [1.3] similarly if the wind vector turned up in this 
[0.2] quadrant here [0.4] we'd have U-negative [0.4] and V- [0.2] positive 
still [0.3] down here both U and V would be negative [1.0] always worth drawing 
a sketch like that [0.3] so that when you've done the calculation you can just 
check that the signs [0.5] check out against that diagram [1.4] if you're using 
a calculator to 
calculate the er vectors [0.8] then [0.4] strictly you should automatically get 
the signs right if you've got this angle theta and you put in an angle of [0.3] 
er [0.3] two-hundred-and-eighty degrees or something like that it should 
automatically [0.3] take care of the signs [0.4] provided you've got your 
calculator set up right [0.5] er [0.2] but even so it's a very good idea when 
you're doing these calculations not just to blindly trust that you've got the 
signs right [0.4] but at each stage [0.3] to check things and a diagram is 
often the er easy way of doing that [2.4] so that's getting the wind components 
right and many of you fell over at that stage because you just said well you 
just call U and V positive no matter which quadrant the wind vector was 
pointing into [1.2] the other [0.7] bit where [0.3] people went wrong and i 
think probably more people went wrong here [0.6] er was when [0.2] we were 
required to take vertical derivatives [1.1] er [0.2] we have things like D-U-by-
D-P [0.2] and D-V-by-D-P [1.5] and what i want to do [0.2] is just to remind 
you how we would estimate [0.3] any 
vertical derivative [0.4] so suppose we have any [0.2] meteorological quantity 
i'll just call it F for the moment [0.4] it might be wind or temperature or 
whatever [1.5] and we're required to estimate D-F-by-D-P [0.3] the vertical 
rate of change of F [1.3] well let's draw a little diagram once again to 
illustrate what we're doing [0.3] very typically we might be [0.3] given the 
value of F at say a hundred kilopascals near the surface [0.4] and the value of 
F [0.4] at fifty kilopascals halfway up through the atmosphere [1.4] and [0.2] 
there's a pressure difference between those two levels [1.0] delta-P which in 
this case is fifty kilopascals [1.3] well [0.2] the way i will estimate 
derivatives always [0.4] is [0.3] by what we call finite differences we just 
take the value at one level [0.3] minus the value at the other level [0.3] 
divided by the delta-P [0.7] and that's an estimate of the derivative [0.6] not 
[0.4] necessarily a terribly accurate one but it is an estimate [1.9] the 
critical bit is getting it the right way round so you get the signs right [0.2] 
and people do get in a muddle here i have 
to say i get into a muddle often if i'm not thinking [2.5] the way it works is 
like this [0.4] we have to take the value of F at the higher pressure [0.2] a 
hundred kilopascals in this case [0.9] minus [0.5] the value of F at the lower 
pressure [1.2] fifty kilopascals [1.1] divided by the pressure difference [1.0] 
delta-P [2.6] and [1.4] this will give us the right sign [0.8] notice that very 
often [0.8] we're dealing with situations where the wind speed increases with 
height [1.0] in other words the wind speed decreases with pressure [0.3] so if 
this was say the U component of wind we might very frequently have the 
situation [0.4] where U at fifty kilopascals is larger [0.4] than U at a 
hundred kilopascals and so [0.4] D-U-by-D-P would be negative [1.1] and people 
get in a real muddle with this because of course the wind increases with height 
[0.6] but that means it decreases with pressure [0.4] so the sign is often [0.
2] perhaps a bit counter-intuitive [0.4] so do remember that [0.3] again [0.2] 
i think the diagram always helps and a little thumbnail diagram [0.3] along 
with your problems [0.3] just to check out what 
it is that you're taking differences between to estimate [0.4] these 
derivatives will not go amiss and again [0.5] don't just trust that you've done 
the algebra correctly when you work out the signs [0.4] just check it against 
the diagram to make sure the sign seems to make sense to you [0.4] and that way 
[0.3] er [0.2] you will get lots of marks and feel gratified and everyone'll be 
happy [1.3] right [1.5] so much for the the the the problems [0.2] let me just 
say as a [0.2] parting shot the problems really aren't meant to be hard if 
you're finding [0.4] that you're spending hours and hours on the problems [0.4] 
i suggest you've rather missed the point [0.4] most of the problems are a 
matter of either picking [0.5] formulae or equations out of the lecture notes 
[0.4] and then [0.2] in a subsequent problem [0.3] er rearranging them maybe [0.
6] or putting some numbers in them [0.2] or what have you it really shouldn't 
take hours and hours and hours to do that [0.6] so [0.2] er [2.9] do bear that 
in mind [1.4] 
right let's get on to today's topic [2.0] er today [0.9] can't find the right 
page in my lecture notes [8.3] oh here we are [5.6] today our topic is called 
[0.2] thermal advection [6.0] which is a rather grand word for a rather simple 
concept [2.5] i'd like to start off by thinking about the weather [0.9] it was 
a cold morning this morning wasn't it 
ss: mm [0.7] 
nm0925: it certainly was [0.4] it was quite a nice afternoon yesterday wasn't 
it [0.3] 
ss: yes 
nm0925: yes [0.6] it was quite warm wasn't it if you went out for a walk in the 
sun [0.8] right anyone like to give me an explanation of why [0.2] it was [0.5] 
five or ten degrees colder [0.3] this morning [0.3] than it was say at three 
o'clock yesterday afternoon [1.1] yeah [0.2] 
sm0926: was a [0.4] clear night last night 
nm0925: yes 
sm0926: a lot of the thermal er a lot of the infrared radiation goes [0.8] 
nm0925: right [0.6] good [0.9] it was a clear night what's more it was dry air 
as well which is important which means there wasn't much water vapour [0.3] and 
so the infrared radiation from the earth's surface could actually escape rather 
more readily [0.6] er not everyone knows of course that water vapour is the 
most important greenhouse gas in the atmosphere [0.3] although it's not the one 
the environmentalists 
make such a fuss about [0.4] but er [0.3] er [0.6] the difference between a [0.
6] a clear night that's got dry air and a [0.2] clear night that's got moist 
air is actually quite dramatic [0.3] last night it was [0.3] clear and dry [0.
6] so yeah [1.1] lots of radiation from the earth's surface the earth's surface 
and the layers of air lying immediately above it got pretty cold [0.5] hence 
our frost this morning [0.8] let me take you back [0.8] to er [0.5] Friday [1.
9] don't know if you can remember Friday i have difficulty remembering events 
before the weekend [0.4] but if we go back to Friday [1.0] er the [0.4] m-, 
minimum [0.3] at about six o'clock in the morning [0.5] was a great deal higher 
it was about four degrees or so [1.0] so it was four or five it was five 
degrees probably six degrees warmer than it was this morning [0.7] so what was 
different [0.9] about [0.3] Thursday night Friday morning from Sunday night [0.
4] Monday morning [1.3] any ideas [2.8] why was it [0.8] fi-, four or six 
degrees warmer [0.5] the same time on Friday morning [0.3] as this morning [0.
2] 
sf0927: [1.2] 
nm0925: a front went past well done [0.3] you've 
got the right answer straight away i was hoping someone was going to witter on 
about clouds and so on [0.5] well [0.2] there are there are two things actually 
[0.3] we might have said well it was a cloudy night and therefore [0.2] the 
radiative effect we've just talked about couldn't operate [2.3] but you're dead 
right [0.8] the difference was [0.4] that a front went through [0.3] we were 
actually sitting in a different air mass [0.3] on [0.2] Thursday morning [0.5] 
and over the weekend [0.3] the front came through it went dramatically colder 
it snowed [0.8] as we shall talk about later on this morning [0.5] in some 
places anyway [0.4] and er [0.4] er so it's altogether colder one mass of air 
[0.2] with one set of thermodynamic properties [0.5] has been replaced by 
another [0.6] and it's that replacement of air with one set of characteristics 
by air with a different set of characteristics [0.6] which is what's meant by 
advection [0.2] and that's what we're going to talk about today [1.3] you see 
if i sit at a particular met station let's just focus our minds on the 
temperature for the 
moment although the same arguments apply to other meteorological quantities [0.
6] if i sit at a meteorological station [0.7] and i observe a change in the 
temperature [1.5] then there are two [0.3] alternative explanations for that 
change in temperature [0.3] and in many cases of course [0.2] both of of them 
operate [1.0] either [1.0] the air sitting at that station has actually changed 
its properties in some sense [0.5] so your explanation that [0.2] there was 
lots of radiation going on last night lots of infrared [0.3] escaping [0.2] and 
therefore the air got colder [0.7] that could be the case in one set of 
situations [2.3] and that's what we sometimes call heating or or diabatic 
processes [2.2] on the other hand [0.2] the temperature at a station [0.3] may 
change [0.8] simply because [0.5] the individual air parcels are are not 
changing they retain their temperature but they move away from your station [0.
4] and they're replaced by another set of air parcels that come along with 
different properties [0.9] so [0.3] during the course of Friday and Saturday [0.
5] warm moist air [0.3] moved 
away from the namex area [0.3] to be replaced by much colder drier air [1.5] 
and that's what we call advection [1.0] so the concept is simple enough what we 
need to do now [0.3] is to [0.2] er put some er quantitative details on it [1.
3] so i'm going to draw a diagram to fix the ideas in our head [0.3] let's draw 
a diagram shall we [0.3] of [0.3] temperature [0.9] and [0.5] let's consider 
one space direction i'll call it X for the moment [0.7] and suppose i have an 
observing site here [1.0] i'll call this [0.2] point A [0.9] and at the 
observing site we observe a particular temperature [0.4] there we are that's 
the temperature [0.5] at [0.5] er point A [5.3] but the point is i'm going to 
suppose that the temperature varies as we go along the X axis so if i was to go 
to another observing site at a different position along X [0.3] i'd observe a 
different temperature [0.6] so [0.3] let's er imagine that the temperatures 
along this X axis [0.5] they go like this i i'll draw a rather simple example 
where they just go along a [0.2] an increasing line like that [0.9] so stations 
to the east of A [0.2] are 
warm [0.6] stations to the west of A in this case are cold [2.6] er let us now 
suppose that a wind is blowing [0.2] okay and i'm going to keep things dead 
simple [0.4] by supposing that the wind is blowing parallel to the X axis [0.6] 
er in general it won't of course in which case we'll have to consider the 
component of the wind parallel to the X axis [0.4] but let's just suppose there 
is a wind [0.4] blowing along the X axis [1.0] there we are wind magnitude U [0.
4] pa-, blowing parallel to the the X-axis [2.4] and [0.3] time passes [2.0] so 
[0.8] after [0.2] some time interval delta-T [1.2] our observing site is no 
longer seeing this parcel of air with a temperature T-A [0.7] what it will 
actually be sampling is an air parcel that started off [0.5] some distance 
upstream [0.7] so [0.2] er [1.8] let's suppose we take that point [0.4] B shall 
we say [0.9] that point B is a distance U [0.7] times the time interval delta-T 
[0.2] upstream [2.7] and so after a time delta-T [0.4] the parcel that started 
off at B [0.9] will have reached [0.3] A [0.9] and so i'll measure a new 
temperature at station A after this time interval [0.5] er i'll 
measure this temperature i'll call that T-B [10.3] let's see if i can write 
down [1.3] a relationship [1.0] between those two temperatures [2.9] well i 
think i can actually i i've a feeling i didn't use quite the same notation in 
the lecture notes if you're following in the lecture notes [0.4] you might want 
to adjust the notation as as we're going along [1.7] what we're interested in 
now is the temperature T-B [0.8] which is the temperature that's now reached my 
observing site [2.1] and we can see here [0.2] that if this [0.6] curve here [0.
7] doesn't depart too much from a straight line between stations A and B [0.3] 
and it won't provided that time interval delta-T is sufficiently small [1.0] er 
[0.2] then it's rather easy to do the calculation we could say that T- [0.3] B 
is going to be T-A [0.8] minus [1.5] the distance delta-X [0.4] to point B [1.
0] multiplied by the gradient [2.7] D-T-by-D-X the rate of change of 
temperature along the X axis [0.5] so that distance there is delta-X and we can 
see that that is equal to [0.4] U times delta-T [4.3] 
so [0.8] if i rearrange that expression [0.4] i can [0.2] get an expression for 
the [0.2] rate of change [0.3] of temperature [0.7] at [0.3] station [0.6] A [6.
8] so rate of change of temperature at [0.3] my station A [0.5] well [0.2] i'm 
i'm going to approximate it by T-B [0.4] minus T-A [0.6] divided by the time 
interval [0.4] delta-T [3.1] and i can write an expression for that from the 
preceding expression by rearranging it if i take the T-A over to that side [0.
6] then i get an expression for T-A [0.4] minus T-B [0.2] and then i've just to 
divide it by delta-T and if i do that [0.2] and don't tell me to keep the signs 
i've got a minus sign here [0.7] i've got a delta-X over a delta-T [1.8] times 
[1.0] D-T-by-D-X [6.4] it's not hard it's just simple algebra here [1.3] what 
i'm going to do now [0.4] is my usual calculus trick [0.7] i'm going to suppose 
[0.4] that delta-T and delta-X become very small [0.2] i'm going to take the 
limit [0.3] in which delta-X and delta-T tend to zero [0.7] if i do that [0.6] 
then that simply becomes the rate of change [0.7] of temperature with respect 
to time [0.7] if you want to 
you can say that's at the point A [2.1] and that's going to be equal to [0.6] 
well in the limit del-, of delta-X and delta-T becoming very small [0.7] delta-
X over delta-T that's just the rate of change of position [0.4] of an air 
parcel [0.4] along the X axis [0.5] it's the [0.2] velocity it's the speed U [1.
3] so i'm going to get minus- [0.2] U [0.7] times [0.5] D-T-by-D-X [3.5] and so 
[1.5] that is an expression which tells us about the rate of change [0.4] of 
temperature [0.5] at [0.5] a fixed point [0.3] in space a fixed observing site 
[0.9] on the assumption [0.6] that the only process that's changing the 
temperature [0.3] is this advection effect in other words [0.3] the replacement 
of one air parcel [0.4] with an air parcel of different properties [4.5] so 
we'll put a box round that 'cause it's important [1.3] so it's the rate of 
change [3.2] due to [1.1] advection [4.2] and the er [3.2] rate of change due 
to advection is given a name [1.5] it's sometimes called [0.4] the Eulerian [0.
3] rate of change [2.0] Euler was a famous [0.8] French mathematician [0.9] who 
had more laws and formulae named after him [0.6] er than most of us have [0.6] 
and this 
is one that's named after him [6.9] put a little a box round that it's an 
important bit of terminology you'll hear referred to [0.4] the Eulerian rate of 
change [0.4] is the rate of change [0.2] at a fixed point in space [0.9] the 
rate of change you would estimate for example [0.5] if you're a meteorologist 
sitting [0.3] at a fixed observing site on the earth's surface [0.4] watching 
the winds blow past you [2.3] course it's not the only rate of change [0.3] 
that we might consider [0.8] suppose we didn't sit at a fixed observing site [0.
4] suppose we attached ourselves to [0.2] some sort of balloon [0.5] which was 
wafted around by the winds [1.4] in that case [0.2] if we had a thermometer 
attached to that balloon [0.8] er we wouldn't see the Eulerian rate of change 
at all [1.1] in fact if we designed the balloon very carefully [0.5] we would 
actually see the rate of change of individual air parcels because our balloon 
[0.4] would effectively be always embedded in the same lump of air as it moved 
around [0.6] actually that's a pretty bad example 'cause it's very hard [0.4] 
to design a 
balloon that does that [0.3] the problem is that the [0.3] balloon usually 
floats at a constant height [0.3] so it follows the winds at that height but it 
doesn't rise and sink with the air parcels [0.6] er anyway that's er [0.2] 
that's a that's a technical difficulty let's suppose we could do that [2.3] so 
we have this other rate of change [0.7] the rate of change following an 
individual fluid element [1.4] and that in fact is identical to the rate of 
change that we mentioned earlier on if if an air parcel cools [0.4] because of 
radiation [0.4] we're actually talking there about this rate of change 
following the fluid element we're thinking of this air parcel [0.3] that's 
sitting there [0.3] radiating infrared radiation to space and thereby changing 
its temperature [1.2] so we need another notation for that [0.6] and so i'm 
going to introduce a notation and this is where some people get a bit confused 
'cause they're a bit sloppy [0.5] with notation on occasions [0.2] so [0.8] er 
[1.7] i'm now going to consider the rate of change [4.5] following [5.2] an 
individual [3.6] fluid parcel [4.9] so what we would do here [0.3] is to 
measure the temperature of our air parcel at some time T [0.5] we would then 
measure the [0.3] er temperature of our air parcel at some later time T-plus-
delta-T shall we say [0.4] we'd subtract the two divide by the time interval [0.
4] take the limit as delta-T tends to zero [0.3] and we'd have a rate of change 
of temperature [0.5] following the fluid parcel [0.7] and we'll use a special 
notation for that [1.2] we will define that as capital-D [2.3] D-T [0.3] i'm 
using too many Ts here but still [2.1] by D-little-T [2.0] and that's the 
notation we will use [0.7] this is sometimes called [2.0] it's the rate of 
change for an individual fluid parcel [0.3] and this is sometimes called the 
Lagrangian [4.4] rate of change [5.7] Lagrange was also [0.3] an eighteenth 
century French mathematician [0.4] who has [0.3] er a vast number of equations 
and formulae named after him and this is just one of them [1.3] so they they 
they were good at getting their names on equations in the eighteenth century 
French republic anyway [0.3] 
er [0.6] rates of change [0.3] so [0.5] we've got these two rates of change [0.
4] and very important they are too [0.5] the rate of change at a fixed point in 
space [0.7] now that's the sort of thing you might say meteorologists are 
interested in 'cause we have [0.4] fixed observing sites [0.4] we're required 
to produce weather forecasts for fixed points on the earth's surface [0.3] the 
forecast for namex or whatever [1.5] and in other words [0.4] the meteorologist 
is interested in the [0.5] Eulerian [0.2] rate of change of air properties [2.
8] but in a sense [0.2] more physically fundamental is the [0.2] Lagrangian 
rate of change if i follow an individual air parcel what happens to it [0.6] 
does heat enter or leave it [0.9] does it rise or sink in the atmosphere 
changes pressure [0.6] whatever [1.8] and if you cast your mind back to [0.8] 
your work last term [0.3] and our work this term [0.9] most of the physical 
laws that we've written down [0.2] that apply to air [0.4] for example [0.3] 
the first law of thermodynamics [0.6] for example Newton's laws of motion [0.9] 
they all apply to air parcels [0.8] we ask how does an air parcel 
accelerate [0.3] how does its temperature change when you pump heat into it [0.
2] and so on [1.1] most of the physical laws [0.3] that govern the atmosphere 
[0.4] are expressed in terms of Lagrangian [0.2] rates of change [0.8] so we 
have this problem [0.7] what we measure and what we want to predict are 
Eulerian rates of change [1.3] what our physics tells us about are Lagrangian 
[0.2] rates of change [0.6] 
so [0.4] what's going to be the relationship [0.6] between [0.5] the [0.2] 
Eulerian and the Lagrangian rate of change [0.6] well it's fairly 
straightforward [0.4] that of course is the rate of change of temperature the 
Eulerian rate of change of temperature [0.4] if i assume air parcels are not 
changing [0.7] it's the rate of change in other words if capital- [0.3] D-T-by-
D-T [0.3] is zero [0.9] all that's happening is that one air parcel is moving 
away [0.3] and another one with different properties is taking its place but 
the individual air parcels aren't changing their properties [2.7] if [0.4] the 
air parcels are actually [0.4] changing their properties at the same time so 
suppose that i had this 
curve that was advecting along but at the same time all the air parcels were 
getting colder [0.4] with time so that curve was dropping down as [0.2] as it 
was moving along [1.0] then [0.3] i'd have to add into that formula [0.5] the 
Lagrangian rate of change [0.6] so [0.4] what i will write is that er [0.6] in 
general [7.0] when [2.4] there are both [2.7] Lagrangian [4.1] and Eulerian [0.
3] changes [4.8] then we can say [0.2] the rate of change at a particular [0.5] 
location [1.7] D-U-by-D-T [0.4] is going to be the Lagrangian rate of change [2.
4] capital- [0.5] D-by-D-T [0.3] minus [0.2] this advective rate of change [0.
4] minus-U times D-T-by-D-X [2.3] and that's really the central result [0.2] of 
today's lecture [4.3] it gives us the relationship between rates of change at a 
position [1.3] and [0.2] the rate of change for an individual parcel [1.2] and 
they're related [0.6] via [0.2] the velocity [0.3] and the temperature gradient 
[5.0] i've deliberately kept things terribly simple in this discussion because 
i supposed that temperature was only varying in the X direction and the wind 
was only blowing parallel to the X 
axis [0.6] it's dead easy to generalize this [0.3] if i have a general wind [0.
4] then that will have components parallel to all three axes X Y and Z [0.9] 
and of course the temperature may vary in all three directions X Y and Z as in 
general it does [0.6] so i can generalize this [0.5] for er three-dimensional 
[0.2] motion [3.0] and i won't go through the arguments of course they just 
work exactly the same way [0.9] but just write a result down [1.1] in three 
dimensions then we can say the rate of change at a fixed position the Eulerian 
rate of change [0.4] is equal to the Lagrangian rate of change [1.0] the rate 
of change [0.4] for an individual air parcel [0.4] minus well the same term [0.
8] U times D-T-by-D-X U remember now is the component of wind parallel to the X 
axis [1.2] minus [0.2] V [0.2] times D-T-by-D-Y [0.7] V is the component of 
wind parallel to the Y-axis [0.5] D-T-D-Y [0.3] is the [0.2] rate of change of 
temperature along the Y-axis [0.4] and then of course we've got the vertical 
one minus-W [0.4] times D-T-by-D-Z [0.6] it looks a bit long [0.6] but that's 
just because [0.7] we've written 
out the same terms three times yes [0.4] 
sm0928: er [0.4] er i might be [1.0] going along the totally wrong lines but 
nm0925: mm 
sm0928: can you not use the [0.3] the [0.2] er 
nm0925: we could [0.4] er [0.2] i'm not going to do that 'cause people'll find 
it so hard their brains will seize up [0.5] but w-, we could [0.6] and er i'll 
er thank you for asking that this has er this has er brought a warm [0.3] glow 
to an old man's heart [0.4] er [0.5] we could write this much more neatly [0.3] 
as D-by-D-T [0.2] minus the vector wind [0.5] U [0.5] dot [0.3] grad [0.5] 
temperature [1.1] okay [0.3] 
sm0928: yeah [0.2] 
nm0925: happy [0.2] right [0.2] write that down [1.1] er [0.5] this is notation 
that we will come to [0.2] much more compact notation [0.4] which we will come 
to [0.3] er [0.4] next year [0.2] next autumn's geophysical fluid dynamics 
course we'll redo this argument [0.5] er using vector notation and it all comes 
out a lot more compact [0.4] i'm not going to use that for this course [0.4] 
because not all of you have met this notation yet [0.4] and er [0.3] er people 
find it hard enough in the second year when they do so [1.1] we'll we'll save 
that for next year [2.9] okay [2.0] well [1.5] what i want to do now is to take 
these ideas these ideas of thermal advection of temperature 
changing at a particular place in the atmosphere [0.5] due to [0.2] advection 
[0.7] replacement of one air parcel by another [0.8] and i [0.4] i'm going to 
make an additional assumption [2.0] suppose we have a situation where the wind 
is essentially [0.3] geostrophic [1.1] that's the situation that's we've been 
er [0.4] er assuming in most of our analysis of synoptic weather charts [2.5] 
then we get [0.2] a very [0.2] useful [0.5] er [0.4] er [0.9] set of 
relationships [1.4] and the reason that we get them you see is that er [0.5] 
thermal wind relationship [0.8] if you er recall [0.4] tells us the 
relationship between the change of temperature in the horizontal things like D-
T-by-D-X and D-T-by-D-Y [0.7] and the change of the wind in the vertical [0.3] 
D-P-by-D-Z [0.2] or sorry D er sorry [0.3] D-U-by-D-P [0.4] or er [0.2] D-V-by-
D-P [1.4] so [0.2] there's an intimate relationship [0.2] between advection [0.
6] and the wind changing with height [1.3] and er [0.3] er [0.4] i want to [0.
5] er [0.9] take a look at that now [1.8] you'll remember that at the end of 
the er lecture two lectures ago when we talked about thermal wind balance [0.4] 
i introduced this concept of the thermal wind [3.4] as a 
reminder the thermal wind [1.7] which [0.2] er i'll just write its magnitude 
down for the moment i called it V-subscript-T for thermal [1.1] and [0.2] it's 
equal to G-over-F [1.1] times the gradient of the thickness so it's D-by-D-Y [1.
0] of [0.5] Z at some high level minus-Z-one at some lower level [1.3] 
typically Z-two would be the fifty kilopascal [0.7] er [0.4] pressures at the 
height of the [0.3] fifty kilopascal surface [0.3] Z-one would be the height of 
the hundred kilopascal [0.5] pressure surface [2.8] so [0.3] we could plot it 
on a chart [0.5] er it bears just the same relationship to lines of thickness 
as the geostrophic wind vector bears to [0.3] lines of constant pressure or 
geopotential height [4.6] and [0.5] we can use these ideas now to [0.2] discuss 
a number of [0.7] typical cases of the wind varying with height [0.6] and i'm 
going to give you three examples and you'll be able to look later on this 
morning at your charts and spot [0.5] what's happening [2.2] let's take the 
simplest case first of all [0.4] where the wind speed [0.2] changes with height 
but the wind direction does not [0.8] so we have a 
geostrophic wind near the surface which is in a particular direction [0.5] the 
wind at say five-hundred millibars is in the same direction but stronger [1.0] 
not an uncommon situation [1.0] let's do a little drawing of what's going on [2.
6] er let me draw thickness contours [0.5] in [1.9] red [1.3] and let me [0.4] 
draw er s-, the er [0.4] er height contours at the lower level [2.2] and [0.7] 
i'll draw them looking like that so that's shall we say the hundred kilopascal 
[2.1] Z [1.4] and the red contours [0.3] show the thickness lines [1.5] so 
that's the [0.5] hundred to fifty kilopascal [1.4] thickness [1.4] if you look 
on the charts in the corridor you'll see that's what we actually plot [0.4] 
people [0.3] tend to [0.2] put the surface pressure say [0.3] but with lines of 
constant thickness on as well [0.8] now in this case the surface geostrophic 
wind [1.4] looks like that [0.6] er i'm going to call that [0.3] V-one [3.8] 
and the thermal wind well remember that's parallel to lines of constant 
thickness [2.3] and inversely proportional to their spacing so [0.9] there's 
going to be [0.3] V-two [0.2] in the same direction [1.8] and if i add the 
two together [0.3] i'll get the wind at the [0.3] second height at er the five-
hundred millibar level [0.5] so the wind at the five-hundred millibar level [0.
3] is just going to be the sum of those two [0.5] it's going to look like that 
[3.9] notice [0.8] that both those wind vectors V-one and V-two are parallel [0.
9] to the thickness contours [0.8] thickness remember is proportional to 
temperature [0.4] that means [0.3] that there are no variations of temperature 
[0.3] in the direction the wind is blowing [1.3] so although the air parcel's 
moving past we're not replacing it by an air parcel with a different 
temperature we're replacing the air parcel [0.4] with one of the same 
temperature [0.7] in other words in this situation [0.6] there is no [0.2] 
thermal advection [0.6] the 
temperature [0.2] change [0.4] i-, [0.2] is not due to replacing air parcels 
'cause we're just replacing one air parcel with another one [0.2] of the same 
temperature [1.3] so we have no [1.1] thermal [0.3] advection [6.3] i have to 
say that's not an uncommon situation but it's a [0.2] boring situation [0.7] 
let's [0.2] consider a much 
more exciting and interesting situation [4.7] what we've had over this weekend 
is a situation which is called cold advection [1.0] that is to say warm air [0.
3] has been displaced by cold air blowing down from the north [0.9] let's have 
a look at what that implies in terms of [0.3] the height and thickness [2.1] 
let me again draw the thousand millibar height [0.2] contours looking like that 
[0.3] and V-one is going to be the same as in the previous diagram i really 
ought to underline these Vs 'cause i i i w-, i ought to stress that i'm talking 
about vectors here [0.4] the direction is important as well as the magnitude [0.
6] so [0.9] there's V-one [0.2] that's the surface geostrophic wind but now 
this time [0.5] i'm going to suppose [0.6] that the [0.2] er height contours [0.
4] are no longer [0.2] is the sorry the thickness contours are no longer 
parallel [0.2] to these height contours [0.4] so i'm going to draw my [0.3] 
thickness contours [0.7] looking like that [1.4] and i'm going to suppose it's 
colder [0.3] here [0.5] so that's low values of thickness [0.4] and it's warmer 
here [0.3] so we have [0.3] large values of 
thickness here [1.5] in this case [0.7] the thermal wind vector [0.7] is going 
to be [0.6] parallel to the thickness contours [1.1] and with [0.2] cold air on 
the left so the thermal wind vector [0.4] is going to [0.2] look like that [2.
3] okay [1.8] now [0.2] what about the wind at five-hundred millibars [1.0] 
well it's simply going to be the vector sum [0.4] of the low level wind [0.5] 
plus the thermal wind [0.7] so if i er [0.5] construct that using my usual [0.
4] er [0.4] triangle [0.8] construction [3.8] i will find that V-two will look 
like that [2.4] very interesting isn't it [0.4] what you will see is there [0.
3] that as you go up from the surface [0.4] to the upper layer [1.5] the wind 
vector [0.4] turns [2.5] it turns in an anticlockwise sense [0.7] and we 
describe this [0.4] as the wind [1.7] backing with height [4.7] and this is a 
situation where we have cold advection [7.7] that's the sort of situation we 
might look out for this weekend and if i can get hold of some charts [0.5] for 
the weekend [0.2] later on we'll we'll we'll take a look at that [4.0] let me 
finish [0.6] by [0.2] taking the opposite case i hope you can guess what's 
going to happen now [0.3] we're now going 
to consider warm advection [8.1] once again i'll put my low level height 
contours in the same direction [0.4] my V-one [0.3] same direction and same 
strength [1.5] that's my surface geostrophic wind and my near-surface 
geostrophic wind [0.6] this time [0.3] i'm going to tip [0.3] the [0.4] hei-, 
thickness contours in the opposite [0.3] sense [0.4] so this time i'm going to 
draw thickness contours that look like this with the warm air down here [1.3] 
cold air [1.3] here [1.5] and [0.7] again the [0.3] thermal wind vector is 
parallel to the thickness contours [0.4] cold air on the left [0.7] so [0.2] 
it's going to look like that [0.9] that's my thermal wind vector [1.7] let's 
get the winds now at the upper level by doing the vector sum [0.2] of the low 
level wind and the thermal wind [1.0] usual parallelogram construction [1.5] 
like so [1.4] that's V-two [4.0] this time [0.2] you will see [0.2] that the 
wind [0.2] has turned in a clockwise sense [0.3] with height [2.2] and so warm 
advection [0.4] is associated with the wind [1.5] veering [3.8] with height [4.
4] i find this very fascinating [0.4] it means that if i have a single 
radiosonde ascent that returns the the the winds [0.5] 
by looking at the way the wind vector is changing [0.4] as the er balloon 
ascends i can actually say something about the spatial distribution of the 
temperature [0.6] can work out what direction the thickness contours [0.3] must 
have [0.5] at that station [7.8] gosh i've drawn all over my lecture notes in 
my excitement [1.0] right [1.3] er you'll have a chance to see these ideas in 
practice as we go through the present case study we'll be looking at the winds 
at different levels we'll be analysing [0.4] the thickness [0.4] and if you all 
say i don't know how to analyse thickness well you've got a treat in store 
because [0.4] er [0.3] er namex's going to tell you all about that in in a 
couple of minutes [0.6] er [0.2] so [0.9] that's the next stage in our 
practicals [0.2] i'm going to hand over er to namex in a moment [0.4] er i just 
want to make one announcement the er [0.2] er the lads down in the lab have 
fixed the radiosonde system [0.3] they have soldered the wire [0.3] back onto 
its connector [laughter] inside the er receiver [0.4] and i'm assured it works 
[0.3] so [0.2] we are 
hoping [0.6] when namex's finished [0.3] that we will have a a second launch 
today [0.3] i don't think we all need to troop down to see the equipment 'cause 
you saw it last time but we will do a launch [0.4] and er we'll try and er plot 
an ascent [0.4] during the course of the morning [0.3] so [0.2] er if the 
people who wanted to volunteer to [0.3] write down some numbers can remember 
who they were [0.3] arms jerking there [0.3] er [0.2] then er [0.2] we'll we'll 
arrange a rota for you to go down [0.2] i i'll let namex finish and then we'll 
sort out the timing [0.6] okay [3.2] namex you've got to be wired for sound 
today