The Weather Equation - Numberphile

[00:00] And this is one of the most famous

[00:01] equations in meteorology. I'm just going

[00:03] to go ahead and and write it down. So we

[00:05] have sigma d^ 2 omega. So where the

[00:08] omega comes from plus f 2 d2 omega by

[00:13] dp^ 2. So this is the omega part and

[00:16] that equals f * the vertical variation

[00:19] of this term here which is the advection

[00:22] of the absolute geostrophic vorticity by

[00:24] the geostrophic wind and the advection

[00:27] of geostrophic wind the temperature

[00:29] gradient.

[00:30] >> So this is this is the qua geostrophic

[00:32] omega equation. So this is a fundamental

[00:35] equation that allows us to through

[00:39] systematic simplifications a lot of

[00:40] simplification actually get the vertical

[00:42] velocity from the adction of this

[00:45] quantity called vorticity and the

[00:47] adction of this quantity called

[00:49] temperature. So thermal adction,

[00:51] vorticity adection. And by looking at

[00:53] horizontal maps of these adections, we

[00:55] can diagnose vertical velocity and

[00:57] vertical velocity is a proxy for

[00:59] development. And development is your

[01:02] developments of your lows, your

[01:03] developments of your highs, high

[01:04] pressure and low pressure. And that

[01:06] relates to the weather forecast.

[01:07] >> So what this tells you what's going to

[01:10] happen in the future?

[01:12] >> No. So that's quite an important

[01:14] distinction about this equation. This

[01:15] does this tells you nothing about the

[01:17] future state of the atmosphere. So this

[01:19] isn't a predictive equation. It's it's

[01:21] it's called a prognostic equation. This

[01:23] is a diagnostic equation. It just allows

[01:26] us to diagnose the vertical velocity

[01:28] from the invection of vorticity and the

[01:30] invection of temperature.

[01:31] >> And why is that an important thing to

[01:33] know?

[01:34] >> Um

[01:35] >> if it doesn't tell you anything about

[01:36] the future. Well, so this I mean we we

[01:39] can go into this in a bit more detail uh

[01:41] later on, but this um essentially is one

[01:44] equation that falls out of a whole

[01:46] system called the quai geostrophic

[01:48] system. And I want to kind of go into a

[01:50] bit about what we mean by quai

[01:53] geostrophic omega. But essentially the

[01:55] quai geostrophic system is a systematic

[01:58] set of simplifications to the main

[02:00] primitive equations which numerical

[02:02] models these days use to forecast the

[02:04] weather. um is a a systematic

[02:06] simplification of those equations that

[02:08] could then be used to make the first

[02:10] kind of workable forecast models that

[02:13] were used on the computers back in the

[02:15] day the the early 1950s. And so this

[02:18] equation is a diagnostic equation that

[02:20] falls out of those and it is part of

[02:22] that forecast process whereby you would

[02:25] calculate some tendencies of various

[02:26] things. You would then get the wind

[02:28] fields which you could calculate wind

[02:30] and thermal fields which you could

[02:31] calculate the vorticity and the thermal

[02:32] invection and from that you could

[02:34] diagnose the vertical velocity and

[02:36] getting to this vertical velocity is

[02:37] actually quite difficult and so this

[02:39] equation gives us a method to to

[02:41] actually do that in a systematic and

[02:45] relatively error-free way

[02:46] >> in even more simple terms then this equ

[02:48] is this equation relating to up and

[02:50] downness of the air what's it like

[02:52] what's what's it

[02:53] >> exactly so when when we're talking about

[02:55] vertical velocity We're talking about

[02:57] the motion of the air in an up and down

[02:59] sense. So on on large scales, so the

[03:02] scales of weather systems, the main the

[03:04] major motion of the atmosphere is in the

[03:06] horizontal. So you've got winds north to

[03:08] south, winds east to west. We don't tend

[03:10] to think on large scales about the winds

[03:13] that are going up and down, but they are

[03:14] there. If you think about a convective

[03:16] cloud, for example, they're moving up

[03:18] into clouds, connects into clouds, and

[03:19] you get rain falling out. Um, but on

[03:21] these large scale, so we we're talking

[03:23] about big scale weather systems here.

[03:25] You know, systems the size of the UK,

[03:27] systems that that that sort of take up a

[03:29] big proportion of the Atlantic.

[03:31] You don't tend to think about vertical

[03:33] motions there, but they are there and

[03:35] they're necessary in order to develop

[03:37] the areas of low pressure and develop

[03:38] the areas of high pressure. So if you

[03:40] have rising motion, you get development

[03:42] of low pressure at the surface. And if

[03:44] you have sinking motion, then you have

[03:45] development of high pressure at the

[03:47] surface. One more basic weather question

[03:50] before we, you know, get our teeth into

[03:52] this. Then I'm pretty familiar with

[03:54] measuring wind speeds horizontally. I've

[03:56] seen those little devices like those

[03:58] little cups that spin around and things

[04:00] like that. Are there devices that

[04:02] measure the up downwind, the vertical

[04:04] wind? Like how do you how do you guys

[04:06] take measurements of that?

[04:07] >> No, it's very difficult. And that's one

[04:09] of the reasons why we kind of look to

[04:11] equations to to diagnose the vertical

[04:14] velocity. It's very hard to measure.

[04:17] it's on a much much much smaller scale

[04:19] than the the horizontal motions. And in

[04:21] fact, you can try and calculate the um

[04:26] vertical motion from the horizontal

[04:28] motions via something called the

[04:29] continuity equation, which we'll have a

[04:31] look at in a little while. Um but that

[04:33] relies on you knowing to a very high

[04:35] degree of accuracy your horizontal

[04:36] winds. And it turns out that the the

[04:38] divergence and the vertical motion

[04:40] errors in that are the same size as the

[04:43] actual vertical motion you're trying to

[04:44] diagnose in the first place. So you can

[04:46] have small areas in the wind that can

[04:47] lead to massive areas in vertical

[04:48] velocity.

[04:49] >> Just quickly, what are the things that

[04:50] measure the wind speed, Cole?

[04:52] >> Anim

[04:53] >> an animometers.

[04:54] >> Animasure

[04:56] wind speed horizontally.

[04:58] >> Yes.

[04:58] >> Are you saying equations the likes of

[05:00] these are kind of what you have instead

[05:03] of those for vertical.

[05:04] >> That's how you get to the vertical

[05:06] velocity.

[05:06] >> Yeah.

[05:06] >> So let's break it down. So um first of

[05:08] all, it's the omega equation. So let's

[05:10] talk about omega. So we got this Greek

[05:12] letter omega here. So omega is just the

[05:14] vertical velocity. So it has a

[05:16] equivalent called W and these two are

[05:19] equivalent but they use different

[05:20] coordinate systems. So for example, if

[05:22] we think about W, this might be a little

[05:23] bit more familiar. We think about a

[05:25] coordinate system that has winds in the

[05:28] XY. So this is like your XY plane and

[05:31] then you've got a vertical plane. If you

[05:32] have some vertical motion or if you have

[05:35] some horizontal motion, we could have a

[05:36] a U wind in the X direction, a V wind in

[05:40] the Y direction. So these are our

[05:41] horizontal winds and then w would be um

[05:45] the wind in the vertical and that would

[05:47] be measured in meters per second. There

[05:49] is a direct equivalent though uh in a

[05:51] different coordinate system which we in

[05:53] meteorology that we like to use because

[05:55] it has some good advantages when coming

[05:58] to equations like this and looking at

[05:59] things like mass continuity. We still

[06:01] have an xy plane but we use pressure as

[06:04] our vertical coordinate. And so if we

[06:06] have vertical motion in the pressure

[06:09] coordinate and we can use pressure as a

[06:11] coordinate because pressure always

[06:13] decreases with height. So it's around

[06:15] about a th00and millibars at the surface

[06:16] 1,000 hector pascals. At the top of the

[06:19] troposphere which is the top of the

[06:20] weather bearing layer is around about

[06:22] 300 hector pascals. So pressure always

[06:24] falls with height. So if you're moving

[06:26] in an upward direction you're moving

[06:29] towards lower pressure. Ascent in the

[06:31] the geometric coordinate system would be

[06:33] W. ascent in the pressure coordinate

[06:35] system would be negative omega because

[06:38] you're going towards lower pressure as

[06:40] you go up whereas in the geometric

[06:42] system you're going towards kind of

[06:44] higher elevations as you go up.

[06:46] >> So does that mean we're using on our x

[06:48] and our y axis here we're using

[06:50] different units because that's

[06:52] presumably still meters/s.

[06:54] That's right. Yep. So this we still have

[06:56] a U wind and a vwind in meters per

[06:58] second

[06:59] >> but we're using a different unit on

[07:00] that.

[07:00] >> Yep. The unit for this one would be

[07:02] pascals per second. It's it's the amount

[07:05] of pressure change that this parcel of

[07:06] air is experiencing per second. So very

[07:08] much like your your speed would be the

[07:11] amount of displacement you're

[07:12] experiencing per second, meters/s. This

[07:14] would be the the pressure change per

[07:16] second, pascals per second. Okay, cool.

[07:19] You can you can link the two uh via

[07:21] something called the hydrostatic

[07:22] equation. So we have this hydrostatic

[07:25] equation which relates the rate of

[07:27] change of pressure in the z direction

[07:29] with this term minus row which is the

[07:32] density and g which is gravity. You can

[07:35] say that um if you treat is the material

[07:38] derivative which you can expand out to

[07:41] this expression here. And um we're going

[07:43] to assume something called a hydrostatic

[07:45] balance. Um, and what hydrostatic

[07:47] balance effectively means is that if we

[07:49] have an atmosphere that's at rest and we

[07:52] have a particle or a parcel of air in

[07:54] the atmosphere, it's got two forces

[07:55] acting upon it. We've already said that

[07:57] pressure decreases with height. So,

[07:58] we've got a pressure gradient in this

[08:00] direction. So, this parcel has

[08:01] experienced a pressure gradient force in

[08:03] that direction. And then we've got a

[08:05] gravitational force in that direction.

[08:07] So, we've got pressure gradient force

[08:09] and gravity. And when these two are in

[08:11] balance, you know, if if there wasn't

[08:13] gravity and there was a pressure

[08:14] gradient force, then this parcel of air

[08:15] would fly up into the atmosphere because

[08:17] it'd be the the higher pre moving from

[08:19] higher pressure to lower pressure and be

[08:20] pushing it up to into the atmosphere.

[08:22] But because it's balanced by gravity,

[08:24] the the particle or the air parcel is at

[08:27] rest and we call this hydrostatic

[08:28] balance. So we assume the atmosphere is

[08:30] at rest. There's no U, there's no V,

[08:31] there's no pressure change with time. So

[08:33] we can say that uh omega is

[08:36] approximately equal to W DP by DZ. And

[08:39] then from this hydrostatic balance

[08:41] relationship which we kind of expanded

[08:43] out here we can say that omega be equal

[08:47] to minus row g which is this term here

[08:51] times w. So what this effectively tells

[08:54] us is that uh omega and w are related

[08:57] but omega is just a scaled version

[09:00] scaled by the density because because

[09:03] the atmosphere is more dense at the

[09:05] surface. So if you have a parcel of air

[09:07] that's moving vertically at fairly low

[09:09] elevations, the pressure is quite high.

[09:12] Because the atmosphere is more dense

[09:14] here, the pressure levels are kind of

[09:16] closer together. And therefore the same

[09:19] vertical speed in height coordinates at

[09:22] lower levels would be crossing more. It

[09:24] be the pressure will be changing more at

[09:26] lower levels than it is at higher levels

[09:28] for the same vertical speed when you're

[09:30] thinking about the change in meters. So

[09:32] this is what this equation tells us.

[09:33] Omega is just a scaled version of the um

[09:37] vertical velocity in geometric height

[09:39] coordinates, but it's scaled by the

[09:41] density to account for this effect.

[09:43] >> Want another piece of paper?

[09:44] >> Yeah, I think so.

[09:45] >> All right.

[09:55] >> The main takeaway about omega is it's

[09:56] the vertical velocity in a pressure

[09:58] coordinate system. Okay. So, there's a

[10:00] few other symbols that we uh we want to

[10:02] look at here. Yeah.

[10:03] >> So again we'll just concentrate on the

[10:05] left hand side at the moment. Right. So

[10:07] this is sigma. Sigma scales the vertical

[10:10] velocity response for a certain size of

[10:12] forcing. This all taken together gives

[10:14] the three-dimensional distribution of

[10:17] omega or the distribution of the

[10:19] curvature of omega. So it doesn't

[10:20] necessarily give us omega itself yet but

[10:22] we'll see how we can retrieve omega from

[10:24] this side. But it tells us something

[10:26] about the threedimensional distribution

[10:27] of omega. Now these terms on the right

[10:29] hand side let's break this down. So this

[10:31] this is a za the G means geostrophic.

[10:34] And I want to delve a little bit more

[10:36] into what geostrophic means in a second.

[10:38] But this Z to G plus F this tells us the

[10:41] absolute vorticity. Vorticity is

[10:43] effectively a measure of the spin in the

[10:46] atmosphere. Any atmospheric process

[10:47] where you've got things spinning around

[10:49] any fluid spinning you have vorticity.

[10:52] And these two different types of spin

[10:54] that we've got make that make up the

[10:55] absolute vorticity are the relative

[10:58] vorticity. So relativeity is just the

[11:01] spin. If you put a paddle wheel in the

[11:03] flow, say you've got some flow coming

[11:05] down here and you've got a lesser flow

[11:07] here and you put a little paddle wheel

[11:09] in the flow. Well, there's more pressure

[11:11] on in the flow on this part of the

[11:13] paddle wheel than there is on this part.

[11:15] So this paddle wheel is going to start

[11:16] to spin. So the fluid at this point

[11:19] would spin in that direction. And that's

[11:21] the vorticity. Now, this would be a

[11:22] sheer vorticity because the we've got a

[11:25] shear. Um, fluid passes are moving

[11:28] faster here than they are here. You

[11:29] could have a curvature vorticity where

[11:31] simply the flow is curved and so if you

[11:34] put a stick in a flow here, it would

[11:36] curve. And so again, you've got

[11:38] vorticity through through curvature. So

[11:40] vorticity or spin can come out through

[11:42] through different mechanisms. Uh, Z to G

[11:44] here just tells us about the vorticity

[11:47] of the flow. Now this f this is the

[11:49] planetary vorticity and this is just

[11:51] vorticity that you have even you have it

[11:53] Brady just by virtue of being on a

[11:56] spinning earth. So anything that's on

[11:58] the earth apart from on the equator has

[12:00] some kind of spin about its local

[12:02] vertical. Yeah. If you think about the

[12:03] globe it's spinning on its axis. So if

[12:06] you were at the north pole you would be

[12:07] experiencing that full planetary

[12:09] vorticity the full spin of the earth.

[12:11] You wouldn't be able you wouldn't know

[12:12] it was there because you're kind of

[12:14] spinning along with it. everything that

[12:15] you can see is spinning along with it.

[12:17] So you wouldn't know it's there, but it

[12:19] is. It's spinning. And you can pick any

[12:21] point on the Earth's axis and you can

[12:23] decompose its spin um into a component

[12:26] about its local vertical. Um and the

[12:29] equation for that F is equal to 2 *

[12:32] another omega but this is the big omega

[12:34] s of the latitude. We calculate what our

[12:37] planetary vorticity here um at the Met

[12:40] headquarters is in exit. If we take this

[12:42] equation here. So f which is our

[12:44] corololis parameter is equal to 2 *

[12:47] omega. Now omega from memory is

[12:49] something like 7.292

[12:52] * 10 - 5. This is in radians/s. So this

[12:55] is just the the number of radians that

[12:57] the earth is turning per second.

[12:59] >> So that's a constant.

[13:00] >> This is a constant. This is a constant.

[13:02] The only thing that varies is the

[13:04] latitude. And we take sign of the

[13:06] latitude. Um I don't know what it is in

[13:08] radians unfortunately. But because we're

[13:09] taking a sign of it, I don't think it

[13:11] matters too much. Uh sin 50°. And that

[13:14] turns out to be, if you go through the

[13:16] calculation, it's approximately uh 1.1 *

[13:20] 10 -4 per second. So that's the

[13:24] planetary vorticity. So go back to the

[13:26] equation. The relative vorticity of flow

[13:28] plus the planetary vorticity, that's the

[13:31] absolute vorticity. But this little

[13:34] quantity here, this little dell symbol

[13:36] means that we take the gradients of

[13:38] that. Gradient means that the absolute

[13:40] velocity is changing in space. We take

[13:42] the gradient of that and we do a dot

[13:45] product with the uh wind velocity and

[13:48] that gives us the advection of the

[13:50] gradient. So it's how the wind is

[13:52] blowing along the vorticity gradient

[13:53] kind of tells us you can imagine that's

[13:56] the advection of the vorticity. If

[13:57] you're going from an area of high

[13:58] vorticity to low vorticity, you're

[14:01] blowing the high volticity towards you

[14:03] and therefore you got positive vorticity

[14:05] in vection.

[14:06] >> I'm beginning to get some appreciation

[14:07] as to why weather is complicated. Well,

[14:10] I that's a really good point. And and

[14:12] when when people say when people say,

[14:13] "Oh, weather forecasting is easy, isn't

[14:15] it?" I just look out the window. Then my

[14:17] temptation is to point them to stuff

[14:18] like this and say, "Well, actually,

[14:20] there's a lot more to it than that." Um,

[14:21] so this is the vorticction part, and

[14:23] we'll come on to kind of what that

[14:24] physically means in a little while. Um

[14:26] this is the temperature vection part. So

[14:27] very much like we had the gradient of

[14:30] temperature. So this is a bit more

[14:31] simple. You can just think of this as

[14:33] temperature. We we but we've got the

[14:34] temperature gradient and again we're

[14:36] vecting that with the geostrophic wind.

[14:39] And these features in in front of all of

[14:41] this well these are this is just the gas

[14:43] constant. This is pressure. This ter

[14:45] this little expression here means not

[14:47] only are we taking the gradient of

[14:49] absolute vorticity and advecting it with

[14:51] a geostrophic wind. We're actually

[14:53] taking the vertical variation of that

[14:55] which is what this uh d by dp term tells

[14:58] us and then it's scaled by the coralous

[15:00] parameter f.

[15:01] >> Even you know this is ridiculous.

[15:03] >> Even you know this is ridiculous. All

[15:05] right.

[15:05] >> Yeah. So before we get to how we use

[15:08] this equation as weather forecasters and

[15:10] believe it or not given all the

[15:12] complexities in a conceptual way, we

[15:14] still do use this equation which is um

[15:18] fascinating to me because if you think

[15:20] about the quai gestroic equation set

[15:22] which was derived to make the first

[15:25] workable numerical models. Well, pretty

[15:27] much all of the rest of that system has

[15:31] not fall well, it has fallen out of use

[15:33] as models have got better, super got

[15:35] faster, but the one kind of enduring

[15:37] thing that we use as forecasters is this

[15:40] QG equation, which is why it's such a

[15:42] famous equation of meteorology. Um, and

[15:45] it's why it's so interesting to me. But

[15:47] we're we're a couple of steps away, I

[15:48] think, from kind of understanding what

[15:51] this physically means. And I think we

[15:53] need to go on to talk about geostrophe

[15:56] and what geostrophic means and then we

[15:58] can talk about what quai geostrophic

[16:00] means. So should we have another piece

[16:02] of paper?

[16:02] >> Yes we shall.

[16:12] >> We've talked about what omega means. Uh

[16:15] I just want to spend a little bit of

[16:16] time about talking what geostrophic

[16:18] means and then by extension we can go to

[16:19] quai geostrophic. We can go back to my

[16:22] lovely rendition of the globe here. We

[16:24] can imagine a weather system on this.

[16:26] And you this might be sort of quite

[16:28] familiar if you imagine this might be

[16:29] like a typical weather map that you

[16:31] would see with an area of low pressure.

[16:32] And when you see the forecasts on the TV

[16:34] and they show the the pressure maps,

[16:36] they often show the winds. If you look

[16:38] at how the winds are blowing, you find

[16:39] that one, they blow counterclockwise

[16:42] around an area over low pressure, and

[16:44] two, they tend to blow parallel to the

[16:46] isobars, which is quite surprising if

[16:48] you think about it because if you think

[16:50] you've got a region of low pressure,

[16:51] that's a bit like a vacuum cleaner. And

[16:53] if you think about what happens when you

[16:55] turn on a vacuum cleaner, it sucks up

[16:56] all the dust and grime and sucks up the

[16:58] tube and it's creating an area of low

[17:00] pressure and the wind is effectively

[17:01] blowing in towards that vacuum. So why

[17:04] doesn't it happen um on our planet? Uh

[17:08] and the reason for this is because um as

[17:10] well as the pressure gradient force

[17:12] there's another force that we have to

[17:13] take into consideration. So if we

[17:15] simplify and we think of our air parcel

[17:18] moving this is like the pressure

[17:20] gradients of the same as on the map and

[17:21] we'll just imagine we've got an area of

[17:23] high pressure here and an area of low

[17:25] pressure here. Now at the moment we

[17:27] don't really know anything about the

[17:28] direction of this but um it it will be

[17:30] blowing in this direction. Um it's B's

[17:33] ballot law actually which says if the

[17:35] wind is in your face so if you were

[17:36] standing here Brady and the wind's in

[17:37] your face uh you would have low pressure

[17:41] on your right

[17:41] >> in both hemispheres.

[17:43] >> Uh I have to think about that one. Um so

[17:46] in the southern hemisphere the wind does

[17:48] blow clockwise around an area of low

[17:49] pressure. So if you were standing if you

[17:53] face the wind it would be to your left.

[17:55] Yes. It would be the opposite.

[17:56] >> That's right.

[17:57] >> Not that we're being hemisphere. Not

[17:59] that we're not we're discriminating

[18:01] against either hemisphere. We we're

[18:02] hemisphere agnostic.

[18:04] >> So we would have a pressure gradient

[18:05] force in this direction and if that was

[18:07] the only force acting then the air

[18:10] parcel instead of going in this

[18:11] direction would go towards the low

[18:12] pressure. But it turns out that's

[18:14] exactly balanced by this other force and

[18:16] this is the corololis force. This is

[18:19] well it's a little bit controversial as

[18:21] to what you describe it. Some people

[18:22] describe them as fictitious forces. Some

[18:25] people will describe them as forces that

[18:28] you have to include in the equations of

[18:31] motion in order to make it appear as if

[18:34] we're not on a rotating planet because

[18:37] although we are on a rotating planet, we

[18:40] can't really judge that. But the motion,

[18:42] the fluid motion, the motion of the air

[18:44] parcels knows we're on a rotating

[18:46] planet. And so you see some odd

[18:47] behaviors. And one of the odd behaviors

[18:48] is if you have an air parcel that starts

[18:51] to move from high pressure to low

[18:53] pressure, it will feel this corololis

[18:55] force acting and pulling it to the

[18:56] right. And so as it moves, it will

[18:58] continue to feel this force until

[19:00] eventually and the pressure gradient

[19:01] force is always acting in this

[19:03] direction. And it will continue to be

[19:04] pulled to the right until the corololis

[19:06] force and the pressure gradient force

[19:07] balance. And then we end up with this

[19:09] what we call geostrophic flow. So

[19:11] geostrophic just basically means there's

[19:13] a balance between the corololis force

[19:14] and the pressure gradient force. That's

[19:16] not fictitious.

[19:18] >> Well, the pressure gradient force isn't

[19:19] fictitious, but the corololis force is

[19:22] kind of fictitious because it's a it's a

[19:24] correction that we have to add to

[19:27] account for what we see. Corololis is a

[19:30] really interesting character and one of

[19:32] the roads around the Met Office is named

[19:33] after him. It's of such fundamental

[19:35] importance to weather prediction. We can

[19:38] look at the equations of motion. Um, so

[19:41] I'm going to write down the equations of

[19:42] motion. Now you you've done the Navy

[19:43] Stokes equations. These equations of

[19:45] motion are sort of analogous to the to

[19:48] those. And if we take the equation of

[19:50] motion in the x direction. So this

[19:51] relates the acceleration of the u wind.

[19:54] So if you remember going back to our

[19:56] coordinates, we had a u wind in the x

[19:59] direction. So how the u wind is changing

[20:01] with time is the acceleration in the u

[20:03] direction and that's equal to this term

[20:05] here which is the pressure gradient

[20:07] force in the x direction. This is the

[20:10] corololis force. And then we've got some

[20:12] frictional terms. And this basically

[20:14] boils down to F= ma Newton's second law.

[20:17] So we've got our forces on one side got

[20:19] our acceleration on the other side. If

[20:21] we apply it to a parcel with mass one

[20:23] unit mass then we end up with basically

[20:25] A= F F= A. So this is Newton's second

[20:28] law in a nutshell and I'll write just

[20:29] quickly write out the one for the V

[20:31] because it follows a very similar

[20:32] pattern. So there's the pressure

[20:33] gradient force in the Y direction. Corus

[20:35] term is minus FU and then plus some kind

[20:38] of frictional forces. When we looking at

[20:40] geostrophic balance, um, we are assuming

[20:44] first of all, we're assuming there's no

[20:45] friction. So, we can't really apply

[20:46] geostrophic balance at the surface. And

[20:49] that is why when you see winds blowing

[20:52] around an area of low pressure and you

[20:54] look at the surface winds, they're not

[20:56] quite parallel to the isobars. They're

[20:58] actually just pointed in slightly

[20:59] towards the low pressure. That's because

[21:01] this frictional force is having a drag

[21:02] effect that's pulling the winds in

[21:04] towards the low pressure. So the

[21:06] geostrophic approximation is is a good

[21:08] first approximation for calculating the

[21:10] winds but it's it is an approximation um

[21:13] partly because of these frictional

[21:14] effects but that's dragging along the

[21:16] ground.

[21:17] >> It's dra exactly that. Yeah it's

[21:19] experiencing a stress against the

[21:20] ground. It's losing it's losing

[21:22] momentum. It's falling slightly down the

[21:25] pressure gradient um rather than just

[21:27] going around it and therefore eventually

[21:29] the winds all kind of curling into the

[21:31] the center of the low. And if there was

[21:32] no process above the load to kind of get

[21:34] rid of air or mass, then that low

[21:37] pressure would just gradually fill up

[21:38] and would become not a low pressure

[21:40] anymore. So we can get rid of these

[21:41] frictional forces because they're in the

[21:43] geostrophic system, we consider them

[21:44] negligible. And also we need to have a

[21:47] non-acelerating atmosphere. So an

[21:49] acceleration could be a speed up or a

[21:51] slow down or it could be a curve because

[21:54] anything that's moving in a curve is

[21:55] being accelerated towards the center of

[21:57] that curve. Anytime you introduce an

[22:00] acceleration and change the strength of

[22:02] the wind, you change this corololis

[22:04] force term because corololis term is a

[22:08] function of the wind speed itself. Is

[22:10] the sun pumping a whole lot of heat into

[22:13] the system not going to cause things to

[22:15] speed up or accelerate?

[22:16] >> So at the moment we're kind of we're

[22:18] neglecting that we're the sun will have

[22:22] uh an impact on smaller scales. So think

[22:25] of that convective cloud we talked about

[22:26] that cumulus cloud. Sun heats the ground

[22:28] and introduces all kinds of motion. But

[22:30] the the these the scales that we're

[22:32] talking about, these very large scales,

[22:34] the the direct heating input from the

[22:37] sun um is

[22:42] not so much of an effect. It obviously

[22:44] has an effect. It creates the conditions

[22:46] necessary to drive weather. Um but it's

[22:48] not incorporated in this simplified

[22:50] system. So yeah, anytime you introduce

[22:52] an acceleration um you will disrupt this

[22:55] force balance because the corololis

[22:58] force is a function of the wind. So if

[23:00] you change the wind, you change the

[23:01] corololis force. These forces are no

[23:03] longer in balance and therefore the flow

[23:06] wouldn't be geostrophic. So basically we

[23:08] just ignore the accelerations as well.

[23:10] And this gives us our geostrophic

[23:11] balance. So it essentially says this

[23:14] pressure gradient force and this

[23:15] coralous force balance. And I can just

[23:17] explicitly show that by taking the

[23:19] pressure gradient force over to the left

[23:21] hand side of the x equation. And

[23:24] similarly to the y equation and then we

[23:26] can divide both sides by f. And this

[23:28] gives us an expression for our

[23:30] geostrophic wind on that diagram of the

[23:32] earth. When I said you know you look at

[23:33] the you look at the winds on the

[23:35] forecast map to a first order

[23:37] approximation or to a good order

[23:38] approximation. The winds are

[23:40] geostrophic. They blow parallel to the

[23:41] isobars and you can calculate those

[23:44] winds. So we're at a point now whereas

[23:47] remember when you said um can you not

[23:49] measure the the vertical velocity? Can

[23:52] you not measure it? Well, we can measure

[23:54] the horizontal winds but the problem and

[23:55] calculate the vertical velocity from

[23:57] that. But if you remember the problem

[23:59] from that is that we we have slight

[24:01] errors in our wind measurements that

[24:02] measurement the wind measurements are

[24:04] quite sparse. So we don't know what's

[24:05] happening in between and the errors

[24:07] introduced by those measurements dwarf

[24:10] the vertical velocity itself. But we've

[24:12] got a situation here where we could

[24:14] actually if we just know the pressure

[24:15] gradients and the corololis parameter

[24:17] which we can calculate we can actually

[24:19] calculate the winds. So we don't need to

[24:22] measure them anymore. We can calculate

[24:23] them exactly.

[24:23] >> Where do you get the pressure gradients

[24:24] from?

[24:25] >> So measurement of pressure at the

[24:26] surface you would just use an instrument

[24:28] called a barometer. So that's the that's

[24:30] the pressure kind of equivalent of an

[24:32] anometer for the wind. So we've got a

[24:34] barometer and an animometer. Measuring

[24:36] pressure through depth. Um that's a

[24:39] slightly different story. We can cover

[24:40] that another time.

[24:42] One of the reasons I wanted to just

[24:44] describe geostrophic wind is because I

[24:48] wanted to show you that you can a

[24:52] connect the vertical motion to a

[24:55] quantity which we call divergence but b

[24:59] the geostrophic wind can't necessarily

[25:01] help us with this right so I want to

[25:02] take the um equation for mass continuity

[25:05] du by dx plus dv by dy plus d omega by

[25:13] dp equals not. So you've heard of the

[25:16] principle of conservation of mass. This

[25:19] is the meteorological equivalent of

[25:21] this. And what it tells us is that if we

[25:23] rearrange this, we say that the rate of

[25:27] change of the u in the x direction. So

[25:29] rate of change of the v wind in the y

[25:31] direction is equal to minus the omega

[25:34] dp.

[25:36] Now we've got some quantity with omega

[25:37] which is vertical velocity. And this

[25:40] expression here is identical to

[25:42] something which we call the horizontal

[25:45] divergence

[25:46] which is this product here. So

[25:48] physically you just think about that as

[25:50] air in a column. If the air is moving

[25:52] apart out of that column you've got

[25:54] divergence of the wind and that

[25:56] generates through this equation some

[25:57] kind of vertical motion.

[25:58] >> Basically the all the energy of the wind

[26:00] has to go somewhere has to do something.

[26:02] >> It's not the energy it's the

[26:03] conservation of mass. So if you're

[26:05] removing the mass in the horizontal

[26:07] direction, then you've got to accumulate

[26:09] the mass from the vertical direction.

[26:11] But the problem with the geostrophic

[26:13] wind is we can calculate the divergence

[26:15] of the geostrophic wind. If we plug this

[26:17] into this equation here, so if we called

[26:19] this du and dvg instead, then we could

[26:22] take d by dx and d by dy of our

[26:25] geostrophic winds. We can plug these

[26:26] geostrophic equations in. So, du by dx d

[26:30] by dx of this equation here -1 / row f

[26:34] dp dy plus and then d by dy of min -1

[26:39] over row f by dx plus one over row f.

[26:42] That's that one there. And you can see

[26:44] here we've got basically a dp a dx dy a

[26:48] dp a dx dy. We've got a plus one over

[26:51] row f and a minus one over row f. So the

[26:54] two terms cancel out and that equals

[26:56] zero. So this is quite an important

[26:58] result that the divergence of the

[27:00] geostrophic wind is zero. But we need

[27:02] the divergence in order to say something

[27:04] about the vertical motion. So this

[27:05] brings us back to our quai geostrophic

[27:08] omega equation. Yeah. All right. So we

[27:10] need a new piece of paper.

[27:11] >> All right. Let's do it.

[27:19] >> Let's talk about the geostrophic system.

[27:20] And one of the assumptions that is made

[27:23] when you go from the full primitive

[27:25] equation set to the quai geostrophic

[27:27] equation set is you assume that the

[27:29] atmosphere is in geostrophic and

[27:30] hydrostatic balance. We can't assume

[27:32] that it's completely in that way because

[27:34] as I already showed you the geostrophic

[27:36] wind is not divergent. So we can't get

[27:38] vertical velocity from the just the pure

[27:40] geostrophic winds. So that's where the

[27:41] quai comes from. You do a systematic

[27:44] simplification of the equations. You

[27:46] have to retain the full wind in just

[27:49] enough terms in order to be able to find

[27:51] vertical velocity but remove it from as

[27:54] much as possible in order to make it as

[27:55] simple as possible. That's where quai

[27:56] geostrophic comes from. But you might

[27:58] ask how good an assumption is it to say

[28:02] that for the types of weather systems

[28:04] that we're considering so the areas of

[28:05] low pressure, high pressure and the

[28:07] fronts and the cloud and the rain. How

[28:09] how relevant or how appropriate is it to

[28:11] say that the atmosphere is in

[28:12] geostrophic balance? And so I just want

[28:14] to introduce this cool number called the

[28:16] Rosby number. It's what we call a

[28:18] non-dimensional number. So it doesn't

[28:21] have dimensions and it's made up of a

[28:23] series of variables, but it is a number

[28:24] and you can use it to show things about

[28:27] geostrophe and and and what have you and

[28:29] the rest of it. So how do we get to the

[28:32] Rosby number? So Rosby comes from KL

[28:34] Rosby. He's a famous meteorologist from

[28:36] the early 1900s. He's as far as I know

[28:39] the only meteorologist to have ever

[28:40] appeared on the cover of Time magazine.

[28:42] Amongst many things, one of his claims

[28:44] to fame, he derived this number called

[28:46] the Rosby number which is essentially

[28:48] the ratio of the acceleration of the

[28:51] flow. So going back to our equations of

[28:53] motion here, the acceleration of the

[28:55] flow to the corololis acceleration. We

[28:58] take the ratio of the acceleration of

[29:00] the flow which in scale analysis terms

[29:03] is so du by dt it's some kind of speed

[29:07] scale over some kind of time scale. So

[29:10] I'm using scales here because eventually

[29:12] I want to think about well what's the

[29:14] typical scale of our weather system and

[29:16] plug and plug this in and get a rosby

[29:18] number. So this this is our acceleration

[29:21] and this is our corololis which is Fus

[29:24] force F times acceleration equivalent F

[29:27] times a speed scale and then we can just

[29:30] simplify that to say V that's the same

[29:33] as V over FTV the V's cancel you get 1

[29:36] over FT it's not quite the quite the

[29:38] rasby number in the form I want to show

[29:39] you because what we want to do is take

[29:42] the kind of speed equals distance over

[29:44] time equation very simple so speed

[29:46] equals distance so a length scale

[29:48] divided by a time scale. I'm just going

[29:50] to substitute t in for that. So that

[29:53] gives t=

[29:55] l / v. And we plug that t into there and

[29:58] we end up with 1 / f * l over v. And we

[30:03] basically just flip that upside down,

[30:04] put it on the top, which gives us v over

[30:07] l roby number. So this is a a

[30:09] dimensionous number. If you plug in the

[30:11] scale, so you got velocity in meters/s,

[30:14] this is in per second, and this is in

[30:15] meters. So you got meters/s over meters

[30:17] per second. The dimensions cancel, but

[30:18] it allows us to say something about

[30:20] geostrophic flow.

[30:21] >> Velocity of what? Of the wind.

[30:22] >> In the meteorological systems, we're

[30:23] talking about the velocity of the wind,

[30:24] but it could be the velocity of wind in

[30:25] a tornado with velocity of a wind in a

[30:27] large area of low pressure. The length

[30:29] scale for a tornado would be of order

[30:32] 100 meters. But the length scale for a

[30:34] large scale weather system would be of

[30:35] the order of a thousand kilometers. So,

[30:37] um, this is the Rosby number. It allows

[30:39] us to say, well, how geostrophic is our

[30:41] flow? So if you just plugged in pure

[30:43] geostrophic flow well the length scale

[30:46] of pure geostrophic flow is infinite or

[30:48] you could another way to look at it is

[30:49] the acceleration is zero. So you end up

[30:52] with zero over this roby number equals

[30:53] zero. So for pure theoretically pure

[30:56] gistrophic flow roby number is zero.

[30:59] What we've shown here is that as the

[31:01] Rosby number gets smaller and smaller,

[31:02] zero is the smallest it can be, but as

[31:04] it gets smaller and smaller, the the

[31:05] more towards geostrophic the flow gets

[31:09] because the corololis is more dominant

[31:11] than the flow acceleration which matches

[31:14] our corololis force here. So for a

[31:16] synoptic scale system, you know, the

[31:18] typical wind speed might be 10 m/s, you

[31:21] know, 1 m per second too slow, 100

[31:23] meters per second, too fast. So typical

[31:26] might be 10 m/s. Typical length scale uh

[31:29] might be of order 1 th00and km said. So

[31:32] >> are you saying for all the all the

[31:34] points along those thousand km the wind

[31:37] is that speed.

[31:38] >> Yeah. Well

[31:39] >> because the wind speed is always like in

[31:41] one position it's a certain speed.

[31:43] >> That's right. Yeah. But we're thinking

[31:44] about orders of magnitude here rather

[31:46] than exact values at the moment. So when

[31:49] thinking about the length scale I'm

[31:50] think about over across entire low

[31:51] pressure is about 1 th00and kilometers

[31:53] across. So low pressures of 100 km

[31:55] across that's quite small for an area of

[31:57] low pressure. 10,000 km is far too big.

[31:59] So th00and is about the order of

[32:01] magnitude. So this is what we think

[32:03] about orders of magnitude here. And we

[32:04] already calculated that the f for our

[32:07] kind of you know rough mid latitude

[32:08] location is of order 1* 10us 4 which we

[32:14] calculated earlier. If we plug them plug

[32:15] these into the Rosby equation then we

[32:17] can say that Rosby number for synoptic

[32:20] scale flow is 10 / 10^ 6 * 10us 4 which

[32:25] equ= 10 / 10^ 2 which is 10 over 100

[32:29] which is 1 over 10 0.1

[32:31] >> Rosby number 0.1. Okay.

[32:33] >> Yeah. So the Rosby number for your

[32:34] typical synoptic scale weather system so

[32:36] your typical areas of low pressure and

[32:38] high pressure is around 0.1. That's very

[32:42] close to zero. So that tells us that to

[32:44] a very good approximation the winds

[32:46] around of low pressure and high pressure

[32:48] can be assumed to be in geostrophic

[32:50] balance which is this balance between

[32:51] pressure gradient and coralis force.

[32:53] >> So a roby number is like a little

[32:54] waiting you add to tell you it's almost

[32:56] like it's almost like an error bar or

[32:58] how close you are to this ideal that you

[33:01] wish you had that would make life easy.

[33:03] >> You can yeah you can think of it like

[33:04] that. Yeah. The smaller it is the closer

[33:06] you are to geostrophic balance. And you

[33:09] could bung in you could bung in for say

[33:11] like um I don't know the eye of a

[33:13] tropical cyclone

[33:15] where you know the wind speed 100 meter

[33:18] pers maybe a little bit too strong but

[33:20] we'll we'll go with it. A typical length

[33:22] scale of the eye of a tropical cyclone

[33:25] maybe 10 km which would be 10^ the 4 m

[33:28] and tropical cyclones tend to form at

[33:30] lower latitudes. The corololis parameter

[33:33] is maybe an order of magnitude smaller.

[33:35] And so if we plung these into our Rosby

[33:38] uh number equation, then we get V 100

[33:41] divided by 10 4 * 10 - 5, which is 100

[33:49] over 10 to the minus one over 10 equals

[33:53] a,000.

[33:54] >> Wow.

[33:55] >> That that you can see is a massively

[33:57] different roby number to a stomp car

[33:59] system. So the type of balance we're

[34:01] looking at when you're thinking about

[34:02] the force balances in the tropical

[34:03] cyclone eye is not geostrophic. It's

[34:05] something we call cyclloic

[34:08] um which is a different balance

[34:09] altogether. But that's what's really

[34:10] cool about this rby number. You can plug

[34:12] in the scales of your system under

[34:14] consideration and it quantifies how good

[34:17] the geostrophic approximation is in your

[34:20] system that you're looking at.

[34:27] Right? So here's our qua geostrophic

[34:30] omega equation. So we've talked about

[34:33] omega that's vertical velocity. We've

[34:34] talked about geostrophic that's some

[34:36] kind of balance between pressure

[34:37] gradient force and coralous force. Now

[34:39] this is quai geostrophic because when

[34:42] you go from the full primitive equation

[34:44] set and you simplify it down using

[34:47] systematic scale analysis and

[34:48] simplification, you need to retain the

[34:51] full wind in some of the terms in order

[34:54] to produce vertical velocity as I showed

[34:56] you through the continuity equation. So

[34:58] you can't have everything geostrophic

[35:00] because if you did you would remove the

[35:01] information about the vertical velocity.

[35:03] Um so that's what makes it quai

[35:05] geostrophic. It's not quite geostrophic

[35:07] but it's it's near enough.

[35:09] >> Now is the Rosby number in here?

[35:11] >> Rosby number is not in here. No no

[35:12] unfortunately not.

[35:13] >> Okay.

[35:14] >> But it is a cool number. So when we

[35:16] think about this, so the whole the whole

[35:18] point of me explaining to this to you is

[35:21] to get to the point where I can show you

[35:23] how we as meteorologists use the

[35:26] principles contained in this equation to

[35:29] help us diagnose things about the

[35:31] weather. Now to get that what we do is

[35:33] we tend to partition these two sides

[35:35] into something called the response and

[35:37] the forcing. So as meteorologists we are

[35:40] basically looking at weather maps of

[35:42] vorticity and maps of temperature or or

[35:45] thickness as we we like to look at it

[35:47] and we look at those as the forcing

[35:49] terms and it's this that's forcing a

[35:52] response in in vertical motion and

[35:54] through vertical motion we can say right

[35:55] where is low pressure likely to develop

[35:57] where is high pressure likely to

[35:58] develop.

[35:59] Although you know you ask me can you

[36:01] predict the weather using using this

[36:03] equation because this is a diagnostic

[36:05] equation it doesn't tell you anything

[36:07] about how the velocity field is changing

[36:09] in time then strictly speaking you can't

[36:13] however you can when we look at

[36:15] subjective assessment of this use it to

[36:19] make inferences about where areas of low

[36:22] pressure are going to develop where

[36:23] we're going to see an increase in shower

[36:24] activity where high pressure is likely

[36:26] to develop and so just looking at this

[36:28] response term here. So this is in the

[36:29] vertical velocity. Unfortunately it's

[36:32] you know this is a very complicated

[36:33] equation and as humans we still can't

[36:35] quite look at this and look at the maps

[36:37] and immediately map one onto the other.

[36:39] Um we have to make some further

[36:41] simplification. So one simplification we

[36:43] make this is this tells you the

[36:45] threedimensional curvature of omega. Now

[36:47] you can do some reasoning and um you

[36:49] know you could say that omega is varying

[36:52] sinosoidally um as a function of

[36:54] pressure. So basically what that means

[36:56] is you could express omega as sin p and

[36:59] if you differentiate this twice you end

[37:01] up with d2 omega by dp^ 2 equals minus

[37:05] sin p which is minus omega. So we can

[37:08] make similar arguments to this side of

[37:10] the equation and we can basically say

[37:11] that all of this is proportional to

[37:15] minus omega. And if you remember minus

[37:18] omega from our um when we looked at the

[37:21] system of coordinates minus omega

[37:22] corresponds to ascent. So we've actually

[37:26] by assuming some kind of wavelike

[37:28] distribution of the vertical velocity in

[37:30] the horizontal and the vertical we can

[37:32] make a good assumption that this

[37:34] response is broadly equivalent to

[37:37] ascent. So what the forcing terms? Well,

[37:38] we've got the vertical variation of

[37:40] voltage vection. So what what does that

[37:42] mean exactly? So want to think about an

[37:45] atmosphere that's got a trough in it. So

[37:47] this might be the 300 mibar surface and

[37:49] this might be the th00and mibar surface.

[37:52] And this is a trough. We're always

[37:54] looking for where the troughs are, where

[37:55] the ridges are, because the troughs and

[37:57] the ridges in the upper air usually

[37:59] correspond to where the high pressures

[38:00] and the low pressures are in the lower

[38:02] atmosphere. Although there's a they're

[38:03] not quite colloccated in vertical,

[38:06] they're displaced one way or the other.

[38:08] But if the geostrophic wind is blowing,

[38:12] vecting this area of positive vorticity.

[38:15] So trough is an area of positive

[38:16] vorticity in this direction. At some

[38:18] later time, this trough is going to be

[38:20] in this position here. Now if we look at

[38:22] what's happened to the thickness of the

[38:25] atmosphere and the thickness is just the

[38:28] distance between the pressure levels and

[38:29] we call this H bar and the thickness is

[38:31] a effectively a measure of the

[38:33] temperature. If the mean temperature of

[38:34] the air is colder then the thickness

[38:36] becomes lower. If the mean temperature

[38:38] is warmer then the thickness becomes

[38:39] higher. But at this location here at

[38:41] some time not we've gone from hn to h1

[38:46] we've gone from a sort of relatively

[38:48] large thickness to a lower thickness.

[38:50] Now what does that mean in terms of the

[38:52] temperature? It means that the

[38:53] atmosphere here is cooled. However,

[38:57] we've got no we've got nothing about

[38:59] temperature invection here that we're

[39:01] not thinking about this term at the

[39:02] moment. There's no mechanism by which we

[39:04] can cool the atmosphere. And if we

[39:06] didn't cool the atmosphere then either

[39:08] it wouldn't be in geostrophic or

[39:10] hydrostatic balance. So how do we cool

[39:12] the atmosphere? Well, we basically have

[39:14] some vertical motion. We have a scent.

[39:16] If you have a parcel of air and it

[39:18] rises, rising air expands and it cools.

[39:21] So this process of vertical motion cools

[39:23] off the atmosphere. And this is what the

[39:25] omega equation tells you. It tells you

[39:27] what is the vertical velocity that I

[39:29] need in order to cool the atmosphere

[39:31] enough to reduce the thickness enough to

[39:34] accommodate this vorticity.

[39:36] And you make you can make a similar

[39:38] argument for this. You can um introduce

[39:40] some thermal adction. And that again, if

[39:43] you have your same pressure levels and

[39:44] you introduce some warm adction, maximum

[39:46] warm adction, you would increase your

[39:48] thickness because you've introduced

[39:49] warmer air that bulges up the contours

[39:52] of the pressure levels at height. And

[39:54] what does this bulge do? Well, it's it's

[39:55] the inverse of a trough. Basically, it's

[39:57] a ridge. It has negative vorticity. If

[39:59] we've got no mechanism to generate

[40:01] vorticity through vorticity action,

[40:04] well, how do we generate vorticity? We

[40:07] diverge. This is the ice skater effect.

[40:09] So, if you have an ice skater who's

[40:10] spinning around

[40:12] and then they pull their arms in,

[40:14] they'll spin faster. Or if they pull

[40:15] their arms out, they'll spin slower.

[40:18] Well, to reduce the vorticity, if the

[40:22] atmosphere pulls its arms out, the

[40:23] vorticity will reduce and we'll create a

[40:26] ridge. And how do we get this atmosphere

[40:28] to do this? We have vertical motion.

[40:30] because vertical motion goes up, it hits

[40:32] the top layer of the atmosphere, it

[40:34] spreads out and it creates the necessary

[40:37] vorticity in order to keep the fields in

[40:39] geostrophic balance. So these are the

[40:41] forcing terms. How do we use them? Why

[40:44] do we use them? Back when the QG system

[40:47] was first developed, this was a really

[40:50] key tool into identifying exactly where

[40:52] the vertical motion was. The best way to

[40:55] get the vertical motion from these

[40:56] forcing terms is to calculate it with a

[40:58] computer. That's kind of how they did

[40:59] it. But as forecasters we we can't do

[41:01] that. We can make these assumptions and

[41:03] we can look at the fields. We identify

[41:05] where the vorticity is. We identify

[41:08] where the thermal adection is. And if

[41:10] you've got positive vorticity, if you've

[41:12] got warmer advction, that corresponds to

[41:15] negative omega, which is ascent. And so

[41:18] by looking at a map of vorticity, you

[41:19] can immediately see where the areas of

[41:22] vertical motion in ascending limb and

[41:24] the descending limb are going to be. And

[41:28] that also comes into its own when the

[41:31] forecast model which can calculate

[41:33] vertical velocity. Now you know high

[41:35] performance supercomputers and really

[41:37] sophisticated models they can calculate

[41:39] vertical velocity but they can also be

[41:41] wrong in the positions of the troughs

[41:43] and the ridges. They can have errors and

[41:45] we can look at satellite imagery and we

[41:47] can diagnose where these errors are and

[41:49] we can say right well if this trough was

[41:50] a bit further back what does that mean

[41:52] for the vertical motion field? we can we

[41:54] can always calculate it in our head and

[41:56] make a forecast based um on that. So

[41:59] although this equation comes from you

[42:01] know deep meteorological history um it's

[42:04] part of an equation set that's no longer

[42:06] used in much simpler times in numerical

[42:08] models. Um it's still used in practice

[42:12] on the bench although with a lot of a

[42:14] lot of assumptions, a lot of

[42:16] simplifications in order to give

[42:17] forecasters this conceptual model of the

[42:19] atmosphere where things are ascending,

[42:21] where things are descending and

[42:23] therefore where areas of low pressure,

[42:24] whereas high pressure are developing.

[42:26] >> But you don't crack out the equation and

[42:28] put numbers in it. You use more this

[42:29] kind of response and forcing situations.

[42:32] >> Exly. Yeah. We look at where the we look

[42:33] at where the vorticity is. We look at

[42:35] where the thermalction is. We diagnose

[42:39] qualit qualitatively whether it's

[42:41] ascending or descending because we can't

[42:44] calculate the magnitudes of these terms

[42:46] in our head. We can't say anything about

[42:48] the size of the response. And there

[42:51] there are occasions where one force in

[42:53] turn will say ascent and one force in

[42:56] turn will say descent and then we have

[42:58] other techniques that we can use to kind

[42:59] of break the tie. But in very

[43:01] qualitative terms, yeah, we're looking

[43:03] at the vorticity inction and the thermal

[43:05] invection. And it tells us whether we

[43:07] can expect upward motion and development

[43:09] of low pressure at the surface, even

[43:10] development of, you know, if it if it

[43:12] links in with a strong frontal gradient,

[43:14] that can lead to development of a potent

[43:16] of low pressure and a a name storm or

[43:18] whether we're looking at descending air

[43:20] and development of high pressure and

[43:21] more settled weather. Well, you made it

[43:23] this far, so well done. If you'd like to

[43:25] hear more from Dan, more of a personal

[43:27] interview, a bit about his life, his

[43:29] nickname at school, his first job as a

[43:32] forecaster, his backyard weather setup,

[43:35] that's something you don't want to miss.

[43:36] Then check out the Number File podcast.

[43:38] It's a great interview and one not to

[43:40] miss.

[43:45] >> Also, just have a tiny little change.

[43:47] Quantifying tiny, but not to blow up to

[43:49] infinity. That wouldn't make sense

[43:51] because I've changed something so so

[43:52] small. Why have I got an entirely

[43:55] different solution?

[43:55] >> We've also been taught that butterflies

[43:57] flapping their wings can cause cyclones.

[44:00] >> The butterfly effect, it's like a chain

[44:01] reaction. It's one thing leads to

[44:02] another leads to another. But in the in

[44:04] the sense of humming an equation, you

[44:07] input something into your equation. It's

[44:08] like a function machine. Input some

[44:10] initial