The Weather Equation - Numberphile

0:00 And this is one of the most famous equations in meteorology.

0:03 I'm just going to go ahead and and write it down.

0:05 So we have sigma d^ 2 omega.

0:08 So where the omega comes from plus f 2 d2 omega by dp^ 2.

0:14 So this is the omega part and that equals

0:17 f* the vertical variation of this term here which is

0:21 the advection of the absolute geostrophic vorticity by the geostrophic

0:25 wind and the advection of geostrophic wind the temperature gradient.

0:30 So this is this is the qua geostrophic omega equation.

0:33 So this is a fundamental equation that allows us

0:37 to through systematic simplifications a lot of simplification actually get

0:42 the vertical velocity from the adction of this quantity

0:45 called vorticity and the adction of this quantity called temperature.

0:49 So thermal adction, vorticity adection.

0:52 And by looking at horizontal maps of these adections,

0:55 we can diagnose vertical velocity and vertical

0:58 velocity is a proxy for development.

1:00 And development is your developments of your lows,

1:03 your developments of your highs, high pressure and low pressure.

1:05 And that relates to the weather forecast.

1:08 So what this tells you what's going to happen in the future?

1:12 No.

1:12 So that's quite an important distinction about this equation.

1:15 This does this tells you nothing about the future state of the atmosphere.

1:18 So this isn't a predictive equation.

1:20 It's it's it's called a prognostic equation.

1:23 This is a diagnostic equation.

1:25 It just allows us to diagnose the vertical velocity

1:28 from the invection of vorticity and the invection of temperature.

1:31 And why is that an important thing to know?

1:35 Um if it doesn't tell you anything about the future.

1:36 Well, so this I mean we we can go into this in a bit more detail uh later

1:41 on, but this um essentially is one equation that falls

1:45 out of a whole system called the quai geostrophic system.

1:49 And I want to kind of go into a bit

1:51 about what we mean by quai geostrophic omega.

1:54 But essentially the quai geostrophic system

1:56 is a systematic set of simplifications

1:59 to the main primitive equations which numerical

2:02 models these days use to forecast the weather.

2:05 um is a a systematic simplification of those equations

2:08 that could then be used to make the first

2:10 kind of workable forecast models that were used

2:13 on the computers back in the day the the early 1950s.

2:17 And so this equation is a diagnostic equation that falls out of those and it

2:22 is part of that forecast process whereby

2:24 you would calculate some tendencies of various things.

2:27 You would then get the wind fields

2:29 which you could calculate wind and thermal fields

2:31 which you could calculate the vorticity and the thermal

2:32 invection and from that you could diagnose

2:34 the vertical velocity and getting to this vertical

2:36 velocity is actually quite difficult and so this equation

2:39 gives us a method to to actually do

2:42 that in a systematic and relatively error-free way

2:46 in even more simple terms then this equ is this equation relating

2:50 to up and downness of the air what's it like what's what's it

2:54 exactly so when when we're talking about vertical velocity We're talking

2:57 about the motion of the air in an up and down sense.

3:00 So on on large scales, so the scales of weather systems,

3:03 the main the major motion of the atmosphere is in the horizontal.

3:07 So you've got winds north to south, winds east to west.

3:10 We don't tend to think on large scales about the winds

3:13 that are going up and down, but they are there.

3:15 If you think about a convective cloud, for example,

3:17 they're moving up into clouds, connects into clouds,

3:19 and you get rain falling out.

3:20 Um, but on these large scale,

3:22 so we we're talking about big scale weather systems here.

3:25 You know, systems the size of the UK,

3:27 systems that that that sort of take up a big proportion of the Atlantic.

3:31 You don't tend to think about vertical motions there,

3:34 but they are there and they're necessary in order to develop

3:37 the areas of low pressure and develop the areas of high pressure.

3:39 So if you have rising motion,

3:41 you get development of low pressure at the surface.

3:43 And if you have sinking motion,

3:45 then you have development of high pressure at the surface.

3:48 One more basic weather question before we, you know, get our teeth into this.

3:52 Then I'm pretty familiar with measuring wind speeds horizontally.

3:56 I've seen those little devices like those little

3:58 cups that spin around and things like that.

4:01 Are there devices that measure the up downwind, the vertical wind?

4:05 Like how do you how do you guys take measurements of that?

4:07 No, it's very difficult.

4:08 And that's one of the reasons why we kind

4:10 of look to equations to to diagnose the vertical velocity.

4:15 It's very hard to measure.

4:17 it's on a much much much smaller scale than the the horizontal motions.

4:21 And in fact, you can try and calculate the um vertical

4:26 motion from the horizontal motions via something called the continuity equation,

4:30 which we'll have a look at in a little while.

4:32 Um but that relies on you knowing

4:34 to a very high degree of accuracy your horizontal winds.

4:37 And it turns out that the the divergence

4:39 and the vertical motion errors in that are

4:41 the same size as the actual vertical motion

4:44 you're trying to diagnose in the first place.

4:45 So you can have small areas in the wind

4:47 that can lead to massive areas in vertical velocity.

4:49 Just quickly, what are the things that measure the wind speed, Cole?

4:53 Anim an animometers.

4:56 Animasure wind speed horizontally.

4:59 Yes.

4:59 Are you saying equations the likes of these are

5:01 kind of what you have instead of those for vertical.

5:04 That's how you get to the vertical velocity.

5:07 Yeah.

5:07 So let's break it down.

5:07 So um first of all, it's the omega equation.

5:10 So let's talk about omega.

5:11 So we got this Greek letter omega here.

5:13 So omega is just the vertical velocity.

5:15 So it has a equivalent called W and these two

5:19 are equivalent but they use different coordinate systems.

5:21 So for example, if we think about W, this might be a little bit more familiar.

5:24 We think about a coordinate system that has winds in the XY.

5:29 So this is like your XY plane and then you've got a vertical plane.

5:32 If you have some vertical motion or if you have some horizontal motion,

5:36 we could have a a U wind in the X direction, a V wind in the Y direction.

5:40 So these are our horizontal winds and then w would be um

5:45 the wind in the vertical and that would be measured in meters per second.

5:49 There is a direct equivalent though uh in a different

5:52 coordinate system which we in meteorology that we like

5:54 to use because it has some good advantages when coming

5:58 to equations like this and looking at things like mass continuity.

6:01 We still have an xy plane but we use pressure as our vertical coordinate.

6:05 And so if we have vertical motion in the pressure coordinate and we

6:10 can use pressure as a coordinate because pressure always decreases with height.

6:14 So it's around about a th00and millibars at the surface 1,000 hector pascals.

6:18 At the top of the troposphere which is the top

6:20 of the weather bearing layer is around about 300 hector pascals.

6:23 So pressure always falls with height.

6:25 So if you're moving in an upward direction you're moving towards lower pressure.

6:30 Ascent in the the geometric coordinate system would be W.

6:34 ascent in the pressure coordinate system would be negative

6:37 omega because you're going towards lower pressure as you

6:40 go up whereas in the geometric system you're going

6:42 towards kind of higher elevations as you go up.

6:46 So does that mean we're using on our x and our y

6:48 axis here we're using different units because that's presumably still meters/s.

6:54 That's right.

6:54 Yep.

6:55 So this we still have a U wind and a vwind in meters per second

6:59 but we're using a different unit on that.

7:01 Yep.

7:01 The unit for this one would be pascals per second.

7:04 It's it's the amount of pressure change

7:05 that this parcel of air is experiencing per second.

7:08 So very much like your your speed would be

7:10 the amount of displacement you're experiencing per second, meters/s.

7:14 This would be the the pressure change per second, pascals per second.

7:17 Okay, cool.

7:19 You can you can link the two uh via something called the hydrostatic equation.

7:23 So we have this hydrostatic equation which relates

7:27 the rate of change of pressure in the z

7:28 direction with this term minus row which is the density and g which is gravity.

7:34 You can say that um if you treat is the material

7:38 derivative which you can expand out to this expression here.

7:43 And um we're going to assume something called a hydrostatic balance.

7:45 Um, and what hydrostatic balance effectively means

7:48 is that if we have an atmosphere that's

7:51 at rest and we have a particle or a parcel of air in the atmosphere,

7:54 it's got two forces acting upon it.

7:56 We've already said that pressure decreases with height.

7:58 So, we've got a pressure gradient in this direction.

8:00 So, this parcel has experienced a pressure gradient force in that direction.

8:04 And then we've got a gravitational force in that direction.

8:07 So, we've got pressure gradient force and gravity.

8:10 And when these two are in balance, you know,

8:12 if if there wasn't gravity and there was a pressure gradient force,

8:14 then this parcel of air would fly up

8:16 into the atmosphere because it'd be the the higher

8:18 pre moving from higher pressure to lower pressure

8:20 and be pushing it up to into the atmosphere.

8:22 But because it's balanced by gravity, the the particle or the air parcel is

8:26 at rest and we call this hydrostatic balance.

8:29 So we assume the atmosphere is at rest.

8:30 There's no U, there's no V, there's no pressure change with time.

8:33 So we can say that uh omega is approximately equal to W DP by DZ.

8:39 And then from this hydrostatic balance relationship which

8:42 we kind of expanded out here we can

8:44 say that omega be equal to minus row g which is this term here times w.

8:53 So what this effectively tells us is that uh

8:55 omega and w are related but omega is

8:58 just a scaled version scaled by the density because

9:02 because the atmosphere is more dense at the surface.

9:05 So if you have a parcel of air

9:07 that's moving vertically at fairly low elevations, the pressure is quite high.

9:12 Because the atmosphere is more dense here,

9:14 the pressure levels are kind of closer together.

9:17 And therefore the same vertical speed in height

9:20 coordinates at lower levels would be crossing more.

9:23 It be the pressure will be changing more at lower levels than it is

9:27 at higher levels for the same vertical speed

9:30 when you're thinking about the change in meters.

9:32 So this is what this equation tells us.

9:33 Omega is just a scaled version

9:36 of the um vertical velocity in geometric height coordinates,

9:40 but it's scaled by the density to account for this effect.

9:43 Want another piece of paper?

9:44 Yeah, I think so.

9:45 All right.

9:55 The main takeaway about omega is it's

9:56 the vertical velocity in a pressure coordinate system.

9:59 Okay.

9:59 So, there's a few other symbols that we uh we want to look at here.

10:03 Yeah.

10:03 So again we'll just concentrate on the left hand side at the moment.

10:06 Right.

10:07 So this is sigma.

10:08 Sigma scales the vertical velocity response for a certain size of forcing.

10:13 This all taken together gives the three-dimensional distribution

10:16 of omega or the distribution of the curvature of omega.

10:20 So it doesn't necessarily give us omega itself yet

10:22 but we'll see how we can retrieve omega from this side.

10:25 But it tells us something about the threedimensional distribution of omega.

10:28 Now these terms on the right hand side let's break this down.

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

10:34 And I want to delve a little bit more into what geostrophic means in a second.

10:38 But this Z to G plus F this tells us the absolute vorticity.

10:42 Vorticity is effectively a measure of the spin in the atmosphere.

10:46 Any atmospheric process where you've got things

10:48 spinning around any fluid spinning you have vorticity.

10:52 And these two different types of spin that we've got

10:54 make that make up the absolute vorticity are the relative vorticity.

10:59 So relativeity is just the spin.

11:01 If you put a paddle wheel in the flow,

11:03 say you've got some flow coming down here and you've got a lesser

11:07 flow here and you put a little paddle wheel in the flow.

11:10 Well, there's more pressure on in the flow on this part

11:13 of the paddle wheel than there is on this part.

11:15 So this paddle wheel is going to start to spin.

11:17 So the fluid at this point would spin in that direction.

11:20 And that's the vorticity.

11:21 Now, this would be a sheer vorticity because the we've got a shear.

11:26 Um, fluid passes are moving faster here than they are here.

11:29 You could have a curvature vorticity where simply the flow is curved

11:33 and so if you put a stick in a flow here, it would curve.

11:37 And so again, you've got vorticity through through curvature.

11:40 So vorticity or spin can come out through through different mechanisms.

11:43 Uh, Z to G here just tells us about the vorticity of the flow.

11:48 Now this f this is the planetary vorticity and this is just vorticity that you

11:52 have even you have it Brady just by virtue of being on a spinning earth.

11:57 So anything that's on the earth apart from on the equator

12:00 has some kind of spin about its local vertical.

12:02 Yeah.

12:02 If you think about the globe it's spinning on its axis.

12:05 So if you were at the north pole you would be

12:07 experiencing that full planetary vorticity the full spin of the earth.

12:11 You wouldn't be able you wouldn't know it was

12:12 there because you're kind of spinning along with it.

12:15 everything that you can see is spinning along with it.

12:17 So you wouldn't know it's there, but it is.

12:19 It's spinning.

12:20 And you can pick any point on the Earth's axis and you

12:23 can decompose its spin um into a component about its local vertical.

12:28 Um and the equation for that F is equal to 2*

12:32 another omega but this is the big omega s of the latitude.

12:36 We calculate what our planetary vorticity here

12:39 um at the Met headquarters is in exit.

12:41 If we take this equation here.

12:43 So f which is our corololis parameter is equal to 2* omega.

12:47 Now omega from memory is something like 7.292* 10- 5.

12:54 This is in radians/s.

12:55 So this is just the the number of radians that the earth is turning per second.

12:59 So that's a constant.

13:00 This is a constant.

13:01 This is a constant.

13:02 The only thing that varies is the latitude.

13:04 And we take sign of the latitude.

13:07 Um I don't know what it is in radians unfortunately.

13:09 But because we're taking a sign of it, I don't think it matters too much.

13:12 Uh sin 50°.

13:13 And that turns out to be, if you go through the calculation,

13:17 it's approximately uh 1.1* 10 -4 per second.

13:23 So that's the planetary vorticity.

13:25 So go back to the equation.

13:27 The relative vorticity of flow plus the planetary vorticity,

13:30 that's the absolute vorticity.

13:32 But this little quantity here,

13:34 this little dell symbol means that we take the gradients of that.

13:39 Gradient means that the absolute velocity is changing in space.

13:42 We take the gradient of that and we do a dot product

13:45 with the uh wind velocity and that gives us the advection of the gradient.

13:51 So it's how the wind is blowing along the vorticity gradient kind

13:54 of tells us you can imagine that's the advection of the vorticity.

13:57 If you're going from an area of high vorticity to low vorticity,

14:01 you're blowing the high volticity towards you

14:03 and therefore you got positive vorticity in vection.

14:06 I'm beginning to get some appreciation as to why weather is complicated.

14:10 Well, I that's a really good point.

14:12 And and when when people say when people say, "Oh,

14:14 weather forecasting is easy, isn't it?" I just look out the window.

14:16 Then my temptation is to point them to stuff like this and say,

14:18 "Well, actually, there's a lot more to it than that." Um,

14:21 so this is the vorticction part,

14:23 and we'll come on to kind of what that physically means in a little while.

14:25 Um this is the temperature vection part.

14:27 So very much like we had the gradient of temperature.

14:31 So this is a bit more simple.

14:32 You can just think of this as temperature.

14:33 We we but we've got the temperature gradient

14:35 and again we're vecting that with the geostrophic wind.

14:39 And these features in in front of all

14:41 of this well these are this is just the gas constant.

14:43 This is pressure.

14:44 This ter this little expression here means not only are we taking

14:48 the gradient of absolute vorticity and advecting it with a geostrophic wind.

14:53 We're actually taking the vertical variation of that which is what this uh d

14:57 by dp term tells us and then it's scaled by the coralous parameter f.

15:01 Even you know this is ridiculous.

15:03 Even you know this is ridiculous.

15:05 All right.

15:06 Yeah.

15:06 So before we get to how we use this equation as weather forecasters

15:10 and believe it or not given all the complexities in a conceptual way,

15:14 we still do use this equation which is um fascinating to me because if you think

15:20 about the quai gestroic equation set which was

15:23 derived to make the first workable numerical models.

15:27 Well, pretty much all of the rest of that system has not fall well,

15:32 it has fallen out of use as models have got better, super got faster,

15:36 but the one kind of enduring thing

15:38 that we use as forecasters is this QG equation,

15:41 which is why it's such a famous equation of meteorology.

15:44 Um, and it's why it's so interesting to me.

15:46 But we're we're a couple of steps away,

15:48 I think, from kind of understanding what this physically means.

15:53 And I think we 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 means.

16:00 So should we have another piece of paper?

16:02 Yes we shall.

16:12 We've talked about what omega means.

16:14 Uh I just want to spend a little bit of time about talking

16:16 what geostrophic means and then by extension we can go to quai geostrophic.

16:21 We can go back to my lovely rendition of the globe here.

16:24 We can imagine a weather system on this.

16:26 And you this might be sort of quite familiar if you imagine this might be

16:29 like a typical weather map that you would see with an area of low pressure.

16:32 And when you see the forecasts on the TV and they show the the pressure maps,

16:36 they often show the winds.

16:37 If you look at how the winds are blowing, you find that one,

16:40 they blow counterclockwise around an area over low pressure,

16:43 and two, they tend to blow parallel to the isobars,

16:47 which is quite surprising if you think about it because

16:49 if you think you've got a region of low pressure,

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

16:53 And if you think about what happens when you turn on a vacuum cleaner,

16:56 it sucks up all the dust and grime and sucks up the tube and it's

16:59 creating an area of low pressure and the wind

17:01 is effectively blowing in towards that vacuum.

17:03 So why doesn't it happen um on our planet?

17:08 Uh and the reason for this is because um as well as the pressure

17:11 gradient force there's another force that we have to take into consideration.

17:15 So if we simplify and we think of our air

17:17 parcel moving this is like the pressure gradients of the same

17:21 as on the map and we'll just imagine we've got an area

17:23 of high pressure here and an area of low pressure here.

17:26 Now at the moment we don't really know anything about the direction

17:29 of this but um it it will be blowing in this direction.

17:32 Um it's B's ballot law actually which says if the wind is in your face so if you

17:36 were standing here Brady and the wind's in your face

17:38 uh you would have low pressure on your right

17:42 in both hemispheres.

17:43 Uh I have to think about that one.

17:45 Um so in the southern hemisphere the wind

17:47 does blow clockwise around an area of low pressure.

17:50 So if you were standing if you face the wind it would be to your left.

17:55 Yes.

17:55 It would be the opposite.

17:57 That's right.

17:57 Not that we're being hemisphere.

17:59 Not that we're not we're discriminating against either hemisphere.

18:02 We we're hemisphere agnostic.

18:04 So we would have a pressure gradient force

18:06 in this direction and if that was the only

18:08 force acting then the air parcel instead of going

18:10 in this direction would go towards the low pressure.

18:13 But it turns out that's exactly balanced

18:15 by this other force and this is the corololis force.

18:18 This is well it's a little bit controversial as to what you describe it.

18:22 Some people describe them as fictitious forces.

18:25 Some people will describe them as forces that you

18:28 have to include in the equations of motion in order

18:32 to make it appear as if we're not on a rotating

18:35 planet because although we are on a rotating planet, we can't really judge that.

18:41 But the motion, the fluid motion,

18:43 the motion of the air parcels knows we're on a rotating planet.

18:46 And so you see some odd behaviors.

18:47 And one of the odd behaviors is if you have an air

18:50 parcel that starts to move from high pressure to low pressure,

18:53 it will feel this corololis force acting and pulling it to the right.

18:57 And so as it moves, it will continue to feel this force

19:00 until eventually and the pressure gradient

19:01 force is always acting in this direction.

19:03 And it will continue to be pulled to the right

19:05 until the corololis force and the pressure gradient force balance.

19:08 And then we end up with this what we call geostrophic flow.

19:11 So geostrophic just basically means there's a balance

19:13 between the corololis force and the pressure gradient force.

19:16 That's not fictitious.

19:18 Well, the pressure gradient force isn't fictitious,

19:20 but the corololis force is kind of fictitious because it's a it's

19:24 a correction that we have to add to account for what we see.

19:29 Corololis is a really interesting character and one

19:32 of the roads around the Met Office is named after him.

19:34 It's of such fundamental importance to weather prediction.

19:36 We can look at the equations of motion.

19:40 Um, so I'm going to write down the equations of motion.

19:42 Now you you've done the Navy Stokes equations.

19:44 These equations of motion are sort of analogous to the to those.

19:48 And if we take the equation of motion in the x direction.

19:51 So this relates the acceleration of the u wind.

19:54 So if you remember going back to our coordinates,

19:56 we had a u wind in the x direction.

20:00 So how the u wind is changing with time is the acceleration in the u direction

20:04 and that's equal to this term here which

20:06 is the pressure gradient force in the x direction.

20:09 This is the corololis force.

20:11 And then we've got some frictional terms.

20:13 And this basically boils down to F= ma Newton's second law.

20:17 So we've got our forces on one side got our acceleration on the other side.

20:21 If we apply it to a parcel with mass one unit

20:24 mass then we end up with basically A= F F= A.

20:27 So this is Newton's second law in a nutshell and I'll write just quickly

20:30 write out the one for the V because it follows a very similar pattern.

20:33 So there's the pressure gradient force in the Y direction.

20:35 Corus term is minus FU and then plus some kind of frictional forces.

20:39 When we looking at geostrophic balance, um,

20:42 we are assuming first of all, we're assuming there's no friction.

20:45 So, we can't really apply geostrophic balance at the surface.

20:48 And that is why when you see winds blowing around

20:52 an area of low pressure and you look at the surface winds,

20:55 they're not quite parallel to the isobars.

20:58 They're actually just pointed in slightly towards the low pressure.

21:00 That's because this frictional force is having a drag

21:02 effect that's pulling the winds in towards the low pressure.

21:05 So the geostrophic approximation is is a good

21:08 first approximation for calculating the winds but it's

21:11 it is an approximation um partly because

21:14 of these frictional effects but that's dragging along the ground.

21:17 It's dra exactly that.

21:19 Yeah it's experiencing a stress against the ground.

21:21 It's losing it's losing momentum.

21:23 It's falling slightly down the pressure gradient

21:26 um rather than just going around it and therefore eventually the winds all kind

21:30 of curling into the the center of the low.

21:31 And if there was no process above the load to kind of get rid of air or mass,

21:36 then that low pressure would just gradually fill

21:38 up and would become not a low pressure anymore.

21:40 So we can get rid of these frictional

21:41 forces because they're in the geostrophic system, we consider them negligible.

21:45 And also we need to have a non-acelerating atmosphere.

21:49 So an acceleration could be a speed up or a slow down or it could be

21:53 a curve because anything that's moving in a curve

21:55 is being accelerated towards the center of that curve.

21:58 Anytime you introduce an acceleration and change the strength of the wind,

22:03 you change this corololis force term because corololis

22:07 term is a function of the wind speed itself.

22:10 Is the sun pumping a whole lot of heat into the system

22:14 not going to cause things to speed up or accelerate?

22:16 So at the moment we're kind of we're neglecting that we're

22:20 the sun will have uh an impact on smaller scales.

22:24 So think of that convective cloud we talked about that cumulus cloud.

22:27 Sun heats the ground and introduces all kinds of motion.

22:30 But the the these the scales that we're talking about, these very large scales,

22:34 the the direct heating input from the sun um is not so much of an effect.

22:43 It obviously has an effect.

22:45 It creates the conditions necessary to drive weather.

22:47 Um but it's not incorporated in this simplified system.

22:51 So yeah, anytime you introduce an acceleration um you will disrupt

22:55 this force balance because the corololis force is a function of the wind.

23:00 So if you change the wind, you change the corololis force.

23:02 These forces are no longer in balance

23:04 and therefore the flow wouldn't be geostrophic.

23:07 So basically we just ignore the accelerations as well.

23:10 And this gives us our geostrophic balance.

23:13 So it essentially says this pressure

23:14 gradient force and this coralous force balance.

23:16 And I can just explicitly show that by taking the pressure

23:20 gradient force over to the left hand side of the x equation.

23:24 And similarly to the y equation and then we can divide both sides by f.

23:28 And this gives us an expression for our geostrophic

23:31 wind on that diagram of the earth.

23:32 When I said you know you look at the you look at the winds

23:35 on the forecast map to a first

23:36 order approximation or to a good order approximation.

23:39 The winds are geostrophic.

23:40 They blow parallel to the isobars and you can calculate those winds.

23:44 So we're at a point now whereas remember when you

23:47 said um can you not measure the the vertical velocity?

23:52 Can you not measure it?

23:53 Well, we can measure the horizontal winds

23:55 but the problem and calculate the vertical velocity from that.

23:57 But if you remember the problem from that is that we we have

24:01 slight errors in our wind measurements

24:02 that measurement the wind measurements are quite sparse.

24:04 So we don't know what's happening in between and the errors

24:07 introduced by those measurements dwarf the vertical velocity itself.

24:12 But we've got a situation here where we

24:13 could actually if we just know the pressure

24:15 gradients and the corololis parameter which we

24:18 can calculate we can actually calculate the winds.

24:21 So we don't need to measure them anymore.

24:22 We can calculate them exactly.

24:23 Where do you get the pressure gradients from?

24:25 So measurement of pressure at the surface you

24:27 would just use an instrument called a barometer.

24:29 So that's the that's the pressure kind

24:31 of equivalent of an anometer for the wind.

24:33 So we've got a barometer and an animometer.

24:36 Measuring pressure through depth.

24:38 Um that's a slightly different story.

24:40 We can cover that another time.

24:42 One of the reasons I wanted to just

24:44 describe geostrophic wind is because I wanted to show

24:49 you that you can a connect the vertical motion

24:54 to a quantity which we call divergence but b

24:59 the geostrophic wind can't necessarily help us

25:01 with this right so I want to take the um

25:03 equation for mass continuity du by dx plus dv

25:08 by dy plus d omega by dp equals not.

25:15 So you've heard of the principle of conservation of mass.

25:18 This is the meteorological equivalent of this.

25:21 And what it tells us is that if we rearrange this, we

25:25 say that the rate of change of the u in the x direction.

25:29 So rate of change of the v wind

25:30 in the y direction is equal to minus the omega dp.

25:36 Now we've got some quantity with omega which is vertical velocity.

25:39 And this expression here is identical to something which

25:43 we call the horizontal divergence which is this product here.

25:48 So physically you just think about that as air in a column.

25:51 If the air is moving apart out of that column you've got divergence

25:54 of the wind and that generates through

25:56 this equation some 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 conservation of mass.

26:04 So if you're removing the mass in the horizontal direction,

26:07 then you've got to accumulate the mass from the vertical direction.

26:11 But the problem with the geostrophic wind is

26:14 we can calculate the divergence of the geostrophic wind.

26:16 If we plug this into this equation here,

26:19 so if we called this du and dvg instead,

26:22 then we could take d by dx and d by dy of our geostrophic winds.

26:25 We can plug these geostrophic equations in.

26:28 So, du by dx d by dx of this equation here -1/ row f dp dy plus

26:37 and then d by dy of min -1 over row f by dx plus one over row f.

26:42 That's that one there.

26:44 And you can see here we've got basically a dp a dx dy a dp a dx dy.

26:49 We've got a plus one over row f and a minus one over row f.

26:54 So the two terms cancel out and that equals zero.

26:56 So this is quite an important result

26:58 that the divergence of the geostrophic wind is zero.

27:02 But we need the divergence in order to say something about the vertical motion.

27:05 So this brings us back to our quai geostrophic omega equation.

27:09 Yeah.

27:09 All right.

27:09 So we need a new piece of paper.

27:11 All right.

27:11 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 equation set

27:25 to the quai geostrophic equation set is you assume

27:28 that the atmosphere is in geostrophic and hydrostatic balance.

27:31 We can't assume that it's completely in that way because

27:34 as I already showed you the geostrophic wind is not divergent.

27:37 So we can't get vertical velocity from the just the pure geostrophic winds.

27:41 So that's where the quai comes from.

27:43 You do a systematic simplification of the equations.

27:46 You 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 much as possible

27:54 in order to make it as simple as possible.

27:56 That's where quai geostrophic comes from.

27:58 But you might 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 low pressure,

28:06 high pressure and the fronts and the cloud and the rain.

28:08 How how relevant or how appropriate is it

28:11 to say that the atmosphere is in geostrophic balance?

28:13 And so I just want to introduce this cool number called the Rosby number.

28:17 It's what we call a non-dimensional number.

28:20 So it doesn't have dimensions and it's made up of a series of variables,

28:23 but it is a number and you can use it to show things

28:26 about geostrophe and and and what have you and the rest of it.

28:30 So how do we get to the Rosby number?

28:32 So Rosby comes from KL Rosby.

28:34 He's a famous meteorologist from the early 1900s.

28:37 He's as far as I know the only meteorologist

28:40 to have ever appeared on the cover of Time magazine.

28:42 Amongst many things, one of his claims to fame,

28:45 he derived this number called the Rosby number which

28:47 is essentially the ratio of the acceleration of the flow.

28:51 So going back to our equations of motion here,

28:54 the acceleration of the flow to the corololis acceleration.

28:58 We take the ratio of the acceleration

28:59 of the flow which in scale analysis terms is

29:04 so du by dt it's some kind of speed scale over some kind of time scale.

29:10 So I'm using scales here because eventually I

29:13 want to think about well what's the typical

29:14 scale of our weather system and plug and plug this in and get a rosby number.

29:18 So this this is our acceleration and this is our corololis which

29:23 is Fus force F times acceleration equivalent F times a speed scale

29:29 and then we can just simplify that to say V that's

29:32 the same as V over FTV the V's cancel you get 1 over

29:36 FT it's not quite the quite the rasby number in the form

29:39 I want to show you because what we want to do is

29:41 take the kind of speed equals distance over time equation very simple

29:45 so speed equals distance so a length scale divided by a time scale.

29:50 I'm just going to substitute t in for that.

29:52 So that gives t= l/ v.

29:57 And we plug that t into there and we end up with 1/ f* l over v.

30:03 And we basically just flip that upside down,

30:04 put it on the top, which gives us v over l roby number.

30:08 So this is a a dimensionous number.

30:10 If you plug in the scale, so you got velocity in meters/s,

30:14 this is in per second, and this is in meters.

30:15 So you got meters/s over meters per second.

30:17 The dimensions cancel, but it allows us to say something about geostrophic flow.

30:21 Velocity of what?

30:21 Of the wind.

30:22 In the meteorological systems, we're talking about the velocity of the wind,

30:24 but it could be the velocity of wind in a tornado

30:26 with velocity of a wind in a large area of low pressure.

30:29 The length scale for a tornado would be of order 100 meters.

30:32 But the length scale for a large scale weather

30:34 system would be of the order of a thousand kilometers.

30:36 So, um, this is the Rosby number.

30:38 It allows us to say, well, how geostrophic is our flow?

30:42 So if you just plugged in pure geostrophic

30:44 flow well the length scale of pure geostrophic

30:47 flow is infinite or you could another way

30:49 to look at it is the acceleration is zero.

30:51 So you end up with zero over this roby number equals zero.

30:54 So for pure theoretically pure gistrophic flow roby number is zero.

30:59 What we've shown here is that as the Rosby number gets smaller and smaller,

31:02 zero is the smallest it can be, but as it gets smaller and smaller,

31:05 the the more towards geostrophic the flow gets because the corololis is

31:11 more dominant than the flow acceleration which matches our corololis force here.

31:15 So for a synoptic scale system, you know,

31:18 the typical wind speed might be 10 m/s,

31:21 you know, 1 m per second too slow, 100 meters per second, too fast.

31:25 So typical might be 10 m/s.

31:28 Typical length scale uh might be of order 1 th00and km said.

31:32 So are you saying for all the all the points

31:35 along those thousand km the wind is that speed.

31:38 Yeah.

31:38 Well because the wind speed is always like in one position it's a certain speed.

31:43 That's right.

31:43 Yeah.

31:43 But we're thinking about orders of magnitude

31:45 here rather than exact values at the moment.

31:47 So when thinking about the length scale I'm think about

31:50 over across entire low pressure is about 1 th00and kilometers across.

31:53 So low pressures of 100 km across that's

31:56 quite small for an area of low pressure.

31:57 10,000 km is far too big.

31:59 So th00and is about the order of magnitude.

32:02 So this is what we think about orders of magnitude here.

32:04 And we already calculated that the f for our kind of you know

32:07 rough mid latitude location is of order 1* 10us 4 which we calculated earlier.

32:14 If we plug them plug these into the Rosby equation

32:17 then we can say that Rosby number for synoptic scale flow

32:20 is 10/ 10^ 6* 10us 4 which equ= 10/ 10^ 2

32:28 which is 10 over 100 which is 1 over 10 0.1 Rosby number 0.1.

32:32 Okay.

32:33 Yeah.

32:33 So the Rosby number for your typical synoptic scale weather system so

32:36 your typical areas of low pressure and high pressure is around 0.1.

32:41 That's very close to zero.

32:42 So that tells us that to a very good

32:45 approximation the winds around of low pressure and high

32:48 pressure can be assumed to be in geostrophic balance

32:50 which is this balance between pressure gradient and coralis force.

32:53 So a roby number is like a little waiting you

32:55 add to tell you it's almost like it's almost like

32:57 an error bar or how close you are to this ideal

33:00 that you wish you had that would make life easy.

33:03 You can yeah you can think of it like that.

33:05 Yeah.

33:05 The smaller it is the closer you are to geostrophic balance.

33:08 And you could bung in you could bung in for say

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

33:13 cyclone where you know the wind speed 100 meter pers maybe

33:19 a little bit too strong but we'll we'll go with it.

33:22 A typical length scale of the eye of a tropical cyclone maybe 10 km which

33:26 would be 10^ the 4 m and tropical cyclones tend to form at lower latitudes.

33:31 The corololis parameter is maybe an order of magnitude smaller.

33:35 And so if we plung these into our Rosby uh number equation,

33:39 then we get V 100 divided by 10 4* 10- 5,

33:46 which is 100 over 10 to the minus one over 10 equals a,000.

33:55 Wow.

33:55 That that you can see is a massively

33:57 different roby number to a stomp car system.

33:59 So the type of balance we're looking at when you're thinking

34:02 about the force balances in the tropical cyclone eye is not geostrophic.

34:05 It's something we call cyclloic um which is a different balance altogether.

34:09 But that's what's really cool about this rby number.

34:11 You can plug in the scales of your system under consideration and it

34:16 quantifies how good the geostrophic approximation is

34:19 in your system that you're looking at.

34:27 Right?

34:28 So here's our qua geostrophic omega equation.

34:31 So we've talked about omega that's vertical velocity.

34:34 We've talked about geostrophic that's some kind

34:36 of balance between pressure gradient force and coralous force.

34:39 Now this is quai geostrophic because when you go from the full primitive

34:44 equation set and you simplify it

34:46 down using systematic scale analysis and simplification,

34:50 you need to retain the full wind in some of the terms

34:53 in order to produce vertical velocity

34:55 as I showed you through the continuity equation.

34:57 So you can't have everything geostrophic because if you

35:00 did you would remove the information about the vertical velocity.

35:03 Um so that's what makes it quai geostrophic.

35:06 It's not quite geostrophic but it's it's near enough.

35:09 Now is the Rosby number in here?

35:11 Rosby number is not in here.

35:12 No no unfortunately not.

35:15 Okay.

35:14 But it is a cool number.

35:15 So when we think about this, so the whole the whole point

35:18 of me explaining to this to you is to get to the point

35:22 where I can show you how we as meteorologists use the principles

35:26 contained in this equation to help us diagnose things about the weather.

35:32 Now to get that what we do is we tend to partition

35:34 these two sides into something called the response and the forcing.

35:38 So as meteorologists we are basically looking

35:40 at weather maps of vorticity and maps of temperature

35:45 or or thickness as we we like to look at it and we look at those as the forcing

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

35:52 in in vertical motion and through vertical motion we

35:55 can say right where is low pressure likely

35:57 to develop where is high pressure likely to develop.

35:59 Although you know you ask me can you predict the weather

36:02 using using this equation because this is a diagnostic equation it doesn't

36:06 tell you anything about how the velocity field is changing in time

36:11 then strictly speaking you can't however you can when we look

36:15 at subjective assessment of this use it to make inferences about

36:21 where areas of low pressure are going to develop where we're going

36:23 to see an increase in shower activity where high pressure is

36:26 likely to develop and so just looking at this response term here.

36:29 So this is in the vertical velocity.

36:31 Unfortunately it's you know this is a very

36:33 complicated equation and as humans we still can't

36:35 quite look at this and look at the maps and immediately map one onto the other.

36:39 Um we have to make some further simplification.

36:42 So one simplification we make this is

36:44 this tells you the threedimensional curvature of omega.

36:47 Now you can do some reasoning and um you know you could

36:50 say that omega is varying sinosoidally um as a function of pressure.

36:55 So basically what that means is you could

36:56 express omega as sin p and if you differentiate

37:00 this twice you end up with d2 omega by dp^

37:03 2 equals minus sin p which is minus omega.

37:07 So we can make similar arguments to this side of the equation

37:10 and we can basically say that all of this is proportional to minus omega.

37:16 And if you remember minus omega from our um when we

37:20 looked at the system of coordinates minus omega corresponds to ascent.

37:24 So we've actually by assuming some kind of wavelike

37:28 distribution of the vertical velocity in the horizontal

37:30 and the vertical we can make a good

37:33 assumption that this response is broadly equivalent to ascent.

37:37 So what the forcing terms?

37:38 Well, we've got the vertical variation of voltage vection.

37:41 So what what does that mean exactly?

37:43 So want to think about an atmosphere that's got a trough in it.

37:47 So this might be the 300 mibar surface

37:49 and this might be the th00and mibar surface.

37:52 And this is a trough.

37:54 We're always looking for where the troughs are, where the ridges are,

37:56 because the troughs and the ridges in the upper air usually correspond

37:59 to where the high pressures and the low pressures are in the lower atmosphere.

38:02 Although there's a they're not quite colloccated in vertical,

38:06 they're displaced one way or the other.

38:08 But if the geostrophic wind is blowing, vecting this area of positive vorticity.

38:15 So trough is an area of positive vorticity in this direction.

38:18 At some later time, this trough is going to be in this position here.

38:22 Now if we look at what's happened to the thickness

38:25 of the atmosphere and the thickness is just the distance

38:28 between the pressure levels and we call this H bar

38:30 and the thickness is a effectively a measure of the temperature.

38:33 If the mean temperature of the air is colder then the thickness becomes lower.

38:37 If the mean temperature is warmer then the thickness becomes higher.

38:40 But at this location here at some time not we've gone from hn

38:45 to h1 we've gone from a sort of relatively large thickness to a lower thickness.

38:50 Now what does that mean in terms of the temperature?

38:52 It means that the atmosphere here is cooled.

38:55 However, we've got no we've got nothing about temperature invection

39:00 here that we're not thinking about this term at the moment.

39:02 There's no mechanism by which we can cool the atmosphere.

39:05 And if we didn't cool the atmosphere then

39:07 either it wouldn't be in geostrophic or hydrostatic balance.

39:11 So how do we cool the atmosphere?

39:13 Well, we basically have some vertical motion.

39:15 We have a scent.

39:16 If you have a parcel of air and it rises, rising air expands and it cools.

39:21 So this process of vertical motion cools off the atmosphere.

39:25 And this is what the omega equation tells you.

39:26 It tells you what is the vertical velocity that I need in order

39:30 to cool the atmosphere enough to reduce

39:32 the thickness enough to accommodate this vorticity.

39:36 And you make you can make a similar argument for this.

39:39 You can um introduce some thermal adction.

39:42 And that again, if you have your same

39:43 pressure levels and you introduce some warm adction, maximum warm adction,

39:47 you would increase your thickness because you've introduced warmer air

39:50 that bulges up the contours of the pressure levels at height.

39:54 And what does this bulge do?

39:55 Well, it's it's the inverse of a trough.

39:56 Basically, it's a ridge.

39:57 It has negative vorticity.

39:59 If we've got no mechanism to generate vorticity through vorticity action,

40:04 well, how do we generate vorticity?

40:06 We diverge.

40:07 This is the ice skater effect.

40:09 So, if you have an ice skater who's spinning around

40:12 and then they pull their arms in, they'll spin faster.

40:15 Or if they pull their arms out, they'll spin slower.

40:18 Well, to reduce the vorticity, if the atmosphere pulls its arms out,

40:23 the vorticity will reduce and we'll create a ridge.

40:26 And how do we get this atmosphere to do this?

40:29 We have vertical motion.

40:30 because vertical motion goes up, it hits the top layer of the atmosphere,

40:33 it spreads out and it creates the necessary vorticity

40:38 in order to keep the fields in geostrophic balance.

40:40 So these are the forcing terms.

40:42 How do we use them?

40:44 Why do we use them?

40:45 Back when the QG system was first developed, this was a really key tool

40:50 into identifying exactly where the vertical motion was.

40:53 The best way to get the vertical motion

40:56 from these forcing terms is to calculate it with a computer.

40:58 That's kind of how they did it.

40:59 But as forecasters we we can't do that.

41:02 We can make these assumptions and we can look at the fields.

41:04 We identify where the vorticity is.

41:07 We identify where the thermal adection is.

41:09 And if you've got positive vorticity, if you've got warmer advction,

41:14 that corresponds to negative omega, which is ascent.

41:17 And so by looking at a map of vorticity,

41:19 you can immediately see where the areas of vertical motion

41:23 in ascending limb and the descending limb are going to be.

41:26 And that also comes into its own when

41:31 the forecast model which can calculate vertical velocity.

41:34 Now you know high performance supercomputers and really

41:37 sophisticated models they can calculate vertical velocity

41:40 but they can also be wrong in the positions of the troughs and the ridges.

41:44 They can have errors and we can look at satellite

41:46 imagery and we can diagnose where these errors are and we

41:49 can say right well if this trough was a bit

41:51 further back what does that mean for the vertical motion field?

41:54 we can we can always calculate it in our head

41:55 and make a forecast based um on that.

41:58 So although this equation comes from you

42:01 know deep meteorological history um it's part

42:04 of an equation set that's no longer

42:06 used in much simpler times in numerical models.

42:09 Um it's still used in practice on the bench

42:13 although with a lot of a lot of assumptions,

42:15 a lot of simplifications in order to give forecasters

42:18 this conceptual model of the atmosphere where things are ascending,

42:21 where things are descending and therefore where areas of low pressure,

42:24 whereas high pressure are developing.

42:26 But you don't crack out the equation and put numbers in it.

42:28 You use more this kind of response and forcing situations.

42:32 Exly.

42:32 Yeah.

42:32 We look at where the we look at where the vorticity is.

42:35 We look at where the thermalction is.

42:37 We diagnose qualit qualitatively whether it's ascending or descending because

42:43 we can't calculate the magnitudes of these terms in our head.

42:47 We can't say anything about the size of the response.

42:50 And there there are occasions where one force

42:53 in turn will say ascent and one force in turn

42:56 will say descent and then we have other techniques

42:58 that we can use to kind of break the tie.

43:01 But in very qualitative terms, yeah,

43:03 we're looking at the vorticity inction and the thermal invection.

43:05 And it tells us whether we can expect upward

43:08 motion and development of low pressure at the surface,

43:10 even development of, you know,

43:11 if it if it links in with a strong frontal gradient,

43:14 that can lead to development of a potent of low pressure and a a name storm

43:18 or whether we're looking at descending air

43:20 and development of high pressure and more settled weather.

43:22 Well, you made it this far, so well done.

43:24 If you'd like to hear more from Dan,

43:26 more of a personal interview, a bit about his life,

43:29 his nickname at school, his first job as a forecaster,

43:33 his backyard weather setup, 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 miss.

43:46 Also, just have a tiny little change.

43:47 Quantifying tiny, but not to blow up to infinity.

43:50 That wouldn't make sense because I've changed something so so small.

43:53 Why have I got an entirely different solution?

43:56 We've also been taught that butterflies flapping their wings can cause cyclones.

44:00 The butterfly effect, it's like a chain reaction.

44:01 It's one thing leads to another leads to another.

44:04 But in the in the sense of humming an equation,

44:07 you input something into your equation.

44:08 It's like a function machine.

44:10 Input some initial