[0:00] And this is one of the most famous
[0:01] equations in meteorology. I'm just going
[0:03] to go ahead and and write it down. So we
[0:05] have sigma d^ 2 omega. So where the
[0:08] omega comes from plus f 2 d2 omega by
[0:13] dp^ 2. So this is the omega part and
[0:16] that equals f * the vertical variation
[0:19] of this term here which is the advection
[0:22] of the absolute geostrophic vorticity by
[0:24] the geostrophic wind and the advection
[0:27] of geostrophic wind the temperature
[0:29] gradient.
[0:30] >> So this is this is the qua geostrophic
[0:32] omega equation. So this is a fundamental
[0:35] equation that allows us to through
[0:39] systematic simplifications a lot of
[0:40] simplification actually get the vertical
[0:42] velocity from the adction of this
[0:45] quantity called vorticity and the
[0:47] adction of this quantity called
[0:49] temperature. So thermal adction,
[0:51] vorticity adection. And by looking at
[0:53] horizontal maps of these adections, we
[0:55] can diagnose vertical velocity and
[0:57] vertical velocity is a proxy for
[0:59] development. And development is your
[1:02] developments of your lows, your
[1:03] developments of your highs, high
[1:04] pressure and low pressure. And that
[1:06] relates to the weather forecast.
[1:07] >> So what this tells you what's going to
[1:10] happen in the future?
[1:12] >> No. So that's quite an important
[1:14] distinction about this equation. This
[1:15] does this tells you nothing about the
[1:17] future state of the atmosphere. So this
[1:19] isn't a predictive equation. It's it's
[1:21] it's called a prognostic equation. This
[1:23] is a diagnostic equation. It just allows
[1:26] us to diagnose the vertical velocity
[1:28] from the invection of vorticity and the
[1:30] invection of temperature.
[1:31] >> And why is that an important thing to
[1:33] know?
[1:34] >> Um
[1:35] >> if it doesn't tell you anything about
[1:36] the future. Well, so this I mean we we
[1:39] can go into this in a bit more detail uh
[1:41] later on, but this um essentially is one
[1:44] equation that falls out of a whole
[1:46] system called the quai geostrophic
[1:48] system. And I want to kind of go into a
[1:50] bit about what we mean by quai
[1:53] geostrophic omega. But essentially the
[1:55] quai geostrophic system is a systematic
[1:58] set of simplifications to the main
[2:00] primitive equations which numerical
[2:02] models these days use to forecast the
[2:04] weather. um is a a systematic
[2:06] simplification of those equations that
[2:08] could then be used to make the first
[2:10] kind of workable forecast models that
[2:13] were used on the computers back in the
[2:15] day the the early 1950s. And so this
[2:18] equation is a diagnostic equation that
[2:20] falls out of those and it is part of
[2:22] that forecast process whereby you would
[2:25] calculate some tendencies of various
[2:26] things. You would then get the wind
[2:28] fields which you could calculate wind
[2:30] and thermal fields which you could
[2:31] calculate the vorticity and the thermal
[2:32] invection and from that you could
[2:34] diagnose the vertical velocity and
[2:36] getting to this vertical velocity is
[2:37] actually quite difficult and so this
[2:39] equation gives us a method to to
[2:41] actually do that in a systematic and
[2:45] relatively error-free way
[2:46] >> in even more simple terms then this equ
[2:48] is this equation relating to up and
[2:50] downness of the air what's it like
[2:52] what's what's it
[2:53] >> exactly so when when we're talking about
[2:55] vertical velocity We're talking about
[2:57] the motion of the air in an up and down
[2:59] sense. So on on large scales, so the
[3:02] scales of weather systems, the main the
[3:04] major motion of the atmosphere is in the
[3:06] horizontal. So you've got winds north to
[3:08] south, winds east to west. We don't tend
[3:10] to think on large scales about the winds
[3:13] that are going up and down, but they are
[3:14] there. If you think about a convective
[3:16] cloud, for example, they're moving up
[3:18] into clouds, connects into clouds, and
[3:19] you get rain falling out. Um, but on
[3:21] these large scale, so we we're talking
[3:23] 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
[3:29] big proportion of the Atlantic.
[3:31] You don't tend to think about vertical
[3:33] motions there, but they are there and
[3:35] they're necessary in order to develop
[3:37] the areas of low pressure and develop
[3:38] the areas of high pressure. So if you
[3:40] have rising motion, you get development
[3:42] of low pressure at the surface. And if
[3:44] you have sinking motion, then you have
[3:45] development of high pressure at the
[3:47] surface. One more basic weather question
[3:50] before we, you know, get our teeth into
[3:52] this. Then I'm pretty familiar with
[3:54] measuring wind speeds horizontally. I've
[3:56] seen those little devices like those
[3:58] little cups that spin around and things
[4:00] like that. Are there devices that
[4:02] measure the up downwind, the vertical
[4:04] wind? Like how do you how do you guys
[4:06] take measurements of that?
[4:07] >> No, it's very difficult. And that's one
[4:09] of the reasons why we kind of look to
[4:11] equations to to diagnose the vertical
[4:14] velocity. It's very hard to measure.
[4:17] it's on a much much much smaller scale
[4:19] than the the horizontal motions. And in
[4:21] fact, you can try and calculate the um
[4:26] vertical motion from the horizontal
[4:28] motions via something called the
[4:29] continuity equation, which we'll have a
[4:31] look at in a little while. Um but that
[4:33] relies on you knowing to a very high
[4:35] degree of accuracy your horizontal
[4:36] winds. And it turns out that the the
[4:38] divergence and the vertical motion
[4:40] errors in that are the same size as the
[4:43] actual vertical motion you're trying to
[4:44] diagnose in the first place. So you can
[4:46] have small areas in the wind that can
[4:47] lead to massive areas in vertical
[4:48] velocity.
[4:49] >> Just quickly, what are the things that
[4:50] measure the wind speed, Cole?
[4:52] >> Anim
[4:53] >> an animometers.
[4:54] >> Animasure
[4:56] wind speed horizontally.
[4:58] >> Yes.
[4:58] >> Are you saying equations the likes of
[5:00] these are kind of what you have instead
[5:03] of those for vertical.
[5:04] >> That's how you get to the vertical
[5:06] velocity.
[5:06] >> Yeah.
[5:06] >> So let's break it down. So um first of
[5:08] all, it's the omega equation. So let's
[5:10] talk about omega. So we got this Greek
[5:12] letter omega here. So omega is just the
[5:14] vertical velocity. So it has a
[5:16] equivalent called W and these two are
[5:19] equivalent but they use different
[5:20] coordinate systems. So for example, if
[5:22] we think about W, this might be a little
[5:23] bit more familiar. We think about a
[5:25] coordinate system that has winds in the
[5:28] XY. So this is like your XY plane and
[5:31] then you've got a vertical plane. If you
[5:32] have some vertical motion or if you have
[5:35] some horizontal motion, we could have a
[5:36] a U wind in the X direction, a V wind in
[5:40] the Y direction. So these are our
[5:41] horizontal winds and then w would be um
[5:45] the wind in the vertical and that would
[5:47] be measured in meters per second. There
[5:49] is a direct equivalent though uh in a
[5:51] different coordinate system which we in
[5:53] meteorology that we like to use because
[5:55] it has some good advantages when coming
[5:58] to equations like this and looking at
[5:59] things like mass continuity. We still
[6:01] have an xy plane but we use pressure as
[6:04] our vertical coordinate. And so if we
[6:06] have vertical motion in the pressure
[6:09] coordinate and we can use pressure as a
[6:11] coordinate because pressure always
[6:13] decreases with height. So it's around
[6:15] about a th00and millibars at the surface
[6:16] 1,000 hector pascals. At the top of the
[6:19] troposphere which is the top of the
[6:20] weather bearing layer is around about
[6:22] 300 hector pascals. So pressure always
[6:24] falls with height. So if you're moving
[6:26] in an upward direction you're moving
[6:29] towards lower pressure. Ascent in the
[6:31] the geometric coordinate system would be
[6:33] W. ascent in the pressure coordinate
[6:35] system would be negative omega because
[6:38] you're going towards lower pressure as
[6:40] you go up whereas in the geometric
[6:42] system you're going towards kind of
[6:44] higher elevations as you go up.
[6:46] >> So does that mean we're using on our x
[6:48] and our y axis here we're using
[6:50] different units because that's
[6:52] presumably still meters/s.
[6:54] That's right. Yep. So this we still have
[6:56] a U wind and a vwind in meters per
[6:58] second
[6:59] >> but we're using a different unit on
[7:00] that.
[7:00] >> Yep. The unit for this one would be
[7:02] pascals per second. It's it's the amount
[7:05] of pressure change that this parcel of
[7:06] air is experiencing per second. So very
[7:08] much like your your speed would be the
[7:11] amount of displacement you're
[7:12] experiencing per second, meters/s. This
[7:14] would be the the pressure change per
[7:16] second, pascals per second. Okay, cool.
[7:19] You can you can link the two uh via
[7:21] something called the hydrostatic
[7:22] equation. So we have this hydrostatic
[7:25] equation which relates the rate of
[7:27] change of pressure in the z direction
[7:29] with this term minus row which is the
[7:32] density and g which is gravity. You can
[7:35] say that um if you treat is the material
[7:38] derivative which you can expand out to
[7:41] this expression here. And um we're going
[7:43] to assume something called a hydrostatic
[7:45] balance. Um, and what hydrostatic
[7:47] balance effectively means is that if we
[7:49] have an atmosphere that's at rest and we
[7:52] have a particle or a parcel of air in
[7:54] the atmosphere, it's got two forces
[7:55] acting upon it. We've already said that
[7:57] pressure decreases with height. So,
[7:58] we've got a pressure gradient in this
[8:00] direction. So, this parcel has
[8:01] experienced a pressure gradient force in
[8:03] that direction. And then we've got a
[8:05] gravitational force in that direction.
[8:07] So, we've got pressure gradient force
[8:09] and gravity. And when these two are in
[8:11] balance, you know, if if there wasn't
[8:13] gravity and there was a pressure
[8:14] gradient force, then this parcel of air
[8:15] would fly up into the atmosphere because
[8:17] it'd be the the higher pre moving from
[8:19] higher pressure to lower pressure and be
[8:20] pushing it up to into the atmosphere.
[8:22] But because it's balanced by gravity,
[8:24] the the particle or the air parcel is at
[8:27] rest and we call this hydrostatic
[8:28] balance. So we assume the atmosphere is
[8:30] at rest. There's no U, there's no V,
[8:31] there's no pressure change with time. So
[8:33] we can say that uh omega is
[8:36] approximately equal to W DP by DZ. And
[8:39] then from this hydrostatic balance
[8:41] relationship which we kind of expanded
[8:43] out here we can say that omega be equal
[8:47] to minus row g which is this term here
[8:51] times w. So what this effectively tells
[8:54] us is that uh omega and w are related
[8:57] but omega is just a scaled version
[9:00] scaled by the density because because
[9:03] the atmosphere is more dense at the
[9:05] surface. So if you have a parcel of air
[9:07] that's moving vertically at fairly low
[9:09] elevations, the pressure is quite high.
[9:12] Because the atmosphere is more dense
[9:14] here, the pressure levels are kind of
[9:16] closer together. And therefore the same
[9:19] vertical speed in height coordinates at
[9:22] lower levels would be crossing more. It
[9:24] be the pressure will be changing more at
[9:26] lower levels than it is at higher levels
[9:28] for the same vertical speed when you're
[9:30] thinking about the change in meters. So
[9:32] this is what this equation tells us.
[9:33] Omega is just a scaled version of the um
[9:37] vertical velocity in geometric height
[9:39] coordinates, but it's scaled by the
[9:41] 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
[9:58] 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